Nonlinear Guidance Law for Three-Dimensional Path-Following Based on Differential Geometry

Abstract

A new path-following guidance law for autonomous vehicles to accurately track a general three-dimensional curve without severe restriction on the initial condition is proposed. For easy implementation of the proposed guidance law, particular emphasis is placed on the design based on geometric intuition and the simplicity of guidance command. The proposed approach has the same structure with classical guidance laws, but it designs a direction of pursuit rather than following a virtual target moving along the desired path.

In the first part of the dissertation, a new three-dimensional nonlinear path-following guidance law is proposed on the basis of elementary differential geometry of space curve. The structure of guidance command is designated to have the same form with a pursuit guidance law which acts to align the velocity vector with the look-ahead vector. Then, a method to design a proper look-ahead vector is proposed, where the look-ahead vector is designed to have cylindrical symmetry with respect to an axis parallel to the tangent vector and shifted by a certain distance from the closest projection point on the desired path to the direction of curvature vector. A necessary condition of normal acceleration command for exact path-following is considered in the design of look-ahead vector. The proposed guidance law can be used to track three-dimensional curves of general shape, and it can be applied without much restriction on the initial position and velocity. Unlike the existing virtual-target-following approach, accurate tracking of general three-dimensional curves besides straight line and circle can be achieved.

In the second part of the dissertation, the stability and transient performance of the proposed guidance law are analyzed. A nonlinear stability analysis on the guidance law is performed considering three-dimensional path-following of a circle. Asymptotic stability for this case is proved by the Lyapunov stability theory without linearization or approximation of the error dynamics. Also, a linear performance analysis on the guidance law is performed considering two-dimensional path-following of a circle. The relation between transient characteristics of the linearized cross-track error dynamics and design parameters is investigated. The proposed guidance law has two design parameters while most existing virtual-target-following methods have only one, which allows more flexibility in adjustment of transient characteristics.

Additional implementation issues such as systematic selection of design parameters, determination of the closest projection point, and command modifications for implementation on a fixed-wing aircraft are discussed in the third part of the dissertation. Finally, the performance and effectiveness of the proposed guidance law is demonstrated by numerical simulations for various types of desired paths.