On the benefits of surrogate Lagrangians in optimal control and planning algorithms

Abstract

Abstract-This paper explores the relationship between numerical integrators and optimal control algorithms. Specifically, the performance of the differential dynamical programming (DDP) algorithm is examined when a variational integrator and a newly proposed surrogate variational integrator are used to propagate and linearize system dynamics. Surrogate variational integrators, derived from backward error analysis, achieve higher levels of accuracy while maintaining the same integration complexity as nominal variational integrators. The increase in the integration accuracy is shown to have a large effect on the performance of the DDP algorithm. In particular, significantly more optimized inputs are computed when the surrogate variational integrator is utilized.

Variational Integrators for Structure-Preserving Filtering

“…Further theoretical studies of the symplectic filtering approach can be performed by backwards error analysis [7] applied to the state-adjoint system. We also note that recent work [18] has shown that it may be possible to construct arbitrarily high-order variational integrators that are still linearizable one-step maps. These integrators are developed through backward error analysis [7] by constructing surrogate Lagrangians.…”

“…The term L d ðq k ; q kþ1 ; hÞ is referred to as the discrete Lagrangian, and f À k ; f þ k are the left and right discrete forces, respectively [6]. Taking variations of the discrete 1 In the conclusion of this document we mention some recent work [18] that describes the generation arbitrary-order VIs that are still one-step maps. These VIs may provide another choice of integrator that satisfies the aforementioned requirements while also yielding a higher order method.…”

Modified equations for variational integrators applied to Lagrangians linear in velocities

“…We have illustrated the construction of modified Lagrangians and the possible issue of parasitic solutions with examples. Our construction is potentially useful to create more accurate variational integrators, for example in the spirit of [2,3].…”

“…This follows by Legendre transformation from the well-known fact that modified equations for symplectic integrators are Hamiltonian. The construction of a modified Lagrangian was combined in [2,3] with the idea of modifying integrators [1] to construct variational integrators of improved convergence order.…”