Stabilization of hybrid periodic orbits with application to bipedal walking

This work describes a general computational framework for robust stabilization of periodic orbits for hybrid systems with impact effects. We demonstrate that for such systems dynamics can be decomposed into the transverse and tangential components if a proper orthogonalizing transform is applied before the decomposition. Subsequently, we show that the robust control synthesis problem can be cast as a semi-definite program which implies that computationally efficient linear matrix inequality (LMI) solvers can be used to find the controllers. The methodology is verified through the simulation on a five-link planar under-actuated biped robot, an example often used by other researchers

[1]  P. Hartman.  Ordinary Differential Equations , 1965 .

[2]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[3]  Marc H. Raibert,et al.  Legged Robots That Balance , 1986, IEEE Expert.

[4]  R. Brach.  Rigid Body Collisions , 1989 .

[5]  Michael Athans,et al.  Analysis of gain scheduled control for nonlinear plants , 1990 .

[6]  Daniel E. Koditschek,et al.  Analysis of a Simplified Hopping Robot , 1991, Int. J. Robotics Res..

[7]  R. Decarlo,et al.  Asymptotic Stability of m-Switched Systems using Lyapunov-Like Functions , 1991, 1991 American Control Conference.

[8]  M. Branicky.  Stability of switched and hybrid systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[9]  G. Lafferriere.  Discontinuous stabilizing feedback using partially defined Lyapunov functions , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

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

[11]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[12]  S. Pettersson,et al.  Stability and robustness for hybrid systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[13]  David E. Orin,et al.  Simulation of contact using a nonlinear damping model , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[14]  S. Kajita,et al.  Experimental study of biped dynamic walking , 1996 .

[15]  Bernard Espiau,et al.  Bifurcation and chaos in a simple passive bipedal gait , 1997, Proceedings of International Conference on Robotics and Automation.

[16]  A. Rantzer,et al.  On the computation of piecewise quadratic Lyapunov functions , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[17]  Matthew D. Berkemeier.  Modeling the Dynamics of Quadrupedal Running , 1998, Int. J. Robotics Res..

[18]  Jong H. Park,et al.  Biped robot walking using gravity-compensated inverted pendulum mode and computed torque control , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[19]  Joel W. Burdick,et al.  Stabilization of Systems with Changing Dynamics , 1998, HSCC.

[20]  Mark W. Spong,et al.  Passivity based control of the compass gait biped , 1999 .

[21]  V. Babitsky.  Theory of Vibro-Impact Systems and Applications , 2013 .

[22]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[23]  Fernando Paganini,et al.  A Course in Robust Control Theory , 2000 .

[24]  Ian A. Hiskens,et al.  Stability of hybrid system limit cycles: application to the compass gait biped robot , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

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

[26]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[27]  Milos Zefran,et al.  A computational approach to dynamic bipedal walking , 2003, Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453).

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

[29]  Christine Chevallereau,et al.  Asymptotically Stable Running for a Five-Link, Four-Actuator, Planar Bipedal Robot , 2005, Int. J. Robotics Res..