Construction of rigid body motions by geometric continuous rational splines

In this talk we present a geometric approach to interpolate given sequence of rigid body positions (i.e. center positions and orientations), which, in contrast to standard approaches, is free of choosing parameter values in advance and it enables the lowest possible degree of the motion. We introduce some interpolation problems and develop interpolation schemes for the construction of rigid body motions by rational splines of a low degree. A slightly different approach to motion construction, namely motion design with Euler-Rodrigues frames of Pythagorean-hodograph curves is also discussed. Two presented schemes for motions of degree three and five  are particularly useful in the applications, where the orientational component is not precisely specified and an algorithm must be formulated to determine a “natural” variation of orientation along the center trajectory. The theoretical results are substantiated with numerical examples.