Spatio-temporal stiffness optimization with switching dynamics

We address the optimal control problem of robotic systems with variable stiffness actuation (VSA) including switching dynamics and discontinuous state transitions. Our focus in this paper is to consider dynamic tasks that have multiple phases of movement, contacts and impacts with the environment with a requirement of exploiting passive dynamics of the system. By modelling such tasks as a hybrid dynamical system with time-based switching, we develop a systematic methodology to simultaneously optimize control commands, time-varying stiffness profiles and temporal aspect of the movement such as switching instances and total movement duration to exploit the benefits of VSA. Numerical evaluations on a brachiating robot driven with VSA and a hopping robot equipped with variable stiffness springs demonstrate the effectiveness of the proposed approach. Furthermore, hardware experiments on a two-link brachiating robot with VSA highlight the applicability of the proposed framework in a challenging task of brachiation.

[1]  Dirk Wollherr,et al.  Ball dribbling with an underactuated continuous-time control phase: Theory & experiments , 2010, 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[2]  Dirk Wollherr,et al.  Ball dribbling with an underactuated continuous-time control phase: Theory & experiments , 2010, IROS.

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

[4]  Emanuel Todorov,et al.  First-exit model predictive control of fast discontinuous dynamics: Application to ball bouncing , 2011, 2011 IEEE International Conference on Robotics and Automation.

[5]  Fumihito Arai,et al.  Swing and locomotion control for a two-link brachiation robot , 1993, IEEE Control Systems.

[6]  Manuel G. Catalano,et al.  Mechanism design for Variable Stiffness Actuation based on enumeration and analysis of performance , 2010, 2010 IEEE International Conference on Robotics and Automation.

[7]  Ikuo Mizuuchi,et al.  Analysis of the 1-Joint Spring-Motor Coupling System and optimization criteria focusing on the velocity increasing effect , 2011, 2011 IEEE International Conference on Robotics and Automation.

[8]  Takashi Emura,et al.  Energy-preserving control of a passive one-legged running robot , 2004, Adv. Robotics.

[9]  K. Rawlik.  On probabilistic inference approaches to stochastic optimal control , 2013 .

[10]  Marc Toussaint,et al.  An Approximate Inference Approach to Temporal Optimization in Optimal Control , 2010, NIPS.

[11]  Sethu Vijayakumar,et al.  Rapid Manufacture of Novel Variable Impedance Robots , 2016 .

[12]  Y. Wardi,et al.  Optimal control of switching times in switched dynamical systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  Kevin M. Lynch,et al.  Stable open-loop brachiation on a vertical wall , 2012, 2012 IEEE International Conference on Robotics and Automation.

[14]  Joel E. Chestnutt,et al.  The Actuator With Mechanically Adjustable Series Compliance , 2010, IEEE Transactions on Robotics.

[15]  Martin Buehler,et al.  Stable control of a simulated one-legged running robot with hip and leg compliance , 1997, IEEE Trans. Robotics Autom..

[16]  Kevin M. Lynch,et al.  Optimal motion planning for a class of hybrid dynamical systems with impacts , 2011, 2011 IEEE International Conference on Robotics and Automation.

[17]  S. Vijayakumar,et al.  Exploiting variable physical damping in rapid movement tasks , 2012, 2012 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM).

[18]  Djordje Mitrovic,et al.  Stochastic optimal control with learned dynamics models , 2011 .

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

[20]  Jun Nakanishi,et al.  Stiffness and temporal optimization in periodic movements: An optimal control approach , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[21]  Alin Albu-Schäffer,et al.  On joint design with intrinsic variable compliance: derivation of the DLR QA-Joint , 2010, 2010 IEEE International Conference on Robotics and Automation.

[22]  Jun Nakanishi,et al.  A brachiating robot controller , 2000, IEEE Trans. Robotics Autom..

[23]  H. Sussmann,et al.  A maximum principle for hybrid optimal control problems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[24]  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.

[25]  J. Meditch,et al.  Applied optimal control , 1972, IEEE Transactions on Automatic Control.

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

[27]  Alin Albu-Schaffer,et al.  Optimal Control for Maximizing Link Velocity of Robotic Variable Stiffness Joints , 2011 .

[28]  Panos J. Antsaklis,et al.  Optimal control of switched systems based on parameterization of the switching instants , 2004, IEEE Transactions on Automatic Control.

[29]  Martin Buss,et al.  Nonlinear Hybrid Dynamical Systems: Modeling, Optimal Control, and Applications , 2002 .

[30]  Nikolaos G. Tsagarakis,et al.  A novel actuator with adjustable stiffness (AwAS) , 2010, 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[31]  Bram Vanderborght,et al.  MACCEPA, the mechanically adjustable compliance and controllable equilibrium position actuator: Design and implementation in a biped robot , 2007, Robotics Auton. Syst..

[32]  Mario W. Gomes,et al.  A five-link 2D brachiating ape model with life-like zero-energy-cost motions. , 2005, Journal of theoretical biology.

[33]  Alin Albu-Schäffer,et al.  Robots Driven by Compliant Actuators: Optimal Control Under Actuation Constraints , 2013, IEEE Transactions on Robotics.

[34]  Jun Nakanishi,et al.  Exploiting Passive Dynamics with Variable Stiffness Actuation in Robot Brachiation , 2012, Robotics: Science and Systems.

[35]  Sethu Vijayakumar,et al.  Optimal variable stiffness control: formulation and application to explosive movement tasks , 2012, Auton. Robots.

[36]  Antonio Bicchi,et al.  Optimality principles in variable stiffness control: The VSA hammer , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[37]  Reza Shadmehr.  Learning Virtual Equilibrium Trajectories for Control of a Robot Arm , 1990, Neural Computation.

[38]  Emanuel Todorov,et al.  Iterative linearization methods for approximately optimal control and estimation of non-linear stochastic system , 2007, Int. J. Control.

[39]  Xuping Xu,et al.  Quadratic optimal control problems for hybrid linear autonomous systems with state jumps , 2003, Proceedings of the 2003 American Control Conference, 2003..

[40]  R. Ham,et al.  Compliant actuator designs , 2009, IEEE Robotics & Automation Magazine.

[41]  Martijn Wisse,et al.  Running with improved disturbance rejection by using non-linear leg springs , 2011, Int. J. Robotics Res..