Differential geometric modeling of guidance problem for interceptors

It is a comparatively convenient technique to investigate the motion of a particle with the help of the differential geometry theory, rather than directly decomposing the motion in the Cartesian coordinates. The new model of three-dimensional (3D) guidance problem for interceptors is presented in this paper, based on the classical differential geometry curve theory. Firstly, the kinematical equations of the line of sight (LOS) are gained by carefully investigating the rotation principle of LOS, the kinematic equations of LOS are established, and the concepts of curvature and torsion of LOS are proposed. Simultaneously, the new relative dynamic equations between interceptor and target are constructed. Secondly, it is found that there is an instantaneous rotation plane of LOS (IRPL) in the space, in which two-dimensional (2D) guidance laws could be constructed to solve 3D interception guidance problems. The spatial 3D true proportional navigation (TPN) guidance law could be directly introduced in IRPL without approximation and linearization for dimension-reduced 2D TPN. In addition, the new series of augmented TPN (APN) and LOS angular acceleration guidance laws (AAG) could also be gained in IRPL. After that, the differential geometric guidance commands (DGGC) of guidance laws in IRPL are advanced, and we prove that the guidance commands in arc-length system proposed by Chiou and Kuo are just a special case of DGGC. Moreover, the performance of the original guidance laws will be reduced after the differential geometric transformation. At last, an exoatmospheric interception is taken for simulation to demonstrate the differential geometric modeling proposed in this paper.