Numerically Stable Maclaurin Approximations for 3D Constant Turn Models in IMM Aircraft Tracking

This paper considers a numerically stable discrete-time representation of the three-dimensional Constant Turn (CT) motion model within the Interacting Multiple Model (IMM) framework for radar tracking of maneuvering aerial targets. Classical discrete CT models used in Kalman-filter-based tracking contain singular expressions in the vicinity of zero and near-zero turn rates, which may degrade estimation accuracy and impair numerical robustness. To address this problem, a Maclaurin-series-based discretization of the three-dimensional CT model is developed, in which the state transition matrix and the process-noise-related matrices are approximated in polynomial form. Linear, quadratic, and cubic approximations are constructed and analyzed. The proposed CT model is integrated into a three-model IMM algorithm together with the Constant Velocity (CV) and Constant Acceleration (CA) models. The study includes both an internal comparison of Maclaurin approximations of different orders and an external comparison with the classical CT discretization and a Padé-based reference discretization. Numerical experiments are performed for representative three-dimensional maneuvering scenarios under radar measurement conditions. The obtained results show that the proposed discretization eliminates singular behavior near zero turn rate while preserving the tracking capability of the IMM estimator. The comparative analysis demonstrates that the quadratic Maclaurin approximation provides the most favorable trade-off between modeling accuracy, numerical stability, and computational cost. It yields tracking performance close to higher-order approximations and competitive with the Padé-based reference approach, while remaining simpler for practical implementation in real-time radar tracking systems. These results indicate that the proposed quadratic approximation is a suitable solution for maneuvering aerial target tracking in three-dimensional radar applications.