Constraint Control of Nonholonomic Mechanical Systems

Abstract

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov's problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction ξ. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov's problem for the rotation group SO(3). We show that it is possible to control the system using the constraint ξ(t) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.

On the dynamics of a rolling ball actuated by internal point masses

“…In Suslov's original formulation [47], ξ was assumed to be fixed in the body frame. In [48] and here, ξ is permitted to vary with time. The Lagrangian for Suslov's problem is its kinetic energy, so that the symmetry-reduced Lagrangian is (Ω) = 1 2 IΩ, Ω and the action integral is S = The reader is referred to [48] for further details.…”

Numerical simulations of a rolling ball robot actuated by internal point masses

“…The derivation of the variational Pontryagin's minimum principle presented here follows [34]. The reader is referred to [34] or Section 2.3 of [35] for a review of the calculus of variations notation needed to understand the derivation of the variational Pontryagin's minimum principle. Let n, m ∈ N. Let a be a prescribed or free initial time and let k1 ∈ N 0 be such that 0 ≤ k1 ≤ n if a is prescribed and 1 ≤ k1 ≤ n + 1 if a is free.…”

“…For example, bvp4c [44], bvp5c [45], sbvp [14], and bvptwp [15] (which encapsulates twpbvp m, twpbvpc m, twpbvp l, twpbvpc l, acdc, and acdcc) are MATLAB Runge-Kutta or collocation ODE TPBVP solvers, while COLSYS [46], COLNEW [24], COLMOD [25], COLCON [31], BVP M-2 [47], TWPBVP [26], TWP-BVPC [27], TWPBVPL [28], TWPBVPLC [29], ACDC [25], and ACDCC [15] are Fortran Runge-Kutta or collocation ODE TPBVP solvers. The reader is referred to the Appendix in [35]…”

BVPs Codes for Solving Optimal Control Problems

“…The paper shows a list of a few interesting problems, for other applications of the same codes, here considered, to more involved optimal control problems, we refer the reader to [25][26][27][28]. Moreover, we highlight that it is not our aim to compare direct and indirect methods, but only to show the efficiency of indirect methods that often are not taken into consideration because users do not know the potentiality of general-purpose codes for BVPs.…”