Modeling and sensitivity analysis methodology for hybrid dynamical systems

Adjoint sensitivity analysis of hybrid multibody dynamical systems

“…Let Z| + t eve and Z| − t eve , with Z ∈ R p , be the sensitivities of the quadrature variable z(t) after and before the event, respectively. It is shown in [17] that, at the time of the event, we have:…”

“…Remark 3. Formalisms for constrained multibody systems may involve Lagrangian coefficients µ : R 1+2n+p → R m that provide the necessary forces to satisfy the kinematic constraints [17]. Our notation encompasses the case where the cost function penalizes the accelerations v and the joint forces via the Lagrangian coefficients µ:…”

“…F q , F v , and F ρ are given in [17]. Similarly, one can compute the adjoint sensitivities of the penalty formulation (31) using (19).…”

“…Performing a sensitivity analysis for a constrained rigid hybrid multibody dynamic system requires finding the jump conditions at the time of event. These jump equations are explained in our previous work [17]. We summarize below the jump equations at the time of event:…”

“…The framework includes both unconstrained and constrained mechanical systems. The direct sensitivity analysis was developed in [17], where a new graphical proof of the jump conditions for direct sensitivity variables was given. Jump conditions for constrained mechanical systems with change of mechanism and dealing with impulsive forces at the time of event was also presented.…”