Likelihood-field
Likelihood Field Alternative
Instead of per-beam ray casting in localization: - Precompute distance transform \(d(\mathbf p)\) to nearest occupied cell. - Likelihood: \[ p(z_t^k \mid \mathbf x_t, m) = \eta \exp\!\left(-\frac{(d(\mathbf e_k))^2}{2\sigma_d^2}\right), \] where \(\mathbf e_k\) is beam endpoint predicted without obstacle truncation.
Pros: faster. Cons: less detailed modeling of short / max phenomena.
Integration in Particle Filter Localization
For particle \(i\) with pose \(\mathbf x_t^{[i]}\): \[ w_t^{[i]} \propto w_{t-1}^{[i]} \prod_{k \in \mathcal K} p(z_t^k \mid \mathbf x_t^{[i]}, m), \] Optionally subsample beams \(\mathcal K\) to reduce computation; use log-sum for stability.
9. Handling Correlation and Beam Selection
Adjacent beams correlated via shared surfaces; independence assumption optimistic. Mitigation:
- Subsample (e.g., every 4th beam).
- Clamp per-beam likelihood to lower bound.
- Use adaptive beam selection near discontinuities.
10. Noise Sources and Effects
| Source | Effect on Distribution |
|---|---|
| Surface reflectivity (sonar) | Increases short / random components |
| Transparent / specular surfaces | Elevates random / max |
| Multi-path | Broadens \(p_{\text{hit}}\) |
| Timing quantization | Slight discretization at small ranges |
| Motion distortion (lidar while moving) | Systematic curvature in scans |
Compensations: deskewing using IMU/odometry, calibrating mixture weights.