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A Measurement Scheduling Method for Multi-vehicle Cooperative

Localization Considering State Correlation

Haigen Min , Yao Li , Xia Wu , Wuqi Wang , Lisha Chen ,

Xiangmo Zhao

PII: S2214-2096(23)00112-2

DOI: https://doi.org/10.1016/j.vehcom.2023.100682

Reference: VEHCOM 100682

To appear in: Vehicular Communications

Received date: 3 June 2023

Revised date: 4 September 2023

Accepted date: 9 October 2023

Please cite this article as: Haigen Min , Yao Li , Xia Wu , Wuqi Wang , Lisha Chen ,

Xiangmo Zhao , A Measurement Scheduling Method for Multi-vehicle Cooperative

Localization Considering State Correlation, Vehicular Communications (2023), doi:

https://doi.org/10.1016/j.vehcom.2023.100682

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A Measurement Scheduling Method for

Multi-vehicle Cooperative Localization

Considering State Correlation

Haigen Min

a

,

b*

, Yao Li

a

, Xia Wu

a

,

b

, Wuqi Wang

a

,

Lisha Chen

a

,

Xiangmo Zhao

a

,

b

a

School of Information Engineering, Chang’an University, Xi’an, China

b

Collaborative Innovation Center for Western China Traffic Safety and Intelligent

Cooperative Control

Abstract

Accurate and robust localization is a fundamental requirement for safe autonomous

driving. Currently, most vehicles rely on single-vehicle localization (SVL) using the

global navigation satellite system (GNSS) for obtaining the position data. However, in

urban canyons or areas with signal interruptions, the GNSS signals may be unreliable,

thus leading to accumulated errors in SVL. In contrast, cooperative localization

provides the advantages of superior accuracy, good fault tolerance, and high

flexibility by leveraging multi-source data from self-sensors and neighboring vehicles.

Nevertheless, establishing communication links with all surrounding vehicles can

introduce a severe communication burden and thus impact real-time data processing

performance. To address these challenges, this paper proposes a cooperative

localization method with loosely-coupled communication that considers the state

correlation. In this study, the communication process is divided into two topologies to

ensure filter consistency, and an approximate variable substitutions technique enables

suboptimal estimation of a cross-covariance matrix. Furthermore, a dynamic period

measurement scheduling (DPMS) method is proposed. The DPMS optimally selects

vehicles for cooperative localization based on the covariance matrix trace values,

considering three factors of location uncertainty and employing a weight function for

period adjustment. Finally, the proposed method is evaluated through multiple

simulations and field tests, assessing the algorithm convergence, localization

accuracy, execution time, and communication burden. The results demonstrate that

compared to the sequential greedy algorithm, the proposed method can significantly

reduce communication and computation burden while achieving higher localization

accuracy than the random measurement scheduling method and the measurement

scheduling method based on a deep neural network.

Keywords: autonomous vehicle, cooperative localization, measurement

scheduling, cross-

covariance matrix

1. Introduction

In recent years, autonomous driving technology [1] has experienced rapid development,

contributing to the resolution of various traffic challenges [2], such as congestion and high energy

consumption. However, high-precision localization, which involves obtaining accurate position

data on vehicles and surrounding traffic participants, is a critical component of autonomous

driving systems. It serves as a foundation for perception[3], path planning and tracking[4], and

control decision-making[5]. The most commonly used localization system is the global navigation

satellite system (GNSS), which estimates vehicle positions by measuring pseudo ranges from

multiple satellites. However, the GNSS's accuracy is limited by certain factors, such as satellite

clock errors, ionospheric delay [6], and multipath effects [7]. As a result, the localization accuracy

of the GNSS can be several tens of meters, which does not meet the precision requirements of

autonomous vehicles [8]. With the development of multi-sensor data fusion technology, various

autonomous vehicle localization technologies have been proposed to address the mentioned

limitation [9]. In the proposed techniques, the fundamental sensors include GNSS sensors and an

inertial measurement unit (IMU) [10]. The IMU is a relative localization sensor capable of

providing information on the three-axis acceleration and angular velocity of a vehicle at a high

frequency. In situations where the GNSS signals are weak or unavailable, the IMU can partially

compensate for the loss of location data. However, the integration of IMU data may lead to error

accumulation. In recent years, map-matching-based localization technology [11], such as

simultaneous localization and mapping (SLAM) [12], has gained increasing attention. This

technique can accurately determine a vehicle’s position in the global coordinate system using lidar

to scan and match a high-definition map in real time [13]. However, visual sensors are sensitive to

environmental factors [14], such as adverse weather conditions, and maintenance and updating of

traffic elements in high-definition maps require high attention [15]Error! Reference source not

found.. Moreover, the cost of sensors is typically high, making it challenging for this technology

to be implemented on a large scale in the short term.

With the rapid development of vehicular ad-hoc network (VANET) [16], cooperative

localization (CL) [17] based on vehicle-to-vehicle (V2V) communication has emerged as a viable

solution. The CL uses communication and perception between vehicles to facilitate multi-vehicle

information sharing, thus overcoming the limitations of autonomous localization methods and

offering promising alternatives [18]. High-precision sensor vehicles (HSVs) equipped with

advanced sensors can obtain highly accurate localization information using SLAM technology.

However, the utilization rate of the localization information is typically low. Still, by sharing the

position information on an HSV with the surrounding ordinary vehicles through V2V

communication and improving their localization estimates using the CL, the advantages of HSV

can be fully leveraged [19]. Based on the data fusion approach, the CL methods can be roughly

categorized into centralized and distributed methods [20]. Centralized CL methods involve

collecting measurement data from all vehicles in a VANET through vehicle-to-everything (V2X)

communication to obtain an optimal solution, which poses a great burden on communication and

computation resources [21]. In contrast, distributed CL methods perform local multi-source data

fusion at each vehicle in a loosely coupled manner, alleviating the computational burden and

providing the advantages of good scalability and robustness. However, in distributed methods,

determining the correlations between vehicle states (i.e., a cross-covariance matrix) can be

challenging [22].

In summary, CL has significant advantages over single-vehicle localization in terms of

localization accuracy and robustness. However, achieving precise CL requires real-time exchange

of state information among vehicles, which comes at the cost of message wireless communication

resources and local computing resources. In scenarios with limited communication and computing

resources, CL faces two possible challenges: (1)Trade-off between state correlation and

localization accuracy: Due to the fusion of different vehicle states and relative measurements in

CL, there exists correlation among vehicle states. This correlation is represented by the covariance

matrix, which indicates the trust level of the information about the collaborating objects. Once

relative measurements are performed between vehicles, a communication link needs to be

established to exchange the states and error information, in order to continuously update the

covariance matrix of both vehicles. At each time step, it is necessary to maintain and update the

covariance matrix between vehicles that are currently performing relative measurements and

communication, as well as propagate the correlation among vehicles that have not performed such

measurements and communication. Continuous covariance matrix updates result in tight coupling

of inter-vehicle communication, leading to significant communication and computational

overhead. However, completely ignoring the correlation among vehicle states can result in

over-trusting phenomenon [23], which introduces bias in the final state estimation. The above

problem can be summarized as: how to effectively update the covariance matrix while relaxing the

coupling of communication measurements among vehicles.(2)Measurement scheduling problem:

In high traffic density urban road scenarios, phenomena such as communication channel

congestion may exist. If vehicles in a cluster still require communication with all surrounding

vehicles, it may severely deteriorate the communication quality of the cluster, such as increased

communication delay and packet loss rate. On the other hand, as distributed CL performs local

fusion estimation of vehicle positions, if vehicles receive excessive redundant neighboring vehicle

data, it not only increases communication burden but also incurs significant computational

overhead. In addition, the contribution of surrounding neighboring vehicles to the localization of

the target vehicle varies. If poorly accurate vehicle data is blindly fused, even with very low

weight assignment, it still introduces uncertainty to the target vehicle’s localization. It is necessary

to develop a measurement scheduling method that allows vehicles to select only a small number of

vehicles for collaborative localization, in order to improve localization accuracy, reduce

communication and computational overhead simultaneously.

Faced with the aforementioned challenges, several CL methods with low resource

consumption characteristics have been developed. Some methods completely ignore the

covariance matrix to address inter-vehicle communication coupling [20], achieving lightweight

CL. However, these filtering methods have the risk of divergence. Other methods perform local

decomposition of the joint covariance matrix [24], allowing each vehicle to update the covariance

matrix locally. However, during the filter update stage, close communication with other vehicles

or a central server is still required to correct their own states. In addressing the measurement

scheduling problem, some approaches select collaboration partners based on signal propagation

quality [25] or distance [26]. However, these selection factors are easily influenced by the

surrounding environment, leading to potential ineffectiveness of the measurement scheduling

method. Measurement scheduling methods based on greedy algorithms accurately select a small

subset of vehicles that are most beneficial to the target vehicle at the current time [27], based on

factors such as vehicle position error. However, these methods often require establishing a fully

connected communication link. Researchers often employ suboptimal greedy strategies, including

limiting communication data volume and frequency [28].

To achieve both high-precision localization and low communication and computational

overhead, the following two key points need to be addressed: (1) Updating the cross-covariance

matrix in a loosely coupled manner. (2) Reducing redundant communication links, lowering

communication frequency, while ensuring high-precision localization. Existing cooperative

localization methods often fail to meet both of these requirements simultaneously, and some

methods still require establishing numerous communication connections when selecting the

optimal cooperating vehicles. This study perfectly satisfies the aforementioned requirements.The

primary objective of this study is to develop a distributed cooperative localization method suitable

for road environments with limited communication and computational resources. In response to

the aforementioned challenges, this study proposes a loosely coupled cooperative localization

method called state correlation-based cooperative localization (SCCL), along with a dynamic

period measurement scheduling (DPMS) method.

The main contributions of this study can be summarized as follows:

(1) A loosely coupled cooperative localization method is proposed. The SCCL method

considers the correlation between vehicle states in a distributed communication environment. It

also incorporates a cross-covariance update strategy based on loosely coupled communication for

different communication scenarios in a relative measurement model. In addition, an approximate

variable substitution technique is used to estimate the cross-covariance matrix sub-optimally,

reducing the state correlation loss and ensuring filter convergence;

(2) A dynamic period measurement scheduling method (DPMS) is developed. To address the

problem of limited communication and computational resources, the DPMS method enables

vehicles to select a small number of optimal vehicles from the neighboring vehicles periodically

for collaborative localization. It considers the uncertainty of the vehicle state and introduces three

localization error rates of change models to evaluate the localization stability of each vehicle in

the cluster at each period;

(3) A period weight function is derived to map the error rate of change to the corresponding

weighting factor. By integrating the three error rate change models, the period is dynamically

adjusted. The Z-Score method is employed to normalize the error rate change models, ensuring

smooth period changes and stable localization;

(4) Extensive simulation experiments and real vehicle tests are conducted. The proposed

method is evaluated through comprehensive simulation experiments from various perspectives,

including algorithm convergence, localization accuracy, execution time, and communication

burden. Real vehicle test scenarios are constructed at a closed connected and automated vehicles

test field of Chang'an University, encompassing ordinary road and tunnel scenes. A thorough

analysis of the localization performance of the proposed method in different scenarios is

performed.

Overall, this study contributes to the development of a distributed cooperative localization

method by addressing the challenge of limited resources in road environments. The proposed

SCCL and DPMS methods provide performance improvements in state correlation, measurement

scheduling, and dynamic period adjustment, thus enhancing localization accuracy and efficiency

and decreasing communication and computational burden.

2. Related Work

To meet the communication and computational requirements of vehicles while maintaining

the correlation between vehicle states, researchers have developed various suboptimal CL methods

that impose low communication and computational burdens. These methods can be roughly

categorized into two groups [29]: fixed communication topology methods and dynamic

communication topology methods. Fixed communication topology methods use a fixed and

unchanging communication topology throughout the localization process [30][31][32]. They

reduce communication and measurement overheads by adjusting the operational and measurement

frequency between vehicles or by compressing the transmitted data. The main aim of these

methods is to minimize the amount of data exchanged between vehicles while maintaining

localization performance. Dynamic communication topology methods dynamically adjust the

communication topology during the localization process [29][33][34]. They select a subset of

vehicles for cooperative localization based on specific selection strategies, eliminating the need

for data exchange with all surrounding vehicles at every time. These methods are commonly

referred to as measurement scheduling methods and aim to reduce communication costs by

optimizing the selection of vehicles involved in the cooperative localization process.

Scholars have conducted extensive research on fixed communication topology methods.

Various filter-based CL methods have been developed with the objective of minimizing

communication data. Paull et al. [35] employs a graph sparsification-based approach to estimate

the vehicle poses within the acoustic communication interval by exploiting the Schur complement,

resulting in a significant reduction in the size of communication packets. The iterative quantized

Kalman filter (IQKF) [36]and Soi-KF method [37] have been developed using bit encodings to

achieve maximum posteriori estimations, thereby minimizing the transmitted data during the

communication process by substituting quantized observations for real-valued sensor

observations.Wang et al. [31] proposed a CL method based on micro message propagation, which

uses multiple pairs of received relative measurement data to obtain partner estimates. The authors

also reconsidered the correlations between the estimation results and used the proposed similarity

transformation method to convert the partner estimates into pseudo-absolute measurements,

thereby reducing the amount of communication data. However, this method still relies on a

fully-connected communication topology. Zhang et al. [38] introduced the joint covariance matrix

of a vehicle cluster as a basis for evaluating vehicle state uncertainty. They adjusted the

measurement frequency of each sensor and the number of measurements available at each time

step through the covariance matrix. Chang et al.[30] proposed a measurement scheduling method

based on update frequency, which separates observation updates and communication updates. This

method dynamically constrains the propagation update, observation update, and communication

update in order to reduce the costs associated with communication and computation. The above

method optimizes the CL method by limiting the amount of communication data and the

measurement frequency. However, the communication coupling between vehicles is not relaxed,

and the establishment and maintenance of communication links still require many resources.

Overall, the previous research has aimed to develop CL methods that achieve a balance between

communication efficiency and localization performance. By considering fixed or dynamic

communication topologies and incorporating constraints on communication and measurement

frequency, these methods can reduce communication costs and resource requirements while

achieving reliable cooperative localization.

Recent studies have explored possible solutions for addressing the limitations of

communication-constrained CL methods that minimize communication data volume through

filtering while maintaining full connectivity between vehicles. However, the process of

maintaining and establishing communication links consumes significant resources, prompting

researchers to investigate relaxing the full-connectivity requirement. As a result, the CL methods

based on non-fixed communication topologies have been proposed for measurement scheduling.

The problem of optimal measurement scheduling is an NP-hard problem. Tzoumas et al.

[39]proposed a suboptimal scheduling method based on a greedy algorithm that can provide a

polynomial complexity scheduling solution, but this method still requires repeated communication

and measurement processes during each local optimal solution search, which is not suitable for

environments with limited communication and computational resources. Zhu et al. [33] developed

a measurement scheduling method based on minimizing the trace of the covariance matrix using

the discorrelated minimum variance (DMV). They derived an upper bound for the state correlation

of each agent and obtained an optimal gain matrix that minimizes the covariance matrix, thus

reducing communication overhead by decreasing the frequency on which agents exchanged

covariance matrices. Nevertheless, the computation of a nonlinear objective function requires

significant computational resources. To address the problem of high computational costs due to

nonlinear optimization and achieve real-time CL, recent studies have suggested learning-based CL

methods [29][40][41]. Chen et al. [40] used machine learning (ML) to optimize the DMV, thus

reducing computation complexity and allowing easy expansion to the other CL measurement

scheduling methods.

Zhu et al. [29] used the DMV framework proposed in [33], but instead of searching for an

optimal solution, they predicted the covariance information of the state using a deep neural

network (DNN) model. Also, they used the trace of the covariance matrix as communication data,

combined it with prior estimates, and fed it to the DNN model to predict the uncertainty of the

estimated state. This served as a basis for selecting neighboring agents for measurement

scheduling, which improved the performance of the DMV method in terms of communication data

volume and optimization complexity. Peng et al. [41] applied deep reinforcement learning (DRL)

to the CL measurement scheduling, treating the CL as a partially observable Markov decision

process (POMDP). Also, they evaluated real-time rewards based on state uncertainty and proposed

the DQN and PG algorithms to solve the POMDP problem, reducing the number of

communication measurements between vehicles. The aforementioned methods can optimize the

CL methods in terms of communication frequency, transmission data volume, and computational

complexity. However, they still struggle to achieve unified communication and computational

overhead and face challenges related to numerous communication links. Yan et al. [44] proposed a

greedy landmark selection method that operates by minimizing an upper-bound on the total

uncertainty of the team. Through rigorous mathematical derivation, they established an upper

bound on the determinant of the team's joint covariance matrix. In this method, each robot only

needs to acquire the local covariance matrix of other robots within communication range to select

the optimal n collaborative targets, effectively addressing the challenge of optimal node selection

in resource-constrained scenarios.However, they neglected the state correlation among different

robots and did not provide an update method for the cross-covariance matrix. Ignoring the

cross-covariance matrix may lead to biased state estimation, thus affecting the accuracy of optimal

node selection. Li et al. [45] proposed a distributed scheduling algorithm based on uncertainty

evolution under the framework of belief propagation (BP). They derived an upper bound for the

determinant of position uncertainty based on Fisher information theory, which served as the basis

for measurement scheduling. Each robot adaptively selects collaboration partners based on the

reduction in expected uncertainty, and determines whether collaborative localization is necessary

based on differences in robot positioning accuracy. This achieves a collaborative localization

algorithm for adaptive measurement scheduling without real-time communication. However, the

algorithm may be negatively affected by low communication frequencies between certain robots

and the neglect of error correlations between different robots' states, leading to a decrease in

localization accuracy. Therefore, a suitable solution for maintaining inter-vehicle covariance

matrices of vehicle states is necessary. The research presented in this paper considers the CL

methods where the communication topology graph can change over time. Moreover, the proposed

cooperative localization solution differs from the previous research in that it can achieve both the

maintenance and update of the cross-covariance matrix and the scheduling of cooperative vehicle

measurements.

3. Vehicle State Model

Assume a vehicle cluster A comprising N vehicles, where each vehicle is equipped with

various sensors, including an IMU, GNSS receiver, radar, lidar, and camera. The GNSS receiver

provides the absolute position of a vehicle, while the radar, lidar, and camera provide the relative

distances between the neighboring vehicles. In addition, each vehicle is equipped with V2V

transceivers, such as dedicated short-range communication (DSRC), for sharing their own

information with other vehicles. The state of vehicle i in the East-North-Up (ENU) coordinate

system is defined as

,,







T

E N N

i i i i

X p p

, where

,,



E N N

i i i

pp

represent the East-coordinate position,

North-coordinate position, and heading angle relative to the North axis of vehicle i, respectively.

At time k, each vehicle obtains its own state estimate

,

ˆik

X

and the corresponding error covariance

matrix

,ii k

P

, which is obtained by self-localization and cooperative localization estimation.

Assume that the belief of a vehicle i at time k is denoted by

 

, , ,

ˆ,

i k i k ii k

bel X P

, and the joint

belief between vehicles i and j at time k is denoted by

 

, , , ,

ˆ ˆ

,

ij k ii k jj k ij k

bel X X P

, where

,ij k

P

represents the cross-covariance matrix between the states of vehicles i and j. At the initial time of

the system operation,

,0

ij k

P

, but after relative measurements and V2V communication,

,0

ij k

P

, indicating the correlation between the states of vehicles i and j.

For a vehicle i,

{1,2,..., }i V N

whose state is

i

X

, the motion model at time k is defined

as follows:

 

, | 1 , 1| 1 , 1| 1

,

i k k i k k i k k

X f X u

    



(1)

where

, | 1i k k

X

represents the prior state at time k,

, 1| 1i k k

X

is the posterior state at time k,

,1ik

u

is

the output of the IMU, wheel speed meter, and other sensors, including the zero mean Gaussian

measurement noise

,ik

w

, and

.ik

Q

is the covariance matrix.

The measurement models of vehicles include two types of data: i) self-state measurement

data

,

ego

ik

Z

obtained from on-board sensors, and ii) relative state measurement data

,

coop

ij k

Z

of

vehicles, which includes the vehicles' relative distance

,

coop

ij k

d

and relative bearing angle

,

coop

ij k



.

The measurement equations are expressed as follows:

 

, , ,

, , , .

,

ego

i k i k i k

coop

ij k i k j k ij k

Z h X

Z h X X













(2)

where

 

h

represents the nonlinear measurement function related to the vehicle state;

,ik



and

.ij k



represent the Gaussian noise obtained by the self-measurement and relative measurement,

respectively. The measurement noise matrix is expressed as:

 

, , ,

T

i k i k i k

RE









，

 

, , ,

T

ij k ij k ij k

RE









.

By linearizing Eq. (2), the measurement error model is obtained as follows:

, , , ,

, . , , .

ˆ

ˆ,

ego

i k i k i k i k

T

coop

ij k ij k i k j k ij k

Z H X

Z H X X















(3)

where

,ik

X

and

,jk

X

are the updated state errors; the measurement matrices

,ik

H

and

.ij k

H

are the Jacobian matrices of

 

h

in

, | 1i k k

X

and

 

, | 1 , | 1

,

i k k j k k

XX



, respectively; thus, it holds

that

, | 1

,

ˆ

i i k k

ego

ik

ik T

ik XX

Z

HX









，

. , | 1 . , | 1

, | 1 . , | 1

,,

.,,

,,

,

i k i k k i k i k k

j j k k j k j k k

coop coop

ij k ij k

ij k TT

XX XX

i k j k

X k X XX

ZZ

HXX



















.

4. Cooperative Localization Scheme

This section focuses on two key aspects. First, Section 4.1 explains the implementation

process of the SCCL method and constructs covariance matrix update strategies under different

measurement models and communication topologies to avoid optimistic estimation. Second,

Sections 4.2 and 4.3 propose further improvements based on the SCCL and introduce the DPMS

method to address measurement scheduling problems. Figure 1 illustrates the cooperative

localization scenario and the framework proposed in this study. In scenarios with limited

communication and computational resources, the SCCL and DPMS methods are employed to

select the optimal vehicles for cooperative localization.

Period-based

measurement

scheduling

Selection

strategy based

on matix trace

Constant period

Dynamic period

Factor 1 Factor 2 Factor 1

Relative

measurement model

Communication

topology 1 Communication

topology 2

Sequential update Self-measurement

model

Fusion estimation

Correlation

update strategy 1 Correlation

update strategy 2

Driving direction

Ego Vehicle

Selected Vehicles

Non-selected Vechiles

DPMS

SCCL

Scenario

Fig. 1. Cooperative localization framework

4.1 CL Considering State Correlation

The prior state of a vehicle i is calculated using the self-sensor data

,

ego

ik

u

and the posterior

state of from the previous time step. Following the principles of Kalman filtering, the update of

the prior state covariance matrix

, | 1i k k

P

is performed as follows:

   

, | 1 , , 1| 1 , , , ,

TT

ego ego ego ego ego ego ego

i k k i k i k k i k i k i k i k

P F P F G Q G

  



(4)

where

, | 1

ego

i k k

P

is a priori estimate of the error covariance matrix at time k;

,,

,

ˆ,0

ego ego ego

ik

i i k

ego

ik

iX X u

f

FX





and

, , ,

,

,ˆ,0

ego ego ego

i k i k i k

ego

ik ego

ik X X u

f

Gw





denote the Jacobian matrices of the state

equation

f

with respect to the vehicle state

ego

i

X

and the sensor measurement error

,

ego

ik

w

,

respectively.

The next step is the update process, which involves dividing the measurement model into a

self-measurement model and a relative measurement model based on the sources of measurement

data. In addition, based on the sources of sensor data acquired by a vehicle, the measurement

model can be divided into three categories: (a) a self-measurement model that can use only its own

GNSS data; (b) a relative measurement model that can use only relative measurement data; (c) a

mixed measurement model that can use both types of data. The processing flow of the three

models is illustrated in Fig. 2, and it is explained in detail in the following.

Ego motion measurement

Predict belief

Is GNSS

detected?

Is ranging

detected?

Relative measurement update

Relative measurement

update as final position

Self-measurement update

Is ranging

detected?

Relative measurement update

Self- measurement update

as final position Fused update

N

YY

Y

 

, | 1 , 1 ,

,

i k k i k i k

bel f bel u





 

, | , | 1 ,

,coop

i k k i k k ij k

bel f bel Z





 

, | , | 1 ,

,ego

i k k i k k i k

bel f bel Z





 

, | , | 1 ,

,coop

i k k i k k ij k

bel f bel Z





,ik

u

Fig. 2. Measurement update process

When a vehicle i is traveling without communicating or performing relative measurements

with the surrounding vehicles and uses only its own GNSS measurement data, then it only needs

to maintain and update its own state and self-covariance matrix information. It is assumed that the

state correlation between vehicles remains unchanged in this scenario, so there is no need to

propagate and update the cross-covariance matrix. The autocovariance matrix

|

ego

kk

P

and

cross-covariance matrix

,ij k

P

are computed using the following expressions:

   

, | 1

1

, , | 1 , , , ,

, | , | 1 , , ,

, | , , , |

, , 1

ˆ

()

i k k

TT

ego ego ego ego ego ego ego

i k i k k i k i k i k i k

ego ego ego ego ego

i k k i k k i k i k i k

ego ego ego ego

ii k k i k i k ii k k

ij k ij k

K P H H P H Q

X X K Z Z

P I K H P

PP













  











(5)

where

,|

ego

i k k

X

is the posterior state of vehicle i's self-state measurement update ;

,|

ego

ii k k

P

is the

posterior state error covariance matrix of vehicle i's self-state measurement update;

,ij k

P

is the

cross-covariance matrix between the states of vehicle i and vehicle j;

,

ego

ik

Z

is the output data of

vehicle GNSS absolute measurements.

If vehicle i fails to obtain the GNSS measurement data at the current time but can maintain

communication with its neighboring vehicle j, only relative measurement updates are performed.

Each vehicle receives the belief from neighboring vehicles (i.e., the state information and

covariance matrix) and uses them to update and correct its own state. In this process, it is

necessary to maintain and update not only the autocovariance matrix but also the cross-covariance

matrix.

To facilitate the discussion, consider the relative measurement update process of a vehicle i

and assume that the update process of other vehicles is the same as that of vechile i. At time k, all

vehicles except for vehicle i can be divided into two sets denoted by

i

k

S

and

i

k

U

, where

i

k

S

represents a set of vehicles that communicate and perform relative measurements with vehicle i at

the current time, and

i

k

U

represents a set of vehicles that neither communicate nor perform

relative measurements with vehicle i at the current time. Vehicle i obtains the relative

measurement data of vehicle j (

i

k

jS

) through the ranging sensor, and vehicle j sends its state

data to vehicle i through V2V communication. At this time, there are two cases of communication

measurement between vehicles j and l (

i

k

lU

), as shown in Fig. 3: (a) vehicles j and l maintain a

communication measurement relationship in the previous time (k – 1) or in the current time k; (b)

vehicles j and l have not maintained the communication measurement relationship recently.

Vehicle i

Vehicle j

Vehicle l

 

,,

coop coop

j j jj ji

bel X P P

 

coop coop

l ll ll lj

bel X ,P ,P

,

coop

ll lj

PP

Vehicle i

Vehicle j

Vehicle l

 

,,

coop coop

ij j jj ji

bel X P P

(a) There is a communication connection (b) There is no communication connection

between vehicles j and l between vehicles j and l

Fig. 3. Communication topology under relative measurement model

In the scenario shown in Fig. 3(a), vehicle j sends its belief

 

, , , | 1

,

coop coop

j k j k j k k

bel X P 



and the

cross-covariance matrix

,1ji k

P

to vehicle i through V2V communication. Upon receiving this

information, vehicle i first combines it with

,1ij k

P

stored locally and compute

,ij k

P

. Next,

,ik

bel

is fused with the computed

,ij k

P

through a local fusion update process of vehicle i to obtain the

posterior state

,|

coop

i k k

X

of vehicles i and j, and the corresponding covariance matrix is obtained as

follows:

     



1

, , | 1 , , | 1 , , , | 1 , ,

T T T

coop coop coop coop coop coop coop coop coop

i k ii k k i k ii k k i k i k ii k k i k i k

K P H P H H P H R 

  



  





(6)

 

, | , | 1 , , , | 1 ,

,

coop coop coop coop coop coop

i k k i k k i k ij k i k k j k

X X K Z h X X



  

(7)

 

, | , , , | 1 , , , | 1

coop coop coop coop coop coop coop

ii k k i k i k ii k k i k ji k ji k k

P I K H P K H P



  

(8)

 

, | , , , | 1 , , , | 1

coop coop coop coop

ij k k i k i k ij k k i k j k jj k k

P I K H P K H P



  

(9)

where the superscript 'coop' indicates that the variable is obtained through collaborative

localization fusion estimation. The matix

,|

coop

ii k k

P

represents the posterior error covariance matrix

of vehicle i; The matix

,|ij k k

P

represents the cross-covariance matrix between vehicle i and vehicle

j; The matix

,

coop

ik

H

and

,

coop

jk

H

are the Jacobian matrices of the measurement equation with

respect to

,

coop

ik

X

and

,

coop

jk

X

, respectively,

 

,,

,

coop coop

i k j k

coop

ik coop

ik

h X X

HX





,

 

,,

,

coop coop

i k j k

coop

jk coop

jk

h X X

HX





.

Vehicle i sends the belief of vehicle j

 

, , ,

,

coop coop

j k j k jj k

bel X P

obtained by its local calculation

to vehicle j so that vehicle j can perform the cooperative localization fusion estimation.

However, as vehicle i has not performed relative measurements and communication with

vehicle l, the covariance matrix between them still needs to be updated, and the update of this

cross-covariance matrix is performed as follows [42]:

 

, , , , 1 , , , 1

coop coop coop coop

il k i k i k il k i k j k jl k

P I K H P K H P



  

(10)

In Eq. (10), there are two unknown variables,

,1il k

P

and

,1jl k

P

. In [42],

,1jl k

P

was

approximated using the cross-covariance matrix

,1il k

P

between two different vehicles and

self-covariance matrix

,

coop

ii k

P

of vehicle i. As a result, the update process defined by Eq. (10) no

longer relies on the communication between vehicles j and l, but instead requires vehicle i to

re-establish the communication link with vehicle l to compute

,1il k

P

and complete the update

process by Eq. (10). However, this method inevitably increases the communication overhead and

reduces the real-time performance of calculation. To address this problem, this paper proposes a

different approach from that presented in [42], which approximates

,1il k

P

using a variable

substitution as follows:

 

1

, 1 , 1 , 1 , 1

coop

il k ii k ji k jl k

P P P P



   



(11)

Eq. (11) does not rely on vehicles i and l to establish a new communication link. In the

current scenario, a vehicle j has established a communication relationship with a vehicle l and

stored

,1jl k

P

locally. Therefore, when vehicle j sends its own belief to vehicle i, it only needs to

send

,1jl k

P

at the same time, as shown in Fig. 2(a). Thus, Eq. (10) can be reconstructed as

follows:

 

1

, , , , 1 , 1 , 1 , , , 1

coop coop coop coop coop

il k i k i k ji k ii k jk k i k i k jl k

P I K H P P P K H P



   

  

(12)

Eq. (12) neither contains the correlation information between vehicles i and l nor requires the

establishment of a communication link between them, thus achieving the goal of reducing the

communication overhead.

In the scenario shown in Fig. 3(b), neither vehicle i nor vehicle j can obtain information

about vehicle l. Therefore, to avoid the establishment of a communication link, at the current time

k, vehicle i obtains the relative state data between vehicle l and itself through a ranging sensor.

Thus, the state estimation formula of vehicle l is as follows:

 

,,

,

[]

, , , ,

,,

cos

sin

coop coop

Eij k ij k

ik

i N coop coop

l k i k ij k ij k

Ncoop

ik ij k

d

p

X p d













  





(13)

where superscript [i] represents the state of vehicle l from the perspective of vehicle i.

According to the error propagation principle [31], the uncertainty of

[]

,i

lk

X

can be estimated as

follows:

 

,

, , , ,

,

coop T

ii k

coop coop coop

il k ll k i k l k

coop

ii k

P

P P H H

R









(14)

The sequential update method [43] is adopted in this study to fuse the information on the

surrounding neighboring vehicles and the self-state information of a vehicle. According to the

order of receiving the neighboring-vehicle data, the data on a neighboring vehicle j are

sequentially fused and updated with the prior state data on vehicle i. The data flow diagram is

shown in Fig. 4.

Estimation

state 1 Estimation

state 2 Estimation

state n

Fusion

estimation

state

···

1's state

i

V

's state

i

V

1' stateV

1's state

i

V

's state

n

V

Fig. 4. The workflow diagram of the sequence update method

When a vehicle is traveling and can receive both the GNSS data and the relative

measurement data, it is necessary to fuse these two types of measurement data. A vehicle i fuses

the data on its n neighboring vehicles using a sequential updating method to obtain the estimation

results of the relative measurement updating process, including the vehicle state

,|

coop

i k k

X

and the

self-covariance matrix

,|

coop

ii k k

P

. Through self-measurement process, vehicle i obtains the state

estimation result

,|

ego

ii k k

X

, with covariance matrix

,|

ego

i k k

P

;

,|

ego

i k k

P

and

,

coop

ii k

P

characterize the

uncertainty of the estimated position and can be used as weights for

,|

ego

ii k k

X

and

,|

coop

i k k

X

,

respectively. The final state estimation result

,|

final

i k k

X

of vehicle i is expressed as a weighted

fusion of

,|

ego

ii k k

X

and

,|

coop

i k k

X

, which can be expressed as follows:

       

1

1 1 1 1

coop

, | , | , | , | , , | , |

final coop ego coop ego ego

i k k ii k k i k k ii k k ii k ii k k ii k k

X P P P X P X



   

   

  

   

   

(15)

4.2 Measurement Scheduling Under Limited Connectivity

Assuming that there are a total of N vehicles around a vehicle i, due to limited

communication and computing resources, vehicle i can communicate only with n (n < N) vehicles

around it and perform relative measurements. The cooperative localization measurement

scheduling problem is how to select an optimal cooperative partner using appropriate filtering

strategies under limited resource conditions. The term “optimal” relates to the minimum

uncertainty of the vehicle state, which is defined by the posterior covariance matrix. The update of

Eqs. (6)–(9) cannot be accomplished locally by vehicle i and requires V2V communication to

obtain the state and covariance information on vehicle j. The matrix computation involved in this

process is usually high-dimensional, resulting in high computational costs. However, in a

resource-limited situation, the uncertainty of the vehicle state can be measured by

 

,ik

trace P

of

the posterior covariance matrix [44] which is a scalar. This value can be used as an evaluation

indicator for the accuracy of the vehicle state estimation, further reducing computational costs.

At time k, define set

i

k

A

as a collection of all vehicles within the communication and

measurement range of vehicle i. Set

i

k

S

as a set of IDs of vehicles selected as cooperative targets

by vehicle i using a specific measurement scheduling method. Conversely, set

i

k

U

consists of IDs

of vehicles not selected by vehicle i, which can be expressed as

i i i

k k k

U A S

. Each vehicle has

the capability to perform up to n measurement processes. The mathematical formulation of

measurement scheduling is as follows:

 

,

min . .

ii

kk

i

i k k

SA trace P s t S n



(16)

This study proposes a measurement scheduling method called constant period measurement

scheduling (CPMS) based on

 

,ik

trace P

for cooperative localization. By solving Eq. (16), an

optimal set of cooperative vehicles

i

k

S

is obtained. The CPMS method uses a non-constant

communication topology, which enables a loosely-coupled approach for cooperative localization.

This approach significantly reduces computational and communication overhead while

minimizing the decrease in vehicle position accuracy.

At the initial time k, the scheduling period is set to

constant

T

, and a vehicle i performs relative

measurement with all vehicles within the communication range, and applies the sequential update

method presented in Section 3.1 to fuse the information from vehicle j and its own vehicle

sequentially. This process generates N estimation results, from which n results with the smallest

traces are selected, and the corresponding vehicle IDs are recorded and added to set

i

k

S

. This set

is used to provide the maximum improvement in the localization accuracy of vehicle i. During the

subsequent

constant

T

period, vehicle i no longer communicates with vehicles in set

i

k

A

but

communicates with vehicles in set

i

k

S

. After

constant

T

, an optimal set of vehicles

i

k

S

for the next

period is determined using the aforementioned strategy.

4.3 Dynamic Period Measurement Scheduling

Considering that the localization accuracy of a vehicle cluster is affected by various factors,

such as GNSS blockage, multipath effect, and cumulative sensor errors, the localization

performance during the driving process is unstable, and vehicles with a high localization precision

at the initial time k may experience a decrease in the precision after driving for a certain period of

time. Therefore, the CPMS may encounter the following situation. Assume that vehicle i performs

communication measurement with all surrounding vehicles at time k and obtains an optimal

vehicle set

i

k

S

. After a short period of time

 

constant

t t T

, some vehicles in

i

k

S

may

experience an increase in the localization error or a decrease in the communication quality (i.e., an

increase in the communication delay or packet loss) due to an increase in the inter-vehicle distance.

During the interval

 

, constant

k t k T

, vehicle i will continue to perform communication with

vehicles in

i

k

S

, but the accuracy of the cooperative localization estimates obtained during this

period will not be optimal.

To address the limitations of the CPMS, this study proposes a DPMS method. It is assumed

that the period should vary dynamically during vehicle travel. At the end time of each period, T is

dynamically adjusted based on the state information on each vehicle in the cluster. The starting

point of the adjustment method is the change in the state uncertainty of the current selected vehicle

set

i

k

S

and the unselected vehicle set

i

k

U

. Assuming that the current time k represents the start

time of period , this study considers the following three factors that affect the state uncertainty

changes:

(1) At start time , vehicle i fuses the data on a vehicle in

i

k

S

with its own data to obtain the

trace of the state covariance matrix

 

,ik

trace P

. At the end time of period (k+T) , vehicle i fuses

the data on the vehicle in

i

k

S

with its own data to obtain the trace of the state covariance matrix

 

,ik

trace P

. The change rate of

 

,ik

trace P

and

 

,i k T

trace P 

is defined as follows:

   

 

,,

1

,

i k T i k

ik

trace P trace P

rtrace P





(17)

(2) At start time k, vehicle i fuses the state data on all vehicles in

i

k

U

with its own state data

to obtain the trace of the state error covariance matrix, which is denoted by

 

,

U

ik

trace P

. At the

end time of period (k+T), vehicle i fuses the state data on all vehicles in

i

k

S

with its own state

data to obtain the trace of the state error covariance matrix, which is denoted by

 

,

S

i k T

trace P 

.

The change rate of

 

,

U

ik

trace P

and

 

,

S

i k T

trace P 

is defined as follows:

   

 

,,

2

,

SU

i k T i k

U

ik

trace P trace P

rtrace P





(18)

(3) At start time k, vehicle  obtains the state and covariance matrix

,jk

P

of vehicle j

(

i

k

jS

) through V2V communication, and the trace of matrix

,jk

P

is denoted by

j

t

, and it

represents a scalar value. Then, the variance of

j

t

is calculated by:

||

1

1i

k

S

j

t

n







(19)

 

|| 2

2

1

1i

k

S

j

t

n









(20)

Similarly, at the end time of period (  ), the trace of the covariance matrix of vehicle j

(

i

k

jS

) is denoted by

j

t

. The mean





and variance

2





of

j

t

are calculated by Eqs. (19)

and (20), respectively. The variance change rate of the trace of the covariance matrix in

i

k

S

is

calculated by:

22

32

r









(21)

When only a single influencing factor is considered, the statistical analysis of historical data

is performed based on the trace of the covariance matrix to determine suitable initial periods

1

T

,

2

T

and

3

T

for three influencing factors. Then, the final period T is determined by weighted

averaging as follows:

1 1 2 2 3 3

T w T w T w T  

(22)

The CRITIC weighting method is employed to determine wi (i = 1, 2, 3), which is an

objective weighting method based on data volatility. This method is based on two indicators,

namely volatility (contrast intensity) and conflict (correlation) indicators. The contrast intensity is

represented by the standard deviation, and the larger the standard deviation of data is, the larger

the volatility and the higher the weight will be. Further, conflict is represented by the correlation

coefficient, and the larger the correlation coefficient between indicators is, the smaller the conflict

and the lower the weight will be. The weights are calculated by multiplying the contrast intensity

and conflict indicators and normalizing the result to obtain the final weights of

1

T

,

2

T

and

3

T

.

The three influencing factors are determined based on the trace value of the covariance matrix, so

there is a certain correlation. Due to the presence of sensor noise, the localization error of vehicles

will fluctuate over time, resulting in certain volatility.

The historical data obtained in intervals

1

T

,

2

T

and

3

T

, which are recorded during the

vehicle driving process, are used as source data in the CRITIC weight method. Assume m is the

length of historical data; then, the changing rate vectors of influencing factors (1)–(3) are given by

 

1 1 1 1

12

, ,..., T

m

r r r r

,

 

2 2 2 2

12

, ,..., T

m

r r r r

, and

 

3 3 3 3

12

, ,..., T

m

r r r r

, respectively. Then,

1

r

,

2

r

,

and

3

r

are normalized and combined to form the input data matrix

1 2 3

,,D r r r





for the

CRITIC method.

Next, compute the information-carrying capacity, including volatility and conflict, where

volatility reflects the stability of data for a single factor, and it is calculated by:

 

2

( ) ( )

11

m

ij j

i

j

DD

an











(23)

where

()j

D

is the average value of all elements in column j of matrix D.

Further, conflict reflects the correlation between multiple influencing factors. The correlation

matrix is obtained by:

  

   

3

( ) ( ) ( ) ( )

,1

33

22

( ) ( ) ( ) ( )

11

ij j ik k

jk

ij j ik k

jk

D D D D

R

D D D D



















(24)

The conflict calculation formula is expressed as follows:

 

3

1

j ij

i

fr







(25)

where

ij

r

represents the correlation coefficient between the ith and jth influencing factors. The

information carrying capacity of an influencing factor j is set to

j

c

, which is calculated by:

j j j

c a f

(26)

The weight of

( 1, 2,3)

j

Tj

is given by:

3

1

j j j

j

w c c





(27)

Further, assume that k is the start time of a certain period, and

1

k

T

,

2

k

T

and

3

k

T

are the

periods corresponding to three influencing factors. At end time of period (k +

i

k

T

), according to

the change rates

1

k

r

,

2

k

r

and

3

k

r

of the three influencing factors, the next period

k

i

kT

T

is

calculated as follows:

 

0 1,2,3

>0

k

i i i i

k k k k

i

kT i i i i

k k k k

T T f r r

Ti

T T f r r



  











，

(28)

The function

   

22

1 exp / 2

ii

kk

f r r





  





is a weight function that controls the increment

i

k

T

and its value range is





0,1

, where



denotes the bandwidth coefficient that controls the

changing rate of

 

i

k

fr

. The smaller the



value is, the smoother the weight changes will be;

also, the larger the  value is, the faster the weight changes will be. When

||

i

k

r

is large,

 

i

k

fr

tends to one, and when

| | 0

i

k

r

, then

 

0

i

k

fr 

. In addition, when

>0

i

k

r

, if

||

i

k

r

and  are

large,

 

i

k

fr

will tend to one indefinitely, resulting in

0

k

i

kT

T

which makes the algorithm

degenerate into a scheduling strategy similar to the SGA and causes tight coupling of

communication. To avoid the mentioned problems, based on the historical changing rate data

before the current time, at end time of period, the Z-score normalization is performed on

||

i

k

r

.

Considering the first influencing factor, when

0

i

k

r

, it can be shown that the trace of the

covariance matrix at the end time of a period is smaller than that at the start time of the period.

The larger the absolute value of

i

k

r

is, the greater the difference between the traces at the two

moments will be. It has been believed that at the end time of s period, vehicle i can still

communicate with a vehicle in

i

k

S

to improve its localization accuracy, so the period should be

increased optimistically. Similarly,

0

i

k

r

indicates that the trace of the covariance matrix at the

end time of a period is larger than the trace of the covariance matrix at the start time of the period.

After a period, after fusing the data on the vehicle in

i

k

S

, the localization uncertainty of the

vehicle i increases, so the period should be reduced.

Considering the second influencing factor, when

0

i

k

r

, the trace of the covariance matrix of

a vehicle in

i

k

S

at the end time of this period is less than the trace of the covariance matrix of a

vehicle in

i

k

U

at the start time of this period. It has been considered that in the next short period

of time, vehicle i can still improve its localization accuracy through communication with vehicles

in

i

k

S

, so the period should be increased optimistically. Similarly,

0

i

k

r

indicates that the trace

of the covariance matrix of a vehicle in

i

k

S

at the end time of this period is greater than the trace

of the covariance matrix of a vehicle in

i

k

U

at the start time of this period, which indicates that

after the period T, the state uncertainty of the vehicle in

i

k

S

increases. In addition, it is likely that

from a certain moment in the middle of the period, the localization uncertainty of the vehicle in

i

k

S

gradually increases, so the period should be reduced.

Considering the third influencing factor, when

0

i

k

r

, the variance of the trace of the

covariance matrix of a vehicle in

i

k

S

at the end time of the period is less than the variance of the

trace of the covariance matrix of the vehicle in

i

k

S

at the start time of this period, and the

difference in vehicle localization uncertainty in

i

k

S

is reduced. In this case, the localization of the

vehicle in

i

k

S

tends to be stable, so the period should be increased. Similarly, when

0

i

k

r

, the

variance of the trace of the covariance matrix of the vehicle in

i

k

S

at the end time of this period is

larger than the variance of the trace of the covariance matrix of the vehicle in

i

k

S

at the start time

of this period, and the difference in vehicle localization uncertainty in

i

k

S

increases. In this case,

the localization of the vehicle in

i

k

S

tends to be unstable, so the period should be reduced.

The pseudo-code of the DPMS algorithm is given in Algorithm 1. At the end time (  ) of

the period, by considering three influencing factors and using the above period adjustment method,

the next period

k

kT

T

can be expressed as follows:

3

1

k

i

k T i k T

i

T w T









(29)

Where

i

w

is the weight calculated by the CRITIC method.

Algorithm 1: The DPMS algorithm

1: Input:

1 2 3

11 i

i,k ii,k|k ii,k k k k k

X ,P ,R ,A ,T ,T ,T



2: Calculate

k

T

using Eq. (28)

3: For any vehicle

i

k

jA

, obtain

11jj,k ji,k

P ,P



4:

1i

k

l ,S  

5: while

ln

do

6: for

i

k

jA

7:

 

[ ] .(8 ) 9

, , 1 , | 1 , , 1 , 1

, , ,

j Eq

ii k i k ii k k ii k jj k ji k

P X P R P P

   



（），

8: Save

[]

,j

ii k

P

9: end for

10:

 

* [ ]

,

argmin

i

k

j

ii k

jA

j trace P







11: Perform relative measurement, obtaining the value of

,

coop

ij k

Z

12:

 

* * * *

.(4)

, , , 1 , | 1 , , 1 , 1 ,

, , , ,

Eqs coop

i k ii k i k ii k k ii k j j k j i k ij k

X P X P R P P Z

 



（8），

13:

**

\{ }, { }

i i i i

k k k k

A A j S S j  

14: Calculate the next period using Eqs. (11)-(25).

15:

1ll

16: end while

17: while time <

k

T

do

18: For any vehicle

i

k

jS

, obtain

11

coop

jj,k ji,k ij,k

P ,P ,Z



19:

 

.(4)

, , , 1 , | 1 , , 1 , 1 ,

, , , ,

Eqs coop

i k ii k i k ii k k ii k jj k ji k ij k

X P X P R P P Z

   



（8），

20: end while

5. Performance Evaluation

5.1 Simulation Test Results

This section analyzes in detail the performance of the proposed method in a simulation test

environment. First, the parameter settings of a simulation environment are introduced. Then, the

proposed method is evaluated and verified from multiple perspectives using various indicators.

In the simulation tests, Carla was used as a simulation test environment, and a cluster was

formed by five vehicles, each of which was equipped with GNSS receivers, IMU, and LiDAR

sensors. The GNSS receiver was used to obtain position and velocity data on a vehicle; the IMU

provided measurement data on acceleration, angular velocity, and heading; the LiDAR sensor was

employed to obtain the relative distance and angle between vehicles. In real road environments,

different vehicles may have variations in sensor performance due to system and environmental

factors, leading to distinct localization results. To simulate a scenario with varying localization

stability among multiple vehicles in a real road environment, different parameters were assigned

to the GNSS data and velocity noise models of each vehicle during different time periods. The

noise followed a zero-mean Gaussian distribution. The variance values of the GNSS and velocity

noise models for the vehicles during different time periods are presented in Table 1.

Table 1. The variance values of the GNSS and velocity noise models

Vechile

Noise parameter (, )/(m,m)

Time period 1

Time period 2

Time period 3

Time period 4

Time period 5

V0

3,2

5,1

V1

3,2

5,1

3,2

5,1

3,2

V2

3,2

5,1

3,2

5,1

3,2

V3

5,1

3,2

5,1

3,2

5,1

V4

5,1

3,2

5,1

3,2

5,1

5.1.1 Consistency Evaluation

In localization problems, the consistency of the state estimate is a measure of how well the

estimate converges to the ground truth (GT). In addition, it is important to assess the consistency

of a filtering method, as a large covariance matrix bias can lead to an incorrect filtering gain.

Therefore, evaluating the consistency of a filtering method is a crucial task. In this study, the

consistency of the filter was evaluated through Monte Carlo statistical tests. The normalized

estimation error squared (NEES)  was employed as an evaluation metric of consistency. The

NEES value was calculated by:

   

1

| | |

ˆ ˆ

T

k k k k k k k k k

X X P X X





  

(30)

where  is the true value of the vehicle state, and 

 is the state estimate of the filter.

Fifty Monte Carlo tests (M = 50) were conducted to verify the consistency of the filter, and

the average NEES value was recorded as



k

. Under the assumption that the filter was uniform

and approximately linear Gaussian ,



k

had a chi-square distribution

2



with

 

dim k

X

degrees of freedom; then,



k

M

had

 

dim 

k

XM

degrees of freedom. When the state

estimation error was consistent with the covariance calculated by the filter,



k

satisfied the

following conditions:

 

,

kab





(31)

where

 

,ab

is the acceptance region of the consistency of the estimation results, and

 

 

, | 1

k

P a b



  

; the significance level was

0.05





, so the acceptance domain was

calculated as

   

22

150 150

(0.025) (0.975)

, , 2.36,3.72ab MM











.

The proposed SCCL method was compared with the Native CL (cross-covariance matrix is

neglected) and DCL[42] method. Figure 5 shows the root mean square error ( RMSE ) and NEES

of different methods over time with 0.5 seconds as a time step. In Fig. 5, it can be observed that in

the first 100 time steps, the RMSE values of the three methods were almost identical. At

approximately 400 time steps, the Naive CL (NCL) method begins to diverge, and the RMSE

index of the DCL method gradually increases, then slowly decreases towards stability. However,

the RMSE of the SCCL method always maintains stability and the highest localization accuracy.

The results in Fig. 5 indicate that after 400 time steps, the NEES value of the NCL method

exceeded the acceptable range, namely,

kb





. This result suggested that the filter trusted the

estimation result excessively, thus causing the actual state uncertainty to be much higher than the

filter-calculated uncertainty, which resulted in a continuous increase in the RMSE value,

indicating inconsistent estimation. In contrast, during the entire simulation, the NEES value of the

proposed SCCL method remained within the acceptable range, demonstrating the consistency of

the estimation method.

(a) RMSE (b) NEES

Fig. 5. The convergence evaluation results of the three methods

The Native CL method ignores the correlation between different vehicle states and is prone to

optimistic estimation. Even if there is a large error in the localization data of an adjacent vehicle,

the Native CL method will mistakenly believe that the localization data are accurate and give a

higher weight, which will result in filter divergence. The SCCL continuously updates and

maintains the correlation of the vehicle status in a loosely coupled manner at each moment, thus

ensuring a correct estimation of the adjacent vehicle localization error.

The DCL method continuously updates the cross-covariance matrix locally at each time step

in a loosely coupled manner, and finally tends to converge, which ensures the consistency of the

method. However, the localization accuracy of the proposed method is higher than that of DCL.

The DPMS method ensures that the data of all vehicles around the vehicle will not be used for

fusion estimation at each moment, but only a small number of vehicles with small localization

error are selected for cooperative localization, and the single vehicle localization results and

cooperative localization results are weighted and fused to achieve the optimal estimation. DCL

does not filter the surrounding vehicles, but fuses and estimates the data of all vehicles within its

communication range, resulting in a decrease in the localization accuracy of the vehicle due to the

fusion of a small number of state data with poor localization quality, as shown in Fig.5 around the

400th time step.

5.1.2 Localization Accuracy Analysis

Further, it was assumed that each vehicle could communicate with up to four other vehicles

since it was not feasible to communicate with all surrounding vehicles due to resource constraints.

Furthermore, it was supposed that each vehicle could establish communication links with only two

vehicles at a time and perform relative measurements, namely,

2

i

k

S

and

4

i

k

A

. The

cumulative distribution function (CDF) curves of the root-mean-square error (MSE) of the four

measurement scheduling methods are shown in Fig. 6, where the orange curve represents the

localization performance obtained by using the single-vehicle localization method (SVL), the

green curve shows the performance of the scheduling method based on a sequential greedy

algorithm (SGA) [39], and the blue curve represents the performance of the random measurement

scheduling method (RMS); the CPMS denotes the proposed constant period measurement

scheduling method with a different period ranging from 20 to 120. As shown in Fig. 6, the RMS

method performed better than the SVL method. The CDF curve of the CPMS method was

completely above those of the random and SVL methods, indicating that the CPMS method was

superior to the RMS and SVL methods in terms of localization accuracy.

Fig. 6. The CDF curves of different methods in the CPMS evaluation

The MSE, root-mean-square error (RMSE), and mean-absolute error (MAE) results of

different localization methods are presented in Table 3, where it can be seen that when a constant

period T was set to 40 time steps instead of 20 time steps, the CPMS achieved the highest

localization accuracy among all methods. This indicated that for a constant period measurement

scheduling method, a smaller constant period would not necessarily provide better localization

performance. When the constant period was small, similar to the RMS method, the main vehicle

would frequently replace the collaborating vehicles, leading to frequent approximate estimation of

the cross-covariance matrix and reduced localization accuracy. In addition, the localization

accuracy of the SGA method was higher than that of the proposed CPMS method, but the

difference in their accuracy was not significant; the SGA method introduced significant

communication and computational overhead.

The SVL method relied only on a single vehicle for localization and could very easily

accumulate localization errors. Therefore, this localization method is usually unreliable. Due to the

randomness of the RMS method in selecting vehicles at each time, the selected vehicles could not

necessarily provide optimal performance in the self-vehicle localization due to the uncertainty in

their own localization accuracy. In addition, the RMS method could encounter the following

situation. The cross-covariance matrix of the cooperation between the ego vehicle and all the

surrounding vehicles could not be effectively and continuously maintained and updated. Due to

the randomness of vehicle selection, the vehicles that were selected in the previous moment might

not be selected in the next moment, resulting in a situation where no other vehicles could help

transmit the covariance matrix to the ego vehicle. As a result, the cross-covariance matrix could

not be continuously maintained and updated, which would reduce localization accuracy.

Prior to the execution of the DPMS, the initial value of the period needed to be determined by

combining the effects of the three influencing factors. Section 4.3 describes how the three factors

affected the selection result of communication vehicles and the final localization result. At the end

time of the current period, the change rates

1

r

,

2

r

and

3

r

that corresponded to the three

influencing factors were calculated. When

1

r

,

2

r

,

3

r

< 0, the fusion of vehicle data in

i

k

S

during

this period could improve the localization accuracy of vehicle i. During the execution of the

DPMS, the values of

1

r

,

2

r

and

3

r

were recorded, and the ratio

0lt

r

of values less than zero

was defined as an evaluation index. The proportion of values less than zero to the total number of

values was calculated for different periods (i.e., 20, 40, 60, 80, 100, and 120 time steps), and the

obtained results are shown in Fig. 7. As shown in Fig. 7, when T was 60, the ratio of values less

than zero to the total number of values was the largest, indicating that the initial period of the

DPMS should be set to T = 60 time steps.

Fig. 7. Changes in  index for different periods

The weights of the three influencing factors were calculated by Eqs. (17)–(24), and the

results are shown in Table 2.

Table 2. Weights of the influencing factors

Factor (1)

Factor (2)

Factor (3)

0.2918273

0.46937859

0.23879411

The localization accuracies of the DPMS, SGA, CPMS (T = 40), and DNN-MS [29] were

compared. The DNN-MS represents a measurement scheduling method based on a deep neural

network, which takes the trace value of the covariance matrix of the surrounding vehicles as a

network input and then predicts the trace value of the next covariance matrix and uses it as a

screening criterion to realize vehicle measurement scheduling. As shown in Fig. 8, the localization

error of the DPMS method was significantly smaller than that of the CPMS (T = 40) method,

slightly better than that of the DNN-MS method, and its CDF curve was very close to that of the

SGA method. The results indicated that dynamic adjustment of the scheduling period could make

the variations in the period more closely approach the variation trend of the localization error of

the vehicle cluster, thereby ensuring localization stability.

Based on the results in Table 3, compared to the CPMS (T = 40) method, the DPMS reduced

the localization error by 32.4%, 18.0%, and 42% regarding the MSE, RMSE, and MAE indicators,

respectively. Further, compared with the DNN-MS method, the DPMS reduced the localization

error by 38.8%, 21.8%, and 29.4% regarding the MSE, RMSE, and MAE indicators, respectively.

The localization error of the CPMS was slightly larger than that of the DNN-MS. However, by

using the proposed period adjustment method, the DPMS method performed better compared to

the CPMS method, and the localization error of the DPMS method was much smaller than that of

the DNN-MS. In addition, compared to the DPMS method, the SGA method reduces the

localization error by 3.9% and 2.0% regarding the MSE and RMSE indicators, respectively.

However, the DPMS method was superior to the SGA method in terms of the MAE index. The

localization advantage of the SGA method relied heavily on communication and computational

pressure. The DPMS achieved almost the same localization accuracy as the SGA method in a

loosely coupled manner.

DPMS method achieves higher localization accuracy than DNN-MS while significantly

reducing communication and computational costs. DNN-MS reduces communication overhead by

transmitting the trace of the state error covariance matrix, and utilizes the state data, trace of

covariance matrix, and noise covariance matrix as input for the DNN model to predict and output

the trace of the posterior covariance matrix, which serves as the basis for node selection.However,

the DNN model relies on accurate noise modeling and covariance matrices provided by the

training data, which poses challenges in accurately predicting in complex and dynamic road

environments. Additionally, DNN-MS still requires maintaining fully connected communication

links at each time step to select the optimal collaboration partners, resulting in higher

communication burden.

Table 3. Localization error index of different methods

Evaluation

indicator

SVL

SGA

RMS

DNN

-MS

CPMS

DPMS

T=20

T=40

T=60

T=80

T=100

T=120

MSE (m2)

9.16

2.18

7.21

3.71

4.20

3.36

4.34

5.50

5.53

7.79

2.27

RMSE (m)

3.02

1.47

2.68

1.92

2.04

1.83

2.08

2.34

2.35

2.79

1.50

MAE (m)

3.51

1.90

3.03

1.80

2.47

2.19

2.62

2.84

2.64

3.39

1.27

Fig. 8. The CDF curves of different methods in the DPMS evaluation

5.1.3 Communication and Computational Costs

The execution time and communication link numbers of different methods are summarized in

Table 4. The trial was done with Intel(R) Core(TM) i7-9700 CPU (8 cores @ 3.00GHz). In terms

of the execution time, the CPMS method outperformed the SGA and DNN-MS methods,

demonstrating its advantage in real-time computation. When T = 20, the DPMS method had a

lower execution time than the CPMS, but for T ≥ 40, the DPMS required more time than the

CPMS. However, the DPMS still had a significantly shorter execution time than the SGA method.

Meanwhile, the DPMS method required additional calculations to determine the size of the next

period at the end of the current period, thus having a slightly longer execution time than the CPMS

method, but could ensure high localization precision and stability. Although the RMS method had

a shorter execution time than the SGA method, its localization performance was considerably poor.

Regarding the communication cost, the CPMS and DPMS methods required half the number of

communication links compared to the SGA and DNN-MS methods. This reduction in the number

of communication links significantly reduced the communication overhead for vehicles while

maintaining loose coupling of communication and maximizing localization accuracy.

Table 4. The number of communication links and execution time of different methods

Evaluation

indicator

SGA

RMS

DNN-

MS

CPMS

DPMS

T=20

T=40

T=60

T=80

T=100

T=120

Execution

time(s)

4.49

3.08

4.11

3.19

2.96

2.92

2.87

2.86

2.87

3.02

Number of

communic

ation links

20944

10472

20944

10995

10733

10646

10602

10576

10559

10578

Remark 1: From the perspective of the convergence of the localization method, the SCCL

method proposed in this paper can effectively ensure the consistency of the localization estimation

results, and can still ensure the stability of the localization accuracy in the scene where the noise

model parameters change dynamically. This depends on three factors : 1) SCCL constructs the

communication topology model shown in Figure 3, and gives an approximate estimation of the

correlation under the two models to avoid optimistic estimation. 2) Three measurement models are

constructed to realize adaptive switching, and the weighted fusion of single vehicle localization

results and cooperative localization results is carried out to further reduce the uncertainty of

localization. 3) Through the DPMS method, the vehicle with large localization error is

dynamically excluded, so as to avoid the reduction of the localization accuracy of the vehicle due

to the fusion of his data. The DCL method in Reference [42] only considers a single

communication topology model. When encountering the scenario shown in Fig. 3 (b), DCL still

uses the solution of the scenario shown in Fig. 3 (a), resulting in redundancy of the

communication link. In addition, DCL fuses the positions with large errors, which will reduce the

localization accuracy of the vehicle, and it performs poorly in the scene where the noise model

parameters change dynamically, because DCL cannot eliminate the data with large errors and

cannot correctly handle the weights between different fusion results. From the perspective of

localization accuracy and time consumption, DPMS truly achieves the unity of high precision and

low communication burden. In addition to SGA, DPMS has obvious advantages in localization

accuracy and communication computational complexity compared with other measurement

scheduling methods. This is due to the three advantages of DPMS: 1) DPMS uses period and

localization uncertainty as the decision-making basis for measurement scheduling, so that vehicles

do not have to perform scheduling algorithms at each moment, reducing the number of

communication links and excluding vehicle data with large errors. 2) Three kinds of localization

uncertainty factors are considered, which are mathematically expressed as Formulas (17), (18) and

(21). At the end of a period, DPMS can evaluate the localization accuracy of each vehicle in the

previous period according to Formulas (17), (18) and (21), so as to accurately adjust the period. 3)

A period weight function (Eq. (28)) is constructed, which converts the three localization

uncertainty factors into the weights that adjust the periodic changes, and prevents the degradation

of periodic scheduling by normalization. The localization accuracy of SGA is comparable to that

of DPMS, but the extremely high communication and computational cost of SGA is intolerable,

which makes DPMS more practical.

5.1.4 Ablation Analysis

To evaluate the effectiveness of each module in the DPMS and their contribution to the

overall localization performance, this study conducted a series of ablation experiments with the

DPMS containing complete modules as a control group. Based on the control group method, the

following experimental groups were defined: (1) the DPMS method with a fixed value of the

weight factor for adjusting the period (DPMS-FWP); (2) the DPMS method with the equal

changing rate weights of the three uncertainty impact factors (DPMS-FWE). The results of the

ablation experiments are presented in Fig. 9, where it can be seen that both the DPMS-FWP and

the DPMS-FWE had significantly higher RMSE values than the DPMS method. Compared to the

DPMS-FWE, the DPMS-FWP had larger errors, which suggested that the contribution of the

period weight factor to the method's accuracy was slightly greater than the contribution of the

uncertainty influence factor's weight changing rate.

Fig. 9. The real-time RMSE curves obtained in the ablation test

The real-time change curves in period T without and with normalization of the changing rate

are presented in Figs. 10(a) and 10 (b), respectively. In Fig. 10(a), it can be seen that without

normalization, T stopped changing after reaching its lowest point at approximately 2,800 time

steps. In terms of scheduling strategy, the CPMS degenerated into the SGA method. Further, in Fig.

10(b), it can be observed that after normalization of the changing rate, period T maintained

dynamic adjustment throughout the entire localization process, indicating that normalizing the

changing rate could ensure the stability of the DPMS method and prevent it from degenerating

into the SGA scheduling method.

(a) Without normalization (b) With normalization

Fig. 10. Variation curves in period T

5.2 Field Test Results

To verify the advantages of the proposed method further, this study performed field tests in

the Connected AV Test Field of Chang'an University. Five vehicles were tested, where one vehicle

was designated as the main vehicle , and the others were marked as –. The main vehicle

 was equipped with a P3-DU localization and direction-finding receiver, an INS550D

high-precision vehicle-mounted navigation system, a laser radar, and DSRC communication

equipment. The other vehicles were equipped only with a P3-DU localization and

direction-finding receiver and DSRC communication equipment. The INS550D provided the true

value of a vehicle's position, and the P3-DU provided the measured value of the vehicle's position.

The self-state data of a vehicle, including the vehicle ID, data sequence number, position (i.e.,

longitude, latitude, and altitude), speed, heading, and satellite time, were broadcasted via the

DSRC. Due to limited communication and computing resources, the maximum number of vehicles

for  to cooperate with during the test period was set to

2

i

k

S

.

Two different test scenarios were designed, as shown in Fig. 11. Scene 1 represented a

segment of an ordinary road with a relatively long distance, and there were tall structures

alongside the road. Vehicles traveling on different segments of Scene 1 experienced varying

degrees of obstruction, causing fluctuations in sensor localization data. Therefore, Scene 1 was

used to evaluate the stability of different localization methods under the condition of sensor data

fluctuations. Scene 2 represented a straight road that included an artificially simulated tunnel. In

the tunnel, the GNSS signals were completely blocked, making it impossible for the P3-DU to

output location information. Scene 2 was used to evaluate the localization performance of

different methods in extreme scenarios.

Fig. 11. The overview of real vehicle test scenarios

5.2.1 Ordinary Road

The localization trajectories obtained by the SGA, RMS, CPMS, and DPMS methods are

presented in Fig. 12, where it can be seen that the RMS method had the largest deviation from the

GT trajectory. When there were fluctuations in the localization data on the neighboring vehicles,

compared to the CPMS method, the DPMS could converge to the vicinity of GT faster, and its

performance was very close to that of the SGA method. The randomness of vehicle selection at

each time step in the RMS method brought greater uncertainty to localization. Compared to the

CPMS, the DPMS could better deal with data fluctuations of the neighboring vehicles by adjusting

the size of the next period in a timely manner at the end time of each period, thus ensuring the

stability of localization.

Fig. 12. Vehicle trajectories of different scheduling methods

The real-time RMSE variations of the DPMS and the CPMS with different period sizes are

presented in Fig. 13. As shown in Figs. 13(b)–13(d), the RMSE of the CPMS showed varying

degrees of spikes, and the duration of spikes increased with the period length. In contrast, as

shown in Fig. 13(a), the RMSE of the DPMS exhibited significantly reduced spikes, which

validated the advantage of the DPMS's dynamic period adjustment strategy in reducing the

localization errors. Since the CPMS method selected optimal neighboring vehicles only based on

the current localization uncertainty at constant intervals, it was highly prone to deviations in

cooperative vehicle localization in each period, and the degree and duration of localization spikes

increased with the period length, all of which might not be perceived by the host vehicle.

(a) DPMS (b) CPMS (T = 40)

(c) CPMS (T = 80) (d) CPMS (T = 160)

Fig. 13. The RMSE index changes of different period scheduling strategies

5.2.2 Tunnel Scene

In the tunnel scene, at the initial moment, all vehicles were located approximately 200 m

from the entrance of the tunnel, and the leading vehicle was positioned at the front of the cluster,

as shown in Fig. 14. Then, all vehicles started to commute simultaneously. The main vehicle 

entered the tunnel first, and its P3-DU temporarily failed. Since the other four vehicles had not

entered the tunnel yet, vehicle  could still receive information from the other vehicles through

V2V communication to realize self-localization. Subsequently, vehicles  and  entered the

tunnel, and vehicle  was about to exit the tunnel when both  and  had entered the

tunnel.

Tunnel

Ego Vehicle

Cooperative Vehicle

Selected Vehicle

Road Section 1

0

V

1

V

2

V

3

V

4

V

Road Section 2

Road Section 3

Road

Section 4

Fig. 14. Illustration of the vehicle driving process in the tunnel scene

Two different localization methods were compared in the tunnel scene, the CPMS method

(relative measuring model) and the DPMS method (relative measuring model + self-measuring

model). The localization trajectories of vehicle  obtained by the two methods are presented in

Fig. 15. The trajectories were divided into road sections corresponding to the location of the

cluster shown in Fig. 14.

When the cluster was located on road section 1 and none of vehicles , ,

, and 

entered the tunnel,  selected  and  as cooperative vehicles using the CPMS or DPMS.

As vehicle  entered the tunnel, both methods maintained a relatively stable localization state

due to the use of the relative measuring model.

When the cluster position was on road section 2, vehicles , , and  all fully entered

the tunnel, and 



 were driving outside the tunnel. As shown in Figs. 15(a) and 15(b), the

localization trajectory obtained by the CPMS exhibited significant fluctuations, while the

localization trajectory of the DPMS remained stable. When vehicles  and  entered the

tunnel, the CPMS could not promptly respond to the increased localization error of some of the

vehicles in the cluster. Its period was constant, causing vehicle to continue to communicate

and cooperate with vehicles  and  for a certain period after they had entered the tunnel,

resulting in position result fluctuations. The DPMS adjusted its period dynamically to perceive the

information on the positional error fluctuations of some of the vehicles in the cluster fast, thus

reducing the period in a timely manner and ensuring the localization stability of vehicle .

When the cluster position was on road section 3, vehicles , , , and  all entered

the tunnel, and vehicle  was about to exit the tunnel. As all cooperative vehicles were unable to

obtain the localization data, the localization trajectory of the CPMS method showed significant

fluctuations. When vehicles  and  exited the tunnel, the localization trajectory of the

CPMS method gradually stabilized. However, the localization trajectory of the DPMS method had

smaller fluctuations. After vehicle  exited the tunnel, autonomous localization was achieved

using the self-measurement model, without relying entirely on the cooperative vehicles' data.

When vehicles  and  exited the tunnel, the localization trajectory further approached the

stability region after using the relative measurement model.

1

2

3

4

2

3

1

4

(a) CPMS (relative measurement model) (b) DPMS (relative measurement model +

self-measurement model)

Fig. 15. The localization trajectories of vehicle  obtained by the two methods in the

tunnel scene

This set of experiments further validated the superiority of the proposed mixed measurement

model and dynamic period adjustment method. When the ego vehicle relied solely on a single

relative measurement model, it became unable to achieve localization if localization data on the

surrounding vehicle was unavailable. Similarly, when the ego vehicle relied solely on a single

self-measurement model for localization, it lost the ability to obtain location information if the

GNSS signal of the ego vehicle was blocked. However, the proposed hybrid measurement model

combines the advantages of both the self-measurement model and the relative measurement model,

which ensures that the ego vehicle is not dependent on a single measurement model, enabling it to

obtain stable position information at all times. In comparison to the DPMS method, the CPMS

method exhibited inferior localization performance in scenarios where there were sudden changes

in the state of the surrounding vehicle. Throughout most time of a period, the vehicle in

i

k

S

might

not necessarily have the lowest localization uncertainty. By considering the rate of change of the

three types of state uncertainty and adjusting the period accordingly, the DPMS could effectively

handle the situations where there were sudden changes in the vehicle state data, ensuring accurate

localization.

6. Conclusion

This study addresses the challenges of cross-covariance matrix update and communication

measurement scheduling in multi-vehicle cooperative localization. Due to limited communication

and computing resources, vehicles cannot maintain fully connected communication. Therefore,

vehicles select a vehicle set from the surrounding vehicles that can contribute the most to their

localization accuracy, and the correlation of vehicle states is updated and maintained. Two relative

measurement update methods are proposed to approximate the estimated cross-covariance matrix

and maintain filter consistency. Additionally, constant and DPMS methods are employed to select

optimal cooperative vehicles based on the covariance matrix, using the dynamic period scheduling

method to enhance the localization stability further. Simulation and real-vehicle test results show

that the proposed methods can achieve similar localization accuracy as the SGA while

significantly reducing communication and computing costs, which makes them highly suitable for

cooperative localization scenarios with limited resources.

Although the covariance matrix can characterize the uncertainty of the vehicle state, this is

based on the assumption that the sensor data are contaminated only by the Gaussian noise.

However, in real road scenes, cooperative localization suffers from non-Gaussian noises. In

addition, V2V communication introduces certain communication delays, causing received data to

lag and potentially introducing localization deviations when fusing the lagging data. Moreover, the

ranging sensor can be blocked to a varying degree, resulting in non-Gaussian noise in the relative

measurement model. Thus, future work could consider the impact of the non-Gaussian noises on

the cooperative localization measurement scheduling method’s result and optimize the proposed

method accordingly.

Reference

[1] Ayoub, J., Zhou, F., Bao, S., & Yang, X. J. (2019, September). From manual driving to

automated driving: A review of 10 years of autoui. In Proceedings of the 11th international

conference on automotive user interfaces and interactive vehicular applications (pp. 70-90).

[2] Neto, J. B. P., Gomes, L. C., Ortiz, F. M., Almeida, T. T., Campista, M. E. M., Costa, L. H.

M., & Mitton, N. (2020). An accurate cooperative positioning system for vehicular safety

applications. Computers & Electrical Engineering, 83, 106591.

[3] Piperigkos, N., Lalos, A. S., & Berberidis, K. (2021). Graph laplacian diffusion localization

of connected and automated vehicles. IEEE Transactions on Intelligent Transportation

Systems, 23(8), 12176-12190.

[4] Belge, E., Altan, A., & Hacıoğlu, R. (2022). Metaheuristic optimization-based path planning

and tracking of quadcopter for payload hold-release mission. Electronics, 11(8), 1208.

[5] Altan, A., & Hacıoğlu, R. (2020). Model predictive control of three-axis gimbal system

mounted on UAV for real-time target tracking under external disturbances. Mechanical

Systems and Signal Processing, 138, 106548.

[6] Srivani, I., Prasad, G. S. V., & Ratnam, D. V. (2019). A deep learning-based approach to

forecast ionospheric delays for GPS signals. IEEE Geoscience and Remote Sensing Letters,

16(8), 1180-1184.

[7] Bao, L., Luo, H., Gao, X., Ning, B., Zhao, F., & Zhu, Y. (2022). MT-e&R: NMEA

Protocol-Assisted High-Accuracy Navigation Algorithm Based on GNSS Error Estimation

Using Multitask Learning. IEEE Transactions on Intelligent Transportation Systems, 23(11),

20464-20475.

[8] Yan, Y., Bajaj, I., Rabiee, R., & Tay, W. P. (2022). A Tightly Coupled Integration Approach

for Cooperative Positioning Enhancement in DSRC Vehicular Networks. IEEE Transactions

on Intelligent Transportation Systems, 23(12), 23278-23294.

[9] Yan, Y., Zhang, B., Zhou, J., Zhang, Y., & Liu, X. A. (2022). Real-Time Localization and

Mapping Utilizing Multi-Sensor Fusion and Visual–IMU–Wheel Odometry for Agricultural

Robots in Unstructured, Dynamic and GPS-Denied Greenhouse Environments. Agronomy,

12(8), 1740.

[10] Adegoke, E. I., Zidane, J., Kampert, E., Ford, C. R., Birrell, S. A., & Higgins, M. D. (2019).

Infrastructure Wi-Fi for connected autonomous vehicle positioning: A review of the

state-of-the-art. Vehicular Communications, 20, 100185.

[11] Hsueh, Y. L., & Chen, H. C. (2018). Map matching for low-sampling-rate GPS trajectories by

exploring real-time moving directions. Information Sciences, 433, 55-69.

[12] Durrant-Whyte, H., & Bailey, T. (2006). Simultaneous localization and mapping: part I. IEEE

robotics & automation magazine, 13(2), 99-110.

[13] Li, N., Tu, W., Ai, H., Deng, H., Tao, J., Hu, T., & Sun, X. (2022). VISEL: A visual and

magnetic fusion-based large-scale indoor localization system with improved high‐precision

semantic maps. International Journal of Intelligent Systems, 37(10), 7992-8020.

[14] Arshad, S., & Kim, G. W. (2021). Role of deep learning in loop closure detection for visual

and lidar slam: A survey. Sensors, 21(4), 1243.

[15] Lai, C., Zhang, M., Cao, J., & Zheng, D. (2019). SPIR: A secure and privacy-preserving

incentive scheme for reliable real-time map updates. IEEE Internet of Things Journal, 7(1),

416-428.

[16] Al-Sultan, S., Al-Doori, M. M., Al-Bayatti, A. H., & Zedan, H. (2014). A comprehensive

survey on vehicular ad hoc network. Journal of network and computer applications, 37,

380-392.

[17] Elazab, M., Noureldin, A., & Hassanein, H. S. (2017). Integrated cooperative localization for

vehicular networks with partial GPS access in urban canyons. Vehicular Communications, 9,

242-253.

[18] Fang, S., Li, H., & Yang, M. (2022). Lidar slam based multivehicle cooperative localization

using iterated split cif. IEEE Transactions on Intelligent Transportation Systems, 23(11),

21137-21147.

[19] Wang, X., Jiang, C., Sheng, S., & Xu, Y. (2023). Stop-Line-Aided Cooperative Positioning

for Connected Vehicles. IEEE Transactions on Intelligent Vehicles.

[20] Li, W., Jia, Y., & Du, J. (2019). Distributed Kalman filter for cooperative localization with

integrated measurements. IEEE Transactions on Aerospace and Electronic Systems, 56(4),

3302-3310.

[21] Wang, S., & Ren, W. (2017). On the convergence conditions of distributed dynamic state

estimation using sensor networks: A unified framework. IEEE Transactions on Control

Systems Technology, 26(4), 1300-1316.

[22] Roumeliotis, S. I., & Bekey, G. A. (2002). Distributed multirobot localization. IEEE

transactions on robotics and automation, 18(5), 781-795.

[23] Li, H., & Nashashibi, F. (2013). Cooperative multi-vehicle localization using split covariance

intersection filter. IEEE Intelligent transportation systems magazine, 5(2), 33-44.

[24] Kia, S. S., Rounds, S., & Martinez, S. (2016). Cooperative localization for mobile agents: A

recursive decentralized algorithm based on Kalman-filter decoupling. IEEE Control Systems

Magazine, 36(2), 86-101.

[25] Rajpoot, V., Tiwari, V., Saxena, A., Chaturvedi, P., Rajput, D. S., Alkahtani, M., & Abidi, M.

H. (2021). RSS-Based Selective Clustering Technique Using Master Node for WSN.

Computers, Materials & Continua, 69(3).

[26] Sharma, H. L., Agrawal, P., & Kshirsagar, R. V. (2014, January). Multipath reliable range

node selection distance vector routing for VANET: Design approach. In 2014 international

conference on electronic systems, signal processing and computing technologies (pp.

280-283). IEEE.

[27] Wang, T., Shen, Y., Conti, A., & Win, M. Z. (2017). Network navigation with scheduling:

Error evolution. IEEE Transactions on Information Theory, 63(11), 7509-7534.

[28] Tasooji, T. K., & Marquez, H. J. (2021). Cooperative localization in mobile robots using

event-triggered mechanism: Theory and experiments. IEEE Transactions on Automation

Science and Engineering, 19(4), 3246-3258.

[29] Zhu, J., & Kia, S. S. (2022). Learning-based measurement scheduling for loosely-coupled

cooperative localization. IEEE Robotics and Automation Letters, 7(3), 6313-6319.

[30] Chang, T. K., & Mehta, A. (2018). Optimal scheduling for resource-constrained multirobot

cooperative localization. IEEE Robotics and Automation Letters, 3(3), 1552-1559.

[31] Wang, L., Zhang, T., & Liang, R. (2016). A small-sized locating message for cooperative

localization under communication constraint. International Journal of Advanced Robotic

Systems, 13(2), 78.

[32] Mourikis, A. I., & Roumeliotis, S. I. (2006). Optimal sensor scheduling for

resource-constrained localization of mobile robot formations. IEEE Transactions on Robotics,

22(5), 917-931.

[33] Zhu, J., & Kia, S. S. (2019). Cooperative localization under limited connectivity. IEEE

Transactions on Robotics, 35(6), 1523-1530.

[34] Singh, P., Chen, M., Carlone, L., Karaman, S., Frazzoli, E., & Hsu, D. (2017, May).

Supermodular mean squared error minimization for sensor scheduling in optimal Kalman

filtering. In 2017 American Control Conference (ACC) (pp. 5787-5794). IEEE.

[35] Paull, L., Huang, G., Seto, M., & Leonard, J. J. (2015, May). Communication-constrained

multi-AUV cooperative SLAM. In 2015 IEEE international conference on robotics and

automation (ICRA) (pp. 509-516). IEEE.

[36] Trawny, N., Roumeliotis, S. I., & Giannakis, G. B. (2009, May). Cooperative multi-robot

localization under communication constraints. In 2009 IEEE international conference on

robotics and automation (pp. 4394-4400). IEEE.

[37] Ribeiro, A., Giannakis, G. B., & Roumeliotis, S. I. (2006). SOI-KF: Distributed Kalman

filtering with low-cost communications using the sign of innovations. IEEE Transactions on

signal processing, 54(12), 4782-4795.

[38] Zhang, L., Tao, X., & Liang, H. (2018, October). Multi AUVs cooperative navigation based

on information entropy. In OCEANS 2018 MTS/IEEE Charleston (pp. 1-10). IEEE.

[39] Tzoumas, V., Atanasov, N. A., Jadbabaie, A., & Pappas, G. J. (2017, May). Scheduling

nonlinear sensors for stochastic process estimation. In 2017 American Control Conference

(ACC) (pp. 580-585). IEEE.

[40] Chen, C., & Kia, S. S. (2022). Cooperative Localization Using Learning-Based Constrained

Optimization. IEEE Robotics and Automation Letters, 7(3), 7052-7058.

[41] Peng, B., Seco-Granados, G., Steinmetz, E., Fröhle, M., & Wymeersch, H. (2019).

Decentralized scheduling for cooperative localization with deep reinforcement learning.

IEEE Transactions on Vehicular Technology, 68(5), 4295-4305.

[42] Luft, L., Schubert, T., Roumeliotis, S. I., & Burgard, W. (2018). Recursive decentralized

localization for multi-robot systems with asynchronous pairwise communication. The

International Journal of Robotics Research, 37(10), 1152-1167.

[43] Zeino, E., Paulik, M., Krishnan, M., Luo, C., Overholt, J., Hudas, G., & Udvare, T. (2014,

June). Ground-truth localization using a sequential-update extended kalman filter. In IEEE

International Conference on Electro/Information Technology (pp. 103-108). IEEE.

[44] Yan, Q., Jiang, L., & Kia, S. S. (2020). Measurement scheduling for cooperative localization

in resource-constrained conditions. IEEE Robotics and Automation Letters, 5(2), 1991-1998.

[45] Li, J., Yang, G., & Cai, Q. (2022). Distributed Scheduling for Cooperative Navigation Based

on Uncertainty Evolution. IEEE Internet of Things Journal, 10(8), 7080-7089.

Declaration of interests

☒ 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.

☐ The authors declare the following financial interests/personal relationships which may

be considered as potential competing interests:

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