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Localization

The environment provides a Particle Filter to estimate the agent's position when its exact location is unknown or when navigating with noisy sensors.

Particle Filter

A Particle Filter (PF) is a Monte Carlo algorithm for estimating the state of a system. In this environment, it is used for self-localization of the agent on the grid.

Mechanics

  1. Initialization: Particles are distributed randomly across the entire grid.
  2. Prediction: When the agent moves, each particle is moved according to the intended action, taking into account the movement uncertainty (slipping).
  3. Update: Each particle's weight is updated based on how well the hypothetical observations at that particle's position match the actual sensor measurements.
  4. Resampling: Particles with low weights are replaced by copies of particles with high weights.

Sensors

The localization relies on two main sensor types:

Color Sensor

The color sensor measures the color of the ground directly beneath the agent. - Accuracy: 80% (correct color). - Noise: 20% (distributed equally among the other two colors). - Colors: White (0), Red (1), Green (2).

CNN Classifier

The CNN classifier runs at every step, regardless of whether an item is present. If no item is present, it predicts the "background" class. - Classes: Dog, Flower, Background. - Usage: The probability assigned by the CNN to the class actually present in a particle's cell is used as the likelihood for that particle.

Sensor Fusion

When using both sensors (sensor_mode='both'), the filter combines the measurements by assuming they are conditionally independent given the state. The joint likelihood is the product of the individual likelihoods:

$$p(z_{\text{color}}, z_{\text{cnn}} | s) = p(z_{\text{color}} | s) \cdot p(z_{\text{cnn}} | s)$$

Assumptions

The Particle Filter makes the following assumptions about the environment:

Map Assumption

The filter has perfect knowledge of the map. It knows: - The dimensions of the grid. - The exact location of all walls. - The ground color of every cell. - The locations of all items (Dogs and Flowers).

Movement Uncertainty

The filter assumes a stochastic motion model matching the environment's slip probability. There are two types of slipping:

  • Perpendicular Slipping: The agent moves in a perpendicular direction with probability $P_{\text{slip}}$ (split equally between the two perpendicular directions).
  • Longitudinal Slipping: The agent moves in the same direction but either stays in place (moves 0 steps) or moves twice (moves 2 steps), each with probability $P_{\text{slip}} / 2$.
  • Intended Movement: The agent moves exactly one step in the intended direction with probability $1 - P_{\text{slip}}$.
  • Walls: If a move would lead into a wall or out of bounds, the agent (and thus the particles) remains in the current cell.

Measurement Uncertainty

The filter assumes the following likelihood models: - Color Sensor: $p(z_{\text{color}} | s) = 0.8$ if the measured color $z$ matches the map color at state $s$, and $0.1$ otherwise. - CNN: $p(z_{\text{cnn}} | s) = \text{CNN_prob}(\text{class at } s)$. The filter assumes that the probability output by the CNN for the true class at a location is the likelihood of that location.