University of Twente

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1 University of Twente EEMCS / Electrical Engineering Control Engineering Variable stiffness for robust and energy efficient 2D bipedal locomotion Matin Jafarian MSc report Supervisors: prof.dr.ir. S. Stramigioli dr.ing. R. Carloni MSc G. van Oort MSc L.C. Visser August 2010 Report nr. 022CE2010 Control Engineering EE-Math-CS University of Twente P.O.Box AE Enschede The Netherlands

2 Abstract Although it is easy for humankind to stably walk on different terrain, it is difficult to achieve a human-like gait for bipedal walking robots due to their complex dynamics. Generally, there are two approaches towards controlling a biped robot: static and dynamic walking. In dynamic walking approach, the walker will have a stable gait over the course of multiple steps, so there is no necessity for the walker to be stabilized in each of the its steps. The simplest example of a dynamic walker is a 2D passive biped, which is powered only by the gravity. It can stably walk over a gentle slope. One of the problems in this field is the narrow area of robustness for a biped. In this project, the steps of modeling and analyzing of the stability and robustness of a passive biped has been done. Also, the effects of applying a passive storage element (spring) along the leg on the energy efficiency and robustness of the robot have been verified. The idea behind applying compliancy to the model is to restore the energy, which is dissipated by the transition impact, using a buffer element like a spring.

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4 Preface This is to report my MSc project that is the final step of my two-year study at the control group of the University of Twente. I would like to take this opportunity to thank Prof. Stefano Stramigioli for his wonderful lecture in Robotics, Prof. J.van Amerongen for his guidance and support during these two years, Dr. Raffaella Carloni for all of her support, helpful advice and motivation. Many thanks to Gijs van Oort for our fruitful discussions, his helpful comments in improving my report, and his enthusiasm. I also would like to thank Dr.Korsten for my scholarship and Ludo Visser for his comments to my work. I am very pleased to have some great friends in Enschede and in CE group and I am very thankful to all of them for making me feel at home and making this project a pleasant time. Last but not least, I would like to thank my family for all of their support and understanding. Without your help I could not accomplish my goals. Matin Jafarian Enschede, August 2010

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6 Contents Chapter1 Introduction Previous work Project goals Report outline... 2 Chapter 2 Passive bipedal walker over slope Description of the basic model Why this model is a good base to study How to model the passive walker sim and the 3D Mechanics Editor Impact of the foot with the ground Compliant model and analysis of the passive walker Energy Analysis Stability Analysis Stability of a limit cycle walker Symmetric and asymmetric gaits Verifying the stability of the passive walker Robustness Analysis Measure of the robustness Comparison of robustness measures Nominal model Chapter 3 Applying compliancy to the model Applying a constant linear spring along the leg The effect of choice of the damper on the stability and robustness of the walker Analyzing the robustness of the passive walker with prismatic compliant foot joint Changing the stiffness of the leg Chapter4 Conclusions and Recommendations Conclusions Recommendations Bibliography Appendix A: Compliant Contact model Appendix B: Initial conditions Appendix C: Numerical Calculation of Eigenvalues Appendix D: Changing the stiffness of the legs University of Twente i

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8 Chapter1 Introduction Bipedal walking is the main form of locomotion for human kind. Our body is flexible and we even can do variations of walking gaits like hand-walking, scrambling, walking on crutches, etc. however making a robot to have a human-like gate is not easy due to the complicated dynamics of the walking. The motivation behind the interest to build human-like walkers is its applications from helping the studies in rehabilitation to the field of entertainment. Generally, walking robots can be divided to two categories: static and dynamic walkers. The research area of dynamic walking which emphasizes on the passive dynamics of the legs, and generally avoids the use of high-gain control and complicated analysis, has received a lot of interest in robotics. Dynamic walking builds on the work of its founding father Tad McGeer, who built the first physical passive dynamic walking machine. There exists a class of two-legged machines for which walking are a natural dynamic mode. Once started on a shallow slope, a machine of this class will settle into a steady gait quite comparable to human walking, without active control or energy input. Interpretation and analysis of the physics are straightforward; the walking cycle, its stability, and its sensitivity to parameter variations are easily calculate. (McGeer, 1990) In dynamic walking approach, the walker will have a stable gait over the course of multiple steps, so there is no necessity for the walker to be stabilized in each of the its steps. On the other hand, static walking approach emphasizes on gaining stability in each walking step that can be usually achieved by actuating all joints to have control over all degrees of freedom. A passive dynamic walker is energy efficient and shows a human-like gait. It can walk faster that static walkers, but it has a narrow robustness. One of the challenging problems in the field of biped walking robots is improving its robustness. This project is to analyze the robustness of a 2D passive walker and study the effects of applying compliancy to the model in order to improve the robustness. 1.1 Previous work After introducing the model of the simplest passive dynamic walker by McGeer in 1988, many researchers have done different studies in this field. Researchers from MIT, SFU, Cornell, and Delft universities have worked on this topic to understand the essentials of dynamic walking and try to improve the pitfalls. Most of these studies have been done on analyzing the simplest passive biped as a base and in order to prevent complicated analysis, as it is one of the advantages of a passive dynamic walker. At the Control group of the University of Twente, also researches have been done in the field on dynamic walking under the supervision of Prof. Stramigioli. Thus far, some PhD projects and several master assignments have been focused on this topic. Also, a 2D passive-based dynamic walker Dribbel and a teen size soccer robot Tulip have been constructed in the control group laboratory. University of Twente 1

9 1.2 Project goals The goal of this project is to know if the robustness of a 2D passive dynamic walker can be improved by applying compliancy to the system. The aim of applying a passive storage element (compliant element) to the robot is to store some part of kinetic energy of the system that is usually dissipated due to the impact of the foot with the ground and return it to the system to have an energy efficient walker. The question here is if this method can have a positive effect on the robustness of the system? Thus far, some studies have been done on applying the compliancy on the hip joint and ankle joint (Hobbelen, 2008). In this project, a spring is placed along the leg to make a prismatic compliant foot-leg joint and its effects on the robustness and velocity of the passive walker has been studied. The project sub goals are: Literature review Modeling a passive 2D compass walker and analysis of its robustness Adopting an appropriate robustness criteria Applying compliancy to the model and analyzing its effect on the robustness of the biped Changing the applied compliancy and study its results 1.3 Report outline In this report, modeling a 2D passive dynamic walker is described and its behavior is analyzed. Also, the effects of applying compliancy along its legs are explained. Chapter 2 describes the steps of modeling the 2D passive walker using 20-sim modeling program. Also, system analysis including: energy, stability and robustness analysis are reported. Different measures for robustness are introduced and a proper measure has been chosen. Finally, a nominal model (slope angle) is selected as the base of further analysis in the next chapter. Chapter 3 explains the effects of placing a passive storage element (spring) along the walker s leg on its robustness, energy efficiency and velocity. Also, the results of changing the stiffness of the legs while walking are reported. Chapter 4 lists conclusions and recommendations. 2 Control Engineering

10 Chapter 2 Passive bipedal walker over slope A passive bipedal walker is a human like walker without knees that is actuated by the gravity while walking on a slope. It is a completely un-actuated and uncontrolled biped that can show a stable gait while walking on a gentle slope. In this chapter, first the basic idea behind the model of the passive walker is described. The basic idea is inspired by the model of the simplest passive walker that is used in the reference literature in this research field (Garcia et al., 1998), (Wisse, Essentials of dynamic walking,phd thesis, 2004). Then, the choice of the basic model is supported by some reasons showing this model can be an appropriate base for the goal of this project. Afterwards, the model that was used in this study is explained. The differences between this model and the basic model are the foot shape and the ground contact model. In all research studies so far, the rigid contact model is used to formulate the interaction of the foot with the ground. In this study, the compliant contact model has been applied. After description of the model, the analysis of the model (including stability, energy and robustness analysis) is reported which shows similar results to the results of analyzing the model of the simplest passive walker reported by other researchers. Finally, a nominal model is chosen, with a specific weight and slope angle to analyze the effects of changing the stiffness of the walker s legs. 2.1 Description of the basic model In this section the model of the simplest 2D passive walker over a gentle slope represented by (Garcia et al., 1998) is described. Figure (2.1) shows an impression of the configuration of the model. The model has two rigid legs connected by a frictionless hinge at the hip. It is composed of three point masses, the mass of the hip (M) and the feet mass (m), the mass of the legs are neglected (very tiny). The hip mass is much bigger than the feet mass and, so the inertia of the swing leg and its related foot will not affect the dynamics of the system, as a result the analysis of the model will be simpler since the involved parameters are reduced. The feet are point feet and they are rigidly connected to the legs. The model walks down a gentle slope (γ) and actuated by the gravity. Figure (2.1): The passive walker over the slope with the model parameters and variables, Note that M>>m. During the walking motion, there is an interaction with the ground (slope) due to the impact of the foot with the ground. The impact results in losing the kinetic energy supplied by the gravity at the end of each step. For this model the following properties are assumed: University of Twente 3

11 Variable stiffness for robust and energy efficient 2D bipedal locomotion o Plastic Impact with the ground (no-bounce, no-slip ), o o Instantaneous double stance phase, Possibility of foot-scuffing: It is a non-physical assumption for the model that let the swing leg to pass through the surface of the slope during the mid-stance phase. Foot scuffing is the result of having straight legs (no knees). The equations of motion of the system during the swing phase are like the equations of a simple double pendulum, representing the angular momentum balance of the whole system around the stance foot, and the angular momentum balance of the swing leg around the hip. The configuration variables of the system are: the angle of the stance leg with respect to the slope normal (θ) and the angle between the stance leg and the swing leg (φ). The equations of motion are: θ = sin(θ γ), φ = sin(φ) θ cos(θ γ) + sin (θ γ). Eq. (2.1) According to the equations above, the position variables and their related velocities are determining the state of the system: θ; θ ; φ; φ. The impact of the foot with the ground, heel strike, happens when the geometrical condition φ (t) 2θ(t) = 0 is satisfied. Although this condition is also satisfied for the mid stance phase, when the legs are parallel, since the foot scuffing is ignored, holding this condition represents an impact. The following equations show the evolution of the system state just after the heel strike, where (+) refers to the system state just after the heel strike and (-) refers to them just before the heel strike. Since, the double stance phase happens instantaneously, the state of the system just after the impact will be dependent on the values of θ, θ just before the impact, so at the start of the step the dimension of system states is reduced to two: [θ ; θ ]. θ = θ, φ = 2 θ, θ = cos(2 θ ) θ, φ = (1 cos(2θ )). Eq. (2.2) 4 Control Engineering

12 P Passive bipedal walker over slope θ θ t=t t=0 Figure (2.2): The evolution of θ; θ during one step. Figure (2.2), shows the evolution of two states of the system θ; θ during one step. The step is started at t=0 (when the solid-leg is on the ground), the swing phase is processing along the curve during which the system is pivoting around the stance leg, the step is ended at t=t (period of the motion), then there is an instantaneous double stance phase in which the role of the legs is exchanged (the previous swing leg becomes the stance leg and vice versa.) University of Twente 5

13 Variable stiffness for robust and energy efficient 2D bipedal locomotion 2.2 Why this model is a good base to study Regarding the various applications of the walking machine from space exploration to prosthetic aids, the basic model should be an appropriate base such that the conclusions can be extended to some different possible configuration. Also, regarding the complexity of the walking motion, it is wise to prevent unnecessary complexity of the model in order to facilitate drawing conclusions. - This model is simple to analyze As can be understood from the previous section, this model is very simple; fixing configuration parameters, the main parameter that can affect the behavior of the passive walker is the angle of the slope. Therefore it can be a good base to study the effects of adding another parameter (for example applying elasticity in the model) to the system. - Generalization of the results are possible o o o The effect of the upper body: In different applications of the walking robots, an upper body is a part of the robot. According to some previous research, adding upper body to this base model, with a certain design, will slightly improve the robustness (Wisse, Essentials of dynamic walking,phd thesis, 2004). Therefore, it is expected that the results obtained by studying the simplest passive walker will be held for a robot with torso, too. The effect of knees: The simplest passive walker does not have knees. Generally, since a normal walking gait starts and ends with the extended legs, the knees do not have an essential effect in the general behavior of the system. According to a study by McGeer (McGeer, 1990) the passive knees can prevent the problem of foot-scuffing during the swing phase and in some cases it may help the stability. Since the problem of foot-scuffing can be ignored in the simulations, adding the knees just increase the complexity of the model, for example the model should prevent the problem of hyper-extension of the leg s length that may happen at the end of the swing phase. The effect of different sizes and weights of the model: From the equation of motion (represented in section 2.1), it can be concluded that the behavior of the model is not dependent on the choice of masses of the robot. In fact, the configuration of the robot (concentrating the mass of the system in its hip and feet) and the ratio β=m/m (where m is the foot mass and M is the hip mass) affects the equation of motion, but if m is chosen much smaller than M, then the term β can be removed from the equations. 6 Control Engineering

14 P Passive bipedal walker over slope 2.3 How to model the passive walker To model the passive walker over slope, 20-sim software (Controllab Products B.V. 20-sim version 4.1., 2010) and its 3D mechanics toolbox was used to implement the model using the port-based approach sim and the 3D Mechanics Editor 20-sim, developed by Controllab Products B.V., is a very powerful modeling and simulation program which contains different modeling libraries, such as block diagrams, mathematical equations and bond graph elements. The energy-based bond graph approach makes it easy to model modules in different domains (electrical, mechanical, thermal, etc). Each module can be implemented in its own domain and be connected to other modules via exchanging energy ports. So, future extension of the model will be easy. The 20-sim library models are open for users to inspect the equations and make required changes. 20-sim offers different features, among them are: 3D mechanics toolbox, animation toolbox, import/export from/to MATLAB, variety of numerical integration methods. 3D mechanics toolbox provides the user with an editor to easily construct the dynamical model of multi-body systems. The rigid bodies can be interconnected by different types of translational and rotational joints. For each body the physical properties including mass, center of mass and moment of inertia can be set. Also, the representation of each of the bodies (shape, size and color) body can be chosen independently of their physical properties. Table (2.1) contains the physical properties of the model of the passive walker constructed in 3D mechanics toolbox. Physical properties of the model Component Mass (kg) size (m) Inertia (kg m ) Hip Leg (length) 10 Foot (radius) 6.10 Table (2.1): Physical properties of the passive walker The 3D Mechanics Editor generates an equation sub-model to be imported in 20-sim. The equations describe the interaction between the different bodies in the model using screw theory (Stramigioli & Bruyninckx, 2001). By default; gravity is the source of possible forces/ torques acting on the system in 3D mechanics toolbox. External forces are needed to be calculated in separated sub-models and interact with the 3D sub-model via ports of the model. To assign ports to the model constructed in 3D mechanics toolbox, actuators can be assigned to the bodies; also there is the possibility to define the system joints as a port. Figure (2.3) shows the block diagram of the implementation of the model in 20-sim. The block diagram consists of two blocks that are connected in a port-based configuration. The dynamics of system is generated by 3D-mechanics toolbox. The interaction between the foot and the ground is a separated sub-model contains the required mathematical equations that have been formulated using the concept of compliant contact model (explained in section 2.3.2). University of Twente 7

15 Variable stiffness for robust and energy efficient 2D bipedal locomotion Model of the walker in 3D-dynamics toolbox of 20-sim Lie-Poisson Reduction: (P ) = adj, (P ) + (W ) v = (Mass Matrix) P Where P is the momentum of the system, W is the wrench acting on the system due to the gravity, T is the twist of the system components, v is the generalized velocity vector of system joints. Interaction between the foot and the Ground Figure (2.3): The block diagram of the implementation. Description of the model in 3D Mechanics toolbox Figure 2.4 shows the construction of a 2D passive walker in 3D mechanics toolbox. The model is composed of rigid bodies (hip, legs, and feet), two translational joints and two rotational joints. The feet are rigidly connected to the legs using weld joints. For the analysis of applying compliancy to the legs, weld joints can be replaced by z-translational joints (explained in chapter 3). The motion of the hip mass is constrained to X and Z directions by being connected to two translational joints. Each leg is connected to two joints: a Y-direction rotational joint and a weld joint to its foot. So, the motion of the legs is constrained to a rotation around the axis (Y-axis) perpendicular to the plane of hip motion (X-Z plane). The reference frame (world coordinate frame) was assigned to the ground level and all the system equations were expressed in this frame. 8 Control Engineering

16 P Passive bipedal walker over slope Figure (2.4): The model of a 2D passive walker in 3D mechanics toolbox of 20-sim. Slope For animation purposes, the slope is modeled as a body that is hinged to the ground. By changing the hinge angle, slope angle can be set. Figure (2.5) shows the animated model of the walker built by 20-sim animation toolbox. Figure (2.5): The animated model of the walker built by 20-sim animation toolbox. Feet The feet are modeled as small masses that their center of mass is rigidly connected to the end side of their related leg. In the implementation of the contact model (the block shown in Figure (2.3)), radius of the feet can be set as a parameter. By setting the radius to zero, point feet are obtained and by setting the radius bigger than zero arc feet are resulted (similar to the feet shape of the passive walker by McGeer (McGeer, 1990)). Arc feet only have one contact point with the ground similar to the point feet in the model of the simplest passive walker, represented in the section 2.1. To model the arc foot, the radius of the foot was added, as a z-position in the direction of the z-axis of the world coordinate frame, to the z- position of the end side of the leg that is rigidly connected to the center of mass of the foot. This University of Twente 9

17 Variable stiffness for robust and energy efficient 2D bipedal locomotion implementation is equivalent to realization of arc feet. Figure (2.6) shows the configuration of the modeled passive walker. The rigid lines represent the legs, the vertical line represents the foot radius (that is added to the z-position of the center of mass of the foot), and the dashed lines represent the equivalent realization of the model with arc feet. Reasons behind modeling the arc feet in this study are: the robustness of the passive walker will be improved (the chance of falling will be decreased) (Wisse et al.,2006), the implementation of the contact model with these feet (since they have just one contact point with the ground) is as simple as the point feet. Although in some research studies (Wisse et al.,2006) has been shown that the bigger foot radius of an arc foot the better the robustness, but since in this study we are interested to verify the role of applying compliancy to the legs, a fixed foot radius has been used in all analysis. The foot radius was chosen based on the proportion of foot-length to leg-length in the human body s scale. For a leg length of 0.7m, a foot length of 0.1m (diameter of the foot) sounds comparable to human leg/foot size. Figure (2.6): The configuration of the model with the arc-feet. A brief explanation about implementation of contact model in 20-sim The interaction with the ground block, shown in figure (2.3), is composed of two other blocks: the coordinate and the floor. The block coordinate assigns a coordinate system to the bottom of the foot (contact point). Since, an arc foot is assumed, with a small radius, all points on the surface of the foot have approximately the same position. This block assigns a coordinate system (homogenous matrix,h ) to the contact point with the orientation of the slope (H ) and position of the bottom of the foot (H ) with respect to the world coordinate system that is assigned to the ground level. The block floor is responsible for recognizing the contact; calculate the contact wrench and applying the wrench, a 6- dimentional variable representing forces and torques, to the center of mass of the foot. Figure (2.7) shows the block diagram of the implementation of the contact model in 20-sim. 10 Control Engineering

18 P Passive bipedal walker over slope H W H H Coordinate Floor Figure (2.7): The block diagram of the implementation of the contact model in 20sim. H As soon as the z-position of the foot and the slope with respect to the world coordinate becomes equal, the contact of the foot with the slope will be started. However, here, the foot scuffing and the impact should be distinguished. As explained in section (2.1), foot scuffing is expected in the model because the walker does not have a knee, consequently during the swing phase it will go through the surface of the slope. In real-world tests, foot scuffing can be avoided by putting some tiles on the surface of the slope. But, in the simulation, foot scuffing should be ignored. Fortunately, according to the configuration of the system, if there is a stable gait, the foot never starts to hit the ground after mid-stance phase, unless there is the end of the step. So, an algorithm can check if there is a strike happened and if the swing leg is in front of the stance leg. With these two conditions, the contact wrench (representing the normal force and friction) will be exerted to foot. The following figure shows the configuration of the blocks. The related codes can be found in Appendix A. figure (2.8) shows the flowchart of the algorithm to distinguish foot scuffing from the impact. Calculate the z-position of the foot w.r.t. the coordinate system assigned to the slope at the point with the same position of the foot (the possible contact point) No If the foot passed the slope level? Yes If the foot is in front of the other foot? No Foot scuffing Yes Impact Figure (2.8): The flowchart of the algorithm to distinguish foot scuffing from the impact. University of Twente 11

19 Variable stiffness for robust and energy efficient 2D bipedal locomotion Impact of the foot with the ground Impact refers to the collision of two bodies during which a relatively large contact force will be exerted over a small period of time on the bodies. The impact process involves different phases including the material deformation and recovery as well as heat generation and energy loss. After the initial contact of two objects, the deformation phase will be started. This phase continues up to the moment that the increase in the contact area is stopped; in the other words the penetration depth reached its maximum value. From this moment, a period of restoration will occur during which the contact area will be decreased to zero. Afterwards, each of the objects starts to move with the new velocities. Impact is a complicated phenomenon and it is dependent on several factors. The aim of modeling the impact is to calculate the force exerted on the bodies and the relative motion of contacting bodies. Elasticity of an impact is dependent on the amount of energy loss. In a completely elastic impact, no energy will be lost and the total kinetic energy of the system (including both objects involved in the collision) is conserved. On the other hand, an impact is called completely plastic, if the whole amount of the kinetic energy is lost during the process. Generally there are two approaches towards modeling the collision of two rigid bodies: rigid and compliant model. Taking the rigid contact approach, the impact is modeled as an impulsive force that has an infinite value during an infinitely short period of time. Rigid model represents the mathematical description of the evolution of the system s state after an impact applying some constraints. For a passive walker, a rigid contact model implies an instantaneously double stance phase. It means as soon as the new stance leg hits the ground, the previous one leaves the ground without having impulsive reaction with the ground. Also, it is assumed that while the impact is taking place all the positions (so the mass matrix that depends on the system s configuration) and non-impulsive forces of the system remain constant and there is an instantaneous dissipation of energy. According to the description of the model (section 2.1), the advantage of a rigid contact model for a passive walker is the representation of a fully plastic and instantaneous impact. In this case, analysis of the model is simple. On the other hand, the implementation of the rigid contact model requires direct changes in the equations of motion. In this project, 20-sim software is used to model the passive walker and as mentioned in the previous section, the dynamics of the system is generated by the 3D-dynamics toolbox of 20-sim. So, acting on the generated codes leads to destroy the modularity of the model that is not generally desired. (Duindam, 2006), (Oort, 2005), (Garcia et al., 1998).The other approach to model the contact between two rigid bodies is the compliant model. The compliant model describes the impact phenomenon as a result of changes in the deformation and compression of the surfaces of the bodies due to their elasticity and damping properties. This model determines the contact force by assumption of small deformation of body surfaces (comparing to the size of the bodies) (Duindam, 2006), however this model is capable to describe the interaction between two rigid (very stiff) bodies where the deformation of the bodies is negligible. In addition to the conformity to the nature of impact, the implementation of the compliant model is much simpler than the rigid model. The related module (codes) can be connected to other modules of the model with a port based configuration. However it is more difficult to analyze the model of a simple passive walker using the compliant contact model (as the interaction of the foot with the ground) due to the noninstantaneous double stance phase. So, some consideration should be taken in to account: - As mentioned in the description of the model, the double stance phase is assumed instantaneous in the basic model, however here because of the assumption of compliancy of the bodies (the feet and the ground), the double stance is a short while in which both of the feet have penetrated into the ground, one of them is about to stay in contact with the ground (the new stance leg) and the other 12 Control Engineering

20 P Passive bipedal walker over slope one is going to leave the ground (the new swing leg). Of course, the stiffer the ground, the smaller the double stance period. Actually, this problem can be overcome by a wise description of system s state during the double stance phase (just for analysis purposes as described in the section 2.5) as well as an appropriate choice for the stiffness and damping coefficients of the contact model (explained later in this section). - To approach a rigid (plastic) contact using compliant model, the bodies in contact (the leg and the ground) should be very stiff. However, increasing the stiffness of the model leads to a faster dynamics of the contact comparing to the rest of the system. So, the integration method should be capable of handling a various range of speeds. This issue will be more prominent when we apply different ranges of stiffness to the model to pursue the main purpose of the project (applying variable stiffness). In this project, we chose the compliant contact model to represent the ground interaction because it is simpler to implement it and the model will have a modular structure. Also, comparing to the rigid model, the compliant model is closer to describe the process of impact in the real world. The compliant contact interaction with the ground may have different components. Here, because of the assumption of a 2D walker with the arc feet that are rigidly connected to the legs, the contact force has just two components: penetrating (normal force) and slipping (friction). The following section is to describe the model of these two forces. Collision and Hunt-Crossley model One way to calculate the penetrating component of the compliant contact force is to model the process as a parallel spring and damper (the Kelvin-Voigt model): F(t) = K z(t) + B z (t) z 0 0 z < 0, Eq. (2.4) where K is the spring (elastic) constant, B is the damping constant and z is the penetration depth. The pitfall of the model using a linear spring and damper is the so called sticky effect. As mentioned before, during the impact the process of deformation and recovery happens. According to the equation above, when the penetration depth ( z ) is very small (at the start of the deformation phase and the end of the recovery phase), the contact force is mainly dependent to the damping property of the model that is not related to the penetration depth in this model. As a result an un-natural force will be applied to the bodies at the moment of impact and the removal (Diolaiti, Melchiorri, & Stramigioli, 2005). To overcome the above problem, the Hunt-Crossley model represents a solution by making the damping coefficient dependent on the penetration depth: F(t) = K z (t) + λ z (t) z (t) z 0, Eq. (2.5) 0 z < 0 where n is a real number, usually close to unity. According this model, a continuous function represents the relation between the penetration depth of the contact bodies and the resulting force that is exerted on the bodies during contact of the bodies (foot and the ground). University of Twente 13

21 Variable stiffness for robust and energy efficient 2D bipedal locomotion Stiffness and damping coefficients As explained before, the choice of stiffness and damping coefficients has an important role in modeling a plastic collision, analyzing the model and avoiding numerical calculation problems. For a non-elastic collision of two rigid bodies when the deformation of the bodies after impact is negligible, the stiffness of the colliding bodies should be very high and also the damping property of the bodies should be high enough to dissipate almost the whole kinetic energy of the foot when it collides with the ground. Figure (2.9) shows the penetration depth of one of the legs during its stance period shown by a rectangle. At the moment of impact the penetration depth is zero, then it reaches its peak value and converges to the steady state condition in which the velocity is negligible. Assuming an appropriate value for the steady state penetration depth, the stiffness of the compliant contact model ( K in the Hunt-Crossley model) can be determined. In steady state (z (t) = 0) from equation (2.5), we have: F (t) =K z (t) and also since F (t) represents the normal force applied on the robot, so: F (t) = (M+2m) g, where M is the hip mass and m is the foot mass. Estimating the proper steady state penetration depth, the stiffness of the model can be calculated: K = ( ). Eq. (2.6) For example, regarding to the size of the foot of our model a steady state penetration depth in the range of (10 m) represents a high stiff collision, using the equation (2.9), K= 10 N/m is chosen for the contact model. Figure (2.9): The plot of the penetration depth for one of the legs during its stance phase. To determine the damping coefficient, the criteria for a critically or over-damped linear system composed of a parallel spring damper connected to a mass, figure (2.10), is used to estimate the range of the damping ratio: ζ = 1, Eq. (2.6) Figure (2.10): The configuration of a linear parallel spring-damper attached to a mass. where m is the mass of the body connected to the spring-damper (here, the total mass of the system, M+2m, is considered), D is the damping coefficient and K is the spring coefficient, and ζ is the damping 14 Control Engineering

22 P Passive bipedal walker over slope ratio. Although Hunt-Crossley model represents a non-linear model in which damping coefficient is dependent on the penetration depth: D = λ z, λ can be estimated by assuming a very small and constant value for the penetration depth. For example, taking the values of z and K (explained in the calculation of K), for K=10, the damping ratio should be bigger than10. To optimize the damping ratio, the measure of obtaining the smallest damping ratio while achieving a plastic impact is taken. The reason behind being interested in an optimal value for damping ratio is to avoid having unnecessary big numbers in the simulation and prevent hardships in analysis of the system while applying compliancy to the legs (described in chapter 3). The dissipated power during impact can be calculated by: P(t)= F(t)* v(t), where P(t) is the exchanged power during the impact, F(t) is the impact force calculated using Hunt-Crossley equation, v(t)= z (t) is the penetration velocity. The dissipated energy ( H) was calculated by integrating the plot of P (t). Figure (2.11), shows the plot of F-x and power exchange according to Hunt-Crossley model. In the following plot, z refers to the maximum penetration depth, t is the time of the deformation (compression phase) of impact. At t the bodies are separated (Diolaiti, Melchiorri, & Stramigioli, 2005). Figure (2.11): the plot of F-x and power exchange for Hunt-Crossely model. Source: (Diolaiti et al., 2005) H and H are the exchanged energies during the compression and recovery phases (described before). The exchanged energy can also be expressed according to the stored and dissipated energy due to the elasticity and damping properties of the contact model. H = kx(t)x (t)dt, H = λ x(t)x (t)dt Eq. (2.7) H is the stored elastic energy and H is the dissipated energy because of the damping term. The exchanged energy due to impact is the summation of these two values. H=H +H =H +H Eq. (2.8) To have a plastic impact the dissipated energy should be very close to the kinetic energy of the body before the impact, so H should be as large as possible. To determine the damping coefficient, the criteria of finding the minimum value of λ that maximizes the absolute value of the dissipated energy ( H) was taken. The test was done by multiple-run tools of 20-sim software. For K=10 N/m, λ= N.s/m ( ) was obtained. Figure (2.12) shows the plot of the dissipated energy per step over a range of λ (N.s/m ) for a slope angle rad. The system shows an unstable behavior for the values of λ that are smaller/bigger than the range showed in the plot. It should be noticed that although the difference between the amounts of the dissipated energy (for different values of λ) is small, the difference between their related λ is big. Also, we expect that increasing of the value of the damping constant leads to an increase in the dissipated energy, but it should be noticed that λ is not the damping coefficient, but λ x. University of Twente 15

23 Variable stiffness for robust and energy efficient 2D bipedal locomotion Figure (2.12): The plot of the dissipated energy due to impact over a range of λ. Friction The friction of the foot with slope (ground) during the contact prevents the foot from slipping on the slope. Having a 2D model in which the feet are rigidly connected to the legs, friction is a translational force along the slope. To model the friction the following formula, that is also the SCVS module in 20-sim, is used. SCVS includes all kinds of friction models (static, viscous, coulomb, stribeck): F = F. ( μ + (μ. abs(tanh(slope. v)) μ ). e. sgn (v) + μ. v, Eq. (2.9) where, F is the normal force, μ is the coulomb coefficient, μ is the static friction coefficient and μ is the viscous friction coefficient. Friction coefficients were estimated by some experimental tests and guesses, for example the static friction coefficient can be determined by estimating how much force is required for a 3 Kg object to start to move on a surface. The following values were set as the friction constants in the model: μ = 0.25, μ =0.5, μ = In equation (2.9), Slope refers to the steepness of the coulomb friction curve Compliant model and analysis of the passive walker In section (2-1), the properties of the basic model (the simplest passive walker) were described assuming a rigid contact with the ground. Now, taking a new approach to model the interaction with the ground, there is a review to the system properties according to the compliant contact model: o o Loss of energy due to impact: By choosing the proper stiffness and damping coefficients, a plastic impact can be achieved. Double stance phase: the double stance phase is not instantaneous, for a very short while both feet are on the slope, however the stiffer the ground, the shorter the double stance phase. 16 Control Engineering

24 P Passive bipedal walker over slope Also, the strategy for switching the role of the legs was defined considering the fact that the double stance phase is a short while due to the property of the system with the compliant contact model. In the basic model as soon as the condition φ (t) = 2θ(t) becomes satisfied, the role of the legs are exchanged. Using the compliant model, however both of the legs are in contact with the ground for a short while, but the condition φ (t) = 2θ(t) just happens when the penetration depth of the legs with the slope is the same. Figure (2.13), shows the plots of θ, φ for a slope angle γ = The plot is symmetric and the condition φ (t) = 2θ(t) holds at the start of each step. Figure (2.13): The plots of θ, φ for a slope angle γ = with the compliant contact model. In the section 2.1, the phase plot of θ; θ for the simplest passive walker (the basic model) was shown. Figure (2.14), shows the plot of θ; θ during one step. As it can be seen, there is a discontinuity in the velocity of the stance foot at the moment that it hits/leaves the ground. The discontinuity upon hitting the ground is much bigger than the other, and also for the stiffer ground, the discontinuities are bigger. θ θ t=t t=0 Figure (2.14): The phase plot of θ; θ with the compliant contact model. University of Twente 17

25 Variable stiffness for robust and energy efficient 2D bipedal locomotion 2.4 Energy Analysis The passive walker over a slope is just energized by the gravity. During each step, the portion of the kinetic energy that is supplied by the gravity is lost at the end of each step, when the impact of the foot with the ground happens. From the dissipative system s theory, the dissipation inequality is: V(t ) < V(t ) + s u(t), y(t) dt, Eq. (2.10) where V(t ) is the system energy at time t, and the integral represents the externally supplied energy (here, supplied by the gravity). V(t ) is the initial energy of the system due to the initial conditions. The system is under-actuated and the loss of energy due to the impact at the end of each step maintains the passivity of the system and makes it stable. The kinetic co-energy of the system is the sum of kinetic co-energies of each of the system s components. The total kinetic co-energy of the system can be expressed by U = (q M(q) q ), where q is vector of the vector of the velocities of the system s components and M(q) is the mass matrix of the system that is a symmetric matrix informing the mass and the configuration of the system components at each time. The potential energy of the system is simply the potential energy due to the gravity: U = M g z + m g z +m g z where M is the mass of the hip, m is the foot mass, and z is the z-position of each of the masses with respect to the world coordinate. If the walker walks stably, the dissipated energy due to impact (the loss of energy due to friction is ignored) is equal to the kinetic energy of the system provided by gravity during the step. The value of the dissipated energy is the dissipated energy ( H) calculated in the section (2-3). Figure (2.15) shows the dissipated energy of the system as well as the potential, kinetic and total energy of the system for some continuous steps. The amount of difference between the levels of total energy at the start of each step is equal to the loss of energy due to the impact (showed by an arrow). Because of loss of energy at each step, the system is dissipative and is not conservative. In the next chapter, a passive storage element will be added to the leg. The comparison of these two cases is reported in chapter Control Engineering

26 P Passive bipedal walker over slope Figure (2.15): The dissipated energy, the potential, kinetic and total energy of the system for some continuous steps. University of Twente 19

27 Variable stiffness for robust and energy efficient 2D bipedal locomotion 2.5 Stability Analysis Generally speaking, walking has a periodic nature. As described in the introduction, walking robots can be divided to two categories: static and dynamic walkers. Here, we focus on the analysis of the motion of a passive dynamic walker. If a passive walker over a slope manages to continue walking without falling down, it means that it is repeating a pattern; in the other words the state of the system starts to repeat itself after some steps. So, each of the system variables has a cyclic behavior. This cyclic behavior is called a limit cycle. Limit cycle is the periodic response of the nonlinear differential equations of the system. By definition a limit cycle is an isolated closed trajectory. Isolated means that other neighboring trajectories either diverge from or converge towards it. Limit cycles can just occur in nonlinear systems because for a linear system if x (t) is a response, then c* x (t) is a response too, so the periodic response will not be isolated (Strogatz, 1994) Stability of a limit cycle walker Definition of stability for walking robots can be as general as not falling down criterion. To verify the stability with this criterion, since simulation/ practical experiments cannot be done for infinite number of steps, we should be more detailed in the definition, for example, not falling down for 80 steps. But nonlinear systems can be very sensitive to the changes in their initial condition, against linear system for which small disturbances in the initial conditions of a stable linear system results in small changes in the output (Schaft, Scherpen, & Jeltsema, 2009). So, if by slightly changing the successful initial conditions, the walker cannot manage to prevent falling down, what can we conclude? Can we be sure about our conclusion about the stability of the system because of not falling down for just 80 steps? One of the solutions to determine the stability can be the linearization of the system equations around an operating point. But linearization can just predict the local behavior of the system. The other solution is to study the stability of the response of the system. As mentioned above, a dynamic walker has a cyclic behavior. The response of the system is a limit cycle. Definition of stability for a limit cycle indicates: If all trajectories in a neighborhood of a limit cycle approaches to it, we call it a stable or attractive limit cycle, otherwise it is unstable from which all neighboring trajectories from both sides are diverging. Figure (2.16) shows a stable and an unstable limit cycle. Figure (2.16): stable and un-stable limit cycle. 20 Control Engineering

28 P Passive bipedal walker over slope In dynamic walking, the definition of the stability of the system is expressed in terms of the stability of its limit cycle. Limit cycle walking is a nominally periodic sequence of steps that is stable as a whole but not locally stable at every instant of time (Hobbelen, 2008). According to the research done for a passive dynamic walker, with the explained configuration, there is one stable limit cycle in for each slope angle in a range of gentle angles (Garcia et al.,1998). Figure (2.17) shows the plots of limit cycles of the modeled passive walker for its variables, θ and φ as, well as the plots of θ and φ over the time for a course of steps with the slope angle is γ=0.012 rad. The horizontal lines in the limit cycles and vertical lines in the time response of θ and φ indicate the moment of the heel strike. Figure (2.17): Limit cycles and plot of θ and φ for a period-one gait with slope angle γ=0.012 rad Symmetric and asymmetric gaits A periodic gait, in which the system states repeat themselves after each p steps, can be symmetric or asymmetric. A gait is symmetric if any two consecutive steps are indistinguishable, that is, all the spatiotemporal variables exactly repeat themselves in each step (Goswamiy et al., 1998).So, for a symmetric gait, p=1 (the so-called period one gait) and for asymmetric gait p>1. For the modeled passive walker with the properties mentioned in the section 2.4 (size, weight and contact model s coefficients), the period-one gait is seen for the slope angles 0<γ< rad. For steeper slopes higher-period gaits are recognized. This result is comparable with the result obtained by other studies (Garcia et al.,1998) that showed higher-period gaits for slope angles steeper than rad. Figure (2.18) plots show the limit cycles and plot of θ and φ for a period-two gait with slope angle γ=0.018 rad. Comparing to the plots of figure (2.15), that represents a period-one gait, the gait is not symmetric for this slope angle. University of Twente 21

29 Variable stiffness for robust and energy efficient 2D bipedal locomotion Figure (2.18): Limit cycles and plot of θ and φ for a period-two gait with slope angle γ=0.018 rad Verifying the stability of the passive walker To find a limit cycle, the initial conditions of the model (positions and velocities) can be varied by trial and error till the walker shows a not falling down for couples of steps behavior. Stability of the limit cycle can be determined by calculating the eigenvalues of the linearized system. Before explaining the linearization, some definitions are given: Poincaré first return map: Suppose that the trajectory L is a response for a nonlinear system, and l is a lower dimensional sub space than the dimension of the system s space state, called the Poincaré section, transversal to the solution. Suppose that L intersects l at point S when t=t and intersects it again at S when t=t >t. There is a vicinity of S such that solutions that are originated from it intersect l another time, the map F: l l, S =F (S ) is called the Poincaré first return map. Figure (2.19) shows an impression of the above definition. Suppose trajectory L intersects l in the points: S, S, S,.. S Where S =F (S ), S =F (S ) S =F (S ). Then the trajectory L is a limit cycle if lim S =S*, F(S*) =S*. (Shiriaev, 2009) 22 Control Engineering

30 P Passive bipedal walker over slope Figure (2.19): The impression of Poincaré first return map. Stride function: The Poincaré map for the passive biped is termed stride function by McGeer. A step can be considered as a map (stride function) that takes the states of the system just after the heel strike and gives the state of the system just after the next heel strike. Fixed point: The gait limit cycle is the fixed point (S*) of the stride function: F(S*) =S* Numerical analysis To verify the stability of the limit cycle, the linearization of the Poincaré map at the fixed point, that is located on the limit cycle and is the initial state of the gait, can be used. In this approach, a small perturbation is applied to the fixed point and the system response will be evaluated. If the states of the system converge to the fixed point, it means that the error caused by the perturbation shrinks and the limit cycle is stable. On the other hand, if the error grows, the limit cycle will be unstable. The equation for the error is: ε =. p, Eq. (2.11) where ε is the error of the system s response with respect to the fixed point after applying the disturbance with the magnitude p. The error is measured at the first heel strike after perturbing the system. Since the system has more than one state, the partial derivative will be a matrix, called Jacobian of the map. The Jacobian can be numerically found by perturbing the state variables one after the other and its eigenvalues determine the stability of the system. If the magnitude of an eigenvalue is larger than one the error will grow and the system is unstable. So for a stable system all magnitudes of the eigenvalues have to be smaller than one (inside the unite circle). As mentioned before, θ; θ ; φ; φ are the independent variables (states) of the system. However, in the numerical calculation of eigenvalues, the state of the system contains θ ; φ; φ. As mentioned in the description of the model, just after the heel strike φ and Ѳ are linearly dependent. The reason behind choosing φ as the independent position variable of the system (instead of Ѳ) just after the heel strike can be intuitively explained considering the fact that the double stance phase is not instantaneous with the compliant model. For a short period of time, there are two stance feet in the sense that the velocity of both of the feet are quite small and they are both in the contact with the ground. So, in this case, the angle between two legs always can be clearly defined, but the angle of the stance leg with the ground is not unique, because both of the legs are in contact with the ground. Table (2.2) shows the calculated eigenvalues for some slope angles. Results are calculated using BDF numerical method, with absolute and University of Twente 23

31 Variable stiffness for robust and energy efficient 2D bipedal locomotion relative tolerance of 10. The calculation of eigenvalues was done for a range of slope angles 0<γ<0.018 rad and it shows that there is a stable limit cycle for this range that contains both period-one (0<γ< rad) and period-two (0.0157<γ<0.018 rad) gaits. These limit cycle are stable and there is an area containing all the neighboring trajectories converging to these limit cycle, with at least the size of the perturbation used to calculate the Jacobian (here, p=0.001). The smaller the eigenvalues, the faster the neighboring trajectories converge toward their limit cycle. As reported in table (2.2), for all of slope angles there is a zero eigenvalue related to the angular velocity of the hip (φ ). The angular velocity of the hip at the start of each step is very tiny; if the perturbation is small enough such that the walker manages to start another step afterwards, φ will have a very tiny value again as if it converges to its nominal value very fast. Comparing to the results of other studies, the same trend in the increase of the magnitude of the eigenvalues was reported (Wisse, 2004), (Garcia et al.,1998). However, instead of the zero eigenvalue, the values for the two other eigenvalues are different in various studies that can be a result of small differences in the models of the system. Slope angle Eigen values of state space: θ; φ; φ γ = [ ] γ = [ ] γ = [ ] γ = [ ] Table (2.2): Eigenvalues of the linearized 2D passive walker for different slope angles. 24 Control Engineering

32 P Passive bipedal walker over slope 2.6 Robustness Analysis In practice, stability is a vulnerable concept; it should be supported by robustness. Disturbances are always present in all environments, so the robustness of the system has a significant importance. For a walking robot, a rough terrain is an example of the cause of disturbances that can disturb the positions and velocities of the system. A passive walker is also very sensitive to the changes in its initial conditions. Definition of stability (section 2.5) of a passive dynamic walker is based on the criterion of converging to a limit cycle. As a result, the definition of the robustness for this system can be how big is the area, which contains the converging trajectories, around the stable limit cycle. On the other hand, some of these possible converging trajectories are more probable to happen in the reality. So, the definition can be limited to the the biggest applied disturbance that the walker s motion will still converge to its stable limit cycle for a range of most probable practical disturbances. In this section robustness measures and their results are discussed Measure of the robustness To study the robustness of the passive walker, an appropriate measure to analyze the effects of the disturbance on the stability of the robot should be taken. In this section, some known measures for the robustness (Hobbelen & Wisse, 2007)of passive walkers are discussed. Basin of attraction By definition, the basin of attraction (BOA) of a fixed point is the intersection of the Poincaré section with the vicinity around the limit cycle (the BOA of the limit cycle) which contains all of the converging trajectories towards it (Wisse, 2004). Poincaré section refers to a lower dimensional sub space than the dimension of the space states. For a dynamic walker this section is defined at the moment of the heel strike, since the system can be described by two state variables at this moment, the dimension of the Poincaré section is two and the BOA contains the all possible states of the system (θ, θ at the start of each step) that if the walker starts to walk with them, its trajectory will converge to the stable limit cycle over the course of some steps. Figure (2.20), shows the phase graph of the walking motion including the Poincaré section and basin of attraction of a fixed point. This phase graph is shown in a stylized 3D; however the real dimension of the system is four (Wisse, 2004). Figure (2.20): The 3D phase plot of the walking motion including the Poincaré section and BOA of the fixed point, Source: (Wisse, 2004). University of Twente 25

33 Variable stiffness for robust and energy efficient 2D bipedal locomotion One of the methods to obtain the basin of attraction around a fixed point (although all points on the limit cycle are the fixed points, as mentioned before in dynamic walking the initial condition of the step is the fixed point that we refer to it) is to vary the initial conditions to verify if the system trajectory will converge to the stable limit cycle or not. Figure (2.21) shows the basin of attraction of the passive walker over the slope angles of γ=0.007 rad. The plot shows the range of the variations in the states, θ, θ around the fixed point. The small dots show the variations that their related trajectories diverge from the stable limit cycle, and the starts show the variations that their related trajectories converge towards the stable limit cycle. The fixed point is marked by a red star inside of the rectangle. The values are scaled by θ =0.4 and θ = 0.8. (θ) =, (θ ) = Eq. (3.1) Figure (2.21): The BOA of the passive walker over the slope angle of rad. Step-down One of the practical measures of the robustness is the ability of the robot to manage to continue its stable gait encountering a lower level of the ground. The measure is the largest step-down size (difference between levels of ground) that the robot can handle and converge to its stable walking pattern after couples of steps. Figure (2.22), shows the step-down experiment for different slope angles. The stepdown size is given some discrete values. As it can be seen from the plot, for angles steeper than rad, which mainly represent the period-two gait, the robot may handle the bigger steps, but they may not manage small steps. The step down results shows an improvement in the robustness of the system by increasing the slope angles in the range of 0<γ<0.012 rad. The system in this range of slope angles shows a peiod-one gait. 26 Control Engineering

34 P Passive bipedal walker over slope Figure (2.22): The Step down result for the passive walker over a range of slope angles. Floquet Multiplier One other measure for the robustness that has been studies in some of the research work (Hobbelen & Wisse, 2007), in this field is the biggest stable eigenvalue (Floquet Multiplier) of the linearized system as explained in 2.5. The measure says the bigger eigenvalue, the less the robustness Comparison of robustness measures In this section three measures of robustness are compared together in order to choose one of them as the base of our analysis. Basin of attraction is the more comprehensive measure and it is correlated with the real world experiences. The problem of this measure is that it needs huge computations that need a long time. On the other hand comparing two basins of attraction is a bit tricky. The basin of attraction of the passive walker is very narrow and also it does not have a simple symmetric shape. One of the ways is to calculate the area of the basin of attraction but in this case it does have no information about the deviation of different states from the fixed point. So, the distance of the fixed point from the border of the BOA can give a better measure for the robustness. But, the problem is that we will obtain many numbers (distance from the fixed point to the boundaries in different directions), and then the question is how can we compare the rows of numbers for two BOAs? because the largest/smallest distance from the fixed point can happen in different directions for two different systems (BOAs). It should be noted that nondimensionalizing of the axes (θ, θ) should be done in order to be able to compare the distances from the fixed point. Using the criteria of how far is the fixed point with respect to the boundaries of the BOA, one measure can be to assign a circle around the fixed point and with the center of the fixed point and obtain the biggest possible radius of it. It can be a good measure, but it depends to our definition of robustness for a system, if we are interested in symmetric deviation from the fixed point, or it is important to know in which direction it grows more. For example, considering the BOA plot for two angles (0.007 and rad) shown in Figure (2.23), the BOA of rad is a little bit more symmetric around the fixed point, but it is more narrow. From the results of step-down, we know that rad can handle larger University of Twente 27

35 Variable stiffness for robust and energy efficient 2D bipedal locomotion disturbances in the form of step-down, so drawing conclusions from the BOA plot is not simple, because it contains all possible information and one should be clear about the specific definition of the robustness in order to use BOA as a basis of the comparison. Although calculating the eigenvalues can be done fast comparing to the two other measures, it does not have a proper relevancy with the reality. The results of the step-down experiment and Floquet multipliers lead to two different conclusions. For example, although the largest eigenvalue for slop angle rad is that is smaller than the same measure for slope angle rad, which is , but according to the step down result, the maximum step down for γ=0.007 rad is less than the maximum step down for γ=0.012 rad. In fact, Floquet multipliers just tell how fast the response will converge to the stable limit cycle. This measure is useful to verify stability (max λ <1) of the limit cycle and in most of the cases it fails to provide valid information about the robustness. The step down is a measure of the category of the largest allowable deterministic disturbance ; it has a good relevancy to the real world and also it takes less calculation time comparing to the BOA. In this project, we mainly discussed the results taking this measure. Figure (2.23): The BOA for two slope angles. 28 Control Engineering

36 P Passive bipedal walker over slope 2.7 Nominal model As described in 2.2, by nondimensionalinzing the equation of motion, the model has only one free parameter, the slope angle. Since the aim of this project is to verify the effects of applying elasticity to the model, we should fix the other free parameters. So, a fixed slope angle, γ=0.009, was chosen as the nominal model that will be the base of our conclusions. The system with this slope angle shows a period one stable gait, and also it is in the range of angles that show a comparably robust behavior according to the step-down result. Table (2.3) shows the properties of the modeled passive walker over the slope angle rad. Size and weight of the model Hip mass Leg mass Foot mass Leg length Foot radius 3 Kg 0.01 Kg 0.1 Kg 0.7 m 0.05 m Compliant contact model parameters Stiffness Damping constant 10 N/m N.s/m Gait Characteristics Step period Step Length Velocity Maximum Ѳ Maximum φ Dissipated energy per step Maximum step down s m m/s rad rad 0.08 J 2.4 mm Table (2.3): Properties of the passive walker over the slope angle rad. University of Twente 29

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38 Chapter 3 Applying compliancy to the model Although a passive dynamic walker can show a human-like gait, there are two main issues that should be studied: cost of energy and robustness. Like all kinds of walking robots, it should be energized, in the model used in this study, and many other studies, the robot is energized by the gravity. But if it walks on a flat surface it should carry its energy source that is, despite the gravity, a limited source. So, it is important to find a solution to decrease the cost of energy, for example by temporarily storing some part of the kinetic energy, which is normally dissipated during walking due to the impact, and returning it to the system. This is the motivation behind applying passive storage elements (springs) to a walking machine. The other critical issue is the capability of the robot to handle the environmental disturbances. So, there is an interesting question here, if we manage to decrease the cost of energy of a passive dynamic walker by applying compliancy (stiffness actuation) in the model, what will happen to the robustness? As studied in the previous section, robustness of a passive walker is narrow and should be improved. In this chapter, first there is a brief refer to the previous studies in this field, then the effects of applying compliancy along the legs of the passive walker will be discussed. So far, some studies have been done in this field, among them; here there is a brief refer to the results of studying the effect of applying compliancy on the hip/ankle joint done by Wisse and Hobbelen (Hobbelen, 2008). Figure (3.1) shows the configuration of the model of the passive dynamic used in these studies. A torsion spring applied to the hip (left side picture) and to the ankle joint (right-side picture). Studies on adding a spring in the hip showed improvement of the robustness by increasing the hip stiffness. This result was reported by changing the stiffness while changing the slope angle to provide the same velocity for all experiments. The other study on adding a torsion spring in the ankle joint of the robot (flat foot shape was assumed), the stiffness was changed piece-wisely (a lower stiffness for landing and a higher for push-off) and the result shows no improvement in the robustness however it has not been decreased too. Figure (3.1): Applying compliancy to the hip and ankle joint. University of Twente 31

39 Variable stiffness for robust and energy efficient 2D bipedal locomotion 3.1 Applying a constant linear spring along the leg In this project, a passive storage element (spring) has been applied along the robot s legs by being located on the joint of the leg and the foot. Figure (3.2) shows an impression of the configuration of this idea. A linear spring connects the leg to the center of mass of the foot. The joint motion is constrained to the movement along the leg. The motivation behind applying compliancy along the leg is to store some part of the kinetic energy, which is dissipated at each step due to the impact of the foot with the ground, and return it to the system. In this section, the slope angle is kept fixed (0.009 rad) and a linear spring with stiffness K is applied to the joint. Figure (3.2): The configuration of the passive walker with a prismatic compliant foot joint. The desired behavior of this configuration for the swing/stance phase of walking motion is: Swing phase: To have a bounce-less (damped) swing that makes it possible for the swing foot to properly find the ground contact point to land, Stance phase: To have an under-damped spring to absorb the impact force, store its energy during the stance phase and return it to the system at the moment that the foot leaves the ground. To achieve the above pattern, the unwanted oscillation during the swing phase, caused due to the mass of the foot, should be damped by adding a damping element parallel to the spring. The damping element also helps to reduce the simulation time considerably (F.Broenink) The effect of choice of the damper on the stability and robustness of the walker To choose a proper damper, the equation (2.6) for calculating the damping ratio of a linear second order system composed of a parallel linear spring-damper attached to a mass (Figure (2.10)) was used. From the equation (2.6) for a critically damped oscillation during the swing phase: ζ = = 1 and for an under-damped stance phase: ζ = < 1, where m is the foot mass, M is the hip mass, D ( ) is the required damper for the swing phase,, D is the required damper for the stance phase and K is the stiffness of the spring. Rewriting the above equations, we obtain: ζ = = 1 D = 2. mk ζ = ( ) < 1 D = 2.. (m + M)K, 0 < < 1 Eq. (3.1) 32 Control Engineering

40 P Applying compliancy to the model As can be understood from equation (3.1), α can be chosen in a range of zero to one to have an underdamped oscillation during the stance phase. The choice of α is important since it affects the range of stiffness with which robot can stably walk. Also, it has an effect on the robustness of the system. Figure (3.3) shows the plot of the system s behavior over the choice of α. Here, step-down experience is the measure of the system s robustness. Also, the range of possible stiffness (with which robot can stably walk) depends on α. We refer to this range as stable stiffness. For 0.13< α < 0.25, the system shows the best robustness comparing to other values of α. This area is pointed out by the expression Fair robustness that refers to the ability of the system to handle the step-down in a continuous range and up to a maximum value that is comparable with the results of the system with rigid legs. The system s robustness for α =0.15, for which the range of stable stiffness starts from 1500N/m, are discussed in the next section of this chapter. By decreasing α, the range of possible stable stiffness will be limited to a range of high stiffness; it is not possible to find a stable gait for low values of stiffness, e.g.2000 N/m. Moreover, if a stable gait can be found for small stiffness, e.g N/m, the step down experience shows poor robustness. For higher stiffness, e.g N/m, a chaotic robustness is resulted. Chaotic robustness refers to the random behavior of the system, for example it can handle a step-down size of 3.5 mm but it fails a step-down size of 1mm. The same behavior was reported in Chapter 2 about the step-down experiment for the system with rigid legs over slope angles larger than rad. The third section of the plot (3.3) shows the system behavior for larger values of α. In this region again the range of stable stiffness is limited. Although the system does not show a chaotic behavior, it shows a comparably poor robustness. Figure (3.3): The plot of the system s behavior over α. According to the above explanation, α =0.15 is chosen as the damping constant for the stance phase. From the equation (3.1), we have: D = 2 mk and D = 0.3 (m + M)K. Since in our model M is 30 times bigger than m, the value calculated for D is very close to D. It means that instead of switching between D and D in the swing and stance phase, a constant damping equation (D ) can be used in both phases. The results of simulations confirm similarity between these two ways of implementation of dampers. Generally, with the described mass configuration (M>>m), the value calculated for D, while providing a under-damped situation for the stance leg, results to a critically-damped or over-damped situation for the swing leg (ζ 1). So, the same damper can be used for both swing and stance phases. University of Twente 33

41 Variable stiffness for robust and energy efficient 2D bipedal locomotion Analyzing the robustness of the passive walker with prismatic compliant foot joint Figure (3.4) shows the result of the analysis of the robustness of the passive walker with prismatic compliant foot joint walking over a slope angle of rad. The stiffness of the leg has been changed from 1500 N/m to N/m. The plot shows an improvement in the robustness by increasing the stiffness. The horizontal lines show the maximum step-down size that a passive walker with rigid legs can handle for different slope angles. It should be noticed that all of these angles and the values of stiffness share the same property: the system shows a period-one behavior and it continuously can handle the step-down experience up to a maximum value. Figure (3.4): The step-down experiment for the passive biped with prismatic compliant joint over a range of stiffness. Comparing the maximum values of the robustness of the compliant legs with rigid legs over the slope rad shows that the robustness of the system starts to improve from K=4500N/m and the improvement continues by increasing the stiffness. For K=9000N/m the robustness is almost 14% improved comparing to slope angle rad. The question is if our comparison makes sense. Both of the systems, rigid and compliant legs walk over the same slope angle and they show period-one behavior. So, their source of energy is the same. But they dissipate different amount of energy and they have different velocities. Comparing two walking machines, the three properties, robustness, velocity and the required energy (energy efficiency) should be considered. To make a comparison between the performances of two walkers, we can choose a value for one of these parameters and compare the two other parameters. For example, for a chosen slope angle, velocity and robustness of the walker with rigid legs can be compared with the velocity and robustness of the walker with compliant legs. By applying a stiff enough spring along the leg (for low stiffness spring the stored energy will be mainly dissipated by the oscillation of the spring) the stored energy during the stance phase will increase the velocity of the motion. Figure (3.5) shows the trend of changing the velocity and the dissipated energy due to the impact (per step) over the leg stiffness. For a passive walker with compliant legs, increasing the stiffness of the leg leads to an increase of both of the velocity and the dissipated energy. From table (2.3), 34 Control Engineering

42 P Applying compliancy to the model the average velocity and dissipated energy for the rigid leg walker over the slope angle rad are 0.274m/s and 0.08J. Comparing the loss of energy, the amount of dissipated energy due to impact for the compliant leg walker is around 10 times smaller than the amount of dissipated energy for the rigid leg walker over the same slope angle. For stiffness bigger than 3000 N/m, the velocity of the compliant legs robot starts to become larger than the velocity of the robot with rigid legs. Figure (3.5): The plot of velocity and dissipated energy for the passive walker with the compliant legs. From figures (3.4) and (3.5) the following conclusions are drawn: - The velocity of the passive walker with compliant prismatic foot joint for stiffness bigger than 3000 N/m is bigger than the same passive walker over the same slope angle. - The robustness of the passive walker with compliant prismatic foot joint for stiffness bigger than 4500 N/m is bigger than the same passive walker over the same slope angle. - The amount of dissipated energy, due to impact of the foot with the ground, of the passive walker with compliant prismatic foot joint is much smaller than the same passive walker over the same slope angle. Figure (3.6) shows the plot of the robustness over the velocity for two cases: the dashed line is related to the rigid leg walker over different slope angles in the range of rad to rad (the slope angle rad is marked by a red star on the plot); the rigid line is related to the compliant leg walker over the fixed slope angle of rad with different stiffness in the range 1500 N/m to 10000N/m. The following conclusions can be drawn from this plot: - For a certain velocity, the robustness of the passive walker with rigid legs is bigger than the robustness of the passive walker with compliant legs. This difference will be smaller by increasing the leg stiffness. - Varying the stiffness of the legs over a fixed slope angle, the velocities of a range of slope angles (smaller and bigger the chosen slope angle) can be obtained. - For a proper choice of leg stiffness (e.g. K=8000 N/m) over the slope angle rad, the velocity of the slope angle rad is obtained, also their robustness is almost the same ( it is 9% University of Twente 35

43 Variable stiffness for robust and energy efficient 2D bipedal locomotion smaller than the robustness of the slope angle rad). It should be noticed that the amount of injected energy for slope angle rad is smaller than rad. So, the gait properties of a larger slope angle are obtained by applying a passive storage element along the walker s legs. Figure (3.6): The plot of robustness over velocity for the passive walker with rigid/compliant legs. 3.2 Changing the stiffness of the leg In this section, the stiffness of the leg has been changed while walking. Changing the stiffness can be realized by a variable stiffness actuator. So far, different structures for a variable stiffness actuator have been proposed. A recent study at the Control group of University of Twente (Visser et al.,2010) represents a conceptual actuator that does not need energy to change the stiffness of the spring. Assuming no consumption of energy, here, the leg stiffness has been ideally changed from one value to another. The strategy is to continuously increase the stiffness from mid-stance to the moment of take off. By increasing the stiffness, some amount of energy will be injected to the system, so the system will be actuated. From the results of the previous section, a high stiffness (K ) can be chosen for the legs, this stiffness will be increased up to a larger one (K ) during the stance period. Equation (3.3) shows the trend of increasing the stiffness after the mid-stance phase and before take-off, K = K + (K K ). ( ), Eq. (3.3) where K is the leg stiffness during the swing phase and part of the stance phase before the mid-stance, K is the stiffness of the stance phase after the mid-stance and before take-off, φ is the angle between the swing-leg and the stance leg, and φ represents the peak value for φ with the fixed stiffness of K. Around the mid stance, the legs are parallel, so the angle between the legs (φ) is very small, it will reach its maximum value at the end of the step. 36 Control Engineering

44 P Applying compliancy to the model Figure (3. 7) shows the change of stiffness from K = 8000 time during which the left leg is the stance leg. to K = The rectangle shows the Figure (3.7): The plot of change in the leg stiffness. Figure (3.8) shows the F- z plot for the leg s spring for the above values, where z is the change of the spring s length (compression). The area confined in the plot shows the amount of power that is injected to the system while changing the stiffness. Changing the stiffness is not a passive operation since the energy that is stored in the spring, due to the non-zero elongation of the spring, will be changed. Figure (3.8): The plot of F- z for the leg spring. University of Twente 37

45 Variable stiffness for robust and energy efficient 2D bipedal locomotion It should be noted that damper should be chosen properly. For the mentioned values of K, K, α=0.15 leads to a chaotic robustness, the results reported here are obtained by α=0.2 for which system shows a continuous robustness. This experiment has been repeated for different values for the stiffness and the results show the same trend. Table (3.1) shows the velocity and robustness of the system with constant stiffness (K) and variable stiffness (K, K ) for three experiments. Constant stiffness Velocity Step-down Variable stiffness Velocity Step-down K (N/m) (m/s) (mm) K 1, K 2 (N/m) (m/s) (mm) 6000 (α=0.15) (α=0.15) (α=0.15) ,8000 (α=0.2) , (α=0.3) ,12000 (α=0.2) Table (3.1): Comparison the velocity and robustness for the constant and variable stiffness. The results show that a big change in the stiffness (from 6000 to N/m) requires increasing the damper in order to have a non-chaotic robustness. For example, by changing K from 6000 to and α=0.2, a step-down of size 3.3 mm can be handled but it fails 1.5 mm. In this case, by increasing the damper, the maximum successful step-down is almost the same as a constant stiffness of 6000 N/m. A smaller change in the stiffness (from 6000 to 8000 N/m) or (8000 to N/m), leads to the velocity and robustness that are comparable to the relevant parameters for the system with a constant stiffness with the upper bound value. For example, the system with variable stiffness from 6000 to 8000 N/m is almost equivalent to the system with a constant stiffness of 8000 N/m. however, the damper used in this case (α=0.2) is larger than the damper in the system with the constant stiffness (α=0.15). It can be concluded that increasing the stiffness results in a larger frequency in the movement of the under-damped stance phase (mass-spring system). To prevent its chaotic behavior, the damper should be increased that will limit the velocity of the system. So, although injection energy to the system before the moment of take-off can intuitively increase the velocity and robustness, the strategy of energy injection has an important role. The damping required to prevent the chaotic behavior of the system, also prevents the system to use the injected energy efficiently. 38 Control Engineering

46 Chapter4 Conclusions and Recommendations 4.1 Conclusions In this Project, the effect of applying compliancy to the model of a 2D passive walker on its robustness has been studied. To accomplish this goal, some sub-goals, described in section 1.2, have been reached. A 2D passive walker has been modeled with compliant ground contact model and results of the analysis showed similarity to other studies in this field using a rigid contact model. Measures of the robustness were discussed and one practical measure (step-down) has been taken as the basis of comparison. It has been verified and reported that the robustness of the passive walker is very narrow. In order to improve the robustness of the passive walker, a passive storage element was added to the system to form a compliant prismatic joint along the robot s legs. The results show that with a proper choice of damper(to damp the unwanted oscillation of the spring during the swing phase)the velocity and robustness of the system can be increased compared to the system with rigid legs over the same slope angle, if the stiffness of the spring is larger than a certain limit. For a fixed mass/size configuration of the robot and with a proper choice of damper, this limit depends on the slope angle. The stiffness of the leg-foot joint has been varied while walking, and the results show no improvement in the robustness comparing to the model with constant stiffness. 4.2 Recommendations There are many things to be studied about the improvement of robustness of a passive dynamic walker. Some suggestions for future work are listed below: To find a definition about what is the expected robustness for a biped according to the probable practical disturbances : in order to improve the measure of the robustness. To apply stiffness on both hip and leg joint: to study if the robustness can be improved. To increase foot radius: Increasing foot radius will improve the robustness (Wisse et al., 2006). The effect of the foot radius can be compared with a compliant element. Also, the effect of combination of these two on the robustness can be verified. To change the foot shape in order to apply ankle stiffness (actuation). To try different strategies for changing the stiffness and actuating the system. To build a 3D model and study the effects of applying compliancy to it. To implement the obtained results for applying a constant compliancy along the leg on a real walking machine. University of Twente 39

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48 Bibliography Controllab Products B.V. 20-sim version 4.1. (2010). Retrieved from Diolaiti, N., Melchiorri, C., & Stramigioli, S. (2005). Contact impedance estimation for robotics systems. IEEE TRANSCATIONS ON ROBOTICS, vol.21, No.5. Duindam, V. (2006). Port-based modeling and control for efficient bipedal walking robots,phd thesis. Enschede, the Netherlands: University of Twente. F.Broenink, J. Numerical methods for complex ODE/DAE systems. Enschede, The Netherlands: University of Twente,EE Department. Garcia, M., Chatterjee, A., Ruina, A., & Coleman, M. (1998). The simplest walking model:stability,complexity,and Scalling. ASME Journal of Biomechanical Engineering. Goswamiy, A., Thuilotz, B., & Espiauy, B. (1998). A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotics Research. Hobbelen, D. (2008). Limit cycle walking,phd thesis. Delft,The Netherlands: Delft University of Technology. Hobbelen, D., & Wisse, M. (2007). A Disturbance Rejection Measure for Limit CycleWalkers: The Gait Sensitivity Norm. IEEE TRANSACTIONS ON ROBOTICS, VOL. 23, NO. 6. McGeer, T. (1990). Passive dynamic walking. International Journal of Robotics Research, 9(2): McGeer, T. (1990). Passive walking with knees. British Columbia,Canada: Simon Fraser University. Oort, G. v. (2005). Strategies for stabilizing a 3D dynamically walking robot. Enschede,The Netherlands: University of Twente. Schaft, A. v., Scherpen, J. M., & Jeltsema, D. (2009). Lecture notes for nonlinear systems theory. Shiriaev, A. (2009). Lectures on Controlling Oscillations in Nonlinear Systems. Stramigioli, S., & Bruyninckx, H. (2001). Geometry and Screw theory for robotics. Strogatz, S. H. (1994). Nonlinear dynamics and chaos with applications to Physics, Biology, Chemistry and Engineering. NewYork,USA: Perseus Book Publishing, L.L.C. Visser, L., Carloni, R., Unal, R., & Stramigioli, S. (2010). Modeling and Design of Energy Efficient Variable Stiffness Actuators. IEEE International Conference on Robotics and Automation, (pp. pp ). Alaska, USA. Wisse, M. (2004). Essentials of dynamic walking,phd thesis. Delft, The Nethelands: Delft University of Technology. Wisse, M., Hobbelen, D. G., Rotteveel, R. J., Anderson, S. O., & Zeglin, G. J. (2006). Ankle springs instead of arc-shaped feet for passive dynamic walkers. 6th IEEE-RAS International Conference on Humanoid Robots. University of Twente 41

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50 Appendix A: Compliant Contact model This appendix shows the block-diagram and codes of the compliant contact model. Compliant model composed of two blocks for right/left foot: Floor right and Floor left. Each of these blocks composed of two other blocks: coordinate and floor (as mentioned in the section 2.3.1). Also, in section A.2, codes of definition of Ѳ,φ is available. It defines the moment that the role of legs is exchanged. A.1: Codes and block-diagrams of the compliant contact model The block-diagram of the compliant contact model The block-diagram of the Floor Left Figure (A-1): Block-diagram of compliant contact model. Codes of the coordinate block Parameters real global Rfoot=0.05; variables real p_c_s[3]; // foot radius // All variables are defined in the world frame // position of the contact point equations // Hpl_0: Homogeneous transition matrix defining the contact point in world frame coordinates // HLeft: the coordinate of the c.o.m of the left-foot w.r.t the world coordinate // Hs: the coordinate system of the slope w.r.t the worl coordinate p_c_s=[hleft[1,4]; HLeft[2,4];HLeft[3,4]]-[0;0;Rfoot]; Hpl_0 = [Hs[1,1], Hs[1,2], Hs[1,3], p_c_s[1]; Hs[2,1], Hs[2,2], Hs[2,3], p_c_s[2]; Hs[3,1], Hs[3,2], Hs[3,3], p_c_s[3]; University of Twente 43

51 Appendix A Codes of the Floor block: parameters real global Kp = 1.0e6, Kd = e7; // Visco-elastic properties of the ground /* Parameters for translational friction (static friction, coulomb friction, viscous friction,steepness of coulomb, friction Stribeck velocity) */ real global Ft\mu_st= 0.5; real global Ft\mu_c = 0.25; real global Ft\mu_v = 0.025; real global Ft\slope = 500.0; real global Ft\v_st = 0.02; real zeros[6]= [0.0;0.0;0.0;0.0;0.0;0.0]; variables real global TranslationJointZ1\pos; real global RotationHingeY1\phi; boolean global contactr; boolean global contactl; boolean global strikel; boolean Impact; // when contactr is true, right-foot is in contact with the ground. // when contactl is true, left-foot is in contact with the ground. // The moment of the contact of the left foot with the ground. //The moment of Impact that can be foot-scuffing or start of the contact. boolean XL; real global errorl; real Tp[6]; real Wp[6]; real Fn; real Fx; real El; // Twist of the foot expressed in the contact point coordinate frame // Wrench on the foot expressed in the contact point coordinate frame // Normal force = Wp[6]. // Translational friction force in x-direction // The energy of the compliant contact model initialequations Impact=false; // initially, it is supposed that right-foot is on the ground. contactl=false; strikel=false; XL=false; equations // The twist of the contact point is obtained by multiplying the twist of the center of mass of the foot with the adjoint matrix of the homogeneous matrix describing the transition from the center of mass to the contact point// Tp = Adjoint(Hs_pl) * P.f; 44 Control Engineering

52 P Appendix A // Determining when the foot hits the ground (height of the contact point in world frame coordinates becomes equal to the local height of the floor (at the moment level floor) errorl= Hpl_0[3,4]-(tan(RotationHingeY1\phi))*(-(Hpl_0[1,4])); Impact = eventdown(errorl); if (Hpl_0[1,4])< (Hpr_0[1,4]) then XL=false; else XL=true; end; strikel=impact and XL; // After the foot has collided with the floor it is in contact with the floor if (strikel) then contactl = true;end; // When the height of the contact point in world frame coordinates is higher then the local height of the floor the foot is no longer in contact with the floor and there is no normal force. if (eventup(errorl)) then contactl=false; end; //The normal force is calculated using the Hunt-Crossley equation. The wrench has 2 non-zero components (no rotation). The z-component of the wrench is the normal force. The other component is the friction in the x-direction modelled by SCVS- friction models. The coefficients used in these models were roughly estimated */ // Tensile force given by the Hunt-Crossley contact model Fn = -Kp*errorL - Kd* -errorl *Tp[6]; if (Fn < 0 ) then end; Fn = 0; // Determining translational friction Fx = -Fn*(((Ft\mu_c+(Ft\mu_st*abs(tanh(Ft\slope*Tp[4]))-Ft\mu_c)*exp(- ((Tp[4]/Ft\v_st)^2)))*sign(Tp[4])+Ft\mu_v*Tp[4])); if (Tp[4] == 0) then Fx = 0; else Fx; end; // Determing resulting wrench due to foot ground interaction Wp = if (contactl) then [0;0; 0; Fx; // translational friction in x-direction 0; Fn ] // translational friction in z-direction else zeros University of Twente 45

53 Appendix A end; // Transforming the wrench present at the contact point into a wrench working at the center of mass of the foot P.e = transpose(adjoint(hs_pl)) * Wp; // To optimize the damping co-efficient, El is the exchanged energy of the compliant contact model El=if (TranslationJointZ1\pos <0.7) then else end; if contactl then else 0 end resint(((-kp*errorl-kd*-errorl *Tp[6])*Tp[6]),0,strikeL,0) A.2: Codes of definition of teta and phi, the codes are located in the variables block that is a separated block in the model. r=errorr-errorl; if (contactr) and (contactl) and eventup(r) then m=1;end; if (contactr) and (contactl) and eventdown(r) then m=0;end; if (m==1) then teta=rotationhingey1\phi-suspensionjoint\phi-hipjoint\phi; tetad=(-suspensionjoint\omega-hipjoint\omega); phi=(-hipjoint\phi); phid=(-hipjoint\omega); else if (m==0) then teta=rotationhingey1\phi-suspensionjoint\phi; tetad=(-suspensionjoint\omega); phi=(hipjoint\phi); phid=(hipjoint\omega); end; end; 46 Control Engineering

54 Appendix B: Initial conditions This appendix contains two tables of the initial conditions for the rigid/compliant leg model. Starting the model with these initial conditions, the walker will converge to its stable limit cycle. Initial position: Right-leg on the floor, floor stiffness: 10, floor damping: , hip joint-angle=0.3,suspension-joint angle=0, Z (initial)=0.7405, X (initial)=1 Slope angle (rad) Initial condition (PDot_int_initial) [ ] [0,0,0,-0.08,-0.17] [ ] [0,0,0,-0.08,-0.18] [ ] [0,0,0,-0.08,-0.185] [ ] [0,0,0,-0.08,-0.19] [ ] [0,0,0,-0.08,-0.195] [ ] [0,0,0,-0.08,-0.17] [ ] [0,0,0,-0.08,-0.18] [ ] [0,0,0,-0.08,-0.185] Table (B-1): Initial conditions for the rigid-leg robot over slope angle. Initial position: Right-leg on the floor, floor stiffness: 10, floor damping: , hip jointangle=0.3,suspension-joint angle=0, Z (initial)=0.7405, X (initial)=1, Slope angle=0.009 rad Stiffness (N/m) Initial condition (PDot_int_initial) [0;0;0;-0;08;-0.127;0;0] [0;0;0;-0;08;-0.129;0;0] [0;0;0;-0;08;-0.133;0;0] [0;0;0;-0;08;-0.138;0;0] [0;0;0;-0;08;-0.14;0;0] Table (B-2): Initial conditions for the compliant-leg robot over stiffness. University of Twente 47

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56 Appendix C: Numerical Calculation of Eigenvalues To numerically calculate the eigenvalues for the passive biped walking over a slope angle, since it is difficult to pick-up all the state of the system at the fixed-point (that is the initial condition of the new gate) and re-enter it to the model as the initial condition, the following steps were followed: Find a not falling-down gait by trial and error. Using the event function and generate an event at the start of one of the steps. Put t=time (at the moment of event) and read the system variables there (Phi, Phidot, tetadot) Do the previous for the next step. Without any perturbations the values for the two consecutive steps should be the same. To read the values tomatlab command has been used to export the data to Matlab. Then each of the system state should be perturbed individually and the state of the system at the start of the next-step (after perturbation) should be read. o To perturb the position state (Phi), the following command in the system equations (generated from3d model) should be changed: HipJoint\phi = if time>t then int (HipJoint\omega) + HipJoint\phiInitial+praf else int (HipJoint\omega) + HipJoint\phiInitial end; // where t is the time of perturbation, praf is the size of perturbation. o To perturb the velocities, the mass matrix of the system at the start of the one of the steps can be saved in a variable. Then the perturbed momentum can be calculated and added to the system momentum at the start of the selected step. The changes in the system behavior should be tracked by reading the states of the next walking step. Here is the related codes: I=(MassMatrix)*[0;0;0;prvt;prvf]; of the hip and suspension joint. // where prvt and prvf are the size of perturbation in the velocities P = int (PDot)+PDotInitial+I*x; // This line should be editied in the coded generated by 3D mechanics Variable x is zero by default and will be equal to one after applying the perturbation. To calculate the Jacobian, the state of the system without perturbation and after applying each of the perturbations can be exported to Matlab. Then the related Jacobian can be calculated by: s0=[tetad;phi;phid]; // s0 is the state of system without purturbation eig([fs-s0]*(inv([s-s0]))); // s is the state of system after purterbation, [s-s0] represents a 3*3 matrix. fs is the state of the system at the next consecutive step after perturbation. [fs-s0] is a 3*3 matrix, as well. University of Twente 49

57 50 Control Engineering

58 Appendix D: Changing the stiffness of the legs This appendix presents the block-diagram of the model with compliant leg-foot joint and the related codes of the variable stiffness joint. Cl=if (MidL) then K1+(K2-K1)*(abs(HipJoint\phi)/(Phi_max)) else K1 end; Cr=if (MidR) then K1+(K2-K1)*(abs(HipJoint\phi)/(Phi_max)) else K1 end; Rl= a*sqrt (3.1*Cl); Rr= 2*a*sqrt (3.1*Cr); // 0<a<1 as described in the section //The related code for mid-stance signal, in the block Variables variables boolean global contactr; boolean global contactl; real global HipJoint\phi; boolean global MidL; boolean global MidR; boolean mid; equations if eventup(hipjoint\phi) then mid=true; end; if eventdown(hipjoint\phi) then mid=false; end; MidL=if (contactl and mid) then true else false end; MidR=if (contactr and not(mid)) then true else false end; University of Twente 51