A discrete adjoint gradient approach for equality and inequality constraints in dynamics

Abstract

The optimization of multibody systems requires accurate and efficient methods for sensitivity analysis. The adjoint method is probably the most efficient way to analyze sensitivities, especially for optimization problems with numerous optimization variables. This paper discusses sensitivity analysis for dynamic systems in gradient-based optimization problems. A discrete adjoint gradient approach is presented to compute sensitivities of equality and inequality constraints in dynamic simulations. The constraints are combined with the dynamic system equations, and the sensitivities are computed straightforwardly by solving discrete adjoint algebraic equations. The computation of these discrete adjoint gradients can be easily adapted to deal with different time integrators. This paper demonstrates discrete adjoint gradients for two different time-integration schemes and highlights efficiency and easy applicability. The proposed approach is particularly suitable for problems involving large-scale models or high-dimensional optimization spaces, where the computational effort of computing gradients by finite differences can be enormous. Three examples are investigated to validate the proposed discrete adjoint gradient approach. The sensitivity analysis of an academic example discusses the role of discrete adjoint variables. The energy optimal control problem of a nonlinear spring pendulum is analyzed to discuss the efficiency of the proposed approach. In addition, a flexible multibody system is investigated in a combined optimal control and design optimization problem. The combined optimization provides the best possible mechanical structure regarding an optimal control problem within one optimization.

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

“…which can be verified using (128). As core of this work, we can now rewrite the kinetic energy (38) in terms of unit quaternions and their velocities as…”

“…Note that we have taken into account the fact that Ω is bilinear in 𝕢 and 𝕧, see (88), and that T (Ω) is a quadratic function, see (38). Moreover, the integrator meets the orthogonality conditions (see [20])…”

“…We analyze a nonlinear spring pendulum in three dimensions, which is a popular finite-dimensional model problem for nonlinear elastic mechanical systems (see, e.g. [38]). We consider spherical coordinates for three degrees of freedom…”