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Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05 Page 97

errors to the control system [22] .

In this paper, a novel discrete adaptive fractional order fast terminal sliding mode controller (AFOFTSMC) is
designed for high-precision gait trajectory tracking tasks. To reduce the difference between theoretical design
andpractical application of digital computer systems, the controller designed in this paper derived the discrete-
time object model based on the Lagrange discretization criterion. In addition, to preserve the global memora-
bility of fractional operators, Gr¥ unwald–Letnikov fractional difference operators are used to construct discrete
sliding mode surfaces. Considering the uncertainty of parameters and the boundedness of disturbances, a new
adaptive terminal sliding mode approach law is proposed to drive the sliding mode dynamics to the region
of finite step size. In this paper, the theoretical analysis of the system entering the stable state in finite time
is given, and the validity of the algorithm is tested on the co-simulation platform of MATLAB and Opensim
software.

The rest of this article is structured as follows. The second part describes the lower extremity exoskeleton
discretemodelbasedontheLagrangesystemdiscretecriterion. The design andstability analysisofthediscrete
adaptive fractional order fast terminal sliding mode controller is presented in Section III. In Section IV, the
simulation results are analyzed to prove the effectiveness of the controller. Section V summarizes the thesis.

2. DYNAMICS MODEL OF LOWER LIMB EXOSKELETON
The swing leg dynamic model is considered according to a two-degree-of-freedom (2-DOF) lower limb ex-
oskeleton diagram shown in Figure 1. Based on the motion mechanism of human lower limbs, the hip and
knee joints are designed as active joints, and the ankle joint is designed as a passive joint. The physical pa-
rameters of the 2-DOF lower limb exoskeleton in Figure 1 are explained as follows. (0, 0) represents the
coordinate origin, ( = 1, 2) denotes the angle of the hip or knee joint, ( = 1, 2) represents the distance
between the centroid of thigh or shank segment and the hip joint or knee joint, ( = 1, 2) corresponds to the
length of the thigh or shank segment, ( = 1, 2) denotes the mass of thigh or shank segment.

To achieve high-precision motion control of the lower limb exoskeleton rehabilitation robot. We established a
dynamic model of the lower limb exoskeleton using the Lagrange equation of motion. The equation of general
forms is expressed as follows [23] :

= − (1)

= ( ) − (2)

¤
where denotes the Lagrangian, and represent kinetic energy and potential energy respectively. repre-
sents the torque of the system.

The equations of motion of a lower limb exoskeleton robot are described according to the Lagrange equation
(2):

( ) ¥ + ( , ¤) ¤ + ( ) + ( , ¤, ¥) = (3)

where , ¤ and ¥ denote the joint angle, angular velocity, and angular acceleration vectors respectively, ( ) ∈

2×2 is the positive definite inertia matrix, ( , ¤) ∈ 2×2 is the Coriolis and centrifugal force matrix, ( ) ∈
[3]
2

2×1 is the gravity matrix , ∈ is the torque vector, ( , ¤, ¥) ∈ 2×1 denotes the uncertainty of model

parameters and external disturbances. ( , ¤, ¥) can be expressed as:

( , ¤, ¥) = Δ ( ) ¥ + Δ ( , ¤) ¤ + Δ ( ) − . (4)

The uncertainty of model parameters and external disturbances should be considered in the actual lower limb
exoskeleton dynamics model. Δ ( ) ∈ 2×2 , Δ ( , ¤) ∈ 2×2 , Δ ( ) ∈ 2×1 denotes the uncertain inertia