Dear Readers,
This is my understanding of the sensor's placement for a balancing system, the ideas are off the top of my head and should not be considered right unless you feel they are correct or can be proven. Feel free to correct me, I am writing my Thesis at the moment and any input would be encouraged. Without further ado, my answer.
I tend to agree with TJL for Gyros
Gyros sense rotation only, not acceleration.
Rotation is a "system wide" variable for a rigid system
They can be placed anywhere you want, even at the very top of the bot
I agree Kas. The basic calculation to vary between translational and rotational movement is v = wr, where the translational velocity (v) can be found by multiplying the angular velocity (w) by the radius it is rotating (r).
As the gyroscope a.k.a angular rate sensor is able to read only rotational velocities, it may be placed anywhere on the robot, assuming that the sensing axis is parallel to the axis of rotation of the mechanical system.
Accelerometers measure both static acceleration (gravity) and dynamic acceleration (movement). They are also very noisy.
Static acceleration is system wide, dynamic acceleration (as speed) is not
As we are interested only in gravity measurement, we want dynamic acceleration as low as possible.
The best place for reducing rotational acceleration is near the the motor axis.
I partly agree with this. Accelerometers sense, as the name suggests, the acceleration along the corresponding sensing axis. However, I stray slightly from your definition Kas.
Your definition of dynamic acceleration seems more like the rate of change of acceleration, which, if anyone is interested it is known as a Jerk. I understand what you were trying to say though!
However, back to the point. In terms of a static system, the only acceleration the accelerometers will sense will the the earth's gravitational pull referenced to each sensing axis. This, for a balancing robot, is what you require from the sensor. However, the situation is complicated when the mechanical system also has translational acceleration. The Accelerometer sense translational accelerations, not angular acceleration. Returning back the to previous equation. As the rate of the angular velocity changes (i.e. angular acceleration) so does the translational acceleration of a particular point of the mechanical system, proportional to the radius of rotation.
So, what does this all mean? placing a gyroscope (theoretically) should be ok to place anywhere, paying close attention to the sensing axes.
Placing the accelerometer as close to the rotating axis is best. This can be proved considering if you placed the accelerometer precisely on the axis ie r = 0. Therefore the translational acceleration of the system will contribute wr = wx0 = 0m/s/s.
Does everyone agree? Too complicated? INCORRECT?
Regards!
P.S. Mentioned previously, I am writing my Thesis and also sitting exams therefore have had no time to program, however my more recent attempts of balancing can be seen here
-> Forward Motion - YouTube (moving forward for a given time stopping while balanced)
and
-> Balanced-unstable.MOV - YouTube (balancing, very unsteady!)
Regards, AGAIN! ;D