Adjoint Gradient Computation For An Extremal Value Of A System Output For The Minimization Of A Maximum

In industry, attempts are made to design tasks efficiently, either to minimize the cycle time or to optimize certain system outputs. In the latter case, it may be of interest to reduce energy consumption or to bring the occurring forces and velocities below certain limit values. The present work illustrates the
derivation of an adjoint gradient for an extremal value of a system output of a non-linear dynamic system, as e.g. the maximum occurring joint reaction force or the velocity at the tool-center-point of a robotic system. Therefore, the objective is defined by the extremal value and the goal is to find an efficient and accurate gradient for an advanced update information in classical gradient-based optimization strategies, as e.g. presented in [1]. Hence, the gradient of the objective is derived here by using the so-called adjoint method following the basic idea tracing back to the pioneering work by Bryson and Ho [2]. The latter work is based on the idea of using adjoint variables in an augmented cost
functional and derive adjoint equations for reducing the computational costs of the gradient update.
Here, the present derivation of the adjoint gradient for an extremal value of a system output follows the basic idea above and uses simplifications due to the interpretation of the behavior of the adjoint variables presented in [3]. The presented adjoint gradient computation will be applied in the example of
a one-mass oscillator in order to show the easy applicability and correctness. Furthermore, this adjoint gradient is applied in the framework of a multi-objective optimization strategy in order to allow several maxima of the objective function.

[1] D. E. Kirk. Optimal Control Theory: An Introduction. Dover Publications, Mineola, New York, 2004.
[2] A.E. Bryson and Y.C. Ho. Applied Optimal Control: Optimization, Estimation and Control. Hemisphere, Washington, DC, 1975.
[3] D. Lichtenecker and K. Nachbagauer. A discrete adjoint gradient approach for equality and inequality constraints in dynamics. Multibody System Dynamics, 61:103-130, 2024.