![](bg6.png)
-
clamp
∆x
∆˙x
Ƭx
N
2
N
1
Controller
Manipulator
∆q
∆ ˙q
∆ ¨q
τ
a
τ
c
−B
T
a
(B
†
p
)
T
τ
p
y =
q
˙q
¨q
r
k
=
x
r
˙x
r
¨x
r
Figure 3: Operational space control of a parallel manipulator.
Table 1: Assumed values for a 3-RPR robot.
Manipulator 1 Manipulator 2 Manipulator 3
q
1,1
= π/3 q
2,1
= 2π/3 q
3,1
= 4π/3
q
1,2
= 1 q
2,2
= 1 q
3,2
= 1
q
1,3
= 0 q
2,3
= −π/3 q
3,3
= −π
x
1,1
= 0.5 x
2,1
= 0.5 x
3,1
= 0.5
x
1,2
= 0.5 x
2,2
= 0.5 x
3,2
= 0.5
x
1,3
= 1 x
2,3
= 1 x
3,3
= 1
The forward kinematics function, F
c
, gives the
following end-effector position for the active joints
1,1,1
T
;
x
end
=
1.498
2.597
1.048
7 CONCLUSIONS
Similar to the analytical Jacobian for a parallel ma-
nipulator, which is a function of joint variables and
relates the velocity of the active joints to the velocity
of the end-effector, the analytical forward kinematics
function is also a function of the joint variables that
relates the position of the active joints to the position
of the end-effector. The generality of the proposed
technique allows the forward kinematics function to
be used in a variety of applications. A control config-
uration is also described in this paper as a prospective
application of the proposed technique.
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ANALYTICAL KINEMATICS FRAMEWORK FOR THE CONTROL OF A PARALLEL MANIPULATOR - A
Generalized Kinematics Framework for Parallel Manipulators
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