Optimal sensor configuration for positioning seafloor geodetic node

Ocean Engineering 142 (2017) 1–9

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Optimal sensor configuration for positioning seafloor geodetic node a,b

Yajing Zou

a,c,⁎

, Chisheng Wang

a

MARK

a

, Jiasong Zhu , Qingquan Li

a

Shenzhen Key Laboratory of Spatial Smart Sensing and Services and the Key Laboratory for Geo-Environment Monitoring of Coastal Zone of the National Administration of Surveying, Mapping and GeoInformation, Shenzhen University, Shenzhen, Guangdong, China b Institution of Oceanographic, School of Geodesy and Geomatics, Wuhan University, Wuhan, Hubei, China c Key Laboratory for National Geographic Census and MonitoringNational Administration of Surveying, Mapping and Geoinformation, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Seafloor observatory Acoustic sensor Fisher information matrix Optimal configuration Underwater positioning

This paper deals with the optimal acoustic sensor configuration for positioning seafloor geodetic node. We use the determinant of Fisher Information Matrix (FIM) to measure the performance of different sensor configuration. This criteria is calculated in both 2D and 3D scenarios on a more practical assumption that range observations have different weights depending on their value, instead of identical weights. Then the uncertainty of initial node position is also taken into consideration for the calculation of determinant of FIM which was often neglected in the optimizing process. We present a kind of optimal configurations based on the maximum of determinant (FIM), which is circular and has different optimal radius at each corresponding scenario. Simulative experiments are carried out to demonstrate the optimal configuration. Practical experiments in South China Sea can further prove the relationship between positioning accuracy, configuration and determinant of FIM.

1. Introduction Last two decades have witnessed a rapid development in seafloor observatory science. Independent seafloor networks are built and maintained by many country in the world, such as Regional Scale Nodes (RSN) by USA, Dense Oceanfloor Network system for Earthquakes and Tsunamis (DONET) by Japan, North East Pacific Time-series Seafloor Networked Experiments (NEPTUNE) and Victoria Experimental Network Under the Sea (VENUS) by Canada, and European Seas Observatory Network (ESONET) by EU (Favali et al., 2010), as an important mean to observe and monitor the ocean environment. Seafloor geodetic node is a fundamental component of these networks, and acts as a reference for seafloor motion and scanning. Therefore, a main effort is to determine the node position, by integrated methods using Global Navigation Satellite System (GNSS) and acoustic sensors, such as Ultra Short Baseline (USBL), Short Baseline (SBL) and Long Baseline (LBL) (Leonard and Bahr, 2016; Seto et al., 2013). In this paper, we look for a group of optimal acoustic sensor configurations to ensure the highest positioning accuracy of seafloor geodetic node. There are two popular indexes in navigation and positioning system to measure the configuration, Geometric Dilution

of Precision (GDOP) and the determinant of FIM (Martínez and Bullo, 2006). GDOP is square root of the trace of Cramer Rao Lower Bound (CRLB), which is the inverse of FIM (Sharp et al., 2009). Moreover, the two indexes are equivalence to some extent that the minimum GDOP and the maximum determinant (FIM) both represent the highest positioning accuracy. The determinant of FIM is selected in this paper because it is easier to calculate without inversing matrix. Nowadays, there are many works on optimizing sensor configuration. Zhang (1992) discussed the relationship between sensor configuration and positioning accuracy in 2D scenario, and gave several conditions with fewer than three sensors. In his later work, the conditions were developed as to satisfy multiple-sensor-configuration (Zhang, 1995). Bishop et al. (2007) gave the conditions using Time of Arrival (TOA) observations in 2D scenario, and showed several detailed configurations composed of three and four sensors respectively. Zhao et al. (2013) expanded this problem to both 2D and 3D scenarios using both bearing and ranging observations, and gave examples for 2D and 3D scenarios using regular polygons and Platonic solids respectively. Methods are also developed for solving dynamic configurations based on genetic algorithm or particle filter, but they more focused on dynamic optimizing algorithm than detailed configuration (Majid and Joelianto, 2012; Ding et al., 2014). There are infinite solutions for this optimizing problem when

⁎ Corresponding author at: Shenzhen Key Laboratory of Spatial Smart Sensing and Services and the Key Laboratory for Geo-Environment Monitoring of Coastal Zone of the National Administration of Surveying, Mapping and GeoInformation, Shenzhen University, Shenzhen, Guangdong, China. E-mail address: [email protected] (C. Wang).

http://dx.doi.org/10.1016/j.oceaneng.2017.06.033 Received 28 October 2016; Received in revised form 6 June 2017; Accepted 17 June 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.

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sensors are more than four, considering rotation and combination of configurations (Xue and Yang, 2017). Researchers tended to give limiting conditions for the problem, instead of providing detailed examples of optimal configurations, unless sensors are equal to or less than four. Moreover, most of the existing works considered that the observations have identical weights, which is not practical in seafloor positioning (Martínez and Bullo, 2006). Furthermore, the optimal configuration is obtained based on an initial node position with uncertainty, so the Probability Density Function (PDF) of the seafloor node should be taken into consideration together. We want to set different weights for observations and make use of the initial PDF in this paper, as to present detailed optimal configuration in more general scenario. Our work are also inspired by researches about optimal GNSS satellites configuration. GDOP and determinant of FIM were used to select satellites for fast positioning (Sharp et al., 2009; Wang, 2005). Optimizing problem was also discussed for multi-GNSS constellations (Han et al., 2013; Teng and Wang, 2014; Teng et al., 2015; Xue et al., 2015). Particularly, a systematical contribution was made by Xue et al. (2015) to solve the optimizing problem analytically, and three kinds of basic configurations were presented in 3D scenario, cone configuration, Descartes configuration, and Walker configuration. Three kinds of solutions for minimizing GDOP depend on the number of sensors, infinite solutions, finite solutions and no solution (Xue and Yang, 2017). But there are some features should be noticed in this project other than the characteristics of GNSS application: a) unknown parameters do not include receiver clock offset; b) the measurement error is assumed as a white Gaussian noise with variance dependent on range; c) acoustic sensors are placed on a horizontal plane, and approximately z = 0. The remainder of this paper is organized as follows. Section 2 analyses the positioning method for seafloor geodetic node, and calculates the value of determinant (FIM) and GDOP. Section 3 provides optimal sensor configuration for 2D and 3D scenarios by maximizing determinant (FIM), and the optimal sensor configuration by introducing the PDF of seafloor node. Some simulative experiments and practical experiments are carried out in Section 4. Finally, conclusions are drawn in Section 5.

Fig. 1. Positioning seafloor geodetic node using GNSS and acoustic sensors.

provide time delay error for the calculation of accurate range, therefore it is not necessary to get Time Difference of Arrival (TDOA) by subtracting two TOAs. 2.2. Determinant of FIM and GDOP Suppose that n sensors are put on sea surface as surface nodes. For the convenience of discussion, we assume that sea surface is quiet, and let pi = (xpi, ypi, 0) denotes the position of i-th surface node. Let q = (xq, yq, zq) denotes the position of seafloor node. Let ri denotes the observed range between i-th surface node and seafloor node, and rˆi denotes the true value of this range. The functional model and random model for the positioning process are expressed respectively as

ri = pi − q

2

+ wi , E (wi ) = 0

(1a)

rˆi ∼ N (ri, σi2 ), σi = a + bri

(1b)

where a and b are the prior information to evaluate the relationship between ri and σi, and can be provided by the performance of LBL sensor in certain underwater environment. Assume that range ri is not correlated with another, and variance σi2 is dependent on ri. Eq. (1) can be linearized as

2. Integrated positioning method using GNSS and acoustic sensors 2.1. Analysis of underwater positioning sensors

ri − pi − q GNSS and acoustic sensors are installed in surface buoys or surveying ships. If the coordinates of GNSS and acoustic sensors in vehicle coordinate system are measured, and the attitudes of the buoys or ships are provided by Gyro, the positions of acoustic sensors can be obtained by some computations based on the GNSS coordinates. The most common acoustic sensors are USBL, SBL and LBL, distinguished according to their baseline, as shown in Table 1(Leonard and Bahr, 2016; Tan et al., 2011). In the case of positioning seafloor geodetic node, LBL sensor is chosen to be installed in the node for its good performance in deep sea environment. The position of underwater LBL sensor is calculated as shown in Fig. 1. Range measurement is calculated by the product of mean sound speed and Time of Arrival (TOA). Many LBL sensors

2

= ei δq + wi

(2)

where δq is the difference between initial position and estimated position of seafloor node, and ei is the line-of-sight vector.

ei =

∂ pi − q

2

∂q

= ( xpi − xq ypi − yq zpi − z q )/ pi − q

2

(3)

The estimated q can be obtained by following iterations until δq reaches a lower bound.

δq = arg min(V T PV )

(4a) (4b)

q = q + δq

T P = diag (1/(a + br1)2 1/(a + br 2)2 ⋯ 1/(a + brn )2 ) .

and where Solving the equation system (2) based on the least square criterion, we can get the expression of FIM and GOOP (Chaffee and Abel, 1994). V = ( w1 w2 ⋯ wn )T

Table 1 Acoustic Sensors.

n

F= Acoustic sensor

Length of Baseline

Measurement

USBL SBL LBL

< 10 cm 20–50 m 50–6000 m

Range and Bearing Range Range

∑ ei eiT /σi2

(5a)

i =1

GDOP =

2

tr (F −1) =

⎡ n ⎤ tr ⎢ ( ∑ ei eiT / σi2 )−1⎥ ⎢⎣ i =1 ⎥⎦

(5b)

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a

c

b

Fig. 2. Configurations based on the first kind of solution, where black triangle represents surface sensor and black rectangle represents seafloor node. (a) 2 sensors. (b) 3 sensors. (c) 4 sensors.

We are looking for a group of sensors to make the determinant of FIM maximal.

{ p1 p2 ⋯ pn } = arg max{det F}

(6)

⎡ n ∂ det(F ) 1 ∂(1/ σui2 ) ⎢ ∑ 1 − cos(2αi ) = ∂ri 2 ∂ri ⎢⎣ i =1 σui2

∑

− sin(2αi )

∑ j =1

n

tan(2αi ) =

∑

sin(2αj ) σuj2

j =1

If the depth of seafloor geodetic node is provided by bathometer, we only need to obtain the plane coordinates of the node and the optimizing case can be constrained on 2-D plane. Otherwise, it should be discussed in 3-D scenario.

n

∑ i =1

Several notations are updated in 2-D case, such as pi = (xpi, ypi), q = (xq, yq). Let ui denotes the new 2-D range measurement, σqz2 denotes the variance of the depth measurement (7)

In general, the accuracy of bathometer is much higher than the ranges, therefore set σqz = 0. Based on the propagation principle of variance-covariance (Ghilani, 2010), the convenience of two different 2-D ranges σuiuj is 0, and the variance of ui is

r 2 (a + bri )2 ∂ui 2 ∂u σi + i σqz2 ≈ i 2 ∂ri ∂z q ri − zq2

(8)

FIM in Eq. (5a) is relatively complicated to calculate. Using polar coordinate to express the surface position can help to simplify FIM. We can assume q is the origin and pi= (ui, αi). Therefore

ei = (cos αi sin αi )

(9a)

n

sin(2αi ) = σui2

∑ i =1

n

⎛ 2 cos αi sin αi ⎞ n 1 cos αi F = ∑i =1 2 ⎜ ⎟ σui ⎝ cos α sin α sin2 α ⎠ i i i ⎛∑n 1 + cos 2αi ∑n sin 2αi ⎞ i =1 2σ 2 ⎜ i =1 2σ 2 ⎟ ui = ⎜ n sin 2αui n 1 − cos 2αi ⎟ i ⎜∑i =1 2 ⎟ ∑i =1 2 2σui 2σui ⎝ ⎠

i =1

(9b)

αi =

j =1

cos(2αj ) σuj2

−

cos(2αi ) σui2

(11b)

cos(2αj )

j =1

σuj2

cos(2αi ) =0 σui2

(12a)

(12b)

n

i =1

(13)

n

∑ j =1

sin(2αj ) σuj2

2πi +γ n−1

(14)

where γ is a rotation angle of the sensor configuration. The configuration still satisfies Eq. (13) after rotating an identical angle around the seafloor node as shown in Fig. 3(a). Regular polygon is an optimal design but not only from Eq. (13). The combination of several regular polygons with identical circumradius still satisfies Eq. (13) as shown in Fig. 3(b). Suppose that n surface sensors are separated by two group n1 and n2 sensors, then Eq. (15) can be obtained. For example, the combination of a regular triangle and a square is still optimal as shown in Fig. 3(b).

(10)

Taking the derivative of determinant F with respect to ri and αi, we can get the critical points for maximizing determinant F. n

sin(2αj ) ⎤⎥ =0 σuj2 ⎥⎦

Referring to Fourier transformation formula (Boyd, 1994), the horizontal angle can be obtained

2 ⎡ n ⎛ n cos(2α ) ⎞2 ⎤ ⎛ n cos(2α ) ⎞2 1 ⎛ 1 ⎞ i ⎟ i ⎟ ⎥ det(F ) = ⎢ ⎜⎜∑ 2 ⎟⎟ − ⎜⎜∑ − ⎜⎜∑ ⎟ ⎟ 2 4 ⎢⎣ ⎝ i =1 σui ⎠ σui ⎠ σui2 ⎠ ⎥⎦ ⎝ i =1 ⎝ i =1

∑

n

/∑

∑ sin(2αi ) = ∑ cos(2αi ) = 0

After some computations and simplifications, we have

∂ det(F ) sin(2αi ) = ∂αi σui2

σuj2

The first kind of solution from Eq. (12a) means that for sensor i and j, |αj - αi| = π/2 or π. It only makes sense when n = 2, 3 and 4 as shown in Fig. 2, otherwise two or more sensors should be placed in the same direction. Previous researches have proved that placing more sensors can improve positioning accuracy (Xue and Yang, 2014). In order to extend the case to n ＞ 4 sensors, the second kind of solution from Eq. (11a) should be more focused on. Infinite solutions can be obtained from Eq. (12b) (Xue and Yang, 2017). However, among the infinite ones, a meaningful solution is highlighted when surface sensors are circular around the seafloor node (Zhao et al., 2016). We assume that range is product of the mean sound speed and TOA, but the mean speed is not accurate due to sound speed error, which is the main cause of inaccurate positioning results. Zhao et al. (2016) proves that if the sensor configuration are circular, the sound speed errors of different range measurements are identical and the effect of sound speed error on plane positioning accuracy can be significantly weakened. As to improve positioning quality, circular configuration is focused on, in which the ranges and their variances in Eq. (12b) are identical and the expression can be simplified as

3.1. 2-D scenario

σui2 =

cos(2αj )

Two different kinds of solutions are obtained from Eq. (11a)

3. Maximizing the determinant

ri2 − zq2

j =1 n

Under the assumption of only Gaussian noise existing in ranging observations, maximizing the determinant of FIM and minimizing GDOP have the same significance in improving positioning accuracy (Xue et al., 2015).

ui =

n

=0 (11a) 3

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radius =

γ

r 2 − zq2

(19)

3.2. 3-D scenario Similar discussion is carried out when the depth of seafloor node is estimated by 3-D ranging intersection as well, instead of being measured by bathometer. Assume q is the origin again and get the polar coordinates of surface node pi= (ri, αi, βi). We have some new updates of notations.

b

a

ei = (cos αi sin βi sin αi sin βi cos βi )

Fig. 3. Two properties of optimal configuration, where black triangle represents surface sensor from triangle, grey triangle represents surface sensor after rotating, black rhombus represent sensor from square, and black rectangle represents seafloor node. (a) Optimal configuration after rotating an arbitrary angle. (b) Optimal configuration after combining a regular triangle and a square.

⎛ cos2 αi sin2 β cos αi sin αi sin2 βi cos αi cos βi sin βi ⎞ i ⎟ 1 ⎜ 2 sin2 αi sin2 βi sin αi cos βi sin βi ⎟ ⎜ cos αi sin αi sin βi σi2 ⎜ ⎟ cos2 βi ⎝ cos αi cos βi sin βi sin αi cos βi sin βi ⎠

n

F=

(20a)

∑ i =1

(20b) Eq. (20b) can be further simplified by substituting cos βi = zq/ri to it.

n

n1

∑ sin(2αi ) = ∑ sin(2αi ) + ∑ i =1

i =1

n

n1

n

sin(2αi ) = 0

F=

(15a)

i = n1+1

i =1

i =1

cos(2αi ) = 0 (15b)

i = n1+1

f1 (ri ) =

Take σui =σu, and substitute the Eq. (13) to Eq. (10),

det(F ) =

n2 4σu2

n

(16)

1 − cos(2αi − 2αj )

j =1

σuj2

S=C−

σi2 ri2

(21b)

(22)

dd T n ∑i =1 f3 (ri )

(23)

n

n

≤ ∑i =1 f3 (ri )det C =

≤

1 n n n ∑ f (ri )[(∑i =1 f1 (ri ))2 − (∑i =1 cos(2αi ) f1 (ri ))2 4 i =1 3 n − (∑i =1 sin(2αi ) f1 (ri ))2 ] 1 n n ∑ f (ri )(∑i =1 f1 (ri ))2 4 i =1 3

(24)

Eq. (24) holds equal if and only if n

n

∑ cos αi f2 (ri ) = ∑ sin αi f2 (ri ) = 0 i =1

n

i =1

i =1

(25a)

n

∑ cos 2αi f1 (ri ) = ∑ sin 2αi f1 (ri ) = 0

(18a)

⎛ 2 ⎛ az 2 ⎞2 ⎜ azq ⎜⎜ q ⎟⎟ − ⎜⎜ ⎟⎟ + ⎜⎜ 2b − ⎟ ⎝ 2b ⎠ ⎝ 3 ⎠ ⎟⎠ ⎝

zq2

det F = ∑i =1 f3 (ri )det S

The first kind of solution from Eq. (17a) means that for sensor i and j, |αj - αi| = π. It only makes sense when n = 2. Therefore Eq. (17b) should be more focused on. Assume for all surface sensors ri = r, and solve Eq. (17b) (Wang and Wang, 2005), then

⎛ 2 ⎜ azq + r=⎜ ⎜ 2b ⎝

σi2 ri2

, f3 (ri ) =

The bound of determinant S is 0≤determinant S ≤determinant C, which can be achieved only when d = 0. According to Schur complement formula, the determinant of F is

(17b)

1 ⎛ 2z 2 ⎞3 ⎞ 3 q ⎟

ri ri2 − zq2

As F is positive definite, the Schur complement of the lower right part of F is (Ouellette, 1981)

(17a)

br 3 − (a + 2br ) z 2 = 0

σi2 ri2

, f2 (ri ) =

⎛C ⎞ d F=⎜ T n ⎟ ∑ d f ( r ) ⎝ i =1 3 i ⎠

=0

∂(1/ σui2 ) =0 ∂ri

ri2 − zq2

(21a)

It is hard to directly calculate the analytical determinant of FIM as well as its derivative with respect to pi. Separate F to four parts as below.

Notice that the increase of n ensures the increase of determinant F. If n is enough large, the positioning accuracy can be significantly improved, and the optimal configuration is a circle with certain radius around seafloor node. If we use a ship instead of a set of buoys to position the seafloor node, this ship can travel along this circle and collect range measurement frequently at different epochs. Therefore, every epochs along the circle represents a surface sensor. Obviously, we can get enough sensor points in this method which can help improve positioning accuracy of seafloor geodetic node significantly. From above discussion, the optimal configuration could be circular. Next key question is to determine the optimal radius of the circle, and it can be deduced by the optimal range between surface nodes to seafloor node based on Eq. (11b). Similarly, two different kinds of solutions are

∑

∑

n

∑ cos(2αi ) = ∑ cos (2αi ) + ∑ i =1

⎛ cos2 αi f (ri ) cos αi sin αi f1 (ri ) cos αi f2 (ri )⎞ 1 ⎜ ⎟ sin2 αi f1 (ri ) sin αi f2 (ri ) ⎟ ⎜ cos αi sin αi f1 (ri ) ⎜ ⎟ sin αi f2 (ri ) f3 (ri ) ⎠ ⎝ cos αi f2 (ri )

n

i =1

(25b)

Similar with discussion in Section 3.1, infinite solutions can be obtained, above which, circular configuration is relatively meaningful. Furthermore, the optimal configuration could be regular or combination of several regular configurations. Eq. (24) is simplified as

1 ⎛ 2z 2 ⎞3 ⎞ 3 q ⎟

⎛ az 2 ⎞2 ⎜⎜ q ⎟⎟ − ⎜⎜ ⎟⎟ ⎟ ⎝ 2b ⎠ ⎝ 3 ⎠ ⎟⎠

(18b)

det F =

In conclusion, in 2D scenario, circular configuration around seafloor sensor is a kind of optimal configuration. 4

⎡ r 2 − z 2 ⎤2 n3zq2 1 q ⎢ ⎥ 4 (a + br )2 r 2 ⎢⎣ (a + br )2r 2 ⎥⎦

(26)

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Optimal r is obtained by taking the derivative of determinant F with respect to r, and optimal radius is obtained based on Eq. (19). Optimal radius is related with the performance of LBL transponder and the depth of seafloor nod. In 3D scenario, cone configuration constructed by circular surface nodes and seafloor node, is a kind of optimal sensor configuration, but the radius of the surface circle is different with that in 2D scenario.

det F = [∑i =1 f3 (ri ) + 1/ Q3]det ⎡ ⎛ ∑n cos2 α f (r ) + 1/ Q i 1 i 1 ⎢ ⎜ i −1 ⎢ ⎜⎝ ∑n cos αi sin αi f1 (ri ) ⎣ i −1 −

dd T

n ∑i −1 cos αi sin αi f1 (ri ) ⎞ ⎟ n ∑i −1 sin2 αi f1 (ri ) + 1/ Q1 ⎟⎠

⎤ ⎥

n ∑i =1 f3 (ri ) ⎥⎦

n

≤ [∑i =1 f3 (ri ) + 1/ Q3] n ⎛ ∑n cos2 αi f (ri ) + 1/ Q1 ∑i −1 cos αi sin αi f1 (ri ) ⎞ 1 ⎟ det ⎜⎜ i −1 n n ∑i −1 sin2 αi f1 (ri ) + 1/ Q1 ⎟⎠ ⎝ ∑i −1 cos αi sin αi f1 (ri ) 1 n n n = [∑i =1 f3 (ri ) + 1/ Q3][(∑i =1 f1 (ri ) + 2/ Q1)2 − (∑i =1 cos(2αi ) f1 (ri ))2 4

3.3. Initial position with uncertainty Above discussion about the optimal configuration is based on the initial position of seafloor node. However, the initial position is not accurate and waiting to be estimated. Therefore, the determinant F in Eq. (26) is not the maximum and might reduce the performance of the optimal configuration. Priori probability density function of the position of seafloor node can be obtained by taking consideration to the initial position when launching LBL sensor (seafloor node), the ocean environment parameters, such as the flow velocity and water density, and the volume and weight of LBL sensor. Therefore, the expected determinant F is

EDET=

∫

det Fψ (q ) dq

n

− (∑i =1 sin(2αi ) f1 (ri ))2 ] ≤

(32) Based on the discussion in Section 3, the optimal configuration should satisfy Eq. (25), which means it could still be cone configuration constructed by circular surface nodes and a seafloor node. Determinant F can be simplified as

det F = (27)

Simulative and practical experiments are carried out to testify our analysis in this section respectively. 4.1. Simulative experiments Assume an LBL sensor is installed on the seafloor with depth 1000 m, and four sensors whose configuration is square on sea surface with circumradius 1500 m. Therefore, coordinates of four sensors are set as (0, 1500, 0), (1500, 0, 0), (0, −1500, 0), (−1500, 0, 0) respectively, and coordinates of seafloor sensor is (x, y, −1000). Let a = 1, and b = 0.001 denote the performance of LBL sensors, and let the variance of bathometer be zero, and let Q1 = Q2 = Q3 = 1 m2. We can obtain the determinant of FIM with respect to the plane coordinates of seafloor sensor (x, y) in three scenarios, 2D scenario, 3D scenario and 3D scenario with uncertain initial seafloor position respectively. Fig. 4 shows that in three scenarios, the biggest determinant of FIM is achieved when the seafloor sensor is under the center of circumcircle constructed by the surface sensors. Similar conclusions can be deduced when surface sensor is a regular polygon with more than four points. Fig. 4 shows the optimal relative position between sensor configuration and seafloor node. It further supports that the optimal configuration of surface sensors should be circular. The biggest difference of the optimal configuration under three scenarios is the radius of circumcircle. Still assume four surface sensors and a seafloor node, let the seafloor node under the center of the surface sensors. Let a=1 m, b=0.001, Q1 = Q2 = Q3 = 1 m2. We can obtain the optimal radius in three scenarios with respect to the water depth. In Fig. 5, three kinds of optimal radius are shown under three scenario. The optimal radius in Fig. 5 is not consistent with previous research that in 2D scenario radius should be as small as possible and in 3D scenario radius = 2 z q (Xue and Yang, 2014; Zhang, 1992). The reason is that we relate the variance of range measurements with the value of range as shown in Eq. (3), and then the weight matrix is not an

Solve Eq. (28) and we get

(29a)

δq = (BT PB + Q−1)−1BT PL

(29b)

In this scenario, new FIM and CRLB is obtained as

FIM = CRLB−1 = BT PB + Q−1

(33)

4. Experiments and discussion

(28)

∂(V T PV + δqT Q−1δq ) = BT PV + Q−1δq ∂δq = BT P (Bδq − L ) + Q−1δq =0

⎤ ⎡ n (r 2 − z 2 ) ⎤2 nzq2 1 ⎡⎢ q + Q3⎥ ⎢ + 2Q1⎥ 2 2 2 2 ⎥⎦ ⎢⎣ (a + br ) r ⎥⎦ 4 ⎢⎣ (a + br ) r

The optimal r is obtained by taking the derivative of determinant F with respect to r, then the optimal radius can be obtained based on Eq. (19).

The new optimal configuration is obtained by taking derivative of EDET with respect to pi. It is very complicated and hard to get analytical solutions. Therefore, existing researches focus on numerical solution. However, when the PDF of the initial position follows normal distribution q ~ N (q0, Q), the analytical solution of the sensor configuration can be obtained as follows. According to least squares criterion, the parameter estimation problem is transferred to

δq = arg min(V T PV + δqT Q−1δq )

1 n n [∑ f (ri ) + 1/ Q3](∑i =1 f1 (ri ) + 2/ Q1)2 4 i =1 3

(30)

By maximizing the determinant of new FIM in Eq. (30), the new optimal configuration can be obtained. We only focus on 3D scenario in this section because it is more general and complicated. We assume Q is a diagonal matrix with elements Q1, Q2 and Q3 on its diagonal line, and let Q1 = Q2 for the purpose of discussion. ⎛∑n cos2 αi f (ri ) + 1/ Q1 ∑n cos αi sin αi f (ri ) ∑n cos αi f (ri ) ⎞ 1 1 2 i −1 i −1 ⎜ i −1 ⎟ n n n F = ⎜ ∑i −1 cos αi sin αi f1 (ri ) ∑i −1 sin2 αi f1 (ri ) + 1/ Q2 ∑i −1 sin αi f2 (ri ) ⎟ ⎜⎜ ⎟ n n n ∑i −1 cos αi f2 (ri ) ∑i −1 sin αi f2 (ri ) ∑i =1 f3 (ri ) + 1/ Q3⎟⎠ ⎝

(31) According to Schur complement formula, we have

5

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Fig. 4. Determinant of FIM with respect to initial plane position of seafloor geodetic node in three scenarios, where black triangle represents surface sensor, and color bar represents the value of determinant. (a) Contour map for 2D scenario. (b) Color map for 2D scenario. (c) Contour map for 3D scenario. (d) Color map for 3D scenario. (e) Contour map for 3D scenario with uncertain initial position. (f) Color map for 3D scenario with uncertain initial position.

6

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identical matrix. Specially, in 2D scenario, the variance of horizontal range measurement is more accurate, because it is calculated based on the propagation principle of variance-covariance. The trend for the third scenario is a little different from another two, because more parameters are influencing the optimal radius, such as the number of sensors n, Q1and Q3 as shown in Eq. (36). Our conclusion about optimal sensor configuration is more accurate, convictive and comprehensive. Additional experiment is carried out to give some examples of different configurations and their determinants of FIM, as shown in Fig. 6. Assume the depth of underwater node is 1000 m, and Let a=1 m, b=0.001, Q1 = Q2 = Q3 = 1 m2. No bathometer and no prior information about node position is provided. The determinants of FIM of different configurations is calculated by Eq. (28). Fig. 6 shows plane projections of different groups of configuration in 3-D scenario. The circumcircle of Set 1 is 860 m, which is the optimal radius deduced by Eq. (28). In Fig. 6(a), Set 1 is a regular triangle, Set 2 scalene triangle, and Set 3 a triangle constructed by three

Fig. 5. Optimal radius with respect to water depth in three scenarios, when seafloor node is under the center of four surface sensors.

Fig. 6. Plane projections of different groups of configurations in 3-D scenario. (a) Configurations distributed along same circle. (b) Configurations distributed along or not along a circle. (c) Configurations along same circle with different numbers of surface sensors. (d) Regular configurations distributed along different circles.

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Table 2 Determinant of FIM of different configurations. Configuration

Set 1

Set 2

Set 3

Set 4

Set 5

Set 6

Set 7

Set 8

Set 9

determinant F

0.0045

0.0037

0.0012

0.0042

0.0040

0.0107

0.0209

0.0026

0.0026

cies of roll and pitch of 0.01° were also adopted in the experiment for providing the vessel-mounted sensor position. The coordinates of all the instrument were measured in vessel coordinate system in advance. The initial plane position (x, y) = (1942.68, 2010.73) of seafloor node was provided by GPS coordinate when launching it on sea surface, and the initial vertical position z = 2045.23 was obtained from terrain map using this plane position. The performance parameters of LBL sensor were set as a = 1.97 m, and b =0.0004, and we obtain the optimal radius of 1995.25 m using Eq. (19) and Eq. (26) in a 3-D scenario. Our vessel travelled along a circle around seafloor node with radius about 2000 m around the initial plane position, and calculate 1317 ranges to seafloor node at different epochs. We can consider every points along the circle as a surface sensor. Pick out three groups of four points from the circle for discussion, namely Set 1, Set 2 and Set 3. And then calculate their determinant of FIM and positioning error with respect to an accurate seafloor position in Table 2. The accurate position is provided by intersection method using all the points along the circle. From above discussion, the number of surface points improve the positioning accuracy. Therefore, the positioning results using 1317 points along the circle is high-precision and reliable. In Fig. 7, Set 1 is regular, Set 2 is not regular but scattered, and Set 3 is on one side. In Table 3, Set 1 have the biggest determinant F and highest accuracy, with Set 2 followed and Set 3 the smallest determinant F and lowest accuracy. It further proves that regular configuration helps improve positioning accuracy of seafloor node. Constrained by the cost and time, we have not carried out additional practical experiments to prove the optimal radius.

Fig. 7. Circle around seafloor node and three sets of points, Set 1, Set 2 and Set 3.

close sensors. In Fig. 6(b), Set 4 and Set 5 are scalene triangles away from the circle. In Fig. 6(c), Set 6 and Set 7 are square and regular pentagon respectively. In Fig. 6(d), Set 8 is a regular triangle with radius of 516 m, and Set 9 with radius of 1376 m. As shown in Table 2, Set 7 has the biggest determinant, which indicated that more surface sensors or measurement times help improve positioning accuracy. The determinant of Set 1 is bigger than that of Set 2 or Set 3, which is a proof of better performance of regular configuration. The comparison of Set 1, Set 4 and Set 5 proves that circular configuration can achieve better positioning result. The result of Fig. 6(d) shows the relationship between radius and determinant of FIM. Above all, regular and cone configuration is a solution of optimal configuration, and optimal radius and more sensors further improve its performance.

5. Conclusions and future work The optimal acoustic sensor configuration has been presented for improving the positioning accuracy of seafloor node in three scenarios. If sensors are placed circular around seafloor node, horizontal positioning accuracy can be improved. Thus, a meaningful and optimal configuration in all the scenarios is circular in 2-D or cone in 3-D scenario，specifically, surface regular polygons or combinations of several regular polygons. Moreover, we can collect vast ranging data if we install an acoustic sensor in a ship which can travel along a circle around seafloor node. Furthermore, the optimal radiuses of the configurations in different scenarios are different. For 2D and 3D case, the radius is related to water depth and the performance of LBL sensors. For 3D scenario with uncertain initial position, the radius is related to the number of sensors and the PDF of seafloor sensor in addition. Our further work will focus on underwater navigation for Autonomous Seafloor Vehicle (AUV), which is a dynamic problem and more complicated.

4.2. Practical experiments Practical experiments are carried out in South China Sea with depth about 2000 m. Sonardyne 6 G LBL system with the ranging accuracy of 0.1% were adopted in the experiments. An LBL senor was mounted under survey vessel and a node sensor on the seafloor. The depth of seafloor node is not provided because of lacking bathometer. Besides, SVP Mar10 ZX21 with the accuracy of 0.25 m/s, Leica 1200 RTK GPS with the accuracy of 2 mm + 1 ppm and Octans III Fiber Optic Gyrocompass with the dynamic heading accuracy of 0.1° and accura-

Table 3 Positioning results, positioning errors and determinants of FIM of 4 configurations, circle Set 1, Set 2 and Set 3 respectively. Configuration

X

Y

Z

dX

dY

dZ

determinant F

Circle Set 1 Set 2 Set 3

1932.74 1931.87 1925.27 1950.8

1999.36 1997.58 2005.38 2009.66

2067.58 2068.25 2073.09 2044.68

0 −0.87 −7.47 18.06

0 −1.78 6.02 10.3

0 0.67 5.51 −22.9

394888.36 0.07 0.05 0

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Acknowledgement This work was supported in part by the Key Laboratory for National Geographic Census and Monitoring, National Administration of Surveying, Mapping and Geoinformation (No. 2015NGCM), the Shenzhen Scientific Research and Development Funding Program (No. SGLH20150206152559032), and the Shenzhen Future Industry Development Funding program (No. 201507211219247860), National Science and Technology Major Project (Coded by 2016YFB0501703), and the Key Laboratory of Surveying and Mapping Technology on Island and Reef, National Administration of Surveying, Mapping and Geoinfomation (coded by 2015B08). References Bishop, A.N., Fidan, B., Anderson, B.D.O., Pathirana, P.N., Doğançay, K., 2007. Optimality analysis of sensor-target geometries in passive localization: part 2 – timeof-arrival based localization. Automatica 46 (3), 479–492. Boyd, S.P., 1994. Linear matrix inequalities in system and control theory. Proc. IEEE 86 (12), 2473–2474. Chaffee, J., Abel, J., 1994. GDOP and the Cramer-Rao bound. In: Proceedings of Position Location and Navigation Symposium, IEEE, Las Vegas, Nevada, USA, pp. 663–668. Ding, S., Chen, C., Chen, J., 2014. An improved particle swarm optimization deployment for wireless sensor networks. J. Adv. Comput. Intell. Intell. Inform. 18 (2), 107–112. Favali, P., Barnes, C.R., Delaney, J.R., Hsu, S.K., 2010. Seafloor observatory science. Ann. Geophys. 49 (2–3), 515–567. Ghilani, C.D., 2010. Error propagation in elevation determination. In: Ghilani, C.D. (Ed.), Adjustment Computations: Spatial Data Analysis.. John Wiley & Sons, Inc, Hoboken, New Jersey, USA. Han, T., Wu, H., Lu, X., Du, J., Zhang, X., 2013. The Mathematical Expectation of GDOP and its Application. In: Proceedings of China Satellite Navigation Conference, Wuhan, China, pp. 501–521. Leonard, J.J., Bahr, A., 2016. Autonomous underwater vehicle navigation. IEEE J. Ocean. Eng. 35 (3), 663–678. Majid, A.S., Joelianto, E., 2012. Optimal sensor deployment in non-convex region using

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