A Virtual Work Based Algorithm for Solving Direct Dynamics Problem of a 3-RRP Spherical Parallel Manipulator

In this paper, we use principle of virtual work to obtain the direct dynamics analysis of a 3-RRP spherical parallel manipulator, also called spherical star-triangle (SST) manipulator (Enferadi et al., Robotica 27, 2009). This manipulator has good accuracy and relatively a large workspace which is free of singularities (Enferadi et al., Robotica, 2009). The direct kinematics problem of this manipulator has eight solution (Enferadi et al., Robotica 27, 2009). Given a desired actuated joint trajectories, we first present an algorithm for selecting the admissible solution. Next, direct velocity and direct acceleration analysis are obtained in invariant form. The concept of direct link Jacobian matrices is introduced. The direct link Jacobian matrix relates motion of any link to vector velocity of actuated joints. Finally, dynamical equations of the manipulator are obtained using the principle of virtual work and the concept of direct link Jacobian matrices. This method allows elimination of constraint forces and moments at the passive joints from motion equations. Two examples are presented and trajectory of moving platform are obtained. Results are verified using a commercial dynamics modeling package as well as inverse dynamics analysis (Enferadi et al., Nonlinear Dyn 63, 2010).