For the velocity subsystem of AHV (17), considering the saturation characteristic of [PHI] (11), and adopting the control law (26) and the adaptive law (37) under the premise of Theorem 1, the closed-loop control system is semiglobally

uniformly ultimately bounded.

Then all of the signals in [V.sub.L] are

uniformly ultimately bounded. This completes the proof.

Then, by selecting the design parameters appropriately, the designed control scheme can guarantee that all signals in the closed-loop system are bounded; for example, the signals [mathematical expression not reproducible] are

uniformly ultimately bounded.

By combining backstepping design and two independent Lyapunov functions, a novel adaptive neural control scheme was presented to guarantee all the signals in the closed-loop system are

uniformly ultimately bounded, while this control scheme achieves predefined transient and steady-state tracking control performances concerning the link angular position and velocity tracking errors.

Therefore, synchronization error e and estimation error [PHI] - [??] are

uniformly ultimately bounded. The size of compact set [parallel][PHI] - [??][parallel] [less than or equal to] [[PHI].sub.m]/2 + [square root of ([[PHI].sup.2.sub.m]/4 + p[[zeta].sub.m]/[k.sub.c])] can be minimized by selecting a large value of [k.sub.c] for a given value of p.

Considering the identification model (6), the identification error (8) will be

uniformly ultimately bounded (UUB) if the weights updating laws are as follows:

Thus, [s.sub.2] and [[??].sub.2] are

uniformly ultimately bounded. Assume that [absolute value of [[??].sup.*.sub.2]] [less than or equal to] [[xi].sub.2], where [[xi].sub.2] is positive.

Given any constant [mu] > 0, for any bounded initial conditions satisfying the prescribed performance (8) and V(0) < [mu], there exists design parameters [C.sub.1], [C.sub.2], [C.sub.3], [C.sub.4], [[GAMMA].sub.2], [[sigma].sub.2], [[GAMMA].sub.4], [[sigma].sub.4], [[tau].sub.1], [[tau].sub.2], and [[tau].sub.3], such that the proposed control scheme guarantees that (1) all the signals in the closed-loop system are

uniformly ultimately bounded and (2) the tracking error converges to a small neighborhood around zero with the prescribed performance (8) in a finite time [T.sub.1].

From proposition 2 in [31], [t.sub.f] = [infinity], it can be concluded that all error signals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the closed-loop system are semiglobally

uniformly ultimately bounded.

Then all the signals involved are semiglobally

uniformly ultimately bounded.

Then, the estimate error x is

uniformly ultimately bounded (UUB).

Moreover, it has been proven that the designed controller is able to guarantee that all signals are semiglobally

uniformly ultimately bounded. Finally, simulation verifies the feasibility and effectiveness of the obtained theoretical results.