# Why do we measure translational accelerations but angular velocities in an IMU aircraft?

I am wondering why we measure for attitude of an aircraft its angular rate (instead of angular acceleration) and for the translational channel the acceleration instead.

The computers can integrate both signals to get the current attitude and position of the aircraft.

In another question on this forum someone stated:

"I think that the biggest problem is that in a coordinated turn, the angular acceleration is 0 and the lateral proper acceleration is also 0, so without measuring angular rate, an IMU would be unable to distinguish a coordinated turn from straight-and-level flight! That's a critical flaw."

To my understanding: if a computer just integrates the signals you will know your attitude and position. So in order to me more precise one should use the acceleration values (and reduce the bias/Drift).

Can you help me in this?

Thank you!

• After further reflection I think the intent of the question is clear enough to answer; deleting comments. If the answer isn't helpful, perhaps the question could be clarified in some way, if it is possible to do that without invalidating the answer. Apr 4 at 14:45
• Working backwards, a control input to create a turn results in constant lateral acceleration and constant rate of turn (angular). But for applications such as aerobatic aircraft, roll axis acceleration may be just as important as (steady state) roll rate. It depends what sensors you have and what you want your software to do. Interesting side bar would be what hydraulicly boosted larger ailerons could do for something like a P-38. May 6 at 12:31

The basic, underlying argument is that an IMU contains a linear accelerometer and angular gyroscope, because these are simple, self-contained instruments which rely on basic physics to obtain measurements. They are the simplest instruments for the job. But there are a couple of aspects which I tried to break up as follows:

1. Pretty much all IMUs measure the translational acceleration and angular velocity because it is easy to do so. Accelerometers and gyrometers are pretty old and also easy to construct as they hinge on basic physical principles. In contrast, measuring translational velocity and translational position or rotational position (attitude) directly, is much much harder, and in general not possible with basic physical principles (at least not in all three dimensions).
2. Measuring rotational acceleration is possible, but again, not as simple as measuring rotational rate. You can arrange several translational accelerometers in a special geometric figure to obtain rotational acceleration, but it is complicated and requires post-processing. There are even rotation accelerometers available for purchase. However, the much bigger problem is that you do not want to use rotation acceleration to obtain attitude. You stated that an acceleration signal can simply be integrated twice to obtain position, but every integration accumulates a bias. This is inevitable with even the most expensive gyroscopes: If you integrate the rotational velocity to obtain attitude, the attitude estimation will accumulate an error (the cheaper the gyroscope, the quicker the error accumulates). Therefore, in general you are happy that you already have rotational velocity, and do not need to obtain it by integrating rotational acceleration which would incoporate even more error.
3. It depends a bit on what you would like to achieve: The goal of pretty much all IMUs is to obtain an estimation of the orientation of the device. The combination of rotational velocity and translational acceleration allows you to do this with basic sensors in a self-contained package. This is ideal for a lot of applications which shall not rely on external sources such as the position of the sun, or GPS satellites for example.

As an important side note: You wrote

"I think that the biggest problem is that in a coordinated turn, the angular acceleration is 0 and the lateral proper acceleration is also 0, so without measuring angular rate, an IMU would be unable to distinguish a coordinated turn from straight-and-level flight! That's a critical flaw.

This is most definitly wrong. A coordinated turn incoporates non-zero angular rates in at least two axis. Therefore, you can most definitly distinguish a coordinated turn from straight-and-level flight with an IMU!

Here is a link to a simple, inexpensive self-contained instrument with no moving parts that measures yaw rate. The nature of piezoelectric rotation sensors is such that no integration is needed to obtain this value. The instrument can be switched on in mid-flight--even while turning-- and will immediately provide reasonably accurate values.

How would it improve accuracy to construct an alternative type of sensor to somehow directly measure angular acceleration -- if that would even be possible-- and then integrate the acceleration values to obtain angular rate?

Similarly, if you are interested in calculating aircraft attitude, wouldn't it be better to measure angular rate and integrate once, than to measure angular acceleration and integrate twice?

The question seems to imply that there is an advantage to measuring acceleration rather than actual rate whenever possible. This is a misconception.

The nature of piezoelectric sensors is such that it is possible to directly measure angular rate. Unfortunately it is not possible to directly measure linear velocity-- if it were, all navigational computations would be greatly simplified. Instead, linear acceleration is measured directly, which then must be integrated to obtain linear velocity, which must be integrated again to determine position.

• Re opening link: these instruments have been built in various versions, some of which go through a calibration phase on start-up, others of which don't. Answer would be improved by changing link to a specific version that is clearly documented not to go through any calibration phase, but I don't have that handy at present. Apr 4 at 14:50
• Additional links to various versions of same, some of which also contain a lateral linear accelerometer driving a slip-skid ball presentation: kitplanes.com/a-modern-turn-coordinator-belite, generalaviationnews.com/2016/09/21/…, flyingmag.com/…, airframer.com/news_story.html?release=13975. As an aside, original units definitely measured yaw rate, not using a canted gyro axis as would an actual "turn coordinator"; can't speak definitively to latest. Apr 4 at 16:54