Dynamic Output Controllers for Exponential Stabilization of Periodic Orbits for Multidomain Hybrid Models of Robotic Locomotion

The primary goal of this paper is to develop an analytical framework to systematically design dynamic output feedback controllers that exponentially stabilize multi-domain periodic orbits for hybrid dynamical models of robotic locomotion. We present a class of parameterized dynamic output feedback controllers such that (1) a multi-domain periodic orbit is induced for the closed-loop system, and (2) the orbit is invariant under the change of the controller parameters. The properties of the Poincaré map are investigated to show that the Jacobian linearization of the Poincaré map around the fixed point takes a triangular form. This demonstrates the nonlinear separation principle for hybrid periodic orbits. We then employ an iterative algorithm based on a sequence of optimization problems involving bilinear matrix inequalities to tune the controller parameters. A set of sufficient conditions for the convergence of the algorithm to stabilizing parameters is presented. Full state stability and stability modulo yaw under dynamic output feedback control are addressed. The power of the analytical approach is ultimately demonstrated through designing a nonlinear dynamic output feedback controller for walking of a 3D humanoid robot with 18 state variables and 325 controller parameters.

[1]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[2]  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.

[3]  R. Braatz,et al.  A tutorial on linear and bilinear matrix inequalities , 2000 .

[4]  Ambarish Goswami,et al.  Postural Stability of Biped Robots and the Foot-Rotation Indicator (FRI) Point , 1999, Int. J. Robotics Res..

[5]  Behçet Açikmese,et al.  Observers for systems with nonlinearities satisfying incremental quadratic constraints , 2011, Autom..

[6]  Murat Arcak,et al.  Certainty-equivalence output-feedback design with circle-criterion observers , 2005, IEEE Transactions on Automatic Control.

[7]  Luca Consolini,et al.  Dynamic virtual holonomic constraints for stabilization of closed orbits in underactuated mechanical systems , 2017, Autom..

[8]  Franck Plestan,et al.  Observer-based control of a walking biped robot without orientation measurement , 2006, Robotica.

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

[10]  Yan Wang,et al.  Feasibility analysis of the bilinear matrix inequalities with an application to multi-objective nonlinear observer design , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[11]  Qu Cao,et al.  Quadrupedal running with a flexible torso: control and speed transitions with sums-of-squares verification , 2016, Artificial Life and Robotics.

[12]  K. Goh,et al.  Robust synthesis via bilinear matrix inequalities , 1996 .

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

[14]  J. Gauthier,et al.  A simple observer for nonlinear systems applications to bioreactors , 1992 .

[15]  Aaron D. Ames,et al.  3D dynamic walking with underactuated humanoid robots: A direct collocation framework for optimizing hybrid zero dynamics , 2016, 2016 IEEE International Conference on Robotics and Automation (ICRA).

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

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

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

[19]  Petar V. Kokotovic,et al.  Nonlinear observers: a circle criterion design and robustness analysis , 2001, Autom..

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

[21]  Dragan Stokic,et al.  Dynamics of Biped Locomotion , 1990 .

[22]  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.

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

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

[25]  Jessy W. Grizzle,et al.  Experimental results for 3D bipedal robot walking based on systematic optimization of virtual constraints , 2016, 2016 American Control Conference (ACC).

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

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

[28]  P. Leung,et al.  Fluid conductance of cancellous bone graft as a predictor for graft-host interface healing. , 1996, Journal of biomechanics.

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

[30]  Hassan K. Khalil,et al.  High-gain observers in the presence of measurement noise: A switched-gain approach , 2009, Autom..

[31]  Jessy W. Grizzle,et al.  A Finite-State Machine for Accommodating Unexpected Large Ground-Height Variations in Bipedal Robot Walking , 2013, IEEE Transactions on Robotics.

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

[33]  Jessy W. Grizzle,et al.  The Spring Loaded Inverted Pendulum as the Hybrid Zero Dynamics of an Asymmetric Hopper , 2009, IEEE Transactions on Automatic Control.

[34]  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..

[35]  Ayush Agrawal,et al.  First Steps Towards Translating HZD Control of Bipedal Robots to Decentralized Control of Exoskeletons , 2017, IEEE Access.

[36]  Jessy W. Grizzle,et al.  Reduced-order framework for exponential stabilization of periodic orbits on parameterized hybrid zero dynamics manifolds: Application to bipedal locomotion , 2017 .

[37]  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..

[38]  A. Teel,et al.  Global stabilizability and observability imply semi-global stabilizability by output feedback , 1994 .

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

[40]  Christine Chevallereau,et al.  Models, feedback control, and open problems of 3D bipedal robotic walking , 2014, Autom..

[41]  A. Germani,et al.  A Luenberger-like observer for nonlinear systems , 1993 .

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

[43]  Aaron D. Ames,et al.  Multi-contact bipedal robotic locomotion , 2015, Robotica.

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

[45]  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..

[46]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[47]  최준호,et al.  On observer-based feedback stabilization of periodic orbits in bipedal locomotion , 2007 .

[48]  Arthur J. Krener,et al.  Linearization by output injection and nonlinear observers , 1983 .

[49]  Jessy W. Grizzle,et al.  Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry , 2014, IEEE Transactions on Robotics.

[50]  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.

[51]  Hassan K. Khalil,et al.  A Nonlinear High-Gain Observer for Systems With Measurement Noise in a Feedback Control Framework , 2013, IEEE Transactions on Automatic Control.

[52]  Manfredi Maggiore,et al.  A separation principle for a class of non-UCO systems , 2003, IEEE Trans. Autom. Control..

[53]  Ricardo G. Sanfelice,et al.  On the performance of high-gain observers with gain adaptation under measurement noise , 2011, Autom..

[54]  H. Khalil,et al.  A separation principle for the stabilization of a class of nonlinear systems , 1997 .

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

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

[57]  R. Rajamani Observers for Lipschitz nonlinear systems , 1998, IEEE Trans. Autom. Control..

[58]  Robert D. Gregg,et al.  Decentralized Feedback Controllers for Robust Stabilization of Periodic Orbits of Hybrid Systems: Application to Bipedal Walking , 2017, IEEE Transactions on Control Systems Technology.

[59]  P. de Leva Adjustments to Zatsiorsky-Seluyanov's segment inertia parameters. , 1996, Journal of biomechanics.

[60]  Aaron D. Ames,et al.  Observer-Based Feedback Controllers for Exponential Stabilization of Hybrid Periodic Orbits: Application to Underactuated Bipedal Walking , 2018, 2018 Annual American Control Conference (ACC).

[61]  Russ Tedrake,et al.  Planning robust walking motion on uneven terrain via convex optimization , 2016, 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids).

[62]  S. Sastry,et al.  Hybrid Geometric Reduction of Hybrid Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[63]  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.

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

[65]  Jessy W. Grizzle,et al.  Interpolation and numerical differentiation for observer design , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[66]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[67]  Franck Plestan,et al.  Step-by-step sliding mode observer for control of a walking biped robot by using only actuated variables measurement , 2005, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems.

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

[69]  M. Spong,et al.  CONTROLLED SYMMETRIES AND PASSIVE WALKING , 2002 .

[70]  Ali Zemouche,et al.  On LMI conditions to design observers for Lipschitz nonlinear systems , 2013, Autom..

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

[72]  Wang Zicai Observer Design for a Class of Nonlinear Systems , 1998 .

[73]  Aaron D. Ames,et al.  Algorithmic Foundations of Realizing Multi-Contact Locomotion on the Humanoid Robot DURUS , 2016, WAFR.

[74]  Y. Aoustin,et al.  Absolute orientation estimation based on high order sliding mode observer for a five link walking biped robot , 2006, International Workshop on Variable Structure Systems, 2006. VSS'06..

[75]  Aaron D. Ames,et al.  Dynamic Humanoid Locomotion: A Scalable Formulation for HZD Gait Optimization , 2018, IEEE Transactions on Robotics.

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

[77]  Katie Byl,et al.  Meshing hybrid zero dynamics for rough terrain walking , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

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

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

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

[81]  O. Toker,et al.  On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.