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.

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” [

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 [

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

Many researches have been made about the formulation of a moving frame of a given trajectory, as recently in [

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:

Force equations:

Kinematic equations:

Moment equations:

Navigation equations:

In practice, we may not necessarily choose

With (

The inner structure of the UAV model.

Here

Before the features of the error functions on SO

According to the screw algebra theory of motions of the rigid body, the definitions and adjoint representations of the 3-dimensional special orthogonal group

(I) The

Any elements belonging to

(II) The standard and adjoint representation of

The standard

(III) The exponential mapping is as follows.

The exponential mapping establishes a connection between

Formula (

Similarly, the exponential mapping from

Special Euclidean group

Furthermore, a

In the beginning of this section an example is introduced to show the features of equations with matrices belonging to

This is different from the previous error functions which are defined on

To choose the tracking error vectors

Let

Let

In addition, by the definition of the trace of a matrix, for any square matrix

By (

For a three-order square matrix

By the definition of the inner product, one has that

According to the rule of finding the derivatives of the rotation matrices with respect to time, we have that

So, we can choose

In getting the above conclusion the following equations are used:

As mentioned above, the error function

If

Detailed discussions of Laplace’s expansion theorem can easily be found in teaching materials of matrix theory or linear algebra, so the proof is omitted here. According to Theorem

Similar to (

Figure

Overview structure of the devised flight control system.

In the simulations, the employed UAV model originates from an improved and trial type of China’s “Sharp Sword” unmanned combat aerial vehicle. Its three-view drawing is shown in Figure

Three-view drawing of the UAV used in the simulations.

The main data of the UAV are shown in Table

Data of the UAV in simulations.

Aerodynamic configuration | Flying-wing configuration | Mass of UAV | |
---|---|---|---|

Inertia | | Wing span of UAV | 8.76 m |

| Max thrust | 500 kgf | |

| Max Mach | 0.85 | |

| Maximum flight altitude | 15 km |

The target trajectory is chosen as an elliptic cylinder helix extending along the horizontal direction. The expression of the given trajectory with regard to a time parameter is defined as

(a) Curve of the target trajectory. (b) A comparison of the forward errors. (c) Curve of the pitch angle. (d) Force of the thrust.

For the similarity of longitudinal and lateral channels of the flight control system, here just take the tracking errors in axis-

Figure

According to the nonlinear model of a UAV, a 3D trajectory tracking method is devised. Efforts have been made to discuss the features about the error functions on SO

The authors declare that there is no conflict of interests regarding the publication of this paper.

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