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A b s t r a c tThis paper presents a new fonnulation of the artificial potential approach to the motion planning prablem for a mobile robot in a global environment. To model the potential (magnetic) field by Maxwell's Equations that completely eliminate the local minima problem, which is exhibited in most artificial potential methods, such (IS Hannonic functions based methods. However, the proposed model is superior to the Hannonic one because it is easily extendable to 30, the time dimension is modeled by default which means it is a suitable model for time-varying enuironments, computations are less and hence faster, and most important it eliminates the "pat-regions" problem. In this work, electrical currents are assumed to be floating in a cluttered environment with obstacles. The obstacles are assigned zero conductivity whereas the goal point is assigned the highest electrical conductivity. The magnetic field induced by the electric currents is used to find a free path between the start and goal points. Simulation results reflects the ualidity and the potential of the proposed model.

SUMMARYA numerical method is presented for the solution of linear systems of differential equations with initialvalue or two-point boundary conditions. For y ' ( x ) = A(x)y(x) + f(x) the domain of interest [ a, 61 is divided into an appropriate number L of subintervals. The coefficient matrix A(x) is replaced by its value Ak at a poirir XA within the kth subinterval, thus replacing the original system by the L discretized systems yB(x) = Akyk(x) + fk(x), k = 1,2, ._., I,. The fundamental matrix solution @k(x, xk) over each subinterval is found by computing the eigenvalues and eigenvectors of each Ak. By matching the solutions yk(x) at the L -I equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different yk(x). The values of yl(a) and y~( b ) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method, ' the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.

A novel approach to generating optimal smooth piecewise trajectories based o n a new energy memure i s proposed. Given the configurations (position and direction) of two points in the plane, we search for the minimal energy trajectory that minimizes the integral of the squared acceleration opposed to curvature, which has been the predominant energy measure studied in the literature. The smoothness of the optimal trajectory depends on how the tangential and normal components of acceleration vary over a n interval of time. A numerical iterative procedure is devised for computing the optimal piecewise trajectory as a solution of a constmined boundary value problem. The resulting trajectories are not only smooth but also safe wtth optimal velocity (acceleration) profiles and therefore suitable for robot motion planning applications.The feasibility of the proposed approach is illustrated by several simulation examples. Besides motion planning, the resulting trajectories may be useful in wmputer graphics and geometric design.

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