where a, are n-dimensional column vectors. The Frobenius norm of A is defined by n m m hemultiplying with Q gives Q A = [ Q a l Qaz Qa,] where Qa; are n-dimensional column vectors. This gives IIQAII; = llQa111; + IlQazII; +...+ IIQamll; =IIalII~+IIazII;+...+IIamII; = IIAII'F which means that the Frobenius norm is invariant under orthogonal transformations. A similar argument gives IIQAZIIF = I l A l l~where Z is an orthogonal m-dimensional matrix. [ 5 ] ) for the matrix AThis gives together with the the singular value decomposition (see ,where p = min{m,n} and U and V are orthogonal matrices of. . u p ] and the rest of the matrix is zero.From the above it follows that, for the 3 x 6 matrix R, the Frobenius n o m is uz I . ? REFERENCES Y. Nakamura and H. Hanafusa, "Inverse kinematic solutions with singularity robustness for robot manipulator control," ASME J. Dynam.Absfruct-We describe an algorithm for navigating a polygonal robot, capable of translational motion, in an unknown environment. The environment contains stationary polygonal obstacles and is bounded by polygonal walls, all of which are initially unknown to the robot. The environment is l a m e d during the navigation process by use of a laser range-finding device, and new knowledge is integrated with previously acquired information. A partial map of the environment is thus obtained. The map contains parts of the obstacles that were "seed' by the robot and the free space between them. The obstacles in the map are transformed into a new set of expanded polygonal obstacles. This enables treating the robot as a point instead of a polygon, and the navigation problem is reduced to point navigation among unknown polygonal obstacles. A navigation graph is built from the transformed obstacles in the map. This is a partial visibility graph of the enlarged obstacles. A search is conducted on the graph for a path to the destination. The path is piecewise linear; at its comers, the robot stops, scans its environment, and updates the map, the obstades, and the planned path. The algorithm is proved to converge to the desired destination in a finite number of steps provided a path to the destination exists. If such a path does not exist, then the navigation process terminates in a finite number of steps with the conclusion that the destination is unreachable.