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

where
W = ∑
i=
M
mi
1
– λ(N – C – B) + q
mi: independent scalar variable of relative joint displacement,
M: total number of independent scalar displacement variable,
λ: total number of independent, scalar loop closure equations,
q: number of redundant connections.
(3.2)
Linking each magnitude of displacement variable mi to one DOF relative joint motion
fi and taking q = 0 in equation (3.2) we get:
j
W = ∑ =
i
1
fi
– λ(N – C – B)
where j is the number of joints connecting n links (j = n – 1).
(3.3)
Using the principle of interchangeability of kinematic pairs, we can describe our
manipulators and mechanisms using just single mobility kinematic pairs. From now on in this
text, ‘joint’ will be used in account for single mobility kinematic pair. Taking W = 0, we get
the equation for structural groups that are indivisible into other structural groups:
j – λ(N – C – B) = 0 (3.4)
The number of independent loops L = 1 + j − l , we can write this as:
j = L + l −1
= L + n
(3.5)
Using equations (3.5), (3.4) and (3.1) we can find a second equation of structural
groups as:
n – (λ – 1)(N – C – B) = 0 (3.6)
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