Some Discussions about the Error Functions on SO(3) and SE(3) for the Guidance of a UAV Using the Screw Algebra Theory

Yi Zhu,Xin Chen,Chuntao Li

doi:10.1155/2017/1016530

Yi Zhu, Xin Chen + Show 1 more

Open Access

https://doi.org/10.1155/2017/1016530

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Abstract

In this paper a new error function designed on 3-dimensional special Euclidean group SE(3) is proposed for the guidance of a UAV (Unmanned Aerial Vehicle). In the beginning, a detailed 6-DOF (Degree of Freedom) aircraft model is formulated including 12 nonlinear differential equations. Secondly the definitions of the adjoint representations are presented to establish the relationships of the Lie groups SO(3) and SE(3) and their Lie algebras so(3) and se(3). After that the general situation of the differential equations with matrices belonging to SO(3) and SE(3) is presented. According to these equations the features of the error function on SO(3) are discussed. Then an error function on SE(3) is devised which creates a new way of error functions constructing. In the simulation a trajectory tracking example is given with a target trajectory being a curve of elliptic cylinder helix. The result shows that a better tracking performance is obtained with the new devised error function.

Highlights

The way of computing the tracking errors plays an important role in the guidance process of a UAV
For the problem of either a 2D tracking in a plane or a 3D tracking in the physical space, many valuable researches have been made about the guidance methods of “trajectory tracking” and “path following” [1]
Efforts have been made to discuss the features about the error functions on SO(3) and SE(3)

Summary

Introduction

The way of computing the tracking errors plays an important role in the guidance process of a UAV. In some literatures the guidance strategy is implemented based on a mixed structure of the attitude loops and guidance loops with controllers of the forces and moments [13, 14]. Another instance is the moving frame guidance method. The designing of the error functions of a moving frame is a difficulty because there is interdisciplinary knowledge involved such as the Lie group theory. The effects of the different error functions are tested in the simulation with a 6-DOF UAV model

Preliminary

Simulations and Analysis

Conclusions