Asymptotic Stabilization Control Based on Trajectory Optimization for Vertical Underactuated Manipulators with the First Joint Actuator

Underactuated system control is a central topic in nonlinear system control. For the three-link vertical underactuated manipulator with only the first joint actuated (APP manipulator), the control objective of swing-up and balancing is challenging. The advantages of this paper are as follows: (i) The proposed method avoids balancing region division in common partitioned control, preventing failure to stabilize at the target position due to improper partitioning. (ii) The time-based switching condition optimized via parameter tuning is easier to satisfy than the state-based condition. (iii) The proposed controller effectively suppresses state fluctuations caused by switching, yielding a smoother transition. (iv) The proposed controller avoids the singularity problem. The main procedures are as follows. First, the dynamic model of the APP manipulator is established. Then, a trajectory is designed to guide the active link from the initial position to the vicinity of the target position. On this basis, to ensure that all links can simultaneously reach the vicinity of the target position, the trajectory parameters are optimized according to the coupling relationship between the links. Next, an NFTSM-based tracking controller is developed to steer the links along the optimized trajectory. After that, an LQR-based stabilization controller is further employed to lock the system at the target position. Finally, the effectiveness of the proposed method is verified through simulations.