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APPENDIX A
TASK GRAPH GENERATION
As mentioned in Section 4, here the pseudo-codes of
the 3D task graph generation are presented. Its input
includes is a sequence of 4×4 location matrices
{ ( ) | 1,2,... }Task i i n
, the discretization densities m
1
and m
2
, and the upper/lower limits of the redundant
variables denoted as
,
and
,
respectively. The algorithm transforms the task
locations
into the joint space.
This procedure contains two steps. Firstly, the
redundant variables are uniformly discretized in the
ranges of
and
, and
m
1
×m
2
×n matrix
1 2 1 2
{ ( , , ), ( , , )}
PL
q k k i q k k i
is
obtained, where
1 1 2 2
1,2,... ; 1,2,...k m k m
and
. Then, at the second step,
,
and
are sequentially applied, and the robot
configuration states for
1 2 1 2 1 2
{ ( , , ), ( , , ) | , , }
PL
q k k i q k k i k k i
are
computed. After checking with the joint limits,
collision and the distance to singularities, the task
graph is finally generated with nodes:
1 2 1 2 1 2 1 2
( , , ) { ( , , ), ( , , ), ( , , )}
P L R
k k i q k k i q k k i k k iLq
where
1 1 2 2
1,2,... ; 1,2,...k m k m
and
.
Input:
{Task(i)|i=1…n} – matrices of task locations.
{q
P
max
(i), q
P
min
(i)|i=1…n}
– upper/lower limits of positioner coordinate.
{q
L
max
(i), q
L
min
(i)|i=1…n}
– upper/lower limits of linear track coordinate.
m1 – discretization density for positioner.
m2 – discretization density for linear track.
u – robot configuration index.
Output:
{L(k1,k2,i)|k1=1…m1;k2=1…m2;i=1…n}
– 3D matrices of task locations:
Notations:
q
L
, q
P
, q
R
– Linear track, positioner and robot joint
coordinates;
P
T
task
– Transformation from positioner base to task
locations;
R
T
task
– Transformation from robot base to task
locations;
Invoked functions:
g
P
(.) – Positioner direct kinematic function;
g
L
(.) – Linear track direct kinematic function;
g
R
-1
(.) – Robot inverse kinematic function;
coll(.)– Collision test function;
cond(.)– Condition number calculation;
Tran(.)– Transformation from robot base to
positioner base.
Optimal Coordination of Robot Motions with Positioner and Linear Track in a Fiber Placement Workcell
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