Page 169 - IJOCTA-15-4

African vultures optimization-based hybrid neural network–proportional-integral-derivative controller...
and the Coriolis terms of all links are given in Table 1. Three-link rigid robotic manipulator
Equations (30)–(32): parameters

Parameter Nominal value
R 1 = −2L a (M c L c sin (ψ 2 + ψ 3 ) + (ψ 2 + ψ 3 ) L a 0.8 m
˙ ˙
× L b sin (ψ 2 )ψ 1 ψ 2 − 2M c L c (L a sin (ψ 2 + ψ 3 ) L b 0.4 m
L c 0.2 m
˙ ˙
+ L b sin (ψ 3 )ψ 2 ψ 3 − 2 M c L c (L a sin(ψ 2 + ψ 3 )
M a 0.1 kg
˙ ˙ M b 0.1 kg
+ L b sin (ψ 3 ))ψ 1 ψ 3
(30) M c 0.1 kg
2
g 9.81 m/s
˙ ˙
R 2 = −2M c L b L c sin (ψ 3 ) ψ 1 ψ 3
(31) 3. The proposed controller structures
˙ ˙
− 2M c L b L c sin(ψ 3 )ψ 2 ψ 3
This section provides a detailed description of
the proposed hybrid controllers. For simplicity,
˙ ˙
R 3 = 2M c L b L c sin(ψ 3 )ψ 1 ψ 2 (32)
the initial analysis focuses on single-input single-
output systems to facilitate primary demonstra-
The term of potential energy P(ψ) is shown
tion. However, the proposed control structures
in Equation (33):
are readily extendable to MIMO systems. The
strategy of constructing the controllers in this
P = [P 1 P 2 P 3 ] T (33) study is different from that used in the previous
study. 39 Moreover, the current study incorporated
and the potential energy terms of all links are
different control structures than those presented
shown in Equations (34)–(36): 39
previously. The first strategy is represented by
the STNN–PID controllers, where the construc-
tion of this controller represents the collabora-
P 1 = (M a + M b + M c ) gL a cos (ψ 1 )
tion between STNNs and PID processes. In other
+ (M b + M c ) gL b cos (ψ 1 + ψ 2 ) (34) words, the controller nature is still a PID con-
+ M c gL c cos(ψ 1 + ψ 2 + ψ 3 ) troller, equipped with self-tuning technology en-
abled by NNs. The second strategy is represented
by the NN–PID controller, which is a hybrid con-
P 2 = (M b + M c ) gL b cos(ψ 1 troller that merges the NN with PID operations.
(35)
+ ψ 2 ) + M c gL c cos(ψ 1 + ψ 2 + ψ 3 ) The previous study introduced different controller
structures that all belong to the second strategy. 39

P 3 = M c gL c cos(ψ 1 + ψ 2 + ψ 3 ) (36) 3.1. Conventional
proportional-integral-derivative
By applying forward kinematics 39 and using controller with filter
the desired joint angles ψ r1 , ψ r2 , and ψ r3 , the co-
ordinates of the required 3-LRRM’s end-effector Among various control strategies, the PID
can be calculated, as shown in Equations (37) and method remains a widely adopted approach due
(38) for the reference trajectory: to its inherent simplicity and effectiveness. A PID
controller combines three fundamental control ac-
tions: proportional, integral, and derivative. The
x r = L a cos (ψ r1 ) + L b cos (ψ r1 + ψ r2 ) proportional component accelerates the system’s
(37)
+ L c cos (ψ r1 + ψ r2 + ψ r3 ) response, while the integral component works to
eliminate steady-state errors. However, the de-
rivative component is highly sensitive to measure-
y r = L a sin (ψ r1 ) + L b sin (ψ r1 + ψ r2 ) ment noise, which can lead to excessive control ac-
(38)
+ L c sin (ψ r1 + ψ r2 + ψ r3 ) tivity in response to small error fluctuations. To
address this issue, a filter is commonly applied to
where ψ r1 , ψ r2 , and ψ r3 are the angles of the de- the derivative term to suppress noise, thereby en-
sired trajectories, and x r , y r are the desired coor- hancing the robustness and reliability of the con-
dinates of the end effector. Table 1 illustrates the troller's performance.
nominal values of the 3-LRRM parameters used The transfer function of a PID controller in-
in this study. 39 corporating a filter is expressed in Equation (39):
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