H R Vanaei

| M Deligant

| M Shirinbayan

| K Raissi

| J Fitoussi

| S Khelladi

| A Tcharkhtchi

A comparative in-process monitoring of temperature profile in fused filament fabrication

Fused filament fabrication (FFF), an additive manufacturing technique, is used to produce prototypes and a gradually more important processing route to get final products. Due to the layer-by-layer deposition mechanism involved, bonding between adjacent layers is controlled by the thermal energy of the material being printed. Thus, it is strongly in conjunction with the temperature development of the filaments during the deposition sequence. This study gives

out an in-process set-up enabling to record temperature profile of two adjacent filaments or a sequence of deposition in various locations during FFF process. The main characteristic of the presented procedure is the possibility of obtaining a global temperature profile resulted from an IR-camera; parallel to those recorded using a K-type thermocouple. Needless to say that a K-type thermocouple accurately records the local temperature at the interface of adjacent filaments. Conversely, an IR-camera signifies the temperature profile on the captured surface. The obtained results showed that there is a remarkable difference between the cooling rate and re-heating peaks. The primary outcome of this study is the consideration of results accuracy and the possibility of working on optimization of the obtained temperature profile. Altogether it helps optimize inter-layer strength while assessing the temperature evolution.

K E Y W O R D S

fused filament fabrication, in-process measurement, IR-camera, local-global approach, thermocouple

| INTRODUCTION

Fused Filament Fabrication (FFF), an extensively Additive Manufacturing (AM) process, [1] involves extrusion of thermoplastic filaments while moving in successive X-Y planes along the Z direction using the mechanism of layer-by-layer deposition. [START_REF] Chennakesava | Fused deposition modelinginsights[END_REF][START_REF] Ambone | [END_REF] As deposited filaments are facing with deposition of new filaments during the process, there is always a cyclic temperature profile resulting in the cooling and reheating of each one. The evolution of the temperature profile of filaments during deposition controls the interlayer bonding. In literature, many works have mentioned the FFF process a thermally driven procedure in which neck growth is stemmed from thermal diffusion of adjacent filaments above the crystallization temperature (for semi-crystalline materials) and the glass transition temperature (for amorphous materials). [4][5][6] Variety of studies have been performed to investigate the mechanical strength of 3D-printed parts, [7][8][9][10] and it have been pointed out that the evolution of the temperature profile is a key parameter that affects the bonding quality. [11,12] More specifically, cyclic cooling and reheating exists during layer deposition of the filaments. The criterion of effective bonding and consequently the mechanical properties are a major concern in FFF. [13,14] In the process of parts fabrication, as the deposition progresses, the hot filament is deposited onto filaments that were previously deposited and/or are being cooled. The contact between the hot filament and the previously deposited filaments causes re-heating of the latter. At the interface of adjacent filaments, temperature rises above the crystallization temperature (T c ) and proper bonding takes place. The evolution of temperature profile could be obtained by employing thermocouples to have local measurements. The deposition; however, might be interrupted while fixing the thermocouple. [15] Kousiatza et al. [16] locally measured the temperature profile. Although they have had an adequate agreement between experimental and theoretical results, the sudden drop of temperature at the head tip of the extruder showed a gap between the recorded and numerically derived temperature peak values. To wipe out this limitation, infrared thermography has been widely used. Albeit it deals with the surface temperature, it still does not record the interface temperature of adjacent layers. [17] Ferraris et al. [18] used IR thermography in determination of the temperature profile of a vertical wall and they observed poor agreement with theoretical predictions. Furthermore, 1D or 2D models have been developed to evaluate the temperature profile of deposited filaments during fabricating a structure. Sun et al. [5] and Jie Zhang et al., [19] tried to evaluate, both numerically and experimentally, the influence of process parameters on the temperature evolution. In another study, Bellini and Güçeri [20] used FEM to model extrusion and cooling rate of FFF process. Rodriguez et al. [21] computed the cooling rate numerically as a criterion for the bonding. In addition, Bellehumer et al. [4] developed a 1D model by taking into account the temperature profile. More recently, Costa et al. [22] developed an analytical approach to predict temperature profile and adhesion quality of 3Dprinted parts.

For the above-mentioned reasons, experimental monitoring of temperature is still challenging in FFF and lack of practical knowledge corresponds to the problem of bonding in this process. To address this limitation, K-type thermocouples (d = 80 μm) were added in parallel with deposition without pausing the process or causing damage. [23] The experimental data were then compared with the predictions obtained by Costa et al. [24] and it was found that there is a sufficient agreement between the experimental and analytical results.

To conclude, research on in process monitoring of temperature profile is still in its infancy. This work presents a comparison between the local and global assessment of temperature profile using both contact and non-contact approach. The aim is to evaluate the nature of both methods, the IR thermography, and small thermocouples (d = 80 μm) in parallel.

| MATERIALS AND METHODS

To track the cooling of filaments and the re-heating peaks of deposition of successive layers during deposition, very small (d = 80 μm) K-type thermocouples were used (see [23] for method description). The schematic of the experiment is presented in Figure 1 containing: the set-up for in-process measurement of temperature profile during the deposition, assembling of two methods together, and thermogram of the printed vertical wall with corresponding layers and locations highlighted for temperature profile. In parallel to the deposition and temperature recording using K-type thermocouples, an Optris PI450 infrared camera was used (at the same points 1-6) with the technical data presented in Table 1. Material emissivity (ε) was obtained by calibrating the absolute difference of the tracks obtained by IR-camera and a thermocouple.

The camera was placed with a specific distance from the extruder to have the plain field of view (FOV) of all the deposited layers. Experiments have been carried out while the camera inspecting (a) the X-Z planes and (b) the Y-Z planes. In the first case (X-Z planes), the printed part is stationary on FOV. It was then recorded temporal temperature variations in the object front plane. Therefore, temperature changes at every location are the consequence of several reheating that stem from new depositions. In-process monitoring was performed on a designed vertical wall sample with geometry of 50×0.2×35 mm 3 using the one-way direction of deposition. Process parameters and related settings to the process are indicated in Table 2.

In the second case (Y-Z planes), the nozzle was fixed and the part was printed by moving the built plate in Y direction. Four points specified as shown in Figure 2 with the following descriptions. Point a stays on extruder during printing (verifying the extruder temperature and the accuracy of measurement). Point b specifies the variation of temperature when material exits from the extruder (diffusion zone of material between two adjacent layers). Point c indicates the temperature of same layer. In other word, it is located on the same layer as that of point b, but with a specific distance from extruder (out of diffusion zone). Point d represents the effect of extruder temperature (or material when exits from extruder) on the previous layer (end of diffusion zone).

Temperature difference (ΔT) between point b and d is an indicator of temperature profile between two adjacent layers. It could be compared with point c for complementary assessment of the inter-layer temperature evolution. Two cases have been shown as (a) deposition from first layer-to-layer 5 and (b) deposition from layer 15 to layer 20 to show the influence of distance from the heat flux of support.

To perform the experimental procedure, polylactic acid (PLA) filament (d = 1.75 ± 0.01 mm) with the density of ρ = 1.24 g/cm 3 has been used (fillamentum supplier). Differential scanning calorimetry (DSC) was done using DSC Q1000 from TA instrument. Sample (6 mg), cut from the printed part, was sealed in an aluminum pan and heated from room temperature to 200 C with heating rate of 5 C/min to determine crystallization and melting temperature, T c and T m , respectively. The related curve and the gathered data are presented in Table 3 The accompanying graphs presented in Figure 3 provide the experimental results (temperature profile) of both IRcamera and K-type thermocouple. They comprise six points in different locations (in different layers) of the sample (see Figure 1). As described, the reported experiments are based on the layer-by-layer deposition of filaments. Under the 3D printing conditions, when a new filament is deposited, the previous one has significantly cooled down. Although there is a notable variance in starting point (when the filament exists from the nozzle) of deposition for each layer, the temperature evolves in the same cooling rate. For post processing, the two signals are synchronized at t = 0, based on the instant of the first peak of temperature (the highest measured value considered as a value at t = 0). As illustrated in Figure 4, the temperature peaks recorded by both methods is described as following. Peak 1 is the re-heating of fifth filament by deposition of sixth filament; peak 2 is the re-heating of fifth filament by deposition of seventh filament; peak 3 is the re-heating of fifth filament by deposition of eighth filament; peak 4 is the re-heating of fifth filament by deposition of ninth filament. Peaks on cooling curves are the return of the extruder to the point of next deposition without feeding of material (not important).

Owing to the nature of thermocouple and the local measurement of temperature at the inter-layer bonding, the temperature peaks recorded by IR-camera are highly overestimated comparing with those recorded by the K-type thermocouple. On the other hand, the sequences of temperature peaks concluded by K-type thermocouple has an acceptable evolution in comparison with those derived from IR-camera. In almost all the conditions, by increasing the distance Presented results in Figure 3 showed that there is a notable difference between the monitored temperature profiles. Despite the acceptable precision of the IRcamera, peak values (particularly in the first 20 seconds) are overestimated. This could be deduced from the radiation of extruder, previously deposited filaments under cooling, or even the heat radiation from support for the first layers. Believably, the precision of the IR-camera is lower than a thermocouple based on the nature of each measurement method. It has some advantages such as straightforwardness of the test and overfilling of data. In future work, numerical validation of both methods will be done for further discussions.

| Upper-limit and peaks evolution

Re-heating peaks decrease with progressively deposition of filaments. As mentioned in section 2, the temperature above T c and below T m is important for inter-diffusion of successive layers. Cyclic evolution of temperature plays an important role in the overall incident. In Figure 5, upper-limit of peak consequences indicates two different observations. Seemingly, the first data recorded by the IR-camera (refer to Figure 5(A) decreases about 23% with a distance increase from support, whereas, those captured by the K-type thermocouple (refer to Figure 5(B) stay around 5%. Apparently, on behalf of the extruder temperature, the thermocouple hands over 4% deviation from real data (T ext = 210 C) while 19% by the IR-camera. As an example, the first data recorded by K-type thermocouple and IR-camera for layer 54 at x = 40 mm from deposition are 202 C and 155 C, respectively.

| Interval of peaks between two approaches

Following observations obtained so far: layers near support are more affected by T supp (higher peaks in IR-camera); layers in the middle distance are less affected by T supp . Whereas T supp does not affect layers far from the support (Recorded peaks by IR-camera are near those recorded by thermocouple).

Table 4 shows the "ΔT=T IR-camera -T thermocouple " at each peak. Worth mentioning that based on described features, such as, support radiation, there is a small difference in correspond peaks at layers far from the support.

The graphs in Figure 6 reveal the difference of upper-limit obtained by both methods as a function of building time. The specified contour for each layer expresses the nature of each measurement method. Apparently, temperature varies between T c and T m in first layers, whereas, the contour drops below T c as the distance from the support is increasing. Their relative change is an important concern in the problem of inter-layers bonding and it should be taken into consideration.

| Four points on IR-camera (second experiment with stationary nozzle)

To have better comprehension, four points have been chosen around the extruder based on Figure 2 (in section 2). The temperature was recorded with the deposition. The mechanism of recording could be explained: the location of points a-d are fixed while the support is moving and it means that at any instance of deposition, the temperature variation of each point is recording. This partially clarifies the temperature of points a-d from beginning of the deposition of a layer and progressively the successive layers.

Figure 7 shows the obtained results. They cover the previous assumptions displaying the effect of distance from support. Presumably from Figure 7(A) and by deposition from layer one to layer five, the temperature of point b shifts between 165 C and 90 C. This is also valid for point c while temperature shifts between 120 C and 80 C.

| Comparison of two applied approaches

A vertical wall of a single filament thickness is considered for the analysis of thermal interaction by employing two methods: IR-camera as a global and applying K-type thermocouples as a local approach. As can be seen from the obtained results and depending on the nature of measurement methods, both local and global in-process monitoring of temperature profile have their own strengths and limitations (see Table 5).

As listed in Table 5, regarding the nature of measurement approach and the obtained experimental results, one can note that the capability of each approach as well as their strengths and limitations has a determinative role in the evaluation of temperature profile. Nevertheless, an entire optimization between two approaches could precisely result in characterization of thermal behavior during fabrication of 3D printed parts.

| CONCLUSIONS AND PERSPECTIVES

This work presents an in-process set-up enabling the record of temperature profile of two adjacent filaments (and/or a sequence of deposition) in various locations during the FFF process. The main characteristic of the presented procedure is the possibility of obtaining a global temperature profile resulted from the IR-camera; parallel to those recorded using a K-type thermocouple (local temperature at interface). Accurate acquisition via local measurement revealed by putting K-type thermocouples in different successive layers. However, IR-camera showed that there is a considerable difference by increasing the distance from support. A comparison through the upperlimit and interval of peaks validated the mentioned difference. The obtained experimental results showed that the optimization of the results obtained with the IR-camera by those achieved using the K-type thermocouples are necessary for the bonding optimization.

Additional experiments with a numerical validation are necessary to the set-up and its usefulness. Future work will focus on applying the results in complex geometry with a controlled-environment (Chamber) temperature and developing a predictive approach. This study is useful in inter-layer bonding optimization of adjacent layers by implementing the temperature evolution of filaments.

  . The temperature range between T c = 103 C and T m = 148 C is an important temperature range in the FFF process in semi-crystalline materials.

F I G U R E 1 2 2 Infill (%) 3 | 1 |

 12231 Representation of (A) in situ measurement of temperature profile during the deposition stage in FFF process, (B) thermogram of a vertical wall and points representing the location of the thermocouples with corresponding layers highlighted for temperature profile [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 1 Technical data of Optris PI 450 camera Process parameters used for printing Parameter Value Liquefier temperature ( C) 210 Support temperature ( C) 50 Printing speed (mm/s) 20 Layer height (mm) 0.RESULTS AND DISCUSSION 3.Local and global temperature profile

F

  I G U R E 2 Schematic representation of temperature variation of point a-d at instance of deposition for (A) layer 1 to 5 and (B) layer 15 to 20 [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 3 Differential scanning calorimetry curve and thermal the support, the temperature profile remains above 50 C. The secondary re-heating (and consequently third, fourth, fifth, and other re-heatings) are increasingly weak enough to enhance and keep the temperature of the previously printed parts. Accordingly, the inter-layer diffusion is limited to adjacent filaments as the secondary re-heating peak is at almost T = 103 (±2) C.

F I G U R E 3

 3 Temperature evolution at six locations during the deposition of a vertical wall consisting of single filaments deposited on top of each other. Point 1-6 corresponds to the fifth, 20th, 37th, 54th, 63rd, and 88th while indicating 30, 20, 35, 40, 25, and 40 mm from start of deposition, respectively [Color figure can be viewed at wileyonlinelibrary.com] F I G U R E 4 Experimental temperature evolution of layer 5 (at x = 30 mm) during the deposition of a vertical wall consisting of single filaments deposited on top of each other [Color figure can be viewed at wileyonlinelibrary.com]

F

  I G U R E 5 Upper-limit and peaks evolution of data recorded at each layer by (A) IR-camera and (B) type K thermocouple [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 4 Data collected from the difference in peak values (Calculated using "ΔT = T IR-camera -T thermocouple " at each peak) Layer IR-camera and thermocouple temperature difference at each peak ( C)

Figure 7 (

 7 B) could explain the general statement by observing the evolution of point b shifts from 150 C and 70 C by progressive deposition of further layers. In fact, one can note that after a specific deposition of layers (in this case after layer 15) points b and c have almost theF I G U R E 6 Temperature contour at six locations during the deposition of a vertical wall consisting of single filaments deposited on top of each other. Point 1-6 corresponds to the fifth, 20th, 37th, 54th, 63rd, and 88th while indicating 30, 20, 35, 40, 25, and 40 mm from start of deposition, respectively [Color figure can be viewed at wileyonlinelibrary.com] F I G U R E 7 Temperature variation of point a-d at instance of deposition for (A) layer 1 to 5 and (B) layer 15 to 20 [Color figure can be viewed at wileyonlinelibrary.com] same temperature. Worth mentioning to say, point d remains constant after third layer. For instance, the distance across point b and d during deposition of the first layer expresses the inter-layer diffusion zone. It varies from 165 C to 100 C that is approximately between T c and T m . This issue thoroughly explains the cyclic temperature profile discussed before.

ORCID H.R. Vanaei https://orcid.org/0000-0003-1501-0482 M. Shirinbayan https://orcid.org/0000-0002-2757-8529


Pierre Leba
email: pierre.leba@tdf.fr

Jean-Yves Baudais

Stéphane Méric #

Matthieu Crussière

Pierre-Yves Jézéquel

Receivers placement for UAV localization in a surveillance area

Keywords: multistatic radar, CRLB, optimum receivers placement, MLE, UAVs localization

Ensuring security of sensitive areas from unmanned aerial vehicles threats has become a major issue. In this context, multistatic radars present significant advantages over monostatic radars due to exploitation of spatial diversity and improved target detectability. In that perspective, sensors placement has an important impact in target detection or localization performance. In this paper, we propose an empirical method to place receivers in a multistatic radar network in order to cover an area of interest representing a sensitive site. The performance is measured with localization error estimation for different numbers of receivers and different target positions.

I. INTRODUCTION

In recent years, Unmanned Aerial Vehicles (UAVs) have become very popular. Their proliferation leads however to the emergence of new threats. Due to their small size and low velocity flight capacity, UAVs detection and localization is very challenging to radar systems [START_REF] Wellig | Radar systems and challenges for C-UAV[END_REF]. Therefore, ensuring security of sensitive areas from UAVs threats has become a major issue. Multistatic radar is a prominent choice for UAVs detection and localization for various reasons. One can cite for example its capacity to exploit targets Radar Cross Section (RCS) diversity [START_REF] Fishler | Spatial diversity in radars-models and detection performance[END_REF], to support slow moving targets by exploiting Doppler estimates from multiple directions [START_REF] Lehmann | MIMO-radar application to moving target detection in homogenous clutter[END_REF], or for the high resolution target localization it features [START_REF] Lehmann | High resolution capabilities of MIMO radar[END_REF].

One issue in multistatic radar is the sensors placement. This placement has an impact on the target detection performance and on the target localization accuracy. Improving this accuracy by the sensors placement can be addressed either by considering the output Signal to Noise Ratio (SNR) [START_REF] Chitgarha | The detector's output SNR as a criterion for receiver placement in MIMO DVB-T based passive coherent location[END_REF], the angle of arrival [START_REF] Doganc | Optimal angular sensor separation for AOA localization[END_REF], the hyperbolic localization by time difference of arrival measurements [START_REF] Lui | A study of two-dimensional sensor placement using time-difference-of-arrival measurements[END_REF], the elliptical localization by time of arrival measurements [START_REF] Rui | Elliptic localization: Performance study and optimum receiver placement[END_REF], or the variance minimization of range and Doppler shift estimates [START_REF] Griffin | Optimal receiver placement in staring cooperative radar networks for detection of drones[END_REF]. The placement of multiple receivers is optimized depending on the target position and velocity [START_REF] Griffin | Optimal receiver placement in staring cooperative radar networks for detection of drones[END_REF], or depending on the target trajectory [START_REF] Gumiero | Using real data for the implementation of multistatic passive radar geometry optimization procedure[END_REF]. However, as far as we know, no study addresses the placement problem in case of an entire area to be covered.

The contribution of this paper consists of receiver placements to cover a given surveillance area. We propose a method to define how many receivers are needed and where to deploy them in a multistatic radar network in order to meet the localization performance requirement. This method is based on the Cramér-Rao Lower Bound (CRLB) of the target position and velocity estimator. This paper is organized as follows. Section II provides the model description and the theoretical tools on which the method is based. Then, the receivers deployment method is described in Section III. Simulations and results are presented in Section IV. Finally, conclusion and perspectives are discussed in Section V.

II. MODEL DESCRIPTION

The receiver placement is derived with a geometrical configuration of the radar network, consisting in the possible positions of the transmitter, the targets and the receivers in the considered area. This placement is also based on a given waveform leading to particular CRLB of the target parameters, used to define optimality of the receivers placement. In this article, the 3D case is not addressed.

A. Geometrical configuration

A 2D active multistatic radar network consists of one transmitter and N omnidirectional receivers. The receivers are assumed to be all constrained at a same fixed distance R r from the center of the site under surveillance, as illustrated in Fig. 1. The possible receivers locus is then a circle. The position of the nth receiver is defined by the angle φ n with respect to the x-axis and the center of the site, with φ n sorted such that φ 1 ≤ φ 2 ≤ ... ≤ φ N . The surveillance area is modeled as a disk centered at the site center, with a fixed radius R a such that R a < R r . The target position is any point in this disk. Let's introduce θ as the target parameters vector defined by θ = (p x p y v x v y ) T , with p x and p y the Cartesian coordinates of the position, v x and v y the velocity coordinates in the 2D-plane. The distance of the center of the site from the transmitter is ∆x 2 + ∆y 2 .

B. Signal model

The studied scenario is assumed to be multipath free. Each receiver of the network receives a backscattered signal which is a noisy, scaled, time delayed and Doppler shifted version of the transmitted signal with complex envelope u(t). The backscattered signal to the nth receiver is expressed in baseband by:

s n (t) = Au(t -τ n )e 2πifnt + w n (t) (1)
where A is the channel attenuation including the target RCS, τ n and f n respectively its delay and Doppler shift, and w n (t) is a complex zero mean additive white Gaussian noise. In our study, the target RCS behaviour is considered as isotropic. Linear frequency modulation (LFM) pulses are commonly used in radar due to the great bandwidth they offer while keeping the pulse duration short and the envelope constant. The LFM is the signal waveform used in this article. This waveform is characterized by its four conventional parameters: T the pulse length, T PRI the pulse repetition interval, M the number of pulses and B the bandwidth.

C. Delay-Doppler estimation for receiver placement

The ambiguity function of the LFM u(t) is used to estimate the delay-Doppler that characterizes the target. With one receiver n identified by its position φ n , the minimum variance unbiased estimations of τ n and f n are given by the CRLB of the joint estimation [START_REF] Griffin | Optimal receiver placement in staring cooperative radar networks for detection of drones[END_REF] [11]:

σ 2 r = c 2 d (T 2 + T 2 PRI (M 2 -1)), σ 2 f = B 2 d , σ rf = cBT d (2)
with d = 1 3 π 2 B 2 T 2 PRI (M 2 -1
) and c the light velocity. Assuming Gaussian error, the corresponding measurement model achieved by the MLE at high SNR regime is [START_REF] Griffin | Optimal receiver placement in staring cooperative radar networks for detection of drones[END_REF]:

τn = τ n (θ) + e τn and fn = f n (θ) + e fn (3)
where the covariance matrix of the error is:

E (e τn e fn ) T (e τn e fn ) = 1 SNR n σ 2 r σ rf σ f r σ 2 f (4)
and SNR n is the SNR at the receiver n.

With one receiver, the CRLB depends on θ, φ n and on the LFM parameters. It depends also on the geometry of the configuration through the bistatic delay, or range, and Doppler equations [START_REF] Skolnik | Radar Handbook[END_REF]. With N receivers, the measurement model combines N delay and Doppler estimations. The CRLB is then derived from the general Gaussian case [START_REF] Kay | Fundamentals of statistical signal processing[END_REF], and depends on the N receiver positions (φ 1 , • • • , φ N ). This CRLB depends on the given positions of the transmitter, the target and the receivers.

Many criteria can be used to optimize the receivers placement [START_REF] Atkinson | Optimum experimental designs[END_REF]. In this study as in [START_REF] Griffin | Optimal receiver placement in staring cooperative radar networks for detection of drones[END_REF], the trace of the CRLB is adopted as the criterion to define receivers placement when a single target configuration is considered, that is for a given position and velocity of the target. The trace of the CRLB, as a function of the receiver positions, is then minimized, leading to the optimal receivers positions, according to the criterion used. However, these optimal positions depend on the target parameter θ and do not ensure optimality for the whole site under surveillance. Instead of the complex N -dimension optimization problem, the proposed algorithm that solves the placement of N receivers is presented in the following section.

III. RECEIVERS PLACEMENT METHOD

The presented method aims at placing N receivers in order to cover the localization of any target present in the surveillance area. The receivers placement procedure is synthetically described in Algorithm 1 and is organized in 3 parts:

1) lines 1 to 6, the distribution of the optimal positions of N = 2 receivers is first evaluated; 2) lines 7 to 11, the quantile regression is applied to set the positions of N ≥ 2 receivers; 3) lines 12 to 17, the RMSE of the asymptotic MLE of θ is computed to evaluate the performance of the proposed algorithm.

Algorithm 1 Receivers placement and validation algorithm.

1: Set N = 2, K ∈ N 2: for j = 1 to K do 3:
Generate the parameter (θ) j of the target 4:

{(φ 1 ) j , (φ 2 ) j } ← arg min trace(CRLB(θ) j ) 5: end for 6: Compute PDF i and CDF i of φ i , i ∈ {1, 2} 7: Set N ∈ 2N 8: for i = 1 to N/2 do 9:

φ i ← CDF -1 1 i N/2+1 10: φ i+N/2 ← CDF -1 2 i N/2+1
11: end for 12: for j = 1 to K do 13:

Generate the parameter (θ) j of the target 14:

( θ) j ← MLE (φ 1 ) j , • • • , (φ N ) j 15: ε 2 j ← (θ) j -( θ) j 2
16: end for 17: Output the error and the RMSE

The first part of the procedure aims at providing empirical Probability Density Function (PDF) and Cumulative Distributive Function (CDF) of optimal positions of N = 2 receivers, for a set of K targets mapping the surveillance area. The algorithm starts with determining optimal locations of N = 2 receivers for each given target located in the surveillance area, with optimality understood as the minimization of the trace of CRLB, as introduced in Section II-C. The process of determining the 2 receivers placement is then repeated for K = 1575 targets uniformly distributed in the surveillance area. Thus, each processed target yields pairs of receivers positions φ 1 and φ 2 that are collected to produce the PDF and CDF of each of the receivers positions, denoted by PDF i and CDF i for the ith receiver, i ∈ {1, 2}. Note that because of the symmetry of the geometrical configuration with respect to the transmitter-site axis, PDF 1 and PDF 2 distributions are symmetrically arranged with respect to the angle Φ a defined by the transmitter-site axis.

In the second part of the algorithm, the placement of N ≥ 2 receivers is based on the PDF and CDF obtained with N = 2. Yet, due to the PDF 1 and PDF 2 symmetry, the case of N receivers placement is considered only for N even. In each of the two PDF i distributions, N/2 angular positions are selected. The (N/2 + 1)-quantiles is the used criterion to select the N/2 receivers angular positions in each CDF i distribution. For N = 4, these are 3-quantiles (or tercile) that are considered to determine the 2 receivers placement in each PDF distribution, PDF 1 and PDF 2 . The sets of N optimal receivers positions are thus defined. Note that the optimality of the receivers position applied to a covered area is defined here in the sense of quantile regression and should not be confused with the optimality as defined in Section II-C when considering a single target configuration.

It is worth noting that target localization with one transmitter usually requires at least 3 receivers, as in the localization by elliptical intersection. However, when additional assumptions are available, one may be able to disambiguate incorrect candidate intersections with 2 receivers only. As in the configuration, such conditions are met when the distances from the target to each of the 2 transmitters are of the same order. Due to the limitation of space, the demonstration of this result is not presented in this paper.

Finally, in the third part of the algorithm, the performance of the N receivers positions obtained in the second part or the algorithm is then evaluated with the error and the RMSE of the estimated parameters θ. To this end, the surveillance area is mapped as in the first part by a set of K targets uniformly distributed within this area. The asymptotic MLE is performed for each target configuration. This estimator is based on the asymptotic parameters estimation given by (2) for each receiver.

IV. RESULTS

In order to materialize the targets in the surveillance area, Monte Carlo simulations are done with target positions uniformly distributed in the area. The targets velocities are also generated with a uniform distribution from 0 to a fixed maximum velocity set at 5.6 m/s as an example and with a uniform distribution of the direction from 0 to 2π. The transmitter power is tuned such that the SNR, as the reference value, is set to 20 dB at a distance of 1000 m from a target placed at the center of the surveillance site. For other target placement, the SNR is evaluated regarding this reference, the target position, and the receiver position. Table 1 provides numerical values of the multistatic network configuration. The Nelder-Mead simplex method is used both for the minimization of the trace of the CRLB, and for the asymptotic MLE.

According to Algorithm 1, the first step is dedicated to calculate the positions (φ 1 ) j and (φ 2 ) j regarding the jth target position. As an example, Fig. 2 displays the trace of CRLB for all combinations of φ 1 and φ 2 with φ 1 ≤ φ 2 , for a given target configuration θ = (1894 m, 3000 m, 1 m/s, 2 m/s) T . This target configuration yields (165°, 295°) as optimal positions (φ 1 ) j and (φ 2 ) j , obtained with the lower trace of the CRLB. These optimal positions and the ones obtained for each random target configuration are used to get the empirical PDF 1 and PDF 2 presented in Fig. 3. The optimal receivers positions are not scattered everywhere from 0 to 2π, but are structured as two clustered PDF distributions. As expected, PDF 1 and PDF 2 are symmetrically arranged with respect to Φ a = 236°. This cluster shape of the distribution can be explained: recall that in the case of a single target, the receivers placement would lead to a single position of each receiver. Even though, there are multiple targets velocities and positions mapping the surveillance area, when the receivers distance to the site is far greater than the surveillance area dimension, this latter is then viewed by the receivers as a single point. This can be seen as the extreme case of a clustered pattern.

Table 2 presents the sets of receivers positions obtained The method can be generalized for higher number N of receivers. According to the minimum estimation accuracy fixed by an acceptable RMSE for the target parameters, N can be deduced, hence fixing one of the sets of N receivers positions. The question of how many receivers to deploy and where to deploy them is thus addressed.

V. CONCLUSION In this article, we propose to study a multistatic radar system for the purpose of detecting UAVs inside a sensitive area. The receivers placement issue is addressed through an empirical method considering a fixed transmitter and several receivers around the area to be covered. We present a method to select the number of receivers and to place them according to the maximum admissible RMSE on target parameters estimation. This method exploits the CRLB of the position and velocity estimation of the target, and defines the optimal receivers position according to a quantile regression.

Fig. 1 .

 1 Fig. 1. Multistatic configuration.

Fig. 3 .

 3 Fig. 3. PDF 1 and PDF 2 of 2 receiver positions.

  Fig. 4 plots the set of the error ε j of each target parameter, for each set of N ∈ {2, 4, 6} receivers positions. It exhibits the fact that the accuracy of the position and velocity estimations also increases with the number of receivers. The maximal value of the position errors can be compared to the range resolution of the LFM, which is c 2B = 150 m: the range resolution is then increased for all the target positions within the surveillance area.

Fig. 4 .

 4 Fig. 4. Error of target parameters estimation.

Table 1 .

 1 Model Parameters.

	Parameter	Value
	Area center position, (∆x, ∆y)	(2000 m, 3000 m)
	Transmitter-area angle, (Φa)	236°S
	urveillance area radius, Ra	150 m
	Maximum target velocity	5.6 m/s
	Receiver to area center distance, Rr	1000 m
	Bandwidth, B	1 MHz
	Pulse Repetition Interval, T PRI	100 µs
	Number of Pulses, M	16
	Pulse Length, T	50 µs
	Wavelength, λ	0.21 m

Fig. 2. Normalized trace of CRLB (dB) for one given target configuration.

Table 2 .

 2 Optimal angular positions and RMSE of N receivers.

				Receivers positions			RMSE
								p	v
	N	1	2	3	4	5	6	(m)	(m/s)
	2	166°306°12.2	5.4
	4	160°170°302°312°8.8	3.8
	6	158°166°173°299°306°314°7.0	3.0

using quantile regression on CDF 1 and CDF 2 . This table shows that the RMSE decreases with the number of receivers used to cover the surveillance area, leading to an increase in accuracy of the target parameters estimation.

ACKNOWLEDGMENT

The authors thank the DGA (Direction Générale de l'Armement) for funding this study through the AMBRA project (Active Multistatic Broadcast RAdar), and especially P. Pouliguen and H. Ruggiero (DGA-AID agency). The authors thank P. Kasser (TDF innovation agency) for assisting them.