Paweł Maciąg

This paper presents a joint–coordinate adjoint method for optimal control of multi-rigid-body systems. Initially formulated as a set of differential-algebraic equations, the adjoint system is brought into a minimal form by projecting the original expressions into the joint’s motion and constraint force subspaces. Consequently, cumbersome partial derivatives corresponding to joint-space equations of motion are avoided, and the approach is algorithmically more straightforward. The analogies between the formulation of Hamilton’s equations of motion in a mixed redundant-joint set of coordinates and the necessary conditions arising from the minimization of the cost functional are demonstrated in the text. The observed parallels directly lead to the definition of a joint set of adjoint variables. Through numerical studies, the performance of the proposed approach is investigated for optimal control of a double pendulum on a cart. The results demonstrate a successful application of the joint-coordinate adjoint method. The outcome can be easily generalized to optimal control of more complex systems.

Summary: The design of multibody systems involves high fidelity and reliable techniques and formulations that should help the analyst to make reasonable decisions. Given that constrained equations of motion for the simplest of multibody systems are highly nonlinear, determining the sensitivity terms is a computationally intensive and complex process that requires the application of special procedures. In this article, two novel Hamiltonian‐based approaches are presented for efficient sensitivity analysis of general multibody systems. The developed direct differentiation and the adjoint methods are based on constrained Hamilton's canonical equations of motion. This formulation provides solutions, which are more stable as compared to the results of direct integration of equations of motion expressed in terms of accelerations due to a reduced differential index of the underlying system of differential‐algebraic equations and explicit constraint imposition at the velocity level.The proposed Hamiltonian based methods are both capable of calculating the sensitivity derivatives and keeping the growth of constraint violation errors at a reasonable rate. The Hamiltonian‐based procedures derived herein appear to be good alternatives to existing methods for sensitivity analysis of general multibody systems.