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Temporally and Spatially Deconflicted Path Planning for Multiple Marine Vehicles. A. Häusler 1 , R. Ghabcheloo 2 , A. Pascoal 1 , A. Aguiar 1 I. Kaminer 3 , V. Dobrokhodov 3 1 Instituto Superior Técnico, Lisbon, Portugal 2 Tampere University of Technology, Tampere, Finland - PowerPoint PPT Presentation
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Temporally and Spatially Deconflicted Path Planning for Multiple Marine Vehicles
A. Häusler1, R. Ghabcheloo2, A. Pascoal1, A. Aguiar1
I. Kaminer3, V. Dobrokhodov3
1Instituto Superior Técnico, Lisbon, Portugal2Tampere University of Technology, Tampere, Finland
3Naval Postgraduate School, Monterey, California
Andreas J. Häusler - MCMC 2009 2
Introduction
Challenges in underwater environments can be overcome through the use of fleets of heterogeneous vehiclesCentral to cooperative control systems are efficient algorithms for multiple vehicle path planningExample: Go-To-Formation maneouvre
September 18th, 2009
Andreas J. Häusler - MCMC 2009 3
The Go-To-Formation Maneouvre
An initial formation pattern must be established before mission startDeploying the vehicles cannot be done in formation (no hovering capabilities)Vehicles can’t be driven to target positions separately (no hovering capabilities)
September 18th, 2009
Mother Ship
Cur
rent
Andreas J. Häusler - MCMC 2009 4
The Go-To-Formation Maneouvre
Need to drive the vehicles to the initial formation in a concerted mannerEnsure simultaneous arrival times and equal speedsEstablish collision avoidance through maintaining a spatial clearance
September 18th, 2009
Mother Ship
Andreas J. Häusler - MCMC 2009 5
Path Planning System
September 18th, 2009
MULTIPLE VEHICLEPATH PLANNING
SYSTEM
Initial Positions
Initial Velocities
Final Positions
Final Velocities
Cost Criterion (e.g. weighted sum of
energies, maneouvring time)
Vehicle dynamical constraints
Collision avoidance constraints
External constraints (e.g.
Obstacles)
Nominal Pathsand
Speed Profiles
Andreas J. Häusler - MCMC 2009 6
Path Planning Foundations
September 18th, 2009
0 0( ), ( )p t v t
( ), ( )f fp t v t
Initial position
Final position
( ( )), ( ( )), ( ( ))i i ip t v t a t
Andreas J. Häusler - MCMC 2009 7
Path Planning Foundations
Dispense with absolute time in planning a path (Yakimenko)
Establish timing laws describing the evolution of nominal speed with Spatial and temporal constraints are thereby decoupled and captured by &Choose and as polynomialsPath shape changes with only varying
September 18th, 2009
( )t
( ) [ ( ), ( ), ( )]p x y z
( )p ( ) d dt
( ) ( )p
f
[0, ]f
Andreas J. Häusler - MCMC 2009 8
Path Planning for Single Vehicles
Polynomial for each coordinate with degree determinded by number of boundary conditionsTemporal speed and acceleration constraints
Choice for timing law with
September 18th, 2009
0( ) N k
xkkx a
0 0ff
τη τ = η + η τ ητ
2 2 2( ) ( ) ' ( ) ' ( ) ' ( ) ( ) '( )v x y z p 2( ) ''( ) ( ) '( ) '( ) ( )a p p
(0) (0)v ( ) ( )f fv t
Andreas J. Häusler - MCMC 2009 9
Path Planning for Single Vehicles
A path is feasible if it can be tracked by a vehicle without exceeding , , and
It can be obtained by minimizing the energy
consumption ,subject to temporal speed and acceleration constraints
September 18th, 2009
minv maxv maxa
33
0
( ) '( )f
f DJ c c p d
max
Andreas J. Häusler - MCMC 2009 10
Optimization for Multiple Vehicles
Instead of trying to minimize the arrival time
interval, we can fix arrival times to be exactly
equal and still get different path shapes
Integrate for
September 18th, 2009
( )
(0) ( ) (0)
( ) (0)( ) (0)
ln( ( ) (0))
i
ii
i
i
i f i f i
i f ifi f i
i f i
t
Andreas J. Häusler - MCMC 2009 11
Optimization for Multiple Vehicles
Then we obtain ,
from which the spatial paths are computedThe result is in turn used to compute velocity and accelerationOptimization is done using direct search
September 18th, 2009
( ) (0)
( ) ( )(0)1 ( ) (0)
( ) ( ) (0)
i i
f
i i
f f i f i
tti i fi
f i f ii f i f i
t t
t
Andreas J. Häusler - MCMC 2009 12
Spatial Deconfliction
September 18th, 2009
1 0 1 0( ), ( )p t v t
2 0 2 0( ), ( )p t v t
1 1( ), ( )f fp t v t
2 2( ), ( )f fp t v t
d E
Initial positions
Final positions(target formation)
Andreas J. Häusler - MCMC 2009 13
Temporal Deconfliction
September 18th, 2009
1 0 1 0( ), ( )p t v t
2 0 2 0( ), ( )p t v t
1 1( ), ( )f fp t v t
2 2( ), ( )f fp t v t
Initial positions
Final positions(target formation)
Andreas J. Häusler - MCMC 2009 14
Deconfliction
Spatial deconfliction: subject to
For temporal deconfliction, the constraint changes to
September 18th, 2009
; 1, , 1
mini
n
i ii n i
w J
2 0i k j lp τ p τ E ;E >
2 0i jp t p t E ;E >
1, 0, fi, j = ,n,i j,t t
1, 0, 0,k l f fi ji, j = ,n,i j, τ ,τ τ τ
Andreas J. Häusler - MCMC 2009 15
Deconfliction
Simultaneous arrival at time
Time-coordinated path following using virtual time with
September 18th, 2009
1 2
opt
, 1
arg minf
n
f i it t t i
t w J
0,1ft t
( ) (0)
ln 1 ( ) (0) 1( ) (0)
ln ( ) (0)
i i
i i
i
i
i f i f i
i f i i ffi f i
i f i
t t
(Ghabcheloo, ACC 2009)
Andreas J. Häusler - MCMC 2009 16
Simulation Results
Spatial deconfliction in 2D for three vehicles – paths and velocity profiles
September 18th, 2009
Andreas J. Häusler - MCMC 2009 17
Simulation Results
Temporal deconfliction in 2D for three vehicles – paths and velocity profiles
September 18th, 2009
Andreas J. Häusler - MCMC 2009 18
Simulation Results
Spatial deconfliction in 3D for three vehicles – paths and velocity profiles
September 18th, 2009
Andreas J. Häusler - MCMC 2009 19
Simulation Results
Temporal deconfliction in 2D, facing a current of 0.5 m/s coming from the eastAdditional optimization criterion: final velocity to match desired value of 1.5 m/s
September 18th, 2009
Andreas J. Häusler - MCMC 2009 20
Conclusions
Multiple vehicle path planning techniques based on direct optimization methodsFlexibility in time-coordinated path following through decoupling of space and timeNo complicated timing laws – constraints are incorporated in spatial description (e.g. initial and final heading)Suitable for real-time mission planning thanks to fast algorithm convergence
September 18th, 2009
Andreas J. Häusler - MCMC 2009 21
Future Work
Show robustness of algorithm through extensive simulationsCompare our results with results from optimal controlEmploy path planning on vehicle hardware (GREX, Co3-AUVs) for sea trialsIncorporate avoidance of fixed obstaclesIntegrate constraints imposed by cooperative path following (e.g. communication links)
September 18th, 2009
Andreas J. Häusler - MCMC 2009 22
Thank you for your attention!
September 18th, 2009
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