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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/).

The problem of determining the optimal geometric configuration of a sensor network that will maximize the range-related information available for multiple target positioning is of key importance in a multitude of application scenarios. In this paper, a set of sensors that measures the distances between the targets and each of the receivers is considered, assuming that the range measurements are corrupted by white Gaussian noise, in order to search for the formation that maximizes the accuracy of the target estimates. Using tools from estimation theory and convex optimization, the problem is converted into that of maximizing, by proper choice of the sensor positions, a convex combination of the logarithms of the determinants of the Fisher Information Matrices corresponding to each of the targets in order to determine the sensor configuration that yields the minimum possible covariance of any unbiased target estimator. Analytical and numerical solutions are well defined and it is shown that the optimal configuration of the sensors depends explicitly on the constraints imposed on the sensor configuration, the target positions, and the probabilistic distributions that define the prior uncertainty in each of the target positions. Simulation examples illustrate the key results derived.

Autonomous vehicles and sensor networks have become ubiquitous and are key tools in the robotics field, due to the versatility and flexibility that they show in a number of scenarios [

From the above, it is clear that the problem of source localization in those areas in which the common GPS systems are useless has become increasingly important in recent years. The localization of a source (or sources) is done through a set of signals acquired by a conveniently designed sensor array. The aim of the work at hand is to determine the sensor positions of the array for which the information obtained about the source or sources is maximized, that is, the sensor placement for which the positioning accuracy is the largest possible for each of the targets involved in the task. The source positions are then defined with the information received by the sensor nodes through a convenient algorithm based on the nature of the measurements. This paper is focused on range-only measurements.

Existent localization techniques depend on the information that is available for the sensor network to define the range distances, and this information may be of a different nature, for example, power level information that consists in measuring the power level of a signal sent from a source to a sensor, known as Received Signal Strength (RSS) [

Interesting results in the area go back to the work of [

From the above, it is clear that the problem of optimal sensor placement is of great interest and importance, and moreover, it plays a key role in different application areas. For example, in [

Motivated by recent results in ground robotics for single target positioning, this paper tackles the

It is important to remark that for the optimization problem, the logarithms of the determinants of the FIMs are used. This makes the functions to be maximized jointly convex in the search parameter space. For a discussion of the convexity of the functions adopted, see, for example, [

For a multi-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), the environment (e.g., ambient noise), the number of targets and their configuration, and the possibly different probability distribution functions that describe the prior uncertainty of each target position. An inadequate sensor configuration may yield large localization errors for some of the targets. Even though the problem of optimal sensor placement for range-based localization is of great importance, not many results are available on this topic yet; even more, the results are only for single target positioning. Some notable exceptions include the work in [

In this paper, in striking contrast to what is customary in the literature, the problem of optimal sensor placement for multi-target localization for an arbitrary number of targets is studied. At this point, it is important to point out that following what is commonly reported in the literature, the work starts by addressing the problem of optimal sensor placement given assumed positions for the targets. It may be argued that this assumption defeats the purpose of devising a method to compute the target positions, for the latter are known in advance. The rationale for the problem at hand stems from the need to first fully understand the simpler situation, where the positions of the targets are known, and to characterize, in a rigorous manner, the types of solutions obtained for the optimal sensor placement problem. In a practical situation, the positions of the targets are only known with uncertainty, and this problem must be tackled directly. However, in this case, it is virtually impossible to develop 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, a notable exception being [

The key contributions of the present paper are threefold: (i) the optimality conditions for single target positioning are derived and characterized analytically in a simple and fast manner, and the maximum accuracy that can be obtained in the localization of a single target working in isolation is well defined; (ii) the optimality conditions that a sensor network must satisfy in order to provide the maximum possible accuracy for multiple targets are analytically determined, and several examples of analytical solutions are offered to illustrate the potential of the methodology developed; and (iii) concepts and techniques from estimation theory and convex optimization are fully exploited to obtain numerical solutions to the optimal sensor configuration problem for multiple targets when the target positions are described by probabilistic distribution functions. This allows one to capture the important fact that the target positions are only known with uncertainty

The paper is organized as follows. Section 2 defines the set-up for single target positioning, and the FIM is properly defined. The optimality conditions for the single target positioning problem are derived in Section 3, and some simulation examples are shown. Section 4 defines the problem formulation and the set-up for the multiple target positioning task. The analytical solution for the two target positioning problem is given in Section 5, and in Section 6, analytical and numerical solutions are defined for the problem of localizing an arbitrary number of targets. In Section 7, the maximization of the average value of the logarithms of the FIM determinants is studied when a sensor network surveys a certain working area or when there is uncertainty in the

Let {_{I}_{I}_{x}_{y}^{T}_{i}_{ix}_{iy}^{T}_{i}_{i}_{i}

Denote by _{i}_{i}_{i}_{1}(_{n}^{T}_{1} ⋯ _{n}^{T}_{i}^{2} ·

In the set-up adopted, the measurement noise model is considered to be distance-independent, in line with common assumptions reported in the literature for theoretical research and systems implementation. Some of the examples that support the model adopted (that is, constant covariance of the measurement errors) include the work of [

In general, the problem of range measurement noise modeling is not trivial. For example, for marine acoustic sensors, range measurements may be strongly affected by multipath effects, Doppler effects, energy attenuation and even uncertainty in the speed of propagation of sound in the physical medium. This uncertainty in the speed of propagation can be reduced by measuring the speed of sound at the operation site. Energy attenuation, with its impact on signal-to-noise (SN) ratio, as well as signal distortion, due to the characteristics of the physical medium, can be dealt with very effectively using acoustic ranging devices that build on spread spectrum techniques and employ cross-correlation techniques for the detection of incoming waves and, therefore, for the measurement of their times of arrival. As a consequence, from short to medium ranges, as long as the SN ratio does not cross a device-dependent limit, the statistics of the measurement errors can be taken as approximately constant. Identical considerations apply to modern devices that can compensate for Doppler effects. The problems caused by multipath effects and/or ray bending, due to the geometry and characteristics of the channel, are of an entirely different breed and will not be considered in the present work. Moreover, it is clearly stated in [

In what follows, it is assumed that the reader is familiar with the concepts of Cramer-Rao Lower Bound (CRLB) and Fisher Information Matrix (FIM); see, for example, [^{n}^{2}, between the observations, ^{2}, where _{q}_{q}_{q}_{q}

Equipped with the above notation and tools of estimation theory, the optimal sensor placement problem is now addressed by solving a related equivalent optimization one: given the FIM for the problem at hand, maximize the logarithm of its determinant by proper choice of the sensor coordinates. This strategy for sensor placement underlies much of the previous work available in the literature; see, for example, [_{q}_{1}, _{2}, …, _{n}^{T}^{2}^{xn}^{−1}. In this context, the optimal sensor placement strategy is obtained by maximizing the logarithm of the determinant of the FIM, which must be computed explicitly. To this effect,

As shown in Section 1, there are many references regarding the 2D target positioning problem. The optimal solutions for sensor placement given in those references are recovered in this paper using a novel methodology, and the results are then extended for the multiple target positioning problem. Furthermore, in this work, not only optimal formations are explicitly defined, the optimality conditions that the formations must satisfy in order to minimize the estimation error are well defined, and therefore, any possible optimal configuration may be defined using them. Optimal sensor placement examples are shown at the end of each section to illustrate the potential of the methodology developed.

In this section, for the sake of completeness, the optimal sensor placement problem for single target positioning is studied. The aim of this section is to recover the results on optimal sensor placement defined in the literature, but with a novel methodology with which the optimality conditions for optimal sensor placement can be defined in a fast and simple manner. The results defined in this section will be used to check the efficacy of the sensor configurations developed for the multiple target positioning problem in the forthcoming sections.

As stated in Section 1, the FIM captures the amount of information that measured data provide about an unknown parameter (or vector of parameters) to be estimated, and the log determinant of the FIM is used for the computation of an indicator of the performance that is achievable with a given sensor configuration. As mentioned above, let _{x}_{y}^{T}_{i}_{ix}_{iy}^{T}_{i}_{i}_{i}_{I}^{n}

As a consequence, the FIM is parametrized by two vectors in ℜ^{n}^{2}. It is also convenient to consider these vectors as elements of the Hilbert space, with elements in ℜ^{n}

The determinant of

To determine the conditions for which

Clearly, sin (

Now, the focus is on the derivatives of the logarithm of

The logarithm of the determinant of the FIM can now be written as:

Thus, the derivative of ^{2} = ^{2} = ^{2} = |ϒ|^{2}. Therefore, the expression of the Fisher Information Matrix that provides the maximum possible (logarithm of the) FIM determinant yields:

It is important to comment that

Thus, from

The D-optimality criterion for the design of optimal sensor configurations is commonly used in the literature. Other indicators, like the A or E-optimality criteria, are also used by many authors. The D-optimality criterion minimizes the volume of the uncertainty ellipsoid for the target estimate, whereas the A-optimality criterion, which consists in minimizing the trace of the CRLB matrix, suppresses the average variance of the estimate, and the E-optimality design, which consists in minimizing the largest eigenvalue of the CRLB matrix, minimizes the length of the largest axis of the same ellipsoid [

An important advantage of using the D-optimality criterion 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, if the global optimal is not obtained, the D-optimality criterion may yield some errors, because the information in one dimension can be improved rapidly, while we may have no practical information in the other. This problem can be avoided with the A/E-optimality criteria, [

The optimal formations can be obtained analytically from the system defined in

Using by now classical terminology, the sensor formation must be first and second moment balanced. Then, from

Thus, the maximum FIM determinant is obtained with the sensor network regularly distributed around the target position. Obviously, an infinite number of solutions may be obtained by rotating the sensors rigidly around the target, that is, by allowing the above angles to become _{c}_{c}

It is important to remark on one important feature of the optimal solutions that can be computed based on the analysis explained above. If two disjoint sets of

In what follows, two examples are shown to illustrate the methodology developed for optimal sensor placement for single target positioning. For both examples, the sensors are considered to be placed at a distance of 20 m with respect to the target.

Clearly, in order for the information about the optimal configurations to be useful, one must check if the logarithm of the determinant of the FIM meets desired specifications. To this effect, and for comparison purposes, the determinant of the FIM obtained for a number of hypothetical target points (based on a fixed optimal sensor configuration corresponding to a well-defined scenario) will, at times, be computed by allowing these points to be on a grid in a finite spatial region,
^{2}. The same comments apply to

^{2}/(^{4}·4) = 16·10^{4}^{−4}. In

In ^{2}·4/^{2}. This correspondence between the minimum trace of the CRB (A-optimum design) and the maximum FIM determinant (D-optimum design) is clear from the fact that the optimal FIM is a diagonal matrix with all the eigenvalues being equal.

^{4}^{−4}.

Notice how for both optimal formations of Examples 1 and 2, the maximum theoretical accuracy is obtained at the target position, and thus, they are equivalent. However, the determinant,

To address the problem of multiple target positioning we start by introducing the notation _{k}_{kx}_{ky}^{T}_{i}_{ix}_{iy}^{T}_{ki}_{k}_{i}_{ki}_{k}_{ik}

As explained before, the determinant of the FIM corresponding to a single target is an indicator of the performance in positioning that can be achieved with a given sensor configuration. To tackle the multiple target problem we adopt, in this paper, an indicator that is the sum of the logarithms of the determinants of the FIMs of each target. Accordingly, the multiple target positioning problem is formulated as that of computing:

At this point it is important to notice that in a number of situations, maximizing this indicator yields sensor formations with the following important property: the determinant of the FIM of each and every target is equal to the maximum value obtained by considering each target in isolation (that is, by solving a single target positioning problem for each of the targets). Stated differently, there exist sensor formations that can be computed efficiently and yield optimal performance for all targets simultaneously. Example 3 below will show how this can be proven analytically for the case of 2 targets and 4 sensors (the same result applies to other relatively simple combinations of targets and sensors). For more complex target/sensor arrangements the analytical proof yields conditions for the sensor configuration that may be extremely hard to interpret geometrically. In these cases, the maximization of the indicator adopted can be done using numerical optimization tools that borrow from convex optimization theory.

It is necessary to point out that although the terms,

In principle, to uniquely compute the position of a target in 2D, three different ranges from noncollinear sensors must be obtained. Notice, however, that the interest of the present study is not in deriving position estimators, but rather, in understanding what is the best performance (in terms of target positioning) that can be achieved with any unbiased estimator; that is, the objective is to understand how the target position impacts on the optimal sensor geometry. Notice now, from a practical standpoint, that if extra information about the target is available, then one can actually compute its position by using a smaller number of sensors. This can be easily understood in the case of a target in 3D using three sensors located in a plane. In this situation, knowledge that the target is above or below the plane is sufficient to find its position using three sensors and not four, as would be necessary if that information was not available. Identical comments apply to positioning in 2D using only two sensors, if extra information will allow us to disambiguate between the two possible solutions. The above reasoning justifies the study of the limits of performance of any estimator that relies on two sensors and, at the same time, incorporates extra information, allowing it to disambiguate between two possible solutions. Because the solutions exhibit symmetry (with respect to the array of sensors), there is no difference in the accuracy (as evaluated by the covariance of the estimation error) with which each of them can be computed. This justifies the demonstration example included.

For this purpose, the FIM for two sensors is defined:

Straightforward computations show that the determinant of _{12} = _{2} − _{1}. The optimal sensor configuration will be defined as the one which maximizes the logarithm of

It is important to remark that the concavity of the logarithm of the FIM determinant is restricted to positive definite matrices [_{12} ∈ ]0, _{12} ∈ ]_{12}, is computed, which yields:

The second derivative yields:

For the sake of clarity, in the notation and demonstration of the forthcoming analysis, index

To shed light into the multiple target positioning problem, the simplest case of two targets and an arbitrary number of sensors (but at least three sensors) is studied first. Then, the methodology is extended to the general problem of an arbitrary number of targets.

The optimal solutions can be searched with convex optimization techniques; however, there exist several situations in which it is possible to define analytical solutions. In this section, the simplest of these situations, the two target positioning problem, is tackled. In this particular case,

The summation of logarithms in _{T}

Thus, from

The optimal solution is defined by computing the derivatives of _{ix}_{iy}_{1} ⋯ _{n}

From the above, the derivative of _{I}

Similarly, the derivative of _{I}

Straightforward computations allow one to rewrite the above derivatives as:

The optimal sensor configuration may be defined by making these equations equal to zero.

Now, _{1}(1)/_{2}(1) = _{1}(2)/_{2}(2), it is:

_{i}_{1} = _{i}_{2} +

It is important to notice from _{1} and _{2} are equal to zero. A closer look at these vectors shows that the only possible condition for them to be zero is that _{ik}

Therefore, to obtain the maximum possible accuracy, the sensor network must be second moment balanced with respect to both targets:

_{1} = [20, 0]^{T}m_{2} = [−20, 0]^{T}m

The optimal sensor formation is defined by the positions listed in ^{2}/(^{4} · 4) ^{−4}, for each target. This optimal formation may also be computed with the gradient optimization algorithm described in Section 6. For the example at hand, the initial guess may be arbitrary, since the only feasible solution provides the theoretical maximum FIM determinant for both targets, as demonstrated above. In ^{2}·4/^{2}, at the target positions.

In _{1} = [20, 0]^{T}m

And for _{2} = [−20, 0]^{T}m

Therefore, it is easy to check that the optimal formation that provides the theoretical maximum FIM determinant, and also the theoretical minimum CRB trace, is obtained when

Once the analytical solution for the two target positioning problem has been defined, the analysis can be extended for an arbitrary number of targets, that is:

Again, ^{2}/^{4}2^{2}, as defined in Section 3.

However,

The above discussion shows that there may be tradeoffs involved in the precision with which each of the targets can be localized; to study them, as mentioned above, techniques that borrow from estimation theory and convex optimization are used. For the latter, the reader is referred to [

Once the gradients defined by

For each target, its FIM is computed for the current sensor formation at iteration

Using _{iξ}

The sensor positions are updated according to the gradients: _{iξ}_{iξ}^{ζ}^{[}^{t}^{]} ∇_{iξ}

If
_{i}_{i}_{ix}_{iy}

If
_{i}

The above cycle is only run once if the targets are stationary. Notice the unrealistic assumption, also made in many of the publications available in this area, that the positions of the targets are known in advance. This is done to simplify the problem and to first fully understand its solution before the realistic scenario where the positions of the targets are known with uncertainty can be tackled. In this respect, see Section 7, which is largely inspired by the work in [

In a practical situation where the targets are in motion, the sensor network must adapt its optimal configuration as the mission unfolds in three different intertwined processes:

The situation where the algorithm described is run during each cycle of the positioning system in (ii) is thus envisioned. Interestingly enough is also the situation where the different iterations of process (ii) can be used to yield set points for the autonomous sensor network to move to, effectively guiding them collectively to the optimal configuration that is being computed.

The advantage of using a gradient optimization method is its simplicity. As it will be seen later, based on the simulations done so far, the method has proven to be quite satisfactory. However, should there be a need for a more refined method, the sensor network positions given by the gradient algorithm can be used as initial estimates in the new method.

The rest of this section contains the results of simulations that illustrate the potential of the methodology developed for optimal sensor placement when multiple targets are involved. A large accuracy can be obtained for each of the targets, but if the mission or task would require different accuracy levels for each of the targets, then weights could be associated with them. The selection of weights would be mission-dependent, but the procedure would be very similar to that explained above.

In what follows, some examples of optimal sensor placement for the multi-target scenario are studied to illustrate the methodology developed.

^{2}/4^{4} = 6.25·10^{4}^{−4}. In ^{2}·4^{2}.

The optimality condition in

The target positions and the optimal sensor formation are listed in

In

The FIM determinants obtained with this formation for each of the targets are stated in ^{2}/(^{4}·4) = 6^{2}/(0.1^{4}·4) = 9·10^{4}^{–4}.

The computation time in this case was 2.2308 s; so, it is clear that the time needed to obtain the optimal configuration increases for a larger number of targets. Despite this, the time required is small, and the optimal network is defined in a fast manner. Therefore, it is possible to design sensor configurations that provide the theoretical maximum FIM determinant for each target or a value very close to the maximum one.

In this section, the situation in which the targets to be positioned are known to lie in well-defined uncertainty regions is addressed. Inspired by the work in [

In what follows, _{iξ}_{i}^{m}^{×2} the probability density functions, with support, ^{2}, that describe the uncertainty in the position of the targets in region _{1} + ⋯ + _{m}

To proceed, one must compute

It now remains to solve the optimization problem defined above. Conceptually, the procedure to determine the optimal sensor configuration is similar to that explained in the previous sections; that is, one must compute the derivatives of

To proceed with the computations, the integral and derivative operations are interchanged: the derivatives are explicitly determined first, and the integration over region

The derivatives can be computed in a recursive way using

Two different examples of multiple target positioning when the target positions are known with uncertainty are studied now. In these examples, an error model defined by

^{2}, whose vertices are given by the points, _{1} = [−120 − 20; −120 20; −80 −20; −80 20]^{T}m_{2} = [80 −20; 80 20; 120 −20; 120 20]^{T}m

It is possible to check in _{max}^{4} m^{−4} and |_{min}^{4} m^{−4}. Notice how the maximum determinant is the theoretical optimal one, |_{opt}^{2}/(4σ^{4}) m^{−4}, and how the minimum determinant is very close to this optimal value, as well, giving a large accuracy for all the points of the uncertainty regions. Moreover, the average FIM determinant inside the regions of interest is |_{avg}^{−4}, showing a very large accuracy for the multiple target positioning task. In a similar manner, in _{min}^{2} and _{max}^{2}. The minimum CRB trace is again the theoretical minimum, _{opt}^{2}/^{2}, and the maximum is also very close to this optimal value.

The computation time for the solution shown was 40.0611 s. This time is larger than in the case of known target positions, although it is still a small computation time, showing how the approach is reliable for practical mission scenarios. Thus, from this example, it is clear that it is possible to define optimal sensor configurations for which the accuracy inside the work areas is very close to the maximum accuracy that would be obtained for a single target with a known position working in isolation.

^{2} defined by the vertices, _{1} = [−100 − 20; −100 20; −60 − 20; −60 20]^{T}m_{2} = [100 − 20; 100 20; 60 − 20; 60 20]^{T}m_{3} = [−20 100; −20 140; 20 100; 20 140]^{T}m

Again, it is possible to notice in

In _{max}^{4} m^{−4} and |_{min}^{4}^{−4}, respectively The average value inside the regions of interest is |_{av}^{4}^{−4}, providing a very large accuracy inside the work areas. Notice how the maximum FIM determinant is the theoretical one that can be obtained for a single target working in isolation, |_{opt}^{2}/(4^{4}) m^{−4}, and the minimum FIM determinant is very close to this theoretical optimal value, so that the accuracy inside the regions of interest is very close to the optimal one. Similarly, in _{min}^{2} and _{max}^{2}, respectively, with the minimum value equal to the theoretical one, ^{2}/^{2}, and the maximum very close to it, as mentioned for the FIM determinant plots. The computation time for this example was 54.4755 s, due to the more complex situation of three uncertain areas.

Because in a practical scenario involving multiple targets, where the sensor network must adapt its configuration to the estimated positions of the targets that can also be moving, it is only required that the optimal sensor locations be computed at a rate that is much smaller than the rate at which control systems are run in real time, the commented computation times show that the algorithm can be easily implemented in practice and yields good accuracy. Moreover, adding parallelism to the computations will further reduce this computation time.

Therefore, with the methodology developed, it is possible to define optimal sensor configurations to localize multiple targets, whose positions are known with uncertainty, with very large accuracy.

The problem of optimal sensor placement for multiple target positioning with range-only measurements in 2D scenarios has been studied. This problem is of key importance in multiple application scenarios in which a variable number of targets must be localized with the largest possible accuracy. It has been shown that in many situations of interest, it is possible to define an analytical solution for which the maximum FIM determinant is obtained for each of the targets involved in the task. Despite this, there exist some complex target formations that do not allow for analytical solutions. For this case, convex optimization tools have been employed to determine the optimal sensor configurations. Moreover, the numerical solutions determined with the optimization algorithm provide accuracies for each of the targets very close to that obtained in the ideal case for a single target working in isolation. Some illustrative examples have been developed to show these issues.

The previous results were extended to the more realistic problem, where the target positions are known with uncertainty. This uncertainty can be defined by any probabilistic distribution function, and the type of function used conditions the optimal sensor formation. An optimization method similar to that previously defined was used to determine the optimal sensor configurations. The main problem to overcome was the resolution of the integrals of the gradient equations, to determine the necessary gradients to increase the average FIM determinant over the work areas in the optimization algorithm. These integrals were solved numerically by a Monte Carlo method, because of the impossibility of solving them analytically. It has been shown through several examples how it is possible to compute optimal formations for which the accuracy obtained inside the uncertainty areas is very close to the ideal one and how it is possible to localize multiple targets with significant accuracy, no matter the configuration of the targets.

Future work will aim to: (i) extend and apply the methodology developed to a real multiple vehicle mission scenario and (ii) study the performance of the algorithms for the optimal sensor configuration computation 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.

The authors declare no conflict of interest.

Target localization problem set-up.

Optimal sensor placement for eight sensors. In (

Optimal sensor configuration for eight sensors formed by the combination of a five sensor regular formation and a three sensor regular formation. In (

Optimal four sensor formation for two target positioning. In (^{2} are shown; and in (

Optimal sensor formation for five sensors and three targets. In (^{2} is shown; and in (

Optimal formation for six sensors and seven targets. In (^{2} is shown; and in (

Optimal sensor placement of five sensors for two target positioning with uncertainty. In (^{2} is shown and, in (

Optimal sensor placement of five sensors for three target positioning with uncertainty. In (^{2} is shown; and in (

Target positions and optimal sensor positions.

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

{_{I} |
20 | −20 | −26.39 | −26.39 | 26.39 | 26.39 |

{_{I} |
0 | 0 | −17.22 | 17.22 | −17.22 | 17.22 |

Target positions and optimal sensor positions.

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

{_{I} |
−25 | −25 | 50 | 42.00 | −17.68 | −36.82 | 7.23 | 37.65 |

{_{I} |
17 | −35 | 0 | 18.48 | 34.57 | 5.99 | −50.50 | −31.09 |

Target positions and optimal sensor positions.

_{1} |
_{2} |
_{2} |
_{4} |
_{5} |
_{6} |
_{7} | |
---|---|---|---|---|---|---|---|

{_{I} |
−25 | −25 | 50 | −75 | 25 | 30 | −40 |

{_{I} |
17 | −35 | 0 | 10 | 65 | −55 | −60 |

| |||||||

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

| |||||||

{_{I} |
59.85 | −12.51 | −66.44 | −63.50 | −13.03 | 58.41 | |

{_{I} |
31.09 | 71.91 | 35.03 | −36.12 | −66.76 | − 22.34 |

Fisher Information Matrix (FIM) determinants for each of the targets.

_{1} |
_{2} |
_{3} |
_{4} |
_{5} |
_{6} |
_{7} | |
---|---|---|---|---|---|---|---|

|^{4}^{−4}) |
8.9964 | 8.9999 | 8.996 | 8.9997 | 8.9974 | 8.9916 | 8.9974 |

Optimal sensor positions.

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

{_{I} |
75.90 | −74.92 | −109.96 | 6.82 | 114.51 |

{_{I} |
100.96 | 96.67 | −92.70 | −51.49 | −97.39 |

Optimal sensor positions.

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

{_{I} |
116.32 | 93.83 | −99.74 | −127.45 | −36.87 |

{_{I} |
−77.46 | 87.18 | 110.65 | −35.18 | −109.61 |