2,425 research outputs found
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
There exist transitive piecewise smooth vector fields on but not robustly transitive
It is well known that smooth (or continuous) vector fields cannot be
topologically transitive on the sphere . Piecewise-smooth vector fields,
on the other hand, may present non-trivial recurrence even on .
Accordingly, in this paper the existence of topologically transitive
piecewise-smooth vector fields on is proved, see Theorem
\ref{teorema-principal}. We also prove that transitivity occurs alongside the
presence of some particular portions of the phase portrait known as {\it
sliding region} and {\it escaping region}. More precisely, Theorem
\ref{main:transitivity} states that, under the presence of transitivity,
trajectories must interchange between sliding and escaping regions through
tangency points. In addition, we prove that every transitive piecewise-smooth
vector field is neither robustly transitive nor structural stable on ,
see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem
\ref{main:general} addressing non-robustness on general compact two-dimensional
manifolds
Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations
An advantageous feature of piecewise constant policy timestepping for
Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation
schemes, and indeed different meshes, can be used for the resulting linear
equations for different control parameters. Standard convergence analysis
suggests that monotone (i.e., linear) interpolation must be used to transfer
data between meshes. Using the equivalence to a switching system and an
adaptation of the usual arguments based on consistency, stability and
monotonicity, we show that if limited, potentially higher order interpolation
is used for the mesh transfer, convergence is guaranteed. We provide numerical
tests for the mean-variance optimal investment problem and the uncertain
volatility option pricing model, and compare the results to published test
cases
Non-Smooth Dynamics and Multiple Equilibria in a Cournt-Ramsey Model with Endogenous Markups
We consider a Ramsey model with a continuum of Cournotian industries where free entry generates an endogenous markup. The model produces two different regimes, monopoly and oligopoly, resulting in non-smooth dynamics. We analyze the global dynamics of the model, demonstrating the model may exhibit heteroclinic orbits connecting multiple equilibria. Small transitory changes in parameters can lead to large permanent effects and there can be a Rostovian poverty trap separating a low-capital and high-markup equilibrium from a high-capital low-markup equilibrium. The paper applies recent results from applied mathematics for non-smooth dynamic systems.Endogenous markups; Non-smooth dynamics; Discontinuity induced bifurcations; Heteroclinic orbits.
Design of of model-based controllers via parametric programming
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