×

Reduced-order framework for exponential stabilization of periodic orbits on parameterized hybrid zero dynamics manifolds: application to bipedal locomotion. (English) Zbl 1377.93130

Summary: This paper shows how controlled-invariant manifolds in hybrid dynamical systems can be used to reduce the offline computational burden associated with locally exponentially stabilizing periodic orbits. We recently introduced a method to systematically select stabilizing feedback controllers for hybrid periodic orbits from a family of parameterized control laws by solving offline optimization problems. These problems search for controller parameters as well as a set of Lyapunov matrices for the full-order hybrid systems. When the method is applied to mechanical systems with high Degrees Of Freedom (DOF), the number of entries in the Lyapunov matrices may render the numerical optimization problems prohibitively slow. To address this challenge, the paper considers a family of attractive and parameterized Hybrid Zero Dynamics (HZD) manifolds in the state space. It then investigates the properties of the associated Poincaré map to translate the full-order optimization framework to a reduced-order one on the parameterized HZD manifolds with lower-dimensional Lyapunov matrices. In addition, the paper provides a systematic approach to numerically compute the Jacobian linearization of the parameterized Poincaré map on the HZD manifolds. The power of the proposed framework is demonstrated by designing a set of stabilizing input-output linearizing controllers for walking gaits of an underactuated \(3D\) bipedal robot with \(13\) DOFs and \(6\) actuators. It is shown that the number of decision variables in the reduced-order optimization problem can be reduced by \(70\%\) compared to the full-order one.

MSC:

93D20 Asymptotic stability in control theory
93D25 Input-output approaches in control theory
93C85 Automated systems (robots, etc.) in control theory
70E60 Robot dynamics and control of rigid bodies
93B18 Linearizations
93C10 Nonlinear systems in control theory

Software:

PENBMI; YALMIP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hurmuzlu, Y.; Marghitu, D. B., Rigid body collisions of planar kinematic chains with multiple contact points, Int. J. Robot. Res., 13, 1, 82-92 (1994)
[2] Grizzle, J.; Abba, G.; Plestan, F., Asymptotically stable walking for biped robots: analysis via systems with impulse effects, IEEE Trans. Automat. Control, 46, 1, 51-64 (2001) · Zbl 0992.93058
[4] Grizzle, J. W.; Chevallereau, C.; Sinnet, R. W.; Ames, A. D., Models, feedback control, and open problems of 3D bipedal robotic walking, Automatica, 50, 8, 1955-1988 (2014) · Zbl 1297.93120
[5] Spong, M.; Bullo, F., Controlled symmetries and passive walking, IEEE Trans. Automat. Control, 50, 7, 1025-1031 (2005) · Zbl 1365.93329
[6] Spong, M.; Holm, J.; Lee, D., Passivity-based control of bipedal locomotion, IEEE Robot. Autom. Mag., 14, 2, 30-40 (2007)
[7] Ames, A., Human-inspired control of bipedal walking robots, IEEE Trans. Automat. Control, 59, 5, 1115-1130 (2014) · Zbl 1360.93460
[8] Ames, A.; Galloway, K.; Sreenath, K.; Grizzle, J., Rapidly exponentially stabilizing control Lyapunov functions and hybrid zero dynamics, IEEE Trans. Automat. Control, 59, 4, 876-891 (2014) · Zbl 1360.93533
[9] Gregg, R.; Righetti, L., Controlled reduction with unactuated cyclic variables: Application to 3D bipedal walking with passive yaw rotation, IEEE Trans. Automat. Control, 58, 10, 2679-2685 (2013) · Zbl 1369.93416
[10] Gregg, R.; Tilton, A.; Candido, S.; Bretl, T.; Spong, M., Control and planning of 3-D dynamic walking with asymptotically stable gait primitives, IEEE Trans. Robot., 28, 6, 1415-1423 (2012)
[14] Manchester, I. R.; Mettin, U.; Iida, F.; Tedrake, R., Stable dynamic walking over uneven terrain, Int. J. Robot. Res., 30, 3, 265-279 (2011)
[15] Shiriaev, A.; Freidovich, L.; Gusev, S., Transverse linearization for controlled mechanical systems with several passive degrees of freedom, IEEE Trans. Automat. Control, 55, 4, 893-906 (2010) · Zbl 1368.93106
[16] Martin, A. E.; Post, D. C.; Schmiedeler, J. P., The effects of foot geometric properties on the gait of planar bipeds walking under HZD-based control, Int. J. Robot. Res., 33, 12, 1530-1543 (2014)
[17] Remy, C. D., Optimal exploitation of natural dynamics in legged locomotion (2011), ETH Zurich, (Ph.D. thesis)
[18] Haddad, W.; Chellaboina, V.; Nersesov, S., Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control (2006), Princeton University Press · Zbl 1114.34001
[19] Goebel, R.; Sanfelice, R.; Teel, A., Hybrid Dynamical Systems: Modeling, Stability, and Robustness (2012), Princeton University Press · Zbl 1241.93002
[20] Bainov, D.; Simeonov, P., Systems With Impulse Effect: Stability, Theory and Applications (1989), Ellis Horwood Ltd. · Zbl 0676.34035
[21] Ye, H.; Michel, A.; Hou, L., Stability theory for hybrid dynamical systems, IEEE Trans. Automat. Control, 43, 4, 461-474 (1998) · Zbl 0905.93024
[22] Parker, T.; Chua, L., Practical Numerical Algorithms for Chaotic Systems (1989), Springer · Zbl 0692.58001
[23] Burden, S.; Revzen, S.; Sastry, S., Model reduction near periodic orbits of hybrid dynamical systems, IEEE Trans. Automat. Control, 60, 10, 2626-2639 (2015) · Zbl 1360.93175
[25] Chevallereau, C.; Grizzle, J.; Shih, C.-L., Asymptotically stable walking of a five-link underactuated 3-D Bipedal Robot, IEEE Trans. Robot., 25, 1, 37-50 (2009)
[26] Ramezani, A.; Hurst, J.; Akbai Hamed, K.; Grizzle, J., Performance analysis and feedback control of ATRIAS, a three-dimensional bipedal robot, J. Dyn. Syst. Meas. Control, 136, 2 (2013)
[27] Akbari Hamed, K.; Grizzle, J., Event-based stabilization of periodic orbits for underactuated 3-D bipedal robots with left-right symmetry, IEEE Trans. Robot., 30, 2, 365-381 (2014)
[28] Sreenath, K.; Park, H.-W.; Poulakakis, I.; Grizzle, J., Embedding active force control within the compliant hybrid zero dynamics to achieve stable, fast running on mabel, Int. J. Robot. Res., 32, 3, 324-345 (2013)
[29] Raibert, M. H., Legged robots, Commun. ACM, 29, 6, 499-514 (1986) · Zbl 0709.70504
[30] Buehler, M.; Koditschek, D. E.; Kindlmann, P. J., Planning and control of robotic juggling and catching tasks, Int. J. Robot. Res., 13, 12, 101-118 (1994)
[31] Carver, S. G.; Cowan, N. J.; Guckenheimer, J. M., Lateral stability of the spring-mass hopper suggests a two-step control strategy for running, Chaos, 19, 2, Article 026106 pp. (2009) · Zbl 1309.70030
[32] Ankarali, M. M.; Saranli, U., Control of underactuated planar pronking through an embedded spring-mass hopper template, Auton. Robots, 30, 2, 217-231 (2011)
[33] Seipel, J.; Holmes, P., A simple model for clock-actuated legged locomotion, Regul. Chaotic Dyn., 12, 5, 502-520 (2007) · Zbl 1229.70028
[34] Seyfarth, A.; Geyer, H.; Herr, H., Swing-leg retraction: a simple control model for stable running, J. Exp. Biol., 206, 15, 2547-2555 (2003)
[35] Akbari Hamed, K.; Buss, B.; Grizzle, J., Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations, Int. J. Robot. Res., 38, 977-999 (2015)
[40] Diehl, M.; Mombaur, K.; Noll, D., Stability optimization of hybrid periodic systems via a smooth criterion, IEEE Trans. Automat. Control, 54, 8, 1875-1880 (2009) · Zbl 1367.93391
[41] Isidori, A., Nonlinear Control Systems (1995), Springer · Zbl 0569.93034
[42] Morris, B.; Grizzle, J., Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots, IEEE Trans. Automat. Control, 54, 8, 1751-1764 (2009) · Zbl 1367.93404
[43] Westervelt, E.; Grizzle, J.; Koditschek, D., Hybrid zero dynamics of planar biped walkers, IEEE Trans. Automat. Control, 48, 1, 42-56 (2003) · Zbl 1364.70015
[44] Gregg, R.; Sensinger, J., Towards biomimetic virtual constraint control of a powered prosthetic leg, IEEE Trans. Control Syst. Technol., 22, 1, 246-254 (2014)
[45] Gregg, R.; Lenzi, T.; Hargrove, L.; Sensinger, J., Virtual constraint control of a powered prosthetic leg: From simulation to experiments with transfemoral amputees, IEEE Trans. Robot., 30, 6, 1455-1471 (2014)
[46] Chevallereau, C.; Abba, G.; Aoustin, Y.; Plestan, F.; Westervelt, E.; Canudas-de Wit, C.; Grizzle, J., RABBIT: a testbed for advanced control theory, IEEE Control Syst. Mag., 23, 5, 57-79 (2003)
[47] Maggiore, M.; Consolini, L., Virtual holonomic constraints for Euler Lagrange systems, IEEE Trans. Automat. Control, 58, 4, 1001-1008 (2013) · Zbl 1369.70026
[53] Shih, C.-L.; Grizzle, J. W.; Chevallereau, C., From stable walking to steering of a 3D bipedal robot with passive point feet, Robotica, 30, 1119-1130 (2012)
[58] Westervelt, E.; Grizzle, J.; Chevallereau, C.; Choi, J.; Morris, B., Feedback Control of Dynamic Bipedal Robot Locomotion (2007), Taylor & Francis/CRC
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.