Adjoint sensitivity index-3 augmented Lagrangian formulation with projections

Abstract

The interest on closed-form analytical sensitivity equations based on the classical forward dynamics formulations, started shortly after the equations were ready and became popular. A vast effort was devoted by the authors in the last years to derive the forward and adjoint sensitivity equations for some state-of-the-art formulations of practical interest. Recently, the forward sensitivity equations for the index-3 augmented Lagrangian formulation with projections (the ALI3-P formulation) have been derived. In this article, the ALI3-P adjoint sensitivity equations are derived and implemented in the MBSLIM library as a general code and tested in one academic example and one real-life multibody system.

Keywords:

• Index-3 augmented Lagrangian method
• multibody system dynamics
• projections
• sensitivity analysis

Conflict of interest

The authors declare that they have no conflict of interest.

Notes

1 This happens if any coordinate does not have associated mass, for example, if additional coordinates are defined in terms of others via constraints

2 Since the “initial conditions” are given at the final time, they can actually be considered terminal conditions.

3 Since $\mathit{\psi }\prime ={\int }_{t_0}^{t_F}\stackrel{.}{\mathit{\psi }}\prime \mathrm{d}t=-{\int }_{t_F}^{t_0}\stackrel{.}{\mathit{\psi }}\prime \mathrm{d}t,$ the forward integration of the adjoint can be done backward with the same expression, taking advantage of the adjoint backward loop, provided a time reversible time-stepping formula like the trapezoidal rule proposed in this work.

4 An equivalent condition is $\text{ker} \mathbf{M} \cap \text{ker} {\mathbf{\Phi }}_{\mathbf{q}}=\mathbf{0}$ which phisically means that any compatible motion cannot be associated with zero kinetic energy.

5 Unlike the index-1 equations in Dopico, Sandu, et al. (2014), the ones proposed here are converted to first order ($\dot{\mathbf{q}}=\mathit{v}$) in order to avoid high order derivatives in the adjoint equations and the explicit dependencies of the objective function on accelerations are removed, in order to avoid incompatible functionals at the final time.

6 This is a model with redundant constraints, therefore the only solvable adjoint system is the one designed to take into account redundant constraints.

7 The direct sensitivity is calculated forward in time while the adjoint system and gradient are solved backward in time.

Additional information

Funding

This study was funded by the Spanish Ministry of Economy and Competitiveness (MINECO) under project DPI2016-81005-P.