Adjoint sensitivity analysis of hybrid multibody dynamical systems

Abstract

Sensitivity analysis of multibody systems computes the derivatives of general cost functions that depend on the system solution with respect to parameters or initial conditions. This work develops adjoint sensitivity analysis for hybrid multibody dynamic systems. Hybrid systems are characterized by trajectories that are piecewise continuous in time, with finitely-many discontinuities being caused by events such as elastic/inelastic impacts or sudden changes in constraints. The corresponding direct and adjoint sensitivity variables are also discontinuous at the time of events. The framework discussed herein uses a jump sensitivity matrix to relate the jump conditions for the direct and adjoint sensitivities before and after the time event, and provides analytical jump equations for the adjoint variables. The theoretical framework for sensitivities for hybrid systems is validated on a five-bar mechanism with non-smooth contacts.

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

“…At the same time, by invoking necessary conditions for the extremum of the functional (10), we came up with a set of DAEs, supplied with the relevant boundary conditions, which must be solved backward in time from time instant t = t f to time instant t = 0. Ultimately, solving these equations for adjoint variables allows for evaluation of the gradient of the cost functional (22). The resultant adjoint system (31) has a comparable size to the underlying set of the equation of motion.…”

“…This test example is commonly used by other authors as a benchmark for multibody optimization procedures. 8,22 Different aspects of direct differentiation and adjoint-based approaches will be investigated in this section.…”

“…The magnitudes of time derivatives of h are relatively large when compared to other quantities in the adjoint system, whereas the factor has an impact only on the derivatives of h (cf. Equation (22)). By scaling the cost functional and setting = 0.01, we bring the magnitudes of these derivatives to the range of other quantities appearing in Equation (26).…”

“…The continuous problem might be reformulated then to a discrete setting by employing various numerical integration routines, such as trapezoidal rule 19 or HHT scheme. 20 Recently, the adjoint approach has been extended to cope with multibody systems involving discontinuous effects (impact or contact forces) 21,22 and bodies' flexibility. 23,24 Another important aspect which has significant impact on efficiency and reliability of the sensitivity analysis for large-scale MBS is associated with formulation of equations of motion.…”

Sensitivity Analysis for Hybrid Systems and Systems With Memory

“…Recent work by Corner et.al. [13] treats ASA for hybrid multibody dynamical systems in the context of non-smooth contact dynamics. In [14], the discrete adjoint method (adjoint of the discretized time-dependent problem) was used for computing derivative information for optimization of power system dynamics.…”