# Does a gyro need to be placed at the center of rotation?

When using a gyro chip to measure angular velocity, does it matter whether it is located at the center of rotation?

I've seen it mentioned online without much explanation that it should be placed at the center, but in my introductory physics courses I learned that the angular velocity is the same at every point on a rigid body. Wikipedia seems to confirm this in the last section of Angular velocity, but I'm not sure I'm interpreting it correctly.

I expect that there is normally some non-ideal coupling of the acceleration terms into the gyro measurement -- is that the main problem?

Is there something obvious that I am missing?

• I wonder if the rigid body theory says the same for the accelerometer...? – SDwarfs Aug 21 '12 at 15:39
• @Stefan - Accelerometers will generally be different, for both of these cases: ① If the object is rotating, even if the angular velocity is constant, each point will see a centripetal acceleration of $\omega^2\cdot r$ pointing toward the center of rotation, where $r$ is the distance from the center. ② If the angular velocity is changing (non-zero angular acceleration), the tangential acceleration is also proportional to the distance from the center. – Reinstate Monica Aug 21 '12 at 17:23
• From what I've seen in RC helicopters, almost always the gyro is not in the center of rotation and they work fine. – AndrejaKo Aug 21 '12 at 17:38
• Thx Justin, thats totally correct. The difference is exacly when rotating. The distance from the rotation center will differ for some points and these will move in different speeds, therefore the acceleration must be different. – SDwarfs Aug 21 '12 at 17:38

I just checked this by "thinking" about that and noticed, that for an ideal rigid body (!) it doesn't matter where you place it.

I just imagined that like this:

• You have a gyro at each side of a stick (axes of the gyros parallel to each other).
• You rotated the stick about one end (e.g. x axis by 90°).
• Observation: the other axes is also rotated 90° about the same axes, same direction, seen from center of the other gyro.

## Theory vs. Practical "Rigid Body":

1. You won't have an ideal rigid body. Your material is at least slightly flexible...
2. If you want to combine data from 2 sensors, e.g. accelerometer / gyro the axis of the two sensors should be pointing in the same direction (or you need even more sophisticated math). And this coupling should be very tight, e.g. not flexible or swinging (else you'll measure a lot of garbage); Rule: the longer the distance the more influence of those effects.

You possibly want to measure the orientation/position of something (e.g. wings of a plane or the "body" of quad copter). So the gyro should be fixed to exactly that.

PS: Thank's for pointing me at this issue. This helps me a lot (need those sensors for research).

Yes, there is a reason for this. For example, consider pitch of an airplane. If the plane pitches up 20 degrees, then the angle at the center of the plane is 20 degrees. If the gyro is mounted in the nose...still 20 degrees. The entire plane is pitched up 20 degrees, so it doesn't matter where you measure this. So this leads many people to assert that the mounting point doesn't matter.

HOWEVER, notice that when the gyro is in the center, it is experiencing 1g all the time, independent of the pitch-up. If the gyro is in the nose, it will experience positive g force while pitching up, and negative g force when the plane is pitching down. Typically gyros will have some sensitivity to both g force and vibration, due to minor asymmetries.

Consider for example this application note from Analog Devices: http://www.analog.com/media/en/technical-documentation/technical-articles/MS-2158.pdf

According to this, the g-sensitivity for cheap gyro (and anything in a model airplane will be "cheap"), can be about 0.3degrees/sec/G. Well, that is not a lot. If your plane nose experiences 3g during a 20-degree up-pitch in 0.5sec (that seems like a rather violent bump), then the rotation is 40deg/sec, and the error term due to g sensitivity would be 3g * 0.3 or about 1 deg/sec. That is only 2.5% error. Well, that is not a lot, but not insignificant either. And as you can see from the application note, other components have less G sensitivity.

So the bottom line is - yes there is some theory behind it. If you want to be conservative, mount it in the center. But I doubt you would notice the difference in typical practice. Also, different MEMS components are different, and it would take some research and calculation to figure out the precise significance in your application.

• The nose won't experience 3g during your pitch. You can actually calculate your tangential acceleration based on the eccentricity from the axis of rotation. The error will be smaller. Of course, during high-g maneuvers, the whole plane will be at hyper-g, and you'll get errors regardless of whether the gyro is at the axis of rotation or not. – Scott Seidman May 23 '17 at 17:58