How to control a kiwi drive robot?

Question

I'm on the FIRST robotics team at my high school, and we are working on developing a kiwi drive robot, where there are three omni wheels mounted in a equilateral triangle configuration, like this:

The problem is programming the robot to drive the motors such that that the robot moves in the direction of a given joystick input. For example, to move "up", motors 1 and 2 would be powered equally, while motor 3 would be off. The joystick position is given as a vector, and I was thinking that if the motors were expressed as vectors too, vector projection might be what I need. However, I'm not sure if this is right, and if it is, how I would apply it. I also have a feeling that there may be multiple solutions to one joystick position. Any help would be greatly appreciated.

Answer 1

As you have recognized, the first part of this will be finding an appropriate equation to represent the resultant motion for any motor settings. Depending on the level of control and feedback you have on your motor speeds, I would suggest the process you go thorough should start with writing a vector equation: (define positive X as straight ahead)

-M1Cos(30)+M2Cos(30)=X (the negative is because 1 and 2 must be powered the same magnitude, but opposite polarities for forward motion)

M1Sin(30)+M2Sin(30)-M3 = Y (as anticlockwise motion on 1 and 2 will result in the robot moving left in the Y and anticlockwise motion on 3 will result in the robot moving to the right)

The other input that you need to add into this is the desired rotation of the robot, thankfully, M1+M2+M3 = W (Rotational velocity)

Your joystick input will give you X,Y and W, so you have 3 equations with 3 unknowns.

From here it is simultaneous equations, so you may end up with multiple solutions, but these can generally be restricted based on possible motor speeds and the like.

An example of this is the rec::robotino::com::OmniDrive Class - the source code for this method is available too...

Answer 2

First let's define some terms. In keeping with the usual convention, the X axis will point to the right and the y axis will point up (so that the thrust of wheel 3 is along the X axis). We'll call the motion of the wheels W₁, W₂ and W₃, each defined so that W_i > 0 means that the wheel rotates in the clockwise direction. In your example, if W₁ < 0, W₂ = W₁ and W₃ = 0, the robot will move in the +Y direction.

If all three wheels rotated at the same rate (W₁ = W₂ = W₃) the robot would rotate in place. I'm guessing you don't want that, so the sum of the rotations must be zero: W₁ + W₂ + W₃ = 0.

The motion of each wheel contributes to the motion of the robot; they add as vectors:
W₁ = -1/2 X - sqrt(3)/2 Y
W₂ = -1/2 X + sqrt(3)/2 Y
W₃ = X

So if you know the desired X and Y from the joystick, you have W₁, W₂ and W₃. As we've already seen, the difference between W₁ and W₂ is what drives Y motion. Their sum drives motion in X.

Answer 3

I've built 9 robots during my time at school (1 FIRST, 8 RoboCup). We used the same omnidrive layout as you do. Beta's answer looks correct but add rotation to all wheels afterwards:

W1 = -1/2 X - sqrt(3)/2 Y + R 
W2 = -1/2 X + sqrt(3)/2 Y + R
W3 = X + R

[This is Beta's formula with some added Rotation]

You need to think about the available ranges for your motors. I am guessing it can take a PWM signal of +/-255, so either the input or the output has to be adjusted somewhat. (It's not that hard...)

A good paper with details

To answer your specific questions: Vector projection is essentially what you are doing here. You apply it by having a matrix M, your input from the joystick I and your output to the motors O. Thus O = M * I;

M = [(-0.5 -sqrt(3)/2    +1)
     (-0.5 +sqrt(3)/2    +1)
     (1     0            +1)]