Mechanical System Kinematics and Dynamics on Differentiable Manifolds

Abstract

Topological and vector space attributes of Euclidean space are employed to create a dierentiable manifold structure for holonomic mechanical system kinematics and dynamics. A local kinematic parameterization is presented that establishes the regular conguration space as a dierentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, d'Alembert variational equations of mechanical system dynamics yield well-posed ordinary dierential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy kinematic conguration, velocity, and acceleration constraints and the variational equations of dynamics. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specied error tolerances, using fourth order Runge-Kutta numerical integration methods.