Section 29.4: Mapping and occupancy grids

"A grid cell is a decision about where the robot may spend risk."

A Loop Closure That Came Back With Receipts

Problem First

A robot map is a probabilistic memory of what sensor rays have implied. Free space, occupied space, and unknown space must stay distinct because a planner should treat unseen cells differently from cells observed as empty.

Occupancy grids update each cell through inverse sensor models, often using log odds for numerical stability. A range ray decreases occupancy probability along free cells and increases it near the measured endpoint. The map is therefore a belief field, not a bitmap.

Formal Model

For occupancy grids, the posterior is a cell-wise belief over free, occupied, and unknown space conditioned on sensor rays and robot pose. The planner consumes inflation, freshness, and uncertainty, not only a black-and-white bitmap.

$$ L_t(c)=L_{t-1}(c)+\log\frac{p(c\mid z_t,x_t)}{1-p(c\mid z_t,x_t)}-L_0(c) $$

The evidence terms are inverse sensor model, ray casting, pose uncertainty, update rate, decay policy, and map resolution. Each one changes whether a cell is safe to traverse or only looks safe in a stale grid.

Worked Diagnostic

Code Fragment 1 is the occupancy update smoke test: one ray should clear free cells, mark an endpoint, and leave unseen space unknown. If that invariant fails, larger maps will fail quietly.

# Apply log-odds updates for free and occupied evidence.
# The probabilities show how repeated rays change a grid cell belief.
import math

def sigmoid(x):
    return 1.0 / (1.0 + math.exp(-x))

log_odds = 0.0
for update in [-0.4, -0.4, 0.9]:
    log_odds += update
    print(f"log_odds={log_odds:.1f}, p_occ={sigmoid(log_odds):.2f}")

Expected output interpretation. The first two updates make the cell increasingly likely to be free, then the occupied endpoint evidence pushes it back toward uncertainty. The final value near 0.5 is exactly the interpretation to watch for: the map has conflicting evidence, so a cautious planner should not treat this cell as confidently traversable.

Tool Workflow

Use the hand grid update as a regression test, then use OctoMap, costmap_2d, Voxblox, or Nav2 layers for scale. The production tool is allowed to be complex; the occupancy invariant should remain simple.