Feasible Trajectories in Task Space from a Singularity for a Nonredundant or Redundant Robot Manipulator

From a singularity with deficiency of rank 1, for any joint velocity the effector velocity in task space is zero along a special direction (the singular direction). However, any motion along the m -1 non singular directions can be produced. Thus, any feasible trajectories from the singularity and with nonzero initial velocity is orthogonal to the singular direction. The feasibility of an effector acceleration along the singular direction is studied. A necessary condition for existence of an effector acceleration along the singular direction is that the joint velocities have to appear explicitly in the expression of this acceleration. This condition is not sufficient, since the joint velocities are related to the effector velocity along the nonsingular directions as well. A simplified expression is found for the effector acceleration along the singular direction whose analysis allows one to characterize the feasible trajectories from the singularity with a nonzero initial acceleration. From some singularities, any path can be tracked. In other cases, the tangent and curvature to the path at the singular point are not arbitrary. In case any path cannot be tracked, it may be interesting to consider trajectories with zero initial acceleration and nonzero initial jerk. Some special examples are studied. A classification of singularities is displayed, and the set of all feasible trajectories is defined in the task space for each type of singularity Various examples are provided. Redundant and nonredundant robots are studied.