Control Strategy Based on Model Reduction and Online Intelligent Calculation for Planar $n$ -Link Underactuated Manipulators

This paper presents a control strategy based on model reduction and online intelligent calculation for a planar <inline-formula> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula>-link underactuated manipulator with a passive first joint (PA<inline-formula> <tex-math notation="LaTeX">$^{{n-1}}$ </tex-math></inline-formula> for short) to realize its control objective, which is to move its end-point from an initial position to a target position. First, two active links of the planar PA<inline-formula> <tex-math notation="LaTeX">$^{{n-1}}$ </tex-math></inline-formula> manipulator are chosen to be the active links of a planar virtual passive–active–active (PAA) manipulator, which guarantees that the geometric reachable range of the planar virtual PAA manipulator is the same as that of the planar PA<inline-formula> <tex-math notation="LaTeX">$^{{n-1}}$ </tex-math></inline-formula> manipulator. The planar PA<inline-formula> <tex-math notation="LaTeX">$^{{n-1}}$ </tex-math></inline-formula> manipulator is reduced to the planar virtual PAA manipulator by keeping the states of two active links in their initial values and controlling the states of the remaining <inline-formula> <tex-math notation="LaTeX">${n-3}$ </tex-math></inline-formula> active links to zero. Then, the planar virtual PAA manipulator is equivalent to two planar virtual Acrobots by adopting two-stage control method. Based on two sets of angle constraint relationships corresponding to two planar virtual Acrobots, an online differential evolution algorithm is employed to obtain all link target angles of the planar virtual PAA manipulator. Next, two Lyapunov functions, each of which is constructed based on the active link angle of one planar virtual Acrobot, are used to design controllers to realize the system control objective. Finally, Simulation results of a planar PAA-active manipulator demonstrate the effectiveness of the proposed control strategy.