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

The sum of virtual work done by internal reduced moment r m and force f r , for
infinitesimal increments of input parameters (rotational: ϕ , ν = 1,
3,
5
ν
and linear:
ν
a , ν = 2,
4,
6)
in a fixed time interval t, is equal to the work done by external force
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f ( Fx
, Fy
, Fz
) by linear displacements δ x, δy,
δz
and external moment m( M x , M y , M z ) by
angular displacements δα , δβ,
δγ on the moving platform. Following convention will be used
here after for convenience: ϕ ϕi
ν =
13 and a13 = si+
3
ν
13
and all i = 1, 2, 3 unless otherwise specified.
The equation of virtual works of external and internal forces and moments can be
written as follows:
3
∑
i=
1
3
∑
i=
1
M δϕ = F δx
+ F δy
+ F δz
+ F δα + F δβ + F δγ
r,
i
r,
i+
3
i
i+
3
x
y
z
x
F δs = F δx
+ F δy
+ F δz
+ M + M δβ + M δγ (5.1)
x
y
z
y
xδα
y z
When operating the manipulator with multi DOF, the virtual work of all external forces
and moments are consequently determined by varying one of the input parameters with others
held constant:
⎡ ∂x
∂y
∂z
∂α
∂β
∂γ
⎤
x = ⎢
⎥ δϕi
⎣∂ϕ
i ∂ϕi
∂ϕi
∂ϕi
∂ϕi
∂ϕi
⎦
[ ] T
δ δy
δz
δα δβ δγ
[ ] T
δ δy
δz
δα δβ δγ
⎡
x = ⎢
⎥ δsi+
3
∂si+
3 ∂si+
3 ∂si+
3 ∂si+
3 ∂si+
3 ∂si+
3
⎣
∂x
∂y
∂z
∂α
∂β
T
z
∂γ
⎤
⎦
T
(5.2)
Fixing the five input parameters, we get a 1-DOF mechanism and for every
combination of fixed input parameters. The values of generalized moment and force equals to
reduced moments and forces. Using equations (5.1) and (5.2) we have:
3
∂x
∂y
∂z
∂α
∂β
∂γ
∑ M r,
i = Fx
+ Fy
+ Fz
+ M x + M y + M z
i= 1 ∂ϕi
∂ϕi
∂ϕi
∂ϕi
∂ϕi
∂ϕi
3 ∂x
∂y
∂z
∑ Fr,
i = Fx
+ Fy
+ Fz
i= 1 ∂si+
3 ∂si+
3 ∂si+
3
+ M
x
∂α
+ M
∂s
∂β
+ M
∂γ
y
z
i+
3 ∂si+
3 ∂si+
3
(5.3)
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