CONTROL OF MULTI-LINK ONE-LEGGED HOPPING LOCOMOTION By Amer Allafi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering – Doctor of Philosophy 2020 CONTROL OF MULTI-LINK ONE-LEGGED HOPPING LOCOMOTION ABSTRACT By Amer Allafi Controlling one-legged hopping locomotion is a challenging problem due to the hybrid dynamics of the hopper and the interaction with ground. The hybrid dynamics of the one-legged hopper consists of mainly two sub-dynamics, one when the hopper is in contact with ground, and the other when there is no contact. The ground model can effect the hopper behavior since the hopper interact with ground when the hopper in contact with ground. Here we investigate the locomotion behavior of the one-legged multi-link hopper hopes on three different ground models, namely, rigid, elastic, and viscoelastic ground. The rigid ground apply an impulsive force to the hopper when the hopper came in contact with ground resulting energy losses. A partial feedback linearization is used to control the internal dynamics of the hopper. A Poincar´e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. The elastic ground, the ground modeled as massless spring, the spring in the ground store some of the energy of the hopper during the contact. A continuous backstepping controller is designed to control the energy level and internal dynamics of the hopper. A Poincar´e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. The viscoelastic ground, the ground modeled as an under- damped mass-spring-damper system, the damper and the impact with ground mass resulting in energy losses and the ground spring store some of the energy of the hopper during the contact. A continuous backstepping controller is designed to control the energy level and internal dynamics of the hopper. A Poincar´e map is used to construct a discrete-time system and a controller with integral action is designed to achieve the control objectives. We considered multiple versions of one-legged hoppers, namely, two-DOF two-mass, two- DOF ankle-knee-hip, and four-link hopper. Simulation results are presented to demonstrate the efficacy of the controllers. I dedicate this work to the memory of my mother, Hessa Almuzaini , who always believed in my and my ability to be successful in the academic arena. You are gone but your belief in me has made this journey possible. My father Lafi Allafi, who first taught me the value of education and the value of hard work. Also, I want to thank my wife, Najla Aleid, who have supported me spiritually throughout the process. iv ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Ranjan Mukherjee for his patience, motivation, vast knowledge, and guidance throughout this work. I would like to thank the other members of my committee, Dr. Hassan K. Khalil, Dr. George Zhu and Dr. Brain Feeny for the assistance they provided. Finally, I would like to thanks the Saudi Ministry of Education and Qassim University for the financial support during my P.h.D program. v TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1 2.2 Dynamics of Two-DOF Hopper Chapter 2 Apex Height Control of a Two-DOF Prismatic Joint Robot Hopping on a Viscoelastic Ground with Inertia . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Flight Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Impact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Contact Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hybrid Control Strategy for Hopping on a Purely Elastic Ground . . . . . . 2.3.1 Change of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hopping on a Viscoelastic Ground with Inertia . . . . . . . . . . . . . . . . . . . . . Simulation Results: Apex Height Control . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Discrete Controller for Stabilization and Control Apex Height 2.4.2 Chapter 3 Apex Height Control of a Two-DOF Ankle-Knee-Hip Robot Hopping on a Rigid Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 Dynamics of AKH Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 System Description and Model . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Impact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Contact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Continuous Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Relative Displacement of COM . . . . . . . . . . . . . . . . . . . . . 3.3.2 Contact Phase Control Design . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Flight Phase Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Closed-Loop System Dynamics 3.4 Discrete Control Design for the Apex Height . . . . . . . . . . . . . . . . . . 3.4.1 Poincar´e Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Discrete Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effect of Continuous Controller Parameters on Hopping Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 6 6 7 7 8 9 10 11 11 12 16 16 16 20 23 25 25 26 26 28 28 29 29 30 30 30 31 32 33 33 35 37 3.5.1 Effect of the Parameters on Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Limitation on Choice of Parameter Values 3.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4 Apex Height Control of a Two-DOF Ankle-Knee-Hip Robot Hopping on an Elastic Ground and a Viscoelastic Ground with Inertia . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dynamics of AKH Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 System Description and Model . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Impact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Contact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hybrid Control Strategy for Hopping on a Purely Elastic Ground . . . . . . 4.3.1 New Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 4.4 Discrete Controller for Stabilizing Hybrid Dynamics and Controlling the Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Elastic Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Viscoelastic Ground with Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Elastic Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Viscoelastic Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Simulation Chapter 5 Four-Link Planar One-Legged Hopping Locomotion . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hybrid System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Impact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Contact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Coordinate Transformation into Normal Form . . . . . . . . . . . . . . . . . 5.3.1 Controlled States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Uncontrollable States - Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Uncontrollable States - Contact Phase 5.4 Partial Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Contact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 40 43 44 44 45 45 47 48 48 49 49 49 50 52 55 56 56 59 62 62 64 67 69 69 70 70 72 72 73 74 74 76 77 78 78 79 80 vii 5.5 Controlling The Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Apex Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 The First Strategy: Negative Damping Based Continuous-Time Con- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . troller 81 81 82 5.5.3 The Second Strategy: Equilibrium Height Based Based Continuous-Time Controller . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Discrete-Time Controller . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Effect of Continuous Controller Parameters on Apex Height . . . . . 5.6 Control of Hopping Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Hopping Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Flight Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Contact Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Poincar´e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Closed-loop Control Design . . . . . . . . . . . . . . . . . . . . . . . 5.7 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 86 91 93 93 93 94 94 95 97 97 Step Size Control in Hopping Locomotion . . . . . . . . . . . . . . . 101 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.7.1 Apex Height Control 5.7.2 Chapter 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 viii LIST OF FIGURES Figure 2.1: (a) Viscoelastic ground with inertia (b) flight phase and (c) contact phase of two-DOF robot hopping on the viscoelastic ground. . . . . . . Figure 2.2: Simulation results for hopping on a viscoelastic ground with inertia. Plot of the position of the upper mass X1, the lower mass X2, COM z, . . . . . . . . . . . . . and the ground mass x3, as a function of time. Figure 2.3: Simulation results for hopping on a viscoelastic ground with inertia: Plot of pE ´ Edesq at the end of the k-th hop (immediately before touch- down), k “ 1, 2, ¨ ¨ ¨ , 12. The actual time scale in the figure is identical to the time scale in Fig.2.2 . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.4: Plot of the Force F1 applied by the prismatic joint. . . . . . . . . . . . Figure 2.5: Simulation results for hopping on a viscoelastic ground with inertia: . . . . Plot of X2, x2, and x3 for the hop between k “ 11 and k “ 12. 8 21 22 22 23 Figure 3.1: The ankle-knee-hip robot. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Figure 3.2: Apex height h is plotted with respect to ν for periodic orbits obtained with rd “ 0.13 m, ζ “ 0.13 and (a) ωn “ 20, (b) ωn “ 25, (c) ωn “ 30, and (d) ωn “ 35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3: Plots of the heights of the upper and lower masses and the COM. . . . Figure 3.4: Plot of the state of the discrete-time system at the beginning of the first nine hops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.5: Plot of the torque applied by the hip actuator. . . . . . . . . . . . . . 41 42 42 43 Figure 4.1: The ankle-knee-hip robot. . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 4.2: Simulation results for hopping on the elastic ground. Absolute height of the two masses y2 and y3, and COM height are plotted as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.3: Simulation results for hopping on the elastic ground: Plots of pE ´Edesq . . . . . . . . . . . . . . . . at the end of the k-th hop, k “ 1, 2 ¨ ¨ ¨ , 6. Figure 4.4: Simulation results for hopping on a viscoelastic ground with inertia. Plot of the position of the upper mass y3, the lower mass y2, COM z, . . . . . . . . . . . . . and the ground mass x1, as a function of time. 63 64 65 ix Figure 4.5: Simulation results for hopping on a viscoelastic ground with inertia: . . . . Plots of pE ´ Edesq at the end of the k-th hop k “ 1, 2, ¨ ¨ ¨ , 21. Figure 4.6: Simulation results for hopping on a viscoelastic ground with inertia: . . . . . . . Plot of y2, and x1 for the hop between k “ 19 and k “ 20. Figure 4.7: Plot of the torque applied by the hip actuator. . . . . . . . . . . . . . Figure 5.1: Four-link planar hopping robot in an arbitrary configuration . . . . . . Figure 5.2: Vertical displacement of the COM during apex-height control η3. . . . Figure 5.3: Discrete-time states ph ´ hdq and pIcm ´ Icmdq at the end of each hop fr apex-height control using negative damping. . . . . . . . . . . . . . . . 65 66 67 71 98 99 Figure 5.4: Vertical displacement of the COM during apex-height control η3. . . . 100 Figure 5.5: Discrete-time states ph ´ hdq and pIcm ´ Icmdq at the end of each hop fr apex-height control using negative damping. . . . . . . . . . . . . . . . 101 Figure 5.6: Position of the toe x during hopping locomotion. . . . . . . . . . . . . 102 Figure 5.7: Variation of actual and desired step sizes over 60 hops. . . . . . . . . . 103 Figure 5.8: Plot of uncontrolled states at the end of each hop k during hopping locomotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 x Chapter 1 Introduction Legged locomotion has certain advantages over wheeled locomotion. Compared to a wheeled robot, a legged robot is better suited for climbing stairs and moving over unstruc- tured terrain. One-legged hopping locomotion is the simplest version of legged locomotion but controlling a one-legged robot is challenging due to the hybrid nature of its dynamics. The dynamics of a one-legged hopping robot typically consists of two phases: a contact phase and a flight phase. In addition, there can be an impact phase if the ground is rigid. The nature of the ground affects the dynamic behavior of the hopping robot significantly. The ground can be modeled as rigid, elastic, or viscoelastic with inertia. A rigid ground results in energy losses when the hopping robot comes in contact, a perfectly elastic ground results in no losses, and a viscoeleastic ground with inertia results in losses due to both impact with the ground and damping. Naturally, the control problem for hopping locomotion is different for different types of ground. Some of the early work on one-legged locomotion can be credited to Seifert [1], Matsuoka [2] and Raibert [3]. Following the work by Raibert, Alexander [4] introduced the spring loaded inverted pendulum (SLIP) model for one-legged hopping and Schwind and Koditchek used a return map of a hop to control locomotion [5]. The leg and hip model, where an actuator is mounted on the hip joint, is an extension of the SLIP model. Cherouvim and Papadopoulos controlled the forward speed and apex height of a robot with a leg and hip, hopping over unknown rough terrain [6]. Mojtaba and Buehler proposed a control strategy for stabilizing and controlling the speed of the leg and hip model [7]. Poulakakis and Grizzle [8], [9] investigated the stability and control of a SLIP model with an asymmetric mass; this model is referred to as the asymmetric spring loaded inverted pendulum (ASLIP). The 1 behavior of the ASLIP model was compared with that of a three-link ankle-knee-hip (AKH) hopper [10]. Saranli et al. controlled an AKH hopper by approximating its model with that of the SLIP model [11]. Zhu et al. controlled the apex height of an AKH hopper [12], Vanderborght et al. investigated the effect of elastic actuation on the apex height [13], and Vu et al. [14] investigated the relationship between energy efficiency, leg stiffness, and stride frequency. Since the dynamics of a hopper typically consists of multiple phases and behavior of the of the hopper is affected by the ground model, researchers have commonly used the Poincar´e map [4, 15–17] to investigate the stability of this hybrid dynamic system. The ground is assumed to be rigid in most investigations: both one- and two-mass models have been used. This results in an impact phase where the robot comes in contact with the ground and there is loss of energy due to the impact. To avoid the impact phase, the one-mass SLIP model assumes the leg to be massless and therefore there is no loss of energy at the time of ground contact. The dynamics of the one-mass SLIP model is qualitatively the same for both rigid and elastic grounds and therefore two-mass models have been investigated. Mikhailova [18] investigated a two-mass hopper; a spring was placed between the lower mass and the ground to prevent loss of energy from impact. Yu and Iida [19] and Mathis and Mukherjee [20, 21] also considered two-mass hoppers but a controller was designed to account for the energy loss due to impact. Saitou et al. [22] used optimal control methods to maximize the height of a two-mass robot hopping on an elastic ground. The ground was assumed to have mass but the controller was implemented in an open-loop fashion. Ishikawa et al. [23] used a port-controlled Hamiltonian method to control the maximum hopping height of a similar two-mass system; the controller is based on feedback but the ground was assumed to be massless. Hutter et al. [24] developed an apex-height controller such that a two-link hopper behaves like a SLIP model by compensating for the energy loss. Here we design control strategies for apex height control of multi-link one-legged hoppers, hopping on rigid, elastic, and viscoelastic ground. Also, we design a control strategy for 2 locomotion control of a six degrees-of-freedom (DOF) four-links hopper, hopping on a rigid ground. For apex height control, the control objective is to achieve a desired value of the maximum height reached by the center-of-mass (COM) of the robot during the flight phase. For locomotion control, the control objective is to achieve a desired step size for each hop. In Chapter 2, we extend the control strategy developed by Mathis and Mukherjee [21, 25] for apex height control of a two DOFs prismatic joint robot hopping on rigid and elastic grounds to hopping on a ground that is viscoelastic and has inertia. We first use backstepping [26] to stabilize the internal dynamics of the robot in the flight and contact phases. The periodic nature of the hybrid dynamic system is then analyzed using a Poincar´e map [27] and the OGY1 method of chaos control [28] is used to adjust a parameter of the backstepping controller discretely and converge the apex height to its desired value. A integral control [26] is required to overcome the energy loss due to impact and damping for the viscoelastic ground with inertia. In Chapter 3, we consider an Ankle-Knee-Hip (AKH) robot, hopping on a rigid ground. The AKH robot has two DOFs in the flight phase and a single DOF in the contact phase. A continuous controller is designed to stabilize the controllable states in the flight phase. In the contact phase, the continuous controller makes the robot behave like a mass-spring- damper system: positive damping is first used to arrest the motion of the COM and negative damping is then used to add energy to the system and compensate for the losses due to impact. A Poincar´e map [27] is constructed to analyze the periodic nature of the hybrid dynamic system. To control the apex height, the OGY method is used to adjust one of the continuous controller parameters discretely. While this works well to stabilize the hybrid dynamics, integral control [26] is required for controlling the apex height. In Chapter 4, we address the problem of apex height control of an AKH robot hopping on an viscoelastic ground with inertia. The apex height is defined as the maximum height reached by the COM of the robot during the flight phase. We use the same control strategy 1A method introduced by Ott, Grebogi and Yorke for achieving stabilization of a periodic orbit [28]. 3 that was developed for a two-mass prismatic-joint hopping robot discussed in Chapter 2. A backstepping controller was developed to stabilize the internal dynamics of the robot in the flight and contact phases for hopping on a purely elastic ground. For hopping on a viscoelastic ground with inertia, the periodic nature of the hybrid dynamic system is analyzed using a Poincar´e map [27]. The OGY method of chaos control [28] is used together with integral action [26] to adjust a parameter discretely and converge the apex height to its desired value. In Chapter 2, we investigated hopping in a prismatic-joint robot and in Chapter 4 we extend the results to a robot with a kinematic structure that is commonly found in bipeds and walking machines. The underlying objective of this transition is to investigate locomotion problems in the commonly studied platforms. The control problem for the AKH hopping robot is more challenging than the prismatic joint robot because of the constraints imposed by the kinematic structure. In Chapter 5, we control the of apex height and hopping locomotion of a four-link an- thropomorphic robot hopping on a rigid ground; the four links correspond to the foot, leg, thigh and hip. The robot has three active joints at the ankle, knee and hip; the toe is not actuated and is therefore passive. One of the active DOFs is used to control the angle of the foot to ensure point contact with the ground; the other two DOFs are used to control the position of the COM. For apex height control, we use two control strategies. The first method is similar to the control strategy that was developed in [29]; this strategy introduces negative damping in the vertical dynamics of the COM during the contact phase. However unlike [29], where the Poincar´e section is chosen at the point of take-off, it is chosen at the point of touch-down - this eliminates the need for numerical search to choose the control gains. Additionally, the results are extended to the hopping locomotion problem. The sec- ond strategy relies on choosing different equilibrium heights of the vertical dynamics of the COM during the flight phase. In this regard, it should be noted that apex height control can be viewed as a special case of the hopping locomotion problem where the step size is zero. The four-link robot has six DOFs in the flight phase and four DOFs in the contact phase. 4 A feedback linearization controller is designed such that the COM and the foot angle have the dynamics of a mass-spring-damper system. The periodic nature of the hybrid system is analyzed using a Poincar´e map [30] and the OGY method of chaos control [31] is used together with integral action [26] to adjust system parameters discretely. By adjusting the parameters, it is possible to achieve a desired apex height while hopping in one location and hop with a desired step size during locomotion. 5 Chapter 2 Apex Height Control of a Two-DOF Prismatic Joint Robot Hopping on a Viscoelastic Ground with Inertia 2.1 Introduction In this chapter, we develop a strategy for controlling the apex height of a two-DOF prismatic-joint robot, hopping on a viscoelastic ground [32]. The apex height is defined as the maximum height reached by the COM of the robot during the flight phase. The problem of apex height control of the two-DOF prismatic-joint robot hopping on a rigid ground and an elastic ground has been studied earlier by Mathis and Mukherjee [21,33]. Here, we extend that work to hopping on a ground that can be modeled as viscoelastic with inertia. Hopping on a viscoelastic ground with inertia introduces an additional DOF and poses challenges due to undesired vibration of the additional DOF and dissipation due to both impact and viscous damping. A hybrid control strategy is developed to converge the apex height of the COM of the two-DOF prismatic-joint robot to a desired value. The hybrid control strategy uses backstepping in continuous time and integral control in discrete time to control the internal dynamics and the total energy. The discrete-time system is constructed using a Poincar´e map at the instant of time just before the impact between the robot and the ground. This 6 Chapter is structured as follows. The dynamics of the robot in the flight, impact and contact phases is presented in Section 2.2. The hybrid control strategy, developed for a purely elastic ground, is presented in Section 2.3; this material is taken from [25]. The hybrid controller is extended to the more general viscoelastic ground with inertia in Section 2.4 and validated with numerical simulations. Section 2.5 contains concluding remarks. 2.2 Dynamics of Two-DOF Hopper We consider a two-mass prismatic-joint robot hopping on a viscoelastic ground with mass m3, stiffness constant Kext, and damping coefficient Cext - see Fig.2.1 (a). The vertical displacement of the mass of the ground is denoted by x3. The robot is shown in its flight and contact phases in Figs.2.1 (b) and (c); it is comprised of an upper mass m1 and a lower mass m2, both of which are constrained to move in the vertical direction. The force applied on the two masses by the prismatic joint actuator is denoted by F1. The absolute position of the two masses are denoted by X1 and X2. The position of mass m1 relative to m2 is denoted by y, and the position of mass m2 relative to m3 is denoted by x2. The height of the COM of m2 from its base is denoted by %. The force of interaction between masses m2 and m3 is denoted by F2. 2.2.1 Flight Phases During the flight phase, the following conditions hold: x2 ą %F 2 “ 0 (2.1) Together, the robot and the ground have three degrees-of-freedom (DOF) but their dynamics 7 y X1 x2 F2 “ 0 g X2 x3 undeformed configuration of foundation m3 m1 F1 m2 % m3 m3 m1 F1 m2 F2 Kext Cext datum (a) (b) (c) Figure 2.1: (a) Viscoelastic ground with inertia (b) flight phase and (c) contact phase of two-DOF robot hopping on the viscoelastic ground. are decoupled. The accelerations of the two masses of the robot are as follows: :X1 “ p:y ` :x2 ` :x3q “ ´g ` :X2 “ p:x2 ` :x3q “ ´g ´ F1 m2 F1 m1 The equation describing the motion of the ground is m3 :x3 ` Cext 9x3 ` Kext x3 “ 0 2.2.2 Impact Phase (2.2) (2.3) The impact phase refers to infinitesimal intervals of time t P rt´, t`s during which the lower mass m2 comes in contact with the ground mass m3 and the condition x2 ą % changes to x2 “ %. We make the following assumptions: 8 Assumption 1. The collision between masses m2 and m3 is inelastic, i.e., m2 and m3 have identical velocities immediately after impact. Assumption 2. The control force F1 between masses m1 and m2 is not impulsive. The above two assumptions, along with the principle of conservation of linear momentum, give the following relations in terms of the position variables: X1pt`q “ X1pt´q ypt`q “ ypt´q X2pt`q “ X2pt´q ñ x2pt`q “ x2pt´q (2.4) x3pt`q “ x3pt´q x3pt`q “ x3pt´q and the following relations in terms of velocity variables: 9X1pt`q “ 9X1pt´q 9X2pt`q “ 9x3pt`q “ m2 9X2pt´q ` m3 9x3pt´q m2 ` m3 9ypt`q “ 9ypt´q ` m3 m2 ` m3 9x2pt´q ñ 9x2pt`q “ 0 9x3pt`q “ 9x3pt´q ` m2 m2 ` m3 9x2pt´q (2.5) ,//. //- 2.2.3 Contact Phases The contact phase commences immediately after the lower mass m2 makes contact with the ground mass m3. We make the following assumption which implies that the masses m2 and m3 cannot stick together: Assumption 3. The force F2 acting between the masses m2 and m3 is non-negative, i.e., F2 ě 0. During contact, x2 “ % and 9x2 “ :x2 “ 0. The system DOF is reduced to two and the 9 equations of motion are as follows: :X1 “ p:y ` :x3q “ ´g ` F1 m1 :x3 “ ´g ´ 1 m2 ` m3 pCext 9x3 ` Kext x3 ` F1q (2.6) The constraint force F2 (F2 ě 0) associated with the constraint x2 “ % and :x2 “ 9x2 “ 0is given by the expression F2 “ m1p:y ` :x3q ` m2 :x3 ` pm1 ` m2qg (2.7) At the instant when the system switches from the contact phase to the flight phase, the reaction force F2 equals zero. 2.2.4 Apex Height If z denotes the height of the COM of the hopping robot in the flight phase, we have z “ 1 m1 ` m2 rm1py ` x2 ` x3q ` m2px2 ` x3qs “ x2 ` x3 ` mf y, mf ∆“ m1 m1 ` m2 (2.8) where mf is the mass fraction. The hopping robot will have multiple flight phases. For each flight phase, the apex height is defined as the maximum value of z, and is denoted by h. 10 2.3 Hybrid Control Strategy for Hopping on a Purely Elastic Ground 2.3.1 Change of Coordinates For a purely elastic ground, we have m3 “ 0, Cext “ 0, Kext ‰ 0 (2.9) There is no impact phase. Furthermore, substitution of (2.9) in (2.3) indicates x3 ” 0 in the flight phase. Therefore, the system has two DOF in the flight phase. The system has two DOF in the contact phase as well since x2 “ % and 9x2 “ 0. For the objective of apex height control, we define r to be the height of the COM of the robot relative to that of the lower mass m2. Using (2.8), it can be shown Next, we define e as r ∆“ pz ´ x2 ´ x3q “ mf y e “ pr ´ rdq (2.10) (2.11) where rd ą 0 is some desired value of r. From (2.10) and (2.11) it can be verified that e ” 0 Ñ 9e ” 0 Ñ 9y ” 0, which implies no relative motion between the two masses. The dynamics of the system in the flight and contact phases can be written in compact form 11 using (2.2), (2.6), (2.8), (2.9), (2.10) and (2.11) as follows: :z “ ´g ´ λ 1 mt Kext x3, mt ∆“ pm1 ` m2q :e “ 1 m2 pF1 ´ λm fKext x3q where 0 : x2 ą % : Flight Phase 1 : x2 “ % : Contact Phase λ “$’& ’% The following choice of the control input F1 “ λm fKextx3 ` m2v “ λm fFext ` m2v, Fext ∆“ Kextx3 results in the hybrid dynamics :z “ ´g ´ λ 1 mt Kext x3 :e “ v (2.12) (2.13) (2.14a) (2.14b) (2.15) (2.16) where v is the new control input. Note that the control input F1 can be chosen according to (2.14a) or (2.14b) depending on whether x3 (displacement of the spring) or Fext (force applied by the spring) is available for measurement. 2.3.2 Backstepping The potential energy of the COM is defined relative to the datum z “ zd zd ∆“ z |px3“0, x2“!, r“rdq “ prd ` %q (2.17) 12 In the absent of relative motion between the masses (e “ 9e “ 0), the total energy can be written as E “ mt„1 2 9z2 ` gpz ´ zdq ` 1 2 λKext pz ´ zdq2 (2.18) For the robot to reach its desired apex height hd, the total energy should be equal to E ” Edes “ mt gphd ´ zdq (2.19) where hd is the desired value of apex height. In addition to e ” 0. The desired equilibrium configuration is therefore given by pE ´ Edes, e, 9eq “ p0, 0, 0q (2.20) With the objective of stabilizing the equilibrium in (2.20), we define the Lyapunov function candidate V1 “ 1 2 ke pE ´ Edesq2 , ke ą 0 (2.21) It should be noted that V1 is a function of λ (since E is a function of λ) but it is continuously differentiable in both the flight phase and contact phase. The Lyapunov function candidates introduced in this section will be used for stability analysis in the two phases separately; therefore, we treat λ as constant and do not make any distinction between the two phases in our derivation. Using (2.11), (2.16), and (2.18), 9V1 can be computed as 9V1 “ ke pE ´ Edesq 9E “ ke pE ´ Edesq 9z rmtp:z ` gq ` λKextpz ´ zdqs “ ke pE ´ Edesq λKext 9ze By choosing e “ t´λke pE ´ Edesq 9zu ∆“ ϕ1 13 (2.22) (2.23) we can make 9V1 negative semi-definite; therefore, integrator backstepping is introduced by defining the new variable q1 “ e ` λke pE ´ Edesq 9z “ pe ´ ϕ1q (2.24) and the composite Lyapunov function V2 “ V1 ` 1 2 q2 1 “ 1 2 ke pE ´ Edesq2 ` 1 2 q2 1 (2.25) Differentiating V2 and substituting (2.22) and (2.24), we get 9V2 “ ke pE ´ Edesq λKext 9ze ` q1 9q1 “ ke pE ´ Edesq λKext 9z rq1 ´ λke pE ´ Edesq 9zs ` q1 9q1 “ ´λ2k2 e Kext pE ´ Edesq2 9z2 ` q1 r 9q1 ` ke pE ´ Edesq λKext 9zs By choosing 9q1 “ t´λkeKext pE ´ Edesq 9z ´ k1q1u ∆“ ϕ2 (2.26) (2.27) where k1 ą 0. We can make 9V2 negative semi-definite. We introduce integrator backstepping again by defining the new variable q2 “ p 9q1 ´ ϕ2q (2.28) and the composite Lyapunov function V3 “ V2 ` 1 2 q2 2 “ 1 2 ke pE ´ Edesq2 ` 1 2 q2 1 ` 1 2 q2 2 (2.29) 14 Differentiating V3 and substituting (2.26) and (2.28), we get 9V3 “ ´λ2k2 eKext pE ´ Edesq2 9z2 ` q1 r 9q1 ` λkeKext pE ´ Edesq 9zs ` q2 9q2 “ ´λ2k2 eKext pE ´ Edesq2 9z2 2 ` q1 r 9q1 ´ ϕ2 ´ k1q1s ` p 9q1 ´ ϕ2qp:q1 ´ 9ϕ2q “ ´λ2k2 eKext pE ´ Edesq2 9z2 ´ k1q2 1 ` q2 r:q1 ´ 9ϕ2 ` q1s (2.30) Our choice of :q1 “ 9φ2 ´ q1 ´ k2q2, k2 ą 0 (2.31) results in a negative semi-definite 9V3 and yields the controller v “ :φ1 ` 9φ2 ´ q1 ´ k2q2 (2.32) The above equation was obtained from (2.31) by substituting the value of :e in (2.16) and (2.24) into (2.31). From the definition of ϕ1, it is clear that :ϕ1 will involve the third derivative of z. This is not a problem since the second derivative of z can be computed easily from (2.15) as ;z “ ´λ 1 mt Kextp 9z ´ 9eq (2.33) 15 2.3.3 Stability Analysis Using (2.10), (2.11), (2.15), (2.18), (2.24) and (2.28) it can be shown that pE ´ Edes, e, 9eq “ p0, 0, 0q ô pE ´ Edes, q1, q2q “ p0, 0, 0q Therefore, V3 in (2.29) is a candidate Lyapunov function for investigating the stability of the equilibrium in (2.20). In both flight and contact phases (λ “ 0 and λ “ 1), the control law in (2.32), results in (2.30) 9V3 ď 0 (2.34) Therefore, pE ´ Edes, q1, q2q “ p0, 0, 0q is stable. Remark 1. The stability of pE ´ Edes, q1, q2q “ p0, 0, 0q in the flight and contact phases do not guarantee stability for the hybrid dynamics. The stability of the hybrid system is analyzed next . 2.4 Hopping on a Viscoelastic Ground with Inertia 2.4.1 Discrete Controller for Stabilization and Control Apex Height To investigate the stability of the hybrid dynamic system, we use a Poincar´e map with the Poincar´e section defined at the point of touch-down, i.e. the time instant prior to impact. To restrict the Poincar´e section to a one-dimensional configuration space, we make the following assumption: 16 Assumption 4. The parameters of the viscoelastic ground (m3, Cext, Kext) are such that the settling time of the system is less than the hopping period of the robot. Remark 2. The above assumption allows us to infer that x3, 9x3 « 0 at the time of touch- down. It will be shown later that the assumption is not overly restrictive. Assuming that x3, 9x3, e, 9e « 0 at the time of touch-down, the modified Poincar´e section is chosen as where X is defined as Z :“ tX P R| z “ zd, e “ 0, 9e “ 0, 9z ă 0u (2.35) X “ 9z (2.36) To use the same set of variables used in the backstepping controller in section 2.3 (designed for the purely elastic ground), namely, pE ´ Edesq, we defined the Poincar´e section using the coordinates Ψ, where Ψ is defined by the coordinate transformation Hp¨q : R ñ R, as follows Ψ “ pE ´ Edesq “ HpXq (2.37) The Poincar´e map QpΨq and the sequence of points Ψk P Hp ¯Zq now satisfy Ψk`1 “ QpΨkq, QpΨq : Hp ¯Zq ÞÑ Hp ¯Zq with periodic point Ψ˚ defined as Ψ˚ “ QpΨ˚q (2.38) (2.39) For the elastic ground, the periodic point which achieves the desired apex height is given by Ψ˚ “ p ¯E ´ Edesq 17 (2.40) where ¯E is the energy of the system in steady state when the backstepping controller is invoked with Ed “ Edes. The value of ¯E is initially unknown but can be determined after the robot has hopped a few times - see section 2.4.2. We define the error state ηk as ηk “ pΨk ´ Ψ˚q “ pEk ´ ¯Eq By linearizing the Poincar´e map about Ψ˚, we have the approximate discrete dynamics given by ηk`1 “ A0 ηk A0 ∆“ The periodic point will be asymptotically stable if and only if dQpΨq dΨ ˇˇˇˇΨ“Ψ˚ ρpA0qă 1 (2.41) (2.42) where ρpAoq is the spectral radius of Ao. To design the discrete controller, we describe Ψ as follows Ψ “ Φ ` u (2.43) Φ ∆“ pE ´ Edq, u ∆“ pEd ´ Edesq where Ed is desired level of energy for a given hop. The new Poincar´e map ¯QpΦ, uq and the sequence of points Φk P Hp ¯Zq satisfy Φk`1 “ ¯QpΦk, ukq, ¯QpΦ, uq : Hp ¯Zq ˆ R ÞÑ Hp ¯Zq with periodic point Φ˚ defined as Φ˚ “ ¯QpΦ˚, u˚q (2.44) (2.45) 18 For the viscoelastic ground with inertia, there exists the following equilibrium point If we define the error state Φ˚ “ p ¯E ´ Edq, u˚ “ 0 µk ∆“ pΦk ´ Φ˚q “ pEk ´ ¯Eq (2.46) (2.47) the Poincar´e map ¯QpΦk, ukq can be linearized about pΦ˚, u˚q to yield the following linear discrete-time system yk “ µk ¯B ∆“ d ¯QpΦ, uq du (2.48) Φ“Φ˚ ˇˇˇˇ u“u˚ µk`1 “ ¯A µk ` ¯Buk, ¯A ∆“ , d ¯QpΦ, uq dΦ ˇˇˇˇ Φ“Φ˚ u“u˚ To converge the system energy from its level at the equilibrium configuration ¯E to the desired value Edes, we propose to use integral control with the integrator defined as θk`1 “ θk ` pE0 ´ ykq, E0 ∆“ pEdes ´ ¯Eq (2.49) where E0 is the desired value of the output variable y. The integrator-augmented discrete system has the form λk`1 “ Aλk ` Buk ` E0,λ , B ∆“» A ∆“» —– —– fi ffifl ´1 1 ¯A 0 k k “„ µT fi ffifl ¯B 0 If tA, Bu is controllable, the input can be chosen as uk “ Kλk 19 θk T (2.50) (2.51) where K satisfies ρpA ` BKqă 1 2.4.2 Simulation Results: Apex Height Control We investigate the behavior of the two-DOF robot hopping on the viscoelastic ground with inertia; the discrete integral controller is used along with the continuous backstepping controller. The mass, stiffness, and damping properties were assumed to be m1 “ 2.6 kg, m2 “ 0.8 kg, m3 “ 0.04 kg % “ 0.06 m, Kext “ 2800 N{m, Cext “ 22 Ns{m (2.52) The mass of the ground was assumed to be 5% of the lower mass m2 and the damping coefficient Cext was chosen such that the ground is slightly overdamped. The desired apex height was chosen to be hd “ 0.25 m and, similar to the last simulation, rd “ 0.0994 m was used. The continuous controller parameters were chosen as ke “ 0.001, k1 “ 200, k2 “ 100 The matrix A, defined in (2.50), was found to have eigenvalues: 1, and 0.004. Since tA, Bu is controllable, the controller gains were chosen as K “„ ´0.35 0.7  20 This places the eigenvalues of the closed loop system at 0.9, and 0.3. The initial conditions were assumed to be x2p0q “ 0.06 m, x3p0q “ ´0.012 m, yp0q “ 0.075 m 9x2p0q “ 0.0 m{s, 9x3p0q “ 0.0 m{s, 9yp0q “ 0.0 m{s (2.53) and the initial value of the integrator state was set to zero. The simulation results are shown in Figs.2.2, 2.3, 2.4 and 2.5. The displacements of the upper mass, lower mass, COM, and ground mass are plotted in Fig.2.2. The contact phases (not explicitly shown in Fig.2.2) are the time intervals during which x2 “ pX2 ´ x3q “ 0.06 m. The value of X2p0q “ 0.048 m indicates that the spring of the viscoelastic ground is initially compressed due to the weight of the robot. The simulation is comprised of two phases. In the initial phase, 0 ď t ď 1.8 s, the discrete controller was switched off and the backstepping controller was used with Ed “ Edes. During this phase (five hops), the apex X1 z 0.25 ) m ( 3 x , z , 2 X , 1 X 6 0 0 . 0.00 -0.10 0.0 x3 1.8 X2 2.5 time (s) 5.0 Figure 2.2: Simulation results for hopping on a viscoelastic ground with inertia. Plot of the position of the upper mass X1, the lower mass X2, COM z, and the ground mass x3, as a function of time. 21 0.0 -1.0 1 ´0.81 t “ 1.8 pE ´ Edesq 3 5 7 k 9 11 Figure 2.3: Simulation results for hopping on a viscoelastic ground with inertia: Plot of pE ´ Edesq at the end of the k-th hop (immediately before touch-down), k “ 1, 2, ¨ ¨ ¨ , 12. The actual time scale in the figure is identical to the time scale in Fig.2.2 height of the robot converges to a constant value that corresponds to E “ ¯E, which was defined in the context of (2.40). The value of ¯E was found to be 2.21 J and E0 “ 0.81 J (the deficit) was computed using (2.49). Using the value of E0, the discrete controller was switched on at t “ 1.8 s. The first element of Ψ is plotted in Fig.2.3; it corresponds to the value of pE ´ Edesq at the end of the k-th hop, k “ 1, 2, ¨ ¨ ¨ , 12 - see Fig.2.3. Figure 2.4 shows the control input (F1), it can be seen that the force is zero when there is no relative displacement between the masses (e “ 9e “ 0). The displacement of the lower mass and the ground is shown in Fig.2.5 for one hop. It can be seen that the lower mass breaks contact with the ground mass below the datum 300 1 F 0 -300 0 2.5 time (s) 5 Figure 2.4: Plot of the Force F1 applied by the prismatic joint. 22 k “ 11 0.12 0.06 0.00 -0.06 contact phase flight phase k “ 12 x2 X2 x3 settling time of foundation mass 4.26 4.405 4.674 time (s) Figure 2.5: Simulation results for hopping on a viscoelastic ground with inertia: Plot of X2, x2, and x3 for the hop between k “ 11 and k “ 12. (x2 ą %, x3 ă 0) at t “ 4.405 s. While the robot is in flight, the ground mass settles to its equilibrium configuration; the response is overdamped, as expected from the choice of parameters in (2.52), and has a 2% settling time of 0.022 s. Since the settling time of the ground mass is much smaller than the flight phase, it becomes clear that Assumption 4 is not overly restrictive. In steady state (k ě 11), the integral controller was found to command the backstepping controller with Ed “ 3.93 J. This is higher than the value of Edes by an amount equal to 0.91 J, which is larger than the value of E0 “ 0.81 J. This is not surprising since a higher value of Ed results in a higher deficit due to greater losses associated with hopping with a larger apex height. A video animation of a two-DOF prismatic joint robot hopping on a viscoelastic ground with inertia has been uploaded as supplementary material. It shows the hopper reaching the desired apex height of hd “ 0.25 m starting from rest. 2.5 Conclusion This chapter presents a method for controlling the apex height of a two-mass robot hop- 23 ping on a viscoelastic ground with inertia. This problem, which has not been investigated earlier, is more challenging than the problems of hopping on rigid and elastic grounds since the system has an extra DOF and there is energy loss due to impact and damping. A continuous-time backstepping controller was used in concert with a discrete-time integral controller to meet the control objective. The backstepping controller regulates the energy of the robot using the internal DOF and simultaneously eliminates this DOF to enable the robot reach its desired apex height. We use the same backstepping controller was devel- oped for elastic ground results is steady-state error. The discrete-time integral controller eliminates this error by commanding the backstepping controller to regulate the energy to a commensurately higher level. Since there is loss of energy in every hop, the backstepping controller has to remain active for all hops. A video animation of apex height control is includes to provide a glimpse of the dynamic behavior. 24 Chapter 3 Apex Height Control of a Two-DOF Ankle-Knee-Hip Robot Hopping on a Rigid Ground 3.1 Introduction In this chapter, we develop a strategy for controlling the apex height of a two-DOF Ankle- Knee-Hip (AKH) robot, hopping on a rigid ground [34]. Although the AKH robot has the same number of DOFs as the prismatic-joint robot, studied earlier [21,33] and in Chapter 2, its kinematic structure is more anthropomorphic and similar to biped robots. The control problem for the AKH is more challenging than the prismatic-joint robot due to the revolute nature of the joints. Hopping on a rigid ground results in energy losses due to impact of the robot with the ground. A continuous and a discrete controller with integral action are developed to drive the apex height of the AKH robot to a desired value. To compensate for the losses, the continuous controller employs negative damping while the velocity of the COM is moving upwards during the contact phase. Using a Poincar´e map at the take- off point, we constructed a discrete-time system. The discrete-time controller adjusts the negative damping parameter used by the continuous controller to control the apex height. The dependence of the apex height on the controller parameters is studied to understand the 25 role of constraints imposed by the robot structure. Simulation results are presented to show the efficacy of the control strategy. This Chapter is structured as follows. The dynamics of the robot is presented in Section 3.2. The continuous and discrete controllers are presented in Sections 3.3 and 3.4. The effect of continuous controller parameters on hopping behavior is discussed in Section 3.5. Numerical simulation results are presented in Section 3.6. Section 3.7 contains concluding remarks. 3.2 Dynamics of AKH Robot 3.2.1 System Description and Model Consider the AKH hopping robot shown in Fig. 3.1. It is comprised of an upper mass mb and a lower mass mf, both of which are constrained to move along the vertical axis. The masses are connected by two links, each of which have mass m and length %. The mass moment of inertia of the two links are equal and denoted by I. The displacements of the masses mb and mf are denoted by y2 and y1, respectively. These two variables correspond to the two degrees of freedom of the robot in the flight phase. The angular displacement of the upper link, measured counter-clockwise with respect to the vertical axis, is denoted by φ. The robot has a single actuator that drives the angular coordinate φ, the torque applied by this actuator is denoted by τ . The vertical force applied on the lower mass mf by the the ground during contact is denoted by Fext. The dynamic model of the robot is obtained using Lagrange’s equations: Mpqq:q ` Npq, 9qq “ Q (3.1) 26 g mb m, % τ, φ m, % mf %0 y2 y1 Fext Figure 3.1: The ankle-knee-hip robot. where q “„ y1 φ T of rMijs2ˆ2 and rNis2ˆ1 are , Q “„ Fext τ T is the vector of generalized forces, and the elements M11 “ mt ∆“ pmb ` mf ` 2mq M12 “ M21 “ ´2%pm ` mbq sin φ M22 “ 1 3 %2 r5m ` 6mb ´ 3pm ` 2mbq cos 2φs N1 “ gmt ´ 2%pm ` mbq cos φ 9φ2 N2 “ 2% r´gpm ` mbq ` %pm ` mbq cos φs sin φ (3.2) The hybrid dynamics of the hopping robot is comprised of the flight phase, the impact phase, and the contact phase. 27 3.2.2 Flight Phase In the flight phase, the robot has two DOF and the following conditions hold: y1 ą %0 Fext “ 0 (3.3) The equation of motion is described by (3.1), where Fext “ 0. 3.2.3 Impact Phase The impact phase refers to infinitesimal intervals of time t P rt´, t`s during which the lower mass mf comes in contact with the ground and the condition y1 ą %0 changes to %0 py1 “ %0q. We make the following assumptions: Assumption 5. The impact between lower mass mf and the ground is inelastic, i.e., 9y1pt`q “ 0. Assumption 6. The control torque τ is not impulsive. The position variables satisfy: y1pt`q “ y1pt´q φpt`q “ φpt´q By integrating (3.1) over the interval rt´, t`s , we get 9y1pt`q “ 0 9φpt`q “ 9φpt´q ` M21 M22 9y1pt´q (3.4) (3.5) 28 3.2.4 Contact Phase The contact phase commences immediately after the lower mass mf makes contact with the ground. We now assume: Assumption 7. The external force Fext acting on the lower mass mf is non-negative, i.e., Fext ě 0. During contact phase, y1 “ %0 and 9y1 “ :y1 “ 0; the system DOF one. Using (3.1), the value of the Fext can be computed as Fext “ N1 ` pM12{M22q pτ ´ N2q (3.6) Substituting (3.5) into (3.1), we get M12{M22 1 Mpqq:q `» —– ñ:q “ M´1pqq» —– fi ffifl M12{M22 1 N2 “» —– fi pτ ´ N2q “» —– ffifl τ fi ffifl fi ffifl pτ ´ N2q M22 (3.7) 0 1 M12{M22 1 At the instant when the system switches from the contact phase to the flight phase, Fext equals zero, which implies 3.2.5 Apex Height N1 ` pM12{M22q pτ ´ N2q “ 0 (3.8) The COM in the flight phase is denoted by 29 z “ 1 mt„mf y1 ` mpy1 ` 1 2 “y1 ` 2%mz cos φ, mz ∆“ % cos φq ` mpy1 ` m ` mb mt 3 2 % cos φq ` mbpy1 ` 2% cos φq (3.9) For each flight phase, the apex height h is defined as the maximum value of z. 3.3 Continuous Control Design 3.3.1 Relative Displacement of COM Using (3.9), we define the displacement of the COM relative to the lower mass as r ∆“ pz ´ y1q “ 2%mz cos φ ñ 9r “ ´2%mz sin φ 9φ (3.10) (3.11) We define the desired equilibrium point of the system to be pr, 9rq “ prd, 0q, where rd ą 0 is the desired value of r. 3.3.2 Contact Phase Control Design During the contact phase, the system has one DOF since y1 ” %0. To transform this dynamics described by (3.7) into the normal form [26], we use (3.10), and (3.11) to define the states ξ1 “ r ´ rd,ξ 2 “ 9r (3.12) 30 The dynamics of the system in normal form is 9ξ1 “ ξ2, 9ξ2 “ ´ 2%mz cos φ 9φ2 ´ 2%mz sin φ :φ (3.13) Substituting the value of :φ in (3.7) into (3.13), and defining the torque τ to be τ “ N2 ` M22 2mz% sin φ p´v ´ 2mz% cos φ 9φ2q 9ξ1 “ ξ2, 9ξ2 “ v (3.14) (3.15) v “ ´ω2 nξ1 ´ 2ζωnαξ2,ζ ă 1 (3.16) we get We choose v as where 1 ν α “$’’& ’’% ξ2 ď 0 ξ2 ą 0 ,ν ă 0 (3.17) which will make the relative displacement r behave like an underdamped mass-spring-damper system when α “ 1 or ξ2 ď 0, and a mass-spring system with negative damping for ξ2 ą 0. The magnitude of ν will determine the amount of energy that will be added to the system during the contact phase to achieve the desired apex height. 3.3.3 Flight Phase Control Design In the Flight phase, the system will have two DOF. To transform the system to the 31 normal form [26], we use the states defined in (3.12) together with η1 “ z ´ zd η2 “ 9z (3.18) where zd “ rd ` %0. The dynamics of the system in normal form can be written as 9η1 “ 9z 9η2 “ :y1 ´ 2mz% cos φ 9φ2 ´ 2mz% sin φ :φ 9ξ1 “ ξ2 9ξ2 “ ´2mz% cos φ 9φ2 ´ 2mz% sin φ :φ (3.19) Substituting the values of :y1 and :φ from (3.1) into (3.19), and defining the torque τ to be τ “ N2 ` p1{M11q« ´ M21N1 ´ pM11M22 ´ M12M21q´ν ` 2mz% cos φ 9φ2¯ 2mz% sin φ ff (3.20) we get 9η1 “ η2 9η2 “ ´g 9ξ1 “ ξ2 9ξ2 “ ν (3.21) We choose v to be given by v “ ´ω2 nξ1 ´ 2ζωnξ2,ζ ă 1 (3.22) This will make the of the relative displacement r like an underdamped mass-spring-damper. 3.3.4 Closed-Loop System Dynamics With the continuous controllers for the contact and flight phases given by (3.14), and (3.20), the hybrid dynamics of the closed-loop system is described by the following equations 32 in the contact phase 9ξ1 “ ξ2, 9ξ2 “ ´ ω2 nξ1 ´ 2ζωnαξ2 (3.23) and the following equations in the flight phase 9η1 “ η2 9η2 “ ´g 9ξ1 “ ξ2 9ξ2 “ ´ω2 nξ1 ´ 2ζωnξ2 (3.24) The contact phase ends when Fext “ 0, which is described by (3.8). By substituting (3.2), (3.9), (3.14), and (3.16) into (3.8), we get the relation g ´ ω2 nξ1 ´ 2ζωnνξ2 “ 0 (3.25) which will be used for analysis in section 3.5. 3.4 Discrete Control Design for the Apex Height 3.4.1 Poincar´e Section A single hop starts at the takeoff point in the flight phase and ends when the lower mass breaks contact with the ground in the contact phase, immediately prior to takeoff. Each hop can be divided into three time intervals: ∆t1 denoting the time during which the robot is in flight phase, ∆t2 denoting the time in the contact phase with 9r ď 0, and ∆t3 denoting the 33 time in the contact phase with 9r ą 0. The total time required for a single hop is T “∆t1 ` ∆t2 ` ∆t3 (3.26) For a single hop, starting at t “ 0, it can be shown that the following conditions hold η1p0q “ ξ1p0q,η 2p0q “ ξ2p0q g ´ ω2 nξ1p0q ´ 2ζωnνξ2p0q “ 0 (3.27) where the last condition is obtained from (3.25). The three constraints in (3.27) imply that the takeoff point is described by one free variable, which we denote by χ. The variable χ belong to the set Ω, where Ω “ tξ2 | η1 “ ξ1,η 2 “ ξ2,ω 2 nξ1 ` 2ζωnνξ2 “ gu (3.28) The first return map between two hops can be described by the discrete system dynamics χpk ` 1q “ P pχpkq, ∆t1, ∆t2, ∆t3q (3.29) where P p.q is the Poincar´e map. The time intervals ∆t1,∆ t2, and∆ t3 can be solved using the following conditions η1p∆t1q “ ξ1,ξ 2p∆t1 ` ∆t2q “ 0 g ´ ω2 nξ1pT q ´ 2ζωnνξ2pT q “ 0 (3.30) Therefore, the return map can be written as χpk ` 1q “ P pχpkqq (3.31) 34 A fixed point χ satisfies χ˚ “ P pχ˚q (3.32) By linearizing the dynamics described by (3.31) about χ˚, we claim asymptotic stability of χ˚ if ρ„BP pχq Bχ |χ“χ˚ ă 1 (3.33) where ρp.q is the spectral radius. 3.4.2 Discrete Controller Design For a fixed set of parameters of the continuous controller, namely, ζ, ωn, ν and rd, a given value of χ results in an unique apex height h. Therefore, given a desired apex height hd, we first solve the inverse problem of determining continuous controller parameter values that result in χ “ χd. We stabilize the point χ “ χd by varying one or more of the continuous controller parameters at the takeoff point. Here, we choose to vary ν only; the dynamics of the controlled system (3.31) can be described by the map χpk ` 1q “ P pχpkq,ν q (3.34) By linearizing the above map about the fixed point χ˚ “ χd, ν˚ “ νd, where νd is the parameter value that results in χd, we get χpk ` 1q “ P pχ˚,ν ˚q ` Apχ ´ χ˚q ` Bpν ´ ν˚q A “„BP pχ, νq Bχ χ “ χ˚ ν “ ν ˚ , B “„ BP pχ, νq Bν χ “ χ˚ ν “ ν ˚ (3.35) 35 By defining the errors we get from (3.35) epkq “ χpkq ´ χ˚, upkq “ νpkq ´ ν˚ (3.36) epk ` 1q “ Aepkq ` Bupkq (3.37) For purpose of integral control, we define the output as ypkq “ epkq (3.38) To converge the apex height to the desired value, we propose to use integral control with the integrator defined as θpk ` 1q “ θpkq ´ ypkq (3.39) The integrator-augmented discrete system has the form λpk ` 1q “ Aλpkq ` Bupkq λpkq “„ epkq θpkq T (3.40) A ∆“» —– A 0 ´1 1 fi ffifl , B ∆“» —– B 0 fi ffifl where A and B is defined in (3.37). If tA, Bu is controllable, the input can be chosen as upkq “ Kλpkq (3.41) such that ρpA ` BKqă 1 36 and the closed-loop system is asymptotically stable. 3.5 Effect of Continuous Controller Parameters on Hopping Behavior 3.5.1 Effect of the Parameters on Apex Height The apex height h depends on the total energy of the system at takeoff point, where (3.25) holds. Substituting (3.12), (3.16) and (3.17) into (3.25), we get pr ´ rdq “ g ´ 2ζνωn 9r ω2 n The total energy of the robot at the takeoff point is E “ 1 2 mt 9r2 ` mtg„ g ´ 2ζνωn 9r ω2 n ` rd (3.42) (3.43) Assuming that the relative motion between the upper and lower masses quickly settles to zero and no significant work is done by the actuator during this time, the apex height h is computed as h “ E mtg “ 1 2g 9r2 ` g ´ 2ζνωn 9r ω2 n ` rd (3.44) To examine the effect of the control parameters (ωn, ν, ζ) and rd on the apex height, we 37 compute the partial derivatives of h in (3.44) with respect to the parameters; they are Bh Bωn Bh Bζ “ ´ “ ´ 2ν 9r ωn 2 pg ´ ζνωn 9rq ω3 n ą 0, ă 0, ă 0 “ ´ 2ζ 9r ωn “ 1 ą 0 Bh Bν Bh Brd (3.45) (3.46) Clearly, the apex height is higher for smaller values of ωn and ν and higher values of ζ, and rd. The choice of the parameters are however subject to the constraint φ ­“ 0, or r ă rmax where rmax “ max φ 2%mz cos φ “ 2%mz The procedure for choosing parameters that satisfy this constraint is discussed next. 3.5.2 Limitation on Choice of Parameter Values Contact Phase Constraint For the contact phase, the solution of the closed-loop system dynamics described by (3.15) and (3.16) is given by sinpωνtq ` ξ10 cospωνtqff ζνξ10 ξ1 “e´ζνωnt« a1 ´ ζ 2ν2 ξ2 “ ´ ζνωne´ζνωntξ10« ζν a1 ´ ζ 2ν2 sinpωνtq ` cospωνtqff ` ωne´ζνωntξ10«ζν cospωνtq ´a1 ´ ζ 2ν2 sinpωνtqff ∆“ ωna1 ´ ζ 2ν2 ων (3.47) where ξ10 is the value of ξ1 when ξ2 “ 0, i.e. when ξ2 switches from a negative value 38 to a positive value and α switches from 1 to ν. To avoid oscillation during the contact phase, the system must satisfy pωna1 ´ ζ 2ν2tqă π at the takeoff point. By substituting (ωna1 ´ ζ 2ν2tqă π in (3.47) and then substituting the expressions for ξ1 and ξ2 in (3.25), we can get the following condition: ´ω2 ne ´πζν ?1´ζ2ν2 ξ10 ą g (3.48) Flight Phase Constraint For the flight phase, the initial conditions must satisfy the takeoff condition in (3.25). The solution of the controllable states in (3.21) is ξ1 “ e´ζωnt rC2 sinpωdtq ` C1 cospωdtqs ξ2 “ ´ ζωne´ζωnt«C2 sinpωdtq ` C1 cospωdtqff ` ωde´ζωnt«C2 cospωdtq ´ C1 sinpωdtqff ωd C1 ∆“ ∆“ ωna1 ´ ζ 2 g ´ 2ζνωnξ20 ω2 n , C2 ∆“ ξ20 ` ζωnC1 ωd (3.49) and ξ20 is the initial value of ξ2 at the takeoff point. Since the controllable states depict an underdamped mass-spring-damper system, ξ1 will reach its maximum value during the first half of the first cycle of oscillation when ξ2 “ 0. By setting ξ2 “ 0 in (3.49), we get t “ 1 ωd tan´1„ ξ20ωd g ´ ζωnξ20 (3.50) The value of ξ1 at t in (3.50) must be less than ξ1,max where ξ1,max “ prmax ´ rdq “ 2%mz ´ rd. Therefore, the following condition must satisfied at the takeoff point e´ζωnt rC2 sinpωdtq ` C1 cospωdtqs ă p2lmz ´ rdq (3.51) 39 3.6 Simulation We present simulation results showing the AKH robot hopping to a desired apex height starting from rest. The robot parameters were assumed to be mb “0.7 kg, mf “ 0.15 kg, m “ 0.4 kg,% “ 0.2 m,% 0 “ 0.05 m (3.52) The apex height was chosen as hd “ 0.35 m (3.53) and the continuous controller parameters were chosen to be ζ “ 0.13, rd “ 0.13 m (3.54) Note that the value of rd was chosen to be approximately half the value of rmax. We find the values of ν˚ and χ˚ that corresponds to hd “ 0.35 through an exhaustive numerical search; the results are shown in Figs. 3.2 (a)-(d) for four different values of controller parameter ωn. The figures show that for a given value of ν, there may be multiple apex heights, each of which will correspond to an unique value of χ. The specific value of h to which the apex height will depend on the initial conditions. To converge to the desired value, we use integral control. In our simulation, the remaining two controller parameters ωn and ν were chosen as follows: ωn was chosen to be 30 since Fig. 3.2 (c). admits a solution for ν corresponding to hd “ 0.35. The values of ν, ωn and χ are provided below: ν “ ´1.19,ω n “ 30,χ ˚ “ 1.669m{s (3.55) 40 0.6 h 0.2 -2 0.8 h 0.2 -2 (a) -1 ν pcq -1 ν 0.5 h 0 0.2 -2 0.6 h 0 0.2 -2 (b) -1 ν pdq -1 ν 0 0 Figure 3.2: Apex height h is plotted with respect to ν for periodic orbits obtained with rd “ 0.13 m, ζ “ 0.13 and (a) ωn “ 20, (b) ωn “ 25, (c) ωn “ 30, and (d) ωn “ 35. The gains of the discrete controller were chosen to place the eigenvalues inside the unit circle. The choice K “„ 0.2 0.5  (3.56) placed the eigenvalues at 0.814 and ´0.87. Initially, the robot was assumed to be at rest on the ground; the initial conditions are y1p0q “ 0.05 m 9y1 “ 0 m{s φp0q “ 0.75 rad 9φ “ 0 rad{s (3.57) The simulation results are shown in Figs. 3.3, 3.4 and 3.5. The height of the upper and lower masses and the COM are plotted in Fig. 3.3; it can be seen that the apex height 41 ) m ( z , 2 y , 1 y 0.5 0.4 0.35 0.3 0.2 0.1 0.05 0 y1 y2 z k “ 1 0 0.5 1 1.5 2 2.5 3 3.5 time (s) k “ 9 4 Figure 3.3: Plots of the heights of the upper and lower masses and the COM. converges to its desired value in seven hops. The contact phase is the period when the value of y1 remains constant at 0.05 and y1 ą 0.05 denotes the flight phase. The state of the discrete-time system is shown in Fig. 3.4; once again it can be seen that it converges to its desired value in seven hops. There is some overshoot but this can be eliminated by properly choosing the controller gains. The control input (hip torque) is shown in Fig.3.5. In this context, it should be pointed out that the control torque can be reduced by 50% if the actuator is moved from the hip to the knee as the angular displacement of the knee is twice that of the hip. A video animation of a two-DOF AKH robot hopping on a rigid ground has been uploaded 0.5 e 0 -0.5 1 3 5 k 7 9 Figure 3.4: Plot of the state of the discrete-time system at the beginning of the first nine hops. 42 ) . m N ( τ 50 0 -50 0 1 2 3 4 5 time (s) Figure 3.5: Plot of the torque applied by the hip actuator. as supplementary material. It shows the hopper reaching the desired apex height of hd “ 0.35 m starting from rest. 3.7 Conclusion This chapter presented a method for controlling the apex height of a two-DOF AKH robot, hopping on a rigid ground. The dynamics of the robot is modeled in flight, impact, and contact phases separately. A continuous-time controller is designed to make the robot behave like a mass-spring-damper system. In the contact phase, the continuous controller employs positive damping to arrest the motion of the COM when it is moving downwards, and negative damping when the COM is moving upwards. The negative damping is introduced to compensate for the energy loss due to impact. A Poincar´e map of the hopping behavior is constructed and asymptotic stability of the hybrid system to the desired apex height is guaranteed by designing a discrete controller with integral action. The constraint imposed by the robot structure and the effect of continuous controller parameters on hopping behavior is discussed. A simulation of the AKH robot, hopping to a desired apex height from rest, is presented. A video animation of apex height control is includes to provide a glimpse of the dynamic behavior. 43 Chapter 4 Apex Height Control of a Two-DOF Ankle-Knee-Hip Robot Hopping on an Elastic Ground and a Viscoelastic Ground with Inertia 4.1 Introduction In this chapter, we develop a strategy for controlling the apex height of the Ankle-Knee- Hip (AKH) robot introduced in Chapter 3 for hopping on an elastic ground and a viscoelastic ground with inertia. The ground is modeled as massless spring for the elastic ground case and a mass-spring-damper system for the viscoelastic ground with inertia. Since the elastic ground is massless, there is no loss of energy when the robot makes contact with the ground. A continuous backstepping controller is designed to control the mechanical energy of the robot to a desired values and stabilize the internal dynamics to reach the desired apex height. For the viscoelastic ground, there will be energy losses due to impact between the robot and the ground and from dissipation in the ground due to damping. These losses are unknown but have to be compensated; this makes the apex height control problem challenging. The continuous backstepping controller developed for the elastic ground is used to converge the energy of the robot to a desired value and stabilize the internal dynamics 44 of the robot. This results in steady-state error, which is eliminated using a discrete-time controller with integral action. The discrete-time controller is constructed using Poincar´e map at the point just before impact. Simulation results are presented to demonstrate the efficacy of the controller. This chapter is structured as follows. The dynamics of the robot and the ground is presented in Section 4.2. The continuous and discrete controllers are presented in Sections 4.3 and 4.4. Numerical simulation results are presented in Section 4.5. Section 4.6 contains concluding remarks. 4.2 Dynamics of AKH Robot 4.2.1 System Description and Model Consider the AKH hopping robot shown in Fig. 4.1. The robot hops on a viscoelastic ground of mass mg, stiffness constant Kext, and damping coefficient Cext. The displacements of the ground mass mg is denoted by x1. The robot is comprised of an upper mass mb and a lower mass mf; these masses are connected by two links, each of which have mass m and length %. The mass moment of inertia of the two links about of their COM are equal and denoted by I. The masses mb and mf are constrained to move along the vertical axis; their displacement are denoted by y3 and y2, respectively. The relative displacement between the lower mass mf and the ground mass mg is denoted by x2. The angular displacement of the upper link, measured counter-clockwise with respect to the vertical axis, is denoted by φ. The robot has a single actuator that drives the angular coordinate φ, the torque applied by this actuator is denoted by τ . The dynamic model of the robot is obtained using Lagrange’s 45 mb m, % τ, φ g mf m, % %0 mg undeformed configuration of spring y3 y2 x2 F1 x1 Kext Cext Kext Cext Figure 4.1: The ankle-knee-hip robot. equations: Mpqq:q ` Npq, 9qq “ Q (4.1) where q “ „ x1 x2 φ T and Q “ „ Fext F1 τ T are the vectors of generalized dis- 46 placement and forces, and the elements of rMijs3ˆ3 and rNis3ˆ1 are M11 “ mt ` mg M12 “ M21 “ M22 “ mt M13 “ M23 “ M31 “ M32 “ ´2%pm ` mbq sin φ M33 “ 1 3 %2ˆ5m ` 6mb ´ 3pm ` 2mbq cosp2φq˙ N1 “ N2 “ gpmf ` mb ` 2mq ´ 2%pm ` mbq cos φ 9φ2 N3 “ 1 2 %ˆ4%pm ` 2mbq cos φ 9φ2 ´ gpmf ` mb ` 2mq˙ sin φ Fext “ ´Cext 9x1 ´ Kextx1 mt ∆“ pmf ` mb ` 2mq (4.2) The hybrid dynamics of the hopping robot is comprised of the flight phase, the impact phase, and the contact phase. 4.2.2 Flight Phase In the flight phase, the robot has three DOF and the following conditions hold: x2 ą %0 F1 “ 0 (4.3) The equation of motion is described by (4.1), where F1 “ 0. 47 4.2.3 Impact Phase The impact phase refers to infinitesimal intervals of time t P rt´, t`s during which mf comes in contact with mg and x2 “ %0. We make the following assumptions: Assumption 8. The impact between mass mf and mg is inelastic, i.e., mf and mg have identical velocities immediately after impact. Assumption 9. The control torque τ is not impulsive. The position variables satisfy: x1pt`q “ x1pt´q x2pt`q “ x2pt´q,φ pt`q “ φpt´q (4.4) By integrating (4.1) over the interval rt´, t`s , we compute the velocity variables 9x1pt`q “ 9x1pt´q ` 9x2pt´q, 9x2pt`q “ 0 mg mf ` mg M13 M33 p 9x2pt´q ´ 9x1pt`qq (4.5) 9φpt`q “ 9φpt´q ` 4.2.4 Contact Phase The contact phase commences immediately after impact and continues for the time du- ration in which mf remains in contact with mg. We assume: Assumption 10. The force F1 acting on both mf and mg is non-negative, i.e. F1 ě 0. During the contact phase, x2 “ %0 and 9x2 “ :x2 “ 0; the system has two DOF. Using 48 (4.1), the value of the F1 can be computed as F1 “N2 ` `pM21M33 ´ M23M31qpFext ´ N1q ` pM11M23 ´ M13M21qpτ ´ N3q˘ pM11M33 ´ M13M31q (4.6) At the instant when the system switches from the contact phase to the flight phase, F1 equals zero. 4.2.5 Apex Height The COM of the robot is denoted by 1 z “ mt“mfpx1 ` x2q ` mpx1 ` x2 ` ` mpx1 ` x2 ` 3 2 1 2 % cos φq “x1 ` x2 ` 2%mz cos φ, (4.7) % cos φq ` mbpx1 ` x2 ` 2% cos φq‰ m ` mb mz ∆“ mt For each flight phase, the apex height is defined as the maximum value of z, and denoted by h. 4.3 Hybrid Control Strategy for Hopping on a Purely Elastic Ground 4.3.1 New Coordinates For a purely elastic ground, we have 49 mg “ 0, Cext “ 0, Kext ‰ 0 (4.8) There is no impact phase. Furthermore, substitution of (4.8) in (4.1) indicates x1 ” 0 in the flight phase. Therefore, the system has two DOF in the flight phase. The system has two DOF in the contact phase as well since x2 “ %0 and 9x2 “ 0. For the objective of apex height control, we define r to be the height of the COM of the robot relative to that of the lower mass mf. Using (4.7), it can be shown r ∆“ pz ´ px1 ` x2qq “ 2 mz % cos φ ñ 9r “ ´2 mz % sin φ 9φ Next, we define the error e as e “ pr ´ rdq (4.9) (4.10) (4.11) where rd ą 0 is some desired value of r. From (4.9) and (4.11) it can be verified that e ” 0 Ñ 9e ” 0 Ñ 9r ” 0, which implies no relative motion between the two masses. 4.3.2 Normal Form For the convenience of control design, we transform the system to the normal form [26] using the following transformations η1 “ z ´ zd zd η2 “ 9zξ 1 “ eξ ∆“ z |px1“0,x2“!0,r“rdq “ rd ` %0 2 “ 9r (4.12) The transformation in (4.12) is diffeomorphic only if φ ‰ 0. The dynamic of the system in 50 normal form can be written as 9η1 9η2 9ξ1 9ξ2 » ———————– fi ffiffiffiffiffiffiffifl “ » ———————– η2 :x1 ` :x2 ´ 2mz% cos φ 9φ2 ´ 2mz% sin φ :φ ξ2 ´2mz% cos φ 9φ2 ´ 2mz% sin φ :φ fi ffiffiffiffiffiffiffifl Substituting (4.1) into (4.13), we get 9η1 9η2 9ξ1 9ξ2 » ———————– fi ffiffiffiffiffiffiffifl “ » ———————– η2 ´g ´“λkextpx1 ` x2 ´ %0q‰{mt ξ2 ´2mz%“ cos φ 9φ2 ` sin φ pM21N1 ` M11pτ ´ N2q{δq‰ fi ffiffiffiffiffiffiffifl where 0 1 : x2 ą %0 : Flight Phase : x2 “ %0 : Contact Phase λ “$’& ’% and δ is the determinant of Mpqq. Define the torque τ to be τ “ N2 ` 1 –´M21N1 ` δ´´ν ´ 2mz% cos φ 9φ2¯ M11 » 2mz% sin φ fi fl (4.13) (4.14) (4.15) (4.16) where v is the new control input. Substituting (4.12), and (4.16) into (4.14) results in “ 9η1 9η2 9ξ1 9ξ2 » ———————– fi ffiffiffiffiffiffiffifl » ———————– fi ffiffiffiffiffiffiffifl η2 ´g ´“λkextpη1 ´ ξ1q‰{mt ξ2 v 51 (4.17) 4.3.3 Backstepping In the absence of relative motion between the masses, the total energy can be written as E “ mt„1 2 9z2 ` gpz ´ zdq ` 1 2 λKext pz ´ zdq2 (4.18) The second term on the right-hand side of (4.18) represents the potential energy stored in the spring when the masses are in their nominal position relative to one another. Indeed, when r “ rd, pz ´ zdq “ px1 ` x2 ` r ´ rd ´ %0q “ px1 ` x2 ´ %0q, which is equal to the spring deformation. For the robot to reach its desired apex height hd, the total energy should be equal to E ” Edes “ mt gphd ´ zdq (4.19) in addition to e ” 0. The desired equilibrium configuration is therefore given by pE ´ Edes, e, 9eq “ p0, 0, 0q (4.20) With the objective of stabilizing the equilibrium in (4.20), we define the Lyapunov function candidate V1 “ 1 2 k1 pE ´ Edesq2 , k1 ą 0 (4.21) It should be noted that V1 is a function of λ (since E is a function of λ) but it is continuously differentiable in both the flight phase and contact phase. The Lyapunov function candidates introduced in this section will be used for stability analysis in the two phases separately; therefore, we treat λ as constant and do not make any distinction between the two phases 52 in our derivation. Using (4.12), (4.18), and (4.21), 9V1 can be computed as 9V1 “ k1 pE ´ Edesq 9E “ k1 pE ´ Edesq η2 rmtp 9η2 ` gq ` λKextη1s “ k1 pE ´ Edesq λKextη2ξ1 By choosing ξ1 “ t´λk1 pE ´ Edesq η2u ∆“ ϕ1 (4.22) (4.23) we can make 9V1 negative semi-definite; therefore, integrator backstepping is introduced by defining the new variable q1 “ ξ1 ` λk1 pE ´ Edesq η2 “ pξ1 ´ ϕ1q (4.24) and the composite Lyapunov function V2 “ V1 ` 1 2 q2 1 “ 1 2 k1 pE ´ Edesq2 ` 1 2 q2 1 (4.25) Differentiating V2 and substituting (4.22) and (4.24), we get 9V2 “ k1 pE ´ Edesq λKextη2ξ1 ` q1 9q1 “ k1 pE ´ Edesq λKextη2 rq1 ´ λk1 pE ´ Edesq η2s ` q1 9q1 “ ´λ2k2 1Kext pE ´ Edesq2 η2 2 ` q1 r 9q1 ` k1 pE ´ Edesq λKextη2s By choosing 9q1 “ t´λk1Kext pE ´ Edesq η2 ´ k2q1u ∆“ ϕ2 (4.26) (4.27) where k2 ą 0. We can make 9V2 negative semi-definite. We introduce integrator backstepping 53 again by defining the new variable and the composite Lyapunov function V3 “ V2 ` 1 2 q2 2 q2 “ p 9q1 ´ ϕ2q (4.28) “ 1 2 k1 pE ´ Edesq2 ` 1 2 q2 1 ` 1 2 q2 2 (4.29) Differentiating V3 and substituting (4.26) and (4.28), we get 9V3 “ ´λ2k2 1Kext pE ´ Edesq2 η2 2 ` q1 r 9q1 ` λk1Kext pE ´ Edesq η2s ` q2 9q2 “ ´λ2k2 1Kext pE ´ Edesq2 η2 2 ` q1 r 9q1 ´ ϕ2 ´ k2q1s ` p 9q1 ´ ϕ2qp:q1 ´ 9ϕ2q “ ´λ2k2 1Kext pE ´ Edesq2 η2 2 ´ k2q2 1 ` q2 r:q1 ´ 9ϕ2 ` q1s (4.30) Our choice of :q1 “ 9ϕ2 ´ q1 ´ k3q2, k3 ą 0 (4.31) results in a negative semi-definite 9V3 and yields the controller v “ :ϕ1 ` 9ϕ2 ´ q1 ´ k3q2 (4.32) The above equation was obtained from (4.31) by substituting the value of 9ξ2 in (4.17) and (4.24) into (4.31). From the definition of ϕ1, it is clear that :ϕ1 will involve the second 54 derivative of η2. This is not a problem since the second derivative of η2 can be computed easily from (4.17) as :η2 “ ´λ 1 mt Kextpη2 ´ ξ2q (4.33) 4.3.4 Stability Analysis Using (4.12), (4.17), (4.18), (4.24) and (4.28) it can be shown that pE ´ Edes,ξ 1,ξ 2q “ p0, 0, 0q ô pE ´ Edes, q1, q2q “ p0, 0, 0q Therefore, V3 in (4.29) is a candidate Lyapunov function for investigating the stability of the equilibrium in (4.20). In both flight and contact phases (λ “ 0 and λ “ 1), the control law in (4.32), results in (4.30) 9V3 ď 0 (4.34) Therefore, pE ´ Edes, q1, q2q “ p0, 0, 0q is stable. Remark 3. The stability of pE ´ Edes, q1, q2q “ p0, 0, 0q in the flight and contact phases do not guarantee stability for the hybrid dynamics. The stability of the hybrid system is analyzed next . 55 4.4 Discrete Controller for Stabilizing Hybrid Dynam- ics and Controlling the Apex Height 4.4.1 Elastic Ground To investigate the stability of the hybrid dynamic system, we use a Poincar´e map with the Poincar´e section defined at the point of touch-down, i.e. the time instant prior to impact. Assuming that η1,ξ 1,ξ 2 « 0 at the time of touch-down, the Poincar´e section is chosen as ¯Z :“ tX P R| η1 “ 0,ξ 1 “ 0,ξ 2 “ 0,η 2 ă 0u (4.35) where X is defined as X “ η2 (4.36) We define the Poincar´e section using the coordinate Ψ, where Ψ is defined by the coordinate transformation Hp¨q : R ñ R, as follows Ψ “ pE ´ Edesq “ HpXq (4.37) It can be shown that the map Hp¨q is a local homeomorphism and therefore locally topo- logically conjugate [27]; this implies that the stability of the Poincar´e maps in Ψ and X coordinates are equivalent. The Poincar´e map P pΨq and the sequence of points Ψk P HpZq satisfy Ψk`1 “ P pΨkq, P pΨq : HpZq ÞÑ HpZq (4.38) 56 with periodic point Ψ˚ defined as Ψ˚ “ P pΨ˚q (4.39) For the elastic ground, the periodic point which achieves the desired apex height is given by Ψ˚ “ 0 (4.40) which follows from (4.19).We define the error state χk as χk “ pΨk ´ Ψ˚q “ Ψk By linearizing the Poincar´e map about Ψ˚, we have the approximate discrete dynamics given by χk`1 “ Aχ k A ∆“ The periodic point will be asymptotically stable if and only if dP pΨq dΨ ˇˇˇˇΨ“Ψ˚ ρpAqă 1 (4.41) (4.42) where ρpAq is the spectral radius of A. Since the condition in (4.42) may not be satisfied, we design a discrete controller to stabilize the closed-loop system; the discrete controller is discussed next. To design the discrete controller, we redefine Ψ as follows Ψ “ Φ ` u, Φ ∆“ pE ´ Edq, u ∆“ pEd ´ Edesq (4.43) where Ed is desired level of energy for a given hop. The new Poincar´e map ¯P pΦ, uq and the sequence of points Φk P HpZq satisfy Φk`1 “ ¯P pΦk, ukq, ¯P pΦ, uq : HpZq ˆ R ÞÑ HpZq (4.44) 57 with periodic point Φ˚ defined as Φ˚ “ ¯P pΦ˚, u˚q (4.45) For the elastic ground, the periodic point which achieves the desired apex height is given by Φ˚ “ 0, u˚ “ 0 (4.46) We define the error state ¯χk as ¯χk “ pΦk ´ Φ˚q “ Φk By linearizing the Poincar´e map about pΦ˚, u˚q, we have the approximate discrete dynamics given by ¯χk`1 “ ¯A ¯χk ` ¯Buk ¯A ∆“ d ¯P pΦ, uq For our choice of input dΦ ˇˇˇˇΨ“Ψ˚,u“u˚ (4.47) , ¯B ∆“ d ¯P pΦ, uq du ˇˇˇˇΨ“Ψ˚,u“u˚ uk “ K ¯χk (4.48) The closed-loop system dynamics takes the form ¯χk`1 “ p ¯A ` ¯BKq ¯χk If t ¯A, ¯Bu is controllable, we can choose K such that ρp ¯A ` ¯BKqă 1 (4.49) and the hybrid dynamical system is asymptotically stable. 58 Remark 4. If the condition in (4.42) is not satisfied and the discrete controller in (4.48) is implemented, the continuous controller will have to be modified. In particular, the fixed desired value of the energy Edes will have to be replaced by the desired value of energy for each hop Ed to account for the change in the Poincar´e map from P pΨq to ¯P pΦ, uq. 4.4.2 Viscoelastic Ground with Inertia The viscoelastic ground results in energy losses due to impact between the lower mass mf and the ground mass mg, and the damping in the ground. We defined the Poincar´e section using the coordinate ¯Ψ, where ¯Ψ is defined by the coordinate transformation Hp¨q : R ñ R, defined as follows ¯Ψ “ pE ´ Edesq “ HpXq The Poincar´e map Qp ¯Ψq and the sequence of points ¯Ψk P Hp ¯Zq now satisfy ¯Ψk`1 “ Qp ¯Ψkq, Qp ¯Ψq : Hp ¯Zq ÞÑ Hp ¯Zq with periodic point ¯Ψ˚ defined as ¯Ψ˚ “ Qp ¯Ψ˚q For the viscoelastic ground, the periodic point is ¯Ψ˚ “ p ¯E ´ Edesq (4.50) (4.51) (4.52) (4.53) where ¯E is the energy of the system in the steady state when the backstepping controller is invoked with Ed “ Edes. The value of ¯E is initially unknown but can be determined after 59 the robot has hopped a few times - see section 4.5.2. We define the error state γk as γk “ pΨk ´ ¯Ψ˚q “ pEk ´ ¯Eq By linearizing the Poincar´e map about ¯Ψ˚, we have the approximate discrete dynamics given by γk`1 “ A0 γk A0 ∆“ The periodic point will be asymptotically stable if and only if dQp ¯Ψq d ¯Ψ ˇˇˇˇΨ“ ¯Ψ˚ ρpA0qă 1 (4.54) (4.55) where ρpAoq is the spectral radius of Ao. To design the discrete controller, we describe ¯Ψas follows ¯Ψ “ ¯Φ ` ¯u, ¯Φ ∆“ pE ´ Edq, ¯u ∆“ pEd ´ Edesq where Ed is desired level of energy for a given hop. The new Poincar´e map ¯Qp ¯Φ, ¯uq and the sequence of points ¯Φk P Hp ¯Zq satisfy ¯Φk`1 “ ¯Qp ¯Φk, ¯ukq, ¯Qp ¯Φ, ¯uq : Hp ¯Zq ˆ R ÞÑ Hp ¯Zq with periodic point ¯Φ˚ defined as ¯Φ˚ “ ¯Qp ¯Φ˚, ¯u˚q For the viscoelastic ground with inertia, there exists the following equilibrium point ¯Φ˚ “ p ¯E ´ Edq, ¯u˚ “ 0 60 (4.56) (4.57) (4.58) If we define the error state µk ∆“ p ¯Φk ´ ¯Φ˚q “ pEk ´ ¯Eq (4.59) the Poincar´e map ¯Qp ¯Φk, ¯ukq can be linearized about p ¯Φ˚, ¯u˚q to yield the following linear discrete-time system yk “ µk ˆB ∆“ d ¯Qp ¯Φ, uq du (4.60) ¯Φ“ ¯Φ˚ ˇˇˇˇ ¯u“¯u˚ µk`1 “ ˆA µk ` ˆB ¯uk, ˆA ∆“ , d ¯Qp ¯Φ, uq dΦ ˇˇˇˇ ¯Φ“ ¯Φ˚ ¯u“¯u˚ To converge the system energy from its level at the equilibrium configuration ¯E to the desired value Edes, we propose to use integral control with the integrator defined as θk`1 “ θk ` pE0 ´ ykq, E0 ∆“ pEdes ´ ¯Eq (4.61) where E0 is the desired value of the output variable y. The integrator-augmented discrete system has the form λk`1 “ Aλk ` Buk ` E0,λ A ∆“» , B ∆“» —– —– fi ffifl ´1 1 ¯A 0 k “„ µk θk T fi ffifl ¯B 0 (4.62) If tA, Bu is controllable, the input can be chosen as ¯uk “ Kλk (4.63) where K satisfies ρpA ` BKqă 1 61 Assumption 11. The parameters of the viscoelastic ground (mg, Cext, Kext) are such that the settling time of the system is less than the hopping period of the robot. Remark 5. The above assumption allows us to infer that x1, 9x1 « 0 at the time of touch- down. It will be shown later that the assumption is not overly restrictive. 4.5 Simulation 4.5.1 Elastic Ground We investigate the behavior of AKH robot hopping on an elastic ground. The parameters of the robot and the ground were assumed to be mf “0.15 kg, m “ 0.4 kg, mb “ 0.7 kg % “0.2m %0 “ 0.05 m, Kext “ 2800 N{m (4.64) The desired apex height hd and the relative displacement rd was chosen to be hd “ 0.3 m, rd “ 0.13 m The parameters of the continuous backstepping controller were chosen as k1 “ 0.001, k2 “ 700, k3 “ 10 The eigenvalue ¯A in (4.47) was found to be 0.21. The discrete controller gain were chosen as K “ ´0.3 62 y3 z y2 0.4 0.3 0.2 0.1 0.05 ) m ( 3 y , 2 y , z 0 k “ 1 k “ 2 0 0.4 k “ 3 k “ 4 k “ 5 k “ 6 0.8 time (s) 1.2 1.6 2 Figure 4.2: Simulation results for hopping on the elastic ground. Absolute height of the two masses y2 and y3, and COM height are plotted as a function of time. This places the eigenvalue of the closed loop system at ´0.053. The initial conditions were assumed to be x1 “ ´0.005 m, x2 “ 0.05 m,φ p0q “ 0.5 rad, 9x1p0q “ 0.0 m{s, 9x2p0q “ 0.0 m, 9φp0q “ 0.0 m{s (4.65) The simulation results are shown in Figs. 4.2, and 4.3. The displacements of mb, mf, and the COM are shown in Fig. 4.2. The initial conditions indicates that the spring of the elastic ground is initially compressed due to the weight of the robot, and the robot starts from the rest. Fig. 4.2 shows that the apex height of the COM converge to the desired value at h “ 0.3 m after three hops. The contact phase is the period where y2 ă %0. The period where y2 ą %0 is the flight phase. The discrete-time system state Ψ is plotted in Fig. 4.3. The state of the discrete system converges to zero after three hops. A video animation of a two-DOF AKH robot hopping on an elastic ground has been uploaded as supplementary material. It shows the hopper reaching the desired apex height of hd “ 0.3 m starting from rest. 63 0.5 0 s e d E ´ E -0.5 1 2 3 4 5 6 k Figure 4.3: Simulation results for hopping on the elastic ground: Plots of pE ´ Edesq at the end of the k-th hop, k “ 1, 2 ¨ ¨ ¨ , 6. 4.5.2 Viscoelastic Ground The parameters AKH robot is given in (4.64). The parameters of the ground assumed to be mg “ 0.015kg, Kext “ 2800N{m, Cext “ 13 Ns{m (4.66) The mass of the ground was assumed to be 10% of the lower mass mf and the damping coefficient Cext was chosen such that the ground is slightly over-damped. The desired apex height hd and the relative displacement rd was chosen to be hd “ 0.3 m, rd “ 0.13 m The continuous controller gains were chosen as k1 “ 0.001, k1 “ 600, k2 “ 50 The eigenvalues of A was found to be: 1, and 0.404. The pair tA, Bu is controllable, and the 64 y3 y2 z 0.4 0.3 0.2 0.05 0 ) m ( 3 y , 2 y , 1 x , z -0.1 k “ 1 0 x1 k “ 11 1 2 2.9 4 5 time (s) k “ 19 6 7 Figure 4.4: Simulation results for hopping on a viscoelastic ground with inertia. Plot of the position of the upper mass y3, the lower mass y2, COM z, and the ground mass x1, as a function of time. controller gains were chosen as K “„ ´0.5 0.6  This places the eigenvalues of the closed loop system at 0.58 ˘ 0.34i. The initial conditions were described by (4.65) and the initial value of the integrator state was set to zero. The simulation results are shown in Figs. 4.4, 4.5 and 4.6. The displacements of the -1.07 0 s e d E ´ E -2 1 5 10 k 15 20 Figure 4.5: Simulation results for hopping on a viscoelastic ground with inertia: Plots of pE ´ Edesq at the end of the k-th hop k “ 1, 2, ¨ ¨ ¨ , 21. 65 0.2 Contact phase Flight phase ) m ( 2 y , 2 x , 1 x 0.05 0 -0.05 4.6 x2 y2 x1 Settling time for the foundation mass time (s) 6.9 Figure 4.6: Simulation results for hopping on a viscoelastic ground with inertia: Plot of y2, and x1 for the hop between k “ 19 and k “ 20. upper mass, lower mass, COM, and ground mass are plotted in Fig. 4.4. The contact phases (not explicitly shown in Fig. 4.4) are the time intervals during which x2 “ py2 ´ x1q “ 0.05 m. The initial value of y2p0q “ 0.045 m indicates that the spring of the viscoelastic ground is initially compressed due to the weight of the robot. The simulation is comprised of two phases. In the initial phase, 0 ď t ď 2.9 s, the discrete controller was switched off and the backstepping controller was used with Ed “ Edes. During this phase (ten hops), the apex height of the robot converges to a constant value that corresponds to E “ ¯E, which was defined in the context of (4.53). The value of ¯E was found to be 0.96 J and E0 “ 1.02 J (the deficit) was computed using (4.61). Using the value of E0, the discrete controller was switched on at t “ 2.9 s. The discrete time system ¯Ψ is plotted in Fig. 4.5 at the end of the k-th hop, k “ 1, 2, ¨ ¨ ¨ , 21. The discrete controller is active for k ą 10 and it results in pE ´ Edesq converging to zero in approximately nine hops, i.e., k “ 19. The displacement of the lower mass and the ground is shown in Fig. 4.6 for one hop. It can be seen that the lower mass breaks contact with the ground mass below the datum (x2 ą %0, x1 ă 0). While the robot is in flight, the ground mass settles to its equilibrium configuration; the response is overdamped, as expected from the choice of parameters in (4.66), and has a 2% settling time of 0.011 s. Since the settling time of the ground mass 66 ) . m N ( τ 60 40 20 0 -20 -40 0 1 2 3 time (s) 4 5 6 7 Figure 4.7: Plot of the torque applied by the hip actuator. is much smaller than the flight phase, it becomes clear that Assumption 4 is not overly restrictive. The control input (hip torque) is shown in Fig.4.7. In this context, it should be pointed out that the control torque can be reduced by 50% if the actuator is moved from the hip to the knee as the angular displacement of the knee is twice that of the hip. A video animation of a Two-DOF AKH robot hopping on a viscoelastic ground with inertia has been uploaded as supplementary material. It shows the hopper reaching the desired apex height of hd “ 0.3 m starting from rest. 4.6 Conclusion This chapter presents a hybrid control strategy to converge the apex height of an AKH robot hopping on an elastic ground and a viscoelastic ground with inertia. A continuous backstepping controller is used in conjunction with a discrete-time controller designed using a Poincar´e map. For the elastic ground, the backstepping controller is used to control the energy level of the hopper and eliminates the relative displacement between the hopper masses. The discrete controller is used to guarantee the stability of the hybrid system and 67 fast convergence of the apex height to the desired value. For the viscoelastic ground, the backstepping controller results in steady state error due to losses from impact between the hopper and the ground and viscous losses in the damper. To overcome these losses, a discrete- time controller with integral action was introduced. Simulation results prove the efficacy of the control strategy for both elastic and viscoelastic grounds. A video animation of apex height control is includes to provide a glimpse of the dynamic behavior. 68 Chapter 5 Four-Link Planar One-Legged Hopping Locomotion 5.1 Introduction In this chapter, we develop a strategy for controlling the apex height of a four-link robot hopping in place with a desired apex height, and moving forward or backward with a desired step size. One-legged hopping locomotion on a rigid ground is a challenging problem due to the energy loss from ground impact. After transforming the dynamic model into normal form, the controllable states are controlled to emulate the dynamics of a mass-spring-damper system with variable damping; the uncontrollable states are shown to remain bounded and well-behaved. The controllable states include the position and velocity of the center-of-mass and the foot angle. We develop two control strategies to compensate for the energy loss due to ground impact. The first strategy introduces negative damping in the dynamics of the mass-spring-damper system that describes the vertical motion of the center-of-mass (COM) in the contact phase. The second strategy alters the equilibrium height of the COM in the vertical direction for both the flight and contact phases. A Poincar´e map is used to construct a discrete-time system at the point of touch-down and a controller with integral action is designed to converge the apex height to a desired value for hopping in place, and for converging the step size to the desired value in the case of locomotion. Simulation results 69 are presented to show the efficacy of the control designs. This chapter is structured as follows. The dynamics of the hybrid system in flight phase, impact phase, and contact phase are presented in Section 5.2. In Section 5.3, we present the dynamical model of the hopper in normal form. A partial feedback linearizing controller is presented in Section 5.4. In Sections 5.5 and 5.6, we present the strategies for apex height control and step-size control during hopping locomotion. Simulations results are presented in Section 5.7 and concluding remarks are presented in Section 5.8. 5.2 Hybrid System Dynamics 5.2.1 System Description Consider the four-link planar hopping robot in Fig.5.1. The hopper is comprised of four links (foot, leg, thigh and body) with link lengths %f, %l, %t, %b, masses mf, ml, mt, mb, and mass moment of inertias If, Il, It, Ib, respectively. The tip of the foot (toe) is denoted by point O and its displacement relative to the ground is denoted by px, yq. The relative angular displacements of the links are denoted by θ1, θ2, θ3, and θ4; the torque inputs applied at the ankle, knee, and hip joints (θ2, θ3, and θ4) are denoted by τ1, τ2, and τ3, respectively. The ground reaction force applied at O (toe) during contact is denoted by Fext. The dynamic model of the robot is obtained using Lagrange equations: 70 mb, Ib, 2%b θ4,τ 3 θ3,τ 2 g mt, It, 2%t ml, Il, 2%l θ2,τ 1 mf, If, 2%f ry Y rx θ1 O ” px, yq Fext X Figure 5.1: Four-link planar hopping robot in an arbitrary configuration Mpqq:q ` Npq, 9qq “ AT ` Fext (5.1) 0 0 0 1 0 0 0 0 0 0 1 0 , T fi ffiffiffiffifl A ∆“» ————– T ∆“„ τ1 0 0 0 0 0 1 τ2 τ3 T , q ∆“r x y θ sT ,θ “ r θ1 θ2 θ3 θ4 sT Fext ∆“„ Fx Fy 0 0 0 0 T (5.2) where q is the vector of generalized coordinates, M P R6ˆ6 denotes the mass matrix, N denotes the vector of centrifugal, Coriolis and forces due to gravity, and Fx, Fy are the components of the reaction force Fext in the x and y directions. The hybrid dynamics of the 71 hopping robot is comprised of three phases: the flight phase, when the toe is not in contact with the ground (y ą 0); the impact phase, the instant when the toe comes in contact with the ground; and the contact phase, during which the toe remains in contact with the ground after impact py “ 0q. 5.2.2 Flight Phase In the flight phase, the robot has six DOF, the following conditions hold: y ą 0 Fx “ Fy “ 0 (5.3) and (5.1) describes the dynamics of the robot. 5.2.3 Impact Phase Let t P rt´, t`s denote the interval of impact. We make the following assumptions: Assumption 12. The impact between the toe and the ground is inelastic. This implies that 9xpt`q “ 9ypt`q “ 0. Assumption 13. The control torques T are not impulsive. Integrating (5.1) over the infinitesimal duration t P rt´, t`s, we get t´ “Mpqq:q ` Npq, 9qq‰dt “ż t` ż t` t´ “AT ` Fext‰dt ∆“ż t` Fext dt t´ ñ Mpqqp 9q` ´ 9q´q “ Iext, Iext (5.4) (5.5) 72 We partition the inverse of the mass matrix as follows M´1 “» —– pM´1q11 pM´1q12 pM´1q21 pM´1q22 fi ffifl From (5.5), we compute the change in the state variables: q` “ q´ 9xpt`q “ 0 9ypt`q “ 0 9θpt`q “ 9θpt´q ´ pM´1q21rpM´1q11s´1» —– 9x´ 9y´ fi ffifl 5.2.4 Contact Phase (5.6) (5.7) The contact phase commences immediately after the toe makes contact with the ground. For this phase, we make the following assumption: Assumption 14. The external forces Fy is non-negative, i.e., Fy ě 0. In the contact phase, the hopper has four DOF, since x “ 9x “ :x “ 0, y “ 9y “ :y “ 0 (5.8) Substituting :x “ :y “ 0 into (5.1), the values of the Fx and Fy can be computed as Fx Fy » —– fi ffifl “» —– N1 N2 fi ffifl ` pM´1q11pM´1q12 » ———————– N3 pN4 ´ τ1q pN5 ´ τ2q pN6 ´ τ3q fi ffiffiffiffiffiffiffifl (5.9) where Ni’s are the entries of the Npq, 9qq vector in (5.1). The dynamical equation of the 73 hopper during this phase can be written as DMpqqDT D :q ` DNpq, 9qq “ DAT (5.10) where D “ » ———————– 0 0 1 0 0 0 0 0 0 1 0 0 fi ffiffiffiffiffiffiffifl 0 0 0 0 1 0 0 0 0 0 0 1 (5.11) The contact phase ends when Fy in (5.9) equals zero. 5.3 Coordinate Transformation into Normal Form 5.3.1 Controlled States To transform the system into normal form [35], we define the position of the COM relative to the position of the toe using the variable r as follows: r “» —– rx ry fi ffifl “» —– fxpθq fypθq fi ffifl (5.12) where rx and ry are the horizontal and vertical components of r and can be expressed as 74 follows fxpqq “ a1 cospθ1q ` a2 cospθ1 ` θ2q ` a3 cospθ1 ` θ2 ` θ3q ` a4 cospθ1 ` θ2 ` θ3 ` θ4q (5.13) fypqq “ a1 sinpθ1q ` a2 sinpθ1 ` θ2q ` a3 sinpθ1 ` θ2 ` θ3q ` a4 sinpθ1 ` θ2 ` θ3 ` θ4q (5.14) where a1 “ “mf ` 2pml ` mt ` mbq‰%f ¯m pmt ` 2mbq%t , a3 “ , ¯m a2 “ “ml ` 2pmt ` mbq‰%l ¯m a4 “ mb%b ¯m ¯m “ mf ` ml ` mt ` mb Differentiating with respect to time, we get 9r “» —– 9rx 9ry fi ffifl “» —– Jxpθq Jypθq fi ffifl D 9q “» —– Jxpθq Jypθq 9θ fi ffifl (5.15) where Jx and Jy are the Jacobian matrices. The output of the system described by (5.1) (flight phase) and (5.9) (contact phase) is chosen to be rx, ry and θ1; it can then be shown that the system has: relative degree “ r2, 2, 2sT (5.16) Accordingly, the six controlled states (in flight and contact phases) are chosen as ζ1 “ prx ´ xdq ζ2 “ pry ´ ydq ζ3 “ pθ1 ´ θdq ζ4 “ 9rx ζ5 “ 9ry ζ6 “ 9θ1 (5.17) 75 5.3.2 Uncontrollable States - Flight Phase In the flight phase, the dynamical system in (5.1) has six DOF or twelve states: these include the controlled states in (5.17) and the following six uncontrollable states: η1 “ xptq ` rx,η 2 “ 9xptq ` 9rx,η 3 “ yptq ` ry, η4 “ 9yptq ` 9ry,η 5 “ HC,η 6 “ Icm (5.18) where η1 and η3 are the Cartesian position and η2 and η4 are the Cartesian velocity of the COM, η5 “ Hc is the angular momentum of the system and η6 “ Icm is its mass-moment of inertia about its center of mass. It can be shown that the uncontrollable states satisfy [35]: Bηipqq Bq “pMpqqq´1AT‰ “ 0, i “ 1, 2, ..., 6. If Tnf denotes the coordinate transformation from the original states pqT , 9qT qT to the states pη1 ¨ ¨ ¨ η6,ζ 1 ¨ ¨ ¨ ζ6qT , it can be shown that Tnf is a diffeomorphism. The dynamics of the 76 system in the normal form can now be written as: “ 9η1 9η2 9η3 9η4 9η5 9η6 9ζ1 9ζ2 9ζ3 9ζ4 9ζ5 9ζ6 » ————————————————————————————————————– fi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifl » —————————————————————————————————————– η2 0 η4 ´g 0 9Icmpη, ζ q 9rx 9ry 9θ1 JxpqqDM´1rAT ´ Npq, 9qqs ` 9Jxpqq 9θ JypqqDM´1rAT ´ Npq, 9qqs ` 9Jypqq 9θ „1 0 0 0 DM´1rAT ´ Npq, 9qqs fi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifl (5.19) 5.3.3 Uncontrollable States - Contact Phase In the contact phase, the dynamical system in (5.9) has four DOF or eight states: these include the controlled states in (5.17) and the following two uncontrollable states: η1 “ HO “ HC ` ¯mprx 9ry ´ ry 9rxq,η 2 “ Icm (5.20) where HO is the angular momentum of the system about the toe - point O in Fig.5.1. Once again, it can be show that the uncontrollable states satisfy [35] Bηipqq Bq “pDMpqqDT q´1DAT‰ “ 0, i “ 1, 2 77 If Tnc denotes the coordinate transformation from the original states pqT , 9qT qT to pη1 η2,ζ 1 ¨ ¨ ¨ ζ6qT , it can be shown that Tnc is a diffeomorphism. The dynamics of the system in the normal form can now be written as: “ 9η1 9η2 9ζ1 9ζ2 9ζ3 9ζ4 9ζ5 9ζ6 » —————————————————————– fi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifl » —————————————————————– ´ ¯m g rx 9Icmpη, ζ q 9rx 9ry 9θ1 JxpqqpDMDT q´1DrAT ´ Npq, 9qqs ` 9Jxpqq 9θ JypqqpDMDT q´1DrAT ´ Npq, 9qqs ` 9Jypqq 9θ „1 0 0 0 pDMDT q´1DrAT ´ Npq, 9qqs fi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifl (5.21) 5.4 Partial Feedback Linearization 5.4.1 Equilibrium Point In this section, we design a continuous feedback controller to control the dynamics of the position of the COM relative to the toe prx, ryq and the angle of the foot θ1. To this end, we use feedback linearization to ensure that pζ1,ζ 2,ζ 3,ζ 4,ζ 5,ζ 6q “ p0, 0, 0, 0, 0, 0q is an equilibrium point of the system. 78 5.4.2 Flight Phase In the flight phase, the dynamics of the controlled states ζ4, ζ5, ζ6 in (5.19) can be expressed as where 9ζ4 9ζ5 9ζ6 » ————– fi ffiffiffiffifl “ JpqqDM´1rAT ´ Npq, 9qqs ` 9Jpqq 9θ (5.22) Jpqq “» ————– Jx Jy C , fi ffiffiffiffifl 9Jpqq “» ————– 9Jx 9Jy 0 , C ∆“„1 0 0 0 (5.23) fi ffiffiffiffifl The control input Tf is chosen as follows Tf “ rJDM´1As´1rvf ` JDM´1N ´ 9J 9θs (5.24) where ´K1ζ1 ´ K4ζ4 ´K2ζ2 ´ K5ζ5 vf “» ————– ´K3ζ3 ´ K6ζ6 fi ffiffiffiffifl (5.25) 79 This results in the following dynamics: 9ζ4 9ζ5 9ζ6 » ————– fi ffiffiffiffifl “» ————– ´K1ζ1 ´ K4ζ4 ´K2ζ2 ´ K5ζ5 ´K3ζ3 ´ K6ζ6 fi ffiffiffiffifl :ζ1 ` K4 9ζ1 ` K1ζ1 “ 0 ñ :ζ2 ` K5 9ζ2 ` K2ζ2 “ 0 (5.26) :ζ3 ` K6 9ζ3 ` K3ζ3 “ 0 The choice of gains Ki, i “ 1, 2, ¨ ¨ ¨ , 6, will be discussed later. 5.4.3 Contact Phase In the contact phase, the dynamics of the controlled states ζ4, ζ5, ζ6 in (5.21) can be expressed as 9ζ4 9ζ5 9ζ6 » ————– “ JpqqpDMDT q´1DrAT ´ Npq, 9qqs ` 9Jpqq 9θ (5.27) fi ffiffiffiffifl The control input Tc is chosen as follows Tc “ rJpDMDT q´1DAs´1rJpDMDT q´1DN ` vc ´ 9J 9θs (5.28) where ´K7ζ1 ´ K10ζ4 ´K8ζ2 ´ αK11ζ5 vc “» ————– ´K9ζ3 ´ K12ζ6 fi ffiffiffiffifl 1, ν, ,α “$’’& ’’% if ζ5 ď 0 if ζ5 ą 0 (5.29) 80 This results in the following dynamics: 9ζ4 9ζ5 » ————– 9ζ6 fi ffiffiffiffifl “» ————– ´K7ζ1 ´ K10ζ4 ´K8ζ2 ´ αK11ζ5 ´K9ζ3 ´ K12ζ6 fi ffiffiffiffifl :ζ1 ` K10 9ζ1 ` K7ζ1 “ 0 ñ :ζ2 ` αK11 9ζ2 ` K8ζ2 “ 0 (5.30) :ζ3 ` K12 9ζ3 ` K9ζ3 “ 0 Once again, the choice of gains Ki, i “ 7, 8, ¨ ¨ ¨ , 12, will be discussed later. 5.5 Controlling The Apex Height 5.5.1 Apex Height The apex height h is defined as the maximum value of the vertical displacement of the COM during the flight phase. It should be mentioned that some fraction of the total energy of the system is lost during impact with the ground. To overcome this loss and reach the desired level of energy and hop to the desired value hd, we developed two control strategies. The first strategy is based on introducing negative damping in the vertical dynamics during the contact phase by choosing the parameter ν to be less than zero. The negative damping is introduced in the vertical dynamics to overcome the energy loss due to the impact. The second strategy is based on choosing different equilibrium heights yd of the vertical dynamics of the COM during the flight phase and contact phase. If the value of yd in the contact phase is equal to ycd, for the flight phase we choose yd “ yf d ă ycd. 81 5.5.2 The First Strategy: Negative Damping Based Continuous- Time Controller In this section we discuss the procedure for choosing gains and the desired values of the continuous controller. In addition, we discuss the behavior of controlled and uncontrolled states. Flight Phase In the flight phase, the gains of the continuous controller are chosen as: K4 “2aK1, K2 “K3, K5 “ 2aK2, K5 “ K6 K6 “ 2aK3 (5.31) This choice results in critically-damped dynamics of the controlled states ζ1, ζ2, ζ3. From (5.19), it can be seen that the uncontrolled states η1, ¨ ¨ ¨ ,η 5 are described by the relations 9η2 “ :η1 “ 0, 9η4 “ :η3 “ ´g, 9η5 “ 0 and are therefore bounded. The uncontrolled state η6 also remains bounded as it denotes the mass-moment of inertia about the COM and all the joints are revolute in nature. The values of xd, yd, and θd were chosen to be constant; a particular set of choices will be shown in our simulations in section 5.7. 82 Contact Phase In the contact phase, the gains of the continuous controller are chosen as: K10 “2aK7, K8 “K9, K11 ă 2aK8, K11 “ K12 K12 ă 2aK9 (5.32) This choice results in critically damped controlled state ζ1 and under-damped controlled state ζ3; the controlled state ζ2 is underdamped when 9ζ2 ă 0 and negatively damped when 9ζ2 ą 0. As mentioned earlier, the negative damping allows for compensation of the losses due to impact. The uncontrolled state η1 and its derivative can be shown to be η1 “ HO “HC ` ¯mprx 9ry ´ ry 9rxq, 9η1 “ 9HO “ ´ ¯mg rx (5.33) The values of yd, and θd were chosen to be constant but xd is chosen as xd “ γHO,γ ą 0 (5.34) This results in 9HO Ñ ´γ ¯mgHO, which in turn results in HO Ñ 0 ñ rx, 9rx Ñ 0 ñ HC Ñ 0 As in the flight phase, the uncontrolled state η2 remains bounded as it denotes the mass- moment of inertia about the COM and all joints of the system are revolute. 83 5.5.3 The Second Strategy: Equilibrium Height Based Based Continuous-Time Controller In this section we discuss the procedure for choosing gains and the desired values of the continuous controller. In addition, we discuss the behavior of controlled and uncontrolled states. Flight Phase In the flight phase, the gains of the continuous controller are chosen as: K4 “2aK1, K2 “K3, K5 “ 2aK2, K5 “ K6 K6 “ 2aK3 (5.35) This choice results in critically-damped dynamics of the controlled states ζ1, ζ2, ζ3. From (5.19), it can be seen that the uncontrolled states η1, ¨ ¨ ¨ ,η 5 are described by the relations 9η2 “ :η1 “ 0, 9η4 “ :η3 “ ´g, 9η5 “ 0 and are therefore bounded. The uncontrolled state η6 also remains bounded as it denotes the mass-moment of inertia about the COM and the joints are revolute. The values of xd, and θd were chosen to be constant; a particular set of choices will be shown in our simulations in section 5.7. 84 Contact Phase In the contact phase, the gains of the continuous controller are chosen as: K10 “2aK7, K8 “K9, K11 ă 2aK8, K11 “ K12 K12 ă 2aK9 (5.36) This choice results in critically damped controlled state ζ1 and under-damped controlled state ζ2 and ζ3. The uncontrolled state η1 and its derivative can be shown to be η1 “ HO “HC ` ¯mprx 9ry ´ ry 9rxq, 9η1 “ 9HO “ ´ ¯mg rx (5.37) The values of yd, and θd were chosen to be constant but xd is chosen as xd “ γHO,γ ą 0 (5.38) This results in 9HO Ñ ´γ ¯mgHO, which in turn results in HO Ñ 0 ñ rx, 9rx Ñ 0 ñ HC Ñ 0 As in the flight phase, the uncontrolled state η2 remains bounded. 85 5.5.4 Discrete-Time Controller Poincar´e Map The hopping motion is comprised of the flight, impact and contact phases. The dy- namics of the controlled states in the flight and contact phases were designed to emulate a mass-spring-damper system. For both continuous controller strategies , the controlled states remain bounded since they have linear dynamics. It was shown that the uncontrolled states are also bounded. To investigate the stability of the hybrid system, we use a Poincar´e map with the Poincar´e section defined at the point of touch-down, i.e. the time instant prior to impact. Assuming at the time of touch-down we have η1 “ xd,η ζ2 “ 0,ζ 2 “ 0,η 3 “ 0,ζ 3 “ yf d,η 4 “ 0,ζ 5 “ 0,ζ 5 “ 0,ζ 1 “ 0, 6 “ 0 (5.39) The Poincar´e section was chosen as Z :“ tX P R2| η1 “ xd,η 2 “ 0,η 3 “ ycd,η 5 “ 0,ζ 1 “ 0, ζ2 “ 0,ζ 3 “ 0,ζ 4 “ 0,ζ 5 “ 0,ζ 6 “ 0u (5.40) where X is defined as X “ r η4 η6 sT (5.41) We defined the Poincar´e section using the coordinate Ψ, where Ψ is defined by the coordinate 86 transformation HpXq : R2 ñ R2, as follows Ψ “ HpXq “ r ph ´ hdq pIcm ´ Icmdq sT (5.42) Assuming that there is no relative motion between the links at touch-down1, the apex height h can be defined as h “ Etotal ¯mg “ 1 2g η2 4 ` yd (5.43) It can be shown that the map Hp¨q is a local homeomorphism; this implies that the stability of the Poincar´e maps in Ψ and X coordinates are equivalent. The Poincar´e map P pΨq and the sequence of points Ψk P HpZq satisfy with periodic point Ψ˚ defined as We define the error state χk as Ψk`1 “ P pΨkq Ψ˚ “ P pΨ˚q χk “ pΨk ´ Ψ˚q (5.44) (5.45) By linearizing the Poincar´e map about Ψ˚, we have the approximate discrete dynamics given by χk`1 “ Aχ k A ∆“ The periodic point will be asymptotically stable iff ρpAqă 1 dP pΨq dΨ ˇˇˇˇΨ“Ψ˚ (5.46) (5.47) 1This is accomplished by choosing the gains in (5.31) for the first strategy and (5.35) for the second strategy, which results in critically damped behavior 87 where ρpAq is the spectral radius of A. Closed-Loop Control Design - The First Strategy We now develop a discrete-time controller to drive the discrete-time state to its desired value Ψ “ 0. By varying two of the continuous controller parameters at touch-down, namely, ν and θd, we will be able to stabilize the desired state Ψ “ 0. The dynamics of the discrete- time system can be described by Ψpk ` 1q “ P pΨpkq,ν,θ dq (5.48) By linearizing the above map about the fixed point Ψ˚ “ 0, ν “ ν˚, and θd “ θ˚d , we get χpk ` 1q “ Aχpkq ` Bupkq, ypkq “ χpkq (5.49) where χpkq “ Ψpkq ´ Ψ˚, BΨ A “„BP pΨ,ν,θ dq B “„BP pΨ,ν,θ dq Bν Ψ“Ψ˚,ν“ν ˚,θd“θ˚ d BP pΨ,ν,θ dq Bθd upkq ““νpkq ´ ν˚ θdpkq ´ θ˚d‰T , Ψ“Ψ˚,ν“ν ˚,θd“θ˚ d (5.50) To converge the apex height to the desired value, we use integral control, with the integrator defined as βpk ` 1q “ βpkq ´ ypkq (5.51) 88 The integrator-augmented discrete system has the form λpk ` 1q “ Aλpkq ` Bupkq,λ A ∆“» —– A 0 ´I I fi ffifl , B ∆“» —– B 0 pkq ““ χT pkq βT pkq ‰T fi ffifl (5.52) where A and B are defined in (5.50). If tA, Bu is controllable, the input can be chosen as upkq “ Kλpkq (5.53) such that ρpA ` BKqă 1 Closed-Loop Control Design - The Second Strategy We now develop a discrete-time controller to drive the discrete-time state to its desired value Ψ “ 0. By varying two of the continuous controller parameters at touch-down, namely, ycd and θd, we will be able to stabilize the desired state Ψ “ 0. The dynamics of the discrete- time system can be described by Ψpk ` 1q “ P pΨpkq, ycd,θ dq (5.54) By linearizing the above map about the fixed point Ψ˚ “ 0, ycd “ y˚cd, and θd “ θ˚d , we get χpk ` 1q “ Aχpkq ` Bupkq, ypkq “ χpkq (5.55) 89 where χpkq “ Ψpkq ´ Ψ˚, BΨ A “„BP pΨ, ycd,θ dq B “„BP pΨ, ycd,θ dq Bycd upkq ““ycd ´ y˚cd θdpkq ´ θ˚d‰T Ψ“Ψ˚,ycd“y˚ cd,θd“θ˚ , d Ψ“Ψ˚,ycd“y˚ cd,θd“θ˚ d BP pΨ, ycd,θ dq Bθd (5.56) To converge the apex height to the desired value, we use integral control, with the integrator defined as βpk ` 1q “ βpkq ´ ypkq (5.57) The integrator-augmented discrete system has the form λpk ` 1q “ Aλpkq ` Bupkq,λ A ∆“» —– A 0 ´I I fi ffifl , B ∆“» —– B 0 pkq ““ χT pkq βT pkq ‰T fi ffifl (5.58) where A and B are defined in (5.56). If tA, Bu is controllable, the input can be chosen as upkq “ Kλpkq (5.59) such that ρpA ` BKqă 1 90 5.5.5 Effect of Continuous Controller Parameters on Apex Height In this section we will study the effect of the controller parameters on the apex height for both strategies. First Strategy: Negative Damping The apex height h depends on the total energy of the system at takeoff point when Fy “ 0. Substituting (5.17), (5.28) and (5.29) into (5.9), we get pry ´ ydq “ g ´ K11ν 9ry K8 The total energy of the robot at the takeoff point is E “ 1 2 ¯m 9r2 y ` ¯mg„g ´ K11ν 9ry K8 (5.60) (5.61) ` yd Assuming that the relative motion between hopper links quickly settles to zero and no significant work is done by the actuator during this time, the apex height h is computed as h “ E ¯mg “ 1 2g 9r2 y ` g ´ K11ν 9ry K8 ` yd (5.62) To examine the effect of the control parameters K8, K11, ν, and yd on the apex height, we compute the partial derivatives of h in (5.62) with respect to the parameters; they are Bh BK8 Bh BK11 pg ´ K11ν 9ryq K 2 8 ă 0, “ ´ “ ´ ν 9ry K8 ă 0, “ ´ K11 9ry K8 ă 0, “ 1 ą 0 Bh Bν Bh Byd (5.63) 91 Clearly, the apex height is higher for smaller values of K8, and ν and higher values of K11, and yd. Second Strategy: Equilibrium Height The apex height h depends on the total energy of the system at takeoff point when Fy “ 0. Substituting (5.17), (5.28) and (5.29) into (5.9), we get pry ´ ycdq “ g ´ K11 9ry K8 The total energy of the robot at the takeoff point is E “ 1 2 ¯m 9r2 y ` ¯mg„g ´ K11 9ry K8 (5.64) (5.65) ` ycd Assuming that the relative motion between hopper links quickly settles to zero and no significant work is done by the actuator during this time, the apex height h is computed as h “ E ¯mg “ 1 2g 9r2 y ` g ´ K11 9ry K8 ` ycd (5.66) To examine the effect of the control parameters K8, K11 and ycd on the apex height, we compute the partial derivatives of h in (5.66) with respect to the parameters; they are Bh BK8 “ ´ pg ´ K11 9ryq K 2 8 , Bh BK11 “ ´ 9ry K8 ă 0, Bh Bycd “ 1 ą 0 (5.67) Clearly, the apex height is higher for smaller values of K11 and higher values of ycd. For K8, the apex height is higher for smaller values of K8 when K11 ă g 9ry 92 (5.68) the apex height is higher for higher values of K8 when K11 ą g 9ry (5.69) 5.6 Control of Hopping Locomotion 5.6.1 Hopping Step Size We define δ to be the hopping step size, i.e., the horizontal displacement of the toe during the flight phase. To achieve a desired value of δ “ δd, the dynamics of the state ζ1 is controlled by varying the desired value xd in both the flight and contact phases. To overcome the energy losses, we will use the negative damping strategy (ν ă 0). 5.6.2 Flight Phase The controller gains for the controlled states are the same as in (5.31) but the value of xd is chosen as 0, if ζ5 ě 0 µ1HC, if ζ5 ă 0 , µ1 ą 0 (5.70) xd “$’’& ’’% Since HC is negative at the time of take-off and remains constant during the flight phase, the above choice of xd results in placement of the toe ahead of the COM in the horizontal direction. 93 5.6.3 Contact Phase In the contact phase, xd is chosen as: xd “ 0, if ζ5 ď 0 µ2,ζ 5 ą 0 and ζ2 ď 0 , µ3 ą µ2 ą 0 (5.71) µ3,ζ 5 ą 0 and ζ2 ą 0 $’’’’’’& ’’’’’’% Since the toe is ahead of the COM in the horizontal direction at the time of touch-down, the impact phase results is a positive angular impulse and instantaneously changes HC from a negative value to a positive value. The above choice of xd in the contact phase ensures that at the time of take-off, HC has a small negative value and the COM has a positive velocity in the horizontal direction. 5.6.4 Poincar´e Map To investigate the stability of the hybrid system, we use a Poincar´e map with the Poincar´e section defined at the point of touch-down. At the time of touch-down, we assume the system states to satisfy: η3 “ yd,ζ 1 “ 0,ζ 2 “ 0,ζ 3 “ 0, ζ4 “ 0,ζ 5 “ 0,ζ 6 “ 0 (5.72) Thus, the Poincar´e section is chosen as ¯Z :“ t ¯Ψ P R5| η3 “ yd,ζ 1 “ 0,ζ 2 “ 0,ζ 3 “ 0,ζ 4 “ 0,ζ 5 “ 0,ζ 6 “ 0u (5.73) 94 where ¯Ψ are the states that define the Poincar´e section ¯Ψ “r δ η2 η4 η5 η6 sT The Poincar´e map P p ¯Ψq satisfies with periodic point ¯Ψ˚ defined as We define the error state ¯χk as ¯Ψk`1 “ P p ¯Ψkq ¯Ψ˚ “ P p ¯Ψ˚q ¯χk “ p ¯Ψk ´ ¯Ψ˚q (5.74) (5.75) (5.76) By linearizing the Poincar´e map about ¯Ψ˚, we have the approximately linear discrete dy- namics given by ¯χk`1 “ ¯A ¯χk ¯A ∆“ The periodic point ¯Ψ˚ will be asymptotically stable iff dP p ¯Ψq d ¯Ψ ˇˇˇˇ ¯Ψ“ ¯Ψ˚ ρp ¯Aqă 1 where ρp ¯Aq is the spectral radius of ¯A. 5.6.5 Closed-loop Control Design (5.77) (5.78) We now develop a discrete-time controller to drive δ to its desired value δd. By varying one of the continuous controller parameters immediately before touch-down, namely µ2, we stabilize the discrete-time states ¯Ψ “ ¯Ψ˚. The dynamics of the discrete-time system can be described by ¯Ψpk ` 1q “ P p ¯Ψpkq, µ2q (5.79) 95 By linearizing the above map about the fixed point ¯Ψ “ ¯Ψ˚, µ2 “ µ˚2, we get ¯χpk ` 1q “ ¯A ¯χpkq ` ¯Bupkq (5.80) where ¯χpkq “ ¯Ψpkq ´ ¯Ψ˚, ¯A “„ BP p ¯Ψ, µ2q B ¯Ψ  ¯Ψ“ ¯Ψ˚ µ2“µ˚ 2 upkq “ µ2 ´ µ˚2 , ¯B “„ BP p ¯Ψ, µ2q Bµ2  ¯Ψ“ ¯Ψ˚ µ2“µ˚ 2 (5.81) To converge δ to its desired value δd, we propose to use integral control with the integrator defined as ¯βpk ` 1q “ ¯βpkq ´ pδ ´ δdq (5.82) The integrator-augmented discrete system has the form ¯λpk ` 1q “ ¯A¯λpkq ` ¯Bupkq, ¯A ∆“» —– ¯A 0 ´1 1 , fi ffifl ¯B 0 ¯B ∆“» —– ¯λpkq ∆““ ¯χpkq ¯βpkq ‰T fi ffifl (5.83) where ¯A and ¯B is defined in (5.81). If t ¯A, ¯Bu is controllable, the input can be chosen as upkq “ ¯K¯λpkq (5.84) such that ρp ¯A ` ¯B ¯Kqă 1 96 5.7 Simulation 5.7.1 Apex Height Control We present simulation results for apex height control for both strategies starting from rest. The robot parameters were assumed to be mf “ 0.15 kg, ml “ 0.3 kg, mt “ 0.3 kg, mb “ 1 kg, %f “ 0.05 m,% l “ 0.15 m,% t “ 0.15 m,% b “ 0.15 m (5.85) and the gains of the continuous controller were chosen to be K1 “ K7 “ 14400 K2 “ K3 “ 6400 K4 “ K10 “ 240, K5 “ K6 “160 K8 “ K9 “ 225, K11 “ K12 “ 6 The robot was assumed to be initially at rest on the ground; the initial conditions are xp0q “ 0 m 9xp0q “ 0 m{s, yp0q “ 0 m 9yp0q “ 0 m{s θ1p0q “ 2.76 rad 9θ1p0q “ 0 rad{s,θ 2p0q “ ´1.84 rad 9θ2p0q “ 0 rad{s θ3p0q “ 1.37 rad 9θ3p0q “ 0 rad{s,θ 4p0q “ ´1.63 rad 9θ4p0q “ 0 rad{s (5.86) The desired apex height and the moment of inertia about the COM were chosen as hd “ 0.6 m, Icmd “ 0.086 kg.m2 (5.87) 97 0.65 0.6 ) m ( 3 η 0.5 0.4 0 5 10 time (s) 15 20 Figure 5.2: Vertical displacement of the COM during apex-height control η3. Negative Damping Strategy For this strategy, we choose the parameters of the continuous controller to be ν “ ´0.65, yd “ 0.485,θ d “ 2.2,γ “ 1 The initial values of the integrator state β were chosen to be zero. Since tA, Bu is controllable, the controller gains were chosen as K “» —– 0.5 0 ´0.3 0 0 ´9 0 5 fi ffifl (5.88) This places the eigenvalues of the closed loop system at 0.797 ˘ 0.533i and 0.847 ˘ 0.071i. Figures 5.2 and 5.3 shows the results of apex height control using negative damping. Figure 5.2 plots the vertical displacement of the COM (η3); it can be seen that the COM converges to the desired apex height. Figure 5.3 shows the states of the discrete-time system; it can be seen that both states converge to their desired values. 98 After the controller achieved its objective, we found the values of ν, and θd to be ν˚ “ ´0.578,θ ˚d “ 2.102 rad (5.89) A video animation of a four-link hopper hop in place has been uploaded as supplementary material. It shows the hopper reaching the desired apex height of hd “ 0.6 m starting from rest. Equilibrium Height Control Strategy For this strategy, we choose the parameters of the continuous controller to be yf d “ 0.49, ycd “ 0.53,θ d “ 2.2,γ “ 1 0.1 0 -0.1 0.01 0 q d h ´ h p q d m c I ´ m c I p -0.01 10 20 k 30 Figure 5.3: Discrete-time states ph´hdq and pIcm´Icmdq at the end of each hop fr apex-height control using negative damping. 99 0.65 0.6 ) m ( 3 η 0.5 0.4 0 5 10 time (s) 15 20 Figure 5.4: Vertical displacement of the COM during apex-height control η3. The initial values of the integrator state β were chosen to be zero. Since tA, Bu is controllable, the controller gains were chosen as K “» —– ´0.1 0 0.1 0 0 ´3 0 3 fi ffifl (5.90) This places the eigenvalues of the closed loop system at ´0.17, 0.3, 0.77, and 0.97. Figures 5.4 and 5.5 shows the results of apex height control using negative damping. Figure 5.4 plots the vertical displacement of the COM (η3); it can be seen that the COM converges to the desired apex height. Figure 5.5 shows the states of the discrete-time system; it can be seen that both states converge to their desired values. After the controller achieved its objective, we found the values of ycd and θd to be y˚cd “ 0.546,θ ˚d “ 2.21 rad (5.91) A video animation of a four-link hopper hop in place has been uploaded as supplementary 100 0.2 0 -0.2 0.01 0 -0.01 q d h ´ h p q d m c I ´ m c I p 10 20 k 30 Figure 5.5: Discrete-time states ph´hdq and pIcm´Icmdq at the end of each hop fr apex-height control using negative damping. material. It shows the hopper reaching the desired apex height of hd “ 0.6 m starting from rest. 5.7.2 Step Size Control in Hopping Locomotion We present simulation results for hopping locomotion starting from rest with a desired step size. The parameters of the continuous controller is chosen to be mf “ 0.15 kg, ml “ mt “ 0.3 kg, mb “ 1.0 kg, %f “ 0.05 m,% l “ %t “ 0.15 m,% b “ 0.25 m and the gains of the continuous controller were chosen as K1 “ K7 “ 2500, K2 “ K3 “ 400, K4 “ K10 “ 100, K5 “ K6 “ 6, K8 “ K9 “ 10000, K11 “ K12 “ 200 101 3.0 ) m ( x 0.0 0 10 time (s) 20 Figure 5.6: Position of the toe x during hopping locomotion. The robot was assumed to be initially at rest and the initial states were xp0q “ 0 m, 9xp0q “ 0 m{s, yp0q “ 0 m, 9yp0q “ 0 m{s,θ 1p0q “ 2.276 rad, θ2p0q “ ´1.397 rad, 9θ2p0q “ 0 rad{s,θ 9θ1p0q “ 0 rad{s, 3p0q “ 1.7 rad, 9θ3p0q “ 0 rad{s,θ 4p0q “ ´1.461 rad, 9θ4p0q “ 0 rad{s The robot was required to hop for a distance of 3.0 m; therefore, the desired value of the uncontrolled state η1 was set to η1d “ 3.0. To reach this configuration, the desired step size was selected according to the relation δd “ 0.1pη1d ´ η1q a1 ` pη1d ´ η1q2 The above relation ensures that the magnitude of the step size do not exceed 0.1 m. The parameters of the continuous controller were chosen as ν “ ´1.1, yd “ 0.54,θ d “ 1.8, µ1 “ 0.3, µ3 “ 1.3µ2 The initial values of µ2 and the integrator state ¯β were chosen to be zero. The matrix ¯A, 102 defined in (5.83), was found to have eigenvalues: 1.0, 0.74, 0.364, 0.154, ´0.034 and ´0.06. Since t ¯A, ¯Bu is controllable, the controller gains were chosen as ¯K “„ ´0.1 0 0 0 0 0.1  This places the eigenvalues of the closed loop system at 0.968, 0.756, ´0.06, ´0.034, 0.154 and 0.365. Figures 5.6 and 5.7 show the results of hopping locomotion. Figure 5.6 shows the hor- izontal position of the toe x; it can be seen that the hopper moves to its desired location η1 “ x “ 3.0. Figure 5.7 shows variation of the actual and desired step sizes with each hop; it can be seen that the step size tracks the desired value and both converge to zero when the robot reaches its desired location. Figure 5.8 plots the remaining states of the discrete-time system; incidentally, these states are also the uncontrolled states of the robot in the flight phase. It can be seen that these states are well-behaved and converge to a constant value within a few hops. The hopper reached its desired position after k “ 60 hops at t “ 26.3 s. A video animation of hopping locomotion has been uploaded as supplementary material. It shows the hopper reaching the desired position of η1d “ 3.0 m starting from rest with step size varying as shown in Fig.5.7. 0.1 ) m ( d δ , δ 0.0 0 δ δd 20 40 60 k Figure 5.7: Variation of actual and desired step sizes over 60 hops. 103 5.8 Conclusion This chapter presents control strategies for hopping in place with a desired apex height and hopping locomotion with a desired step size for a four-link one-legged hopper. The hybrid dynamics of the hopper is comprised of flight, impact, and contact phases and each phase is converted into normal form to identify the controllable and uncontrollable states. A partial feedback linearization controller constrains the controllable states to behave like a mass-spring-damper system. To overcome the energy losses due to the impact with ground, we used two strategies. The first strategy introduces negative damping in the dynamics of the mass-spring-damper system that corresponds to the vertical displacement of the COM during the contact phase. The second strategy varies the equilibrium height corresponding 0.20 0.00 -0.10 -0.20 0.10 -0.10 0.16 0.12 η2 η4 η5 η6 20 40 60 k Figure 5.8: Plot of uncontrolled states at the end of each hop k during hopping locomotion. 104 to the vertical displacement of the COM during the flight phase. For apex height control and hopping locomotion, separate Poincar´e maps are constructed at the point of touch- down. These maps are used to obtain a discrete-time model of the system and controllers are designed to meet the two control objectives. Simulation results are presented to validate the efficacy of the controllers. A video animation of hopping in place and hopping locomotion are includes to provide a glimpse of the dynamic behavior. 105 Chapter 6 Conclusion In this dissertation we presented several control strategies for motion control of multi-link hoppers. Two and six degree-of-freedom hoppers were considered and the ground model was assumed to be rigid, elastic, and viscoelastic with inertia. For the two degree-of-freedom hoppers, the control objective was to converge the apex height for hopping on different ground models. For the six degree-of-freedom hopper, locomotion with a desired step size was the main control objective. We first presented a control strategy for controlling the apex height of a two-mass prismatic-joint robot hopping on a viscoelastic ground. This problem is challenging due to the losses in the total energy of the system due to impact and damping in the ground. A continuous-time backstepping controller was used in concert with a discrete- time integral controller to meet the control objective. The backstepping controller regulates the energy of the system using the internal degrees of freedom and is useful for apex height control on an elastic ground; for the viscoelastic ground with inertia, it results in steady state error. The discrete-time integral controller eliminates this error by commanding the backstepping controller to regulate the energy to a commensurately higher level. Since there is loss of energy in every hop, the backstepping controller has to remain active for all hops. As the next step, we considered apex height control of the two degree-of-freedom ankle- knee-hip (AKH) robot. The AKH robot is more anthropomorphic but the control problem is more challenging due to the revolute nature of the joints. We considered apex height control of the AKH robot hopping on a rigid ground. The dynamics of the robot is modeled in flight, impact, and contact phases separately. A continuous-time controller is designed to make the robot behave like a mass-spring-damper system. In the contact phase, the continuous 106 controller employs positive damping to arrest the motion of the center-of-mass when it is moving downwards, and negative damping when the center-of-mass is moving upwards. The negative damping is introduced to compensate for the energy loss due to impact. A Poincar´e map of the hopping behavior is constructed and asymptotic stability of the hybrid system to the desired apex height is guaranteed by designing a discrete controller with integral action. The constraint imposed by the robot structure and the effect of continuous controller parameters on hopping behavior is discussed. A simulation of the AKH robot, hopping to a desired apex height from rest, is presented. Following the control strategy for the two-DOF AKH robot hopping on a rigid ground, we presented a control strategy to converge the apex height of an AKH robot hopping on an elastic ground and a viscoelastic ground with inertia. Similar to the two-DOF prismatic- joint robot, a continuous-time backstepping controller is used in conjunction with a discrete- time controller; the discrete-time controller is designed by linearizing a Poincar´e map. For the elastic ground, the backstepping controller is used to control the energy level of the hopper and eliminate the relative displacement between the hopper masses. The discrete- time controller is used to guarantee the stability of the hybrid system and fast convergence of the apex height to the desired value. For the viscoelastic ground, the backstepping controller results in steady state error due to losses from impact between the hopper and the ground as well as viscous losses in the damper. To overcome these losses, the discrete-time controller is used with integral action. Simulation results prove the efficacy of the control strategy for both the elastic ground and the viscoelastic ground. We extend the results of the two-DOF AKH robot hopping on a rigid ground to a four- link six-DOF robot with a body, thigh, leg and foot. We consider two separate problems, namely, controlling the apex height while hopping in one location and locomotion with a fixed step size on a rigid ground. The hybrid dynamics of the hopper is comprised of flight, impact, and contact phases and each phase is converted into normal form to identify the controllable and uncontrollable states. A partial feedback linearization controller constrains 107 the controllable states to behave like a mass-spring-damper system. To overcome the energy losses due to the impact with ground, we used two strategies. The first strategy introduces negative damping in the dynamics of the mass-spring-damper system corresponding to the vertical displacement of the COM during the contact phase. The second strategy alters the equilibrium height corresponding to the vertical displacement of the COM during the flight phase. For both control strategies, it is established that both controllable and uncontrollable states remain bounded. For apex height control and hopping locomotion, separate Poincar´e maps are constructed at the point of touch-down. These maps are used to obtain a discrete- time model of the system and controllers are designed by linearizing the map to meet the control objectives. Simulation results are presented to validate the efficacy of the controllers. Future research will focus on experimental verification of the control strategies developed for the four-link hopping robot. This is a challenging design problem. Hopping requires large torques to be applied by the actuators over relatively small intervals of time. Mechanical advantage such as a gearbox can amplify the torque generated but reduces the speed of response. On the other hand, direct-drive motors that can generate the desired torques are bulky and increase the weight of the robot. This compounds the design problem further as the motors in the lower links have to support the weight of the motors in the upper links. Another extension of this work is the development of control strategies for performing more complex maneuvers such as flipping in the flight phase. Flipping requires high angular momentum in the flight phase and the control torques should be able to generate this momentum at the time of takeoff. The speed of flipping during the flight phase depends on the moment of inertia about the COM, which can be controlled. The moment of inertia has to be controlled such that the robot completes the flip in the flight phase and has the correct orientation prior to touchdown. The design of a control strategy for flipping motion requires nontrivial extensions of the continuous controller discussed in Chapter 5. 108 APPENDIX 109 Appendix A Computing the Dynamical Equations For Four-Link Hopper To drive the dynamical of equations described in (5.1), and (5.9), we use the lagrangian equations, where B Bt p BL B 9q q ´ BL Bq “ Q, L ∆“ T ´ U (A.1) where T , and U are the total kinetic and potential energy of the system, respectively. This resulting the dynamical of equations described in (5.1), and (5.9), where rMijs6ˆ1, and rNis6ˆ6 and M11 “M22 “ ¯m ∆“ mf ` ml ` mt ` mb M12 “M21 “ 0 M13 “M31 “ ´lf“mf ` 2pml ` mt ` mbq‰ sin θ1 ´ ll“ml ` 2pmt ` mbq‰ sinpθ1 ` θ2q M14 “M41 “ ´ll“ml ` 2pmt ` mbq‰ sinpθ1 ` θ2q ´ ltpmt ` 2mbq sinpθ1 ` θ2 ` θ3q ´ ltpmt ` 2mbq sinpθ1 ` θ2 ` θ3q ´ lbmb sinpθ1 ` θ2 ` θ3 ` θ4q ´ lbmb sinpθ1 ` θ2 ` θ3 ` θ4q M15 “M51 “ ´ltpmt ` 2mbq sinpθ1 ` θ2 ` θ3q ´ lbmb sinpθ1 ` θ2 ` θ3 ` θ4q M16 “M61 “ ´lbmb sinpθ1 ` θ2 ` θ3 ` θ4q M23 “M32 “ lf“mf ` 2pml ` mt ` mbq‰ cos θ1 ` ll“ml ` 2pmt ` mbq‰ cospθ1 ` θ2q ` ltpmt ` 2mbq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q 110 M24 “M42 “ ll“ml ` 2pmt ` mbq‰ cospθ1 ` θ2q ` ltpmt ` 2mbq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q M25 “M52 “ ltpmt ` 2mbq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q M26 “M62 “ lbmb cospθ1 ` θ2 ` θ3 ` θ4q M33 “If ` Il ` It ` Ib ` l2 ` l2 t mb ` l2 bmb ` l2 t mt ` 4l2 l“ml ` 4pmt ` mbq‰ f“mf ` 4pml ` mt ` mbq‰ ` 4lfll“ml ` 2pmt ` mbq‰ cos θ2 ` 4ltpmt ` 2mbq“ll cos θ3 ` lf cospθ2 ` θ3q‰ ` 4lbmb“lt cos θ4 ` ll cospθ3 ` θ4q ` lf cospθ2 ` θ3 ` θ4q‰ ` 2lfll“ml ` 2pmt ` mbq‰ cos θ2 ` 4llltpmt ` 2mbq cos θ3 ` 2lbmb“2lt cos θ4 ` 2ll cospθ3 ` θ4q ` lf cospθ2 ` θ3 ` θ4q‰ t mt ` mb“4pl2 l ml ` 4l2 l mt ` l2 ` 2lfltpmt ` 2mbq cospθ2 ` θ3q b‰ l ` l2 t q ` l2 M34 “M43 “ Il ` It ` Ib ` l2 M35 “M53 “ It ` Ib ` l2 t pmt ` 4mbq ` l2 bmb ` 2llltpmt ` 2mbq cos θ3 ` 2lbmb“2lt cos θ4 ` ll cospθ3 ` θ4q ` lf cospθ2 ` θ3 ` θ4q‰ ` 2lfltpmt ` 2mbq cospθ2 ` θ3q M36 “M63 “ Ib ` l2 M44 “Il ` It ` Ib ` l2 bmb ` 2lbmb“lt cos θ4 ` ll cospθ3 ` θ4q ` lf cospθ2 ` θ3 ` θ4q‰ M45 “M54 “ It ` Ib ` l2 bmb ` l2 t pmt ` 4mbq ` l2 l“ml ` 4pmt ` mbq‰ ` 4“llltpmt ` 2mbq cos θ3 ` lbmbplt cos θ4 ` ll cospθ3 ` θ4qq‰ ` 2lbmb“2lt cos θ4 ` ll cospθ3 ` θ4q‰ t pmt ` 4mbq ` 2llltpmt ` 2mbq cos θ3 bmb ` l2 bmb ` 2lbmb“lt cos θ4 ` ll cospθ3 ` θ4q‰ bmb ` 4lbltmb cos θ4 t pmt ` 4mbq ` l2 M46 “M64 “ Ib ` l2 M55 “It ` Ib ` l2 M56 “M65 “ Ib ` l2 bmb ` 2lbltmb cos θ4 111 M66 “Ib ` l2 bmb N1 “ ´“lf`mf ` 2pmb ` ml ` mtq˘ cos θ1 ` ll`ml ` 2pmb ` mtq˘ cospθ1 ` θ2q ` 2ltmb cospθ1 ` θ2 ` θ3q ` ltmt cospθ1 ` θ2 ` θ3q 1 ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´“llpml ` 2`mb ` mtq˘ cospθ1 ` θ2q ` ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´ 2ltmb cospθ1 ` θ2 ` θ3q 9θ2 3 ´ ltmt cospθ1 ` θ2 ` θ3q 9θ2 2 3 ´ lbmb cospθ1 ` θ2 ` θ3 ` θ4q 9θ2 3 ´ 2lbmb cospθ1 ` θ2 ` θ3 ` θ4q 9θ3 9θ4 ´ lbmb cospθ1 ` θ2 ` θ3 ` θ4q 9θ2 4 9θ4 9θ4 9θ2 9θ3 ´ 2“ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´ 2“lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´ 2“ll`ml ` 2pmb ` mtq˘ cospθ1 ` θ2q ` ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 ´ 2“ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 ´ 2“lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 ´“lf`mf ` 2pmb ` ml ` mtq˘ sin θ1 ` ll`ml ` 2pmb ` mtq˘ sinpθ1 ` θ2q ` lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´“llpml ` 2`mb ` mtq˘ sinpθ1 ` θ2q ` ltp2mb ` mtq sinpθ1 ` θ2 ` θ3q ` lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ` 2ltmb sinpθ1 ` θ2 ` θ3q ` ltmt sinpθ1 ` θ2 ` θ3q ´ 2ltmb sinpθ1 ` θ2 ` θ3q 9θ2 3 ´ ltmt sinpθ1 ` θ2 ` θ3q 9θ2 9θ3 1 2 N2 “gmb ` gmf ` gml ` gmt 3 ´ lbmb sinpθ1 ` θ2 ` θ3 ` θ4q 9θ2 3 ´ 2lbmb sinpθ1 ` θ2 ` θ3 ` θ4q 9θ3 9θ4 ´ lbmb sinpθ1 ` θ2 ` θ3 ` θ4q 9θ2 4 112 9θ2 9θ4 9θ3 9θ3 9θ4 ´ 2“ltp2mb ` mtq sinpθ1 ` θ2 ` θ3q ` lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´ 2“lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 ´ 2“ll`ml ` 2pmb ` mtq˘ sinpθ1 ` θ2q ` ltp2mb ` mtq sinpθ1 ` θ2 ` θ3q ` lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 ´ 2“ltp2mb ` mtq sinpθ1 ` θ2 ` θ3q ` lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 ´ 2“lbmb sinpθ1 ` θ2 ` θ3 ` θ4q‰ 9θ1 N3 “g“lf`2mb ` mf ` 2pml ` mtq˘ cos θ1 ` llp2mb ` ml ` 2mtq cospθ1 ` θ2q ` ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ ´ 2lf“ll`ml ` 2pmb ` mtq˘ sin θ2 ` ltp2mb ` mtq sinpθ2 ` θ3q ` lbmb sinpθ2 ` θ3 ` θ4q‰ 9θ2 ´ 2“llltp2mb ` mtq sin θ3 ` lfltp2mb ` mtq sinpθ2 ` θ3q ` lbmbpll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4qq‰ 9θ2 ´ 2lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q‰ 9θ2 ´ 4lf“ll`ml ` 2pmb ` mtq˘ sin θ2 ` ltp2mb ` mtq sinpθ2 ` θ3q ` lbmb sinpθ2 ` θ3 ` θ4q‰ 9θ1 ´ 4“llltp2mb ` mtq sin θ3 ` lfltp2mb ` mtq sinpθ2 ` θ3q ` lbmb`ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q˘‰ 9θ1 ´ 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q‰ 9θ1 ´ 4“llltp2mb ` mtq sin θ3 ` lfltp2mb ` mtq sinpθ2 ` θ3q ` lbmb`ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q˘‰ 9θ2 ´ 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q‰ 9θ2 ´ 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q‰ 9θ3 9θ3 9θ3 9θ4 9θ4 9θ4 2 3 9θ2 4 113 3 9θ3 9θ3 4 1 9θ4 N4 “g“llp2mb ` ml ` 2mtq cospθ1 ` θ2q ` ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ ` 2lf“ll`ml ` 2pmb ` mtq˘ sin θ2 ` ltp2mb ` mtq sinpθ2 ` θ3q ` lbmb sinpθ2 ` θ3 ` θ4q‰ 9θ2 ´ 2ll“ltp2mb ` mtq sin θ3 ` lbmb sinpθ3 ` θ4q‰ 9θ2 ´ 2lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q‰ 9θ2 ´ 4ll“ltp2mb ` mtq sin θ3 ` lbmb sinpθ3 ` θ4q‰ 9θ1 ´ 4lbmb“lt sinpθ4q ` ll sinpθ3 ` θ4q‰ 9θ1 ´ 4ll“ltp2mb ` mtq sin θ3 ` lbmb sinpθ3 ` θ4q‰ 9θ2 ´ 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q‰ 9θ2 ´ 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q‰ 9θ3 ` 2“ltll`mt ` 2mbq sin θ3 ` ltlfp2mb ` mtq sinpθ2 ` θ3q ` lbmb`ll sinpθ3 ` θ4q ` lf sinpθ2 ` θ3 ` θ4q˘‰ 9θ2 ` 2ll“ltp2mb ` mtq sin θ3 ` lbmb sinpθ3 ` θ4q‰ 9θ2 ` 4ll“ltp2mb ` mtq sin θ3 ` lbmb sinpθ3 ` θ4q‰ 9θ1 ´ 2lbmblt sin θ4 N5 “g“ltp2mb ` mtq cospθ1 ` θ2 ` θ3q ` lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ 9θ2 4 9θ2 9θ4 9θ4 1 2 ´ 4lbmblt sin θ4 9θ4p 9θ1 ` 9θ2 ` 9θ3q N6 “g“lbmb cospθ1 ` θ2 ` θ3 ` θ4q‰ ` 2lbmb“ll sinpθ3 ` θ4q ` lt sin θ4 ` lf sinpθ2 ` θ3 ` θ4q‰ 9θ2 1 2 ` 2lbmblt sin θ4 9θ2 3 ` 2lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q‰ 9θ2 ` 4lbmb“lt sin θ4 ` ll sinpθ3 ` θ4q‰ 9θ1 ` 4lbmblt sin θ4 9θ3 9θ2 9θ2 ` 4lbmblt sin θ4 9θ1 9θ3 (A.2) 114 BIBLIOGRAPHY 115 BIBLIOGRAPHY [1] H. 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