## Euler Quaternions and the Trouble with Y-Up Space, Part 2

Welcome to my series of blogs about Eulers and Quaternions! This is part 2/2 of the series. See part 1 here. This series aims to talk about some of the difficulties that I’ve had in dealing with Euler <-> Quaternion transformations. In part 1, I talked about the difficulties around finding a transformation from Quaternion to Euler given a non-standard order of operations in the Euler and what it takes to actually derive the equations required to perform that transformation. We left off with equations that were useable; they did not however take into account the rotational singularities at the poles. For the unfamiliar, Euler rotations introduce two singularities into the space of rotations at the north and south poles of the rotation globe. As the Pitch approaches +90 or -90 degrees, the Yaw and Roll components degenerate further and further into a single rotational axis. This is known as gimbal lock.

As can be seen, the quickest way to detect for a singularity is to check if the Pitch component is +/-90. As detailed here (section Singularities), we can efficiently detect this by reusing our equation for pitch:

- Θ, pitch: asin(-RPY
_{12}) = asin(2(q_{0}q_{3}– q_{1}q_{2}))

Given the fact that at +/-90 degrees, sin(Θ) = 1/-1:

1 = 2(q_{0}q_{3} – q_{1}q_{2}), -1 = 2(q_{0}q_{3} – q_{1}q_{2})

0.5 = q_{0}q_{3} – q_{1}q_{2}, -0.5 = q_{0}q_{3} – q_{1}q_{2}

So test = q_{0}q_{3} – q_{1}q_{2
}

If test is less than -0. 4999999 then we can assume a pitch of -90; if test is greater than 0.4999999 then we can assume a pitch of 90. We have a choice for what we could do with the degenerated yaw/roll rotational axis when a singularity is detected. Following in the footsteps of euclideanspace, I chose to put all rotation into the yaw. This means that we can assume an angle of 0 for roll. Applying all of this through our previous equations, we get:

**When test > 0.4999999:
**

- Φ, roll: 0

- Θ, pitch: 90

- Ψ, yaw: 2 * Math.atan2(q
_{3}, q_{1})

**When test < -0.4999999:
**

- Φ, roll: 0

- Θ, pitch: -90

- Ψ, yaw: -2 * Math.atan2(q
_{3}, q_{1})

You can get a better feel for this by following the derivation on the euclideanspace page. We now have a complete chain to convert a quaternion into a Y-Up Euler rotation! What’s nice about this is that we can now transform any order-of-operations Euler rotation into any other order-of-operations Euler rotation by transforming the first Euler representation into a quaternion then deriving the equations for getting an Euler rotation back from a quaternion with the second Euler representation. ie Roll, Pitch, Yaw into Yaw, Roll, Pitch, etc…

Hopefully this helps somebody out there because it sure as hell stumped me when I initially ran into this problem. Thank you for reading.