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According to d’Alembert’s principle the contribution of all the inertial forces must equal the
contribution of all the non-inertial forces (11), this applied at the joint level leads to:
τ + τ + τ = τ + τ
Gn Gb
Eq. 4.24
where τ is the vector of torques that have to be applied to the three actuated joints. τn is
containing the term n X& & which can be expressed in joint space by Eq. 4.16. This gives
which can be written as:
T
T
τ = ( Ib + mnt
J J)
& θ&
+ ( J mntJ&
) & θ −(
τ Gn + τ Gb)
Eq. 4.25
τ = A ( θ)
& θ&
+ C(
θ,
& θ)
& θ + G(
θ)
Eq. 4.26
where A is the inertia matrix, C describes the accounting of the centrifugal and Coriolis forces
and G contains the gravity forces acting on the manipulator.
4.9 SYSTEM DYNAMICS
Figure 4.9, block scheme of manipulator with linear decoupled and non-linear coupled part.
As shown in section 4.8.3 the motion of a manipulator can be described by
τ = A( θ)
& θ&
+ C(
θ,
& θ)
& θ + F &
m θ + G(
θ)
Eq. 4.27
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Public Report ELAU GmbH, Marktheidenfeld
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n