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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 99

3. CONTROLLER DESIGN AND STABILITY ANALYSIS
In the previous section, we modeled and analyzed the dynamics model of the lower limb exoskeleton robot. In
this section, we will design an appropriate controller for the lower limb exoskeleton robot. The design of the
controller is divided into five parts, and the process of controller construction, stability proof, and parameter
selection are explained completely. Firstly, before introducing the fractional order sliding mode surface, the
definitionoffractionalorderneedstobeintroduced. Fractionalorderhastheadvantageofglobalmemorability
and plays an important role in the construction of AFOFTSMC. In the second part, the fractional order fast
terminal sliding mode surface and adaptive terminal control law is proposed to construct an AFOFTSMC
controller for the lower limb exoskeleton robot. In the third part, the stability of the controller is proved in
detail, and it is proved that both the sliding mode variables and the system errors can converge in the bounded
region. The fourth part gives some guiding opinions on the parameter selection of the controller. Finally,
conventional sliding mode control (CSMC) and fast terminal sliding mode control (FTSMC) controllers are
designed to compare with AFOFTSMC.

3.1. Preliminaries
In order to design a discrete adaptive fractional order fast terminal sliding mode controller for the lower limb
exoskeleton system, the sliding mode surface function should be designed according to the properties of the
fractional order operator, and the adaptive sliding mode control law should be designed to form the controller.
Therefore, we will elaborate on the basic definition and related nomenclatures of the fractional operator in
detail.

Definition 1: The Gr¥ unwald–Letnikov fractional order operator is defined as follows [22] :

[( − )/ℎ]

Õ
1 (11)

( ) = (−1) ( − ℎ)
ℎ
=0
where is the arbitrary order of function ( ), the value of will affect the calculus properties of fractional
order operators. When > 0, the fractional order operator is a differentiator, while < 0, the fractional
order operator is an integrator [19] . is the initial value of the integral, and generally, zero initial condition can
be assumed, that is, = 0. ℎ is the sampling time interval, and is the binomial coefficient. The specific
calculation method is as follows:

(

1 = 0
= . (12)
( −1)···( − +1) = 1, 2, 3, . . .
!
However, storing all motion data to calculate fractional integrals in practical engineering applications con-
sumes hardware resources and makes the calculation inefficient. Therefore, to improve the operation efficiency
and ensure the global memory of the fractional operator, the fractional integral can be calculated by storing
part of the motion data, as shown below:

1 Õ
(13)

( ) = (−1) ( − ℎ)
ℎ
=0
where represents the limited amount of data stored.
Lemma 1 [22] : The sum of binomial coefficients in equation (13) can be expressed by gamma function Γ( ) =
¯
∞ − −1
as:
0

Õ Γ( + 1 − )
(14)
(−1) = .
Γ(1 − )Γ( + 1)
=0