Physics of robot base vs. width

One of the basic techniques to improve 6 wheels arcade drive is to lower the middle wheels to allow the robot to “rock.” I understand that this is to effectively reduce the robot base (distance from back wheel to front wheel) and improve turning.

My question is how do you explain this phenomenon using physics? The way I think of it is that force is applied by the wheels producing a torque about (assuming) the centre of all the wheels. But whether if it is 4 wheels or 6 wheels in contact, the perpendicular distance from the force applied by the wheels to the axis of rotation remains the same and the centre of mass stays relatively the same when you are “rocking”. So how does this improve turning?

The outer wheels have to slide on the carpet during a turn, hence the term “skid steer”. By having a drop center wheel set, you reduce the amount of weight on those outer wheels, and thus less friction you have to overcome.

The idea behind the rocking 6wd is to shorten the wheelbase while maintaining the same stability as the longer wheelbase.

To turn a “skid steer” system the wheels have to slip. Imagine a robot with 4 wheels on the corners, a level 6wd will behave pretty similar to this system. The force trying to turn the robot is applied at a distance equal to 1/2 the robot width. The friction force preventing the robot from turning is applied at a distance equal to 1/2 the robot width.

If the length is longer than the width, and the wheels have similar coefficients of friction in the two different directions, the torque about the center of rotation being applied by the wheels trying to spin is not enough to overcome the torque being applied by the friction of the wheels being pushed sideways.

Chris Hibner wrote a great paper on 4wd turning.
http://www.chiefdelphi.com/media/papers/1443

I would recommend reading through it. Basically, a 6wd drop center is essentially 2 4wds (as others have stated).

An interesting phenomenon can occur with 6wd with a reasonably flexible chassis. Even with stiff rails, if the two sides are not torsionally stiff relative to each other, then diagonal outer wheels can end up touching at the same time. This will cause it to act like a longer wheel base which can reduce maneuverability.

We used this paper too way back to explain why our 4 wheel robot had trouble turning in 2003. I recommend it too.

On a flat surface with drop center 6 wheel drive. Only 4 wheels can touch the ground, 2 have to be the center wheel. So it turns a robot with over 30 inch wheel base, to one that is about 15 inches in length. By reducing the wheel base length, you are reducing the moment around the axis of rotation resulting from wheel friction. You are essentially making a smaller lever arm lowering the force. That is why its easier for wide robots to turn with 4 wheels the long robots running 4 wheels.

On the carpet, because the robot sinks in, the effect merged with description the vikesrock stated above. Really all 6 wheels are touching the carpet, some wheels carry more weight than others. Its the combination of these two effects that make this system work so well in FIRST.

In addition to the decrease in skidding with the dropped center, the reasoning behind this can also be explained with vectors. With only the center wheels and one pair of the side wheels touching in the dropped center design, the robot is turning about the center of those four wheels rather than the center of the robot. This means that if the force of the wheels are broken up into vectors, the rotational component of the turning wheels (portion of the vector perpendicular to the imaginary segment connecting the center of rotation and the wheel) is much larger in the dropped center than it is in the non-dropped center six wheel base. The other component of the force from the wheel is wasted because it is canceled out by a force from the opposite wheel on the bot.

Diagram:
http://s3.postimage.org/1dyd83hrp/6wcd.jpg

In this diagram, the red arrows are the rotational component of the force provided by the wheels and X marks the center of rotation. In this particular moment, the top 4 wheels on the dropped center bot are contacting the floor and the bottom two are not. Note that the rotational vectors on the dropped center diagram are significantly larger than those on the non-dropped center bot, especially at the corners.