Series Expansions for the Evolution of Mechanical Control Systems

This paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time-varying external force. Mechanical systems are presented as second-order systems on a configuration manifold via the notion of affine connections. The series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula. A convergence analysis is obtained using some analytic functions and combinatorial analysis results. This expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems.

[1]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[2]  D. Sattinger,et al.  Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics , 1986 .

[3]  R. Carter.  Lie Groups , 1970, Nature.

[4]  George E. Andrews,et al.  Number theory , 1971, Bicycle or Unicycle?.

[5]  Naomi Ehrich Leonard,et al.  Motion Control of Drift-Free, , 1995 .

[6]  Naomi Ehrich Leonard,et al.  Motion control for underactuated mechanical systems on lie groups , 1997, 1997 European Control Conference (ECC).

[7]  Andrew D. Lewis,et al.  Simple mechanical control systems with constraints , 2000, IEEE Trans. Autom. Control..

[8]  Gerardo Lafferriere,et al.  A Differential Geometric Approach to Motion Planning , 1993 .

[9]  Naomi Ehrich Leonard,et al.  Motion control of drift-free, left-invariant systems on Lie groups , 1995, IEEE Trans. Autom. Control..

[10]  W.S. Gray,et al.  Hankel operators and Gramians for nonlinear systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  Ilya Kolmanovsky,et al.  Stabilizing feedback laws for internally actuated multibody systems in space , 1996 .

[12]  Richard M. Murray,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997, SIAM Rev..

[13]  S. Lang.  Differential and Riemannian Manifolds , 1996 .

[14]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[15]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[16]  Francesco Bullo,et al.  A series describing the evolution of mechanical control systems , 1999 .

[17]  Eduardo D. Sontag,et al.  Time-optimal control of manipulators , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[18]  R. Bianchini,et al.  Graded approximations and controllability along a trajectory , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[19]  S. Krantz.  Function theory of several complex variables , 1982 .

[20]  Andrei A. Agrachev,et al.  Local controllability and semigroups of diffeomorphisms , 1993 .

[21]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[22]  Héctor J. Sussmann,et al.  Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory , 1997 .

[23]  Matthias Kawski.  Nonlinear Control and Combinatorics of Words , 1997 .

[24]  H. Sussmann.  A Sufficient Condition for Local Controllability , 1978 .

[25]  A. Bloch,et al.  Nonholonomic Control Systems on Riemannian Manifolds , 1995 .

[26]  Francesco Bullo,et al.  Averaging and Vibrational Control of Mechanical Systems , 2002, SIAM J. Control. Optim..

[27]  Naomi Ehrich Leonard,et al.  Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups , 2000, IEEE Trans. Autom. Control..

[28]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[29]  John Baillieul.  Stable average motions of mechanical systems subject to periodic forcing , 1993 .

[30]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[31]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[32]  H. Sussmann.  A general theorem on local controllability , 1987 .

[33]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[34]  R. Gamkrelidze,et al.  THE EXPONENTIAL REPRESENTATION OF FLOWS AND THE CHRONOLOGICAL CALCULUS , 1979 .

[35]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[36]  M. Djemai,et al.  Exact steering and stabilization of a PVTOL aircraft , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[37]  Willard Miller,et al.  Lie Groups and Algebras , 1984 .

[38]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .