Physics help.

So there is a physics problem I can probably solve but almost certainly need help with. I cannot put my Gyro/Accelerometer(IMU) at the center of rotation of the ball robot, If i could the accelerometer would be useless and the gyro would need no translation. What I need is the matrix transform for if the IMU assuming it is not at the center of the ball. How do I translate normalized gyro measurements into robot rotation? How do I translate normalized accelerometer measurements into robot rotation? Is the derivitive of one the other? I am looking to make a PD or PID controller.

Gyro working now.

I finally got some time and now not only is the accelerometer working but so is the gyro, so now I should be able to write a PD controller very easily and add the I if I need to.

BOM

Videos

Conceptual design

To save on motor cost and for power efficiency I went with a  holonomic motor setup. Translation in X and Y and rotation about Z are possible with most land rovers. This can be accomplished with 3 motors if designed correctly but must use mecanum or omni wheels.

The inverse kinematics of a ball balancing robot are little different from a three omni wheel robot that does not ballance, equations for inverse kinematics are basically the same. This article provides a decent overview of those inverse kinematics .

This is a blog about creating a robot to balance on a ball and hold my beer. The idea comes from this video  and others like it. The source code and bill of materials live on github.