2,374 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
High order variational integrators in the optimal control of mechanical systems
In recent years, much effort in designing numerical methods for the
simulation and optimization of mechanical systems has been put into schemes
which are structure preserving. One particular class are variational
integrators which are momentum preserving and symplectic. In this article, we
develop two high order variational integrators which distinguish themselves in
the dimension of the underling space of approximation and we investigate their
application to finite-dimensional optimal control problems posed with
mechanical systems. The convergence of state and control variables of the
approximated problem is shown. Furthermore, by analyzing the adjoint systems of
the optimal control problem and its discretized counterpart, we prove that, for
these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-
C1-continuous space-time discretization based on Hamilton's law of varying action
We develop a class of C1-continuous time integration methods that are
applicable to conservative problems in elastodynamics. These methods are based
on Hamilton's law of varying action. From the action of the continuous system
we derive a spatially and temporally weak form of the governing equilibrium
equations. This expression is first discretized in space, considering standard
finite elements. The resulting system is then discretized in time,
approximating the displacement by piecewise cubic Hermite shape functions.
Within the time domain we thus achieve C1-continuity for the displacement field
and C0-continuity for the velocity field. From the discrete virtual action we
finally construct a class of one-step schemes. These methods are examined both
analytically and numerically. Here, we study both linear and nonlinear systems
as well as inherently continuous and discrete structures. In the numerical
examples we focus on one-dimensional applications. The provided theory,
however, is general and valid also for problems in 2D or 3D. We show that the
most favorable candidate -- denoted as p2-scheme -- converges with order four.
Thus, especially if high accuracy of the numerical solution is required, this
scheme can be more efficient than methods of lower order. It further exhibits,
for linear simple problems, properties similar to variational integrators, such
as symplecticity. While it remains to be investigated whether symplecticity
holds for arbitrary systems, all our numerical results show an excellent
long-term energy behavior.Comment: slightly condensed the manuscript, added references, numerical
results unchange
A Discrete Geometric Optimal Control Framework for Systems with Symmetries
This paper studies the optimal motion control of
mechanical systems through a discrete geometric approach. At
the core of our formulation is a discrete Lagrange-d’Alembert-
Pontryagin variational principle, from which are derived discrete
equations of motion that serve as constraints in our optimization
framework. We apply this discrete mechanical approach to
holonomic systems with symmetries and, as a result, geometric
structure and motion invariants are preserved. We illustrate our
method by computing optimal trajectories for a simple model of
an air vehicle flying through a digital terrain elevation map, and
point out some of the numerical benefits that ensue
Discrete mechanics and optimal control: An analysis
The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
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