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

Abstract

SUMMARYIn 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).

Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism

“…The virtual displacements of the system at each time t and at each position X are the vectors in 3 n ∈ X   such that [2] ( ) ( ),,…”

Some Applications of Quasi-Velocities in Optimal Control

“…These equations are also obtained in [20,21] for the case E = T M. When we consider the forces as control variables, since we are assuming that the con- where v a are the acceleration coordinates, i.e. our control variables.…”

“…Nonholonomic constraints are very relevant and appear in many problems in physics and engineering, and in particular in control theory. Such nonholonomic constraints restrict possible virtual displacements and when taking into account such constraints d'Alembert-Lagrange principle leads to Boltzmann-Hamel equations [7,9,[20][21][22].…”

Regularity properties of fiber derivatives associated with higher-order mechanical systems

“…These "higher-order variational problems" are of great interest for their useful applications in aeronautics, robotics, computer-aided design, air traffic control, trajectory planning, and, more generally, problems of interpolation and approximation of curves on Riemannian manifolds. These kinds of problems have been studied in [4,5,7,30,37,41,44] and more recently, in [22,23,24,43] the development of variational principles which involve higher-order cost functions for optimization problems on Lie groups and their application in template matching for computational anatomy have been studied. These applications have produced a great interest in the study and development of new modern geometric tools and techniques to model properly higher-order variational problems, with the additional goal of obtaining a deepest understanding of the intrinsic properties of these problems.…”