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

Zhu, Yi; Chen, Xin; Li, Chuntao

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

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2017 | Article ID 1016530 | 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

Yi Zhu,¹Xin Chen,¹and Chuntao Li¹

Academic Editor: Stephen C. Anco

Received02 Aug 2016

Revised03 Nov 2016

Accepted29 Nov 2016

Published04 Jan 2017

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.

Here , , and are the angular deviations of the elevator, ailerons, and rudder, is the opening degree of throttle, and Mach represents Mach number. As shown in Figure 1, the actuators are , , , and rather than forces and moments. Then, it is viable to construct a simulation model of the given aircraft. By (1)~(4) a nonlinear state-space system can be obtained with state variables defined by and control input vector as .

2.2. Adjoint Representations of Lie Algebras

Before the features of the error functions on SO and SE are discussed, the features of SO and SE themselves should be made clear. In the following part, some basic concepts of the screw algebra and Lie group theory are discussed in detail.

According to the screw algebra theory of motions of the rigid body, the definitions and adjoint representations of the 3-dimensional special orthogonal group , the 3-dimensional special Euclidean group , and the corresponding Lie algebras and are presented as follows.

(I) The matrix adjoint representation of is as follows.

is the Lie algebra of the 3-dimensional special orthogonal group , the adjoint representation of which has a form of a skew-symmetric matrix:Since the Lie algebra is represented as a skew-symmetric matrix, one basis of the 3-dimensional vector space in the form of matrices iswhere , , and .

Any elements belonging to can be represented as a linear combination of this basis.

(II) The standard and adjoint representation of is as follows.

is the Lie algebra of the 3-dimensional special Euclidean group . , also denoted , is defined to describe rigid body motions including translations and rotations, which is based on an identity of a rigid body motion and a curve in the Euclidean group.

The standard matrix representation of is where and . These elements belong to a space which is a subset of . The generators of this 6-dimensional vector space areAlso there is a matrix adjoint representation of defined aswhere the operator is isomorphic to and the generators areWe can see that the Lie algebra is a subspace of , where the symbol means isomorphic.

(III) The exponential mapping is as follows.

The exponential mapping establishes a connection between and , and as well. According to the Rodrigues equation, when the rotation axis and the revolute joint are given, the rotation matrix can be obtained as where , , and .

Formula (13) presents the exponential mapping from to , the proof of which can be found in literature [22].

Similarly, the exponential mapping from to is defined as where is the standard matrix representation of , and(IV) The relationship between and is as follows.

Special Euclidean group is a closed subgroup of 3-dimensional affine group . can be represented as a semidirect product of the special orthogonal group and the translation group ; that is,The geometric meaning of above semidirect product is a rotation motion acting on a translation.

Furthermore, a finite displacement screw matrix is defined aswhere rotation matrix and is a skew-symmetric matrix of translation action. Then we can see thatSo the finite displacement screw matrix belongs to the special linear group , which is a subgroup of the general linear group . Also is an element of the Lie group .

3. Error Functions Defined on SO(3) and SE(3)

3.1. General Situation

In the beginning of this section an example is introduced to show the features of equations with matrices belonging to . The following equations are given:where and . Introduce the matrices , defined bywhere , are both matrices. The following equation holds:Also, there is matrix representation of elements of :These two adjoint representations of and matrices, that is, and , are isomorphic to each other. , the Lie algebra of , is isomorphic to as well. It is convenient to choose different forms we need in different situations. However, it is not difficult to verify that the matrix representations of do not satisfy (21).

This is different from the previous error functions which are defined on , which is the subgroup of , such asIn this paper a trial has been made to define an error function straightforwardly on , so that the error function will include both the information of the rotation matrix and the position vectors or the speed vectors. The new error function on is defined byActually, (24) has a close relationship with . SincethusHence one can see that with an initial position and a trajectory , as long as a negative feedback of position signals is guaranteed, the position error is certain to be a decreasing function when it tends to the steady state. So the error function is bounded. Let ; then the domain of attraction of is regarded as a linear manifold of . That means some features about the error function defined on will still be helpful.

3.2. Error Function on SO(3)

To choose the tracking error vectors and reasonably, letBy finding the derivative of (27) we haveBefore further discussion, the following properties of 3-order skew-symmetric matrices are presented:Some proofs of (29)~(36) are clearly given [23, 24]; hence only the proofs of (29), (30), and (36) are given here. First of all, Theorem 1 is presented.

Theorem 1. Let , be and matrices, respectively; then the following equation holds:

Proof. Letand thus
is a square matrix of order, where .
is a square matrix of order, where :So , proof finished.

In addition, by the definition of the trace of a matrix, for any square matrix , obviously we haveThen the proof of (29) is presented as follows.

Proof of (29). By (37), (40), and the property of skew-symmetric matrix thatwe haveProof finished.

Proof of (30). For a three-order square matrix , it is easy to see that is a skew-symmetric matrix. DenotingthenProof finished.

By the definition of the inner product, one has thatBy (29), (30), and (45), the following equation holds:With regard to (46), (28) can be rewritten aswhere is a skew-symmetric matrix, is the inverse mapping of the hat mapping . Thus, the tracking error function of attitude can be defined asThen the proof of (36) is presented as follows.

Proof of (36). According to the rule of finding the derivatives of the rotation matrices with respect to time, we have thatAccording to (35)Proof finished.

So, we can chooseas the tracking error function of angular velocity vector. Actually, is the angular velocity of the rotation matrix , which is represented in the body frame, because of the following formulation:The proof of (52) is given as follows.

Proof of (52). Proof finished.

In getting the above conclusion the following equations are used:

3.3. Error Function on SE(3)

As mentioned above, the error function is defined as (24) by the adjoint matrix representation of elements on . The benefit of this adjoint representation rests with the simplicity of defining the semidirect product. However, the form of matrix representation is adopted here for the convenience of calculation. For a matrix , according to the principle of Chasles motion decomposition, one haswhere , , and being the Frobenius norm of matrices. By the definition of Frobenius norm, for a matrix For a three-order skew-symmetric matrix which is obtained by a hat mapping, its Frobenius norm isSometimes when it is necessary to change the pitch parameter of a screw and the variable is added, , thenwhere . See (59) for details and it is can be seen that because . Before further discussion of , another theorem is presented.

Theorem 2 (the Laplace’s expansion theorem). If rows (or columns) of -order determinant are selected, where , the sum of products of all the -order subdeterminants of elements of the rows (or columns) and the corresponding algebraic cofactors equals the value of the determinant .

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 2, for a block lower (upper) triangular matrixor in all the subdeterminants of the first rows of , only one of them is nonzero. Thus, by expansion of the first rows, the following deduction is obtained:So , and we can see that (the special linear group), which is a subgroup of general linear group .

Similar to (23) and (24), a new error function is defined bywhere, with regard to (59), , , is the Darboux vector of the frame and , , and , substituted into (63), and we getLet and ; then we have . Obviously, , are both skew-symmetric matrices belonging to . Since elements of have closure property with additive operation, thus . Another vector is defined where and another error function is defined byBy (30)

4. Simulations and Analysis

Figure 2 shows an overview structure of the whole flight control system. It is can be seen that there are inner loops (attitude loops and trajectory loops) since the 6-DOF model of the UAV is used. However, the designing of the inner loops is independent of the designing of error functions. So the inner loop controllers are all chosen as ordinary ones.

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 3.

The main data of the UAV are shown in Table 1. From Table 1 it is can be seen that the UAV has a big size of 2300 kg; thus the target trajectory is also a large curve.

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 aswhere are all the coefficients. The beginning of the tracking action is from the moment the UAV arrives in a small neighborhood of any point of the target trajectory. The initial speed vector of body frame is and the attitude angles vector is , where the units are, respectively, “m/s” and “°.” According to the initial condition of the UAV, the values of the coefficients of (67) are finally chosen byThe curve of the target trajectory is shown as in Figure 4(a).

(a)

(b)

(c)

(d)

For the similarity of longitudinal and lateral channels of the flight control system, here just take the tracking errors in axis- for example. In Figure 4(b), the magnitude of tracking error is largest with the ordinary error function. When the error function on SO is added, the tracking error decreases significantly. When the error function on SE is added, the tracking error decreases more. It should be noted that in Figure 4(b) the condition “error function on SE” means “error function on SE is added” so the previous error functions are still being used.

Figure 4(c) shows the tracking curve of the pitch angle . The pitch angle is limited in the reasonable ranges with amplitude limits of 20° and 30°. The pitch angle ranges within ±8°. Some slight buffeting near the vertex of the curve of the pitch angle has something to do with the inner loop controllers. Figure 4(d) shows the curve of the single thrust and there are two engines. In most literatures there is a supposition that the thrust of the aircraft is large enough. Actually a too large thrust means a difficulty in rapid reduction of the forward speed. The UAV mass is about 2300 kg and in simulation the maximum thrust weight ratio is about 0.32. The first 30 s of Figure 4(d) indicates a process of rapid convergence of the forward tracking error. The thrust is controlled directly by the opening degree of the throttle which is constrained in the closed interval .

5. Conclusions

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 and SE. The tracking effect of the flight control system is tested in the numerical simulation. The result of the simulations shows a satisfactory tracking performance so that the error functions designed in this paper are feasible in the UAV tracking process. The designing of the new error function on SE provides a new way of error functions constructing in solving the guidance problem of the UAV.

Competing Interests

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

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Copyright

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.

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