## Euler Quaternions and the Trouble with Y-Up Space, Part 1

Yesterday I did something that I didn’t think I was capable of. It involved quaternions, matrices and a decent amount of math. Let’s start with the problem definition:

- We have a system that tracks rotations using quaternions
- We would like people to be able to use Euler coordinates (Tait-Bryan angles if you’re picky)
- Our world is Y-UP

Doesn’t sound too hard. In fact, half of it was fairly easy. Turning a set of Euler coordinates into a quaternion is as simple as creating three quaternions using angle axis pairs derived from the incoming Euler rotation. For us, the three Quaternions were as follows:

- [
**Angle:**Yaw,**Axis:**[0, 1, 0]] - [
**Angle:**Pitch,**Axis:**[0, 0, 1]] - [
**Angle:**Roll,**Axis:**[1, 0, 0]]

We create a combined rotation quaternion by conjugating the three quaternions together in the following order:

Roll * (Pitch * (Yaw) * Pitch^{-1}) * Roll^{-1}

This is all that is required to produce a quaternion from a set of Euler coordinates. Now, we also had to retrieve Euler coordinates from a given quaternion. This is more difficult than taking Euler coordinates and converting them into a quaternion as the task requires a fairly solid understanding of the math. This too would be simple, if it weren’t for the fact that *the entire Internet* agrees that Z-Up is the way to go for Euler/Quaternion conversions. From what I could tell, the equations for converting a rotation quaternion into a Y-Up Euler coordinate do not exist! (On the internet).

To my shock, horror and dismay, this meant that I had to derive the equations myself. Needless to say, this was a daunting task. But, after an arm and a leg… and the other arm… and the other leg… and parts of my face/body and an assortment of pieces from whichever organs remain… I was victorious. My glorious victory over the maths was quite euphoric.

First, let’s talk about why Y-Up Euler coordinates are so different from Z-Up Euler coordinates. The idea behind Euler coordinates is that given three angular values, you apply three rotational transformations in a predetermined order around three dynamic or constant axes (depending on convention) and arrive at a destination rotation.

Rotational transformations are not commutative, that means that applying them in a different order will produce a different rotation all-together. I won’t get into *why* that is but it has to do with the fact that the space of all 3D rotations is something akin to the surface of a 4 dimensional hypersphere and applying a rotation is like *traveling* from one point on the surface of that sphere to another surface point while always remaining on the surface. (Feel like going for a dip in some advanced math? Read here).

The effect is that a rotation quaternion composed from rotations around axes in the order ZYX is fundamentally different from one composed from rotations in the order YZX, which is what we want. So, the equations must be derived. I won’t delve very much into *why* the following works as that could take several blogs. First, we must build a rotation matrix that will compose our Euler coordinates in the order that we want. This is as simple as getting the angle-axis matrix formula and plugging some numbers into it. That formula is the following:

Where c is cos(θ), s is sin(θ) and C is 1-c, θ is the angle of rotation and (x,y,z) are the vector components of the axis. This is taken from here . By plugging in axis and angle, we construct three rotation matrices:

**Yaw, about [0, 1, 0], angle sign Ψ (Psi):**

**Pitch, about [0, 0, 1], angle sign Θ (Theta):**

**Roll, about [1,0,0], angle sign Φ (Phi):**

**Now, we multiply the matrices together in the order (Roll)(Pitch)(Yaw), producing the following:**

**Next step is to refer to the Quaternion -> Rotation matrix formula:
**

This formula was taken from here. The quaternion components are [w, x, y, z] -> [0, 1, 2, 3]

Finally, to retrieve the Euler coordinates we must match the matrix form of the rotation quaternion to our RPY rotation matrix exploiting cells where the values are easy to retrieve. In this case, the values are as follows (Notation, Matrix _{Row Column} where row and column are in the range [1,3]):

- Φ, roll: atan2(RPY
_{32}, RPY_{22}) = atan2(2(q_{0}q_{1}+ q_{2}q_{3}), q_{0}q_{0}– q_{1}q_{1}+ q_{2}q_{2}– q_{3}q_{3}) - Θ, pitch: asin(-RPY
_{12}) = asin(2(q_{0}q_{3}– q_{1}q_{2})) - Ψ, yaw: atan2(RPY
_{13}, RPY_{11}) = atan2(2(q_{0}q_{2}+ q_{1}q_{3}), q_{0}q_{0}+ q_{1}q_{1}– q_{2}q_{2}– q_{3}q_{3})

This is almost enough. What is left is to test for pitch singularities (+90, -90) and treat them specially. Since this blog has gone on for far too long, I will stop now as what has been given is certainly enough to get a similar system working. In my next blog, I will detail how we could cheaply test for the singularities and what we could do in that case. You can do some advance reading on that here.