[Paper Review] Energy conserving nonholonomic integrators

Jorge Cortés|ArXiv.org|Sep 24, 2002

Control and Dynamics of Mobile Robots6 references3 citations

TL;DR

This paper introduces energy-conserving nonholonomic integrators by extending discrete mechanics to nonautonomous systems via a discrete version of the Lagrange-d'Alembert principle. The method constructs variational integrators that preserve symplectic structure, nonholonomic momentum, and energy simultaneously—overcoming a theoretical obstruction by allowing variable time steps.

ABSTRACT

We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a discrete version of the Lagrange-d'Alembert principle for nonautonomous systems.

Motivation & Objective

• Address the challenge of constructing numerical integrators for nonholonomic Lagrangian systems that preserve key geometric properties of the continuous flow.
• Overcome the theoretical obstruction that fixed-step variational integrators cannot simultaneously conserve energy, symplecticity, and momentum.
• Develop a discrete formulation of the Lagrange-d'Alembert principle for nonautonomous nonholonomic systems to enable energy conservation.
• Ensure the resulting integrators inherit discrete versions of symplectic structure evolution and nonholonomic momentum conservation.
• Establish conditions under which the discrete energy is conserved, particularly in autonomous systems.

Proposed method

• Formulate a discrete version of the Lagrange-d'Alembert principle for nonautonomous nonholonomic systems on the extended configuration space $\overline{Q} = \mathbb{R} \times Q$.
• Define the extended discrete Lagrangian $L_d$ on $\overline{Q} \times \overline{Q}$, incorporating time evolution and constraints.
• Derive the extended discrete Lagrange-d'Alembert (EDLA) equations as the variational principle governing the integrator dynamics.
• Introduce variable time steps to circumvent the theoretical impossibility of fixed-step energy-symplectic-momentum preservation.
• Apply reduction theory to the EDLA equations when symmetries are present, leading to reduced extended discrete Lagrange-d'Alembert (REDLA) equations.
• Express forces in bundle coordinates using the local connection form $\mathcal{A}_{\text{loc}}$, enabling computation in symmetric settings.

Experimental results

Research questions

• RQ1 Can a discrete integrator for nonholonomic systems preserve energy, symplectic structure, and momentum simultaneously?
• RQ2 What is the role of variable time steps in enabling energy conservation within variational integrators for nonholonomic systems?
• RQ3 How can the discrete Lagrange-d'Alembert principle be extended to nonautonomous systems with nonholonomic constraints?
• RQ4 What geometric properties are inherited by the resulting integrators, particularly regarding symplecticity and momentum maps?
• RQ5 Under what conditions does the discrete energy remain conserved in the nonholonomic setting?

Key findings

• The proposed extended discrete Lagrange-d'Alembert (EDLA) integrators preserve the symplectic form and the nonholonomic momentum map, ensuring geometric fidelity.
• Energy is conserved in the discrete system when the discrete Lagrangian and constraints are invariant under time translations, i.e., in autonomous systems.
• The discrete energy $E_{L_d}^+$ remains constant along the flow: $E_{L_d}^+(t_k, q_k, t_{k+1}, q_{k+1}) = E_{L_d}^+(t_{k-1}, q_{k-1}, t_k, q_k)$, as shown in Proposition 4.
• The method overcomes the theoretical obstruction of Theorem 1 by allowing variable time steps, enabling simultaneous conservation of energy, symplecticity, and momentum.
• Reduced EDLA equations (REDLA) are derived for symmetric systems, with forces expressed via the local connection form $\mathcal{A}_{\text{loc}}$.
• The integrators are shown to project onto solutions of the reduced equations, ensuring consistency in symmetric configurations.