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State Key Laboratory of Complex Electromagnetic Environment Effects on Electronics and Information System，Luoyang 471003，P.R.China

Abstract: When a group of mobile agents track a target，they can locate themselves and the target in a cooperative manner.To maximize the group advantage，a parallel integration strategy of cooperative target-localization（CTL）and cooperative self-localization（CSL）is designed.Firstly，a global cost function containing the agents’positions and the target’s position is established.Secondly，along with the agents’positions being re-estimated during CTL，the Utransform is employed to propagate the error covariance of the position estimations among the agents.The simulation results show that，the proposal exploits more information for locating the target and the agents than the cases where CTL and CSL run separately，and the global optimal position estimations of the agents and the target are obtained.

Key words：cooperative self-localization；cooperative target-localization；non-classical multi-dimensional scaling；majoring function

0 Introduction

Multi-agent（such as unmanned aerial vehicle，robot）system has great potential on target monitoring，disaster rescuing and so on.Multi-agent can cooperate for different goals，such as cooperative selflocalization（CSL）［1-4］，cooperative target-localization（CTL），cooperative searching，and cooperative communication.In this paper，CSL and CTL are highlighted.

CSL and CTL have different goals.Usually，CSL and CTL are researched as two topics.CSL aims at improving the localization accuracy of each agent by utilizing the relative information among the agents.Now，the CSL based on Kalman filtering（KF-CSL） or its extension has been heavily researched［5-9］.However，CTL aims at improving the target localization accuracy by fusing the relative information between the agents and the target.

Traditionally，CSL and CTL run separately.CSL can provide each agent’s position estimation for CTL.When these estimations are used for CTL，they are usually assumed to be deterministic and independent with each other.Based on this assumption，some algorithms have been proposed for CTL，such as least squares，D-S evidence theory，and neural networks［10］.In Ref.［11］，the Bayesian estimation method was proposed to deal with multisource uncertainty data.In Ref.［12］，a data aggregation algorithm based on the convolutional neural network model was proposed.In Ref.［13］，an improved data fusion based on the D-S evidence theory was proposed.

However，in the KF-CSL，the position estimations of the agents are mutually referred.As a result，the position estimations are non-deterministic characterized in variance and relevant characterized in covariance［5-8］.When these estimations are further used for CTL，the non-deterministic and relevant characteristics must be considered.

In this paper，CTL and CSL are together regarded as a global non-classical multi-dimensional scaling（nMDS）problem，that is，at one moment，the agents，the target and the distances（contain noise）among them constitute a snap-shot.For the nMDS，a distributed nMDS algorithm was proposed in Ref.［14］，where the majoring function（MF）is adopted for estimating the static nodes.In Ref.［15］， the MF-based optimization algorithm was used for mobile multi-agents.

In the typical studies，the followings are assumed：（1）The agents’position estimations are independent each other.（2）CSL and CTL are respectively solved.This paper is characterized as follows：

（1）A global cost function including the position variables of the agents and the target is established，where the position uncertainty of each agent and the position correlation among them are introduced.

（2）During the optimization iteration of the established cost function，the U-transform is employed to propagate the error covariance of the position estimations among the agents.

1 Problem Statement

The symbols used throughout this paper are defined in Table 1.

Table 1 Symbols used in text and derivations

Assume thatNagents move in a 2-D area to track a target.A fixed reference frame is set，where the actual position of the agentiat the time stepkis denoted asxi(k)=[xi(k)，yi(k)].The outputs of the KF-CSL，i.e.，the position estimations of the agents，are expressed as {x?i(k)}N i=1，and the corresponding error covariance is expressed as

In Fig.1，the agents（numbered in 1，2，…，N）track the target“o”.The dashed circles denote the position estimations of the agents through CSL.The solid circles denote the true positions of the agents.

Fig.1 Application demonstration of CSL and CTL

Whether one agent can find the target is changing for its varying position or view.At the timek，assume thatn(n≤N) agents（constitute a collectionC）find the target“o”and obtain the relative measurements to the target.The measurements are denoted aszio(k)(i∈C).Assume that the measurements are independent each other andandzio(k)，we attempt to re-estimate the agents and the target.In this process，the followings need to be solved.

（1）Under the case of the KF-CSL，the factors that affect CTL are the performance of the exteroceptive sensor which measures the relative measurement，and the agents’positions that are non-deterministic and relevant.

（2）Since the agents’positions estimated by the KF-CSL are non- deterministic，they are re-estimated during CTL.Then the re-estimated resultsshould include the uncertainty and the correlation to support the periodic CSL.

2 CTL Based on nMDS

2.1 KF-CSL and CTL

The KF-CSL is a premise of this paper.The KF-CSL is employed for its excellent recursion mechanism including the predictive and posteriori update［16-17］.Using the agent’s linear velocity measurements and the relative measurements，etc.，the position estimations and the corresponding error covariance are continually updated.Provide that the output form of the KF-CSL is unified as the posteriori estimations.The input/output of the KF-CSL and CTL is listed in Table 2.It can be found that the uncertainty and correlation are essential for CSL.In the traditional serial mode where the position estimations of the agents from CSL are constant input for CTL，the agents’positions are estimated in CSL and unchanged during CTL.

Table 2 Comparison of input/output between KF-CSL and CTL

In this paper，the operational logic of CSL and CTL is converted from the serial mode to the integration one.The following two facts are considered.Firstly，all the agents are cooperative while the target is not.Hence，the frequency of the data supplying for CSL is higher than that for CSL.Secondly，the re-estimated positions of the agents in CTL need to support CSL smoothly.The relationship between CSL and CTL are demonstrated in Fig.2.

Fig.2 Operation relationship between CSL and CTL

2.2 Object function modeling

When multiple agents find the same static target at the same time，the agents，the target and the relative measurements constitute a snapshot，where the agents are the anchors with a priori knowledge（from the KF-CSL），the target is unknown，and the relative measurements among them are constraint［18］.According to the nMDS，a global loss function with respect to the position variables of the agents and the target，is established as①Thereafter, the time index k is omitted to simplify the notation.

wherex=[x1，…，xN]represents the position vector of all agents②For the correlation of all the agents, the states of agent which does not find the target is also changed in the deep integration mode.So the states of all agents are included.，the distance function between the agentmand the targeto，and‖ ‖· the norm operation.Eq.（1）can be further rewritten as

The correlation and uncertainty of the elements(i=1，…，N) are introduced to the loss functionSby the block element(i≠j)and.

Let (?)-1=A.Each block elementAij∈R2×2represents the uncertainty of the position estimation(i=j) and the correlation among the estimations(i≠j).Then Eq.（1）can be further rewritten as

Usually，the optimization method is employed for Eq.（3）.During optimizing，the agents’position estimations are iterated，and the corresponding error covariance is changing synchronously.To propagate the covariance，the majorizing function（MF）is employed，which can provide an analytic representation between two-step iterations.

A majorizing functionT(X，Y) ofS(X) is a function that satisfies：（1）?Y，T(X，Y)≥S(X)，（2）T(X，X)=S(X).

In the MF，the iteration is described as follows：letY=X0and substitute it toT(X，Y) as the initial value.The iteration is repeated until the convergence condition is satisfied，i.e.

The upper-notationsis the mark of iteration step.Following Refs.［14-15］，Eq.（3）is rewritten as

where

Define the MF ofS(X)asT(X，Y)

Through the MF，minimizingS(X) is now a task of finding the minimum ofT(X，Y).

2.3 Optimizing derivation

The optimal solution can be obtained through taking the derivative ofT(X，Y)，shown as

According to the iteration principle，letxj=，Then the iteration models are given as

whereThe iteration coefficients are given according to two cases.

（1）Ifi∈C，the iteration coefficients are given as

（2）Ifi?C

For the targeto

In the case whereC=?，it means that no target is found and CTL doesn’t run，then Eqs.（11，13）are not required.At this moment，the proposed algorithm is degraded into a common CSL.

2.4 Covariance propagation

When the position estimateis iterated to，its covariance also changes fromIn this paper，the U transformation is utilized to propagate the position estimations and their covariance.The process is given as：

（1）Calculate（4N+1）σpoints

whereλis constant，and ()pthep'th column of the matrixThe detail can be seen in Ref.［19］.

（2）Propagate theσpoints as

where

2.5 Performance analysis

The fisher information matrix（FIM）and its determinant are adopted to evaluate the information value.FIM is defined as

where the Jacobian matrix of the relative measurement with respect to the whole stateX=[x1，…，xN，xo]is

Whenx1，…，xNare constant

Whenx1，…，xNare variable

ForB1BT1andB2BT2are the symmetrical matrix，then

It means that by the proposal，more information is exploited from the relative measurements for estimating the stateX.

3 Simulation and Discussion

The simulation parameters are listed in Table 3.Additionally，when the distance between the agent and the target is less than 400 m，the agent finds the target in a probability of 0.8.When the distance between the agent and the target is less than 200 m，the simulation is stopped.

Table 3 Simulation setting

Three aspects are verified by the simulation.（1）Whether the proposal is effective.（2）Whether the proposal has contributed to the self-localization.（3）Whether the proposal has contributed to the target-localization.

3.1 Validating for effectiveness

In order to verify the effectiveness of the proposal，the trajectories generated by different algorithms are contrasted in Fig.3.Assume that fouragents move to the same target.The initial positions of four agents are known and marked asA1，A2，A3 andA4.The black line is the true trajectory.The red line is the trajectory from the independent localization（IL）where no relative measurements are used.The green line is the trajectory from the proposed CSL-CTL.It can be seen that the estimated trajectories based on the CSL-CTL are closer to the real trajectories than that of IL.The effectiveness of the proposed algorithm is intuitively demonstrated in Fig.3.

Fig.3 Estimated trajectories of four agents under IL and CSL-CTL

In Fig.4，the boundaries determine the 3σconfidence region of the distance error.They are calculated according to the variance ofx，ydirections.The red boundaries are from IL.The green boundaries are from CSL-CTL.The blue line represents the Euclidean distance error in the case of the CSLCTL.In the proposal，for the target acting as the node，the more constraint information is added.As a result，the 3σregion determined by the proposed CSL-CTL is smaller than that of IL，which means that the localization uncertainty is reduced.Additionally，it can be found that the distance errors are always within the 3σregion.Thus，the proposed algorithm is effective.

Fig.4 3σ boundaries and error curves of four agents’position estimates

3.2 Validating for self-localization

With the parallel integration or not，the average accuracy of the self-localization is contrasted in Fig.5，where the curves represent the average distance error between the estimated positionand the true position， i.e.Average error=The following conclusions are obtained.

Fig.5 Comparison of average estimate accuracy of four agents under CSL and IL

（1）In the case of CSL-CTL，the self-localization accuracy is improved，as the green line in Fig.5.It can be explained as when multiple agents obtain the relative measurements to the same target，these measurements as the new constraints are added.Consequently，the error of the position estimation is reduced.

（2）Without the parallel integration，the localization accuracy of IL（red line）is inferior to CSL（blue line） as shown in Fig.5（a）.However，as Fig.5（b），by integrating with CTL，the localization accuracy from IL is higher than that from CSL.

3.3 Validating for target-localization

With the parallel integration or not，five sets of the target estimations are given in Table 4.Each item is obtained under different noise setting.From Table 4，it can be found that under the parallel integration mode，the target-localization accuracy is improved.

Table 4 Comparison of target estimate under IL and CSL

4 Conclusions

CSL and CTL are two tasks for multi-agent system.In this paper，two tasks are integrated in parallel to further improve their accuracy.The following problems are solved：（1）Use the position estimations generated from the KF-CSL for CTL.（2）During CTL，the agents regarded as the variables are re-estimated.Then ensure the re-estimated results support the consecutive KF-CSL.

The CSL-CTL algorithm is proposed.Essentially， the proposed algorithm utilizes the data which are respectively prepared for CSL and CTL at the same time.The proposed algorithm can work properly even if one type of data is absent.The algorithm is robustness.

The effectiveness is verified by the simulation.The results show that the accuracy of two tasks in the parallel integration mode is higher than the case where CSL and CTL are executed one by one.It should be noted that the study is based on the case of one static target.In the future，the research can be extended to the case of the dynamic target or multiple targets.