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

solutions for arbitrary values of input parameters efficiently. The algorithm should be started
from the initial conditions by supplying the input velocities, for each arbitrary actuator value.
In this thesis, numerical integration is used to verify the results acquired utilizing screw
theory.
4.1 Forward Displacement Analysis Using Screw Theory and Function Minimization
Approach
To form the mathematical for the forward displacement analysis of spatial parallel
structure manipulator, we will create a set of equations to be solved simultaneously for the
unknown variables. First of all, we will describe the structure of the manipulator (Figure 4.1).
4.1.1 Structural Definition of the Manipulator
The manipulator shown in figure 4.1 is a six degree of freedom, spatial parallel
manipulator. Following the definitions given in chapter 3, it is of class 1, type 6, kind 0, order
6, mod 2. The primary revolute pairs are placed on the x-y plane. Their axes intersect at O, the
center of the frame. Following the initial revolute pair, another revolute pair is placed
perpendicularly. Note that these two joints comprises a universal joint.The center of the
universal joint is on the x-y plane. Between the spherical joint on the mobile platform and the
universal joint, a prismatic joint is placed. For the inner branches, which have an offset ra
from the center, the actuated joints are the initial revolute pairs(rotary actuators/motors).
However, for the outer branches, which have an offset rb from the center, the actuated joints
are the prismatic pairs (linear actuators).
4.1.2 Definition of Screw Axis
where
The first two known screws are defined as follows:
v
E 1 (0,0,1, svrv, -cvrv,0)
v
E 2 (cv,sv,0,0,0,0)
c = c = C( π / 2)
, s = s = S( π / 2)
, c = c = C( 7π
/ 6)
1
2
1
2
s = s = S( 7π
/ 6)
, c = c = C( 11π
/ 6)
, s = s = S( 11π
/ 6)
3
4
5
r1= r3= r5= ra , r2= r4= r6= rb , v = 1,2,…,6
6
3
5
4
6
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