Re: Math for constant speed ellipse

Todd,

I took a crack at your problem because it is interesting. I'm not
much of a mathematician so I analyzed it graphically using my trusty
TI calculator. I think I have a possible solution.

(1) Draw an ellipse whose X-axis dimension is 5 times the Y-axis. No
special reason for 5, I just needed to pick a number.

x = sqrt ( 25 - 25*y^2) would define the ellipse. (EQ 1)

(2) Inscribe a circle with a raduis of 1 inside the ellipse. Sweep a
projected radius at a constant angular rate (I picked 9 degrees with
an initial offset of 4.5 degrees).

(3) Take the sine for each increment. That will give you the value
for "y", which you plug into EQ 1 to get the corresponding "x" on the
elipse.

(4) Calculate the segment lengths between each pair of x,y co-
ordinates. First conclusion: The segment length (thus velocity)
around x=0 is 5 times the velocity around y=0 for a constant angular
rate. From this one can conclude that the angular rate around y=0
should be 1/5 the rate around x=0. What is the relationship f(x) for
this 5:1 range in velocity?

(5) Sum the segment lengths. Normalize the results by dividing the
lengths by 5. Subtract each from "1". The resulting table now looks
suspiciously like a sine table for every 9 degrees. It is in fact.

(6) Fooling with the calculator suggests the angular velocity for a
constant path velocity should be:

The relationship seems to be x/y -(x/y -1)*cos theta over the span of
+/- pi/2. The result must range between 1 to 5 from +/- pi/4 and is
the multiplier for "delta" summed to "theta" angle that sets the path
velocity.

Mariss

None of (6) has been carefully checked.







--- In CAD_CAM_EDM_DRO@yahoogroups.com, "Todd Fleming" <todd@f...>
wrote:

> > Dear Todd:
> >
> > Your request is an interesting one. This afternoon I made an
> > ellipse on my CNC "just for the heck of it". It came out well but
> > the tangential speeds were related to the height (2) and width

(10).

> > I started with a circle and scaled down the outout from the Y
> > interpolator.
> >
> > In considering what was going on, I consluded that if one were to
> > scale the clock frequency according to:
> >
> > f =

_____________fo_____________________

> > SQR((COS theta)^2 + (SIN theta/R)^2)
> >
> > This will result in constant instantaneous velocity. Theta is the
> > angle which an equivalent circle would have (height = width), f

is

> > the new clock frequency, R is the ratio of Width to Height.
> >
> > I haven't tried the frequency portion yet so I can't prove it OK,
> > but the numbers on paper work out.
>
> If I understand your equation, f = d theta / dt. Since I'm

interested in

> theta as a function of t, this is a D.E. similar to the one I

found; we're

> back to square one again.
>
> I tried finding a power series expansion of the D.E., but I can't

find a way

> to get the D.E. into a suitable form.
>
> > I would be interested in details of your requirement. What speed
> > will you run, what resolution, what ratio Width to Height, what

is

> > your increment of motion?
>
> All of these are variables; I want to design a controller that can

handle

> any reasonable variation.
>
> > Regards,
> > Jack C. Cain
>
> Todd Fleming
> flemingcnc.com