ISSN:
0044-2275
Keywords:
Key words. Dynamics, rigid body, Euclidean group, multibody system.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract. In this article we formulate, in a Lie group setting, the equations of motion for a system of n coupled rigid bodies subject to holonomic constraints. A mapping $f: { \cal {M} \rightarrow \cal {N}$ is constructed, where $\cal M$ is the m-dimensional configuration manifold of the system, and $\cal{N} = \rm{SE(3)} \times \cdots \times \rm{SE(3)}$ (n copies) is endowed with the left-invariant Riemannian metric h corresponding to the total kinetic energy of the system, where SE(3) is the special Euclidean group. The generalized inertia tensor of the system is given by the pullback metric $f^*h$ ; the equations of motion are then the geodesic equations on $\cal M$ with respect to this metric. We show how this coordinate-free formulation leads directly to a factorization of the generalized inertia tensor of the form ${\cal S}^T {\cal L}^T {\cal H} {\cal L} {\cal S}$ , where $\cal S$ is a constant block-diagonal matrix consisting only of kinematic parameters, $\Cal H$ is a constant block-diagonal matrix consisting only of inertial parameters, and ${\cal L}$ is a block lower-triangular matrix composed of Adjoint operators on se(3). Such a factorization is useful for various multibody system dynamics applications, e.g., inertial parameter identification, adaptive control, and design optimization. We also show how in many practical situations ${\cal N}$ can be reduced to a submanifold, thereby considerably simplifying the derivation of the equations of motion. Our geometric formulation not only suggests ways to choose the best coordinates for analysis and computation, but also provides high-level insight into the structure of the equations of motion.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00001521
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