Page 168 - IJOCTA-15-4

Oleiwi et al. / IJOCTA, Vol.15, No.4, pp.706-727 (2025)

Figure 1. Three-link rigid robotic manipulator model

And the Euler–Lagrange expression is written
as in Equation (13): 2 2 2
Q 22 = M b L + M c L + M c L + 2M c L b L c cos(ψ 3 )
c
b
b
(20)

d ∂LD ∂LD
− = Fψ i (13)
˙
dt ∂ψ i ∂ψ i Q 23 = M c L + M c L b L c cos(ψ 3 ) (21)
2
c
where Fψ i or τ i is the control torque applied to
2
the i-th link. Q 31 = M c L + M c L a L c cos(ψ 2
c
(22)
The robotic manipulator dynamics are given + ψ 3 ) + M c L b L c cos(ψ 3 )
39
in Equations (14)–(24) : 2
Q 32 = M c L + M c L b L c cos(ψ 3 ) (23)
c
¨
˙ 2
˙ ˙
Q(ψ)ψ + F(ψ, ψ ) + R(ψ, ψ i ψ j ) + P(ψ) = τ Q 33 = M c L 2 c (24)
(14)
The term of centrifugal forces F ψ, ψ ˙ 2 is
where Q(θ) is the inertia matrix.
shown in Equation (25):
 
Q 11 Q 12 Q 13 T
Q = Q 21 Q 22 Q 23  (15) F = [F 1 F 2 F 3 ] (25)

Q 31 Q 32 Q 33 The centrifugal forces of all links are given in
Equations (26)–(28):
2
Q 11 = (M a + M b + M c ) L + (M b + M c ) L 2
a
b
F 1 = −L a (M c L c sin (ψ 2 + ψ 3 ) + M b L b sin (ψ 2 )
2
+ M c L + 2M c L a L c cos (ψ 2 + ψ 3 ) + 2(M b
c
˙ 2
+ M c L b sin (ψ 2 ) ψ −M c L c (L a sin (ψ 2 + ψ 3 )
+ M c )L a L b cos (ψ 2 ) + 2M c L b L c cos(ψ 3 ) 2
(16) + L b sin (ψ 3 ))ψ ˙ 2
3
(26)
2
2
Q 12 = (M b + M c ) L + M c L + M c L a L c cos(ψ 2 + ψ 3 )
c
b
F 2 = L a (M c L c sin (ψ 2 + ψ 3 ) + M b L b sin (ψ 2 )
+ (M b + M c ) L a L b cos (ψ 2 ) + 2M c L b L c cos(ψ 3 )
˙ 2
(17) + M c L b sin (ψ 2 )ψ − M c L b L c sin(ψ 3 )ψ ˙ 2
1
3
(27)
2
Q 13 = M c L + M c L a L c cos (ψ 2 + ψ 3 )
c
(18) F 3 = M c L c (L a sin (ψ 2 + ψ 3 ) + L b sin (ψ 3 )) ψ ˙ 2
+ M c L b L c cos(ψ 3 ) 1
+ M c L b L c sin(ψ 3 )ψ ˙ 2 2
2 2 2 (28)
Q 21 = M b L + M c L + M c L + M c L a L c cos(ψ 2
b
b
c
˙ ˙
The term of Coriolis R(ψ, ψ i ψ j ) is shown in
+ ψ 3 ) + M b L a L b cos (ψ 2 ) + M c L a L b cos (ψ 2 ) Equation (29):
+ 2M c L b L c cos(ψ 3 )
(19) R = [R 1 R 2 R 3 ] T (29)
710