Abstract
In this paper we present new results for nonlinear tracking control design based on the search for a “control contraction metric” (CCM). We show that for fully-actuated robots, CCMs can be found analytically that correspond to classical methods of robot control including sliding and energy-based designs. We also show that, for underactuated robots, the CCM approach extends more recent optimization-based methods such as LQR-Trees to tracking of arbitrary feasible trajectories. Therefore the CCM methodology represents a bridge between these methods. We illustrate our results with calculations and simulations on the single and double inverted pendulums and the underactuated cart-pole system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Vukobratović, M., Borovac, B.: Zero-moment point — thirty five years of its life. Int. J. Humanoid Robot. 1(1), 157–173 (2004)
Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, USA (1991)
Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control, vol. 3. Wiley, New York (2006)
Mellinger, D., Michael, N., Kumar, V.: Trajectory generation and control for precise aggressive maneuvers with quadrotors. Int. J. Robot. Res. 31(5), 664–674 (2012)
Collins, S., Ruina, A., Tedrake, R., Wisse, Martijn: Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712), 1082–1085 (2005)
Albu-Schäffer, A., Ott, C., Hirzinger, G.: A unified passivity-based control framework for position, torque and impedance control of flexible joint robots. Int. J. Robot. Res. 26(1), 23–39 (2007)
Fantoni, I., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, Berlin (2001)
Tedrake, R.: Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation (Course Notes for MIT 6.832) (2014)
Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Prentice-Hall, USA (1990)
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology (2000)
Tedrake, R., Manchester, I.R., Tobenkin, M., Roberts, John W.: LQR-trees: feedback motion planning via sums-of-squares verification. Int. J. Robot. Res. 29(8), 1038–1052 (2010)
Majumdar, A., Ahmadi, A.A., Tedrake, R.: Control and verification of high-dimensional systems via DSOS and SDSOS optimization. In: 53rd IEEE Conference on Decision and Control, Los Angeles, CA (2014)
Henrion, D., Korda, M.: Convex computation of the region of attraction of polynomial control systems. IEEE Trans. Autom. Control 59(2), 297–312 (2014)
Majumdar, A., Vasudevan, R., Tobenkin, M.M., Tedrake, R.: Convex optimization of nonlinear feedback controllers via occupation measures. Int. J. Robot. Res. 33(9), 1209–1230 (2014)
Manchester, I.R., Slotine, J.-J.E.: Control contraction metrics and universal stabilizability. In: Proceedings of the IFAC World Congress, Cape Town, South Africa (2014)
Manchester, I.R., Slotine, J.-J.E.: Control contraction metrics: convex and intrinsic criteria for nonlinear feedback design, IEEE Transactions on Automatic Control, in press (2017)
Lohmiller, W., Slotine, J.-J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)
Spong, M.W.: The swing up control problem for the acrobot. Control Systems, IEEE, 15(1):49–55, (1995)
Åström, K.J., Furuta, K.: Swinging up a pendulum by energy control. Automatica 36(2), 287–295 (2000)
Ijspeert, A.J., Nakanishi, J., Hoffmann, H., Pastor, P., Schaal, S.: Dynamical movement primitives: learning attractor models for motor behaviors. Neural Comput. 25(2), 328–373 (2013)
Manchester, I.R., Slotine, J.-J.E.: Transverse contraction criteria for existence, stability, and robustness of a limit cycle. Syst. Control Lett. 62, 32–38 (2014)
Do Carmo, M.P.: Riemannian Geometry. Springer, Berlin (1992)
Hespanha, J.P.: Linear Systems Theory. Princeton University Press, USA (2009)
Aylward, E.M., Parrilo, P.A., Slotine, J.-J.E.: Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming. Automatica 44(8), 2163–2170 (2008)
Wang, W., Slotine, J.-J.E.: On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92(1), 38–53 (2005)
Jouffroy, J., Slotine, J.-J.E.: Methodological remarks on contraction theory. In: 43rd IEEE Conference on Decision and Control, The Bahamas (2004)
Slotine, J.-J.E., Li, W.: On the adaptive control of robot manipulators. Int. J. Robot. Res. 6(3), 49–59 (1987)
Bullo, F., Murray, R.M.: Tracking for fully actuated mechanical systems: a geometric framework. Automatica 35(1), 17–34 (1999)
Leung, K., Manchester, I.R.: Nonlinear stabilization via control contraction metrics: a pseudospectral approach for computing geodesics. In: Proceedings of the 2017 American Control Conference, Seattle, WA (2017)
Spong, M.W.: Energy based control of a class of underactuated mechanical systems. In: Proceedings of the IFAC World Congress, San Francisco, CA (1996)
Löfberg, J.: Yalmip: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Manchester, I.R., Tang, J.Z., Slotine, JJ.E. (2018). Unifying Robot Trajectory Tracking with Control Contraction Metrics. In: Bicchi, A., Burgard, W. (eds) Robotics Research. Springer Proceedings in Advanced Robotics, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-60916-4_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-60916-4_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-60915-7
Online ISBN: 978-3-319-60916-4
eBook Packages: EngineeringEngineering (R0)