Cooperative Task AssignmentDonato Di Paola

Institute of Intelligent Systems for Automation National Research Council

Talk Outline

Talk Outline

Cooperative Mission Planning and Task Assignment

The Multi-Robot Task Assignment Problem

Decentralized Task Assignment Algorithms

Task Allocation in Complex Scenarios

Cooperative Mission Planning and Task Assignment

The big picture Cooperative Multi-Robot Systems

Heterogeneous robots

Local sensing

Communication Network

The big picture Cooperative Multi-Robot Systems

Heterogeneous robots

Local sensing

Communication Network

Achieve Global Objectives

The role of task assignment Mission Control Architecture

Planning Task Assignment

Task Execution

Mission Actions

The role of task assignment Mission Control Architecture

Planning Task Assignment

Task Execution

Mission ActionsTasks

Robots

The Multi-Robot Task Assignment Problem

The Task Assignment Problem

J = {1, . . . ,m}

Given a set of robot

Given a set of tasks

David W. Pentico, Assignment problems: A golden anniversary survey, European Journal of Operational Research, Volume 176, Issue 2, 16 January 2007, Pages 774-793

The Task Assignment Problem

J = {1, . . . ,m}

Given a set of robot

Given a set of tasks

David W. Pentico, Assignment problems: A golden anniversary survey, European Journal of Operational Research, Volume 176, Issue 2, 16 January 2007, Pages 774-793

The Task Assignment Problem

J = {1, . . . ,m}

Given a set of robot

Given a set of tasks

GoalFind a conflict free assignment of tasks to robots, such that the global cost (score) is minimized (maximized)

David W. Pentico, Assignment problems: A golden anniversary survey, European Journal of Operational Research, Volume 176, Issue 2, 16 January 2007, Pages 774-793

Taxonomy of Task Assignment problems

G. A. Korsah, A. Stentz, and M. B. Dias. A comprehensive taxonomy for multi-robot task allocation. International Journal of Robotics Research, 32(12):1495–1512, 2013.

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

Robot capacity constraint

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

Robot capacity constraint

Conflict-free constraint

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

Robot capacity constraint

Conflict-free constraint

Assignment variables

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

8i 2 I

8j 2 J

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

8i 2 I

8j 2 J

Cost Matrix Solution: minimized global cost

Example

Mathematical formulation for MR-ST-IA

Objective Function

Assignment Constraints

8i 2 I

8j 2 J

Cost Matrix Solution: minimized global cost

Example

Optimal solutiont1 assigned to r2 t2 assigned to r3 t3 assigned to r1

Optimal valueCost = 13

Mathematical formulation for MR-ST-TA

Objective Function

Assignment ConstraintsmX

j=1

xij Lt 8i 2 I

nX

i=1

xij 1 8j 2 J

Mathematical formulation for MR-ST-TA

Objective Function

Assignment ConstraintsmX

j=1

xij Lt 8i 2 I

nX

i=1

xij 1 8j 2 J

Mathematical formulation for MR-ST-TA

Objective Function

Assignment ConstraintsmX

j=1

xij Lt 8i 2 I

nX

i=1

xij 1 8j 2 J

Centralized Approach

Centralized Vs Decentralized approaches

Pros

Centralized Approach

Centralized Vs Decentralized approaches

Optimal SolutionWell-studied problems

ConsPros

Centralized Approach

Centralized Vs Decentralized approaches

Optimal SolutionWell-studied problems

Not robust against failures

ConsPros

Centralized Approach

Centralized Vs Decentralized approaches

Decentralized Approach

Optimal SolutionWell-studied problems

Not robust against failures

Robust to failuresScale with the size of the robotic network

ConsPros

Centralized Approach

Centralized Vs Decentralized approaches

Decentralized Approach

Optimal SolutionWell-studied problems

Not robust against failures

Robust to failuresScale with the size of the robotic network

Sub-Optimal solution

Decentralized Task Assignment Algorithms

Consensus Based Bundle Algorithm

H.-L. Choi, L. Brunet, J. P. How, Consensus-Based Decentralized Auctions for Robust Task Allocation, IEEE Transactions on Robotics, Vol. 25, No. 4, pp. 912-926, August 2009

Decentralized Task Assignment The CBBA

Consensus Based Bundle Algorithm

It solves MR-ST-TA problems

Decentralized algorithm (leverages the consensus theory)

Iterative algorithm (known number of iteration for convergence)

Performance guarantee

H.-L. Choi, L. Brunet, J. P. How, Consensus-Based Decentralized Auctions for Robust Task Allocation, IEEE Transactions on Robotics, Vol. 25, No. 4, pp. 912-926, August 2009

Synchronized communication

Decentralized Task Assignment The CBBA

CBBA: The two phases approach

Each robot does

CBBA: The two phases approach

Bundle Construction Phasethe robot assigns to itself the tasks with the maximum score, until the bundle if full

Each robot does

CBBA: The two phases approach

Bundle Construction Phasethe robot assigns to itself the tasks with the maximum score, until the bundle if full

Each robot does

Communicationthe robot exchanges information with its neighbors

CBBA: The two phases approach

Bundle Construction Phasethe robot assigns to itself the tasks with the maximum score, until the bundle if full

Each robot does

Communicationthe robot exchanges information with its neighbors

Conflict Resolution PhaseThe assigned tasks are already assigned to other robots?

if the previous check is true

the robot maintains the tasks with the higher score

CBBA: The two phases approach

Bundle Construction Phasethe robot assigns to itself the tasks with the maximum score, until the bundle if full

Each robot does

Communicationthe robot exchanges information with its neighbors

Conflict Resolution PhaseThe assigned tasks are already assigned to other robots?

if the previous check is true

the robot maintains the tasks with the higher score

R1

R2

R3

T1 T2 T3 T4 T5

Bundle Construction PhaseIteration 1

CBBA: An example

R1

R2

R3

T1 T3

T1 T2 T3 T4 T5

Bundle Construction PhaseIteration 1

CBBA: An example

R1

R2

R3

T1 T3

T1 T2

T1 T2 T3 T4 T5

Bundle Construction PhaseIteration 1

CBBA: An example

R1

R2

R3

T1 T3

T1 T2

T3 T5

T1 T2 T3 T4 T5

Bundle Construction PhaseIteration 1

Conflict !!!One or more tasks (T1 and T3) are assigned to more than one robot

CBBA: An example

R1

R2

R3

T1 T3

T1 T2

T3 T5

T1 T2 T3 T4 T5

Conflict Resolution PhaseIteration 1

CBBA: An example

COMMUNICATION

R1

R2

R3

T1 T3

T1 T2

T3 T5

T1 T2 T3 T4 T5

Conflict Resolution PhaseIteration 1

CBBA: An example

COMMUNICATION

R1

R2

R3

T1 T3

T1 T2

T3 T5

T1 T2 T3 T4 T5

Conflict Resolution PhaseIteration 1

CBBA: An example

Conflict Removal

T1: score of R1 > score of R2 T3: score of R1 > score of R3

COMMUNICATION

R1

R2

R3

T1 T3

T3 T5

T1 T2 T3 T4 T5

Conflict Resolution PhaseIteration 1

CBBA: An example

Conflict Removal

T1: score of R1 > score of R2 T3: score of R1 > score of R3

COMMUNICATION

R1

R2

R3

T1 T3

T1 T2 T3 T4 T5

Conflict Resolution PhaseIteration 1

CBBA: An example

Conflict Removal

T1: score of R1 > score of R2 T3: score of R1 > score of R3

R1

R2

R3

T1 T3

T1 T2 T3 T4 T5

CBBA: An example

Bundle Construction PhaseIteration 2

R1

R2

R3

T1 T3

T2 T5

T1 T2 T3 T4 T5

CBBA: An example

Bundle Construction PhaseIteration 2

R1

R2

R3

T1 T3

T2 T5

T4 T2

T1 T2 T3 T4 T5

CBBA: An example

Bundle Construction PhaseIteration 2

Conflict !!!One or more tasks (T2) are assigned to more than one robot

R1

R2

R3

T1 T3

T2 T5

T4 T2

T1 T2 T3 T4 T5

CBBA: An example Iteration 2Conflict Resolution Phase

COMMUNICATION

R1

R2

R3

T1 T3

T2 T5

T4 T2

T1 T2 T3 T4 T5

CBBA: An example Iteration 2Conflict Resolution Phase

Conflict Removal

T2: score of R2 > score of R2

COMMUNICATION

R1

R2

R3

T1 T3

T2 T5

T4

T1 T2 T3 T4 T5

CBBA: An example Iteration 2Conflict Resolution Phase

Conflict Removal

T2: score of R2 > score of R2

R1

R2

R3

T1 T3

T2 T5

T4

T1 T2 T3 T4 T5

CBBA: An example Iteration 2Conflict Resolution Phase

R1

R2

R3

T1 T3

T2 T5

T4

T1 T2 T3 T4 T5

CBBA: An example Iteration 2Conflict Resolution Phase

Assignment CompletedAll the tasks are assignedNo conflictsMaximization of score

Task Assignment in Complex Scenarios

Problem Setting

Open challenges

Multi Robot Transportation Problems with Capacity

Task Assignment

Multi-Robot Routing

Balancing of vehicle within the given area

Heterogeneous robots

Each robot picks up a task in a location

Load capacity constraint

Distributed coverage of a given area

Multi Robot Transportation Problems with Capacity Problem Setting

Multi Robot Transportation Problems with Capacity Problem Setting

Maximize a Network Score

subject to

Multi Robot Transportation Problems with Capacity Problem Setting

Maximize a Network Score

subject to

Routing Constraints }

Multi Robot Transportation Problems with Capacity Problem Setting

Maximize a Network Score

subject to

Routing Constraints

Assignment constraints

}}

Multi Robot Transportation Problems with Capacity Problem Setting

Maximize a Network Score

subject to

Routing Constraints

Assignment constraints

Balancing constraints

}}}

Multi Robot Transportation Problems with Capacity Problem Setting

Maximize a Network Score

subject to

Routing Constraints

Assignment constraints

Balancing constraints

}}}

NP-Hard

Decentralized Assignment with Load Balancing 2 Phases Iterative Algorithm

A. Acquaviva, D. Di Paola, A. Rizzo, Decentralized Optmization-based Load Balancing in Mobility On-Demand Systems, in preparation

Each robot does

Decentralized Assignment with Load Balancing 2 Phases Iterative Algorithm

Auction Phasethe robot assigns to itself the pair (task,route) with the max score

A. Acquaviva, D. Di Paola, A. Rizzo, Decentralized Optmization-based Load Balancing in Mobility On-Demand Systems, in preparation

Each robot does

Decentralized Assignment with Load Balancing 2 Phases Iterative Algorithm

Auction Phasethe robot assigns to itself the pair (task,route) with the max score

A. Acquaviva, D. Di Paola, A. Rizzo, Decentralized Optmization-based Load Balancing in Mobility On-Demand Systems, in preparation

Each robot does

Communicationthe robot exchanges information with its neighbors

Decentralized Assignment with Load Balancing 2 Phases Iterative Algorithm

Auction Phasethe robot assigns to itself the pair (task,route) with the max score

A. Acquaviva, D. Di Paola, A. Rizzo, Decentralized Optmization-based Load Balancing in Mobility On-Demand Systems, in preparation

Each robot does

Communicationthe robot exchanges information with its neighbors

Balancing PhaseThe starting or the destination area is balanced?

if the previous check is false and the robot is one of those exceeding the area capacity

the robot drops the assignment

Decentralized Assignment with Load Balancing 2 Phases Iterative Algorithm

Auction Phasethe robot assigns to itself the pair (task,route) with the max score

A. Acquaviva, D. Di Paola, A. Rizzo, Decentralized Optmization-based Load Balancing in Mobility On-Demand Systems, in preparation

Each robot does

Communicationthe robot exchanges information with its neighbors

Balancing PhaseThe starting or the destination area is balanced?

if the previous check is false and the robot is one of those exceeding the area capacity

the robot drops the assignment

Results Simulation Setup

Simulated City from citibike web site

Real Statistics

Subset of stations of the NY citibike bike sharing system

Rate of arrival of customers

Average number of vehicle at a given station

Average travel time for a given pair of stations

Results

Optimality (6 stations and 30 vehicles)

Convergence 10 stations and 50 vehicles

Always DLB > 70% OPT

20 30 40 50 60 70 80 905000

6000

7000

8000

9000

10000

11000

12000

13000

Number of Customers

Scor

e

0 50 100 1500

50

100

150

200

250

300

Number of Customers

Num

ber o

f Ite

ratio

ns

OPT

DLB

70%OPT

Conclusions

Conclusions Solving Task Assignment in Multi-Robot Systems

Conclusions Solving Task Assignment in Multi-Robot Systems

Composed by 2 or more phases

Hierarchical Iterative Algorithms

Local decision/evaluation

Decentralized Auction/ElectionNearest neighbors communication

Conclusions Solving Task Assignment in Multi-Robot Systems

Composed by 2 or more phases

Hierarchical Iterative Algorithms

Local decision/evaluation

Decentralized Auction/ElectionNearest neighbors communication

Communications with time delays

Open Issues

Protocols for secure communications

Asynchronous communications

Malicious / Misbehaving robot

Cooperative Task AssignmentDonato Di Paola

Institute of Intelligent Systems for Automation National Research Council