Trajectory selection for a rigid/flexible manipulator

Dynamic analysis and trajectory planning for a rigid-flexible manipulator is considered here. The manipulator under consideration consists of two links, the first link is rigid while the second link is flexible. Euler-Bernoulli´s beam theory is utilized to model the elastic link where shear deformation and rotary inertia can be neglected if the cross sectional area is small compared to the length of the beam. The tangential coordinate frame is employed to describe the elastic deflection of the flexible link. The equations of motion of the manipulator are derived using the Extended Hamilton´s Principle. The joints trajectory will be designed based on two different trajectories and the joints torques required to move the manipulator joints according to a prescribed trajectory will be obtained through the solution of the inverse dynamics problem. To validate the applicability of soft motion trajectory for flexible manipulators, a comparison with fifth-order polynomial trajectory is carried out for the first three mode shapes of the flexible link.