Time-optimal solutions of Parallel navigation and Finsler geodesics

M Rafie-Rad

doi:10.1016/j.nonrwa.2010.02.010

Abstract

A geometric approach to kinematics in control theory is illustrated. A non-linear control system is derived for the problem and the Pontryagin maximum principle is used to find the time-optimal trajectories of the Parallel navigation. It is proved that the time-optimal relative trajectories of the Parallel navigation are geodesics of a Finsler metric. It is notable that the approach has the advantages using feedback.

Highlights

The historical development of what became the Calculus of Variations is closely linked to certain minimization principles in the majority subjects in mechanics, namely, the principle of least distance, the principle of least time and the principle of least action [7]
The application of Finsler geometry in Physics, seismology and Biology is a subject of numerous papers such as [1], [2],[3], [5], [9], [13], [15], [18], etc
Its time derivative r = rT − rM = vT − vM is the relative velocity between the two objects, and vT and vM are the velocities of T and M, respectively

Summary

Introduction

The historical development of what became the Calculus of Variations is closely linked to certain minimization principles in the majority subjects in mechanics, namely, the principle of least distance, the principle of least time and the principle of least action [7]. Back in history, T could be a merchantman and M a pirate ship Control problems typically concern finding a (not necessarily unique) control law δ(.) , which transfers the system in finite time from a given initial state xi = r(0) , to a given final state xf = r(tf ). This transition is to occur along an admissible path, i.e. r(.) and respects all kinematic constraints imposed on it. Theorem 1.2 Given any time-optimal solution (r, δ) of P-navigation, the curve r is a geodesic of the Finsler metric (1). One may find various techniques in missile guidance and control in [17]

Preliminaries

The kinematics of Parallel navigation