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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, we address the problem of determining the optimal geometric configuration of an acoustic sensor network that will maximize the angle-related information available for underwater target positioning. In the set-up adopted, a set of autonomous vehicles carries a network of acoustic units that measure the elevation and azimuth angles between a target and each of the receivers on board the vehicles. It is assumed that the angle measurements are corrupted by white Gaussian noise, the variance of which is distance-dependent. Using tools from estimation theory, the problem is converted into that of minimizing, by proper choice of the sensor positions, the trace of the inverse of the Fisher Information Matrix (also called the Cramer-Rao Bound matrix) to determine the sensor configuration that yields the minimum possible covariance of any unbiased target estimator. It is shown that the optimal configuration of the sensors depends explicitly on the intensity of the measurement noise, the constraints imposed on the sensor configuration, the target depth and the probabilistic distribution that defines the prior uncertainty in the target position. Simulation examples illustrate the key results derived.

The last decade has witnessed tremendous progress in the development of marine technologies that are steadily affording scientists advanced equipment and methods for ocean exploration and exploitation. Recent advances in marine robotics, sensors, computers, communications and information systems are being applied to develop sophisticated technologies that will lead to safer, faster and far more efficient ways of exploring the ocean frontier, especially in hazardous conditions. As part of this trend, there has been a surge of interest worldwide in the development of autonomous underwater vehicles (AUVs) capable of roaming the oceans freely, collecting relevant data at an unprecedented scale. In fact, for reasons that have to do with autonomy, flexibility and the new trend in miniaturization, AUVs are steadily emerging as tools par excellence to replace remotely operated vehicles (ROVs) and also humans in the execution of many demanding tasks at sea. Furthermore, their use in collaborative tasks allows for the execution of complex missions, often with relatively simple systems; see [

Many AUV mission scenarios call for the availability of good underwater positioning systems to localize one or more vehicles simultaneously based on acoustic-related range or angle information received on board a support ship or an autonomous surface system (e.g., a number of autonomous surface vehicles equipped with acoustic receivers, moving in formation). The info thus obtained can be used to follow the state of progress of a particular mission or, if reliable acoustic modems are available, to relay it as a navigation aid to the navigation systems existent on board the AUVs. Similar comments apply to a future envisioned generation of positioning systems to aid in the tracking of one or more human divers. Inspired by related work and similar developments in ground robotics, in this paper, we address the problem of single target positioning based on measurements of the azimuth (bearing, in 2D scenarios) and elevation angles between an underwater target and a set of sensors, obtained via acoustic devices. Thus, the target position is determined with bearing-only measurements, in contrast to what is customary in marine systems, where range measurements are often used. In what follows, we will refer to these angle measurements in 3D as AE (azimuth-elevation) measurements or, for simplicity, with an obvious abuse of notation, simply as bearing measurements. Speaking in loose terms, we are interested in determining the optimal configuration (formation) of a sensor network that will, in a well-defined sense, maximize the AE-related information available for underwater target positioning. To this effect, we assume that the AE-measurements are corrupted by white Gaussian noise, the variance of which is distance-dependent. The computation of the target position may be done by resorting to triangulation algorithms, based on the nature of the measurements. See, for example, [

Given a target localization problem, the optimal geometry of the sensor configuration depends strongly on the constraints imposed by the task itself (e.g., maximum number and type of sensors that can be used) and the environment (e.g., ambient noise). An inadequate sensor configuration may yield large localization errors. It is important to remark that even though the problem of optimal sensor placement for bearing- and/or range-based localization is of great importance, not many results are available on this topic yet. Exceptions include a series of interesting results that go back to the work of [^{2}. In these two references, the optimal Fisher Information Matrix is defined for a constant variance error model; again, the goal is to compute optimal sensor configurations. Finally, in [

Motivated by previous results published in the literature, in this paper, we address the problem of finding the optimal geometric configuration of a sensor formation for the localization of an underwater target, based on AE-only measurements. The optimality conditions for a generic sensor formation are defined, and the explicit optimal geometric configuration of a sensor formation based on AE-only measurements is studied for two different scenarios:

The case in which the sensors lie on a sphere centered at the target position, which provides a simple example of how to define optimal sensor configurations for a given set of (physical or mission-related) constraints imposed on the sensor formation.

The application scenario in which a surface-based sensor formation is defined for the localization of an underwater target. Notice that in this scenario, the sensors are restricted to lie at the sea surface. A problem of this type was previously studied in [

Given a localization strategy, the optimal sensor configuration can be ascertained by examining the corresponding Fisher Information Matrix (FIM) or its inverse, the so-called Cramer-Rao Bound (CRB) matrix. In this paper, we use the trace of the CRB matrix (A-optimality criterion [

It is important to point out that following what is commonly reported in the literature, we start by addressing the problem of optimal sensor placement given an assumed position for the target. It may be argued that this assumption defeats the purpose of devising a method to compute the target position, for the latter is known in advance. The rationale for the problem at hand stems from the need to first fully understand the simpler situation, where the position of the target is known, and to characterize, in a rigorous manner, the types of solutions obtained for the optimal sensor placement problem. In a practical situation, the position of the target is only known with uncertainty, and this problem must be tackled directly. However, in this case, it is virtually impossible to make a general analytical characterization of the optimal solutions, and one must resort to numerical search methods. At this stage, an in-depth understanding of the types of solutions obtained for the ideal case is of the utmost importance to compute an initial guess for the optimal sensor placement algorithm adopted. These issues are rarely discussed in the literature, with the exception of [

The key contributions of the present paper are threefold: (i) global solutions to the optimal sensor configuration problem in 3D are obtained analytically in the cases where the sensor network is restricted to lie on a sphere centered at the target position or on a plane, the latter capturing the situation, where the sensors are deployed at the sea surface; (ii) in striking contrast to what is customary in the literature, where zero-mean Gaussian stochastic processes with fixed variances are assumed for the measurements, the variances are now allowed to depend explicitly on the distances between target and sensors. This allows us to explicitly address the important fact (rooted in first physics principles) that the measurement noise may increase in a nonlinear manner with distance; finally, (iii) the solutions derived are extended to the case, where

The document is organized as follows. Section 2 offers the reader a brief overview of the most common underwater positioning systems. Section 3 derives the FIM when the measurement noise is Gaussian, with distance-dependent variance. The optimal Fisher Information Matrix that minimizes the trace of the corresponding CRB matrix is computed in Section 4. The optimal sensor configuration is explicitly defined for the case in which the sensors lie on a sphere centered at the target position in Section 5. In Section 6, the optimal sensor placement is computed in the context of a sensor network restricted to lie on a plane, and two illustrative scenarios are shown as examples. In Section 7, the optimal sensor placement problem is solved for the case where the prior knowledge about the target in 3D is given in terms of a probability density function. Finally, the conclusions and a brief discussion of topics for further research are included in Section 8.

To better motivate and help understand the problem that is at the core of the present paper, this section affords the reader a brief overview of the most common underwater, acoustic-based positioning systems that are available commercially. The latter stand in sharp contrast to the techniques that are used on land or in the air, such as Global Positioning Systems (GPSs).

The most important features of a GPS system are its wide area coverage, the capability to provide navigation data seamlessly to multiple vehicles, the relatively low power requirements, the miniaturization of receivers and the fact that it is environmentally friendly, because its signals do not interfere significantly with the ecosystem. Typical acoustic underwater positioning systems are quite the opposite: they have reduced area coverage; they do not normally scale well for multiple vehicle operations; and they may have high power requirements, with a subsequent moderate to high impact on the environment in terms of acoustic pollution. The applications of underwater acoustic positioning systems include a wide range of scientific and commercial activities, such as biological and archaeological surveying, marine habitat mapping and gas and oil pipeline inspections, to name just a few. The systems available are quite diverse and suited for a number of different tasks. Most of them are based on the computation of the ranges or bearings (azimuth and elevation angles) between the target to be localized and a set of acoustic sensors with known positions. This is done by measuring the times of arrival (TOA) or the time differences of arrival (TDOA) of the acoustic signals that arrive at an array of sensors; see, for example, [

The main elements of a complete USBL system consist of an acoustic unit that works both as an emitter and receiver (

The accuracy with which an underwater target can be positioned is highly dependent on the installation and calibration of the transceiver, as well as on the accuracy with which the inertial position and attitude of the support ship can be determined using a GPS system and an advanced attitude and heading reference unit, respectively. In this sense, advanced signal processing techniques are required in these systems. Correct calibration of the system is crucial, for any error due to poor calibration of the USBL system will translate directly into errors on the target position estimates. USBL systems are widely used, because they are simple to operate and have relatively moderate prices. However, the resulting position estimation errors are usually greater that those found with longer baseline systems, are very sensitive to attitude errors on the transducer head and increase with the slant range between the transducer and the transponder. Thus, under certain operational conditions, USBL systems can yield large absolute target position errors.

The principle of operation of a SBL systems is identical to that of an USBL system in that it relies on the emission of acoustic signals by a unit installed on board a support ship, followed by the detection of the replies emitted by a transponder installed on the underwater target. However, unlike in the case of USBL systems, the receiving units have larger baselines that can be on the order of hundreds of meters [

Traditionally, LBL systems have been the most widely used for underwater target positioning. The key elements in an LBL system are a set of sea-floor mounted baseline transponders, with a spacing between them that can be on the order of a few kilometers. In this set-up, a target to be positioned carries a transceiver. In a typical application, the emitter interrogates the transponders sequentially. Upon detection of the incoming acoustic signal, each of the transponders replies to the target. The latter, in turn, measures the round trip travel time for each acoustic emission and, therefore, its range to each of the transponders. The position of the target can then be computed by using one of many available algorithms, which, in their essence, amount to performing some kind of trilateration [

The GIB system was first introduced in [

The commercial systems for underwater target positioning described above share the fact that they rely on the propagation of acoustic signals and the computation of their times of arrival—or time differences of arrival—at a number of receivers. Using these principles, other non-standard positioning systems can, of course, be envisioned. Examples are available in the literature that show clearly that there is tremendous interest in the development of underwater positioning systems based on sensor networks. This is a very active area of research. The interested reader will find in [

As a contribution to the above goal, the present paper offers a solution to the problem of optimal sensor placement for underwater target positioning with bearing (AE) measurements only. When compared with other possible techniques commonly used for underwater target positioning, the problem of determining the optimal sensor placement for target localization using AE-only measurements is of special interest, because no information flows from the sensor network to the target, and therefore, its solution does not require the exchange of information between the target and the sensor network. Furthermore, the clocks of the surface platforms do not need to be synchronized with those of the underwater targets. Thus, AE-based strategies allow for the sensor network to observe without being detected itself. A problem of this type was studied in [

For the sake of clarity, the work is at first motivated by a specific positioning system that holds promise for practical applications and seeks inspiration from the original GIB system described before: the surface buoys are replaced by autonomous surface vehicles (ASVs), and the pinger installed on board the underwater target does not have to be synchronized with GPS time prior to system deployment. With this set-up, the system can not only compute the position of the underwater target, but also adaptively reconfigure its formation (geometric arrangement) in accordance with the estimated position of the target, target depth and the noise measurement characteristics of the acoustic sensors, so as to yield good positioning accuracy. The sensor placement solution described is, therefore, of key importance to adequately reconfigure the sensor formation in response to on line detected changes in the mission conditions. At the root of the algorithms to determine the geometric formation to be adopted as a specific mission unfolds are the optimality conditions determined analytically in the present work.

In what follows, {_{I}_{I}_{I}_{x}_{y}_{z}^{T}_{i}_{ix}_{iy}_{iz}^{T}_{i}_{i}_{i}_{i}

To each of the acoustic sensors at the surface, we attach a parallel translation of {_{i}_{i}_{i}^{T}_{i}_{i}_{I}y_{I}_{I}y_{I}_{I}_{i}_{i}_{i}_{i}_{αi}_{βi}^{T}

For analytical tractability, it is commonly assumed that measurement errors can be described as Gaussian, zero-mean additive noise with constant covariance. See, for example, [_{i}_{α}_{β}_{β}_{0} and _{α}_{0} are zero-mean Gaussian stochastic processes described by the probability density function, _{0}) with Σ_{0} = ^{2} · _{i}_{i}

Define _{1}(^{T}_{n}^{T}^{T}^{2}^{nx}^{2}^{n}

We assume that the reader is familiar with the concepts of Cramer-Rao Lower Bound (CRLB) and Fisher Information Matrix (FIM); see for example [

Formally, let ^{n}^{3}, between the observations, ^{3}, where _{q}

From the above notation, following standard procedures, the FIM is computed from the likelihood function:
^{2}^{nx}^{3}, and ^{−1}. In this context, the optimal sensor placement strategy for a single vehicle localization problem is obtained by minimizing the trace of the CRLB; this is the so-called

An important advantage of D-optimality is that it is invariant under scale changes in the parameters and linear transformations of the output, whereas A-optimality and E-optimality are affected by these transformations. However, as commented upon in Section 1, the D-optimality criterion can yield to some errors, as the information in one dimension can be improved rapidly, providing a very large FIM determinant, while we can have no information in other dimensions. This problem can be avoided with the A-E-optimality criteria [

For the sake of simplicity, and without loss of generality, hereinafter, the target is considered to be placed at the origin of the inertial coordinate frame. To compute the trace of the CRB matrix, it is convenient to introduce the following three vectors in ℜ^{2}^{n}

The latter should be viewed as vectors of a Hilbert space with elements in ℜ^{2}^{n}^{2}^{n}^{2} = < X, X > and < X, ϒ > = |X||ϒ| cos(_{Xϒ}), from which it follows that the angle, _{Xϒ}, between vectors, X and ϒ, is given by _{Xϒ} = cos^{−1}(< X, ϒ > /(|X||ϒ|)).

With this notation, the FIM becomes:
_{Xϒ}, θ_{XZ} and θ_{ϒZ} are the angles defined by vectors X and ϒ, X and Z, and ϒ and Z, respectively, and |FIM| denotes the determinant of the FIM. Straightforward computations show that:

Notice how _{Xϒ}, _{XZ} and _{ϒ}_{Z}_{i}, β_{i}_{i}_{i}_{i}_{i}

Consider, now, the problem of minimizing _{i}, β_{i}_{i}^{2}^{n}

Then, as it will be shown next,

The above inequality is equivalent to:

Notice, however, that because: cos^{2} (_{XZ}) + cos^{2} (_{Xϒ}) ≥ 2 cos (_{XZ}) cos (_{Xϒ}) and 0 ≤ |cos (_{ϒZ})| ≤ 1, it follows that
_{XY}_{YZ} must be equal to

Formally, the conditions that an optimal sensor configuration must satisfy may now be obtained by computing the derivatives of _{i}_{i}_{i}_{i}_{z}_{iz}_{z}_{z}_{iz}

This section shows how the incorporation of physical or mission-related constraints on the positions of the sensors leads to a methodology to determine a solution to the optimal sensor placement that eschews tedious computations and lends itself to a simple geometric interpretation. To this effect, we consider the situation where all the sensors are placed on a sphere centered at the target position, that is, the distances from the sensors to the target are equal. With this assumption, _{i}

It is important to notice that for this scenario, the optimal solutions corresponding to constant or distance-dependent measurement noise covariances are identical. In fact, the solutions depend only on the azimuth and elevation angles of each sensor with respect to the target location, and the distance between the target and sensors does not affect the solutions (distance is the constraint parameter). This fact does not hold true in the practical scenario of surface sensor networks, as will be shown in Section 6, where the optimal solutions depend explicitly on the range distances between the target and sensors and on the noise model. At this point, the derivatives of _{i}_{i}_{i}) = 0; (ii) sin (α_{i}) = 0; (iii) A*^{2} = B*^{2}. Similarly, _{i}) = 0. The last condition is not studied in detail, because, if all the sensors are placed, such that sin (β_{i}) = 0, it can be shown that the condition yields a local maximum for tr(CRB). Thus, in what follows, we consider that the optimality condition for

If cos (_{i}_{I}z_{I}

The above equation only holds for a single value of cos^{4} (_{i}^{2}, ^{2} and ^{2} are constant for a given optimal configuration, and _{i}_{i}

Consider, now, the case where sin (_{i}_{I}z_{I}

A reasoning similar to that used in the previous case allows for the conclusion that this solution must also be discarded.

Finally, if cos (_{i}_{i}

It must be noticed that _{i}_{z} − p_{iz}

To define _{t}

_{Xϒ}) = cos (_{XZ}) = cos (_{ϒZ}) = 0); that is:

A simple and elegant solution that satisfies the two above extra conditions is obtained by noticing the orthogonality relations for sines and cosines from Fourier analysis [

Therefore, we can take a regularly distributed formation on the circumferences, with the sensors placed along one or both of them. Using classical terminology, the sensor formation must be first and second moment balanced. Therefore, with this configuration, the minimum trace of the CRB is obtained for this scenario.

In real situations, the sensors cannot be placed at will, either due to physical or mission constraints. As an interesting application scenario, we tackle the case where the sensors are restricted to lie in the horizontal plane,

It is clear that the angles, _{i}_{i}_{i}_{i}_{z}_{i}_{z}_{i}_{i}_{i}_{i}

We now examine _{i}_{i}

Following a procedure similar to that of the previous section, the analysis of _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{z}

Based on _{z}_{z}_{
}. In this paper,
^{2}.

Analyzing

The value of _{i}_{i}

Straightforward computations yield:

Clearly, the optimal elevation angle, ^{2} at the target depth (_{D}

In

Following the reasoning of the previous example, the radius of the circumference can be obtained easily by adequately manipulating

The only valid solution of

It is important to remark that the above values of

At this point, it is important to point out that following what is commonly reported in the literature, we have started by addressing the problem of optimal sensor placement given an assumed position for the target. In a practical situation, the position of the target is only known with uncertainty, and this problem must be tackled directly. However, in this case, it is virtually impossible to make a general analytical characterization of the optimal solutions, and one must resort to numerical search methods. At this stage, an in-depth understanding of the types of solutions obtained for the ideal case is of the utmost importance to compute an initial guess for the optimal sensor placement algorithm adopted.

The objective is to obtain a numerical solution when the target is known to lie in a well-defined uncertainty region. We assume that the uncertainty in the target position is described by a given probability distribution function, and we seek to minimize, by proper sensor placement, the average value of the trace of the CRB matrix for the target.

In what follows, _{i}_{ξ}; ^{3}a probability density function with support, ^{3}, that describes the uncertainty in the position of the target in region

To proceed,

The seemingly complex form of the derivatives, shown in the

In what concerns the computation of the triple integral over the region,

The methodology developed is now illustrated with the help of several examples that address the problem of optimal surface sensor placement for uncertain underwater target positioning. Therefore, the main constraint imposed to the problem is that the range distances depend explicitly on the elevation angles _{i}_{i}_{z}_{i}_{z}

In this work, important practical issues related to the time required to compute the optimal sensor placement for targets lying on a region of uncertainty have not been explicitly addressed, since it is not within the scope of this work. This is a problem of considerable importance, in view of the need to compute a triple integral over a region of interest using a Monte Carlo method. For this reason, although the objective of this work is to accurately define the configurations of the optimal sensor networks, the main characteristics of the Monte Carlo computations are defined next.

For the triple integrals of each of the following examples, a set of 50,000 samples are used. The computations are carried out in a laptop Intel Core i7, with 8 Gb RAM and running an MS Windows 7 Operative System. The computation times were similar for the examples, with an average time of 128.57 s and a standard deviation of 29.31 s, for the different simulations carried out. Moreover, adding parallelism to the computations will further reduce the computing time. The above indicates that the methodology proposed for optimal sensor placement is computationally feasible.

^{2} and 7.73 ^{2}, respectively. Despite the difference between the maximum and minimum values of the CRB trace, the average value inside the work area is 8.63 ^{2}; so, the average accuracy is close to the optimal one, and thus, for most points, the accuracy is closer to the minimum value of the CRB trace. Notice how the optimal radius becomes larger than in Example 1, where the target position was known without uncertainty.

^{2} and 124.25 ^{2}, respectively. However, a homogeneous accuracy over the area of interest is obtained, with an average value of 161.15 ^{2}, which shows that for most of the points of the area of interest, an accuracy close to the minimum one is obtained.

^{3} centered at the origin of the inertial coordinate frame and its center placed at 50 meters under the ocean surface and 50 meters over the ocean bottom, but there is no additional knowledge about the target position; so, the probability distribution function is a step-like distribution. The target is positioned by a six-sensor network at the sea surface, as shown in the set-up of

The minimum and maximum CRB trace values obtained inside the volume are _{min}^{2} and _{max}^{2}, respectively, with an average value of _{avg}^{2}, providing a large accuracy for most points inside the region of interest.

We may notice how the formation is smaller than that of Example 5 to reduce the impact of the distance-dependent added error, with the network keeping a formation similar to a circumference of an approximate radius of 37 meters. The minimum and maximum CRB traces inside the volume of interest are _{min}^{2} and _{max}^{3}^{2}, respectively, with an average value of _{avg}^{2}, which shows that, in this example, the accuracy is dramatically affected by the added distance-dependent error component.

Notice that the formation shape, although split into two formations, is very similar to the one obtained in the previous scenario, but with an approximate radius of 26 meters. However, in this case, the minimum and maximum CRB traces are _{min}^{2} and _{max}^{2}, respectively, with an average value of _{avg}^{2}, which shows how the accuracy, for the constant covariance case, increases when the formation consists of two formations, one at the sea surface and another at the sea bottom. Notice how the maximum value of the CRB trace is smaller with respect to Example 5 and how the average CRB trace is very close to the minimum value.

Again, the formation shape is similar to that obtained in Example 6, but with an approximate radius of 22 meters. The minimum and maximum CRB trace are now _{min}^{2} and _{max}^{2}, respectively, with an average value of _{avg}^{2}. We can notice how the maximum CRB trace is significantly reduced with respect to the value obtained in Example 6. The average value is, again, smaller, showing that a very good average accuracy is obtained inside the volume of interest. Finally, the minimum value of the CRB trace is also smaller. Thus, a more homogeneous accuracy inside the region of interest, with a significantly smaller error, is obtained when the sensors are split into two formations, one at the sea surface and the other at the sea bottom.

Therefore, for an unknown target location, it is clear that the average accuracy inside the working area is improved if we can place the sensors in two different parallel planes. This fact shows the importance of the constraints that are imposed on the sensor placement in order to define the sensor configuration that provides the largest possible accuracy in the volume of interest.

We studied the problem of determining optimal configurations of sensor networks that will, in a well-defined sense, maximize the AE-related information available for underwater target positioning. To this effect, we assumed that the measurements were corrupted by white Gaussian noise, the variance of which is distance-dependent. The Fisher Information Matrix and the minimization of the trace of the CRB matrix were used to determine the optimal sensor configurations. Explicit analytical results were obtained for both distance-dependent and distance-independent noise. In the application scenario of underwater target positioning by a surface sensor network, we have shown that the optimal formation lies on a circumference around the target projection and that a “regularly distributed formation” around this target provides an optimal configuration, the size of which depends on the measurement noise model and the target depth. The methodology was then extended to deal with uncertainty in the target location, because in a practical situation, the target position is only known with uncertainty. Simulation examples illustrated the concepts developed in different application scenarios, showing that the optimal configuration of the sensors depends explicitly on the intensity of the measurement noise, the constraints imposed on the sensor configuration, the target depth and the probabilistic distribution that defines the prior uncertainty in the target position.

Future work will aim at: (i) extending the methodology developed to deal with more than one target simultaneously; and (ii) studying the performance of the algorithms for optimal sensor configuration placement developed herein, together with selected algorithms for target tracking and cooperative sensor motion control.

The authors wish to thank the Spanish Ministry of Science and Innovation (MICINN) for support under project DPI2009-14552-C02-02. The work of the second author was partially supported by the EU FP7 Project, MORPH, under grant agreement No. 288704.

This Appendix contains the derivatives of the trace of the CRB with respect to the angles, _{i}_{i}^{2}^{nx}^{3}, Σ ∈ ℜ^{2}^{nx}^{2}^{n}^{−}^{1}. The sensors are considered to be placed at the sea surface, so that the range distance, _{i}_{i}_{z}_{i}_{z}_{i}_{i}_{i}

To finalize with the analysis of the derivatives of the trace of the CRB matrix with respect to the angles, _{i}_{i}_{i}_{i}

Therefore, the derivatives of the CRB trace with respect to _{i}_{i}

The authors declare no conflict of interest.

Elevation and azimuth angles measured in the inertial coordinate frame used in marine systems.

Optimal surface sensor formations for a target depth of 50 meters, _{D}^{2} are shown—lighter regions indicate higher accuracy—and on the right-hand side, their magnitudes in 3D for

Fisher Information Matrix (FIM) determinant

Optimal surface sensor formation for an uncertain target position at a depth of 50 meters, _{D}^{2} are shown and in (

Optimal surface sensor formation for an uncertain target position at a depth of 50 meters, _{D}^{2} are shown and in (

Sensor formations for an uncertainty volume of 60 × 60 × 60 ^{3} (

Optimal sensor positions for constant covariance.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

{_{I} |
35.48 | 0.07 | −35.33 | −35.3 | 0.07 | 35.48 |

{_{I} |
20.37 | 40.80 | 20.37 | −20.52 | −40.96 | −20.52 |

{_{I} |
50 | 50 | 50 | 50 | 50 | 50 |

Optimal sensor positions for

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

{_{I} |
32.76 | 0.04 | −32.69 | −32.68 | 0.04 | 32.76 |

{_{I} |
18.91 | 37.80 | 18.91 | −18.87 | −37.77 | −18.87 |

{_{I} |
50 | 50 | 50 | 50 | 50 | 50 |

Optimal sensor positions for constant covariance.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

{_{I} |
21.98 | −0.04 | −22.15 | −22.23 | −0.08 | 22.14 |

{_{I} |
12.841 | 25.68 | 12.84 | −12.7 | −25.38 | −12.74 |

{_{I} |
−50 | 50 | −50 | 50 | −50 | 50 |

Optimal sensor positions for

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} | |
---|---|---|---|---|---|---|

{_{I} |
19.74 | 0.14 | −19.28 | −19.41 | 0.21 | 19.70 |

{_{I} |
11.16 | 22.66 | 11.14 | −11.16 | −22.66 | −11.14 |

{_{I} |
−50 | 50 | −50 | 50 | −50 | 50 |