Hao Wu,Peng-fei Wu,Zhang-song Shi,Shi-yan Sun,Zhong-hong Wu

School of Ordnance Engineering, Naval University of Engineering, Wuhan, 430033, China

Keywords:Highly-dynamic topographic structure MDS Relative ranging Aerial ammunition ad hoc network

ABSTRACT In the internet of battlefield things,ammunition is becoming networked and intelligent,which depends on location information.Therefore,this paper focuses on the self-organized network collaborative localization of munitions with an aerial three-dimensional (3D) highly-dynamic topographic structure under a satellite denied environment.As for aerial networked munitions,the measurement of munitions is objectively incomplete due to the degenerated and interrupted link of munitions.For this reason,a cluster-oriented collaborative localization method is put forward in this paper.Multidimensional scaling(MDS) was first integrated with a trilateration localization method (TLM) to construct a relative localization algorithm for determining the relative location of a mobile cluster network.The information related to relative velocity was then combined into a collaborative localization framework to devise a TLM-vMDS algorithm.Finally,an iterative refinement algorithm based on scaling by majorizing a complicated function (SMACOF) was employed to effectively eliminate the influence of incomplete link observation on localization accuracy.Compared with the currently available advanced algorithms,the proposed TLM-vMDS algorithm achieves higher localization accuracy and faster convergence for a cluster of extensively networked munitions,and also offers better numerical stability and robustness for highspeed motion models.

1.Introduction

Networked munitions,also known as networked intelligence sub-munitions (abbreviated as sub-munitions),refer to munitions that are equipped with a flight power device to process the grid information of operations and collaboratively complete the detection,tracking,localization,attack and damage assessment of targets while flying in the air,and also have the capability of highaccuracy collaborative localization for networked sub-munitions[1].As a new concept in ammunition management,networked munitions have sufficiently combined the advantages of ammunition and unmanned aerial vehicle technologies to fulfill a number of operational purposes including target reconnaissance and detection,target indication,accurate attack,damage effectiveness assessment,communications,relay,and aerial alert.These networked munitions may be a group of cannonballs or loitering munitions,or a cluster of missiles or unmanned aerial vehicles[2].In particular,a communication data link is added to sub-munitions to achieve information sharing between sub-munitions.The shared information mainly includes information on targets such as the location of a detected target and its model and classification.Additionally,the state information of sub-munitions including their location can also be shared with all other sub-munitions in the communication coverage by virtue of a real-time data link.In this way,a super large attack network with sub-munitions as its nodes is constructed,and contains more than 1000 nodes [3].

The key to collaborative operations is the sharing of data between clusters by virtue of a data link.Collaborative localization with shares locations plays an important role in such data sharing.In recent years,the Defense Advanced Research Projects Agency(DARPA)of the US Department of Defense has conducted the most extensive study on collaborative localization.Back in 2013,DARPA had initiated collaborative operations in denied environment(CODE) [4].In the program,denied environment means a harsh environment in which navigational signals from satellites are unavailable because of strike and interference,spoofing attack and some less possible disastrous events.As for collaborative localization,a highly automatic and dynamic aerial ad hoc network is greatly different from a static or lowly dynamic wireless network on the ground.The ground systems often have a relatively fixed topographic structure,but still quickly obtain external information support even if some networks are not thoroughly connected.Additionally,their resources are also replenished in a relatively convenient way.In contrast,aerial ad hoc networks [5] have difficulty accessing such support under the same circumstances.For this reason,a universal localization algorithm is developed in the present study for aerial networking to eliminate the limitation of localization.

Munitions are often equipped with an omnidirectional antenna for mutual ranging and velocity measurement.The measurement results are then shared by virtue of the data link between munitions[6].Presently,the mainstream ranging methods include time difference of arrival (TDOA),received signal strength indicator(RSSI),and ultra wideband (UWB),and so on [7-9].The methods used to measure velocity include Doppler shift and frequency difference of arrival (FDOA),and others [10,11].Localization algorithms based on ranging can be classified into the following two types: single-target and collaborative localization.

Single-target localization algorithms are based on the trilateration localization method (TLM) and least squares method,which are simple and fully distributed,and require only local communications.Nevertheless,these algorithms involve problems such as difficult initiation [12],premature shutdown [13],and iterative error accumulation [14].Among them,the algorithms based on a rigidity diagram [15-17]can lower the requirements for singletarget localization,which resolves the problem of difficult initiation to some extent.Nevertheless,rigidity diagram theory is mainly applicable to two-dimensional(2D)networks,but not very suitable for three-dimensional (3D) networks [18].In practice,many networks not applicable to single-target localization algorithm can depend on the information sharing between nodes for collaborative localization.Collaborative localization is advantageous because the relevant information for measurement can be used in the global solution for resource localization (RL).Collaborative localization may be modeled by solving the pressure function in Eq.(2).It is commonly achieved in three ways as follows:a.Eq.(2)is regarded as a nonlinear least squares problem that can be solved by Levenberg Marquardt(LM);b.Eq.(2)is taken as a nonlinear problem that can be solved by a quasi-Newton algorithm;c.Eq.(2)is regarded as a non-convex optimization problem that can be solved by semidefinite programming (SDP) and multidimensional scaling (MDS).

Among them,the key of LM method [19] is to use the model functionfto make a linear approximation to the parameter vector p to be estimated in its neighborhood,while ignoring the derivative terms above the second order,and transforming it into a linear least squares problem,which has the advantage of fast convergence,but the disadvantage is that the solution process requires several iterations,which will produce a large error accumulation [20],and is not applicable under the condition of large-scale networking.The quasi-Newton method [21] is a classical algorithm for solving nonlinear systems of equations,and generally the BFGS algorithm is used to solve convex minimization problems by using Wolfe's linear search with global convergence,which has the advantage of simple computation and high search accuracy,but the disadvantage is that the global search ability is poor,the convergence speed is slow,and the processing ability for complex problems is poor[22],so it is not applicable under the research conditions of this paper.The method based on SDP[23]follows the basic idea that an anon-convex optimization problem is relaxed into a convex optimization problem,and then solved by simulated annealing or the interior point method,and so on [24-26].However,its computation is extremely complex.Ideally,the computational complexity reachesO（n3）-O（n4）.Meanwhile,MDS is a data analysis method.Shang et al.[27] promoted MDS into the localization of a wireless sensor network,and put forth the MDS-MAP algorithm.The MDSbased algorithms can be divided into two types as follows: classic MDS and iterative MDS [28].Among them,the classic MDS algorithms have the distance of the shortest path in place of the actual distance to construct a distance matrix.However,this may lead to a large localization error in the aerial ammunition ad hoc network highly-dynamic topographic structure.In order to address this problem,some iterative MDS algorithms have been proposed[29-32].Among them,dwMDS [29] and vMDS [30] rely on the algorithm,that is,scaling by majorizing a complicated function(SMACOF) [31],to solve anon-convex optimization problem,but they take into account only the slow-speed motion network in a2D environment.As a nonlinear optimization method,vBFGS [32]utilizes a Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm under the strong Wolfe conditions to solve problems Eq.(27).When the information from multiple sources is combined to solve a localization problem in the observation of a synchronicity and misses,BFGS is more extensible and applicable than SMACOF.While BFGS is more suitable for sparse networks,it is not very effective in extensive networking.The single SMACOF algorithm features high accuracy,but is more sensitive to initial inputs.Meanwhile,it is very easy for the classic MDS algorithms to obtain an initial solution,but the accuracy is not satisfactory because it uses the distance of the shortest path instead of the actual distance to construct a distance matrix.

Meanwhile,the classical MDS is easy to solve the initial solution.but the accuracy is not satisfactory because it uses the distance of the shortest path instead of the actual distance to construct a distance matrix.And the MDS is a fully centralized computing architecture,which requires harsh network conditions,and the failure of one node can lead to the failure of the whole algorithm,which will also have a great impact on the positioning accuracy.The trilateration localization methods is a distributed algorithm,which can exactly compensate for the localization problem in the case of partial node failure or interference.The idea is to implement relative localization for a part of fully connected nodes using MDS,and then use the submunitions that have been localized as anchor nodes,so as to carry out trilateral localization for other nodes,and then expand layer by layer until the localization of the whole network is completed.This method effectively solves the error problem caused by the shortest path instead of multi-hop distance,and also reduces the number of error expansion layers compared with using all trilateration localization methods for positioning,which greatly improves the positioning accuracy.Finally,the improved algorithm is combined with the SMACOF algorithm to achieve further complementary advantages.This paper mainly establishes the following conditions.

(1) In this paper,the study of collaborative localization is mainly conducted for an aerial ammunition ad hoc network,and does not depend on localization algorithms that require satellite navigation.The high-speed and high-rotation motion of projectiles result in some objective phenomena,e.g.the degeneration of the link between munitions causing data packet loss,and the destruction of projectiles resulting in an interrupted communication link and incomplete information for ranging between munitions (increasing and unobservable error).Moreover,the conventional MDS localization algorithms use the distance of the shortest path in place of the actual distance,which worsens the accuracy of localization.For this reason,a relative localization method is proposed for networked munitions,that is,TLM-MDS,specifically,an aerial three-dimensional multi-point localization algorithm integrating a trilateration localization method(TLM)and multidimensional scaling(MDS).A hybrid localization architecture based on an observable link is constructed to address the influence of incomplete link observation on localization accuracy.A 3D collaborative localization model for the proposed method has been rarely studied at present.

(2) The measured value of relative velocity is introduced into the TLM-MDS algorithm to obtain the TLM-vMDS algorithm.The SMACOF algorithm is employed to optimize its collaborative localization.The TLM-vMDS algorithm prevents the occurrence of differential and step size problems in the order reduction method,so as to considerably enhance the accuracy of localization.

(3) The localization framework developed in Reference [33] for 2D space is expanded into 3D space,and the Cramer-Rao lower bound (CRLB) is derived under the increasing observation error or the unobservable conditions.Moreover,the proposed algorithm is tested and compared with other algorithms such as MDS-MAP,dwMDS,vBGFS and EKF,and others.The test results reveal that the proposed algorithm offers higher localization accuracy and faster convergence while greatly fitting into the environment of extensively networked munitions.

The other sections of this paper are arranged as follows: the second section mainly introduces the system models developed in this paper;the third section mainly describes the TLM-vMDS algorithm used for collaborative localization;the fourth section presents the test and the analysis of its results;the fifth section provides the conclusions.

2.System models

2.1.Collaborative localization model based on ranging

where ωijis the weight coefficient,which will be discussed in detail in the subsequent section.

2.2.Classic MDS localization model

A classic MDS is essentially a method used for data analysis.It uses the similarity between objects to construct coordinate and rigidity diagrams in multidimensional space.Higher similarity between objects results in a shorter distance between nodes in the relevant coordinate and rigidity diagrams.This feature is used by MDS in the collaborative localization of networked munitions.However,distance takes place of similarity to achieve the relative localization of nodes between munitions[34].Note that MDS does not need a beacon node.Based on the mutual ranging of nodes between munitions,it can output the coordinate relationships among all sub-munitions by virtue of a matrix,which meets the requirements for localization in this paper.Eq.(3) states

where STREE stands for the stress coefficient in MDS;pijindicates the matrix of similarity between objects;dijis the distance between the nodesiandjin the coordinate diagram or rigidity diagram;Trepresents the coordinate matrix of nodes in the space;f:pij→dij（T）is the function for mappingpijtodij,and may be also written asf（pij）=dij（T）.In practical application,there is normally a measurement error,so thatf（pij）is not absolutely equal todij（T）.For this reason,a stress coefficientSTREEis added and made to be zero as often as possible.

In a classic MDS,the geometrical distance between nodes is measured to replace the similarity between munitions.Basically,the goal is to perform the mapping between similar or dissimilar matrices under the constraint of a stress coefficient,so as to obtain the actual values for convenient observation.

The localization of a classic MDS localization algorithm is derived in the following process:

First,the distance square matrix of ammunition groups is determined with the ranging information of sub-munitions

wheredij,i=1,2,3,…,n,j=1,2,3,…,n indicates the distance between the nodes i and j;and n stands for the number of nodes in the observation area.

It is assumed that the coordinates of the nodeiare（xi，yi，zj）.Then

Two matrices R and X are constructed as follows:

Among them,X is the coordinate matrix of the node.Then there isBoth sides of D（2）are multiplied by the central matrix J for double centralization.The matrix B is used to represent the doublecenter form of D（2）.Then we have

Meanwhile,it is very easy to obtain RJ=0，JRT=0.

where U and V represent the diagonal matrix formed by the eigenvalues of B and the matrix formed by its eigenvectors obtained after eigenvalue decomposition,respectively;and JXTstands for the relative coordinates of sub-munitions after centralization.

Above all,the classic MDS algorithm is calculated in the following steps:

Step 1: The ranging informationdijof each node in the ad hoc network is used to obtain the distance square matrix D（2）.Next,centralization is carried out to solve B;

Step 2: The eigenvalues of the matrix B λ1≥λ2≥…≥λNare calculated together with their corresponding standard orthogonal eigenvectors v1，v2，…，vN;

Step 3: Thekvalue is determined according to the number of dimensions in the space.The firstklargest positive eigenvalues are selected from the eigenvalues of B to form the diagonal matrix U.The diagonal matrix is multiplied by the corresponding eigenvector matrix to obtain the relative coordinates ofNnodes in thek-dimensional Euclidean space.

As revealed in the above analysis,classic MDS has some limitations.As a concentrated calculation method,it has rigid requirements for the relative relationship of all nodes in the ad hoc network.However,it is difficult to satisfy such requirements because of the degenerated link between munitions and other unpredictable objective phenomena.For this reason,other localization methods must be employed to overcome the influence of incomplete measurements on MDS.Normally,if two nodes in a wireless sensor network on the ground cannot realize mutual ranging,their shortest path is calculated with the Dijkstra or Floyd algorithm [35],and taken as an approximate substitute for the Euclidean distance between them.In this way,the distance matrix of all nodes in the network can be obtained.As presented in Fig.1,the ranging link is interrupted between the nodes A and D,and their actual distance is 6.However,if the shortest path is used,the measured distance is 9.The error is quite large.This method is more applicable to the ground networks having lower requirements for localization accuracy,but cannot meet the requirements for highaccuracy localization in an aerial highly-dynamic ad hoc network.

Fig.1.Schematic diagram of ranging with the shortest path.

3.A TLM-vMDS cluster collaborative localization algorithm

3.1.A TLM-MDS relative localization algorithm

An interrupted data link between munitions or data packet loss will often cause difficulty in easily measuring the relative measured distance of all nodes,which results from the objective complexity of a battlefield environment and the high-speed and high-rotation motion of projectiles.The commonest solution is to use the shortest path instead of the distance between multi-hops,but the uncertainty and large deviation involved in this method may severely undermine the accuracy of localization.Additionally,a fullycentralized computational architecture has very strict requirements for network conditions.If any node fails,it may result in the failure of the entire algorithm.For this reason,a trilateration localization method(TLM)and MDS are integrated into a TLM-MDS algorithm for multi-node collaborative localization.

As a distributed algorithm,a TLM is basically intended to solve binary quadratic equations.A 2D plane is taken as an example to illustrate the basic principles of its implementation.If three circles with known radii intersect at one point,there is only one intersection.As shown in Fig.2,it is assumed that the nodes A,B,and C are the anchor nodes with a known location,and O is an unknown node to be measured.The location of O is measured using three nodes A,B,and C,respectively,to obtaindA,dB,dC.Subsequently,each of these nodes,A,B,and C,is taken as a center to make a circle with the radii ofdA,dB,dC,respectively.It is evident that the node O must be at the intersection of three circles.The coordinates of O are set to(x,y),and the known coordinates of the nodes A,B,and C areA（x1，y1），B（x2，y2），C（x3，y3）,respectively.The following equations are obtained:

Fig.2.Schematic diagram of a trilateration localization algorithm.

The above equations are then solved to obtain the coordinates for the location of the node O as follows:

Based on the above analysis,it is found that,if MDS is employed alone in localization for the entire dynamic network,it may experience an interruption of the algorithm or the loss of nodes because of a degenerated link between munitions or the incomplete information of ranging between munitions caused by the enemy’s electromagnetic interference.Moreover,if localization is conducted with TLM alone for the entire network,some nodes with known location must be arranged at the edge of the network,and then inwards layer by layer to achieve the localization inside the network.This may result in the progressive transfer of location and measurement errors layer by layer,and eventually lead to the dramatic drop of localization accuracy.When MDS is combined with TLM for localization,it is first implemented in the relative localization for some fully-linked nodes.Subsequently,these localized sub-munitions are taken as the anchor nodes to perform the trilateration localization for the other nodes,and can be expanded layer by layer until localization is completed for the entire network.The specific steps are given as follows:

Step 1: The measured Euclidean distance between submunitions is used to obtain the distance matrix as follows:

If the distance between two nodesiandjcannot be measured,or the data is jammed or lost because of environmental factors,we can directly letdij=0 and do not use the shortest path in place of the measured distance.

Step 2: The area containing the largest number of nodes is determined.All the nodes in the area are then interlinked (which allows the mutual ranging of all nodes).The number of fully-linked nodes is set tom.From the matrix D,we selectmrows andmcolumns with the same numberings.The intersections of these selected rows and columns are not zero except their diagonals in the matrix D.We select the rows and columns that have the largest value ofm,and extract their intersections in the same sequence as arranged in the matrix D,so as to construct a new distance matrix as follows:

The MDS algorithm is implemented for the matrix D to obtain the relative location coordinates X for thesemnodes.

The main purpose of this step is to reveal the largest fully-linked structure in a cluster,and determine its relative location.However,the algorithm for searching for the largest fully-linked structure is practically complicated and time-consuming.Therefore,a threshold ε can be set based on the number of sub-munitions contained in the cluster.The search is halted as soon as ε fullylinked nodes are found.Then,the TLM algorithm can be implemented.

Step 3:The remaining nodes are searched to further identify any node that has two or more links to the localized area,and trilateration localization is then implemented for such a node.After localization,the node is allocated to the localized area.The remaining nodes are further searched until all the nodes are localized.

When localization expands to the nodes in the outermost layer,a node in the outermost layer may be linked to only one node in the localized area.If any technical support is unavailable,the shortest path may substitute for the distance between two nodes.In this case,a deviation from the localization results may happen to some nodes,but has little effect on the whole.

If a node in the outermost layer links to two or more nodes in the localized area,and the coordinates of the linked nodes are（x1，y1，z1），（x2，y2，z2）…（xn，yn，zn）,the coordinates of the node to be localized are（xc，yc，zc）.The following equations are constructed:

As revealed in the above equations,the least squares method can be used to estimate the coordinates of the node to be localized.The estimation has better accuracy and a smaller error than the result of the trilateration localization that uses only two nodes.

Here is an example in Fig.3,Each node in the area formed by ABCDE is connected to each other,and can measure distance.There are two or more connected nodes between nodes F and H and the nodes in ABCDE area.Therefore,the first five fully connected nodes are positioned using MDS,and then two nodes F and H are positioned using trilateration localization method.Node G is connected to two nodes F and D.Node G can be positioned after two nodes D and F are positioned,and node I obtains the ranging information by using the shortest path,thus reducing the positioning error between nodes of the overall network system(The TLM-MDS relative location algorithm is shown in Fig.4).

Fig.3.MDS-TLM positioning schematic.

Fig.4.Process flow of implementing the multidimensional scaling multi-point localization algorithm with trilateration localization.

Fig.5.Calculation of the relative velocity between nodes.

Fig.6.Process flow of the SMACOF algorithm.

3.2.Registration in absolute coordinates

After the network is relatively localized,if relative coordinates need to be transformed into absolute coordinates,the absolute coordinates of the anchor nodes that can acquire the absolute location in the cluster are set to X1,and their relative coordinates are set to X2.The rotation matrix is first calculated with X1as the reference coordinate system.Thus,there is

MatrixAis obtained by subtracting the first row in X2from the rows starting from the second row and ending until thelth row

It is obtained by subtracting the first row in X1from the rows starting from the second row and ending until thelth row:

The rotation matrix is set to Q,so that AQ=b.The least squares method is employed to solve the rotation matrix Q and obtain

The shift matrix is then calculated.The shift is set as S,and thus it is

If the localization results of all the nodes in the network under the relative coordinate system are denoted by,and the localization results under the absolute coordinate system are Xreal,there is

3.3.Collaborative localization model with ranging information

3.4.Iterative refinement algorithm based on SMACOF

Considering the objective existence of measurement errors and abnormal data,the results of the localization with MDS are not absolutely accurate,and can be further optimized.However,the environment for localization in this study contains only two observed quantities,specifically,distance and velocity information.The localization can be iteratively optimized by using the differences between their observed and calculated results.In this section,SMACOF is introduced to optimize the results of the localization with MDS.

The SMACOF algorithm avoids the differential and step length problems in the order reduction method,so that it is applied in the optimization of localization results in this paper.

After completing the localization of nodes in the network,the coordinates of these nodes are used to calculate the distance and velocity between nodes.The optimization of coordinates can be transformed into Eq.(27).

The stress function in Eq.(27) is rewritten into the following compact form:

where

Eq.(27)is a non-convex planning problem.If it is directly solved to obtain the global optimal solution,the algorithm will be very complicated,and consume a tremendous amount of computational resources.Moreover,its computation will take too much time,and cannot meet the requirements for real-time implementation in the study of topics.For this purpose,some scholars have proposed an iterative optimization algorithm,such as a majorizationminimization (MM) [36],to solve the non-convex optimization problem.A non-convex optimization problemf（x）is difficult to solve,sothat anMMalgorithmfindsaneasilysolvedfunctiong(x,z),wherezis a known constant.The functiong(x,z)is regarded as the majorizing function off（x）if it meets both of the following conditions:

(a) The functiong(x,z)is an optimization problem that is easier to solve thanf（x）;

(b) To ?x,f（x）≤g（x，z）holds.The equality holds if and only ifx=z,that is,f（x）=g（x，x）.

We solve ming(x,z) to obtainx=x1,so that there is:

We further solve ming(x,x1) to obtainx=x2,so that there is

Iteration is performed constantly to obtain the column of points｛xk｝,which makesf（xk）monotonically non-increasing.

The SMACOF algorithm can obtain the majorizing function of Eq.(27),which is constantly iterated with the MM algorithm to cause Eq.(27) generate a monotonically non-increasing sequence.

The Cauchy-Schwarz inequality can be applied into the nonconvex term in Eq.(27) to derive the main factors ofParticularly,to any z∈Rp,there is

The items different from Eq.(28) can be rewritten into

whenk≥1.This attribute ensures that the centroid of coordinates remains in a bounded domain.

As shown in the above diagram,in the input is the initial relative coordinates obtained by the MDS algorithm with TLM.Therefore,the localization framework in this paper can be described as Algorithm 1,where ε andKmaxstand for the criteria for ending the iteration of the algorithm,and Xkis the estimated coordinates at thekth iteration.

Algorithm 1.TLM-vMDSLocalization Framework

3.5. Convergence of the proposed algorithm

For convenience of notation,the superscript(t)in symbols such

Convergence is achieved by virtue of square error

Lemmas.The following conclusions hold:

(1)To allXk andZ,ΩZ（Xk）determines the upper limit for the value of f（Xk）;

(2)The function fk decreases monotonically,i.e.fk+1≤fk holds when k∈N;

(3)When k→∞,ξk→0.

Based on the monotone convergence theorem,Lemma (1) ensures that the stress functionfkconverges monotonically to σ∞≥0,while Lemma(2)reveals that the convergence of the error function ξk(which is not certainly monotone) is zero.The lemmas are proved as follows:

Proof (1).The total stress function at Xkmay be written as

It is found by the Cauchy-Schwarz inequality

If all satisfy the condition Z∈R2N×p,there is

It is found with Eq.(39) and Eq.(41)

The equality is obtained when Xk=Z,i.e.f（Z）=ΩZ（Z）.Thus,the function ΩZ（Z）is the upper limit for the value of the functionf（Xk）.

Proof(2).First,ΩZ（Xk）is minimized at Xkwith Eq.(35)to obtain Xk=ΩZ（Xk）.Then,we get the inequalityf（Γ（Z））≤ΩZ（Γ（Z））.Lastly,the following inequality is obtained:ΩZ（Γ（Z））=C-η2（Γ（Z））

Subsequently,the Cauchy-Schwarz inequality can be applied to Eq.(37) to obtain

3.6.Complexity of the proposed algorithm

It is assumed that an aerial ammunition ad hoc network containsNnodes,includingmfully-linked nodes,and s anchor nodes.First,the MDS algorithm is implemented for m fully-linked nodes,and its complexity isO（m3）.The TLM algorithm is adopted forN-m other nodes.Its time complexity isO（1）,and computational complexity isO（N-m）.Therefore,TLM-MDS has a total complexity ofO（m3）.

After the relative velocity is added to measurement,the most important step of Algorithm 1 is to solve Eq.(35),which mainly involves the multiplication of matrices and vectors.Its computational complexity isO（m3p3）,where p is the number of dimensions in the space.It is assumed that the maximum number of iterations through SMACOF iskmax,so that the total complexity of TLM-vMDS isO（kmax·m3p3）.

4.Test and results analysis

4.1.Simulation settings

The simulation was conducted in an area covered by a 100 × 100 × 50 r3network.In this paper,we set r=100 m.The motion velocity of each node was a random value in 0-100 m/s,and its direction varied randomly in 0-360°.It was assumed that the maximum communication and measurement ranges for nodes were both R=50 r [37].From the nodes of the network,six nodes that were able to obtain absolute coordinates were selected as anchor nodes.Theoretically,only four anchor nodes were needed under 3D conditions.However,six nodes were selected here to prevent the anchor nodes from appearing in the same plane,which might lead to a very large error in the coordinate transformation.It was assumed that its ranging error was subject to Gaussian distribution,that is,uij～N（0，102）,and its velocity measurement error matched with the distributionev～N（0，42）.A simple model for the aerial motion of munitions was constructed.The maximum velocity component of a node was denoted by vmax.The coordinates of nodes at the time t could be defined as

where rand is an evenly distributed random number and rand～U（-1，1）.

4.2.Derivation of CRLB under 3D conditions

All the results were the averages taken from 300 stochastic simulation experiments.Meanwhile,the normalized root mean square error (NRMES) was used to assess the localization results,and defined by

The combined probability density function for all nodes may be expressed as

The combined probability density function for the nodes ofunder the velocity measurement error can be derived in the same way.The process of derivation is omitted here.

Moreover,CRLB depends on the Fisher information matrix(FIM)F.The definition of F is

To alli,j=1,…,N,CRLB is defined as the reciprocal of FIM,and expressed as

In 3D space,F may be rewritten into a matrix as follows:

where ［FYX］=［FXY］T，［FZX］=［FXZ］T，［FZY］=［FYZ］T.Thus we can obtain

Based on the above definition of NRMES,it is derived that

Hence,CRLB with NRMES may be written as

4.3.Influence of λ and relative velocity measurement noise on NRMSE

In order to verify the influence of λ on NRMSE,λ was set from 0 to 10,and the interval of the value was o.5 in various experiments.We made vmax=100,andN=200.The velocity measurement noise distribution was set toev～N（0，3）andev～N（0，20）in various experiments to verify the robustness of the TLM-vMDS algorithm against the relative velocity measurement noise.Among them,the value of the velocity measurement noise was referenced in Ref.[39].The simulation results are given in Fig.7:

Fig.7.Influence of λ on localization accuracy under different velocity measurement noise levels: (a) ev～N（0，3）;(b) ev～N（0，20）.

As revealed in Fig.6(a),when λ=0,the relative velocity information is not used to assist localization,so that the NRMSE of the TLM-vMDS algorithm is 0.029.When λ=1,the NRMSE of the TLMvMDS algorithm is 0.015,approximately 48.28% lower than the result at λ=0.In other words,relative velocity can significantly improve the localization accuracy when it is used to assist the localization.In Fig.6(b),much higher noise levels are taken into account in the velocity measurement,but the relative velocity information can still provide effective assistance by using a smaller value of λ,e.g.λ=0.6.

4.4.Performance of the proposed algorithm under two networks subject to random

In order to verify the applicability of the proposed algorithm to dynamic networks,two different configured networks were prepared.One network was subject to a uniform random distribution.In the network,N dynamic nodes (N=200) were randomly deployed.The other network was a U-shaped network withNnodes (N=150).Moreover,λ=0.6 in this network.We made vmax=100,uij～N（0，102）,andev～N（0，16）.The localization results are given in Fig.8 and Fig.9.

Fig.8.Localization results of TLM-MDS and MDS-MAP for two different networks: (a) The network subject to a random distribution;(b) The network subject to a U-shaped distribution.

Fig.9.Localization results of TLM-MDS and TLM-vMDS for two different networks: (a) The network subject to a random distribution;(b) The network subject to a U-shaped distribution.

While Fig.8(a) presents the results of the localization with a traditional MDS-MAP using the shortest path in place of the multihop distance,Fig.8(b)gives the results of the localization with the proposed TLM-MDS algorithm,but velocityvis not introduced to assist the localization,and iterative refinement is also not conducted.As clearly revealed in Fig.8,the proposed algorithm achieves much better localization accuracy than the traditional MDS-MAP in the area subject to random distribution,and it does not cause any dramatic deviation.It is shown in Fig.8(b) that the proposed algorithm can still provide a location consistent with the actual location in the U-shaped network.Moreover,its deviation is small.However,the traditional MDS algorithm shows a larger deviation in the localization results for the U-shaped network compared to the network subject to random distribution.·······

The localization results of TLM-MDS and TLM-vMDS are presented with the nodes subject to different distributions as presented in Fig.9(a) and (b),respectively.In this case,TLM-vMDS includes velocityvfor assistance,and is improved by iterative refinement.As revealed in the localization results,the location given by the TLM-vMDS algorithm is closer to the actual location of nodes after it includes the velocity v and the MDS localization results are optimized by SMACOF.The optimization achieves a better effect for the nodes with ranging abnormalities.

In order to further understand the overall localization in the networks,the error distribution of TLM-MDS and TLM-vMDS in the localization of all nodes in both networks was generalized when the ranging error distribution was 0.25R%,and uniform.The results are given in Fig.10 and Fig.11,respectively.

Fig.10.Error distribution of localization results with (a) TLM-MDS and (b) TLM-vMDS for the network subject to a U-shaped distribution.

Fig.11.Error distribution of localization results with (a) TLM-MDS and (b) TLM-vMDS for the network subject to a random distribution.

In Fig.10,it is found that localization error is less than 0.7%R for more than 70% of the nodes,which is closer to the average localization error of the network,and only around 10%of the nodes have the localization error greater than 0.9%R.The error distribution in Fig.11 is similar to that in Fig.9.Most nodes in Fig.11 have a localization error close to the average localization error of the network,and there is not any major error.In other words,the TLMvMDS algorithm is greatly stable and rarely causes any abnormality in localization.

4.5.Performance of NRMSE

In order to further verify the superiority of the proposed algorithm,the TLM-vMDS algorithm was compared with vBFGS,EKF[40],dwMDS,and the traditional MDS-MAP in terms of the performance of NRMSE with different communication radii.We setRto 2500,3000,3500,4000,4500,and 5000,in different experiments and made λ=1.For the network subject to a random distribution,we took N randomly distributed nodes (N=200,120),vmax=100,maximum iterationskmax=50,uij～N（0，102）,andev～N（0，16）.The data were recorded starting from t=3.

The value of CRLB is 0.0031 whenN=200,but 0.0042whenN=120(Fig.12).More nodes lead to more data from observations,while few nodes result in a lower CRLB.Compared with Fig.12(a)and (b) shows the increase of NRMSE for these five algorithms.In Fig.12(b),NRMSE decreases with an increase in the communication radius for all these algorithms.The proposed algorithm has basically kept the lowest NRMSE,which is close to that of vBFGS,but vBFGS is slightly better than the proposed algorithm in someRranges.However,the proposed algorithm performs better than vBFGS in Fig.12(a),which means that vBFGS is more suitable for localization in a sparse network.This paper focuses on the localization in a network of extensively networked aerial munitions for the battlefield environment in the future,which normally involves hundreds or thousands of nodes.For this reason,the proposed algorithm is more applicable to extensive networks.

Fig.12.Fluctuation of NRMSE with communication radius for different numbers of nodes: (a) N=200;(b) N=120.

4.6.Convergence comparison of algorithms

In this section,we will compare the convergence of the algorithms including TLM-vMDS,vBFGS and dwMDS.We setR=50 rm,vmax=100,N=200,λ=1t∈［1，100］,kmax=20,uij～N（0，102）,andev～N（0，16）.The main results are shown in Fig.13.

Fig.13.Convergence comparison of three algorithms.

The TLM-vMDS algorithm took TLM-MDS as its input,while vBFGS and dwMDS used the traditional MDS-MAP as its initial value.As revealed in Fig.13,all three algorithms can converge quickly and achieve global convergence.The TLM-vMDS algorithm converges faster than vBFGS and dwMDS by 10-15 intervals.

4.7.Run time comparison of algorithms

The algorithms including TLM-vMDS,vBFGS,dwMDS,EKF,and MDS-MAP were compared with different numbers of nodes in terms of run time in MATLAB.We tookR=50 rm,and vmax=100.The network subject to a random distribution was used withN=200,λ=1,kmax=20,uij～N（0，102）,andev～N（0，16）.The main results are given in Fig.14.

Fig.14.Run time comparison of five algorithms.

Nevertheless,it must be noted that each algorithm is run independently in MATLAB,and its run time can be only regarded as an approximate value for the calculation of complexity.WhenNincreases constantly,the run time of TLM-vMDS increases gradually,but the proposed algorithm has major advantages when compared with other algorithms.Among them,the run time of MDS-MAP increases significantly with the increasing number of nodes,since MDS-MAP is a concentrated algorithm.Moreover,the run time of TLM-vMDS is longer than that of vBFGS after the number of nodes exceeds 90,but the difference is not large.

4.8.Processing capability of abnormal data

A dynamic network is vulnerable to environmental disturbances because of its mobile nodes,which results in abnormalities in ranging.The TLM-vMDS algorithm uses TLM to separately localize the nodes with abnormal data,lowering the adverse effects of abnormal data.Therefore,it has some capability of coping with abnormalities.As revealed in the previous experiments,the topographic structure of a network does not significantly affect the TLM-vMDS algorithm in localization.An experiment for abnormal data was then conducted with a network having 200 nodes subject to a random distribution.

In the experiment,ranging error was taken as an example.It was assumed that the error distribution of ranging between nodes wasuij～N（0，（0.0025R）2）,but changed touij～N（0，（0.1R）2）if there were some abnormal data of ranging.

The percentage of abnormal data refers to the proportion of abnormal measurements in the measurements for the nodes in the entire network.The localization results with unprocessed abnormalities are for the MDS-MAP algorithm,but those with processed abnormalities are for the TLM-vMDS algorithm.

The results given in Fig.15 show that localization error increases abruptly if some abnormalities exist in ranging and are not processed for fault tolerance.The proposed algorithm relies on TLM for the separate localization of the nodes that are not fully-linked,and removes the abnormal ranging information and the localization results with a large error caused thereby.Therefore,the proposed algorithm can maintain a high level of accuracy of localization with a small percentage of ranging abnormalities.

Fig.15.Localization results with abnormal data.

5.Conclusions

This paper focuses on the relative ranging and velocity measurement between munitions in a satellite denied environment to achieve the collaborative localization of the 3D aerial networked munitions in the ad hoc network,and proposes the TLM-vMDS algorithm.Compared with some popular algorithms including vBFGS,EKF,and dwMDS and with the traditional MDS-MAP algorithm,the proposed algorithm shows higher localization accuracy and faster convergence for extensively networked munitions.Moreover,it offers better numerical stability and robustness for the networks of a highly-dynamic topographic structure.In the future,we will pay attention to the actual limitations of resources on the nodes in an aerial network,and use a clustering algorithm to balance the computational pressure on nodes and reasonably allocate the resources in the network.In this way,the nodes with more resources can undertake more computational tasks to enhance the stability of the network.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.