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Set-membership localization of intermittent RF sources using a fleet of collaborating UAVs
Jacques Bois, Raul de Lacerda, Michel Kieffer
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%
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Summary
This paper proposes a Set-Membership Approach (SMA) for localizing intermittent, periodic Radio Frequency (RF) sources using a fleet of collaborating Unmanned Aerial Vehicles (UAVs). The method evaluates set estimates of source locations and certifies regions free of sources by analyzing observation time intervals and emission periods. Simulation results demonstrate that the proposed SMA outperforms a Bayesian baseline in localization accuracy and convergence speed.
Entities (8)
Relation Signals (6)
UAV → usedby → Set-Membership Approach
confidence 98% · localize radio frequency (RF) sources observed by a collaborating fleet of Unmanned Aerial Vehicles (UAVs).
Set-Membership Approach → usedfor → RF Source
confidence 98% · This paper proposes a set-membership approach (SMA) to localize radio frequency (RF) sources
Jacques Bois → affiliatedwith → Université Paris-Saclay
confidence 95% · address: Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes
Raul de Lacerda → affiliatedwith → Université Paris-Saclay
confidence 95% · address: Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes
Michel Kieffer → affiliatedwith → Université Paris-Saclay
confidence 95% · address: Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes
Set-Membership Approach → outperforms → Bayesian Baseline
confidence 95% · Simulation results show that SMA outperforms a Bayesian baseline approach in terms of localization accuracy and convergence speed.
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Abstract
Abstract:This paper proposes a set-membership approach (SMA) to localize radio frequency (RF) sources observed by a collaborating fleet of Unmanned Aerial Vehicles (UAVs). Considering frequency-separable RF transmitters with intermittent and periodic emission patterns, %and unknown but bounded periods the SMA evaluates set estimates of the source locations and a set free of sources. Simulation results show that SMA outperforms a Bayesian baseline approach in terms of localization accuracy and convergence speed.
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- Source: https://arxiv.org/abs/2609.17797v1
- Canonical: https://arxiv.org/abs/2609.17797v1
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Set-membership localization of intermittent RF sources using a fleet of collaborating UAVs Jacques Bois Raul de Lacerda Michel Kieffer Abstract This paper proposes a set-membership approach (SMA) to localize radio frequency (RF) sources observed by a collaborating fleet of Unmanned Aerial Vehicles (UAVs). Considering frequency-separable RF transmitters with intermittent and periodic emission patterns, the SMA evaluates set estimates of the source locations and a set free of sources. Simulation results show that SMA outperforms a Bayesian baseline approach in terms of localization accuracy and convergence speed. Index Terms: Distributed estimation, Intermittent sources, RF source localization, Set-membership estimation, UAVs †address: Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91192, Gif-sur-Yvette, France. 1 Introduction Localizing radio frequency (RF) sources with periodic emissions using a fleet of collaborating Unmanned Aerial Vehicles (UAVs) is critical in many applications, including post-disaster search and rescue [1], border and critical infrastructure surveillance [2], electromagnetic intelligence [3], and IoT asset monitoring [4]. Sources with periodic transmissions are commonly encountered in civil aviation emergency beacons [5], cellular synchronization channels [6], GNSS repeaters [7], and dedicated tracking systems [8]. Signal detection is a prior to RF source localization. While matched filtering [9] or time-frequency approaches [10] perform properly when waveform signatures are well known, their performance degrades significantly under unknown parameter fluctuations [11, 12, 13]. More fundamentally, emission intermittency complicates signal association and source separation [14, 15, 16]. Standard probabilistic formulations either assign empirical emission probabilities and address localization via maximum likelihood schemes [17], or introduce fixed missed-detection rates [18]. RF localization classically exploits Time Difference of Arrival [19], Received Signal Strength Indicator (RSSI) [20, 21], Frequency Difference of Arrival [22], or Angle of Arrival [23, 24]. Measurement fusion via Bayesian frameworks [15, 25] or Probability Hypothesis Density filters [26] allows source localization and tracking under clutter. However, when deployed to localize intermittent sources, probabilistic approaches struggle to certify the absence of sources within an observed region. Absence of detection during an observation window may indicate either an empty zone or an emitter remaining in a silent period. State-machine heuristics [14] and Bernoulli-based intermittency tracking [15] provide confidence probabilities, yet they fail to deliver deterministic guarantees of clearance. Deploying mobile UAV fleets mitigates spatial occlusion and enhances coverage through collaborative sensing. Multi-agent coordination requires robust information fusion and path planning, typically realized via distributed particle or Kalman filtering [27], rule-based collision-avoiding flocking [28], Fisher information and Cramér-Rao Lower Bound optimization [29, 30, 31], distributed multi-agent optimization [32], or reinforcement learning [33]. As a robust alternative to stochastic filtering, set-membership approaches (SMA) [34] are able to characterize sets guaranteed to contain all value of the uncertain parameters consistent with the measurement model and noise bounds. SMA have been applied for mobile network positioning [35], RSSI filtering [36, 37], and inter-UAV cooperative localization [38]. While vision-based UAV setups have applied SMA to occluded sources [39], apparent intermittency in these contexts arises solely from line-of-sight visual masking [40] rather than genuine temporal RF emission characteristics. To address these limitations, this paper introduces a distributed SMA for localizing static and frequency-separable RF sources with intermittent but periodic emissions using a fleet of collaborating UAVs. The remainder of this paper is structured as follows. Section 2 details the system model and problem formulation. Section 3 details the iterative SMA for source localization and absence of source certification. Section 4 compares the performance of with proposed SMA with a Bayesian baseline approach. 2 Hypotheses and problem formulation Consider a unknown number NsN^s of static sources emitting RF signals and situated in a bounded 22-D area 0⊂ℝ3X_0 ^3 to which a frame ℱF is attached. The source locations are jt x^t_j, j∈t=⟦1,Nt⟧j ^t= 1,N^t . Source j emits periodically RF signals of duration δj _j, with period τj∈[δj,τmax] _j∈[ _j, _ ], where τmax _ is the maximum emission period, and emission offset t0,jt_0,j, defined with respect to a global temporal reference. The emission duration, period, and offset are not known. The signals emitted by two sources are considered as separable. A fleet of NuN^u UAVs explores 0X_0 to detect the sources and estimate their location. Time is slotted in intervals of duration T. The state of UAV i at time kTkT is iu(k) x^u_i(k) and consists of the location and orientation of the origin of ℱiF_i, the frame attached to UAV i in ℱF. We assume that each UAV knows its own state. Each UAV i∈u=⟦1,Nu⟧i ^u= 1,N^u is equipped with an antenna and signal processing system. As in [41], we assume that during the time interval [(k−d)T,kT][(k-d)T,kT], UAV i is only able to detect the presence of a source in its field of detection (FoD) i(k)F_i (k ), which shape and location depends on iu(k) x^u_i(k). If a source j emits at time t such that [t,t+δj]∩[(k−1)T,kT]≠∅[t,t+ _j]∩[(k-1)T,kT]≠ and jt∈i(k) x^t_j _i (k ), then source j is always detected by UAV i. The internal clocks of all UAVs of the fleet are assumed to be perfectly synchronized. The communication latency and errors are neglected, while the communication range rcomr_com is assumed to be limited: the list of one-hop neighbors of UAV i during the time interval [kT,(k+1)T][kT,(k+1)T] is denoted i(k)N_i(k). 2.1 Set estimates and problem formulation The set i(k)I_i(k) gathers the information available to UAV i up to time kTkT, including iu x^u_i(k) (k ), the list i(k)K_i(k) of indexes of already known sources (detected up to kTkT), all its measurements, and similar information obtained from its neighbors via communications. As will be detailed in Section 3, during the time interval [kT,(k+1)T][kT,(k+1)T], using i(k)I_i(k) and information obtained from its neighbours, UAV i evaluates a list i(k)=i,j(k)j∈i(k)X_i(k)= \X_i,j(k) \_j _i(k) of set estimates i,j(k)X_i,j(k) of jt x^t_j for all sources with index in i(k)K_i(k) and a set ¯i(k) X_i(k) containing all locations in 0X_0 where UAV i can state that there is no source. Then UAV i evaluates the source localization uncertainty Φiu(k)=ϕ(⋃j∈i(k)i,j(k)), ^u_i(k)=φ ( _j _i(k)X_i,j(k) ), (1) where ϕ()φ(X) is a measure of the set X. UAV i also evaluates the size of the set potentially containing sources Φia(k)=ϕ(0/¯i(k)). ^a_i(k)=φ (X_0/ X_i(k) ). (2) The objective of the fleet of UAVs is to minimize Φ(k)=1Nu∑i=1Nu(Φiu(k)+λΦia(k)), (k )= 1N^u_i=1^N^uΣ ( ^u_i(k)+λ ^a_i(k) ), (3) where λ⩾0λ 0 adjusts the trade-off between the source localization uncertainty and the uncertainty about the absence of sources. 3 Iterative Set-Membership Estimation Assuming that a given UAV i, immediately before time kTkT, has access only to i(k−1)K_i(k-1), i(k−1)=i,j(k−1)j∈i(k−1)X_i(k-1)= \X_i,j(k-1) \_j _i(k-1), and ¯i(k−1) X_i(k-1), with these sets initialized as i(0)=∅K_i(0)= , i(0)=∅X_i(0)= , and ¯i(0)=∅ X_i(0)= , the list i(k)K_i(k) is updated at time kTkT based on the list i(k)D_i(k), which contains the indexes of the sources detected by UAV i during the time interval [(k−1)T,kT] [ (k-1 )T,kT ] i(k)=i(k−1)∪i(k).K_i(k)=K_i(k-1) _i(k). (4) For a newly detected source, i.e., such that j∈i(k)j _i(k) and j∉i(k−1)j _i(k-1), the set estimate of the location j x_j is evaluated as i,j(k)=i(k)∖¯i(k−1),X_i,j(k)=F_i(k) X_i(k-1), (5) as ¯i(k−1) X_i(k-1) is free of sources. For a previously detected source j that is detected again, i.e., such that j∈i(k)∩i(k−1)j _i(k) _i(k-1), i,j(k)=(i,j(k−1)∩i(k))∖¯i(k−1),X_i,j(k)= (X_i,j(k-1) _i(k) ) X_i(k-1), (6) since the location of such source is known to belong to i,j(k−1)X_i,j(k-1), to i(k)F_i(k), and not to ¯i(k−1) X_i(k-1). If source j is not detected again, i.e., j∉i(k)j _i(k) and j∈i(k−1)j _i(k-1), (6) boils down to i,j(k)=i,j(k−1)∖¯i(k−1).X_i,j(k)=X_i,j(k-1) X_i(k-1). (7) 3.1 Proving the absence of a source The intermittent emission of sources makes it difficult to be certain of the absence of sources in subsets of 0X_0. A basic solution is to keep UAV i static for a duration larger than τmax _ . If no emission is perceived, the corresponding FoD can be considered free of sources, since the emission period is less than τmax _ and RF emissions are always detected from sources in the FoD of UAVs. This strategy prioritizes the reduction of Φia(k) ^a_i(k) in (3) over a reduction of Φiu(k) ^u_i(k) and is not necessarily the best to quickly detect sources so that UAVs are likely to move continuously. Consequently, some ∈0 x _0 may be observed by various UAVs during non-consecutive time intervals, usually of duration less than τmax _ . Even if the cumulated observation duration is larger than τmax _ , a source may be missed. Our aim in what follows is to determine under which condition related to the observation time intervals of a given subset of 0X_0, this subset may be considered free of sources. 3.1.1 Time intervals without detection and source detectability Consider a location ∈0 x _0 and i(,k)=[tq]q∈⟦1,No,i(,k)⟧O_i ( x,k )= \ [t_q ] \_q∈ 1,N_o,i ( x,k ) , the list of No,i(,k)N_o,i ( x,k ) observation time intervals during which no emission from x has been detected by UAV i. i(,k)O_i ( x,k ) is evaluated iteratively as i(,k)=i(,k−1)∪[(k−1)T,kT]O_i ( x,k )=O_i ( x,k-1 )∪ \ [ (k-1 )T,kT ] \ (8) for all ∈i(k)∖⋃j∈i(k)i,j(k), x _i (k ) _j _i (k )X_i,j(k), and i(,k)=i(,k−1)O_i ( x,k )=O_i ( x,k-1 ) else. Our aim in what follows is to determine using i(,k)O_i ( x,k ) whether there is no source at x. A source j located at x with emission period τj∈[τmin,τmax] _j∈[ _ , _ ] is detectable by UAV i using i(,k)O_i ( x,k ) if, whatever the initial emission offset t0∈[0,τj]t_0∈ [0, _j ], at least one of the periodic emissions would occur during at least one of the observation time intervals, i.e., if ∀t0∈[0,τj[,∃ℓ∈ℕ,∃[t]∈i(,k) s. t. t0+ℓτj∈[t],∀ t_0∈ [0, _j [,∃ ,∃ [t ] _i ( x,k ) s. t. t_0+ _j∈ [t ], (9) where [t]=[t¯,t¯][t]=[ t, t] represents an interval with lower bound t¯ t and upper bound t¯ t. Proposition 1. Consider ∈0 x _0 and i(,k)O_i ( x,k ). If for all τ∈[τmin,τmax]τ∈ [ _ , _ ], (9) is satisfied, then no source can be located in x. Proof. Assume that a source with emission period τ0 _0 and offset t0t_0 is present at x. Since for all τ∈[τmin,τmax]τ∈ [ _ , _ ], (9) is satisfied, for the period τ0 _0 and offset t0t_0, there exists ℓ∈ℕ and [t]∈i(,k) [t ] _i ( x,k ) such that t0+ℓτ0∈[t]t_0+ _0∈ [t ]. Thus, the source at x has emitted during the time interval [t] [t ]. We assumed that all emissions are detected, which leads to a contradiction. Consequently, no source is present at x. ∎ 3.1.2 Proving the absence of a source at a point of 0X_0 Proving the absence of sources at some x requires determining whether (9) is satisfied for all τ∈[τmin,τmax]τ∈ [ _ , _ ]. In what follows, we relax this criterion by considering a discrete set of periods d⊂[τmin,τmax]T_d⊂ [ _ , _ ]. This result is extended to a continuous set of emission periods in Appendix A.1. Prop. 2 introduces a criterion to verify whether (9) is satisfied for a given τ∈dτ _d. It involves the modulo of an interval defined as [t]modτ=tmodτ∣t∈[t]. [t ]modτ= \t τ t∈ [t ] \. (10) If the width w([t])w ( [t ] ) of the interval [t] [t ] is such that w([t])⩾τw ( [t ] ) τ, (10) is evaluated as [t]modτ=[0,τ[ [t ]modτ= [0,τ [ and if w([t])<τw ( [t ] )<τ, as [t]modτ=[t¯modτ,t¯modτ+t¯−t¯[if t¯modτ⩽t¯modτ,[0,t¯modτ[∪[t¯modτ,τ[else. [t ]modτ= cases [ tmodτ, tmodτ+ t- t [&if tmodτ tmodτ,\\ [0, tmodτ [∪ [ tmodτ,τ [&else. cases (11) Proposition 2. Consider ∈0 x _0, i(,k)O_i ( x,k ), and τ∈dτ _d. If [0,τ[=⋃[t]∈i(,k)[t]modτ [0,τ [= _ [t ] _i ( x,k ) [t ]modτ (12) then (9) is satisfied. Proof. To show that (12) implies (9), consider τ∈dτ _d with τ≠0τ≠ 0 and t0∈[0,τ[t_0∈ [0,τ [. Then (12) implies that ∃[t]∈(,k)∃ [t ]∈$ O$$ ( x,k )$ such that t0∈[t]modτt_0∈ [t ]modτ. Three cases are considered. First, if w([t])⩾τw ( [t ] ) τ, [t]modτ=[0,τ[ [t ]modτ= [0,τ [, so t0∈[0,τ[t_0∈ [0,τ [. The sequence uℓ:ℓ↦t0+ℓτu_ : t_0+ τ is increasing, so ∃ℓ∈ℕ∃ such that t0+ℓτ⩾t¯⩾t0+(ℓ−1)τ.t_0+ τ t t_0+( -1)τ. (13) Adding τ to the second term of (13), one gets t¯+τ⩾t0+ℓτ t+τ t_0+ τ. As t¯−t¯⩾τ t- t τ, one has t¯⩾t0+ℓτ t t_0+ τ. Using the first term of (13), one has t¯⩾t0+ℓτ⩾t¯ t t_0+ τ t. Therefore, ∃ℓ∈ℕ such that t0+ℓτ∈[t]∃ such that t_0+ τ∈ [t ]. Second, if w([t])<τw ( [t ] )<τ and t¯modτ⩽t¯modτ tmodτ tmodτ, then [t]modτ=[t¯modτ,t¯modτ+t¯−t¯[ [t ]modτ= [ tmodτ, tmodτ+ t- t [. As t0∈[t]modτt_0∈ [t ]modτ, t¯−ℓ¯τ⩽t0⩽t¯−ℓ¯τ+t¯−t¯ t- τ t_0 t- τ+ t- t, with ℓ¯=⌊t¯/τ⌋ = t/τ . Thus, ∃ℓ¯∈ℕ such that t0+ℓ¯τ∈[t]∃ such that t_0+ τ∈ [t ]. Third, if w([t])<τw ( [t ] )<τ and t¯modτ>t¯modτ tmodτ> tmodτ then [t]modτ=[0,t¯modτ[∪[t¯modτ,τ[ [t ]modτ= [0, tmodτ [∪ [ tmodτ,τ [. Two sub-cases are now considered. First subcase: t0∈[0,t¯modτ[t_0∈[0, tmodτ[ implies t0∈[0,t¯−ℓ¯τ[t_0∈[0, t- τ[, with ℓ¯=⌊t¯/τ⌋ = t/τ . Hence, ℓ¯τ⩽t0+ℓ¯τ⩽t¯. τ t_0+ τ t. (14) Moreover, as t¯modτ>t¯modτ tmodτ> tmodτ, ∃ℓ¯∈ℕ∃ such that t¯−ℓ¯τ>t¯−ℓ¯τ t- τ> t- τ. Hence (ℓ¯−ℓ¯)τ>t¯−t¯>0 ( - )τ> t- t>0 and ℓ¯>ℓ¯ > (because OPENτ⩾0)τ 0). This last result gives ℓ¯τ⩽t¯<ℓ¯τ τ t< τ. This, with (14) yields t¯⩽t0+ℓ¯τ⩽t¯ t t_0+ τ t. Second subcase: t0∈[t¯modτ,τ[t_0∈ [ tmodτ,τ [ implies t0∈[t¯−ℓ¯τ,τ[t_0∈ [ t- τ,τ [, so t¯⩽t0+ℓ¯τ⩽(ℓ¯+1)τ t t_0+ τ ( +1 )τ. With ℓ¯=⌊t¯/τ⌋ = t/τ and ℓ¯>ℓ¯ > , one has t¯⩽t0+ℓ¯τ⩽(ℓ¯+1)τ⩽ℓ¯τ⩽t¯ t t_0+ τ ( +1 )τ τ t. Thus, in both sub-cases, ∃ℓ∈ℕ such that t0+ℓτ∈[t]∃ such that t_0+ τ∈ [t ]. Finally, in all of the three cases, we have shown that ∃ℓ∈ℕ such that t0+ℓτ∈[t]∃ such that t_0+ τ∈ [t ]: (9) is then verified. ∎ Proving the absence of sources with emission period in dT_d at x requires thus verifing (12) for all τ∈dτ _d. The set ¯j(k) X_j(k) is then iteratively updated as ¯i(k)=¯i(k−1)∪ X_i(k)= X_i(k-1)∪\ ∈i(k)∣ x _i (k ) ∀τ∈d,[0,τ[=⋃[t]∈i(,k)[t]modτ. -28.45274pt∀τ _d, [0,τ [= _ [t ] _i ( x,k ) [t ]modτ\. (15) This update is only performed for points in i(k)F_i (k ) as i(,k)O_i ( x,k ) is only updated for those points using (8). 3.1.3 Proving the absence of sources in subsets of 0X_0 If (12) is satisfied for some 0∈0 x_0 _0 and i(0,k)O_i( x_0,k), then it also holds true for all ∈0 x _0 such that i(,k)=i(0,k)O_i( x,k)=O_i( x_0,k). To prove the absence of sources in subsets of 0X_0, 0X_0 is partitioned in boxes []q [ x ]_q, q=1,…,Ncq=1,…,N_c gathered in a list 0X_0. The lists of observation time intervals are then considered for each box []∈0[ x] _0 and updated as follows. If []⊂i(k)∖⋃j∈i(k)i,j(k) [ x ] _i (k ) _j _i (k )X_i,j(k), i([],k)=i([],k−1)∪[(k−1)T,kT].O_i ( [ x ],k )=O_i ( [ x ],k-1 )∪ \ [ (k-1 )T,kT ] \. (16) Prop. 2 is easily extended to verify whether (12) is satisfied for all ∈[] x∈[ x] considering i([],k)O_i([ x],k). The size of the cells []∈0 [ x ] _0 determines the compromise between evaluation complexity and estimation accuracy. If [] [ x ] is too large, i([],k)O_i ( [ x ],k ) may remain empty. 3.2 Communications and information fusion To extend the previous framework to a fleet of communicating UAVs, we assume that during the time interval [kT,(k+1)T][kT,(k+1)T], UAV i broadcasts iu(k) x^u_i(k), i(k)K_i(k), i(k)X_i(k), and ¯i(k) X_i(k). The lists i([],k)O_i ( [ x ],k ) for all []∈0∖¯i(k) [ x ] _0 X_i(k) are also transmitted. After reception of similar information from its neighbors, UAV i evaluates i(k)=i(k)∪⋃ℓ∈i(k)ℓ(k)K_i(k)=K_i(k)∪ _ _i(k)K_ (k) and ¯i(k)=¯i(k)∪⋃ℓ∈i(k)¯ℓ(k) X_i(k)= X_i(k)∪ _ _i(k) X_ (k). The set estimates of the source locations for all j∈i(k)j _i(k) are updated as i,j(k)=i,j(k)∩⋂ℓ∈i(k),j∈ℓ(k)ℓ,j(k).X_i,j(k)=X_i,j(k)∩ _i(k),j _ (k) X_ ,j(k). 4 Simulation and results For the numerical simulation, we consider a square search area 0X_0 of 100100 m wide partitioned into square cells of 22 m wide in which 1010 sources are randomly placed following a uniform distribution. Each source j emits a periodic signals with period τj _j uniformly distributed in [τmin,τmax]=[2,16] [ _ , _ ]= [2,16 ] s and initial offset t0,jt_0,j uniformly distributed in [0,τj] [0, _j ]. The initial locations of UAVs are set randomly, following a uniform distribution over 0X_0. UAVs move at a constant speed and follow a lawn-mower trajectory, see Figure 2. This enables a fair comparison without control-induced effects. More sophisticated control techniques may be employed as in [39]. Figure 1: Lawn-mower trajectory followed by UAVs. The FoD i(k)F_i (k ) is the intersection of two disks of centers the orthogonal projection of u(k−1) x^u (k-1 ) and u(k) x^u (k ) on 0X_0 and radius 1010 m. The communication radius is rcom=30r_com=30 m, typical of the Bluetooth Low Energy protocol. The sampling period for signal acquisition and estimate updates is T=1T=1 s. 4.1 Bayesian baseline algorithm (BBA) An adaptation to several UAVs of the Bayesian estimator introduced in [18] serves as baseline for comparisons. This Bayesian baseline algorithm is denoted BBA. During the time interval [(k−1)T,kT] [ (k-1 )T,kT ], the variable zi,j(k)=1z_i,j (k )=1 indicates that UAV i has detected source j in i(k)F_i (k ). If zi,j(k)=0z_i,j (k )=0, the source j has not been detected. We assume that the detection probability of a signal emitted by the j-th source at j x_j is ℓj(u(k),j)=pD _j ( x^u (k ), x_j )=p_D if j∈i(k) x_j _i (k ) and ℓj(u(k),j)=pF _j ( x^u (k ), x_j )=p_F if j∉i(k) x_j _i (k ), where pDp_D and pFp_F are the detection and false alarm probabilities. BBA also consider the partition of 0X_0 in the list of boxes 0X_0. For UAV i located at iu(k) x^u_i (k ), the likelihood of zi,j(k)z_i,j (k ) for the cell [] [ x ] is taken as the upper-bound of the likelihoods considering all ∈[] x∈[ x] p(zi,j(k)∣iu(k),[]) p (z_i,j (k ) x^u_i (k ), [ x ] ) =max∈[]ℓj(iu(k),)zi,j(k) = _ x∈[ x] _j ( x^u_i (k ), x )^z_i,j (k ) (1−ℓj(iu(k),))1−zi,j(k), (1- _j ( x^u_i (k ), x ) )^1-z_i,j (k ), (17) which is easily evaluated by testing whether [][ x] intersects i(k)F_i (k ). As in [18], UAV i evaluates pi,j([]∣zi,j(1:k),iu(1:k))p_i,j ( [ x ] z_i,j (1:k ), x^u_i (1:k ) ), the a posteriori probability mass function (pmf) of presence of source j in a cell []∈0 [ x ] _0 from the measurements zi,j(1:k)z_i,j (1:k ) obtained for the UAV locations iu(1:k) x^u_i (1:k ). The number of sources NsN^s has to be known and NsN^s pmfs are managed in parallel, one for each source, and initialized at p0=1/|0|p_0=1/ |X_0 | for all []∈0 [ x ] _0. An estimate ^i,jt,B(k) x^t,B_i,j (k ) of the source location is taken as the mode of pi,j([]∣zi,j(1:k),iu(1:k))p_i,j ( [ x ] z_i,j (1:k ), x^u_i (1:k ) ) and the estimation uncertainty is the set i,jB(k)X^B_i,j(k) of all []∈0 [ x ] _0 with a pmf larger than some threshold puBp^B_u. The set ¯iB(k) X^B_i(k) considered clear of sources in the BBA consists of all []∈0 [ x ] _0 such that the average over all sources i of pi,j([]∣zi,j(1:k),iu(1:k))p_i,j ( [ x ] z_i,j (1:k ), x^u_i (1:k ) ) is less than some threshold paBp^B_a. We choose pF=10−6p_F=10^-6 and pD=1/τavgp_D=1/ _avg, with τavg _avg the average of τmin=1s _ =1~s and τmax=16s _ =16~s, to account the periodicity of emissions. The thresholds puBp^B_u and paBp^B_a are respectively set to puB=102p0p^B_u=10^2p_0 and paB=10−2p0p^B_a=10^-2p_0. Each UAV i updates a probability map i,j(k)P_i,j (k ) for each detected target j: i,j(k)=pi,j([]∣zi,j(1:k),iu(1:k))[]∈0.P_i,j (k )= \p_i,j ( [ x ] z_i,j (1:k ), x^u_i (1:k ) ) \_ [ x ] _0. (18) UAVs share their probability maps i,j(k)P_i,j (k ) for all detected targets with their neighbors in list i(k)N_i(k). For the sake of simplicity, let us note i,j(k)=zi,j(1:k),iu(1:k)Z_i,j(k)= \z_i,j (1:k ), x^u_i (1:k ) \. The probability maps of two UAVs i and ℓ are the merged using p([]|i,j(k),yℓ)=a+1p ([ x]|Z_i,j(k),y_ )= aa+1 (19) with a=p([]|i,j(k))p([]|ℓ,j(k))(1−p([]|i,j(k)))(1−p([]|ℓ,j(k)))1−p0p0,a= p ([ x]|Z_i,j(k) )p ([ x]|Z_ ,j(k) ) (1-p ([ x]|Z_i,j(k) ) ) (1-p ([ x]|Z_ ,j(k) ) ) 1-p_0p_0, (20) see Appendix A.2. This approach assumes that the observations of UAVs i and ℓ are independent, which may not be always verified in practice. 4.2 Metrics For UAV i, we consider (i) the proportion of detected sources at time kTkT di(k)=|i(k)|/Nt,d_i(k)= |K_i(k) |/N^t, and (i) once a source has been detected, the average source localization error εi(k)=1|i(k)|∑j∈i(k)‖^i,jt(k)−jt‖2, _i (k )= 1 |K_i(k) | _j _i(k) \| x^t_i,j (k )- x^t_j \|_2, (21) where ^i,jt(k) x^t_i,j (k ) is the estimate of jt x^t_j obtained using SMA (barycenter of the set estimates) or BBA (mode of the probabilities of presence evaluated for all boxes in 0X_0), and (i) the uncertainty on the absence of sources Φia(k) ^a_i(k) introduced in (2). We evaluate and represent the average of these metrics over all UAVs and averages over 1010 independent simulation runs. 4.3 Results Fig. 2(a) shows the impact of the speed of UAVs On the proportion of detected targets when 2 UAVs are used. Increasing the exploration speed allows a faster detection of all sources. Fig. 2(b) shows the impact of the fleet size when the UAV speed is 1.25m/s1.25\,m/s. As expected, the time decreases to localize all targets when the fleet size increases. As the UAV trajectory is the same for SMA and BBA, the curves are the same for both algorithms. Figs 3(a) and (b) show the evolution of the average source localization error for SMA and BBA, for different UAV speeds with a fleet of 2 UAVs. The SMA provides a reduced error compared to BBA. Reducing the UAV speed reduces the error, as more measurements of the same sources can be acquired during a single sweep. (a) (b) Figure 2: Proportion of detected targets for 2 UAVs with various speeds (a) and for various UAV numbers at a speed of 1.25m/s1.25\,m/s (b). (a) (b) (c) (d) Figure 3: Average source localization error for various displacement speeds (2 UAVs), SMA (a) and BBA (b); Average size of the set potentially containing sources (2 UAVs), SMA (c) and BBA (d). (a) (b) Figure 4: Average error on source localization (a) and Average size of the set potentially containing sources (b) for various fleet sizes evolving at 1.25m/s1.25\,m/s, SMA (solid lines) and PBA (dashed lines). Figs 3(c) and (d) show the evolution of the average size of the set potentially containing sources, again for different UAV speeds with a fleet of 2 UAVs for SMA and BBA. While a displacement at a high speed allows a quick detection and first localization of sources, the best speed to show the absence of targets in large areas are respectively 1.251.25 and 2.5m/s2.5\,m/s for the SMA and the BBA. The optimal speed results then in a compromise between delay of all source detection, detection accuracy, and ability to show the absence of targets. Figs 4 shows the impact of the size of the fleet on the average localization error (a) and on the average size of the set potentially containing sources (b), when the UAV velocity is 1.25m/s1.25\,m/s. The simulations confirm that the larger the fleet, the faster all sources are detected and the faster the size of 0∖¯i(k)X_0 X_i(k) decreases. SMA performs consistently better than BBA. The evolution of the size of ¯iB(k) X^B_i(k) using BBA varies depending on the values of threshold paBp^B_a, but we showed that regardless of the values of this threshold (between 10−1p010^-1p_0 and 10−8p010^-8p_0 in the simulation results presented in Figure 5), SMA performs better than BBA, for the scenario presented in this paper. Indeed, our probabilistic definition of absence uncertainty only quantifies the area for which the probability of containing a target is below an arbitrary threshold (2), set to paB=p0⋅10−2=4⋅10−6p^B_a=p_0· 10^-2=4· 10^-6 in the simulations leading to the results presented in this Section 4.3. Consequently, 1) the uncertainty level highly depends on the chosen threshold (see Figure 5), 2) this probabilistic uncertainty definition does not have the same meaning as the deterministic uncertainty, which relies on the certainty that no target can be located in a certain area (under an assumption about the maximal emission period). Figure 5: Average size of the set potentially containing sources for a fleet of 2 UAVs, comparison between the deterministic algorithm (SMA) and the Bayesian baseline (PBB) with several values (10−1p010^-1p_0,10−2p010^-2p_0, 10−3p010^-3p_0, 10−4p010^-4p_0) of threshold paBp^B_a. Videos of the simulation can be found here. 5 Conclusion The proposed SMA is able to evaluate set estimates for the location of sources emitting periodically RF signals. It also provides subsets of the search area clear of sources, provided that the hypotheses on the source emission period and the detection probability are satisfied. Simulations shows that the SMA slightly outperforms the BBA in terms of localization accuracy and convergence speed. The ability to deterministically localize and certify the absence of intermittent RF sources has direct applications in search and rescue, surveillance, electromagnetic intelligence, and connected object monitoring. By providing mathematically rigorous guarantees, this work offers a compelling alternative to probabilistic methods in safety-critical applications. Among open challenges, one may cite an extension to moving sources with sporadic but non-periodic emissions. Appendix A Appendix A.1 Extending the criterion of target absence from discrete periods to period intervals Consider the set of observation time intervals (,k)O ( x,k ) and assume that for some τ∈dτ _d, one has [0,τ[=⋃[t]∈(,k)([t]modτ). [0,τ [= [t ] ( x,k ) ( [t ]modτ ). (22) Our aim in what follows is to determine an interval [τ′] [τ ] with τ∈[τ′]τ∈ [τ ] such that (22) is satisfied for all τ′∈[τ′]τ ∈ [τ ]. The main idea is to consider small variations of τ and to determine conditions under which (22) is still satisfied. Several cases depending on |(,k)| |O ( x,k ) | are considered and illustrated in Figure 6. Figure 6: Different cases for which [0,τ[=⋃[t]∈(,k)([t]modτ) [0,τ [= [t ] ( x,k ) ( [t ]modτ ). A.1.1 When |(,k)|=1 |O ( x,k ) |=1 In this case, one has (,k)=[t]O ( x,k )= \ [t ] \. From (22), [0,τ[=[t]modτ [0,τ [= [t ]modτ. With w([t])w ( [t ] ) the width of interval [t] [t ], for all τ′⩽w([t])τ w ( [t ] ), one has [0,τ′[=[t]modτ′ [0,τ [= [t ]modτ . Consequently, (22) is satisfied for all τ′∈[0,w([t])]τ ∈ [0,w ( [t ] ) ]. This corresponds to the first case in Figure 6. A.1.2 When |(,k)|=2 |O ( x,k ) |=2 In this case, (,k)=[t1],[t2]O ( x,k )= \ [t_1 ], [t_2 ] \. Without loss of generality, assume that t¯1<t¯2 t_1< t_2, i.e., that [t2] [t_2 ] is above [t1] [t_1 ]. • Case a: If w([t1])⩾τw ( [t_1 ] ) τ or w([t2])⩾τw ( [t_2 ] ) τ, then, a direct application of the result of the case |(,k)|=1 |$ O$ ( x,k ) |=1 yields that (22) holds true for all τ′∈[0,maxw([t1]),w([t2])]τ ∈ [0, \w ( [t_1 ] ),w ( [t_2 ] ) \ ]. This corresponds to the second case in Figure 6. • Case b: When [t¯1,t¯1]modτ [ t_1, t_1 ]modτ and [t¯2,t¯2]modτ [ t_2, t_2 ]modτ consist of a different number of intervals, several subcases have to considered. – case b-1: [t¯1,t¯1] mod τ=[0,t1′[∪[t1′,τ[ [ t_1, t_1 ] mod τ= [0,t_1 [∪ [t_1 ,τ [ and [t¯2,t¯2] mod τ=[t2′,t2′[ [ t_2, t_2 ] mod τ= [t_2 ,t_2 [, see the third case in Figure 6. Since (22) is satisfied for τ,τ,one has t1′⩾t2′,t2′⩾t1′. array[]ct_1 t_2 ,\\ t_2 t_1 . array (23) Introducing ℓ1=⌊t¯1/τ⌋ _1= t_1/τ , ℓ2=⌊t¯2/τ⌋ _2= t_2/τ and t1′(τ′)=t¯1−ℓ1τ′,t2′(τ′)=t¯2−ℓ2τ′,t1"(τ′)=t¯1−(ℓ1−1)τ′,t2"(τ′)=t¯2−ℓ2τ′, array[]ct_1 (τ )= t_1- _1τ ,\\ t_2 (τ )= t_2- _2τ ,\\ t_1" (τ )= t_1- ( _1-1 )τ ,\\ t_2" (τ )= t_2- _2τ , array (24) one has t1′(0)=t1′,…,t2"(0)=t2"t_1 (0 )=t_1 ,…,t_2" (0 )=t_2". Since t¯1<t¯2 t_1< t_2 and t1′⩾t2′t _1 t _2, one has ℓ1<ℓ2 _1< _2. One has to determine the possible values of τ′τ such that the union of [t¯1,t¯1]modτ′ [ t_1, t_1 ]modτ and [t¯2,t¯2]modτ′ [ t_2, t_2 ]modτ still yields [0,τ′[ [0,τ [. A sufficient condition is obtained by considering all τ′τ such that t2′(τ′)⩽t1′(τ′),t1"(τ′)⩽t2"(τ′),t2′(τ′)⩾0,t2"(τ′)⩽τ′, array[]ct_2 (τ ) t_1 (τ ),\\ t_1" (τ ) t_2" (τ ),\\ t_2 (τ ) 0,\\ t_2" (τ ) τ , array (25) as then, one has ([t¯1,t¯1]modτ′)∪([t¯2,t¯2]modτ′) ( [ t_1, t_1 ]modτ )∪ ( [ t_2, t_2 ]modτ ) =[0,t1′(τ′)[∪[t2′(τ′),t2′(τ′)[∪[t1′(τ′),τ′[ = [0,t_1 (τ ) [∪ [t_2 (τ ),t_2 (τ ) [∪ [t_1 (τ ),τ [ =[0,τ′[. = [0,τ [. Using (24) in (25), one can easily show that (22) is satisfied for all τ′τ in [maxt¯2ℓ2+1,t¯2−t¯1ℓ2−ℓ1,mint¯2−t¯1ℓ2−ℓ1+1,t¯2ℓ2]. [ \ t_2 _2+1, t_2- t_1 _2- _1 \, \ t_2- t_1 _2- _1+1, t_2 _2 \ ]. (26) – case b-2: [t¯1,t¯1]modτ=[t1′,t1′[ [ t_1, t_1 ]modτ= [t_1 ,t_1 [ and [t¯2,t¯2]modτ=[0,t2′[∪[t2′,τ[ [ t_2, t_2 ]modτ= [0,t_2 [∪ [t_2 ,τ [, see the fourth case in Figure 6. Since (22) is satisfied for τ, one has t1′⩽t2′t2′⩽t1′. array[]ct_1 t_2 \\ t_2 t_1 array. (27) Consider ℓ1=⌊t¯1/τ⌋ _1= t_1/τ , ℓ2=⌊t¯2/τ⌋ _2= t_2/τ , and t1′(τ′)=t¯1−ℓ1τ′,t2′(τ′)=t¯2−ℓ2τ′,t1"(τ′)=t¯1−ℓ1τ′,t2"(τ′)=t¯2−(ℓ2−1)τ′. array[]ct_1 (τ )= t_1- _1τ ,\\ t_2 (τ )= t_2- _2τ ,\\ t_1" (τ )= t_1- _1τ ,\\ t_2" (τ )= t_2- ( _2-1 )τ . array (28) Again, one has t1′(0)=t1′,…,t2"(0)=t2"t_1 (0 )=t_1 ,…,t_2" (0 )=t_2". Now, for all τ′τ such that t2′(τ′)⩾t1′(τ′),t1"(τ′)⩾t2"(τ′),t1′(τ′)⩾0,t1"(τ′)⩽τ′, array[]ct_2 (τ ) t_1 (τ ),\\ t_1" (τ ) t_2" (τ ),\\ t_1 (τ ) 0,\\ t_1" (τ ) τ , array (29) one has ([t¯1,t¯1]modτ′)∪([t¯2,t¯2]modτ′)=[0,τ′[. ( [ t_1, t_1 ]modτ )∪ ( [ t_2, t_2 ]modτ )= [0,τ [. (30) Since t¯1<t¯2 t_1< t_2 and t1′⩾t2′t_1 t_2 , one has ℓ1+1<ℓ2 _1+1< _2. Using (28) in (29), one can again easily show that (22) is satisfied for all τ′τ in [max(t¯1ℓ1+1,t¯2−t¯1ℓ2−ℓ1−1),min(t¯2−t¯1ℓ2−ℓ1,t¯1ℓ1)] [ ( t_1 _1+1, t_2- t_1 _2- _1-1 ), ( t_2- t_1 _2- _1, t_1 _1 ) ] (31) • case c: When [t¯1,t¯1]modτ [ t_1, t_1 ]modτ and [t¯2,t¯2]modτ [ t_2, t_2 ]modτ both consist of two intervals, one has [t¯1,t¯1]modτ=[0,t1′[∪[t1′,τ[ [ t_1, t_1 ]modτ= [0,t_1 [∪ [t_1 ,τ [ and [t¯2,t¯2]modτ=[0,t2′[∪[t2′,τ[ [ t_2, t_2 ]modτ= [0,t_2 [∪ [t_2 ,τ [. Then, consider ℓ1=⌊t¯1/τ⌋ _1= t_1/τ , ℓ2=⌊t¯2/τ⌋ _2= t_2/τ (with ℓ2>ℓ1, _2> _1,else [t¯1,t¯1] [ t_1, t_1 ] and [t¯2,t¯2] [ t_2, t_2 ] would overlap) and t1′(τ′)=t¯1−ℓ1τ′,t2′(τ′)=t¯2−ℓ2τ′,t1"(τ′)=t¯1−(ℓ1−1)τ′,t2"(τ′)=t¯2−(ℓ2−1)τ′, array[]ct_1 (τ )= t_1- _1τ ,\\ t_2 (τ )= t_2- _2τ ,\\ t_1" (τ )= t_1- ( _1-1 )τ ,\\ t_2" (τ )= t_2- ( _2-1 )τ , array (32) one has again t1′(0)=t1′,…,t2"(0)=t2"t_1 (0 )=t_1 ,…,t_2" (0 )=t_2". Two cases have to be considered. – case c-1: t2′>t1′t_2 >t_1 , see the fifth case in Figure 6. Then, for all τ′τ such that t2′(τ′)⩾t1′(τ′),t1′(τ′)⩾0,t2"(τ′)⩽τ′, array[]ct_2 (τ ) t_1 (τ ),\\ t_1 (τ ) 0,\\ t_2" (τ ) τ , array (33) one has ([t¯1,t¯1]modτ′)∪([t¯2,t¯2]modτ′)=[0,τ′[. ( [ t_1, t_1 ]modτ )∪ ( [ t_2, t_2 ]modτ )= [0,τ [. (34) Introducing (32) in (33), one easily shows that (22) is satisfied for all τ′∈[t¯2ℓ2,min(t¯1ℓ1,t¯2−t¯1ℓ2−ℓ1+1)].τ ∈ [ t_2 _2, ( t_1 _1, t_2- t_1 _2- _1+1 ) ]. (35) – case c-2: t1′>t2′t_1 >t_2 , see the sixth case in Figure 6. Then, for all τ′τ such that t1′(τ′)⩾t2′(τ′),t2′(τ′)⩾0,t1"(τ′)⩽τ′, array[]ct_1 (τ ) t_2 (τ ),\\ t_2 (τ ) 0,\\ t_1" (τ ) τ , array (36) one has ([t¯1,t¯1]modτ′)∪([t¯2,t¯2]modτ′)=[0,τ′[. ( [ t_1, t_1 ]modτ )∪ ( [ t_2, t_2 ]modτ )= [0,τ [. (37) Introducing (32) in (36), one shows that (22) is satisfied for all τ′∈[max(t¯1ℓ1,t¯2−t¯1ℓ2−ℓ1−1),t¯2ℓ2].τ ∈ [ ( t_1 _1, t_2- t_1 _2- _1-1 ), t_2 _2 ]. (38) A.1.3 When |(,k)|⩾3 |O ( x,k ) | 3 In this case, since [0,τ[=⋃[t]∈(,k)([t]modτ) [0,τ [= [t ]∈$ O$$ ( x,k )$ ( [t ]modτ ), there exists at least one subset ′(,k)O ( x,k ) of (,k)O ( x,k ) of smallest cardinality such that [0,τ[=⋃[t]∈′(,k)([t]modτ) [0,τ [= [t ] ( x,k ) ( [t ]modτ ) (39) and (39) is not true if a single interval is removed from ′(,k)O ( x,k ), i.e. ∀[t]∈′(,k)∀ [t ] ( x,k ), (39) is not true for ′(,k)∖[t]O ( x,k ) \ [t ] \. Then, one partitions ′(,k)O ( x,k ) as ′(,k)=1′(,k)∪2′(,k)O ( x,k )=O _1 ( x,k ) _2 ( x,k ) (40) where 1′(,k)O _1 ( x,k ) is the list of intervals [t]∈′(,k) [t ] ( x,k ) such that [t]modτ [t ]modτ consists of one interval and 2′(,k)O _2 ( x,k ) is the list of intervals [t]∈′(,k) [t ] ( x,k ) such that [t]modτ [t ]modτ consists of two intervals. Figure 7 illustrates a general case with |(,k)|⩾3 |O ( x,k ) | 3 and (39) satisfied. The gray and purple intervals are not in ′(,k)O ( x,k ). The orange, yellow, and red intervals are in 1′(,k)O _1 ( x,k ). The pink and blue intervals are in 2′(,k)O _2 ( x,k ). Figure 7: General case for (,k)O ( x,k ) where [0,τ[=⋃[t]∈(,k)([t]modτ) [0,τ [= [t ] ( x,k ) ( [t ]modτ ). Proposition 3. 2′(,k)O _2 ( x,k ) contains at most two intervals. Proof. For all [ti]∈2′(,k)[t_i] _2 ( x,k ), [ti]modτ=[0,ti′[∪[ti′,τ[ [t_i ]modτ= [0,t_i [∪ [t_i ,τ [. ∃ℓ∈⟦0,|2′(,k)|⟧∃ ∈ 0, |O _2 ( x,k ) | such that tℓ′=maxi∈⟦0,|′2(,k)|⟧ti′.t_ = i∈ 0, |O _2 ( x,k ) | \t_i \. (41) If moreover, tℓ"=mini∈⟦0,|′2(,k)|⟧ti′,t_ "= i∈ 0, |O _2 ( x,k ) | \t_i \, (42) then ⋃[t]∈2′(,k)([t]modτ)=[tℓ]modτ [t ] _2 ( x,k ) ( [t ]modτ )= [t_ ]modτ (43) and all intervals [ti] [t_i ] with i≠ℓi≠ cannot belong to 2′(,k)⊂′(,k)O _2 ( x,k ) ( x,k ) since ′(,k)O ( x,k ) is a smallest subset of (,k)O ( x,k ) for which (39) is satisfied. Now assume that ∃h∈⟦0,|2′(,k)|⟧∃ h∈ 0, |O _2 ( x,k ) | , h≠ℓh≠ such that th"=mini∈⟦0,|′2(,k)|⟧ti′,t_h"= i∈ 0, |O _2 ( x,k ) | \t_i \, (44) then ⋃[t]∈2′(,k)([t]modτ)=[tℓ]modτ∪[th]modτ [t ] _2 ( x,k ) ( [t ]modτ )= [t_ ]modτ∪ [t_h ]modτ (45) and all intervals [ti] [t_i ] with i≠ℓi≠ or i≠hi≠ h cannot belong to 2′(,k)O _2 ( x,k ). ∎ In what follows, we consider the case |2′(,k)|=2 |O _2 ( x,k ) |=2. Results are easily extended to the case |2′(,k)|=1 |O _2 ( x,k ) |=1. Let n′=|’(,k)|n = |$ O$' ( x,k ) | and the following indexing of intervals in 1′(,k)O_1 ( x,k ) and 2′(,k)O_2 ( x,k ) 1′(,k) _1 ( x,k ) =[t2],…,[tn′−1] = \ [t_2 ],…, [t_n -1 ] \ (46) 2′(,k) _2 ( x,k ) =[t1],[tn′] = \ [t_1 ], [t_n ] \ (47) such that t2′ t_2 <⋯<tn′−1′ <…<t_n -1 (48) t1′ t_1 =maxt1′,tn′ = \t_1 ,t_n \ (49) and tn′"=mint1",tn′".t_n "= \t_1",t_n " \. (50) An example of this indexing is provided in Figure 7. Consider ℓi=⌊t¯i/τ⌋ _i= t_i/τ , i=1,…,n′i=1,…,n and t1′(τ′)=t¯1−ℓ1τ′,ti′(τ′)=t¯i−ℓiτ′, i=2,…,n′−1tn′(τ′)=t¯n′−ℓn′τ′,t1"(τ′)=t¯1−(ℓ1−1)τ′,ti"(τ′)=t¯i−ℓiτ′, i=2,…,n′−1,tn′"(τ′)=t¯n′−(ℓn′−1)τ′. array[]ct_1 (τ )= t_1- _1τ ,\\ t_i (τ )= t_i- _iτ , i=2,…,n -1\\ t_n (τ )= t_n - _n τ ,\\ t_1" (τ )= t_1- ( _1-1 )τ ,\\ t_i" (τ )= t_i- _iτ , i=2,…,n -1,\\ t_n " (τ )= t_n - ( _n -1 )τ . array (51) For all τ′τ such that t1′(τ′)⩾t2′(τ′)t1′(τ′)⩽τ′ti′(τ′)⩾ti+1′(τ′), i=2,…,n′−2tn′−1′(τ′)⩾tn′τ′tn′(τ′)⩾0 array[]ct_1 (τ ) t_2 (τ )\\ t_1 (τ ) τ \\ t_i (τ ) t_i+1 (τ ), i=2,…,n -2\\ t_n -1 (τ ) t_n τ \\ t_n (τ ) 0 array (52) one has ([t1]modτ′)∪⋯∪([tn′]modτ′)=[0,τ′[. ( [t_1 ]modτ )∪…∪ ( [t_n ]modτ )= [0,τ [. (53) Combining (52) and (51), one gets t¯1−t¯2⩾(ℓ1−ℓ2)τ′t¯1⩽ℓ1τ′t¯i−t¯i+1⩾(ℓi−ℓi+1)τ′, i=2,…,n′−2t¯n′−1−t¯n′⩾(ℓn′−1−ℓn′+1)τ′t¯n′⩾ℓn′τ′. array[]c t_1- t_2 ( _1- _2 )τ \\ t_1 _1τ \\ t_i- t_i+1 ( _i- _i+1 )τ , i=2,…,n -2\\ t_n -1- t_n ( _n -1- _n +1 )τ \\ t_n _n τ . array (54) The conditions to be satisfied by τ′=τ′τ =τ to ensure that (53) holds true depend on the signs of ℓi−ℓi+1 _i- _i+1, i=1,…,n′−2i=1,…,n -2 and of the sign of ℓn′−1−ℓn′+1 _n -1- _n +1. Consider the sets of indices ℐ−=i=1,…,n′−2∣ℓi<ℓi+1I^-= \i=1,…,n -2 _i< _i+1 \ and ℐ+=i=1,…,n′−2∣ℓi>ℓi+1.I^+= \i=1,…,n -2 _i> _i+1 \. Then, if ℓn′−1<ℓn′−1 _n -1< _n -1, τ′ τ ∈[max(t¯1ℓ1,maxi∈ℐ−t¯i−t¯i+1ℓi−ℓi+1,t¯n′−1−t¯n′ℓn′−1−ℓn′+1), ∈ [ ( t_1 _1, i ^- t_i- t_i+1 _i- _i+1, t_n -1- t_n _n -1- _n +1 ), . min(t¯n′ℓn′,mini∈ℐ+t¯i−t¯i+1ℓi−ℓi+1)], 56.9055pt . ( t_n _n , _i ^+ t_i- t_i+1 _i- _i+1 ) ], (55) else, if ℓn′−1>ℓn′−1 _n -1> _n -1, τ′ τ ∈[max(t¯1ℓ1,maxi∈ℐ−t¯i−t¯i+1ℓi−ℓi+1), ∈ [ ( t_1 _1, i ^- t_i- t_i+1 _i- _i+1 ), . min(t¯n′ℓn′,mini∈ℐ+t¯i−t¯i+1ℓi−ℓi+1,t¯n′−1−t¯n′ℓn′−1−ℓn′+1)]. . ( t_n _n , _i ^+ t_i- t_i+1 _i- _i+1, t_n -1- t_n _n -1- _n +1 ) ]. (56) A.2 Fusion of probability maps between UAVs With the notations introduced in Section 4.1, this section shows how to update the probability maps (18) exchanged between two UAVs. For the sake of simplicity, the following notations are introduced: p([])p ( [ x ] ) is the probability that a target is located in cell [] [ x ] and p(Ø)=1−p([])p ( Ø )=1-p ( [ x ] ) is the probability that the cell [] [ x ] is empty of targets. Assuming that the observations and trajectories of UAVs i and ℓ are independent, p(i,j(k),ℓ,j(k)|[])=p(i,j(k)|[])p(ℓ,j(k)|[]).p (Z_i,j(k),Z_ ,j(k)| [ x ] )=p (Z_i,j(k)| [ x ] )p (Z_ ,j(k)| [ x ] ). (57) p([]|i,j(k))p ( [ x ]|Z_i,j(k) ) and p([]|ℓ,j(k))p ( [ x ]|Z_ ,j(k) ) are known. One has p([]|i,j(k),ℓ,j(k)) p ( [ x ]|Z_i,j(k),Z_ ,j(k) ) =p(i,j(k),ℓ,j(k)∣[])p(i,j(k),ℓ,j(k))p([]) = p (Z_i,j(k),Z_ ,j(k) [ x ] )p (Z_i,j(k),Z_ ,j(k) )p ( [ x ] ) (58) =p(i,j(k)|[])p(ℓ,j(k)|[])p(i,j(k),ℓ,j(k))p([]) = p (Z_i,j(k)| [ x ] )p (Z_ ,j(k)| [ x ] )p (Z_i,j(k),Z_ ,j(k) )p ( [ x ] ) and a a =p([]|i,j(k),ℓ,j(k))1−p([]|i,j(k),ℓ,j(k)) = p ( [ x ]|Z_i,j(k),Z_ ,j(k) )1-p ( [ x ]|Z_i,j(k),Z_ ,j(k) ) (59) =p(i,j(k)|[])p(ℓ,j(k)|[])p(i,j(k)|Ø)p(ℓ,j(k)|Ø)p([])p(Ø) = p (Z_i,j(k)| [ x ] )p (Z_ ,j(k)| [ x ] )p (Z_i,j(k)| Ø )p (Z_ ,j(k)| Ø ) p ( [ x ] )p ( Ø ) (60) =p([]∣i,j(k))p([]∣ℓ,j(k))p(Ø∣i,j(k))p(Ø∣ℓ,j(k))p(Ø)p([]) = p ( [ x ] _i,j(k) )p ( [ x ] _ ,j(k) )p ( Ø _i,j(k) )p ( Ø _ ,j(k) ) p ( Ø )p ( [ x ] ) (61) =p([]|i,j(k))p([]|ℓ,j(k))(1−p([]|i,j(k)))(1−p([]|ℓ,j(k)))1−p0p0 = p ([ x]|Z_i,j(k) )p ([ x]|Z_ ,j(k) ) (1-p ([ x]|Z_i,j(k) ) ) (1-p ([ x]|Z_ ,j(k) ) ) 1-p_0p_0 (62) thus p([]=1|i,j(k),ℓ,j(k))=a+1.p ( [ x ]=1|Z_i,j(k),Z_ ,j(k) )= aa+1. (63) References [1] E. 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