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.
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 [
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 [
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 [
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 [
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
Illustration of the moving frame tracking method.
It can be seen from Figure
By mathematical ways, it is implementable to form a moving frame of a 3D Curve. In the early stages of classical differential geometry, the Frenet frame was adopted to formulate a curve in the three-dimensional Euclidean space
Today the Frenet frame has been improved to become the Bishop frame. The type 1 Bishop frame is defined by
From the definition of the Bishop frame and the relationship between it and the Frenet frame we can see that, roughly speaking, the Bishop frame is a frame which has the same original point with the Frenet frame and aligns one of its axes with one of the axes of the Frenet frame. Then the rest of the two axes of the Bishop frame rotate at a certain angle
Let the inertial frame be defined by
Assuming the Bishop frame of the target curve is defined by
Let
To be more precise about the meaning and resolving method of the angular velocity vector just mentioned above, some discussions of the Darboux vector are given in the following. The Darboux vector is named after Gaston Darboux who discovered it. It is also called angular momentum vector, because it is directly proportional to angular momentum. For a rigid body moving absolutely along a general curve which is expressed by a moving frame, its motions can be divided into two kinds of motions as the translational motion and the rotational motion. The rotational motion is formulated by the Darboux vector
The planform and the actuators distribution of the flying-wing UAV.
Assume there is an initial position
The mass of the UAV, which is used in the simulation, is 2300 kg and its wingspan equals 8.76 m with a typical flying-wing configuration. This kind of UAV has no vertical or horizontal tails, so its aerodynamic characteristic is unordinary in various aspects. That leads to a worse static heading stability. The actuators of a flying-wing UAV are not the conventional elevators, ailerons, or rudders. Thus the allocation has to be made to transform the signals of actuators of a flying-wing UAV into the signals of the virtually ordinary actuators, such as
Figure
Simulation structure of the UAV flight control system.
A UAV nonlinear state-space system can be obtained with state variables defined by
Particularly, the controllers of the attitude and trajectory loops are all the ordinary controllers for simplicity. Also when a better tracking result is obtained, the success can be clearly attributed to the devised method rather than the other advanced and complex controllers of the innerloops.
The target curve is chosen as an elliptic cylinder helix extending along the horizontal direction. The expression of the target curve with regard to a time parameter is defined as
By (
In simulation, according to (
(a) Trajectory of target curve, (b) the target curve and its Frenet frame, (c) the target curve and its Bishop frame, and (d) the target curve and its moving frames.
In Figure
For the lengths of the intervals of three axes of the moving frames are different, there are some zoom in and out of these axes in Figure
The beginning of the tracking action starts from the moment the UAV arrives in a small neighborhood of an arbitrary point of the target curve. The initial value of the speed vector of the body frame is chosen at
(a) 3D trajectory tracking result, (b) tracking curves in the vertical direction, (c) tracking curves in the lateral direction, and (d) tracking curves in the forward direction.
Figure
The moving frame tracking method devised in this paper based on the classical differential geometry is interdisciplinary. It is a trial to solve the trajectory tracking problems in a new direction avoiding the complex designing of the innerloop controllers. Although the innerloop controllers are simple, the whole flight control system in the simulation is much more complicated since a 6 DOF model of a large flying-wing UAV is employed. From the simulation it can be seen that a rapid, accurate, and safe trajectory tracking result is obtained.
The authors declare that there are no competing interests regarding the publication of this paper.