Page 115 - Mechatronics for Safety, Security and Dependability in a New Era

Ch21-I044963.fm Page 99 Tuesday, August 1, 2006 3:30 PM
Page 99 Tuesday, August 1, 2006
3:30 PM
Ch21-I044963.fm
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using other remaining joints. In this new method, these three analytical syntheses are carried out
iteratively until sufficient convergence is obtained.
For example, the case that 14 DOF robot (already shown in Fig. 2) passes through two holes is
assumed. It is possible to synthesize the end effector's position analytically by using 3 joints (#,)
among 14 joints, and it is possible to synthesize analytically the end effector's orientation by using
wrist 3 joints (0 4 ). Also, it is possible to synthesize the configuration analytically, which avoids two
cylindrical holes by using other remaining joints (0, ,0 2 ), where 0,, <P 2 , #,, * 4 are as follows:

0 2 =(0 6 AAAo/ (2)
<P 3=(<2> 3 ,*„,<*>„)'' ( 3 )
0 4 = ( W « 1 4 ) 7 (4)
Let the restraint condition for passage through the first hole be described as //,, which is concretely
the cluster of four equations regulating the position of J5 axis and L 6 link (see Fig. 3). Also, let the
restraint condition for passage through the second hole be described as H 2, which is the cluster of
four equations regulating the position of J10 axis and In link (also see Fig. 3). And, let the objective
0
position of the end effector (workpiece) be r' ', the objective orientation be E'J" . They are functions
of 0 i , 0 2, • • •, 0 14 and let they be expressed as follows:
#,=/(<*>, ,4> 2,4> 3,0 4) (5)
H 2=f 2{0 2,0,,0,,0 4) (6)
'•r=/3(*3;*..*2.*4) (7)
K" = M*4 ; * P * 2 . * 3 ) (8)
In these expressions, / ( 0 , ; 0 2,0 1,4> 4) means 0, is dominant for realizing the objective result
compared with 0 2 ,0 3 ,0 4 . These equations can be analytically solved by fixing non-dominant variables
( 0,,0 3 ,0 4 in this case) to appropriate constant values. Let these solutions be expressed as follows:
<t> {=f;\H {,<t> 2,0 3,0 A) (tf> 2,4> 3,<Z> 4=const.) (9)
0 2=f;'(H 2-0 l,0 3,& i) (<2V<2> 3,<Z> 4=const.) (10)
1
&3=fi (C ) \4>\4>i4>d (*!,*;,,* 4 =const.) (11)
'«° ; ( t f P P ) (12)
In this new method, these four analytical syntheses are carried out iteratively until sufficient
convergence is obtained. If the convergence is not enough, the combination of jonts for positioning
end effector and avoiding collision (combination of joints for orientating end effector is fixed to the
wrist 3 joints) is changed and the iterative analytical synthesis is carried out again.
The features of this method are as follows: 1) iteration number is very small and computing time is
reduced to about one tenth of that computed by Newton method, 2) the initial value of iterative
calculation is defined analytically (it is surely in the vicinity of true value), 3) it gives all solutions
while Newton method gives only one solution, so one can choose the most adequate solution for his
task among them.

Setting of Middle Configurations

The Path (the transition of configuration of the robot) is generated from goal position and orientation.
Wrist 3 joints of 0 12, 0 13, 0 14 are used only for realizing desired goal orientation. Five joints are
necessary before a wall to pull away a link through a hole, since four DOF shown in Fig. 1 and one