Joint–coordinate adjoint method for optimal control of multibody systems

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Abstract
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.
Funding Information
  • Politechnika Warszawska (Excellence Initiative, Research University in the field of Artificial Intelligence and Robotics, grant no. 1820/25/Z01/POB2/2021)

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