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Predefined-time fractional-order terminal SMC for robot dynamics
promising approach for precise and resilient con- the robot model, control design of FoPtSMC,
trol of robotic manipulators, particularly in sce- and stability investigations using Lyapunov the-
narios involving uncertainties and disturbances. 23 ory. Section 4 includes simulations to validate our
Subsequent sections will examine further the the- approach, and Section 5 discusses it. In the end,
ory of PtSMC, present simulation results demon- Section 6 provides a conclusion of this work.
strating its effectiveness, and explore its implica-
tions for controlling robotic systems. 2. Preliminaries
Finite-time control methods guarantee state
convergence within a set timeframe, but the sys- The important Lemmas are given in this section.
tem’s initial conditions can influence the speed Lemma 1. For P(t), the Lyapunov function
of convergence. 24,25 Fixed-time SMC ensures con- with an initial value of P(0), predefined stability
vergence within a predefined time, independent is deduced by 29
of the initial state. 26 However, this bound may
˙
not always match real-world settling times, and P(t) ≤ −d 1 P(t) 1+ h 2 − d 2 P(t) 1− h 2
achieving the desired performance can be chal-
where d i > 0 and 0 < h < 1. The predefined-time
lenging for certain systems due to design pa-
T can be obtained as
rameter dependencies. To address these limi-
π
tations, predefined time control allows design- T = p (1)
1 2
ers to specify a desired settling time bound in h d d
advance, offering greater flexibility and poten- Lemma 2. We provide the significant in-
tially faster convergence than fixed-time meth- equalities for k i as 30
ods. Various predefined time control techniques n n 1+h
2
have been developed for nonlinear systems, such P |k i | 1+h ≥ P |k i | 2 , when 0 < h < 1
as predefined time synchronization for chaotic i=1 i=1
n n h
systems using fast TSM controllers and prede- P |k i | ≥ n 1−h P |k i | , when h > 1.
h
fined time parameters. 27 Another example is a i=1 i=1
unique sliding surface design based on a sig- (2)
moid function, ensuring robust predefined-time Definition 1: The fractional-order derivative
convergence for second-order nonlinear systems with function z(t) ∈ R, we have
with matched disturbances. 28 This approach en- Z t
ϱ
ables precise predefined-time contour tracking D z(t) = 1 d z(T) ϱ dT
t
for robotic manipulators without requiring exact Γ(1 − ϱ) dt b (t − T)
knowledge of the robot parameters. 29 where 0 ≤ ϱ < 1, we get the fractional derivative
This study explores predefined-time conver- of signum function as 34
gence control for robotic manipulators in the pres-

ence of uncertainties and disturbances. To the ϱ > 0 when z(t) > 0 and t > 0
D sign(z(t))
best of the authors’ knowledge, there is currently t < 0 when z(t) < 0 and t > 0 .
no literature on the fractional-order control with
predefined-time SMC technique for robotic sys-
tems using the predefined-time lemma. Thus, we
propose a predefined-time fractional-order sliding 3. Design of predefined-time control
mode control (FoPtSMC) scheme, which offers approach
the following:
The equation represents the robotic manipulator
(1) Enhanced trajectory tracking to ob- 30
dynamics utilized in this study :
tain high performance by combining
predefined-time SMC with a fractional-
order control. m(x)¨x + c(x, ˙x) ˙x + g(x) = v(t) + v d + v u (3)
(2) Improved closed-loop system response where x, ˙x, ¨x ∈ R is position, velocity, and ac-
n
through the use of fractional order. celeration, respectively. m(x) ∈ R n×n is pos-
(3) Uncertain dynamics are mitigated using itive definite inertia matrix with the condition
the robust SMC scheme. 0 < ¯x 1 (m(x)) ≤ ∥m(x)∥ ≤ ¯x 2 (m(x)) , where
(4) Guarantee of predefined-time convergence ¯ x 1 and ¯x 2 expresses the min and max eigenval-
of the closed-loop system through Lya- ues. c(x, ˙x) ∈ R n×n denotes the coriolis and cen-
punov stability analysis. tripetal forces, g(x) ∈ R represents the gravi-
n
n
This work is organized as follows: Section 2 tational force vector. In addition, v u ∈ R are
n
consists of the preliminaries. Section 3 presents uncertain dynamics. v(t) ∈ R denotes control
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