ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 54)

Motion Library Tutorial Switch Radius Calculation
MAN-MLT (Ver 2.0)
2-33
(C
1
) 2 + (C
2
) 2 – 1 = [(X
2
– X
1
)/d] 2 + [(Y
2
– Y
1
)/d] 2 – 1 = d 2 /d 2 – 1 = 0
( 2.3.1 -11)
as
d 2 = ρ 2 (P
1
,P
2
) = (X
2
– X
1
) 2 + (Y
2
– Y
1
) 2 .
and if noting
C
6
= 2(C
1
C
3
+ C
2
C
4
+ R
2
), C
7
= (C
3
) 2 + (C
4
) 2 – (R
2
) 2
The results for r are as follows:
r = –C
7
/C
6
(2.3.1-12)
where
C
1
= (X
2
– X
1
)/d
C
2
= (Y
2
– Y
1
)/d
C
3
= X
1
– X
c2
(2.3.1-13)
C
4
= Y
1
– Y
c2
C
6
= 2(C
1
C
3
+ C
2
C
4
+ R
2
)
C
7
= (C
3
) 2 + (C
4
) 2 – (R
2
) 2
The maximum switch radius can be also estimated as a solution of the alternate system
(X
o
– X
1
)/(X
o
– X
c1
) = r/(R
1
+ r) ( 2.3.1-14 )
(Y
o
– Y
1
)/(Y
o
– Y
c1
) = r/(R
1
+ r) ( 2.3.1-15 )
(X
o
– X
c2
) 2 + (Y
o
– Y
c2
) 2 = (R
2
– r) 2 ( 2.3.1-16 )
Its advantage is that there is no need to calculate coordinates of the point P2.
From (2.3.1-14),
X
o
= [r(X
1
– X
c1
) + X
1
R
1
]/R
1
= [r(X
1
– X
c1
)/R
1
] + X
1
= rC
1
+ X
1
( 2.3.1-17 )
where
C
1
= (X
1
– X
c1
)/R
1
and from (2.3.1-15)
Y
o
= [r(Y
1
– Y
c1
) + Y
1
R
1
]/R
1
= [r(Y
1
– Y
c1
)/R
1
] + Y
1
= rC
1
+ Y
1
( 2.3.1-18 )
where
C
2
= (Y
1
– Y
c1
)/R
1
By substituting (2.3.1-17) and (2.3.1-18) with (2.3.1-16), the results are
[rC
1
+ X
1
– X
c2
] 2 + [rC
1
+ Y
1
– Y
c2
] 2 = (R
2
– r) 2 (2.3.1-19)
or