Extended Kalman Filter (EKF)

A nonlinear state estimation algorithm that uses a deterministic sampling technique called the unscented transform to capture the mean and covariance of a probability distribution propagated through nonlinear functions. Unlike the EKF, the UKF avoids linearization entirely, instead selecting a minimal set of sigma points to represent the distribution. This eliminates the need to compute Jacobian matrices and often yields higher accuracy for highly nonlinear systems while maintaining comparable computational cost.

A sequential Monte Carlo method that represents the posterior probability distribution using a set of weighted random samples called particles. Unlike the EKF's Gaussian assumption, particle filters can represent arbitrarily complex, multi-modal distributions, making them ideal for highly non-Gaussian and nonlinear environments. Key characteristics:

• Handles global localization and the kidnapped robot problem
• Computational cost scales with the number of particles
• Often used when EKF linearization errors become unacceptable

A graphical model framework that represents state estimation as a bipartite graph of variable nodes (states) and factor nodes (probabilistic constraints). Unlike the EKF's recursive, causal filtering approach, factor graph optimization performs batch smoothing over a window of historical data, relinearizing and resolving the entire trajectory. This provides superior accuracy for SLAM applications by revisiting past estimates with future information, at the cost of higher computational demand.

A data fusion algorithm for combining state estimates when their cross-correlation is unknown or intractable. In decentralized sensor networks, EKFs often produce correlated estimates that cannot be naively fused without risking overconfident results. Covariance intersection computes a consistent fused covariance by finding a weighted average that guarantees the result is not overconfident, even under unknown correlations. This is critical for scalable multi-sensor architectures where maintaining full correlation matrices is infeasible.

A mathematical assessment of whether a dynamic system's internal states can be uniquely inferred from available sensor measurements over time. For an EKF to converge, the system must be observable. Analysis involves:

• Computing the observability matrix and checking its rank
• Identifying degenerate motion profiles that cause unobservability (e.g., constant velocity with only a monocular camera)
• Informing sensor placement and trajectory design to ensure the EKF maintains consistent estimates

The computational process of determining which sensor measurements originate from which physical objects. The EKF assumes known measurement-to-target correspondences, but in cluttered environments with multiple targets, this association is ambiguous. Incorrect associations cause filter divergence. Advanced techniques like Joint Probabilistic Data Association (JPDA) and Multiple Hypothesis Tracking (MHT) extend EKFs to handle this uncertainty by evaluating association probabilities or deferring decisions until ambiguity resolves.