Virtual Passive-Joint Space Based Time-Optimal Trajectory Planning for a 4-DOF Parallel Manipulator

The 4-DOF (3T1R) 4PPa-2PaR parallel manipulator is developed for high-speed pick-and-place operations. However, conventional trajectory planning methods in either active-joint space or Cartesian space have some shortcomings due to its high nonlinear kinematics. Owing to its unique four-to-two leg structure, the middle link that connects to the two proximal parallelogram four-bar linkages in each side only generates 2-DOF translational motions in a vertical plane. By treating each of the middle link as a 2-DOF virtual passive joint, a new trajectory planning method in the 4-DOF virtual passive-joint space is proposed, which not only simplifies the kinematic analysis, but also decreases the kinematics nonlinearity. By introducing the virtual passive joints, both displacement and velocity analyses are readily investigated. The Lagrangian method is employed to formulate the closed-form dynamic model. The quintic B-spline is utilized to generate trajectories in the virtual passive-joint space, while the Genetic Algorithm is implemented to search for the time-optimal trajectory. The simulation results show that the motion time planned in the virtual passive-joint space is decreased by 2.8% and 8.1% compared with the active-joint space method and Cartesian space method respectively. The average and peak jerks of the moving platform are decreased by 14.6% and 37.6% compared with the active-joint space method.