Omni Wheel Math
When I posted my Omnibot page, I said I'd make a video attempting to make some of the counter intuitive math easier to understand. The attached video is my first attempt at illustrating the relationship between wheel angle, wheel speed and robot speed.
The following text was copied from the Omnibot page.
Omni Bot Math
One of the strange things about these omni wheels is the the way they need to slow down as the angle between the direction of travel and the direction the wheel is rolling increases. The greater the angle, the slower the wheel needs to turn without causing the robot to rotate. A wheel at 90 degrees to the direction of travel needs to stop in order for it not to rotate the robot.
The amount of slowing can be expressed in the following equation:
SpeedOfWheel = SpeedOfRobot * cos(angleOfWheelFromDirectionOfTravel)
If you use zero degrees for the angle, you see the speed of the wheel and the speed of the robot are the same. At ninety degrees, the formula accurately predicts the need for zero speed of the wheel.
A kind of bazaar effect of this is, it is possible to increase the speed of the robot by aligning the wheels away from the direction of travel.
The relationship described in the equation quickly breaks down if used to predict robot speed. Rearranging the above equation, one gets:
SpeedOfRobot = SpeedOfWheel / cos(angleOfWheelFromDirectionOfTravel)
The above equation does hold reasonably true for relatively shallow angles but when the wheels are at large angles to the direction of travel, the force of friction soon plays a large role to limit the speed of the robot. If not for friction, a robot could be driven at very high speeds just by angling the wheels away from the direction of travel. At reasonable angles, the speed of the robot does increase in a very noticeable way. You can see how the wheels need to slow down as they point away from the direction of travel in both of the attached videos. I used the above equation to compute the wheel speeds.