ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 64)

Motion Library Tutorial Switch Radius Calculation
MAN-MLT (Ver 2.0)
2-43
Consider the case that the sweep angle of the first circle is
β
1
< 90
and the sweep angle of
the second circle is
β
2
< 90 .
Draw a line L
1
defined by two points: circle C
1
center
(X
c1
,
Y
c1
)
and circle C
1
start point (X
1
,Y
1
) and line L
2
defined by circle C
2
center point (X
c2
,
Y
c2
) and circle C
2
end point. If line
L
1
does not intersect circle arc C
2
and line L
2
does not
intersect the circle arc C
1,
then the intersection point of two lines is point (X
3
, Y
3
) – Figure
4.3. Note: l
1
the length of the line L
1
: l
1
= ρ((X
1
,Y
1
),(X
3
,Y
3
)) and
l
2
the length of the
line L
2
: l
2
= ρ((X
2
,Y
2
),(X
3
,Y
3
)).
If l
1
>l
2
for the maximum switch radius r calculation, use the following system:
(X
o
– X
1
)/(X
1
– X
c1
) = r/R
1
(2.3.3-1)
(Y
o
– Y
1
)/(Y
1
– Y
c1
) = r/R
1
(2.3.3-2)
(X
o
– X
c2
)
2
+ (Y
o
–Y
c2
)
2
= (R
2
+ r)
(2.3.3-3)
Equations (2.3.3-1)-(2.3.3-2) can be written in the following format
X
o
= X
1
+ r(X
1
– X
c1
)/R
1
= X
1
+ rC
1
, C
1
= (X
1
– X
c1
)/R
1
(2.3.3-4)
Y
o
= Y
1
+ r(Y
1
– Y
c1
)/R
1
= Y
1
+ rC
2
, C
2
= (Y
1
– Y
c1
)/R
1
(2.3.3-5)
Substituting into (2.3.3-3), the results are:
(X
1
+ rC
1
– X
c2
)
2
+ (Y
1
+ rC
2
– Y
c2
)
2
= (rC
1
+ C
3
)
2
+ (rC
2
+ C
4
)
2
= (R
2
+ r)
2
(2.3.3-6)
where C
3
= X
1
– X
c2
C
4
= Y
1
– Y
c2
.
Simplifying (2.3.3-6), the results are:
r
2
[1 – (C
1
)
2
– (C
2
)
2
] + r[2R
2
– 2C
1
C
3
– 2C
2
C
4
] +[(R
2
)
2
– (C
3
)
2
– (C
4
)
2
] =