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 kine- matics 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 dy- namic model. The quintic B-spline is utilized to generate trajecto- ries in the virtual passive-joint space, while the Genetic Algorithm is implemented to search for the time-optimal trajectory. The sim- ulation 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.