New Results on Three-dimensional Differential Geometric Guidance and Control Problem

Abstract

This paper presents a novel approach to the three-dimensional (3D) differential geometric (DG) guidance and control problem. The DG guidance commands are firstly studied in the time domain using the classical DG formulations, and the guidance curvature command is modified in order to meet the requirement of practical onboard computation. Then, DG flight control system is introduced using classical PID controller. Finally, a new necessary initial condition is derived to intercept a high-speed target, as well as to guarantee both the DG guidance commands are well defined throughout the engagement. Simulations results demonstrate that the DG guidance law works well in a realistic engagement, and performs better than conventional proportional navigation guidance law in the case of intercepting a maneuvering target. Moreover, the terms of the target’s information in the DG guidance curvature command could be ignored without sacrificing the performance 2 studied 3D proportional navigation (PN) guidance law in terms of the geodesic and normal curvatures of the missile’s path on the surface generated by the line-of-sight (LOS), there haven’t been many attempts on the application of differential geometric (DG) formulations to the missile guidance problems. Chiou and Kuo 3-5 proposed a PN type DG guidance commands for 2D and 3D missile guidance problems using the Frenet formulas 11 . Li 6-7 examined its applications in a realistic missile defense engagement, together with its iterative solution using classical Newton iterative algorithm. Ariff 8-9 presented a novel DG guidance algorithm using the information of the involute of target’s trajectory. White 10 studied the application of DG formulations to a planar interception engagement, whose kinematics equations are developed and expressed in DG terms. The results of their papers indicated that the DG guidance algorithm performed better than the conventional PN guidance law in most cases. This paper presents the applications of 3D DG guidance commands 3 using classical differential geometry formulations and its initial condition. The performance of DG guidance curvature command is studied and modified to facilitate easy implementation in a realistic engagement. Moreover, a new necessary initial condition is developed based on an assumption that the target/missile speed ‡ ratio is bigger than one. This paper also presents numerical simulations to show the influence of the term of target’s information to the performance of the DG guidance law.