## System Modelling

## Itō’s Insight

Consider the stochastic process where is standard Brownian motion (or the Wiener process) and is a twice-differentiable function. Ito’s lemma states that The first term is recognizable from the chain rule in classical calculus, but why the second term? If is truly infinitesimal, it doesn’t even seem possible that . To understand Ito’s lemma intuitively, […]

## Zeno Bouncing

If you drop a rigid ball onto a rigid surface, it eventually stops bouncing without ever bouncing the last time! This is called Zeno behavior, after Zeno’s most famous paradox (Achilles and the tortoise). Suppose we drop the ball from a height of 1m, and the coefficient of restitution is 0.9. So every bounce is […]

## 6DoF Visual Positioning

It is possible to position and orientate an object relative to a camera, given only three of its points (e.g. LEDs) in image-space. In fact, this is the minimal visual positioning system. It infers depth from the object’s size, rather than using a stereo camera system. It also infers the object’s 3-DOF orientation (with a […]

## 6DoF Rigid Body Dynamics

If you throw an arbitrarily-shaped rigid object into the air with some random rotational motion, the motion can proceed semi-chaotically, unless it happens to be spinning purely around one of its “principle axes”. (You could try your mobile phone, for example – give most spin around its tall axis for interesting results.) General rigid body […]

## Expressing 3DoF Rotation

The pose of a rigid body is a 6-DOF quantity, consisting of translational position (3-DOF) and rotational position, or orientation (3-DOF). Translational position is straightforwardly expressed by a triplet (x, y, z). Rotational position, or orientation, is not so simple. Rotation Vector Rotations, like translations, can be represented by a 3D vector. The rotation vector’s […]