Effective Tracking of Unknown Clustered Targets Using A Distributed Team of Mobile Robots

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Effective Tracking of Unknown Clustered Targets Using A Distributed Team of Mobile Robots

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Authors Chen, Jun; Dames, Philip; Park, Shinkyu

Citation Chen, J., Dames, P., & Park, S. (2023). Effective Tracking of Unknown Clustered Targets Using A Distributed Team of Mobile Robots. https://doi.org/10.21203/rs.3.rs-3054854/v1

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DOI 10.21203/rs.3.rs-3054854/v1

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Effective Tracking of Unknown Clustered Targets Using A Distributed Team of Mobile Robots

Jun Chen (  [email protected] )

King Abdullah University of Science and Technology Philip Dames

Temple University Shinkyu Park

King Abdullah University of Science and Technology

Research Article

Keywords: Multiple target tracking, sensor-based control, distributed multi-robot systems, swarm robotics, task and motion planning

Posted Date: June 13th, 2023

DOI: https://doi.org/10.21203/rs.3.rs-3054854/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License.

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Additional Declarations: No competing interests reported.

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Effective Tracking of Unknown Clustered Targets Using A Distributed Team of Mobile Robots

Jun Chen

^1*

, Philip Dames

²

and Shinkyu Park

¹

1*Computer, Electrical, and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia.

2College of Engineering, Temple University, 1947 North 12th Street, Philadelphia, 19122, Pennsylvania, USA.

*Corresponding author(s). E-mail(s): [email protected];

Contributing authors:[email protected];[email protected];

Abstract

Distributed multi-target tracking is a canonical task for multi-robot systems, encompassing appli- cations from environmental monitoring to disaster response to surveillance. In many situations the unknown distribution of the targets in a search area is non-uniform, e.g., herds of animals moving together. This paper develops a novel distributed multi-robot multi-target tracking algorithm to effectively search for and track clustered targets. There are two key features. First, there are two parallel estimators, one to provide the best guess of the current states of targets and a second to provide a coarse, long-term distribution of clusters. Second, robots use the power diagram to divide the search space between agents in a way that effectively trades off between tracking detected targets within high density areas and searching for other potential targets. Extensive simulation experiments demonstrate the efficacy of the proposed method and show that it outperforms other approaches in tracking accuracy of clustered targets while maintain good performance for uniformly distributed targets.

Keywords:Multiple target tracking, sensor-based control, distributed multi-robot systems, swarm robotics, task and motion planning

1 Introduction

Multi-target tracking using distributed multi- robot systems (MRSs) has drawn increasing atten- tion over the past decades as robots become more powerful and low-cost. In a large number of real- world scenarios, targets are likely to distribute in clusters, such as the acquisition of image data from coral reef for high-precision 3D reconstruction of its habitats (Arain et al, 2019), waste detection and collection in a desert area (Majchrowska et al, 2022), flaw inspection and repair on surfaces of

buildings and large machinery (Jung et al,2018), detection and sample collection of vegetation in a nature reserve (Torres-S´anchez et al, 2015).

In such cases, detecting a target indicates that some other targets are likely to appear nearby.

Most existing multi-robot multi-target tracking (MR-MTT) algorithms underperform in trading- off between having robots to search for undetected targets and tracking detected targets when targets are not evenly distributed across the search space given no prior knowledge (Xin and Dames,2022).

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This paper aims to develop effective distributed tracking algorithms for unknown targets that are likely to be distributed in clusters. There are two key components to a multi-robot multi- target tracking (MR-MTT) system: an estimation system to model and track objects as they are detected and a control system to drive the motion of individual robots in the team towards areas that are likely to contain useful information.

1.1 Multi-Target Tracking

Different from single target tracking, the main challenge of MTT is matching detections to target tracks, a process known as data association. This becomes especially challenging in the presence of false negative and false positive detections. There are a number of standard MTT algorithms, each of which solve data association in a different way:

global nearest neighbor (GNN) (Konstantinova et al, 2003), joint probabilistic data association (JPDA) (Hamid Rezatofighi et al, 2015), multiple hypothesis tracking (MHT) (Blackman,2004), and particle filters (Doucet et al, 2002). Each of these trackers propagates the posterior of target states over time and solves the data association problem prior to tracking. Another class of MTT techniques, derived from random finite set (RFS) statistics (Mahler, 2007), simultaneously solves data association and tracking. We use one of these, the probability hypothesis density (PHD) filter (Mahler, 2003), which tracks the spatial density of targets. This makes it well suited to situations where each target is not required to have a unique identity. We recently developed a distributed PHD filter that is provably equivalent to the centralized solution (Dames, 2020).

1.2 Coverage Control

A representative class of MR-MTT methods uti- lize Lloyd’s algorithm (Lloyd, 1982), one of the best-known coverage control algorithms which represents target states by a weighting function over the task space and to recursively drive each robot to the weighted centroid of its Voronoi cell (Cortes et al,2004).Pimenta et al(2009) develop a decentralized controller based on Lloyd’s algorithm in which the weighting function is a linear combination of a constant term to encourage coverage and of radial basis functions centered at each target location to encourage tracking.Dames

(2020) sets the weighting function as the probability hypothesis density (PHD) filter to allow simultaneous multi-robot multi-target state estimation via noisy sensors and moving of the team to search for and track targets in a distributed manner.

Compared to other MR-MTT algorithms (Ahmad et al, 2013; Sung et al, 2020; Banfi et al,2015;Corah and Michael, 2021; Zhou et al, 2022), Dames’ method accommodates unknown and time-varying number of targets due to the probabilistic representation of target state, and naturally guarantees collision-free between robots since task area for each robot is non-overlapping at each time horizon. Moreover, it drives more robots to areas where targets are more likely to appear, while allows fewer robots to search for targets in the rest of the areas based on the weighting function, performing both area coverage and target tracking behavior. However, when no target is within a robot’s Voronoi cell, the robots move erratically, reacting to any false positive detections as well as the dynamically changing shape of their Voronoi cells. As a result, robots often stay within empty sub-regions instead of purpose- fully seeking out un-tracked targets, reducing the overall target tracking accuracy of the team. This problem is further exacerbated when a majority of targets gather within some small subsets of the environment.

1.3 Contributions

This paper presents an estimation and control policy to improve tracking accuracy of clustered targets with comprehensive qualitative and quantitative simulation results. This builds upon our previous conference paper (Chen et al, 2022), in which we introduced the cumulative target state to represent the long-term belief of target locations through historical detecting outcomes, and a novel state estimation strategy is proposed incor- porating both instantaneous, i.e., the multi-target state posterior propagated by a filtering algorithm, and cumulative target states. While the instantaneous states allow robots to track detected targets precisely through noisy measurement, the cumulative states enable them to learn coarse distribution of targets by updating the probability of target appearance at each location using past

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observations. We then implement the power diagram in Lloyd’s algorithm to dynamically reassign task space to each robot through weighting each cell by the estimated number of target each robot can found in its cell. We presented a similar idea in (Chen and Dames,2021) but use a power weighting function based on the expected number of targets in a robot’s cell. Thus, by recursively driving the robots to the weighted centroids of their power cells, the team is able to explore or track targets more effectively. Lastly, Chen et al(2022) demonstrate both qualitatively and quantitatively that with the proposed method, the team finds and tracks targets more effectively than using the previous algorithms fromDames(2020) when targets move at a lower speed within some static clusters in the task space.

In this paper, we make several evolved contributions over our previous work (Chen et al, 2022). First, we introduce a counter in the distributed control algorithm to reduce to chance of losing tracked targets due to false negative measurements. Second, this paper conducts an extensive series of simulation to present a comprehensive evaluation of the proposed algorithm, including 1) comparison results of using our proposed target state estimation algorithm paired with the Voronoi diagram for distributed control in addition to the three algorithms compared in (Chen et al, 2022) for tracking targets moving within static small sub-regions, 2) tracking targets moving within dynamic small sub-regions, 3) tracking targets that are uniformly distributed across the entire task space, and 4) tracking targets that are moving faster than the robots. On this basis, several conclusions are further drawn that our proposed algorithm yields the best target tracking accuracy of moving targets clustered within both static and dynamic sub-regions comparing with other methods while not deteriorating the performance of tracking uniformly distributed targets.

2 Problem Formulation

At each time step t, a varying numbermt of targets with poses X = {x1, . . . , xmt} are located within a convex open task space denoted byE ⊂ R². A team of N robots R = {r1, . . . , rN} are tasked with determiningmtandX, both of which are unknown and may vary over time. We assume

that each robotriknows its locationqiin a global reference frame (e.g. from GPS), though our proposed method can be immediately extended to handle localization uncertainty using the algorithms from our previous work (Chen and Dames, 2020). At each time step, a robotrireceives a set of noisy measurements Zi = {zi, z2, . . . , z|Zi|} of target poses in a map frame within the field of view (FoV)Fiof its onboard sensor. Note that the sensor may experience false negative or false positive detections so the number of detections may not match the true number of targets.

2.1 Lloyd’s Algorithm

Lloyd’s algorithm, which is defined by the following functional, determines the movement of the robots:

H(Q,W) =

N

X

i=1

Z

Wi

f ∥x−qi∥

ϕ(x)dx, (1)

whereWiis dominance region of robotri(i.e., the region that robotri is responsible for),∥ · ∥is the Euclidean distance,x∈ E,ϕ(x) is the weighting function for allx∈E, andf(·) is a monotonically increasing function. The role of f is to quantify the cost of sensing due to degradation of a robot’s ability to measure events with increasing distance.

The dominance regionsWi form a partition over E, meaning the regions have disjoint interiors and the union of all regions isE (Cortes et al,2004).

The goal is for the team to minimize the functional in Equation (1), both with respect to the partition set W and the robot positions Q. Minimizing H with respect to W induces a partition on the environment Wi = {x | i = arg min_k=1,...,Nf(∥x−qk∥)}. In other words, Wi

is the collection of all points that are the nearest neighbors ofriaccording to the distancef. When f(x) =x², this is the Voronoi partition, and these Vi are the Voronoi cells, which are convex by construction. We call qi the generator point of Wi. MinimizingHwith respect toQleads each sensor to the weighted centroid of its Voronoi cell (Cortes et al,2004), that is

q_i^∗= R

Wixϕ(x)dx R

Wiϕ(x)dx . (2)

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The discrete time version of Lloyd’s algorithm sets the control input for robotritour(q^∗_i), where

ui(g) = min dstep,∥g−qi∥ g−qi

∥g−qi∥, (3) where gis an arbitrary goal location, and dstep >

0 is the maximum distance a robot can move during one time step. Robots follow this control law online, i.e., recursively move to the tempo- rary weighted centroids of their Voronoi cells, re-construct their cells based on their new positions, and compute the new weighted centroids to move to. As a result, the robots asymptoti- cally converge to the weighted centroids of their Voronoi cells, causing the team towards reaching a local minimum of Equation (2). This still holds true whenϕ(x) varies with time.

3 Distributed Multi-target Tracking

3.1 Instantaneous State Estimation

The sets X andZi from above contain a random number of random elements, and thus are realiza- tion of random finite sets (RFSs) (Mahler, 2007).

The first order moment of an RFS is known as theProbability Hypothesis Density (PHD) (which we denote v(x)) and takes the form of a density function over the state space of a single target or measurement. By assuming that the RFSs are Poisson, meaing the number of targets follows a Poisson distribution and the spatial distribution of targets is i.i.d., the PHD filter recursively updates this target density function in order to track the distribution over target sets (Mahler,2003).

The PHD filter uses three models to describe the motion of targets: 1) The motion model,f(x| ξ), describes the likelihood of an individual target transitioning from an initial state ξ to a new statex. 2) The survival probability model,ps(x), describes the likelihood that a target with state xwill continue to exist from one time step to the next. 3) The birth PHD, b(x), encodes both the number and locations of the new targets that may appear in the environment.

The PHD filter also uses three models to describe the ability of robots to detect targets: 1) The detection model,pd(x|q), gives the probability of a robot with state q successfully detecting

a target with state x. Note that the probability of detection is identically zero for all x outside the sensor FoV. 2) The measurement model,g(z| x, q), gives the likelihood of a robot with state q receiving a measurement z from a target with state x. 3) The false positive (or clutter) PHD, c(z|q), describes both the number and locations of the clutter measurements.

Using these target and sensor models, the PHD filter prediction and update equations are:

¯

vt(x) =b(x) + Z

E

f(x|ξ)ps(ξ)vt−1(ξ)dξ (4a) vt(x) = (1−pd(x|q))¯vt(x) + X

z∈Zt

ψz,q(x)¯vt(x) ηz(¯v^t)

(4b) ηz(v) =c(z|q) +

Z

E

ψz,q(x)v(x)dx (4c) ψz,q(x) =g(z|x, q)pd(x|q), (4d) where ψz,q(x) is the probability of a sensor at q receiving measurementzfrom a target with state x.

The PHD represents the best guess of target states at current time step and is utilized as the instant state estimation of target density, denoted by I(x), i.e., I(x) = v(x), x ∈ E. In Dames (2020) a distributed PHD filter is formulated, with each robot maintaining the PHD within a unique subset,Vi, of the environment. Three algorithms then account for motion of the robots (to update the subsets Vi), motion of the targets (in Equation (4a)), and measurement updates (in Equation (4b)). As a result, each robot recursively re-constructs its Voronoi cell online based on current relative locations of neighboring robots and maintains PHD locally by communicating with neighbors to estimate target states, yielding identical results to running a centralized PHD filter over the task space. In this paper, we apply the same strategy, using the dominance regionsWito propagateI(x) in a distributed manner.

3.2 Cumulative State Estimation

The PHD filter prediction step computes the best guess of target density based on prior knowledge of target birth model, survival model, and motion model over the state space. However, in many

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cases the prior knowledge of these models is com- pletely unknown and therefore the predicted state posterior provides no effective information of the target distribution. As a result, individuals do not actively search for a target when no target is detected and could often spend a long time locat- ing targets that appeared in underexplored regions of the environment, degrading the tracking performance especially when targets are clustered. To improve this, we need to estimate online the probability of target appearance over a long period of time, i.e., the cumulative stateC(x).

In a variety of scenarios, targets move randomly in some relatively fixed clusters, e.g., animals cluster around water sources. In such cases, the frequency of target appearance at each point x∈E can be regarded as time-invariant and the cumulative state estimation is a density distribution that quantifies the best guess of the number of expected targets at each location based on accumulated observation.

3.2.1 Conjugate Prior

In Bayesian inference, if the posterior belongs to the same family as the prior given a likelihood function, the prior and posterior are said to be conjugate distributions, and the prior is called a conjugate prior. A conjugate prior provides an algebraic convenience for propagating the closed-form posterior of target densities, since it avoids the intractability of computing the numeri- cal integration in Bayes’ rule by using the previous posterior as the new prior and updating it with new observations. Therefore, the posterior of target density at each location of the task space can be coarsely learned as observations accumulate, under the assumption that the target densities are relatively static over time.

The PHD filter assumes that the number of targets follows a Poisson distribution. The conjugate prior of this is a Gamma distribution, which describes the distribution of the expected number of targetsy. This is given by

Γ(y|α, β) = y^α−1e^−βyβ^α

(α−1)! , y >0, α, β >0, (5) where αis the shape parameter andβ is the rate parameter. In our case, we wish to estimate the number of targets at a specific location in space,

i.e., y(x). Thus, both α and β depend on the locationxsince we aim to estimate the expected number of targets at each location x, with α(x) describing the total rewards obtained from the observations at locationxandβ(x) describing the inverse of the number of historical observations sensors collected at locationx.

3.2.2 Parameter Update

To propagate the posterior of the cumulative state over time, robots must update the parameters for the gamma distributions at each location x over the task space. We assume that at most one target can occupy each locationx. Initially,α0(x) = β0(x) = _∞¹(= 0). When a robot ri receives a measurement setZi at timet >0, we define X˜ =

(x

∥x−zi∥< ϵ, x∈E, zi∈Zi , Zi ̸=∅

∅, Zi =∅.

(6) whereϵis a small constant to account for the phys- ical size and measurement uncertainty of targets, and ˜X is the set of all points in the task space that are within a distance of ϵ of a measured target.

Then the robot updatesα(x), β(x) α(x) =α(x) + 1,∀x∈X˜

β(x) =β(x) + 1,∀x∈X˜ ∪Fi, (7) whereFi={x|pd(x|qi)>0}is the field of view of robot ri. In Equation (7), the shape parameter α is augmented for each location close to or at an observed target, indicating the increase of expected target density at that location. Since at most one target can be found at each location,α is augmented by 1 for each measurement. Mean- while, for each location within the sensor FoV and/or around an observed target, the rate parameter is increased by 1 to account for the increase of total number of measurements taken at that location.

Then, for each x ∈ E, the cumulative state estimation over the task spaceC(x) is given by the mode of the distribution Γ(α(x), β(x)), obtained via finding the zero of the first order derivative of Equation (5). Therefore,

C(x) =

(_α(x)−1

β(x) , α(x)≥1

0, α(x)<1. (8)

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Remark 1. C(x) ∈ [0,1) since α(x) ≤ β(x) according to the parameter update principle in Equation (7). The value ofC(x)indicates the current belief of the future presence of a target at x, with a value near 1 indicating a high likelihood and a value near 0 indicating a low likelihood.

3.3 Distributed Control

3.3.1 Optimized Space Assignment In the case of uniform distribution of targets, the probability of the robots finding a target to track is similar, and each robot has the same amount of tasks for target tracking. However, the number of targets that each robot in the team actively tracks is significantly different when targets are clustered. Therefore, the search area assigned to each robot must vary with the expected amount of target tracking tasks in this area, with robots that expect to track a large number of targets being given a smaller search space. Our previous work (Chen and Dames, 2021) applies the power diagram, a variant of Voronoi diagram in Lloyds algorithm and sets a normalized unused sensing capacity, a quantification of a sensor’s current capability to track more targets, as the power radius of its assigned power cell for optimal space assignment of a heterogeneous sensor network.

This paper adopts a similar idea, using a quantification of the expected number of targets to be found in a cell as the weights for power diagram construction.

In order to optimize the assigned coverage area, each robotri is endowed a weightρ²_i in con- structing its Voronoi cell, and the monotonically increasing function in Equation (1) becomes

f(∥x−qi∥) =∥x−qi∥²−ρ²_i, (9) where ρi is the power weight of robot ri. The resultingWis an additively weighted Voronoi diagram, or power diagram, a variant of the standard Voronoi diagram, andWiis the power cell. Recall that R

WiC(x)dx quantifies the expected number of targets to be found inWiand that the optimal partition Wi = {x | i = arg min_k=1,...,nf(∥x− qk∥)}. Therefore, for each robotri, we setρi as

ρi= Z

Wi

C(x)dx −¹₂

, (10)

so that the area of Wi is approximately inversely proportional to the estimated number of targets that can be found inWi. Thereby, robots in areas with lower expected target density are assigned larger area to explore, which takes advantage of the search and tracking capability of the team.

3.3.2 Algorithm Outline

In this paper, we use uniform grids to representI andCover the task space. Therefore, the integrals in the original definitions become sums over the values within each grid cell, and the value ofI(x) andC(x) represent the expected number of targets in grid cellx.

The distributed target search and tracking algorithm is outlined as Algorithm 1. For each robot, both instantaneous and cumulative states are initialized to null as there is no prior knowledge about target states, and α(x) and β(x) are initialized (lines 2-4). As robots start to explore and receive sensor measurements (line 7), each of them exchange its location with neighbors to compute its power cell Wi (lines 8-9).

Then each robot updates and broadcast the tuple {α(x), β(x),I(x),C(x)} (lines 10-12).

Since sensor measurements are noisy, false negative measurements (i.e., missed detection) are especially harmful as they may directly lead to loss of tracking of detected targets. To avoid this, we consider a robot to be idle only after it consis- tently finds no targets over the past k time steps (lines 13-16). Therefore, a robot with false negative ratepfnin a single measurement is guaranteed not to lose track of a target with a probability of 1−p^k_fn if the target is inside its field of view for the lastk time steps.

If a robot has not seen a target for a sufficiently long time, then it enters the idle state and uses the cumulative target density to search for targets in areas where they are more likely to be found based on accumulated experience (line 17-18). On the other hand, when a robot detects a target then it enters the busy state and uses the instantaneous target density to better track the detected targets (lines 19-20). This switching behavior allows the team to trade off between taking advantage of historical measurements and further improving the tracking accuracy of targets. Finally, robots use Lloyd’s algorithm to calculate the goal location and move towards it (lines 21-23).

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Algorithm 1: Distributed Search and Tracking

1 forri ∈Rdo

2 I(x)←0,C(x)←0∀x∈E

3 α(x)← _∞¹, β(x)← _∞¹ ∀x∈E

4 emptyCounti←0

5 whiletruedo

6 forri ∈Rdo

7 Receive measurement setZi 8 Exchange stateqi,ρi with

neighbors Ni

9 Compute power cellWi

10 Updateα(x), β(x)∀x∈ Wi using (7)

11 UpdateI(x),C(x)∀x∈ Wi using (4) and (8), respectively

12 Broadcast

{α(x), β(x),I(x),C(x)} ∀x∈ Wi 13 if Zi=∅then

14 emptyCounti← emptyCounti+ 1

15 else

16 emptyCounti←0

17 if emptyCounti> kthen

18 ϕi(x)← C(x)∀x∈ W ▷Idle

19 else

20 ϕi(x)← I(x)∀x∈ W ▷Busy

21 Set goalgi=q^∗_i using (2)

22 Computeui(gi) using (3)

23 qi ←qi+ui ▷ Move towards goal

Algorithm1boosts the performance of the previous method of Dames (2020) in that it allows the team to actively explore the environment and learn the characteristics of the target distribution. In particular, the PHD updates the myopic estimation of targets’ current locations, while α, β decode the farsighted likelihood of target distribution estimated from past measurements. Therefore, the robots are now able to use a combination of detailed instantaneous information (coming from the PHD) and coarse cumulative information (coming from α, β) to inform their actions.

4 Simulations

We will use a series ofMatlabsimulations to test the following hypotheses.

Hypothesis 1. We believe that Algorithm1 will yield a similar tracking performance compared with the method of Dames (2020) for evenly distributed targets becauseC will be close to uniform across the task space and therefore W will be similar to a standard Voronoi diagram.

Hypothesis 2. We believe that the advantages of our method will be more pronounced when targets are not uniformly distributed in the space but are instead grouped together within small regions. Under these circumstances, idle robots will be especially helpful in learning the difference in target density among sub-regions and optimiz- ing the assignment of tracking effort in different sub-regions.

We will conduct all tests within the same basic environment, detailed below. The task space is a 100 m×100 m square. All robots begin each trial at randomized locations within a 20 m×10 m box at the bottom center of the environment. Robots have a maximum speed of 5 m/s and are equipped with isotropic sensors with a sensing radius 5 m.

Both instantaneous and cumulative estimation are approximated by uniform grid implementation, with a grid size of 0.1 m×0.1 m. In the PHD filter, the robots use a Gaussian random walk with σ= 0.35 m/s for the motion modelsf, set the survival probability to 1, and the birth rate to 0. We randomly selected these target models because no prior knowledge of targets is learned in all tests, which may degrade the performance of the PHD prediction step if they are reasonably different from true values. For concise design of the simulations, we use the same sensor models for all the robots as inDames(2020). However, our proposed method is readily is compatible with heterogeneous sensor models (Chen and Dames,2021). In all tests, we setϵto be 5 m.

4.1 Qualitative Comparison

We first show how our proposed algorithms improve multi-target tracking performance using a single trial deploying 70 robots. Targets are distributed in clusters, where 30 are located in a 33×33 m square sub-region at the lower-left corner of E, and another 30 targets in a 33×33 m

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(a) Dames’ Method (b) Our Method

Fig. 1 Figures show 70 robots and 60 targets distribution in a 100 m×100 m squared task space after 300 s of tracking using the method from Dames(2020) and our method, respectively. Orange diamonds plot locations of targets. Green squares and circles show locations and field of views of robots, respectively. Dashed lines plot boundaries of robots’ current assigned space.

(a)I(x) Clustered (b)C(x) Clustered

(c)I(x) Uniform (d)C(x) Uniform

Fig. 2 Figures plot surface ofI(x) andC(x) distributed over the 100 m×100 m task spaces after tracking clustered targets (Figures2aand2b), and uniformly distributed targets (Figures2cand2d) for 300 s, respectively.

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squared sub-region at the top-right corner. Targets move within the sub-regions following a Gaus- sian random walk at a maximum speed of 3 m/s.

To introduce the possible target born and disap- pearance while maintaining a consistent density of target clusters, without loss of generality, we allow targets moving out of their clustered sub-regions to disappear automatically, and the true target birth rate over each sub-region is adjusted in a way that the number of targets in each cluster is fluctuating around 30. One may consider the corresponding scenarios such as an open area with the possible presence of animals enclosed by shadows (e.g., trees) observed from drones and pollutants emitted from a source site that are continuously diluted around the source until un-detectable.

Figure 1 shows the locations of robots and targets after 300 s using both the method from Dames (2020) and our method. When targets gather within only a small portion of the environment, only a few robots have found targets and are tracking them while the majority of the robots are idle and move erratically, following false positive detections. This demonstrates the inefficiency of the previous algorithm in using the total sensing capability of the team to search for untracked targets. Our method improves this by driving a larger number of robots to gather at the clusters of targets while a handful of other robots continue to search unexplored areas, as the exact locations and motion models of targets are unknown. The result indicates the efficacy of our method in that the distribution of the clusters is learned over time and the power diagram distributes more balanced workload to individuals.

Figure 2 shows the values of I(x) and C(x) across the environment after 300 s, illustrating the different characteristics of the instantaneous and cumulative target estimates. The instantaneous estimate, i.e., the PHD, provides the best guess of the exact locations of targets at the current time through a set of sharp peaks, shown as Figure2a.

The PHDI(x) returns to near zero rapidly when targets are no longer found at x, assuming a low expected target birth rate, exhibiting high accuracy of estimating target exact locations but short

“memory” of historical target distribution. On the contrary, the cumulative estimate C(x) presents a relatively smooth and continuous distribution over the task space, with higher values distributed

over the entire target clusters and near-zero values over the rest of the space, shown as Figure2b.

Therefore, the instantaneous and the cumulative estimation are utilized to drive busy and idle robots respectively, as the former robots require targets’ exact locations for accurate tracking while the latter robots are in need of coarse distribution of clusters for optimized deployment.

4.2 Quantitative Comparison

To test the efficacy of our proposed approach, we conduct a series of trials using a range of team sizes (from 60 to 90 robots) and quantify the tracking accuracy of targets when different strategies are applied. We will test four configurations:

1) static clusters: targets move around within fixed sub-regions of the environment, 2) dynamic clusters: targets move around within moving sub- regions of the environment, 3) uniform distribution: targets start uniformly distributed across the environment and move randomly, and 4) fast targets: targets move at speeds faster than the robots within fixed sub-regions of the environment. These first two tests will address Hypothesis2while the second two tests will address Hypothesis1. Note, the targets move slower than the robots in tests 1-3.

4.2.1 OSPA

We use the first order Optimal SubPattern Assign- ment (OSPA) metric (Schuhmacher et al,2008), a commonly used MTT performance measure. The OSPA error between two setsX, Y, where|X|= m≤ |Y|=nwithout loss of generality, is

d(X, Y) = 1

n min

π∈Πn

c^p(n−m) +

m

X

i=1

dc(xi, yπ(i))^p

!!1/p

,

(11) wherecis a cutoff distance,dc(x, y) = min(c,∥x−

y∥), and Πn is the set of all permutations of the set {1,2, . . . , n}. This gives the average error in matched targets, where OSPA considers all possible assignments between elements x ∈ X and y ∈ Y that are within distance c of each other.

This can be efficiently computed in polynomial time using the Hungarian algorithm (Kuhn,1955).

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We use c = 10 m, p = 1, and measure the error between the true and estimated target sets. Note that a lower OSPA value indicates a more accurate tracking of the target set.

4.2.2 Static Clusters

We compare four strategies to show the effects of different components within our system: 1) the method of Dames (2020) (“D” method) which uses only instantaneous estimation for tracking and Voronoi diagram for control, 2) our tracking strategy which estimates both instantaneous and cumulative target state, and Voronoi diagram for control (“V” method), 3) tracking through instantaneous estimation only with our proposed control algorithm using power diagram (“P” method), with the power weight in Equation (12) depending on PHD instead, given by

ρi= Z

Wi

I(x)dx −¹₂

, (12)

and 4) our tracking and control method (“O”

method), which uses both instantaneous and cumulative estimation for tracking, and power diagram for space assignment and control, outlined in Algorithm1. For each team size, we run 10 trials with 1000 s duration for each trial. The test set- tings are identical to those used in the qualitative comparison in Section4.1.

Figure 3a shows the average steady-state OSPA value across the 10 trials. We do this by measuring the OSPA over the final 700 s of each trial, as this allows the team to reach or be close to a steady state and the average smooths out the effects of spurious measurements that cause the OSPA to fluctuate. “D” method shows the worst performance in tracking highly clustered targets regardless of team sizes, due to the extreme unbalanced workload assigned to each robot which fails to drive sufficient number of robots to the clusters. The increment of cumulative state estimation slightly improve the tracking accuracy by allowing robots to estimate the location of the clusters, shown by the results of “V”. Yet, the inef- ficient space assignment using Voronoi diagram without accounting for unbalanced workload lim- its this improvement. “D” method can be further improved by assigning optimized task spaces to robots and driving them to closely space around

detected targets, as depicted by the results of “P”

method. However, in this case robots still do not move to the locations where targets are likely to be clustered if no target is tracked instantly. Our proposed method further improves this flaw and shows the best tracking performance on clustered targets, as suggested by the OSPA values of “O”

method. In summary, these trials show the benefit both of distributing search across based on current workload and of the cumulative state estimate to provide guidance when no targets are visible, validating Hypothesis 2.

Figure 3b shows the average OSPA values of all 40 trials of four different team sizes using the four methods for 1000 s. It is shown that the OSPA error drops at similar rates for the first 200 s as robots start moving from the starting area and explore the entire search space, despite the applied MR-MTT strategy. After that, “D”

method no longer improves the tracking accuracy and the team reaches a steady state, while the OSPA error of the other two algorithms continues to decrease. “V” and “P” methods reach a steady state at around 400 s, after which the team has completed updating a steady-state PHD and space assignment. Meanwhile, the OSPA error of “O”

method decreases until approximately 600 s. This is because the cumulative estimation of the target number continuously grows as more observations are gained and more robots congregate in target clusters. The simulation results illustrate that our proposed method allows the robots to explore more areas and continuously learn the cluster distribution which improves the tracking accuracy compared to the two other algorithms.

4.2.3 Dynamic Clusters

In some real world scenarios, moving targets gather at some slowly changing rather than fixed areas, such as migrations of animal clusters and spreading forest fire ignition sites. The next batches of simulation trials evaluate the performance of our proposed algorithm in searching and tracking such dynamic target clusters, where the maximum speed of targets is still slower than that of robots. Two clusters of targets are initially located at the lower-left and upper-right corner of the 100 m×100 m squared task space, respectively, each occupies a 33 m×33 m area, depicted as Figure 4a. The targets move at the

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(a) Steady State OSPA (b) OSPA VS Time

Fig. 3 Figures show OSPA errors of group of trials. Figure 3adisplays boxplots of average OSPA errors from 300 s to 1000 s using the four methods (“D”, “V”, “P”, and “O”) with four team sizes, each over ten trials.|R|denotes the number of robots. Figure3bplots average OSPA errors of the total of 40 trials (10 for each team size) using the four methods over the entire 1000 s.

(a) Motion of Clusters (b) 0.15 m/s

(c) 0.5 m/s (d) 1.0 m/s

Fig. 4 Figure4ademonstrates the initial and final locations of two clusters, distinguished by blue and green, respectively.

Each cluster, initially spaced in the solid square, moves slowly towards the dashed square, with targets moving inside of it. Figure4aplots average OSPA errors of the final 300 s using Dames’ method and our method with four team sizes, each over ten trials.

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(a) 30 Targets (b) 60 Targets

(c) 90 Targets (d) 120 Targets

Fig. 5 Figures plot the steady state average OSPA error of 60, 70, 80, and 90 robots tracking approximately 30, 60, 90, and 120 uniformly distributed targets in the task space using both Dames’ algorithm (“D” method) and our algorithm (“O”

method), respectively, each over ten trials.

maximum speed of 3 m/s within their respective clusters while each cluster moves slowly towards the diagonals of the task space at three speeds:

0.15 m/s, 0.5 m/s, ane 1.0 m/s, respectively. For each team size we run 10 trials with 500 s duration for each trial. The rest of the simulations compare the OSPA error of the “D” and “O” methods only in order to focus on the enhancement effect of our algorithm compared to the baseline. As shown in Figures 4b, 4c, 4d, our method “O” achieves higher accuracy in tracking slowly moving clusters of targets than “D”. The tracking accuracy increases as does the number of robots since the coarse prior of the target distribution, i.e.C(x), is more frequently updated which allows the robots to track the clusters swiftly.

Nevertheless, the performance of our method deteriorates when target clusters are moving since there is a delay in updating the time-varying cumulative state from historical observation. This effect grows the faster the cluster moves, as seen

by the shrinking gap between the “D” and “O”

methods. Therefore, the cumulative state estimation approach may not be effective when target clusters are moving fast across the search space in which case the cumulative state might be close to uniform over the space and providing no useful information of target distribution. However, for slowly moving clusters, our method attains significant improvement in cluster tracking, validating Hypothesis2.

4.2.4 Uniformly Distributed Targets We verify whether the performance of our proposed method degrades when tracking uniformly distributed targets. Consider that targets within the task space may move outside of it, and new targets may appear at any location within the task space with a certain birth rate. We tune the target birth rate for each batch of trials in a way that the total number of targets inside the task space is fluctuating at approximately 30, 60, 90, and 120

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(a) 5 m/s (b) 6 m/s

(c) 8 m/s (d) 10 m/s

Fig. 6 Figures plot the steady state average OSPA error of 60, 70, 80, and 90 robots tracking targets that are moving at the speed of 5 m/s, 6 m/s, 8 m/s, and 10 m/s in the task space using both Dames’ algorithm (“D” method) and our algorithm (“O” method), respectively, each over ten trials.

for each batch. We record the OSPA error for 700 s for each trial and plot the steady state OSPA error for the last 400 s.

As shown in Fig 5, for both algorithms, the OSPA error increases as does the number of targets since, with the increased number of targets, it becomes harder for the team to find and track the targets. While a few sets of result show that the OSPA error is slightly higher using our method

“O” compared to “D”, however, no evidence of statistical significance is observed. Overall, our method does not show any significant performance loss in target tracking even if the targets are uniformly distributed across the space, validating Hypothesis1.

4.2.5 Tracking Faster Targets

We further investigate the performance of our method when targets are moving faster than the robots. We run batches of trials with target maximum speed ranging from 6 m/s to 10 m/s. For

comparison, we also test the case where targets move at identical speed as robots (maximum speed of 5 m/s). For all trials, we use the same target distribution and motion models as in the static cluster scenarios. Comparing the graphs in Figures6a,6d, and3a, we observe that the OSPA error decreases as the targets move faster, regardless of which algorithm is applied. To explain the reason behind such observation, note that as targets move faster in a cluster of high density, the chance of the targets moving into the field of view of a robot located in the same cluster increases.

Therefore, an idle robot is likely to find a target more quickly and the total idle time of the team decreases, resulting in lower OSPA errors.

On the other hand, comparing with Dames’

method, the simulation results suggest that our algorithm underperforms when the targets are moving faster than the robots. When the target speed is twice as high as the robot speed, such degradation becomes substantial, especially when

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the number of robots is small. This is because as the targets move faster than the robots, a robot can no longer follow a detected target over long period of time, and thus the tracking accuracy relies largely on the chance of targets moving into a robot’s field of view. The fast random walk of a dense crowd of targets tends to drive robots to quickly spread out across the target cluster when all robots apply Dames’ method and attempt to follow the moving targets. However, our algorithm tends to “trap” the idle members of the team at the locations where targets were previously observed, slowing down the process of spreading out the team to some degree, and therefore reducing the chance of coming across more moving targets slightly. Apparently, the faster the targets are moving, the more significantly such phenomenon can be observed until saturated, i.e., targets move fast enough so that there are hardly idle robots.

When targets move at the same or lower speed than the robots, our method outperforms.

5 Conclusions

Among existing MR-MTT methods, Lloyd-based algorithms are a useful method in many appli- cations ranging from area coverage to target tracking. However, the tracking accuracy of Lloyd- based algorithms significantly decays when targets are clustered instead of evenly distributed across the task space. In this paper, we propose a novel distributed multi-target tracking algorithm that allows a team of robots to effectively track clustered targets, despite given no prior knowledge of target poses and motion models. Each robot estimates both the instantaneous and cumulative target density and the team uses this information to dynamically optimize space assignment using a power diagram implementation of Lloyd’s algorithm. As a result, robots can track detected targets precisely while congregating at target clusters by learning the coarse cluster distribution from past observations. Simulation results suggest that our algorithm is superior compared to other approaches in effectively tracking the targets that are moving at a slower speed than robots in relatively static clusters, while attaining the same performance for tracking uniformly distributed targets.

Acknowledgments. This work was supported by funding from King Abdullah University of Sci- ence and Technology (KAUST). Philip Dames was supported by NSF grant CNS-2143312.

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