Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations

This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.

[1]  Anne E. Martin,et al.  Predicting human walking gaits with a simple planar model. , 2014, Journal of biomechanics.

[2]  M. A. Aizerman,et al.  DETERMINATION OF STABILITY BY LINEAR APPROXIMATION OF A PERIODIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES , 1958 .

[3]  Aaron D. Ames,et al.  Three-Dimensional Kneed Bipedal Walking: A Hybrid Geometric Approach , 2009, HSCC.

[4]  Koushil Sreenath,et al.  Embedding active force control within the compliant hybrid zero dynamics to achieve stable, fast running on MABEL , 2013, Int. J. Robotics Res..

[5]  Daniel E. Koditschek,et al.  Hybrid zero dynamics of planar biped walkers , 2003, IEEE Trans. Autom. Control..

[6]  Katie Byl,et al.  Switching policies for metastable walking , 2013, 52nd IEEE Conference on Decision and Control.

[7]  Luca Consolini,et al.  Virtual Holonomic Constraints for Euler–Lagrange Systems , 2013, IEEE Transactions on Automatic Control.

[8]  Aaron D. Ames,et al.  A geometric approach to three-dimensional hipped bipedal robotic walking , 2007, 2007 46th IEEE Conference on Decision and Control.

[9]  B. Anderson,et al.  Output feedback stabilization and related problems-solution via decision methods , 1975 .

[10]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[11]  Jessy W. Grizzle,et al.  Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3D bipedal robot , 2014, 53rd IEEE Conference on Decision and Control.

[12]  Timothy Bretl,et al.  Control and Planning of 3-D Dynamic Walking With Asymptotically Stable Gait Primitives , 2012, IEEE Transactions on Robotics.

[13]  Jessy W. Grizzle,et al.  Walking gait optimization for accommodation of unknown terrain height variations , 2015, 2015 American Control Conference (ACC).

[14]  D. Henrion,et al.  Solving polynomial static output feedback problems with PENBMI , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[15]  Russ Tedrake,et al.  Optimizing robust limit cycles for legged locomotion on unknown terrain , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[16]  Christine Chevallereau,et al.  Models, feedback control, and open problems of 3D bipedal robotic walking , 2014, Autom..

[17]  N. Cowan,et al.  Lateral stability of the spring-mass hopper suggests a two-step control strategy for running. , 2009, Chaos.

[18]  H. Sebastian Seung,et al.  Stochastic policy gradient reinforcement learning on a simple 3D biped , 2004, 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566).

[19]  Shuuji Kajita,et al.  Legged Robots , 2008, Springer Handbook of Robotics.

[20]  Philip Holmes,et al.  Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps , 2004, Int. J. Robotics Res..

[21]  Aaron D. Ames,et al.  Rank deficiency and superstability of hybrid systems , 2012 .

[22]  S. Shankar Sastry,et al.  Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems , 2013, IEEE Transactions on Automatic Control.

[23]  Christine Chevallereau,et al.  RABBIT: a testbed for advanced control theory , 2003 .

[24]  W. Haddad,et al.  Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , 2008 .

[25]  Daniel E. Koditschek,et al.  Planning and Control of Robotic Juggling and Catching Tasks , 1994, Int. J. Robotics Res..

[26]  M. A. Pai,et al.  Hybrid systems view of power system modelling , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[27]  Moritz Diehl,et al.  Stability Optimization of Hybrid Periodic Systems via a Smooth Criterion , 2009, IEEE Transactions on Automatic Control.

[28]  Christine Chevallereau,et al.  From stable walking to steering of a 3D bipedal robot with passive point feet , 2012, Robotica.

[29]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1998, IEEE Trans. Autom. Control..

[30]  J. Grizzle.  Remarks on Event-Based Stabilization of Periodic Orbits in Systems with Impulse Effects , 2006 .

[31]  Jessy W. Grizzle,et al.  Hybrid Invariant Manifolds in Systems With Impulse Effects With Application to Periodic Locomotion in Bipedal Robots , 2009, IEEE Transactions on Automatic Control.

[32]  Jonathan W. Hurst,et al.  THE DESIGN OF ATRIAS 1.0 A UNIQUE MONOPOD, HOPPING ROBOT ∗ , 2012 .

[33]  Dongjun Lee,et al.  Time-Scaling Trajectories of Passive-Dynamic Bipedal Robots , 2007, Proceedings 2007 IEEE International Conference on Robotics and Automation.

[34]  Aaron D. Ames,et al.  Human-Inspired Control of Bipedal Walking Robots , 2014, IEEE Transactions on Automatic Control.

[35]  Koushil Sreenath,et al.  A Compliant Hybrid Zero Dynamics Controller for Stable, Efficient and Fast Bipedal Walking on MABEL , 2011, Int. J. Robotics Res..

[36]  Christine Chevallereau,et al.  Asymptotically Stable Walking of a Five-Link Underactuated 3-D Bipedal Robot , 2009, IEEE Transactions on Robotics.

[37]  Leonid B. Freidovich,et al.  Transverse Linearization for Controlled Mechanical Systems With Several Passive Degrees of Freedom , 2010, IEEE Transactions on Automatic Control.

[38]  Hartmut Geyer,et al.  Swing-leg retraction: a simple control model for stable running , 2003, Journal of Experimental Biology.

[39]  Franck Plestan,et al.  Asymptotically stable walking for biped robots: analysis via systems with impulse effects , 2001, IEEE Trans. Autom. Control..

[40]  Jessy W. Grizzle,et al.  Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry , 2014, IEEE Transactions on Robotics.

[41]  Jonathon W. Sensinger,et al.  Virtual Constraint Control of a Powered Prosthetic Leg: From Simulation to Experiments With Transfemoral Amputees , 2014, IEEE Transactions on Robotics.

[42]  Jessy W. Grizzle,et al.  Preliminary walking experiments with underactuated 3D bipedal robot MARLO , 2014, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[43]  Jonathon W. Sensinger,et al.  Towards Biomimetic Virtual Constraint Control of a Powered Prosthetic Leg , 2014, IEEE Transactions on Control Systems Technology.

[44]  M. Spong,et al.  CONTROLLED SYMMETRIES AND PASSIVE WALKING , 2002 .

[45]  Ian R. Manchester,et al.  Stable dynamic walking over uneven terrain , 2011, Int. J. Robotics Res..

[46]  Justin Seipel,et al.  A simple model for clock-actuated legged locomotion , 2007 .

[47]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[48]  A. Isidori.  Nonlinear Control Systems , 1985 .

[49]  C. Canudas-de-Wit,et al.  Motion planning and feedback stabilization of periodic orbits for an Acrobot , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[50]  Mustafa Mert Ankarali,et al.  Control of underactuated planar pronking through an embedded spring-mass Hopper template , 2011, Auton. Robots.

[51]  Nasser Sadati,et al.  Stabilization of Periodic Orbits for Planar Walking With Noninstantaneous Double-Support Phase , 2012, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[52]  Ian A. Hiskens,et al.  Trajectory Sensitivity Analysis of Hybrid Systems , 2000 .

[53]  C. David Remy,et al.  Optimal exploitation of natural dynamics in legged locomotion , 2011 .

[54]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[55]  E. Westervelt,et al.  Feedback Control of Dynamic Bipedal Robot Locomotion , 2007 .

[56]  O. Toker,et al.  On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[57]  Martijn Wisse,et al.  A Disturbance Rejection Measure for Limit Cycle Walkers: The Gait Sensitivity Norm , 2007, IEEE Transactions on Robotics.

[58]  M.W. Spong,et al.  Reduction-based control with application to three-dimensional bipedal walking robots , 2008, 2008 American Control Conference.

[59]  Leonid B. Freidovich,et al.  A Passive 2-DOF Walker: Hunting for Gaits Using Virtual Holonomic Constraints , 2009, IEEE Transactions on Robotics.

[60]  Jessy W. Grizzle,et al.  Performance Analysis and Feedback Control of ATRIAS, A Three-Dimensional Bipedal Robot , 2014 .

[61]  C. B. Soh,et al.  Schur stability of convex combination of matrices , 1990 .

[62]  Koushil Sreenath,et al.  Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics , 2014, IEEE Transactions on Automatic Control.

[63]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[64]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[65]  [5] , Geometrical Methods in the Theory of Ordinary Differential , 2022 .

[66]  Nicholas Kalouptsidis,et al.  Stability of discrete-time bilinear systems with constant inputs , 1986 .

[67]  R. Braatz,et al.  A tutorial on linear and bilinear matrix inequalities , 2000 .

[68]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[69]  Aaron D. Ames,et al.  Planar multi-contact bipedal walking using hybrid zero dynamics , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[70]  Dan B. Marghitu,et al.  Rigid Body Collisions of Planar Kinematic Chains With Multiple Contact Points , 1994, Int. J. Robotics Res..