GENERALIZED VERSION OF DUPORCQ’S EQUATION

Abstract

Ernest Duporcq proposed in 1898 a theorem having as object the rigid-body motions with spherical trajectories. The theorem says that if five points situated on a moving plane  move on five fixed spheres with centers on a fixed plane , then there exists on  a sixth point which also describes such a sphere. More clearly, given five points in  and five points in , then there exists anadditional pair of points which also describes a sphere. The original scientific contribution of this paper consists in a generalized version of this theorem: Given five points on a moving plane and five points in a fixed plane , that are moving on the super-ellipsoidal trajectories  then there exists an additional pair of points which describe such a super-ellipsoidal trajectory. According to this version, the trajectories of a moving body defined by six distance or/and angular constraints between six pairs of points, lines and/or planes, are identified by intersections of two super-ellipsoids. These motions are of interest for simulation, control and performance design of Stewart-Gough platforms in order to prevent their destruction due to the chaotic behavior. The dynamics of such platforms can be extremely complicated due to the riddling bifurcation and the appearance of the hyperchaotic attractors.