SHI Haoran,LU Faxing,WANG Hangyu,and XU Junfei

College of Weaponry Engineering,Naval University of Engineering,Wuhan 430033,China

Abstract:This article investigates the optimal observation configuration of unmanned aerial vehicles(UAVs)based on angle and range measurements,and generalizes predecessors’researches in two dimensions into three dimensions.The relative geometry of the UAVs-target will significantl affect the state estimation performance of the target,the cost function based on the Fisher information matrix(FIM)is used to derive the FIM determinant of UAVs’observation in three-dimensional space,and the optimal observation geometric configu ation that maximizes the determinant of the FIM is obtained.It is shown that the optimal observation configu ation of the UAVs-target is usually not unique,and the optimal observation configu ation is proved for two UAVs and three UAVs in three-dimension.The long-range over-the-horizon target tracking is simulated and analyzed based on the analysis of optimal observation configu ation for two UAVs.The simulation results show that the theoretical analysis and control algorithm can effectively improve the positioning accuracy of the target.It can provide a helpful reference for the design of over-the-horizon target localization based on UAVs.

Keywords:target state estimation,optimal observation configu ration,Fisher information matrix(FIM),Cramer-Rao lower bound(CRLB).

1.Introduction

In recent years,unmanned aerial vehicles(UAVs)have been widely used in military and civilian applications due to their diverse sensors,concealed actions,controllable costs,and no need for personnel to come forward.In particular,they have the characteristics of aerial reconnaissance,so that UAVs have the potential to provide precise target position for long-range strike weapons[1,2].

Target localization and tracking have been studied extensively in numerous references.Xu and Shao et al.[3,4]studied the factors that affect the precise localization by UAVs,the main influencing factors include the position of the UAV,the navigation accuracy of the attitude angle and the relative position relationship between the UAV and the target.In the research of multi-UAV coordinated localization and tracking,in order to achieve the purpose of effectively using observation information[5],Campbell et al.[6]used unscented Kalman filtering combined with information filtering to fuse target state estimation,Wang et al.[7]used distributed unscented information filtering to achieve estimation of targets,and Yu et al.[8]used the two-level localization mode of coarse and fine to obtain the optimal solution of the target position.

However,the fusion filtering method is the only way to improve the target state estimation based on the observation data,the validity of the observation data varies according to the geometry of the UAV relative to the target.Subsequent research of relevance focuses mainly on developing methods for optimizing observation configuration to achieve the best target estimation performance.A common approach is to use objective functions derived from the determinant of the Fisher information matrix(FIM)[9–11].FIM characterizes the amount of information about state parameters contained in a given observation sequence.The larger the value,the smaller covariance of the state estimation and the more accurate of the target state estimation.Tichavsky et al.[12]proposed a recursive FIM calculation method.Adrian et al.[13]studied the optimal sensor-target geometries for range-only,time-ofarrival-based and bearing-only localization.Sameera et al.[14]gave an FIM expression based on angle-measuring of UAVs system in three dimensions but did not simplify it.For a target-tracking application with range sensors,the determinant of the FIM is computed in the two-dimension(2-D)and three-dimension(3-D)cases by Sonia et al.[15].Eric et al.[16]researched the optimal observation configuration of double UAVs based on angle and range measurements in two-dimensional space.Wang[17]gave the determinant of FIM based on angle and range information in two dimensions and proposed the optimal observation configuration of UAVs.Zhong et al.[18]analyzed the optimal configuration of sensor-target geometries for bearing-only passive localization in three-dimensional space and theoretically derived it.Similar optimization functions are considered in some researches.Logothetis et al.[19]used mutual information as the objective function to model the optimal observational motion problem as a part of a considerable Markov process.Frew[20]explored the problem of trajectory planning for a 2-D ground robot equipped with vision sensors,using the determinant of the error covariance matrix as the objective function.The target tracking problem was solved by using the trace of the FIM as the objective function by Zhou et al.[21].

At present,most researches about the problem of coordinated localization and tracking of UAVs are based on short-range tracking,but there is little research on overthe-horizon target tracking by UAVs.Some researches with the optimal configuration of UAVs are in two-dimensional space,and some only consider the measurements of range only or bearing-only in three-dimensional space,which is difficult to meet the requirements of accurate localization of the over-the-horizon target.For those reasons,this work builds upon previous literature by developing a cost function based on the determinant of FIM for designing UAVs observation configuration to localize a target using range and angle measurement sin three-dimensional space.The purpose of this article is to focus on analyzing the optimal configuration of UAVs based on range and angle measurements,increasing the information provided by the measurements,reducing the uncertainty of target state estimation,and improving the accuracy in localization of the target.The optimal observation control algorithm of two UAVs is designed and simulated based on the analysis of optimal observation configuration for two UAVs.The simulation results verify the effectiveness of the theoretical analysis and control algorithm in this paper.

2.Determinant of FIM

2.1 Definition of FIM

Error covariance is a common performance function of state estimation,which represents the uncertainty of state estimation,so the error covariance should be as small as possible.

The Cramer-Rao lower bound(CRLB)provides a lower bound on the covariance of the estimation.Moreover,CRLB is related to the inherent properties of the system and has nothing to do with the specific estimation algorithm.The mathematical expression of CRLB is

wherePk|kdenotes the covariance matrix of the estimation error,Xkis the state to be estimated,andis the estimated value ofXkunder the given measurement data.Frepresents FIM,which is the inverse of CRLB,and it represents the amount of information about the state parameters contained in a given observation sequence.

The determinant of FIM det(F(T))is inversely proportional to the uncertainty area ofT,therefore the optimality analysis of sensor-target geometry is equal to maximizing det(F(T)).

2.2 Calculation for the determinant of FIM

In three-dimensional space, the target locates inT(xt,yt,zt),UAVilocates inUi(xi,yi,zi),UAVjlocates inUj(xj,yj,zj),the angle of sight between UAVs and the target isθij,which is shown in Fig.1.

Fig.1 Target localization by two UAVs

Fig.1 shows the horizontal angle of the target observed by UAVjisβj,and the pitch angle of the target observed by UAVjis?j.is the target state at timek,Uk=[xu,yu,zu]Tkis the UAV state at timek,and the related vector between the UAV and the target iswhererx=xu?xt,ry=yu?yt,rz=zu?zt.We can obtain The target information observed by the UAV is

The Jacobian matrix of the observation function is expressed as

eri,eβiande?iconstitute the observation error matrixRiof UAVi.eri～N(0),eβi～N(0),e?i～N(0).The observation error matrix of UAViis

The FIM of this observation system[14]is

In the optimal observation configuration,only the current position of the target is considered,and it is assumed that the current position of the target is fixed.

Therefore,for the current UAV observation platforms,the total FIM of the system[17]can be expressed as

Using the Cauchy-Binet[22]formula,det(J)is composed of the following ten parts,and the case ofN=2 is explained at the same time.

i)Asi=j=k,that isi,j,kcolumns of observation come from the same UAV,this sub-part is equal to

whereS11={(i,j,k)|1i=j=kN},N(S11)=C1N.

ii)Asi=jk,both columnsiandjcome from the same UAV,columnkcomes from another UAV,this subpart is equal to

whereS12={(i,j,k)|1i=jkN},N(S12)=C1NC1N?1.

iii)Asi=kj,both columnsiandkcome from the same UAV,columnjcomes from another UAV,this sub-part is equal to

whereS13={(i,j,k)|1i=kjN},N(S13)=

whereS14={(i,j,k)|1ij=kN},N(S14)=

whereS15={(i,j,k)|1ijkN},N(S15)=.In the case ofN=2,this part does not exist.S1=S11∪S12∪S13∪S14∪S15,the first part is equal to the sum of(11)–(15).

i)Asi=j,both columnsiandjcome from the same UAV,this sub-part is equal to

whereS21={(j,k)|1j

ii)Asj=k,both columnsjandkcome from the same UAV,this sub-part is equal to

whereS22={(i,j)|1i

whereS23={(i,j,k)|1i

Similar to the analysis of the second part,the third part is also divided into three situations.The third part is as follows:

Similar to the analysis of the second part,the fourth part is also divided into three situations.The fourth part is as follows:

where

S43={(i,j,k)|1i

Similar to the analysis of the second part,the fifth part is also divided into three situations.The fifth part is as follows:

where

S53={(i,j,k)|1j

Similar to the analysis of the second part,the sixth part is also divided into three situations.The sixth part is as follows:

Similar to the analysis of the second part,the seventh part is also divided into three situations.The seventh part is as follows:

This part is equal to zero.

whereS9={(i,j,k)|1i

whereS10={(i,j,k)|1i

2.3 Four equations used to compute the determinant of FIM

The following four equations are needed to calculate the FIM determinant of a system.

Givenr(?i,βi)=(cos?isinβi,cos?icosβi,sin?i),we can get the followings:

(i)According to the definition of cosθijin Fig.1,

where 1i

(ii)From(35),we can get

where 1i

(iii)Calculate the mixed product ofr(?1,β1),r(?2,β2)andr(?3,β3):

(iv)There are the following relationships:

3.Analysis of optimal observation configuration

The optimal observation configuration refers to UAVs observation configuration which minimizes the error of target state estimation.The quality of the measurement data is closely related to the observation position of the UAV relative to the target.The target information provided by the measurement data obtained from different observation positions is different.The “good”or“bad”of the measured data directly affects the target state estimation.This part mainly analyzes the observation configuration of the UAV,so that the UAV is in a better observation position and get a better target state estimation.

From the derivation process of FIM determinant,it can be seen that when moving UAVifromUi=(xui,yui,zui)to=(2xt?xui,2yt?yui,zui),the total determinant value of FIM does not change,so the optimal observation configuration of the UAVs is not unique.

3.1 Analysis of optimal observation configuration for N=1

WhenN=1,that is,a single UAV observes the target.

The total determinant of FIM is

This shows that the smaller distance between the UAV and the target is,the larger determinant of the FIM is,and the more information UAV observes on the target,especially0,det(J)→∞.Therefore,when a single UAV observes,it will shorten the distance between the UAV and the target as soon as possible.There is a minimum observation distancerminin practical application,the maximum determinant value of the system FIM is det(J)=Especiallydet(J)→0,there is no unbiased estimator.

3.2 Analysis of optimal observation configuration for N=2

WhenN=2,that is,two UAVs observe a single target,the total determinant of FIM is

From(41),we can get when two UAVs observe,Ri=rmin(i∈{1,2})is a necessary condition for the maximum determinant of FIM.WhenR1=R2=rmin,then,

Obviously det(J)is maximized if cosθ12=0,when the two UAVs measure the range and angle of the target in three-dimensional space,two UAVs should be as close to the target as possible and keep the line of sight angleθ12= π/2,so that the determinant of FIM gets the maximum and the target tracking error is the smallest.Fig.2 shows the optimal observation geometry configuration of two UAVs.

Fig.2 Optimal observation geometry configuration of two UAVs

Equation(41)is proved as follows.

ProofAccording to the analysis of Section 2.2,letN=2,the total determinant of FIM is computed as follows:

Letr(?i,βi)=(cos?isinβi,cos?icosβi,sin?i),we can get

From(46),we can get

From(48),we can get

Substituting(44),(45),(47)and(49)into(43),we obtain(41)as follows:

3.3 Analysis of optimal observation configuration for N=3

LetF(i,j)be a function of(i,j)(i,j∈N+)and

LetE(i,j,k)be a function of(i,j,k)(i,j,k∈N+)and

WhenN=3,that is,three UAVs observe the target,the total determinant of FIM is

The process of proof refers toN=2.

From(52),we can get when three UAVs observe,Ri=rmin(i∈{1,2,3})is a necessary condition for the maximum determinant of FIM.WhenR1=R2=R3=rmin,

The problem is transformed into finding the extreme value of(53).It can be obtained that(53)achieves the maximum value whenθ12=θ13=θ23= π/2,so when the three UAVs observe a single target,the optimal observation configuration between the three UAVs isθ12=θ13=θ23= π/2.Fig.3 shows the optimal observation geometry configuration of three UAVs.When the distance between the UAV and the target isr,the distance between UAVs is.

Fig.3 Optimal observation geometry configuration of three UAVs

3.4 Analysis of optimal observation configuration for N4

The total determinant of FIM forN4 is

When there is a minimum observation distanceRi=rmin(i∈{1,2,...,N}),maximizing(54)is equivalent to minimizing

whereaandbare constants aboutrmin,σrandσ[23].

When the sea surface target is measured byNUAVs,the problem becomes how to distributeNUAVs on the hemisphere with the target as the center to minimize(55).Some studies on the problem of optimally arranging points on a sphere that maximize or minimize some quantity are still unsolved[24],so it is a worldwide problem.

Through observation,we can see that(55)is an expression with strict symmetry,and it is conjectured that the geometric configuration maximizing(55)is also strictly symmetrical.One possible geometric configuration is thatNUAVs are uniformly distributed on the hemispheric surface.Unfortunately,the strict mathematical proof cannot be given yet,which is the direction of future work.

4.Cooperative target tracking

The problem of two UAVs cooperating on target tracking is considered[25],and the UAV flies at the same altitude.According to the analysis of the optimal observation configuration of two UAVs,we can get from(42)that

Equation(56)shows that when two UAVs are as close as possible to the safe observation distance and maintain a line of sight angle of π/2,the target tracking error is the smallest,and the target state estimation is more accurate.

4.1 Optimal observation control for cooperative target tracking

The optimal control method is designed to make the angle between UAVs and the target close to or maintain at 90?in three-dimensional space,so as to improve the accuracy and real-time performance of target tracking.

It is only necessary to design the control method in the approaching phase because the UAV is far away from the target and is close to the target motion for over-the-horizon target tracking.In order to reach the line of sight angle of 90?,the heading angle control of UAVs[26–28]is designed as follows:

According to(35)

Theβ1andβ2components guide the UAV to fly to the target and shorten the distance to the target,the sin(θ12?π/2)component makes the line-of-sight angle between the two UAVs and the target approach 90?.Adjust the degree of tending to 90?through proportional coefficientsk1(θ12)andk2(θ12).

whereK1andK2are proportional coefficients.

whereDis a distance factor;D1andD2are the distance from two UAVs to the target respectively;andrminrepresents the minimum safe observation distance.When it is far from the target,increase the control amount of UAV in the direction of the target movement to approach the target.When it is close,increase the control amount of the angle between the UAV and the target line of sight to achieve the optimal observation angle.

4.2 Fusion filtering algorithm

In order to make effective use of observation information and reduce the impact of observation errors on positioning accuracy,a certain filtering algorithm is needed.In this paper,the classical extended Kalman filter(EKF)algorithm is used to filter the observation data of UAV.

Assume that two UAVs get two trajectoriesiandjafter EKF with state estimatesiand,error covariancesPiandPj.The purpose of estimation fusion is to find the best estimatexand error covariance matrixP.

5.Numerical simulation and analysis

It is assumed that the target is moving with a constant speed at 8 m/s and direction at π/4,the initial position of the target is at(0 m,200 000 m,0 m).The speeds of UAVs are 100 m/s with the fly attitude at 2 500 m.The initial positions of UAV1and UAV2are(12 500 m,48 412 m,2 500 m)and(–12 500 m,48 412 m,2 500 m)respectively.The random and system errors of sensors are(1 m,0.05?,0.05?)and(3 m,0.1?,0.1?)with random error obeying Gauss distribution.The minimum safe distance isrmin=50 km and the entire simulation lasts 1 250 s.The simulation results are shown in Figs.4–7.In Fig.4,?and Δ represent the initial position of the UAV and the target,respectively.

Fig.4 Trajectory of UAVs and target

Fig.5 Angle of sight between UAVs and target

Fig.6 Determinant of FIM

Fig.7 Tracking error of the target

From the simulation results,it can be seen that the two UAVs separate a certain angle when tracking as shown in Fig.4,and the line-of-sight angle between the UAVs and the target increases as shown in Fig.5.As the UAVs approach the target and the line-of-sight angle increases,the determinant of the system FIM becomes larger as shown in Fig.6,the error of the target state which the UAV estimates is smaller as shown in Fig.7,and the target positioning is more accurate.

Through the analysis of Fig.6,the change rate of the determinant of FIM is getting larger as a result of the UAV is getting closer to the target,and it can be derived from(41).Combined with Fig.5,although the line-of-sight angle between the UAVs and the target increases slowly with time,it has little effect on the growth rate of the determinant of FIM.Therefore,it is necessary to make the UAV close to the target as soon as possible under the condition that the line-of-sight angle approaches 90?.

As seen from Fig.7,the target error observed by UAV1is 196.2m at the initial time,and the target error of cooperative tracking observation by two UAVs is 133.4 m,which improves the error accuracy by 32%.Therefore,compared with the single UAV,it can effectively improve the positioning accuracy of the target by two UAVs’observations.Moreover,the cooperative tracking error of the target is 133.4 m at the initial time in Fig.7,and the cooperative tracking error is 56.2 m at 1 250 s,which improves the tracking error accuracy by 58%.It can be concluded that the theory analyzed in this paper and the proposed control algorithm can effectively improve the positioning accuracy of the over-the-horizon target.

6.Conclusions and future work

In this paper,the problem of optimal configuration based on range and angle measurements is extended from twodimensional space to three-dimensional space,which can be widely used in military and engineering fields.This paper focuses on the observation model of sensors based on range and angle measurements and takes the FIM determinant of UAV observation in three-dimensional space as the cost function.The expression of optimal observation geometry and the conditions of optimal configuration solution of UAVs observation are given.It is pointed out that the optimal observation configuration is usually not unique.The optimal control method of cooperative target tracking by two UAVs is proposed based on the analysis of optimal observation configuration of the two UAVs.The simulation results show that the theoretical analysis and the control algorithm in this paper can effectively improve the positioning accuracy of the over-the-horizon target.

However,only an optimal control method for cooperative tracking of two UAVs is proposed,how to realize the optimal observation control of three or more UAVs is the focus of the next research.Moreover,when using UAVs as the observation platform to estimate the target state,we should also consider the influence of the UAV attitude angle accuracy on the tracking error.How to improve the accuracy of the UAV attitude angle is also the next research direction.