Adjoint gradient computation for an extremal value of a system output

Extremal values of a system output pose major issues in various disciplines, e.g., the maximum velocity in human-robot collaboration results in high contact forces in the event of a collision, or force and stress peaks cause faster crack growth or fatigue of components. Reducing these extremal values implies a reduction in the risks to humans and an increase in the durability of the components. Therefore, the present paper focuses on minimizing an extremal value of a system output of dynamical system, whereby a gradient-based solution strategy based on the adjoint method is proposed. Since several local extremal values can occur in the time evolution of the system output, it is necessary to apply multi-objective optimization, whereby in particular the largest value is to be minimized. One promising approach in this regard is found in the goal attainment method, which is implemented in the MATLAB function fminimax, or alternatively, the so-called minimax problem can be investigated in a smoothed objective open for any nonlinear programming software package. In the scope of these minimax problems, the maximum reaction force of a one-mass oscillator and the maximum velocity of the tool center point of a two-axis robot during a rest-to-rest maneuver are minimized efficiently using the proposed adjoint gradient.