Single controller design based on integrated trajectory for three-link vertical underactuated manipulators with first active joint

For a three-link vertical underactuated manipulator with first active joint, few methods can swing its endpoint up from the vertical downward equilibrium point and stabilise it at the vertical upward equilibrium point. This paper proposes a single controller design method based on integrated trajectory to realise the above control objective effectively. To obtain the trajectories of each link from the starting states to ending states, we first plan a forward and a reverse trajectory with adjustable parameters for the active link, where the forward trajectory is from the starting states to intermediate states and the reverse one is from the ending states to intermediate states. Based on the system coupling characteristic, the adjustable parameters are optimised by the intelligent optimisation algorithm, so getting the trajectories of the passive links from the starting states to ending states. To ensure that the system with the gravity can remain at the ending states, we plan the constant trajectory for each link. Then, we integrate the optimised forward trajectory, the optimised reverse trajectory and the constant trajectory of each link into the continuous trajectory. Next, according to the integrated trajectory, we construct a linear time-varying error system, and design the single controller to make the system follow the integrated trajectory. Thus, the swing-up and stable control of the system is realised. Finally, simulation results demonstrate the validity of the control method.