Motion Planning for Multiple Autonomous Vehicles: Chapter 4b - Dynamic Distributed Lanes using Graph...

5/27/2018 Motion Planning for Multiple Autonomous Vehicles: Chapter 4b - Dynamic Distributed Lanes using Graph Search

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School of Systems, Engineering, University of Reading rkala.99k.orgApril, 2013

Motion Planning for Multiple

Autonomous Vehicles

Rahul KalaDynamic Distributed Lanes

Presentation of the paper: R. Kala, K. Warwick (2014) DynamicDistributed Lanes: Motion Planning for Multiple AutonomousVehicles, Applied Intelligence, DOI: 10.1007/s10489-014-0517-1


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Motion Planning for Multiple Autonomous Vehicles

Why Graph Search?

Completeness

Optimality

Issues

Computational Complexity

Key Idea State Reduction: Select which states to expand

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Key Contributions

The generalized notion of lanesis defined.

The generalized notion is used for planningand coordinationof multiple

vehicles.

A pseudo centralizedcoordination technique is designed which uses the

concepts of decentralized coordinationfor iteratively planning differentvehicles but empowers a vehicle to move around the other vehicles. The

coordination is hence better in terms of optimality and completeness than

most approaches (discussed so far) while being somewhat computationally

expensive in the worst cases.

The concept of one vehicle waitingfor another vehicle coming from the

other direction is introduced, when there may be space enough for only one

vehicle to pass.

Heuristicsare used for pruningthe expansions of states, which result in a

significant computational efficiency while leading to a slight loss of

optimality.

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Defining (Virtual) Lanes

Properties

Distributed The number of lanesand their widthsmaychange in different segments of the road

Dynamic The lanes change along with timeforevery pathway as the vehicles pass by.

Single Vehicle No two vehicles may occupy a lane side-by-side

Variable Width Every lane has a different width

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Solution Approach

A general graph (here Uniform Cost Search) search can

be used with:

High resolution maps

Rich action set for the vehicle

To solve within computational time, the number of

states expanded needs to be restricted

Coordination strategy needs to be devised

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State Reduction

Reduction hypothesis:

Vehicles maximize lateral separations

Vehicles prefer to travel at same lateral positions along

the road

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State Expansion

Consider expansion of a state L

Problem is selecting child states L formed as a result of

the expansion

Only expansionstep of the graph search algorithm is

discussed in this work, while the rest of the algorithm is

same as a general graph search.

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CONSIDERING ONLYA SINGLE VEHICLE

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State Reduction

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Use asweeping

line (Z) forobstacle

analysis inlarge steps

Selectregions of

line(Pathways)in obstaclefree regions

Select bestpoint inside

everypathway

(L3)

Selectsimilar

point (L2) atan earliersweepingline (Z2)

Feasibilityanalysis

Make thesechildren

andconnect toparent (L)

by asmooth

trajectory

Vehicle PlacementHypotheses

(i) Maximize lateralseparations

(ii) Attempt to drive insame relative position at

the road



State Reduction

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Sweeping line used for

obstacle analysis (Z)

L

All pathways

(3 expansions to be done)

A pathway

Vehicle position at

extremity of pathwayY

Earlier sweeping line for

expanded node (Z2)



Why use L2and not L3?

Or why select a state not at the sweeping line but earlier?

Or why Z2and not Z?

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Sweeping Line

Best point L3Obstacle

But if vehicle lands at L3, it cannot steer sharp enough and

avoid the obstacle

ZL



Why L2and not L3?

Vehicle expected to align itself at L2for obstacle seen at

L3

If vehicle can go from L to L3via L2, it is safe at L2

Curve from L to L2may be traversed if optimal Curve from L2to L3is only for connectivity check/obstacle

analysis

L3acts as a roadmap for L2informing it about any

obstacle well in advance before the vehicle actually lands

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State Selection

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L

L2 L3

Z2 Z

PlannedTrajectory Trajectory onlyfor analysis



State Selection

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Every turn is on the basis of obstacle assessed at the next sweepingline while states are selected on the basis of the set hypotheses.



Curve Generation

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Connect L

to L2 by aclothoidcurve

ConnectL2 to L3 by

curvealong theroad

length

Checkfeasibility

Ifinfeasible,

performlocaloptimizati

on

If stillinfeasible,

reduce

speed/sweeping linestep size

and repeat



Small Obstacle Problem

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L

Z2(original)L3(original)

L2(original)

Z (original)Z (modified)Z2(modified)

L3

(modified)L2

(modified)small obstacle

Z (original) cannot be used for obstacle analysis as it cannot detect the small obstacleand hence always return infeasibility. Hence it is modified to Z (modified) byreduction in step size of the sweeping line. Corresponding Z2s are Z2(original) and

Z2(modified)



Lane

Graph search gives the shortest path as output

Result consists of trajectory and all the expanded states

The expanded states together with available lateral

bounds constitute a Lane Trajectory may be modified by moving the expanded

states laterally

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Graph generated

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Results

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CONSIDERING MULTIPLE VEHICLES

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Problem

A number of vehicles with planned trajectories and lanes

are given

Construct plan of the next vehicle entering the scenario

The trajectories of the other vehicles may be modified bychanging their lanes (moving their expanded states

laterally)

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Expansion Strategy

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Expansion

StrategyFree StateExpansion

VehicleFollowing

Wait forVehicle


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Free State Expansion

Expand using maximum speed, overtaking expansion

strategy

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Use sweepingline for

obstacleanalysis

For everypathway

Computevehiclesrequiring

independentlane

Distributepathwaywidth

amongst thevehicles

Vehicle being followed orgoing ahead by a larger speed

not considered

To skip details go to slide 32


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Strategy

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Find vehicles(Rb) affectedby navigation

of vehicleunderplanning (Ra)

to Z and hencerequire a

separate lane

Find vehicles(Rc) affected

by altered

navigation ofRaor Rbandhence alsorequire a

separate lane

Find vehicles(Rd) affected

by altered

navigation ofRa, Rbor Rcand

hence alsorequire

separate lane

And so on till

no vehicle isfurther

affected


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Region of independent lane

Region within which all vehicles have independent lanes

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Region ofindependent

lanes (S)

Vehicles

requiringindependent

lanes (H)

Check if avehicle in H

while travellingin S affects any

other vehicle Rj

Calculateregion (S2) in

which Rjassumes a new

lane

Add S2

Add Rj


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Calculating new affected region Let Ribe the vehicle being planned

Let Rjbe the vehicle affected

Let Rkbe any other vehicle which might be affected by Rj

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Rjmoves tonew lane as

soon aspossible (a)

Rjreturns tooriginal

planned laneafter crossing

navigation

region of Ri (Lto Z) , say (b)

New lane iseffective in(a) to (b)

While Rjtravels from (a) to (b), it should not affect any othervehicle (Rk). If yes, it is also added, and so on


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Calculating affected region

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Ri Ri

Rj Rj RjRi

Rj RjRi

Rj

Motion from L to Z

(a) (b) (b) (a)

Same side Opposite sides


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While Rjtravels from (a) to (b) , some Rkalso travels as

per its trajectory

Does Rjeffect Rk, in which case the affected region andvehicles are updated and the process repeats?

Since motion of Rjfrom (a) to (b) is yet to be computed,

approximations are used

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Rj

(a) (b)Rk

Rj

(a) (b)

Rk

Same SideIf Rkis safe with Rjat (b), it is not

affected, distances will only increasewith Rkmoving forward

Opposite SidesIf Rkmoved by a threshold (proportional

to speed) is safe with Rjat (b), it is notaffected. With time Rkwill get closer to Rj

and may get affected. Maximum

threshold assumed.


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Lane Distribution

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Get vehicles

requiringindependen

t lane

Getpreferred

position forvehiclebeing

planned

Check ifplacement

possiblewithoutmoving

othervehicles

If not, checkif placement

possiblewith

maintenance of

maximumseparationthreshold

If not,computetotal free

space anddistributebetweenvehiclesevenly


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Free State Expansion

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Z2 Z (b)(a)

RjRi

Trajectory of Rjalready covered(this point and before)Trajectory of Ri

Trajectory of Rj

planned

Distribution of lanes amongst vehicles

Generated trajectories for vehicles

Computing lane vehicles

Original laneof Rj

Modified lane of RjLane of Ri Planned position of Rj

Planned position

of Ri


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Vehicle Following

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Select thevehicle to

follow

Follow theselectedvehicle

Laterally closestvehicle is selected

Speed is fixed toallow following

with somedistance

Z is taken as pernormal speed

Z2is taken asper reduced

speed

Rest expansionsame as free

state expansion


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Vehicle Following

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Trajectory of Rjalready covered(this point and before)Trajectory of Ri

Trajectory of Rjplanned

Z2 Z


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Wait for Vehicle

Consider a narrow bridge with a stream of vehicles

coming from one side

Plan to wait for some of the vehicles in the stream to

clear and hence go forward

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Wait for Vehicle

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Decide thevehicle to

wait for

Compute theposition/

trajectory towait

Modify othervehicles

trajectories

Computestate after

wait

Attempt for allvehicles in the

crossing stream

Number of stateexpansions =

number ofvehicles in stream


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Computing the position of wait

Assume obstacle analysis line (Z) lies inside the narrow

bridge

Assume previous line (Z2) lies outside the narrow bridge

in a wide region Maximum separation threshold is left

Two options to wait: Left of the leftmost vehicle in the

stream, or Right from the rightmost vehicle in the stream

Whichever has minimum deviation from current position

is selected

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Modification of Trajectories

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Modifytrajectory

for thevehiclebeing

waited for

Calculatetime whenit crosses

the vehiclebeing

planned

Computetrajectory

of thevehiclebeing

planned

Modifytrajectorie

s of allother

vehicles till

this time

Get asnapshotof state

after waitand use as

the

expandedstate

Calculatepositionsaroundwaiting

vehicle at Z2

Set the lateralposition asconstant inthis region

Hence allovertakingand like

behavioursget cancelled

Z2Z2


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Wait for Vehicle Strategy

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22

Z Z2Ri

Rk

Detection of suddennarrowing of width of road

Lane of Rk

Planned

position of Ri

(a) (b)

Modification of lane of Ri

Trajectory of Rk

Trajectory of Ri

The generated trajectory for vehicles


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Expansion strategy selection

All 3 strategies used for all state expansions results in too

many expanded states

To limit expansions following heuristic rule is used

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Strategy PreconditionFree StateExpansion

Parent state was expanded by free stateexpansion and no new vehicleencountered/left, a new vehicleencountered/left

VehicleFollowing Parent state was expanded by vehiclefollowing and no new vehicleencountered/left, a new vehicleencountered/left

Wait for vehicle Sudden change in road width used forobstacle assessment

Once decided toovertake/ follow avehicle, continue

doing so till a new

vehicle approachesor leaves (changingovertake scenario)


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Results

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Results

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Results

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Results

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Analysis

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0

50

100

150

0 1 2 3 4 5 6Numberofnon-

c

onnectingstates

Number of obstacles (n)

0

500

1000

1500

0 1 2 3 4 5 6

Numb

erof

connectingstates

Number of obstacles (n)


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Analysis

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00.5

1

1.5

2

2.5

3

3.5

0 1 2 3

N

umberofnon-

connectingstates

Number of vehicles in same direction (a)

0

20

40

60

80

0 1 2 3Numberofcn

necting

states

Number of vehicles in same direction (a)


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Thank You

Acknowledgements:

Commonwealth Scholarship Commission

in the United Kingdom

British Council

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