ElmoMC Multi-Axis Motion Controller-Maestro Motion Control - Manual (Page 27)

Motion Library Tutorial Switch Radius Calculation
MAN-MLT (Ver 2.0)
2-6
2.2 Circle – line intersection
Note: C – circle arc, L – line, R – circle radius, r – switch arc radius, (X
c
,Y
c
) - circle center,
(X
i
, Y
i
) – intersection point, (X
last
, Y
last
) – last point on the circle (X
first
, Y
first
) – first point on
the line L, d – distance from point (X
i
, Y
i
) to point (X
first
, Y
first
), (X
o
,Y
o
) – switch arc center
coordinates.
There are three possible cases that influence the calculation of parameters that define a
switch arc: initial circle center and switch arc center belong to the same half-plane, initial
circle center and switch arc center belong to different half-planes (defined by the line
L
),
and when the line (or continued line) moves through the center of the initial circle. For each
of these three cases, two sub cases are possible: from the point of intersection the line goes
either outside or inside the circle.
C ircle – line intersection geometry must satisfy some necessary conditions for the switch
arc to be built.
On the first stage of calculations we define switch arc radius. It can be predefined by the user
or calculated as
V
2
/(vae*AC
v
) , where V
–
end velocity and
AC
v
vector acceleration, vae
-admissible acceleration error
.
If the intersection was defined by the distance d
than the
switch arc radius can be calculated as described in 2.2.2
No matter how a switch arc radius was defined it must be coordinated with a circle and
a line parameter
2.2.1 Line goes inside the circle
2.2.1.1 Switch arc center and circle center belong to two different
half planes defined by the line L
The switch arc radius must obey
(vse)
2
/(vac*vae) < vsr < (R – h)/2 (2.2.1.1-1)
where R - radius of the initial circle, h
–
the length of the perpendicular dropped from the
circle center on the line L, vsr
-
parameter that defines switch arc radius, vae – admissible
acceleration error.
This condition is necessary but not always sufficient. It’s sufficient only in the case that the
projection point
P
1
of the circle center on the line belongs to the line segment (P
1
∈
L)
and the point of intersection of the continued perpendicular with the circle (point P
2
in
Figure 2-2) belongs to the circle arc (P
2
∈
C) .
If one of these conditions is not fulfilled, the maximum possible switch radius r must be re-
calculated due to intersection geometry. Possible cases are considered below.