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Page 344 Zhang et al. Intell. Robot. 2025, 5(2), 333-54 I http://dx.doi.org/10.20517/ir.2025.17
while the value of affects the contribution of the nonlinear term to the rate of convergence. Furthermore,
from Lemma 1, it can be concluded that the system’s tracking error can be driven to s = 0 within finite time.
1− +1
1 2 4 2 + 2 2 3
≤ ln (26)
4 (1 − ) +1
2 2 3
Next, Equation (22) is substituted into Equation (6) to derive the sliding motion equations. Subsequently, a
stabilityanalysisoftheslidingmotionisconducted. Giventheconsiderationsof ¤ = 0and = 0, thefollowing
˜
can be derived:
¥ = − 3 ( ) − 4 + 5 sgn( ) (27)
In finite-time control, the objective is to ensure that the error converges to zero within a finite time interval,
rather than merely driving the derivative of the error to zero. Therefore, ¤ = 0 itself is not an attractor. Instead,
the control law is designed such that both the error and its derivative ¤ converge to zero in finite time [46] .
4. SIMULATION
The simulation is distributed into two parts: one is the trajectory planning based on the motion switching
function, and the other part is the control method for tracking the generated trajectory. The control method
simulation uses two passive rehabilitation training control methods: one based on a linear SMC algorithm and
the other based on a non-singular TSMC algorithm with a perturbation observer. The joint tracking error data
were recorded during the training process.
In order to verify the performance of the proposed control framework, a series of simulation experiments
are conducted using MATLAB. A point-to-point arrival task is selected for the experiment, where task points
1 = 0 −0.1 −0.3 0.6 −0.7 and the farthest position 2 = 0.9 4 5.5 5.5 5 .
4.1. System modeling
In this paper, the simulation uses a 5 degrees of freedom (5-DOF) upper-limb exoskeleton model [45] . The
schematic diagram of the exoskeleton robot is shown in Figure 6, which illustrates the structure and key com-
ponents of the device. The robot dynamics can be referred to Equation (4). The details are as follows:
11 12 13 0 0
21 22 23 0 0
( ) = 31 32 33 0 35
0 0 0 44 0
0 0 53 0 55
1 ¤ 2 2 ¤ 3 + 3 ¤ 2 4 ¤ 2 + 5 ¤ 3 0 0
6 ¤ 1 7 ¤ 3 8 ¤ 3 0 0
( , ¤) = 5 ¤ 1 9 ¤ 2 0 0 0
10 ¤ 2 0 11 ¤ 2 0 0
0 12 ¤ 2 0 0 0
( ) = 0 2 3 0 5
4.2. Trajectory planning
Consider a fifth-order polynomial for trajectory planning based on the optimization of the minimum-jerk
principle, where the initial position 1 and the farthest position 2. The motion trajectory path is a reciprocal
motion first from 1 to 2 and then back to 1 as shown in Figure 7. It should be noted that the reference
