# Converting from quaternion to angular velocity then back to quaternion

I am trying to write a small physics simulator. I have q(t), the quaternion orientation in the inertial frame.

I would like to simulate the gyroscope input, so I need to get the moving frame angular velocity.

My first question is, how to calculate the temporal derivative of a quaternion by sampling q(t).

I thought about two solution $(q(t+dt) - q(t))/dt$ and $q(t+dt)*q(t)^{t}$ where the superscript t is for the conjugate. I found the second one by thinking that it would give me the change of $q(t)$ during $dt$ in the inertial reference.

The first one seems weird to me for a rotation. I am not sure if they are both correct or equivalent. If not, why ?

My second question is, assuming I have the correct $dq(t)/dt$, How do I get:

• the body frame angular velocity (to simulate gyroscope input) $\omega_B$
• And then from the angular velocity, how do I retrieve the body frame quaternion temporal derivative. $dq_B(t)/dt$
• Then how to integrate that temporal derivative to get the new attitude $q_B(t+dt)$ with respect to the body frame
• I assume that from there, I could retrieve $q(t+dt)$ by composing with the rotation $q(t)$: $q(t+dt) = q_B(t+dt)*q(t)$ ? Is that correct ?

I have made some attempt by using https://www.astro.rug.nl/software/kapteyn/_downloads/attitude.pdf . But I was very unsuccessful so far http://paste.awesom.eu/lzKh

Thank you in advance for your help

• Something worth noting: The derivative of a rotation quaternion isn't another rotation quaternion; instead, it's a purely imaginary quaternion which essentially denotes the instantaneous axis of rotation. Ken Shoemaker has some notes about this from SIGGRAPH in the early '90s that you should be able to find with a little bit of hunting. – Steven Stadnicki May 16 '17 at 3:48
• Hestenes populariztion of Grassman/Clifford "Geomtric Algebra" articulating Rotors/Bivectors/Multivectors with the familiar Cartesian 3D Vectors of the current universaly taught Gibb's Vector Algebra seems better suited to a modern Physics Classical Mechanics simulation environment than Hamilton's Quaternion Algebra or Gibb's later Dyadics – f5r5e5d May 16 '17 at 4:28

## 1 Answer

You have a curve $q\colon I \to \mathbb S^3$, where $I$ is a time interval and $\mathbb S^3\subseteq\mathbb R^4$ is the set of all unit quaternions. Here, we represent quaternions as 4-dimensional vectors. Note that if a quaternion should encode an orientation or rotation, it has to be a unit quaternion.

Since $q(t)\in\mathbb S^3\subseteq \mathbb R^4$ lives in the linear space $\mathbb R^4$ you can calculate its time derivative $\dot q(t) = \tfrac d{dt}q(t)$ by \begin{align}\tag{1} \dot q(t) = \tfrac d{dt}q(t) = \lim_{h\to0} \frac{q(t+h) - q(t)}h, \end{align} meaning that $\bigl(q(t+\delta)-q(t)\bigr)/\delta$ is an approximation to $\dot q(t)$ for small $\delta$. We can see that this approximation lives somewhere in $\mathbb R^4$. More specifically, it is an element of $T_{q(t)}\mathbb S^3$, the tangent space of the sphere at the element $q(t)\in\mathbb S^3$, because $q(\tau)\in\mathbb S^3$ for all $\tau\in I$.

Since $\mathbb S^3$ is a Lie group it is favourable to represent the velocity of $q(t)$ by a vector $\Omega(t)\in\mathbb R^3$ that fulfills $$\dot q(t) = \frac12 q(t) * \begin{bmatrix}0\\\Omega(t)\end{bmatrix}.$$ This can be thought of as mapping the tangent space $T_{q(t)}\mathbb S^3$ to the tangent space $T_e\mathbb S^3 = \{[0,x^T]^T\in\mathbb R^4\}$ with the neutral element $e=[1,0,0,0]^T$. The $\Omega(t)$ is actually the angular velocity in the body frame. We can calculate $\Omega(t)$ from $\dot q(t)$ by $$\tag{2} \Omega(t) = \operatorname{Im}\bigl(2 \overline{q(t)}*\dot q(t)\bigr),$$ where the overline represents quaternionic conjugation (which is actually the inversion on $\mathbb S^3$) and $\operatorname{Im}$ extracts the imaginary part, hence just drops the first component (which has to be zero here). Now we can put the limit expression for $\dot q(t)$ in here an get \begin{align*} \Omega(t) &= \operatorname{Im}\bigl(2 \overline{q(t)}*\dot q(t)\bigr)\\ &= \operatorname{Im}\left(2 \overline{q(t)}*\lim_{h\to0} \frac{q(t+h) - q(t)}h\right) \\ &= \operatorname{Im}\left(2 \lim_{h\to0} \frac{\overline{q(t)}*q(t+h) - \overline{q(t)}*q(t)}h\right) \\ &= \operatorname{Im}\left(2 \lim_{h\to0} \frac{\overline{q(t)}*q(t+h) - e}h\right) \\ &= \lim_{h\to0} 2\operatorname{Im}\frac{\overline{q(t)}*q(t+h)}h, \end{align*} since $\operatorname{Im}e = [0,0,0]^T$. This means that $2\operatorname{Im}\overline{q(t)}*q(t+\delta)/\delta$ is an approximation to $\Omega(t)$, the body frame angular velocity, for small $\delta$.

With equations (1) and (2) from above, you can transform the derivative $\dot q(t) = \tfrac d{dt}q(t)$ into the body frame angular velocity $\Omega(t)$ and back.

Note that $2\operatorname{Im}q(t+\delta)*\overline{q(t)}/\delta \approx \omega(t)$ gives an approximation to the angular velocity with respect to the inertial frame. It simply holds $\omega(t)=\Omega(t)^{q(t)}$, where $x^{q(t)}\in\mathbb R^3$ is the application of the rotation $q(t)$ to $x$ and is defined by $$\begin{bmatrix}0\\ x^{q(t)}\end{bmatrix} = q(t)*\begin{bmatrix}0\\ x\end{bmatrix}*\overline{q(t)}.$$

The simplest way to numerically integrate the attitude is to use a forward Lie-group Euler method. If you somehow calculate $\dot q(t)$ or $\Omega(t)$ you can get the new attitude $q(t+\delta)$ by $$q(t+\delta) = q(t)*\widetilde{\exp}(\delta\cdot \Omega(t)),$$ where $\widetilde{\exp}\colon\mathbb R^3\to\mathbb S^3$ is essentially a Lie group exponential function which is defined by a power series, but can, for unit quaternions, be written in closed form as $$\widetilde{\exp}(v) = \cos(\tfrac12 \|v\|) + \frac{v}{\|v\|}\sin(\tfrac12 \|v\|)$$ for $v\in\mathbb R^3$, where $\|\bullet\|$ is the standard Euclidean norm.

If you're interested in more sophisticated Lie group time integration, you can check out one of my papers [Arnold, Hante 2016] or a nice paper (without quaternions though) of my supervisor [Arnold, Cardona, Brüls 2016] or the preliminary version.

Hope I could clear things up a little.

• Excellent answer, thank you. I've been looking all over for a good explanation of how to estimate $\omega$ from a sequence of quaternions and this is the only one I've found that is helpful enough to write code from. – adamconkey Sep 27 '17 at 20:36
• I would if it were my question :) – adamconkey Oct 5 '17 at 17:44