Holonomic and Nonholonomic constraints

BillDarby wrote:

>
> Was just sitting here wondering if anyone has used any CNC lathe software that would allow you to turn shapes, other then just
> circular. It occures to me that under CNC control, a lathe should be easily capable of turning all sorts of shapes (squares, hex,
> lobs, cams)????
>
> Bill Darby
>

Yes, I frequently fiddle with various odd geometries for CNC machines,
at least mentally.

One sort of robot I've done more than mentally fiddle with, and which
is a good design for placing an effector at a point in a large volume (like
"somewhere inside this football stadium") is a "slack wire" robot.

Put up 3 poles. Put a pulley on a swivel on the top of each pole. Cut 3 cables.
Run a cable through each
pulley and tie one end of each togather (call this the "effector").
Build 3 winches with steppers or servos. Run the free end of each cable through
a winch.

Now, by pulling in and out on the 3 winches you can move the effector anywhere in
a triangular cylinder. In practice, it works best in about the lower half of this
cylinder and within the central area.

Clearly one doesn't want to use this for CNC, although I considered it once as a specialized
solution to carve a large amount of foam for a big club model RR layout.
I built one of these with a puppet on the effector.

An "odd" milling machine could be made with three vetical rotary tables. Bolt #1 to the bench.
Bolt #2 to #1's table. Bolt #3 to #2's table. Now mount a spindle (it doesn't need a quill)
somewhere to extend a tool into the volume.
You actually can move the tool tip to anywhere in a volume of an object mounted on #3's table.
Before you run out and start pricing rotary tables, let me caution you that You can't really
control the tool angle, which will move in bizarre ways. Of course, if you mount the spindle
on 2 more rotary tables you can remove this and have a design for a 5 axis machine.

However you arrange the axes, however, if you look at how the effector (the tool) moves, you find
it takes one axis for each linear degree of freedom and one axis for each rotational degree of freedom.

We call these kinds of systems "holonomic", because the "path" taken through the axis space is irrelevant,
if the axes are at certain settings the tool will be in a certain place and orientation.

There is another sort of constraint possible, the "nonholonomic" type. A car is a nonholonomic system.
Suppose I go out in a parking lot and set my car at 0,0, facing north. Lets say for ease my wheels
have a 5' circumference. I drive one hundred feet north, my wheels turn 20 times. Now I turn my wheels right,
and drive (say another 5 turns) until I'm facing east. I straighten my wheels and then back up 25 turns.
Now if we look at the net motion of each axis (the wheels and the steering wheel) it is 0, but the car
is NOT at the starting point.

Nonholonomic systems have the advantage that they can "multiply" axes. With two axes, steering wheel
and wheels, I can orient my car in 3 axes (2D surface of parking lot, and 1D direction).

Could you build a nonholonomic milling machine? Sure. Imagine a sort of "caterpiller", built very
rigidly, which would drive about on a metal plate. Put a normal quill type spindle above this and
you could have a milling machine & rotary table with only two axes.