Abstract
Underactuation is ubiquitous in human locomotion and should be ubiquitous in bipedal robotic locomotion as well. This chapter presents a coherent theory for the design of feedback controllers that achieve stable walking gaits in underactuated bipedal robots. Two fundamental tools are introduced, virtual constraints and hybrid zero dynamics. Virtual constraints are relations on the state variables of a mechanical model that are imposed through a time-invariant feedback controller. One of their roles is to synchronize the robot’s joints to an internal gait phasing variable. A second role is to induce a low-dimensional system, the zero dynamics, that captures the underactuated aspects of a robot’s model, without any approximations. To enhance intuition, the relation between physical constraints and virtual constraints is first established. From here, the hybrid zero dynamics of an underactuated bipedal model is developed, and its fundamental role in the design of asymptotically stable walking motions is established. The chapter includes numerous references to robots on which the highlighted techniques have been implemented.
Similar content being viewed by others
References
K. Akbari Hamed, J.W Grizzle, Event-based stabilization of periodic orbits for underactuated 3-d bipedal robots with left-right symmetry. IEEE Trans. Robot. 30(2), 365–381 (2014)
K. Akbari Hamed, B.G. Buss, J.W. Grizzle, Continuous-time controllers for stabilizing periodic orbits of hybrid systems: application to an underactuated 3D bipedal robot, in 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), Dec 2014, pp. 1507–1513
A.D. Ames, Human-inspired control of bipedal walking robots. IEEE Trans. Autom. Control, 59(5), 1115–1130 (2014)
A.D. Ames, AMBER-Lab (2016). https://www.youtube.com/user/ProfAmes
A.D. Ames, K. Galloway, K. Sreenath, J.W. Grizzle, Rapidly exponentially stabilizing control lyapunov functions and hybrid zero dynamics. IEEE Trans. Autom. Control 59(4), 876–891 (2014)
B.G. Buss, Systematic Controller Design for Dynamic 3D Bipedal Robot Walking. PhD thesis, University of Michigan, Ann Arbor, May 2015
B.G. Buss, A. Ramezani, K. Akbari Hamed, B.A. Griffin, K.S. Galloway, J.W. Grizzle, Preliminary walking experiments with underactuated 3D bipedal robot MARLO, in 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2014), Sept 2014, pp. 2529–2536
C. Byrnes, A. Isidori, Asymptotic stabilization of nonlinear minimum phase systems. IEEE Trans. Autom. Control 376, 1122–37 (1991)
C. Chevallereau. Time scaling control for an underactuated biped robot. IEEE Trans. Robot. Autom. 19(2), 362–368 (2003)
C. Chevallereau, G. Abba, Y. Aoustin, F. Plestan, E.R. Westervelt, C. Canudas, J.W. Grizzle, RABBIT: a testbed for advanced control theory. IEEE Control Syst. Mag. 23(5), 57–79 (2003)
C. Chevallereau, D. Djoudi, and J.W. Grizzle, Stable bipedal walking with foot rotation through direct regulation of the Zero Moment Point. TRO 25(2), 390–401 (2008)
C. Chevallereau, J.W. Grizzle, C.-L. Shih, Asymptotically stable walking of a five-link underactuated 3D bipedal robot. IEEE Trans. Robot. 25(1), 37–50 (2009)
H. Dai, R. Tedrake, Optimizing robust limit cycles for legged locomotion on unknown terrain, in 2012 IEEE 51st Annual Conference on Decision and Control (CDC), 2012, pp. 1207–1213
R.D. Gregg, High-performance control of a powered transfemoral prosthesis with amputee subjects (2016). https://www.youtube.com/watch?v=sl1IXs0j4Ww
R.D. Gregg, T. Lenzi, L.J. Hargrove, J.W. Sensinger, Virtual constraint control of a powered prosthetic leg: from simulation to experiments with transfemoral amputees. IEEE Trans. Robot. 30, 1455–1471 (2014)
R.D. Gregg, E.J. Rouse, L.J. Hargrove, J.W. Sensinger, Evidence for a time-invariant phase variable in human ankle control. PLoS One 9(2), e89163 (2014)
B. Griffin, J.W. Grizzle, Nonholonomic virtual constraints for dynamic walking, in Preprint Submitted to IEEE Conference on Decision and Control, 2015
B. Griffin, J.W. Grizzle, Walking gait optimization for accommodation of unknown terrain height variations, in American Control Conference, 2015
J.W. Grizzle, Dynamic Leg Locomotion (2016) www.youtube.com/user/DynamicLegLocomotion
J.W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Autom. Control 46, 51–64 (2001)
J.W. Grizzle, C. Chevallereau, A. Ames, R. Sinnet, Models, feedback control, and open problems of 3D bipedal robotic walking. Automatica 50(8), 1955–1988 (2014)
K.A. Hamed, B.G. Buss, J.W. Grizzle, Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: application to bipedal locomotion with ground height variations. Int. J. Robot. Res. 35(8), 977–999 (2016)
A. Hereid, S. Kolathaya, M.S. Jones, J. Van Why, J.W. Hurst, A.D. Ames, Dynamic multi-domain bipedal walking with ATRIAS through SLIP based human-inspired control, in Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control (ACM, 2014), pp. 263–272
J. Horn, J. Reher, Zhao, V. Paredes, A.D. Ames, AMPRO: translating robotic locomotion to a powered transfemoral prosthesis, in International Conference on Robotics and Automation, ICRA, May 2015
Y. Hürmüzlü, D.B. Marghitu, Rigid body collisions of planar kinematic chains with multiple contact points. Int. J. Robot. Res. 13(1), 82–92 (1994)
J. Lack, M.J. Powell, A.D. Ames, Planar multi-contact bipedal walking using hybrid zero dynamics, in 2014 IEEE International Conference on Robotics and Automation (ICRA) (IEEE, 2014), pp. 2582–2588
I.R Manchester, Transverse dynamics and regions of stability for nonlinear hybrid limit cycles. IFAC Proc. 44(1), 6285–6290 (2011)
A.E. Martin, D.C. Post, J.P. Schmiedeler, 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)
B. Morris, J. W. Grizzle, A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: application to bipedal robots, in Proceedings of the 2005 IEEE International Conference on Decision and Control European Control Conference, Seville, 2005, pp. 4199–206
B. Morris, J. W. Grizzle, Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots. IEEE Trans. Autom. Control 54(8), 1751–1764 (2009)
H.-W. Park, A. Ramezani, J.W. Grizzle, A finite-state machine for accommodating unexpected large ground height variations in bipedal robot walking. IEEE Trans. Robot. 29(29), 331–345 (2013)
I. Poulakakis, J.W. Grizzle, The spring loaded inverted pendulum as the hybrid zero dynamics of an asymmetric hopper. IEEE Trans. Autom. Control 54(8), 1779–1793 (2009)
M.J. Powell, A.D. Ames, Hierarchical control of series elastic actuators through control lyapunov functions. Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, 2986–2992
M.J. Powell, H. Zhao, A.D. Ames, Motion primitives for human-inspired bipedal robotic locomotion: walking and stair climbing, in 2012 IEEE International Conference on Robotics and Automation (ICRA), 2012, pp. 543–549
A Ramezani, J.W. Hurst, K. Akbari Hamed, J.W. Grizzle, Performance analysis and feedback control of ATRIAS, a three-dimensional bipedal robot. J. Dyn. Syst. Meas. Control 136(2), 0210112-1–0210112-12 (2014)
H. Razavi, A.M. Bloch, C. Chevallereau, J.W. Grizzle, Restricted discrete invariance and self-synchronization for stable walking of bipedal robots, in American Control Conference, 2015
N. Sadati, G.A. Dumont, K.A. Hamed, W.A. Gruver, Hybrid Control and Motion Planning of Dynamical Legged Locomotion. In: MengChu Zhou (ed.), IEEE Press Series on Systems Science and Engineering. (Wiley, 2012), 272 pages. ISBN: 978-1-118-31707-5
J. Schmiedeler, ERNIE robot walking with different feet (2016). https://www.youtube.com/watch?v=T2x3VvPaacA
C.-L. Shih, J.W. Grizzle, C. Chevallereau, From stable walking to steering of a 3D bipedal robot with passive point feet. Robotica 30(7), 1119–1130 (2012)
A.S. Shiriaev, L.B. Freidovich, I.R. Manchester, Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems. Ann. Rev. Control 32(2), 200–211 (2008)
A.S. Shiriaev, L.B. Freidovich, S.V. Gusev, Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Trans. Autom. Control 55(4), 893–906 (2010)
K. Sreenath, H.-W. Park, I. Poulakakis, J.W. Grizzle, A compliant hybrid zero dynamics controller for stable, efficient and fast bipedal walking on MABEL. Int. J. Robot. Res. 30(9), 1170–1193 (2011)
K. Sreenath, H.-W. Park, I. Poulakakis, J.W. Grizzle, Embedding active force control within the compliant hybrid zero dynamics to achieve stable, fast running on MABEL. Int. J. Robot. Res. 33, 988–1005 (2014)
P. van Zutven, Control and Identification of Bipedal Humanoid Robots: Stability Analysis and Experiments. PhD thesis, University of Technology, Eindhoven, 2014
M. Vukobratović, B. Borovac, D. Surla, D. Stokic, Biped Locomotion (Springer, Berlin, 1990)
T. Wang, C. Chevallereau, D. Tlalolini, Stable walking control of a 3D biped robot with foot rotation. Robotica FirstView:1–20, (2014)
E.R. Westervelt, G. Buche, J.W. Grizzle, Experimental validation of a framework for the design of controllers that induce stable walking in planar bipeds. Int. J. Robot. Res. 24(6), 559–582 (2004)
E.R. Westervelt, J.W. Grizzle, C. Chevallereau, J.-H. Choi, B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. (CRC Press, Boca Raton, 2007)
E.R. Westervelt, J.W. Grizzle, D.E. Koditschek, Hybrid zero dynamics of planar biped walkers. IEEE Trans. Autom. Control 48(1), 42–56 (2003)
T. Yang, E.R. Westervelt, A. Serrani, J.P. Schmiedeler, A framework for the control of stable aperiodic walking in underactuated planar bipeds. Auton. Robot. 27(3), 277–290 (2009)
H.-H. Zhao, W.-L. Ma, A.D. Ames, M.B. Zeagler, Human-inspired multi-contact locomotion with AMBER2, in 2014 ACM/IEEE International Conference on Cyber-Physical Systems (ICCPS), 2014, pp. 199–210
Acknowledgements
The work of J.W. Grizzle has been generously supported by NSF grants EECS-1525006, ECCS-1343720, and CNS-1239037. The work of C. Chevallereau is supported by ANR Equipex Robotex project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media B.V.
About this entry
Cite this entry
Grizzle, J.W., Chevallereau, C. (2017). Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion. In: Goswami, A., Vadakkepat, P. (eds) Humanoid Robotics: A Reference. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7194-9_47-1
Download citation
DOI: https://doi.org/10.1007/978-94-007-7194-9_47-1
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-7194-9
Online ISBN: 978-94-007-7194-9
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering