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

The screws are placed on the axis of kinematic pairs subsequently. Location of the
moving platform can be defined fully if we know the coordinates of three points on it. Since
two intersecting screws define a point, we’ll place
be able to find the center of the spherical pair using
connection point on the moving platform.
v
E5
⎡
⎢
⎢
⎢
= ⎢
⎢
⎢
⎢
⎢
⎣
v
E4
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ν
E5 on the first axis of the spherical pair to
ν
E4 and
v v v
sv
Cα13
Sα
24 + cv
Cα
24
v v v
− cv
Cα13
Sα
24 + sv
Cα
24
v v v
sv
Cα13
Sα
24 + cv
Cα
24
v v
svrv
Sα13
Sα
24
v v
− cvrv
Sα13
Sα
24
0
v v v
cv
Sα
24 − sv
Cα13
Cα
24
v v v
sv
Sα
24 + cv
Cα13
Cα
24
v v
− Sα
Cα
ν
E 5 . This point coincides with the
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
13 24
v
[ ( 2c
Cα
v
13 Sα
v
24 − Cα
v
v
sv
24 ) rv
+ sva35
]
v
( 2s
Cα
v
Sα
v
+ c Cα
v ) r − c a
[ v
v
]
13
24
− a
v
35
24
Cα
13
v
v
v
Sα13
v
35 Sα13
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4.1)
To solve the forward task for the parallel manipulator, we will separately handle each
branch as a serial manipulator. Using the recurrent equations (2.28) and (2.30), the
expressions for the components of
ν
E4 and
ν
E 5 are found as given in (4.1). Note that, to find
these expressions, a simple recursive algorithm is used. (Appendix B).
4.1.3 Construction of the Set of Equations to be Solved
For two intersecting unit screws ν E4 and ν 5 E in space (a45 = 0), the equations of screw
axis in Plücker coordinates can be written as:
40