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

• Published in: International journal of robust and nonlinear control Vol. 21; no. 4; pp. 373 - 386
• Main Authors: Maruskin, Jared M., Bloch, Anthony M.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 10.03.2011
• Subjects: Bolts; Boltzmann-Hamel equations; Differential equations; Kinematics; Mathematical analysis; Mechanical systems; nonholonomic constraints; Optimal control; Optimization; quasi-velocities; Rigid-body dynamics; variational principles
• ISSN: 1049-8923; 1099-1239; 1099-1239
• DOI: 10.1002/rnc.1598

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. 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). Copyright © 2010 John Wiley & Sons, Ltd.