Gourab Ghatak
email: gourab.ghatak@iiitd.ac.in

Antonio De Domenico
email: antonio.de-domenico@cea.fr

Marceau Coupechoux
email: marceau.coupechoux@telecom-paristech.fr

Relay Placement for Reliable Ranging in Cooperative mm-Wave Systems

We study millimeter wave based ranging of randomly located terminal nodes (TN) using fixed relay nodes (RN) deployed around a central node (CN). This setting may correspond to a disaster-relief scenario where the rescuers require positioning information in the absence of a global positioning system (GPS). We derive the Bayesian Cramer-Rao lower bound (BCRLB) for the TNs range estimation from the CN as well as from the RNs in this network using a stochastic geometry framework. Contrary to existing studies, we take the effect of link-blockages into account while deriving the BCRLB, and thereby present a more accurate bound on the ranging error. For the special case of no blockages, we formulate a convex problem for obtaining the optimal relay positions. Our results provide the operator a guideline for initial deployment planning, in terms of number and location of RNs to be deployed in order to achieve an accurate ranging.

I. INTRODUCTION

Reliability-constrained wireless applications, especially those pertaining to disaster-relief are becoming more important every day. Such use cases often require cyber-physical systems capable of operating with wireless links of high resilience, for which today's wireless networks are not optimized. One challenging issue in deploying such systems is the accurate positioning of the terminal nodes (TNs) of the network [START_REF] Popovski | Ultra-reliable communication in 5G wireless systems[END_REF]. In contrast to the traditional positioning techniques [START_REF] Yassin | A survey of positioning techniques and location based services in wireless networks[END_REF], localization using millimeter wave (mm-wave) signals offers a more attractive solution, precisely due to the higher temporal resolution and high directivity [START_REF] Lin | 3-D Indoor Positioning for Millimeter-Wave Massive MIMO Systems[END_REF]. However, signals at such high frequencies suffer from high transmission losses and high sensitivity to blockages [START_REF] Rappaport | Millimeter wave mobile communications for 5G cellular: It will work![END_REF]. In case the communication link state changes from line of sight (LOS) to non-LOS (NLOS), severe degradation of localization performance may occur, which may prove detrimental. The aim of this letter is to study the enhancement of the mm-wave ranging performance of the TNs through cooperation. This is facilitated by anchoring some of the TNs and employing them as relay nodes (RNs) to perform the positioning of other randomly located TNs.

Lemic et. al. [START_REF] Lemic | Localization as a Feature of mmWave Communication[END_REF] have shown that mm-wave signals can be used for very accurate positioning of nodes, even with a limited number of anchor nodes. Although extensive literature is available pertaining to RNs for communication reliability enhancement in ad-hoc wireless networks, only a few focus on RNs for localization (e.g., see Dardari et al. [START_REF] Dardari | Enhanced localization coverage with nonregenerative UWB relays[END_REF]). In this regard, Zhang et al. [START_REF] Zhang | Cooperative localization in 5G networks: A survey[END_REF] provided a brief survey of cooperative localization in 5G networks, where they discussed the benefit of exploiting the location information from additional localization measurements between TNs. However, the authors only provided a schematic for performance evaluation in terms of localization accuracy and did not provide any quantifiable gain. Coluccia et al. [START_REF] Coluccia | RSS-based localization via Bayesian ranging and iterative least squares positioning[END_REF] have recently investigated RSSbased localization via Bayesian ranging and iterative least squares positioning. They have compared several range-based techniques in terms of their accuracy and computational cost. The evaluation of the statistical localization performance for random positions of TNs has relatively sparse literature. One recent work by O'Lone et al. [START_REF] Christopher | A statistical characterization of localization performance in wireless networks[END_REF] has statistically characterized the localization performance of a single user placed anywhere throughout a wireless network, where anchors are distributed according to a Poisson point process (PPP).

The contributions of this work can be summarized as follows. We study the ranging accuracy of randomly located TNs, characterized by their Bayesian Cramer-Rao lower bound (BCRLB) of ranging from the perspective of a central node (CN), and the nearest RN. We have accounted for the pathloss and the signal degradation due to physical blockage on the resulting bounds, which are generally ignored in the literature [START_REF] Dardari | Enhanced localization coverage with nonregenerative UWB relays[END_REF]. Leveraging on this, and using tools from stochastic geometry, we mathematically characterize the enhancement in the ranging performance of the TNs by employing some TNs as RNs. Finally, we optimize the RN positions for the case without blockages and study the effect of the number of RNs in the network on the joint ranging accuracy of the TNs. Thus, our framework can be used for initial deployment planning for such a network.

Notations: In this letter, the positions of the nodes are depicted by bold font (e.g., x), and the scalar variables are depicted by lowercase and uppercase normal math font (e.g., x and R). Sets are denoted by math calligraphic font (e.g., S). The symbols ||.|| refer to the norm of the argument (e.g., ||x n -x 0 || is the distance between x n and x 0 ). II. SYSTEM MODEL We consider the deployment area to be a closed bounded subset S of the two-dimensional Euclidean plane. For simplicity, we can consider S to be a diskof radius R. A controlling CN is assumed to be located at the center of S. Furthermore, we assume that there are N T TNs distributed uniformly in S. Generally, search and rescue operations consist of a fixed and limited number of nodes [START_REF] Malone | Wireless search and rescue: Concepts for improved capabilities[END_REF]. Accordingly, we assume N T to be fixed. Thus, the positions of the TNs: {x i }, i = 1, . . . N T , form a binomial point process (BPP).

To increase the reliability of ranging through cooperation, N R (≥ 2) TNs are employed as RNs, which feedback local ranging measurements to the CN using a reliable channel. Consequently, N = N T -N R are the remaining TNs. The RNs are assumed to be located at known positions r j ∈ S, j = 1, . . . N R , each at a distance ||r j || = p (which we optimize in our work) from the CN and separated from each other by an angle of 2π N R . The placement of the CN at the center follows the assumption of uniform distribution of the users in the area. In a real deployment, depending on the actual TN positions, the network operator needs to further fine-tune the CN and TN positions for that network. It must also be noted that the presence of a single CN relates to a single point of failure for the network operation which may prove to be detrimental in case of search and rescue operations. It is thus important to note that our work can be easily extended to multiple interconnected CNs located at designated positions of the network. Naturally, the ranging performance would improve with multiple CNs. Although our model allows for the non-isotropic placement of RNs, due to the symmetry of the problem, we focus on a single sector and we define the x-axis as the straight line between the CN and the RN of the sector. Consequently, we simplify the notation and let (p, 0), 0 ≤ p ≤ R be the coordinate of the RN on the x-axis in the studied sector. Accordingly, we use the terms RN position and RN distance interchangeably.

We assume that the mm-wave communication links are blocked by physical objects locations of which are modeled as atoms of a point process independent of the TN BPP (see e.g., [START_REF] Bai | Analysis of Blockage Effects on Urban Cellular Networks[END_REF] for discussions on blockages and their effects on data-communication). The probability of a TN located at x i to be in LOS with respect to a reference point x 0 , is denoted by p [START_REF] Bai | Analysis of Blockage Effects on Urban Cellular Networks[END_REF], where β is the blockage density. Since highly resilient signals are required for accurate ranging and mm-wave signals can suffer a high degree of penetration loss due to blockages [START_REF] Rappaport | Millimeter wave mobile communications for 5G cellular: It will work![END_REF], we assume that ranging over blocked links is unfeasible. Let G C , P C and G R , P R be the directional antenna gains and the transmit powers of the CN and RN, respectively. Our aim is to study the optimal value of p for a given N R and N T from the perspective of ranging accuracy of the TNs. The ranging accuracy is calculated in terms of theoretical bound on the error of estimation (i.e., the BCRLB) of the distance of the TNs from either the RN or the CN.

L (||x i -x 0 ||) = exp(-β||x i -x 0 ||)
Before deriving our main results, we would like to state that in this work, we derive the BCRLB by considering the order statistics of the TNs rather than assuming uniform distribution of each TN. The main motivation for using the order statistics of the TN locations is to characterize individual ranging performance of the TNs as a function of their order. This is useful when the operator wants to guarantee ranging performance for individual nodes.

III. BCRLB FORMULATION

In this section, first we derive the relevant distance distributions of the TNs from the CN and the RN. Let us assume that the indexes of the TNs are arranged in increasing order of their distances from the CN, i.e., ||x i || < ||x k || for i < k. From the perspective of the CN, we have the following lemma.

Lemma 1. The pdf of the distances of the n th closest TN (with location x n ) to the CN is given by:

f C xn (x) = 2 R Γ n + 1 2 Γ (N + 1) Γ (n) Γ N + 3 2 B x 2 R 2 ; n + 1 2 , N -n + 1 ,
where Γ(•) and B(•) are the gamma, and the beta functions, respectively. Proof. This follows from the void probabilities of BPP [START_REF] Srinivasa | Distance distributions in finite uniformly random networks: Theory and applications[END_REF].

Then, for a TN located at a distance ||x n || from the CN, the distribution of its distance from the RN is derived in the following Lemma.

Lemma 2. For a TN located at a distance ||x n || to the CN at an angle ψ from the x-axis, the probability density function (PDF) of its distance to the RN, conditioned on ψ, is:

f R xn (y|ψ) = F 1 (y, ψ) f C xn p cos ψ + y 2 -p 2 sin 2 ψ + f C xn p cos ψ -y 2 -p 2 sin 2 ψ
where

F 1 (y, ψ) = y √ y 2 -p 2 sin 2 ψ
. Here, ψ is uniformly distributed between 0 and π N R . Proof. For a TN present at a distance x and at an angle ψ from the CN, its squared distance from the RN is:

y 2 = x 2 + p 2 -2xp cos (ψ).
As a result the cumulative density function (CDF) can be calculated as:

F Y (y|ψ) = P (Y ≤ y|ψ) = P x 2 + p 2 -2xp cos ψ ≤ y 2 = P (x -p cos ψ) 2 ≤ y 2 -p 2 sin 2 ψ
We now consider two cases 1 . Case I: x ≥ p cos ψ, for which:

F Y (y|ψ) = P x ≤ p cos ψ + y 2 -p 2 sin 2 ψ = F C X (p cos ψ + y 2 -p 2 sin 2 ψ).
And case II: x < r cos ψ, for which: F Y (y|ψ) = P p cos ψ -p < y 2 -p 2 sin 2 ψ = P p > r cos ψ -y 2 -p 2 sin 2 ψ = 1 -F C X (p cos ψy 2 -p 2 sin 2 ψ). The result then follows from adding the two possibilities and differentiating with respect to y.

Leveraging on the distance distributions, we calculate the BCRLB of the TNs as given in the following theorem.

Theorem 1. The BCRLB for the estimation of the distance of the n th TN from the CN or the RN using averaged received signal strength indicator (RSSI) is calculated as:

B t (x n ) = E dn,t σ -2 t (x n ) + d dx log f t xn (x) -1 , (1)
where, d n,t := ||x n -x t || and t = {C, R} depends on whether the estimating node is the CN or an RN. The distance distribution is obtained as f dn,t (y, ψ) = f t xn (y|ψ)f ψ (ψ), where f ψ (ψ) follows uniform distribution of ψ between 0 and π N R . In this expression, σ 2 t (x n ) is given by (2), which is the minimum variance of an unbiased estimator of the range of a TN from x t (a CN or an RN).

Proof. Let the RSSI measurement at the locating node (RN or CN) of a TN be perturbed by a zero mean additive white Gaussian noise (AWGN), N , of variance σ 2 n . The average RSSI thus can be written as:

Y = KP t G t d -α n,t exp (-βd n,t ) + N , (3)
σ 2 t (x n ) = d -α-2 n,t exp (-βd n,t ) 2σ 2 n β 2 KP t G t d 2 n,t + 2αβKP t G t d n,t + α 2 KP t G t -1 ; (2)
where α is the path-loss exponent and K is the path-loss coefficient.

Let µ = Y -KP t G t d -α n,t exp (-βd n,t ) that is distributed as N (0, σ 2 n ). The likelihood is thus:

L (µ) = 1 √ 2πσ 2 n • exp - y -KP t G t d -α n,t exp (-βd n,t ) 2 2σ 2 n
As a result, the Fischer information can be derived as:

J = ∂ 2 ln (L (µ|x n )) ∂d 2 n,t = d -α-2 n,t exp (-βd n,t ) 2σ 2 n • β 2 KP t G t d 2 n,t + 2αβKP t G t d n,t + α 2 KP t G t . (4)
The second term in (1) is the prior information of the distances of the TNs. Inverting the sum of the prior information and J, we obtain the BCRLB.

Note that given a number of relays being deployed in the network area, the expected value of the BCRLB for the estimation of the angle of arrival (AoA), ψ, given by

E ψ σ 2
ψ is a multiple of σ 2 d [START_REF] Shahmansoori | 5G Position and Orientation Estimation through Millimeter Wave MIMO[END_REF], [START_REF] Garcia | Location-aided mm-wave channel estimation for vehicular communication[END_REF]. As a result the ranging performance metric can be easily extended to a localization performance metric by re-writing the objective function as a linear combination of σ 2

d and E ψ σ 2 ψ .

IV. RN PLACEMENT PROBLEM The BCRLB provides us a convenient objective for the RN placement problem, as it is the expected value of the minimum variance of an unbiased estimator of the TN ranges. However, for reliable applications, merely a constraint on the expected ranging performance of the TNs does not suffice, and performance guarantees on the ranging performance of each TN must be addressed. Accordingly, we formulate the optimal RN placement problem as:

p * = min p N i=1 min{B C (x i ), B R (x n )} subject to 0 ≤ p ≤ R ∀i ∈ {1, . . . , N }, P (min{B R (x i ), B C (x i )} ≤ ) ≥ η (5)
The objective is the sum of the point-wise minimum of the ranging error bounds of the TNs from the CN and the nearest RN, respectively. The constraints on the individual BCRLBs give a performance guarantee of η that the ranging accuracy is at least for any realization of {x n }.

Evidently, the problem ( 5) is difficult to solve due to several issues. The objective function contains the prior terms, E log f t xn (x) which are non-convex. Moreover, the pointwise minimum in the final n constraints are also not convex, thereby making the overall problem difficult to tackle. As a result, we present a modified problem that uses the expected distances of the TNs for acquiring a coarse estimate of the RN ranges. For the special case of β = 0, the modified problem becomes a convex optimization problem. For the general case, the modified problem avoids the calculation of the point-wise minimum in the objective function. Let the expected locations of the TNs be given by xn . Then, the corresponding distances from the CN can be denoted as i.e,

|| xn || = R 0 xf C xn (x).

A. Certainty Equivalent (CE) Formulation

As the BCRLB increases with increasing distance of the CN and the TN (this can be easily seen by differentiating (4) with respect to d n,t ), we note that there exists an index k such that we have B C ( xi ) < , ∀i ≤ k. Thus, for all TNs that are nearer to the CN than || xk ||, the BCRLB for the estimation of their ranges from the CN is less than . In other words, the distances of these TNs are efficiently estimated by the CN. Accordingly, we propose a modified problem that considers the placement of RNs for the TNs whose distances from the CN are larger than || xk ||. The modified problem given as:

p * = min p N i=k B R ( xi ) subject to 0 ≤ p ≤ R ∀i = k, . . . , N B R ( xi ) ≤ (6)
where, k = argmax n 1 (B C (x n ) < ). Thus, the modified problem (6) essentially calculates the optimal RN placements with respect to the average ranges of the TNs k, k+1, . . . , N T .

Corollary 1. For β = 0, each term of the objective function in (6) and the last N-k constraints are of the form 2σ 2 n α 2 KP t Gd α+2 n,t , which is convex with respect to p. Thus, the problem (6) is convex for the special case of no blockages.

The solution of (6) may not be optimal for several realizations of the TN positions, as they are governed by their distance distributions rather than the expected values. However, following the solution of (6), the operator can perform a robust estimate for the RN placement. In other words, the solution to (6) acts as a guideline for an operator for initial planning of such a network. This can be further fine-tuned according to the specific environment and use case.

Although a detailed discussion of the stochastic placement of the RNs is out of scope of this letter, in the next section, we present some numerical results that show how the placement of RN following the solution of (6) even for the case without blockages, improves the localization performance of the TNs. Furthermore, we also show that depending on the area of the deployment region, the gain in localization performance by adding more RNs saturates after a certain number of RNs.

V. NUMERICAL RESULTS AND DISCUSSION For evaluating the system performance, we have assumed P C = 30 dBm, P R = 20 dBm, G C = 20 dBi, and G R = 10 dBi. The path-loss coefficient K is derived from the 3GPP specifications [START_REF] Gpp | TR 38.900[END_REF]. The path-loss exponent are assumed to be α = 2 for LOS propagation. We assume that ranging is infeasible for NLOS links. The blockage parameter is assumed to be β = 0.007 m -1 [START_REF] Bai | Coverage and Rate Analysis for Millimeter-Wave Cellular Networks[END_REF]. We assume a carrier frequency of 60 GHz and a bandwidth of 1 GHz. In Fig. 1a we plot the optimal RN placement for N R = 3 and N R = 5. Interestingly, we see that as the number of RNs increase, the optimal placement of the RNs from the center increases. This is because a smaller N R results in a larger sector. As an analogy, we can recall that the center of mass of a sector of a uniformly dense disk is nearer to the center of the disk if the sector is larger. Similarly, with uniformly distributed TNs, for larger sectors (i.e., with less number of RNs), the RNs should be placed nearer to the center to guarantee good ranging performance of the TNs located farther from the x-axis in the representative sector. In Fig. 1b, we plot the BCRLB of the sum of the mean ranging errors of the TNs with optimally placed relays, with respect to the deployment radius. As the deployment radius increases, the ranging accuracy decreases due to increasing number of TNs located farther from the CN. More interestingly, in case of small deployment radius (e.g., R = 1 km), the gain with more number of RNs (5 as compared to 3) is not appreciable. In Fig. 1c we plot the percentage gain in ranging performance by cooperation as a function of the number of RNs in a 1 km radius. Mathematically, it is represented as:

%Gain = B * -B 0 B 0 × 100 (7)
where B * is the sum of BCRLBs of ranging errors from the RNs, i.e., the the objective function in [START_REF] Dardari | Enhanced localization coverage with nonregenerative UWB relays[END_REF] for the case when relays are optimally placed, whereas, B 0 is the BCRLB of the ranging errors of the TNs for the case when only CN performs the ranging. First we note that with cooperation, the ranging performance increases significantly (by upto 80%).

Furthermore, the additional gain with more than 4 RNs is not appreciable. Thus, as long as four quadrants are covered by the RNs, the limits of gain for localization with cooperation is achieved. In case the operator wants to improve the ranging performance even further, she/he must deploy more effective cooperative algorithms, since deploying more RNs simply saturate the gain. It must be noted that for a larger deployment area, the number of RNs to achieve the gain limit may increase. Thus, a study to account for the cost of using more TNs as RNs, from the deployment perspective is necessary, which we will address in a future work. VI. CONCLUSION We have investigated the placement of relay nodes to augment the ranging accuracy of the terminal nodes in a mmwave network. Appropriate placement of the relays for local ranging may improve the ranging performance by up to 80 percent as compared to utilizing only the central node. The ranging accuracy gain saturates when using a large number of relays. Our results provide the operator a guideline for initial planning in terms of the number and placement of relays to support given ranging accuracy requirements. In this regard, a parallel study of the cost of relay provisioning is essential, which we plan to carry out as future work.

Fig. 1 .

 1 Fig. 1. (a) Optimal RN placement for different number of RNs, (b) Bound on the ranging error vs deployment radius, and (c) Percentage gain in localization accuracy with cooperation as a function of the number of RNs.

Note that y is always greater than or equal to p sin ψ as the later is the perpendicular distance from the TN to the x-axis.


J Graffeuil
email: graffeui@laas.fr

L Escotte
email: escotte@laas.fr

J G Tartarin
email: tartarin@laas.fr

Low Frequency Noise Deviation from Schottky theory in p-n junctions

Keywords: Low frequency noise, diffusion noise, noise frequency dispersion

Theories on linear white noise sources such as thermal noise or shot noise are well established and massively used for low noise device modeling and circuit design. However, it has been experienced that diffusion noise in a large variety of pn diodes (transistors) can deviate from the expected value given by the Schottky theorem or by the Van der Ziel representation commonly used. In this work, more than ten types of pn junctions have been investigated, all featuring an increase of the diffusion noise floor in the low frequency band when operated under low d.c current conditions. These specific conditions certainly explain why such phenomenon has not been reported earlier; however, this noise degradation becomes a problem as many systems make use of pn diodes for low signal photodetection (PPD or CCD), operating at very low (dark) current. For the first time, we report current spectral densities deviations from the Shottky theorem at low frequency; a focus on the experimental workbench is given to remove any doubt regarding the opportunity to analyze data under concern. Then, low frequency noise spectra are presented for various diodes and pn junctions, and a model is proposed. Impedance spectroscopy is also used to support this study.

I. INTRODUCTION

Low frequency noise of junctions is still of great interest as it is massively used for the noise modelling of solid state devices such as diodes and transistors. The well-established theory formulated by W. Schottky one hundred years ago (and first demonstrated in ideal vacuum tubes [1]) still applies and lays down the basic principles of electronic sensitivity for receivers, detectors or any electronic system facing the case of minimum detectable signal. From this theory, a d.c current I flowing through a barrier of potential will generate a current noise spectral density as Si=2q(I+2Is), as established by Van der Ziel [START_REF] Van Der Ziel | Noise[END_REF]. In this equation, the contribution of saturation current IS is added to the main d.c current I initially considered by Schottky in its formulation. Some exceptions have been investigated, due to generation-recombination processes occurring in the space charge region. This mechanism can be expressed by the Fano factor γ = Si/(2q(I+2Is)) where 0.5 ≤ γ < 1, depending on the overall current, on the frequency and on the temperature. In this paper, a new low frequency spectral signature is presented, never previously reported to the authors' knowledge. Still using the normalized γ expression of the noise floor (but not referred to as Fano factor), this factor deviates from unity beyond a given frequency, and stabilizes at a value ranging from one to more than six as depicted in figure 1 (and more than ten in specific conditions). This noise degradation phenomenon has only been observed on silicon p-n diodes at low frequency and for very low biasing conditions. Various emitter-base or collector-base junctions have been investigated in silicon transistors such 2N2222, 2N2905, BC557 for the most commonly used devices. In section 2, the Low Frequency Noise (LFN) experimental workbench is presented, and measurements are performed with various pre-amplification schemes in order to validate reported data. Given the very low noise spectral densities involved, particular attention is paid to the instrumentation used for the characterization of the LFN, and to the deembedding of the diode noise from various contributions (amplifiers termination versus frequency, noise contribution and correlation of amplifier noise sources) to the total noise. This warrants a rigorous characterization of the Devices Under Test (DUT) noise. Calibration steps and measurements are presented in this section. The third section is dedicated to the modeling and analysis of the observed phenomenon versus d.c current. Diode impedance spectroscopy is used in order to substantiate LFN data from a newly proposed model. Then it is easy to express Si as in equation 2. It can be noticed that the term in red from equation 1 (and equation 2) corrects the fraction of the current flowing to the DUT instead of the preamplifier (it vanishes if Rin is very low).

Finally different challenges must be considered simultaneously in order to be able to observe the frequency deviation from the accepted formulation of Si derived from the d.c current I+2Is.

1-The 1/f contribution should be negligible or easily corrected at low frequency. This is the reason why we were able to investigate only silicon devices. No GaAs diode has been found sufficiently rid of flicker noise in order to warrant the diffusion noise floor characterization.

2-The low d.c biasing conditions at which the phenomenon appears implies low Si noise current levels. Therefore high sensitivity and high gain are needed for the voltage-current preamplifier, at the cost of a reduced bandwidth. Low amplifier noise, that warrants accuracy, and wide frequency bandwidth cannot be achieved simultaneously (as a consequence of the well-known properties of any current amplifier), and tradeoff with various experimental conditions are developed and compared. Moreover, the selection of a high gain (and a high sensitivity that corresponds to a low noise current Sia) is associated with an increase in the input impedance of the preamplifier EG&G 5182. This means that a lower fraction of the diode noise current is collected by the amplifier, but fortunately Rd also increases with I decreasing that makes the measurement still possible. It must be noticed that, even if the amplifier gain selection for the better accuracy over a given bandwidth at a given bias is always a challenge, it can nevertheless be handled (see the recovery of spectra obtained for different frequencies and gains in Figure 1). Figure 3 compares measurements performed with different preamplifier configurations (current amplifier, voltage amplifier and two current amplifiers for cross-correlation measurements) at a not too low current (in order to warrant voltage amplifier measurements) on the same device. Although the DUT noise deembedding from the measurements is very different for each technique, the final results are similar and demonstrate the same noise enhancement. Voltage amplifiers are not suitable for those measurements as the DUT impedance must remain much lower than that of the amplifier, and this is only achieved at elevated d.c current where the investigated phenomenon is no more present. Correlation technique has also been used with no noticeable improvement.

3-The impedances used in equation 1 and 2 must be accurately characterized versus frequency for each gain selection of the preamplifier (Rin and Cin) or d.c biasing of the DUT (Rd and Cd, obtained from impedance spectroscopy, see section III) in order to warrant a correct noise deembedding. Furthermore, noise of parallel R-C cells using discrete elements is measured as a test; resistance and capacitor values are chosen in the range of the measured diode Rd-Cd values. As expected, measured γ normalized values of the R-C cell are nearly equal to unity over all the measured bandwidth, as depicted in Figure 3 for a given R-C configuration. This last calibration test gives consistency to the measurements presented in Figure 1 and in Figure 4 where γ is reported versus frequency for three different silicon devices (and for various d.c current in Figure 4).

B. LFN deviation from diffusion noise floor

More than ten silicon devices have been investigated, with a special focus on many 2N2222, 2N2905 and BC557 devices.

Figure 4 plots the evolution of the γ values versus frequency, and for many d.c currents. Whatever the considered DUT, the same trend is observed: the corner frequency shifts towards higher frequencies when increasing d.c current. The plateau magnitude also increases with I, but then the frequency band limitation does not allow to develop any conclusions about the probable decrease of this plateau beyond a second frequency corner.

Similar trends on the variation of γ versus frequency have been observed for different technologies. Deviation from the unity expected value can be large (up to more than 10). As a result of this noise increase, significant errors can appear on CCD detector equipment's if this phenomenon is not properly accounted for. In the next section, a model is developed that correctly describes LFN measurements. Junction impedance spectroscopy is performed in order to substantiate this model.

III. BEHAVIOUR VERSUS D.C CURRENT

From the LFN spectra measurements, a model is proposed that fits all the measured noise spectra of the devices under test, as illustrated by the dotted line representation from Figure 4 a) b) c). This model is depicted in Figure 5: serial Radd-Cadd elements are added to the conventional diode equivalent network in order to match both the LFN and impedance spectroscopy measurements. Moreover, the variation of these additional electrical elements with I shows similar features for the three devices under test as shown in Figure 6. An inversely proportional relation of Radd with I for all the tested devices suggests that Radd is closely related to the material, as for the main diode resistance Rd (=nUT/I) in figure 5 (crosses in Figure 6).

The capacitances share the same behavior with the biasing d.c current in Figure 6 (in spite of a slightly higher value for the 2N2222 diode): once again, it must be mentioned that the LFN measurements have been performed over different frequency bands, still featuring the same increase from the conventional noise floor (diffusion noise).

Moreover, it must be noticed that Radd produces thermal noise in the proposed model. Thus Radd-Cadd cannot be interpreted as a time constant or as a trap. Therefore, the spectral current noise Si(f) can be modelled by Equation 3: 5 that can therefore be used to describe accurately the diffusion noise enhancement in silicon diodes, even if no data allows us to conclude about the higher frequency behavior (as a decrease of γ is expected!).

IV. CONCLUSIONS

In this paper, a deviation from the conventional diffusion noise theory has been evidenced below 1 MHz in various silicon junction devices operating below µA. As the noise level under consideration is very low, its measurement needs special considerations that have been evoked in this paper. An electrical and noise model is proposed for this noise floor degradation. The model is well substantiated since elements provided by the diode impedance spectroscopy closely fit those obtained from noise data. This newly observed excess white noise can be of great impact on the noise floor of photodetectors considering that the dark current in silicon devices range between 0.001 and 100 nA, i.e. within the biasing range under concern in this study. However, some issues still need to be solved; if the excess diffusion noise grows up from a specific frequency, it should be expected to return to the conventional level above a second specific frequency. Despite sustained efforts, no experimental result has been obtained on this last point so far.

  II. LOW FREQUENCY DEVIATION FROM THE EXPECTED DIFFUSION NOISE FLOOR.A. Experimental setupVarious experimental workbenches for LFN measurements have been developed over the past five decades [3][4]; the experimental workbench allows LFN measurement of a 2-port DUT current (or voltage) noise sources from 1 Hz to 1 MHz. For the study case under concern, only diodes are measured; the current spectral density Si of the DUT is obtained from a transimpedance low noise preamplifier (model 5182 from EG&G Instruments) or also a voltage low noise preamplifiers (model 5184 from Perkin Elmer Instruments or model SR560 from Stanford Research Systems). According to the selected preamplifier, different appropriate noise deembedding techniques are used to get the

Fig. 1 :

 1 Fig. 1 : Evidence of the diffusion noise floor increase at low frequencies (deviation of the normalized representation from unity), for three various silicon technologies. Nota: emitter-base diodes (collector open) are investigated

Fig. 3 :

 3 Fig. 3 : LFN spectra of the emitter-base junction of silicon 2N2222 device, making use of 3 different techniques (red transimpedance amplifier, green voltage amplifier, black correlation technique). Purple constant plot over the whole frequency band represent a calibration test with parallel RC discrete elements.

Fig. 2 :

 2 Fig. 2 : LFN experimental setup for diode measurement with currentpreamplifier.

  emitter-base junction biased at d.c current of 0.18 nA, 1.2 nA, 6 nA, 12 nA, 27 nA and 100 nA. 2N2905 emitter-base junction biased at d.c current of 0.54 nA, nA, 7,7 nA, 103 nA and 669 nA. BC557 emitter-base junction biased at d.c current of 0.46 nA, 0.63 nA, 1.2 nA, and 6.2 nA.

Fig. 4 :

 4 Fig. 4 : γ normalized LFN diffusion noise for three different silicon diodes. Dotted lines represent the fitting model for the plateau magnitude and specific corner frequency extraction.

Figure 7 compares

 7 Figure7compares Radd and Cadd versus I, either obtained from noise measurements or from impedance spectroscopy measurements, for the emitter-base diode of the 2N2222 transistor. Good agreement is found between the two techniques. It validates the proposed model of Figure5that can therefore be used to describe accurately the diffusion noise enhancement in silicon diodes, even if no data allows us to conclude about the higher frequency behavior (as a decrease of γ is expected!).

Fig. 6 :

 6 Fig. 6 : Radd Cadd variations versus d.c current I for three silicon E-B diodes (from 2N222, 2N2905, BC557 transistors). The diode resistance Rd for the emitter-base diode of 2N222 is also reported for comparison (+).

Fig. 7 :

 7 Fig. 7 : Comparison of Radd and Cadd variation either derived from noise (data from Fig. 4) or from impedance spectroscopy data.

ACKNOWLEDGMENT

Pr. J. Graffeuil and Pr. J.G. Tartarin would like to express their deep gratitude to Pr. L. Escotte for the discussions and fruitful work done together and for the moments of shared friendships. R.I.P Laurent.


Claire Fave
email: claire.fave@univ-paris-diderot.fr

Bernd Schöllhorn
email: bernd.schollhorn@univ-paris-diderot.fr

Electrochemical activation of halogen bonding

Keywords: Halogen bonding, Noncovalent interactions, Redox switching, Molecular recognition, Molecular electrochemistry, Anion detection

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Halogen bonding (XB), an attractive interaction between a halogen atom acting as an electrophilic species (called XB donor) and a Lewis base (called XB acceptor) (Figure 1), is nowadays fully recognized by the scientific community as an important noncovalent interaction and an interesting tool for applications either in crystal engineering or materials science but also for (bio)supramolecular chemistry in solution and at the liquidesolid interface. [START_REF] Gilday | Halogen bonding in supramolecular chemistry[END_REF][START_REF] Cavallo | The halogen bond[END_REF] In 2013, the term 'halogen bonding' was officially defined by International union of pure and applied chemistry (IUPAC) [START_REF] Desiraju | Definition of the halogen bond[END_REF], stressing its complementarity with hydrogen bonding (HB), and is well documented in the solid state [4e9]. This is not the case for systems in homogeneous solution or at the electrode surface despite the interest of emerging applications in molecular recognition, anion sensing and transport, medicinal chemistry, and organocatalysis. [START_REF] Gilday | Halogen bonding in supramolecular chemistry[END_REF][START_REF] Cavallo | The halogen bond[END_REF], [10e12] Most published studies of XB in solution rely on spectroscopic techniques such as ultraviolet-visible (UV-vis), infrared (IR), Raman, and nuclear magnetic resonance (NMR) spectroscopy [START_REF] Erdélyi | Halogen bonding in solution[END_REF]. In the seventies, dielectric polarization measurements were conducted for the characterization of organohalides in organic solvents allowing for a distinction of the observed XB from other noncovalent interactions [START_REF] Dumas | Molecular interactions involving organic halides[END_REF]. Surprisingly, and despite their low cost and relatively simple implementation, electrochemical methods remained absent until 2014 [14]. The challenge to explore the electrochemical activation of XB for molecular recognition in solution and in surface confined systems is, in our view, a new and attractive topic, and some questions have only recently been addressed. Tuning the strength of a redox active XB donor via a reversible electrochemical reaction should change its affinity toward XB acceptors and vice versa (Figures 1 and3). Such type of redox switching is of great interest because the electrochemical probing and controlling of XB will help to better understand the corresponding supramolecular properties and reactivity in complex systems such as liquid electrolytes involving multiple competing noncovalent interactions. The present article reports on the most recent studies in this field.

Redox switching of XB acceptors: reductive XB activation

In aprotic solvents, quinones can be reversibly reduced in two sequential one-electron transfer (ET) steps [15,[START_REF] Lund | Organic electrochemistry[END_REF] generating the corresponding anion radical (semiquinone radical) and dianion, with a consequently significant increase of the respective Lewis base strength. The influence of HB donors and Lewis acids on the electrochemical behavior of quinones has been extensively studied in the past [17e21]. The observed shifts of the quinone reduction potentials were attributed to the stabilization of the corresponding anions because of their association with HB donors (in the absence of proton transfer reactions) and Lewis acids (eg. alkali and earth-alkali metal cations) [START_REF] Jr | The effect of lithium ion on the mechanism of the polarographic reduction of benzil in dimethylformamide[END_REF][START_REF] Desbene-Monvernay | Ionpair effects on the electroreduction and electrochromic properties of ortho-chloranil in dipolar aprotic solvents[END_REF]. An analogous behavior has been demonstrated recently for p-quinones such as tetrachloroquinone (TCQ, Figure 2) or dichlorodicyanoquinone (DDQ, Figure 2), validating for the first time the concept that the strength of XB can be controlled by the redox switching of the Lewis basic XB acceptor (Figures 2 and3) [14]. Cyclic voltammetry (CV) was thus identified as an analytical technique well adapted to probe and control XB interactions in liquid electrolytes. Iodo-perfluorocarbons (I-R f ) induced high standard potential shifts exclusively of the second TCQ reduction wave (corresponding to the TCQ dianion formation) of up to 140 mV in polar electrolytes, such as acetonitrile containing 0.1 M of tetrabutylammonium hexafluorophosphate (TBAPF 6 ) as the supporting electrolyte salt. A stoichiometry of 1:1 was deduced from Job plots, and binding enhancements (ratio of the equilibrium constants K 3 /K 2 of the respective XB complexes, Figure 3) of up to 10 3 could be determined by simulation and fitting of the voltammograms to the experimental results. Competition experiments involving two different Lewis bases (dianionic quinone and chloride) were conducted evidencing reversible equilibra between the formed XB complexes and allowing for the indirect determination of affinity constants between the involved nonelectroactive species (halide anions and I-R f ).

Redox switching of XB donors

Two different ways to electrochemically switch XB donors have been described in recent literature (Figure 2). In the first case, the donor strength is increased upon oxidation of a neutral or cationic redox center, thus activating XB. In a second case, cationic XB donors are reduced involving XB deactivation. Both switching modes have been applied to selective anion detection, an important issue in analytical chemistry [START_REF] Beer | Anion, cation and ion-pair recognition by macrocyclic and interlocked host systems[END_REF][START_REF] Gale | Anion sensing by small molecules and molecular ensembles[END_REF].

Oxidative XB activation

Iodinated tetrathiafulvalene (TTF) and ferrocene (Fc) derivatives have been used as redox moieties for XB activation upon electrochemical oxidation. In general, the XB-donor strength of organohalides increases with the size and polarizability of the halogen atom (I > Br >> Cl >> F) and the presence of electron withdrawing groups.

Iodo-alkynes and iodoperfluorocarbons are known to be particularly strong XB donors. Besides these neutral compounds, more recently, it has been proven by NMR spectroscopy that cationic halogenated compounds (halo-imidazolium and triazolium derivatives) [START_REF] Schulze | Beyond click chemistrysupramolecular interactions of 1,2,3-triazoles[END_REF][START_REF] Dreger | The role of charge in 1,2,3-triazol(ium)-based halogen bonding activators[END_REF] also possess strong XBdonor properties. The use of positively charged receptors that benefit from a Coulombic attraction to anions is recognized as a promising general strategy for anion recognition. Electrochemical oxidation represents a well-adapted tool for conveniently generating positive charges in molecular systems under ambient conditions. Since 1995, electrocrystallization was used to prepare various solid-state donor-acceptor complexes based on charge-assisted XB involving iodo-tetrathiafulvalene (I-TTF) derivatives [START_REF] Fourmigué | Activation of hydrogen-and halogenbonding interactions in tetrathiafulvalene-based crystalline molecular conductors[END_REF]. Recently, the XB-donor strength of iodinated TTF derivatives (I-TTF and IMe 3 -TTF, Figures 2 and3) has been electrochemically modulated and controlled in DMF solution [START_REF] Oliveira | Electrochemical activation of a tetrathiafulvalene halogen bond donor in solution[END_REF]30]. A significant binding enhancement to Cl À upon oxidation of I-TTF ('XB OFF') to I-TTF þ ('XB ON') was observed and attributed to the strong inductive effect of the generated positive charge polarizing the C-I bond. The significant contribution of XB to the observed stabilization of I-TTF þ was proven, and other noncovalent interactions such as HB or p-anion interactions could be excluded. First spectroelectrochemical measurements in the Uv-vis range along with Time-dependent density functional theory (TD-DFT) calculations confirmed XB formation upon generation of the cation radical. Depending on the nature of the anion (Cl À , Br À , TfO À ), oxidation potential shift values in the range of 0e50 mV have been observed and affinity constants of up to 425 M À1 could be determined for the XB adducts, corresponding to a binding force of around À12 to À15 kJ mol À1 . Moreover, competition between I-TTF 1þ and other XB donors (iodo-perfluorocarbons) as well as HB donors (alcohols, water) for the recognition of Cl À allowed for the quantitative comparison of the relative donor strength. This competitive approach is an interesting electroanalytical tool for probing noncovalent XB and HB interactions in solution. Thanks to the second oxidation step affording, the more electrophilic dication I-TTF 2þ , it was possible to detect XB even in protic media (water/DMF and ethanol/DMF mixtures).

The redox properties of Fc have been exploited in a similar way for the electrochemical detection of anions in solution. Beer and colleagues have prepared several Fc derivatives bearing two iodo-triazoles or cationic iodo-triazolium units (Figure 2). Some of these bidentate XB donors showed interesting anion receptor properties, suggesting a significant contribution of XB. In CV and square wave voltammetry (SWV) measurements, standard potential shifts of the ferrocenyl probes were observed in the presence of various anions, in particular halides (Fc-1, Figure 2) [31] and azide (Fc-2, Figure 2) [32]. In all cases, the respective affinity constants (K a ) of the iron (II) (Fc) containing XB donors could be determined by 1 H NMR which was not possible for the activated paramagnetic iron (III) complexes (ferricinium). Thus, electrochemical techniques have not been used yet to quantify, via titration experiments and fitting to an appropriate model, the affinity constants for the activated receptors. However, when comparing the potential shift values of analogs XB (iodinated triazoliums) and HB receptors, XB proved to be equivalent or even stronger in the investigated liquid electrolytes. A tri-ferrocenylbis(iodotriazole) (Fc-3, Figure 2), conceived by the group of Molina, showed two reversible oxidation steps involving a two-electron and a one-electron transfer and proved to be remarkably selective toward oxoanions, affecting either the oxidation potential of the second ET (sulfate, acetate) or of both (dihydrogenophosphate and hydrogenopyrophosphate), whereas halides and nitrate showed no effect Reductive XB deactivation N,N 0 -disubstituted viologens constitute an important class of redox mediators widely applied in catalysis and the elaboration of supramolecular complexes [START_REF] Beer | Anion, cation and ion-pair recognition by macrocyclic and interlocked host systems[END_REF]. Iodinated dicationic viologen derivatives were used as potent XB donors in the thermodynamically stable redox state. Reduction to the cation radical or the neutral state proved to significantly decrease the respective XB-donor strength. Based on this principle, reversible reduction of a diquat bis-iodotriazole derivative (I-Diquat, Figure 2) was used for the electrochemical sensing of halide anions [35]. A comparative study of different noncovalent interactions between cationic N-phenylviologens and halides was performed (I-PV, Figure 2) [36]. XB contributed to a stabilization of the initially dicationic redox state of the iodo-tetrafluorophenyl viologen derivative in the presence of chloride and bromide, although the role of other competing noncovalent interactions (HB, ion pairing and peanion interactions) is not fully understood. CV allowed the quantification of the overall affinity of halide anions toward N-phenylviologens depending on their redox state. Local probing of the various interactions was possible, thanks to NMR spectroscopy of the diamagnetic phenyl viologen dications.

Interfacial redox switching of halogen bonding

Noncovalent interactions play an important role in interfacial supramolecular electrochemistry [37]. Recently, the concept of electrochemical XB activation has been transferred onto the electrode surface via the immobilization of diiodinated TTF dithiolane derivatives in form of self-assembled monolayers (SAMs) on gold surface. Thus electrochemically driven interfacial charge-assisted halogen bonding between redoxactive SAMs and halide anions has been explored for the first time [38]. The XB-donor properties of the surface adsorbates (I 2 -TTF, Figure 4A) were activated by electrochemical switching their oxidation state. A particularly high and selective binding enhancement of 565 (ratio of affinity constants for the two TTF oxidation states) upon oxidation of the SAM toward chloride anions was estimated from electrochemical simulation, suggesting a significant surface chelate effect of the assembled 2D material. Strong evidence is provided for a significant contribution of XB as the dominant noncovalent interaction in the investigated system. A high sensitivity (limit of detection = 6•10ˉ6 M) was determined for chloride detection which is a promising finding considering a potential application of the concept for future development of novel selective anion sensors, capture/release systems, or molecular electronics.

XB promoted ET reactions

The group of Rosokha described halogen bond-assisted ETreactions of aliphatic bromo-substituted electrophiles [39,40]. The isolation of halogen-bonded prereactive complexes between aliphatic organobromides and Lewis bases (phenylene diamine and iodide) at low temperature and the kinetics of a subsequent ET are consistent with an inner-sphere mechanism involving strongly coupled redox centers (Figure 4B). Electronic coupling of XB donor and XB acceptor within the prereactive complex was suggested. Up to date, we are aware of only one example exploiting this potentiality for the regeneration of a photo-oxidized halogenated dye via iodide oxidation, thus impacting the open-circuit voltage of a solar cell (Figure 4C) [41]. Parlane et al [42] used X-ray absorption spectroscopy to directly detect interfacial XB between a homologous soluble halide species (Cl À ) and the activated dye cation demonstrating that these interactions occur after electron injection into TiO 2 .

Conclusions and perspectives

Most of the work on the electrochemical investigation of XB has been published during the past 5 years, and many potentialities in this domain are still to be explored. However, the fundamental concept of electrochemical XB activation/deactivation for molecular recognition and detection is now well established. Several reversible redox probes have been used, covering the activation of XB donors (TTF, Fc) and XB acceptors (quinones) as well as the deactivation of XB donors (viologen cations). Examples of electrochemical deactivation of XB acceptors for XB are not described yet. The possibility of reversible switching between three different redox states (0/AE1/AE2) represents a decisive advantage for detecting a wide panel of analytes. It is noteworthy that besides some first conclusive results, there is still an important lack of spectroelectrochemical measurements.

Recognition of neutral Lewis bases [43], instead of anions, involving redox-active XB donors would also be an important and challenging issue. It is likely that the design of tailored oligo-dentate receptors will be required eventually benefitting from the cooperative interplay of multiple complementary noncovalent interactions. Another issue, already approached in some of the above cited examples, is the attempt to promote the high potential of XB to complement HB in the selective recognition and sensing of biologically and industrially relevant anions under aqueous conditions. Besides selective detection, also the specific capture/release of anions should be considered for biomedical applications in particular. The principle of redox switching of XB has been successfully transferred from solution onto an electrode surface. This approach opens the way for the development of new electrochemical sensors or sponge captors for depollution applications.

Finally, we believe that XB promoted ET reactions involving the activation of covalent bonds represents a completely new domain for exploration. Evidence for prereactive XB complexes in the dissociative reduction of C-X bonds involving an inner-sphere ETstep provides a new perspective on electrochemical reduction of organic halides in general [44]. At the same time, it seems to be a pivotal question paving the way, not only to a better mechanistic understanding of other halogen involving electrochemical-chemical (EC) or electrochemical-chemical-electrochemical (ECE)

[45] mechanisms but also helping to identify new electrochemically assisted catalytic reactions. In the first encouraging example concerning solar cells, it has been shown challenging but possible to experimentally prove how weak XB interactions can affect interfacial ET reactions.

In conclusion, we strongly believe that XB should be considered as a powerful tool in molecular electrochemistry at a similar level as complementary HB. The stronger directionality of XB should be an important advantage for the conception of supramolecular systems and property prediction. Furthermore, future work in this domain should be accompanied more often by molecular modeling as a computational tool for confirming, optimizing and predicting structure/property relationships.

Figure 1 a)

 1 Figure 1

Figure 2

 2 Figure 2

  [33]. Electrochemical activation of Fc-based XB donors could also be performed in up to 10% of watercontaining solvent mixtures[31]. Recently, a tetraiodinated monocationic[START_REF] Cavallo | The halogen bond[END_REF]-Rotaxane bearing a ferrocenyl moiety as an interlocked host molecule showed weak perturbation of the voltammetric signal (in a acetone/acetonitrile/water mixture) but still selective recognition of bromide versus chloride or linear thiocyanate[34].

Figure 3

 3 Figure 3

Figure 4

 4 Figure 4

  series of different halo-phenyl viologens has been studied by cyclic voltammetry and NMR spectroscopy analyzing the competition of the involved non-covalent interactions (XB, HB and p-anion). Halogen bond-assisted electron transfer reactions of aliphatic bromosubstituted electrophiles. Phys Chem Chem Phys 2014, 16:1809-1813. Evidence for pre-reactive XB complexes in the dissociative reduction of C-X bonds involving an inner-sphere ET step provides a new perspective on electrochemical reduction of organic halides. 40 . Rosokha SV: Electron-transfer reactions of halogenated electrophiles: a different look into the nature of halogen bonding. Faraday Discuss 2017, 203:315-332. This article analyses kinetic data confirming a decrease of the ET barrier between halogen-bonded reactants suggesting that halogen bonding should be considered in reaction mechanisms of halogenated species. 41. Swords WB, Simon SJC, Parlane FGL, Dean RK, Kellett CW, Hu K, Meyer GJ, Berlinguette CP: Evidence for interfacial halogen bonding. Angew Chem Int Ed 2016, 55:5956-5960. 42 . Parlane FGL, Mustoe C, Kellett CW, Simon SJ, Swords WB, Meyer GJ, Kennepohl P, Berlinguette CP: Spectroscopic detection of halogen bonding resolves dye regeneration in the dye-sensitized solar cell. Nat Commun 2017, 8:1-8. Direct evidence for halogen bonding at the dye-electrolyte interface using X-ray absorption spectroscopy suggesting that weak intermolecular interactions between photo-oxidized dyes and the electrolyte can impact device photovoltages. 43. Laurence C, Graton J, Berthelot M, El Ghomari MJ: The diiodine basicity scale: toward a general halogen-bond basicity scale. Chem Eur J 2011, 17:10431-10444. 44. Savéant JM: Elements of molecular and biomolecular electrochemistry. An electrochemical approach to electron transfer chemistry. John Wiley; 2006. 45. Bard AJ, Faulkner LR: Electrochemical methodsfundamental applications. chapter IIelectrode reactions with coupled homogeneous chemical reactions. 2nd ed. John Wiley; 2001:429.

	37. Cui K, Dorner I, Mertens SFL: Interfacial supramolecular elec-
		trochemistry. Curr Opin Electrochem 2018, 8:156-163.
	38	. Hijazi H, Vacher A, Groni S, Lorcy D, Levillain E, Fave C,
		Schöllhorn B: Electrochemically driven interfacial halogen
		bonding on self-assembled monolayers for anion detection.
		Chem Commun 2019, 55:1983-1986.
	The concept of reversible redox switching XB for molecular anion
	recognition and detection has been successfully transferred from so-
	lution onto an electrode surface.
	39	. Rosokha SV, Vinakos MK:
		. Oliveira R, Groni S, Vacher A, Barrière F, Lorcy D,
		Fourmigué M, Maisonhaute E, Schöllhorn B, Fave C: Electro-
		chemical activation of TTF-based halogen bond donors: a
		powerful, selective and sensitive analytical tool for probing
		a weak interaction in complex media. Chemistry 2018, 3:
		8874-8880.
		Supplementary proof and systematic study of various parameters and
		methods influencing the electrochemical activation of iodinated TTF
		XB-donors treating also the following aspects: supporting electrolyte
		effects, spectro-electrochemistry, microelectrodes, XB/HB competition.
		31 . Lim JYC, Cunningham MJ, Davis JJ, Beer PD: Halogen
		bonding-enhanced electrochemical halide anion sensing by
		redox-active ferrocene receptors. Chem Commun 2015, 51:
		14640-14643.
		First electrochemical activation of a bidentate XB-donor for the detec-
		tion of halides. Affinity constants of the initial oxidation state are
		determined by NMR and shift values of the measured ferrocene
		oxidation potentials are suggesting binding enhancement due to XB
		formation.
		32 . Lim JYC, Beer PD: A halogen bonding 1,3-disubstituted
		ferrocene receptor for recognition and redox sensing of
		azide. Eur J Inorg Chem 2017:220-224.
		Host-guest shape and size complementarity of a novel bidentate XB
		receptor accounts for the selective electrochemical detection of azide
		over a range of anions with differing geometries.
		33. Zapata F, Caballero A, Molina P: Ferrocene-triazole combi-
		nation as a benchmark for the electrochemical detection of
		noncovalent halogen-bonding interactions. Eur J Inorg Chem
		2017:237-241.
		34 . Lim JYC, Beer PD: Electrochemical bromide sensing with a
		halogen bonding [2]Rotaxane. Eur J Org Chem 2018, https://
		doi.org/10.1002/ejoc.201801571.
		This article reports on an original redox active tetradentate XB-donor.
		35 . Mullaney BR, Cunningham MJ, Davis JJ, Beer PD: Acyclic
		halogen and hydrogen bonding diquat-containing receptors
		for the electrochemical sensing of anions. Polyhedron 2016,
		116:20-25.
		First example for cathodic deactivation of XB. The affinity constants of
		the dicationic bidentate XB-donors towards anions were determined by
		NMR spectroscopy.
		36 . Creste G, Groni S, Fave C, Branca M, Schöllhorn B: Compara-
		tive study of non-covalent interactions between cationic N-
		phenylviologens and halides by electrochemistry and NMR:
		the halogen bonding effect. Faraday Discuss 2017, 203:
		301-313.

A

Acknowledgements

The authors gratefully acknowledge the financial support from the University Sorbonne Paris Cite ´(IDEX HalBond) and the Centre National de la Recherche Scientifique (CNRS).

Conflict of interest

Nothing declared.


Eric Marchand

François Chaumette

Thomas Chabot

Keyvan Kanani

Alexandre Pollini

Alexandre Pollini Re

RemoveDebris Vision-Based Navigation preliminary results

Keywords: VBN, RemoveDebris, Visual tracking, Detection, Localization

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Since the beginning of the space era in 1957, the number of missions has incredibly increased. Unfortunately, it has also progressively generated a huge amount of debris that are still orbiting Earth. Today, space agencies feel more and more concerned about Earth orbital environment cleaning since they realize those debris represent a big threat due to the high risk of collision that could, among safety issues, damage operational satellites [START_REF] Bonnal | Active debris removal: Recent progress and current trends[END_REF]. To investigate solutions to this issue, various projects aim to actively remove debris objects by catching them. Among these projects, is the RemoveDebris project co-funded by the European Union and members of European space industry. In a first stage, the RemoveDebris mission aimed at catching two miniaturized satellites (CubeSats DSat-1 and DSat-2 produced by the Surrey Space Centre) previously ejected from a microsatellite called RemoveSat, itself released from the International Space Station (ISS). The CubeSats were used as targets instead of actual debris. This debris removal demonstration included net capture, harpoon capture and vision-based navigation using standard camera and LiDAR [START_REF] Forshaw | The active space debris removal mission Re-moveDebris. Part 1: from concept to launch[END_REF][START_REF] Aglietti | The active space debris removal mission RemoveDebris. Part 2: in orbit operations[END_REF].

In this paper we describe the results of the Visionbased navigation experiments. Along with the development of VBN Harware [START_REF] Chabot | Vision-based navigation experiment onboard the removedebris mission[END_REF][START_REF] Forshaw | The active space debris removal mission Re-moveDebris. Part 1: from concept to launch[END_REF], it consists in tracking and localizing the DSat-2 using two model-based tracking algorithm provided by Inria and Airbus, and using the Lidar provided by CSEM.

In the literature, several works propose approaches based on active sensors, such as LiDAR, for space rendezvous and docking missions [START_REF] Howard | Orbital express advanced video guidance sensor[END_REF]. Those approaches have proved to be efficient in situations where lighting conditions can be tricky due to directional sunlight resulting in high specularities and shadows. So far, Li-DAR were known to be generally expensive and rather heavy sensors. Lidar has a limited range and accuracy, and it cannot take the advantage of using potential texture information available on the target surface in order to improve the navigation process. Based on the knowledge of the 3D model of a target and from the images acquired by the camera, estimating the pose (i.e., the 3D position and orientation) between the camera and the target has always been an ongoing issue in computer vision and robotics [START_REF] Marchand | Pose estimation for augmented reality: a hands-on survey[END_REF]. For instance, regarding space applications, [START_REF] Oumer | Vision-based localization for on-orbit servicing of a partially cooperative satellite[END_REF][START_REF] Petit | Visionbased detection and tracking for space navigation IAC-19-6[END_REF] use model-based tracking approaches for space rendezvous with space target or debris. More generally, common approaches address this problem by using either texture [START_REF] Bleser | Real-time 3d camera tracking for industrial augmented reality applications[END_REF], edge features [START_REF] Drummond | Real-time visual tracking of complex structures[END_REF][START_REF] Comport | Real-time markerless tracking for augmented reality: the virtual visual servoing framework[END_REF][START_REF] Petit | Visionbased detection and tracking for space navigation IAC-19-6[END_REF][START_REF] Oumer | Vision-based localization for on-orbit servicing of a partially cooperative satellite[END_REF] or color and IAC-19-6.6.7 intensity features [START_REF] Panin | An efficient and robust real-time contour tracking system[END_REF][START_REF] Prisacariu | Pwp3d: Real-time segmentation and tracking of 3d objects[END_REF][START_REF] Petit | A robust model-based tracker for space applications: combining edge and color information[END_REF].

In this paper, we focus on the vision-based navigation part of the RemoveDEBRIS mission using a single standard B&W or RGB camera on one hand, and a LiDAR on the other hand.

The paper is organized as follow: in a first stage, the model-based tracking methods developed by Inria and Airbus are recalled. Then, we shortly give an overview of the Lidar experiment. Finally, results are presented, first on synthetic image sequences to assess the proposed methods, then on the actual data acquired during the VBN experiment achieved in October 2018. A comparison of these results is then proposed.

3D MODEL-BASED TRACKING

Within the Removedebris projet, two localization algorithms have been proposed by Inria and by Airbus respectively. Both are model-based methods, as described in the next subsections and have been tested on various space robotics scenarii. They consist in determining the pose that minimizes the error between observations in the image and the forward-projection of the model of the target in this image (see Fig. 1 for an overview of the localization process). The applications are miscellaneous, from debris removal to in-orbit servicing or air-to-air re-fueling (see [START_REF] Petit | A robust model-based tracker for space applications: combining edge and color information[END_REF][START_REF] Kanani | Dual technologies to mature vision based navigation: the example of space rendezvous and air to air refueling[END_REF]). Airbus and Inria's approaches are very similar. The main differences lay in their implementation. Indeed, Airbus version of the tracking is compatible with space missions (HW limitations, trajectories, background,...). Furthermore, Airbus method also features successive detections of the target.

r = arg min r ∆(r) [1] with ∆(r) = i ρ(e i (r)) [2]
∆(r) accounts for errors e i (r) between a set of visual features extracted from the image and the forward projection of their 3D homologues in the image plane according to the pose. ρ is a robust estimator [START_REF] Malis | Experiments with robust estimation techniques in real-time robot vision[END_REF], which reduces the sensitivity to outliers. This is a non-linear minimization problem with respect to the pose parameters r, and we follow a Gauss-Newton minimization framework to handle it, by iteratively updating the pose r.

At each iteration k, a displacement is performed in the parameter space which is SE(3), the displacement δr is computed through:

δr = -λ(DJ) + De [3]
with J = ∂e(r) ∂r [START_REF] Brox | High accuracy optical flow serves 3-D pose tracking: exploiting contour and flow based constraints[END_REF] and where (DJ) + is the pseudo inverse of DJ, J being the Jacobian matrix of the error vector e(r) with respect to the pose. λ is a proportional gain and D is a weighting matrix associated to the Tukey robust estimator. The pose r, represented by its homogeneous matrix M k+1 , can be updated as follows, using the exponential map [START_REF] Ma | An invitation to 3-D vision[END_REF]:

M k+1 = exp([δr]) M k . [5]
where exp([δr]) is the exponential map enabling to express the rigid motion δM generated by δr.

Cost function. For a better accuracy and robustness, we have chosen to integrate, within this framework, geometrical edge-based features with geometrical features: keypoints extracted from the image and tracked through the KLT algorithm [START_REF] Shi | Good features to track[END_REF]. The objective is to fuse in the criterion ∆ to be optimized a geometrical information provided by distances between edges with distances between keypoints. The goal is to benefit from the complementarity of these features and to overcome the limitations of classical single cue approaches. By combining these features, ∆ can be rewritten as:

∆ = w g ∆ g + w p ∆ p [6]
∆ g refers to the geometrical edge-based error function and ∆ p their corresponds to the keypoint feature. w g and w p are the respective weighting parameters.

Edge-based visual features

From the knowledge on the 3D model of the target, a potential option is to use its silhouette to define features to rely on. Thus and as defined in [START_REF] Chabot | Vision-based navigation experiment onboard the removedebris mission[END_REF], the first features we consider for this method are edge-based features. As in IAC-19-6.6.7 [START_REF] Bonnal | Active debris removal: Recent progress and current trends[END_REF][START_REF] Brox | High accuracy optical flow serves 3-D pose tracking: exploiting contour and flow based constraints[END_REF], the edges that are considered correspond to the projection of the CAD model of the target, using the current camera pose r. More precisely, the edges corresponding to the visible polygons of the model are considered in the minimization process. That is why this approach requires a simplified version of the 3D model that only contains polygons with visible edges (in term of gradient in the image).

The edge-based function ∆ g is computed in a similar way to [START_REF] Petit | A robust model-based tracker for space applications: combining edge and color information[END_REF]. From the model edge points we perform a 1D search (see Fig. 8) along the normal of the underlying edge of each x i (r k ). A common approach is to choose on the scan line the pixel with the maximum gradient as the matching edge point x i in the new image.

Once correspondences between the set of control points {x i } Ng i=1 and the set of image edge points {x i } Ng i=1 are established, our approach considers the distance between the projected 3D line l i (r) underlying the projected control point x i (r) (projected from the 3D point X i ) and the selected matching point x i in the image. The error function ∆ g (r) to be minimized with respect to the pose r can be written as:

∆ g (r) = 1 N g i ρ g (e g i (r)) [7]
where e g i (r) = d ⊥ (l i (r), x i )) and d ⊥ (l i (r), x i )) is the distance between the point x i and the corresponding line l i (r). ρ g is a Tukey robust estimator.

Geometrical keypoint based features

Another class of visual features which can be used are keypoints tracked across the image sequence. As previously suggested by [START_REF] Brox | High accuracy optical flow serves 3-D pose tracking: exploiting contour and flow based constraints[END_REF] or by [START_REF] Vacchetti | Combining edge and texture information for real-time accurate 3d camera tracking[END_REF][START_REF] Pressigout | Hybrid tracking approach using optical flow and pose estimation[END_REF] within their hybrid approaches, the idea is to design a texture-based objective function ∆ p accounting for geometrical distances between keypoints extracted and tracked over successive images. But in contrast to [START_REF] Pressigout | Hybrid tracking approach using optical flow and pose estimation[END_REF], which process 2D-2D point correspondences to estimate the 2D transformation from I k to I k+1 of planar local regions underlying the points, we use 2D-3D correspondences to directly minimize ∆ p w.r.t. the pose r.

More specifically, let us denote {x i } Np i=1 a set of keypoints detected in frame I k . Assuming the pose r k has been properly estimated, we can restrict these points to be lying on the projected 3D model with respect to r k . Since we rely on a complete 3D model, the depth of the points in the scene can be accurately retrieved, and using r k , we can back-project these points on the 3D model, giving a set

∆ p (r) = 1 N p Np i ρ p (e p i ) [8]
with e p i = (x i (r) -x i ) and x i (r) = pr(X i , r)). ρ p is the Tukey robust estimator associated to these errors.

Target detection

Target pose estimation is divided into two phases: detection of the target and tracking. Detection enables to find the target in the image, without any initialization (or IAC-19-6.6.7 a very coarse one) and to assess the target pose. It is generally used to initialize the tracking, which is faster and more accurate. Tracking, on its side, assesses the pose of the target from one image to the other. It generally requires a small motion of the target between two consecutive images. The main difficulty of detection is to find the descriptor which limits the size and the number of elements in the database, while covering all the possible expected poses of the target. For RemoveDebris mission, we have chosen to use a template matching technique previously used for asteroid pose estimation (see [START_REF] Chapuy | Neoshield-2: design and end-to-end validation of an autonomous closed-loop gnc system for asteroid kinetic impactor missions[END_REF]). This solution enables to estimate the pose of the target for a various set of attitudes, knowing the sun phase angle, and with a removed background. It has been applied to RemoveDebris RGB camera images, which could not be successfully processed by tracking given their low acquisition rate (0.1Hz).

REMOVEDEBRIS LIDAR

LiDAR is considered as an enabling technology for missions with rendezvous phases. The LiDAR architecture used for RemoveDebris is based on the flash principle where the object or the scene of interest is illuminated at once without any form of scanning. There is a long term strategy to develop this type of LiDAR for space applications to facilitate not only rendezvous but also rover navigation and precise landing.

While today the keyword LiDAR is known by many people due to its use in sensor solutions for level 3, 4 and 5 autonomous vehicles, it was not like that at the time of the project elaboration. A limited number of commercial critical components (e.g. VCSEL laser diodes array and time-of-flight detector) existed and no qualified ones were available. Unconventionally, the hardware TRL was at 2 at the start of the project (paper concept) and at the end it achieved TRL8, 4 years later with the in-orbit demonstration. This fast development pace was only possible by using a "cubesat" philosophy consisting in using commercial off-the-shelf components and assessing them for low earth orbit operation. The overall development budget is estimated to have been divided by 10-20 and the cost of one unit is around 20-40% of the cost of LiDAR developed according to the traditional stepwise space industry approach.

VBN EXPERIMENT USING SIMULATED DATA

To assess the VBN experiment and prepare the actual in-orbit mission we first considered synthetic images. In the sequence depicted on Fig. 5, the considered target, the foreseen RemoveDebris 2U Cubesat, is on a geosynchronous orbit. It has an angular speed that creates rotational motions in the camera frame. However, the distance to the camera remains the same over the sequence. We have also considered different lighting conditions to simulate day and night (where the background also changes) in order to prove the robustness of our methods whatever the situation (see Fig. 5). The full video result is available at: https://youtu.be/Gr78pmRPKOA. The proposed method has been able to perfectly track the target. The estimated trajectory is accurate and smooth IAC-19-6.6.7 (see Fig. 7). Furthermore, according to the collected ground truth data, the vision-based navigation provides precise pose estimation with an RMSE that is less than 1cm on the translation and about 2deg on the rotation. Finally, on Fig. 6, we can see the norm of the translational and rotational errors to the ground truth, which also shows where the algorithms encounter difficulties. Around frame 2200 for example, the precision of the estimated pose decreases a bit. The reason is that the panels of the target are orthogonally aligned with the focal axis of the camera (see Fig. 5.c), which results in increasing the incertitude on the target orientation.

VBN EXPERIMENT FROM ON-ORBIT DATA

The vision-based navigation (VBN) experiment was carried out in October 2018. The VBN hardware and software is considered as a tool to observe and quantify relative dynamics between an uncooperative debris and the platform preparing for its retrieval. To test the VBN, the RemoveDebris spacecraft released a CubeSat. The experiment goal was to localize a 2U CubeSat using optical images acquired by cameras and a LiDAR on-board the spacecraft.

RemoveDebris VBN Sensors

The VBN is made of 2 vision-based sensors: a color camera and a flash-imaging LIDAR (see Fig. 8). It weights 1.5 kg and is 10x10x15 centimeters large. The device includes a laser source that lights up the object in the field-of-view (FoV). The FoVs are respectively 8 o x6 o for the LiDAR and 21 o x15 o for the color camera. 3D (160x120 pixels), color (2048x1536 pixels) and B&W (1280x1024 pixels, 12mm lens with 12mm square pixels) images were acquired at respectively 0.1, 0.08, and 1Hz, for the B&W camera with successively 3 different aperture times, meaning that successive images at same aperture were acquired at 0.3Hz. The VBN captures images independently after a start trigger from the satellite computer according to pre-programmed sequences defining the respective series of sampling times and integration times for the LiDAR and the camera. Both 3D and color images are stored in non-volatile flash memory. There are downloaded by the satellite computer whenever possible and forwarded down to Earth. The VBN generates 3 images: standard RAW or JPEG color images with the camera, and gray-level and depth map images with the LiDAR. The images on Fig. 9 were captured during the on-orbit calibration phase above Australia.

In-orbit operations overview

The first fact to be highlighted is that DSAT2 remained in the cameras field of view over the entire sequence, meaning that the attitude guidance profile was correctly generated and followed by RemoveSat, and that disper-IAC-19-6.6.7 sions at ejection remained at an acceptable level. This is illustrated on Fig. 10. The planned acquisition timeline was perfectly followed and images were acquired as planned. A late decision to increase SV camera acquisition frequency allowed the use of 3 different integration times, maximising the likelihood of correct exposure all along the demonstration.

Different VBN algorithms provide 3-DoF (position only) and 6-DoF (position and attitude) relative navigation depending on the rendezvous phase:

• At far and medium range, only 3-DoF (position only) relative navigation is needed and performed

• At short range, 6-DoF (position and attitude) relative navigation is required for successful capture. In this paper we only focused on these approaches.

Examples of image processing algorithms and corresponding transitions are illustrated on Fig. 11.

Tracking results

Both Airbus and Inria tracking solutions rely on a 3D model of the DSAT2 target (see Fig. 12).

Results on the actual RemoveDebris data using

Inria algorithm The experiment validated the VBN algorithm (along with the hardware) through ground-based processing of Images (1280x1024 pixels, 12mm lens with 12mm square pixels) were acquired at 3 Hz with successively 3 different aperture times, meaning that successive images at same aperture were acquired at 1 Hz. In particular, the CubeSat has been successively tracked and localized (see Fig. 13 and Fig. 14) on 210 successive images (that is, 210 s or 3.5 minutes). Note that the time processing of each frame is less than 100 ms. After the DSAT2 target is to small in the image to be localized (position and orientation) by the model-based tracking approach.

Results on the actual RemoveDebris data using Airbus algorithm

The Airbus in-house tracking solution has been tested on the VBN camera images. This solution had to be improved in order to deal with the low acquisition frequency of the VBN camera (0.1 Hz), showing good results as long as the background is dark as illustrated on Fig. 15. The green contours illustrate the 3D model of the target which has been matched by the algorithm to the actual DSAT2 image.

Results of the localization using the Lidar

After several trade-offs mostly between satellite safety and experiments operation conditions requirements, inorbit experiments time windows were chosen to favor a strong illumination of the target by the Sun. These opera-IAC-19-6.6.7 According to the camera picture it is difficult to say whether the cubesat (on the left picture's right hand side) has been or will be catched by the NET. Looking at the Li-DAR image it is instantaneously clear that it is just before the capture as the NET (the green structure in the right picture) is on average at 12 m while the cubesat (the blue structure on the right hand side of the right picture with 2 arms of the sail deployed) is at 16 m. At this distance, there is not a complete overlap of the respective FoV of the camera and the LiDAR, this is why the images appear to have been taken for two different positions. Fig. 17 The un-saturated areas provides measurements allowing to get correct TOF and distances. From these measurements several secondary figures can be calculated such as: 3D target image, relative attitude, velocity and rotation of the target, etc.

In relation with long term flash imaging LiDAR development strategy, the RemoveDebris experience is unvaluable as it shows what it takes in a real situation with strong Sun and Earth background perturbations to get proper visual measurements that can be considered with confidence to control the approach of space debris by waste collection crafts or more generally for in space rendezvous between two objects. The RemoveDebris VBN captured color and LiDAR images over more than 100 m of a 3U cubesat size target (10x10x30 cm) within the expected distance accuracy of a few tens of cm. The strong light background either due directly to the Sun and the sunlight reflected by the Earth is the most challenging factor. It can be addressed through several ways including faster image capture rate, shorter integration time, smaller bandwidth optical filter, matched filtering, etc.

Comparison of the different methods

The results from the image processing algorithms based on SV0 images using the Inria approach (section 5.3.1) and VBN camera images using the Airbus approach (section 5. As can be seen SV0 and VBN camera measurements are consistent (2.3 cm/s ejection) using both Inria and Airbus approaches. It is also consistet with the expected trajectory. Both approaches allow the estimation of the position and orientation for almost 300s. Thanks to its tracking by detection ability, Airbus allows a longer estimation of the target position. Although not reported in this paper, it has to be noted that GPS measurements seem not reliable for short range (<10m) and that LiDAR measurements (provided by CSEM) are also close to expected trajectory (2 cm/s ejection).

CONCLUSION

Main objectives were mostly met with the VBN demonstration, therefore considered a success by Airbus / CSEM / Inria. Indeed, a very rich and useful set of images and LiDAR data was gathered, allowing preliminary characterization of algorithms performance and identification of possible improvements. Future activities should focus on:

• Analysis of the remaining data, in particular, medium/long range data

• Improvement of ground truth data, including investigation of sensors cross-correlation

• Work on algorithms current limitations and improvements e.g. medium range behaviour, influence of background, navigation aiding for improved tracking, robustness to 3D model errors, LiDAR/camera fusion.

IAC-19-6.6.7

For a future ADR demonstration, next steps would include real-time, onboard VBN algorithms and eventually closed-loop GNC.

ACKNOWLEDGEMENT

This research is supported by the European Commission FP7-SPACE-2013-1 (project 607099) "RemoveDebris -A Low Cost Active Debris Removal Demonstration Mission", a consortium partnership project consisting of: Surrey Space Centre (University of Surrey), SSTL, Airbus GmbH, Airbus SAS, Airbus Ltd, Ariane Group, Innovative Solutions in Space (ISIS), CSEM, Inria and Stellenbosch University.

Fig. 1 :

 1 Fig. 1: Overview of the localization process

Fig. 2 :

 2 Fig. 2: Model-based tracking principle considering the case of edge and keypoint features

  {X i } Np i=1 of 3D points of the 3D model. We employ the Harris corners detector inside the silhouette of the projected model in the image to extract {x i } Np i=1 in I k . Then, the KLT tracking algorithm, for which Harris corners are optimal [21], enables to track this set of points in frame I k+1 , resulting in a corresponding set {x i } Np i=1 (see Fig. 3, right image, red and blue (tracked) dots).

Fig. 3 :

 3 Fig. 3: Determination of correspondences between tracked keypoints and their 3D homologues. On the left are represented Harris points detected on a frame (blue dots). These keypoints are back-projected on the 3D model (pink dots) using the pose computed for the frame. The keypoints are then tracked on a new frame (right image, blue dots) for determining the new pose.

Fig. 4 :

 4 Fig. 4: Target detection with template matching approach

Fig. 5 :

 5 Fig. 5: Vision-based navigation of a CubeSat DS-2 for the first sequence of a geosynchronous orbit, using the CPU-based approach.

Fig. 6 :

 6 Fig. 6: Error to ground truth.

Fig. 7 :

 7 Fig. 7: 3D poses estimated for the first sequence of a geosynchronous orbit (translation in meters and rotation in radians).

Fig. 8 :

 8 Fig. 8: Vision-based sensor (VBN). Top right: color camera optics. Bottom center: laser illumination head and above LiDAR receiver optics.

Fig. 9 :

 9 Fig. 9: from left to right: color camera JPEG image, graylevel and depth map/time-of-flight LiDAR images.

Fig. 10 :

 10 Fig. 10: DSAT2 within VBN camera field of view over the reference trajectory

Fig. 11 :Fig. 12 :

 1112 Fig. 11: Different image processing algorithms are used depending on relative distance and navigation needs

Fig. 13 :Fig. 14 :

 1314 Fig. 13: Vision-nased navigation of the Cubesat for the actual Remodebris sequence. The Cubesat has been successively tracked and localized on 210 successive images (that is during 201s, ie, 3.5 minutes)

Fig. 15 :

 15 Fig. 15: Example of tracking performed during the VBN demonstration using the Airbus tracking approach.

Fig. 16 :

 16 Fig. 16: Left, cropped color camera image, right, time-offlight LiDAR image

  Fig. 17: Left, cropped color camera image, right, time-offlight LiDAR image, captured respectively 272 s and 268 s after ejection.

  3.2) together with LiDAR data), GPS measurements and the expected nominal DSAT2 trajectory are shown hereafter on Fig. 18.

Fig. 18 :

 18 Fig. 18: Tracking results synthesis: DSAT2 assessed distance (expected distance in blue).