Measurement Model

The observation function (often denoted h(x)) is the deterministic, mathematical core of the model. It describes the ideal, noise-free sensor reading given a perfect knowledge of the system's state x. Its form is dictated by sensor physics and geometry.

• For a camera: It projects 3D world points onto the 2D image plane using a pinhole model.
• For a LiDAR: It calculates the expected range and bearing to a surface given the robot's pose and a map.
• For a GPS receiver: It converts the estimated global position into an expected latitude, longitude, and altitude.

The noise model accounts for the stochastic uncertainty in every real sensor measurement. It defines the probability distribution of the discrepancy between the ideal observation h(x) and the actual sensor reading z. This is typically modeled as additive Gaussian noise with zero mean and a covariance matrix R.

• Covariance Matrix R: This matrix quantifies the magnitude of noise (diagonal elements) and correlations between different measurement dimensions (off-diagonal elements). A high value in R tells the estimator to trust that particular measurement less.
• Non-Gaussian Noise: For sensors with outliers or heavy-tailed distributions (e.g., vision-based feature matching), more robust models like a Huber or Cauchy distribution may be used.

Measurement models are classified by the linearity of their observation function h(x).

• Linear Models: h(x) is a linear matrix operation (z = Hx + noise). These are optimal for filters like the Kalman Filter. Example: measuring position with a perfect linear encoder.
• Nonlinear Models: h(x) is nonlinear. This is the norm in robotics (e.g., camera projection, ranging sensors). They require approximations:
  • Extended Kalman Filter (EKF): Linearizes h(x) around the current state estimate using the Jacobian matrix.
  • Unscented Kalman Filter (UKF): Propagates carefully chosen sample points through the nonlinear function.
  • Particle Filter: Evaluates the full nonlinear function for each particle.

For nonlinear models used in an EKF, the Jacobian matrix H is the linear approximation of the observation function. It is the matrix of partial derivatives H = ∂h/∂x, evaluated at the current state estimate.

• Role: It maps a small change in the state δx to the predicted change in the measurement δz. It is critical for the filter's update step, telling it how sensitive the measurement is to each state variable.
• Example: For a bearing-and-range sensor to a landmark, the Jacobian relates changes in the robot's (x, y) position to changes in the measured range and angle. An incorrect Jacobian leads to filter inconsistency and divergence.

Data association is the process of determining which part of the observed sensor data corresponds to which known feature or object in the state estimate. It is a prerequisite for applying the measurement model.

• Challenge: In cluttered environments, a sensor may see many unlabeled features (e.g., visual keypoints, LiDAR clusters).
• Methods:
  • Nearest Neighbor with Mahalanobis distance (accounts for uncertainty).
  • Joint Probabilistic Data Association (JPDA) for multiple hypotheses.
  • Scan Matching algorithms like Iterative Closest Point (ICP) for dense point clouds.
• Incorrect association is a primary source of catastrophic filter failure.

The measurement model works in tandem with the process model (or motion model) within the Bayesian filtering cycle.

1. Prediction: The process model f(x) predicts the state forward in time using control inputs.
2. Update: The measurement model h(x) predicts what the sensors should see given the predicted state. The difference between this prediction and the actual sensor data—the innovation—is used to correct the state estimate.

This cycle forms the core of estimators like the Kalman filter, where the measurement model's accuracy directly determines the precision of the final state correction. A well-specified measurement model with realistic noise parameters is essential for optimal fusion.

Also known as a motion model, this is the mathematical function that predicts how a system's state evolves over time based on its previous state and any control inputs. It defines the state transition probability. In a Kalman filter, this is the 'predict' step.

• Key Role: Projects the state estimate forward in time.
• Contrast with Measurement Model: The process model predicts the state; the measurement model predicts what the sensors should observe from that predicted state.

The overarching probabilistic framework for state estimation. It recursively applies Bayes' theorem to update the belief about the system's state as new measurements arrive. The update step directly uses the measurement model to compute the likelihood of the observation.

• Core Equation: Posterior ∝ Likelihood × Prior.
• Measurement Model's Role: Provides the likelihood function p(zₜ | xₜ), which is the probability of receiving measurement zₜ given the state xₜ.

A square matrix that quantifies the uncertainty and correlations between all elements of the estimated state vector. The measurement model directly influences the measurement noise covariance matrix (R), which encodes the confidence in each sensor reading.

• In the Update Step: The Kalman gain is computed using both the process covariance and the measurement covariance from R.
• Practical Impact: A poorly tuned R matrix (incorrect noise assumption in the model) will lead to the filter over- or under-weighting new sensor data.

The process of determining the intrinsic and extrinsic parameters of a sensor. These parameters are embedded directly within the measurement model.

• Intrinsic Calibration: Determines parameters like focal length and lens distortion for a camera, which define how 3D points project onto the 2D image plane.
• Extrinsic Calibration: Determines the sensor's position and orientation (pose) relative to the robot's body frame, which is used to transform measurements into a common coordinate system for fusion.

A system-theoretic property that determines whether the internal state of a system can be uniquely inferred from a sequence of its outputs (measurements). The structure of the measurement model is fundamental to observability.

• Unobservable States: If the measurement model cannot provide information to distinguish between two different states, those states are unobservable. For example, a monocular camera cannot observe absolute scale without additional sensors.
• Design Implication: Sensor suite and placement are chosen to ensure the combined measurement models make the critical states (e.g., full 6D pose) observable.

Algorithms to identify and discard sensor measurements that are statistically inconsistent with the current state estimate and the expected behavior defined by the measurement model. This is a critical robustness layer.

• Common Methods: Mahalanobis distance gating within a Kalman filter, or RANSAC for vision-based models.
• Mechanism: A measurement z is compared to the predicted measurement h(x). If the innovation (z - h(x)) is too large relative to the expected innovation covariance, it is flagged as an outlier.