Singularity of nonlinear feedback control scheme for robots

The author investigates the singular problem of a robot-control scheme with u sin θ as the orientation error, where u and θ are, respectively, the unit vector of the rotational axis and the rotational angle from the current orientation of the end effector to the desired one. It is rigorously found that the singularity points of the control scheme are at θ=±π/2 and ±π, and, for a step into θ, π/2<θ<π, the orientation error converges to θ=π instead of θ=0. Therefore, the applicable domain of the control scheme is only -π/2<θ<π/2. The theory is also verified by simulations run on the Stanford manipulator