I'm using a 3-omni-wheeled bot whose wheels are whose wheels are at an angle to the direction in which they are moving.
Suppose you wanted to move in the forward direction and the two adjacent wheel are rotated at a 60 degree angle, if the wheel has traveled forward x feet, how can you calculate the forward distance?
Alternatively, if you wanted to measure the distance y degrees off forward direction (i.e. angle between direction we want to travel in - angle of wheel orientation), how could we calculate the distance traveled in that direction?
I may be trivializing the problem, but wouldn't the distance moved forward be equal to the distance traveled by the wheel parallel to the direction of travel?
To find the necessary velocity of each wheel, because they must be powered independently to travel in a straight line, is a trickier problem, but we're still missing information.
You would have to approach the problem in a way that first sums the vectors in the x-direction to equal zero. This would cause the two right wheels to be moving in the same direction. The direction, which we'll call negative (based on the drawn arrows) attempts to spin the omni-bot counterclockwise. To then have a net-torque equal to zero, the remaining wheel (on the left) has to spin in the positive direction twice at twice the rate of the first two wheels. (Assuming all the wheels are the same size).
Then, you would be able to sum the components in the y-direction. The omni-bot will be moving forward with a velocity of $2y+2ysin(60^o)$ where $y$ is the tangential velocity of the wheels, which cannot be determined from the given information. This depends on the RPM of the wheels and the diameter of the wheels.
I'm not entirely confident in this answer, nor am I sure it answers the question. Please do not hesitate to clarify any of my assumptions or to point out mistakes in my response. Thank you.