Energy optimization of a planar parallel manipulator with kinematically redundant degrees of freedom

In this paper, a planar parallel manipulator with kinematically redundant degrees of freedom is investigated. First of all, the manipulator is illustrated and then formulation for trajectory generation for the end-effector is described and a trajectory is formulated for a special case. Jacobian matrices are obtained by using constraints equations at the next section. After that, inverse dynamics problem is solved by Kane's method and equation of motion is obtained. Then it is shown that for a non-redundant mechanism, there is singularity in the prescribed trajectory of the end-effector and it causes the torques become infinity. After that, energy optimization and singularity avoidance optimization are done for a 6-DOF manipulator and torque and force of all actuators of the manipulator are obtained.