Multi-Robot Task Scheduling

Multi-Robot Task Scheduling

Yu ("Tony") Zhang Lynne E. Parker

Distributed Intelligence Laboratory

Electrical Engineering and Computer Science Department University of Tennessee, Knoxville TN, USA

IEEE International Conference on Robotics and Automation, 2013

Multi-robot task scheduling

Individual robots may not have all the required capabilities

==⇒

Scheduling:

A set of robots,R={r1, ...,ri, ...} A set of tasks,T ={t1, ...,tl, ...}

Multi-robot task scheduling

To represent a general scheduling problem:P|T|func

Multi-robot task scheduling:

P→Multi-purpose processor T →Multi-processor task Restrictions:

Execution is non-preemptive Robots are non-divisible

Complexity of

MPM MPT

Two types of multi-robot tasks:

Loosely coupled: reducible to single robot tasks (MPM MPT becomesMPM)

Scheduling for tightly coupled multi-robot tasks

Steps:

1 ReduceMPM MPT toMPM 2 Solve theMPMproblem

When considering acoalitionas arobot,MPM MPT becomesMPM.

However, coalitions caninterferewith each other:

Coalition 1:{r1,r4,r5} Coalition 2:{r4,r6}

Contributions

Considers the scheduling problem for multi-robot tasks at the

coalition level

Proposes fourefficientheuristics to address the problem with

provable solution bounds

Providesformal analysesandsimulation resultsto demonstrate and compare their performances

Notations

Table:NOTATIONS USED

R Set of robots ri

C Set of coalitions cj

T Set of tasks tl

pjl Processing time oftl bycj el End time of tasktl

We considerfunc=P

MinProcTime

Definition (MinProcTime)

At each step:

1 _{Find the assignment that has the smallestpjl} 2 _{Schedule the task at the earliest possible time}

Theorem

The MinProcTime heuristic yields a solution quality bounded by|T|₂+1.

MinStepSum

Definition (MinStepSum)

At each step:

1 _{Find the assignment that increases}P

lel the least

Theorem

The MinStepSum heuristic yields a solution quality bounded by |T|₂+1.

InterfereAssign

To consider the interference between coalitions:

Definition (Coalition Interference)

For any two coalitionscj andcj0 (j6=j0),c_j interferes (or conflicts) with

cj0 if and only ifc_j∩c_j0 6=∅.

Consider the impact of an assignmentcj →tl onPlel:

1 _{The assignment’s processing timep} jl 2 _{Tasks that are scheduled onc}

j aftertl

3 _{Tasks scheduled on coalitions that interfere withc} j

InterfereAssign

Forcj →tl:

1 _{The assignment’s processing timep} jl 2 _{Tasks that are scheduled onc}

j aftertl

Together, contributeIjl·pjl

Ijl: scheduling position fortl oncj

For example: c2:t2⇒t1⇒t3

InterfereAssign

Forcj →tl:

3 _{Tasks scheduled on coalitions that influence withc} j

Upper bound is| ∪c∈FjNc| ·pjl

Fj: coalitions that interfere withcj Nc: set of tasks thatccan accomplish

InterfereAssign

ConvertMPM MPTtoMPMby constructing an assignment problem:

Create a task node for each tasktl

Create a coalition-position node for each coalitioncj and position pair, with positions ranging from 1 toNcjfor coalitioncj

If a coalitioncj can accomplish a tasktl, connecttl with all coalition-position nodes forcj, and set the weights to be (| ∪c∈FjNc|+Ijl)·pjl, respectively, based onIjl

InterfereAssign

Lemma

There exists a schedule that is no worse than the solution of the assignment problem.

Theorem

The schedule that is constructed from the solution of the assignment problem yields a solution quality bounded bymaxj| ∪c∈Fj Nc|+1.

Quality dependent oncomplex structureof the problem instance Less coalition interference, better quality

MinInterfere

jl and choose the assignment that minimizes it:

β_jl=ejl+| ∪c∈FjNc\Mjl| ·pjl

Summary

Table:SUMMARY OF DISCUSSED HEURISTICS

Name Solution Bound Complexity

Optimal 1 O((|C||T|)|T||T|!)

MinProcTime |T|₂+1 O(|C||T|3₎ MinStepTime |T|₂+1 O(|C||T|3₎ InterfereAssign maxj| ∪c∈FjNc|+1 O(|C|

3_|T|3₎ MinInterfere Not Determined O(|C||T|3₎

A simple scenario

Task Robots Required Process Time 1) Object 1 One gripper, one localizer 6

2) Object 2 One gripper, one localizer 6

A simple scenario

Figure:Schedules created by our heuristics

Parameters

Table: PARAMETERS USED IN THE SIMULATIONS

Parameter Description

nc No. of coalitions

nt No. of tasks

nf Average no. of conflicting coalitions per coalition ne Average no. of executable tasks per coalition nmin,nmax Minimum and maximum processing time

Varying

n

The average solution quality is better than the proven bounds

Varying

n

Similar observations

Varying

n

Varying

n

Varying

n

, with large

n

and

n

Time analysis. Left: Varying

n

. Right: Varying

n

Conclusions

When there is less interference between coalitions, use InterfereAssign

Contributions

Considers the scheduling problem for multi-robot tasks at the

coalition level

Proposes fourefficientheuristics to address the problem with

provable solution bounds

Providesformal analysesandsimulation resultsto demonstrate and compare their performances

References

Gerkey, B. and Mataric, M. (2004).

A formal analysis and taxonomy of task allocation in multi-robot systems.

The International Journal of Robotics Research, 23(9):939–954.

Zhang, Y. and Parker, L. (2010).

IQ-ASyMTRe: Synthesizing coalition formation and execution for tightly-coupled multirobot tasks.