Hybrid Zero Dynamics of Bipedal Robots Under Nonholonomic Virtual Constraints

This letter investigates the hybrid zero dynamics for planar bipedal robots with one degree of underactuation subject to nonholonomic virtual constraints (NHVCs). We first derive the closed-form expression of the bipedal robot zero dynamics under NHVCs. We next present conditions that make the NHVCs invariant with respect to rigid impacts with the ground. Lastly, a reduced dimensionality test, which is independent of the number of degrees of freedom of the bipedal robot, is proposed for checking existence and exponential stability of hybrid periodic orbits under NHVCs. Simulation results using the RABBIT biped robot demonstrate the robustness of the proposed NHVCs against a randomized horizontal push disturbance. A statistical significant difference between the mean number of steps until failure is shown between the NHVC and virtual holonomic constraint control schemes.

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

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

[3]  Katie Byl,et al.  Metastable Walking Machines , 2009, Int. J. Robotics Res..

[4]  Christine Chevallereau,et al.  Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion , 2017, Humanoid Robotics: A Reference.

[5]  Robert D. Gregg,et al.  Stable, Robust Hybrid Zero Dynamics Control of Powered Lower-Limb Prostheses , 2017, IEEE Transactions on Automatic Control.

[6]  Ludovic Righetti,et al.  Controlled Reduction With Unactuated Cyclic Variables: Application to 3D Bipedal Walking With Passive Yaw Rotation , 2013, IEEE Transactions on Automatic Control.

[7]  Jessy W. Grizzle,et al.  Nonholonomic virtual constraints for dynamic walking , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[8]  Sushant Veer,et al.  Local input-to-state stability of dynamic walking under persistent external excitation using hybrid zero dynamics , 2016, 2016 American Control Conference (ACC).

[9]  Aaron D. Ames,et al.  Multicontact Locomotion on Transfemoral Prostheses via Hybrid System Models and Optimization-Based Control , 2016, IEEE Transactions on Automation Science and Engineering.

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

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

[12]  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.

[13]  Robert D. Gregg,et al.  Continuous-Phase Control of a Powered Knee–Ankle Prosthesis: Amputee Experiments Across Speeds and Inclines , 2018, IEEE Transactions on Robotics.

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

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

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

[17]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

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

[19]  Jessy W. Grizzle,et al.  Nonholonomic virtual constraints and gait optimization for robust walking control , 2017, Int. J. Robotics Res..