Extended Kalman Filter (EKF)

The EKF's defining mechanism is its use of first-order Taylor expansion to linearize nonlinear process and observation models around the current state estimate. This is done by calculating the Jacobian matrices—the matrices of all first-order partial derivatives—of these functions. The Jacobian of the process model (F) linearizes how the state evolves, and the Jacobian of the observation model (H) linearizes how the state maps to measurements. This local approximation allows the EKF to apply the standard Kalman filter equations in a locally valid region.

Like the standard Kalman filter, the EKF operates in a two-step recursive loop:

• Prediction (Time Update): Uses the nonlinear process model f(x) to project the state estimate and its covariance forward in time. The covariance is propagated using the linearized Jacobian F.
• Update (Measurement Update): When a new sensor measurement arrives, the EKF computes the Kalman Gain—a weighting factor that balances the predicted state with the new observation. The state and covariance are then corrected. This cycle runs continuously, fusing predictions with asynchronous sensor data.

The EKF maintains and propagates a covariance matrix (P) that represents the estimated uncertainty of the state. This matrix is crucial because:

• It quantifies confidence in the estimate.
• It dictates the Kalman Gain, determining trust in the model vs. the sensor.
• It grows during prediction (due to process noise) and shrinks during a measurement update (as uncertainty is reduced). The EKF's linearization directly affects how this covariance is transformed, making accurate Jacobians critical for valid uncertainty bounds.

The EKF is designed for mild nonlinearities. Its performance is best when the system's dynamics and sensor models are smooth and the linearization point (the current estimate) is close to the true state. For systems with high nonlinearity (e.g., aggressive vehicle dynamics) or highly non-Gaussian noise, the linearization error can cause the filter to diverge. This limitation is why alternatives like the Unscented Kalman Filter (UKF) or Particle Filter are used for more severe nonlinearities.

The EKF provides a natural framework for sensor fusion. Different sensor measurements (e.g., IMU, GPS, lidar) are incorporated sequentially in the update step, each with its own observation model h(x) and Jacobian H. The filter's state vector can be extended to include calibration parameters (like sensor biases), enabling online estimation of these parameters alongside the primary state. This makes the EKF a cornerstone of Visual-Inertial Odometry (VIO) and Lidar-Inertial Odometry systems.

For many real-time applications, the EKF strikes a favorable balance between accuracy and computational cost. Its complexity is O(n^3) where n is the state dimension, primarily due to the matrix inversion in the Kalman Gain calculation. This is often manageable for state vectors in robotics (e.g., pose, velocity, biases). Engineers use techniques like state cloning for sliding window filters or Schur complement tricks to maintain real-time performance in systems like SLAM.

The foundational linear recursive algorithm for optimal state estimation. It operates on two core equations: a process model to predict the next state and a measurement model to correct the prediction with new sensor data. It assumes:

• A linear system dynamics model.
• Gaussian-distributed process and measurement noise. The Kalman filter's elegance lies in its efficient propagation of the state's mean and covariance matrix, making it the theoretical basis for the EKF.

A modern alternative to the EKF for handling nonlinearities. Instead of linearizing models with a Jacobian, the UKF uses the unscented transform. This technique selects a minimal set of sample points (sigma points) that capture the true mean and covariance of the state distribution. These points are propagated through the nonlinear functions, and a new Gaussian is reconstructed from the transformed points. The UKF often provides superior accuracy and stability compared to the EKF, especially for systems with strong nonlinearities or discontinuous models.

A sequential Monte Carlo method for state estimation in systems that are both nonlinear and non-Gaussian. It represents the posterior probability distribution with a large set of random samples called particles, each with an associated weight. The algorithm cycles through:

• Prediction: Particles are propagated according to the motion model.
• Update: Weights are updated based on sensor likelihood.
• Resampling: Low-weight particles are replaced by copies of high-weight particles to avoid degeneracy. While computationally intensive, it is highly flexible and does not rely on Gaussian assumptions.

The overarching probabilistic framework that unifies Kalman, EKF, UKF, and particle filters. It formulates state estimation as a recursive application of Bayes' theorem. The goal is to compute the posterior probability distribution of the current state, given all previous measurements. The framework consists of two steps:

1. Prediction (Chapman-Kolmogorov): Propagates the state belief forward using the process model.
2. Update (Bayesian Update): Incorporates the latest measurement. All specific filters are implementations that make different approximations to make this computationally tractable.

A critical application domain for the EKF, historically implemented via EKF-SLAM. SLAM is the problem where a robot must build a map of an unknown environment while simultaneously tracking its own location within it. In EKF-SLAM, the state vector contains both the robot's pose and the estimated positions of all observed landmarks. The filter's covariance matrix explicitly models the correlations between the robot and every landmark, which grows quadratically with map size. This approach demonstrates the EKF's capability but also highlights its scalability challenges in large environments.

A graphical model representation that has largely supplanted the EKF for large-scale optimization problems like SLAM. It represents the estimation problem as a bipartite graph of variables (robot poses, landmarks) and factors (sensor measurements, motion constraints). Unlike the EKF, which maintains a single linearization point and a monolithic covariance matrix, factor graphs enable batch optimization (e.g., via Levenberg-Marquardt) over the entire history. This approach is more robust to linearization errors and allows for efficient incremental smoothing rather than pure filtering.