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

where
Ii : moments of inertia for input links relative to axis of rotation of revolute joints
wi : angular velocity of input links; m is the mass of moving platform
.
.
.
x , y,
z : projections of the velocity, of the acting point of the external force
Ix, Iy, Iz : moments of inertia relative to main reference axes of inertia
wx, wy, wz : projections of angular velocity onto the reference axes
mi : masses of translational input links
Vi : linear velocity of translational input links.
The position of the acting point of the external force and angular orientation of the
moving platform are functions of three angular and three linear displacements of input links
as:
x x ϕ , ) y y ϕ , ) z z ϕ , )
= ( i si+
3
= ( i si+
3
= ( i si+
3
α α ϕ , ) β β ϕ , ) γ γ ϕ , )
(5.6)
= ( i si+
3
= ( i si+
3
= ( i si+
3
Differentiation of equation (5.6) wrt. time yields the projections of the velocity for the
acting point of external force and projections of instantaneous angular velocity onto main
reference axes:
where
. ⎡V
⎤ ⎡
⎢ ⎥
=
⎣ ⎦
⎢
x
w ⎣
.
y
.
z
w
x
w
y
⎡ ⎤
⎢ ⎥
=
⎣ ⎦
J
V
w
T
⎤
wz
⎥
⎦
[ ] T
V w
[ ] [ ] T
T
w V = w w w V V V
i
⎡∂p
J = ⎢
⎣∂q
i+
3
∂φ⎤
∂q
⎥
⎦
T
1
2
p ( x,
y,
z)
, φ ( α,
β,
γ ) , q
ϕ , ϕ , ϕ , s , s , s )
3
4
( 1 2 3 4 5 6
5
i
6
i+
3
(5.7)
58