Patrolling the Border: . . . How to Describe . . . Constraints on . . . Simplification of the . . . Optimizing Trajectories for Detection at Crossing . . . Unmanned Aerial Vehicles Strategy Selected by . . . Towards an Optimal . . . (UAVs) Patrolling the An Optimal Strategy: . . . Taking Fuzzy . . . Border Home Page Chris Kiekintveld 1 , Vladik Kreinovich 1 , 2 , and Title Page Octavio Lerma 2 ◭◭ ◮◮ 1 Department of Computer Science ◭ ◮ 2 Computational Sciences Program Page 1 of 13 University of Texas at El Paso 500 W. University Go Back El Paso, TX 79968, USA contact email: vladik@utep.edu Full Screen Close Quit
Patrolling the Border: . . . How to Describe . . . 1. Patrolling the Border: a Practical Problem Constraints on . . . • Remote areas of international borders are used by the Simplification of the . . . adversaries: to smuggle drugs, to bring in weapons. Detection at Crossing . . . Strategy Selected by . . . • It is therefore desirable to patrol the border, to mini- Towards an Optimal . . . mize such actions. An Optimal Strategy: . . . • It is not possible to effectively man every single seg- Taking Fuzzy . . . ment of the border. Home Page • It is therefore necessary to rely on other types of surveil- Title Page lance. ◭◭ ◮◮ • Unmanned Aerial Vehicles (UAVs): ◭ ◮ – from every location along the border, they provide Page 2 of 13 an overview of a large area, and Go Back – they can move fast, w/o being slowed down by clogged roads or rough terrain. Full Screen • Question: what is the optimal trajectory for these UAVs? Close Quit
Patrolling the Border: . . . How to Describe . . . 2. How to Describe Possible UAV Patrolling Strate- Constraints on . . . gies Simplification of the . . . • Let us assume that the time between two consequent Detection at Crossing . . . overflies is smaller the time needed to cross the border. Strategy Selected by . . . Towards an Optimal . . . • Ideally, such a UAV can detect all adversaries. An Optimal Strategy: . . . • In reality, a fast flying UAV can miss the adversary. Taking Fuzzy . . . Home Page • We need to minimize the effect of this miss. Title Page • The faster the UAV goes, the less time it looks, the more probable that it will miss the adversary. ◭◭ ◮◮ • Thus, the velocity v ( x ) is very important. ◭ ◮ Page 3 of 13 • By a patrolling strategy, we will mean a f-n v ( x ) de- scribing how fast the UAV flies at different locations x . Go Back Full Screen Close Quit
Patrolling the Border: . . . How to Describe . . . 3. Constraints on Possible Patrolling Strategies Constraints on . . . 1) The time between two consequent overflies should be Simplification of the . . . smaller the time T needed to cross the border: Detection at Crossing . . . Strategy Selected by . . . – the time during which a UAV passes from the loca- tion x to the location x +∆ x is equal to ∆ t = ∆ x Towards an Optimal . . . v ( x ); An Optimal Strategy: . . . – thus, the overall flight time is equal to the sum of Taking Fuzzy . . . these times: Home Page dx � T = v ( x ) . Title Page ◭◭ ◮◮ 2) UAV has the largest possible velocity V , so we must have v ( x ) ≤ V for all x . ◭ ◮ 1 def Page 4 of 13 It is convenient to use the value s ( x ) = v ( x ) called slow- Go Back ness , so � � � = 1 Full Screen def T = s ( x ) dx ; s ( x ) ≥ S . V Close Quit
Patrolling the Border: . . . How to Describe . . . 4. Simplification of the Constraints Constraints on . . . • Since s ( x ) ≥ S , the value s ( x ) can be represented as Simplification of the . . . def S + ∆ s ( x ), where ∆ s ( x ) = s ( x ) − S . Detection at Crossing . . . Strategy Selected by . . . • The new unknown function satisfies the simpler con- Towards an Optimal . . . straint ∆ s ( x ) ≥ 0. An Optimal Strategy: . . . • In terms of ∆ s ( x ), the requirement that the overall Taking Fuzzy . . . � time be equal to T has a form T = S · L + ∆ s ( x ) dx . Home Page • This is equivalent to: Title Page � ◭◭ ◮◮ T 0 = ∆ s ( x ) dx, where: ◭ ◮ • L is the total length of the piece of the border that Page 5 of 13 we are defending, and Go Back def • T 0 = T − S · L . Full Screen Close Quit
Patrolling the Border: . . . How to Describe . . . 5. Detection at Crossing Point x Constraints on . . . • Let h be the width of the border zone from which an Simplification of the . . . adversary (A) is visible. Detection at Crossing . . . Strategy Selected by . . . • Then, the UAV can potentially detect A during the Towards an Optimal . . . time h/v ( x ) = h · s ( x ). An Optimal Strategy: . . . • So, the UAV takes ( h · s ( x )) / ∆ t photos, where ∆ t is Taking Fuzzy . . . the time per photo. Home Page • Let p 1 be the probability that one photo misses A. Title Page • It is reasonable to assume that different detection er- ◭◭ ◮◮ rors are independent. ◭ ◮ • Then, the probability p ( x ) that A is not detected is Page 6 of 13 p ( h · s ( x )) / ∆ t , i.e., p ( x ) = exp( − k · s ( x )) , where: 1 Go Back = 2 h def k ∆ t · | ln( p 1 ) | . Full Screen Close Quit
Patrolling the Border: . . . How to Describe . . . 6. Strategy Selected by the Adversary Constraints on . . . • Let w ( x ) denote the utility of the adversary succeeding Simplification of the . . . in crossing the border at location x . Detection at Crossing . . . Strategy Selected by . . . • Let us first assume that we know w ( x ) for every x . Towards an Optimal . . . • According to decision theory, the adversary will select An Optimal Strategy: . . . a location x with the largest expected utility Taking Fuzzy . . . u ( x ) = p ( x ) · w ( x ) = exp( − k · s ( x )) · w ( x ) . Home Page Title Page • Thus, for each slowness function s ( x ), the adversary’s gain G ( s ) is equal to ◭◭ ◮◮ G ( s ) = max u ( x ) = max [exp( − k · s ( x )) · w ( x )] . ◭ ◮ x x Page 7 of 13 • We need to select a strategy s ( x ) for which the gain Go Back G ( s ) is the smallest possible. Full Screen G ( s ) = max u ( x ) = max [exp( − k · s ( x )) · w ( x )] → min s ( x ) . x x Close Quit
Patrolling the Border: . . . How to Describe . . . 7. Towards an Optimal Strategy for Patrolling the Constraints on . . . Border Simplification of the . . . • Let x m be the location at which the utility u ( x ) = Detection at Crossing . . . exp( − k · s ( x )) · w ( x ) attains its largest possible value. Strategy Selected by . . . Towards an Optimal . . . • If we have a point x 0 s.t. u ( x 0 ) < u ( x m ) and s ( x 0 ) > S : An Optimal Strategy: . . . – we can slightly decrease the slowness s ( x 0 ) at the Taking Fuzzy . . . vicinity of x 0 (i.e., go faster in this vicinity) and Home Page – use the resulting time to slow down (i.e., to go Title Page slower) at all locations x at which u ( x ) = u ( x m ). ◭◭ ◮◮ • As a result, we slightly decrease the value ◭ ◮ u ( x m ) = exp( − k · s ( x m )) · w ( x m ) . Page 8 of 13 • At x 0 , we still have u ( x 0 ) < u ( x m ). Go Back • So, the overall gain G ( s ) decreases. Full Screen • Thus, when the adversary’s gain is minimized, we get Close u ( x ) = u 0 = const whenever s ( x ) > S. Quit
Patrolling the Border: . . . How to Describe . . . 8. Towards an Optimal Strategy (cont-d) Constraints on . . . • Reminder: for the optimal strategy, Simplification of the . . . Detection at Crossing . . . u ( x ) = w ( x ) · exp( − k · s ( x )) = u 0 whenever s ( x ) > S. Strategy Selected by . . . u 0 Towards an Optimal . . . • So, exp( − k · s ( x )) = w ( x ), hence An Optimal Strategy: . . . s ( x ) = 1 k · (ln( w ( x )) − ln( u 0 )) and ∆ s ( x ) = 1 Taking Fuzzy . . . k · ln( w ( x )) − ∆ 0 . Home Page Title Page = 1 def • Here, ∆ 0 k · ln( u 0 ) − S. ◭◭ ◮◮ • When s ( x ) gets to s ( x ) = S and ∆ s ( x ) = 0, we get ◭ ◮ ∆ s ( x ) = 0. Page 9 of 13 • Thus, we conclude that Go Back � 1 � Full Screen ∆ s ( x ) = max k · ln( w ( x )) − ∆ 0 , 0 . Close Quit
Patrolling the Border: . . . How to Describe . . . 9. An Optimal Strategy: Algorithm Constraints on . . . • Reminder: for some ∆ 0 , the optimal strategy has the Simplification of the . . . form Detection at Crossing . . . � 1 � Strategy Selected by . . . ∆ s ( x ) = max k · ln( w ( x )) − ∆ 0 , 0 . Towards an Optimal . . . An Optimal Strategy: . . . • How to find ∆ 0 : from the condition that Taking Fuzzy . . . � Home Page ∆ s ( x ) dx = Title Page � � 1 � ◭◭ ◮◮ max k · ln( w ( x )) − ∆ 0 , 0 dx = T 0 . ◭ ◮ • Easy to check: the above integral monotonically de- Page 10 of 13 creases with ∆ 0 . Go Back • Conclusion: we can use bisection to find the appropri- Full Screen ate value ∆ 0 . Close Quit
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