[Paper Review] Discrete Mechanics and Optimal Control: an Analysis

S. Ober-Bloebaum, Oliver Junge+1 more|ArXiv.org|Oct 8, 2008

Numerical methods for differential equations5 citations

TL;DR

This paper introduces Discrete Mechanics and Optimal Control (DMOC), a structure-preserving method that discretizes the variational principles of mechanical systems directly to solve optimal control problems. By formulating the discrete optimal control problem via a discrete variational principle, DMOC inherits symplecticity, momentum conservation, and energy behavior from the continuous system, leading to superior convergence and reduced computational cost compared to collocation methods.

ABSTRACT

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.

Motivation & Objective

• To develop a numerical optimal control framework that preserves the intrinsic geometric and structural properties of mechanical systems during discretization.
• To address the limitations of standard collocation and indirect methods by directly discretizing the variational structure of the system’s dynamics.
• To reduce computational cost in optimization by leveraging configuration-level parameterization instead of configuration-velocity or configuration-momentum formulations.
• To establish a theoretical foundation for convergence and equivalence to symplectic partitioned Runge-Kutta schemes.
• To enable efficient solution of optimal control problems in applications such as orbital transfers and robotic manipulation while maintaining physical consistency.

Proposed method

• Discretize the Lagrange-d’Alembert principle using a discrete variational formulation to derive structure-preserving time-stepping equations.
• Formulate the optimal control problem as a finite-dimensional nonlinear programming problem using the discrete action sum and discrete Euler-Lagrange equations.
• Ensure the discrete solution inherits symplecticity, momentum maps, and energy behavior from the continuous system via the discrete variational structure.
• Establish equivalence between DMOC and a symplectic partitioned Runge-Kutta scheme, enabling theoretical convergence analysis.
• Implement the method using sequential quadratic programming (SQP) solvers with configuration-level variables to reduce the number of optimization variables.
• Apply the method to benchmark problems such as low-thrust orbital transfers and two-link manipulators to validate performance and structure preservation.

Experimental results

Research questions

• RQ1 How can the variational structure of mechanical systems be preserved in discrete optimal control formulations?
• RQ2 What is the relationship between DMOC and classical symplectic integrators such as partitioned Runge-Kutta schemes?
• RQ3 To what extent does DMOC reduce the number of optimization iterations compared to collocation-based methods?
• RQ4 Can DMOC maintain structural invariants like momentum maps and symplecticity in practical optimal control problems?
• RQ5 How does the configuration-level formulation in DMOC affect computational efficiency and convergence speed?

Key findings

• DMOC is mathematically equivalent to a symplectic partitioned Runge-Kutta discretization of Hamilton’s equations, ensuring long-term stability and geometric fidelity.
• The DMOC formulation reduces the number of optimization variables by approximately 67% compared to collocation methods, leading to a 1.5× reduction in SQP iterations.
• Numerical experiments show that DMOC achieves faster convergence rates in both configuration and control trajectories compared to collocation approaches.
• The method successfully preserves momentum maps and symplectic structure in examples such as low-thrust orbital transfers and two-link manipulators.
• DMOC enables efficient solution of optimal control problems with reduced computational effort, particularly when symmetry and structure preservation are critical.
• Future extensions using DMOC primitives and adaptive time-stepping show promise for global optimization and dynamic adaptation in complex systems.