129 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
The power dissipation method and kinematic reducibility of multiple-model robotic systems
This paper develops a formal connection between the power dissipation method (PDM) and Lagrangian mechanics, with specific application to robotic systems. Such a connection is necessary for understanding how some of the successes in motion planning and stabilization for smooth kinematic robotic systems can be extended to systems with frictional interactions and overconstrained systems. We establish this connection using the idea of a multiple-model system, and then show that multiple-model systems arise naturally in a number of instances, including those arising in cases traditionally addressed using the PDM. We then give necessary and sufficient conditions for a dynamic multiple-model system to be reducible to a kinematic multiple-model system. We use this result to show that solutions to the PDM are actually kinematic reductions of solutions to the Euler-Lagrange equations. We are particularly motivated by mechanical systems undergoing multiple intermittent frictional contacts, such as distributed manipulators, overconstrained wheeled vehicles, and objects that are manipulated by grasping or pushing. Examples illustrate how these results can provide insight into the analysis and control of physical systems
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