Legendre-Gauss Pseudo-spectral Method on Special Orthogonal Group for Attitude Motion Planning

Legendre-Gauss pseudo-spectral method is extended to the special orthogonal group for attitude motion planning of an object in three-dimensional space. First, on the special orthogonal group, a complete left-invariant rigid-body dynamical model for the object in body-fixed frame is established, including kinematical model and dynamical model. Second, for kinematical model, equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of local coordinate map. Considering the Lie algebra space is isomorphism to Euclidean space, Legendre-Gauss pseudo-spectral method is directly used to discretize the dynamical model. Finally, an algorithm for transcribing the attitude motion planning problem is proposed based on LG-PS method on the special orthogonal group. The simulation results illustrate that the proposed method is effective for planning the trajectory of an object rotating in three-dimensional space.