Static balancing of rigid-body linkages and compliant mechanisms

Transcription

1 Static balancing of rigid-body linkages and compliant mechanisms A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF ENGINEERING by Sangamesh Deepak R Department of Mechanical Engineering Indian Institute of Science Bangalore 5612, INDIA MAY 212

2 c Copyright by Sangamesh Deepak R 213 All Rights Reserved ii

3 Abstract Static balance is the reduction or elimination of the actuating effort in quasi-static motion of a mechanical system by adding non-dissipative force interactions to the system. In recent years, there is increasing recognition that static balancing of elastic forces in compliant mechanisms leads to increased efficiency as well as good force feedback characteristics. The development of insightful and pragmatic design methods for statically balanced compliant mechanisms is the motivation for this work. In our approach, we focus on a class of compliant mechanisms that can be approximated as spring-loaded rigid-link mechanisms. Instead of developing static balancing techniques directly for the compliant mechanisms, we seek analytical balancing techniques for the simplified spring loaded rigid link approximations. Towards that, we first provide new static balancing techniques for a spring-loaded four-bar linkage. We also find relations between static balancing parameters of the cognates of a four-bar linkage. Later, we develop a new perfect static balancing method for a general n- degree-of-freedom revolute and spherical jointed rigid-body linkages. This general method distinguishes itself from the known techniques in the following respects: 1. It adds only springs and not any auxiliary bodies. 2. It is applicable to linkages having any number of links connected in any manner. 3. It is applicable to both constant (i.e., gravity type) and linear spring loads. 4. It works both in planar and spatial cases. This analytical method is applied on the approximated compliant mechanisms as well. Expectedly, the compliant mechanisms would only be approximately balanced. iii

4 We study the effectiveness of this approximate balance through simulations and a prototype. The analytical static balancing technique for rigid-body linkages and the study of its application to approximated compliant mechanisms are among the main contributions of this thesis. iv

5 Acknowledgements The motivating idea for this work is that static balancing techniques for compliant mechanisms can be developed by extracting insights from static balancing techniques for rigid-body linkages. The originality of this idea belongs to my advisor, Prof. G. K. Ananthasuresh. Further, I had regular meetings with my advisor where he offered his comments and suggestions on my work. Some of his suggestions helped me in directing this work to this form. I also appreciate his emphasis on prototypes and the support system he has developed for making them. He has spent significant time in sharing his experiences and opinions on technical writing. These were valuable in documenting my work. In making prototypes presented in the thesis, I had received help from Mr. A. Ravikumar, Mr. Ramu G., Mr. B. M. Vinod Kumar and Mr. Praveenraj H. K. I deeply value their knowledge and skills. I also acknowledge the co-operation that I received from Mr. A. Raja in accessing the workshop. Almost all the fabrication aspects of the prototype presented in Appendix G was handled by Mr. Amrith Hansoge. I deeply value his industrial experience, which were crucial in making a rather impressive prototype. Prof. Ashitava Ghosal, advisor for my master s thesis, was very helpful during my transition from master s to PhD. I also appreciate the remarkable care he took to see that I choose a topic of research that suits me. My course-work in IISc has been one of the most exciting and satisfying aspects of my life in IISc. I immensely thank all the instructors for their effort in getting us exposed to challenging concepts. Prof. C. R. Pradeep s class on Topology ranks as the best class that I ever had in my academic career. Courses such as these have v

6 helped me a lot. AnotherexcitingaspectofmystayinIIScwastheweeklygroupmeetingthatProf. Ananthasuresh conducts. Through these meetings, my advisor and my colleagues in the lab gave me an exposure to diverse areas of knowledge and research. In an association spanning a little less than five years, I probably have many things to say about my advisor. However, I will sum up this association like this it has been mostly a pleasure and definitely a privilege. My lab-mates have been invariably nice, co-operative and understanding. I feel very privileged to be part of such a good set of people. A lot of people formed the source for many exciting academic exchanges that I had. Among them, I would like to make a special mention of Narayana Reddy, Kalidas, Meenakshi Sundaram, Hariharan and Sreenath. These exchanges have helped me to put my attention on some of the fundamental principles in mechanics and mathematics in general. While my interaction with people in IISc is somewhat less, I must nevertheless acknowledge that in general people have been kind and understanding towards me. I would like to specially thank many mess workers who served me with food even if I was late many times. The physical training that I received from Master Stephen Kumar, Master Manjunath and my senior Rahul S., has been one of the best things to happen in my life. This training helped me to maintain my mental balance even in depressing times. On personal front, my mother absorbed all the shocks, pulls and pressure to ensure a free ambience for my growth. It was because of this free ambience that my understanding of science and mathematics matured over the years. Patience, perseverance, hope, courage, endurance and wisdom displayed by my mother are inspirational. I also acknowledge the care shown by my father towards my well-being. vi

7 Contents Abstract Acknowledgements iii v 1 Introduction What is static balance? Static balance of rigid-body linkages Compliant mechanisms and its static balance Static balance: rigid-body linkage vs. compliant mechanisms Motivation for the thesis Scope of the thesis Three methods to statically balance a zero-free-length springloaded four-bar linkage Static balancing of cognates Static balancing of revolute-jointed linkages without auxiliary links Static balancing of spatial and/or revolute jointed linkages without auxiliary links Static balancing of flexure-based compliant mechanisms by addition of springs Literature Survey Rigid-body linkages under gravity loads Counter-weight balancing vii

8 2.1.2 Balancing by addition of springs Rigid linkages under spring loads Static balance of compliant mechanisms Prior Art Preliminaries Plagiograph Derivation of the balancing solution in [1] Static balancing of a four-bar Introduction Static balancing of a given spring-loaded four-bar linkage Technique Technique Technique A prototype with technique Static balance of the cognates Introduction Cognates Static balancing parameters and the cognates A geometric problem and its solution Case (i): a, b and c are parallel Case (ii): a, b and c are concurrent Case (iii): a, b, and c are neither parallel nor concurrent Finding the focal pivot The result A parameterization of the balancing parameters of the three cognates that has cognate triangle related invariants Static balancing of planar linkages Introduction Balancing a lever viii

9 5.2.1 Potential energy as a function of the configuration variable Invariance of potential energy with respect to the configuration variable Balancing of a rigid body in a plane New static balancing techniques for revolute-jointed linkages The potential energy of loads on a body transformed as a function of another body Proposition 2 as the recursive relation of an iterative static balancing algorithm Static balancing of any revolute-jointed linkages with any kind of zero-free-length spring and constant load interaction within the linkage A note on prismatic joint Static balancing of spatial linkages Introduction The class of functions in feature Joints that can potentially satisfy feature Spherical joint has feature Revolute joint has feature Algorithm to synthesize static balancing solution of a spatial revolute/sphericaljointed tree-structured linkage having zero-free-length spring and/or gravity loads exerted by a reference link Illustrative example Static balance of any kind of spatial revolute and/or spherical jointed linkage with constant load and zero-free-length spring load interaction A note Static balance of compliant mechanisms Balancing a flexure beam The flexure beam Rigid-body model for the flexure beam ix

10 7.1.3 Approximation of torsional spring by zero-free-length spring Static balancing by addition of a zero-free-length spring Balancing springs on the flexure beam Framework Flexure-based compliant four-bar mechanism Description of the mechanism Step Step 2 The effort function Step Step 4 Static balance of the linkage under zero-free-length spring load Step 5 Approximate static balance of flexure-based four-bar linkage A way to improve the static balance of the flexure-based fourbar linkage Another flexure-based four-bar linkage Description of the mechanism Step Step Step Step Step First order correction Prototype Flexure-based 2R compliant mechanism Description of the compliant mechanism Step 1: Step 2 Identification of effort function Step Step Step x

11 7.6 Discussion Static balancing of compliant mechanisms by individually balancing flexures Static balancing of compliant mechanisms using rigid-body linkages Conclusion A summary of new static balancing techniques for spring and/or gravity-loaded rigid-body linkages Static balancing of a four-bar linkage loaded by a spring on its coupler link Static balancing parameters and the cognates of a four-bar linkage Static balancing without auxiliary bodies planar case Static balancing without auxiliary bodies spatial case A framework for designing statically balanced compliant mechanisms The novelty of the contribution in the context of the current literature Future work A Proofs on finding the focal pivot 169 A.1 If I a,b and I c,a circles are coincident, then the given lines a, b and c has to be concurrent A.2 When M and A are distinct, M is the focal pivot. (Refer to section (4.2.4 and figure (4.11).) B An elementary theorem 172 C Normal springs 173 D Satisfying Constraints 175 D.1 Satisfying constraints (5.12), (5.13) and (5.14) D.2 Satisfying constraints (5.19), (5.2) and (5.21) E Solving balancing constraints spatial case 178 xi

12 F Virtual work calculations 181 F.1 Calculation of stiffness in case 3a based on case 2a F.1.1 Obtaining F x vs. u x in case 2a using virtual work balance F.1.2 Obtaining F x vs. u x in case 3a using virtual work balance F.2 Verification of static balance in case 3b through virtual work balance 195 F.3 A first order correction to balancing springs on the flexure-based fourbar linkage G Another compliant mechanism balancing 27 H Springs between successive links 212 Bibliography 216 xii

13 List of Tables 3.1 Summary of different techniques to statically balance a spring-loaded four-bar linkage presented in this chapter Deduction of various quantities in equations ( ) from the cognate triangle Ratio of spring constants of balancing spring 1 for three cognates Potential energy of the weight and the zero-free-length component of the spring acting on the lever is a linear combination of cosθ, sinθ, and Potential energy of weight and spring acting on a link moving in a plane Details of the loads in figure (5.5). It may be checked that the loads satisfy equations ( ) The potential in the general form shown in figure (6.3), can be expressed as a linear combination of the basis functions shown in the table. Each basis function in the table is a function of translational variable r and the Z-X-Z Euler angle α, β and γ Relevant quantities to calculate the torsional stiffness of the spring Virtual work calculation for lever in case (2a) Virtual work calculation and slope of F x vs. u x Verification of equations ( ) being satisfied Verification of static balance through virtual work calculations xiii

14 7.6 The origin and the slope of F x vs. u x being matched between case 2a and case 3a Verification of equations ( ) for the spring-loaded 2R linkage of figure (7.14) Details of the flexure and calculation of torsional spring constant Matching value and slopes at the origin of the effort function Details of the original spring and balancing spring First order correction of balancing spring parameters Details of flexure and calculation of torsional spring constant The value and the first derivative of F vs. u at u = Verification of static balance of springs in case 3b xiv

15 List of Figures 1.1 A lever having two discrete equilibrium configurations, namely (b) and (c) while (a) is not in equilibrium A lever having a continuous set of equilibrium configurations; it is in static balance A statically balanced lever as a manually operated road barrier. (Source: Counterweight balancing in a two degree-of-freedom linkage Spring-based balancing in a two degree-of-freedom linkage Spring-based balancing of a lever Effects of free-length and pre-tension on force-distance plot of a linear extension spring A compliant crimper A spring-loaded four-bar linkage to be statically balanced A static balancing solution for the spring-loaded four-bar linkage by addition of auxiliary links and springs A compliant four-bar mechanism A counter balancing technique for serial revolute jointed linkages Use of auxiliary links (colored in grey) along with springs in an existing technique for balancing a n-degree-of-freedom linkage under constant load A basic spring force balancer A lever with ordinary springs Balanced parallelogram xv

16 2.6 Balanced two degree-of-freedom parallelogram linkage Composition of two zero-length-spring into an equivalent one. The virtual, equivalent spring between E and D is shown in grey color A plagiograph or a skew pantograph linkage A balanced parallelogram with one its spring along a diagonal decomposed into two springs The parallelogram linkage with a duplicate that maintains a constant angle of α from the original Synchronization of motion between two parallelogram linkages Scaling the duplicate parallelogram linkage Removal of a spring and compensating it with increase in stiffness of another spring Removal of two more links The current literature and our contributions A four-bar linkage with a zero-free-length spring anchored from its coupler to the ground Possibility of the four-bar linkage being statically balanced as it is A balanced parallelogram on the load spring Technique 1: Static balancing with two auxiliary links and one balancing spring Forming a parallelogram using two auxiliary links A virtual spring between opposite vertices of the parallelogram AF ED Adding balancing spring 2 across opposite vertices of the parallelogram to balance the load spring and balancing spring Option 2 for balancing the four-bar linkage without requiring auxiliary links Technique Plagiograph with base pivot at B xvi

17 3.11 A prototype of four-bar linkage that is statically balanced using option 2. The four-bar linkage seen in the above figure is a Watt s straight-line mechanism Realization of a zero-free length spring The cognates of a four-bar mechanism taking a load spring along their common coupler curve A cognate triangle and the ground anchor point of a load spring Technique 3: the base pivot of the plagiograph (U) and the base of the balanced parallelogram (W) for every cognate are not coincident Technique 3: the base pivot of the plagiograph (U) and the base of the balanced parallelogram (W) for every cognate are coincident Technique 2: the base of the balanced parallelogram (W) at the indicated vertices of the cognate triangle A problem in planar geometry Finding S a, S b, and S c in case (i) Finding S a, S b, and S c in case (ii) Description of focal pivot Points A 2, B 2 and C 2 form a solution to S a, S b and S c Geometric construction to find the focal pivot Lack of methods for spring-based n-body linkage balancing without the usage of auxiliary bodies A lever under a constant load and a spring load Static balancing of a weight by a spring A body that is free to move in a plane A rigid body moving freely in a plane under a constant load is made to have θ-independent potential energy by addition of two zero-free-length springs The gravity-loaded serial 4R linkage to be statically balanced Details of Iteration Details of Iteration xvii

18 5.9 Details of Iteration Details of Iteration Statically balanced gravity-loaded serial 4R linkage Statically balanced serial 3R linkage Statically balanced serial 2R linkage Details of static balance of a 2R linkage under spring load Details of static balance of a 4R tree-structure linkage under a constant load and a spring load Potential Energy variation of spring loads, constant loads, and their sum Breaking a problem as a superposition of several problem with each problem being static balance of revolute-jointed tree-structured linkage with loads exerted by the root body Static balance of a tree-structured linkage with inter-body load interactions Balancing the lever lever loads in the second load set of figure (5.18) Details of static balancing of six degree-of-freedom spatial balancing under gravity loads Details of the flexure beam Our attention is on reducing horizontal force for a range of horizontal displacements of point P Small-length-flexure model applied to the flexure beam Approximation of the torsional spring by a zero-free-length spring The approximate match in F x vs. u x relation between case 2a and case 3a Static balance of the approximated zero-free-length spring-loaded lever by addition of a zero-free-length spring All the cases related to the flexure and its approximation by the springloaded lever F x vs. u x relation obtained from finite element analysis of the flexure beam xviii

19 7.9 A consolidated plot of F x vs. u x for flexure beam and its rigid-body models A flexure-based four-bar linkage The quadrilateral formed by the centers of the flexures Approximation of the flexure-based four-bar linkage as a rigid-body four-bar linkage with torsional springs Approximation of torsional springs by zero-free-length springs Static balancing by addition of zero-free-length springs A consolidated figure of all the cases Finite element simulation results for case 1a and case 1b F x vs. u x after first order correction to the stiffness of balancing springs F X vs. u x plot for all the cases shown in figure (7.15) Consolidated figure containing all the cases Effort function in all the cases A prototype to demonstrate reduction in effort A consolidation of all the cases Details of springs F x vs. u in two views F y vs. u in two views Plot of effort function when flexure length is increased three-fold Ideal circular arc-like and non-ideal deformation of flexures Static balancing of each of the flexures, independently of one another The current literature and our contributions A.1 I a,b and I c,a circles are coincident A.2 I a,b is at M F.1 l, the length of a diagonal of the quadrilateral of four-bar bar linkage is used as a convenient configuration defining parameter G.1 A compliant gripper compensated by a small spring loaded 2R linkage. (Basement board dimension: 2.5 feet 2.5 feet) xix

20 G.2 A statically balanced parallelogram and its modification G.3 A compliant gripper H.1 A tree-structured linkage under gravity load xx

21 Chapter 1 Introduction Overview The concept of static balance. The importance of statically balancing a rigid-body linkage. Zero-free-length springs, their importance and their practical realization. The importance of statically balancing a compliant mechanism. The motivation of the thesis. The contributions of the thesis. 1.1 What is static balance? A system is said to be in static balance if it can undergo quasi-static motion without any external effort when any dissipative force interactions in the system are removed. In the general motion of a system, at any configuration, inertia forces of the system, conservative and dissipative force interactions within the system and external forces acting on the system are in equilibrium. However, in the motion of a statically 1

22 CHAPTER 1. INTRODUCTION 2 balanced system, the conservative force interactions have to be in equilibrium by themselves. Consider the lever shown in figure (1.1a). The conservative force interactions on it are the gravity between the lever and the ground and the constraint force between the lever and the pivot-post (fixed to the ground). If, at a configuration, these forces could be in equilibrium by themselves then we call the configuration as equilibrium configuration. This system has only two discrete equilibrium configurations shown in figures (1.1b c). It is impossible to have quasi-static motion that does not pass through any configuration other than these two discrete equilibrium configurations. Hence, the system in not in static balance. In contrast, consider figure (1.2), which is the same lever with an extra mass added so that the overall centre of gravity is at the pivot. Here, every configuration is an equilibrium configuration. Hence, one can have quasi-static motion passing through only equilibrium configurations. Therefore, this system is in static balance. The effortless motion of this system has been utilized for a long time in manually operated road barriers such as the one shown in figure (1.3). If a system is not in static balance, it may be possible to add extra conservative force interactions to the system such that all the conservative forces are in equilibrium. This process of addition is called static balancing. An example of static balancing is the addition of extra mass to the system in figure (1.1) to obtain the system in figure (1.2) Static balance of rigid-body linkages In the example of figure (1.3), static balancing meant nullifying the effect of gravity forces on the material making up the barrier. In fact, much of the past research on static balance was concerned with nullifying the effects of gravity on the material making up a rigid-body linkage. The motivation for such research efforts was that many practical systems such as leg-orthosis, robots and flight-simulators are made up of rigid-body linkages and nullifying the gravity effects in them would significantly reduce the torque or the force requirement from the actuators. Static balance of rigid-body linkages may be broadly classified into two groups:

23 CHAPTER 1. INTRODUCTION 3 g g g (a) (b) (c) Figure 1.1: A lever having two discrete equilibrium configurations, namely (b) and (c) while (a) is not in equilibrium. m o c o +m a c a = g m a c a c o m o The centres of gravity of the original lever and that of the extra mass are collinear with the pivot c o and c a : Coordinates of the centres of gravity of the original lever and the additional mass about the pivot m o and m a : Masses of the original lever and the additional mass Figure 1.2: A lever having a continuous set of equilibrium configurations; it is in static balance.

24 CHAPTER 1. INTRODUCTION 4 Figure 1.3: A statically balanced lever as a manually operated road barrier. (Source: counterweight-addition and spring-addition. Figure (1.4) shows the example of a counterweight balancing technique. The original weight could be that of, say, a lamp head in a table lamp. Figure (1.5) shows the spring-based balancing of a two degreeof-freedom system. Most of the counterweight balancing methods are derived from the lever-balancing principle shown in figure (1.2). For example, in figure (1.4), the center of mass of the original weight and that of the counterweight are always collinear with the ground pivot. Further, the ratio of distances between these points is always the same and the magnitude of the counterweight added is such that it obeys the same equation as in figure (1.2). Similarly, the genesis of most of the spring-based balancing techniques could be traced to the spring-based balancing of a lever shown in figure (1.6). We now explain how the lever in figure (1.6) is balanced. In figure (1.6), by geometry, we have sŝ = l+hĵ (1.1) By denoting the free-length of the spring by s, the spring force on the lever is given

25 CHAPTER 1. INTRODUCTION 5 Original weight Counter weight Figure 1.4: Counterweight balancing in a two degree-of-freedom linkage Original weight Figure 1.5: Spring-based balancing in a two degree-of-freedom linkage

26 CHAPTER 1. INTRODUCTION 6 s ŝ h k sŝ = l+hĵ ĵ w weight θ l Figure 1.6: Spring-based balancing of a lever by f =k(s s )ŝ =k(s s ) 1 s ( ) l+hĵ, from eqn. (1.1) (1.2) The moment of the spring force about the lever is given by M spring = l f = k(s s ) s = k(s s ) h s ( l l+hĵ ( ) l ĵ ), from eqn. (1.2) (1.3) Similarly, the moment of the gravity load is M weight = l wĵ (1.4) The net moment becomes M spring +M weight = ( kh(s s ) ) +w l ĵ (1.5) s

27 CHAPTER 1. INTRODUCTION 7 If we want static balance, then the net moment has to be zero over a continuous range of θ. Given that l ĵ is zero at only discrete values of theta, we can only expect its coefficient to be zero over a continuous set of configurations. However, when s, the coefficient could become zero, again, at only a discrete set of configurations (Note that s is a function of θ). Nevertheless, if we can have s =, i.e., if we can have a spring of zero free-length, then the coefficient becomes a constant over every θ and, in particular, zero if kh = w. Thus, the conditions under which the lever shown in figure (1.6) attains static balance are zero free-length and kh = w. While achieving kh = w is not hard, it is not popularly known that zero-free-length can also be achieved in practice. The credit for recognizing the importance of zero free-length in static balance and also its practical implementation goes to George Carwardine [2] and Lucien LaCoste [3]. Having understood the necessity of zero-free-length springs for perfect static balance, we now focus on one of the ways of its practical realization. Figure (1.7) shows the spring force (f) versus the relative distance (d) of anchor points of a linear extension spring for various cases. The plot for zero-free-length spring is expected to be collinear with the origin, as shown in figure (1.7a). A normal extension spring not only has a finite positive free-length but also what is called as pre-tension. In a pre-tensioned spring, even if there is no external force, the coils press against each other. The coils separate and the spring extends only when the external force is more than the pre-tension. The effect of free-length l is to shift the force-distance plot along l axis by l and the effect of pre-tension f p is to shift the plot along the f-axis by f p. When f p is kl where k is slope of the plot (i.e., the stiffness of the spring), the plot becomes collinear with the origin. Thus, even with unavoidable free-length, by inducing appropriate pre-tension, one can have force-distance relation to be the same as that of a zero-free-length spring for l > l. Therefore, there is no practical hindrance in realizing a zero-free-length spring. There are several other ways of realizing a zero-free-length spring, as discussed in [4] and [5].

28 CHAPTER 1. INTRODUCTION 8 f (force) f l (distance) l l (a) zero free length (b) positive free length without pre-tension f f p l l (c) positive free length with pre-tension Figure 1.7: Effects of free-length and pre-tension on force-distance plot of a linear extension spring

29 CHAPTER 1. INTRODUCTION Compliant mechanisms and its static balance In recent years, compliant mechanisms have emerged as a plausible design option, especially in micro mechanical systems. A compliant mechanism, in contrast to a rigid-body linkage, is made up of a single monolithic piece that transmits force and displacement through elastic deformation. They are quite amenable to microfabrication techniques apart from being free of friction and backlash. Analysis of rigid-body mechanisms can often be carried out analytically. This is possible since it is the geometry of triangles, quadrilaterals and other polygons that underlies the kinematics of rigid-body mechanisms. Furthermore, many graphical and analytical synthesis methods have also been developed for the design of rigid-body linkages. Accurate analysis of compliant mechanisms, on the other hand, generally require numerical finite element analysis. In an attempt to bring the analytical and graphical techniques developed for the synthesis of rigid-body mechanisms into the realm compliant mechanisms, Midha and Howell ([6], [7]) developed pseudo-rigid-body model for a class of compliant mechanisms. This model allows a certain class of compliant mechanisms to be approximately represented by spring-loaded rigid-body mechanisms. Flexure-based compliant mechanism form an important subclass of compliant mechanisms that can be represented as spring-loaded rigid-body mechanisms. Howell and Midha [7] showed that the flexure can be approximated by a revolute joint with a torsional spring having linear torque angle characteristics. This type of approximation is popularly know as small-length flexure approximation. Thus, with these models, one can bring in several analytical and graphical rigid-body mechanism design techniques into the realm of compliant mechanisms. Static balance of compliant mechanisms While compliant mechanisms are superior to rigid-body mechanisms in terms of friction and backlash, they have a feature that could be disadvantageous. Figure (1.8 a) shows a compliant mechanism. Figure (1.8 b) shows the mechanism acting on a workpiece. Figure (1.8 c) shows the same mechanism requiring effort even though it

30 CHAPTER 1. INTRODUCTION 1 is not acting on any workpiece. Mere actuation of the compliant mechanism requires effort. The source of this effort is due to the spring-like behaviour of the compliant mechanism arising out of its inherent elasticity. (a) (b) (c) Figure 1.8: A compliant crimper Statically balancing the elastic forces, i.e., nullifying the spring-like behaviour of compliant mechanisms is under increasing attention. The reason for that is not just the reduction in the effort to operate a compliant mechanism but also the prospect

31 CHAPTER 1. INTRODUCTION 11 of having tools that offer good force feedback. If a person handling a tool gets a good sense of the force that the tool is applying on a workpiece, then the tool is said to offer good force feedback. In certain tools consisting of linkages, such as laparoscopic grippers, the force feedback is very bad for the reasons attributed to friction in the joints. Compliant mechanisms on the other hand do not have joint-friction but their spring-like behaviour can affect the force feedback. If this spring-like behaviour can be removed, then compliant mechanisms can offer good force feedback Static balance: rigid-body linkage vs. compliant mechanisms In rigid-body linkages, for static balance, we often want all possible motion that the linkage can take to be effortless. This implies that conservative forces in all its configurations are in equilibrium. As per our definition, all possible motion is not necessary to qualify as static balance. Nevertheless, in this thesis, to comply with the popular notion, we have struck to this all possible motion as far as static balancing of rigid-body linkages are concerned. In compliant mechanisms, which have infinite degree-of-freedom, in contrast to finite degree-of-freedom of rigid-body linkages, it is not feasible to expect all possible motion to be effortless. Only one or two modes of motion (or deformation) in which the compliant mechanism normally operates is sought to be made effortless. In other words, weaimtomakeconfigurations only along one or two paths tobeinequilibrium. 1.2 Motivation for the thesis The motivating idea of this thesis is to make use of rigid-body approximations, such as pseudo-rigid-body model, for static balancing of compliant mechanisms. This motivation was first proposed in [8]. Pseudo-rigid-body model was successful in the synthesis of compliant mechanisms because of the existence of simple analytical techniques for the synthesis of rigid-body mechanisms. Similarly, for pseudo-rigid-body model to be successful in making statically balanced compliant mechanisms, there

32 CHAPTER 1. INTRODUCTION 12 has to be simple analytical techniques for static balancing of spring-loaded linkages rather than gravity-loaded linkages. Priortotheworkofthisthesis,therewasonlyonework,carriedoutbyJustHerder ([1]), that recognized the importance of static balancing of spring-loaded rigid-body linkages. The context of Herder s work ([1]) was compensating elastic forces of a cosmetic glove in a hand prosthesis. Herder approximated the motion of the hand prosthesis as the motion of a four-bar linkage and the elastic forces of the glove as a zero-free-length spring attached to the coupler link of the four-bar linkage, as shown in figure (1.9). Herder then derived a static balancing solution to it as shown in figure (1.1). As can be seen in figure (1.1), the balancing solution involves addition of two auxiliary bodies and two balancing springs. The result of incorporating this balancing solution into an actual hand prosthesis was presented in [9], where reduction in the effort to operate the prosthesis was demonstrated. Coupler (Link 2) E Link 1 C B Link 3 D A K Fixed pivots Anchor point Loading zero-free-length spring (k l ) Figure 1.9: A spring-loaded four-bar linkage to be statically balanced When we thought of using Herder s balancing solution ([1]) through a rigid-body approximation on a compliant four-bar mechanism with flexure pivots, such as the one shown in (1.11), we noticed that incorporating additional bodies of the balancing solution of [1] (see figure (1.1)) into the compliant mechanism could be a little

33 CHAPTER 1. INTRODUCTION 13 Balancing spring 2 E C F T D k l B G Auxiliary links k t Loading spring K A Anchor point k b Anchor point Balancing spring 1 H Figure 1.1: A static balancing solution for the spring-loaded four-bar linkage by addition of auxiliary links and springs cumbersome. This motivated us to take a closer look at the principles of static balancing to see if there are other ways of statically balancing the same spring-loaded four-bar linkage (of figure (1.9)), preferably, without using auxiliary bodies. We did find other ways of balancing the four-bar linkage and one such way does not use auxiliary bodies. We could also get some new general static balancing techniques without using auxiliary bodies. We eventually gave a framework for making use of these static balancing solutions to make statically balanced flexure-based compliant mechanisms through small-length flexure model. An overview of these contributions, which make up the bulk of the chapters in the thesis is presented next.

34 CHAPTER 1. INTRODUCTION 14 R Figure 1.11: A compliant four-bar mechanism 1.3 Scope of the thesis Three methods to statically balance a zero-free-length spring-loaded four-bar linkage For the static balancing problem shown in figure (1.9), apart from the method shown in figure (1.1), which was already in the literature [1], we give two more methods where the number of additional springs and the number of auxiliary links are less than or equal to that of the method shown in figure (1.1). Further, we also recognized a variant of the method given in figure (1.1). Among the methods that we give, one method does not use any auxiliary link. These results are elaborated in Chapter Static balancing of cognates In kinematics, there is a well-known theorem called Roberts-Chebyshev cognate theorem (see [1], for example). According to the theorem, for every four-bar linkage with a specified coupler point on it, one can find two more four-bar linkages and coupler points on them such that the coupler curves of the three four-bar linkages are the same. These three four-bar linkages are termed as cognates. The triangle

35 CHAPTER 1. INTRODUCTION 15 formed by the ground anchor points of cognates, called the cognate triangle, plays a central role in a few other elegant relations that the theorem states. With the intent to extend the theorem to static balancing, we present some relations among static balancing parameters (spring constants and anchor points of balancing springs) of different cognates in which the cognate triangle again plays a central role. These results are elaborated in Chapter Static balancing of revolute-jointed linkages without auxiliary links Building upon a method discussed in Section 1.3.1, we present a general method that can statically balance any revolute-jointed linkage having zero-free-length spring force and constant force interactions between the bodies constituting the linkage. This method adds only zero-free-length springs but not auxiliary links. This result is elaborated in Chapter Static balancing of spatial and/or revolute jointed linkages without auxiliary links Further extending the planar method of Section to spatial linkages, we show that any spatial revolute and/or spherical jointed linkages with zero-free-length spring and constant force interactions can be statically balanced. Again, the method of balancing requires only addition of zero-free-length springs but not auxiliary links. This result is elaborated in Chapter Static balancing of flexure-based compliant mechanisms by addition of springs As pointed in Section 1.2, the motivation to find new methods for static balancing of spring-loaded rigid-body linkages was to make statically balanced compliant mechanisms through pseudo-rigid-body model. Having found the new methods, we give a

36 CHAPTER 1. INTRODUCTION 16 simple step-by-step framework for static balancing of a flexure-based compliant mechanisms through small-length flexure model. We give four examples to illustrate the framework. A prototype of one of the examples is also made. Both simulations and the fabricated prototype show more than 7 % reduction in the effort. These results are described in Chapter 7. Summary Static balance implies equilibrium among conservative forces over a continuous set of configurations. Static balancing of rigid-body mechanisms generally focus on compensating gravity forces. Primary focus in static balancing of a compliant mechanism is to compensate the inherent elastic forces of the compliant mechanism. The motivating idea of the thesis is to use rigid-body models of compliant mechanisms to design statically balanced compliant mechanisms. We recognize that for the idea to be successful, it is necessary to have new analytical static balancing methods for spring-loaded rigid-body linkages. This thesis presents new static balancing methods for spring-loaded linkages without using auxiliary bodies. The thesis also presents a simple framework to use these methods through small-length flexure model for static balancing of compliant mechanisms. The last section of this chapter defined the scope of the thesis and its organization in the remaining chapters.

37 Chapter 2 Literature Survey Overview Literature on static balancing of gravity-loaded rigid-body linkages by adding counter-weights or springs. Literature on the importance of static balancing of spring-loaded rigid-body linkages and the existing methods to handle such problems. Literature on various approaches that have been pursued to address the design of statically balanced compliant mechanisms. 2.1 Rigid-body linkages under gravity loads There are two dominant ways of balancing a linkage under gravity loads. One is by addition of counter weights and the other is by addition of springs. The literature in these fields is described next Counter-weight balancing The simplest of rigid-body linkages is a lever. The conditions for static balance of a lever under gravity loads was first given by Archimedes ([11]). It relies on making the 17

38 CHAPTER 2. LITERATURE SURVEY 18 overall center of gravity of the system to be a constant. This lever-balancing principle has been adapted to pantograph linkages, as in counterweight balanced lamps ([12]). Further, even though we do not have specific references, counterweight balancing of a serial revolute jointed linkage such as the one shown in figure (2.1) seems to have been known for a long time. Such a balancing of serial linkages has been a part of many balancing schemes, such as in [13]. 1 2 W 3 W 3 W 4W 1 2 (a) 2W (b) Figure 2.1: A counter balancing technique for serial revolute jointed linkages Balancing by addition of springs George Carwardine, a British engineer, is the pioneer in the area of static balancing of gravity loaded linkages by addition of springs. In a series of patents he obtained ([14], [15], [16], [17] and [18]) he gave the art of statically balancing a gravity load supported on a two-revolute jointed linkage. The balancing method involved addition of two zero-free-length springs and auxiliary links. The patent for statically balanced Anglepoise lamp ([2]), which is still popular today, is among these patents. At around the same time as Carwardine, American physicist Lucien LaCoste ([3]) recognized that a pendulum could be in perfect static balance when a zero-free-length spring is attached to it. LaCoste is credited with first recognizing the role of zero-freelength springs in perfect static balancing of a gravity-loaded lever. This discovery was made in the context of devising a pendulum with a very long period of oscillation. Such a pendulum is apparently useful in seismographs. Streit and Gilmore [5] made a thorough study of a lever under spring loads. They

39 CHAPTER 2. LITERATURE SURVEY 19 discussed achieving a set of discrete as well as a set of continuous equilibrium configurations. Some of the ways to realize zero-free-length springs were also discussed. Nathan [19] gave a way to extend the spring-based lever-balancing to two-degreeof-freedom linkages using auxiliary parallelogram linkages. Pracht et al. [2] made a slightly different extension where all the springs are anchored from the ground to different parts of the linkage. Streit and Shin [21] made another extension using pantograph linkages. They obtained different design variants by varying input points and input motion. They also suggested that such a gravity-balancer could be used in walking machines so that the torque requirement of the motors is reduced. The attempt of Wongrathanaphisan and Cole [22] to statically balance a load on the coupler of a four-bar linkage led to a solution that is conceptually not different from LaCoste s solution. Nathan [19] also extended his two-degree-of-freedom linkage-balancing to a n- degree-of-freedom revolute-jointed linkage. Figure (2.2) illustrates the method in [19] where, to balance a gravity load on a 3R linkage shown in figure (2.2a), auxiliary linkages (colored in grey) and extra springs are added. Streit and Shin [23] similarly extended the work of Pracht et al. [2] to a n-degree-of-freedom revolute-jointed linkage. Reference [23] termed the extension of Nathan [19] as vertical link systems and their extension of reference [2] as parallel link systems. It also presented static balancing of a linkage having a series of revolute-prismatic pairs of joints W W (a) (b) Figure 2.2: Use of auxiliary links (colored in grey) along with springs in an existing technique for balancing a n-degree-of-freedom linkage under constant load.

40 CHAPTER 2. LITERATURE SURVEY 2 Herder s PhD thesis [4] brought new approaches to the field of spring-based static balancing. Herder obtained a variety of multi-degree-of-freedom gravity-balancing linkages using 1) a few modification rules, 2) a few basic statically balanced linkages, such as the balanced lever of LaCoste, and 3) the properties of auxiliary parallelogram and pantograph linkages. Herder also introduced the concept of a floating suspension. Whatever reaction forces that a pivot exerts on a gravity-loaded lever, the same forces could be exerted by a floating suspension, which is nothing but springs anchored from the ground to the lever. Thus, under quasi-static conditions, a floating suspension could form a friction-less replacement for a pivot. Walsh et al. [24] showed a way to balance a spatial rigid-body attached to the ground by a two degree-of-freedom joint formed by two revolute joints of intersecting axes. Streit et al. [25]), in a similar work, dealt with two degree-of-freedom Hooke s joint in a somewhat different way. Wongratanaphisan and Chew [26] gave a way to balance a more general revolute-jointed two-degree-of-freedom serial spatial manipulator using auxiliary links. Agrawal and Fattah [27] provided an interesting method to balance a spatial gravity-loaded linkage. In the method, by adding auxiliary parallelogram linkages, a physical point that is also the center of mass of the overall system is first identified. Then, depending on the kind of motion this center of mass undergoes, springs are added to compensate the gravity. The work of Rahman et al. [28] gives a straight forward extension of Nathan s vertical link systems to spatial n body revolute-jointed linkages. While Rahman et al. [28] used simple parallelograms in their extension, Lin et al. [29] used spatial RSSR (revolute-spherical-sphericalrevolute) parallelogram linkages to provide a more comprehensive extension. There is a lot of literature on the static balancing of parallel manipulators such as [13], [3], [31], [32], [33], [34], [35], [36], [37] and [38]. While the techniques in the above literature use auxiliary bodies in addition to springs, we (Sangamesh and Ananthasuresh [39]) showed how to balance a n-degreeof-freedom (n 1) revolute-jointed spring and/or gravity linkage using only springs but not auxiliary bodies. Lin et al. [4], using what they called as stiffness matrix approach, also provided balancing methods without auxiliary bodies for two and three revolute-jointed serial gravity-loaded linkages. Shieh and Chen [41] showed how to

41 CHAPTER 2. LITERATURE SURVEY 21 statically balance revolute-jointed planar one-degree-of-freedom closed-loop linkages without using auxiliary links. Our work [39] was extended in [42], which gave more general static balancing conditions. All the techniques discussed above use zero-free-length springs for perfect static balance of the gravity loads. While there are a few works that use other kinds of springs as balancing elements, they are either approximate balancing techniques or use cams to modulate the spring behaviour. Gopalswamy et al. [43] gave an approximate static balancing technique where torsional springs are used as balancing elements. Agrawal and Agrawal [44] presented an approximate static balancing method using non-zero-free-length springs. The balancing techniques that modulate the behaviour of springs include the technique in [45], where a pulley of varying radius was used, and the technique in [46] and [47], where a cam was used. There are a few works dealing with biomedical applications of static balancing. The ones dealing with leg include [48], [49], [5], [51], [52], [53], [54], [55], and [56]. The ones that deal with upper limb include [57], [58], [59], [6], [61], [62] and [63]. 2.2 Rigid linkages under spring loads Herder swork[1]wasthefirsttorecognizetheimportanceofaclassofproblemswhere springs forces, which could be an approximation for more complex elastic forces, need to be compensated. The motivation for the work was to compensate the elastic forces of the cosmetic glove of a hand prosthesis. In this work, the motion of the fingers of a hand prosthesis was modelled as the motion of the coupler link of a four-bar linkage. The elastic forces of the cosmetic glove were lumped into to a zero-free-length spring attached to a point on the coupler. The work then gives a perfect balancing solution for statically balancing this spring-loaded four-bar linkage. The obtained solution is based on extending the balancing of a skew lever to a skew pantograph. Incorporation of the pantograph results in auxiliary bodies being added to the four-bar linkage. Visser and Herder [9] used the solution of [1] in an actual hand prosthesis and gave quantitative data on the reduction of effort in actuation of the fingers.

42 CHAPTER 2. LITERATURE SURVEY 22 Our work [8] was the first to recognize that the compensation methods for springloaded linkages are useful in approximately compensating flexure-based compliant mechanisms as well. The recognition was based on small-length flexure model [64] where flexure-based compliant mechanisms could be replaced by rigid-body linkages under torsional-spring loads. The recognition was also the motivation for our work [8] where we found that a larger set of methods can address the problem of spring-force compensation enunciated in [1]. One of the methods in [8] led us to a more general class of spring-force compensation methods [39]. Reference [39] showed that any n-rigid-body linkage under zerofree-length spring and/or constant force can be compensated by addition of springs. This method does not require addition of auxiliary bodies. The same method was generalized and also presented in a more systematic manner in [42]. In [42], we argued that the method is applicable not only for planar linkages but also for spatial linkages. The small-length flexure model gives a rigid-body linkage loaded with torsional springs. Hence, it may seem natural that there be a focus on finding static balancing methods for rigid-body linkages loaded with torsional springs rather than extension springs, as in [1], [8], [39] and [42]. However, our own attempts at finding an analytical solution to static balancing of torsional spring-loaded linkages did not yield any results. A similar attempt by Radaelli et al. [65] relies on genetic-algorithm-based numerical optimization or a manual search. To facilitate the manual search, which could potentially give some insights, they developed an interactive interface called interactiveparams. In, this thesis, we however approximate the torsional springs by zero-free-length springs through matching a few terms in the Taylor series expansion and then continue to use the analytical methods for balancing zero-free-length spring-loaded linkages. 2.3 Static balance of compliant mechanisms One of the earliest papers that recognized the importance of statically balanced compliant mechanisms is [66]. Design of laparoscopic graspers that can give good force

43 CHAPTER 2. LITERATURE SURVEY 23 feedback to the operator has been the main motivation in the field of statically balanced compliant mechanisms. The papers that specifically focussed on laparoscopic graspers include [67], [68] and [69]. Stapel and Herder [67] showed that a fully compliant, statically balanced grasper is feasible. Tolou and Herder [68] separated a statically balanced compliant grasper into grasper part and balancer part. They focused on obtaining a negative stiffness balancer that can compensate the positive stiffness of compliant grasper. De Lange et al. [7] also used the concept of grasper and balancer. They used topology optimization to match the stiffness characteristics of a grasper and a balancer. Various synthesis strategies for statically balanced compliant mechanisms, not restricted to just graspers, has been addressed in [71], [72] and [73]. Morsch and Herder [71] gave a compliant joint that is statically balanced so that when a linkage is built out of these compliant joints, the linkage would be in static balance. Building block approach is one of the known strategies in the synthesis of compliant mechanisms and Hoetmer et al. [72] explored its extension to static balancing. Rosenberg et al. [73] made use of the results of Radaelli et al. [65] to design flexure-based statically balanced compliant mechanisms. Research in static balance has ventured in tensegrity structures well, as could be seen in [74] and [75]. The work of Guest et al. [76] where they present a zero-stiff elastic shell is worth taking note of. 2.4 Prior Art Priortoourwork, theworkin[1]wastheonlyworkthatdealtwithanalyticalsolution to static balancing of a spring-loaded rigid-body linkage. We now describe the way the solution was arrived at in [1] along with necessary preliminaries that can also be found in [4].

44 CHAPTER 2. LITERATURE SURVEY Preliminaries Statically balanced lever under spring loads Consider a lever under the action of two zero-free-length springs, as shown in figure (2.3). The spring on the left hand side is anchored between points D and P. Let the spring force on the lever at D be a positive constant times the relative displacement of point P with respect to D, i.e., k 1DP. Similarly, the spring force due to right hand spring is k 2DN. Let these forces be resolved along DA and PN. When the constants and anchor points are chosen such that k 1AP = k2an, the resolved forces along PN become zero and the remnant forces along DA gives zero moment on the lever about A. This signifies equilibrium. Moreover, this is true for any configuration of the lever. Hence, we have static balance here. P A k 1 DA k 2 DA k 1AP k 2 AN k 1 D k 2 φ N Figure 2.3: A basic spring force balancer To appreciate the critical role played by zero-free-length of the springs, consider a case where the free-length of the spring is not zero. Then, the spring force on D from A is actually l 1 l 1 l 1 DA where l1 is the free-length and l 1 is the relative distance of the anchor points. The resolved components of the forces are as shown in figure (2.4). Similarly, the resolved components of the right hand side spring are also shown. It may be noted that no matter what strictly positive values l 1, k 1, l 2 and k 2 take and where P and N are placed, the resolved forces along PN cannot cancel at every configuration. The contrast between figures (2.3) and (2.4) in terms of static balance should highlight the necessity of zero-free-length character for perfect static balance.

45 CHAPTER 2. LITERATURE SURVEY 25 l 1 = PD l 2 = ND P k 1 l 1 l 1 l 1 k 1 AP l 1 l 1 l 1 φ A k 1DA l 2 l 2 k 2DA l 2 D l 2 l 2 l 2 k 2 k 2 AN N Figure 2.4: A lever with ordinary springs Statically balanced parallelogram linkage Figure (2.5) shows a parallelogram linkage, ADEN, with two springs attached diagonally between the joints. It may be verified that, for the same φ, the potential energy of this system is the same as that in figure (2.3). If one is statically balanced, so is the other. D E A φ k k N Figure 2.5: Balanced parallelogram Figure (2.6) shows again a parallelogram linkage that now has two degrees of freedom. For the same φ, the potential energy of this system is the same as that in

46 CHAPTER 2. LITERATURE SURVEY 26 figure (2.5). If one is statically balanced, so is the other. E k N D φ A k θ Figure 2.6: Balanced two degree-of-freedom parallelogram linkage Composition of springs In [4], a useful concept of composing two zero-free-length springs into an equivalent one is presented. By referring to figure (2.7), suppose that there is a spring with one end anchored at A and the other end at E. Also suppose that we desire the spring to be anchored at point D rather than at A but modification of the spring is not allowed. In such a case, another spring of spring constant, say k 2, is connected between point E and a point on the ground, say B, such that k 1DA = k2db. When the forces at E are resolved along AD and DE, it can be noticed that forces along AD always cancel out and the net force is (k 1 + k 2 ) ED. The same force is obtained if there were to be a spring of spring constant (k 1 +k 2 ) anchored at D and the other end at E. Hence, the net effect or composition of the two springs anchored at A and B

47 CHAPTER 2. LITERATURE SURVEY 27 is a virtual spring anchored at D. This is an important concept for the techniques presented in this chapter. k 1 DA k 1 ED E k 2 ED k 2 DB A k 1 k 2 D B k 1DA = k2db Figure 2.7: Composition of two zero-length-spring into an equivalent one. The virtual, equivalent spring between E and D is shown in grey color Plagiograph In a plagiograph linkage, the path traced at the output point is a scaled and rotated replica of the path traced at the input point. If the linkage shown in figure (2.8) satisfies the following conditions: Condition 1: PQ = SR and PS = QR so that PQRS is a parallelogram, Condition 2: RQM = NSR (this angle is labelled as α) and RQ = NS MQ RS labelled as m) so that NSR is similar to RQM, (this ratio is then the linkage is called a plagiograph and it can be proved to have the following property: output point N follows the input point M through a scaling and rotation transformation about point P with the scale factor and the rotation angle being m and α. That is, PN = mr α ( PM) (2.1)

48 CHAPTER 2. LITERATURE SURVEY 28 where, R α is the rotation operator, which operates on a planar vector to rotate it by angle α. α M R α N α S Q P Figure 2.8: A plagiograph or a skew pantograph linkage The point, P, about which the scaling and rotation transformations happen, is referred to as the base pivot of the plagiograph. A detailed treatment of plagiographs can be found in [77] and [78] Derivation of the balancing solution in [1] In the derivation, Herder starts with a simple statically balanced parallelogram linkage shown in figure (2.6) and performs a series of modifications to arrive at the statically balanced four-bar linkage (see figure (1.1)) without losing static balance. We now discuss those modifications. In the balanced two degree-of-freedom parallelogram linkage shown in figure (2.6), the spring between A and E is replaced by two springs as per the principle of spring composition/decomposition of Section This replacement is depicted in figure (2.9). As per the principle of composition, the two springs have to satisfy

49 CHAPTER 2. LITERATURE SURVEY 29 k 2 AH k 2 EA E k 1 EA k 1 AK k 2 k 3 N k 1 H A D φ θ K π Figure 2.9: A balanced parallelogram with one its spring along a diagonal decomposed into two springs. k 1 AK = k2 AH (2.2) so that the net effect of the two springs is along AE. Further, the effective spring constant (k 1 +k 2 ) should equal k 3, the spring constant of the spring along DN, in order to have static balance (see figure (2.6)). Next, a parallelogram linkage AFGM is formed that is a duplicate of the linkage ANED. The linkage AFGM undergoes the same motion as the linkage ANED, except for maintaining a constant angle of α between the two, as shown in figure (2.1). We show in figure (2.11) how the duplicate linkage synchronizes its motion with the original linkage. A spring of spring constant k 2 is now attached to the duplicate linkage. The ground anchor point H of the spring is also rotated so that its length remains the same upon shifting to the duplicate linkage. The spring of spring constant k 3 that was in the original linkage along the diagonal not containing A, is also duplicated into the duplicate linkage. However, its spring constant in the original and the duplicate linkage is changed to k 4 and k 5, such that k 4 + k 5 = k 3. Since, (k 1 +k 2 ) is also k 3, for convenience, let us take k 1 = k 4 and k 2 = k 5. It may be

50 CHAPTER 2. LITERATURE SURVEY 3 noted that the potential energy of the spring in figure (2.1) is the same as that in figure (2.9). Hence, the static balance in figure (2.9) implies that the linkage of figure (2.1) is also statically balanced. It may be noted that because of this step, equation (2.2) gets modified to ( ) k 1 R α AK = k 2AH (2.3) where R α is a rotation operator that rotates a vector by an angle α. E G F k 5 α D φ θ k 4 N k 1 K M A π α k 2 H Figure 2.1: The parallelogram linkage with a duplicate that maintains a constant angle of α from the original Figure (2.11) shows how the parallelogram linkages ANED and AFGM can be made to have the same motion except for maintaining a constant angle between them. Here, essentially, a plagiograph has been introduced into the parallelogram linkages. The following conditions have to be satisfied to have the synchronized motion: ED = DC, EDC = α, CF = FG, CFG = α. The parallelogram linkage AFGM is scaled by a factor of m, about point A, as shown in figure (2.12). The plagiograph that maintains the synchronization between

51 CHAPTER 2. LITERATURE SURVEY 31 C E G α F k 5 α D α k 4 N k 1 K M A π α k 2 H Figure 2.11: Synchronization of motion between two parallelogram linkages the two parallelogram linkages gets modified accordingly, as shown in the figure. The two springs attached to it are also scaled by the same factor. This necessitates that the anchor point H be also scaled by the same factor about A. Because of the spatial scaling, the potential energy of the two springs scales by the square of the scaling factor m. To restore the potential energy to that of figure (2.11), the spring constantsofthetwospringsarescaledbyafactorof1/m 2. Sincethepotentialenergies in figure (2.11) and (2.12) are the same, the static balance in figure (2.11) implies the static balance in figure (2.12) too. Because of these modifications, equation (2.3) gets modified to ( ) k 1 R α AK = m 2 1 k 2 AH = mk 2AH (2.4) m As far as the length is concerned, the spring between M and F is an exact replica of the spring between D and N, except for a scale factor m. Hence, one can eliminate the spring between D and N and increase the stiffness of the spring between M and F from k 2 to (k 2 +k 1 /m 2 ) so that the potential energy remains unaltered.

52 CHAPTER 2. LITERATURE SURVEY 32 α E C α F G k 5 = k 2 M α D A k 4 = k 1 N k 1 K π α k 2 H Figure 2.12: Scaling the duplicate parallelogram linkage This alteration is shown in figure (2.13) where two links AN and NE are no longer necessary. Again, static balance has remained unaltered due to unaltered potential energy. As shown in figure (2.14), two more links AM and MG can also be removed by relocating the spring between M and F to be between A and T where T is a point on link GFC such that GF = FT. Since MF = AT, this relocation does not change the potential energy. Hence, the static balance remains intact. Again, adding a link BC, as shown in the same figure does not disturb the static balance. The intent of this addition is to make the four-bar linkage BCDA. This is how the static balancing solution given in figure (1.1) was arrived at in [4]. In this thesis, we take a different approach to arrive at a static balancing solution to the spring loaded four-bar linkage, leading to many more solutions. Further, we give a static balancing technique that is applicable to general n-degree-of-freedom revolute and/or spherical jointed linkages.

53 CHAPTER 2. LITERATURE SURVEY 33 E C α G α F D k 1 k 5 = k 2 + k 1 m 2 M A π α K k 2 H Figure 2.13: Removal of a spring and compensating it with increase in stiffness of another spring Balancing spring 2 E C F T D k l B G Auxiliary links k t Loading spring K A Anchor point k b Anchor point Balancing spring 1 H Figure 2.14: Removal of two more links.

54 CHAPTER 2. LITERATURE SURVEY 34 Summary There is a wide array of literature on static balancing of gravity-loaded rigidbody linkages by addition of counter-weights as well as springs. Prior to our work, Herder s derivation has been the only work in the area of static balancing of spring-loaded linkages. Herder s derivation is based on cascading modification of simple statically balanced mechanism into the final mechanism without losing static balance in any of the modifications. Literature shows that some of the strategies explored for designing statically balanced compliant mechanisms include topology optimization, designing generic zero-stiffness compliant joint and the building block approach. Apart from Chapter 7, the bulk of the thesis is based on our work presented in [8] and [42]. Hence, it may be apt to pictorially summarize our work and the literature in a way that brings out the distinct features of our work. Figure (2.15) shows one such representation.

55 CHAPTER 2. LITERATURE SURVEY 35 Rigid linkages under gravity load Rigid linkages under spring load Compliant mechanism By addition of mass Approximate balancing and cam using methods By addition of springs Perfect balancing methods Torsion load balanced by torsion loads Approximate methods Radaelli et al. (211) Extension spring balanced by extension springs Perfect methods By addition of springs Chapter 7 : Extension of Deepak & Ananthasuresh (212) to compliant mechanisms using pseudo-rigid body model Other strategies Topology optimization (see De Lange et al. (28)) Building block approach (see Hoetmer, et al. (21)) Extension of Radaelli et al. (211) to compliant mechanisms as seen in Rosenberg et al (211) Gopalswamy et al. (1992) Agrawal & Agrawal (25) Ulrich & Kumar (1991) Koser (29) For specific linkages For n - dof linkages For specific linkages Basic spring force balancer (Herder (21)) Herder (1998) Deepak & Ananthasuresh (212) For n -dof linkages LaCoste (1934) Shin & Streit (1991) Walsh et al. (1991) Herder (21) Lin et al. (21) Using auxiliary bodies Without auxiliary bodies Without auxiliary bodies For planar revolute jointed and spatial revolute and/or spherical jointed linkages Streit & Shin (1993) Agrawal & Fattah (24) For planar revolute jointed and spatial revolute and/or spherical jointed linkages Limiting case Original or novel contributions of this thesis dof: degree of freedom Figure 2.15: The current literature and our contributions

56 Chapter 3 Static balancing of a four-bar linkage loaded by a spring Overview Static balancing of a spring-loaded four-bar linkage by incorporating a parallelogram linkage. Static balancing without auxiliary bodies. Static balancing with the incorporation of a plagiograph linkage. A prototype demonstrating static balancing without auxiliary bodies. 3.1 Introduction The problem that is addressed in this chapter is static balancing of a four-bar linkage loaded with a zero-free-length spring connected between its coupler point and a point on the ground. Figure (3.1) shows one such four-bar linkage. There is already one method [1] whose derivation was discussed in detail in Section The motivation for attempting to find more methods, as discussed in the introduction, is to have a suitable set of balancing methods that could be used in making statically balanced 36

57 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 37 compliant mechanisms using pseudo-rigid-body models. All the results of this chapter are documented in our paper [8]. Coupler (Link 2) E Link 1 C B Link 3 D A K Fixed pivots Loading zero-free-length spring (k l ) Anchor point Figure 3.1: A four-bar linkage with a zero-free-length spring anchored from its coupler to the ground 3.2 Static balancing of a given spring-loaded fourbar linkage Consider a four-bar linkage loaded by a zero-free length spring at its coupler point as shown in figure (3.1). Before adding auxiliary links and balancing springs to balance it, let us see if there is a possibility of it being in static balance as such. Figure (3.2) shows the coupler curve traced by point E. If the linkage has to be in equilibrium at every configuration, then the potential energy of the spring has to be the same at every configuration. This is possible only when every point of the coupler curve is at a constant distance from K, i.e., when the coupler curve is a circle centered at K. In

58 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 38 general this is not true except in extreme cases such as when E coincides with D (or C) and K coincides with A (or B). E C B D A k l K Figure 3.2: Possibility of the four-bar linkage being statically balanced as it is Next, various possibilities of static balancing of a four-bar linkage are considered by adding auxiliary links and springs. Only those possibilities where the number of additional balancing springs and auxiliary links is less than or equal to that of [1] are explained. As can be seen in figure (1.1), [1] used two auxiliary links and two balancing springs Technique 1 Motivated by the observation in figure (2.6), the four-bar linkage can be balanced by creating a balanced parallelogram as shown in figure (3.3). Here, four auxiliary links and one balancing spring are used. But the number of auxiliary links can be reduced to two as explained next. Reducing the number of auxiliary links As shown in figure (3.4), if T is a point on the (extended) link EP, such that EP = PT, then TKQP forms a parallelogram. Relocation of the spring between P and Q to be between T and K does not change its potential energy and hence static balance is undisturbed. The advantage of the relocation is that the two auxiliary links KQ

59 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 39 E C Q k l D P k l B A K Figure 3.3: A balanced parallelogram on the load spring and QE are unnecessary. This way of balancing forms this chapter s technique 1 to statically balance a four-bar linkage. E C Q D P k l B A K k l (balancing spring) T l 1 = PE l 2 = PK Balancing parameters Figure 3.4: Technique 1: Static balancing with two auxiliary links and one balancing spring The characteristics of technique 1 can be summarized as follows: 1. This way of balancing does not induce any stress in any of the links of the four-bar linkage.

60 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 4 2. The balancing variables are l 1 and l 2, i.e., lengths of PE and PK, as shown in figure (3.4). l 1 and l 2 can be of any convenient value Technique 2 Instead of adding four links to make up a parallelogram as in figure (3.3), if links ED anddaareusedtoformtheparallelogramasshowninfigure(3.5), thentwoauxiliary links can be avoided. This parallelogram can balance a spring connected between its opposite vertices. While the load spring is not between its opposite vertices, another spring(labelled as balancing spring 1) is added so that the resultant of its composition with the load spring is between the opposite vertices of the parallelogram, as shown in figure (3.6). Recall the composition property of springs noted in Section E C F D k l K B A Figure 3.5: Forming a parallelogram using two auxiliary links. The virtual equivalent spring between A and E in figure (3.6) is balanced by adding another spring (labelled as balancing spring 2) across the other two opposite vertices of the parallelogram as shown in figure (3.7). Thus, the equivalent spring and balancing spring 2 are in static balance. Equivalently, the load spring, balancing spring 1 and balancing spring 2 are in static balance. With this, the linkage is in static balance with two auxiliary links and two balancing springs. The auxiliary links can be eliminated, as explained next. As shown in figure (3.8), if T is a point on link DE (extended if necessary) satisfying ED = DT, then AFDT is always a parallelogram and DF = TA. Hence,

61 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 41 Balancing spring 1 C Equivalent spring k t F E B H k b k b AH = kl KA, D A kt = k l + k b k l K Figure 3.6: A virtual spring between opposite vertices of the parallelogram AF ED. Balancing spring 1 Balancing spring 2 E C k t F B H k b k b AH = kl KA, D A kt = k l + k b k l K Figure 3.7: Adding balancing spring 2 across opposite vertices of the parallelogram to balance the load spring and balancing spring 1.

62 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 42 relocation of the spring between D and F in figure (3.7) to be between T and A as in figure (3.8), does not change its potential energy. Consequently, the static balance is also undisturbed. Furthermore, auxiliary links AF and F E are no longer required. This way of balancing shown in figure (3.8), requiring two balancing spring but no auxiliary link, constitutes the second technique to balance a four-bar linkage. Balancing spring 1 C Balancing spring 2 F E B H k b D A k l K T k t TD = DE k bah = klka, kt = k l + k b Figure 3.8: Option 2 for balancing the four-bar linkage without requiring auxiliary links This technique has two options. In figure (3.5), a parallelogram was completed out of links AD and DE. One could have completed the parallelogram out of links BC and CE as well and proceed in a similar manner. In the former case we say that the balanced parallelogram is based at A and in the later case, it is based at B. Among the balancing parameters of this technique, the anchor point of balancing spring 2 is always one of the fixed pivots of the four-bar linkage. The remaining balancing parameters: (i) H, the anchor point of balancing spring 1, (ii) k b, the spring constant of balancing spring 1, and (iii) k t the spring constant of balancing spring 2, have to be solved from equations included in figure (3.8). Those equations

63 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 43 can be rewritten in a general form so that they are applicable to both the options of this technique. If W is where the balanced parallelogram is based (i.e., W is A or B) and K is the anchor point of load spring, then k b = k l KW WH k bwh = klkw (3.1) (positive if WH is along KW, and negative otherwise) (3.2) k t = k b +k l (3.3) Equation(3.1)saysthatH hastolieonalinestartingfromw andalongthedirection of KW. The line is called the straight-line locus of H. Once a point on this line is chosen as H, k b and k t get determined as per equations (3.2) and (3.3). Thus, the set of solutions to (H,k b,k t ) is a one-parameter family. This choice can be used to fulfill practical considerations without compromising static balancing Technique 3 As mentioned earlier, the only known method in literature to statically balance a spring-loaded four-bar linkage was described by [1]. In [1], a plagiograph is first statically balanced and then it is modified to obtain balanced four-bar linkage with two auxiliary links and two balancing springs as shown in figure (1.1). It is now shown that the balancing arrangement obtained by [1] can also be obtained by combining technique 2 of this chapter with the concept of plagiograph. Whereas this later approach provides four options, the approach of [1] provides only two of these options. The later approach constitutes the third technique of this chapter. The technique and its options are described next. In technique 2, the balancing spring 1 was connected to the same point as that of the loading spring the coupler point E. If the coupler point is not accessible to the balancing spring 1, then another point to anchor the spring has to be found. It is also desirable that the motion of the other point is related to the coupler point. By taking a cue from Section 2.4.2, a plagiograph can be completed out of two links of the given four-bar linkage, as shown in figure (3.9a) so that point G follows a scaled

64 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 44 and rotated locus of E. ADCF is a parallelogram CFG = EDC = GFC CDE CD = GF ED CF } E E B G C F Additional links AG = mr α ( The fixed point of the plagiograph AE) (scaling and rotation transformation) D A k l K B (kl) m 2 G F C AL = mr α ( AK), L D A m = CD is the scale factor ED α = EDC (anticlockwise) is the rotation angle A is the reference point for scaling and rotation (a) (b) Technique 2 Final balancing arrangement E E B G F C L T kt D A k l K B G F C L T kt D A k b k b AL = mr α ( AK) kl m LA = 2 kbah k t = kl m FT = + k 2 b GF H (d) kl m LA = 2 kbah k t = kl FT = + k m 2 b GF H (c) Figure 3.9: Technique 3 Whatever scaling and rotation transformation of E that G follows, the same transformationisappliedtok toobtainl,i.e., AL = mr α ( AK),asshowninfigure(3.9b). If a spring is attached between L and G, as in figure (3.9b), then the spring is a scaled (byfactorm)androtatedcopyoftheloadspringinfigure(3.9a). Further,ifthespring constantofthespringinfigure(3.9b)is 1 m 2 timesthespringconstantoftheloadspring spring, i.e., k l m 2, then its potential energy k l 2m 2 (LG) 2 = k l 2m 2 (m(ke)) 2 = k l 2 (KE)2 is

65 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 45 the same as that of the load spring. Therefore, the potential energies of the spring loads in figures (3.9a) and (3.9b) are the same. As a consequence of this, any extra spring and auxiliary link addition that balances the spring load in figure (3.9b) would also balance the spring load in figure (3.9a). In figure (3.9b), one may think of links BC, CF, and FA to constitute a four-bar linkage loaded with a spring at its coupler point G. Application of technique 2 of static balancing to this spring-loaded linkage leads to addition of two extra springs, as shown in figure (3.9c). The same two springs would statically balance the spring load of figure (3.9a) also. Balancing the given spring load of figure (3.9a) by adding the balancing spring loads of figure (3.9c), as shown in figure (3.9d), is the third technique to balance a spring-loaded four-bar linkage. It may be noted in figure (3.9d) that none of the balancing springs are connected to the coupler point E. When the four-bar linkage of figure (3.9b) was balanced using technique 2, there were two options: balanced parallelogram based at A or based at B (see the section (3.2.2) describing technique 2). Furthermore, when a plagiograph was completed out of the four-bar linkage in figure (3.9a), the base pivot of the plagiograph was at A. We can also complete a plagiograph out of the same four-bar linkage so that the base pivot is at B, as shown in figure (3.1). Thus, we have 2 2 = 4 options base pivot of plagiograph at A or B, and balanced parallelogram based at A or B. In the technique of [1], the base pivot of the plagiograph and the balanced parallelogram always coincide. Therefore, only two of the above four options are derivable from the technique of [1]. The following balancing parameters: the anchor point of balancing spring 1 (H), spring constant of balancing spring 1 (k b ) and spring constant of balancing spring 2 (k t ), have to be solved from the equations provided in figure (3.9d). If U is the base pivot of the plagiograph (instead of A) and W is the point where balanced parallelogram is based(instead of A), then the equations can be generalized as follows,

66 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 46 BFDC is a parallelogram ECD = DFG = DFG ECD CD CE = FG FD Additional links B } C F G D A k l E K The fixed pivot of the plagiograph BG = mr α ( BE), where m = CD CE and α = ECD clockwise Figure 3.1: Plagiograph with base pivot at B

67 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 47 so that they are applicable to any of the four options of this technique. UL = mr α ( UK) L is the anchor point of the equivalent load spring (3.4) WH = k l LW m 2 k b k b = k l(lw) m 2 (WH) k t = k b + k l m 2 (3.5) (positive if WH is along LW, negative otherwise) (3.6) (3.7) m and α in the above equations are found from figure (3.9a) or (3.1) depending and whether U is A or B. Equation (3.5) indicates that H can lie anywhere on a line along LW and passing through W. The balancing spring constants k b and k t are functions of the position of H on this line as per equations (3.6) and (3.7). Therefore, the set of solutions for (H,k b,k t ) is a one-parameter family of solutions. The main features of the three techniques presented in this section as well as the Herder s method are summarized in Table 3.1. Technique 1 Technique 2 Technique 3 Herder s Technique Number of auxiliary links Number of balancing springs Variable balancing parameters l 1, l 2 H, k b, k t H, k b, k t H, k b, k t Family of feasible balancing two- one- one- one- parameters parameter parameter parameter parameter Number of options one two four two Table 3.1: Summary of different techniques to statically balance a spring-loaded fourbar linkage presented in this chapter

68 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 48 Figure 3.11: A prototype of four-bar linkage that is statically balanced using option 2. The four-bar linkage seen in the above figure is a Watt s straight-line mechanism. 3.3 A prototype with technique 2 A prototype of a spring-loaded four-bar linkage balanced using technique 2 is shown in figure (3.11). The zero-free-length springs used in the prototype were realized using pulley and string arrangement described in [4]. In the pulley and string arrangement (see figure (3.12)), a string attached to a spring is passed over a small pulley anchored to the ground such that when the spring deflection is zero, the end of the string is at the pulley. Because of this, the distance between the end of the string and the pulley is the same as the deflection of the spring. Since force exerted by the string-end is along the line joining the pulley and the string-end, and of magnitude equal to the deflection of the spring, a zero-free-length spring anchored at the pulley is realized.

69 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 49 Pulley k Undeflected spring F = kx x Figure 3.12: Realization of a zero-free length spring x The prototype design is such that the ratio of spring constants of loading spring, balancing spring 1 and balancing spring 2 is 1 : 1 : 2. To realize these three springs, three identical springs were taken. While two springs were as it is used within springpulley arrangement, only half the length of the spring was used for the remaining spring, as can be seen in the right hand side of figure (3.11). It may be noted that the anchor points of loading spring and balancing spring 1 on the coupler link are the same. Further, the anchor point of loading spring and the anchor point of balancing spring 2 on the coupler link are located symmetrically with respect to a revolute joint on the coupler link. It may further be verified that the prototype satisfies equations (3.1) (3.3). To test the prototype, the linkage is first manually constrained to be in a configuration. Then the behaviour of the linkage is observed after removing the constraint. These type of trials were conducted at several configurations in two sets. In the first set, one or both the balancing springs were removed. In the second set, the original as well as the two balancing springs were in place. In the first set, it was noted that the linkage does not sustain its configuration after the manual constraint is removed. It springs back to one or two specific configurations. In the second set, the linkage sustains its configuration even after the removal of the manual constraint. Inspite of the presence of friction, the difference in the behaviour of the two sets of trial strongly suggest that the prototype with the balancing springs is in static balance.

70 CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 5 Summary In general, a four-bar linkage loaded by a spring at its coupler point is not in static balance. Three new techniques to static balance the four-bar linkage were recognized. In the first technique, a balanced parallelogram linkage was made out of the loading spring followed by relocation of a balancing spring to reduce the number of auxiliary bodies. In the second technique, composition of springs was used to make a balanced parallelogram linkage out of the given four-bar linkage itself. Rest of the steps are the same as those of the first technique. In the third technique, by incorporating a plagiograph linkage, a new statically equivalent virtual-spring-loaded four-bar linkage was obtained. This was followed by applying the second technique to this equivalent four-bar linkage. A prototype demonstrating the second technique was made. The number of balancing springs and auxiliary bodies, respectively, for the three techniques are {one, two}, {two, zero}, and {two, two}.

71 Chapter 4 Static balance of the cognates of a four-bar linkage Overview Cognates of a four-bar linkage and its properties Quest for a unified parameterization of static balancing variables for the three cognates. A geometric problem and its solution leading to a unified parameterization that may be considered as an extension of Roberts-Chebyshev cognate theorem. 4.1 Introduction Cognates In kinematics of rigid-body linkages, it is well known that for a given four-bar linkage and a coupler point on it, one can find two more four-bar linkages and corresponding coupler points such that the coupler curves of all the three four-bar linkages are the same. The three four-bar linkages are called cognates of each other. Figure (4.1) shows a set of three cognates: B 1 C 1 D 1 B 2, B 2 C 2 D 2 B 3 and B 3 C 3 D 3 B 1. In the 51

72 CHAPTER 4. STATIC BALANCE OF THE COGNATES 52 configuration shown in the figure, the coupler points are coincident at E. The triangle formed by the ground anchor points, B 1 B 2 B 3, is called the cognate triangle. There are a few interesting properties involving this triangle, as could be seen in [1]. One such property is that C 1 D 1 E, EC 2 D 2, and D 3 EC 3 are similar to the cognate triangle B 1 B 2 B 3. The cognates and their properties are summarized in what is called Roberts-Chebyshev Cognate theorem (see [1]). C 3 B 3 K γ 1 γ 3 E k l D 3 γ 2 γ 1 γ 2γ3 C 2 γ1 D 2 γ 2 γ 3 C 1 D 1 B 1 B2 Figure 4.1: The cognates of a four-bar mechanism taking a load spring along their common coupler curve Static balancing parameters and the cognates A designer seeking to statically balance a given spring-loaded four-bar linkage has to evaluate a family of feasible balancing parameters for each of the options under the three techniques (see table 3.1) before choosing the one that best meets the design requirements, including practical considerations. A more general design problem

73 CHAPTER 4. STATIC BALANCE OF THE COGNATES 53 wouldbetodesignafour-barlinkagethatwouldguidetheloadspringalongaspecified path and then statically balance the linkage so that the load spring can be moved along the path effortlessly. If a four-bar linkage with its coupler curve matching the specified path is found, then by Roberts-Chebyshev cognate theorem, it follows that there are two more four-bar linkages whose coupler curves also match the specified path. In order to make the best design choice, the designer has to evaluate feasible balancing parameters of options under techniques 2 and 3, described in Chapter 3, on all the three cognates. This chapter presents a result that will aid the designer to evaluate these feasible balancing parameters on all the cognates in a unified manner. As far as technique 1 is concerned, balancing parameters can be evaluated independent of the four-bar linkage. Since three four-bar linkages are considered in this chapter, balancing parameters of techniques 2 and 3, such as H, k b, k t, W, U, and L, are subscripted with 1, 2, or 3, to correspond to the first, the second or the third cognate. Since all the cognates are evaluated for the same load spring, K and k l are the same for all the cognates. Among the balancing parameters, it is sufficient to focus only on H i, k bi, and k ti, i = 1, 2, or 3, since the remaining parameters are not variables. Nevertheless, to determine the locus of H i and the relation between H i, k bi and k ti (see equations ( ) and equations ( )), it is necessary to know the remaining parameters. While the remaining parameters can be deduced from the details of loaded i th cognate linkage, table 4.1 shows that the detail is not necessary if the cognate triangle and the ground anchor point of the loading spring are known. Figure (4.2) shows the anchor point of a loading spring and a cognate triangle where each side corresponds to a cognate. Sides corresponding to cognates 1, 2 and 3 are labelled as s 1, s 2 and s 3, respectively. When an option of technique 2 or 3 is applied on all the cognates with the cyclic symmetry, the straight-line loci of H 1, H 2 and H 3, which are labelled as l 1, l 2 and l 3, could be drawn, as illustrated in figures (4.3), (4.4) and (4.5). Note that for both techniques, the origin for l 1, l 2 and l 3 is one of the vertices of the cognate triangle and there is one-one and onto relation between the origins and the vertices. Because of this feature, it is now shown that H 1, H 2 and H 3, can be varied as if they are at the vertices of the cognate triangle that is

74 CHAPTER 4. STATIC BALANCE OF THE COGNATES 54 Table 4.1: Deduction of various quantities in equations ( ) from the cognate triangle Quantities other than H, k b and k t in equations ( ): K, U, W, m, α, L K Anchor point of the loading spring W Base pivot of the balanced parallelogram Coincident with a fixed pivot of the four-bar linkage One of the vertices of the cognate triangle fixed pivots of cognates form the cognate triangle (see figure (4.1)) U Base pivot of the plagiograph One of the vertices of the cognate triangle (for the same reasons as of W) m Ratio of two sides of the coupler triangle (see figures (3.9a) and (3.1)) Ratio of two sides of the cognate triangle the cognate triangle is similar to the coupler triangle (see figure (4.1)) α An angle of the coupler triangle (see figures (3.9a) and (3.1)) An angle of the cognate triangle ( cognate coupler ) L Anchor point of the equivalent of the loading spring (see figure (3.9b)) Found using the equation UL = mr α ( UK). (U, m, α and K are as above.) undergoing scaling and rotation transformation about a fixed point, while remaining at the same time on their respective loci l 1, l 2 and l 3. When H 1, H 2 and H 3 are varied as noted above, k b1, k b2 and k b3 also vary, but it would be shown that their ratio remains constant. This result follows from a solution that we give in this chapter to a problem in geometry. The way we formulate the problem and find its solution are described next. 4.2 A geometric problem and its solution Given a triangle and three straight lines originating from its three vertices, find three points on the three lines so that they from a triangle that is similar to the given triangle. With respect to figure (4.6), the problem may be stated as: find points S a, S b and S c on straight lines a, b and c respectively, so that S a S b S c is similar to ABC.

75 CHAPTER 4. STATIC BALANCE OF THE COGNATES 55 B 3 s 2 s 3 B 1 s 1 B 2 Cognate triangle Common anchor point of load springs K Figure 4.2: A cognate triangle and the ground anchor point of a load spring l 1 B 3, W 2, U 2 L 2 l 2 s 3 s 2 L 3 B 1, W 3, U 3 s 1 B 2, W 1, U 1 L 1 l 3 U 1, W 1 : B 2 U 2, W 2 : B 3 U 3, W 3 : B 1 K Figure 4.3: Technique 3: the base pivot of the plagiograph (U) and the base of the balanced parallelogram (W) for every cognate are not coincident.

76 CHAPTER 4. STATIC BALANCE OF THE COGNATES 56 U 1 : B 2 ; W 1 : B 1 U 2 : B 3 ; W 2 : B 2 U 3 : B 1 ; W 3 : B 3 l 3 B 3 L 2 s 2 L 3 s 3 l 1 B 1 s 1 B 2 L 1 l 2 K Figure 4.4: Technique 3: the base pivot of the plagiograph (U) and the base of the balanced parallelogram (W) for every cognate are coincident.

77 CHAPTER 4. STATIC BALANCE OF THE COGNATES 57 l 2 l 1 l 3 B 3, W 2 s 2 s 3 B 1, W 3 s 1 B 2, W 1 W 1 : B 2 W 2 : B 3 W 3 : B 1 K Figure 4.5: Technique 2: the base of the balanced parallelogram (W) at the indicated vertices of the cognate triangle.

78 CHAPTER 4. STATIC BALANCE OF THE COGNATES 58 a B b C A c Figure 4.6: A problem in planar geometry The method to obtain S a, S b and S c varies for the following three cases, as described next Case (i): a, b and c are parallel. S a, S b and S c can be obtained by translating A, B and C along the parallel lines, as shown in figure (4.7), so that S a, S b and S c stay on lines a, b and c while S a S b S c is congruent to ABC. b S b a c S c S a C B A Figure 4.7: Finding S a, S b, and S c in case (i)

79 CHAPTER 4. STATIC BALANCE OF THE COGNATES Case (ii): a, b and c are concurrent. S a, S b and S c can be obtained by scaling A, B and C about the point of concurrence, as shown in figure (4.8), so that S a, S b and S c stay on lines a, b and c while S a S b S c is similar to ABC. a b c S b B S a S c C A Figure 4.8: Finding S a, S b, and S c in case (ii) Case (iii): a, b, and c are neither parallel nor concurrent. If the geometric problem does not fall under case (i) or case (ii), such as the one in figure (4.6), then it falls under case (iii). In this case, as described later in section (4.2.4), it is possible to find a non-zero angle η such that when lines a, b and c are rotated by the same angle η about points A, B, and C, respectively, the lines become concurrent at a point, as depicted in figure (4.9). The concurrent point is named as the focal pivot. In figure (4.9) the focal pivot is denoted as P. In order to find a solution to S a, S b and S c, rotate triangle ABC, about point P, by some angle δ, to obtain A 1 B 1 C 1, as shown in figure (4.1). If the respective intersection points of lines PA 1, PB 1, and PC 1 with the lines a, b, and c, are A 2, B 2, and C 2, as shown in figure (4.1), then it is proved in the following paragraph that A 2 B 2 C 2 is similar to ABC. Points A 2, B 2 and C 2 form a solution to S a, S b and S c, because, in addition to similarity between A 2 B 2 C 2 and ABC, by definition, A 2, B 2 and C 2 lie on a, b and c respectively. Different values of angle δ leads to different solutions.

80 CHAPTER 4. STATIC BALANCE OF THE COGNATES 6 b a η B η C η A c P Figure 4.9: Description of focal pivot b a S b S a η B B 1 A 1 η C η A c S c C 1 δ δ δ P Figure 4.1: Points A 2, B 2 and C 2 form a solution to S a, S b and S c

81 CHAPTER 4. STATIC BALANCE OF THE COGNATES 61 Proof of similarity of A 2 B 2 C 2 and ABC: A 2 PA, B 2 PB and C 2 PC are similar to each other since the angles δ and η are common to all of them. Hence, PA 2 PA = PB 2 PB = PC 2 PC (4.1) Substitution of PA = PA 1, PB = PB 1 and PC = PC 1 ( A 1 B 2 C 1 is rotation of ABC about P) in equation (4.1), leads to PA 2 PA 1 = PB 2 PB 1 = PC 2 PC 1, which implies that A 2 B 2 C 2 is a scale of A 1 B 1 C 1 (about P), which in turn is a rotation of ABC (about P). Hence, A 2 B 2 C 2 is similar to ABC and can be visualized as a rotated scaling of ABC about the focal pivot P Finding the focal pivot Let lines a, b and c be rotated by the same angle, say β, about points A, B and C, respectively. If for some β, the three lines become concurrent, then by definition, β = η and the point of concurrence is the focal pivot. If I a,b is the intersection point of a, b and I c,a is the intersection point of c, a, then at the concurrence, I a,b and I c,a meet. I c,a and I a,b can meet at only the intersection of paths traced by I a,b and I c,a, during rotation. The path traced by I a,b is a circle passing through A, B and the original position of I a,b (see the theorem in appendix B). Similarly, the locus of I c,a is a circle passing through C, A and the original position of I c,a. The two loci are shown in figure (4.11). Locating the focal pivots in all the possible types of intersection between the loci is addressed as follows. I a,b or I c,a will not exist when the a and b, or c and a are parallel and hence the I a,b and I c,a circles cannot be drawn: Finding the focal pivot is necessary only in case (iii), where all the three of a, b and c are not parallel to each other. Hence, it is possible to find at least two pairs among a, b and c, which are not parallel to each other. Those pairs may be taken as {a,b} and {c,a}, for which I a,b and I c,a exists. I a,b circle and I c,a circle are coincident: It is shown in appendix (A.1) that if I a,b circle and I c,a circle are coincident then a, b and c before rotation (at β = )

82 CHAPTER 4. STATIC BALANCE OF THE COGNATES 62 have to be concurrent which means that the problem falls under case (ii). Since the procedure to find the focal pivot is used for only case (iii), the possibility of coincidence of I a,b and I c,a circles does not arise. I a,b circle and I c,a circle intersect at two distinct points: This is the generic possibility and is illustrated in figure (4.11). One of the intersection points is always A. The other intersection point is denoted as M. In appendix (A.2), it is shown that if rotation angle β is such that I a,b is at M, then for the same β, I c,a is also at M. On the other hand, if β is such that I a,b is at A, then for the same β, I c,a cannot be at A. Hence, it can be concluded that M is the one and only one focal pivot. b a I a,b B I b,c C I c,a A c M Figure 4.11: Geometric construction to find the focal pivot I a,b circle and I c,a circle touch each other at a single point: This is just a limiting case of two distinct intersection points A and M, of the previous possibility, merging into one. Here also, M is the only focal pivot but it happens to coincide with A. Thus, to find the focal pivot for case (iii), assuming that non-parallel pair of lines are {a,b} and {c,a}, one should draw two circles: one passing through A, B and I a,b (at β = ) and the other passing through C, A, I c,a (at β = ). The two circles either

83 CHAPTER 4. STATIC BALANCE OF THE COGNATES 63 intersect at two distinct points: M and A, or touch at the point A. In the former case, the focal pivot is M and in the later case it is A. 4.3 The result The common properties of solution to S a, S b and S c in all the possible cases presented in sections 4.2.1, and 4.2.3, are summarized below. Property 1: The set of possibilities for S a, S b and S c constitute a one-parameter set. A convenient parameter parameterizing the set in case(i) is the translation, in case (ii) is the scale factor, and in case (iii) is the rotation angle δ. Property 2: The ratio AS a : BS b : CS c is the same throughout the oneparameterset,eventhoughas a,bs b andcs c themselvesvaryovertheset. The ratio is 1 : 1 : 1 in case (i), PA : PB : PC in case (ii), and again PA : PB : PC in case (iii) (see equation (4.1) where A 2, B 2 and C 2 are solutions to S a, S b and S c ). These properties are now applied to the cognate triangle and loci l 1, l 2, and l 3, as described next A parameterization of the balancing parameters of the three cognates that has cognate triangle related invariants The solutions given in section (4.2) are now applied to the cognate triangle and loci of H shown in figures ( ), by taking A, B, C, a, b, c, S a, S b, S c as W 1, W 2, W 3 (the vertices of the cognate triangle), l 1, l 2, l 3, H 1, H 2, H 3, respectively. Then, the properties at the end of section (4.2) take the following form: There is a one-parameter parameterization of the balancing parameters of the three cognates where H 1 H 2 H 3 is similar to W 1 W 2 W 3 (the cognate triangle), and

84 CHAPTER 4. STATIC BALANCE OF THE COGNATES 64 the ratio W 1 H 1 : W 2 H 2 : W 3 H 3 = W 1 P : W 2 P : W 3 P or 1 : 1 : 1 is the same for the entire one parameter set of possibilities. The second property is used in table (4.2) to rewrite the ratio of spring constants of the first balancing spring in the three cognates. It is seen that the ratio involves only constants and hence the ratio itself is invariant for this one-parameter set. It may further be noted that all these constants are derivable just with the knowledge of the cognate triangle and the location of anchor point of the loading spring. With this follows the final result of this section: The balancing parameters of the three cognates, when an option of technique 2 or 3 is applied on all of them, form a one-parameter family where 1. The triangle of anchor point of balancing spring 1 ( H 1 H 2 H 3 ) of the three cognates is proportional to the cognate triangle over the entire family. The two triangles are related by a combination of scaling, rotation and translation transformations. 2. The ratio of spring constants of balancing spring 1 in the three cognates (k b1 : k b2 : k b3 ) is the same throughout the one-parameter family. To calculate this ratio, the knowledge of the location of the anchor point of the loading spring and the cognate triangle is enough. Table 4.2: Ratio of spring constants of balancing spring 1 for three cognates Technique k b1 : k b2 : k b3 k b1 : k b2 : k b3 rewritten using W 1 H 1 : W 2 H 2 : W 3 H 3 = W 1 P : W 2 P : W 3 P or 1 : 1 : 1 3 k l (L 1 W 1 ) : k l (L 2 W 2 ) : k l (L 3 W 3 ) m 2 1 (W 1H 1 ) m 2 2 (W 2H 2 ) m 2 3 (W 3H 3 ) (see equation (3.6)) or (L 1W 1 ) m 2 1 (L 1 W 1 ) : (L 2 W 2 ) : (L 3 W 3 ) m 2 1 (W 1P) m 2 2 (W 2P) m 2 3 (W 3P) : (L 2W 2 ) m KW k 1 KW l W 1 H 1 : k 2 KW l W 2 H 2 : k 3 l W 3 H 3 KW W 1 K W 2 K (see equation (3.2)) Case (ii) of section (4.2) applies : (L 3W 3 ) m 2 3 = 1 : 1 : 1 W 3 K K is the same as P in this case It is believed that the above parameterization would help a designer to visualize balancing parameters of all the cognates together, when an option of techniques 2 or 3

85 CHAPTER 4. STATIC BALANCE OF THE COGNATES 65 is being evaluated. This result may be seen as an extension of the Roberts-Chebyshev cognate theorem to static balancing. Summary Cognates of a four-bar linkage have the same coupler curve. When static balancing parameters of the three cognates are considered together, one can have a unified parameterization with properties related to the cognate triangle. As the unified parameter varies, the triangle formed by the anchor points of the balancing springs remain similar to the cognate triangle. Under the unified parameterization, the ratio of spring constants of a set of corresponding balancing springs is constant. The constant ratio can be geometrically determined from the knowledge of the cognate triangle and the anchor point of the balancing spring. In summary, the results of this chapter may be considered as an extension of the classical theorem on cognates to static balancing.

86 Chapter 5 Static balancing of planar linkages without auxiliary bodies Overview Statically balancing a lever under spring loads and gravity loads. Statically balancing the rotational motion of a rigid-body that is free to move in a plane by addition of springs. Transforming potential energy dependence from one body to another body through a common point of the bodies. Recursive application of the transformation to statically balance any revolutejointed linkage that is under gravity and or spring loads. Examples demonstrating the recursive method. 5.1 Introduction Figure (5.1a) shows a 4R linkage under a gravity load on its last link. This load is balanced by counterweights added all through the linkage. Note the pattern of weights; infact this balancing method extends to a general n-body revolute-jointed 66

87 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 67 linkage as well. The important fact for us is that there are no auxiliary bodies added unlike some of the techniques seen in [12] (see figure (1.4)). The fact that this method of balancing a general n-body linkage is known can be gleaned from the video [ Suppose that instead of weights being the counter balancer, we require springs to be the counter balancer with the other features applicability to n-body linkage and non-usage of auxiliary bodies remaining. As is evident from the chapter on literature survey, there are no methods in the literature to fulfil this requirement. Similarly, there are no methods in the literature on balancing of spring loads through addition of other spring loads. This chapter fills these gaps in the literature. The relevance of these methods in gravity balancing as well as balancing elastic forces is already explained in the introductory chapter of the thesis. For simplicity, we use constancy of the potential energy as the basis in deriving the methods. This also helps us to give a rigorous proof on the need for zero-free-length springs. We consider balancing conditions for (1) a lever, (2) a rigid body in a plane and (3) eventually for a general n-body revolute-jointed linkage. 5.2 Balancing a lever Consider a lever pivoted to the ground, as shown in figure 5.2. The configuration of the lever with respect to the global frame of reference (X Y) can be described by θ, which is the angle from the global frame to the local frame of reference of the lever. Figure 5.2 also shows two kinds of loads: (1) a spring attached between a point of the lever and a point of the global frame, and (2) a constant force acting at a point on the lever. Here, a constant force means that the force has a constant direction with respect to the global frame and a constant magnitude. A complete specification of the spring load would involve (1) the spring constant, denoted by k, (2) the local [ ] T, coordinates of the anchor point on the lever, denoted by a = a x a y and (3) the global coordinates of the anchor point on the global reference frame. A complete specification of the constant force would involve(1) ] the force components with respect to the global frame, denoted by f = [f x f y, and (2) the local coordinates of the

88 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 68 (a) (b) (c) Original load Original load Original load Original load (black colored) balanced by counterweights (grey colored) Balancing loads : Springs Balancing loads : Springs?? Figure 5.1: Lack of methods for spring-based n-body linkage balancing without the usage of auxiliary bodies point of action of the force on the lever, denoted by p = [p x p y ] Potential energy as a function of the configuration variable By referring to figure (5.2), the potential energy of the constant load is PE c = f T (r +R(θ)p) (5.1)

89 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 69 f p y x θ X a d k Y r b where r = X Figure 5.2: A lever under a constant load and a spring load [ r x r y ] T is the coordinates of the origin of the local frame on the lever with respect to the global frame and R is the rotation matrix function given by R(ψ) = [ ] cosψ sinψ sinψ cosψ The potential energy of the spring is for any angle ψ (5.2) PE s = k 2 (l l ) 2 = k 2 l2 kl l+ k 2 l2 (5.3) wherel isthefreelengthofthespringandlisthemagnitudeofd, thedisplacementof one-end point of the spring with respect to the other. This d, referring to figure (5.2), is: d = (r +R(θ)a) b (5.4) Since l 2 = d T d, the potential energy in expression (5.3) may be rewritten as PE s = k 2 dt d kl d T d+ k 2 l2 (5.5)

90 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 7 If the free-length of the spring is zero, then only the first term in equation (5.5) remains and hence we call it as PE s,zero, i.e., PE s,zero = k 2 dt d = k 2 ((r +R(θ)a) b)t ((r +R(θ)a) b) = k 2 (rt r +a T R T (θ)r(θ)a+b T b 2r T b+2r T R(θ)a 2b T R(θ)a) = k ( r T r +a T a+b T b 2r T b+2r T R(θ)a 2b T R(θ)a ) (5.6) 2 R T (θ)r(θ) = I Since the remaining last two terms of the potential energy in equation (5.5) are nonzero only if free-length l is non-zero, we name these terms as PE s,nonzero, i.e., PE s,nonzero = kl d T d+ k 2 l2 (5.7) From equation (5.6), it follows that d T d = 2 k PE s,zero. Substituting this in equation (5.7) leads to the following expression for PE s,nonzero. PE s,nonzero = l 2kPEs,zero + k 2 l2 (5.8) In the expressions of the potential energy in equations (5.1), (5.6) and (5.8), as the configuration of the lever varies, f, p, a, b, k, l remain constants and r is made a constant by choosing the origin of the local frame on the lever to coincide with the pivot point. The dependency of the expressions on the configuration is due the matrix R(θ), which by examining the definition of R in equation (5.2), can be split as: [ ] 1 R(θ) = cosθ+ 1 [ ] 1 1 sinθ = Icosθ +R( π )sinθ (5.9) 2 This form of R(θ) indicates that PE c in equation (5.1) and PE s,zero in equation (5.6) can be written as a linear combination of sinθ, cosθ, and 1 (for constants). The coefficients of sinθ, cosθ and 1 are presented, for clarity, in a tabular form in

91 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 71 Table 5.1: Potential energy of the weight and the zero-free-length component of the spring acting on the lever is a linear combination of cosθ, sinθ, and 1. Basis Coefficients Weight Zero-free-length component of spring load cosθ f T p = (f y p y +f x p x ) k(r b) T a = k(a y r y +a x r x a y b y a x b x ) sinθ f T R( π) p k(r b) T R( π) a 2 2 = (f x p y f y p x ) = k(a ( x r y a y r x a x b y +a y b x ) 1 f T r + ki 2 r T r +a T a+b T b 2r T b ) = f y r y f x r x = + k i 2 (r2 y 2b y r y +rx 2b 2 x r x +b 2 y+b 2 x+a 2 y+a 2 x) table 5.1. Thus, we now have potential energy of constant and spring loads expressed as functions of configuration variable θ Invariance of potential energy with respect to the configuration variable Trivial conditions The potential energy of the spring on the lever can have constant potential energy only under trivial conditions: (1) the spring stiffness is zero (k = ), (2) the anchor point on the lever is at the hinge point (a = ), and (3) the anchor point on the global frame is at the hinge point (b = r). Similar trivial conditions for the constant loads are: (1) the load is zero (f = ), and (2) the load acts at the pivot point (p = ). It is only under these trivial conditions that the coefficients of cosθ and sinθ become zero in table 5.1. The discovery of Lucien LaCoste Even though a non-trivial spring and a non-trivial constant load cannot be individually in static balance, they together can be so, as demonstrated in figure (5.3). This was first recognized by Lucien LaCoste (see [3]) in the context of having a pendulum of infinite period. Figure (5.3) shows a lever under the action of a weight W that is balanced by a zero-free-length spring of spring constant k anchored above the pivot

92 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 72 of the lever at a height of h. As shown in the figure, under the condition W = kh, the potential energy is invariant with respect to configuration variable θ. h b = y [ ] h Y x θ X k zero-free-length spring f = [ W l W ] weight a = p = [ ] l r = [ ] PE c = () cosθ + (Wl) sinθ + constant + PE z = () cosθ + ( khl) sinθ + constant PE net = (l (W kh)) sinθ + constant when W = kh Figure 5.3: Static balancing of a weight by a spring Several zero-free-length springs and constant loads ThebalancingconditionW = khoftheexampleinfigure(5.3)willnowbegeneralized to a lever under several constant loads and zero-free-length spring loads. Since several loads are now being considered, let both constant loads and zero-free-length spring loads be ordered to allow indexing. The notation a i, b i k i has the same meaning as a, b and k in figure (5.2) other than that it corresponds to i th spring. f i and p i also have similar meaning. Further, let the number of constant loads be n c and the

93 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 73 number of zero-free-length spring loads be n s. Since the potential energy of each of the constant loads and the zero-free-length spring loads is a linear combination of cosθ, sinθ and 1, their net potential energy is also a linear combination of cosθ, sinθ and 1. Further, since cosθ and sinθ and 1 are linearly independent functions of θ, their linear combination is a constant if and only if the coefficients of non-constant functions, i.e., cos θ and sin θ, are zero. Writing, with the help of table 5.1, the coefficients of cosθ and sinθ of the net potential energy of all the loads and equating them to zero lead to the following equations: n c (f y,i p y,i +f x,i p x,i )+ n c i=1 i=1 n s k i (a y,i r y +a x,i r x a y,i b y,i a x,i b x,i ) = (5.1) i=1 (f x,i p y,i f y,i p x,i )+ n s i=1 k i (a x,i r y a y,i r x a x,i b y,i +a y,i b x,i ) = (5.11) which are the conditions for constant potential energy (or static balance) of several constant and zero-free-length spring loads on a lever. These conditions are applicable to all the three categories: 1) balancing weights by weights, 2) balancing weights by springs, and 3) balancing springs by springs. Further, by choosing appropriate load parameters, it is possible to satisfy the conditions in practice, as was the case in the example of figure (5.3). Normal positive-free-length springs As far as normally available positive-free-length springs are concerned, the square root term in equation (5.8) poses a severe restriction on static balancing, as explained in detail in appendix C. Hence, for the remainder of this chapter, all the spring loads are of zero free-length with the understanding that a positive-free-length spring can be brought into the ambit of zero free-length by combining it with an appropriate

94 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 74 f 2 f nc b i k i b 2 p 2 p nc a ns y a i x θ X a 1 a 2 k 2 p 1 f1 b ns k ns Y r k 1 b 1 X Figure 5.4: A body that is free to move in a plane negative-free-length spring. Our next aim is to derive a set of conditions for the static balance of a revolutejointed multi-body linkage loaded by constant loads and zero-free-length spring loads. Before that, it is useful to consider the static balance of a single rigid body moving freely in a plane. 5.3 Balancing of a rigid body in a plane Consider the rigid body shown in figure (5.2). An appropriate set of configuration variables for the body is {r,θ}. It may be noted that r in figure (5.4), in contrast to figure (5.2), is an independent variable because the body is free to move in the plane. The loads on the body are a set of zero-free-length spring loads and constant loads, and both sets of loads are exerted by the global frame of reference as shown in figure (5.4). The notation n c, n s, a i, b i, k i, f i, and p i has the same meaning as in Section 5.2. The potential energy of the loads is also the same as in Section 5.2

95 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 75 Table 5.2: Potential energy of weight and spring acting on a link moving in a plane. Basis Coefficients of the basis Weight Spring load Generalized potential (see eqn. (5.15) cosθ (f y p y + k(a y b y +a x b x ) (q y w y +q x w x ) f x p x ) sinθ +(f x p y k(a x b y a y b x ) (q x w y q y w x ) f y p x ) r x cosθ ka x v x r y cosθ ka y v y r x sinθ ka y v y r y sinθ ka x v x rx 2 k κ 2 ry 2 k κ 2 r x f x kb x u x r y f y kb y u y 1 + k 2 (a2 x +a 2 y +b 2 x + b 2 y) c except that r x and r y are now independent variables. In table 5.1 of Section 5.2, when linearly independent functions of {r, θ} are pulled out as basis functions, table 5.2 is obtained. As is evident from table 5.2, the potential energy of the loads is now a linear combination of the following basis functions: cosθ, sinθ, r x cosθ, r y cosθ, r x sinθ, r y sinθ, rx, 2 ry, 2 r x, r y, and 1. The net potential energy of n c constant loads and n s spring loads is also a linear combinations of the same basis functions. Furthermore, cosθ, sinθ, r x cosθ, r y cosθ, r x sinθ, r y sinθ, rx, 2 ry, 2 r x, r y, and 1 are linearly independent functions of {r,θ}. Hence, from a reasoning similar to the one in Section 5.2, for the net potential energy to be independent of the configuration variables, the coefficients of all the basis functions other than 1 have to be zero. However, it is not practical to make the coefficients of all these functions as zeros because of the following reasons: There are only gravity loads: Gravity is the most important practically seen instance of a constant load. When all the constant loads are gravity loads,

96 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 76 f i = m i g i, where m i is the mass and g is( the acceleration due to gravity. n g ) ( n g Further, the coefficient of r x and r y become g x m i and g y m i ). Since m i >, i, n g i=1 m i >. Also, since the acceleration due to gravity is non-zero, both g x and g y cannot be zero. Hence, the coefficients of both r x and r y cannot be zero. There are zero-free-length spring loads, possibly with gravity loads: In this case the coefficients of both rx 2 and ry 2 are ns k i. Since the spring constants of all the springs considered here are positive (k i >, i), ns k i cannot be zero. i=1 Hence, the coefficients of r 2 x and r 2 y cannot be zero. However, as shown in appendix D.1, there is no such practical difficulty in making the coefficients of all θ-dependent functions, i.e., cosθ, sinθ, r x cosθ, r y cosθ, r x sinθ, and r y sinθ, to be zeros. Setting θ-dependent terms to zero amounts to the following set of independent constraints: n c n s ((f y,i p y,i +f x,i p x,i )) (k i (a y,i b y,i +a x,i b x,i )) = (5.12) i=1 n c i=1 n s + ((f x,i p y,i f y,i p x,i )) (k i (a x,i b y,i a y,i b x,i )) = (5.13) i=1 i=1 n s i=1 i=1 i=1 i=1 (k i a i ) = (5.14) It is shown in Appendix D.1 that if these constraints are not satisfied by the loads, then by adding not more than two zero-free-length springs, these constraints can be satisfied. A numerical example to demonstrate the same is given in figure (5.5) with the details of the loads in table 5.3. Inspite of being able to make the potential energy of the loads on the link independent of θ, the dependency on r still remains. In Section 5.4, we show that if the body is joined to an appropriate linkage, then by adding extra loads to other parts of the linkage, the r-dependent terms of the potential energy can be balanced out. Before we proceed to Section 5.4, it may be noted that the potential energy of a

97 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 77 [ 1 2,3]T C 1 Z 1 [1,] T [ 1 2,]T θ X Z 2 [ 1 2,1]T Y Global frame X Figure 5.5: A rigid body moving freely in a plane under a constant load is made to have θ-independent potential energy by addition of two zero-free-length springs Table 5.3: Details of the loads in figure (5.5). It may be checked that the loads satisfy equations ( ) Z 1 Z 2 Spring Loads a b k [ 1 2,]T [ 1 2,3]T 1 [ 1 2,]T [ 1 2,1]T 1 Constant Loads p f C 1 [1,] T [, 1] T kb T a kb T R ( ) pi a ka f T p f T R ( pi 2) p 1 [ 1 2,]T [ 1 2,]T

98 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 78 constant load or a zero-free-length spring load fall under the following general form: Φ = r T u+κr T r +r T R(θ)v +w T R(θ)q +c (5.15) where θ and r are the configuration variables of the rigid body on which the load acts. In the case of constant loads, by comparing equation (5.1) with equation (5.15), we have u = f, κ =, v =, w = f, q = p, and c = (5.16) and in the case of zero-free-length spring loads, by comparing equation (5.6) and (5.15), we have u = kb, κ = k 2, v = ka, w = b, q = ka, and c = k ( b T b+a T a ) 2 (5.17) Later in the chapter, we encounter potential energy functions that are of the form given in equation(5.15), but they cannot be attributed to zero-free-length spring loads or constant loads acting on the body. Hence, there is a need to generalize constraints (5.1), (5.11), (5.12), (5.13), and (5.14) to the form given in equation (5.15). Such a generalization is possible because, as can be seen in the last column of table 5.2, the potential given in equation (5.15) is a linear combination of the basis functions given in table 5.2 just as in the case of constant and zero-free-length spring loads. The following proposition states the generalization. Proposition 1. If there are n functions of the form Φ i = r T u i +κ i r T r +r T R(θ)v i +w T R(θ)q i +c i, i = 1 n (5.18) with r and θ as the variables, then n Φ i is independent of θ if and only if the following i=1

99 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 79 constraints are satisfied: n (q y,i w y,i +q x,i w x,i ) = (5.19) i=1 n (q x,i w y,i q y,i w x,i ) = (5.2) i=1 n v i = (5.21) i=1 When these constraints are satisfied, n Φ i depends only on r in the following form i=1 n Φ i = i=1 n ( ) ( r T u i +κ i r T r +c ) n i = r T u i + ( r T r ) n κ i + i=1 i=1 i=1 n c i (5.22) i=1 Furthermore, if r happens to be a constant (as in a lever) with only θ being the variable, then n Φ i is independent of θ (and hence a constant) if and only if the i=1 following constraints are satisfied: n (v y,i r y +v x,i r x +q y,i w y,i +q x,i w x,i ) = (5.23) i=1 n (v x,i r y v y,i r x +q x,i w y,i q y,i w x,i ) = (5.24) i=1 Proof. The proof is along the same lines as the derivation of equations (5.1), (5.11), (5.12), (5.13), and (5.14). It may be noted that inspite of considering a general form of potential in equation (5.18), the inability to make the net potential energy independent of r remains because of the following reason. In all the cases that we consider next, κ i and n κ i > for atleast one value of i. Hence, the r-dependent term, κ i r T r, cannot be zero in the expression for n Φ i. i=1 i=1

100 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES New static balancing techniques for revolutejointed linkages If there is a single rigid body with loads exerted by a reference frame, then the net potential energy of the loads depends on the configuration of the body with respect to the reference frame. If there are several such bodies, then the net potential energy of all the loads on all the bodies depends on the configuration of all the bodies. This dependency on the configuration of all the bodies can be reduced to that of a single body provided the bodies are connected by revolute joints (to begin with, say, in a serial or a tree-structured manner) and the loads are zero-free-length spring loads and constant loads. If this single body is the reference frame itself, then the net potential energy is a constant (implying static balance) since the configuration of the reference frame with respect to itself is always fixed. This result follows as a consequence of the proposition that is presented next The potential energy of loads on a body transformed as a function of another body We are now considering several rigid bodies, each of them with its own r, θ, n c, n s, a i, b i, k i, p i, etc. To distinguish these quantities belonging to different rigid bodies, we number the rigid bodies and put the number as a superscript to these symbols. Hence r, θ, n c, n s, a i, b i, k i, p i, and f i of body j are now represented as r j, θ j, n j c, n j s, a j i, bj i, kj i, p i j, etc. Proposition 2. The net sum of a set of functions of the configuration variables of a body l in the form given in equation (5.15), i.e., Φ l i = r lt u l i + κl i rlt r l + r lt R ( θ l) v l i + wlt R ( θ l) q l i + cl i, i = 1 nl (5.25)

101 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 81 can be expressed as a function of the same form but of body j, i.e., nl i=1 ) ) Φ l i = Φ j i = r jt u j i + κj i rjt r j + r jt R (θ j v j i + wjt R (θ j q j i + cj i (5.26) provided the following conditions are satisfied: Condition 1: There is a point that is rigidly fixed to both body l and body j. Such a point is called as a common point of bodies l and j. Condition 2: The origin of the local coordinate frame of body l is at the common point. Condition 3: The sum of the set of functions of body l is dependent only on r in the form given in equation (5.22). Proof. Let the local coordinates of the common point required by condition 1 in body l be s l j and in body j be sj l. The commonality of the point can be written as follows. r l +R ( θ l) ) s l j = rj +R (θ j s j l (5.27) Condition 2 implies that s l j =. Substituting sl j = into equation (5.27) leads to ) r l = r j +R (θ j s j l (5.28) Condition 3 implies that the sum of the set of functions of body l can be written as nl nl Φ l i = rlt i=1 i=1 ( ) u l i + nl r lt r l i=1 κ l i (5.29) The constant term is omitted in equation (5.29) since it is inconsequential for the discussion. Substitution of r l from equation (5.28) into equation (5.29) and simplification

102 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 82 using the fact that R T (θ)r(θ) is identity lead to the following expression for nl i=1 Φ l i : r jt u j i {}} { {}} {{ }} { nl nl ) nl u l i +r jt r j κ l i +r jt R (θ j 2s j l κ l i i=1 κ j i i=1 w j T i v j i i=1 {}} { T q j i nl ){}}{ ( ) nl + R (θ j = Φ l i = Φj i (5.3) i=1 Again, the constant term is omitted in equation (5.3). It may be readily recognized nl that the sum of the set of functions on body l, Φ l i, as seen in equation (5.3) is indeed of the form given in equation (5.26). u l i i=1 s j l i= Proposition 2 as the recursive relation of an iterative static balancing algorithm We now show that proposition 2 can be treated as a recursive relation that can be incorporated into an iterative procedure to achieve static balance of a linkage. For the purpose of this subsection, we restrict the linkage on which the iterative procedure can be applied to have the following features: 1. The linkage should be tree-structured (i.e., no closed loops). This feature is necessary since a recursive relation requires a tree-structure to propagate. 2. All the joints of the tree structure should be revolute joints. This feature is necessary to satisfy condition 1 of proposition We want all the loads to have potential energy functions of the form given in equation (5.25) of proposition 2. While we know that zero-free-length springs and constant loads do have this form (see equations (5.16) and (5.17)) the fact

103 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 83 that there are several bodies involved requires a little attention. The configuration variables ( r l,θ l) of different bodies (i.e., of different l) should be with respect to a common global frame of reference. Hence, constant loads on all the bodies should be constant with respect to a common global reference frame and any zero-free-length spring should have its one anchor point on the same common global frame while the other anchor point can be on any of the bodies constituting the linkage. 4. The common reference frame should be one of the bodies of the linkage, i.e., it should join to body/bodies of the linkage by revolute joint/joints. The iterative static balancing algorithm We now present the iterative algorithm and prove that it leads to static balance. Preparatory steps 1. Assign the reference body as the root node of the tree-structure (bodies are represented as nodes and joints as lines joining the nodes). With this assignment, for every link/body other than the root, there is a parent body. Further, every links other than the terminal links has one or more children. 2. On every link, choose a local frame that coincides with the center of the revolute joint between the link and its parent. For every link k, r k and θ k decide the configuration of its local frame with respect to the frame of the root. 3. Give this tree-structure with the given constant and zero-free-length spring loads (together referred to as original loads) as an input to the following iterative procedure. Iterative procedure Entry condition: If the tree-structure contains only the root node, then exit from the iterative procedure. Otherwise, proceed to step 1.

104 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 84 Step 1: Any terminal node l, has associated with it the following three kinds of potential energy functions: 1) due to original loads on body l, 2) due to association that happened in step 3 of previous iterations, and 3) additional loads on body l. Let the number of such functions be represented by n l o, n l c, n l a, respectively. The first two kinds of functions are known from the given problem and previous iterations, respectively, and the task in this step is to find the additional loads so that case a: equations ( ) are satisfied if l is not a child (i.e., not first generation descendant) of the root. case b: equations ( ) are satisfied if l is a child of the root. Note that in this case r l is a constant because of the way local frame is chosen in the preparatory steps. This task makes sense only if the all kinds of potential energy functions fall under the form of equation (5.15) with (r,θ) being ( r l,θ l). The first and the third kind of potential energy functions do fall under the form because of the kind of loads we are considering (see equations (5.16) and (5.17)). The second kind of potential energy functions conform to the form because of step 2, which is in accordance with proposition 2. The critical role of proposition 2 in enabling this iterative procedure may be noted. Further, Appendix D.2 asserts that the task of this step is always feasible. It may be noted that there are several sets of additional loads that satisfy these equations. This non-uniqueness calls for discretion of the designer in choosing a suitable set of additional loads. Step 2: In case (a), express n l o+n l c+n l a i=1 Φ l i in the form given in equation (5.15) where r and θ are the configuration variables of the parent of node l. This is possible since condition 1 (because of revolute joint), condition 2 (because of preparatory steps) and condition 3 (because of step 1) of proposition 2 are n l o+n l c+n l a satisfied. In case (b), recognize that Φ l i is a constant as per proposition 1. i=1

105 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 85 Step 3: Associate n l o+n l c+n l a i=1 Φ l i with the parent link of l and for energy conservation, disassociate Φ l i, i = 1 n l o +n l c +n l a from node l. Because of this association, n p(l) c (p(l) denotes parent node of l, and n k c denotes the number of potential energy functions associated with node k so far at step 3.) gets incremented by one and the associated function can be written as: n l o+n l c+n l a Φn p(l) c = Φ l i (5.31) Step 4: With all the potential energy functions disassociated from node l to its parent, delete this terminal node l. Iterator: Once steps 1 to 4 are completed, a new trimmed tree-structure results where the parents of the nodes deleted in step 4 has additional potential energy functions associated with them. Follow this iterative procedure again with this trimmed tree-structure as the input. With every iteration, the tree-structure shrinks and it eventually gets reduced to the single root node. Any of the n c potential energy functions (of the second kind) associated with this reduced root is from one of the children of the root. As per step 1, this association is through case (b). Any function associated through case (b) is a constant as recognized in step 2. Thus, the sum of these n c potential energy functions is also constant. Further, the sum of these n c potential energy functions is actually the total potential energy of original loads and additional loads on all the descendants of the root. This can be verified by recursive substitution in equation (5.31) as exemplified in equation (5.32). Therefore, the original loads are in static balance with the additional loads. i=1 Illustration of the algorithm on a 4R linkage under constant loads Figure (5.6) shows a 4R linkage where four revolute joints connect the ground and four other bodies serially. The ground exerts constant gravitational force on each of the four bodies. Hence, we take the ground as the root and number the bodies accordingly as

106 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 86 shown in figure (5.6). A local frame of reference is located on each of the bodies as per the preparatory step 2. The constant loads on bodies 1, 2, 3 and 4 are represented as C1, 1 C1, 2 C1 3 and C1, 4 respectively. Their details (point of action p and force vector f) are presented in item number 1, 4, 7 and 1 of the tables in figures ( ). 4 3 y 3 C 3 1 x 4 y 4 x 2 C 4 1 C 1 1 C 2 1 y 1 y y 2 1 x 2 Figure 5.6: The gravity-loaded serial 4R linkage to be statically balanced. Now, we give the tree-structure to the iterative procedure. The terminal node of thetreeis4. Thereisonlyoneoriginalload, C1, 4 onthenodeanditspotentialenergyis represented as Φ 4 1. To emphasize that potential energy function Φ 4 1 is because of load C1, 4 we write it as Φ 4 1(C1). 4 There is no potential energy function of the second kind (n 4 c = ). Two zero-free-length springs Z1 4 and Z2 4 are added so that the functions Φ 4 1(C1), 4 Φ 4 2(Z1), 4 and Φ 4 3(C2) 4 satisfy equations ( ) as per case (a) of step 1. All the details of the springs, constant loads as well as their potential energies in the standard form (see equation (5.15)) are presented in the table of figure (5.7). Now,

107 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 87 as per step 2, the sum of Φ 4 1(C1), 4 Φ 4 2(Z1), 4 and Φ 4 3(C2) 4 is transformed as a Φ 3 2(r 3,θ 3 ) in accordance with equation (5.3) of proposition (2). This is followed by making of a new tree-structure obtained by deleting node 4 and associating Φ 3 2 with node 3 of the new tree-structure. This completes the first iteration. Parameters of potential energy functions associated with node 4 (j = 4) Step 1: add springs/ Eqs. ( ) no. Φ j i a j k /pj k b j k /fj k k j k u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] [ ] [ ] ( ) 1 Φ 4 1 C eq.(5.16) [ ] [ 1 ] [ 1 ] [ ] [ 1 ] [ ] ( ) 2 Φ 4 2 Z eq.(5.17) [ ] [ ] [ 4 ] [ ] [ ] [ ] ( ) 3 Φ 4 3 Z eq.(5.17) [ ] i=1 u 4 i = 3i=1 23 κ 4 i =.8 eqs. ( ) true 2 4 ( ) ( ) ( ) Φ 4 1 C 4 1, Φ 4 2 Z 4 1, Φ 4 3 Z 4 2 n 4 o = 1 n4 a = ( ) Φ 3 1 C 3 1 n 3 o = 1 ( ) Φ 2 1 C 2 1 n 2 o = 1 ( ) Φ 1 1 C 1 1 n 1 o = 1 From equation (5.3) [ ] 1 s 3 4 = Association to node 3 Step 2 and 3 j = 3 Φ j i u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] Φ Step 4: deletion Φ 3 1( C 3 1 ), Φ 3 2, n 3 o = 1 n3 c = 1 ( ) Φ 2 1 C 2 1 n 2 o = 1 ( ) Φ 1 1 C 1 1 n 1 o = 1 Iteration 1 Figure 5.7: Details of Iteration 1 The second iteration acts on the new tree-structure. The tables in figures ( ) give all the details of all the iterations. At the end of four iterations, we are left with a single root node having constant function Φ 1 associated with it. The springs added in these iterations, along with the original gravity loads are shown in figure

108 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 88 (5.11). By following the dashed arrowed line of figures ( ) in the reverse order, it may be verified that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Φ 1 = Φ 1 1 C 1 1 +Φ 2 1 C 2 1 +Φ 3 1 C 3 1 +Φ 4 1 C 4 1 +Φ 4 2 Z 4 1 +Φ 4 3 Z 4 2 +Φ 3 3 Z 3 1 +Φ 2 3 Z 2 1 (5.32) Hence, C1, 4 C1, 3 C1, 2 and C1 1 are in static balance with Z1, 4 Z2, 4 Z1, 3 and Z1. 2 Step 1: add springs/ Eqs. ( ) from iteration 1 no. Φ j i a j k /pj k b j k /fj k k j k u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] [ ] [ ] ( ) 4 Φ 3 1 C eq.(5.16) 1 [ 1 ] [ ] [ 1 ] [ ] 5 Φ From iteration [ ] [ ] [ 2 ] [ ] [ 23 2 ] [ ] ( ) 6 Φ 3 3 Z eq.(5.17) [ ] i=1 u 3 i = 3i=1 9 κ 3 i = 1.6 eqs. ( ) true ) ( ) Φ1( 3 C 3 1, Φ 3 2, Φ 3 3 Z 3 1 n 3 o = 1 n3 c = 1 n3 a = 1 ( ) Φ 2 1 C 2 1 n 2 o = 1 ( ) Φ 1 1 C 1 1 n 1 o = 1 From equation (5.3) [ ] 1 s 2 3 = Association to node 2 Step 2 and 3 no. Φ j i u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] 8 Φ Step 4: deletion Φ 2 1( C 2 1 ), Φ 2 2, n 2 o = 1 n2 c = 1 ( ) Φ 1 1 C 1 1 n 1 o = 1 Iteration 2 Figure 5.8: Details of Iteration 2 To verify the static balance, this linkage along with the loads was modelled in

109 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 89 Step 1: add springs/ Eqs. ( ) from iteration 2 no. Φ j i a j k /pj k b j k /fj k k j k u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] [ ] [ ] ( ) 7 Φ 2 1 C eq.(5.16) 1 [ 1 ] [ ] [ 1 ] [ ] 8 Φ From iteration [ ] [ ] [ 1 ] [ ] [ 9 1 ] [ ] ( ) 9 Φ 2 3 Z eq.(5.17) [ ] i=1 u 2 i = 3i=1 2 κ 2 i = 3.2 eqs. ( ) true ) ( ) Φ1( 2 C 2 1, Φ 2 2, Φ 2 3 Z 2 1 n 2 o = 1 n2 c = 1 n2 a = 1 ( ) Φ 1 1 C 1 1 n 1 o = 1 From equation (5.3) [ ] 1 s 1 2 = Association to node 1 Step 4: deletion 2 Φ 1 1( C 1 1 ), Φ 1 2, Step 2 and 3 1 n 1 o = 1 n1 c = 1 n1 a = no. Φ j i u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] Φ Iteration Figure 5.9: Details of Iteration 3

110 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 9 Step 1: springs unnecessary/ eqn. ( ) of propn. 1 are satisfied as it is. from iteration 3 no. Φ j i a j k /pj k b j k /fj k k j k u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] [ ] [ ] ( ) 1 Φ 1 1 C eq.(5.16) [ 1 ] [ 1 ] [ ] [ 1 ] [ ] Φ 1 2 s 1 2 = eq.(5.3) [ 2 5 ] eqs. ( ) is true with r 1 = ) Φ1( 1 1 C 1 1, Φ 1 2, n 1 o = 1 n1 c = 1 n1 a = Step 4: deletion Step 2 and 3 1 Φ 1 n c = 1 A constant (from propn. 1) Hence it is associated with the root Iteration 4 Figure 5.1: Details of Iteration 4

111 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 91 Z y 3 C 3 1 x 4 y 4 Z 4 1 Z 3 1 x 2 C 4 1 C 1 1 C 2 1 y 1 y y 2 1 x Z Figure 5.11: Statically balanced gravity-loaded serial 4R linkage.

112 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 92 ADAMS ( With zero damping, a pulse of energy was initially introduced to the system. When the dynamic simulation of the system was carried out it was noticed that the net kinetic energy was constant over time. This implies that there was no potential gradient along the path that the linkage took in the dynamic simulation. In figure (5.12), joint 1 and body 1 are eliminated to modify this 4R [ example ] into a 3R example. Joint 2 now joins body 2 with the ground at r 2 =. The rest of the bodies and their numbering remain unchanged. The first two iterations for this example are identical to the 4R example. The third iteration is the last since node 2 now is a child of the root. As required at this [ iteration, ] it may be verified that equations ( ) are satisfied with r 2 =. One can have a similar modification of 4R example into a 2R example as shown in figure (5.13) Illustration of the algorithm on a 2R linkage under a zero-free-length spring load Just as figures ( ) have all the details of the serial 4R example, figure (5.14) has all the details of this example. The explanation is also along the same lines as that of the previous example. In this example, the given original load is Z1 2 and the balancing loads are Z2 2 and Z3. 2 To reaffirm the fact that zero-free-length springs are practical, a prototype of this example was made, as shown in figure (5.14b). To realize zero-free-length springs, pulley-string arrangement was used, the details of which were explained in figure (3.12). Illustration of the algorithm on a 4R tree-structure linkage under both constant load and zero-free-length spring load While the previous two examples had serial architecture, this example has branches emanating form the same node, as shown in figure (5.15a). The original loads acting on it are C1 3 and Z1. 4 Instead of taking original loads to be exclusively constant loads or exclusively zero-free-length spring loads, here we have taken a combination of both types of loads. These original loads are balanced by adding springs Z1, 3 Z2, 3 Z2, 4 Z1, 2 and Z1 1 at various iterations

113 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 93 Z C y 3 y 4 x 4 2 C 4 1 Z 3 1 Z 4 1 y 2 y C 2 1 x r 2 = [ 5 32 ] Figure 5.12: Statically balanced serial 3R linkage

114 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 94 4 Z 4 1 y 4 x 4 Z 4 2 C 4 1 y 3 x 3 C y x r 3 = [ ] Figure 5.13: Statically balanced serial 2R linkage

115 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 95 x 2 Z Step1: add springs Eqs. ( ) Φ 2 ( 1 Z 1 2 ), Φ 2 ( 2 Z 2 2 ), Φ 2 ( 3 Z 3 2 ) n 2 o = 1 n2 a = 2 y x 1 1 Z 2 3 (b) Iteration 1 Steps 2 and 3 Z 2 2 (a) no. Φ j i a j k b j k k j k u j i κ j i v j i w j i q j i [ ] [ ] [ ] [ ] [ ] [ ] ( ) 1 Φ 2 1 Z eq.(5.17).5 [ ] [ 12 ] [ 12 ] [ ] [ 12 ] [ ] ( ) 2 Φ 2 2 Z eq.(5.17).5 [ ] [ 12 ] [ 12 ] [ ] [ 12 ] [ ] ( ) 3 Φ 2 3 Z eq.(5.17) 1 [ ] 3i=1 u 2 i = 3i=1 κ 3 i = 2 eqs. ( ) true [ ] [ ] [ ] [ ] [ ] Φ 1 1 s 1 2 = eq.(5.3) 2 [ ] eqs. ( ) is true with r 1 = 2 Step 4: Deletion 1 1 Step 4: Deletion Φ 1 Step1: 1 n 1 Eqs. ( ) c = 1 No extra springs Steps 2 and 3 Iteration 2 (c) Φ Constant from 1 proposition 1 n c = 1 Figure 5.14: Details of static balance of a 2R linkage under spring load in the iterative algorithm. A pictorial depiction of the iterations on this linkages is given in figure (5.15b). All the remaining details are given in the table of the same figure. To verify the static balance, θs of bodies 1, 2, 3 and 4 are varied in the following form: θ 1 = π 4 + πsin(2πt), θ2 = π 12 + πsin(2πt), θ3 = π πsin(2πt), θ4 = π πsin(2πt). The potential energy variation of original loads C 3 1 and Z 4 1 as well as the balancing loads, i.e., Z 3 1, Z 3 2, Z 4 2, Z 2 1, and Z 1 1, are plotted in figure (5.16). The sum of all these variations is also plotted and it has turned out to be a constant. This verifies the static balance.

116 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 96 Z 3 2 C 3 1 Z 3 1 Original loads Z Z1 4 1 Z2 4 Step1: add springs Eqs. ( ) ( ) ( ) ( ) Φ 3 1 C 3 1, Φ 3 2 Z 3 1, Φ 3 3 Z 3 2 n 3 o = 1 n3 a = 2 Step 2 & 3 ( ) ( ) 3 4 Φ4 1 Z 4 1, Φ 4 2 Z 4 2 n 4 o = 1 n4 a = Step 2 & 3 Step1: add springs Eqs. ( ) Iteration 1 Step1: add springs Eqs. ( ) ( ) 2 Φ2 1, Φ 2 2, Φ 2 3 Z 2 1 n 2 c = 2 n2 a = 1 (b) Y X Z 1 1 (a) 1 Step 2 & 3 Iteration 2 no. Φ j i a j k /pj k b j k /fj k k j k u j i κ j i v j i w j i q j i [ ] [ ( ) 1 Φ 4 1 Z ] [ ] [ ] [ ] [ ] eq.(5.17) [ ] [ ( ) 2 Φ 4 2 Z ] [ ] [ ] [ ] [ ] eq.(5.17) [ ] i=1 7 u 4 i = 2i=1 κ 3 4 i = 1 eqs. ( ) true [ ] [ ] [ ] [ ] [ ] [ ] ( ) 3 Φ 3 1 C eq.(5.16) [ ( ) 1 ] [ 1 ] [ ] [ 4 Φ 3 2 Z ] [ ] [ 1 1 eq.(5.17) ] [ ] [ 3 ( ) 5 Φ 3 3 Z ] [ ] [ ] [ ] [ ] eq.(5.17) [ ] 1 3i=1 1 u 3 i = 3i=1 κ 3 3 i = 1 eqs. ( ) true [ ] [ ] [ ] [ ] [ ] Φ 2 1 s 2 3 = 3 eq.(5.3) [ 5 ] [ 3 ] [ 5 ] [ 3 ] [ 5 ] Φ 2 2 s 2 4 = 3 eq.(5.3) [ ] [ 5 ] [ 3 ] [ 5 ] [ 3 ] [ 5 ] ( ) 9 Φ 2 3 Z eq.(5.17) [ ] i=1 12 u 2 i = 3i=1 κ 14 2 i = 3 eqs. ( ) true [ ] [ ] [ ] [ ] [ ] Φ 1 1 s 1 2 = eq.(5.3) 3 [ ( ) 1 ] [ 5 ] [ ] [ ] [ ] [ ] 11 Φ 1 2 Z eq.(5.17) [ ] eqs. ( ) is true with r 1 = Φ ( ), Φ 1 2 Z 1 1 n 1 c = 1 n1 a = 1 Φ 1 Step1: add springs Eqs. ( ) Step 2 & 3 Iteration 2 A constant Figure 5.15: Details of static balance of a 4R tree-structure linkage under a constant load and a spring load Static balancing of any revolute-jointed linkages with any kind of zero-free-length spring and constant load interaction within the linkage In the static balancing method for linkages provided in Section 5.4.2, other than the fact that the linkage to be balanced has to be revolute-jointed and that load interactions are of zero-free-length spring or constant loads, there were two more restrictions:

117 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES Net Potential Energy 2 Z 4 1 PE 15 Z 1 1 Z Z C1 Z 3 Z t Figure 5.16: Potential Energy variation of spring loads, constant loads, and their sum 1. It should be possible to consider that the loads on all the bodies are exerted by a common reference body (or frame) of the linkage. 2. The linkage should have a tree-structure (i.e., without closed loops). When the first restriction is violated, as in figure (5.17a), it is always possible to break the load interactions into a superposition of several load sets with each set complying with the first restriction. For example, the load interaction in figure(5.17a) is broken into two load sets in figures (5.17b) and (5.17c). The reference body in each of these sets is indicated by an asterisk symbol (*) in their respective figures. Furthermore, in a load set, if there are closed loops, then the closed loops can be broken by relaxing certain joint constraints. Figures (5.17c) and (5.17d) illustrate breaking of closed loops in figures (5.17b) and (5.17c), respectively. With closed

118 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 98 Breaking load interactions * + * (a) (b) (c) Breaking closed loops Joint relaxed Joint relaxed (c) (d) Figure 5.17: Breaking a problem as a superposition of several problem with each problem being static balance of revolute-jointed tree-structured linkage with loads exerted by the root body loops broken, each of the load sets comply with the two restrictions and they can be statically balanced by adding balancing loads as per Section Once each of the load sets is balanced, the joint constraints that were relaxed for breaking closed loops can be reimposed without disturbing the static balance. In other words, when the potential energy that is a function of the configuration space is a constant, it remains as the constant even when the configuration space is restricted (due to re-attachment of the broken joints). Once the constraints are reimposed, the linkages in all the load sets are the same as the original linkage and the loads on all the sets can be superposed. Since each load set is in static balance, the superposition is also in static balance. In other words, the sum of several constant potential energy functions due to several load sets is also a

119 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 99 constant. This superposition contains all the original loads on the given linkage. The remnant loads in this superposition are the additional loads that balance the original loads. In this way, additional loads that statically balance any revolute-jointed linkage with zero-free-length spring and constant load interactions between the bodies of the linkage can always be found. An example is presented next to illustrate this. A numerical example Figure (5.18a) shows a linkage which is the same as that in figure (5.15) except for an additional zero-free-length spring between the three-lobed body and the left-most body. Unlike in figure (5.15), where the ground can be considered as the reference body that exerts loads on all the bodies, in figure (5.18a) it is not possible to find a single reference body that exerts loads on all the bodies. Hence the loads are split into two sets as in figures (5.18b) and (5.18c). The first set shown in figure (5.18b) is the same as that in figure (5.15) whose balancing loads are already found using iterative algorithm detailed in figure (5.15). In the second set shown in figure (5.18c), the three-lobed body is treated as the reference frame. The bodies are labelled accordingly with the number assigned to the reference body. The only spring in the figure is labelled as Z 1 1. In this figure, there is only one load interaction, which is between body 1 and the reference frame. Further, body 1 can be considered as lever on the reference body. An extra spring Z 1 2 between body 1 and is added, the details of which are given figure (5.19), to satisfy the conditions of static balance of a lever (equations (5.1) (5.11)). With this the second load set is balanced. The balanced first and second load sets are shown in figures (5.18d) and (5.18e), which are further superposed as in figure (5.18f) to obtain the balancing solution to the original load set shown in (5.18a). 5.5 A note on prismatic joint InSection5.3, itwasnotedthatbymerelyaddingsprings, thedependencyofabody s potential energy on r cannot be annihilated. As shown in the later sections, in the

120 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 1 Set 1 Set 2 (a) (b) (c) (f) Splitting original loads Superposition of balanced load sets The ground (d) Z 3 1 Z 3 2 C C Z Z Z 4 1 Reference body 4 Z 4 1 Z Static balancing See figure (5.15) + (e) Z Reference body Static balancing See figure (5.19) Figure 5.18: Static balance of a tree-structured linkage with inter-body load interactions Z 1 2 Z case of the body having a revolute joint with another body, this dependency on r can be transmitted as a dependency on the coordinates of the other body when r points to the common point of the revolute joint. This allows us to think that the dependency of the potential energy on r is conceptually removed by transmission. The static balancing strategy outlined in this chapter is based on carrying out this transmission in a cascading manner. If the body has a prismatic joint instead of the revolute joint, there is no scope for the conceptual removal by transmission since r cannot be expressed in terms of the coordinates of the other body. This un-annihilated potential energy dependence of r implies that in a tree-structured chain, if there is a prismatic joint, there is no way static balance can be achieved by merely adding springs. In such situations, it is necessary to add revoluted-jointed chain as auxiliary chain and carry out the

121 CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 11 [ 3 4 ] T 1 [ 3 4 ] T Z 1 2 Z 1 1 [ 1 2 Y X ] 3 T 1 Body 1 is a lever with respect to body Spring details a b Z 1 1 Z 1 2 [ 3 4 ] T [ 1 2 [ 3 4 ] T [ 1 2 k ] 3 T 1 1 ] 3 T 1 1 k(r b) T a k(r b) T R ( π 2) a Reference frame r = [ 1 3 5] T is a constant Equations (5.1) and (5.11) satisfied Figure 5.19: Balancing the lever lever loads in the second load set of figure (5.18) cascading transmission through this auxiliary chain. In closed loop linkages, such auxiliary chains may be inherently present as in the case of a slider crank linkage which can be thought as a 3R chain closing its loop with a prismatic joint. Summary We presented a technique to statically balance any planar revolute-jointed linkage having zero-free-length spring and constant load interactions between the bodies of the linkage. The technique involves only addition of zero-free-length springs but not any extra link, unlike spring-aided perfect static balancing techniques currently in the literature. The technique relies on a recursive relation to iteratively remove the dependence of the potential energy on the configuration variables of the bodies of the linkage. Recognizing the recursive relation along with the minimal conditions that enable it constitutes the contribution of this chapter.

122 Chapter 6 Static balancing of spatial linkages without auxiliary bodies Overview Just as a recursive relation in the planar case enabled an iterative algorithm, there is a recursive relation in the spatial case also. The recursive relation holds when the loads are constant loads and/or zero-freelength spring loads and the joints are revolute and/or spherical joints. Similar to the planar case, even though the iterative algorithm works on a treestructured linkage, extension to an arbitrary linkage is straightforward. 6.1 Introduction The results of Chapter 5 hinged on Proposition 2, which was recursively applied in the iterative algorithm of Section (5.4.2). If we can have a similar proposition in the spatial case, then we can have an iterative static balancing algorithm for spatial case also. The features of the proposition that allowed its recursive application are: Feature 1: The allowable loads on a body, say body j, should have a potential energy that belongs to a class of functions of configuration variables of the body. 12

123 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 13 Let the class of functions be represented by χ(j). In Section 5.4.2, the allowable loads are zero-free-length spring and constant loads, and the class of functions are as in equation (5.15). Feature 2: A joint between two bodies, say body j and body l, should provide a relation between configuration variables of j and l that can be used to express the sum of a set of functions in χ(l) as a function in χ(j). This is essential for steps 1 to 3 of the iterative procedure in Section For spatial linkages, a class of functions and the joints having the two features are presented next. 6.2 The class of functions in feature 1 If a body in space has its configuration with respect to a reference frame defined by r and R, the position vector and rotation matrix of the local frame with respect to the global frame, then the potential energy of a constant load and a positive-stiffness zerofree-length spring and a general form of potential energy function that can express potential energies of both constant load and zero-free-length spring loads are: PE c = ( f) T r +( f) T Rp (6.1) PE s = ( kb) T r +( kb) T Ra+r T R(ka)+ k 2 rt r + k ( a T a+b T b ), k > 2 (6.2) Φ = u T r +w T Rq +r T Rv +κr T r +c, κ (6.3) The derivation and the notation used in equations (6.1), (6.2), and (6.3) are the same as those in equations (5.1), (5.6), and (5.15), respectively, except that they are all spatial quantities now. The class of functions referred in feature 1 would be the functions of the form given in equation (6.3). We do not consider a broader class of functions that includes potential energies of positive zero-free-length springs since the extra terms associated

124 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 14 with positive-free-length spring (see equation (5.8)) cannot be balanced out for the reasons same as those given in Appendix C. It may be noted that R, unlike in planar case, is not a function of just a single variable. Among many possibilities, R may be considered as a function of three Euler angles. We use body fixed Z X Z system of Euler angles with the three angles represented as α, β and γ so that R becomes c 1 c 3 s 1 c 2 s 3 c 1 s 3 s 1 c 2 c 3 s 1 s 2 R = s 1 c 3 +c 1 c 2 s 3 s 1 s 3 +c 1 c 2 c 3 c 1 s 2 (6.4) s 2 s 3 s 2 c 3 c 2 where c 1 = cosα, s 1 = sinα, c 2 = cosβ, s 2 = sinβ, c 3 = cosγ, and s 3 = sinγ. Just as in table 5.1, the function in equation (6.3) can be expressed as a linear combination a finite number of linearly independent functions where each function is a function of r and {α, β, γ}. The functions, which are also a basis for the function in equation (6.3), and the corresponding coefficients are presented in table Joints that can potentially satisfy feature 2 Consider the following proposition, which is along the same same lines as in Section 5.3. This proposition helps us to narrow down on spatial joints that can potentially have feature 2, as discussed after the proof of the proposition. Proposition 3. For a given set of functions of the form Φ i = r T u i +w T i Rq i +r T Rv i +κ i r T r +c i, κ i and i = 1 n f (6.5) that are associated with a rigid-body defined by r and R, 1. the net sum of these functions along with the potentials of one or more positivestiffness zero-free-length springs cannot be independent of r.

125 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 15 Table 6.1: The potential in the general form shown in figure (6.3), can be expressed as a linear combination of the basis functions shown in the table. Each basis function in the table is a function of translational variable r and the Z-X-Z Euler angle α, β and γ No. Basis functions Coefficient 1 r x u x 2 r y u y 3 r z u z 4 rx 2 +ry 2 +rz 2 κ 5 r x (c 1 c 3 s 1 c 2 s 3 )+r y (s 1 c 3 +c 1 c 2 s 3 )+r z (s 2 s 3 ) v x 6 r x ( c 1 s 3 s 1 c 2 c 3 )+r y ( s 1 s 3 +c 1 c 2 c 3 )+r z (s 2 c 3 ) v y 7 r x (s 1 s 2 )+r y ( c 1 s 2 )+r z (c 2 ) v z 8 c 1 c 3 s 1 c 2 s 3 w x q x 9 s 1 c 3 +c 1 c 2 s 3 w y q x 1 s 2 s 3 w z q x 11 c 1 s 3 s 1 c 2 c 3 w x q y 12 s 1 s 3 +c 1 c 2 c 3 w y q y 13 s 2 c 3 w z q y 14 s 1 s 2 w x q z 15 c 1 s 2 w y q z 16 c 2 w z q z 17 1 c

126 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES it is possible to find a finite set of positive-stiffness zero-free-length spring loads such that the net sum of the functions and potentials of springs is independent of orientation variable R. Proof. Let (k i,a i,b i ), k i > and i = 1 n a be any set of n a 1 zero-free-length spring loads on the rigid-body. Their potential energy can be written in the general form shown in equation(6.3), where the parameters of the general form, corresponding ton a springs, arerepresentedas{u i, w i, q i, v i, κ i, c i }, κ i >, i = n f +1 n f +n a. The net sum of the given functions and the potential energies of the springs take the following form. n f +n a i=1 r T R Φ i = r T n f +n a i=1 n f +n a i=1 v i +r T r u i + n f +n a i=1 n f +n a i=1 κ i + w T i Rq i + n f +n a i=1 c i (6.6) The coefficient of r T r in equation (6.6) is a non-zero positive number since κ i for n f +n a i = 1 n f, κ i > for i = n f +1 n f +n a and n a 1. Hence, the net sum Φ i, i.e., the net sum of functions along with the potentials cannot be independent of r. Thus the first part of the proposition is proved. Now, consider the second part of the proposition. The net sum is also a linear combination of the basis functions given in table 6.1 with the corresponding coefficients being the same as in the table except that the coefficients are subscripted with i and summed over i from 1 to n f +n a. The net sum is independent of orientation if and only if all the coefficients of the basis functions that are dependent on α or β or γ are zero. The coefficients of the basis function nos. 5, 6 and 7 in table 6.1 being constrained to zero can be compactly written as i=1 n f +n a i=1 v i = (6.7) The elements in the rows 8 16 in the right most column of table 6.1 are actually

127 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 17 nine elements of the matrix qw T. Hence, the coefficients for the net sum of the basis function nos in table 6.1 being constrained to zero can be written as [q 1 q nf+na ][ w 1 w nf +n a ] T = (6.8) ItisshowninAppendixEthatnomatterwhattheparameters{u i, w i, q i, v i, κ i, c i } for i = 1 n f are, there are always four zero-free-length springs having such a set of parameters (k i,a i,b i ), k i > and i = 1 4 that the equations (6.7) and (6.8) are satisfied. Hence, the second part of the proposition is also true. If a joint should have feature 2, then the joint should transform the dependence on r, which cannot be balanced out as per the proposition, to dependence on the configuration variables of another body between which the joint exists. In other words, r, i.e., position vector of local origin on the body, should be a function solely of the configuration variables of the other body. Among the basic joints, such a thing is possible only if the other body is connected to the body by a revolute or spherical joint at the origin. Thus, spherical joints and revolute joints can potentially have feature 2 and we next prove that they indeed have. From here onwards, we deal with more than one body and each body is assigned a distinct number. Further, if quantities such as r, R, u i, k i are associated with body j, then the corresponding symbols are superscripted with j. With this, quantities associated with different bodies can be distinguished from one another. 6.4 Spherical joint has feature 2 Here is the proposition with its proof that essentially says that for the class of functions given in equation(6.3), the spherical joint has feature 2 under certain conditions. Proposition 4. For a given set of functions Φ l i = r lt u l i +w l it R l q l i +r lt R l v l i +κ l ir lt r l +c l i, κ l i and i = 1 n l (6.9)

128 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 18 that is associated with body l, if: Condition 1: There is a spherical joint between body j and body l such that a point fixed to body j is coincident with the local origin of body l. Let the local coordinates of the point on body j be s j l. Condition 2: Equations (6.7) and (6.8) are satisfied with n f +n a being n l. then where n l i=1 Φ l i = r jt u j i +wj it R j q j i +rjt R j v j i +κj i rjt r j (6.1) nl u j i = nl u l i, w j i = u l i, q j i = sj l, i=1 i=1 v j i = 2sj l n l nl κ l i, κ j i = i=1 i=1 κ l i (6.11) Proof. The equation form of condition 1 is r j +R j s j l = r l (6.12) The equations referred in condition 2 are the necessary and sufficient constraints to make the net sum nl independent of R l, as shown in the proof of proposition 3. i=1 Hence, under condition 2, in the expression for nl Φ l i, the R l dependent terms can be omitted as shown next n l i=1 nl Φ l lt i = r i=1 i=1 nl u j i +rlt r l κ l i (6.13) i=1 n l i=1 Φ l i represents potential energy and our interest is in the gradient of the potential energy with respect to the configuration variables. Hence, the constant terms, which

129 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 19 do not affect gradients, are inconsequential. This is the reason for omitting constant terms as well in equation (6.13). On the same lines as in proposition 2, substitution for r l from equation (6.12) into equation (6.13), simplification and omission of the constant term leads to equations (6.1) and (6.11). 6.5 Revolute joint has feature 2 Proposition 4 holds for revolute joints as well. However, the fact that in a revolute joint, a line, rather than a point, fixed to two bodies are coincident allows us to make condition 2 of proposition 4 less stringent as shown next. Proposition 5. For given a set of functions that is associated with body l, if: Φ l i = r lt u l i +w l it R l q l i +r lt R l v l i +κ l ir lt r l +c l i, κ l i and i = 1 n l (6.14) Condition 1: There is a revolute joint between body j and body l such that a point fixed to body j is coincident with the local origin of body l and a unit vector fixed to body j is always parallel to local z-axis of body l. Let the local coordinates of the point on body j be s j j reference be ˆk j l. Condition 2: Following equations are satisfied. and the unit vector fixed to body j in local frame of n l i=1 v l i = (6.15) [ q l1 q ln l ][ w l 1 w l n l ] T = (6.16) where v = [v x v y ] T and q = [q x q y ] T.

130 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 11 then n l i=1 Φ l i = Φ j i +Φj i+1 (6.17) where φ j i and Φ j i+1 are functions of the form in equation (6.14) but associated with body j. u j i, wj i, qj i, vj i, κj i are as given in equation (6.11), and uj i+1, wj i+1, qj i+1, vj i+1, κ j i+1 are as given next. nl u j i+1 =, wj i+1 = qz,iw l l i, q j i+1 = ˆk j l i=1 v j i+1 = ˆk j l n l i=1 Proof. From condition 1 of the proposition, we have v l z,i, κ j i = (6.18) r j +R j s j l = r l (6.19) R jˆk j l = R l [ 1] T = [s l 1s l 2 c l 1s l 2 c l 2] T (6.2) What equation (6.2) says is that terms s l 1s l 2, c l 1s l 2 and c l 2 can be considered as functions of configuration variables of body j. Hence, for the net sum, it is not necessary to constrain the coefficients of basis functions involving only these functions and r l (function nos. 7, 14, 15 and 16 in table 6.1) to zero. Instead, one can transform the dependence on these function and r l to dependence on the configuration variables of body j using equations (6.19) and (6.2). This is the reason for truncating v and q in equations (6.15) and (6.16). This truncation omits coefficients corresponding to basis function nos. 7, 14, 15 and 16 in table 6.1. The contribution of basis functions nos. 1, 2, 3, and 4 (as in table 6.1), after transformation to configuration of body j, is already written in equations (6.1) and (6.11). Φ j i inequation(6.17)representthesame. Thecontributionfrombasisfunction no.7 is nl r lt [s l 1s l 2 c l 1s l 2 c l 2] T vz,i l (6.21) i=1

131 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 111 The contribution from basis function nos. 14, 15 and 16 is n l i=1 q l z,iw l i T [s l 1s l 2 c l 1s l 2 c l 2] T (6.22) Substitution for r l and [s l 1s l 2 c l 1s l 2 c l 2] from equations (6.19) and (6.2) into the sum of expressions (6.21) and (6.22), followed by simplification, yields r jt R j ˆk j n l l n l + vz,i l qz,iw l l i i=1 i=1 T R jˆk j l (6.23) This term is represented as Φ l i+1 in equation (6.17) and is the contribution from basis function nos. 7, 14, 15 and 16, after transformation to configuration variables of body j. The reason for not combining two functions on the right hand side of equation (6.17) into one is that the terms w j it R j q j i and wjt i+1r j q j i+1 cannot be combined. 6.6 Algorithm to synthesize static balancing solution of a spatial revolute/spherical-jointed tree-structured linkage having zero-free-length spring and/or gravity loads exerted by a reference link This is along the same lines as in Section Preparatory steps 1. Assign the reference body as the root node of the tree-structure. 2. If the joint between a body and its parent body is a revolute joint, then choose the local frame of reference of the body such that the z-axis of the local frame

132 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 112 is aligned along the axis of the revolute joint. If the joint between a body and its parent is spherical joint, then choose the local frame of reference of the body such that the origin of the local frame coincides with the center of the spherical joint. 3. Give this tree-structure with the given constant and zero-free-length spring loads as an input to the following iterative procedure. Iterative procedure Entry condition: If the tree-structure contains only the root node, then exit from the iterative procedure. Otherwise, proceed to step 1. Step 1: For every body/node, there are two kinds of potential energy functions associated with it. One kind is due to loads acting on the body and the other kind is due to the association that happened in step 3 of previous iterations. From equations ( ) (which holds for spatial case also), proposition 4 and proposition 5, respectively, both kinds of potential energy are of the form given in equation (6.3) where r and R are configuration variables of the body. For every terminal node l of the tree-structure, add extra n l a zero-free-length springs such that the potential energy functions due to 1) n l o original loads on the body, 2) n l c associations that happened with node l in step 3 of previous iterations, and 3) n l a extra zero-free-length spring loads satisfy condition 2 of proposition 4 in the case of spherical joint and condition 2 of proposition 5 in the case of revolute joint. Appendix E shows that such a thing is possible. Step 2: Express n l o +nl c +nl a i=1 Φ l i as function of configuration variable of the parent link. This is possible since condition 1 (because of preparatory steps), condition 2 (because of step 1) of proposition 4 in the case of spherical joint and proposition 5 in case revolute joint are satisfied. Step 3: Associate n l o +nl c +nl a i=1 Φ l i with the parent link of l and for energy conservation, disassociate Φ l i, i = 1 n l o +n l c +n l a from node l.

133 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 113 Step 4: With all the potential energy functions disassociated from node l to its parent, delete this terminal node l. Iterator: Once steps 1 to 4 are completed, a new trimmed tree-structure results where the parents of the nodes deleted in step 4 has additional potential energy functions associated with them. Follow this iterative procedure again with this trimmed tree-structure as the input. The arguments that this algorithm indeed leads to static balance is also the same as that in Section The distinct cases (a) and (b) of Section exist in spatial case also. We indicate case (b) in the last iteration of the following example Illustrative example Figure(6.1-(ii)) shows a six degree-of-freedom spatial linkage consisting of four bodies that are serially connected to the ground. Starting from the ground, the first three joints are revolute and the last joint is spherical. The architecture is the same as PUMA robot except that the wrist of the PUMA robot is replaced by a spherical joint. Starting from for the ground, the bodies are serially numbered as 1, 2, 3 and 4. Bodies 2, 3 and 4 are under the influence of constant loads C1, 4 C1, 3 and C1 2 as shown in figure (6.1). The constant loads are due to gravitational force on the bodies. In order to statically balance this spatial linkage, we follow the iterative algorithm described in Section 6.6. Local frames of reference are assigned as per the preparatory steps of the algorithm. A D-H table as well as S j l and ˆk j l parameters describing relative position of the frames are given in tables (c) and (d) of figure (6.1). In the following iterative procedure, six springs Z1, 4 Z2, 4 Z3, 4 Z1, 3 Z1, 2 and Z1 1 are added to balance the three constant forces. Tables (a) and (b) of figure (6.1) provide parameters of the constant forces as well as zero-free-length springs. Further, by comparing equations (6.1) and (6.2) with equation (6.3), the potential energies in the generalized form for these constant forces and zero-free-length springs are tabulated in table (e) of figure (6.1). The generalized form of potential energies that are obtained in steps 2 and 3 of the

134 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 114 (i) Step 1 of iteration 1: Spring addition Cond. 2 of prop. 4 ( ) ( ) ( ) ( ) 4 Φ 4 1 C 4 1, Φ 4 2 Z 4 1, Φ 4 3 Z 4 2, Φ 4 4 Z 4 3 Φ 3 1 n 4 o = 1 n4 a = 3 ( C 3 1 ) ( ) Φ 2 1 C 2 1 Step 2 of iteration 1: = Φ 3 2 (as per eqn. 6.1) Step 3 of iteration 1: association with node 3 (ii) z1 y 1 z 2 x 1 y 2 x 2 C 2 1 Z y 3 z 3 x 3 3 C 3 1 Z1 3 Z2 4 z 3 y 3 x 3 C4 1 4 Step 4 of iteration 4 1 Deletion of node 4 3 ) ( ) Φ1( 3 C 3 1, Φ 3 2, Φ 3 3 Z 3 1 Step 1 of iteration 2: Spring addition Cond. 2 of prop. 5 1 Z 4 3 Z n 3 o = 1 n3 c = 1 n3 a = 1 Φ 2 1 ( C 2 1 ) Step 2 of iteration 2: = Φ 2 2 +Φ 2 3 (aspereqn. 6.17) 1 Step 4 of iteration 2 Deletion of node ) ( ) Φ1( 2 C 2 1, Φ 2 2, Φ 2 3, Φ 2 4 Z 2 1 Spring addition Step 4 of iteration 3 2 Eqs. (6.24) and Deletion of node 2 (6.25) ( ) 1 Φ 1 1, Φ 1 2, Φ 1 3 Z 2 1 a constant (c) S 1 S 1 2 S 2 3 S 3 4 (e) S j l and kj l [ ] parameters T k [ ] T 1 1 [ ] T k 1 [ ] T 2 1 [ ] T k 2 [ ] T 3 1 [ ] T.6.. z y x Z 1 1 (a) Parameters of constant forces C j i p j i f j i [ ] C1 4 T [ ] T [ ] C1 3 T [ ] T [ ] C1 2 T [ ] T (b) Parameters of zero-free-length springs Z j i a j i b j i k j i [ ] Z1 4 T [ ] T [ ] Z2 4 T [ ] T [ ] T [ ] T Z 4 3 Z 3 1 Z 2 1 Z 1 1 [.2..5 ] T [ [ ] T [ (d) D-H parameters i α i 1 a i 1 d i θ i θ 1 π θ θ 3 4 π 2.6 3] T 9 24] T 36 [ ] T [ T 24] 48 (Ground) Further tranformation of frame 4 is described by 321 body fixed euler angles: e 4 1, e 4 2, and e 4 3. Parameters of potentials due to springs, constant forces and transformation cum association in steps 2 and 3 Φ j i u j i q j i w j i v j i κ j i ) [ ] Φ1( 4 C 4 T [ ] T [ ] T [ ] T ( ) [ ] Φ 4 2 Z 4 T [ ] T [ ] T [ ] T ( ) [ ] 2 Φ 4 3 Z 4 T [ ] T [ ] T [ ] T ( ) [ ] 2 Z 4 T [ ] T [ ] T [ ] T Φ 4 4 Step 3 of iteration 2: association with node 2 Step 1 of iteration 3: Spring addition Cond. 2 of prop. 5 n 2 o = 1 n2 c = 2 n2 a = 1 Step 2 of iteration 3: = Φ 1 1 +Φ 1 2 (aspereqn. 6.17) Step 3 of iteration 3: association with node 1 ) Φ1( 3 C 3 1 [ ] T...9 [ ] T.2.. [ ] T...9 [ ] T... Φ 3 2 = 4 Φ 4 i [ ] T [ ] T.6.. [ ] T [ ] T i=1 ( ) [ ] Φ 4 3 Z 3 T [ ] T [ T [ ] T 3] ) [ ] Φ1( 2 C 2 T [ ] T [ ] T [ ] T [ ] Φ 2 2 +Φ 2 3= 3 T [ ] T [ ] T [ ] T Φ 3 6 i [ ] T [ ] T [ ] T [ ] T i= ( ) [ ] Φ 2 4 Z 2 T [ T [ ] T [ ] T 24] [ ] Φ 1 1 +Φ 1 2= 4 T [ ] T [ ] T [ ] T Φ 2 24 i [ ] T [ i= T [ ] T [ ] T 4 4 8] ( ) [ ] Φ 1 3 Z 1 T [ T [ ] T [ ] T 24] Figure 6.1: Details of static balancing of six degree-of-freedom spatial balancing under gravity loads 2

135 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 115 iterative procedure are also tabulated in the same table. The table is intended to facilitate verification of the constraints that we satisfy in the iterative procedure. The iterative procedure is pictorially depicted in figure (6.1-(i)). The terminal body of the tree-structure is body 4. n 4 o = 1 and it is due to load C 4 1. n 4 c = since this is the first iteration. In the first iteration, three springs Z 4 1, Z 4 2, and Z 4 3 are added so that potential energies Φ 4 1(C 4 1) (i.e., potential energy due to load C 4 1), Φ 4 2(Z 4 1), Φ 4 3(Z 4 2), and Φ 4 4(Z 4 3) satisfy condition 2 of proposition 4. Further, in step 2, the summation of these potential energies is transformed as Φ 3 2, as per equation (6.1). Steps 3 and 4 involve association of the summed potential with node 3 of a new tree-structure formed by deleting node 4. All these steps are depicted in figure (6.1-(i)). Similarly, the details of iteration 2 and 3 are also depicted in the same figure. In the fourth iteration, instead of carrying out the same four steps, one can take advantage of the fact that r 1 is a constant and the first two of the Euler angles describing R 1 are zeros. Among the configuration variables describing body 1, only the last Euler angle, i.e., γ 1 is a variable. In this scenario, examination of table 6.1 reveals that any sum of potential energies of the form in equation (6.3) is a linear combination sinγ 1 (i.e., s 3 ) and cosγ 1 (i.e., c 3 ) and 1. By setting the coefficients of sinγ 1 and cosγ 1 to zero, i.e., n 1 ( ) r 1 y,i vx,i 1 rx,iv 1 y,i 1 +wy,iq 1 x,i 1 wx,iq 1 y,i 1 = (6.24) i=1 n 1 ( ) r 1 x,i vx,i 1 +ry,iv 1 y,i 1 +wx,iq 1 x,i 1 +wy,iq 1 y,i 1 = (6.25) i=1 one gets the condition for the sum of potential energies associated with node 1 to be a constant. In the fourth iteration, a spring Z1 1 is added so that potential energies Φ 1 1, Φ 1 2 and Φ 1 3(Z1) 1 satisfy constraints (6.24) and (6.25). Hence Φ Φ Φ 1 3(Z1) 1 is a constant. Further Φ Φ Φ 1 3(Z1) 1 is the net sum of potential energies of all the constant loads and the zero-free-length springs added in the iterative process. (This may be verified by retracing the arrow lines in figure (6.1-(i))). Since the net

136 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 116 potential is constant, static balance is attained. 6.7 Static balance of any kind of spatial revolute and/or spherical jointed linkage with constant load and zero-free-length spring load interaction The restriction for algorithm of Section cited in Section is true for the spatial algorithm of Section 6.6 also. A general spatial and/or revolute jointed linkage with constant forces and zero-free-length spring may not fall under these restrictions. Nevertheless, one can split the loads into load sets followed by breaking of closed loops, as in figure (5.17), so that the resulting loaded linkages fall within the ambit of the algorithm. Statically balancing these resulting linkages would balance the original linkage for the same reasons that are presented in Section In this way, the algorithm of Section 6.6 can be used to balance any revolute and/or spherical jointed linkage with any kind of constant load and zero-free-length spring load interactions between the bodies. 6.8 A note Interference is an important issue in a practical implementation of the method presented in this chapter. In the planar case, the elements that can potentially interfere can be placed in different parallel planes. Such a leeway is generally not available in the spatial case. Nevertheless, balancing solutions from the method are not unique. This non-uniqueness could possibly be exploited to see that the expected motion of the linkage being balanced falls within the workspace that is devoid of interference.

137 CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 117 Summary For a set of potential energies of a body that includes constant loads and zerofree-length springs loads, we derived the conditions under which the sum of the potential energies can be transformed as potential energies of another body through revolute joint or spherical joint. The conditions for the revolute joint is less stringent in comparison to the spherical joint, reflecting the difference between the joints. Recursive application of the transformation of potential energies from one body to the other leads to static balance. Interference could be an issue in a practical implementation of the method presented in this chapter.

138 Chapter 7 Towards static balance of compliant mechanisms Overview Proposal of a framework for analytical, albeit approximate, static balancing of a flexure-based compliant mechanism that makes use of the analytical static balancing techniques for spring-loaded rigid-body linkages developed in previous chapters. Static balancing of a flexure-beam as a prelude to understanding the framework. Three examples based on the framework with encouraging results. A prototype to demonstrate static balancing of a flexure-based compliant mechanism. A discussion on the limitations of the framework. 118

139 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Balancing a flexure beam The flexure beam The details of the flexure beam are shown in figure (7.1). It has two portions: a rigid portion and a flexure. The dimensions of the flexure, geometrically a cuboid, are as shown in the figure. The flexure beam is considered to have linear isotropic elasticity. The Young s Modulus and the Poisson s ratio are also given in the figure. Material Properties: Young s modulus: 25e9 Pa Poisson ratio :.28 b = 2mm h =.5mm l = 3mm L = 2mm Rigid portion L l 2 h Flexure l h b Dimension of the flexure beam Flexure cross-section Figure 7.1: Details of the flexure beam The continuous set of configurations over which static balance is sought The flexure, being an elastic body, can have arbitrary deformation. Figure(7.2) shows different types of loading and deformation. Among these, the rigid body model of the flexure beam can accurately model the statics and kinematics of the situation in figure

140 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 12 (7.2a). Here, the interest is to know what horizontal force at point P is necessary for a range of horizontal displacements of point P. Our static balancing will focus on considerably reducing this horizontal force over the range of horizontal displacements. If this force is reduced to zero, it would become perfect static balancing but, as we will see, it is not the case Rigid-body model for the flexure beam When the flexure beam model [7] is applied to the flexure beam of figure (7.1) a torsional spring-loaded lever results, as shown in figure (7.3). In this model, the flexure is replaced by a revolute joint with a linear torsional spring at the joint. The joint is usually placed at the centre of the flexure s undeformed configuration. Further, in the undeformed configuration, the moment exerted by the torsional spring is zero. To obtain the spring constant, the flexure is modelled as an Euler-Bernoulli beam (see [79]). An Euler-Bernoulli beam under pure moment will have a constant ratio between the moment and the relative change in the tangent-angle of the end-points of the beam. This ratio is evaluated for the flexure and it is taken as the torsional spring constant. If E is the Young s modulus of the flexure, I is the area moment of inertia of the cross-section of the flexure and l is the length of the flexure, then the ratio is given by k t = EI l The numerical values of the relevant quantities are tabulated in table 7.1. Table 7.1: Relevant quantities to calculate the torsional stiffness of the spring Numerical value for torsional spring E Young s modulus [Pa] h Height of the flexure cross-sections [m] b Width of the flexure cross-sections [m] I Area moment of inertia = bh [m 4 ] 12 l Length of the flexures [m] k t Torsional spring constant = EI 1.424[N m/rad] l L Lever arm length.2[m] (7.1)

141 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 121 F P The flexure beam under bending (a) F P The flexure beam under a loading The flexure beam under axial tension (b) (c) Figure 7.2: Our attention is on reducing horizontal force for a range of horizontal displacements of point P

142 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 122 Case 1a Case 2a k t = EI l Figure 7.3: Small-length-flexure model applied to the flexure beam Approximation of torsional spring by zero-free-length spring We now approximate torsional springs by zero-free-length springs. This approximation is necessary since we know how to analytically static balance a linkage loaded by zero-free-length springs but not a linkage with torsional springs. Our eventual goal is to reduce horizontal force F x over a range of horizontal displacementsu x ofpointp. So, azero-free-lengthspring, shownincase3aoffigure(7.4) can be considered to approximate the torsional spring of case 2a if the replacement of the torsional spring by the zero-free-length spring does not significantly perturb F x vs. u x relation. Case (2a) in figure (7.4) shows the lever with the torsional spring. This has to be approximated by the lever loaded by a zero-free-length spring as shown in case 3a of the same figure. The task now is to find the parameters of the zero-free-length spring (spring constant and anchor points) so that F x vs. u x curve for the two cases match as closely as possible. Towards that, we find the zero-free-length spring parameters

143 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 123 so that F x and dfx du x at u x = match. With this, the two F x vs. u x curves have the same Taylor s expansion upto the first order and we expect them to match atleast in a range around u x =. The details of this are presented next. F P u P k 1 =? y y θ θ x x k t Case 2a Case 3a Positive rotational direction: anticlockwise Figure 7.4: Approximation of the torsional spring by a zero-free-length spring F x vs. u x in the torsional spring-loaded lever (case 2a) There may be ways to find F x vs. u x relation that are simple in comparison to what is presented next. However, for the sake of uniformity across the examples, we use the principle of virtual work to get a balance equation form which F x vs. u x relation can be deduced. Let θ and δθ be the variables that parameterize the configuration and its virtual change. The quantities related to the calculation of the virtual work are as in table (7.2). For static equilibrium at configuration θ, the net virtual work should be zero, i.e., δθ(k t θ+f x Lcosθ) = (7.2)

144 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 124 Table 7.2: Virtual work calculation for lever in case (2a) Sl. no. description of quantities at the pivot at point P [ ] Lsinθ 1 generalized displacement θ u = [ Lcosθ ] Fx 2 generalized force k t θ F = [ ] cosθ 3 generalized virtual displacement δθ δθl sinθ 4 virtual work δθk t θ δθf x Lcosθ Equation (7.2) can be used to get the following expression for F x as F x = k tθ Lcosθ (7.3) In table 7.2 we had already written u x as a function of θ, i.e., u x = Lsinθ (7.4) Equations (7.3 and 7.4) give relation between F x and u x in the parametric form. The derivative or the slope of F x vs. u x is given by df x du x = df x dθ du x dθ k ( t θsinθ L cos = 2 θ + 1 ) cosθ Lcosθ (7.5) The reference configuration (corresponding to the undeformed configuration of the flexure beam) corresponds to θ =. At θ =, F x and dfx du x are given as F x = and df x du x = k t L 2 (7.6) F x vs. u x in the lever loaded by zero-free-length spring (case 3a) The lever loaded by zero-free-length spring is shown as case (3a) in figure (7.4). One end of the spring is anchored at P and in the reference configuration (θ = ) the

145 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 125 spring is undeformed. The various quantities for calculating virtual work balance are tabulated. Table 7.3: Virtual work calculation and slope of F x vs. u x Sl. no. description of quantities at P (spring force) at P (force F x ) [ ] [ ] Lsinθ Lsinθ 1 generalized displacement u = u = [ Lcosθ ] [ Lcosθ ] sinθ Fx 2 generalized force k 1 L F = (1 cosθ) [ ] [ ] cosθ cosθ 3 generalized virtual displacement δθl δθl sinθ sinθ 4 virtual work δθk 1 L 2 sinθ δθf x Lcosθ Virtual work balance: δθ(k 1 L 2 sinθ+f x Lcosθ) = (7.7) F x and u x in parametric form F x = k 1Lsinθ cosθ (7.8) u x = Lsinθ (7.9) Slope of F x vs. u x df x du x = df x dθ du x dθ = ( ) sin k 1 L 2 θ +1 cos 2 θ Lcosθ (7.1) F x and slope of F x vs. u x at θ = F x =, and df x du x = k 1 (7.11)

146 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 126 The parameters of the zero-free-length spring in case 3a The anchor points of the zero-free-length spring (see figure (7.4)) were chosen such that in the reference configuration (θ = ), F x is zero. Thus, at θ =, F x for case 2a and case 3a trivially match (see equations (7.6) and (7.11)). The remaining unknown parameter of the zero-free-length spring is its spring constant k 1. This is found by equating df x du x at θ = for the two cases. From equations (7.6) and (7.11), matching of the slopes lead to the following expression for k 1. k 1 = k t L 2 (7.12) Using the numerical values for k t and L, given in table 7.1, the numerical value for k 1 is 35.6 [N/m]. With this value of k 1, we plot F x vs. u x relation for both case 2a and case 3a (from equations (7.3), (7.8) and (7.4)) for a range of 5cm displacement of point P around its reference position, as shown in figure (7.5). The match is not perfect since there are deviations, which grow towards either end. Nevertheless, for the range of u x plotted, one can say, by examining the plot, that the deviations are less than 5% Static balancing by addition of a zero-free-length spring Case 3a of figure (7.4) is the simplest of revolute-jointed linkages and it has only zero-free-length load. Hence, case 3a in figure (7.4) is apt for application of static balancing method of Chapter 5. Case 3a of figure (7.4) is a lever. Hence, the loads on it have to satisfy equations ( ) for it to be in static balance. Towards that, we add a zero-free-length spring of stiffness k 2 = k 1, as shown in figure (7.6). It can be verified through table 7.4 that the original zero-free-length spring and the additional zero-free-length spring together satisfy equations ( ). As a cross-check, one may also do virtual work analysis, similar to the way it is in table 7.3. The virtual work calculation table is the same as that of table 7.3 except for an additional column corresponding to the balancing spring, as shown in table 7.5.

147 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Force vs. deflection at the reference point 1..5 Horizonatl force at P (effort) case 2a case 3a Horizonatal displacement of point P Figure 7.5: The approximate match in F x vs. u x relation between case 2a and case 3a

148 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 128 k 1 L x 1 y y 1 x k 2 = k 1 L Balancing spring Case (3b) Figure 7.6: Static balance of the approximated zero-free-length spring-loaded lever by addition of a zero-free-length spring

149 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 129 Table 7.4: Verification of equations ( ) being satisfied k i a i b i r k i (r b i ) T a i k i (r b i ) T R( π)a 2 i [ ] [ ] [ ] L 1 st spring k 1 k L 1 L 2 [ ] [ ] L 2 nd spring k 1 k L 1 L 2 Summation over two springs Table 7.5: Verification of static balance through virtual work calculations no. quantities Original spring balancing spring force F [ ] [ ] [ x ] Lsinθ Lsinθ Lsinθ 1 displacement u = u = u = [ Lcosθ ] [ Lcosθ ] [ Lcosθ ] sinθ sinθ Fx 2 force k 1 L k (1 cosθ) 2 L F = [ ] ( 1 cosθ) [ ] [ ] cosθ cosθ cosθ 3 virtual displ. δθl δθl δθl sinθ sinθ sinθ 4 virtual work δθk 1 L 2 sinθ +δθk 2 L 2 sinθ δθf x Lcosθ Along the same lines as Section 7.1.3, we obtain Virtual work balance: δθ((k 1 k 2 )L 2 sinθ+f x Lcosθ) = (7.13) F x and its slope at θ = F x = (k 1 k 2 )Lsinθ cosθ (7.14) df x du x = k 1 k 2, at θ = (7.15) From equation (7.14), it may be noted that when k 1 = k 2, the effort is zero for any configuration.

150 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Balancing springs on the flexure beam Case 3a approximates case 2a (see figure (7.4) and case 2a in turn approximates case 1a (see 7.3). Further, a perfectly statically balanced case 3b was obtained by adding a balancing spring (see figure (7.6)). Case 1b, shown in figure (7.7), obtained by adding the same balancing spring to the flexure of case 1a is expected to be at least approximately statically balanced because of the approximation between case 1a and case 3a. Figure (7.7), by the way, is a consolidation of all the cases. To check the efficacy of balancing in case 1b, we performed finite element analysis of the flexure beam using commercially available software COMSOL( The flexure beam was modelled as a plane stress problem with geometric nonlinearity. Point P was given a horizontal displacement constraint of u x (see figure (7.2)). F x was obtained as the reaction force of this constraint. By varying u x, F x for a range of values of u x was obtained. This analysis was carried out both with and without balancing spring force. This data is plotted in figure (7.8). It is important to note that prior to this, all the calculations to obtain the approximately balancing spring are analytical. The numerical simulation has been used only to judge the effectiveness of this approximately balancing spring. In figure (7.8), while the curve for f 1b is not the ideal zero, it has reduced to less than 2% when compared with unbalanced case 1a. A way to improve the balance Having carried out the numerical analysis to judge the effectiveness of the approximate balance, we are in a position in to find deviation from perfect balance. In particular we can know what deviations are there for F x and its slope at the origin. Let us call these quantities as c and m (these symbols are intended to allude to standard equation of a straight line: y = mx + c). From the plot, there is no deviation in c while the deviation in m is 5.4 [N/m]. Further, suppose that we can make changes to parametersofthebalancingspring, saytoitsstiffnessk 2. Ifwecanfindthesensitivity of m with respect to k 2, then we can make a first order correction to k 2. While one can find this sensitivity from finite element analysis, as a matter curiosity, we have

151 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 131 P P P (1a) (2a) (3a) P P P (1b) (2b) (3b) Figure 7.7: All the cases related to the flexure and its approximation by the springloaded lever

152 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS F x vs. u x at point P 2 f 1a 1 F x (N) f 1b f 1c u x (m) Figure 7.8: F x vs. u x relation obtained from finite element analysis of the flexure beam

153 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 133 used the sensitivity from the rigid-body model, in particular, from equation (7.15). From the equation, it is evident that the sensitivity is 1. The first order correction k 2 should satisfy: m desired m current = (sensitivity of m) k = 1 k 2 Hence, the correction k 2 is 5.4 [N/m] and F x vs. u x curve for the flexure with the corrected k 2 is shown as f 1c in figure (7.8). Without the correction, F x of case 1b had decreased to less than 2% of case 1a whereas with the correction F x has decreased to less than 2% of case 1a. We end this section with a consolidated f x vs. u x plot for all cases shown in figure (7.9). Intermediate nature of cases 2a and 2b The intermediate nature of cases 2a and 2b is worth noting. One can compose smalllength flexure model and the first order Taylor s approximation into a single approximation model to avoid the intermediate cases 2a and 2b. However, by using small-length flexure model without any modification or composition, we are only acknowledging the motivational role played by small-length flexure model, which is already documented in literature. Alternatively, one can use numerical methods such as Finite Element Methods to directly establish approximations between cases 1a and 3a and between cases 3b and 1b. However, the use of numerical methods defeats our intent to obtain a simple analytical static-balancing framework that is described next. 7.2 Framework Based on the example of balancing the flexure beam, we now formally propose a framework to use the static balancing techniques developed for rigid-body linkages loaded with zero-free-length springs for static balancing of flexure-based compliant mechanisms.

154 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS F x vs. u x at point P 2 1a 2a, 3a F x (N) b, 3b, 1c case 1a case 2a case 3a case 1b case 2b case 3b u x (m) 1b case 1c (corrected) Figure 7.9: A consolidated plot of F x vs. u x for flexure beam and its rigid-body models

155 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 135 Step 1: Apply small-length-flexure model to the given flexure-based compliant mechanism to obtain a revolute-jointed rigid-body linkage loaded by torsional springs. The given flexure-based compliant mechanism may be labelled as case 1a and the linkage loaded by torsional springs as case 2a. Step 2: Identification of an effort function. Identify a function whose closeness to zero implies closeness of the system to static balance. The definition of the function should preferably be such that it is convenient to compute it both in flexure-based compliant mechanisms and rigid-link mechanisms. Step 3: Torsional springs to zero-free-length springs. In this step, a new case, labelled case 3a is formed where the torsional springs are substituted by zerofree-length springs. The parameters of the zero-free-length springs should be such that the effort function in case 2a and case 3a should have a close match. In this thesis, to obtain the match, we ensure that there is a perfect match in terms of the value of the function and the slope of the function at a reference configuration. It is known that if two functions have the same value and slope at a configuration then in a small range around the configuration, the two functions match closely. This is the consequence of the match in the Taylor s expansion of the two functions upto the first order. Step 4: Static balancing of the linkage with zero-free-length springs. Statically balance the spring-loaded linkage of case 3a to obtain case 3b. In this thesis, we propose the use of static balancing techniques of Chapter 5, where there is no addition of auxiliary bodies. The previous step was necessary since, till now, there are analytical static balancing principles only for rigid-body linkages under zero-free-length spring loads and not under torsional spring loads. Step 5: Whatever balancing zero-free-length springs were added in case 3b, add the same springs to the compliant mechanism of case 1a to obtain case 1b. The assumption here is that with cases 1a, 2a and 3a being approximately close to each other, whatever additional load balances one case would balance other cases also. Note that there are instances where this assumption may break

156 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 136 down and so does the whole framework. Breakdown of this step is discussed in Section 7.6. A way to improve the static balance In case 1b, the effort function and its slope are ideally expected to be zero. However, due to approximations, generally there would be deviations. The deviations could be found either by finite element analysis or by practical measurements. Based on the sensitivity of the function value and its slope with respect to the balancing spring parameters, one can do a first order correction to balancing spring parameters. This generally brings effort function closer to zero. While the sensitivity should ideally be evaluated for case 1b using finite element methods, in this thesis, out of curiosity, we analytically obtain sensitivity based on case 3b. We label this corrected version of case 1b as case 1c. We now illustrate this framework on three more examples where the results are encouraging. 7.3 Flexure-based compliant four-bar mechanism Description of the mechanism Figure(7.1) shows a flexure-based four-bar linkage. In the undeformed configuration, all the flexures are vertical. All the flexures have the same geometrical dimensions as well as elastic properties. Each flexure is a cuboid. The in-plane dimensions are.5 mm 1 mm and the out-of-plane dimension is 4 mm. In the plane of motion and in the undeformed configuration, the quadrilateral formed by the centers of the flexures is a quadrilateral of sides a, b, c, and d with a diagonal of length l, as shown in figure (7.11). In the flexure at the vertex between sides c and d (see figure (7.11)), the bottom-most point is labeled as P (see figure (7.1)). From the figure, it may also be noted that P is at the centre of the interface between the flexure and the rigid portion. This point is later used to define the effort function.

157 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 137 Case 1a Flexures P Flexures Undeformed configuration Figure 7.1: A flexure-based four-bar linkage. l = 14mm c = 13mm b = 1mm l d = 12mm a = 15mm (drawing not to the scale) Figure 7.11: The quadrilateral formed by the centers of the flexures

158 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Step 1 Application of small-length flexure model leads to torsional spring-loaded four-bar linkage, as shown in figure (7.12). All the quantities required to calculate the spring constant of the torsional spring are shown in the table of the same figure. P k t c P k t b d k t a k t Case 1a Case 2a E Young s modulus 25e9[Pa] h Height of the flexure cross-sections 5e-6[m] b Width of the flexure cross-sections 4e-2[m] I Area moment of inetria = bh e-13[m 4 ] l Length of the flexures 1e-2[m] k t Torsional spring constant [N-m/rad] Figure 7.12: Approximation of the flexure-based four-bar linkage as a rigid-body four-bar linkage with torsional springs Step 2 The effort function When point P is constrained to have a horizontal displacement of u x, there is a corresponding horizontal constraint force F x. The function F x vs. u x over a range [.25m,+.25m] of u x is taken as the effort function.

159 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Step 3 We next propose to approximate the torsional springs of case 2a by two zero-freelength springs as in figure (7.13). We further propose to fix the anchor points of the springs at the coordinates shown in figure (7.13). We find the remaining parameters, i.e., the spring constants, so that there is a close match in the effort function between case 2a and case 3a. Towards that, at the reference configuration, the effort function (F x at u x = ) and its slope (df x /du x at u x = ) for the two cases are analytically found and tabulated in table 7.6. The details of this analytical calculation, which is again based on virtual work balance as in Section 7.1, are presented as WXMAXIMA ( worksheet in Appendix F.1. To achieve a close match in effort function between case 2a and case 3a, we equate F x at u x = and df x /du x at u x = between the two cases and solve the stiffness from the equations. The stiffness values so obtained are shown in table 7.6. k t P k t y 2 x 2 P Z 2 1 k t k t x 1 y 1 y Z 1 1 Case 2a x Case 3a Spring a b k local global stiffness m m ] N/m Z 2 1 Z 1 1 [ [.8 ] [.5 ] [ ] k 2 1 k 1 1 Figure 7.13: Approximation of torsional springs by zero-free-length springs

160 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 14 Table 7.6: The origin and the slope of F x vs. u x being matched between case 2a and case 3a at reference configuration case 2a case 3a F x.146k k 1 1 df x du x k k 1 1 Matching F x and df x du x leads to k 2 1 = and k 1 1 = Step 4 Static balance of the linkage under zero-freelength spring load The four-bar linkage with the zero-free-length spring is now statically balanced using the principles of Chapter 5. As shown in figure (7.14), the unloaded body on the right side is disregarded. This leads to a 2R linkage with springs Z 1 1 and Z 2 1. Based on the static balancing algorithm of Chapter 5, two more zero-free-length springs Z 2 2 and Z 1 2 are added to perfectly balance the linkage. The additional springs are added in two iterations to satisfy equations ( ). Details of the quantities required to verify these equations, including spring details, are presented in table 7.7. This balancing is also cross-checked using virtual work calculations in Appendix F.2. Table 7.7: Verification of equations ( ) for the spring-loaded 2R linkage of figure (7.14) u v w q κ kb ka b ka k 2 [ ] [ ] [ ] [ ] Φ (Z2 1 ) [ ] [ ] [ ] [ ] Φ (Z2 2 ) [ ] 3 [ ] 3.1 u = s e 2 2 = κ = [ ] [ ] [ ] [ ] Φ 1.1 2e 2.1 1e 2 1 (Z2 1,Z2 2 ) [.12 ] [ ] [.12 ] [ ] Φ 1 2 (Z1 1 ) e e [ ] [ ] [ ] [ ] Φ e e 2 3 (Z1 2 )

161 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 141 Case 3b Z 2 2 Z 2 1 y 2 x 2 2 x 1 y 1 1 Z 1 1 Z 1 2 y x Spring a b k local global m m N/m [ ] [ ].13.5 Z [ 3 ] [.6 ].13.5 Z [ 3 ] [.6 ].8.25 Z [ ] [.5 ].8.92 Z Figure 7.14: Static balancing by addition of zero-free-length springs

162 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Step 5 Approximate static balance of flexure-based four-bar linkage Becauseofsucessiveapproximationsfromcase1atocase2aandcase2atocase3a, we expect that whatever balances case 3a balances approximately case 1a too. Based on this, the balancing springs Z2 2 and Z2 2 are added on to flexure-based four-bar linkage as shown in figure (7.15 1b). Fx vs. u x plot for case 1a and case 1b, using finite element simulation, is given in figure (7.16). From the plot, it may be seen that F x in case (1b) has reduced to less that 2% of case 1a. P P P Z 2 1 (1a) (2a) (3a) Z 1 1 Z 2 2 Z 2 2 Z 2 2 Z 2 1 Z 1 2 Z 1 2 (1b) (2b) (3b) Z 1 2 Z 1 1 Figure 7.15: A consolidated figure of all the cases A way to improve the static balance of the flexurebased four-bar linkage However, from the plot of figure (7.16), for case 1b, at u x = there is a deviation for F x and df x /du x from their expected zero values. At u x =, let us call F x and

163 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS a 1b 15 F x [N] u x [m] Figure 7.16: Finite element simulation results for case 1a and case 1b

164 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 144 df x /du x as c and m and the deviations as δc and δm. We can make a first order correction to these deviations by changing k 2 2 and k 1 2 as below: [ c m ] = c k 2 2 m k 2 2 c k 1 2 m k 1 2 [ ] k 2 2 k 1 2 (7.16) While the partial derivative in the matrix in equation (7.16) should ideally be found by finite element method based sensitivity analysis, we analytically find them based on the approximate rigid-body model of case 3b. The details of this calculation are in Appendix F.3, where it turns out that the corrections are k 2 2 = 39.4 and k 1 2 = One more finite element analysis of the flexure-based four-bar linkage is carried out using the corrected k 2 2 and k 1 2 and F x vs. u x relation turns out be as in figure (7.17) Case 1a Case 1b Corrected F x [N] u x [m] Figure 7.17: F x vs. u x after first order correction to the stiffness of balancing springs

165 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 145 cases. We end this example with figure (7.18), which is a consolidated plot of all the F x [N] a 1b 1c (corrected) 2a 2b 3a 3b u x [m] Figure 7.18: F X vs. u x plot for all the cases shown in figure (7.15) 7.4 Another flexure-based four-bar linkage Description of the mechanism This compliant mechanism is the same as in Section 7.3 except that the flexures are longer as could be seen in case 1a of figure (7.19). Point P is also defined as in Section 7.3. Its location however is different because of the longer flexure.

166 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 146 R R R y 1 x 1 1 y (1a) (2a) x (3a) R R Z 1 R 1 (1b) (2b) (3b) Z 1 2 Figure 7.19: Consolidated figure containing all the cases

167 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 147 Table 7.8: Details of the flexure and calculation of torsional spring constant Step 1 E Young s modulus 25e9[Pa] h Height of the flexure cross-sections 5e-6[m] b Width of the flexure cross-sections 4e-2[m] I Area moment of inetria = bh e-13[m 4 ] 12 l Length of the flexures 3e-2[m] k t Torsional spring constant 2.847[Nm/rad] Small-length-flexure model is applied on case 1a to obtain a torsional spring-loaded rigid-body four-bar linkage (case 2a) as in figure (7.19). The details of the flexures and the torsional spring constants are given table Step 2 The effort function is again F x vs. u x where F x is the horizontal reaction force corresponding to a horizontal displacement constraint of u x on point P Step 3 We propose to approximate the torsional springs of case 2a by a single zero-free-length spring as shown in case 3a of figure (7.19). Here again, the anchor point of the spring is fixed and we assign such a spring constant so that there is a close match between effort functions in case 2a and case 3a. The local coordinates of the anchor point on body 1 (see case 3a in figure (7.19)) is [.632,.31] T m and the global coordinates of the anchor point on the ground is [,.7] T m. Table 7.9 shows, at u x =, F x and df x /du x. F x is zero in both cases. df x /du x is a linear multiple of the stiffness in case 3a. By equating df x /du x between the two cases, we get the stiffness as shown in the same table. With this stiffness, the extent of match between case 2a and case 3a could be seen in figure (7.2). The contents of the table were arrived at analytically (using WXMAXIMA ( in a manner similar to Section 7.3.

168 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 148 Table 7.9: Matching value and slopes at the origin of the effort function at u x = case 2a case 3a F x df x /du x k Matching the slope leads to k = a 1b 1c 2a 2b 3a 3b 1 generalized force [N] generalized displacment [m] Figure 7.2: Effort function in all the cases

169 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Step 4 Table 7.1: Details of the original spring and balancing spring Spring a b k local global stiffness m m N/m [ ] [ ].632 Z 1 1 Z 1 2 [.31 ] [.7 ] For the purpose of static balancing in case 3a, one only needs to consider the body labelled as 1 (in figure (7.19)) along with the ground. With such a consideration, this becomes a spring-loaded lever, similar to case 3a in Section 7.1. We balance the spring Z 1 1 with additional spring Z 1 2 as shown in case 3b of figure (7.19). The details of each of the springs is given in table 7.1. From the table, it may be verified that the spring parameters satisfy the lever-balancing equations ( ) Step 5 The balancing spring of case 3b is incorporated into case 1a to obtain case 1b, as shown in figure (7.19). The extent of balance in case 1b can be judged through the plots in figure (7.2). Just by visual judgement, decrease in F x to less than 3% of case 1a can be noted First order correction From the plot for case 1b in figure (7.2) it may be noted that at the origin, i.e., u x =, the slope of the plot deviates from the ideal zero-slope. The deviation (calculated using finite element analysis data) is shown table Using sensitivity of the slope with respect to stiffness of the balancing springs(calculated from case 3a), the correction is found in the same table. The effort function after this correction is labelled as 1c in figure (7.2) and it may be noted that the effort function has indeed become closer to zero.

170 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 15 Table 7.11: First order correction of balancing spring parameters Slope error Slope sensitivity w.r.t. k2 1 k2 1 N/m N/m Prototype Basedonthisexample, aprototypewasmadetodemonstratethereductionineffortof a flexure-based compliant four-bar mechanism, as shown in figure (7.21). The image in the figure is taken from the backside. Hence correlation between this and case 1b of figure (7.19) has to be done through a flipping transformation. The prototype deviates from the example in one important respect. It uses normal springs having finite free-length as balancing springs. In fact four balancing springs are used, which are mostly in parallel, except for a little offset between anchor points. The balancing springs were chosen such that the net force of all of them when extended between anchor points is approximately same as what is exerted by the balancing spring of case 1c of the example of this section. With further tuning in terms of varying the number of active coils in springs, we could demonstrate a perceptible difference in the effort to deflect the prototype mechanism. Almost all people who tried their hand on this found the difference to be significant. Force measurements showed a reduction in the effort to less than 25% over a range of 2 cm on either side of the reference configuration. What we want to suggest through this prototype is that the theoretical solution serves as a guideline and one can deviate from the theoretical solution (in this case, deviation was in free-length) with some compromise on static balance. This compromise may be small enough to be acceptable in many practical situations.

171 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 151 Figure 7.21: A prototype to demonstrate reduction in effort

172 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Flexure-based 2R compliant mechanism Description of the compliant mechanism The flexure-based compliant mechanism shown in case 1a of figure (7.22) is so conceived that when small-length-flexure approximation is applied to it, it becomes a 2R linkage, as shown in case 2a of the same figure. Case 1a of figure (7.22) shows points P, F 1 and F 2. F 1 is the center of the first flexure and F 2 is the center of the second flexure. P is a point on the rigid portion that comes after the second flexure. In the undeformed configuration, F 1 P is horizontal, F 1 P = 8cm, PF 2 = 6cm, and F 1 F 2 = 1cm. F 1 P P Z 2 1 F 2 Z 1 1 (1a) (2a) (3a) Z 1 2 Z 2 2 (1b) (2b) (3b) Figure 7.22: A consolidation of all the cases Step 1: The small-length-flexure model is applied to case 1a to obtain torsional spring-loaded rigid linkage as shown in figure (7.22 2a). The details of the flexures and the calculation of torsional spring constants from it are given in table 7.12.

173 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 153 Table 7.12: Details of flexure and calculation of torsional spring constant. E Young s Modulus 25e9[P a] h height of the flexure cross-section 5e-6[m] b width of flexure cross-section 4e-2[m] I area moment of inertia= bh e-13[m 4 ] 12 l a length of first flexure 1e-2[m] l b length of second flexure 2e-2[m] kt a first torsional spring constant= EI l a [Nm] kt b second torsional spring constant= EI l b [Nm] Step 2 Identification of effort function The effort function is F = [F x F y ] T vs. u = [u x u y ] T where F x and F y are horizontal and vertical reaction forces when point P is constrained to have a horizontal and vertical displacements of u x and u y. We focus our interest in the range of [.15,.15]m for both u x and u y Step 3 We propose to approximate the torsional springs in case 2a by two zero-free-length springs, Z1 2 and Z1, 1 as shown in case 3a of figure (7.22). The details of these springs, along with the springs that would be added in step 4, are shown in figure (7.23). Similar to Section 7.3, one can calculate F and its derivative F with respect to u at u =. Note that in case 3a, these quantities are functions of spring constants that are yet to be determined. In order to have a close match in the effort function between case 2a and case 3a, we find the stiffness of the spring constants so that at u =, F and F are the same for the two cases. The spring constants turn out be as in table With these springs constants, the extent of match between the effort function of the two cases could be seen in figures ( ) Step 4 The zero-free-length spring-loaded 2R linkage of case 3a is statically balanced by adding two more zero-free-length springs, Z 2 2 and Z 1 2, as shown in figure (7.22 3b).

174 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 154 x Z 2 1 x 1 Z 2 2 Z 1 2 Z 1 1 x 2 (3b) spring a b k local global stiffness m m N/m [ ] [ ] 6e 2 8e 2 Z [ ] [ ] 6e 2 8e 2 Z coordinates of pivot[ of (1 to 2) ] revolute joint 1e 2 s 1 2 = [ ] [ ] 8e 2 6.4e 2 Z 1 1 Z 1 2 [ 8e 2 ] [ 4.8e 2 ] 8e 2.35e Figure 7.23: Details of springs Table 7.13: The value and the first derivative of F vs. u at u = F F case 2a case 3a [N/m] [N/m] [ ] [ ] [ ] [ k k1 2 +k1 1 k1 2 = and k1 1 = ]

175 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 155 Fx-3a Fx-3b Fx-2a Fx-2b Fx-1a Fx-1b F x [N] u x [m] u -.5 y [m] Fx-3a Fx-3b Fx-2a Fx-2b Fx-1a Fx-1b 1 F x [N] Figure 7.24: F x vs. u in two views

176 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS F y [N] Fy-3a Fy-3b Fy-2a Fy-2b Fy-1a Fy-1b u x [m] u y [m] Fy-3a Fy-3b Fy-2a Fy-2b Fy-1a Fy-1b Figure 7.25: F y vs. u in two views

177 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 157 Table 7.14: Verification of static balance of springs in case 3b u v w q κ kb ka b ka k 2 Φ 2 [ ] [ ] [ ] [ ] 8e 2 6e 2 8e 2 6e 2 1 (K 2,1 ) Φ 2 [ ] [ ] [ ] [ ] 8e 2 6e 2 8e 2 6e 2 1 (K 2,2 ) [ ] 16e 2 u = s 1 [ ] 1e 2 2 = κ = Φ 1 [ ] [ ] [ ] [ ] 16e 2 2e 2 16e 2 1e 2 1 (K 2,1,K 2,2 ) Φ 1 [ ] [ ] [ ] [ ] 2 (K 1,1 ) e 2 8e 2 6.4e 2 8e e 2 4.8e 2 2 Φ 1 [ ] [ ] [ ] [ ] 3 (K 1,2 ) e 2 8e 2 8e 2 8e e 2.23e 2 2 The details of all the springs can be seen in figure (7.23). The static balance is done in two iterations as per the balancing algorithm described in Section To facilitate verification of equations that are required to be satisfied by the algorithm of Section 5.4.2, the spring details are written in appropriate form in table Step 5 ThebalancingspringsZ2 2 andz2 1 presentincase3bareaddedonthecompliantsystem of case 1a to obtain case 1b as shown in figure (7.22). For the reasons discussed in Section 7.2, we expect the effort function to be closer to zero in case 1b, in comparison to case 1a. That it is indeed so can be seen in plots of figures ( ) where the plots for case 1a and 1b were obtained through a finite element analysis. In most places, the effort has got reduced to less than 3% of that of case 1a. 7.6 Discussion The proposed framework of Section 7.2 could fail if

178 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Rigid-body model does not very well approximate the actual flexure-based compliant mechanism. 2. The range of configuration changes (in the examples, the range of u x or u that was in focus) is very large around the undeformed configuration. The second point is understandable because the rigid-body model of flexure-based compliant mechanism are themselves valid over a finite but limited range around the undeformed configuration. Secondly, the approximation between case 2 and 3 is based on considering the Taylor s series terms upto first order. As far as the first point is considered, the deficiencies are the same as that of the small-length flexure model. The model gets more accurate when the length of the flexure becomes smaller. Sometimes, it may so happen that for a given flexure length, the approximation between case 1a and case 2a/3a is acceptable but approximation between case 1b and case 2b/3b is unacceptable. This happens because the additional load that the flexures experiences due to balancing spring loads could be high enough to make the flexure deform in way that is deviant from ideal circular arc like deformation. We noticed this in the example of Section 7.3 when the flexure length was made longer. In the example, let the length of all the flexures be increased three times. Inspite of this change, cases 2 and 3 remain the same except that the spring constants of all the springs change to one third of their values. Figure (7.26) shows the effort function for cases 1a, 3a, 3b and 1b. It may be noted that the plot for case 1b is quite different from what was noticed in previous examples. The effort function of case 1b in some places around the origin is not even less that that of case 1a. The way we reason is that in case 1a, without any balancing springs, the deformation of the flexures resembled ideal arc like deformation, as shown in figure (7.27 a). However, with the balancing springs, the load on the flexure, especially the one at point O in figure (7.27 b), was high enough to have a non-ideal non-arc like deformation. Because of this non-ideal deformation, coupled with longer length of the flexure, the kinematics and statics of the flexure-based compliant mechanism deviated largely from the rigid-body model of case 3b.

179 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS Case 1a Case 1b Case 3a Case 3b 1 F x [N] u x [m] Figure 7.26: Plot of effort function when flexure length is increased three-fold Static balancing of compliant mechanisms by individually balancing flexures In flexure-based compliant systems, the elastic resistance to deformation arises due to flexures. Hence, it is feasible to statically balance each of the flexures independently. As an illustration, consider figure (7.28). The given compliant system is shown in figure (7.28a) which is the same as that of the example in Section 7.5. In figure (7.28c), the flexure between rigid portions A and B is statically balanced by adding spring K A based on the principle that is the same as that used in figure (7.7), where the flexure between the rigid-link and the ground is balanced by adding a spring. Similarly, spring K B statically balances the flexure between the portion B and the ground in figure (7.28c). The parameters of the spring K A are determined only by the flexure between portions A and B and the spring reduces the effort only for the relative motion between the rigid portions A and B. Similarly comments apply to spring K B as well. This is what we mean when we say that each of the flexures are balanced independently of each other. When each of the flexures in a flexure based compliant system is balanced, the whole compliant system is balanced. In the above scheme of balancing, the balancing springs were added only between

180 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 16 (a) O (b) Figure 7.27: Ideal circular arc-like and non-ideal deformation of flexures

181 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 161 F 1 P K B B F 2 B A (a) K A A (c) Figure 7.28: Static balancing of each of the flexures, independently of one another successive rigid portions. However, such addition of springs only between successive rigid links is infeasible for statically balancing gravity-loaded tree-structured rigidbody linkages, as shown in appendix H Static balancing of compliant mechanisms using rigidbody linkages In this chapter, we developed a framework for approximately balancing compliant mechanisms. The balancing elements used were springs, zero-free-length springs in particular. Since, springs are also compliant elements, the eventual statically balanced compliant mechanisms consist of only compliant elements. Note that it is not necessary to have a pivot at points where springs anchor with the compliant mechanisms since springs have negligible bending stiffness. However, one can also statically balance compliant mechanisms using springs in conjunction with rigid-body linkages, as shown in Appendix G. We have not emphasized this option since some of the advantages that compliant mechanisms have over rigid-body linkages will be compromised in this option.

182 CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 162 Summary A way to make use of analytical rigid-body static balancing methods for approximate analytical static balancing of flexure-based compliant mechanism was demonstrated through a simple flexure-beam. A framework for approximate analytical static balancing of flexure-based compliant mechanisms was proposed. Three more examples based on the frame work have shown encouraging static balance. That there is scope for further improvement through simple first order correction was demonstrated. A prototype based on one of the examples was made and it did show perceptible reduction in operating effort. The prototype also demonstrated that one can pragmatically deviate from the theoretical solutions with a little compromise on the extent of static balance. The situations where the proposed framework might fail was also discussed.

183 Chapter 8 Conclusion Overview A summary of the static balancing techniques of rigid-body linkages that this thesis presented. A summary of the framework that this thesis gives for designing statically balanced flexure-based compliant mechanisms. Comparison of the contributions of the thesis with the existing literature to show that the contributions are novel. Compliant mechanisms are superior to rigid-body mechanisms in terms of lack of friction and backlash. However they have inherent elastic stiffness that could be undesirable in certain situations. Static balancing of the compliant mechanisms would eliminate or reduce this drawback. Furthermore, because of the elastic deformation of the body of the compliant mechanism, there is an inherent diversion of input energy into the mechanism. A statically balanced compliant mechanism, which is suitably pre-loaded, does not suffer from this deficiency in the efficient transmission of input energy to the output. This thesis proposed using pseudo-rigid-body models, where a few classes of compliant mechanisms are modelled as spring-loaded rigid-body linkages, for designing statically balanced compliant mechanisms. Towards that, the 163

184 CHAPTER 8. CONCLUSION 164 contributions of the thesis are two fold: 1) the development of new analytical techniques for statically balancing spring-loaded rigid-body linkages and 2) a framework to use the techniques through small-length flexure model for designing statically balanced compliant mechanisms. 8.1 A summary of new static balancing techniques for spring and/or gravity-loaded rigid-body linkages Static balancing of a four-bar linkage loaded by a spring on its coupler link Chapter 3 presented three techniques to statically balance a zero-free-length springloaded four-bar linkage. The first technique requires two auxiliary bodies and one balancing spring. The second technique requires two balancing springs without using any auxiliary bodies. The third technique requires two auxiliary bodies and two balancing springs. The second technique was demonstrated through a prototype Static balancing parameters and the cognates of a fourbar linkage Chapter 4 presented a unified parameterization for the triplet of the family of static balancing solutions of the three cognates where the anchor points of the balancing springs always form a triangle that is similar to the cognate triangle and the ratio of spring constants of a set of corresponding balancing springs is a constant. This result came out of a graphical solution that the chapter gave for a problem in geometry. While the result of this chapter cannot be stated as a new static balancing method, it helps in exploring various design possibilities as stated in the chapter.

185 CHAPTER 8. CONCLUSION Static balancing without auxiliary bodies planar case Chapter 5, through systematic arguments based on making potential energy a constant, gave an algorithm to statically balance any zero-free-length spring-loaded and/or gravity-loaded revolute jointed rigid-body linkage. In the algorithm, the net potential energy dependence on the rotational motion of the bodies making up the linkage are recursively eliminated. The chapter gave several examples to illustrate the algorithm Static balancing without auxiliary bodies spatial case Chapter 6 showed that the recursive static balancing algorithm of Chapter 5 also works for spatial revolute and/or spherical jointed linkages under zero-free-length spring and/or gravity loads. The chapter highlighted that revolute joints would require less stringent static balancing conditions. The chapter also gave an example to illustrate the recursive algorithm in the spatial case. 8.2 A framework for designing statically balanced compliant mechanisms Chapter 7 provided a framework for using small-length-flexure model in conjunction with the new static balancing methods of Chapter 5 for designing approximately statically balanced compliant mechanisms. Flexure model of compliant mechanisms gives rise to torsional spring-loaded rigid-body linkages whereas Chapter 5 deals with static balance of zero-free-length (length extending) spring-loaded rigid-body linkages. Therefore, the framework proposed approximating torsional springs by zerofree-length springs. Instead of using least squares approximation for this approximation, the framework proposed matching the value and the slope of the effort function at a reference configuration as a way to approximate the two kinds of springs. This allowed the framework to avoid numerical optimization. The chapter also presented

186 CHAPTER 8. CONCLUSION 166 four examples along with a prototype of one of the examples to illustrate the framework. 8.3 The novelty of the contribution in the context of the current literature The contributions of the thesis covers the following areas of static balance. 1. Rigid linkages under gravity load 2. Rigid linkages under spring load 3. Compliant mechanism under their inherent elastic forces Figure (8.1), a recapitulation of figure (8.1), pictorially depicts the current literature as well our contribution (in grey boxes) under these broad areas. Our contribution for static balancing of gravity-loaded rigid linkages falls within a category where the methods (1) are perfect (in contrast to approximate methods), (2) add springs for balancing, and (3) places no restriction on the degree-of-freedom of the linkages. While there are methods in literature ([23], [27], and [29]) under the category, these methods use auxiliary links in contrast to our method. Currently, under this category, our contribution presented in chapters 5 and 6 is the only work that adds only springs but not auxiliary links. Hence this is a novel contribution. The literature is relatively sparse for the static balancing techniques of rigid linkages under spring loads. Apart from our contribution in this area that is presented in Chapters 3, 5 and 6, the works in this area are [1], [4] and [65]. While [1] and [4] deal with balancing zero-free-length spring loads by adding zero-free-length springs, [65] stands apart as the one that balances, although approximately, torsional spring loads by adding torsional springs. [4] and [1] give ways to statically balance a lever and a four-bar linkage under zero-free-length spring loads. In the contribution stated in Chapter 3, using the concepts not different from the one presented in [4], we give more methods for the problem addressed in [1] (also see figure (1.9)). As far as the methods that deal with a n-degree-of-freedom linkage under zero-free-length spring

187 CHAPTER 8. CONCLUSION 167 Rigid linkages under gravity load Rigid linkages under spring load Compliant mechanism By addition of mass Approximate balancing and cam using methods By addition of springs Perfect balancing methods Torsion load balanced by torsion loads Approximate methods Radaelli et al. (211) Extension spring balanced by extension springs Perfect methods By addition of springs Extension of Deepak & Ananthasuresh (212) to compliant mechanisms using pseudo-rigid body model Other strategies Topology optimization (see De Lange et al. (28)) Building block approach (see Hoetmer, et al. (21)) Extension of Radaelli et al. (211) to compliant mechanisms as seen in Rosenberg et al (211) Gopalswamy et al. (1992) Agrawal & Agrawal (25) Ulrich & Kumar (1991) Koser (29) For specific linkages For n - dof linkages For specific linkages Basic spring force balancer (Herder (21)) Herder (1998) Deepak & Ananthasuresh (212) For n -dof linkages LaCoste (1934) Shin & Streit (1991) Walsh et al. (1991) Herder (21) Lin et al. (21) Using auxiliary bodies Without auxiliary bodies Without auxiliary bodies For planar revolute jointed and spatial revolute and/or spherical jointed linkages Streit & Shin (1993) Agrawal & Fattah (24) For planar revolute jointed and spatial revolute and/or spherical jointed linkages Limiting case Original or novel contributions of this thesis dof: degree of freedom Figure 8.1: The current literature and our contributions

188 CHAPTER 8. CONCLUSION 168 loads, we haven t found any literature. Hence, our contribution stated in Chapters 5 and 6 for a n-degree-of-freedom linkage is new. A few reported works in the literature on static balance of compliant mechanisms are listed in figure (8.1). As far as the ones that rely on spring-loaded rigid linkage balancing techniques, using flexure model as the interface between rigid mechanisms and compliant mechanisms, there is a work by Rosenberg et al. [73]. However, they rely on numerical optimization and their method is not completely analytical. This should signify that our contribution in Chapter 7 is distinct from the existing literature. Furthermore, that rigid-link balancing techniques could be used for flexurebased compliant mechanisms was first enunciated in our work [8] (reported in 29). 8.4 Future work We now briefly outline the prospects of continuing the work from the thesis. In this thesis, though the main algorithm developed in Chapter 5 was for treestructured linkages, we showed how to extend it to closed loop linkages. The concept behind this extension was straightforward and the design obtained was conservative. An investigation on overcoming this conservative design is a prospect for future work. The investigation could be based on the work of Shieh and Chen [41] which exploits closed loop equations. Addition of springs during static balancing leads to increase in internal stresses of the material making up the linkage. This is true in our compliant mechanism methodology as well. It is useful to investigate ways of reducing this internal stress without sacrificing the static balance. Summary A summary of previous chapters. A comparison with the existing literature to show the distinctness of the contributions of this thesis from the existing literature.

189 Appendix A Proofs on finding the focal pivot A.1 If I a,b and I c,a circles are coincident, then the given lines a, b and c has to be concurrent In the following proof, a, b, c, I a,b and I c,a refer to their original position before rotation by β. Figure (A.1) shows I a,b and I c,a circles that are coincident. Also shown are lines a and b and their intersection point I a,b. The point I c,a lie on this common circles, since the circles is its locus. By the definition of I c,a as the intersection of c and a, it should also line on line a. Hence I c,a should lie on the intersection of the circle with the line a. Since I c,a and I a,b circles, by definition pass through A and line a also by definition pass through A, A is always an intersection point between the line and the circle. The following types of intersection between the circle and line a are possible: 1. WhenI a,b isdistinctfroma, thelineaintersectsthecircleattwodistinctpoints :A and I a,b, as shown in figure (A.1). 2. When I a,b coincides with A, the line a has to be tangent to the circle at A. (see the theorem in appendix (B)). In the first possibility, I c,a has be either at I a,b or at A. If it is at A, then line a has to be tangent to the circle (see the theorem in appendix (B)), which contradicts 169

190 APPENDIX A. PROOFS ON FINDING THE FOCAL PIVOT 17 Before rotation (at β = ) a Ia,b B b C A Coincident I a,b and I c,a circles Figure A.1: I a,b and I c,a circles are coincident the earlier observation that line a intersects the circle at two distinct points. Hence it I c,a cannot be at A and by elimination, has to be at I c,a. This implies that I a,b and I c,a coincide, which further imply that a, b, and c are concurrent. Inthesecondpossibility,sinceAistheonlyintersectionbetweenlineaandlocusof I c,a, I c,a has be at A. Thus, here also both I a,b and I c,a coincide implying concurrence of lines a, b and c. Hence, if I a,b and I c,a circles are coincident, then the given lines a, b and c has to be concurrent. A.2 When M and A are distinct, M is the focal pivot. (Refer to section (4.2.4 and figure (4.11).) Letrotationangleβ besuchthatpointi a,b isata, sothatbythetheoreminappendix B, line b aligns along the segment AB and line a becomes tangent to I a,b circle. If for the same β, point I c,a is also at A, then by the theorem in appendix B, line a becomes tangent to I c,a circle. This implies that line a is a common tangent to both I a,b and I c,a circles at the A, which is one of the two distinct points of intersection of the two circles. This contradicts the fact that for two circles intersecting at two distinct points, there cannot be common tangent at the point of intersection. Hence

191 APPENDIX A. PROOFS ON FINDING THE FOCAL PIVOT 171 δ B C δ A M a = I a,b b Figure A.2: I a,b is at M there is no β, for which both I a,b and I c,a are at A. Therefore A is not a focal pivot. Let rotation angle β be such that I a,b is at M as shown in figure (A.2). It may further be seen that not only circle I a,b but even line a intersects the I c,a circle at two distinct points: A and M. Since point I c,a has to lie on both its locus I c,a circle and on line a (by definition), it should be at one of the intersection points (A or M) of the circle and the line. If point I c,a is at A, then by the theorem in appendix B, line c aligns along CA and line a becomes tangent to I c,a circle at A. This contradicts the earlier noted fact that line a intersects at two distinct points. Hence, point I c,a cannot be at A and by elimination of choices, it has to be at M Thus for the angle of rotation β, I c,a as well as I a,b lies on M, implying that M is a focal pivot.

192 Appendix B An elementary theorem of geometry A chord of a circle subtends the same angle at any point on the circumference of the circle. The converse of the theorem is as follows: If two non-parallel lines passing through two ends of a line segment are rotated equally about the respective endpoints of the segment, then the intersection point of the two lines traces a circle. The circle has the line segment as its chord. During rotation, when the intersection point reaches one end of the chord, one of the lines will align along the chord while the other becomes tangent to the circle. 172

193 Appendix C Perfect static balance and positive-free-length springs This is an appendix to section 5.2. Here the difficulty in achieving perfect static balance of a lever by using normally available positive free-length springs is discussed. The d T d term in the potential energy expression of a spring given in equation (5.5) was seen to be a linear combination of sinθ, cosθ and 1 when expanded as in equation (5.6). Hence, by writing d T d as αsinθ + βcosθ + γ, the potential energy expression of the spring becomes: ( ) k 2 (αsinθ +βcosθ+γ) kl k (αsinθ+βcosθ +γ)+ 2 l2 (C.1) The first term in equation (C.1) is the zero-free-length part and the second term is the free-length part. If the free-length is positive, i.e., l >, then the free-length part is negative. The free-length part of the spring is non-constant (i.e., k and not both α and β is zero) except for trivial situations where spring constant is zero or the spring is attached to the pivot of the lever. When there are several but finite positive-free-length and non-trivial springs, the net contribution of the free-length part is negative, and it is also not known have the possibility of being a constant, unlike the zero-free-length part. Furthermore, the free-length part is also not known 173

194 APPENDIX C. NORMAL SPRINGS 174 to be in the function space spanned by sinθ and cosθ. Hence, the possibility of freelength part cancelling (modulo a constant) with zero-free-length part is also ruled out. Thus, with several positive-free-length springs, there is no way the net potential energy could become a constant.

195 Appendix D Constraints can be satisfied, if not as it is, by addition of extra zero-free-length spring loads D.1 Satisfying constraints (5.12), (5.13) and(5.14) This appendix demonstrates how by adding extra zero-free-length spring loads constraints (5.12), (5.13) and (5.14) can be satisfied. To differentiate between original loads and balancing loads, let n s,o and n s,b respectively represent the number of original and balancing zero-free-length spring loads with n s,o + n s,b = n s. Also, let the spring loads be indexed such that the first n s,o loads are original loads with the remaining being balancing loads. Similar meaning applies for n c,o and n c,b. Case 1: Original loads violate the constraint (5.14), and balancing loads are only zero-free-length springs. Let us try to satisfy all the constraints by adding a single zero-free-length spring. Asperthenotation, thisspringgetstheindexi = n s,o +1. Theconstraint(5.14) can be written as follows: n s,o k ns,o+1a ns,o+1 = k i a i i=1 (D.1) 175

196 APPENDIX D. SATISFYING CONSTRAINTS 176 where the known quantities related to original loads are on the right hand side. Equation (D.1) gives the unique solution of k i a i for i = n s,o +1 to the constraint (5.14). Furthermore, the constraints (5.12) and (5.13) can be rewritten as ([ ][ ]) ki a x,i k i a y,i bx,i = k i a y,i k i a x,i b y,i i=n s,o+1 nc n s,o ((f yi p y,i +f x,i p x,i )) (k i (a y,i b y,i +a x,i b x,i )) i=1 i=1 n s,o (D.2) ((f x,i p y,i f y,i p x,i )) (k i (a x,i b y,i a y,i b x,i )) + nc i=1 The 2 2 matrix on the left hand side of the equations is known since k i a i for i = n s,o + 1 is already solved in equation (D.1). Furthermore, the matrix is non-singular since the right hand side of (D.1) that is the same as k [ ] ns,o+1a ns,o+1 T is non-zero as per the description this case. We take b x,i b y,i for i = ns,o +1 as the inverse of the 2 2 matrix times the right hand side of the equation (D.2) so that the constraints (5.12) and (5.13) can also be satisfied. Thus, theoretically, with a single additional zero-free-length spring, all three constraints (5.12), (5.13) and (5.14) can be satisfied. Case 2 Original loads satisfy (5.14), but violate atleast one of the constraints (5.12) and (5.13). Balancing loads are only zero-free-length springs. If we proceed along the same lines as the previous case, then in equation (D.2) the 2 2 matrix on the left hand side becomes singular zero-matrix whereas the right hand side is non-zero by the description of the case. Thus, in this case, with a single balancing zero-free-length spring, it is not possible to satisfy the related constraint. However, it may be verified that by adding two balancing springs, all the constraints can be satisfied. The cases 1 and 2 cover all possible types of constraint violation. Hence we assert that if the constraints are not satisfied as it is, then by adding a minimum of one zerofree-length spring in case (1) (the component related to original loads in constraint (5.14) is non-zero) and two zero-free-length spring in case (2) (the component related i=1

197 APPENDIX D. SATISFYING CONSTRAINTS 177 to original loads in constraint (5.14) is zero), the constraints can be satisfied. D.2 Satisfying constraints (5.19), (5.2) and(5.21) In appendix B1, if the constraints(5.12), (5.13) and(5.14) are respectively substituted by constraints (5.19), (5.2) and (5.21), there is going to be no change except for the righthandsideofequation(d.1),whichtakestheform no v i andtherighthandside i=1 ofequation(d.2),whichtakestheform [ n n (q y,i w y,i +q x,i w x,i ) (q x,i w y,i q y,i w x,i )] T. i=1 There is no change in the left hand side since here also, the additional loads are zerofree-length springs. Hence, analogous to the conclusion of appendix B1, we conclude that if the constraints (5.19), (5.2) and (5.21) are not satisfied as it is, then by addition of a minimum of one zero-free-length spring in case the constraint (5.21) is originally violated and a minimum of two zero-free-length springs in case the constraint (5.21) is not originally violated, the three constraints can be satisfied. i=1

198 Appendix E Solving constraints on spatial orientation independence for the parameters of additional zero-free-length springs This is an appendix to proof of proposition 3. Let the number of zero-free-length springs, n a, be 3. With this equations (6.7) and (6.8) can be split as [ ][ ] n T f v nf +1 v nf +2 v nf = i=1 v i (E.1) [ ][ q nf +1 q nf +2 q nf +3 w nf +1 w nf +2 w nf +3 ] T (E.2) = [q 1 q nf ][w 1 w nf The left hand side of equation are due zero-free-length springs and using relation (5.17), the parameters u i, w i and q i on the left hand side can be written in terms of (k i,a i,b i ), k i > and i = 1 3 as ] T ][ ] n T f [k 1 a 1 k 2 a 2 k 3 a = i=1 v i (E.3) 178

199 APPENDIX E. SOLVING BALANCING CONSTRAINTS SPATIAL CASE 179 ] T [k 1 a 1 k 2 a 2 k 3 a 3 ][b 1 b 2 b 3 ] T (E.4) = [q 1 q nf ][w 1 w nf Solution to equations (E.3) and (E.4) Suppose that such that n f i=1 n f i=1 v i. In the three dimensional space, one can always find a plane v i is not along the plane. Let m 1 and m 2 be two two linearly independent( vectors along such a plane. As a result, the matrix n f )] [m 1 m 2 m 1 m 2 v i is a full-rank invertible 3 3 matrix and also a i=1 ] solution to the matrix [k 1 a 1 k 2 a 2 k 3 a 3 in equation(e.3). With this as the solution ] [ ] T for [k 1 a 1 k 2 a 2 k 3 a 3, the matrix b 1 b 2 b 3 can be solved by pre multiplying ] equation (E.4) with the inverse of [k 1 a 1 k 2 a 2 k 3 a 3. With this, one has a solution for k 1 a 1, k 2 a 2, k 3 a 3, b 1, b 2, and b 3. By choosing a convenient positive value for k 1, n f k 2 and k 3, one can obtain a 1, a 2 and a 3 as well. Thus, when v i there exist a non-unique solution for {k i, a i, b i }, i = 1 3 in equations (6.7) and (6.8). General case including the possibility of When n f i=1 n f i=1 i=1 v i = v i = four zero-free-length spring are sufficient to satisfy (6.7) and (6.8), as shown next. Just as equations (6.7) and (6.8) took the form of equations (E.3) and (E.4) for three springs, the equations take the following form for four spring. ][ ] n T f [k 1 a 1 k 2 a 2 k 3 a 3 k 4 a = i=1 v i (E.5) ] T [k 1 a 1 k 2 a 2 k 3 a 3 k 4 a 4 ][b 1 b 2 b 3 b 4 ] T (E.6) = [q 1 q nf ][w 1 w nf

200 APPENDIX E. SOLVING BALANCING CONSTRAINTS SPATIAL CASE 18 Let m 1, m 2 and m 3 be any linearly ( independent vectors in three dimensional space. n f )] The matrix [m 1 m 2 m 3 m 1 m 2 m 3 v i is a solution for ] i=1 [k 1 a 1 k 2 a 2 k 3 a 3 k 4 a 4 in equation (E.5) and at the same time is of full-rank. ] With [k 1 a 1 k 2 a 2 k 3 a 3 k 4 a 4 being full rank, there exists solution, though nonunique, for b 1 b 2 b 3 b 4 in equation (E.6) not matter what the right hand [ ] T side of the equation is. Now we have a solution for {k i a i, b i }, i = 1 4. By choosing a convenient positive value for k i, one can solve for a i from the known k i a i. Thus, there is always a solution for spring parameters {k i, a i, b i }, i = 1 4 so that equation (6.7) and (6.8) are satisfied.

201 Appendix F Maxima code on flexure-based four-bar mechanism - two spring approximation In the code below, the length l of a diagonal of the quadrilateral shown in figure (F.1) is used as configuration defining variable. The choice is motivated by the fact that the angles of the quadrilateral can be found from l based on cosine rule. N c M b l d O a L Figure F.1: l, the length of a diagonal of the quadrilateral of four-bar bar linkage is used as a convenient configuration defining parameter. COMMENTS: Angles LON, NLO, LNO, LMN, NLM and MNL are 181

202 APPENDIX F. VIRTUAL WORK CALCULATIONS 182 written below as functions of l. (%i1) theta_l_ab : acos( (a^2 + b^2 - l^2)/(2*a*b) ); ( ) l 2 +b 2 +a 2 (%o1) acos 2ab (%i2) theta_b : acos( (a^2 + l^2 - b^2)/(2*a*l) ); ( ) l 2 b 2 +a 2 (%o2) acos 2al (%i3) theta_a : acos( (l^2 + b^2 -a^2)/(2*b*l) ); ( ) l 2 +b 2 a 2 (%o3) acos 2bl (%i4) theta_l_cd : acos( (c^2 + d^2 - l^2)/(2*c*d) ); ( ) l 2 +d 2 +c 2 (%o4) acos 2cd (%i5) theta_c : acos( (l^2 + d^2 - c^2)/(2*l*d) ); ( ) l 2 +d 2 c 2 (%o5) acos 2dl (%i6) theta_d : acos( (l^2 + c^2 - d^2)/(2*l*c) ); ( ) l 2 d 2 +c 2 (%o6) acos 2cl COMMENTS: The angles and their derivatives with respect to l of the four corners of the quadrilateral of pivots shown in figure (F.1). (%i7) psi_ab : theta_l_ab;

203 APPENDIX F. VIRTUAL WORK CALCULATIONS 183 ( ) l 2 +b 2 +a 2 (%o7) acos 2ab (%i8) d_psi_ab : diff(psi_ab, l); l (%o8) ab 1 ( l2 +b 2 +a 2 ) 2 4a 2 b 2 (%i9) psi_bc : theta_a + theta_d; ( ) ( ) l 2 d 2 +c 2 l 2 +b 2 a 2 (%o9) acos +acos 2cl 2bl (%i1) d_psi_bc : diff(psi_bc,l); 1 l2 d 2 +c 2 c 1 l2 +b 2 a 2 b 2cl (%o1) 2 2bl 2 1 (l2 d 2 +c 2 ) 2 1 (l2 +b 2 a 2 ) 2 4c 2 l 2 4b 2 l 2 (%i11) psi_cd : theta_l_cd; ( ) l 2 +d 2 +c 2 (%o11) acos 2cd (%i12) d_psi_cd : diff(psi_cd, l); l (%o12) cd 1 ( l2 +d 2 +c 2 ) 2 4c 2 d 2 (%i13) psi_da : theta_c + theta_b; ( ) ( ) l 2 +d 2 c 2 l 2 b 2 +a 2 (%o13) acos +acos 2dl 2al (%i14) d_psi_da : diff(psi_da, l);

204 APPENDIX F. VIRTUAL WORK CALCULATIONS l2 +d 2 c 2 d 2dl (%o14) 2 1 (l2 +d 2 c 2 ) 2 4d 2 l 2 1 l2 b 2 +a 2 a 2al 2 1 (l2 b 2 +a 2 ) 2 4a 2 l 2 COMMENTS: Angle of each side of the quadrilateral with respect to the horizontal is calculated below. (%i15) phi_a : ; (%o15) (%i16) d_phi_a : diff(phi_a, l); (%o16) (%i17) phi_b : theta_l_ab; ( ) l 2 +b 2 +a 2 (%o17) acos 2ab (%i18) d_phi_b : diff(phi_b, l); l (%o18) ab 1 ( l2 +b 2 +a 2 ) 2 4a 2 b 2 (%i19) phi_c : phi_b - %pi + psi_bc; ( ) ( ) ( ) l 2 d 2 +c 2 l 2 +b 2 a 2 l 2 +b 2 +a 2 (%o19) acos +acos +acos π 2cl 2bl 2ab (%i2) d_phi_c : diff(phi_c, l); 1 l2 d 2 +c 2 c 2cl (%o2) 2 1 (l2 d 2 +c 2 ) 2 4c 2 l 2 1 l2 +b 2 a 2 b 2bl 2 1 (l2 +b 2 a 2 ) 2 4b 2 l 2 + l ab 1 ( l2 +b 2 +a 2 ) 2 4a 2 b 2

205 APPENDIX F. VIRTUAL WORK CALCULATIONS 185 (%i21) phi_d : %pi - psi_da; ( ) ( ) l 2 +d 2 c 2 l 2 b 2 +a 2 (%o21) acos acos +π 2dl 2al (%i22) d_phi_d : diff(phi_d, l); (%o22) 1 l2 +d 2 c 2 d 2dl 2 1 (l2 +d 2 c 2 ) 2 4d 2 l l2 b 2 +a 2 a 2al 2 1 (l2 b 2 +a 2 ) 2 4a 2 l 2 COMMENTS: Numerical values for the sides of the quadrilateral. (%i23) subss(x):= [a=15/1, b=1/1, c=13/1, d = 12/1, l=x]; (%o23) subss(x) := [a = ,b =,c =,d =,l = x] COMMENTS: The value of l corresponding to the reference configuration, i.e., the undeformed configuration of the flexure-based four-bar mechanism. (%i24) l_ : 14/1; (%o24) 7 5

206 APPENDIX F. VIRTUAL WORK CALCULATIONS 186 F.1 Calculation of stiffness in case 3a based on case 2a F.1.1 Obtaining F x vs. u x in case 2a using virtual work balance COMMENTS: Details of the flexure and calculation of torsional spring constant of the small-length flexure model. (%i25) E : 25e9; (%o25) (%i26) flex_width : 5e-6; (%o26) (%i27) flex_thick : 4e-2; (%o27).4 (%i28) flex_length : 1/1; (%o28) 1 1 (%i29) area_i : 1/12*flex_thick* (flex_width)^3; (%o29) (%i3) kt : E*area_I/flex_length; (%o3)

207 APPENDIX F. VIRTUAL WORK CALCULATIONS 187 (%i31) subst(subss(l), psi_ab); ( ( (%o31) acos l2) ) 3 COMMENTS: Virtual work of the torsional springs. (%i32) virt_wrk_tor_springs : subst( subss(l), -1*( (psi_ab - subst([l=l_], psi_ab))*d_psi_ab + (psi_bc - subst([l=l_], psi_bc))*d_psi_bc + (psi_cd - subst([l=l_], psi_cd))*d_psi_cd + (psi_da - subst([l=l_], psi_da))*d_psi_da ) )$ COMMENTS: Rotation matrix. (%i33) R_mat(theta) := matrix( [cos(theta),-sin(theta)], [sin(theta),cos(theta)] ); ( ) cos(θ) sin(θ) (%o33) R mat(θ) := sin(θ) cos(θ)

208 APPENDIX F. VIRTUAL WORK CALCULATIONS 188 (%i34) float( subst( subss(l_), matrix([a], []) + R_mat( phi_d). matrix( [d], []) + matrix([], [-*flex_length/2]) ) ); (%o34) ( ) COMMENTS: Local coordinate of point P on the frame attached to the link associated with side d. (%i35) ref_coor_d : float( subst( subss(l_), invert(r_mat( phi_d)). ( R_mat( phi_d). matrix( [d], []) + matrix([], [-flex_length/2]) ) ) ); (%o35) ( ) COMMENTS: u x as a function of l (%i36) ref_displ : matrix([1], []). ( subst( subss(l), R_mat(phi_d).ref_coor_d) - subst( subss(l_), R_mat(phi_d).ref_coor_d) )$ COMMENTS: The range of parameter l, on which we focus our interest.

209 APPENDIX F. VIRTUAL WORK CALCULATIONS 189 (%i37) l_min:94/1; l_max:17/1; 47 (%o37) 5 17 (%o38) 1 (%i39) l_range : [l,l_min,l_max] ; (%o39) [l, 47 5, 17 1 ] (%i4) plot2d([ref_displ], l_range, [xlabel, "l (m)"], [ylabel, "u_x (m)"], [gnuplot_term,ps],[gnuplot_out_file,"./for_thesis_img/ref_displ.eps"]) u x (m) l (m) COMMENTS: Virtual work contribution of the horizontal force F x at point P.

210 APPENDIX F. VIRTUAL WORK CALCULATIONS 19 (%i41) virt_wrk_ref_force : matrix([1], []). subst( subss(l), diff( R_mat(phi_d).ref_coor_d, l) )$ COMMENTS: Explicit expression for F x in case 2a. (%i42) f_2a_func_l : kt*virt_wrk_tor_springs/-virt_wrk_ref_force$ (%i43) plot2d([parametric,ref_displ, f_2a_func_l, l_range], [xlabel, "u_x (m)"], [ylabel, "F_x in case 2a (N)"], [gnuplot_term,ps],[gnuplot_out_file,"./for_thesis_img/f_2a.eps"] )$ F x in case 2a (N) u x (m) COMMENTS: Slope of F x vs. u x at l = l in case 2a. (%i44) f_2a_derivative_l_ : float( subst( subss(l_), diff(f_2a_func_l, l)/diff(ref_displ,l) ) ); (%o44)

211 APPENDIX F. VIRTUAL WORK CALCULATIONS 191 F.1.2 Obtaining F x vs. u x in case 3a using virtual work balance COMMENTS: Coordinates of anchor points, spring displacement, spring force, and virtual work contribution of spring K 2 1. (%i45) spring_c_1_glb_anch : matrix( [-.5], [.6]); ( ).5 (%o45).6 (%i46) spring_c_1_local : subst( subss(l_), matrix( [c/1], [-c/3]) ); ( ) 13 1 (%o46) 13 3 (%i47) spring_c_1_displ : subst( subss(l), R_mat(phi_b). matrix([b],[]) + R_mat(phi_c).spring_c_1_local )$ (%i48) spring_c_1_force : -spring_c_1_k*(spring_c_1_displ - spring_c_1_glb_anch)$ (%i49) spring_c_1_virt_wrk : ( spring_c_1_force.diff(spring_c_1_displ, l) )$ COMMENTS: Coordinates of anchor points, spring displacement, spring force, and virtual work contribution of spring K 1 1. (%i5) spring_b_1_glb_anch : matrix( [.25], [-.5] );

212 APPENDIX F. VIRTUAL WORK CALCULATIONS 192 ( ).25 (%o5).5 (%i51) spring_b_1_local : matrix( [8e-3], [ ] ); ( ).8 (%o51) (%i52) spring_b_1_displ : subst( subss(l), R_mat(phi_b). spring_b_1_local ); ( 13 l2) 4 (%o52) 13 1( l2 ) 2 9 (%i53) spring_b_1_force : (%o53) -spring_b_1_k*(spring_b_1_displ - spring_b_1_glb_anch); ( ( 13 l2).25 ) spring b 1 k ( 4 ) 13 1( l2 ) spring b 1 k 9 (%i54) spring_b_1_virt_wrk : spring_b_1_force. diff(spring_b_1_displ, l)$ COMMENTS: By virtual work balance, F x in case 3a is explicitly written as below. Note that it is a function of l with spring constants of K1 1 (spring b 1 k) and K1 2 (spring c 1 k) being unknowns. (%i55) f_3a_func_l_unit_k : (spring_c_1_virt_wrk + spring_b_1_virt_wrk )/-virt_wrk_ref_force$ COMMENTS: F x and slope of F x vs. u x at l = l.

213 APPENDIX F. VIRTUAL WORK CALCULATIONS 193 (%i56) f_3a_func_l_unit_k_origin : expand( float( subst( subss(l_), f_3a_func_l_unit_k) ) ); (%o56) spring c 1 k spring b 1 k (%i57) f_3a_func_l_unit_k_derivative : expand( float( subst(subss(l_), diff( f_3a_func_l_unit_k, l) / diff(ref_displ,l ) ) ) ); (%o57) spring c 1 k spring b 1 k COMMENTS: Equating F x and slope of F x vs. u x at l = l between case 2a and case 3a, the stiffness of springs K1 2 (spring c 1 k) and K1 1 (spring b 1 k) are solved below. (%i58) spring_cnst_sub: float( solve( [f_3a_func_l_unit_k_origin=, f_3a_func_l_unit_k_derivative=f_2a_derivative_l_ ], [spring_c_1_k, spring_b_1_k]) ); rat : replaced by 1625/663 = rat : replaced rat : replaced by /77 = rat : replaced by 1526/117 = rat : replaced (%o58) [[spring c 1 k = ,spring b 1 k = ]]

214 APPENDIX F. VIRTUAL WORK CALCULATIONS 194 (%i59) plot2d([[parametric,ref_displ, f_2a_func_l, l_range], [ parametric, ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k), l_range] ], [xlabel, "u_x (m)"], [ylabel, "F_x (N)"], [legend, "case 2a", "case 3a"], [gnuplot_term,ps], [gnuplot_out_file,"./for_thesis_img/f_2a_f_3a.eps"] ); 5 4 case 2a case 3a 3 F x (N) (%o59) u x (m)

215 APPENDIX F. VIRTUAL WORK CALCULATIONS 195 F.2 Verification of static balance in case 3b through virtual work balance (%i6) spring_c_2_glb_anch : spring_c_1_glb_anch; ( ).5 (%o6).6 (%i61) spring_c_2_local : -spring_c_1_local$ (%i62) spring_c_2_displ : subst( subss(l), R_mat(phi_b). matrix([b],[]) + R_mat(phi_c).spring_c_2_local )$ (%i63) float( subst( subss(l_), spring_c_2_displ ) ); ( ) (%o63) (%i64) spring_c_2_force : -spring_c_2_k*(spring_c_2_displ - spring_c_2_glb_anch)$ (%i65) spring_c_2_virt_wrk : ( spring_c_2_force.diff(spring_c_2_displ, l) )$ (%i66) spring_b_2_glb_anch : matrix( [ ], [ ] ); (%o66) ( )

216 APPENDIX F. VIRTUAL WORK CALCULATIONS 196 (%i67) spring_b_2_local : matrix( [8e-3], [ ] ); ( ).8 (%o67) (%i68) spring_b_2_displ : subst( subss(l), R_mat(phi_b). spring_b_2_local )$ (%i69) float( subst( subss(l_), spring_b_2_displ ) ); ( ).344 (%o69) (%i7) spring_b_2_force : -spring_b_2_k*(spring_b_2_displ - spring_b_2_glb_anch)$ (%i71) spring_b_2_virt_wrk : spring_b_2_force. diff(spring_b_2_displ, l); ( (%o71) l ( l ( 13 4 l2) (%i72) bal_sprng_cnst : ( 13 4 l2 ) 2 ( ) 13 4 l2 ) ) ( 13 4 l2 ) 2 9 spring spring b 2 k [spring_c_2_k= , spring_b_2_k= ]; (%o72) [spring c 2 k = , spring b 2 k = ]

217 APPENDIX F. VIRTUAL WORK CALCULATIONS 197 (%i73) f_bal_func_l_unit_k : (spring_c_2_virt_wrk + spring_b_2_virt_wrk )/-virt_wrk_ref_force$ (%i74) plot2d([[parametric,ref_displ, ( subst(spring_cnst_sub, f_3a_func_l_unit_k) + subst( bal_sprng_cnst, f_bal_func_l_unit_k) ), l_range], [ parametric, ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k), l_range], [ parametric, ref_displ, subst( bal_sprng_cnst, f_bal_func_l_unit_k), l_range] ], [xlabel, "u_x (m)"], [ylabel, "F_x (N)"], [legend, "case 3b", "case 3a", "balancing springs"], [gnuplot_term,ps], [gnuplot_out_file,"./for_thesis_img/f_2a_f_3a_balnce_contri.eps"] ); (%o74) COMMENTS: The following plot shows that virtual work contribution of original springs and balancing springs in case 3b is zero over the range of l that is plotted. This signify that static balance has indeed been attained.

218 APPENDIX F. VIRTUAL WORK CALCULATIONS case 3b case 3a balancing springs F x (N) u x (m) F.3 A first order correction to balancing springs on the flexure-based four-bar linkage COMMENTS: u x (cms ref displ), F x (cms f 3a) in case 1a and F x (cms f 3b) in case 1b are respectively imported from the finite element analysis of the flexure-based four-bar mechanism.

219 APPENDIX F. VIRTUAL WORK CALCULATIONS 199 (%i75) cms_ref_displ : [ -.25, , , , , , , , , , , , , , , , , , , , ,.25 ]$

220 APPENDIX F. VIRTUAL WORK CALCULATIONS 2 (%i76) cms_f_3a : [ , , , , , , , , , , , , , , , , , , , , , ]$

221 APPENDIX F. VIRTUAL WORK CALCULATIONS 21 (%i77) cms_f_3b : [ , , , , , , , , , , , , , , , , , , , , , ]$ COMMENTS: Deviation in F x (%i78) f_3b_orig_error : ( )/2; (%o78) COMMENTS: Deviation in slope of F x vs. u x.

222 APPENDIX F. VIRTUAL WORK CALCULATIONS 22 (%i79) f_3b_slope_error : ( )/ ( ); (%o79) COMMENTS:Twolinearequationsdefiningthefirstordercorrectiononk2 2 (spring c 2 k) and k2 1 (spring b 2 k). (%i8) correc_eq_1 :float( expand( subst( subss(l_), f_bal_func_l_unit_k) ) ) = -f_3b_orig_error; (%o8) spring c 2 k spring b 2 k = (%i81) correc_eq_2 : float( expand( subst( subss(l_), diff(f_bal_func_l_unit_k,l)/ diff(ref_displ,l ) ) ) ) =-f_3b_slope_error; (%o81) spring c 2 k spring b 2 k = COMMENTS: The two equations are solved for the correction in stiffness of the balancing springs. (%i82) float( solve( [correc_eq_1, correc_eq_2], [spring_c_2_k, spring_b_2_k]) ); rat : replaced by 7977/23527 = rat : replaced

223 APPENDIX F. VIRTUAL WORK CALCULATIONS by 889/76676 = rat : replaced by443/ rat : replaced by48645/139 = rat : replaced by 9979/1493 = rat : replaced by 1415/546 = (%o82) [[spring c 2 k = ,spring b 2 k = ]] COMMENTS: Finite element analysis data of F x after the correction. (%i83) cms_3b_post_correc : [ , , , , , , , , , , , , , , , , , , , , , ]$

224 APPENDIX F. VIRTUAL WORK CALCULATIONS 24 (%i84) plot2d([[parametric,ref_displ, f_2a_func_l, l_range], [ parametric, ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k), l_range], [ parametric, ref_displ, f_2a_func_l+subst( bal_sprng_cnst, f_bal_func_l_unit_k), l_range], [ parametric, ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k) + subst( bal_sprng_cnst, f_bal_func_l_unit_k), l_range], [discrete, cms_ref_displ, cms_f_3a], [discrete, cms_ref_displ, cms_f_3b], [discrete, cms_ref_displ,cms_3b_post_correc ]], [legend, "f2a", "f3a", "f2b", "f3b", "f1a", "f1b", "f1c" ], [xlabel, "generalized displacement [m]"], [ylabel, "generalized force [N]"], [gnuplot_term,ps],[gnuplot_out_file,"case_plots.eps"]); (%o84) (%i85) load(draw)$ (%i86) case_1a : points(cms_ref_displ, cms_f_3a)$ (%i87) case_1b : points(cms_ref_displ, cms_f_3b)$

225 APPENDIX F. VIRTUAL WORK CALCULATIONS 25 (%i88) case_1c : points( cms_ref_displ,cms_3b_post_correc)$ (%i89) case_2a : parametric(ref_displ, f_2a_func_l, l, l_min, l_max)$ (%i9) case_2b : parametric(ref_displ, f_2a_func_l+ subst( bal_sprng_cnst, f_bal_func_l_unit_k), l, l_min, l_max)$ (%i91) case_3a : parametric(ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k), l, l_min, l_max)$ (%i92) case_3b : parametric( ref_displ, subst(spring_cnst_sub, f_3a_func_l_unit_k) + subst( bal_sprng_cnst, f_bal_func_l_unit_k), l, l_min, l_max)$

226 APPENDIX F. VIRTUAL WORK CALCULATIONS 26 (%i93) draw2d( xlabel="u_x [m]", ylabel="f_x [N]", points_joined=true, point_size=1, line_width=1, color=red, key="1a", point_type=2,case_1a, color=magenta, key="1b", point_type=1,case_1b, color=brown, key="1c (corrected)", point_type=3,case_1c, color=royalblue, key="2a", case_2a, color=blue, key="2b",case_2b, color=green,key="3a", case_3a, color=dark-green, key="3b", case_3b, terminal = eps, file_name = "case_plots" ); (%o93) [gr2d(points, points, points, parametric, parametric, parametric, parametric)] F x [N] a 1b 1c (corrected) 2a 2b 3a 3b u x [m]

227 Appendix G Static balancing of compliant mechanisms using rigid-body linkages and springs Figure (G.1) shows a compliant gripper that is statically balanced with a small 2R linkage loaded by a zero-free-length spring. The zero-free-length spring is realized by a pulley-string arrangement. In order to understand the principle behind the static balance in figure (G.1), consider figures (G.2) and (G.3). Figure (G.2a) shows a statically balanced parallelogram (for details see figure (2.6)). The spring between between B and D in figure (G.2a) is relocated to be between E and A, as shown in figure (G.2b), where BC = EB. From geometry, it follows that this relocation does not change the length of the spring and hence does not change the potential energy. Therefore the linkage of figure (G.2b) is also statically balanced. Figure (G.3) shows a compliant gripper, which is actuated by pulling down the actuator rod shown in the figure. The actuator rod is guided to maintain symmetric actuation. Because of the guide and the inherent elasticity of the compliant mechanism, the actuator rod behaves as a one-dimensional spring. We found that for the range of operation of the compliant mechanism, the force-displacement relation is linear for practical purposes. 27

228 APPENDIX G. ANOTHER COMPLIANT MECHANISM BALANCING 28 Figure G.1: A compliant gripper compensated by a small spring loaded 2R linkage. (Basement board dimension: 2.5 feet 2.5 feet)

229 APPENDIX G. ANOTHER COMPLIANT MECHANISM BALANCING 29 D C k k A l B (a) C k A l B k l E (b) Figure G.2: A statically balanced parallelogram and its modification

230 APPENDIX G. ANOTHER COMPLIANT MECHANISM BALANCING 21 Figure G.3: A compliant gripper