Perfect Static Balance of Linkages by Addition of Springs But Not Auxiliary Bodies

Transcription

1 Sangamesh R. Deepak e-mai: G. K. Ananthasuresh e-mai: Department of Mechanica Engineering, Indian Institute of Science, Bangaore 56001, India Perfect Static Baance of Linkages by Addition of Springs But Not Auxiiary Bodies A inkage of rigid bodies under gravity oads can be staticay counter-baanced by adding compensating gravity oads. Simiary, gravity oads or spring oads can be counterbaanced by adding springs. In the current iterature, among the techniques that add springs, some achieve perfect static baance whie others achieve ony approximate baance. Further, a of them add auxiiary bodies to the inkage in addition to springs. We present a perfect static baancing technique that adds ony springs but not auxiiary bodies, in contrast to the existing techniques. This technique can counter-baance both gravity oads and spring oads. The technique requires that every joint that connects two bodies in the inkage be either a revoute joint or a spherica joint. Apart from this, the inkage can have any number of bodies connected in any manner. In order to achieve perfect baance, this technique requires that a the spring oads have the feature of zerofree-ength, as is the case with the existing techniques. This requirement is neither impractica nor restrictive since the feature can be practicay incorporated into any norma spring either by modifying the spring or by adding another spring in parae. [DOI: / ] 1 Introduction A inkage is said to be staticay baanced if it is in static equiibrium in a its configurations. In this paper, a the oads on a inkage are assumed to be conservative. Hence, static baance is equivaent to invariance of the net potentia energy of a the oads with respect to a configurations of the inkage. This paper gives a new technique to staticay baance a revoute-jointed inkage oaded by constant forces (e.g., gravity) and/or zero-free-ength springs. Athough the technique is detaied in this paper for panar inkages, it extends to spatia spherica and/or revoute-jointed inkages. The need for static baance of gravity oads in structures and machines is we known. Hence, a number of techniques are deveoped for static baance of gravity oads [1 ]. The need for static baance of inherent spring oads is not as common. This need is particuary fet in compiant mechanisms where an easticay deformabe structure is used but its stiffness is not aways desired. This work is motivated by such practica appications. Techniques in the iterature for static baancing a oaded inkage may be cassified into approximate baancing techniques and perfect baancing techniques. The perfect baancing techniques can be further subdivided into Category 1: Both origina oads and baancing oads are constant weights. Category : Origina oads are constant weights and baancing oads are zero-free-ength spring oads. Category 3: Both origina and baancing oads are zero-freeength spring oads. These categories are iustrated in Fig. 1. The top row of Fig. 1 shows these categories for a ever. Whie the static baance of a ever in category 1 is known for a ong time, the baancing of a ever under categories and 3 was discovered reativey recenty, as woud be evident from the iterature survey given ater. The bottom row of Fig. 1 shows these categories for a mutibody inkage. For mutibody revoute-jointed inkages, static baancing techniques are known ony for categories 1 and. Furthermore, for mutibody inkages under category, beyond 3R seria inkage, a the methods reported so far in the iterature use auxiiary bodies. The bottom row of Fig. 1 under category iustrates one such reported method [] where auxiiary bodies are highighted in gray coor. This paper deas with perfect static baancing of mutibody revoute-jointed inkages. It shows that just as baancing under category 1 can be done without auxiiary bodies, baancing under categories and 3 can aso be done without using auxiiary bodies. The background for this work is presented next. 1.1 Zero-Free-Length Springs and Perfect Static Baancing. Zero-free-ength springs, in contrast to norma springs, have zero-ength between its endpoints when the spring force is zero. When a spring is anchored to two bodies having reative motion, the spring force as a function of its two anchor points is of interest. As iustrated in Fig., this function happens to be inear in a zero-free-ength spring but noninear in a positive-free-ength spring inspite of both springs having a inear force-defection reationship. Appendix A shows that the noninearity associated with nonzero-free-ength springs prevents Contributed by the Mechanisms and Robotics Committee of ASME for pubication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received February 6, 011; fina manuscript received February, 01; pubished onine Apri 5, 01. Assoc. Editor: Frank C. Park. Fig. 1 Three categories of perfect static baancing techniques shown on a ever and a mutibody inkage Journa of Mechanisms and Robotics Copyright VC 01 by ASME MAY 01, Vo. /

2 Fig. Difference between zero-free-ength spring and norma spring perfect static-baance when ony normay avaiabe positive-freeength springs are present. Herder [5] has documented a few practica arrangements to decrease the free-ength of a norma spring a the way to zero and even to a negative vaue. If a norma positive-free-ength spring cannot be incorporated into any of the arrangements of Herder [5], as is the case with oading springs, then by adding an appropriate negative-freeength spring in parae to it, a zero-free-ength spring can be reaized out of both of them. In this way, even norma spring oads can be brought under the ambit of techniques under category 3. Thus, the technique presented in this paper is practica and genera enough to hande norma spring oads in addition to weights and zero-free-ength spring oads. 1. Literature Survey 1..1 Perfect Static Baancing of a Lever. The three categories of perfect static baancing of a ever can be seen in Fig. 1. The technique under category 1 is known from historic times and is popuary known as the ever principe of Archimedes. The discovery of zero-free-ength springs and the technique under category are credited to Lucien LaCoste (see Ref. [1]). We consider this discovery to be ground-breaking since it showed for the first time that a weight can be staticay baanced by a spring. Recenty, using a differentia beve gear, this technique is adapted for static baancing a body having the spatia ro and pitch motion about a point [6]. The techniques under category 3 for a ever are discussed in detai by Herder [5]. 1.. Perfect Static Baancing of a Mutibody Linkage. Most of the static baancing techniques for mutibody inkages under category 1 have been known for a ong time. The technique shown in Fig. 1 under category 1 is one such. Another exampe of the techniques under this category is in Ref. [7]. Under category, Streit and Shin [] showed that in principe any panar inkage oaded by gravity oads can be baanced using zero-free-ength springs. They aso provided a different technique to staticay baance seria revoute-jointed panar inkages. It is this technique that is iustrated using three-revoute-jointed inkage in Fig. 1 under category. The same work aso provides another technique for a inkage with revoute-sider joint pairs. Athough Streit and Shin [] provide techniques for panar inkages, the techniques can be extended to spatia inkages. Rahman et a. [8] provide one such extension to anthropomorphic robots. Recent work on static baancing of spatia inkages incudes that of Lin et a. [9]. References [3] and [] provide a different cass of techniques under category for revoute-jointed inkages. A these techniques use auxiiary bodies in addition to extra zerofree-ength springs. Under category 3, for a mutibody inkage, there is a technique which is appicabe ony for a four-bar inkage and a two-revoute-jointed inkage [10]. Later, we recognized two more methods for the same inkages in Ref. [11]. Under categories and/or 3, Refs. [1] and [13] as we as a method in Ref. [11] do not use auxiiary inks. The current paper has evoved out of Ref. [1] and aows a more genera cass of soutions in comparison to Ref. [1]. Reference [13] derives equations governing static baance of gravity oaded R and 3R inkages and provides soutions to the equations without using auxiiary inks Other Static Baancing Techniques. Among the baancing techniques outside the ambit of the aforementioned three categories, most are approximate baancing techniques and a few, athough perfect baancing techniques using ordinary springs, use cams and pueys to moduate the behavior of springs. Further, a those techniques baance against gravity oads. Agrawa and Agrawa [1] presented an approximate static baancing method using nonzero-free-ength springs. Gopaswamy et a. [15] gave an approximate static baancing technique where torsiona springs were used as baancing eements. There is a ot of iterature on static baancing of parae manipuators and one such work is Ref. [16]. The baancing techniques that moduate the behavior of springs incude the techniques in Refs. [17] and [18], where a puey of varying radius was used, and the technique in Ref. [19], where a cam was used. 1.3 Practica Reevance. The utiity of techniques under category is we recognized. These techniques are appied in static baancing of robots, angepoise amps, and fight simuators. If a robot or a fight simuator is staticay baanced, then the actuators do not have to work against the gravity oads acting on the inks of the robot or the cockpit of fight simuator. This greaty reduces the force/torque requirement of the actuators and aso supposedy makes the actuator contro easy. Further, an advantage of baancing techniques in category over category 1 is that the inertia added to the inkage is minima. The new technique that this paper presents under this category wi provide one more option to a designer seeking to staticay baance inkages under gravity oads using springs. The utiity of any technique under category 3 is not direct. There are hardy any practica probems where a inkage under zero-free-ength springs is required to be staticay baanced against it. However, there are situations where a inkage under eastic oads other than the zero-free-ength spring oads is required to be baanced. For exampe, the baancing of the eastic forces of a cosmetic covering in a hand prosthesis (see Refs. [10] and [0]) and the inherent eastic forces in a compiant mechanism (see Ref. [1]) is desired. Unike in a hand prosthesis, there is no inherent inkage in a compiant mechanism, which is a monoithic eastic piece that transmits force or motion by virtue of eastic deformation. However, it is estabished that compiant mechanisms with fexura joints and certain kinds of sender segments can be modeed as rigid-inkages with torsiona springs and tension springs (see Refs. [] and [3]). Since the perfect static baance of these types of eastic forces on inkages is not demonstrated, one woud ook for a good approximate static baance. If such eastic forces are approximatey modeed as zero-free-ength springs, then the techniques under category 3 woud offer insights and aso a starting point for optimization techniques to baance such eastic oads. 1. Organization of the Paper. In order to show the features of zero-free-ength springs and constant oads that make perfect static baance possibe with them, perfect static baance of one of the simpest inkages: a rigid body on a fucrum, i.e. a ever, is discussed in Sec.. Section 3 shows that even though the principes of static baance of a ever can be extended to a rigid body freey moving in a pane, static baancing the transation component of the rigid body is not possibe in most practica conditions. Based on a resut in Sec. 3, it is shown in Sec. that an assembage of / Vo., MAY 01 Transactions of the ASME

3 Since ¼ d T d, the potentia energy in expression (3) may be rewritten as PE s ¼ k pffiffiffiffiffiffiffiffi dt d k 0 d T d þ k 0 (5) If the free-ength of the spring is zero, then ony the first term in Eq. (5) remains and hence, we ca it as PE s,zero, i.e. Fig. 3 A ever under a constant oad and a spring oad PE s;zero ¼ k dt d ¼ k ððr þ RðÞa h Þ bþt ððr þ RðÞa h Þ bþ ¼ k ðrt r þ a T R T ðhþrðhþa þ b T b r T b þ r T RðhÞa b T RðhÞaÞ ¼ k rt r þ a T a þ b T b r T b þ r T RðhÞa b T RðhÞa * R T ðhþrðhþ ¼I (6) rigid bodies in a pane with zero-free-ength spring and constant oad interactions between the bodies can aways be staticay baanced if the assembage forms a revoute-jointed inkage. Section 5 argues that the technique for panar revoute-jointed inkages extends for spatia spherica and/or revoute-jointed inkages. Concuding remarks are in Sec. 6. Baancing a Lever Consider a ever pivoted to the ground, as shown in Fig. 3. The configuration of the ever with respect to the goba frame of reference (X Y) can be described by h, which is the ange from the goba frame to the oca frame of reference of the ever. Figure 3 aso shows two kinds of oad: (1) a spring attached between a point of the ever and a point of the goba frame and () a constant force acting at a point on the ever. Here, a constant force means that the force has a constant direction with respect to the goba frame and a constant magnitude. A compete specification of the spring oad woud invove (1) the spring constant, denoted by k, () the oca coordinate of the anchor point on the ever, denoted by a ¼½a x a y Š T, and (3) the goba coordinate of the anchor point on the goba reference frame, denoted by b ¼½b x b y Š T. A compete specification of the constant force woud invove (1) the force components with respect to the goba frame, T, denoted by f ¼ f x f y and () the oca coordinate of the point of action of the force on the ever, denoted by p ¼ ½p x p y Š T..1 Potentia Energy as a Function of the Configuration Variabe. By referring to Fig. 3, the potentia energy of the constant oad is PE c ¼f T ðr þ RðÞp h Þ (1) T where r ¼ r x r y is the coordinate of the origin of the oca frame on the ever with respect to the goba frame and R is the rotation matrix function given by cos w sin w RðwÞ ¼ for any ange w () sin w cos w The potentia energy of the spring is PE s ¼ k ð 0Þ ¼ k k 0 þ k 0 (3) where 0 is the free ength of the spring and is the magnitude of d, the dispacement of one-end point of the spring with respect to the other. This d, referring to Fig. 3,is d ¼ ðr þ RðÞa h Þb () Since the remaining ast two terms of the potentia energy in Eq. (5) are nonzero ony if free-ength 0 is nonzero, we name these terms as PE s,nonzero, i.e. pffiffiffiffiffiffiffiffi PE s;nonzero ¼k 0 d T d þ k 0 (7) From Eq. (6), it foows that d T d ¼ k PE s;zero. Substituting this in Eq. (7) eads to the foowing expression for PE s,nonzero. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PE s;nonzero ¼ 0 kpe s;zero þ k 0 (8) In the expressions of the potentia energy in Eqs. (1), (6), and (8), as the configuration of the ever varies, f, p, a, b, k, 0 remain constants and r is made a constant by choosing the origin of the oca frame on the ever to coincide with the pivot point. The dependency of the expressions on the configuration is due the matrix R(h) which, by examining the definition of R in Eq. (), can be spit as RðhÞ ¼ 1 0 cos h þ 0 1 sin h ¼ I cos h þ R p sin h (9) This form of R(h) indicates that PE c in Eq. (1) and PE s,zero in Eq. (6) can be written as a inear combination of sin h, cos h, and 1 (for constants). The coefficients of sin h, cos h and 1 are presented, for carity, in a tabuar form in Tabe 1. Thus, we now have potentia energy of constant and spring oads expressed as functions of configuration variabe h.. Invariance of Potentia Energy With Respect to the Configuration Variabe..1 Trivia Conditions. The potentia energy of the spring on the ever can have constant potentia energy ony under trivia conditions: (1) the spring stiffness is zero (k ¼ 0), () the anchor point on the ever is at the hinge point (a ¼ 0), and (3) the anchor point on the goba frame is at the hinge point (b ¼ r). Simiar trivia conditions for the constant oads are (1) the oad is zero (f ¼ 0) and () the oad acts at the pivot point (p ¼ 0). It is ony under these trivia conditions that the coefficients of cos h and sin h become zero in Tabe 1... The Discovery of Lucien LaCoste. Even though a nontrivia spring and a nontrivia constant oad cannot be individuay in static baance, they together can be, as demonstrated in Fig.. This was first recognized by Lucien LaCoste (see Ref. [1]) in the context of having a penduum of infinite period. Figure shows a ever under the action of a weight W that is baanced by a zero- Journa of Mechanisms and Robotics MAY 01, Vo. /

4 Tabe 1 Potentia energy of the weight and the zero-free-ength component of the spring acting on the ever is a inear combination of cos h, sin h, and 1. Coefficients Basis Weight Zero-free-ength component of spring oad cos h f T p ¼(f y p y þ f x p x ) k(r b) T a ¼ k(a y r y þ a x r x a y b y a x b x ) sin h f T R p p ¼ðfx p y f y p x Þ kðr bþ R p a ¼ kðax r y a y r x a x b y þ a y b x Þ 1 f T r ¼f y r y f x r x þ k rt r þ a T a þ b T b r T b ¼þ k ðr y b yr y þ rx b xr x þ b y þ b x þ a y þ a x Þ free-ength spring of spring constant k anchored above the pivot of the ever at a height of h. As shown in the figure, under the condition W ¼ kh, the potentia energy is invariant with respect to configuration variabe h...3 Severa Zero-Free-Length Springs and Constant Loads. The baancing condition W ¼ kh of the exampe in Fig. wi now be generaized to a ever under severa constant oads and zero-free-ength spring oads. Since severa oads are now being considered, et both constant oads and zero-free-ength spring oads be ordered to aow indexing. The notation a i, b i, k i, has the same meaning as a, b, and k in Fig. 3 other than that it corresponds to ith spring. f i and p i aso have simiar meaning. Further, et the number of constant oads be n c and the number of zero-free-ength spring oads be n s. Since the potentia energy of each of the constant oads and the zero-free-ength spring oads are a inear combination of cos h, sin h and 1, their net potentia energy is aso a inear combination of cos h, sin h and 1. Further, since cos h and sin h and 1 are ineary independent functions of h, their inear combination is a constant if and ony if the coefficients of nonconstant functions, i.e., cos h and sin h are zero. Writing, with the hep of Tabe 1, the coefficients of cos h and sin h of the net potentia energy of a the oads and equating them to zero ead to the foowing equations: Xnc X nc þ Xns f y;i p y;i þ f x;i p x;i k i a y;i r y þ a x;i r x a y;i b y;i a x;i b x;i ¼ 0 (10) f x;i p y;i f y;i p x;i þ Xns k i ða x;i r y a y;i r x a x;i b y;i þ a y;i b x;i Þ¼0 (11) which are the conditions for constant potentia energy (or static baance) of severa constant and zero-free-ength spring oads on a ever. These conditions are appicabe to a the three categories of Fig. 1. Further, by choosing appropriate oad parameters, it is possibe to satisfy the conditions in practice, as was the case in the exampe of Fig.... Norma Positive-Free-Length Springs. As far as normay avaiabe positive-free-ength springs are concerned, the square root term in Eq. (8) poses a severe restriction on static baancing, as expained in detai in Appendix A. Hence, for the remainder of this paper, a the spring oads are of zero-free-ength with the understanding that a positive-free-ength spring can be brought into the ambit of zero-free-ength by combining it with an appropriate negative-free-ength spring. Our next aim is to derive a set of conditions for the static baance of a revoute-jointed mutibody inkage oaded by constant oads and zero-free-ength spring oads. Before that, it is usefu to consider the static baance of a singe rigid body moving freey in a pane. 3 Baancing of a Rigid Body in a Pane Consider the rigid body shown in Fig. 3. An appropriate set of configuration variabes for the body is fr,hg. It may be noted that r in Fig. 5, in contrast to Fig. 3, is an independent variabe because the body is free to move in the pane. The oads on the body are a set of zero-free-ength spring oads and constant oads, and both sets of oads are exerted by the goba frame of reference as shown in Fig. 5. The notations n c, n s, a i, b i, k i, f i, and p i have the same meaning as in Sec.. The potentia energy of the oads is aso the same as in Sec. except that r x and r y are now independent variabes. In Tabe 1 of Sec., when ineary independent functions of fr,hg are pued out as basis functions, Tabe is obtained. As is evident from Tabe, the potentia energy of the oads is now a inear combination of the foowing basis functions: cos h, sin h, r x cos h, r y cos h, r x sin h, r y sin h, rx, r y, r x, r y, and 1. Fig. Static baancing of a weight by a spring Fig. 5 A body that is free to move in a pane / Vo., MAY 01 Transactions of the ASME

5 Tabe Potentia energy of weight and spring acting on a ink moving in a pane. Coefficients of the basis Basis Weight Spring oad Generaized potentia (see Eq. (15) cos h (f y p y þ f x p x ) k(a y b y þ a x b x ) (q y w y þ q x w x ) sin h þ(f x p y f y p x ) k(a x b y a y b x ) (q x w y q y w x ) r x cos h 0 ka x v x r y cos h 0 ka y v y r x sin h 0 ka y v y r y sin h 0 ka x v x rx k 0 j ry k 0 j r x f x kb x u x r y f y kb y u y 1 0 þ k ða x þ a y þ b x þ b y Þ c The net potentia energy of n c constant oads and n s spring oads is aso a inear combinations of the same basis functions. Furthermore, cos h, sin h, r x cos h, r y cos h, r x sin h, r y sin h, rx, r y, r x, r y, and 1 are ineary independent functions of fr,hg. Hence, from a reasoning simiar to the one in Sec., for the net potentia energy to be independent of the configuration variabes, the coefficients of a the basis functions other than 1 have to be zero. However, it is not practica to make the coefficients of a these functions as zeros because of the foowing reasons: There are ony gravity oads: Gravity is the most important practicay seen instance of a constant oad. When a the constant oads are gravity oads, f i ¼ m i g i, where m i is the mass and g is the acceeration due to P gravity. Further, the coefficient of r x and r y become g x m P ng ng i and gy m i. Since m i > 0, 8i, P n g m i > 0. Aso, since the acceeration due to gravity is nonzero, 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-ength spring oads, possiby with gravity oads: In this case, the coefficients of both r P x and r y are ns k i. Since the spring constants of a the springs considered here are positive, (k i > 0, 8i), P n s k i cannot be zero. Hence, the coefficients of rx and r y cannot be zero. However, as shown in Appendix B.1, there is no such practica difficuty in making the coefficients of a h-dependent functions, i.e., cos h, sin h, r x cos h, r y cos h, r x sin h, and r y sin h as zero. Setting h-dependent terms to zero amounts to the foowing set of independent constraints: Xnc X ns ðf y;i p y;i þ f x;i p x;i Þ k i ða y;i b y;i þ a x;i b x;i Þ ¼ 0 (1) þ Xnc X ns ðf x;i p y;i f y;i p x;i Þ k i ða x;i b y;i a y;i b x;i Þ ¼ 0 (13) X ns ðk i a i Þ ¼ 0 (1) It is shown in Appendix B.1 that if these constraints are not satisfied by the oads, then by adding not more than two zero-freeength springs, these constraints can be satisfied. A numerica exampe to demonstrate the same is given in Fig. 6. Inspite of being abe to make the potentia energy of the oads on the ink independent of h, the dependency on r sti remains. In Sec., we show that if the body is joined to an appropriate inkage, then by adding extra oads to other parts of the inkage, the r-dependent terms of the potentia energy can be baance out. Before we proceed to Sec., it may be noted that the potentia energy of a constant oad or a zero-free-ength spring oad fas under the foowing genera form: U ¼ r T u þ jr T r þ r T RðhÞv þ w T RðhÞq þ c (15) where h and r are the configuration variabes of the rigid body on which the oad acts. In the case of constant oads, by comparing Eq. (1) with Eq. (15), we have u ¼f ; j ¼ 0; v ¼ 0; w ¼f ; q ¼ p; and c ¼ 0 (16) and in the case of zero-free-ength spring oads, by comparing Eqs. (6) and (15), we have Fig. 6 A rigid body moving freey in a pane under a constant oad is made to have h-independent potentia energy by addition of two zero-free-ength springs Journa of Mechanisms and Robotics MAY 01, Vo. /

6 u ¼kb; j ¼ k ; v ¼ ka; w ¼b; q ¼ ka; and c ¼ k bt b þ a T a (17) Later in the paper, we encounter potentia energy functions that are of the form given in Eq. (15), but they cannot be attributed to zero-free-ength spring oads or constant oads acting on the body. Hence, there is a need to generaize constraints (10) (1) to the form given in Eq. (15). Such a generaization is possibe because as can be seen in the ast coumn of Tabe, the potentia given in Eq. (15) is a inear combination of the basis functions given in Tabe just as in the case of constant and zero-free-ength spring oads. The foowing proposition states the generaization. Proposition 3.1. If there are n functions of the form U i ¼ r T u i þ j i r T r þ r T RðhÞv i þ w T i RðÞq h i þ c i ; i ¼ 1 n (18) with r and h as the variabes, then P n U i is independent of h if and ony if the foowing constraints are satisfied: X n X n q y;i w y;i þ q x;i w x;i ¼ 0 (19) q x;i w y;i q y;i w x;i ¼ 0 (0) X n v i ¼ 0 (1) When these constraints are satisfied, P n U i depends ony on r in the foowing form: X n U i ¼ Xn X n ¼ r T r T u i þ j i r T r þ c i X n u i þ r T r j i þ Xn c i () Furthermore, if r happens to be a constant (as in a ever) with ony h being the variabe, then P n U i is independent of h (and hence a constant) if and ony if the foowing constraints are satisfied: X n X n v y;i r y þ v x;i r x þ q y;i w y;i þ q x;i w x;i ¼ 0 (3) v x;i r y v y;i r x þ q x;i w y;i q y;i w x;i ¼ 0 () Proof. The proof is aong the same ines as the derivation of Eqs. (10) (1). It may be noted that inspite of considering a genera form of potentia in Eq. (18), the inabiity to make the net potentia energy independent of r remains because of the foowing reason. In a the cases that we consider next, j i 0 and j i > 0 for ateast one vaue of i. Hence, the r-dependent term, P n j ir T r, cannot be zero in the expression for P n U i. New Static Baancing Techniques for Revoute-Jointed Linkages If there is a singe rigid body with oads exerted by a reference frame, then the net potentia energy of the oads depends on the configuration of the body with respect to the reference frame. If there are severa such bodies, then the net potentia energy of a the oads on a the bodies depends on the configuration of a the bodies. This dependency on the configuration of a the bodies can be reduced to that of a singe body provided the bodies are connected by revoute joints (to begin with, say, in a seria or a treestructured manner) and the oads are zero-free-ength spring oads and constant oads. If this singe body is the reference frame itsef, then the net potentia energy is a constant (impying static baance) since the configuration of the reference frame with respect to itsef is aways fixed. This resut foows as a consequence of the proposition that is presented next..1 The Potentia Energy of Loads on a Body Transformed as a Function of Another Body. We are now considering severa rigid bodies, each of them with its own r, h, n c, n s, a i, b i, k i, p i, etc. To distinguish these quantities beonging to different rigid bodies, we number the rigid bodies and put the number as a superscript to these symbos. Hence r, h, n c, n s, a i, b i, k i, p i, and f i of body j are now represented as r j, h j, n j c, nj s, aj i, bj i, kj i, pj i, etc. Proposition.1. The net sum of a set of functions of the configuration variabes of a body in the form given in Eq. (15), i.e. U i ¼ rt u i þ j i rt r þ r T R h v i þ w T i R h q i þ c i ; i ¼ 1 n (5) can be expressed as a function of the same form but of body j, i.e. X n U i ¼ Uj i ¼ rjt u j i þ jj i rjt r j þ r jt R h j v j i þ wjt i R h j q i þ c j i (6) provided the foowing conditions are satisfied: Condition 1: There is a point that is rigidy fixed to both body and body j. Such a point is caed as a common point of bodies and j. Condition : The origin of the oca coordinate frame of body is at the common point. Condition 3: The sum of the set of functions of body is dependent ony on r in the form given in Eq. (). Proof. Let the oca coordinate of the common point required by condition 1 in body be s j and in body j be sj. The commonaity of the point can be written as foows: r þ R h s j ¼ r j þ R h j s j (7) Condition impies that s j ¼ 0. Substituting s j ¼ 0 into Eq. (7) eads to r ¼ r j þ R h j s j (8) Condition 3 impies that the sum of the set of functions of body can be written as X n U Xn i ¼ rt u i þ X n rt r j i (9) The constant term is omitted in Eq. (9) since it is inconsequentia for the discussion. Substitution of r from Eq. (8) into Eq. (9) and simpification using the fact that R T (h)r(h) is identity ead to the foowing expression for P n U i : / Vo., MAY 01 Transactions of the ASME

7 r jt u j i zffffffffff} ffffffffff{ þ X n u i! þ r jt r j w jt i zffffffffffff} ffffffffffff{! T Xn u i j j i zffffffffff} ffffffffff{ X n j i q j i zff} ff{ R h j s j! þ r jt R h j ¼ Xn v j i zffffffffffffffff} ffffffffffffffff{! X s j n j i U i ¼ Uj i (30) Again, the constant term is omitted in Eq. (30). It may be readiy recognized that the sum of the set of functions on body, P n U i, as seen in Eq. (30) is indeed of the form given in Eq. (6).. Proposition.1 as the Recursive Reation of an Iterative Static Baancing Agorithm. We now show that Proposition.1 can be treated as a recursive reation that can be incorporated into an iterative procedure to achieve static baance of a inkage. For the purpose of this section, we restrict the inkage on which the iterative procedure can be appied to have the foowing features: (1) The inkage shoud be tree-structured (i.e., no cosed oops). This feature is necessary since a recursive reation requires a tree-structure to propagate. () A the joints of the tree-structure shoud be revoute joints. This feature is necessary to satisfy condition 1 of Proposition.1. (3) We want a the oads to have potentia energy functions of the form given in Eq. (5) of Proposition.1. Whie we know that zero-free-ength springs and constant oads do have this form (see Eqs. (16) and (17)), the fact that there are severa bodies invoved requires attention. The configuration variabes (r,h ) of different bodies (i.e., of different ) shoud be with respect to a common goba frame of reference. Hence constant oads on a the bodies shoud be constant with respect to a common goba reference frame and any zero-free-ength spring shoud have its one anchor point on the same common goba frame whie the other anchor point can be on any of the bodies constituting the inkage. () The common reference frame shoud be one of the bodies of the inkage, i.e., it shoud join to body/bodies of the inkage by revoute joint/joints...1 The Iterative Static Baancing Agorithm. We now present the iterative agorithm and prove that it eads to static baance. Preparatory Steps (1) Assign the reference body as the root node of the treestructure (bodies are represented as nodes and joints as ines joining the nodes). With this assignment, for every ink/ body other than the root, there is a parent body. Further, every ink other than a termina ink has one or more chidren. () Choose a oca frame of reference on every ink to coincide with the center of revoute joint between the ink and its parent. For every ink k, r k and h k decide the configuration of its oca frame with respect the frame of the root. (3) Give this tree-structure with the given constant and zerofree-ength spring oads (together referred to as origina oads) as an input to the foowing iterative procedure. Iterative Procedure. Entry condition: If the tree-structure contains ony the root node, then exit from the iterative procedure. Otherwise, proceed to step 1. Step 1: Any termina node, has associated with it the foowing three kinds of potentia energy functions: (1) due to origina oads on body, () due to association that happened in step 3 of previous iterations, and (3) additiona oads on body. Let the number of such functions be represented by n o, n c, n a, respectivey. The first two kinds of functions are known from the given probem and previous iterations, respectivey, and the task in this step is to find the additiona oads so that Case (a): Equations (19) (1) are satisfied if is not a chid (i.e., not first generation descendant) of the root. Case (b): Equations (3) () are satisfied if is a chid of the root. Note that in this case r is a constant because of the way oca frame is chosen in the preparatory steps. This task makes sense ony if the a kinds of potentia energy functions fa under the form of Eq. (15) with (r, h) being (r, h ). The first and the third kind of potentia energy functions do fa under the form because of the kind of oads we are restricting to (see Eqs. (16) and (17)). The second kind of potentia energy functions conform to the form because of step which is a direct consequence of Proposition.1. The critica roe of Proposition.1 in enabing this iterative procedure may be noted. Further, Appendix B. asserts that the task of this step is aways feasibe. It may be noted that there are severa set of additiona oads that satisfy these equations. This nonuniqueness cas for discretion of the designer in choosing a suitabe set of additiona oads. Step : In case (a), express P n o þn c þn a U i in the form given in Eq. (15) where r and h are the configuration variabes of the parent of node. This is possibe since condition 1 (because of revoute joint), condition (because of preparatory steps) and condition 3 (because of step 1) of Proposition.1 are satisfied. In case (b), recognize that P n o þn c þn a U i is a constant as per Proposition 3.1. Step 3: Associate P n o þn c þn a with the parent ink of and for U i energy conservation, disassociate U i, i ¼ 1 n o þ n c þ n a from node. Because of this association, n pðþ c (pðþ denotes parent node of, and n k c denotes the number of potentia energy functions associated with node k so far at step 3) gets incremented by one and the associated function can be written as n U pðþ o þn n c ¼ X c þn a U i (31) Step : With a potentia energy functions robed from node to its parent, deete this termina node. Iterator: Once steps 1 are competed, a new trimmed treestructure resuts where the parents of the nodes deeted in step has additiona potentia energy functions associated with them. Foow this iterative procedure again with this trimmed treestructure as the input. With every iteration, the tree-structure shrinks and it eventuay gets reduced to the singe root node. Any of the n 0 c potentia energy functions (of the second kind) associated with this reduced root is from one of the chidren 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. Thus, the sum of these n 0 c potentia energy functions is aso constant. Further, the sum of these n 0 c potentia energy functions is actuay the sum potentia energy of origina oads and additiona oads on a the descendants of the root. This can be verified by recursive substitution in Eq. (31) as exempified in Eq. (3). Therefore, the origina oads are in static baance with the additiona oads. Iustration of the agorithm on a R inkage under constant oads: Figure 7 shows a R inkage where four revoute joints connect the ground and four other bodies seriay. The ground exerts constant gravitationa force on each of the four bodies. Hence we take the ground as the root and number the bodies accordingy as shown in Fig. 7(b). The oca frame of reference are ocated on each of the bodies as per the preparatory step. The constant oads on bodies 1 are represented as C 1 1, C 1, C3 1, and C 1, respectivey. Their detais (point of action p and force vector f) are presented in item number 1,, 7, and 10 of the tabe in Fig. 7. Now we give the tree-structure to the iterative procedure. The termina node of the tree is. There is a potentia energy function associated with the node due to C 1 which is represented as Journa of Mechanisms and Robotics MAY 01, Vo. /

8 Fig. 7 Detais of staticay baanced gravity oaded R inkage and its modification into 3R and R inkage. U 1 C 1. There is no potentia energy function of the second kind ðn c ¼ 0Þ. Two zero-free-ength springs Z 1 and Z are added so that the functions U 1 C 1, U Z 1, and U 3 C satisfy Eqs. (19) (1) as per case (a) of step 1. A the detais of the springs, constant oads as we as their potentia energy in the standard form (see Eq. (15)) are presented in the tabe offig. 7. Now, as per step, the sum of U 1 C 1, U Z 1, and U 3 Z is transformed as a U 3 ðr3 ; h 3 Þ in accordance with Eq. (30) of Proposition (.1). This is foowed by making of a new tree-structure by deeting node and associating U 3 with node 3 of the new treestructure. This competes the first iteration. The second iteration acts on the new tree-structure. The tabe in Fig. 7 and Fig. 7(a) give a the detais of a the iterations. At the end of four iterations we are eft with a singe root node having constant function U 0 1 associated with it. By foowing the dashed arrowed ine of the Fig. 7(a) in the reverse order, it may be verified that U 0 1 ¼ U1 1 C1 1 þ U 1 C 1 þ U 3 1 C 3 1 þ U 1 C 1 þ U Z 1 þ U 3 Z þ U 3 3 Z 3 1 þ U 3 Z 1 (3) Hence C 1, C3 1, C 1, and C1 1 are in static baance with Z 1, Z, Z3 1, and Z 1. To verify the static baance, this inkage aong with the oads was modeed in ADAMS. With zero damping, a puse of energy was initiay introduced to the system. When the dynamic simuation of the system was carried out it was noticed that the net kinetic energy was constant over time. This impies that there was no potentia gradient aong the path that the inkage took in the dynamic simuation. In Fig. 7(c), joint 1 and body 1 are eiminated to modify this R exampe into a 3R exampe. Joint now joins body with the ground at r ¼ 0 5. Rest of the bodies and their numbering is 3 unchanged. The first two iterations for this exampe are identica to the R exampe. The third iteration is the ast since node now is a chid of the root. As required at this iteration, it may be verified that Eqs. (3) and () are satisfied with r ¼ 0 5. One can 3 have a simiar modification of R exampe into a R exampe as shown in Fig. 7(d) / Vo., MAY 01 Transactions of the ASME

9 Fig. 8 Detais of static baance of a R inkage under spring oad Fig. 9 Detais of static baance of a R tree-structure inkage under a constant oad and a spring oad Iustration of the agorithm on a R inkage under a zero-freeength spring oad: Just as Fig. 7 has every detai of the R exampe, Fig. 8 has every detai of this exampe. The expanation is aso aong the same ines of the previous exampe. In this exampe, the given origina oad is Z 1 and the baancing oads are Z and Z 3. To buttress the fact that zero-free-ength springs are practica, a prototype of this exampe was made, as shown in Fig. 8(b). To Journa of Mechanisms and Robotics MAY 01, Vo. /

10 Fig. 10 Potentia Energy variation of spring oads, constant oads, and their sum reaize zero-free-ength springs, puey-string arrangement was used, the detais of which can be found in Herder [5]. Iustration of the agorithm on a R tree-structure inkage under both constant oad and zero-free-ength spring oad: Whie the previous two exampes had seria architecture, this exampe has branches emanating form the same node, as shown in Fig. 9(a). The origina oads acting on it are C 3 1 and Z 1. Instead of taking origina oads to be excusivey constant oads or excusivey zero-free-ength spring oads, here we have taken a combination of both types of oads. These origina oads are baanced by adding springs Z 3 1, Z3, Z, Z 1, and Z1 1 at various iterations in the iterative agorithm. A pictoria depiction of the iterations on these inkages is given in Fig. 9(b). A the remaining detais are given in the tabe of the same figure. To verify the static baance, h s of bodies 1 are varied in the foowing form: h 1 ¼ p þ psin ð pt Þ, h ¼ 1 p þ psin ð pt Þ, h3 ¼ 1:7 p þpsinðptþ, h ¼ p 1:3 þ psin ð ptþ. The potentia energy variation of origina oads C 3 1 and Z 1 as we as the baancing oads, i.e., Z 3 1, Z3, Z, Z 1, and Z1 1, are potted in Fig. 10. The sum of a these variations is aso potted and it has turned out to be a constant. This verifies the static baance..3 Static Baancing of Any Revoute-Jointed Linkages With Any Kind of Zero-Free-Length Spring and Constant Load Interaction Within the Linkage. In the static baancing method for inkages provided in Sec.., other than the fact that the inkage to be baanced has to be revoute-jointed and that oad interactions are of zero-free-ength spring or constant oads, there were two more restrictions as foows: (1) It shoud be possibe to consider that the oads on a the bodies are exerted by a common reference body (or frame) of the inkage. () The inkage shoud have a tree-structure (i.e., without cosed oops). When the first restriction is vioated as in Fig. 11(a), it is aways possibe to break the oad interactions into a superposition of severa oad sets with each set compying to the first restriction. For exampe, the oad interaction in Fig. 11(a) is broken into two oad sets in Figs. 11(b) and 11(c). The reference body in each of these sets is indicated by an asterisk symbo (*) in its respective figure. Furthermore, in a oad set, if there are cosed oops, then the cosed oops can be broken by reaxing certain joint constraints. Figures 11(c) and 11(d) iustrates breaking of cosed oops respectivey in Figs. 11(b) and 11(c). With cosed oops broken, each of the oad sets compy with the two restrictions and they can be staticay baanced by adding baancing oads as per Sec... Fig. 11 Breaking a probem as a superposition of severa probem with each probem being static baance of revoute-jointed tree-structured inkage with oads exerted by the root body Once each of the oad sets is baanced, the joint constraints that were reaxed for breaking cosed oops can be reimposed without disturbing the static baance. In other words, when the potentia 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 inkages in a the oad sets are the same as the origina inkage and the oads on a the sets can be superposed. Since each oad set is in static baance, the superposition is aso in static baance. In other words, the sum of severa constant potentia energy functions due to severa oad sets is aso a constant. This superposition contains a the origina oads on the given inkage. The remnant oads in this superposition are the additiona oads that baance the origina oads. In this way, additiona oads that staticay baance any revoute-jointed inkage with zero-free-ength spring and constant oad interactions between the bodies of the inkage can aways be found. 5 Static Baance of Spatia Linkages Having Zero-Free-Length Spring and Constant Load Interactions Within the Linkage The static baancing technique of Sec. was based on Proposition.1, which was presented for two panar bodies, with the revoute joints ony serving to satisfy the condition 1 of the proposition. We can have anaogous static baancing technique for spherica and revoute-jointed spatia inkages with zero-freeength spring and constant oad interactions between the bodies of the inkage provided that (1) the potentia energy for zero-free-ength spring and constant oads has the same form as given in Eq. (5), () Proposition.1 is true even if the two bodies ( and j) are free to move in space, (3) spherica and revoute joints ensure condition 1 of Proposition.1, and () anaogous to constraints (19) (1) which enabe satisfying condition 3 of the Proposition.1, there are constraint equations for spatia case that a body can satisfy in practice, possiby by addition of extra zero-free-ength spring oads. The first one is true since in the derivation of the potentia energy of constant oads in Eq. (1) and zero-free-ength spring oads in Eq. (6) woud require no modification even if r were to be considered as a spatia goba coordinate (3 1 matrix), a, b, p were to be considered as spatia oca coordinates, and R(h) were to be considered as the spatia rotation matrix of the bodies with h possiby representing Euer anges. That R T R is the identity / Vo., MAY 01 Transactions of the ASME

11 matrix, which was used in the derivation, is true in the spatia case aso. The second one is true since Proposition.1 reies on Eq. (7), that R T R is the identity matrix, and matrix-agebraic manipuations. A of them are vaid in the spatia case aso. The third one is true since a spherica joint ensures a common point between the two bodies it joins and a revoute joint ensures a common ine between the two bodies it joins. In the fourth one, the constraints (19) (1) were obtained by expressing the potentia energy as a inear combination of a set of basis functions invoving r and h and setting the coefficient of h- dependent basis functions to zero. In a simiar vein, in the spatia case aso, the potentia energy woud be a inear combination of a set of basis functions invoving r and rotation defining, say, Euer anges. By setting the coefficients of Euer ange-dependent terms to zero, the anaogous spatia constraints can be obtained. The caim that we are not substantiating in this paper, for the sake of brevity, is that these anaogous constraint equations can aso be satisfied in practice, if necessary with the addition of extra zerofree-ength springs. Thus, the static baancing technique presented in the paper for revoute-jointed panar inkages extends to spherica and revoutejointed spatia inkages. 6 Concusion We presented a technique to staticay baance any panar revoute-jointed inkage having zero-free-ength spring and constant oad interactions between the bodies of the inkage. The technique invoves ony addition of zero-free-ength springs but not any extra ink, unike spring-aided perfect static baancing techniques currenty in the iterature. The technique extends to spatia spherica and revoute-jointed inkages as we. The technique reies on a recursive reation to iterativey remove the dependence of the potentia energy on the configuration variabes of the bodies of the inkage. Recognizing the recursive reation aong with the minima conditions that enabe it constitutes the main contribution of this paper. Appendix A: Perfect Static Baance and Positive-Free-Length Springs This is an appendix to Sec.. Here, the difficuty in achieving perfect static baance of a ever by using normay avaiabe positive-free-ength springs is discussed. The d T d term in the potentia energy expression of a spring given in Eq. (5) was seen to be a inear combination of sin h, cos h and 1 when expanded as in Eq. (6). Hence, by writing d T d as a sin h þb cos h þ c, the potentia energy expression of the spring becomes: k a sin h þ b cos h þ c ð Þk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ða sin h þ b cos h þ cþþ k 0 (33) The first term in Eq. (33) is the zero-free-ength part and the second term is the free-ength part. If the free-ength is positive, i.e., 0 > 0, then the free-ength part is negative. The free-ength part of the spring is nonconstant (i.e., k = 0 and not both a and b is zero) except for trivia situations where spring constant is zero or the spring is attached to the pivot of the ever. When there are severa but finite positive-free-ength and nontrivia springs, the net contribution of the free-ength part is negative, and it is aso not known have the possibiity of being a constant, unike the zerofree-ength part. Furthermore, the free-ength part is aso not known to be in the function space spanned by sin h and cos h. Hence, the possibiity of free-ength part canceing (moduo a constant) with zero-free-ength part is aso rued out. Thus, with severa positive-free-ength springs, there is no way the net potentia energy coud become a constant. Appendix B: Constraints Can be Satisfied, If Not as It Is, by Addition of Extra Zero-Free-Length Spring Loads Appendix B.1: Satisfying Constraints (1) (1). This appendix demonstrates how by adding extra zero-free-ength spring oads constraints (1) (1) can be satisfied. To differentiate between origina oads and baancing oads, et n s,o and n s,b respectivey represent the number of origina and baancing zerofree-ength spring oads with n s,o þ n s,b ¼ n s. Aso, et the spring oads be indexed such that the first n s,o oads are origina oads with the remaining being baancing oads. Simiar meaning appies for n c,o and n c,b. Case 1: Origina oads vioate the constraint (1), and baancing oads are ony zero-free-ength springs. Let us try to satisfy a the constraints by adding a singe zero-free-ength spring. As per the notation, this spring gets the index i ¼ n s,o þ 1. The constraint (1) can be written as foows: k ns;oþ1a ns;oþ1 ¼ Xns;o k i a i (3) where the known quantities reated to origina oads are on the right hand side. Equation (3) gives the unique soution of k i a i for i ¼ n s,o þ 1 to the constraint (1). Furthermore, the constraints (1) and (13) can be rewritten as k i a x;i k i a y;i ¼ 6 Xnc þ Xnc k i a y;i bx;i k i a x;i b y;i i¼n s;oþ1 3 X ns;o ðf yi p y;i þ f x;i p x;i Þ k i ða y;i b y;i þ a x;i b x;i Þ X ns;o 7 5 ðf x;i p y;i f y;i p x;i Þ k i ða x;i b y;i a y;i b x;i Þ (35) The matrix on the eft hand side of the equations is known since k i a i for i ¼ n s,o þ 1 is aready soved in Eq. (3). Furthermore, the matrix is nonsinguar since the right hand side of Eq. (3) that is the same as k ns,o þ 1 a ns,o þ 1 is nonzero as per the description this case. We take ½b x;i b y;i Š T for i ¼ n s,o þ 1 as the inverse of the matrix times the right hand side of the Eq. (35) so that the constraints (1) and (13) can aso be satisfied. Thus, theoreticay, with a singe additiona zero-free-ength spring, a three constraints (1) (1) can be satisfied. Case : Origina oads satisfy (1), but vioate ateast one of the constraints (1) and (13). Baancing oads are ony zero-freeength springs. If we proceed aong the same ines as the previous case, then in Eq. (35), the matrix on the eft hand side becomes singuar zero-matrix whereas the right hand side is nonzero by the description of the case. Thus, in this case, with a singe baancing zero-free-ength spring, it is not possibe to satisfy the reated constraint. However, it may be verified that by adding two baancing springs, a the constraints can be satisfied. The cases 1 and cover a possibe types of constraint vioation. Hence we assert that if the constraints are not satisfied as it is, then by adding a minimum of one zero-free-ength spring in case (1) (the component reated to origina oads in constraint (1) is nonzero) and two zero-free-ength spring in case () (the component reated to origina oads in constraint (1) is zero), the constraints can be satisfied. Journa of Mechanisms and Robotics MAY 01, Vo. /