Real Time Adaptive Scheduling for a Swarm ... - Platform Lab · Feb 7, 2019 Platform Annual Review 4 Fully distributed Local information Limited communication Reactive control and

Real Time Adaptive Scheduling for a Swarm of Manufacturing Robots

Mac SchwagerAssistant Professor

Aeronautics and AstronauticsStanford University

Feb 7, 20191

Platform Annual Review

MSL

Flexible Manufacturing

Feb 7, 2019 Platform Annual Review 2

Credit: ECM

Credit: Martin Sehr, Siemens Credit: Martin Sehr, Siemens

Flexible Manufacturing Scenario


A

B

C

DProcess Stations

C, D, AQueues

A, B

C, A, C

A, B, C D, C, A

B, A

Out Bin

Robots

Data Center

How to coordinate, control, and plan for large teams of manufacturing robots in dynamic, stochastic

environments?


Fully distributedLocal informationLimited communicationReactive control and sensing

Fully centralizedGlobal information

Communication intensiveOptimal planning and scheduling

Credit: Business-opportunities.bizCredit: Farelli.info

Our Approach


Individual robots Centralized planner

Local-Global Map

Local sensor information

Global-Local Online replanning

Global state

• High level assignment scheduling

• Low-level path planning

• Local-global map






Assignment Scheduling


Patrick Washington

Widget State MachineAlgorithm 2 Widget j State Machine

Inputs:

Oj , Tj , ⌧dj (0)

Initialize:

sj 1⌧dj ⌧dj (0)

if rj = 0 and ai = j and ⌧i = 0 and ⌧ sj 0 then . Pick Uprj isj sj + 1

end if

if rj > 0 and ⌧rj = 0 then . Drop O↵rj 0⌧ sj Tj(sj)

end if

if sj = length(Oj) and ⌧rj = 0 then . Widget FinishedFinished

end if

⌧ sj �1⌧dj �1

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Algorithm 1 Robot i State Machine

Inputs:

bmax

i

, D, vi

, rc

, rd

Initialize:

bi

bmax

i

ai

0

⌧i

0

wi

0

pi

0

if ⌧i

= 0 and wi

= 0 then . Set Time to Goal

if ai

> 0 then . Time to Widget

⌧i

D(pi

, Oai(sai))/vi

else if ai

= 0 then . Time to Charger

⌧i

D(pi

, 0)/vi

end if

end if

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if ⌧i

= 0 and ai

> 0 then . Arrived with/at Widgetpi

Oai(sai) . Record Position

if wi

= 0 and ⌧ sai 0 then . Pick Up and Go

wi

1⌧i

D(Oai(sai), Oai(sai + 1))

else . Drop O↵/Idlew

i

0end if

end ifif ⌧

i

> 0 then . Moving/Discharging⌧i

�1bi

�rd

else if ai

= 0 then . Chargingpi

0⌧i

0if b

i

= bmax

i

then . Do Not Overchargebi

0else

bi

rc

end ifelse . Idle

⌧i

0bi

0end if

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Robot State Machine

Problem Setup

Unexpectedly Complex!

Optimization Formulation

min

a(t)

TZ

0

NX

i=1

˙bouti

(t)dt

subject to

timer dynamics

charge dynamics

bi

(t) > 0 8i, 8t⌧dj

(t) � 0 8j, 8t<latexit sha1_base64="jkpzMZdk6euPFEL9I9eUN902eP4=">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</latexit><latexit sha1_base64="jkpzMZdk6euPFEL9I9eUN902eP4=">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</latexit><latexit sha1_base64="jkpzMZdk6euPFEL9I9eUN902eP4=">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</latexit>

• Minimize robot battery discharge

• Dynamics set in state machines

• Robots must not die

• Deadlines must be met


Intractable!

Greedy Assignment Heuristic


1

2

3

4

5

Robots

1

2

3

4

5

Tasks

c11

c12

c13

c14

c15

c11 c12 c13 c14 c15c21 c22 c23 c24 c25c31 c32 c33 c34 c35c41 c42 c43 c44 c45c51 c52 c53 c54 c55

1 2 3 4 5

1 2345

Tasks

Robots

subject to Ax b

x 2 Znx � 0

Integer Linear Program

minimize c

Tx

Poly time Algorithm: Hungarian Algorithm

Resolve at each time step, real time for 1000s of robots

• The Black Magic: Hand-tuned assignment costs• Incorporates:

– Travel distance– Time deadlines– Heuristic look-ahead penalty– Lots of intuition

• Working to formalize cost design mechanisms


Assignment Cost Design

c11 c12 c13 c14 c15c21 c22 c23 c24 c25c31 c32 c33 c34 c35c41 c42 c43 c44 c45c51 c52 c53 c54 c55

1 2 3 4 5

1 2345

Tasks

Robots







Multi-Robot Routing in Graph


Oriana PeltzerA

B

C

D

C, D, AJob Queue

A, B

C, A, C Out Bin

Transform to Network Flow Problem


One edgeOne robot

Accomplished work

X Adaptation of Jingjin Yu’s network flow model to time-varying edges

t=1t=1

t=1

t=1

t=2

Expanded StructureDirected Network Flow Expanded Network

u

v

Technique from: J. J. Yu and S. M. LaValle, T-RO 2016

Accomplished work


Flow constraints: 2 per node per time step

Green and blue edges are constrained by the capacity of a node

Black edges (robots traversing an edge) are Constrained by the capacity of the edge

For edges with travel times larger than 1, the sum at each time step of all robots currentlytraversing the edge must be small than the edge capacity

Solve as Integer Linear Program


Accomplished work


Flow constraints: 2 per node per time step

Green and blue edges are constrained by the capacity of a node

Black edges (robots traversing an edge) are Constrained by the capacity of the edge

For edges with travel times larger than 1, the sum at each time step of all robots currentlytraversing the edge must be small than the edge capacity

• Poly time for interchangeable jobs (reduces to LP)

• Flexible capacity limits on edges• Wait times on nodes

maximize c

Tx

subject to Ax b

x 2 Znx � 0

Expanded Network Integer Linear Program





Local-Global Map


A

B

C

D

C, D, AJob Queue

A, B

C, A, C Out Bin

Kyle Brown

From Observed Interruptions to Trajectory Prediction


Data Center

From Observed Interruptions to Trajectory Prediction


Data CenterBayesian filter for trajectory estimationModel regression for intent predictionUpdate edge travel times for re-routing

Looking Forward

• Formalize assignment costs in Hungarian heuristic

• Frame multi-robot routing problem as single (or few) commodity flow for computational efficiency

• Build estimation and intent prediction framework

• Integrate in simulation environment


Thanks!

Platform Annual ReviewFeb 7, 2019 24

Prof. Mykel Kochenderfer

Patrick Washington Oriana Peltzer

Dr. Martin Sehr

Kyle Brown

Documents

Real Time Adaptive Scheduling for a Swarm ... - Platform Lab · Feb 7, 2019 Platform Annual Review 4 Fully distributed Local information Limited communication Reactive control and