LiDAR

LiDAR measures distance using laser light.

• A laser sends out a short light pulse.
• It reflects off objects in the environment.
• A photodetector measures the time of flight (how long the light takes to return).
• Light travels extremely fast, resulting in very accurate range measurements

Each laser provides a point in space $(x,y,z)$ and sometimes may also provide an intensity. A full scan produces a 3D map of the surroundings.

LiDAR Types

Mechanical spinning LiDAR:

• Rotates 360° continuously
• Sends out many beams at different vertical angles
• Creates a 2D or 3D point cloud
• Common example: Velodyne VLP-16

Solid-State LiDAR (no moving parts)

• Uses MEMS mirrors or phased light array
• Smaller, cheaper, robust, but narrower FOV
• Example: Intel RealSense L515

Robot Pose and LiDAR Data

At time $k$, the robot receives a LiDAR scan $z_k$, which consists of $N$ range measurements, $z_k^1,\dots ,z_k^N$ , each with a corresponding bearing angle ${\theta }_{\text{sen}}^1,{\theta }_{\text{sen}}^N$.

Each point of $z_k^i$ first needs to be coordinate transformed to $(x_{\text{sen}}^i,y_{\text{sen}}^i)$ in the LiDAR sensor frame:

$$\left[\begin{array}{c}{x}_{\text{sen}}^i\\ y_{\text{sen}}^i\end{array}\right]=z_k^i\left[\begin{array}{c}\cos {\theta }_{\text{sen}}^i\\ \sin {\theta }_{\text{sen}}^i\end{array}\right]$$

Then, we can convert to $(x_{z_k^i},y_{z_k^i})$ in the global frame:

$$\left[\begin{array}{c}{x}_{z_k^i}\\ y_{z_k^i}\end{array}\right]=\left[\begin{array}{c}{x}_k\\ y_k\end{array}\right]+\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}{x}_{\text{sen}}^i\\ y_{\text{sen}}^i\end{array}\right]$$

Beam Sensor Model

How do we get a posterior for the measurement, $p(z_k^i\mid {\xi }_k,m)$? How likely is this LiDAR scan, assuming the robot is at pose ${\xi }_k$ looking at map $m$?

For each ray, we consider 3 main sources of noise:

• Correct range with local measurement noise: nominal noise, etc. This would give a Gaussian centered at the expected range.
• Sensor failure/max-range error: Missed data, light absorbing objects. This would give a distribution with a spike at $z$ max
• Uniformly-distributed noise: Phantom readings, cross-talk, etc. This would give a uniform distribution obviously.

The posterior can be found as a weighted sum of these:

$$p(z_k^i\mid {\xi }_{k,m)}={\eta }_1{p}_{\text{hit}}(z_k^i\mid {\xi }_k,m)+{\eta }_2{p}_{\text{max}}(z_k^i\mid {\xi }_k,m)+{\eta }_3{p}_{\text{rand}}(z_k^i\mid {\xi }_k,m)$$

• ${\eta }_1+{\eta }_2+{\eta }_3=1$

Assuming independence of each beam ($z_k^i$ for each $i$), this would be a multiplication:

$$p(z_k\mid {\xi }_k,m)=\prod _{i=1}^{N}p(z_k^i\mid {\xi }_k,m)$$