Advances in Mathematical Physics

Volume 2017 (2017), Article ID 1016530, 11 pages

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

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

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Yi Zhu

Received 2 August 2016; Revised 3 November 2016; Accepted 29 November 2016; Published 4 January 2017

Academic Editor: Stephen C. Anco

Copyright © 2017 Yi Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

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].

To solve the tracking problems, different researchers hold different opinions. The early methods somewhat originate from the target tracking of missiles such as proportional navigation, way point, and vector field method [2–4]. Then the body-mass point model is usually used so that the direct relationship between the position deviation and speed (or acceleration) can be concerned. Sometimes the influence of the attitude angles and the angular velocity of body frame is also taken into consideration [5]. On the contrary, the features of the inner loops of the aircraft system are often clearly figured out for lager aircrafts [6, 7]. For a 2D tracking issue there are some novel navigation methods and guidance strategies emerging in the light of geometrical intuition and physical interpretation [8–10]. Also, for the curves of 3D trajectories, the constraint of time parameter can be transformed into an arc length parameter by theory of differential geometry [11, 12].

Actually, when a 6-DOF model of an aircraft is concerned, there are at least three basic coordinate frames included which are the inertial frame, the aircraft-body frame, and the airspeed frame. So the coordinate transformations between these different coordinate frames are directly related to the accuracy of the tracking errors computing, that is, where the error functions on SO are used. For examples, 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. This guidance method changes the ordinary error functions from the inertial frame to a moving frame by orthogonal matrices which belong to SO [15].

Many researches have been made about the formulation of a moving frame of a given trajectory, as recently in [16–19] and previously in [20, 21]. However, 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. Some literatures indicate that the analyses about the Lie group can be simplified by the screw algebra theory [22]. Theses analyses are important particularly in the tracking process of aircrafts [23, 24]. So in this paper some discussions have been made to provide clear relationships between Lie groups SO and SE and their Lie algebras and . Then some features of the error functions on SO are proved before a new designed error function on SE is proposed. Thus a new way of error functions constructing is presented. The effects of the different error functions are tested in the simulation with a 6-DOF UAV model.

#### 2. Preliminary

##### 2.1. UAV Model

A flight control system is a bit more complicated than ordinary control systems. The analytic expressions of 6-DOF motion of an aircraft, that is, the 12 nonlinear differential equations, are formulated as follows:(I)Force equations:(II)Kinematic equations:(III)Moment equations:where , , , , , , , , and .(IV)Navigation equations:orwhere is the mass of the aircraft, represents the force of axes of aircraft-body coordinate frame, respectively, , , and are moments of body frame, , , and are speed components of body frame, , , and represent pitch angle, yaw angle, and bank angle, respectively, , , and are angular velocity from body frame to inertial frame resolved in body frame, , , and represent the position of the aircraft in inertial frame, , , and are rotary inertias of axes of body frame, is true airspeed, and , are flight-path angles between the first/second axis of wind coordinate frame and inertial frame, respectively.

In practice, we may not necessarily choose , , and as the state variables of an aircraft model concerning different requirements and we usually choose the true air speed and aerodynamic angles, such as angle of attack and sideslip angle , instead. According to the rotation matrix from wind frame to body frame, along with the relationship , one has the equation

With (1)~(4) the 12 differential equations are obtained; however it is not adequate to establish a complete nonlinear model of a UAV. More additional parts are needed. Figure 1 shows the inner structure of the UAV dynamic model.