Reduced equations for nonholonomic mechanical systems with dissipative forces

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This paper addresses the problem of computing reduced equations for mechanical systems with nonholonomic constraints, Lie group symmetries, and dissipative forces. Results are presented for two important cases: the unconstrained case, for both body and spatial representations, and the constrained (mixed kinematic and dynamic) case. For unconstrained systems, we show that the structure of the reduced Lagrangian almost transparently reveals two useful components in the reduction process, namely the local forms of the locked inertia tensor and the mechanical connection. For the case when nonholonomic constraints are present, we develop an extension to the nonholonomic momentum equation of Bloch, Krishnaprasad, Marsden, and Murray [5] that includes general forcing functions. We then provide an alternate proof to give additional intuitive insight into the origin and structure of these equations. Finally, we present a method for modeling (invariant) damping forces using a Rayleigh dissipation function. These techniques are illustrated with examples from robotics, including the snakeboard example.