Computing reduced equations for robotic systems with constraints and symmetries

Abstract

Develops easily computable methods for deriving the reduced equations for mechanical systems with Lie group symmetries. These types of systems occur frequently in robotics, and are found generically in robotic locomotion, wheeled mobile robots, and satellites or underwater vehicles with robotic arms. Results are presented for two important cases: (1) the unconstrained case, for both body and spatial representations; and (2) the constrained (mixed kinematic and dynamic) case. In each case, the dynamic equations for these nonholonomic mechanical systems are given, and illustrated by the appropriate calculations for an example system. A primary result of the paper is to show that the spectrum of possible constraints-ranging from no constraints to fully constrained systems-can be expressed within a single unifying principle for calculating the reduced equations. In this process, the structure of the reduced Lagrangian directly reveals two useful components in the reduction process, namely the local forms of the locked inertia tensor and the mechanical connection. Finally, it is shown that the reduced dynamics decouple from any explicit dependence on the group configuration variables.