Walking With Confidence: Safety Regulation for Full Order Biped Models

Safety guarantees are valuable in the control of walking robots, as falling can be both dangerous and costly. Unfortunately, set-based tools for generating safety guarantees (such as sums-of-squares optimization) are typically restricted to simplified, low-dimensional models of walking robots. For more complex models, methods based on hybrid zero dynamics can ensure the local stability of a pre-specified limit cycle, but provide limited guarantees. This letter combines the benefits of both approaches by using sums-of-squares optimization on a hybrid zero dynamics manifold to generate a guaranteed safe set for a ten-dimensional walking robot model. Along with this set, this letter describes how to generate a controller that maintains safety by modifying the manifold parameters when on the edge of the safe set. The proposed approach, which is applied to a bipedal RABBIT model, provides a roadmap for applying sums-of-squares techniques to high-dimensional systems. This opens the door for a broad set of tools that can generate flexible and safe controllers for complex walking robot models.

[1]  Aaron D. Ames,et al.  FROST∗: Fast robot optimization and simulation toolkit , 2017, 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[2]  C. David Remy,et al.  The Energetic Benefit of Robotic Gait Selection—A Case Study on the Robot RAMone , 2017, IEEE Robotics and Automation Letters.

[3]  Paulo Tabuada,et al.  First steps toward formal controller synthesis for bipedal robots with experimental implementation , 2017 .

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

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

[6]  Koushil Sreenath,et al.  Optimal Robust Control for Bipedal Robots through Control Lyapunov Function based Quadratic Programs , 2015, Robotics: Science and Systems.

[7]  Sergey V. Drakunov,et al.  Capture Point: A Step toward Humanoid Push Recovery , 2006, 2006 6th IEEE-RAS International Conference on Humanoid Robots.

[8]  Koushil Sreenath,et al.  3D dynamic walking on stepping stones with control barrier functions , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[9]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[10]  Ian R. Manchester,et al.  Invariant funnels for underactuated dynamic walking robots: New phase variable and experimental validation , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

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

[12]  Russ Tedrake,et al.  Convex optimization of nonlinear feedback controllers via occupation measures , 2013, Int. J. Robotics Res..

[13]  Didier Henrion,et al.  Convex Computation of the Region of Attraction of Polynomial Control Systems , 2012, IEEE Transactions on Automatic Control.

[14]  Pierre-Brice Wieber.  On the stability of walking systems , 2002 .

[15]  Sercan Yildiz,et al.  Sum-of-Squares Optimization without Semidefinite Programming , 2017, SIAM J. Optim..

[16]  Tedrake Russ,et al.  Balance control using center of mass height variation: Limitations imposed by unilateral contact , 2016 .

[17]  Sushant Veer,et al.  Composing limit cycles for motion planning of 3D bipedal walkers , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[18]  Shankar Mohan,et al.  Convex computation of the reachable set for hybrid systems with parametric uncertainty , 2016, 2016 American Control Conference (ACC).

[19]  Frank Allgöwer,et al.  CONSTRUCTIVE SAFETY USING CONTROL BARRIER FUNCTIONS , 2007 .

[20]  Xingye Da,et al.  Combining trajectory optimization, supervised machine learning, and model structure for mitigating the curse of dimensionality in the control of bipedal robots , 2017, Int. J. Robotics Res..

[21]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[22]  Amir Ali Ahmadi,et al.  DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization , 2014, 2014 48th Annual Conference on Information Sciences and Systems (CISS).

[23]  Aaron D. Ames,et al.  Control barrier function based quadratic programs with application to bipedal robotic walking , 2015, 2015 American Control Conference (ACC).

[24]  Twan Koolen,et al.  Balancing and Step Recovery Capturability via Sums-of-Squares Optimization , 2017, Robotics: Science and Systems.

[25]  Koushil Sreenath,et al.  Exponential Control Barrier Functions for enforcing high relative-degree safety-critical constraints , 2016, 2016 American Control Conference (ACC).

[26]  Jessy W. Grizzle,et al.  Feedback Control of a Cassie Bipedal Robot: Walking, Standing, and Riding a Segway , 2018, 2019 American Control Conference (ACC).

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

[28]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[29]  Sushant Veer,et al.  Safe Adaptive Switching among Dynamical Movement Primitives: Application to 3D Limit-Cycle Walkers , 2018, 2019 International Conference on Robotics and Automation (ICRA).

[30]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[31]  Pengcheng Zhao,et al.  Optimal Control for Nonlinear Hybrid Systems via Convex Relaxations , 2017, 1702.04310.

[32]  Ruzena Bajcsy,et al.  Convex computation of the reachable set for controlled polynomial hybrid systems , 2014, 53rd IEEE Conference on Decision and Control.

[33]  Colin Neil Jones,et al.  Controller design and value function approximation for nonlinear dynamical systems , 2015, Autom..