Page 56 - IJOCTA-15-3

S. Ahmed et.al. / IJOCTA, Vol.15, No.3, pp.426-434 (2025)
n
torque, and v d ∈ R is bounded external distur- robotic manipulators. This control law is de-
bances. signed to be resilient to disturbances and uncer-
Equation (3) is provided as follows: tainties.
¨ x = m −1 (x)(v(t) + v d + v u − c(x, ˙x) ˙x − g(x)) 3.2. Design of FoPtSMC scheme
(4)
The control law FoPtSMC is designed to ensure
The tracking error is given as:
the robust operation of perturbed robotic manip-
¨ ε = m −1 (x)(v(t) + v d + v u − c(x, ˙x) ˙x − g(x)) − ¨x d ulators under bounded uncertain dynamics:
(5)  
m −1 (x)(c(x, ˙x) ˙x + g(x)) + ¨x d
where ε = x − x d is tracking error, and x d is de- 2
 −(℘ 1 + ℘ 2 ∥x∥ + ℘ 3 ∥ ˙x∥ )sign(σ(t)) 
sired input.  η η η − η 


This section examines the development of an  −a 1 (1 + )|ε| 2 ˙ε − a 2 (1 + )|ε| 2 ˙ε 
2
2
 −a 3 D
v(t) = m(x)  α+1 sign(ε) 
innovative control strategy for perturbed robotic µ 

manipulators in the existence of uncertainty.  −b |σ(t)| 1+ 2 sign(σ(t)) 
1

First, it analyzes the key characteristics of the −b |σ(t)| 1− µ
2 sign(σ(t))
2
proposed predefined-time fractional-order sliding (10)
mode control (FoPtSMC) scheme. where |ε| − η 2 = 0 if ε = 0, b 1 , b 2 are positive con-
stants, and 0 < µ < 1.
By substituting v(t) into (9), ˙σ(t) can be com-
3.1. Predefined-time based puted as
fractional-order sliding surface η η
η
η
˙ σ(t) = a 1 (1 + )|ε| 2 ˙ε + a 2 (1 − )|ε| − 2 ˙ε
2
In light of SMC’s rapid convergence and robust- +a 3 D α+1 sign(ε) 2
ness, researchers have explored various sliding   m −1 (x)c(x, ˙x) ˙x  
surfaces to enhance control performance. The −1
  +m (x)g(x)  
demonstrated ability of the recommended sliding   2  
  −(℘ 1 + ℘ 2 ∥x∥ + ℘ 3 ∥ ˙x∥ )  
surface to deliver precise predefined-time control    
  ×sign(σ(t))  
for n-degree-of-freedom (DOF) robotic manipula-   η  
η
  −a 1 (1 + )|ε| 2 ˙ε  
tors makes it particularly advantageous. Conse- +m −1  m(x)  2 η − η  

quently, the proposed sliding surface is formulated (x)   −a 2 (1 + )|ε| 2 ˙ε  
2
 

  −a 3 D α+1  

as  sign(ε) + ¨x d  
 −b |σ(t)| 2 sign(σ(t))
  1+ µ  
 1  
η
η
σ(t) = a 1 |ε| 1+ 2 sign(ε) + a 2 |ε| 1− 2 sign(ε) (6)  −b |σ(t)| 1− µ 2 sign(σ(t)) 


2
α
+a 3 D sign(ε) + ˙ε −c(x, ˙x) ˙x − g(x)
n
where sliding surface is represented by σ(t) ∈ R , +℘(x, ˙x, ¨x) − ¨x d
a 1 , a 2 , a 3 ∈ R + are positive constants, and (11)
Simplifying (11), ˙σ(t) is expressed as
0 < η < 1.
2
Then, ˙σ(t) can be obtained as ˙ σ(t) = −(℘ 1 + ℘ 2 ∥x∥ + ℘ 3 ∥ ˙x∥ )sign(σ(t))
µ
1+
+℘(x, ˙x, ¨x) − b |σ(t)| 2 sign(σ(t))
η
η
1
η
η
˙ σ(t) = a 1 (1 + )|ε| 2 ˙ε + a 2 (1 − )|ε| − 2 ˙ε −b |σ(t)| 1− µ 2 sign(σ(t))
2
2
+a 3 D α+1 sign(ε) + ¨ε (7) 2
(12)
From (5) into (7) yields We have finalized the design of the control scheme
η
η
η
˙ σ(t) = a 1 (1 + )|ε| 2 ˙ε + a 2 (1 − )|ε| − η 2 ˙ε and sliding surface, and we are now ready to move
2 2
+a 3 D α+1 sign(ε) + m −1 (x)(v(t) + v d + v u (8) forward with the stability analysis. The complete
proposed diagram is given in Figure 1.
−c(x, ˙x) ˙x − g(x)) − ¨x d
Equation (8) can be given as
η
η
η
˙ σ(t) = a 1 (1 + )|ε| 2 ˙ε + a 2 (1 − )|ε| − η 2 ˙ε 3.3. Stability analysis
2
2
+a 3 D α+1 sign(ε) + m −1 (x)(v(t) − c(x, ˙x) ˙x − g(x))
In this section, we conduct an analysis of the
+℘(x, ˙x, ¨x) − ¨x d
closed-loop system’s stability using the Lyapunov
(9)
where ℘(x, ˙x, ¨x) = m −1 (x)(v u +v d ), it is bounded theorem. Equation (3) is utilized to represent the
2 system dynamics.
by ∥℘(x, ˙x, ¨x)∥ ≤ ℘ 1 +℘ 2 ∥x∥+℘ 3 ∥ ˙x∥ , and ℘ 1 , ℘ 2
If σ(t) = 0 in (6), one has
and, ℘ 3 > 0.
η 1− η
1+
Using the provided sliding surface, we will ˙ ε = −a 1 |ε| 2 sign(ε) − a 2 |ε| 2 sign(ε) (13)
α
now develop the FoPtSMC method for n-DOF −a 3 D sign(ε)
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