Multimicroprocessor-based cartesian space control techniques for a mechanical manipulator

Rotation Due to the rotation property of the two-dimensional Fourier transform rotation of f (x , y) through an angle & also rotates the spectrum by the same amount. It is also well known that the Fourier spectrum (modulus) is invariant to translation. Now, suppose that we scan the Fourier spectrum IF,(along a circle centered at the origin. If circular sampling of 1 F,I * is used, then by taking the Fourier transform of these samples, extracting the Fourier phase, and subtracting this phase from the reference phase, the rotation angle q50 can be obtained. In general, few circles may be required to obtain unique and accurate value for the rotation angle. The number of circles can be minimized if the radius and the corresponding samples are properly selected. The method can be summarized as follows. Step I: For each specific choice of radius pi, extract K proper samples over the circle centered at the origin with radius pi, where p , = (U * + U 2) " 2 .