Multi-Robot Patrolling with Sensing Idleness and Data Delay Objectives

Abstract

Multi-robot patrolling represents a fundamental problem for many monitoring and surveillance applications and has gained significant interest in recent years. In patrolling, mobile robots repeatedly travel through an environment, capture sensor data at certain sensing locations and deliver this data to the base station in a way that maximizes the changes of detection. Robots move on tours, exchange data when they meet with robots on neighboring tours and so eventually deliver data to the base station. In this paper we jointly consider two important optimization criteria of multi-robot patrolling: (i) idleness, i.e. the time between consecutive visits of sensing locations, and (ii) delay, i.e. the time between capturing data at the sensing location and its arrival at the base station. We systematically investigate the effect of the robots’ moving directions along their tours and the selection of meeting points for data exchange. We prove that the problem of determining the movement directions and meeting points such that the data delay is minimized is NP-hard. For this purpose, we define a structure called tour graph which models the neighborhood of the tours defined by potential meeting points. We propose two heuristics that are based on a shortest-path-search in the tour graph. We provide a simulation study which shows that the cooperative approach can outperform an uncooperative approach where every robot delivers the captured data individually to the base station. Additionally, the experiments show that the heuristic which is computational more expensive performs slightly better on average than the less expensive heuristic in the considered scenarios.

Appendices

Appendix A: MILP formulation of MDTD

The mixed integer linear programming (MILP) model of MDTD is based on a multi-commodity flow formulation for trees on a graph G = (V, A) with n + 1 vertices V (including a vertex 0 for a virtual base station tour) and arc set A [22]. The base station is the source of a commodity flow $f^c_e$ for each vertex (constraint (8)). A flow of commodity c represents the path of the data from robot c towards the base station (though the flow originates at the base station in this formulation). For each vertex the sum of incoming flows is equal to the sum of outgoing flows for each commodity not dedicated to that vertex (constraint (9)), and each vertex c consumes the commodity of type c (constraint (10)). There can be only a flow on an edge if this edge is selected in the tree (constraint (11)) and the sum of the edges must be n (constraint (12)).

$$ \begin{array}{@{}rcl@{}} \sum\limits_{(v,0) \in A}{f_{v0}^{c}} - \sum\limits_{(0,v) \in A}{f_{0v}^{c}} &=& -1 \quad \forall c \in V \setminus \{0\} \end{array} $$

(8)

$$ \begin{array}{@{}rcl@{}} \sum\limits_{(w,v) \in A}{f_{wv}^{c}} - \sum\limits_{(v,w) \in A}{f_{vw}^{c}} &=& 0 \quad \forall v \in V \setminus \{0, c\}, \forall c \in V \end{array} $$

(9)

$$ \begin{array}{@{}rcl@{}} \sum\limits_{(w,c) \in A}{f_{wc}^{c}} - \sum\limits_{(c,w) \in A}{f_{cw}^{c}} &=& 1 \quad \forall c \in V \setminus \{0\} \end{array} $$

(10)

$$ \begin{array}{@{}rcl@{}} {f_{e}^{c}} &\leq& x_{e} \quad \forall e \in A, \forall c \in V \setminus \{0\} \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} \sum\limits_{e \in A}{x_{e}} &=& n \end{array} $$

(12)

$$ \begin{array}{@{}rcl@{}} x_{e} &\in& \{0, 1\} \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} {f_{e}^{c}} &\geq& 0 \quad \forall e \in A, \forall c \in V \setminus \{0\} \end{array} $$

(14)

The data which robot j gets at the meeting point between i and j and is forwarded at meeting point between j and k has to travel the distance $l_{ik}^{j,ccw}$ or $l_{ik}^{j,cw}$ on tour j, depending on the direction robot j traverses its tour. Therefore, two flow variables $f_{ij}^c$ and $f_{jk}^c$ are involved in the cost calculation in constraint (15) for data originating from c and traversing the tour j. The separation of the flows in this formulation allows the definition of a min-max objective. For each commodity c, z_c models the delay of data originating at robot c and the objective is to minimize z. The decision variables $u_j^{ccw}$ and $u_j^{cw}$ determine the direction robot j traverses its tour.

$$ \begin{array}{@{}rcl@{}} z_{c} &=& u_{j}^{ccw} \sum\limits_{(j,c)\in A}{f_{jc}^{c}(l_{c}-{l_{c}^{S}}(p_{c}^{meet}(j), ccw))} + \\ && u_{j}^{cw} \sum\limits_{(j,c)\in A}{f_{jc}^{c}(l_{c}-{l_{c}^{S}}(p_{c}^{meet}(j), cw))} + \\ && \sum\limits_{(i,j),(j,k)\in A}{f_{ij}^{c} f_{jk}^{c} u_{j}^{ccw} l_{ik}^{j,ccw}} + f_{ij}^{c} f_{jk}^{c} u_{j}^{cw} l_{ik}^{j,cw} \quad \forall c \in V \setminus \{0\} \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} z_{c} &\leq& z \quad \forall c \in V \setminus \{0\} \end{array} $$

(16)

$$ \begin{array}{@{}rcl@{}} u_{j}^{ccw} + u_{j}^{cw} &=& 1 \quad \forall j \in V \setminus \{0\} \end{array} $$

(17)

$$ \begin{array}{@{}rcl@{}} u_{j}^{ccw}, u_{j}^{cw} &\in& \{0, 1\} \quad \forall j \in V \setminus \{0\} \end{array} $$

(18)

The products, e.g. $f_{ij}^c f_{jk}^c u_j^{ccw}$, can be linearized (likewise $f_{ij}^c f_{jk}^c u_j^{cw}$) with an additional variable $f_{ijk}^{c, ccw}$ and the constraints:

$$ \begin{array}{@{}rcl@{}} f_{ijk}^{c, ccw} &\leq& f_{ij}^{c} \end{array} $$

(19)

$$ \begin{array}{@{}rcl@{}} f_{ijk}^{c, ccw} &\leq& f_{jk}^{c} \end{array} $$

(20)

$$ \begin{array}{@{}rcl@{}} f_{ijk}^{c, ccw} &\leq& u_{j}^{ccw} \end{array} $$

(21)

$$ \begin{array}{@{}rcl@{}} f_{ijk}^{c, ccw} &\geq& f_{ij}^{c} + f_{jk}^{c} + u_{j}^{ccw} - 2 \end{array} $$

(22)

Appendix B: List of symbols

Symbol :

Meaning

X :

set of points of environment

P ^S :

sensing locations

Y :

communication relation

R = {1,…,n}:

set of n robots/tours

π ∈π:

patrolling strategy/schedule (from the set of all strategies π)

π ⁺ :

repeated schedule (repetition of schedule π)

$\mathbb {R}_{\geq 0}$ :

set of real numbers larger or equal 0

$I_t^{\pi }(x)$ :

instantaneous idleness of x at time t (using π)

$D_t^{\pi }(x, t^{\prime }, t^{\prime \prime })$ :

instantaneous delay of x at time t (using π)

$WI_t^{\pi }(x)$ :

instantaneous worst idleness at time t (using π)

$WD_t^{\pi }(x)$ :

instantaneous worst delay at time t (using π)

WI, WD:

worst idleness, worst delay

G = (V, E):

(tour) graph with vertex set V and edge set E

G = (V, A):

(tour) graph with vertex set V and arc set A

T = (V, E):

(tour) tree with vertex set V and edge set E

[v, w] ∈ E:

(undirected) edge between v and w

(v, w) ∈ A:

(directed) arc from v to w

$G^{\prime }=(V^{\prime },E^{\prime },W)$ :

converted graph of tour graph G

v _{k l} :

vertex of converted tour graph

v₀ or 0:

base station

l _v :

minimum traversal time (without stops) of tour v

$l_v^S$ :

minimum time any sensing location can be reached from a point on tour v

L :

$\max \limits _{v\in V}\{l_v\}$

d _v :

direction (cw or ccw) robot v traverses its tour (cw or ccw)

p_r(t):

position of robot r at time t

time_v(p, q, d):

minimum travel time on tour v from point p to point q with direction d

$p_v^{start}$ :

start position of robot v on its tour v

$p_v^{meet}(w)$ :

meeting point of robot v on tour v with robot w

wait_v(p):

waiting time for robot v on meeting point p

Δ_wv:

time difference between the starts of robots v and w at their start positions, $wait_w(p_w^{start})-wait_v(p_{v}^{start})$

P(v):

set of all positions on tour v

τ _v :

traversal time plus waiting times on tour v

dist_G(s, d):

length of shortest path between vertices s and d in (weighted) graph G