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Emerging applications of collaborative autonomy, such as Multi-Target Tracking, Unknown Map Exploration, and Persistent Surveillance, require robots to 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.
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1
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 RAINs 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
2
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. RTPs 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 RTPs 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 (τ1, while τ00);
wN(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,t1, ui,t1), 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(yt1) + wt, t = 1,2,..., (2)
where ytdenotes the target’s state at time t, and wtdenotes
gaussian process noise; we consider wtN(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.
3
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). RTPs pseudo-code
is presented in Algorithm 2.RTPs performance is quantified
in Section V. We next give an intuitive description of RTP.
Intuitive Description. RTPs 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. RTPs 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 (RTPs 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 (RTPs 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 RTPs 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, 0J?
{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 Vs 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 Vs 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, vs 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,1min
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 Vs 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, hs
total curvature is quantified as
ch,1min
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 1κJ·,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 1s
assumptions). More broadly, when κJ·,t+1:t+TPLAN <1or
cJ·,t+1:t+TPLAN <1,RTPs 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 1cJ·,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(1cJ·,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 RTPs approximation per-
formance when the robots in Vsolve optimally the (attack-
free) information acquisition problems in RTPs 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 RTPs performance when the robots in Vsolve the
problems in RTPs 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 1κJ·,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 RTPs 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 RTPs 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 RAINs 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 RTPs line 2, line 6, and line 8): In the multi-
robot case (pertained to RTPs 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 RTPs 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 RAINs 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, wtN0, 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, vtN(0, V (xt, yt,m));
h(x, ym) = r(x, ym)
α(x, ym),p(y1x1)2+ (y2x2)2
tan1((y2x2)(y1x1)) 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)(y1x1) (y2x2) 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), RAINs
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 RTPs 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(z0, σ2), d < zmin;
N(zzmax, σ2), d > zmax;
N(zd, 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 RTPs
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, wtN(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, vtN(0, V (xt, yt,m ));
h(x, ym) = r(x, ym),v
u
u
t
3
X
i=1
(y1x1)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 RAINs 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, . . . , bi1}).(25)
In addition, Definition 5of total curvature implies:
f(bi|A ∪ {b1, b2, . . . , bi1})(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 0cf1.
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, . . . , ij1})(1
cf)f(ij|B ∪ {i1, i2, . . . , ij1}). 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 0cf1, 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:i1, 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:i1, 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)1cJ
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:i1, u?
i+1:n)]
=J(ucd
1:n) +
n
X
i=1
[J(ucd
1:i1, u?
i:n)(35)
J(ucd
1:i1, u?
i+1:n)]
=J(ucd
1:n) +
n
X
i=1
J(u?
i|{ucd
1:i1, u?
i+1,n})(36)
J(ucd
1:n) +
n
X
i=1
J(u?
i|ucd
1:i1)(37)
J(ucd
1:n) +
n
X
i=1
J(ucd
i|ucd
1:i1)(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:i1)(for any i= 1,2, . . . , n) as
J(u?
i, ucd
1:i1)J(ucd
1:i1); 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:
(1cJ)J(u?
1:n)
= (1 cJ)J(ucd
1:n)+
(1 cJ)
n
X
i=1
J(u?
i|{ucd
1:i1, u?
i+1,n})
J(ucd
1:n) + (1 cJ)
n
X
i=1
J(u?
i|{ucd
1:i1, u?
i+1,n})
(41)
J(ucd
1:n) +
n
X
i=1
J(u?
i|ucd
1:i1)(42)
J(ucd
1:n) +
n
X
i=1
J(ucd
i|ucd
1:i1)(43)
=J(ucd
1:n) + J(ucd
1:n)(44)
2J(ucd
1:n),(45)
where, ineq. (41) holds since 0cJ1; 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:i1)(for any i= 1,2, . . . , n) as
J(u?
i, ucd
1:i1)J(ucd
1:i1); 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 1κJof 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 1κJof 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 1κJof 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
JuV\L,t+1:t+TPLAN )(53)
min
¯
L⊆V,
|¯
L| ≤ α
max
¯ui,t ∈ Ui,t , i ∈ V,
t0=t+ 1 : t+TPLAN
JuV\ ¯
L,t+1:t+TPLAN )
(54)
15
max
¯ui,t ∈ Ui,t , i ∈ V,
t=t+ 1 : t+TPLAN
min
¯
L⊆V,
|¯
L| ≤ α
JuV\ ¯
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
JuV\ ¯
L,t+1:t+TPLAN )JuV\ 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
JuV\ ¯
L,t+1:t+TPLAN )
min
¯
L⊆V,
|¯
L| ≤ α
JuV\ ¯
L,t+1:t+TPLAN )
min
¯
L⊆V,
|¯
L| ≤ α
max
¯ui,t ∈ Ui,t , i ∈ V,
t0=t+ 1 : t+TPLAN
JuV\ ¯
L,t+1:t+TPLAN )
max
¯ui,t ∈ Ui,t , i ∈ V,
t0=t+ 1 : t+TPLAN
min
¯
L⊆V,
|¯
L| ≤ α
JuV\ ¯
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 1κJof 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 0η1. 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 0η1,
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. 0η1:We first prove η0,
and then η1:η0, since ηJ(A?
2|V \ A?)/J(V \ L),
and Jis non-negative; and η1, 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?
2|J(L+)
αJ(L+),(63)
since |A?
2| ≤ α. Overall,
J(L+)1
αJ(A?
2|L+)(64)
1
αJ(A?
2|L+(V \ L)+)(65)
=1
αJ(A?
2|V \ A?)(66)
=η1
αJ(V \ L),(67)
where ineq. (64) to eq. (67) hold for the following reasons:
ineq. (64) follows from ineq. (63); ineq. (65) holds since J
is submodular and L+⊆ L+(V \ L)+; eq. (66) holds due
to the definitions of the sets L+,(V \ L)+and A?; finally,
eq. (67) holds due to the definition of η. Overall, the latter
derivation concludes the proof of ineq. (60).
d) Proof of ineq. (61):Let b= 1. We complete the
proof first for the case where (1 η)ηb, and then for the