A Passive 2-DOF Walker: Hunting for Gaits Using Virtual Holonomic Constraints

A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.

[1]  Bernard Espiau,et al.  A Study of the Passive Gait of a Compass-Like Biped Robot , 1998, Int. J. Robotics Res..

[2]  Ian R. Manchester,et al.  Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems , 2008, Annu. Rev. Control..

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

[4]  Bernard Espiau,et al.  Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws , 1997, Auton. Robots.

[5]  Ning Liu,et al.  Passive walker that can walk down steps: simulations and experiments , 2008 .

[6]  Arthur D. Kuo,et al.  Stabilization of Lateral Motion in Passive Dynamic Walking , 1999, Int. J. Robotics Res..

[7]  Kentaro Hirata,et al.  Stability analysis of linear systems with state jump - motivated by periodic motion control of passive walker , 2003, Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003..

[8]  J. Dingwell,et al.  ApJ, in press , 1999 .

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

[10]  Mahyar Naraghi,et al.  Passive dynamic of the simplest walking model: Replacing ramps with stairs , 2007 .

[11]  Y. Aoustin,et al.  Design of reference trajectory to stabilize desired nominal cyclic gait of a biped , 1999, Proceedings of the First Workshop on Robot Motion and Control. RoMoCo'99 (Cat. No.99EX353).

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

[13]  Ian A. Hiskens,et al.  Stability of hybrid system limit cycles: application to the compass gait biped robot , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

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

[15]  Tad McGeer,et al.  Passive Dynamic Walking , 1990, Int. J. Robotics Res..

[16]  Carlos Canudas-de-Wit,et al.  Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach , 2005, IEEE Transactions on Automatic Control.

[17]  Eric R. Westervelt,et al.  Analysis results and tools for the control of planar bipedal gaits using hybrid zero dynamics , 2007, Auton. Robots.

[18]  Zhiwei Luo,et al.  Asymptotically stable gait generation for biped robot based on mechanical energy balance , 2007, 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[19]  Ian R. Manchester,et al.  Stability Analysis and Control Design for an Underactuated Walking Robot via Computation of a Transverse Linearization , 2008 .

[20]  Arthur D. Kuo,et al.  Choosing Your Steps Carefully , 2007, IEEE Robotics & Automation Magazine.

[21]  Russ Tedrake,et al.  Efficient Bipedal Robots Based on Passive-Dynamic Walkers , 2005, Science.

[22]  A. Shiriaev,et al.  Periodic motion planning for virtually constrained Euler-Lagrange systems , 2006, Syst. Control. Lett..

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

[24]  Joachim Haß,et al.  Optimal Mass Distribution for Passivity-Based Bipedal Robots , 2006, Int. J. Robotics Res..

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

[26]  M. Coleman,et al.  The simplest walking model: stability, complexity, and scaling. , 1998, Journal of biomechanical engineering.

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

[28]  B. Morris,et al.  Sample-Based HZD Control for Robustness and Slope Invariance of Planar Passive Bipedal Gaits , 2006, 2006 14th Mediterranean Conference on Control and Automation.