Measurement Model

Sensors do not measure the robot’s pose directly. Instead, sensors produce readings $z_k$, such as:

• Range to a wall or landmark (LiDAR / sonar)
• Bearing / angle to an object (camera)
• GPS coordinates
• Magnetometer heading
• Visual feature matches

So the measurement model is a function

$$z_k=h({\xi }_k,m)$$

where ${\xi }_k$ is the robot’s pose and $m$ is the map.

Unlike motion models which are probabilistic, measurement models are typically static.

Example

Consider a robot that observes landmark at map coordinate $(x_{\ell }^i,y_{\ell }^i)$. The robot measures three things for each landmark:

• $r_k^i$: Range/distance to landmark (noisy)
• ${\varphi }_k^i$: Bearing/angle to landmark (noisy)
• $s_k^i$: Signature (e.g. average color) to landmark

Then, the measurements looks like

$$z_k=\left(\left[\begin{array}{c}{r}_k^1\\ {\varphi }_k^1\\ s_k^1\end{array}\right],\left[\begin{array}{c}{r}_k^2\\ {\varphi }_k^2\\ s_k^2\end{array}\right],\dots \right)$$

These measurement values depend on the robot’s pose and the map, so they can be written as functions of ${\xi }_k$ and $m$, i.e. $r_k^i({\xi }_k,m),{\varphi }_k^i({\xi }_k,m)$ and $s_k^i({\xi }_k,m)$.

Probabilistic Measurement Model

How do we incorporate probabilistic noise into sensor models?

$$z_k=h({\xi }_k,m)⟶z_k\sim p(z_k\mid {\xi }_k,m)$$

• Note that the map $m$ is an input

Unlike motion models, we usually need only the posterior (or generative) model $z_k\sim p(z_k|{\xi }_k,m)$, i.e. the sample model is not normally needed.