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

Thus, according to Lagrange, the reduced moments and forces equals to the sum of
possible work done by external forces and moments, applied to the moving platform when
varying just one input parameter.
5.2 Equation of Motion of the Manipulator
Let’s consider the equation of motion for the manipulator in figure 4.1. The Lagrange-
Euler’s equation of motion of second order will be setup in the following form:
where
d
dt
d
dt
⎛ ∂T
⎞ ∂T
⎜ − = M
w ⎟
⎝ ∂ i ⎠ ∂ϕi
o ⎛
⎜ ∂T
⎜ ∂V
⎝
i+
3
o ⎞
⎟ ∂T
⎟
−
∂s
⎠
i+
3
r,
i
= F
r,
i+
3
T is the sum of kinetic energy of the platform and rotating input links;
o
T is the sum of kinetic energy of the platform and the translating input links.
(5.4)
The kinetic energy of the system will be determined on the assumption that all links
are counterbalanced and inertia masses and moments of each branch can be neglected since
they are located symmetrically relative to the moving platform. The reference axes are chosen
so that all mass moments of inertia are equal to zero.
The equation of kinetic energy for the moving platform and for input links with linear
and angular parameters of manipulator is formed as:
2T
o
=
2T
=
3
∑
i=
1
3
∑
i=
1
I
m
i
w
2
i
2
i+
3 Vi+
3
+ m(
x
.
2
+ m(
x
+ y
.
2
.
2
+ y
+ z
.
2
.
2
+ z
) + I
.
2
x
) + I
w
x
2
x
w
+ I
2
x
y
+ I
w
y
2
y
w
+ I
2
y
z
w
z
2
z
+ I w
2
z
(5.5)
57