Institute for Systems Research Technical Reports

Odometry information alone isnot sufficient for minimizing path length if the terrain is uneven orif it includes obstacles. We generalize existing results on a simplecontrol law, also known as "local pursuit," which is appropriate inthe context of formations and which requires limited interactionbetween vehicles.

Our algorithm is iterative and converges to alocally optimal path. We include simulations and experimentsillustrating the performance of the proposed strategy.

Evaluating the demagnetizing field directly requires work of O(N^2) for a grid of N cells and thus it is the bottleneck in computational micromagnetics. A fast hierarchical algorithm using multipole approximation is developed to evaluate the demagnetizing field. We first construct a mesh hierarchy and divide the grid into boxes of different levels. The lowest level box is the whole grid while the highest level boxes are just cells. The approximate field contribution from the cells contained in a box is characterized by the box attributes, which are obtained via multipole approximation. The algorithm computes field contributions from remote cells using attributes of appropriate boxes containing those cells, and it computes contributions from adjacent cells directly. Numerical results have shown that the algorithm requires work of O(NlogN) and at the same time it achieves high accuracy. It makes micromagnetic simulation in three dimensions feasible.

A hierarchical algorithm using multipole approximation speeds up the evaluation of the demagnetizing field, reducing computational cost from O(N^2) to O(NlogN). A hybrid 3D/1D rod model is adopted to compute the magnetostriction: a 3D model is used in solving the LLG equation for the dynamics of magnetization; then assuming that the rod is along z-direction, we take all cells with same z-cordinate as a new cell. The values of the magnetization and the effective field of the new cell are obtained from averaging those of the original cells that the new cell contains. Each new cell is represented as a mass-spring in solving the motion equation.

Numerical results include: 1. domain wall dynamics, including domain wall formation and motion; 2. effects of physical parameters, grid geometry, grid refinement and field step on H-M hysteresis curves; 3. magnetostriction curve.

First, we consider the problem of computing the controllability energyfunction without numerically solving the family of optimal control problems,or the associated Hamilton-Jacobi-Bellman equation, implied in its definition.Stochastically excited systems play a major role in our methodology. Wepresent a stochastic method for computing an estimate of the controllabilityfunction, and show that in certain situations the method provides an exactsolution. The procedure is tested on applications via Monte-Carlo experiments.

Then, we address the problem of numerically determining a Morse transformationfor a function with non-degenerate critical point at 0. We develop analgorithm for computing the desired nonlinear transformation and estimatingthe neighborhood on which the transformed controllability function isquadratic.

In the literature, examples of applied nonlinear balancing have been limited topseudo-balancing of 2-dimensional gradient systems and noting that in the caseof linear systems the energy functions approach reduces to the usual setting ofgramians. We apply our approach to numerically derive, for the first time,balanced representations of nonlinear state-space models. In particular, wepresent applications to a forced damped pendulum system and a forced dampeddouble pendulum system.

An important feature of the facility we describe is that it uses severaltypes of sensing modalities including position sensing, tactile sensingand more conventional vision sensing. It can interact with objectsof different complexity and is subject to communication constraints arising in a completelynatural and generic way. In constructing it we have used off-the-shelfcomponents wherever possible and made choices with an eye towardflexibility and reliability.

Object-Oriented Programming techniques for solvingBVPs are introduced in this work. Object classes are created toencapsulate trial function sequences, discretized differential andquadrature operators, and other data structures used for spectraldiscretization and projection operations. Operator/functionoverloading subsequently is used to numerically implement theGalerkin projection method. Emphasis is placed on developingnumerical methods suitable for discretizing 2- and 3-dimensionalproblems, integrating the resulting ODE/AE systems in time, andreconstructing the solutions in the physical space. A detailed model of anisothermal carbon-carbon chemical vapor infiltration (CVI) systemwas studied as a true test of the ability of the numerical methods.

Numericalexperiments on a queueing model with high-dimensional parameter vectorsdemonstrate orders of magnitude faster convergence using thesealgorithms over related $(N+1)$-Simulation finite difference analoguesand another two-simulation finite difference algorithm that updates incycles.