The Boltzmann–Hamel equations for the optimal control of mechanical systems with nonholonomic constraints. (English) Zbl 1213.49029
Summary: We generalize the Boltzmann-Hamel equations for non-holonomic mechanics to a form suited for the kinematic or dynamic optimal control of mechanical systems subject to non-holonomic 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. By using an appropriate choice of quasi-velocities, one is able to reduce the required number of differential equations by \(m\) and \(3m\) for the kinematic and dynamic optimal control problems, respectively, where \(m\) is the number of nonholonomic constraints. 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 \(SO(3)\).
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 70Q05 | Control of mechanical systems |
| 34H05 | Control problems involving ordinary differential equations |
Keywords:
nonholonomic constraints; optimal control; variational principles; Boltzmann–Hamel equations; quasi-velocitiesReferences:
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