2,471 research outputs found
On the Benefits of Surrogate Lagrangians in Optimal Control and Planning Algorithms
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
Contact variational integrators
We present geometric numerical integrators for contact flows that stem from a
discretization of Herglotz' variational principle. First we show that the
resulting discrete map is a contact transformation and that any contact map can
be derived from a variational principle. Then we discuss the backward error
analysis of our variational integrators, including the construction of a
modified Lagrangian. Throughout the paper we use the damped harmonic oscillator
as a benchmark example to compare our integrators to their symplectic
analogues
Geometric, Variational Integrators for Computer Animation
We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important
computational tool at the core of most physics-based animation techniques. Several features make this
particular time integrator highly desirable for computer animation: it numerically preserves important invariants,
such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy
behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite
simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key
properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during
an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a
factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the
implementation of the method. These properties are achieved using a discrete form of a general variational principle
called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate
the applicability of our integrators to the simulation of non-linear elasticity with implementation details
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