Iterative solution to differential geometric guidance problem

Abstract

PurposeTo study the application of three‐dimensional differential geometric (DG) guidance commands to a realistic missile defense engagement, and the application of the Newton's iterative algorithm to DG guidance problems.Design/methodology/approachThe classical differential geometry theory is introduced firstly to transform all the variables in DG guidance commands from an arc length system to the time domain. Then, an algorithm for the angle‐of‐attack and the sideslip angle is developed by assuming the guidance curvature command and guidance torsion command equal to its corresponding value of current trajectory. Furthermore, Newton's iteration is utilized to develop iterative solution of the stated algorithm and the two‐dimensional DG guidance system so as to facilitate easy computation of the angle‐of‐attack and the sideslip angle, which are formulated to satisfy the DG guidance law.FindingsDG guidance law is viable and effective in the realistic missile defense engagement, and it is shown to be a generalization of gain‐varying proportional navigation (PN) guidance law and performs better than the classical PN guidance law in the case of intercepting a maneuvering target. Moreover, Newton's iterative algorithm has sufficient accuracy for DG guidance problem.Originality/valueProvides further study on DG guidance problem associated with its iterative solution.

Analysis of 3D Geometric Guidance Problem

“…16) Referring to Table 1, the VPN guidance law refers to a special optimal guidance algorithm designed for a groundto-air interceptor; 4) MMD denotes the mean miss distance; MPCA denotes the mean peak commanded acceleration; MPAOA denotes the mean peak commanded AOA; MPSSA denotes the mean peak commanded sideslip angle; Time means the engagement time. Case I denotes a non-maneuvering target, Case II denotes a maneuvering target, where the angles (AOA and sideslip angle) of the target are time-varying.…”

“…The iterative algorithm is prescribed elsewhere. 15,16) In particular, the target's velocity and acceleration vectors must be available when solving Eq. (15).…”

“…Adler 10) introduced 3D PPN guidance law first in terms of the geodesic and normal curvatures of the missile's path on the surface generated by the LOS. Chiou and Kuo [11][12][13] proposed DG guidance commands for missile guidance problems using the Frenet formulas, 14) while Jing and Li 15,16) examined their applications. Ariff presented a novel DG guidance algorithm using the information of the involute of the target's trajectory.…”

Application of PID controller to 2D differential geometric guidance problem

“…Chiou and Kuo [9∼11] proposed DG guidance commands for missile guidance problems using the Frenet formulas [12]. Jing and Li [13,14] examined the applications of the DG guidance commands to a realistic missile defense engagement. The resulted DG guidance law was shown to be a generalization of the PN guidance law.…”

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

“…C.Y. Li, et al [3][4][5] translated the DGGC of the arc-length system into the time domain, and studied scenarios of DGGC interception of a tactical ballistic missile, as well as the initial capture condition of the case. K.B.…”