Discrete adjoint method for variational integration of constrained ODEs and its application to optimal control of geometrically exact beam dynamics

• Published in: Multibody system dynamics Vol. 60; no. 3; pp. 447 - 474
• Main Authors: Schubert, Matthias, Sato Martín de Almagro, Rodrigo T., Nachbagauer, Karin, Ober-Blöbaum, Sina, Leyendecker, Sigrid
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2024; Springer Nature B.V
• Subjects: Automotive Engineering; Constraints; Control; Dynamical Systems; Dynamics; Electrical Engineering; Engineering; Mechanical Engineering; Optimal control; Optimization; Vibration; Variational integrators; Geometrically exact beam; Optimal control; Discrete adjoint method; Holonomically constrained system
• ISSN: 1384-5640; 1573-272X
• DOI: 10.1007/s11044-023-09934-4

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE.