363 research outputs found
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
Variational integrators are well-suited for simulation of mechanical systems
because they preserve mechanical quantities about a system such as momentum, or
its change if external forcing is involved, and holonomic constraints. While
they are not energy-preserving they do exhibit long-time stable energy
behavior. However, variational integrators often simulate mechanical system
dynamics by solving an implicit difference equation at each time step, one that
is moreover expressed purely in terms of configurations at different time
steps. This paper formulates the first- and second-order linearizations of a
variational integrator in a manner that is amenable to control analysis and
synthesis, creating a bridge between existing analysis and optimal control
tools for discrete dynamic systems and variational integrators for mechanical
systems in generalized coordinates with forcing and holonomic constraints. The
forced pendulum is used to illustrate the technique. A second example solves
the discrete LQR problem to find a locally stabilizing controller for a 40 DOF
system with 6 constraints.Comment: 13 page
Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control
Variational integrators are well-suited for simulation of mechanical systems
because they preserve mechanical quantities about a system such as momentum, or
its change if external forcing is involved, and holonomic constraints. While
they are not energy-preserving they do exhibit long-time stable energy
behavior. However, variational integrators often simulate mechanical system
dynamics by solving an implicit difference equation at each time step, one that
is moreover expressed purely in terms of configurations at different time
steps. This paper formulates the first- and second-order linearizations of a
variational integrator in a manner that is amenable to control analysis and
synthesis, creating a bridge between existing analysis and optimal control
tools for discrete dynamic systems and variational integrators for mechanical
systems in generalized coordinates with forcing and holonomic constraints. The
forced pendulum is used to illustrate the technique. A second example solves
the discrete LQR problem to find a locally stabilizing controller for a 40 DOF
system with 6 constraints.Comment: 13 page
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
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
Rigid body dynamics with a scalable body, quaternions and perfect constraints
In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton-Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrator
A Brief Survey on Non-standard Constraints: Simulation and Optimal Control
In terms of simulation and control holonomic constraints are well documented and thus termed standard. As non-standard constraints, we understand non-holonomic and unilateral constraints. We limit this survey to mechanical systems with a finite number of degrees of freedom. The long-term behavior of non-
holonomic integrators as compared to structure-preserving integrators for holonomically constrained systems is briefly discussed. Some recent research regarding the treatment of unilaterally constrained systems by event-driven or time-stepping schemes for time integration and in the context of optimal control problems is outlined
Nonsmooth Lagrangian mechanics and variational collision integrators
Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.
Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated
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