# Gyroscope Measurement and Angular Rates

by MHR-Love
Tags: angles, frame of reference, gyro, sensor, velocities
 P: 15 Do gyroscopes give the angular rates about its local axes or about a frame that is fixed and initially coincides with its local frame? I mean: would this sensor give the rate of change of the Euler's angles or the angular velocity ω (about the inertial frame)? One more question: I know that Gyroscopes measure the rate of change of rotation (angular velocities), so I think that they actually don't give any information about the angle values and these angles can later be found by differentiating these rates. Is what I think true? Thnx
 Mentor P: 13,625 A three-axis gyro gives angular velocity with respect to inertial but expressed in case frame coordinates. You can't integrate angular velocity, nor can you integrate Euler angle rates. Rotation in three dimensional space doesn't work that way. What you can do is use the angular velocity to find the derivative of the transformation matrix or quaternion and integrate this. Even this is tricky. Typical numerical integration techniques will yield something that isn't an orthonormal matrix or isn't a unit quaternion. A standard trick is to force the result to be an orthonormal matrix or to normalize the quaternion. A better approach is to use Lie group integration techniques, but that's pretty advanced stuff.
P: 15
 A three-axis gyro gives angular velocity with respect to inertial but expressed in case frame coordinates.
What do you mean by case frame coordinates? Is it same as the global frame which initially coincide with the local frame?

 You can't integrate angular velocity, nor can you integrate Euler angle rates. Rotation in three dimensional space doesn't work that way. What you can do is use the angular velocity to find the derivative of the transformation matrix or quaternion and integrate this. Even this is tricky. Typical numerical integration techniques will yield something that isn't an orthonormal matrix or isn't a unit quaternion. A standard trick is to force the result to be an orthonormal matrix or to normalize the quaternion. A better approach is to use Lie group integration techniques, but that's pretty advanced stuff.
Well, what I understood from you is that we need to find the relationship between the angular velocities ωx, ωy and z and the rates of Euler's angles (which are not orthogonal). This is handled by finding the matrix mapping these rates. After doing so, we can integrate to find position or differentiate to find acceleration.

My knowledge in quaternion actually is not that good.

Mentor
P: 13,625

## Gyroscope Measurement and Angular Rates

The case frame is the frame defined by the manufacturer of the gyro. It's some kind of gyro-fixed frame, specified by the manufacturer. If you mount the gyro in some goofy orientation, the gyro doesn't know that. You do. You need to transform case frame outputs of the gyro to the body frame of the object to which the gyro is attached. If the attachment isn't perfect (nothing is perfect), there are going to be misalignment errors between the case frame and what you think the case frame is. The gyro doesn't know these misalignment errors, and neither do you. You can deduce these misalignment errors, either by testing or if you have a smart enough Kalman filter.

You can't integrate angular velocity. Well, you can, but the results of that integration are not physically meaningful. You similarly can't integrate Euler rates. Using angular velocity to deduce how orientation has changed is non-trivial. If you don't know the subject at all, you unknowingly are verging on asking me to write a book.

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