Target Tracking Using Kalman Filter for Circular Motion

In this document, we discuss the process of target tracking using Kalman filter for objects following circular motion patterns. The implementation involves multiple stages including system modeling for circular motion, Kalman filter initialization, and trajectory visualization. Key programming aspects include defining state transition matrices that account for circular kinematics, implementing measurement update functions, and handling process-noise covariance. We generate predicted signals based on the circular motion model, simulate observation signals with added Gaussian noise, and apply Kalman filtering to obtain refined trajectory estimates. The code includes functions for plotting three distinct trajectories: the original predicted path, noisy observations, and the filtered results demonstrating noise reduction. The documentation explains the mathematical foundation of Kalman filtering, particularly its recursive prediction-correction mechanism that optimally combines model predictions with real measurements. We detail how the filter's gain calculation minimizes estimation error covariance, making it particularly effective for tracking problems with periodic motion patterns. Furthermore, we examine Kalman filter advantages such as computational efficiency, real-time processing capability, and handling of systems with incomplete observations. The implementation uses matrix operations for state propagation and employs numerical techniques for covariance updates, ensuring stability during long tracking sequences.