The Boltzmann–Hamel equations for the optimal control of mechanical systems with nonholonomic constraints

In this paper we generalize the Boltzmann-Hamel equations for nonholonomic mechanics to a form suited for the kinematic or dynamic optimal control of mechanical systems subject to nonholonomic constraints. In solving these equations one is able to eliminate the controls and compute the optimal trajectory from a set of coupled first order differential equations with boundary values. We compare our approach to standard approaches on a sample of mechanical problems. In particular we derive a set of differential equations that yields the optimal reorientation path of a free rigid body. In the special case of a sphere, we show that the optimal trajectory coincides with the cubic splines on SO(3).

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