Differential geometric modeling of guidance problem for interceptors

Abstract

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. kinematic equations of LOS, instantaneous rotation plane of LOS, proportional navigation, augmented true proportional navigation, LOS angular acceleration guidance law, differential geometric guidance commands Citation:

Performance analysis of three-dimensional differential geometric guidance law against low-speed maneuvering targets

“…This approach might complicate the description of the relative motion due to the cross-coupling effects and introduce some auxiliary variables to the relative motion analysis. Establishing the kinematic equations in LOS rotating coordinate could ease the complexity in the description of the 3D relative motion [14][15][16][17][18]. The relative motion in the LOS rotating coordinate system can be divided into two decoupled submotions: (1) the relative motion in the instantaneous osculating plane of the missile and the target spanned by the relative position and velocity vectors, which is also called the engagement plane; (2) the rotation of this plane.…”

“…(1) and (2), the reader is referred to Refs. [14][15][16][17][18]. According to (A1)-(A4), the flight trajectories of the missile and the target can be considered as continuous smooth curves in 3D space.…”

“…Li et al [6,9] transformed it into the time domain, without attenuation of the complexity. Li et al [15][16][17][18] proposed the simplified explicit expression of 3D DGGL in the time domain recently, viz.,…”

“…For 3D DGGL, both the expression in the arc-length system proposed by Chiou et al [3] and the expression in the time domain given by Li et al [6] were not intuitive and convenient enough for performance analysis and application to practical interception scenarios. Based on the LOS rotating coordinate system [14], Li et al [15][16][17][18] proposed a simplified 3D DGGL which gave simple and explicit expression of the commanded acceleration in the time domain. Although many studies on 3D DGGL have been conducted, the results were focused on issues such as initial capture conditions, applications to practical engagements, and estimation of target maneuvering acceleration.…”

Intelligent guidance method based on differential geometric guidance command and fuzzy self-adaptive guidance law

“…K.B. Li, et al [10][11][12][13] analyzed DGGC according to the relative kinematic equations established in the line of sight (LOS) rotation reference system, and proposed a new, simplified DGGC formation. The primary contribution of DGGC is the derivation of a new direction of command acceleration, which better controls the LOS rate compared to traditional 3D PPN.…”