Anton's Research Ramblings

Debunking 'Gimbal Lock'

The Two Types of Maths Chapter

When you read texts on 3d graphics you always get to what writers call the 'dreaded maths chapter'. I don't like this. As a pragmatist, I was never stimulated by pure maths and calculus, but found myself learning it later when it started to become genuinely useful for engineering and computer graphics applications. Writers in computer graphics texts tend to fall into two categories:

  1. Massively theoretical; providing extended proofs, with little or no discussion, leaning on other mathematical concepts that the reader may not have encountered before.
  2. Overly familiar and chummy, somewhat patronising the reader not to be 'scared' of the maths chapter, and brushing all of the details under the carpet.

The first approach only appeals to those with a pure mathematics background, or perhaps, more accurately - the narrow band of personality types to which that appeals. I don't like this because it cuts out 3/4 of a class from progressing, even though they are capable of understanding the material if presented differently. There is a certain level of elitism amongst computer scientists, who would more than happily pooh-pooh any students not up to that level...which would be fine if modern computer science programmes actually re-enforced calculus, and science fundamentals - which they don't. These days computer science degrees are basically glorified IT vocational training with very little depth at the undergraduate level, and expecting even fourth-level students to have this knowledge when they finally get to the graphics class is kind of uncool. It's actually not hard to learn the mathematics required for 3d transformations - I suspect even the most efficient way to learn because you have real tangible visual rewards when it goes correctly. I got a lot of traction from finding practical, hands-on ways to explain these concepts to students - the same way that it appealed to me. Now, that's not my only problem with approach number 1. Some very popular texts in the field tend to dump pages of proofs that don't always seem to be terribly applicable to the problems at hand - the "dazzle them with maths" approach to hide the gaps in the writer's knowledge.

The problem with the second approach is that, even at the expert level, writers in games and graphics often have a very poor grasp of the mathematics themselves, and lean on analogies instead of stepping breaking down and stepping through the problem logically. This is a bunch of nonsense. Enter: 'Gimbal Lock'.

Frightening Students

One of the most terrifying mathematical concepts to pop-up to graphics students is a concept used for rotation - Hamilton's quaternions. At some point, the student will get their head around matrix rotation, and will try to make either a free-moving 3d camera, or some sort of space-ship game, or a vehicle moving on slopes. It gets really hard to work out by combining fixed-axis X,Y, and Z rotation matrices. What you really want to is to derive a local X, Y, and Z axis, and perform a local (from the point of view of the object or camera being rotated) pitch/yaw/roll instead.

Typically, the student will try to combine a series of X, Y, and Z rotations. It will sort-of work, in some ranges of orientation, but come unstuck when you really need 3 axes to rotate at once. You can change the order around, but then it won't work in the original ranges of angle. If you open any graphics or games maths textbook you'll come to the Quaternions section, and get one of the two styles of explanation, as discussed above. If you're a pragmatic person, you're not going to be satisfied by either of them. Firstly, the proofs don't give you the practical functions that you're looking for, and raise more questions than they answer. You may even get a diagram in 2d explaining how a hypersphere in 4d expresses rotation, pulsating in size as the angle changes. The inter-dimensional horror! Secondly, in the 'buddy' style maths chapter you'll get some fear-mongering about gimbal lock, absolutely no analytical example of where or how this happens, and a copy-and-paste function to "solve" the problem without thinking about it.

Totally Backwards Gyroscope Videos

The go-to explanation of "gimbal lock" is a crudely-constructed 3d scene of 3 gimbals rotating with a miniature aeroplane in the middle of them. At some point the narrator will make 2 of the gimbals line up, and then the aeroplane can only pitch and roll, and the "yaw" gimbal also pitches, and it can no longer yaw. This is ignorant in so many different dimensions!

  1. This is not how a gyroscope works
  2. The gimbals are not locking
  3. The axes were never clean local pitch/yaw/roll because they are in a hierarchy
  4. The aircraft should be on the outside of the gyroscope - they are missing a whole extra orientation

The primary confusion is that the narrator has tricked themselves into thinking that their Z * Y * X combination of matrices are the same as a local yaw/pitch/roll, except that it doesn't work in certain conditions. Wrong. If you build one yourself, you'll discover that actually the axes are still global X, Y, and Z. The local axes are never respected. All they've done is build a parent-child relationship with the rotation matrices:

Only the bottom-level rotation is local to the miniature aeroplane. It's the same as a skinned character animation where the ankle bone has its own local orientation, but it also inherits the knee, and the knee inherits the hip. If the hip rotates 45 degrees around Z then the knee around Y is going to give you something half-way between a roll and a yaw. The 3d "gimbals" drawn here are a bunch of hokum. I made one, to try it out:

This kind of "gimbal lock" demo is rubbish: the problem is actually that the rotations are in a hierarchy of rotations around global axes - they were never around local axes to begin with. Gyroscopes don't work like this in the real world - they do the opposite.

Yeah, that's not how a gyroscope works. First of all - the aircraft or ship is on the outside of the gimbals, not in the middle, obviously. It has its own 3d orientation, independent of all 3 gimbals. The object in the centre of the gimbals - usually a flywheel, but it can be a billiards table or a pot in a ship's galley - is supposed to always stay facing the exact same way. Whenever the aircraft rolls or pitches or yaws there is an axis that can move to adjust, so that the billiards table isn't upset. Now, in a real gyroscope, the ship might turn exactly 90 degrees which makes two of the gimbals line up. That might upset the billiard balls because you lose an axis entirely. This is easily solved by adding a fourth gimbal. We can't do that with the rotation matrices. Our rotations are also no longer pitch/yaw/roll as soon as the parent 'gimbals' are not perpendicular i.e. right away, not just when just they line up to 90 degrees. So the analogy is wrong, as well as the understanding of the problem.

Authors and narrators will blame Euler and gimbal lock...um no...you've just misunderstood what you're doing. These matrix rotations are all correct, but they are around global axes. The problem here is actually that what you really want is 3 rotations around arbitrary axes which we will make perpendicular to the object's current facing orientation. You can find functions to create a rotation matrix about an arbitrary line (axis). I think it's very unlikely that you've used a variant of this that is subject to an incomplete domain, and really does undefined or locked orientations. By far the most elegant way to compute this is with Hamilton's 4d quaternions. The fourth dimension is used to resolve the 3d rotation very neatly.


Why is this whole topic so badly explained and mis-informative? Are the maths really that hard or 'scary'? Are the hard-core mathematicians really hiding from explaining the problem properly? Is Hamilton rolling in his grave or pitching?