Omniwheel vehicle speed consistency

I am currently working on an omniwheel drive vehicle with a rather... unique control system. I am estimating it will be done by January. It involves wireless Arduino serial communication via XBEE, water-jet cut custom aluminum parts, h-bridge PWM DC motor controllers, and mutilated IKEA furniture, haha.

This post is about some speed control considerations on a vector-based control algorithm. Depending on what direction you going in, the speed will differ, because the wheel orientations change. Obviously, for a good human-operated vehicle this speed difference must not be too great. If the direction is aligned with the wheel's normal rolling direction, the speed will be at a maximum. The wheel most aligned with the direction of movement will dictate the top speed of the vehicle. All other wheels will provide moment balance and extra torque.

Below are some plots showing the normalized (meaning min is 0, max is 1) speed vs. direction, in radians. The different plots are for different numbers of motors/wheels installed on the vehicle.

As you can see, for 3 and 4 motors the speed difference is quite significant. For three motors, the lowest speed depending on the direction of vehicle movement can be about 14% lower than the normal speed (in the graphs it would be 1). Four motors fares even worse, which can be surprising at first glance. Four motors give a minimum speed around 30% lower than the normal speed.

A few interesting patterns can be seen here. The number of dips equals the number of wheels/motors if the number of wheels/motors is odd, and twice the number of wheels/motors if the number is even. Also, locally the speed difference is minimized if the number of wheels/motors is odd. 5 wheels/motors is a reasonable number that gives an acceptable speed difference, <5%. Of course, all of this can be fixed to give virtually no speed difference consistent speed in the controller's software.

This was a very interesting investigation! The code to generate the plots above was written in MATLAB and is included below (MATLAB is not free, but SCILAB is! And it is very similar! It will probably run the code with little or no modification!):

%FPS bot testing

offsetAngles = [0 0 0 0 0];%Just the difference in angle between frontmost motor and forward direction.

motorCounts = [3 4 5 6 7];

motorAngles = 2*pi./motorCounts;

for j = 1:numel(motorCounts)

mvec = zeros(motorCounts(j),2);

for k = 1:motorCounts(j)

mvec(k,:) = [cos(offsetAngles(j)+(k-1)*motorAngles(j)) -sin(offsetAngles(j)+(k-1)*motorAngles(j))];

end

resolution = 10000;

mag = zeros(resolution,1);

theta = linspace(0,2*pi,resolution)';

directionVector = zeros(resolution,2);

directionVector(:,1) = cos(theta);

directionVector(:,2) = sin(theta);

currentMag = zeros(motorCounts(j),1);

for k=1:resolution

for i=1:motorCounts(j)

currentMag(i) = abs(mvec(i,:)*directionVector(k,:)');

end

mag(k) = max(currentMag);

end

figure;

plot(theta,mag);

ylabel('Normalized speed magnitude');

xlabel('Direction (radians)');

title(['Normalized speed magnitude vs. direction, ' num2str(motorCounts(j)) ' motors']);

end