Adjoint Gradient Computation for an Extremal Value of a System Output

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 in the dynamics of a multibody system. Therefore, the objective is defined by the extremal value and the goal is to find an efficient and accurate gradient information for an advanced update in classical gradient-based optimization strategies. Hence, the gradient of the objective is derived here by using the adjoint method following the basic idea tracing back to Bryson and Ho. The presented adjoint gradient computation will be applied in two examples. An academic example of an one-mass oscillator shows the easy applicability and a more complex application of a two-arm robot will show the use-case for a multi-objective optimization scenario.