Bayesian Occupancy Grids

Probabilistic obstacle mapping.

Introduction 

Occupancy grids represent the environment as a 2D grid where each cell contains the probability of being occupied.

Each cell stores \(P(m)\) - the probability the cell is occupied.

For numerical stability, we use log-odds:

\[l(m) = \log\frac{P(m)}{1 - P(m)}\]

Conversion:

\[P(m) = 1 - \frac{1}{1 + e^{l(m)}}\]

Given a sensor measurement \(z\):

\[l(m|z_{1:t}) = l(m|z_{1:t-1}) + l(m|z_t) - l_0\]

Where \(l_0 = \log\frac{p_0}{1-p_0}\) is the prior.

Simplified (with uniform prior):

\[l_{new} = l_{old} + l_{sensor}\]

To handle dynamic environments, apply decay:

\[l_{t+1} = \gamma \cdot l_t\]

Where \(\gamma < 1\) causes old observations to fade.

Elfes, A. (1989). “Using occupancy grids for mobile robot perception and navigation.”
Thrun, S. (2002). “Robotic mapping: A survey.”