Killough Center of Rotation

[gpetilli]

We have been experimenting with an Asymmetric Killough chassis off season. The intent is that by using a toe-in off 30 degrees instead of 45, we can trade off more lateral traction loss for less forward traction loss.

As part of that, we would like to be able to implement a "bump and run" where we rotate around the corner of the bumper. We have developed a spreadsheet to calculate the relative wheel speeds. The the wheel locations and toe-in are parameterized so it is usable for other configurations.

Please review and recommend corrections to the math. the intent was to calculate the radius and tangent for the circle traversed by each wheel contact point with respect to the center of rotation. That velocity was then rotated by the difference of the tangent angle and the wheel angle to find the wheel velocity with the correct tangential component.

Use at your own risk; standard mileage may vary warnings apply.

[Ether]

There's a problem with your calculations.

If you make the rotation point (26.57,16), the speed of both right wheels (FR & RR) should be zero, and the speed of both left wheels (FL & RL) should be non-zero.

[Ether]

Here's a 32-bit Windows console app which computes the omni wheel speeds for an arbitrary center of rotation.

Scroll to the bottom of the list of attachments:

http://www.chiefdelphi.com/media/papers/2390

[gpetilli]

Corrected a sign error.

Ether, I did not run your Win32 yet, but does it allow you to change the toe-in from the standard 45 deg?

[gpetilli]

Ether,

I see that you do indeed use toe-in - thank you. I also see the purpose of your specific test case. Your answer does look correct (wheels perpendicular to tangent are near zero) and I do not get the same answer. Back to the drawing board.

[Ether]

In case anyone reading this thread is interested, here's an explanation how the omni rotation equations were derived. Scroll to the bottom of the list of attachments.

[gpetilli]

All,

I have updated the xls to use Ether's algorithm and a cleaner user interface.

Ether,

I think I duplicated your calculations, and I get the same final wheel speed results, but the intermediate alpha angle results for the rear wheels don't look appropriate to me. I have checked the equations several times and I think they match yours. Do they look correct to you?

[Ether]

Originally Posted by gpetilli

All,

I have updated the xls to use Ether's algorithm and a cleaner user interface.

Ether,

I think I duplicated your calculations, and I get the same final wheel speed results,

Yes, except for the (8,12.25) center of rotation test case. You have a division-by-zero error. You need to put a wrapper around Excel's atan2 function to test the XY input parameters and return a value of zero if both X and Y are zero. That will fix it.

but the intermediate alpha angle results for the rear wheels don't look appropriate to me.

What about them doesn't look appropriate? You do realize they are in radians, right?

I have checked the equations several times and I think they match yours. Do they look correct to you?

They don't match mine. You've got a cosine in there. The equations I posted don't have a cosine anywhere.

Are they correct? They seem to be, for the test cases you have run. Whether or not they are equivalent for all test cases would require further review. If you want to put together a labeled diagram explaining your equations similar to what I posted¹ I'd be happy to review it.


¹attachment "omni rotation derivation of equations" in this CD-media paper contains figures and a working Octave script showing the equations.

[gpetilli]

Added div0 checks. Hopefully final version.