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Resilient Active Information Acquisition with

Teams of Robots

Brent Schlotfeldt,1Student Member, IEEE, Vasileios Tzoumas,2Member, IEEE, George J. Pappas,1Fellow, IEEE

Abstract—Emerging applications of collaborative autonomy,

such as Multi-Target Tracking,Unknown Map Exploration, and

Persistent Surveillance, require robots plan paths to navigate

an environment while maximizing the information collected via

on-board sensors. In this paper, we consider such information

acquisition tasks but in adversarial environments, where attacks

may temporarily disable the robots’ sensors. We propose the

ﬁrst receding horizon algorithm, aiming for robust and adaptive

multi-robot planning against any number of attacks, which

we call Resilient Active Information acquisitioN (RAIN). RAIN

calls, in an online fashion, a Robust Trajectory Planning (RTP)

subroutine which plans attack-robust control inputs over a look-

ahead planning horizon. We quantify RTP’s performance by

bounding its suboptimality. We base our theoretical analysis on

notions of curvature introduced in combinatorial optimization.

We evaluate RAIN in three information acquisition scenarios:

Multi-Target Tracking,Occupancy Grid Mapping, and Persistent

Surveillance. The scenarios are simulated in C++ and a Unity-

based simulator. In all simulations, RAIN runs in real-time, and

exhibits superior performance against a state-of-the-art baseline

information acquisition algorithm, even in the presence of a high

number of attacks. We also demonstrate RAIN’s robustness and

effectiveness against varying models of attacks (worst-case and

random), as well as, varying replanning rates.

Index Terms—Autonomous Robots; Multi-Agent Systems; Re-

active Sensor-Based Mobile Planning; Robotics in Hazardous Fi-

elds; Algorithm Design & Analysis; Combinatorial Mathematics.

I. INTRODUCTION

Active information acquisition has a long history in robotics

[1], [2]. The “active” characterization captures the idea that

robots which move purposefully in the environment, acting

as mobile sensors instead of static, can achieve superior

sensing performance. Indeed, active information acquisition

has recently been extended to a variety of collaborative

(multi-robot) autonomy tasks, such as Multi-Target Tracking

[3], [4], Exploration and Mapping (including Simultaneous

Localization and Mapping) [5]–[7], Monitoring of Hazardous

Environments [3], [8], and Persistent Surveillance [9]–[11].

Robotics research has enabled the solution of several classes

of information acquisition problems, both for single-robot and

multi-robots scenarios, with methods that include search-based

[3], [12], sampling-based [13], [14], and gradient-based plan-

ning [15]. The methods can also differ on problem parameters,

1B. Schlotfeldt and G. J. Pappas are with the Department of Electrical and

Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104,

USA. {brentsc, pappasg}@seas.upenn.edu

2V. Tzoumas is with the Department of Aerospace Engineering, University

of Michigan, Ann Arbor, MI 48109, USA. vtzoumas@umich.edu

This work was partially supported by ARL CRA DCIST W911NF-17-2-

0181 and the Rockefeller Foundation.

Fig. 1: Persistent Surveillance under Attacks. Unity simulation

environment depicting a 5-robot team engaging in a Persistent

Surveillance task for monitoring a set of buildings. Some robots are

under attack. The attacks can disable the sensing capabilities of the

robots, at least temporarily. Each blue disc indicates the ﬁeld of view

of a (non-attacked) robot, while each red disc indicates an attacked

robot. In this adversarial environment, the robots must resiliently

plan trajectories to (re-)visit all the building landmarks to continue

acquiring information despite the attacks.

such as the length of the planning horizon —cf. myopic (or

one-step-ahead) planning [16] versus non-myopic (or long-

horizon) planning [3]— and the type of target process (e.g.,

Gaussian [17], Gaussian mixture, [5], [18], occupancy grid

map [7], [19], and non-parametric [20]).

But most robotics research places no emphasis on robus-

tifying information acquisition against attacks (or failures)

that may temporarily disable the robots’ sensors; e.g., smoke-

cloud attacks that can block temporarily the ﬁeld of view

of multiple sensors. For example, [21] focuses on formation

control, instead of information acquisition; [22] focuses on

state estimation against byzantine attacks, i.e., attacks that

corrupt the robots’ sensors with adversarial noise, instead of

disabling them; and [23] focuses on a trajectory scheduling in

transportation networks when travel times can be uncertain,

instead on trajectory planning for information acquisition. An

exception is [16], which however is limited to multi-target

tracking based on myopic planning, instead of non-myopic.

Related work has also been developed in combinatorial op-

timization [24], [25], paving the way for robust combinatorial

optimization against attacks: [26] proposes algorithms for sub-

modular set function optimization against worst-case attacks,

but under the assumption the attacks can remove (disable) only

a limited number of elements from the optimized set. Instead,

[27] proposes algorithms for optimization against any number

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of removals. And, recently, [28] extended the algorithms to

matroid constraints, enabling the application of the algorithms

in robotics since multi-robot path planning can be cast as a

matroid-constrained optimization problem [7].

Contributions. In this paper, in contrast to the aforemen-

tioned works, we extend the attack-free active information

acquisition to account for attacks against the robots while

simultaneously performing non-myopic multi-robot planning.

We make the following three key contributions.

1. Receding Horizon Formulation and Algorithm. Sec-

tion II formalizes the attack-aware active information acquisi-

tion problem as a ﬁnite horizon control optimization problem

named Resilient Active Information acquisitioN (P-RAIN) —

in the acronym, the “P” stands for “Problem.” (P-RAIN) is

a sequential mixed-integer optimization problem: it jointly

optimizes the robots’ control inputs such that the robots’

planned paths are robust against worst-case attacks that may

occur at each time step. The upper-bound to the number of at-

tacks is assumed known and constant. (P-RAIN)’s information

objective function is assumed non-decreasing in the number of

robots (a natural assumption, since the more robots the more

information one typically can collect).

Section III proposes RAIN, the ﬁrst receding horizon al-

gorithm for the problem of resilient active information ac-

quisition (P-RAIN). RAIN calls in an online fashion Robust

Trajectory Planning (RTP), a subroutine that plans attack-

robust control inputs over a planning horizon. RTP’s planning

horizon is typically less than the acquisition problem’s ﬁnite

horizon, for computational reasons. For the same reason, RTP

assumes constant attacks. RTP is presented in Section IV.

2. Performance Analysis. Although no performance guar-

antees exist for the non-linear combinatorial optimization

problem (P-RAIN), Section Vprovides suboptimality bounds

on RTP’s performance, i.e., on the algorithm used by RAIN

to approximate a solution to (P-RAIN)locally, in a receding

horizon fashion. The theoretical analysis is based on notions of

curvature introduced for combinatorial optimization; namely,

the notions of curvature [29] and total curvature [30]. The

notions aim to bound the worst-case complementarity of the

robots’ planned paths in their ability to jointly maximize

(P-RAIN) information acquisition objective function.

3. Experiments. Section VI evaluates RAIN across three

multi-robot information acquisition tasks: Multi-Target Track-

ing,Occupancy Grid Mapping, and Persistent Surveillance.

All evaluations demonstrate the necessity for attack-resilient

planning, via a comparison with a state-of-the-art baseline

information acquisition algorithm, namely, coordinate de-

scent [6]. Speciﬁcally, RAIN runs in real-time and exhibits

superior performance in all experiments. RAIN’s effectiveness

is accentuated the higher the numbers of attacks is (Sec-

tion VI-A). RAIN remains effective even against non-worst-

case attacks, speciﬁcally, random (Section VI-B). Even when

high replanning rates are feasible (Section VI-C), in which

case coordinate descent can adapt at each time step against the

observed attacks, RAIN still exhibits superior performance. The

algorithm is implemented in C++ and a Unity-based simulator.

Comparison with Preliminary Results in [31]. This paper

extends the results in [31], by introducing novel problem

formulations, algorithms, and numerical evaluations, as well

as, by including all proofs (Appendix), which were omit-

ted from [31]. Particularly, the receding horizon formulation

(P-RAIN) is novel, generalizing the (P-RTP) formulation ﬁrst

presented in [31]. The algorithm RAIN is also ﬁrst presented

here. Additionally, the simulation evaluations on Occupancy

Grid Mapping and Persistent Surveillance are new, and have

not been previously published. They also include for the ﬁrst

time a sensitivity analysis of RAIN against varying models of

attacks (worst-case and random), as well as, varying replan-

ning rates.

II. RESILIENT ACTIVE INFORMATION ACQUISITION

(RAIN) PRO BL EM

We present the optimization problem of Resilient Active

Information acquisitioN (RAIN) (Section II-B). To this end, we

ﬁrst formalize the (attack-free) active information acquisition

problem (Section II-A). We also use the notation:

•φV, τ :τ+τ0,{φi,t}i∈ V , t ∈[τ,...,τ +τ0], for any variable

of the form φi,t, where Vdenotes a set of robots (i.e.,

i∈ V is robot i), and [τ , . . . , τ +τ0]denotes a discrete

time interval (τ≥1, while τ0≥0);

•w∼N(µ, Σ) denotes a Gaussian random variable wwith

mean µand covariance Σ.

A. Active Information Acquisition in the Absence of Attacks

Active information acquisition is a control input optimiza-

tion problem over a ﬁnite-length time horizon: it looks to

jointly optimize the control inputs for a team of mobile robots

so that the robots, acting as mobile sensors, maximize the

acquired information about a target process. Evidently, the

optimization must account for the (a) robot dynamics, (b)

target process, (c) sensor model, (d) robots’ communication

network, and (e) information acquisition objective function:

a) Robot Dynamics: We assume noise-less, non-linear

robot dynamics, adopting the framework introduced in [6]:

xi,t =fi(xi,t−1, ui,t−1), i ∈ V, t = 1,2,..., (1)

where Vdenotes the set of available robots, xi,t ∈Rnxi,t

denotes the state of robot iat time t,1and ui,t ∈ Ui,t denotes

the control input to robot i;Ui,t denotes the ﬁnite set of

admissible control inputs to the robot.

b) Target Process: We assume any target process

yt=g(yt−1) + wt, t = 1,2,..., (2)

where ytdenotes the target’s state at time t, and wtdenotes

gaussian process noise; we consider wt∼N(µwt,Σwt).

c) Sensor Model: We assume measurements of the form

zi,t =hi(xi,t, yt) + vi,t (xi,t), t = 1,2,..., (3)

where zi,t denotes the measurement by robot iat time t,

and vi,t denotes measurement noise; we consider vi,t ∼

N(µvi,t ((xi,t),Σvi,t ((xi,t )). Both the noise and sensor func-

tion hidepend on xi,t, as it naturally is the case for, e.g.,

bearing and range measurements (cf. Section VI-A).

1xi,t in eq. (1) belongs to an appropriate state space, such as SE (2) or

SE(3), depending on the problem instance.

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d) Communication Network among Robots: We assume

centralized communication, i.e., all robots can communicate

with each other at any time.

e) Information Acquisition Objective Function: The in-

formation acquisition objective function captures the acquired

information about the target process, as collected by the robots

during the task via their measurements. In this paper, in

particular, we consider objective functions of the additive form

JV,1:TTASK ,

TTASK

X

t=1

J(yt|uV,1:t, zV,1:t),(4)

where TTASK denotes the duration of the information acquisi-

tion task, and J(yt|uV,1:t, zV,1:t)is an information metric

such as the conditional entropy [6] (also, cf. Section VI-A)

or the mutual information [7] (also, cf. Section VI-B), where

we make explicit only the metric’s dependence on uV,1:tand

zV,1:t(and we make implicit the metric’s dependence on the

initial conditions y0and xi,0, and on the noise parameters, i.e.,

the means and covariances of wtand vtfor t= 1,...,TTASK ).

Problem ((Attack-Free) Active Information Acquisition).At

time t= 0, ﬁnd control inputs uV,1:TTASK by solving the

optimization problem

max

uV,t ∈ UV,t

t=1:TTASK

JV,1:TTASK.(5)

Eq. (5) captures a control input optimization problem where

across a task-length horizon, the control inputs of all robots

are jointly optimized to maximize JV,1:TTASK.

Solving eq. (5) can be challenging, mainly due to (i) the

non-linearity of eqs. (1)-(3), (ii) the duration of the task,

TTASK, which acts as a look-ahead planning horizon (the

longer the planning horizon is, the heavier eq. (5) is in

computing an optimal solution), and (iii) that at t= 0 no

measurements have been realized yet.

To overcome the aforementioned challenges, on-line so-

lutions to eq. (5) have been proposed [6], similar to the

Receding Horizon Control solution —also known as Model

Predictive Control (MPC)— for the ﬁnite-horizon optimal

control problem [32, Chapter 12]. Speciﬁcally, per the receding

horizon approach, one aims to solve eq. (5) sequentially in

time, by solving at each t= 1,...,TTASK an easier version

of eq. (5), but of the same form as eq. (5), where (i) the

look-ahead horizon TTASK is replaced by a shorter TPLAN

(TPLAN ≤TTASK), and (ii) eqs. (1)-(3) are replaced by their li-

nearizations given the current xV,t and current estimate of yt.

B. Active Information Acquisition in the Presence of Attacks

Eq. (5) may suffer, however, from an additional challenge:

the presence of attacks against the robots, which, if left

unaccounted, can compromise the effectiveness of any robot

plans per eq. (5). In this paper, in particular, we consider the

presence of the following type of attacks:

f) Attack Model: At each t, an attack Atcan remove at

most αrobots from the information acquisition task (At⊆ V

and |At| ≤ α), in the sense that any removed robot i(i∈ At)

cannot acquire any measurement zi,t. In selecting the attack,

the attacker has perfect knowledge of the state of the system.

The attacker can select the worst-case attack (cf. Problem 1).

Nevertheless, the attacker cannot necessarily prevent the robots

from moving according to their pre-planned path, nor can

cause any communication loss among the robots.

Problem 1 (Problem of Resilient Active Information acquisi-

tioN (P-RAIN)).At time t= 0, ﬁnd control inputs uV,1:TTASK

by solving the optimization problem

max

uV,t ∈ UV,t

t=1:TTASK

min

At⊆ V,|At| ≤ α

t=1:TTASK

JV\At,1:TTASK (P-RAIN)

(P-RAIN) goes beyond eq. (5) by accounting for the attacks

At(t= 1,...,TTASK). This is expressed in (P-RAIN) with the

minimization step, which aims to prepare an optimal solution

uV,t against any worst-case attack that may happen at t.

Remark 1 (Need for (P-RAIN)).Reformulating the (attack-

free) eq. (5)as in (P-RAIN)may seem unnecessary, since we

consider that the attacker cannot cause any communication

loss among the non-attacked robots (cf. the attack model

deﬁned above): indeed, if the non-attacked robots can instan-

taneously observe the attacks at each t, and instantaneously

replan at the same moment t, then (P-RAIN)is unnecessary.

However, replanning instantaneously in practice is impossible,

due to (i) computationally-induced and algorithmic delays [6],

as well as (ii) delays induced by the temporal discretization

of the robot and target dynamics. Thus, for the duration

replanning is impossible, the plans need to account for attacks.

III. RECEDING HORIZON APP ROXIMATION:

RAIN ALGORITHM

In solving (P-RAIN), one has to overcome not only the

challenges involved in eq. (5) (cf. Section II) bult also the

additional challenge of the worst-case attacks At(which are

unknown a priori). We develop an on-line approximation

procedure for (P-RAIN), summarized in Algorithm 1.

Intuitive Description. RAIN proposes a receding horizon

solution to (P-RAIN), that enables on-line reaction to the

history of attacks, and, thus, is resilient, by executing the steps:

a) Initialization (line 1):At t= 0, the acquisition task

has not started and no attacks are assumed possible (A0=∅).

b) Receding Horizon Planning (lines 2-17):At each t=

1,...,TTASK,RAIN executes the receding horizon steps:

•Robust Trajectory Planning (RTP) (lines 3-7):Given

the current estimate ˆytof the target process, all robots

jointly optimize their control inputs by solving Problem 2,

presented next, which is of the same form as (P-RAIN)

but where (i) the look-ahead horizon TTASK is replaced

by a shorter TPLAN (TPLAN ≤TTASK), and (ii) the attack

is considered ﬁxed over the look-ahead horizon:

Problem 2 (Robust Trajectory Planning (RTP)).At time t,

ﬁnd attack-robust control inputs uV, t+1:t+TPLAN by solv-

4

ing the optimization problem

max

uV,t0∈ UV,t0

t0=t+1 : t+TPLAN

min

A⊆V,|A| ≤ αJV\A, t+1:t+TPLAN .

(P-RTP)

Both aforementioned (i) and (ii) intend to make (P-RTP)

computationally tractable, so (P-RTP) can be solved in

real time for the purposes of receding horizon planning.

Particularly, we assume that the the algorithm we propose

for (P-RTP), RTP, is called in RAIN every TREPLAN steps.2

Remark 2 (Role of TREPLAN).TREPLAN is chosen so

that a receding horizon plan can always be generated

in the duration it takes to compute a solution to (P-RTP)

via RTP;e.g., if one timestep —real-time interval from

any tto t+ 1— has duration 0.5s, and solving (P-RTP)

via RTP requires 2s, then TREPLAN = 4 steps. Generally,

TREPLAN ≥1steps. Factors that inﬂuence the required

time to solve (P-RTP)include the size of the robot team,

the length of the planning horizon TPLAN [3], the need

for linearization of eqs. (1)-(3), and the number of

possible attacks α—evidently, the latter factor is unique

to (P-RAIN), in comparison to the attack-free eq. (5).

•Control Execution (lines 8-10):Each robot iuses their

computed ui,t to make the next step in the environment

(in the meantime, the real time changes from tto t+ 1

by the completion of the step).

•Attack Observation (line 11):RAIN observes the current

attack, which affects the robots while they execute ui,t.

•Measurement Collection (lines 12-14):The measure-

ments from all non-attacked robots are collected.

•Estimate Update (line 15):Given all received measure-

ments up to the current time, the estimate of ytis updated.

•Time Update (line 16):RAIN updates the time counter to

match it with the real time.

IV. ROBUST TR AJ EC TO RY PLANNING (RTP) ALGORITHM

We present RTP, which is used as a subroutine in RAIN, in a

receding horizon fashion (cf. Section III). RTP’s pseudo-code

is presented in Algorithm 2.RTP’s performance is quantiﬁed

in Section V. We next give an intuitive description of RTP.

Intuitive Description. RTP’s goal is to maximize (P-RTP)’s

objective function JV\A, t+1:t+TPLAN , despite a worst-case at-

tack Athat removes up to αrobots from V. In this context,

RTP aims to fulﬁll (P-RTP)’s goal with a two-step process,

where (i) RTP partitions robots into two sets (the set of robots

L, and the set of robots V \ L; cf. RTP’s lines 1-4), and, then,

(ii) RTP appropriately selects the robots’ control inputs in each

of the two sets (cf. RTP’s lines 5-8). In particular, RTP picks

Laiming to guess the worst-case removal of αrobots from

V,i.e., to guess the optimal solution to the minimization step

in (P-RTP). Thus, intuitively, Lis aimed to act as a “bait” to

the attacker. Since guessing the optimal “bait” is, in general,

intractable [33], RTP aims to approximate it by letting Lbe

the set of αrobots with the αlargest marginal contributions

to J·, t+1:t+TPLAN (RTP’s lines 5-7). Then, RTP assumes the

2RTP’s pseudo-code is presented in Algorithm 2, and described in more

detail Section IV. We quantify RTP’s performance guarantees in Section V.

Algorithm 1 Resilient Active Information acquisitioN (RAIN).

Input: RAIN receives the inputs:

•Ofﬂine: Duration TTASK of information acquisition task;

look-ahead horizon TPLAN for planning trajectories

(TPLAN ≤TTASK); replanning rate TREPLAN (TREPLAN

≤TPLAN); model dynamics fiof each robot i’s state

xi,t, including initial condition xi,0(i∈ V ); sensing

model hiof each robot i’s sensors, including µvi,t and

Σvi,t ; model dynamics gof target process, including

initial condition y0, and µwtand covariance Σwt; ob-

jective function J; number of attacks α.

•Online: At each t= 1,...,TTASK, observed (i) attack

At(i.e., robot removal At⊆ V), and (ii) measurements

zi,t from each non-attacked robot i∈ V \ At.

Output: At each t= 1,...,TTASK, estimate ˆytof yt.

// Initialize //

1: t= 0;ˆyt=y0;At=∅;zt=∅;

// Execute resilient active information acquisition task //

2: while t < TTASK do

//(Re)plan robust trajectories for all robots //

3: if tmod TREPLAN = 0 then

4: It={t, {fi, xi,t, hi, µwt,Σwt, µvi,t ,Σvi,t }i∈V , g, ˆyt,

5: zt,TPLAN, α};// Denote by Itthe information

needed by the RTP algorithm, called in the next line.

6: uV, t+1:t+TPLAN =RTP(I0:t);// Plan robust trajec-

tories for all robots with look-ahead planning horizon

TPLAN.

7: end if

// Execute current step of trajectory computed by RTP//

8: for all i∈ V do

9: xi,t+1 =fi(xi,t, ui,t );

10: end for

11: Observe At+1;// Determined by environment/attacker.

// Integrate measurements from non-attacked robots //

12: for all i∈ V \ At+1 do

13: Receive measurement zi,t+1;// Only measurements

from non-attacked robots are received.

14: end for

15: update Estimate ˆyt+1 of yt+1 given z1 : t+1;// z1 : t

collects all available measurements up to the time t,i.e.,

z1 : t,{zi,τ :i∈ V \ Aτ, τ = 1, . . . , t}.

16: t=t+ 1;// Time update.

17: end while

robots in Lare non-existent, and plans the control inputs for

the remaining robots (RTP’s line 8).

V. PE RF OR MA NC E GUARANTEES O F RTP

Performance guarantees are unknown for RAIN, and, corre-

spondingly, (P-RAIN) ((P-RAIN) is a mixed-integer, sequential

control optimization problem, with limited a priori information

on the measurements and attacks that are going to occur during

the task-length, look-ahead time horizon). Nevertheless, in this

section we quantify RTP’s performance, which is used by

RAIN in a receding horizon fashion to approximate a solution

to (P-RAIN) locally (cf. RAIN’s lines 2-17), by picking sequen-

5

Algorithm 2 Robust Trajectory Planning (RTP).

Input: Look-ahead horizon TPLAN for planning trajectories;

current time t; set of robots V; model dynamics fiof

each robot i’s state, including current state xi,t; sensing

model hiof each robot i’s sensors, including µvi,t and

Σvi,t ; model dynamics gof target process yt, including

current estimate ˆyt, and µwtand covariance Σwt; objective

function J; measurement history z1:t; number of attacks α.

Output: Control inputs ui,t0, for all robots i∈ V and all times

t0=t+ 1, . . . , t +TPLAN .

// Step 1: Generate bait robot set (to approximate a worst-

case attack assumed constant ∀t0∈[t+ 1, t +TPLAN ]) //

1: for all i∈ V do // Compute the value of (P-RTP) assuming

(i) only robot iexists and (ii) no attacks will happen.

2: J?

{i}, t, 0,max

ui,t0∈ Ui,t0

t0=t+1 : t+TPLAN

J{i}, t+1:t+TPLAN ;

3: end for

4: Find a subset Lof αrobots such that J?

{i}, t, 0≥J?

{j}, t, 0

for all i∈ L and j∈ V \ L;// Lis the bait robot set

(L⊆V and |L| =α).

5: for all i∈ L do // Assign to each robot i∈ L the

trajectory that achieves J?

{i}, t, 0.

6: ui, t+1:t+TPLAN = arg max

ui,t0∈ Ui,t0

t0=t+1 : t+TPLAN

J{i}, t+1:t+TPLAN ;

7: end for

// Step 2: Remaining robots, V \L, plan assuming (i) only

robots in V \ L exists and (ii) no attacks will happen //

8: uV\L, t+1:t+TPLAN = arg max

uV\L,t0∈ UV \L,t0

t0=t+1 : t+TPLAN

JV\L, t+1:t+TPLAN .

tially in time control inputs given (i) a shorter, computationally

feasible look-ahead time horizon (cf. Section III), and (ii) the

history of the so far observed measurements and attacks.

Particularly, in this section we bound RTP’s approximation

performance and running time. We use properties of the

objective function JV\A, t+1:t+TPLAN in (P-RTP) as a function

of the set of robots; namely, the following notions of curvature.

A. Curvature and Total Curvature

We present the notions of curvature and total curvature for

set functions. We start with the notions of modularity, and of

non-decreasing and submodular set functions.

Deﬁnition 1 (Modularity [34]).Consider a ﬁnite (discrete)

set V. A set function h: 2V7→ Ris modular if and only if

h(A) = Pv∈A h(v), for any A ⊆ V.

Hence, if his modular, then V’s elements complement each

other through h. Speciﬁcally, Deﬁnition 1implies h({v}∪A)−

f(A) = h(v), for any A⊆V and v∈ V \ A.

Deﬁnition 2 (Non-decreasing set function [34]).Consider a

ﬁnite (discrete) set V.h: 2V7→ Ris non-decreasing if h(B)≥

h(A)for all A ⊆ B.

Deﬁnition 3 (Submodularity [34, Proposition 2.1]).Consider

a ﬁnite (discrete) set V.h: 2V7→ Ris submodular if h(A ∪

{v})−h(A)≥h(B ∪ {v})−h(B)for all A⊆B and v∈ V.

Therefore, his submodular if and only if the return

h(A ∪ {v})−h(A)diminishes as Agrows, for any v. If

his submodular, then V’s elements substitute each other, in

contrast to hbeing modular. Particularly, consider hto be

non-negative (without loss of generality): then, Deﬁnition 3

implies h({v} ∪ A)−h(A)≤f(v). Thereby, v’s contribution

to f({v}∪A)’s value is diminished in the presence of A.

Deﬁnition 4 (Curvature [29]).Consider a ﬁnite (discrete) V,

and a non-decreasing submodular h: 2V7→ Rsuch that

h(v)6= 0 for any v∈ V , without loss of generality. Then, h’s

curvature is deﬁned as

κh,1−min

v∈V

h(V)−h(V \ {v})

h(v).(6)

Deﬁnition 4implies κh∈[0,1]. If κh= 0, then h(V)−

h(V \{v}) = h(v), for all v∈ V,i.e.,his modular. Instead, if

κh= 1, then there exist v∈ V such that h(V) = h(V \ {v}),

that is, vhas no contribution to h(V)in the presence of V\{v}.

Overall, κhrepresents a measure of how much V’s elements

complement (and substitute) each other.

Deﬁnition 5 (Total curvature [30, Section 8]).Consider a

ﬁnite (discrete) set Vand a monotone h: 2V7→ R. Then, h’s

total curvature is quantiﬁed as

ch,1−min

v∈V min

A,B⊆V\{v}

h({v}∪A)−h(A)

h({v}∪B)−h(B).(7)

Deﬁnition 5implies ch∈[0,1], similarly to Deﬁnition 5

for κh. When his submodular, then ch=κh. Generally, if

cf= 0, then his modular, while if ch= 1, then eq. (7) implies

Deﬁnition 5’s assumption that his non-decreasing.

B. Performance Analysis for RTP

We quantify (i) suboptimality bounds on RTP’s approxima-

tion performance, and (ii) upper bounds on the running time

RTP requires. We use the notation:

•J?

V, t, α is the optimal value of (P-RTP):

J?

V, t, α ,max

uV,t0∈ UV,t0

t0=t+1 : t+TPLAN

min

A⊆V,|A| ≤ αJV \A, t+1:t+TPLAN ;

•A?is an optimal removal of αrobots from Vper (P-RTP):

A?,arg min

A⊆V,|A|≤αJV \A, t+1:t+TPLAN .

We also use the deﬁnitions:

Deﬁnition 6 (Normalized set function [34]).Consider a

discrete set V.h: 2V7→ Ris normalized if h(∅)=0.

Deﬁnition 7 (Non-negativeness [34]).Consider a discrete

set V.h: 2V7→ Ris non-negative if h(A)≥0for all A.

Theorem 1 (Performance of RTP).Consider an instance of

(P-RTP). Assume the robots in Vcan solve optimally the

(attack-free) information acquisition problem in eq. (5).

•Approximation performance: RTP returns control inputs

uV,1:t+TPLAN such that (i) if J·,t+1:t+TPLAN : 2V7→ Ris

6

non-decreasing, and, without loss of generality, normal-

ized and non-negative, then

JV\A?, t+1:t+TPLAN

J?

V, t, α

≥(1 −cJ·,t+1:t+TPLAN )2;(8)

(ii) If, in addition, J·,t+1:t+TPLAN is submodular, then

JV\A?, t+1:t+TPLAN

J?

V, t, α

≥max 1−κJ·,t+1:t+TPLAN ,1

1 + α.

(9)

•Running time: If ρupper bounds the running time for

solving the (attack-free) information acquisition problem

in eq. (5), then RTP terminates in O(|V|ρ)time.

Theorem 1’s bounds in eqs. (8)-(9) compare RTP’s selection

uV,1:t+TPLAN against an optimal selection of control inputs that

achieves the optimal value J?

V, t, α for (P-RTP). Particularly,

eqs. (8)-(9) imply that for (i) non-decreasing and (ii) non-

decreasing and submodular functions J·,t+1:t+TPLAN ,RTP guar-

antees a value for (P-RTP) which can be close to the optimal.

For example, eq. (9)’s lower bound 1/(1 + α)is non-zero for

any ﬁnite number of robots |V|, and, notably, it equals 1in

the attack-free case (RTP is exact for α= 0, per Theorem 1’s

assumptions). More broadly, when κJ·,t+1:t+TPLAN <1or

cJ·,t+1:t+TPLAN <1,RTP’s selection uV,1:t+TPLAN is close to

the optimal, in the sense that Theorem 1’s bounds are non-

zero. Functions with κJ·,t+1:t+TPLAN <1include the log det

of positive-deﬁnite matrices [35]; objective functions of this

form are the conditional entropy and mutual information when

used for batch-state estimation of stochastic processes [36].

Functions with cJ·,t+1:t+TPLAN <1include the average min-

imum square error (mean of the trace of a Kalman ﬁlter’s

error covariance across a ﬁnite time horizon) [37].

Theorem 1’s curvature-dependent bounds in eqs. (8)-(9)

also make a ﬁrst step towards separating the classes of (i)

non-decreasing and (ii) non-decreasing and submodular func-

tions into functions for which (P-RTP) can be approximated

well, and functions for which it cannot. Indeed, when either

κJ·,t+1:t+TPLAN or cJ·,t+1:t+TPLAN tend to zero, RTP becomes

exact. For example, eq. (8)’s term 1−cJ·,t+1:t+TPLAN increases

as cJ·,t+1:t+TPLAN decreases, and its limit is equal to 1for

cJ·,t+1:t+TPLAN →0. Notably, however, the tightness of Theo-

rem 1’s bounds is an open problem. For example, although for

the attack-free problem in eq. (5) a bound O(1−cJ·,t+1:t+TPLAN )

is known to be optimal (the tightest possible in polynomial

time and for a worst-case J·,t+1:t+TPLAN ) [30, Theorem 8.6],

the optimality of eq. (8) is an open problem.

Overall, Theorem 1quantiﬁes RTP’s approximation per-

formance when the robots in Vsolve optimally the (attack-

free) information acquisition problems in RTP’s line 2, line 6,

and line 8. Among those, however, the problems in line 6

and line 8are computationally challenging, being multi-robot

coordination problems; only approximation algorithms are

known for their solution. Such an approximation algorithm

is the recently proposed coordinate descent [6, Section IV].

Coordinate descent has the advantages of having a provably

near-optimal approximation performance. Therefore, we next

quantify RTP’s performance when the robots in Vsolve the

problems in RTP’s line 6, and line 8using coordinate descent.3

Proposition 1 (Approximation Performance of RTP via Co-

ordinate Descent).Consider an instance of (P-RTP). Assume

the robots in Vsolve the (attack-free) information acquisition

problem in eq. (5)suboptimally in the case of multiple robots

(|V| ≥ 2)via coordinate descent [6, Section IV], and opti-

mally in the case of a single robot (|V| = 1). Then:

•Approximation performance: RTP returns control inputs

uV,1:t+TPLAN such that (i) if J·,t+1:t+TPLAN : 2V7→ Ris

non-decreasing, and, without loss of generality, normal-

ized and non-negative, then

JV\A?, t+1:t+TPLAN

J?

V, t, α

≥1

2(1 −cJ·,t+1:t+TPLAN )3;(10)

(ii) If J·,t+1:t+TPLAN is also submodular, then

JV\A?, t+1:t+TPLAN

J?

V, t, α

≥1

2max 1−κJ·,t+1:t+TPLAN ,1

1 + α.

(11)

•Running time: If ρCD upper bounds the running time for

solving the information acquisition problem in eq. (5)via

coordinate descent, then RTP terminates in O(ρCD)time.

Proposition 1’s suboptimality bounds are discounted ver-

sions of Theorem 1’s bounds: (i) eq. (10) is the discounted

eq. (8) by the factor (1 −cJ·,t+1:t+TPLAN )/2; and (ii) eq. (11)

is the discounted eq. (9) by the factor 1/2. The source of the

discounting factors is the requirement in Proposition 1that

the robots in Vcan solve only suboptimally (via coordinate

descent) the information acquisition problem in eq. (5) (and,

in effect, the problems in RTP’s line 6and line 8). In more

detail, in Lemma 5, located in Appendix 2, we prove that

(i) for non-decreasing objective functions, coordinate descent

guarantees the suboptimality bound (1 −cJ·,t+1:t+TPLAN )/2for

eq. (5) (which is the discounting factor to eq. (8), resulting

in eq. (10)), while (ii) for non-decreasing and submodular

functions, coordinate descent is known to guarantee the sub-

optimality bound 1/2for eq. (5) (which is the discounting

factor to eq. (9), resulting in eq. (11)) [6].

Proposition 1also implies that if the robots in Vuse coordi-

nate descent to solve the (attack-free) information acquisition

problems in RTP’s line 6and line 8, then RTP has the same

of order of running time as coordinate descent. The proof of

Proposition 1is found in Appendix 4.

VI. AP PL IC ATIO NS A ND EXPERIMENTS

We present RAIN’s performance in applications. We present

three applications of Resilient Active Information Acquisi-

tion with Teams of Robots: (i) Multi-Target Tracking (Sec-

tion VI-A), (ii) Occupancy Grid Mapping (Section VI-B), and

(iii) Persistent surveillance (Section VI-C). We conﬁrm RAIN

effectiveness, even as we vary key parameters in (P-RAIN):

(i) the number of attacks α, among the permissible values

{0,1,...,|V|}, to test RAIN’s performance to both small

and high attack numbers (Section VI-A);

3We refer to Appendix 2 for a description of coordinate descent.

7

(ii) the attack model, beyond the worst-case model prescribed

by (P-RAIN)’s problem formulation, to test RAIN’s sensi-

tivity against non worst-case failures; particularly, random

failures (Section VI-B).4

(iii) the replanning rate TREPLAN, among the permissible val-

ues {1,2,...,TPLAN}, to test RAIN’s performance even

when the replanning rate is high (Section VI-C).5

Common Experimental Setup across Applications:

a) Approximation Algorithm for (Attack-Free) Informa-

tion Acquisition problem in eq. (5)(and in effect for the

problems in RTP’s line 2, line 6, and line 8): In the multi-

robot case (pertained to RTP’s line 6, and line 8), the algorithm

used for approximating a solution to eq. (5) is the coordinate

descent [6] (also, cf. Appendix 2). Evidently, coordinate

descent does not account for the possibility of attacks, and for

this reason, we also use it as a baseline to compare RAIN with.

In the single robot case (pertained to RTP’s line 2), eq. (5)

reduces to a single-robot motion planning problem, and for

its solution we use reduced value iteration (ARVI algorithm

[4]), except for the application of Occupancy Grid Mapping

(Section VI-B) where we use forward value iteration [3].

b) Worst-Case Attack Approximation: Computing the

worst-case attack requires brute-force, since the minimization

step in (P-RAIN) is NP-hard [38]. The consequence is that solv-

ing for the worst-case attack requires solving an exponential

number of instances of the information acquisition problem

in eq. (5), prohibiting real-time navigation performance by

the robots, even for small teams of robots (|V| ≥ 5). In

particular, the running time required to solve eq. (5), even via

coordinate descent, can be exponential in the number of robots

and task length horizon, namely, O(|U||V |TTASK)[6] (Udenotes

the set of admissible control inputs to each of the robots in V,

assumed the same across all robots). Hence, we approximate

the worst-case attacks by solving the minimization step in

(P-RAIN) via a greedy algorithm [24].

c) Computational Platform: Experiments are imple-

mented in C++, and run on an Intel Core i7 CPU laptop.

A. Resilient Multi-Target Tracking

In Resilient Multi-Target Tracking, a team of mobile robots

is tasked to track the locations of multiple moving targets, even

in the presence of a number of attacks against the robots. For

the purpose of assessing RAIN’s effectiveness against various

number of attacks, we will vary the number of attacks across

scenarios, where we will also possibly vary the number of

robots and targets. In more detail, the experimental setup and

simulated scenarios are described below.

Experimental Setup. We specify the used (a) robot dynam-

ics, (b) target process, (c) sensor model, and (d) information

acquisition objective function:

4For random failures, one would expect RAIN’s performance to be the

same, or improve, since RAIN is designed to withstand the worst-case.

5When the replanning rate tends, ideally, to inﬁnity, in that the robots can

instantaneously observe all attacks and replan (which, however, is in practice

impossible due to algorithmic, computational, and communication delays),

then it is expected that RAIN’s advantage over a non-resilient algorithm with

the same replanning rate, such as coordinate descent, would diminish.

Fig. 2: Resilient Multi-Target Tracking Scenario. 10 robots are

depicted tracking 10 targets, while 4 of the robots are be attacked

(causing their sensing capabilities to be, at least temporarily, dis-

abled). The robots are depicted with their conic-shaped ﬁeld-of-

view, colored light blue for non-attacked robots and light red for

attacked robots. The targets are depicted with red disks. Planned robot

trajectories are shown as solid blue lines. Predicted target trajectories

are shown as solid red lines. Each light-green ellipse represents the

covariance of the target’s location estimate.

a) Robot Dynamics: Each robot ihas unicycle dynamics

in SE(2), discretized with a sampling period τ, such that

x(1)

i,t+1

x(2)

i,t+1

θi,t+1

=

x(1)

i,t

x(2)

i,t

θi,t

+

νisinc(ωiτ

2) cos(θi,t +ωiτ

2)

νisinc(ωiτ

2) sin(θi,t +ωiτ

2)

τωi

,(12)

where (νi, ωi)is the control input (linear and angular velocity).

b) Target Dynamics: The targets move according to

double integrator dynamics, which are assumed corrupted with

additive Gaussian noise. Speciﬁcally, if Mdenotes the number

of targets, then yt= [y>

t,1, . . . , y>

t,M ]>, where yt,m is the planar

coordinates and velocities of target m, and

yt+1,m =I2τI2

0I2yt,m +wt, wt∼N0, q τ3/3I2τ2/2I2

τ2/2I2τI2.

with qbeing a noise diffusion parameter.

c) Sensor Model: The robots’ sensor model consists of

a range and bearing for each target m= 1, . . . , M:

zt,m =h(xt, yt,m) + vt, vt∼N(0, V (xt, yt,m));

h(x, ym) = r(x, ym)

α(x, ym),p(y1−x1)2+ (y2−x2)2

tan−1((y2−x2)(y1−x1)) −θ.

Since the sensor model is non-linear, we linearize it around

the currently predicted target trajectory. Particularly, given

∇yh(x, ym) = 1

r(x, ym)(y1−x1) (y2−x2) 01x2

−sin(θ+α(x, ym)) cos(θ+α(x, ym)) 01x2,

the observation model for the joint target state can be ex-

pressed as a block diagonal matrix containing the linearized

observation models for each target along the diagonal:

H,diag (∇y1h(x, y1),...,∇yMh(x, yM)) .

8

Fig. 3: Resilient Multi-Target Tracking Results. Performance com-

parison of RAIN with coordinate descent (noted as NonResilient in the

plots), for two conﬁgurations and two performance metrics: top row

considers 10 robots, 10 targets, and 2 attacks; bottom row considers

the same number of robots and targets but 6 attacks. The left column

depicts mean entropy per target, averaged over the robots; the right

column depicts the position Root Mean Square Error (RMSE) per

target, also averaged over the robots.

The sensor noise covariance grows linearly in range and in

bearing, up to σ2

r, and σ2

b, where σrand σbare the standard

deviation of the range and the bearing noise, respectively. The

model here also includes a limited range and ﬁeld of view,

denoted by the parameters rsense and ψ, respectively.

d) Information Acquisition Objective Function: For the

information objective function we use the time averaged log

determinant of the covariance matrix, which is equivalent to

conditional entropy for Gaussian variables [3]. This objec-

tive function is non-decreasing, yet not necessarily submod-

ular [39]. Overall, we solve an instance of (P-RAIN) per the

aforementioned setup and the objective function:

JV,1:TTASK ,1

TTASK

TTASK

X

t=1

log det(ΣV, t),

where ΣV, t is the Kalman ﬁltering error covariance at t, given

the measurements collected up to tby the robots in V[3].6

Simulated Scenarios. We consider multiple scenarios of

the experimental setup introduced above: across scenarios, we

vary the number of robots, n, the number of targets M, and the

number of attacks, α; cf. ﬁrst column of Table I. Additionally,

the robots and targets are restricted to move inside a 64×64m2

environment (Fig. 2). The admissible control input values to

each robot are the U={1,3}m/s × {0,±1,±3}rad/s. At

the beginning of each scenario, we ﬁx the initial positions of

both the robots and targets, and the robots are given a prior

distribution of the targets before starting the simulation. The

targets start with a zero velocity, and in the event that a target

6The multi-target tracking scenarios are dependent on a prior distribution

of the target’s initial conditions y0and Σ0|0, assumed here known. Yet, if a

prior distribution is unknown, then an exploration strategy can be incorporated

to ﬁnd the targets by placing exploration landmarks at the map frontiers [6].

Mean RMSE Peak RMSE

NonRes RAIN NonRes RAIN

n= 5,M= 10

α= 1 0.28 0.19 9.62 2.09

α= 2 1.47 0.68 26.07 15.71

α= 4 10.67 4.9 225.47 103.82

n= 10,M= 5

α= 2 0.35 0.14 57.65 1.87

α= 4 0.39 0.28 6.66 3.17

α= 6 2.07 0.65 93.27 15.63

n= 10,M= 10

α= 2 0.13 0.08 1.4 1.32

α= 4 0.24 0.23 4.19 2.66

α= 6 4.39 1.2 69.77 26.4

TABLE I: Resilient Multi-Target Tracking Results. Performance

comparison of RAIN with coordinate descent (noted as NonRes in the

table), for a variety of conﬁgurations, where ndenotes the number of

robots (n=|V|), Mdenotes the number of targets, and αdenotes the

number of failures. Two performance metrics are used: mean Root

Mean Square Error (RMSE), and peak RMSE, both per target, and

averaged over the robots in the team.

leaves the environment its velocity is reﬂected to remain in

bounds. Finally, across all simulations: TPLAN =TREPLAN =

25 steps, τ= 0.5s,rsense = 10m,ψ= 94◦,σr=.15m,

σb= 5◦, and q=.001. We use TTASK = 500.Compared

Techniques. We compare RAIN with coordinate descent. We

consider two performance measures: the average entropy, and

average Root Mean Square Error (RMSE) per target, averaged

over the robots in the team.

Results. The results, averaged across 10 Monte-Carlo runs,

are depicted in Fig. 3and Table I. In Fig. 3,RAIN’s per-

formance is observed to be superior both in terms of the

average entropy and the RMSE. Particularly, as the number

of attacks grows (cf. second rows of plots in Fig. 3), RAIN’s

beneﬁts are accentuated in comparison to the non-resilient

coordinate descent. Similarly, Table Idemonstrates that RAIN

achieves a lower mean RMSE than coordinate descent, and,

crucially, is highly effective in reducing the peak estimation

error; in particular, RAIN achieves a performance that is 2 to

30 times better in comparison to the performance achieved by

the non-resilient algorithm. We also observe that the impact

of Algorithm 2is most prominent when the number of attacks

is large relative to the size of the robot team.

B. Resilient Occupancy Grid Mapping

We show how (P-RAIN)’s framework for resilient active in-

formation acquisition can be adapted to exploring an environ-

ment when the map and objective are deﬁned via occupancy

grids. In this section, we also assess RAIN’s sensitivity against

non worst-case attacks, in particular, random.

Experimental Setup. We specify the used (a) robot dy-

namics, (b) target process, (c) sensor model, (d) information

acquisition objective function, and (e) algorithm for solving

the optimization problem in RTP’s line 2:

a) Robot Dynamics: The robots’ dynamics are as in the

multi-target tracking application (Section VI-A).

9

Fig. 4: Resilient Occupancy Grid Mapping Scenarios. Two scenarios are considered: a square obstacle map (top row), and a corridor

map (bottom row), where free space is colored white, occupied space is colored black, and unexplored/unknown space is colored gray. The

non-attacked robots are shown with their ﬁeld-of-view colored blue, whereas the attacked robots are shown with ﬁeld-of-view colored red.

The left-most column shows the considered ground truth maps; the middle column shows the map estimate halfway through the task horizon;

the right-most column shows the map estimate near completion.

b) Target Process: We deﬁne the target process yt,

which we will denote henceforth as Mfor consistency with

common references on occupancy grid mapping [40], where

we also drop the time subscript since the process ytdoes not

evolve in time. The occupancy grid Mis a 2-dimensional

grid with nrows and mcolumns, discretized into cells

M={C1, . . . , Cnm }, which are binary variables that are

either occupied or free, with some probability. Cell occupancy

is assumed to be independent, so that the probability mass

function can be factored as P(M=m) = Qnm

i=1 P(Ci=ci),

where ci∈ {0,1}, and where m∈ {0,1}nm is a particular

realization of the map.

c) Sensor Model: We express the sensor model as a

series of Bbeams, such that zt= [z1

t, . . . , zB

t], where zb

k

is the random variable of the distance that a beam btravels

to intersect an object in the environment. We next deﬁne the

distribution for a single beam, determined by the true distance

dto the ﬁrst obstacle that the beam intersects:

p(zb

k=z|d) =

N(z−0, σ2), d < zmin;

N(z−zmax, σ2), d > zmax;

N(z−d, σ2),otherwise;

(13)

zmin and zmax are the minimum and maximum sensing ranges.

d) Information Acquisition Objective Function: The

used information objective function is the Cauchy Schwarz

Quadratic Mutual Information (CSQMI), which is shown in

the literature to be computationally efﬁcient, as well as,

sufﬁciently accurate for occupancy mapping [5]. We denote

the CSQMI gathered at time tby ICS (m;zV,t ), given the

measurements collected at tby the robots in V. Then,

JV,1:TTASK ,1

TTASK

TTASK

X

t=1

ICS (m;zV,t ),

which is non-decreasing and submodular [7].

Remark 3 (Evaluation of CSQMI).Details on the evaluation

of the CSQMI objective are beyond the scope of this paper;

we refer the reader to [19]. We note that it relies on a ray-

tracing operation for each beam, computed over the current

occupancy grid map belief to determine which cells each beam

from the LIDAR will observe when the sensor visits a given

pose. CSQMI is approximated by assuming the information

computed over single beams is additive, but not before pruning

approximately independent beams. This removal of approxi-

mately independent beams encourages robots to explore areas

where their beams will not overlap. In coordinate descent,

once a set of prior robots has planned trajectories, future

robots must check their beams to see if they are approximately

independent from the ﬁxed beams, before their individual beam

10

Fig. 5: Resilient Occupancy Mapping Results. Comparison of achieved entropy by RAIN against coordinate descent (noted as NonResilient

in the plots) for increasing time and for two types of attacks, worst-case and random: (left plot) result for the square obstacles map (top row

of Fig. 4); (right plot) result for the corridor map (bottom row of Fig. 4).

contributions may be added to the joint CSQMI objective.

e) Algorithm for Solving Optimization Problem in RTP’s

line 2:The single-robot motion planning is performed via a

full forward search over a short planning horizon TPLAN = 4,

since scaling beyond short horizons is challenging in occu-

pancy mapping problems; for details, cf. [19]. We remark

that the performance guarantees do not explicitly hold for the

single-robot planner since the measurement model is highly

non-linear and the cost function depends on the realization

of the measurements, so open-loop planning is not optimal

as it is with the Gaussian case [3]. Nonetheless, approaches

similar to what we adopt here have been successfully used for

(attack-free) occupancy grid mapping [5], [7].

Simulated Scenarios. To evaluate the performance of the

resilient occupancy mapping algorithm, we compare the results

with different attack types. Namely, we consider an attack

model where the attacks on robots may be uniformly random,

rather than the worst case attack assumption of the previous

section and the algorithm itself. In the experiment, the robots

choose trajectories composed of TPLAN = 4 steps of duration

τ= 1 sec, with motion primitives U={(v, ω) : v∈

{0,1,2}m/s, ω ∈ {0,±1,±2}rad/s}. The maximum sensor

range rsense is 1.5m, with a noise standard deviation of

σ=.01m. The experiment considers a team of six robots,

subject to two attack models, described in the following

paragraph. The attacked set of robots is re-computed at the

end of each planning duration. We evaluate the performance

on two map environments, which we will refer to as the square

obstacles map (Fig 4, top), and the corridor map (Fig 4,

bottom). We use TTASK = 50 and TTASK = 100 for the squares

and corridor map respectively, and TREPLAN =TTASK.

Compared Attack Models. We test RAIN’s ability to be

effective even against non worst-case failures. To this end,

beyond considering the worst-case attack model prescribed

by (P-RAIN)’s problem formulation (cf. red and blue curves

in Fig. 5), we also consider random attacks, chosen with

uniformly random assignment among the robots in V, given

the attacks number α(cf. green and yellow curves Fig. 5).

Results. The results, averaged across 50 Monte-Carlo runs,

are shown in Fig. 5. The plots indicate RAIN always improves

performance. Speciﬁcally, RAIN improves performance both

when (i) worst-case attacks are present (cf. blue and red curves

in Fig. 5), and when (ii) random attacks are present (cf. green

and yellow curves in Fig. 5); in both cases, RAIN attains lesser

map entropy against coordinate descent (noted as NonResilient

in the plots). Both the square obstacles map (left plot in

Fig. 5) and the corridor map (right plot in Fig. 5) support

this conclusion. Moreover, Fig. 5supports the intuition that

since RAIN is designed to withstand the worst-case attacks,

RAIN’s performance will improve when instead only random

failures are present (cf. blue and green curves in Fig. 5).

C. Resilient Persistent Surveillance

In Resilient Persistent Surveillance, the robots’ objective is

to re-visit a series of static and known landmarks, while the

robots are under attack. The landmarks represent points of

interest in the environment. For example, a team of robots

may be tasked to monitor the entrances to buildings for

intruders [41]; the task becomes especially interesting as the

number of entrances becomes more than the number of robots.

In this section, we choose the landmarks to be a set of

buildings in an outdoor Camp environment (Fig. 6). We use the

simulated scenarios to determine the effect of the replanning

rate on RAIN’s performance (cf. Footnote 5).

Experimental Setup. The environment used is a 3D

environment provided by ARL DCIST (Fig. 6). It contains

a set of outdoor buildings, over which we place landmarks to

encourage visitation (one landmark per building). To have the

robots (re-)visit the landmarks, we add artiﬁcial uncertainty

to the location of each landmark by proportionally increasing

the uncertainty with time passed since the last observation.

The software simulation stack used is based on the Robot

Operating System (ROS); the back-end physics are based on

Unity. In all experiments, the map is assumed to be known.

Localization is provided by the simulator. We next specify the

used (a) robot dynamics, (b) target process, (c) sensor model,

11

Fig. 6: Resilient Persistent Surveillance Scenario. Camp Lejeune

3D environment. The robots’ trajectories are shown in pink. The

blue lines along the trajectories indicate the velocity proﬁle generated

along the trajectory. The blue discs show the ﬁeld-of-view of the non-

attacked robots. A red colored ﬁeld-of-view indicates an attacked

robot. The cyan spheres represent the relative uncertainty on the

landmarks’ locations. The landmarks are depicted as the red spheres.

and (d) information acquisition objective function:

a) Robot Dynamics: The robot motion model is adapted

from the 2D in eq. (12) to the following 3D:

x1

t+1

x2

t+1

x3

t+1

θt+1

=

x1

t

x2

t

x3

t

θt

+

νsinc(ωτ

2) cos(θt+ωτ

2)

νsinc(ωτ

2) sin(θt+ωτ

2)

0

τω

.

That is, we assume the quadrotors to ﬂy at a ﬁxed height

over the environment (x3

t+1 =x3

t)always).

b) Target Process: The targets are assumed static in

location, but corrupted with uncertainty that increases over

time to encourage (re-)visitation by the robots, according to

a noise covariance matrix qkt,mI3, where qis the rate of

uncertainty increase, and kt,m denotes the number of time

steps since target mwas last visited.

yt+1,m =yt,m +wt, wt∼N(0, qkt,m I3).

c) Sensor Model: We assume the robots operate a 360◦

ﬁeld-of-view downward facing sensor. In particular, we as-

sume a range sensing model that records information as long

as the robots are within some radius rsense from a landmark;

otherwise, no information is granted to the robot. The range

based model for detecting the buildings is as follows:

zt,m =h(xt, yt,m) + vt, vt∼N(0, V (xt, yt,m ));

h(x, ym) = r(x, ym),v

u

u

t

3

X

i=1

(y1−x1)2.

d) Information Acquisition Objective Function: We use

the same information acquisition objective function as in the

Multi-Target Tracking scenarios (Section VI-A).

Simulation Setup. The admissible control inputs are the

U={(ν, ω) : ν∈ {1,3}m/s, ω ∈ {0,1,2}}.rsense is

10 meters, the task duration is TTASK = 300 steps, and the

planning horizon is TPLAN = 10 steps. Speciﬁcally, each

timestep has duration τ= 1s. The noise parameter is q=.01.

Fig. 7: Resilient Persistent Surveillance Results. Comparison of av-

erage time between consecutive observations of the same landmarks

by RAIN and coordinate descent (noted as NonResilient in the plot)

for increasing replanning values TREPLAN.

Performance Metric. We measure RAIN’s performance by

computing the average number of timesteps that a building

goes unobserved for. For example, if a landmark is observed

at timestep k, and not observed again until timestep k+l,

we record las the number of timesteps the landmark was

unobserved. Particularly, we average these durations across all

targets and timesteps for a given experimental trial.

Results. The results, averaged across 50 Monte-Carlo runs,

are shown in Fig. 7. In Fig. 7, we observe even for the highest

replanning rate (TREPLAN = 1), RAIN offers a performance

gain of '24% in comparison to coordinate descent (noted

as NonResilient in Fig. 7). The gain increases on average,

the lower the replanning rate becomes, as expected (cf. Foot-

note 5). More broadly, Fig. 7supports the intuition that a

higher replanning rate allows even a non-resilient algorithm,

such as coordinate descent, to respond to attacks rapidly, and

thus perform well. Still, in Fig. 7RAIN dominates coordinate

descent across all possible replanning rate values.

VII. CONCLUSION

We introduced the ﬁrst receding-horizon framework for

resilient multi-robot path planning against attacks that dis-

able robots’ sensors during information acquisition tasks

(cf. (P-RAIN)). We proposed Resilient Active Information

acquisitioN (RAIN), a robust and adaptive multi-robot planner

against any number of attacks. RAIN calls, in an online

fashion, Robust Trajectory Planning (RTP), a subroutine that

plans attack-robust control inputs over a look-ahead planning

horizon. We quantiﬁed RTP’s performance by bounding its

suboptimality, using notions of curvature for set function

optimization. We demonstrated the necessity for resilient

multi-robot path planning, as well as RAIN’s effectiveness,

in information acquisition scenarios of Multi-Target Track-

ing,Occupancy Grid Mapping, and Persistent Surveillance.

In all simulations, RAIN was observed to run in real-time,

and exhibited superior performance against a state-of-the-art

baseline, (non-resilient) coordinate descent [6]. Across the

12

three scenarios, RAIN’s exhibited robustness and superiority

even (i) in the presence of a high number of attacks, (ii)

against varying models of attacks, and (iii) high replanning

rates. Future work includes extending the proposed framework

and algorithms to distributed settings [7], [16].

APPENDICES

In the appendices that follow, we prove Theorem 1(Ap-

pendix 3) and Proposition 1(Appendix 4). To this end, we ﬁrst

present supporting lemmas (Appendix 1) and the algorithm

coordinate descent [6] (Appendix 2). We also use the notation:

Notation. Consider a ﬁnite set Vand a set function f:

2V7→ R. Then, for any set X ⊆ V and any set X0⊆ V,

the symbol f(X |X 0)denotes the marginal value f(X ∪ X 0)−

f(X0). We also introduce notation emphasizing that subsets

of robots may use different algorithms to compute their

control inputs: we let J(ua

A,1 : T, ub

B,1 : T),JAa∪Bb,1 : T

indicate that the robots in Acontribute their measurements

to Jand their control inputs are chosen with algorithm a

(e.g., coordinate descent), while robots in Balso contribute

their measurements to Jbut their inputs are chosen with

another algorithm b. We also occasionally drop the subscript

for time indices, since all time indices in the appendices

are identical, (namely, t+ 1 : t+TPLAN). Similarly, when

only the set of robots is important, we use the notation

J(A),JA, t+1:t+TPLAN , for any A⊆V. Lastly, the notation

J(∅)refers to the information measure evaluated without any

measurements from the robot set.

APPENDIX 1. PRELIMINARY LEM MA S

The proof of the lemmas is also found in [27], [42].

Lemma 1. Consider a ﬁnite set Vand a non-decreasing and

submodular set function f: 2V7→ Rsuch that fis non-

negative and f(∅) = 0. For any A⊆V:

f(A)≥(1 −κf)X

a∈A

f(a).

Proof of Lemma 1:Let A={a1, a2, . . . , a|A|}. We prove

Lemma 3by proving the following two inequalities:

f(A)≥

|A|

X

i=1

f(ai|V \ {ai}),(14)

|A|

X

i=1

f(ai|V \ {ai})≥(1 −κf)

|A|

X

i=1

f(ai).(15)

We begin with the proof of ineq. (14):

f(A) = f(A|∅)(16)

≥f(A|V \ A)(17)

=

|A|

X

i=1

f(ai|V \ {ai, ai+1 , . . . , a|A|})(18)

≥

|A|

X

i=1

f(ai|V \ {ai}),(19)

where ineqs. (17) to (19) hold for the following reasons:

ineq. (17) is implied by eq. (16) because fis submodular

and ∅ ⊆ V \ A; eq. (18) holds since for any sets X ⊆ V

and Y ⊆ V we have f(X |Y ) = f(X ∪ Y )−f(Y), and also

{a1, a2, . . . , a|A|}denotes the set A; and ineq. (19) holds since

fis submodular and V \ {ai, ai+1, . . . , aµ} ⊆ V \{ai}. These

observations complete the proof of ineq. (14).

We now prove ineq. (15) using the Deﬁnition 4of κf, as

follows: since κf= 1 −minv∈V f(v|V \{v})

f(v), it is implied that

for all elements v∈ V it is f(v|V \ {v})≥(1 −κf)f(v).

Therefore, adding the latter inequality across all elements a∈

Acompletes the proof of ineq. (15).

Lemma 2. Consider any ﬁnite set V, a non-decreasing and

submodular f: 2V7→ R, and non-empty sets Y,P ⊆ V such

that for all y∈ Y and all p∈ P f(y)≥f(p). Then:

f(P|Y)≤ |P |f(Y).

Proof of Lemma 2:Consider any y∈ Y (such an element

exists since Lemma 2considers that Yis non-empty); then,

f(P|Y) = f(P ∪ Y )−f(Y)(20)

≤f(P) + f(Y)−f(Y)(21)

=f(P)

≤X

p∈P

f(p)(22)

≤ |P | max

p∈P f(p)

≤ |P|f(y)(23)

≤ |P|f(Y),(24)

where eq. (20) to ineq. (24) hold for the following reasons:

eq. (20) holds since for any sets X ⊆ V and Y ⊆ V,f(X |Y ) =

f(X ∪Y)−f(Y); ineq. (21) holds since fis submodular and,

as a result, the submodularity Deﬁnition 3implies that for any

set A ⊆ V and A0⊆ V ,f(A∪A0)≤f(A)+ f(A0); ineq. (22)

holds for the same reason as ineq. (21); ineq. (23) holds since

for all elements y∈ Y and all elements p∈ P f(y)≥f(p);

ﬁnally, ineq. (24) holds because fis monotone and y∈ Y.

Lemma 3. Consider a ﬁnite set Vand a non-decreasing f:

2V7→ Rsuch that fis non-negative and f(∅)=0. For any

set A⊆V and any set B ⊆ V such that A∩B=∅:

f(A∪B)≥(1 −cf) f(A) + X

b∈B

f(b)!.

Proof of Lemma 3:Let B={b1, b2, . . . , b|B|}. Then,

f(A∪B) = f(A) +

|B|

X

i=1

f(bi|A ∪ {b1, b2, . . . , bi−1}).(25)

In addition, Deﬁnition 5of total curvature implies:

f(bi|A ∪ {b1, b2, . . . , bi−1})≥(1 −cf)f(bi|∅)

= (1 −cf)f(bi),(26)

where the latter equation holds since f(∅)=0. The proof is

completed by substituting (26) in (25) and then taking into

account that f(A)≥(1 −cf)f(A)since 0≤cf≤1.

13

Lemma 4. Consider a ﬁnite set Vand a non-decreasing f:

2V7→ Rsuch that fis non-negative and f(∅)=0. For any

A ⊆ V and any B ⊆ V such that A \ B 6=∅:

f(A) + (1 −cf)f(B)≥(1 −cf)f(A∪B) + f(A∩B).

Proof of Lemma 4:Let A\B={i1, i2, . . . , ir}, where

r=|A − B|. From Deﬁnition 5of total curvature cf, for any

i= 1,2, . . . , r, it is f(ij|A ∩ B ∪ {i1, i2, . . . , ij−1})≥(1 −

cf)f(ij|B ∪ {i1, i2, . . . , ij−1}). Summing these rinequalities,

f(A)−f(A∩B)≥(1 −cf) (f(A∪B)−f(B)) ,

which implies the lemma.

Corollary 1. Consider a ﬁnite set Vand a non-decreasing

f: 2V7→ Rsuch that fis non-negative and f(∅) = 0. For

any A ⊆ V and any B ⊆ V such that A∩B=∅:

f(A) + X

b∈B

f(b)≥(1 −cf)f(A∪B).

Proof of Corollary 1:Let B={b1, b2, . . . , b|B|}.

f(A) +

|B|

X

i=1

f(bi)≥(1 −cf)f(A) +

|B|

X

i=1

f(bi)) (27)

≥(1 −cf)f(A∪{b1}) +

|B|

X

i=2

f(bi)

≥(1 −cf)f(A∪{b1, b2}) +

|B|

X

i=3

f(bi)

.

.

.

≥(1 −cf)f(A∪B),

where (27) holds since 0≤cf≤1, and the rest due

to Lemma 4since A ∩ B =∅implies A \ {b1} 6=∅,

A∪{b1}\{b2} 6=∅,. . .,A∪{b1, b2, . . . , b|B|−1}\{b|B|} 6=∅.

APPENDIX 2. COORDINATE DE SC EN T

We describe coordinate descent [6, Section IV], and gen-

eralize the proof in [6] that coordinate descent guarantees an

approximation performance up to a multiplicative factor 1/2

the optimal when the information objective function is the

mutual information. In particular, we extend the proof to any

non-decreasing and possibly submodular information objective

function; the result will support the proof of Proposition 1.

The algorithm coordinate descent works as follows: con-

sider an arbitrary ordering of the robots in V, such that

V ≡ {1,2, . . . , n}, and suppose that robot 1ﬁrst chooses its

controls, without considering the other robots; in other words,

robot 1solves the single robot version of Problem 2,i.e.,

J{1},t+1:t+TPLAN := J(u1), to obtain controls u{1}such that:

ucd

1, t+1:t+TPLAN = arg max

ui,t0∈ Ui,t0

t0=t+1 : t+TPLAN

J(ui, t+1:t+TPLAN )

(28)

Afterwards, robot 1communicates its chosen control se-

quence to robot 2, and robot 2, given the control sequence of

robot 1.computes its control input as follows, assuming the

control inputs for robot 1 are ﬁxed:

ucd

2, t+1:t+TPLAN = arg max

ui,t0∈ Ui,t0

t0=t+1 : t+TPLAN

J(ucd

1, ui, t+1:t+TPLAN )

(29)

This continues such that robot isolves a single robot prob-

lem, given the control inputs from the robots 1,2, . . . , i −1:

ucd

i, t+1:t+TPLAN = arg max

ui,t0∈ Ui,t0

t0=t+1 : t+TPLAN

J(ucd

1:i−1, ui, t+1:t+TPLAN )

(30)

Notably, if we let u∗

ibe the control inputs for the i-th robot

resulting from the optimal solution to the nrobot problem,

then from the coordinate descent algorithm, we have:

J(ucd

1:i−1, u?

i)≤J(ucd

1:i)(31)

Lemma 5 (Approximation performance of coordinate de-

scent).Consider a set of robots V, and an instance of prob-

lem (P-RTP). Denote the optimal control inputs for problem

(P-RTP), across all robots and all times, by u?

V,t+1 : t+TPLAN .

The coordinate descent algorithm returns control inputs

ucd

V, t+1 : t+TPLAN , across all robots and all times, such that:

•if the objective function Jis non-decreasing submodular

in the active robot set, and (without loss of generality) J

is non-negative and J(∅) = 0, then:

J(ucd

1:n)

J(u?

1:n)≥1

2.(32)

•If the objective function Jis non-decreasing in the active

robot set, and (without loss of generality) Jis non-

negative and J(∅)=0, then:

J(ucd

1:n)

J(u?

1:n)≥1−cJ

2.(33)

Proof of Lemma 5:

•if the objective function Jis non-decreasing and sub-

modular in the active robot set, and (without loss of

generality) Jis non-negative and J(∅) = 0, then:

J(u?

1:n)≤J(u?

1:n) +

n

X

i=1

[J(ucd

1:i, u?

i+1:n)(34)

−J(ucd

1:i−1, u?

i+1:n)]

=J(ucd

1:n) +

n

X

i=1

[J(ucd

1:i−1, u?

i:n)(35)

−J(ucd

1:i−1, u?

i+1:n)]

=J(ucd

1:n) +

n

X

i=1

J(u?

i|{ucd

1:i−1, u?

i+1,n})(36)

≤J(ucd

1:n) +

n

X

i=1

J(u?

i|ucd

1:i−1)(37)

≤J(ucd

1:n) +

n

X

i=1

J(ucd

i|ucd

1:i−1)(38)

14

=J(ucd

1:n) + J(ucd

1:n)(39)

≤2J(ucd

1:n),(40)

where ineq. (34) holds due to monotonicity of J; eq. 35)

is a shift in indexes of the ﬁrst term in the sum; eq. (36)

is an expression of the sum as a sum of marginal

gains; ineq. (37) holds due to submodularity; ineq. (38)

holds by the coordinate-descent policy (per eq. (31));

eq. (39) holds due to the deﬁnition of the marginal

gain symbol J(u?

i|ucd

1:i−1)(for any i= 1,2, . . . , n) as

J(u?

i, ucd

1:i−1)−J(ucd

1:i−1); ﬁnally, a re-arrangement of

the terms in eq. (40) gives J(ucd

1:n)/J(u∗

1:n)≥1/2.

•If Jis non-decreasing in the active robot set, and (without

loss of generality) Jis non-negative and J(∅) = 0, then

multiplying both sides of eq. (36) (which holds for any

non-decreasing J) with (1 −cJ), we have:

(1−cJ)J(u?

1:n)

= (1 −cJ)J(ucd

1:n)+

(1 −cJ)

n

X

i=1

J(u?

i|{ucd

1:i−1, u?

i+1,n})

≤J(ucd

1:n) + (1 −cJ)

n

X

i=1

J(u?

i|{ucd

1:i−1, u?

i+1,n})

(41)

≤J(ucd

1:n) +

n

X

i=1

J(u?

i|ucd

1:i−1)(42)

≤J(ucd

1:n) +

n

X

i=1

J(ucd

i|ucd

1:i−1)(43)

=J(ucd

1:n) + J(ucd

1:n)(44)

≤2J(ucd

1:n),(45)

where, ineq. (41) holds since 0≤cJ≤1; ineq. (42) holds

since Jis non-decreasing in the set of active robots, and

Deﬁnition 5of total curvature implies that for any non-

decreasing set function g: 2V7→ R, for any element

v∈ V, and for any set A,B ⊆ V \ {v}:

(1 −cg)g(v|B)≤g({v}|A); (46)

ineq. (43) holds by the coordinate-descent algorithm;

eq. (44) holds due to the deﬁnition of the marginal

gain symbol J(u?

i|ucd

1:i−1)(for any i= 1,2, . . . , n) as

J(u?

i, ucd

1:i−1)−J(ucd

1:i−1); ﬁnally, a re-arrangement of

terms gives J(ucd

1:n)/J(u∗

1:n)≥(1 −cJ)/2.

APPENDIX 3. PROO F OF TH EO RE M 1

We ﬁrst prove Theorem 1’s part 1 (approximation perfor-

mance), and then, Theorem 1’s part 2 (running time).

A. Proof of Theorem 1’s Part 1 (Approximation Performance)

The proof follows the steps of the proof of [27, Theorem 1]

and [42, Theorem 1]. We ﬁrst prove eq. (9), then, eq. (8).

To the above ends, we use the following notation (along with

the notation introduced in Theorem 1and in Appendix A):

given that using Algorithm 2the robots in Vselect control

inputs uV,t+1:t+TPLAN , then, for notational simplicity:

•let A?,A?(uV, t+1:t+TPLAN );

•let L+,L \ A?,i.e.,S1be the remaining robots in L

after the removal of the robots in A?;

•let (V \ L)+,(V \ L)\ A?,i.e.,S2be the remaining

robots in V \ L after the removal of the robots in A?.

Proof of ineq. (9):The proof follows the steps of the

proof of [27, Theorem 1]. Consider that the objective function

Jis non-decreasing and submodular in the active robot set,

such that (without loss of generality) Jis non-negative and

J(∅)=0. We ﬁrst prove the part 1−κJof the bound in the

right-hand-side of ineq. (9), and then, the part h(|V|, α)of the

bound in the right-hand-side of ineq. (9).

To prove the part 1−κJof the bound in the right-hand-

side of ineq. (9), we follow the steps of the proof of [27,

Theorem 1], and make the following observations:

J(V \ A?)

=J(L+∪(V \ L)+)(47)

≥(1 −κJ)X

v∈L+∪(V\L)+

J(v)(48)

≥(1 −κJ)

X

v∈(V\L)\(V \L)+

J(v) + X

v∈(V\L)+

J(v)

(49)

≥(1 −κJ)J{[(V \ L)\(V \ L)+]∪(V \ L)+}(50)

= (1 −κJ)J(V \ L),(51)

where eq. (47) to (51) hold for the following reasons: eq. (47)

follows from the deﬁnitions of the sets L+and (V \ L)+;

ineq. (48) follows from ineq. (47) due to Lemma 1; ineq. (49)

follows from ineq. (48) because for all elements v∈ L+and

all elements v0∈(V \ L)\(V \ L)+we have J(v)≥J(v0)

(note that due to the deﬁnitions of the sets L+and (V \ L)+,

|L+|=|(V \ L)\(V \ L)+|, that is, the number of non-

removed elements in Lis equal to the number of removed

elements in V \ L); ﬁnally, ineq. (50) follows from ineq. (49)

because the set function Jis submodular and, as a result,

the submodularity Deﬁnition 3implies that for any sets S ⊆ V

and S0⊆ V,J(S) + J(S0)≥J(S ∪ S 0)[34, Proposition 2.1].

We now complete the proof of the part 1−κJof the bound in

the right-hand-side of ineq. (9) by proving that in ineq. (51):

J(V \ L)≥J?

,(52)

when the robots in Voptimally solve the problems in Algo-

rithm 2’s step 8, per the statement of Theorem 1. In particular,

if for any active robot set R ⊆ V, we let ¯uR,{¯ui,t0: ¯ui,t0∈

Ui,t0, i ∈ R, t0=t+ 1, . . . , t +TPLAN }denote a collection

of control inputs to the robots in R, then:

J(V \ L)≡max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

J(¯uV\L,t+1:t+TPLAN )(53)

≥min

¯

L⊆V,

|¯

L| ≤ α

max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

J(¯uV\ ¯

L,t+1:t+TPLAN )

(54)

15

≥max

¯ui,t ∈ Ui,t , i ∈ V,

t=t+ 1 : t+TPLAN

min

¯

L⊆V,

|¯

L| ≤ α

J(¯uV\ ¯

L,t+1:t+TPLAN )(55)

≡J?

,(56)

where (53)-(56) hold true since: the equivalence in eq. (53)

holds since the robots in Vsolve optimally the problems in

Algorithm 2’s step 8, per the statement of Theorem 1; (54)

holds since we minimize over the set L; (55) holds because

for any set ˆ

L⊆Vand any control inputs ˆuR,t+1:t+TPLAN ,

{ˆui,t : ˆui,t ∈ Ui,t , i ∈ R, t0=t+ 1, . . . , t +TPLAN }:

max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

J(¯uV\ ¯

L,t+1:t+TPLAN )≥J(ˆuV\ ˆ

L,t+1:t+TPLAN ),

which implies

min

¯

L⊆V,

|¯

L| ≤ α

max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

J(¯uV\ ¯

L,t+1:t+TPLAN )≥

min

¯

L⊆V,

|¯

L| ≤ α

J(ˆuV\ ¯

L,t+1:t+TPLAN )⇒

min

¯

L⊆V,

|¯

L| ≤ α

max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

J(¯uV\ ¯

L,t+1:t+TPLAN )≥

max

¯ui,t ∈ Ui,t , i ∈ V,

t0=t+ 1 : t+TPLAN

min

¯

L⊆V,

|¯

L| ≤ α

J(¯uV\ ¯

L,t+1:t+TPLAN ),

where the last one is eq. (55); ﬁnally, the equivalence in

eq. (56) holds since J?(per the statement of Theorem 1)

denotes the optimal value to Problem 2. Overall, we proved

that ineq. (56) proves ineq. (52); and, now, the combination of

ineq. (51) and ineq. (52) proves the part 1−κJof the bound

in the right-hand-side of ineq. (9).

We ﬁnally prove the part 1/(1 + α)of the bound in the

right-hand-side of ineq. (9), and complete this way the proof

of Theorem 1. To this end, we follow the steps of the proof

of [27, Theorem 1], and use the notation introduced in Fig. 8,

along with the following notation:

η,J(A?

2|V \ A?)

J(V \ L)(57)

Later in this proof, we prove 0≤η≤1. We ﬁrst observe that:

J(V \ A?)≥max{J(V \ A?), J (L+)};(58)

in the following paragraphs, we prove the three inequalities:

J(V \ A?)≥(1 −η)J(V \ L),(59)

J(L+)≥η1

αJ(V \ L),(60)

max{(1 −η), η 1

α} ≥ 1

α+ 1.(61)

Then, if we substitute ineq. (59), ineq. (60) and ineq. (61) to

ineq. (58), and take into account that J(V \ L)≥0, then:

J(V \ A?)≥1

α+ 1J(V \ L),

which implies the part 1/(1 + α)of the bound in the right-

hand-side of ineq. (9), after taking into account ineq. (52).

We next complete the proof of the part 1/(1 + α)of the

bound in the right-hand-side of ineq. (9) by proving 0≤η≤1,

V

LA?

1A?

2

Fig. 8: Venn diagram, where the set Lis the robot set deﬁned in

step 4of Algorithm 2, and the set A?

1and the set A?

2are such that

A?

1=A?∩L, and A?

2=A?∩(V \ L)(observe that these deﬁnitions

imply A?

1∩ A?

2=∅and A?=A?

1∪ A?

2).

ineq. (59), ineq. (60), and ineq. (61).

a) Proof of ineq. 0≤η≤1:We ﬁrst prove η≥0,

and then η≤1:η≥0, since η≡J(A?

2|V \ A?)/J(V \ L),

and Jis non-negative; and η≤1, since J(V \ L)≥J(A?

2),

due to monotonicity of Jand that A?

2⊆ V \ L, and J(A?

2)≥

J(A?

2|V \A?), due to submodularity of Jand that ∅ ⊆ V \ A?.

b) Proof of ineq. (59):We complete the proof of

ineq. (59) in two steps. First, it can be veriﬁed that:

f(V \ A?) = f(V \ L)−

J(A?

2|V \ A?) + J(L|V \ L)−J(A?

1|V \ A?

1),(62)

since for any sets X ⊆ V and Y ⊆ V ,J(X |Y )≡

J(X ∪ Y )−J(Y). Second, eq. (62) implies ineq. (59), since

J(A?

2|V \ A?) = ηJ(V \ L), and J(L|V \ L)−J(A?

1|V \

A?

1)≥0; the latter is true due to the following two obser-

vations: J(L|V \ L)≥J(A?

1|V \ L), since Jis monotone

and A?

1⊆ L; and J(A?

1|V \ L)≥J(A?

1|V \ A?

1), since Jis

submodular and V \ L ⊆ V \ A?

1(see also Fig. 8).

c) Proof of ineq. (60):To prove ineq. (60), since A?

26=∅

(and, as a result, also L+6=∅), and for all elements a∈ L+

and all elements b∈ A?

2,J(a)≥J(b), from Lemma 2we

have:

J(A?

2|L+)≤ |A