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where Fm is added and describes the friction coefficients of the manipulator.
By observing that the diagonal elements of A(θ) are formed by constant terms and functions of
sine and cosine for revolute joints, one can set
A( θ) = A + ∆A(
θ)
Eq. 4.28
where A is the diagonal matrix whose constant elements representing the average inertia at
each joint (9). Substituting Eq. 4.17 - Eq. 4.18 into Eq. 4.27 yields
where
τ
m
=
−1
AKr
& θ&
+ F & θ + d
−1
Kr m
Eq. 4.29
m m
−1
−1
Fm = Kr ( Fv
+ Fs
) Kr
Eq. 4.30
describes the matrix of viscous friction and static friction about the motor axis, and
−1
−1
−1
−1
∆A(
θ) Kr & θ&
+ Kr C(
θ,
& θ)
Kr & θ + Kr G(
θ)
−1
d = Kr
m
m
Eq. 4.31
is the vector which represents the coupling effect between the three arms. θm is the angular
variable of the motor and θ is the angular variable of the robot arm position.
As illustrated by Figure 4.9 the system of the manipulator with coupling effects can be
constituted by two subsystems, one that has τm as input and θm as output and the other with θm,
θm & , θm & as input and d as output. The first subsystem is a linear and decoupled system, since
each component of τm only influences the corresponding component of θm. The other subsystem
corresponds for all those non-linear and coupling terms between the three arms.
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Public Report ELAU GmbH, Marktheidenfeld
27