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Page 336 Zhang et al. Intell. Robot. 2025, 5(2), 333-54 I http://dx.doi.org/10.20517/ir.2025.17

sliding-mode dynamics [37] . Wen et al. proposed an adaptive sliding-mode control scheme [38] . The mismatch
uncertainty considered by the above methods must be H2 paradigm bounded, which is an unreasonable as-
sumption for practical systems. Utkin et al. proposed the integral SMC (I-SMC), which accumulates the error
by an integral term so as to keep the error of the system zero even in the steady state, which can effectively
suppress the effect of external perturbations on the system and reduce the jitter [39] . However, all of the above
methods deal with the mismatch uncertainty by degrading the performance. Wei et al. incorporate the syner-
gistic effect of SMC feedback and feedforward compensation based on disturbance estimation to construct a
control law that improves the robustness and dynamic performance of the system [40] .

This study provides insights into exoskeleton control methods using a non-singular fast terminal sliding mode
controller (NFTSMC) based on the nonlinear disturbance observer (NDO) framework. The main contribu-
tions of this manuscript are summarized below:
(1) A trajectory planning method based on polynomial interpolation and optimization of the minimum jerk
model has been proposed for exoskeleton rehabilitation robots. This method can generate a desired trajec-
tory that conforms to the natural motion characteristics of the human body, achieving smooth and natural
joint movements during rehabilitation training;
(2) A motion-dependent switching function has been designed to achieve a smooth transition between normal
trainingmodeandsafestopmode. Thisfunctioniscrucialforprotectingpatientsafetyduringrehabilitation
training and avoiding discomfort or injury caused by sudden changes;
(3) Non-singularfastterminalslidingmode(NFTSM)controlbasedonNDO:ANFTSMcontrolmethodbased
on a NDO has been proposed for precise tracking of the desired joint angles, with the aim of achieving a
tracking error that tends to zero in a finite time. This method is robust against unknown bounded external
disturbances and addresses the singularity and chattering issues associated with traditional SMC.

The rest of the paper is organized as follows: Section 2 describes the dynamic model of the upper limb reha-
bilitation robot. Section 3 describes the robot-assisted rehabilitation control strategy, and Section 4 validates
the designed control algorithm through numerical simulation. Finally, Section 5 discusses and concludes this
study.

2. PROBLEM FORMULATION
2.1. Notations, definitions, and lemmas

Notations: R, R , R × represent the set of real numbers, dimensional vectors space, and × real ma-
trices space, respectively. The maximum and minimum eigenvalues of the matrix Φ ∈ R × are defined by

max (Φ) and min (Φ), respectively. For ∈ R , denote ( ) = ( ( 1 ) , . . . , ( )), ( ) =

sgn ( ) | | ( = 1, . . . , ), and the signum function is given by
 1, > 0

 
sgn ( ) = −1, < 0 (1)

 0, = 0

Lemma 1 [41] Assume that there exists a continuous positive definite function ( ) which satisfies the following
inequality:

¤
( ) + ( ) + ( ) ≤ 0 (2)
Subsequently, the function ( ) approaches the equilibrium position within a finite time :
1 1− ( 0 ) +
≤ ln (3)
(1 + )
where > 0, > 0 and 0 < < 1.