pic: OmniSwerve chassis Bottom View and Concept Discussion

[Aren Siekmeier]

Force vectors add. Velocity vectors do not. Rather they are constrained by slip conditions and the relative motion of bodies.

It's sort of a continuously variable transmission.

Say you are constrained to go in a direction 0 (by a wall or omnis opposing each other, etc.), but an omni wheel is pointed in a direction theta relative to your heading, and spinning at speed w about its axis. The overall translation velocity vector v is identically the velocity of the center of the wheel relative to the ground, which can be broken into the rotation of the wheel (w), and the rotation of the rollers (s), assuming no slippage. Then v must project onto the wheel speed w, as w=v*cos(theta). The roller speed is s=v*sin(theta) and has no limit. The overall speed is v = w/cos(theta), resulting in a higher speed.

To do this, you could have a four wheel drive with steerable omnis in each corner. Start with them directed forward, (theta=0), and spin the wheels at speed w. Then v = w / cos(0) = w. Then turn the left and right wheels toward the middle by an angle theta. Now v = w / cos(theta) will give an increase in top speed.

However, the force available in the forward direction drops. Each wheel supplies F in its own direction. When theta = 0 then the total force is 4*F, all straight forward with no sideways component. When theta is increased, the sideways components cancel, but at each wheel, the sideways component must add with the forward component to get F in the direction theta. So the forward component from each wheel is F*cos(theta), and the overall force available is 4F*cos(theta).

In this way, steering the wheels toward the middle by an angle theta acts as another stage of reduction, increasing the speed and decreasing the force by a factor of cos(theta), assuming no wheel slippage.

Vector diagram to help