# Are solid-state attitude indicators subject to gimbal lock?

With all the shiny new glass cockpits it would seem that the days of the spinning mechanical gyro (and associated tumbling due to gimbal lock) should be over: Sparing everyone the boring math it should suffice to say that solid-state gyros can be engineered and built in such a way that gimbal lock is impossible, but I'm not certain that's how they're actually designed.

Do modern AHRS systems with solid-state gyros (or replacement electronic horizons like the RC Allen 2600 series) still suffer from gimbal lock, or do they provide true 3-dimensional freedom?

I'm interested primarily in answers from a light General Aviation standpoint, but answers about electronic gyros on commuter and transport category aircraft would be interesting too.

Yes and no. Gimbal lock is a physical AND mathematical phenomenon. Physically, gimbals sharing the same axis can cause a lock, and mathematically, a representation of a multi-axis rotation can be in a condition which is singular, depending on how it is represented and calculated internally.

Normally, we discuss and think about aircraft attitude intuitively as set of angles: roll, pitch and heading. In mathematical circles, this is known as a set or 3-2-1 Euler angles. If you look into the boring (or exciting, if you are that kind of person) math, you can see a vector-matrix equation relating the change in Euler angles given a set of angular rates given in the frame of the aircraft, by common convention these are referred to as p, q and r for roll rate, pitch rate, and yaw rate respectively. The way these rates affect the change in the angles depends on the current state of the the angles, e.g. when the aircraft is pitched, a body-frame yaw rate will alter the pitch angle. Mathematically, this means that the "state matrix" is state-dependent and must be constantly updated using the freshest set of states available. Any set of Euler angles will have a set of states which causes the state matrix to become poorly conditioned, resulting in numerical instability at those conditions. For the roll-pitch-heading set of angles, this happens when pitch is + or - 90 degrees.

The way that we escape the singularities when writing the software for an AHRS is by representing the attitude in a non-singular form. This done by using more numbers to create the representation, then adding constraints which preserve uniqueness. A 3x3 direction cosine matrix (DCM) can do this, It has constraints which cause it's eigenvalues to lie on a unit circle in the complex plane. My favourite representation, however, is a quaternion constrained to have unit length. The quaternion can be thought of in terms of Euler's rotation theorem. There's that Euler guy again, he must have been some kind of genius. Anyhow, the idea is that you can define an axis of rotation with three numbers, and use a fourth to represent an angle about that axis to rotate. This gives you a system free of "mathematical gimbal lock" The state matrix remains adequately conditioned under all states. All you need to do to keep things friendly is to normalize the quaternion from time to time, and you always have a "background" state that you can translate into aviator-friendly Euler angles.

• Does this "gimbal lock-less" system manifest on the instrument display in a glitchy way as it transits/avoids these singularities, a-la the black cat passing by in The Matrix. i.e. "Deja vu is a glitch in the matrix. It happens when they change something."? Sep 20, 2015 at 5:13
• @radarbob, no, because there are no transitions involved in avoiding the singularities. The system simply always uses a more complicated representation that does not have them. Sep 20, 2015 at 21:56

AHRS cannot suffer from gimbal lock (as there are no gimbals that can share axes):

Conventional gyros are also susceptible to gimbal lock under certain conditions. The AHRS is an all-attitude system and is free from such problems. — from A Layman's Guide to AHRS

That said, AHRS systems can still be tumbled - I know that for a while an air show routine was flown in a (then) Columbia 400 with G1000; the pilot remarked that the system Xed out only briefly while executing inverted maneuvers. In comparison, the Avidyne AHRS at the time required a full 2+ minutes of straight and level flight before resetting itself.

• The G1000 has built-in attitude limitations beyond which it will not read correctly. I will try to look them up next time I'm near a technical manual. But like you said, since they can't tumble it's entirely possible to make an AHRS capable of accuracy at any attitude. Feb 11, 2014 at 18:53
• Yeah, I don't recall what they are either - I'll try to take a look at the manual I've got at home this evening.
– egid
Feb 11, 2014 at 19:26
• Specific attitude limitations would lead me to believe they're using X/Y/Z (Euler) coordinates directly instead of deriving them (so "It's not gimbal locked" (because there are no gimbals), but the effect is largely the same (big red X on the attitude display when you exceed the limitations until the gyro can be reset)) Feb 11, 2014 at 20:23
• Hm, I still don't understand how they get the limits. The gyros are always orthogonal, so the input has the necessary information at any attitude. Rotating the accumulated vectors is possible in any representation given correct formulas and with incorrect formulas (e.g. just adding the pitch axis to the inclination directly) will noticeably loose precision even at shallow bank angles. So it seems strange to me they could get away with incorrect formulas (and the "correct" formulas are not that complicated anyway). Feb 11, 2014 at 20:35
• @JanHudec the G1000 AHRS also uses the magnetometer for some information (it's part of why the in-flight realign takes so little time) so it's possible that outside of a certain realm the data isn't verifiable between the sensors. A dual AHRS system might be able to compare the two, but what most GA aircraft have maybe cannot?
– egid
Feb 11, 2014 at 23:57