International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 3406256, 9 pages

http://dx.doi.org/10.1155/2016/3406256

## A Moving Frame Trajectory Tracking Method of a Flying-Wing UAV Using the Differential Geometry

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

Received 27 July 2016; Accepted 19 October 2016

Academic Editor: Mahmut Reyhanoglu

Copyright © 2016 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

The problem of UAV trajectory tracking is a difficult issue for scholars and engineers, especially when the target curve is a complex curve in the three-dimensional space. In this paper, the coordinate frames during the tracking process are transformed to improve the tracking result. Firstly, the basic concepts of the moving frame are given. Secondly the transfer principles of various moving frames are formulated and the Bishop frame is selected as a final choice for its flexibility. Thirdly, the detailed dynamic equations of the moving frame tracking method are formulated. In simulation, a moving frame of an elliptic cylinder helix is formulated precisely. Then, the devised tracking method on the basis of the dynamic equations is tested in a complete flight control system with 6 DOF nonlinear equations of the UAV. The simulation result shows a satisfactory trajectory tracking performance so that the effectiveness and efficiency of the devised tracking method is proved.

#### 1. Introduction

The main difficulties of a UAV trajectory tracking issue differing from the ordinary tracking problems lie in the strict real-time requirement and the complexity of 3-dimensional curves. Some literature only attempts to deal with the reduced situations such as the “path following” problem without a constrained time parameter [1–3] or the simplified target curves in 2D plane [4–6]. What is more, different types of curves usually call for different guidance strategies. In contrast, the moving frame tracking method is employed to give a more universal solution for various kinds of complex curves in the 3D space.

There are two procedures of the moving frame tracking method.

The first procedure of the moving frame tracking method is to form a moving frame of the given target curve. A moving frame is a moving coordinate axis system with its three axes being the curve’s tangent vector, principal normal vector, and secondary normal vector. There are many ways to obtain a moving frame of a given curve [7, 8]. The Bishop frame which is also called the parallel frame is the most famous one [9, 10]. In recent years, papers tend to concentrate on the improvements of the Bishop frame [11, 12]. In this paper by comparisons of the Frenet frame and the type 1 Bishop frames, we can see some insights of forming a moving frame and find out the reason why the Bishop frame is suitable for solving a UAV trajectory problem.

The second procedure of the moving frame tracking method is to formulate the dynamic error equations of the whole tracking process. In this procedure a quite unusual tracking process is presented. We no longer need to care about the individual shape of the target curve [13, 14]. That is to say, whether the target curve is a straight line, a circle, or an arc is unimportant. Instead, the moving frame of the curve with its tangent vector, principal normal vector, and secondary normal vector is the objectives to follow. Some knowledge of the theoretical mechanics about the motions of rigid bodies is necessary. In essence, the objective dynamic error equations are the result of a series of coordinate transformations and vector operations [15, 16]. In this procedure, the final aim is to obtain the kinematics errors in the moving frame of the target curve rather than the inertial coordinate frame.

The previous papers treating the trajectory tracking problem with the moving frame method are not sufficient. The reason is that the moving frame tracking method is very interdisciplinary for the theorists of guidance, navigation, and control. However, the analysis of geometrical properties of curves actually plays an important role in practical applications. Thus, the moving frame tracking is an effective and practical method for the UAV 3D trajectory tracking problem. In the previous literature some trials have been made to apply this tracking method to the tiny UAVs or quadrotors [15, 17]. Some other sources of literature are concerned about the mathematical features of this tracking method [18, 19]. All these sources of literature tackle the particle models of the microaircrafts with very low flight speed. For example, in literature [15], the velocity of the aircraft is 0.4 m/s. In this paper a much larger and heavier UAV is used and its velocity is about 200 m/s so that the situation is different. Also, it is well known that when a particle model is used, the angular motions inside the rigid body are omitted. So another distinction with the previous literature is that in this paper a 6 DOF UAV model is employed instead of a particle model.

#### 2. Formulation of the Moving Frame Tracking Method

##### 2.1. General Illustration

The core idea of the moving frame tracking method lies in the conception that the position errors are resolved in the moving coordinate frame of the target curve instead of the inertial frame in ordinary tracking methods. In this method the time constraint is well satisfied since the target curve is formulated with a time parameter . The primary aim is aligning the heading direction to the tangent vector of the target curve with almost no position deviation. The principle of the moving frame tracking method is illustrated in Figure 1.