Decentralized feedback controllers for exponential stabilization of hybrid periodic orbits: Application to robotic walking

This paper presents a systematic algorithm to design time-invariant decentralized feedback controllers to exponentially stabilize periodic orbits for a class of hybrid dynamical systems arising from bipedal walking. The algorithm assumes a class of parameterized and nonlinear decentralized feedback controllers which coordinate lower-dimensional hybrid subsystems based on a common phasing variable. The exponential stabilization problem is translated into an iterative sequence of optimization problems involving bilinear and linear matrix inequalities, which can be easily solved with available software packages. A set of sufficient conditions for the convergence of the iterative algorithm to a stabilizing decentralized feedback control solution is presented. The power of the algorithm is demonstrated by designing a set of local nonlinear controllers that cooperatively produce stable walking for a 3D autonomous biped with 9 degrees of freedom, 3 degrees of underactuation, and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg.

[1]  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.

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

[3]  Lubomír Bakule,et al.  Decentralized control: An overview , 2008, Annu. Rev. Control..

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

[5]  Seungmoon Song,et al.  A neural circuitry that emphasizes spinal feedback generates diverse behaviours of human locomotion , 2015, The Journal of physiology.

[6]  Robert D. Gregg,et al.  Hybrid invariance and stability of a feedback linearizing controller for powered prostheses , 2015, 2015 American Control Conference (ACC).

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

[8]  David C. Post,et al.  The effects of foot geometric properties on the gait of planar bipeds walking under HZD-based control , 2014, Int. J. Robotics Res..

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

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

[11]  Dragoslav D. Šiljak,et al.  Decentralized control of complex systems , 2012 .

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

[13]  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..

[14]  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.

[15]  P. Leva.  Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters. , 1996 .

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

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

[18]  Jessy W. Grizzle,et al.  Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations , 2016, Int. J. Robotics Res..

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

[20]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

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

[22]  Robert D. Gregg,et al.  A Robust Parameterization of Human Gait Patterns Across Phase-Shifting Perturbations , 2017, IEEE Transactions on Neural Systems and Rehabilitation Engineering.

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

[24]  Luca Consolini,et al.  Virtual Holonomic Constraints for Euler-Lagrange Systems , 2010 .

[25]  Dongjun Lee,et al.  Passivity-Based Control of Bipedal Locomotion , 2007, IEEE Robotics & Automation Magazine.

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

[27]  Katie Byl,et al.  Approximate optimal control of the compass gait on rough terrain , 2008, 2008 IEEE International Conference on Robotics and Automation.

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

[29]  Kimberly A. Ingraham,et al.  Configuring a Powered Knee and Ankle Prosthesis for Transfemoral Amputees within Five Specific Ambulation Modes , 2014, PloS one.

[30]  E. Davison,et al.  On the stabilization of decentralized control systems , 1973 .

[31]  W C Flowers,et al.  Stance phase control of above-knee prostheses: knee control versus SACH foot design. , 1987, Journal of biomechanics.

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

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

[34]  Michael Goldfarb,et al.  Upslope Walking With a Powered Knee and Ankle Prosthesis: Initial Results With an Amputee Subject , 2011, IEEE Transactions on Neural Systems and Rehabilitation Engineering.

[35]  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).

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

[37]  Russ Tedrake,et al.  L2-gain optimization for robust bipedal walking on unknown terrain , 2013, 2013 IEEE International Conference on Robotics and Automation.

[38]  Jessy W. Grizzle,et al.  Iterative Robust Stabilization Algorithm for Periodic Orbits of Hybrid Dynamical Systems: Application to Bipedal Running , 2015, ADHS.

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