ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 58)

Motion Library Tutorial Switch Radius Calculation
MAN-MLT (Ver 2.0)
2-37
r 2 C
12
+ (2C
1
C
3
)r + C
32
+ r 2 C
22
+ (2C
2
C
4
)r + C
42
= r 2 + (2R
2
)r + R
22
(2.3.1-37)
or
r 2 (C
12
+ C
22
– 1) + 2[C
1
C
3
+ C
2
C
4
– R
2
]r + (C
32
+ C
42
– R
22
) = 0
(2.3.1-38)
But
C
12
+ C
22
= [(X
c1
– X
1
) 2 + (Y
c1
– Y
1
) 2 ]/R
12
= 1
(2.3.1-39)
So (4.1-38) transforms into
rC
6
+ C
7
= 0
(2.3.1-40)
where
C
6
= 2[C
1
C
3
+ C
2
C
4
– R
2
]
(2.3.1-41)
and
C
7
= C
32
+ C
42
– R
22
(2.3.1-42)
So the an upper limit estimation for r is:
r = – C
7
/ C
6
(2.3.1-29)
Another case when the line O
1
O
2
does not intersect circle arc C
2
is shown in the Figure 2-
32. Draw a line O
2
P
2
from the center of the circle C
2
to its end point. If noticing that d =
ρ(O
2
P
2
), the switch arc radius r is:
(X
2
– X
o
)/(X
2
– X
c2
) = r/d
(2.3.1-30)
(Y
2
– Y
o
)/(Y
2
– Y
c2
) = r/d
(2.3.1-31)
(X
o
– X
c1
)
2
+ (Y
o
– Y
c1
)
2
= (R
1
+ r)
2
(2.3.1-32)
From (2.3.1-30)
X
2
d – X
o
d = r(X
2
– X
c2
)
X
o
= X
2
– r(X
2
– X
c2
)/d = X
2
+ r(X
c2
– X
2
)/d = X
2
+ C
1
r (2.3.1-33)
From (2.3.1-31)
Y
2
d – Y
o
d = r(Y
2
– Y
c2
)
Y
o
= Y
2
– r(Y
2
– Y
c2
)/d = Y
2
+ r(Y
c2
– Y
2
)/d = Y
2
+ C
2
r
(2.3.1-34)