control and robotics lab electrical engineering department, technion search of targets by multiple...

Control and Robotics Lab Electrical Engineering Department , Technion

Search of targets by multiple UAVs using a probability map

By

Alexander Strizhiver Michael Shamis

Supervised by

Mark Moulin

Presentation outline

• Definition of probability map • Case #1: Single target search using single UAV (STSU problem)• Greedy and K-Shortest path algorithms• Variation of K-Shortest path algorithm for STSU.• Case #2 : Single target search using multiple UAVs (STMU problem) – using STSU case solution.• Case #3 : Multiple target search using multiple UAVs (MTMU problem) – using STMU case solution.

Definition of probability map

• A given map of an area can be divided into hexagons

• For each hexagon assigned a probability to find target in it.

• Why hexagon ?

Case #1: Single target search using single UAV (STSU problem)

• Definitions: P – probability map ( probability function ).

- Location of UAV base

- Location of target drawn according to P.

F - size of UAV fuel tank.

- Legal search path, that is an ordered set of point on probability map which satisfy following properties:

x yB=(B ,B )

x yT=(T ,T )

1 2 3

i

i i+1

i 1 1

1

{ , , ,...}

Where C ( , )

C is a neighbour of C - the continuity of the path

B {C , ,..., } - the fuel constraint.

- starting point is the base

P(T )=1 - the path covers all

i i

i i F

C C C

x y

i

i C C

C B

possible target locations

Case #1: Single target search using single UAV (STSU problem)

• Legal sub path

• Essential sub path length:

• STSU problem can then be defined as:

( ,T) inf( )

legal path S

k S

minimize ( ( , ))

.

is legal search path

E k T

S T

1 2, nS = { C ,C ...,C }

n

T S

Case #1: Single target search using single UAV (STSU problem)

• STSU problem is a very hard problem (seems to be NP-hard problem ).

• The optimal solution for NP hard problems is computational heavy, and not used for real problems.

• STSU problem should be solved by approximate solutions.

Redefinition of STSU problem

• Alternative definition of STSU problem ( simplified – not equivalent ):

• This problem is still hard problem, but have good approximate solutions. All the solutions will try to find .

''

' ' '1 2 k

' ' '1 2 k

' ' '1 2 j-1

C

'

, ,..., be subsets of legal path :

= ... means ordered chaining.

maximize ( ) given P(C) = 0 C ...

.

jj

j

Let

here

j P C

S T

F

' '1 2, ,...

Representation of probability map as non directed graph

• Each hexagon in the probability map becomes a vertex on a graph

• There is an edge (V,U) iff the hexagons U and V on probability map are neighbors.

Solving STSU problem using greedy algorithm

• Greedy algorithm: C=B

clear collectedPathwhile |collectedPath|<F{

S = all closest neighbors C’ of C with non zero probability, for which F-|collectedPath|-dist(C,C’)-dist(C’,B)>=0 C’ = coordinate with largest probability in S add path form C to C’ to collectedPathC = C’

}output collectedPath.

Greedy algorithm advantages and disadvantages

• Advantages:– Computationally light.– Gives good results when the probability distribution

close to uniform.– Covers well areas around the base.

• Disadvantages:– Can go on “wrong path” when probabilities are slightly

larger than the surrounding.– Does not work well when large probabilities

concentrated far from base

K-Shortest path algorithm

Given: G – Graph. V - Vertex in graph GK – number of edges

Goal:Find weight of “lightest” (shortest) path from V to all the vertices in graph G with K edges exactly.

Lightest path here means – that sum of all the weights is

smallest. K-Longest path algorithm - algorithm that finds heaviest

( longest ) path instead of lightest path.

K-Longest path algorithmHere is a version of K-longest path that matches our needs better(find longest path with K or less edges) :

For each vertex uPath(0,u) = -infinity

Path(0,V) = 0For k=1 to max_k{

{for each vertex v

path(k,v) = path(k-1,v)}for each vertex v

for each u neighbor of vpath(k,u) = max(path(k,u),path(k-1,v)+cost(u))

}

K-Longest path algorithm: example

K-Longest path algorithm

• K-Longest path algorithm has major drawback – the path it finds can get stuck on local maxima.

• The variation we use solves this problem by “remembering” all the vertices which were visited on heaviest path to each vertex.

Case #2 : Single target search using multiple Case #2 : Single target search using multiple UAVs (STMU) – using STSU UAVs (STMU) – using STSU

solution.solution.

• Non formal definition of the STMU problem:Given:

Same inputs as for STSU problemN – Number of UAVs

Goal:Find path for each UAV, so that the

mean sum of distances all UAVs pass until finding the target is lowest.

• Our solution to STSU problem concentrated on finding subsets of :

• The extension for multiple UAVs is trivial – UAVs run on those subsets in parallel.

' '1 2, ,...

Case #3 : Multiple target search using multiple Case #3 : Multiple target search using multiple UAVs (MTMU) – using STMU solution. UAVs (MTMU) – using STMU solution.

Non formal definition of the MTMU problem:• Given:

Same inputs as for STMU problemP(T2|T1) - conditional probability function for second

target location given first target location.• Goals:

Find path for each UAV, such that the mean sum of distances all UAVs pass until finding one of the targets is lowest.

When one of the targets is found then find a path for each UAV, such that the mean sum of distances all UAVs pass until

finding the second target is lowest.

Case #3 : Multiple target search using multiple Case #3 : Multiple target search using multiple UAVs (MTMU) – using STMU solution. UAVs (MTMU) – using STMU solution.

• Using Bayes formulas we can build new probability map:

• Then normalize P’, and supply it as a legal probability map input to the STMU algorithms.

1 2 1 2 1 2

1 2 1 2 1

2 1 1'

'( ) ( =C or T ) ( ) ( ) ( =C and T )

:

( =C and T ) ( ) ( | )

( 2 ) ( =C| =C') ( ') C

P T C P T C P T C P T C P T C

Where

P T C P T C P T C T C

P T C P T T P T C

Case #3 : Multiple target search using multiple Case #3 : Multiple target search using multiple UAVs (MTMU) – using STMU solution. UAVs (MTMU) – using STMU solution.

• MTMU problem solved in 2 steps:

1) Run STMU algorithms on P’.

2.a) If target #1 found - run STMU algorithms again with probability map P(T2|T1)

2.b) If target #2 found – run STMU algorithms

again with probability map P(T1|T2) which is given by: 1 2 1

2

P(T =C')×P(T =C|T =C')P(T1=C'|T2=C)=

P(T =C)

Conclusions

• In this project we have reviewed different solutions for the problem of finding targets, given their probabilities map.

• K-Longest path approach in most of the cases gives better results than greedy algorithms( it finds targets earlier ).

• There is still place for improvement of the solution of the original problem since the problems we solved are far from the “ideal solution” for the problems we looked for.

• It seems to be possible to adjust the K-Longest path algorithm for dynamic targets

• Hybrid of K-Longest path and greedy algorithms can be used to try solving the problem better.