Graphics Reference
In-Depth Information
For these reasons, one is willing to put up with some crinkling or folding; however,
the amount of such crinkling or folding that is allowed is something that the user
should be able to specify. Therefore, the ability to
(1) specify the actual path and direction angle of the tape, and
(2) predict any gaps, overlaps, and laminate thickness,
would be important to anyone doing off-line programming of the process. In an auto-
mated environment a robotic arm that is laying the tape needs to be able to allow for
sideways movement and turning.
Note that in the case of filament winding only the path g(t) is important so that
the flatness of objects is not so critical. However, filament tows often consists of several
filaments and in that case one runs into problems similar to those with tapes because,
if a surface is not flat, then adjacent filaments may bulge and move apart if they do
not travel equal distances.
From this discussion we see that both in the case of tapes and filaments (those
that are sticky or those that are part of multi-filament tows) one is looking for curves
that are close, but not necessarily equal to geodesics. Let us show how, given a coef-
ficient of friction m, it is possible to define
generalized geodesics
([Crai88]) that are con-
trolled by a “steering function” s(t) satisfying
-<
()
<
m
st
m
which will do the job. To find these generalized geodesics we can use the mathemat-
ics developed in Section 15.3.1. All we have to do is replace the function
b
(t) in equa-
tion (15.7) by
(
()
)
()
n
n
g
g
t
t
()
-
()
b
tst
(15.10)
(
)
and solve these new differential equation. We can also think of the function s(t) as
allowing for stickiness of a filament.
15.5
Dropping Curves on Surfaces
Drawing curves on surfaces is important in robotics applications that involve gener-
ating paths for tools to follow. In fact, in those cases one also wants to generate frames
which are tangent to the surface along the path. These would be needed by the robot
arm for orientation purposes. Mathematically, the easy part of the problem is finding
the point where a ray pierces a surface. This is something we already discussed in
Section 13.4.1.
Let
S
be a surface parameterized by a function p(u,v). Picking points
p
i
on
S
with
a mouse would also give us a sequence of points
q
i
= (u
i
,v
i
) in the parameter space
X
of
S
, where p(
q
i
) =
p
i
. The goal is to generate the frames F
i
= (
u
1i
,
u
2i
,
u
3i
,
p
i
). See Figure
15.12. From the points
q
i
we can generate a curve
[]
Æ
XR
2
s :,
01
.
