Three-Dimensional Cooperative Localization via Space-Air-Ground Integrated Networks

Wenxuan Li,Yuanpeng Liu,Xiaoxiang Li,Yuan Shen,*

1 Department of Electronic Engineering,Tsinghua University,Beijing 100084,China

2 Beijing National Research Center for Information Science and Technology,Tsinghua University,Beijing 100084,China

Abstract: The space-air-ground integrated network(SAGIN) combines the superiority of the satellite,aerial, and ground communications, which is envisioned to provide high-precision positioning ability as well as seamless connectivity in the 5G and Beyond 5G (B5G) systems.In this paper, we propose a three-dimensional SAGIN localization scheme for ground agents utilizing multi-source information from satellites, base stations and unmanned aerial vehicles(UAVs).Based on the designed scheme, we derive the positioning performance bound and establish a distributed maximum likelihood algorithm to jointly estimate the positions and clock offsets of ground agents.Simulation results demonstrate the validity of the SAGIN localization scheme and reveal the effects of the number of satellites, the number of base stations, the number of UAVs and clock noise on positioning performance.

Keywords:space-air-ground integrated network(SAGIN); three-dimensional (3D) localization; clock noise;multi-source information

I.INTRODUCTION

Position information is important for emerging applications in the Internet of Things (IoT) such as inventory management, intelligent transportation systems(ITS), autonomous driving [1-4], etc.As an essential part of the space network, the Global Navigation Satellite System(GNSS)is widely used to obtain positions in outdoor environments.GNSS offers positioning, navigation, and timing services on a global or regional basis.Especially, advanced GNSS technologies including real-time kinematic(RTK)and precise point positioning(PPP)can realize high-precision positioning with centimeter-level.However, existing GNSS technologies cannot fully satisfy the accuracy and real-time requirements of emerging applications like autonomous vehicles,and they are not reliable in harsh environments such as indoor and city canyons[5].

The ground network is often applied to enhance the GNSS positioning.The ground network includes the cellular network, the mobile ad hoc network and the wireless local area network, etc.Among them, the cellular network has been standardized to provide localization services for mobile terminals from the first generation(1G)to the fifth generation(5G).For example, the Long Term Evolution system specifies three positioning methods including enhanced cell-ID, observed time difference of arrival and assisted GNSS[6].However,it is infeasible to deploy the ground network for coverage in some areas due to geospatial restrictions,and the cellular signals are easily blocked in urban areas.

In these scenarios,the air network can provide larger coverage and better positioning abilities [7].The air network mainly consists of airships,balloons and unmanned aerial vehicles (UAVs) [8].Compared with satellites, the air network with lower altitudes can achieve a higher signal-to-noise ratio (SNR).Compared with base stations in the ground network, the high mobility of the air network enables more lineof-sight (LOS) links [9].Emerging device-to-device technologies in the 5G system allow agents to exchange signals directly with each other,which enables the air network to provide localization information for ground agents[10-12].

Space-Air-Ground Integrated Network(SAGIN)integrates the above three networks, which aims to achieve ubiquitous network accessibility as well as ultra-reliable and low-latency communications in the 5G and B5G systems[13].By incorporating the merits of the satellite,aerial and ground communications,the SAGIN is envisioned to achieve significant performance gain in coverage, reliability, flexibility and throughput,enabling numerous applications including ITS, search-and-rescue operations and military sensing [14, 15].In particular, the satellites and base stations provide high capacities while the air network augments coverage and SNR,and it is highly expected to enhance localization by exploiting the advantages of each network in the SAGIN.

Motivated by the enormous potential,increasing attention from academia to the industry has been paid to the research of the SAGIN.In [16], the hybrid satellite-aerial-terrestrial network is proposed as an emergency network when the terrestrial network is not available.In[17],the space-air-ground integrated vehicular network is investigated employing softwaredefined networking to exploit heterogeneous resources in an agile and flexible manner.In [18], network functions virtualization and service function chaining technologies are implemented in the SAGIN to enable bidirectional mission offloading.Deep learning techniques are employed in [19] to improve the performance of the SAGIN.Additionally, many companies have made extensive efforts on exploiting the SAGIN in industries in recent years.For example, Google’s Loon and AT&T’s Flying COW deploy balloons and UAVs to provide air-assisted wireless Internet access,and SpaceX is developing Starlink to build a satellitebased access network.

Alongside the enormous connectivity potential,location-aware services are also essential in the SAGIN [20-22].A range-free localization mechanism for sensors in wireless sensor networks using flying anchors is presented in [23].In [24], the localization scheme based on received signal strength is developed where UAVs are used as anchors.The impacts of the UAV altitude, hovering time, the number of waypoints and path length are considered in[25].In[26],a novel UAV-assisted IoT network is proposed,where the UAV platform is employed as both a mobile data collector and an aerial anchor node to assist terrestrial base stations in data collection and device positioning.

However, most of the current studies focus on either a single network or the integration of the air and ground network in the SAGIN.Therefore,to utilize the integrated information from all the components in the SAGIN,we investigate the 3D localization for ground agents in terms of performance bound and algorithm design.We propose a localization scheme exploiting multi-source information in the SAGIN to obtain absolute and relative positions of ground agents.The Cram´er-Rao Lower Bound(CRLB)is derived for the proposed scheme, and a distributed maximum likelihood algorithm is developed to jointly estimate the positions and clock offsets of ground agents.

The rest of this paper is organized as follows.Section II explains the key concepts and presents the system model of the SAGIN localization scheme.In Section III,we derive the expression of Fisher information matrix(FIM)for the ground agents’positions and analyze the performance bound of the proposed scheme.In Section IV,we propose a distributed maximum likelihood algorithm to estimate the positions and clock offsets of ground agents.Simulation results are given in Section V to evaluate the performance of the proposed scheme.Finally,conclusions are drawn in Section VI.

Notation: [A]kjdenotes the element in thek-th row andj-th column of matrixA;[A]k:j,m:ndenotes a submatrix composed of the elements from rowkto rowjand columnmto columnnof matrixA; ⊗denotes Kronecker product and tr{·}denotes the trace function.

II.PROBLEM DESCRIPTION

In this section, we first present the key concepts in the proposed localization scheme,concerning the SAGIN, 3D localization, absolute and relative localization.Then,we describe the system model and the relative localization theory.

2.1 Key Concepts

As illustrated in Figure 1,the SAGIN is composed of three network layers: the space network, the air network,and the ground network.The space network includes satellites classified into three categories: geostationary orbit (GEO), medium earth orbit (MEO),and low earth orbit (LEO) satellites based on the altitude[27].The GEO satellite networks,such as GPS,GLONASS, and BEIDOU navigation systems, can provide broad coverage and basic navigation services.The MEO satellites significantly complement the GEO network since they can provide positioning information for the high latitudes and equatorial regions that GEO satellites can not well cover.The LEO satellite networks support higher communication rate, richer Doppler information and more flexible network for positioning;the air network consists of aerial communication devices including UAVs,balloons,etc.;and the ground network is mainly composed of the terrestrial communication networks represented by the cellular network evolved from the 1G to the 5G wireless network.

Figure 1. The architecture of the Space-Air-Ground Integrated Network.

A single network in the SAGIN cannot provide all users with fair and high-quality services.The space network can provide global coverage on the earth but has long propagation latency.The ground network has the lowest transmission delay,while it is expensive to deploy in rural areas and it is vulnerable to natural disasters or artificial infrastructure damages.The air network has the advantages of low latency and wide coverage, but suffers from limited capacity and unstable links.In the SAGIN,the high cost of the 5G network can be considerably reduced since the largely rural areas can be adequately covered by different satellite systems.In parallel, reaping the flexibility and controlled mobility of UAVs, many spatiotemporally dynamic network functions and applications can be enabled.

Compared with the two-dimensional(2D)localization,the height of the agent must be determined to realize the 3D positioning.The 3D localization is crucial for applications such as determining the floor for a mobile terminal or distinguishing between superposed tracks for road and rail use cases.Since the SAGIN is a multi-level heterogeneous network consisting of different network segments with different heights,it naturally supports the 3D localization for ground agents.

In terms of the positioning accuracy metrics, absolute position errors and relative position errors are often used[28-30].Absolute position errors evaluate the distance between the estimated position and the real position, which is widely used in most applications.As shown in Figure 2,relative position errors refer to the “shape” error of a network geometry [31], which is often more relevant in tasks such as overtaking and formation control [32-34].In this paper, both absolute and relative localization will be considered in the SAGIN.

Figure 2. The illustration of the absolute and relative positioning errors where p is the real position, ˆp is the estimated absolute position and ˆpr is the estimated relative position.

2.2 System Model

In the SAGIN localization scheme, we consider a 3D network containingNaground agents,NuUAVs,Nbbase stations andNssatellites, and the total number of nodes isNT=Na+Nu+Nb+Ns.To simplify the statement, the base stations and satellites are referred to as anchors with known positions.The ground agents and UAVs are referred to as cooperative nodes with unknown positions.Cooperative nodes can obtain pseudo-range measurements from the satellites and base stations.Furthermore,the cooperative nodes can exchange ranging signals and position information with each other.All the anchors are assumed to be synchronized and small clock offsets exist between the cooperative nodes and anchors.Based on all the information collected from different sources in the SAGIN,the purpose is to estimate the ground agents’absolute positions,relative positions and clock offsets.

The index sets of ground agents, UAVs, base stations and satellites denoted byNa,Nu,NbandNs.The total set of nodes is defined asNTand the set of cooperative nodes isNc=Na∪Nu.The position of the nodekin the SAGIN is denoted bypk= [xk,yk,zk]T∈R3.The position vector of the cooperative nodes isp=∈R3(Na+Nu).The clock offset of nodek ∈ NTisνk ∈R andνk=0 fork ∈Nb∪Ns.The distance between nodek ∈Ncandj ∈NTisdkj=‖pk-pj‖.The received signal at nodek ∈Ncfrom nodej ∈NTcan be modeled as

in which nkj(t)is the zero-mean Gaussian noise with the power spectral densityN0/2,sj(t) is a known waveform with Fourier transformSj(f),αkjandτkjare the amplitude and time of arrival (TOA), andTobis the observation time.The relationship of TOA,distance,and clock offsets can be written as[35,36]

wherecis the signal transportation velocity andbk=cνkrepresents the distance offset corresponding to the clock offset.The channels between nodes are assumed to be reciprocal, which meansdkj=djkandαkj=αjk.

Besides the signal noise,the clock of the cooperative nodes will also be affected by random noise.Considering the influence of clock noise, the measured distance at nodek ∈Ncfrom neighboring nodej ∈NTis given by

where wkjis the equivalent zero-mean Gaussian error introduced by signal noise nkj(t)and vkjis the clock noise modeled as a zero-mean Gaussian noise.The system model can be extended with angle measurement,where the received signals and channel parameters can be extended accordingly.

2.3 Relative Localization

Relative geometry represents the relative position relationships of nodes which is usually referred to as the“shape” of the network [37, 38].Hence, the performance of relative localization is not affected by translation, rotation and in some cases scaling of the network.Since only the TOA is available in the proposed SAGIN localization scheme, the scaling of the network is fixed while the rotation and translation need to be considered[31].Given an estimation ˆpofpobtained from some estimator, the transformation of ˆp,which includes rotation and translation,can be written as

where 1(Na+Nu)denotes a column vector with(Na+Nu)ones and

The transformation is denoted byω= [Γφ,t],wheret= [x,y,z]Tdenotes the translation parameter inX,YandZaxes and Γφ ∈R3×3denotes the orthogonal rotation (including reflection) matrix.For relative geometry,the goal is to choose anωminimizing the Euclidean distance between the transformed estimationTω()and the true positionp.

Therefore, the relative position estimation can be obtained based on the estimation of the absolute position and the optimization of the transformation.Given the optimal transformationω∗and the absolute position estimationthe relative position estimation ofpis denoted by

and then the relative positioning error of cooperative nodes is defined as the distance between the true position and the relative position estimation.

III.PERFORMANCE BOUND

In this section,we provide the theoretical analysis for the proposed SAGIN localization scheme in terms of CRLB,which determines the lower bound on the variance of an unbiased estimator.In terms of positioning,the CRLB is widely used to evaluate the theoretical optimal performance of the localization system.

According to the system model in Section II, the channel parameter of the signals in the proposed SAGIN localization scheme can be defined asη=whereτandαrepresent the TOA parameter and the amplitude parameter respectively, given by

whereτk= [τk1,...,τkj,...τkNT]Tandαk=[αk1,...,αkj,...αkNT]Tforj ∈NT{k}.

The parameter vector of positions and clock offsets of cooperative nodes in the SAGIN can be written asζ=whereν= [ν1,ν2,··· ,νNa+Nu]T.The channel parameter can be estimated using signals received at cooperative nodes.Based on the estimated channel parameter,the absolute positions,relative positions and clock offsets of ground agents in the SAGIN can be determined.

The log-likelihood ratios of the signals received at nodek ∈Ncfrom neighboring nodej ∈NTcan be written as[39].

whereukj(t) =αkjsj(t-τkj) and rkjis the Karhunen-Loeve expansion of rkj(t).

Since the signal noise is generated at different nodes in the system, the observations of different signals are independent.By the additivity of FIM and the property of the Schur-complement, we can derive the equivalent Fisher information matrix (EFIM) for the parameters related to TOA.The EFIM is the submatrix of FIM that only contains the parameter of interest,which is sufficient to derive the positioning CRLB by information inequality.The EFIM for TOA parameter vectorτis

where diag(X) generates a diagonal matrix using each row inXas elements and [B]kj=λkj, whereλkjcan be derived as

Theλkjcan be regarded as the ranging information which is related to the SNR and bandwidth.The EFIM forζcan be obtained using the transformation betweenζandτ

wheredenotes the Jacobian matrix.Based on the diagonal structure ofJe(τ)in(13),we can derive the EFIM for the positions and clock parameters

where

whereEkjis theNadimensional square matrix with all zeros except for an one at thekth row andjth column.The matrixesG2andG3have the same structure asG1by replacingKkjwithmkjandnkj, respectively,where

whereukj=(pk-pj)/‖pk-pj‖.

With the EFIMJe(ζ), the absolute localization CRLB for ground agentican be written as

The EFIM for agents’ positionsJe(p) can be obtained fromJe(ζ) by using the property of Schurcomplement.Note that the relative positioning CRLB describes the error of a network geometry instead of a single agent in (17).The relative position estimation is equivalent to addingkconstraintsh(ˆp)=0 on the absolute estimation,and the corresponding relative CRLB is given by

whereUCis defined in[37],the columns ofUCform the orthonormalized basis for the null space of the gradient matrixUNC=∂hT(/∂andCUC=0.

IV.DISTRIBUTED POSITIONING ALGORITHM

In this section,we design a distributed maximum likelihood algorithm,which considers different characteristics of all the components in the SAGIN to estimate the ground agent’s positions and clock offsets, thus realizes positioning and synchronization simultaneously.In centralized algorithms,all the measurements need to be collected into a central point, resulting in high computation complexity and low real-time performance.In the proposed distributed maximum likelihood localization algorithm,each node constructs its local cost function and broadcasts its estimation of position and clock offset.The estimation is obtained by minimizing the local cost functions of cooperative nodes.

Given all the ranging measurements information in(3),the aim is to estimate the ground agents’absolute positions, relative positions and clock offsets.Given the likelihood function of all the measurements,the local cost function for the cooperative nodekis defined as

The cost function contains all the ranging information from satellites, base stations and cooperative nodes.The optimization problem can be solved iteratively by the gradient descent, and the position and clock offset parameters can be updated at each cooperative node.One of the terms inHk(pk,νk)is denoted byand the gradient ofhkcan be written as

The gradient of other terms inHk(pk,νk) can be given similarly.The steps of the distributed maximum likelihood algorithm are described in algorithm 1.In each iteration,each cooperative node constructs its cost function and estimates the position and clock offset until convergence or the maximum number of iterations.

V.SIMULATION RESULTS

Since the SAGIN has not been deployed at present,theoretical analysis and numerical simulation are often used to validate the effectiveness of the technologies in the SAGIN [].In this section, we present simulation results of the proposed SAGIN localization scheme in terms of CRLB and root mean squared error(RMSE).CRLB is the most useful metric in wireless localization due to its properties of generalization and simple implementation.RMSE is used to evaluate the average performance of a localization algorithm,which can be written as

Algorithm 1. Distributed maximum likelihood localization algorithm in the SAGIN.Input: All the measurements, the positions of anchors and the initial estimated positions Output: The estimated positions and clock offsets of the ground agents 1: repeat 2: In the kth iteration:3: for node i=1 to Na+Nu do 4: Broadcasts its estimate ˆpki and ˆνki;5: Receives ˆpkj and ˆνkj of neighboring nodes;6: Constructs local cost function Hi(pi,νi)and estimates ˆpk+1i and ˆνk+1 i of the (k+1)th iteration.7: end for 8: until convergence or the maximum number of iterations

whereNis the number of Monte Carlo simulations andis the estimated position ofpkin thenth simulation.

In the simulation scenario, eight satellites are visible and their positions are based on the ephemeris,three base stations are placed using the Urban Macro scenario according to 3GPP 38.901,six UAVs are randomly placed in[-1000 m,1000 m]×[-1000 m,1000 m]×[100 m,500 m]and four ground agents are randomly placed in[-1000 m,1000 m]×[-1000 m,1000 m] × [0 m, 20 m].The noise spectral density is -174 dBm/Hz and the transmit power is 1 mW.The carrier frequency is 5.9 GHz for links between ground agents and UAVs,and 2 GHz for satellites and base stations.The signal bandwidth is 5 MHz for all links and the channel model is WINNER II [40].We analyze the effects of the number of base stations, the number of UAVs and clock noise on localization performance for ground agents.

5.1 SAGIN Localization Performance

The localization performance of the proposed SAGIN localization scheme is evaluated in Figure 3.It can be observed that when the transmit power increases,both the absolute and relative localization errors decrease since the SNRs in the receivers are improved.The performance gain tends to be saturated when the transmit power reaches 10 mW,and both the absolute and the relative localization can achieve sub-meter accuracy.The proposed localization algorithm is effective for both absolute and relative positioning as the RMSE is close to the CRLB, especially with high transmit power.Compared with a single network, the SAGIN can provide better accuracy as the number and the variety of anchors increases.

Figure 3. The 3D localization CRLB and RMSE of the proposed SAGIN localization scheme.

5.2 Effect of Satellites

We then investigate the effect of the number of satellites, which is shown in Figure 4.The performance is evaluated with three or without base stations and the number of satellites varies from 4 to 10.With three base stations,the absolute and relative positioning performances slightly improve as the number of UAVs increases.When without base stations and only satellites are utilized,the positioning performance improves more significantly as the number of satellites increases and the ground agents can only obtain coarse results with meter-level positioning accuracy.While with three base stations,the performance does not depend much on the number of satellites and sub-meter level accuracy is achievable.Therefore,in SAGIN localization applications,a ground agent usually obtains a rough position by satellite positioning,and then utilizes the base stations to acquire a more accurate positioning result.

Figure 4. The 3D absolute localization CRLB and RMSE with respect to the number of satellites.

5.3 Effect of Base Stations

We next consider the effect of the number of base stations.As the number of base stations increases from 0 to 1, the CRLB and RMSE for both 3D and 2D positioning significantly decrease by more than half.Thus,the base station can alleviate the error of satellite positioning to a great extent.The comparison between the red and blue lines in Figure 5 indicates that the 3D positioning error is higher than the 2D positioning error because the former includes the error in height.As the number of base stations increases, the gap between the two lines becomes smaller since the error in height gradually declines.For instance, without base stations,the 3D CRLB is 2.16 times of the 2D CRLB while with three base stations,the ratio is 1.48.Compared with satellites, base stations with much lower height can provide more precise resolution in height.Therefore,base stations can greatly reduce the 3D positioning error,especially for errors in height.

Figure 5. The 3D and 2D absolute localization CRLB and RMSE with respect to the number of base stations.

5.4 Effect of UAVs

We also explore the effect of the number of UAVs on positioning performance as shown in Figure 6.The effect is analyzed with three or without base stations and the number of UAVs varies from 4 to 10.With three base stations,the absolute and relative positioning performances slightly improve as the number of UAVs increases.Without base stations, the absolute positioning CRLB declines more obviously.For example, when the number of UAV increases from 4 to 10, the absolute CRLB is decreased by 0.84 m without base stations while it is decreased by 0.41 m with three base stations.Therefore, for pre-deployment of UAVs in the SAGIN,in some areas where the ground network cannot be deployed, the air network such as UAVs can be a useful complement.In addition, the number of UAVs can be appropriately determined according to the cost and the required positioning accuracy.

Figure 6. The 3D absolute and relative localization CRLB with respect to the number of UAVs.

5.5 Effect of Clock Noise

Clock noise will degrade the positioning performance as it will impair the ranging measurement.We analyze the impact of clock noise on the proposed SAGIN localization scheme.In Figure 7, it can be observed that when the standard variance of clock noise increases,the absolute and relative positioning performances both deteriorate seriously from the sub-meter level to the meter level.To achieve sub-meter absolute positioning accuracy, the required standard variance of clock noise is less than 2 ns.For example,the clock needs to synchronize every 4 s to suppress the influence of clock noise if it drifts 0.5 ns per second.Therefore,to reduce the negative effect of clock noise,frequent clock synchronization is needed.

Figure 7. The 3D localization CRLB and RMSE with respect to the clock noise.

VI.CONCLUSION

In this paper, we proposed a 3D SAGIN localization scheme for ground agents.The CRLB for the ground agents’ absolute and relative position errors were obtained using the information inequality and subspace projection.A distributed maximum likelihood localization algorithm was developed to estimate the ground agents’positions and clock offsets.Simulation results demonstrated that base stations can reduce the height error significantly and UAVs can be a useful complement when the ground network is not available.The effect of clock noise illustrated that frequent clock synchronization is required for sub-meter level positioning accuracy.Compared with existing methods, the introduced SAGIN localization system incorporates the merits of the space, air and ground network in terms of localization.The proposed SAGIN localization scheme can provide insight for applications in the future 5G and B5G systems.