Abstract
Geometric mechanics techniques based on Lie brackets provide high-level characterizations of the motion capabilities of locomoting systems. In particular, they relate the net displacement they experience over cyclic gaits to area integrals of their constraints; plotting these constraints thus provides a visual “landscape” that intuitively captures all available solutions of the system’s dynamic equations. Recently, we have found that choices of system coordinates heavily influence the effectiveness of these approaches. This property appears at first to run counter to the principle that differential geometric structures should be coordinate-invariant. In this paper, we provide a tutorial overview of the Lie bracket techniques, then examine how the coordinate-independent nonholonomy of these systems has a coordinate-dependent separation into nonconservative and noncommutative components that respectively capture how the system constraints vary over the shape and position components of the configuration space. Nonconservative constraint variations can be integrated geometrically via Stokes’ theorem, but noncommutative effects can only be approximated by similar means; therefore choices of coordinates in which the nonholonomy is primarily nonconservative improve the accuracy of the geometric techniques.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Shapere, F. Wilczek. J. Fluid Mech. 198, 557 (1989)
P.S. Krishnaprasad, D.P. Tsakiris, G-snakes: Nonholonomic kinematic chains on lie groups. In 33rd IEEE Conference on Decision and Control (Lake Buena: Vista, Florida, 1994)
S.D. Kelly, R.M. Murray, J. Robotic Syst. 12, 417 (1995)
J.P. Ostrowski, J. Burdick, Int. J. Robotics Res. 17, 683 (1998)
A.D. Lewis, IEEE Trans. Automatic Control. 45, 1420 (2000)
P.S. Krishnaprasad, D.P. Tsakiris, Dynamical Syst. 16, 347 (2001)
M. Anthony, Bloch, et al., Nonholonomic Mechanics and Control (Springer, 2003)
J.B. Melli, C.W. Rowley, D.S. Rufat, SIAM J. Appl. Dynamical Syst. 5, 650 (2006)
R.L. Hatton, H. Choset, Approximating displacement with the body velocity integral. In Proceedings of Robotics: Science and Systems V (Seattle, WA USA, 2009)
R.L. Hatton, H. Choset, Optimizing coordinate choice for locomoting systems. In Proc. IEEE Int. Conf. Robotics and Automation (Anchorage, AK USA, 2010), p. 4493
R.L. Hatton, H. Choset, Int. J. Robotics Res. 30, 988 (2011)
A. Shapere, F. Wilczek, Am. J. Phys. 57, 514 (1989)
R. Abraham, J.E. Marsden, Foundations of Mechanics (Addison Wesley, 1985)
J.E. Marsden, R. Montgomery, T.S. Ratiu, Mem. Am. Math. Soc. 436 (1990)
J.E. Marsden, Introduction to Mechanics and Symmetry (Springer-Verlag, 1994)
J.P. Ostrowski, J.P. Desai, Vijay Kumar. Int. J. Robotics Res. 19, 225 (2000)
F. Bullo, K.M. Lynch, IEEE Trans. Robotics Automation 17, 402 (2001)
E.A. Shammas, H. Choset, A.A. Rizzi, The Int. J. Robotics Res. 26, 1075 (2007)
R.L. Hatton, H. Choset, Proc. IEEE BioRobotics Conf. 451 (2008)
S.D. Kelly, The mechanics and control of driftless swimming (in press)
J.E. Avron, O. Raz, New J. Phys. 9 (2008)
K. McIsaac, J.P. Ostrowski, Robotics Automation 637 (2003)
K.A. Morgansen, B.I. Triplett, D.J. Klein, IEEE Trans. Robotics 23, 1184 (2007)
R. Mukherjee, D.P. Anderson, IEEE Int. Conf. Robotics Automation, 802 (1993)
G.C. Walsh, S. Sastry, Robotics and Automation, IEEE Transactions 11, 139 (1995)
G.C. Walsh, S. Sastry, In Proc. 33th Conf. Decision Control, 1190 (1991)
E.A. Shammas, K. Schmidt, H. Choset, In IEEE Int. Conf. Robotics Automation (2005)
E.A. Shammas, H. Choset, A.A. Rizzi, Int. J. Robotics Res. 26, 1043 (2007)
R.L. Hatton, H. Choset. In Proceedings of the ASME Dynamic Systems and Controls Conference (DSCC) (Cambridge, Massachusetts, USA, 2010)
R.L. Hatton, H. Choset, Y. Ding, D.I. Goldman, Phys. Rev. Lett. 110, 078101 (2013)
R.M. Murray, Z. Li, S.S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, 1994)
R.M. Murray, S.S. Sastry, IEEE Trans. Automatic Control. 38, 700 (1993)
R.L. Hatton, H. Choset, IEEE Trans. Robotics 29, 615 (2013)
E.A. Shammas, Ph.D. thesis, Carnegie Mellon University, 2006
J.E. Radford, J.W. Burdick, In Proc. Int. Symposium on Mathematical Theory of Networks and Syst. (Padova, Italy, 1998)
W. Magnus, Comm. Pure Appl. Math. VII, 649 (1954)
R.L. Hatton, H. Choset, Proc. Robotics: Sci. Syst. VII (Los Angeles, CA USA, 2011)
M. Santander, Am. J. Phys., 782 (1992)
M.A. Travers, R.L. Hatton, H. Choset, Proc. Am. Controls Conf. (ACC) (2013)
F. Bullo, A.D. Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems (Springer, 2004)
D. Bachman, A Geometric Approach to Di↑erential Forms (Birkhäuser, 2006)
W.L. Burke, Appl. Di↑erential Geometry (Cambridge University Press, 1985)
W.M. Boothby, An Introduction to Di↑erentiable Manifolds and Riemannian Geometry (Academic Press, 1986)
G.B. Arfken. Mathematical Methods for Physicists, 6th edition (Elsevier, 2005)
Q. Guo, M.K. Mandal, M.Y. Li, Pattern Recognition Lett. 26, 493 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hatton, R., Choset, H. Nonconservativity and noncommutativity in locomotion. Eur. Phys. J. Spec. Top. 224, 3141–3174 (2015). https://doi.org/10.1140/epjst/e2015-50085-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2015-50085-y