Object State

A quaternion is a compact way of representing orientation (via rotation from a standard position).

Quaternion $q = [ q_0\ q_1\ q_2\ q_3 ]$ represents rotation by angle $\theta$ about an axis $a$: $$q_0 = \cos{\theta \over 2}$$ $$a = \sin{\theta \over 2}\ [ q_1\ q_2\ q_3 ]^T$$

We can convert between quaternions and equivalent $3 \times 3$ rotation matrices: $$\left[\begin{array}{c} q_0 \\ q_1 \\ q_2 \\ q_3 \end{array}\right] \ \ \Longleftrightarrow\ \ 2\ \left[\begin{array}{cccc} q_3\ q_3 + q_0\ q_0 - \frac12 & q_0\ q_1 - q_3\ q_2 & q_0\ q_2 + q_3\ q_1 \\ q_0\ q_1 + q_3\ q_2 & q_3\ q_3 + q_1\ q_1 - \frac12 & q_1\ q_2 - q_3\ q_0 \\ q_0\ q_2 - q_3\ q_1 & q_1\ q_2 + q_3\ q_0 & q_3\ q_3 + q_2\ q_2 - \frac12 \end{array}\right]$$

Thinking geometrically, since $|q| = 1$, $q$ is a point on a 4D sphere. Each point on this 4D sphere represents a particular orientation (as a rotation from a standard position).

By the way: This is the key to interpolating between two orientations, $q_a$ and $q_b$: Write each orientation as a quaternion (i.e. a point on the 4D sphere) and find the great circle arc (in 4D!) that joins $q_a$ and $q_b$. Moving along that great circle arc interpolates smoothly (and "naturally") between the orientations.

An object's state, $(x,v,q,\omega)$, consists of its position and velocity, both translational $$\textrm{position } x = [ x_0\ x_1\ x_2 ]^T$$ $$\textrm{velocity } v = [ v_0\ v_1\ v_2 ]^T$$

and rotational: $$\textrm{orientation } q = [ q_0\ q_1\ q_2\ q_3 ]^T$$ $$\textrm{angular velocity } \omega = [ \omega_0\ \omega_1\ \omega_2 ]^T$$

For angular velocity $\omega$:

• the rotation axis is $\omega$ and
• the rotation speed is $|\omega|$ radians per second. The positive rotation direction is determined with the right-hand rule: Point the right thumb in the direction of $\omega$ and curl the fingers to get the positive rotation direction.

For the time derivative of $p$ we will use the notation $\dot{p} = {dp \over dt}$.

Then $v = \dot{x}$.

But $w \neq \dot{q}$. Actually, $$\dot{q} = Q\ \omega$$

or $$\dot{q} \;=\; \left[\begin{array}{c} \dot{q_0} \\ \dot{q_1} \\ \dot{q_2} \\ \dot{q_3} \end{array}\right] \;=\;\; \underbrace{\frac12\ \left[ \begin{array}{rrr} -q_1 & -q_2 & -q_3 \\ q_0 & -q_3 & q_2 \\ q_3 & q_0 & -q_1 \\ -q_2 & q_1 & q_0 \end{array}\right]}_{Q} \ \ \left[\begin{array}{c} \omega_0 \\ \omega_1 \\ \omega_2 \end{array}\right]$$

So, in summary, we have: $$\begin{array}{ccl} \textrm{state} & \textrm{derivative} & \\ x & \dot{x} & \textrm{velocity} \\ \dot{x} & \ddot{x} & \textrm{acceleration} \\ q & \dot{q} & = Q\ \omega \\ \omega & \dot{\omega} & \textrm{angular acceleration} \end{array}$$

Consider the velocity, $\dot{x}(t)$ as it changes from time $t$ to time $t+\Delta t$:

In this same interval, the position, $x$, changes as $$\Delta x = \int_{t}^{t + \Delta t} \dot{x}(t)\ dt$$

In other words, the change in position is the area (both positive and negative) under the velocity curve.

The same applies to the other state variables: $\dot{x}, q, \omega$.

Integral approximation

Let $y$ be the state vector, which contains the state of each of the $n$ objects being animated: $$y = [ x_1\ q_1\ \dot{x}_1\ \omega_1\ \ \ x_2\ q_2\ \dot{x}_2\ \omega_2\ \ \ \dots\ \ \ x_n\ q_n\ \dot{x}_n\ \omega_n ]^T$$

Its derivative is $$\dot{y} = [ \dot{x}_1\ \dot{q}_1\ \ddot{x}_1\ \dot{\omega}_1\ \ \ \dot{x}_2\ \dot{q}_2\ \ddot{x}_2\ \dot{\omega}_2\ \ \ \ldots\ \ \ \dot{x}_n\ \dot{q}_n\ \ddot{x}_n\ \dot{\omega}_n ]^T$$

and we update the state vector with $$y(t+\Delta t) = y(t) + \int_t^{t+\Delta t} \dot{y}(t)\ dt$$

But we need the values in $\dot{y}$ to do this integration:

• $\dot{x}_i$ is known from the state, $y$.
• $\dot{q}_i$ can be computed from $y$ as $Q_i\ \omega_i$.
• $\ddot{x}_i$ is unknown.
• $\dot{\omega}_i$ is unknown.

$\ddot{x}_i$ and $\dot\omega_i$ can be determined from physics.