Robust optimal planning and control of non-periodic bipedal locomotion with a centroidal momentum model

This study presents a theoretical method for planning and controlling agile bipedal locomotion based on robustly tracking a set of non-periodic keyframe states. Based on centroidal momentum dynamics, we formulate a hybrid phase-space planning and control method that includes the following key components: (i) a step transition solver that enables dynamically tracking non-periodic keyframe states over various types of terrain; (ii) a robust hybrid automaton to effectively formulate planning and control algorithms; (iii) a steering direction model to control the robot’s heading; (iv) a phase-space metric to measure distance to the planned locomotion manifolds; and (v) a hybrid control method based on the previous distance metric to produce robust dynamic locomotion under external disturbances. Compared with other locomotion methodologies, we have a large focus on non-periodic gait generation and robustness metrics to deal with disturbances. This focus enables the proposed control method to track non-periodic keyframe states robustly over various challenging terrains and under external disturbances, as illustrated through several simulations.

[1]  V. Borkar,et al.  A unified framework for hybrid control: model and optimal control theory , 1998, IEEE Trans. Autom. Control..

[2]  Ambarish Goswami,et al.  Momentum-based reactive stepping controller on level and non-level ground for humanoid robot push recovery , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[3]  Ye Zhao,et al.  Stabilizing Series-Elastic Point-Foot Bipeds Using Whole-Body Operational Space Control , 2016, IEEE Transactions on Robotics.

[4]  Yevgeniy Yesilevskiy,et al.  Selecting gaits for economical locomotion of legged robots , 2016, Int. J. Robotics Res..

[5]  Andreas G. Hofmann Robust execution of bipedal walking tasks from biomechanical principles , 2006 .

[6]  David E. Orin,et al.  3D-SLIP steering for high-speed humanoid turns , 2014, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[7]  Anirudha Majumdar,et al.  Robust online motion planning with reachable sets , 2013 .

[8]  Pierre-Brice Wieber,et al.  A robust linear MPC approach to online generation of 3D biped walking motion , 2015, 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids).

[9]  Ye Zhao,et al.  Phase space planning and robust control for data-driven locomotion behaviors , 2013, 2013 13th IEEE-RAS International Conference on Humanoid Robots (Humanoids).

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

[11]  Hannes Sommer,et al.  Quadrupedal locomotion using hierarchical operational space control , 2014, Int. J. Robotics Res..

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

[13]  Alin Albu-Schäffer,et al.  Trajectory generation for continuous leg forces during double support and heel-to-toe shift based on divergent component of motion , 2014, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[14]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[15]  Christopher G. Atkeson,et al.  Push Recovery by stepping for humanoid robots with force controlled joints , 2010, 2010 10th IEEE-RAS International Conference on Humanoid Robots.

[16]  Taku Komura,et al.  A Feedback Controller for Biped Humanoids that Can Counteract Large Perturbations During Gait , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[17]  Ian R. Manchester,et al.  Real-time planning with primitives for dynamic walking over uneven terrain , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[18]  Olivier Stasse,et al.  A versatile and efficient pattern generator for generalized legged locomotion , 2016, 2016 IEEE International Conference on Robotics and Automation (ICRA).

[19]  Twan Koolen,et al.  Capturability-based analysis and control of legged locomotion, Part 1: Theory and application to three simple gait models , 2011, Int. J. Robotics Res..

[20]  Kris K. Hauser,et al.  Fast interpolation and time-optimization with contact , 2014, Int. J. Robotics Res..

[21]  Yuval Tassa,et al.  Synthesis and stabilization of complex behaviors through online trajectory optimization , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[22]  William D. Smart,et al.  Bipedal walking on rough terrain using manifold control , 2007, 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems.

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

[24]  J. Hedrick,et al.  Control of multivariable non-linear systems by the sliding mode method , 1987 .

[25]  David A. Winter,et al.  Human balance and posture control during standing and walking , 1995 .

[26]  Chee-Meng Chew,et al.  Virtual Model Control: An Intuitive Approach for Bipedal Locomotion , 2001, Int. J. Robotics Res..

[27]  F E Zajac,et al.  Human standing posture: multi-joint movement strategies based on biomechanical constraints. , 1993, Progress in brain research.

[28]  David Silver,et al.  Monte Carlo Localization and registration to prior data for outdoor navigation , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[29]  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).

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

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

[32]  Uluc Saranli,et al.  Reactive Planning and Control of Planar Spring–Mass Running on Rough Terrain , 2012, IEEE Transactions on Robotics.

[33]  Johannes Englsberger,et al.  Biologically inspired deadbeat control for running on 3D stepping stones , 2015, 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids).

[34]  James P. Schmiedeler,et al.  A framework for the control of stable aperiodic walking in underactuated planar bipeds , 2009, Proceedings 2007 IEEE International Conference on Robotics and Automation.

[35]  Ye Zhao,et al.  A three dimensional foot placement planner for locomotion in very rough terrains , 2012, 2012 12th IEEE-RAS International Conference on Humanoid Robots (Humanoids 2012).

[36]  Z. Popovic,et al.  Terrain-adaptive bipedal locomotion control , 2010, ACM Trans. Graph..

[37]  B. Fajen,et al.  Humans exploit the biomechanics of bipedal gait during visually guided walking over complex terrain , 2013, Proceedings of the Royal Society B: Biological Sciences.

[38]  Yoshihiko Nakamura,et al.  Kinodynamic Planning in the Configuration Space via Admissible Velocity Propagation , 2013, Robotics: Science and Systems.

[39]  Koushil Sreenath,et al.  Dynamic Walking on Randomly-Varying Discrete Terrain with One-step Preview , 2017, Robotics: Science and Systems.

[40]  Karsten Berns,et al.  Biologically motivated push recovery strategies for a 3D bipedal robot walking in complex environments , 2013, 2013 IEEE International Conference on Robotics and Biomimetics (ROBIO).

[41]  John Lygeros,et al.  Hybrid Systems: Modeling, Analysis and Control , 2008 .

[42]  Albert Wu,et al.  The 3-D Spring–Mass Model Reveals a Time-Based Deadbeat Control for Highly Robust Running and Steering in Uncertain Environments , 2013, IEEE Transactions on Robotics.

[43]  Kris Hauser,et al.  Generalizations of the capture point to nonlinear center of mass paths and uneven terrain , 2015, 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids).

[44]  Yuan F. Zheng,et al.  Trajectory generation for dynamic walking in a humanoid over uneven terrain using a 3D-actuated Dual-SLIP model , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[45]  Gordon Cheng,et al.  Full-Body Compliant Human–Humanoid Interaction: Balancing in the Presence of Unknown External Forces , 2007, IEEE Transactions on Robotics.

[46]  A. Hof The 'extrapolated center of mass' concept suggests a simple control of balance in walking. , 2008, Human movement science.

[47]  Twan Koolen,et al.  Balance control using center of mass height variation: Limitations imposed by unilateral contact , 2016, 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids).

[48]  Marc H. Raibert,et al.  Legged Robots That Balance , 1986, IEEE Expert.

[49]  Zoran Popovic,et al.  Discovery of complex behaviors through contact-invariant optimization , 2012, ACM Trans. Graph..

[50]  Alin Albu-Schäffer,et al.  Bipedal walking control based on Capture Point dynamics , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[51]  Jonathan P. How,et al.  Robust motion planning using a maneuver automation with built-in uncertainties , 2003, Proceedings of the 2003 American Control Conference, 2003..

[52]  Auke Jan Ijspeert,et al.  Robust and Agile 3D Biped Walking With Steering Capability Using a Footstep Predictive Approach , 2014, Robotics: Science and Systems.

[53]  Luis Sentis,et al.  Motion planning of extreme locomotion maneuvers using multi-contact dynamics and numerical integration , 2011, 2011 11th IEEE-RAS International Conference on Humanoid Robots.

[54]  Eiichi Yoshida,et al.  Model preview control in multi-contact motion-application to a humanoid robot , 2014, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[55]  Ruben Grandia,et al.  Hybrid direct collocation and control in the constraint-consistent subspace for dynamic legged robot locomotion , 2017, Robotics: Science and Systems.

[56]  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).

[57]  Sangbae Kim,et al.  High-speed bounding with the MIT Cheetah 2: Control design and experiments , 2017, Int. J. Robotics Res..

[58]  Martijn Wisse,et al.  A Disturbance Rejection Measure for Limit Cycle Walkers: The Gait Sensitivity Norm , 2007, IEEE Transactions on Robotics.

[59]  Oussama Khatib,et al.  Compliant Control of Multicontact and Center-of-Mass Behaviors in Humanoid Robots , 2010, IEEE Transactions on Robotics.

[60]  Abderrahmane Kheddar,et al.  Multi-contact walking pattern generation based on model preview control of 3D COM accelerations , 2016, 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids).

[61]  Manoj Srinivasan,et al.  Computer optimization of a minimal biped model discovers walking and running , 2006, Nature.

[62]  Katie Byl,et al.  Reachability-based control for the active SLIP model , 2015, Int. J. Robotics Res..

[63]  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.

[64]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[65]  Aaron D. Ames,et al.  Human-inspired multi-contact locomotion with AMBER2 , 2014, 2014 ACM/IEEE International Conference on Cyber-Physical Systems (ICCPS).

[66]  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..

[67]  Alin Albu-Schäffer,et al.  Three-Dimensional Bipedal Walking Control Based on Divergent Component of Motion , 2015, IEEE Transactions on Robotics.

[68]  Kirill Van Heerden,et al.  Real-Time Variable Center of Mass Height Trajectory Planning for Humanoids Robots , 2017, IEEE Robotics and Automation Letters.

[69]  Pierre-Brice Wieber,et al.  Trajectory Free Linear Model Predictive Control for Stable Walking in the Presence of Strong Perturbations , 2006, 2006 6th IEEE-RAS International Conference on Humanoid Robots.

[70]  Scott Kuindersma,et al.  Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot , 2015, Autonomous Robots.

[71]  Arthur D Kuo,et al.  Energetics of actively powered locomotion using the simplest walking model. , 2002, Journal of biomechanical engineering.

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

[73]  Ufuk Topcu,et al.  High-level planner synthesis for whole-body locomotion in unstructured environments , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[74]  Kazuhito Yokoi,et al.  Biped walking pattern generation by using preview control of zero-moment point , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[75]  Ye Zhao,et al.  Robust Phase-Space Planning for Agile Legged Locomotion over Various Terrain Topologies , 2016, Robotics: Science and Systems.

[76]  Katie Byl,et al.  Robust Policies via Meshing for Metastable Rough Terrain Walking , 2014, Robotics: Science and Systems.

[77]  Kazuhito Yokoi,et al.  Balance control based on Capture Point error compensation for biped walking on uneven terrain , 2012, 2012 12th IEEE-RAS International Conference on Humanoid Robots (Humanoids 2012).

[78]  Takashi Matsumoto,et al.  Real time motion generation and control for biped robot -2nd report: Running gait pattern generation- , 2009, 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[79]  E. Feron,et al.  Robust hybrid control for autonomous vehicle motion planning , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[80]  Ambarish Goswami,et al.  A Biomechanically Motivated Two-Phase Strategy for Biped Upright Balance Control , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[81]  Katie Byl,et al.  Approximation and Control of the SLIP Model Dynamics via Partial Feedback Linearization and Two-Element Leg Actuation Strategy , 2016, IEEE Transactions on Robotics.

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

[83]  Virginia Peragallo-Dittko,et al.  In Balance, in Control , 1988 .

[84]  Nikolaos G. Tsagarakis,et al.  Fall Prediction of legged robots based on energy state and its implication of balance augmentation: A study on the humanoid , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

[85]  Christopher G. Atkeson,et al.  Optimization‐based Full Body Control for the DARPA Robotics Challenge , 2015, J. Field Robotics.

[86]  Alexander Spröwitz,et al.  ATRIAS: Design and validation of a tether-free 3D-capable spring-mass bipedal robot , 2016, Int. J. Robotics Res..

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

[88]  Russ Tedrake,et al.  Whole-body motion planning with centroidal dynamics and full kinematics , 2014, 2014 IEEE-RAS International Conference on Humanoid Robots.

[89]  Russ Tedrake,et al.  A direct method for trajectory optimization of rigid bodies through contact , 2014, Int. J. Robotics Res..

[90]  Zoran Popović,et al.  Terrain-adaptive bipedal locomotion control , 2010, SIGGRAPH 2010.

[91]  Gordon Cheng,et al.  Disturbance Rejection for Biped Humanoids , 2007, Proceedings 2007 IEEE International Conference on Robotics and Automation.

[92]  Takashi Matsumoto,et al.  Real time motion generation and control for biped robot -1st report: Walking gait pattern generation- , 2009, 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[93]  A. Isidori Nonlinear Control Systems: An Introduction , 1986 .

[94]  Aaron Hertzmann,et al.  Robust physics-based locomotion using low-dimensional planning , 2010, SIGGRAPH 2010.

[95]  Shuuji Kajita,et al.  Pattern Generation of Biped Walking Constrained on Parametric Surface , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.