The Theory of Perpendicular Curve in Rotation Group Aiming at Incomplete Orientation Constraint Problems

In many robot applications, the complete orientation constraint in 3D Euclidean space is not necessary to fulfill given tasks. In these tasks, only one direction of the target orientation is constrained. These problems are called incomplete orientation constraint problems. Aiming at such problems, the definition and the analytical expressions are given for the foot of perpendicular and the perpendicular curve of geodesic in rotation group. Based on the perpendicular curve theory, a point-to-point trajectory generating algorithm by following the perpendicular curve is proposed. Finally, trajectory planning of manipulators is solved by simulation experiments. And the PUMA 560 with 6 DOFs (degrees of freedom) is taken as the platform, of which the 6-th and the 4-th joints are fixed successively to get 5-DOF and 4-DOF manipulators. The proposed algorithm can not only be applied to 6-DOF manipulators with function redundancy, but also the 5-DOF or 4-DOF manipulators without function redundancy. The experimental results demonstrate that the proposed method is highly universal in solving incomplete orientation constraint problems. Moreover, the algorithm can avoid the representational singularity problem in parametrization methods.