Bayesian Filtering

At its core, Bayesian filtering is a two-step recursive process:

• Prediction Step: Uses a process model (or motion model) to propagate the current state estimate forward in time, increasing uncertainty.
• Update Step: Incorporates a new sensor measurement using a sensor model (or observation model) to correct the prediction, reducing uncertainty. This cycle of predict → update is repeated for each new measurement, enabling real-time estimation. The algorithm maintains a full probability distribution over the possible states (the posterior), not just a single point estimate.

Bayesian filters do not output a single 'best guess.' Instead, they maintain a complete probability distribution p(state | measurements) that quantifies uncertainty. This distribution encodes:

• The most likely state (the mean).
• The confidence in that estimate (the spread or covariance).
• Correlations between different state variables. For example, in a robot tracking problem, the distribution would specify not just the estimated (x, y) position, but also the ellipsoidal region where the robot is likely to be with 95% probability. This explicit uncertainty representation is critical for robust decision-making in autonomous systems.

The Kalman Filter is the optimal closed-form solution to the Bayesian filtering equations when two key assumptions hold:

1. The process and sensor models are linear.
2. All noise is Gaussian (normally distributed). Under these conditions, the state's probability distribution remains Gaussian, fully described by a mean vector and a covariance matrix. The Kalman Filter's equations provide the exact, computationally efficient update for this mean and covariance. It is foundational for aerospace, guidance, and countless engineering applications where linearity is a valid approximation.

Real-world systems (like robot motion) are often nonlinear. The Extended Kalman Filter (EKF) handles this by linearizing the nonlinear models around the current state estimate using a first-order Taylor expansion. While effective for mild nonlinearities, linearization errors can cause divergence. The Unscented Kalman Filter (UKF) uses a more robust approach: the unscented transform. It selects a minimal set of sample points (sigma points) that capture the mean and covariance of the distribution, propagates these points through the true nonlinear functions, and then recomputes the mean and covariance from the transformed points. The UKF often provides better accuracy and stability than the EKF for highly nonlinear systems.

When systems are both highly nonlinear and non-Gaussian, the Particle Filter (a Sequential Monte Carlo method) is used. It represents the probability distribution with a set of random samples called particles, each with an associated weight.

• Each particle is a hypothesis about the system's true state.
• The algorithm propagates particles through the process model, updates their weights based on sensor likelihood, and resamples to focus computation on high-probability regions. This approach is extremely flexible and can represent complex, multi-modal distributions (e.g., tracking an object that might be in one of several possible rooms), but is computationally more expensive than Kalman-family filters.

A key challenge in multi-sensor systems is data association: determining which measurement came from which tracked object (or if it is clutter). Bayesian filtering frameworks formally address this. For example, the Probabilistic Data Association Filter (PDAF) calculates the probability that each measurement is associated with a track and performs a weighted update. More advanced methods like Multiple Hypothesis Tracking (MHT) explicitly maintain multiple association histories over time. This probabilistic treatment of association ambiguity is what allows Bayesian filters to maintain robust tracks in cluttered environments with false alarms and missed detections.

Kalman and particle filters are core to the localization stack in self-driving cars. They fuse high-frequency data from inertial measurement units (IMUs), wheel odometry, GNSS (Global Navigation Satellite System), and lidar to maintain a precise, real-time estimate of the vehicle's pose (position and orientation). This continuous state estimation is essential for path planning and control, especially during GNSS signal loss in urban canyons or tunnels. The filter's covariance matrix provides a direct measure of localization uncertainty, which is used for safety-critical decision-making.

Extended Kalman Filters (EKF) and graph optimization techniques form the backbone of many SLAM systems. As a robot explores an unknown environment, the filter recursively estimates both its own trajectory (localization) and the positions of observed landmarks (mapping). It processes streams from visual odometry, lidar scans, and IMU data to build a consistent spatial model. The factor graph representation allows for efficient bundle adjustment, correcting drift by optimizing over all past poses and measurements when loop closures are detected.

Kalman filtering is the standard method for INS/GNSS integration. A low-cost MEMS IMU provides high-rate acceleration and angular velocity but suffers from significant drift. The Kalman filter uses periodic, absolute position fixes from GNSS to estimate and correct for the IMU's bias and scale factor errors. This tightly-coupled integration provides smooth, continuous navigation output. The prediction step uses the process model (kinematic equations), while the update step incorporates GNSS measurements, with the Mahalanobis distance used to reject outlier satellite signals.

Bayesian filters enable multi-object tracking (MOT) in cluttered sensing environments like air traffic control radar or underwater sonar. Algorithms like the Interacting Multiple Model (IMM) filter handle targets with switching dynamics (e.g., an aircraft maneuvering vs. cruising). The core challenge is data association—determining which of many noisy detections belongs to which existing track. Methods like joint probabilistic data association (JPDA) or multiple hypothesis tracking (MHT) work within the Bayesian framework to maintain probabilistic track histories, even through occlusions and false alarms.

Precise robotic manipulation, such as in assembly or surgery, relies on fusing force/torque sensing with vision and joint encoders. A Bayesian filter estimates the contact state between a robot's end-effector and an object. This allows for hybrid force-position control. For example, while inserting a peg into a hole, vision provides coarse alignment, but the filter fuses force feedback to detect jams and adjust the approach. The filter's ability to model and separate process noise (from robot motion) from measurement noise (from sensor quantization) is critical for smooth, compliant operation.

A key application of Bayesian filtering is fault detection and isolation (FDI). By comparing the filter's innovation sequence (the difference between predicted and actual measurements) against its expected statistical distribution, the system can identify sensor failures. A sudden, sustained increase in innovation for one sensor indicates a potential fault. More advanced methods use multiple model filters to hypothesize different fault conditions (e.g., bias drift, complete failure). This enables robust estimation by de-weighting or excluding faulty sensor data, maintaining system functionality despite hardware degradation.