Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry

Models of robotic bipedal walking are hybrid, with a differential equation that describes the stance phase and a discrete map describing the impact event, that is, the nonstance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both continuous and discrete (event-based) actions. This paper concentrates on the event-based portion of the feedback design problem for 3-D bipedal walking. The results are developed in the context of robustly stabilizing periodic orbits for a simulation model of ATRIAS 2.1, which is a highly underactuated 3-D bipedal robot with series-compliant actuators and point feet, against external disturbances as well as parametric and nonparametric uncertainty. It is shown that left-right symmetry of the model can be used to both simplify and improve the design of event-based controllers. Here, the event-based control is developed on the basis of the Poincaré map, linear matrix inequalities and robust optimal control. The results are illustrated by designing a controller that enhances the lateral stability of ATRIAS 2.1.

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

[2]  Milos Zefran,et al.  Underactuated dynamic three-dimensional bipedal walking , 2006, Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006..

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

[4]  Nasser Sadati,et al.  Exponential stabilisation of periodic orbits for running of a three-dimensional monopedal robot , 2011 .

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

[6]  Robert D. Gregg,et al.  Reduction-based Control of Three-dimensional Bipedal Walking Robots , 2010, Int. J. Robotics Res..

[7]  Manfred Morari,et al.  Model predictive optimal averaging level control , 1989 .

[8]  Carlos Canudas de Wit,et al.  Switching and PI control of walking motions of planar biped walkers , 2003, IEEE Trans. Autom. Control..

[9]  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).

[10]  Alireza Ramezani,et al.  Performance Analysis and Feedback Control of ATRIAS , A 3 D Bipedal Robot , 2012 .

[11]  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.

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

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

[14]  Jessy W. Grizzle,et al.  Atrias 2.0, A new 3D bipedal robotic walker and runner , 2012 .

[15]  Riccardo Scattolini,et al.  Robustness and robust design of MPC for nonlinear discrete-time systems , 2007 .

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

[17]  Yasuhisa Hasegawa,et al.  Multi-Locomotion Control of Biped Locomotion and Brachiation Robot , 2006 .

[18]  Jessy W. Grizzle,et al.  Robust event-based stabilization of periodic orbits for hybrid systems: Application to an underactuated 3D bipedal robot , 2013, 2013 American Control Conference.

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

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

[21]  Frank Allgöwer,et al.  Assessment and Future Directions of Nonlinear Model Predictive Control , 2007 .

[22]  Franck Plestan,et al.  Stable walking of a 7-DOF biped robot , 2003, IEEE Trans. Robotics Autom..

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

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

[25]  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).

[26]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[27]  Francesco Bullo,et al.  Controlled symmetries and passive walking , 2005, IEEE Transactions on Automatic Control.

[28]  Christine Chevallereau,et al.  3D Bipedal Robotic Walking: Models, Feedback Control, and Open Problems , 2010 .

[29]  Carlos Canudas de Wit,et al.  A new model for control of systems with friction , 1995, IEEE Trans. Autom. Control..

[30]  Ryan W. Sinnet,et al.  3 D Bipedal Robotic Walking : Models , Feedback Control , and Open Problems , 2010 .

[31]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

[32]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

[33]  Koushil Sreenath,et al.  Design and experimental implementation of a compliant hybrid zero dynamics controller with active force control for running on MABEL , 2012, 2012 IEEE International Conference on Robotics and Automation.

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

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

[36]  Jessy W. Grizzle,et al.  Experimental Validation of a Framework for the Design of Controllers that Induce Stable Walking in Planar Bipeds , 2004, Int. J. Robotics Res..

[37]  Carlos Canudas de Wit,et al.  Theory of Robot Control , 1996 .

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

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

[40]  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.

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

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