Kinematic and Dynamic Analysis of Spatial Six Degree of Freedom ...

When we differentiate the equation of kinematic energy (5.8) wrt. the angular and
linear velocity of input links and impose simple mathematical transformation, we obtain:
∂T
∂w
o
∂T
∂V
i
i+
3
=
=
3
∑
k=
1
3
∑
k=
1
( I w + 0.
5I
V )
k,
i
⎛
⎜
⎜0.
5
⎝
k
o
I k,
i+
3
w
k
k,
i+
3
+
i+
3
o
I k+
3,
i+
3
V
k+
3
⎞
⎟
⎠
(5.10)
Differentiating (5.10) wrt. time and substituting those differentiations into Lagrange-
Euler’s equation (5.4) along with equation(5.9) we obtain:
M
r,
i
=
1
3
3
3
3
k , i
k+
3,
i+
3
ε I k , i+
3ai
+ 3 + ∑ wk
wi
− ∑ Vk
+ 3Vi
+ 3
k= 1 2 k=
1 2 k = 1 ∂ϕi
2 k= 1 ∂ϕi
∑Ik, i k + ∑
1
o
o
3
3
1 o
o
3
3
1 ∂ I k + 3,
i+
3 1 ∂ I k,
i
Fr , i+
3 = ∑Ik, i+
3 ε k + ∑Ik+
3,
i+
3 ak
+ 3 + ∑ k+
3Vi+
3 − ∑
2 k= 1
k=
1
2 k= 1 ∂si+
3 2 k = 1 ∂si+
3
∂I
1
∂I
V w w (5.11)
for i = 1,2,3
Equations (5.11) are the equations of motion for the 6-DOF manipulator with parallel
structure having variable inertia factors, depending on the input angular and linear
displacements of six input links.
k
i
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