Retraction maps in optimal control of nonholonomic systems

TL;DR

The paper develops geometric integrators for the PMP-based optimal control of fully actuated nonholonomic systems by exploiting discretization (retraction) maps and their cotangent lifts to construct symplectic integrators on $T^*\nD$. It applies the framework to the controlled Chaplygin sleigh, deriving an explicit retraction-based symplectic scheme that preserves nonholonomic constraints and analyzes its performance against high-order Runge–Kutta methods in terms of constraint preservation and energy behavior. The results indicate that constraint fidelity can be strong with retraction-based schemes, while energy preservation benefits from higher-order methods, motivating further study of step-size effects and backward-error considerations. The work points to future extensions to nonholonomic control on Lie groups, homogeneous spaces, and underactuated systems, broadening the applicability of geometric integrators in optimal control contexts.

Abstract

In this paper, we compare the performance of different numerical schemes in approximating Pontryagin's Maximum Principle's necessary conditions for the optimal control of nonholonomic systems. Retraction maps are used as a seed to construct geometric integrators for the corresponding Hamilton equations. First, we obtain an intrinsic formulation of a discretization map on a distribution $\mathcal{D}$. Then, we illustrate this construction on a particular example for which the performance of different symplectic integrators is examined and compared with that of non-symplectic integrators.

Figures (6)

• Figure 1: Definition of the cotangent lift of a discretization.
• Figure 2: Commutative diagram defining $R_{\mathcal{D}, d}=(\mathcal{P}\times \mathcal{P})\circ R_{d} \circ Ti_{\mathcal{D}}.$
• Figure 3: Plot of the discrete constraint function $\phi_{d}$ as the number of iterations increases. The plot depicts standard second and fourth-order Runge-Kutta methods, against the retraction integrator with $\delta=1/2$.
• Figure 4: Plot of the discrete constraint function $\phi_{d}$ as the number of iterations increases. The plot depicts the two (symplectic) integrators based on retractions.
• Figure 5: Plot of the Hamiltonian function (energy) as the number of iterations increases. The plot depicts a symplectic fourth order method against a standard fourth-order Runge-Kutta method.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Example 1
• Proposition 1
• Corollary 1
• Example 2
• Definition 5: BlCoGuMdD
• Remark 1