
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72247
10.1038/s41598-024-72247-9
Article
An identification method of LBL underwater positioning systematic error with optimal selection criterion
Xing Yao
Wang Jiongqi
He Zhangming
Chen Yuyun
Zhou Xuanying julia_chow07@163.com

https://ror.org/05d2yfz11 grid.412110.7 0000 0000 9548 2110 College of Science, National University of Defense Technology, Changsha, 410073 China
13 9 2024
13 9 2024
2024
14 2143218 4 2024
5 9 2024
© The Author(s) 2024
2024
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Unlike the space targets such as satellites whose observation distance is about the order of 100 km, the measurement distance of long baseline (LBL) underwater positioning system is much shorter. If the noncoincidence between the multiple beacons and the target center is not considered, this systematic error will cause a little larger positioning error. Therefore, aiming at the situation that multiple beacons are installed outside the target and the distance between the beacons and the target center is significant, a multi-beacon positioning model of the LBL underwater system is constructed in this paper. Furthermore, for the sound speed systematic error and the time of arrival (TOA) systematic error in the multi-beacon LBL system positioning measurement data, three identification models of systematic error are constructed. According to the positioning process of the LBL underwater system, the expression of the optimal test statistic D is derived based on the TOA residual. Finally, the optimal selection criterion is given, and the underwater target position and systematic error can be accurately estimated. Both of the numerical simulation and the underwater positioning test show that the appropriate identification model of systematic error in the measurement data can be selected according to the optimal selection criterion, and the underwater target positioning accuracy of the LBL system can be improved.

Subject terms

Applied mathematics
Statistics
National Natural Science Foundation of ChinaNo.62203458 issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

In the development and utilization of the marine resources1,2, the measurement, navigation and positioning of underwater target are the most basic and critical issues3,4. Due to the strong attenuation of the electromagnetic wave in water, the navigation and positioning for the underwater targets cannot be achieved through the global navigation satellite system (GNSS) or other radio navigation technologies. Acoustic wave has the characteristic of long-distance transmission underwater, so the acoustic measurement has become a powerful tool for underwater target positioning5–7.

According to the baseline length, underwater acoustic positioning system can be divided into long baseline (LBL) positioning system8,9, short baseline (SBL) positioning system10 and ultra-short baseline (USBL) positioning system11,12, etc. Among them, the baseline length of LBL positioning system is generally several hundred meters to some thousand meters, which can obtain higher-positioning accuracy in a large range. Therefore, LBL positioning system is widely used in high-accuracy positioning for the underwater targets13,14.

The LBL system measures the propagation time of the acoustic signal from the underwater target to the LBL system station, that is, the time of arrival (TOA). According to the propagation time of the acoustic signal and the underwater sound speed, the straight distance between the underwater target and the station can be obtained. If the station positions of the LBL system is known, the target position parameters can be solved by the optimal estimation method15,16.

There are mainly two kinds of systematic errors that affect the positioning accuracy of the LBL system, one is the structural systematic error called in this paper, the other is the measurement data systematic error. Different from space targets such as satellites with the measurement distance of more than hundreds of kilometers, the measurement distance of the LBL system is only hundreds of meters to several kilometers, while the traditional LBL underwater positioning model simplifies the target into a particle. When multiple acoustic beacons are installed outside the target and the size of the target cannot be regarded as a particle, the positions of the beacon are quite different from the center of the target. This systematic error is called the position inconsistency, which belongs to the structural systematic error. The measurement data systematic errors mainly include three kinds, namely the station position error, the sound speed error and the TOA error. The station position error is the error between the station position measurement value and the real position; the sound speed error refers to the error between the measurement value of the underwater sound speed and the true value caused by the inaccurate sound speed measurement equipment or the change of the underwater sound speed caused by the underwater environment; the TOA error is the error between the measurement value and the true value of the TOA of the acoustic signal due to the time of the station and the beacon is not strictly synchronized, or due to the signal conversion delay and other reasons.

The existence of the systematic error will seriously affect the positioning accuracy of LBL system. The main solution to deal with the systematic error is to find the expression form of the error through experiments and analysis, and then to use the parameter models to represent the systematic error, so as to eliminate or correct it17.

For the structural systematic error, the commonly method is to analyze the geometric relationship between the target center and the beacon. Sun et al.18 proposed a long baseline high-precision positioning method for underwater volume target. The method eliminated the model error and realized the approximate reduction of the geometric center of the volume target through the joint estimation of the attitude and position coordinates. For the positioning test in which the target (approximate cylinder) moves approximately vertically underwater and multiple beacons are installed at the same horizontal and equidistance outside the target, the geometric compensation is required to deal with the problem of position inconsistency in this paper, that is, according to the geometric relationship between multiple beacons and the target center, an underwater positioning model of multi-beacon LBL system is constructed. Different with the traditional positioning model, which simplifies the target into a particle, the underwater positioning model of multi-beacon LBL system considers the target’s radius and its rotation angle, so as to avoid the structural systematic error in the traditional positioning model.

For the measurement data systematic error, the commonly used error identification method is the error model best estimate of trajectory (EMBET). This method establishes the equations of the target position and the station positions according to the geometric relationship, constructs the measurement data systematic error identification model, and then uses the measurement data of the LBL system to calculate the target trajectory and the measurement data systematic error, that is, the redundant measurement data is required to estimate the target position parameters and the measurement data systematic error at the same time19,20. Further, in order to reduce the number of the parameters to be estimated, by using the kinematics characteristics of the target trajectory at multiple moments, Xu et al.21 and Liu et al.22 respectively construct the EMBET model based on spline constraints or dynamic constraints, that is, the target trajectory is parameterized, and the measurement data at multiple moments are combined to calculate the target position parameters and the systematic error model parameters.

However, whether the EMBET model or the EMBET model under spline or dynamic constraints, it is necessary to clarify the type and quantity of the systematic errors. In fact, the measurement data systematic errors of LBL positioning system are not known in advance. If they are all added to the underwater target positioning model of the multi-beacon LBL system, the model complexity will be increased, resulting in the waste of the computing resources23. Moreover, if the systematic errors with little or no significant influence are included in the positioning model, too many parameters to be estimated will also affect the positioning accuracy24,25. Therefore, how to select the optimal systematic error identification model is the premise of the effective identification for the systematic errors.

In order to solve the optimal selection problem for the systematic error identification model of multi-beacon LBL system constructed in this paper, the TOA residual is calculated, and the estimation error is taken as the objective function. The optimal selection criterion for the systematic error identification model is derived. The optimal model selection criterion selects the model with the smallest statistic as the optimal model. According to the model selection criterion, the appropriate model can be selected to achieve the effective estimation of underwater target position parameters and systematic error. The main contributions of this paper are as follows: Construct the systematic error identification model for the multi-beacon LBL positioning system: In order to eliminate the structural systematic error caused by the position inconsistency, the straight distance is established as a function of the positions of two underwater points (underwater beacon, LBL system station), the target radius and the rotation angle, and then the traditional positioning model is converted into the positioning model of multi-beacon LBL system. Further, aiming at the possible measurement data systematic errors such as sound speed error or TOA error, based on the positioning model of multi-beacon LBL system, three systematic error identification models are constructed, namely, sound speed error identification model, TOA error identification model, sound speed and TOA error identification model.

Propose the optimal selection criterion for the systematic error identification model: Since we do not know what systematic errors exist in the measurement data in the actual situation, the optimal test statistic of the identification model is constructed by analyzing the TOA residual in this paper, and the optimal selection criterion for the systematic error identification model is proposed to achieve the effective estimation for the underwater target position parameters and the reasonable identification of the systematic error. The rest structure of this paper is as follows. The principle of LBL underwater positioning model is introduced in “LBL underwater positioning model”. Aiming at the position inconsistency of the structural systematic error, the positioning model of multi-beacon LBL system is constructed in “Systematic error identification model of multi-Beacon LBL system”, and based on this three measurement data systematic error identification models are constructed. The optimal selection criterion for the systematic error identification model is given in “Optimal model selection criterion”. The optimal model selection criterion proposed in this paper is verified by the numerical simulation in “Numerical simulation” and the underwater positioning test in “Underwater positioning test”. The conclusion is provided in “Conclusion”.

LBL underwater positioning model

The coordinate system is established as Fig. 1. A point within the measurement range of the LBL system is set as the origin O0,0,0. The ox axis and the oy axis toward the east and the north respectively, and the oz axis completes the triad.Fig. 1 The LBL positioning system.

In the LBL positioning system, the station is generally fixed at the seabed or installed at the bottom of the ship. The layout of the LBL positioning system is shown in Fig. 1, there are n1 stations fixed on the seabed and n2 stations on the sea surface.

The beacon is installed on the target, and the beacon and the LBL system are time-synchronized. In the process of underwater target positioning, the beacon X emits the acoustic signal and the emission time is known, the LBL system receives the acoustic signal, so as to the TOA ti,i=1,…,n of the acoustic signal propagating from the beacon X to the station Si can be obtained. The straight distance Ri from the target to the station can be calculated by using Eq. (1).1 Ri=cti,

where c is the underwater sound speed.

The position of the sea surface station (the ship) can be obtained by GNSS, and the underwater station position can be obtained by calibration before the positioning test26. According to the geometric relationship between the target X=x,y,zT and the station Si=xi,yi,ziTi=1,⋯,n, the straight distance Ri can be obtained by using the following equation:2 Ri=x-xi2+y-yi2+z-zi2.

Combined with Eqs. (1) and (2), the underwater positioning model of the LBL system is as follows:3 cti=x-xi2+y-yi2+z-zi2.

Model (3) can be solved by Gauss–Newton method27,28, and the position of the underwater target can be obtained.

The Jacobian Matrix of model (3) is:4 J1=x-x1R1y-y1R1z-z1R1x-x2R2y-y2R2z-z2R2⋯x-xnRny-ynRnz-znRn,

where Ri is the straight distancefrom the target to the station.

Systematic error identification model of multi-beacon LBL system

Due to the inconsistency between the beacon and the target center and the short measurement distance, the traditional underwater target positioning model has a little larger positioning error. Furthermore, if the systematic errors exist in the measurement data, it can also affect the positioning accuracy of the traditional positioning model. In this section, aiming at the structural systematic error caused by the position inconsistency, the underwater positioning model of multi-beacon LBL system is first constructed. In addition, there may be systematic errors in the measurement data, such as the station position error, the sound speed error, the TOA error and so on. The station position can be corrected during the calibration of the underwater station, so the sound speed error and the TOA error are mainly considered in this paper. Assume that the underwater sound speed is the same at the same moment, and the systematic errors in the measurement data are considered as a constant value, the systematic error identification models are constructed on the basis of the underwater positioning model of the multi-beacon LBL system for the existence of only the sound speed error or the TOA error, and the simultaneous existence of the sound speed error and the TOA error.

Underwater positioning model of multi-beacon LBL system

As shown in Fig. 2, there are KK≥2 beacons, which are installed equidistant outside the target (the target is approximately a cylinder), the center of the target is at the same depth as the beacons, and the LBL system and the beacons are time-synchronized. The beacons Xj,j=1,…,K emit the acoustic signals, and the LBL system receives the acoustic signals to obtain the TOA tij of the acoustic signal propagating from the beacon Xj to the station Si,i=1,…,n. The straight distance Rij from the beacon Xj to the station Si can be calculated by using Eq. (5).5 Rij=ctij

Due to the occlusion of the target itself and the limited propagation angle of the acoustic signal emitted by the beacon, one station only outputs one receiving time of the acoustic signal at one moment during the positioning test, and then calculates the TOA of the acoustic signal propagating underwater.Fig. 2 The multi-beacon LBL positioning system.

As shown in Fig. 2, according to the geometric relationship between the beacon Xj=xj,yj,zjT and the station Si=xi,yi,ziT, the following equation can be constructed:6 Rij=xj-xi2+yj-yi2+zj-zi2

Assume that the angle between the direction from the target center X to beacon 1 and the east is the rotation angle θ, because the beacons are equidistant installed on the outside of the target, the relationship between the target center X and the beacons Xj can be expressed as:7 xj=x+rcosθ+j-12πKyj=y+rsinθ+j-12πKzj=z

where r is the radius of the cylinder.

Combining with Eqs. (6) and (7), the underwater positioning model of multi-beacon LBL system is as follows:8 ctiji=x-xi+rcosθ+ji-12πK2+y-yi+rsinθ+ji-12πK2+z-zi2

where ji is the beacon number corresponding to the ith station.

Let:9 αji=θ+ji-12πKRiji=x-xi+rcosαji2+y-yi+rsinαji2+z-zi2

The model (8) can be solved by Gauss-Newton method, and the Jacobian Matrix is:10 J2=x-x1+rcosαj1R1j1y-y1+rsinαj1R1j1z-z1R1j1y-y1rcosαj1-x-x1rsinαj1R1j1x-x2+rcosαj2R2j2y-y2+rsinαj2R2j2z-z2R2j2y-y2rcosαj2-x-x2rsinαj2R2j2⋯x-xn+rcosαjnRnjny-yn+rsinαjnRnjnz-znRnjny-ynrcosαjn-x-xnrsinαjnRnjn

Sound speed error identification model of multi-beacon LBL system

The sound speed error is the main factor that affects the underwater target positioning accuracy. Assume that in the measurement process of the LBL system, the measurement value of the sound speed is v, the TOA measurement value is τij, and the systematic error of the sound speed is Δc, regardless of the TOA systematic error.

According to Eq. (5), we can obtain:11 Rij=R^ij+ΔRij=vτij=c+Δcτij

The straight distance error caused by the sound speed error is:12 ΔRij=τijΔc

According to Eq. (12), when the sound speed error is constant, the larger the TOA measurement value, the larger the straight distance error.

Let the measurement value of the straight distance be Rij and the true value be R^ij. Eq. (13) can be obtained according to the geometric relationship between the beacon Xj and the station Si.13 Rij=xj-xi2+yj-yi2+zj-zi2+τijΔc

According to Eqs. (11) and (13), the sound speed error identification model can be written as follows:14 vτiji=τijiΔc+x-xi+rcosθ+ji-12πK2+y-yi+rsinθ+ji-12πK2+z-zi2

where τiji is the time of the acoustic signal propagating from the jith beacon to the ith station.

Let:15 A1=τ1j1τ2j2…τnjnT

The Jacobian Matrix of model (14) is:16 J3=A1J2

TOA error identification model of multi-beacon LBL system

The TOA error is another major factor that affects the positioning accuracy of LBL system. Assume that the TOA measurement value of LBL system is τij, and the systematic error of the TOA is Δt, regardless of the sound speed systematic error.

According to Eq. (5), we can obtain:17 Rij=R^ij+ΔRij=cτij=ctij+Δt

The straight distance error caused by the sound speed error is:18 ΔR=cΔt

According to the geometric relationship between the center of the target to be measured and the stations, the expression of Rij can be obtained, that is:19 Rij=x-xi2+y-yi2+z-zi2+cΔt

According to Eqs. (17) and (19), the TOA error identification model is:20 cτiji=cΔt+x-xi+rcosθ+ji-12πK2+y-yi+rsinθ+ji-12πK2+z-zi2

Let:21 A2=cc…cT

where the dimension of A2 is n×1.

The Jacobian Matrix of model (20) is:22 J4=A2J2

Sound speed and TOA error identification model of multi-beacon LBL system

Assume that the measurement value of the sound speed is v, and the measurement value of the TOA is tij. If there are both sound speed error Δc and TOA error Δt in the measurement data, according to Eq. (5):23 Rij=R^ij+ΔRij=vτij=c+Δctij+Δt

where is R^ij the true value of the straight distance, c is the true value of sound speed and tij is the true value of TOA.

According to Eq. (23), the straight distance error ΔRij is:24 ΔRij=cΔt+tijΔc+ΔtΔc=vΔt+τijΔc-ΔtΔc

The measurement value of straight distance Rij can be expressed as:25 Rij=x-xi2+y-yi2+z-zi2+vΔt+τijΔc-ΔtΔc

According to Eqs. (23) and (25), the sound speed and TOA error identification model for multi-beacon LBL system is as follows:26 vτiji=vΔt+τijiΔc-ΔtΔc+x+rcosθ+ji-12πK-xi2+y+rsinθ+ji-12πK-yi2+z-zi2

Let:27 A3=v-Δcv-Δc…v-Δcτ1j1-Δtτ2j2-Δt…τnjn-ΔtT

The Jacobian Matrix of model (26) is:28 J5=A3J2

Similar to the solution to model (8), models (14), (20) and (26) can also be solved by Gauss-Newton method. When using different positioning models to estimate unknown parameters, the number of stations should be larger than the number of parameters to be estimated. In the process of solving the underwater target position parameters, we usually do not know which measurement data systematic errors exist in the positioning process. The improper selection of the systematic error identification model will increase the complexity of the model and reduce the positioning accuracy for the underwater target. Therefore, how to select the appropriate systematic error identification model is the key problem to improve the underwater target positioning accuracy.

Optimal model selection criterion

For the processing of LBL measurement data, the positioning accuracy of different systematic error identification models will be different, to select the appropriate systematic error identification model can effectively improve the positioning accuracy of LBL system. Denote t=t1,t2,…,tnT and τ=τ1,τ2,…,τnT are the true value of the TOA at a certain moment and the TOA measurement data obtained by the LBL system, respectively. It is assumed that the measurement data model is:29 τ=t+ee∼0,σ2I,Eek1ek2=σ2

where e are random errors.

Suppose that the parameters β∈RN×1 to be estimated and the design matrix H∈Rn×N of the systematic error identification model are defined as follows:30 β=β1,β2,…,βNTH=ξ11ξ1N⋱ξn1ξnN

where ξij is the coefficient of the j-th parameter βj in the i-th equation.

Let b=b1,b2,…,bnT be the residual between t and its approximation using the best combination of the parameters β and the design matrix H.31 t=Hβ+b

According to Eq. (31), Eq. (29) can be rewritten as:32 τ=Hβ+b+e

Using the least square method, according to Eq. (32), the estimation of β can be obtained as:33 β^=HTH-1HTτ=β+HTH-1HTb+HTH-1HTe

According to Eqs. (31) and (33), Hβ^ is the estimation of t, and the estimation error is:34 EHβ^-t2=EHβ^-Hβ+b2=b-HHTH-1HTb2+EHHTH-1HTe2=b-HHTH-1HTb2+Nσ2

Let:35 HX=HHTH-1HT

By substituting HX into Eq. (34), we can obtain:36 EHβ^-t2=I-HXb2+Nσ2

According to Eq. (31):37 I-HXb2=I-HXt-Hβ2=t-Hβ+HHTH-1HTHβ-HHTH-1HTt2=I-HXt2

The mean squared error is:38 EHβ^-t2=I-HXt2+Nσ2

According to the expression of the residual square sum and combining Eqs. (36) - (38), we can obtain:39 Eτ-Hβ^2=Et+e-HHTH-1HTt+e2=I-HXt2+EI-HXe2=I-HXt2+n-Nσ2=I-HXb2+n-Nσ2

When σ2 is known,40 E2N-nσ2+τ-Hβ^2=Nσ2+I-HXb2

Combining Eqs. (38) - (40) we can obtain:41 D=τ-Hβ^2+2N-nσ2

Let:42 RSS=τ-Hβ^2

The optimal test statistic D of the systematic error identification model can be obtained:43 D=RSS+2N-nσ2

If σ2 is unknown, when the statistic Q=n-N is large, the estimation of σ2 can be given by the residual square sum of the TOA. Let the corresponding residual square sum of the q systematic error identification models are RSS1,RSS2,…,RSSq, and the corresponding statistic Q are Q1,Q2,…,Qq. Then the estimation of σ2 is:44 σ∗2=minRSS1Q1,RSS2Q2,…,RSSqQq

σ∗2 can be calculated by using Eq. (44), and the statistic D of each systematic error identification model can be calculated by substituting σ∗2 into Eq. (43).

The key to accurately calculate the underwater target position using the measurement data of LBL system is to select an appropriate systematic error identification model to minimize the residual square sum and the number of variables. When there are several systematic error identification models in the positioning system, only the statistic D needs to be compared, and the model with the smallest statistic D is the optimal systematic error identification model.

Remark: As for the optimal selection criterion of the systematic error identification model described in Eq. (43), when the parameter number of the models is the same, only the residual square sum of these models need to be calculated, and the model with the smallest residual square sum is optimal; When the parameter number of these models is different, it is necessary to calculate the statistic D for different models, and the model with the smallest statistic D is the best.

Numerical simulation

Scenario design

Simulation design: The underwater target is a cylinder with a radius of 1m, and 6 beacons are fixed at equal intervals outside the target. Assume that the beacons are always at the same depth during the movement of the target. 10 stations are arranged under the water, and one station is arranged on the water surface. The beacons and stations are strictly time-synchronized.

The simulation sets up four scenarios, and different systematic errors are added to the measurement data to verify the applicability of the optimal model selection criterion in each scenario, that is, do not add the systematic error in the measurement data, only add the sound speed systematic error, only add the TOA systematic error, and add both the sound speed and TOA systematic error. It is assumed that the systematic errors do not change with time and target position.

For the above four simulation scenarios, models (3), (8), (14), (20) and (26) are used to calculate the underwater target position, the rotation angle and the corresponding systematic error respectively. Set the true value and the initial value: The target moves approximately vertically in the water, and the target positions and rotation angle at kth (k=1,...,m) moment are Xk=xk,yk,zkT and θk respectively, as shown in Figs. 3 and  4. According to the geometric relationship between the stations and the beacons, the serial number of the beacon corresponding to the station is shown in Table 1. The true value of underwater sound speed is c=1500m/s, and the frequency of the acoustic signal emitted by beacons is 10Hz. According to Eq. (8), Fig. 4 and Table 1, the TOA t of the acoustic signal propagating from the beacons to the corresponding stations can be calculated. The initial value of the target position is X0=0,0,-10T(unit: m), the initial value of the rotation angle is θ=0∘, the initial value of the TOA systematic error is Δt0=0s, and the initial value of the sound speed systematic error is Δc0=0m/s.

Generate the measurement data: The standard deviation of the station site random error is σX=0.05m, the standard deviation of the TOA random error is σt=50us. According to the above simulation scenarios, the systematic errors are added to the true value of the sound speed and TOA respectively, and the sound speed measurement value v and the TOA measurement value τ can be obtained. The straight distance measurement value Rc can be calculated by Eq. (5).

Calculation: Let the positioning model without the identification systematic error of single beacon system be model 1 (M1), the positioning model without the identification systematic error of multi-beacon LBL system be model 2 (M2), the sound speed error identification model of multi-beacon LBL system be model 3 (M3), the TOA error identification model of multi-beacon LBL system be model 4 (M4), and the sound speed and TOA error identification model of multi-beacon LBL system be model 5 (M5). In each simulation scenario, Gauss-Newton iteration is used to solve the target position parameters according to the principle of M1 - M5: First, set the iteration accuracy εmin and the maximum iterations kmax, and then substitute the initial value into the Jacobian Matrix J, calculate the descent direction, and iterate continuously until the termination conditions satisfy. Finally, the underwater target position X^=x^,y^,z^T, the rotation angle θ^, the TOA systematic error Δt and the sound speed systematic error Δc can be obtained, and further the position error ΔX=X^-X2, the rotation angle error Δθ and the statistic D of each model can be calculated.

Repeat the calculation: Using Monte Carlo method, repeat the calculation for 100 times according to Step (2)–(3) above, record the results of each calculation.

Fig. 3 The station positions and the target trajectory. (a) The station positions and the target trajectory. (b) The target trajectory under close observation.

Fig. 4 The rotation angle. The cylinder rotates slightly during the movement.

Table 1 Corresponding relationship between the stations and beacons.

Station number	1	2	3	4	5	6	7	8	9	10	11	
Beacon number	1	2	4	2	4	3	3	5	6	4	1	

Simulation results

In simulation scenario 1, there is no systematic error in the measurement data, and the simulation results are shown in Table 2 and Figs. 5, 6. In simulation scenario 2, only the sound speed measurement data has a systematic error of -2m/s, and the simulation results are shown in Table 3 and Figs. 7, 8. In simulation scenario 3, only the TOA measurement has a systematic error of -500us, and the simulation results are shown in Table 4 and Figs. 9, 10. In simulation scenario 4, the sound speed systematic error is − 2 m/s, and the TOA systematic error is − 500 us. The simulation results are shown in Table 5 and Figs. 11, 12.

According to the target position parameters X^=x^,y^,z^T estimated by M1-M5, the positioning accuracy can be calculated by using:45 |Δx|¯=1m∑x-x^|Δy|¯=1m∑y-y^|Δz|¯=1m∑z-z^‖ΔX‖¯=1m∑x-x^2+y-y^2+z-z^2

where X=x,y,zT is the true value of the underwater target position.

Similarly, the results of |Δθ|¯, |Δc|¯, and |Δt|¯ can be obtained. According to the optimal model selection criterion, the statistic D of each model can be obtained.Table 2 Solution results of simulation scenario 1. When there is no systematic error in the measurement data, the positioning accuracy of the model without the identification systematic error of multi-beacon LBL system is the highest.

	|Δx|¯(m)	|Δy|¯(m)	|Δz|¯(m)	ΔX¯(m)	|Δθ|¯(∘)	|Δc|¯(m/s)	|Δt|¯(us)	D	
M1	0.3335	0.0475	1.8857	1.9164	–	–	–	4.8955	
M2	0.0317	0.0340	0.1433	0.1595	1.9645	–	–	0.0361	
M3	0.0354	0.0376	0.1439	0.1627	2.1728	− 0.0043	–	0.0420	
M4	0.0325	0.0364	0.1471	0.1641	2.1713	–	− 2.0019	0.0417	
M5	0.0390	0.0377	0.1526	0.1727	2.1700	0.0095	− 4.5149	0.0478	

Fig. 5 Target position error in simulation scenario 1. (a) Positioning error in x direction. (b) Positioning error in y direction. (c) Positioning error in z direction. (d) Total positioning error. When there is no systematic error in the measurement data, M2 has the highest positioning accuracy and is about 0.15 m.

Fig. 6 Comparison of statistic D and positioning error of each method in simulation scenario 1. (a) Comparison of statiatic D. (b) Comparison of positioning error. The positioning error of M2 is the smallest, and the statistic D of M2 is also the smallest.

Table 3 Solution results of simulation scenario 2. When there is only the sound speed systematic error, the positioning accuracy of the sound speed error identification model of multi-beacon LBL system is the highest.

	|Δx|¯(m)	|Δy|¯(m)	|Δz|¯(m)	ΔX¯(m)	|Δθ|¯(∘)	|Δc|¯(m/s)	|Δt|¯(us)	D	
M1	0.9241	0.504	2.2511	2.4864	–	–	–	16.5864	
M2	0.6258	0.5525	0.5446	1.0092	4.5145	–	–	7.3320	
M3	0.0354	0.0376	0.1439	0.1627	2.1728	− 2.0043	–	0.0415	
M4	0.3860	0.1728	0.5861	0.7358	2.5258	–	− 531.4097	1.4136	
M5	0.0390	0.0377	0.1526	0.1727	2.1700	− 1.9905	− 4.5149	0.0483	

Fig. 7 Target position error in simulation scenario 2. (a) Positioning error in x direction. (b) Positioning error in y direction. (c) Positioning error in z direction. (d) Total positioning error. When there is only the sound speed systematic error, M3 has the highest positioning accuracy and is about 0.16 m.

Fig. 8 Comparison of statistic D and positioning error of each method in simulation scenario 2. (a) Comparison of statiatic D. (b) Comparison of positioning error. The positioning error of M3 is the smallest, and the statistic D of M3 is also the smallest.

Table 4 Solution results of simulation scenario 3. When there is only the TOA systematic error, the positioning accuracy of the TOA error identification model of multi-beacon LBL system is the highest.

	|Δx|¯(m)	|Δy|¯(m)	|Δz|¯(m)	ΔX¯(m)	|Δθ|¯(∘)	|Δc|¯(m/s)	|Δt|¯(us)	D	
M1	0.5299	0.3081	2.7719	2.8396	–	–	–	12.3038	
M2	0.2277	0.3546	1.0504	1.1347	3.3216	–	–	5.3507	
M3	0.2537	0.0758	0.6537	0.7136	2.9946	− 1.5389	–	1.0674	
M4	0.0325	0.0364	0.1471	0.1641	2.1713	–	− 502.0019	0.0412	
M5	0.0390	0.0377	0.1526	0.1727	2.1700	0.0095	− 504.5149	0.0483	

Fig. 9 Target position error in simulation scenario 3. (a) Positioning error in x direction. (b) Positioning error in y direction. (c) Positioning error in z direction. (d) Total positioning error. When there is only the TOA systematic error, M4 has the highest positioning accuracy and is about 0.16 m.

Fig. 10 Comparison of statistic D and positioning error of each method in simulation scenario 3. (a) Comparison of statiatic D. (b) Comparison of positioning error. The positioning error of M4 is the smallest, and the statistic D of M4 is also the smallest.

Table 5 Solution results of simulation scenario 4. When there are systematic errors of sound speed and TOA, the positioning accuracy of the TOA error identification model of multi-beacon LBL system is the highest.

	|Δx|¯(m)	|Δy|¯(m)	|Δz|¯(m)	ΔX¯(m)	|Δθ|¯(∘)	|Δc|¯(m/s)	|Δt|¯(us)	D	
M1	1.1105	0.8426	3.0399	3.3453	–	–	–	35.3535	
M2	0.8336	0.8990	1.4729	1.9221	5.8700	–	–	23.8463	
M3	0.2537	0.0758	0.6537	0.7136	2.9946	− 3.5389	–	1.0666	
M4	0.3860	0.1728	0.5861	0.7358	2.5257	–	− 1031.4097	1.4128	
M5	0.0390	0.0377	0.1526	0.1727	2.1700	− 1.9905	− 504.5149	0.0490	

Fig. 11 Target position error in simulation scenario 4. (a) Positioning error in x direction. (b) Positioning error in y direction. (c) Positioning error in z direction. (d) Total positioning error. When there are systematic errors of sound speed and TOA, M5 has the highest positioning accuracy and is about 0.16 m.

Fig. 12 Comparison of statistic D and positioning error of each method in simulation scenario 4. (a) Comparison of statiatic D. (b) Comparison of positioning error. The positioning error of M5 is the smallest, and the statistic D of M5 is also the smallest.

According to the simulation results of the above scenarios, the following conclusions can be drawn: In each simulation scenario, the underwater target position parameter accuracy of the positioning model without the identification systematic error of single beacon system is the worst, and the statistic D of this model is also the largest of all models.

When there is no systematic error in the measurement data, the accuracy of the target position parameters and the rotation angle of the positioning model without the identification systematic error of multi-beacon LBL system is the highest, and the statistic D of this model is also the smallest.

When there is only one systematic error in the measurement data, the positioning accuracy of the corresponding error identification model of multi-beacon LBL system is the highest, and the average positioning error is about 0.16 m. The model can also estimate the systematic error most accurately, and the statistic D of this model is also the smallest. If the systematic error identification model is selected improperly, the position accuracy will be reduced, and the non-existent systematic error will be calculated, and the statistic D of this model will also increase. For example, if there is only the sound speed systematic error in the measurement data and the TOA error identification model of multi-beacon LBL system is selected, the average position solution error is 0.74 m, and the TOA error is calculated at the same time, but the TOA error is not added to this simulation scenario; If the sound speed and TOA error identification model of multi-beacon LBL system is selected, the target position solution error is 0.17 m, which increases by 6.15% relative to the sound speed error identification model of multi-beacon LBL system, and the TOA error is also calculated.

When there are both the sound speed systematic error and the TOA systematic error in the measurement data, the solution error of the sound speed and TOA error identification model of multi-beacon LBL system is the smallest, the average position solution error is about 0.17 m, the deviation between the calculated sound speed error and the real sound speed error is 0.48%, the deviation between the calculated TOA error and the real TOA error is 0.90%, and the statistic D is also the smallest. If the model that only identifies one systematic error is selected, the two systematic errors existing in the measurement data will be calculated into one systematic error. The average deviation between the calculated position and the true value is more than 0.7 m. The calculated systematic error has a large deviation from the true value, and the statistic D of the model is also large.

From the results of Figs. 6, 8, 10 and 12, it can be seen that the statistic D of the positioning model is positively correlated with the solution error of the target position. The smaller the statistic D is, the smaller the solution error of target position is, which reflects that the degree of the systematic error identification model fitting the real model is better, and the higher the positioning accuracy is. According to the above analysis of the results: When the random error is reasonably small relative to the systematic error and several systematic error identification models can be selected in the process of underwater target positioning, the optimal systematic error identification model can be determined by calculating the statistic D of each model according to the optimal model selection criterion. In addition, we also tested the effectiveness of the optimal model selection criteria at different station numbers (from 4 to 10) and beacon numbers (from 4 to 8). All simulation results show that when the statistic D of the positioning model is the smallest, the target position estimation error of the model is the smallest, which directly verifies the effectiveness of the optimal model selection criterion.

Underwater positioning test

Test scenario

In order to further verify the effectiveness of the optimal model selection criterion, we had a long baseline system underwater positioning experiment in a certain sea region. The range of the long baseline system is 1500 × 2000 m, with 7 measurement stations deployed underwater and 1 measurement station installed on a small boat on the surface, as shown in Fig. 13. A large ship on the surface uses a crane to control the cylinder.Fig. 13 The target trajectory and the station positions. (a) The target trajectory. (b) The station positions.

During the underwater positioning test, the two ships are 100 m apart and 6 beacons are fixed at equal intervals on the outerside of the cylinder, as shown in Fig. 14.Fig. 14 The cylinder and beacons. (a) Schematic diagram of the cylinder. (b) Picture of the cylinder. The radius of the cylinder is 1 m, and 6 beacons are installed on the outside of the cylinder.

At the beginning of the test, the large ship uses a crane to place the cylinder into the sea region, as shown in Fig. 15. The cylinder remains stationary when it reaches a depth of about 10 m, and the beacons fixed on the outside of the cylinder begin to transmit the acoustic signals. The cylinder is pulled to the water surface at a constant speed of about 0.15 m/s by the crane, and the rise time lasted for about 1 min. During this period, both the large and small boats drift and move in the horizontal direction on the sea surface.Fig. 15 The lifted cylinder.

Due to the extremely fast attenuation of electromagnetic wave underwater, the GPS in this underwater test located directly above the place where the cylinder into the water. During the process of pulling the cylinder to the water surface, the GPS can provide a reference for the horizontal direction movement trend of the underwater cylinder. The SVP installed on the cylinder can provide depth data, which can be used as the depth reference. In the underwater LBL system positioning test, a measurement station can only receive the acoustic signals from one beacon. The corresponding relationship between the stations and the beacons is shown in Table 6.Table 6 Corresponding relationship between the stations and beacons.

Station number	1	2	3	4	5	6	7	8	
Beacon number	6	6	5	1	2	6	6	6	

The measurement data of the LBL system is shown in Fig. 16.Fig. 16 The straight distance measurement data of LBL system.

Test results

The positioning models and solution methods are the same as the “Numerical simulation”, using a total of 5 models from M1 to M5 to calculate the target position parameters, the rotation angle, the sound speed error, the TOA error, and the statistic D.

The results of the target position parameters and the rotation angle are shown in Fig. 17. The results of the statistic D are shown in Fig. 18.Fig. 17 The results of position and rotation angle. (a) Positioning error in x direction. (b) Positioning error in y direction. (c) Positioning error in z direction. (d) The rotation angle θ.

Fig. 18 Comparison of statistic D. The statistic D of M5 is the smallest.

According to the target position parameters, the positioning accuracy σ can be calculated by using GDOP and σR.46 σ=GDOP∗σR

where GDOP=diagJTJ-1, and σR2=R-Rc2/n.Table 7 Solution results of underwater positioning test.

	σx(m)	σy(m)	σz(m)	σX(m)	θ¯(∘)	|Δc|¯(m/s)	|Δt|¯(us)	D	
M1	0.2451	0.4663	2.3900	2.4482	–	–	–	2.8760	
M2	0.2074	0.3942	2.0173	2.0666	51.8458	–	–	2.0813	
M3	0.1493	0.2884	1.6864	1.7178	45.7196	− 0.7036	–	1.2339	
M4	0.1802	0.3432	1.7882	1.8302	49.3688	–	− 137.2043	1.6708	
M5	0.0485	0.0957	0.7439	0.7517	42.9791	− 2.9710	801.8304	0.3810	

According to the results of the underwater LBL system positioning test, the following conclusions can be drawn: The depth data provided by SVP is accurate and can therefore be used as the reference values. It can be seen that the positioning result in z direction of the M5 is closest to the depth reference values. The positioning accuracy of M5 is 0.7517 m, which is the highest, and the statistic D of this model is also the smallest of all models, as shown in Fig. 18.

Due to the factors such as the water flow, the horizontal position reference provided by GPS deviates from the target is about 1m, which is consistent with the results in Fig. 17a,b. The movement trend in the horizontal positioning results of each model is the same as that of GPS.

As shown in Table 7, the smaller the statistic D is, the smaller the positioning error and the higher the underwater positioning accuracy will be. Due to the depth of the sea region being only a few tens of meters and the range of the LBL system being at the kilometer level, the geometry of the station layout is quite poor, so the target positioning error is mainly concentrated in the z direction.

According to the above analysis of the results: When there are unknown systematic errors in the measurement data in the process of LBL system underwater target positioning test, multiple system error identification models can be constructed. Based on the optimal model selection criterion and the statistic D of the systematic error identification model, the most suitable solution model can be selected to improve the estimation accuracy of the target position parameters.

In underwater positioning tests, the drastic underwater environment changes will result in very poor quality of measurement data. Due to the influence of underwater environment, the underwater sound speed varies, and the measurement error of acoustic signal reception time is complex, and the systematic error of each measurement station is different. The measurement systematic error identification models constructed in this paper are aimed at the situation where the systematic error of each measurement station is the same. Therefore, it is necessary to study the systematic error identification method in complex underwater environment in the future.

Conclusion

The systematic errors such as the position inconsistency, the sound speed error and the TOA error in the LBL underwater positioning system will affect the underwater target positioning accuracy. Therefore, multiple systematic error identification models of multi-beacon LBL system are constructed in this paper to improve the underwater target positioning accuracy. Aiming at the position inconsistency of the structural systematic error, an underwater positioning model of the multi-beacon LBL system is constructed. For the systematic error of the measurement data, the systematic error identification models are constructed based on the positioning model of multi-beacon LBL system. In the positioning process, we do not know which systematic errors exist in the measurement data, and the improper systematic error identification model will also affect the positioning accuracy. Therefore, by analyzing the TOA residual, the optimal test statistic D for the systematic error identification model is derived in this paper, and the optimal model selection criterion is given. Finally, both of the numerical simulation and the underwater positioning test are designed to verify the criterion. In each simulation scenario, different systematic errors are added to the measurement data, and the target position parameters are solved by using the positioning models in this paper, and the model with the smallest statistic has the highest positioning accuracy. In LBL system underwater positioning test, the solution result of the systematic error identification model with the smallest statistic D is closest to the depth reference values provided by SVP in z direction and has the highest estimation accuracy. Both of the simulation and the test results show that when the statistic D of the positioning model is the smallest, the target position solution accuracy of the model is the highest, which directly verifies the effectiveness of the optimal model selection criterion. It should be noted that the separability of systematic errors is related to the random errors of measurement data. When the measurement environment or measurement equipment is terrible, resulting in large random errors, the optimal model selection criterion may be invalid. Therefore, in order to ensure the effectiveness of the optimal model selection criterion in practical applications, the underwater tests need to be carried out under good sea conditions and adopt high-precision measurement equipment. In summary, the systematic error identification models and the optimal model selection criterion proposed in this paper can provide the theoretical and technical support for the high-accuracy underwater target measurement, navigation and positioning.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.62203458).

Author contributions

Y.X.: original draft, review, editing, visualization. J.W.: review, editing, conceptualization, formal analysis, funding acquisition. Z.H.: methodology, review, editing, funding acquisition. Y.C.: review, editing, funding acquisition. X.Z.: review, editing, formal analysis, funding acquisition. All authors reviewed the manuscript. All authors agree to publication.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Hou M Wu H Peng J Li K Long-range and high-precision localization method for underwater bionic positioning system based on joint active-passive electrolocation Sci. Rep. 2023 13 21475 10.1038/s41598-023-48957-x 38052848
Hou, M., Wu, H., Peng, J. & Li, K. Long-range and high-precision localization method for underwater bionic positioning system based on joint active-passive electrolocation. Sci. Rep. 13, 21475 (2023).38052848 10.1038/s41598-023-48957-x
2. Prateek Reddy TS Chandra S Arya R Verma AK Malicious anchor node extraction using geodesic search for survivable underwater wireless sensor network Sci. Rep. 2022 12 13691 10.1038/s41598-022-17956-9 35953697
Prateek Reddy, T. S., Chandra, S., Arya, R. & Verma, A. K. Malicious anchor node extraction using geodesic search for survivable underwater wireless sensor network. Sci. Rep. 12, 13691 (2022).35953697 10.1038/s41598-022-17956-9
3. Zhang T Tang J Qin S Wang X Review of navigation and positioning of deep-sea manned submersibles J. Navig. 2019 72 1021 1034 10.1017/S0373463319000080
Zhang, T., Tang, J., Qin, S. & Wang, X. Review of navigation and positioning of deep-sea manned submersibles. J. Navig. 72, 1021–1034 (2019).10.1017/S0373463319000080
4. Xing Y TOA positioning algorithm of LBL system for underwater target based on PSO J. Syst. Eng. Electron. 2023 34 1319 1332 10.23919/JSEE.2023.000107
Xing, Y. et al. TOA positioning algorithm of LBL system for underwater target based on PSO. J. Syst. Eng. Electron. 34, 1319–1332 (2023).10.23919/JSEE.2023.000107
5. Li T Zhao J Ma J A precise underwater positioning method by considering the location difference of transmitting and receiving sound waves Ocean Eng. 2022 247 110480 10.1016/j.oceaneng.2021.110480
Li, T., Zhao, J. & Ma, J. A precise underwater positioning method by considering the location difference of transmitting and receiving sound waves. Ocean Eng. 247, 110480 (2022).10.1016/j.oceaneng.2021.110480
6. Sun D Zheng C Zhang J Han Y Cui H Development and prospect for underwater acoustic positioning and navigation technology Bull. Chin. Acad. Sci. (Chin. Version) 2019 34 331 338
Sun, D., Zheng, C., Zhang, J., Han, Y. & Cui, H. Development and prospect for underwater acoustic positioning and navigation technology. Bull. Chin. Acad. Sci. (Chin. Version) 34, 331–338 (2019).
7. Wang Y Zhang X Sun S Wang J Underwater navigation using a single beacon based on the time delays of the direct signals and the surface-reflected signals Appl. Acoust. 2022 187 108503 10.1016/j.apacoust.2021.108503
Wang, Y., Zhang, X., Sun, S. & Wang, J. Underwater navigation using a single beacon based on the time delays of the direct signals and the surface-reflected signals. Appl. Acoust. 187, 108503 (2022).10.1016/j.apacoust.2021.108503
8. Qin X Yang Y Sun B The refined resilient model for underwater acoustic positioning Ocean Eng. 2022 266 112795 10.1016/j.oceaneng.2022.112795
Qin, X., Yang, Y. & Sun, B. The refined resilient model for underwater acoustic positioning. Ocean Eng. 266, 112795 (2022).10.1016/j.oceaneng.2022.112795
9. Yang H Gao X Huang H Li B Xiao B An LBL positioning algorithm based on an EMD-ML hybrid method Eurasip J. Adv. Signal Process. 2022 2022 1 20 10.1186/s13634-022-00869-0
Yang, H., Gao, X., Huang, H., Li, B. & Xiao, B. An LBL positioning algorithm based on an EMD-ML hybrid method. Eurasip J. Adv. Signal Process. 2022, 1–20 (2022).10.1186/s13634-022-00869-0
10. Xin M A TOA/AOA underwater acoustic positioning system based on the equivalent sound speed J. Navig. 2018 71 1431 1440 10.1017/S037346331800036X
Xin, M. et al. A TOA/AOA underwater acoustic positioning system based on the equivalent sound speed. J. Navig. 71, 1431–1440 (2018).10.1017/S037346331800036X
11. Wang Y Hu R Chen Y Huang S Adaptive noise cancelling for an AUV-mounted passive inverted USBL array Ocean Eng. 2023 288 115998 10.1016/j.oceaneng.2023.115998
Wang, Y., Hu, R., Chen, Y. & Huang, S. Adaptive noise cancelling for an AUV-mounted passive inverted USBL array. Ocean Eng. 288, 115998 (2023).10.1016/j.oceaneng.2023.115998
12. Fan S Liu C Li B Xu Y Xu W AUV docking based on USBL navigation and vision guidance J. Mar. Sci. Technol. 2019 24 673 685 10.1007/s00773-018-0577-8
Fan, S., Liu, C., Li, B., Xu, Y. & Xu, W. AUV docking based on USBL navigation and vision guidance. J. Mar. Sci. Technol. 24, 673–685 (2019).10.1007/s00773-018-0577-8
13. Huang J Yan S An improvement of long baseline system using particle swarm optimization to optimize effective sound speed Mar. Geodesy 2018 41 439 456 10.1080/01490419.2018.1487352
Huang, J. & Yan, S. An improvement of long baseline system using particle swarm optimization to optimize effective sound speed. Mar. Geodesy 41, 439–456 (2018).10.1080/01490419.2018.1487352
14. Yan W Chen W Cui R Moving long baseline positioning algorithm with uncertain sound speed J. Mech. Sci. Technol. 2015 29 3995 4002 10.1007/s12206-015-0845-z
Yan, W., Chen, W. & Cui, R. Moving long baseline positioning algorithm with uncertain sound speed. J. Mech. Sci. Technol. 29, 3995–4002 (2015).10.1007/s12206-015-0845-z
15. J.Aragon, F. & A.Goberna, M. Nonlinear Optimization (Springer, 2019).
16. Wang, Z., Yi, D., Duan, X., Yao, J. & Gu, D. Theory and Method of Fusion Processing for Shooting Range Measurement Data (CRC Press, 2016).
17. He, Z., Zhou, X. & Wang, J. Data Modeling and Analysis (Science Press, 2021).
18. Sun D Li Z Zheng C A high-precision long-baseline positioning method for underwater volume target J. Electron. Inf. Technol. 2023 42 592 599
Sun, D., Li, Z. & Zheng, C. A high-precision long-baseline positioning method for underwater volume target. J. Electron. Inf. Technol. 42, 592–599 (2023).
19. Wang Z Estimation of constant systematic errors of continuous wave radar system Chin. Space Sci. Technol. 1996 16 11 19
Wang, Z. Estimation of constant systematic errors of continuous wave radar system. Chin. Space Sci. Technol. 16, 11–19 (1996).
20. Qian, K. & Wan, Y. Simulation analysis of multi-source measurement data fusion based on EMBET. In 2021 IEEE 4th International Conference on Electronics Technology (ICET) (2021).
21. Xu, H., Wang, Z., Ma, X., Cao, W. & Li, Y. Application of embet method with spline constraint in systematic error self-calibration of pulse radar. In 2016 First IEEE International Conference on Computer Communication and the Internet (2016).
22. Liu, L., Wu, B. & Yang, P. Orbit Precision Determination & Self-calibration Technique of Spacecraft (National Defense Industry Press, 2005).
23. Kashani M Arashi M Rabiei M Urso PD Giovanni LD A fuzzy penalized regression model with variable selection Expert Syst. Appl. Int. J. 2021 175 114696 10.1016/j.eswa.2021.114696
Kashani, M., Arashi, M., Rabiei, M., Urso, P. D. & Giovanni, L. D. A fuzzy penalized regression model with variable selection. Expert Syst. Appl. Int. J. 175, 114696 (2021).10.1016/j.eswa.2021.114696
24. Miller & Alan, J. Subset Selection in Regression (CRC Press, 2021).
25. Osborne N Peterson CB Vannucci M Latent network estimation and variable selection for compositional data via variational em J. Comput. Graph. Stat. 2022 31 163 175 10.1080/10618600.2021.1935971 36776345
Osborne, N., Peterson, C. B. & Vannucci, M. Latent network estimation and variable selection for compositional data via variational em. J. Comput. Graph. Stat. 31, 163–175 (2022).36776345 10.1080/10618600.2021.1935971
26. Han Y Zheng C Sun D A high precision calibration method for long baseline acoustic positioning systems Chin. J. Acoust. 2017 41 489 500
Han, Y., Zheng, C. & Sun, D. A high precision calibration method for long baseline acoustic positioning systems. Chin. J. Acoust. 41, 489–500 (2017).
27. Zhou, H., Wang, J., Meng, Q., Zhou, X. & He, Z. Theory and Method of Fusion Processing for Shooting Range Measurement Data (Science Press, 2019).
28. Hendrix, E. Introduction to Nonlinear and Global Optimization (Springer, 2010).
