The control objective of the vertical plane underactuated manipulator (VPUM) is generally to swing up the end point (EP) from the vertical down equilibrium point (VDEP) to the vertical up equilibrium point (VUEP), and finally, to stabilize it at the VUEP. The zone control methods are usually used to achieve this control objective, but the problem of these methods is that they cannot find a unified zone range, and it is easy to appear singularity in the motion process. In this article, for two-link VPUMs, we propose a control strategy based on trajectory planning and optimization, which does not need to divide the zone range, avoids the singularity problem, and can achieve the control objective quickly and smoothly. We design the trajectory by using the time scale method for the active link according to the initial and target states. Then, we obtain the trajectory parameters by employing differential evolution algorithm to ensure that the passive link can also move from the initial states to the target states simultaneously. We design a sliding mode tracking controller to make the active link move along the designed trajectory, so the EP is moved to VUEP. Meantime, we design a sliding mode stabilization controller to ensure that the system is stable at VUEP. Finally, the universality and robustness of the proposed method are proved by simulations.

For a three-link vertical underactuated manipulator (TVUM) with only one active joint, the control target is to swing up its endpoint from the straight-down equilibrium point (SDEP) and to stabilize the endpoint at the straight-up equilibrium point (SUEP) eventually. Up to now, there are few effective control strategies to achieve the above control target. In this article, we propose an effective control method based on the trajectory optimization to realize the system control target, and the main steps of this article are: 1) a continuous trajectory that consists of two segments with design parameters is planned for the actuated link, along which the actuated link can be swung up from the initial states to the final states; 2) the design parameters are optimized by using the intelligent optimization algorithm to guarantee that the states of the underactuated links are continuous at the junction. In this way, the underactuated links are also moved to their final states with the actuated link simultaneously; 3) a tracking controller is designed by using the sliding-mode method to track the trajectory with optimized design parameters, so the endpoint is swung up from the SDEP to the SUEP directly; and 4) a stabilizing controller is further devised through the LQR method to keep the endpoint being stable at the SUEP. Finally, simulation results show that the proposed control method achieves the swing-up and stable control target of the system, and the control performance of the proposed method is superior than that of the existing control methods through the comparisons.

An iterative contraction stability control strategy is proposed to solve the end point position control problem of the planar prismatic-rotational (PR) underactuated robot. We establish the dynamic model of the planar PR underactuated robot, and use the algebra method to derive the nilpotent approximation model (NAM) with similar characteristics to the original system. According to the robot structure and control objectives, we propose a two-stage control idea: (1) Design a sliding mode variable structure controller to ensure that the active link can back to the target position after subsequent iterative operations; (2) Design an open-loop iterative control strategy based on the previous NAM to control the active link for iterative operation. The underactuated link is indirectly controlled and stabilized at any different accessible position under the action of the coupling relationship between links by controlling the active link. Finally, the simulation results prove that the end point of the robot can be effectively controlled to any positions.

In this paper, we propose a general control method via the intelligent algorithm for a planar R-type underactuated robot. This control method solves the unified control problem of R-type underactuated manipulator. Meanwhile, the proposed method is also applicable to cases of nonzero initial velocity and interference rejection. Our total control program includes two stages. In the first stage, we design the trajectory based on the states of the actuated link, and then the controller is designed to track the planned trajectory to realize the objective of the actuated link. In the second stage, the trajectory with adjustable parameters is planned for the actuated link. Then, the adjustable parameters are calculated by the intelligent algorithm based on the underactuated constraints. Subsequently, the controller is designed to track the second trajectory to realize the objective of the actuated manipulator and the underactuated manipulator. Finally, the performance of the proposed method is verified through simulations.