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Zhang et al. Intell. Robot. 2025, 5(2), 333-54 I http://dx.doi.org/10.20517/ir.2025.17 Page 341

where is a defined auxiliary variable for the convenience of NDO design and 1, 2 are the gain matrixes to
be designed, 1 = { 11 , . . . 1 }, 2 = { 21 , . . . 2 }, 3 = { 31 , . . . 3 } and 0 < < 1.

Theorem 1: In consideration of the exoskeleton system Equation (5), the DOC is designed in accordance
with Equation (14). Subsequently, the disturbance approximation error of the proposed DOC is shown to be
convergent in a finite time.

Proof: Let us define = − . Then, from Equations (5) and (14) it follows that:
˜
ˆ

˜ −1 −1

= − 1 − 2 ( ) − 3 sgn ( ) − ¥ − ¤ + +

= ¤ − ¥ (15)
= ¤
Therefore, it can be concluded that the convergence of the disturbance error is consistent with the convergence
of the auxiliary variables 1. Choose the Lyapunov function candidate as:

1
1 = (16)
2
The time derivative of 1 is given by

1 = ¤
¤

= (¤ − ¥)

−1

= ¤ + −1 ¤ + − −
(17)
= (− 1 − 2 ( ) − 3 sgn ( ) − )

≤ − 1 − 2 ( ) − 3 k k + k k k k

≤ − 1 − 2 ( )
+1 +1
≤ −2 1 1 − 2 2 2 1 2
where 1 = min ( 1 ) and 2 = min ( 2 ), 3 = min ( 3 ) and 3 ≥ k k.

According to Lemma 1 , the auxiliary variable 1 converges to the equilibrium point in the finite time 1 defined
by
1− +1
1 2 1 2 + 2 2 2
1 ≤ ln (18)
1 (1 − ) +1
2 2 2
In light of the finite time convergence property of the auxiliary variable 1, it can be concluded that the distur-
bance approximation error of the proposed disturbance observer is convergent in a finite time.

3.3. NFTSMC design
Furthermore, a NFTSM control method based on NDO is proposed as a means of enhancing the performance
of the control system. In the NFTSMC design, the singular value problem is circumvented and rapid conver-
gence is attained through a modified terminal sliding mode surface. The overview of the NFTSMC framework
is illustrated in Figure 5, which provides a detailed view of the structure and key components of the control
strategy.