Vehicle Routing Algorithms to Intercept Escaping Targets

Transcription

1 Vehicle Routing Algorithms to Intercept Escaping Targets Pushkarini Agharkar 1 and Francesco Bullo Abstract We study a dynamic ehicle routing problem for moing targets Our setup is as follows: (i targets appear uniformly distributed on a unit disk ia a temporal Poisson process and moe radially outward with a constant speed, (ii a single ehicle with speed greater than that of the targets aims to intercepts them before they escape the disk With the goal of maximizing the fraction of captured targets in the steady state, we propose three policies: Capturable Nearest Neighbor (CNN, Sector Wise (SW and Stay Near Boundary (SNB policy We derie lower bounds on the fraction of targets captured by the CNN, SW and SNB policies The CNN policy is shown to be optimal for arrial rate λ below a critical alue For arrial rates aboe another critical alue, the fraction of targets captured by the SW policy is more than that of the CNN policy Finally, the SNB policy is within a constant factor of optimal in the limiting regime of + and λ + We present numerical simulations to illustrate our results I INTRODUCTION Vehicle motion planning arises in many important autonomous ehicle applications such as sureillance and perimeter defense In this paper we consider a ehicle routing problem in which a ehicle must intercept targets before they escape a bounded enironment which is a disk of unit radius This problem setup is motiated by applications such as perimeter defense, wherein the demands could be isualized as moing targets trying to escape a region under sureillance by a ehicle Vehicle routing problems (VRP are concerned with planning optimal ehicle routes for proiding serice to a gien set of customers The traeling salesperson problem (TSP is a specific instance of the VRP and has been extended to include ariations such as the deadline-tsp [] Recently, researchers hae looked at the TSP with moing objects In [6] the authors consider objects moing on straight lines and consider the case when the objects are slower than the ehicle and when the ehicle is constrained to moe parallel to the horizontal or ertical axes The same problem is studied in [7], but with arbitrary ehicle motion, and it is called the translational TSP The authors of [7] propose a polynomial-time approximation scheme to catch all objects in minimum time Other ariations of the problem are studied in [8] and [1] In the VRP, the routes are planned with complete information of the customers, and thus the optimization is This material is based upon work supported in part by NSF Award and in part by ARO Award W911NF Pushkarini Agharkar is with the Department of Mechanical Engineering, Uniersity of California Santa Barbara, CA , USA agharkar@umailucsbedu Francesco Bullo is with Faculty of Mechanical Engineering, Uniersity of California Santa Barbara, CA , USA bullo@engineeringucsbedu static, and typically combinatorial [15] In contrast, in the dynamic ehicle routing (DVR ehicles must plan paths through serice demand locations that arrie sequentially oer time DVR problems hae been reiewed in [5], [13] and [1] In [4] and [14] DVR problems inoling targets moing with a constant elocity are studied In [4], a single ehicle intercepts the targets eentually In [14], the ehicle is required to intercept the targets before they reach a deadline [1] study a DVR problem in which demands disappear if not sericed within a time window In this paper, we introduce a DVR problem in which demands appear uniformly distributed on a disk and moe radially outward until they escape the disk The arrial process is Poisson with rate λ and the demand speed (,1 is fixed This problem is referred to as the Radially Escaping Targets (RET problem for conenience The goal of this paper is to design routing policies for a single ehicle in order to maximize the fraction of demands captured in the steady state The contributions of this paper can be summarized as follows For any policy for the RET problem, we determine a policy independent upper bound on the fraction of targets captured We also propose a method to establish upper and lower bounds on the path through radially escaping targets Next, we formulate three policies: Capturable Nearest Neighbor (CNN, Sector Wise (SW and Stay Near Boundary (SNB policy Lower bounds on the fraction of targets captured using the CNN, SW and SNB policies are obtained The CNN policy is shown to be optimal for arrial rate below a critical alue for all target speeds For arrial rates aboe another critical alue, the capture fraction of the SW policy is more than that of the CNN policy The SW policy is thus more effectie than the CNN policy in that regime In the limit of low target speed and high arrial rate, the SNB policy is within a constant factor of optimal and the fraction of targets captured by the ehicle using the SNB policy asymptotically tends to 1 We present numerical simulations which erify our results The paper is organized as follows In Section II, we formally introduce the problem In Section III, we present some background results The CNN, SW and SNB policies are introduced in Section IV Results from numerical simulations are presented in Section V II PROBLEM FORMULATION We start with introducing a DVR problem in which targets appear independently and uniformly distributed in an enironment E := (r,θ : r 1 θ [,π} with uniform spatial density and according to a temporal Poisson process

2 with rate λ Uniform spatial distribution of the targets is realized through probability density functions f 1 (r = r and f (θ = 1/π where r and θ are random ariables describing the location of appearing targets in radial coordinates Once the targets appear, they moe radially outwards with a constant speed (,1 and eentually intercept the boundary of the enironment A ehicle with speed of 1 and confined to moe in E intercepts the targets and captures them before they escape the enironment We refer to this problem as the Radially Escaping Targets (RET problem for conenience The parameters of the RET problem are the target speed and arrial rate λ Let the generation of targets commence at t = Let the ehicle start sericing the targets at t = as well Let n capt (t and n esc (t be the number of targets that hae been captured and hae escaped respectiely at time t In this paper, the goal is to find policies which maximize the fraction of targets c [,1] captured by the ehicle where Fig 1 c := limsupe t + ] n capt (t n capt (t + n esc (t [ Schematic of the Radially escaping targets (RET problem III PRELIMINARY RESULTS In this section, we proide some results which will be used to analyze policies for the RET problem A Constant bearing principle The optimal strategy (ie taking minimum time for a ehicle to capture a target moing at a speed less than that of the ehicle is to moe in a straight line with maximum speed to intercept the target based on the constant bearing principle[9] Accordingly, the time taken by the ehicle starting from (x, to reach a demand located at (r,θ and moing radially outward with constant speed is T (x,r,θ = (xcosθ r + ( (xcosθ r (1 (rxcosθ x r 1/ 1 To distinguish static targets from moing targets, we introduce some terminology Definition 31: (Escaping target A target moing radially outward is referred to as an escaping target A target is said to hae been captured by the ehicle if the ehicle reaches the target before it escapes the enironment Fig Optimal ehicle location x and the maximum probability ρ of capturing an escaping target starting from (x, as a function of target speed B Optimal placement of ehicle By optimal placement, we mean the location at which the ehicle should be placed in order for it to hae the highest probability of capturing a target To determine optimal placement, we start by defining the capturable set Definition 3: (Capturable set A ehicle located at (x, can only reach demands located in the capturable set C(x, := (r,θ : r < r c θ [,π} using the constant bearing principle, where ( r c (x,,θ = max,1 1 + x xcosθ These are the locations for which r + T (x,r,θ 1 The expression for r c is obtained by setting r c + T (x,r c,θ = 1 The probability that a target is in the capturable set of a particular ehicle location (x, is ρ(x, := = π 1 π rc [P(r,θ C(x,] f 1 (r f (θdrdθ f 1 (r f (θdrdθ When the ehicle is at location (x (, where x ( := arg max ρ(x,, (1 x 1 the probability of it capturing a target is maximum The ehicle location x is referred to as the optimal location Let ρ ( := ρ(x, The ariation of x ( and ρ ( is shown in Fig For target speed 5, x = and the ehicle location (, maximizes the probability of the ehicle being able to capture a target before it escapes For higher speeds, this location is closer to the boundary There is a qualitatie difference between these two cases In the former case, (, is the unique ehicle location which maximizes ρ whereas for the later case, the set of corresponding optimal locations is (x (,θ θ [,π} We also define the expected time T e taken by a ehicle starting from (x, to serice capturable targets appearing with a uniform spatial distribution in the enironment T e (x, := π Further, T e := T e (x, rc T (x,r,θ f 1(r f (θdrdθ ρ(x,

3 C Quantification of targets inside the enironment The number of targets in an unsericed region in the enironment is quantified now We distinguish between targets originating and accumulating in a certain region Targets are said to hae accumulated in a region when after appearing, they spend time in the region in the course of their trajectories Lemma 33: (Expected number of targets The expected number of targets at steady state in an unsericed (i disk of radius s 1 centered at (, is s 3 λ/3 with probability density function f a (r = λr / and (ii enironment E is λ/3 Proof: Firstly, steady state is assumed, meaning that the initial transient has already passed, hence t 1/ Also, by unsericed, we mean that the ehicle has not sericed targets in the region under consideration for at least time 1/ before the time instant under consideration Let us examine the number of demands originating in the ring of radius p (,1] and accumulating in the annulus B r of inner radius r [p,1 r] and thickness r The intensity of the Poisson arrial process λ p on this ring is directly proportional to its area Hence, λ p = pλ P(B r contains n demands originating from p = P(n demands originated from p in time interal [t,t + r/] = P(n demands originated from p in time interal r/ ( λp r n = e λ p r/ n! where t = (r p/ Now, since Poisson processes are additie, the Poisson distribution of all the targets inside B r has strength r p= (λ p r/d p = (r λ r/ It should be noted that the thickness r 1 and the aboe expression does not include targets originating inside B r Thus the expected number of targets in an unsericed annulus of inner radius r and thickness r due to targets originating from the disk of radius r is r λ r/ Next, consider a disk of radius s 1 Since Poisson processes are additie, the expected number of targets in this disk is the sum of expected number of targets inside disjoint annuli like B r In integral form, the expected number of targets in the disk is s (r λ/dr = s 3 λ/3 Let f a (r be the probability density function of the accumulating targets in the disk, where r denotes the random ariable which is the radial coordinate of a target Since r f a(sds = r 3 λ/3, we get f a (r = λr / D Bounds on paths and tours through static and escaping targets The following results are used to estimate and bound the length of the path/tour through targets in the enironment Theorem 34: (Upper bound on path through escaping targets Let targets starting from (r i,θ i, i 1,,N} moe radially outward with speed Let T be the length of the path through these escaping targets in some arbitrary order γ : 1,,N} 1,,N} Let T s be the length of the path Fig 3 The thick line labeled T i+1 indicates the trajectory of the ehicle starting from the target i to serice the target i + 1 through static targets located at (r i + T,θ i, i 1,,N} processed in order γ and T T Then, T 1 Proof: Without loss of generality, let the targets be labeled in the order in which they are processed Let the ehicle coer distance T j to serice the jth escaping target after haing sericed the ( j 1th escaping target Consider the ith escaping target starting from (r i,θ i The ehicle serices this target after haing coered the distance i j=1 T j It then starts for the escaping target i + 1 and reaches it in time T i+1 Let T i+1 be the distance between (r i + i+1 j=1 T j,θ i and (r i+1 + i+1 j=1 T j,θ i+1 Also, let T i+1 be the distance between (r i + T,θ i and (r i+1 + T,θ i+1 while T s,i+1 is the distance between (r i + T,θ i and (r i+1 + T,θ i+1 Since the distance between two targets moing radially outward with the same speed is a non-decreasing function of time, T i+1 T i+1 T s,i+1 Referring to Fig 3, from the triangle inequality, T i+1 + T i+1 T i+1, ie T i+1 (T i+1 /(1 (T s,i+1 (1 Extending this to all the targets in the path, T = n i=1 T i+1 T s n i=1 T s,i+1 1 = T s 1 Theorem 35: (Lower bound on path through escaping targets Let targets starting from (r i,θ i, i 1,,N} moe radially outward with speed Let T be the length of the path through these escaping targets in some arbitrary order γ : 1,,N} 1,,N} Let T be the length of the path through static targets located at (r i,θ i, i 1,,N} processed in order γ Then, T T 1 + Proof: The proof is similar to that of Theorem 34 and is omitted for breity Gien a set D of n points, the Euclidean traeling salesperson problem (ETSP is to determine the shortest tour, ie, a closed path that isits each point exactly once

4 Algorithm 1: Capturable Nearest Neighbor (CNN policy Gien: x ( 1 if C((x,, contains outstanding demands then serice demand closest to (x,θ and in C(x, using constant bearing control; 3 else 4 return to (x (, 5 end Theorem 36: (Asymptotic ETSP length,[3] If a set D of n points is distributed independently and identically in a compact set Q, then there exists a constant β T SP, such that E[ET SP(D] lim = β T SP, ϕ(q 1/ dq, n + n Q where ϕ is the density of the absolutely continuous part of the point distribution The constant β T SP has been estimated numerically as β T SP 71 ± [11] IV POLICIES In this section, we propose two policies for the RET problem We start by proiding an upper bound on the capture fraction c for any policy for the RET problem Theorem 41: (Capture fraction upper bound For eery policy P for the RET(,λ problem, c(p ρ ( We now introduce two policies: the capturable nearest neighbor (CNN and the sector-wise (SW policy A Capturable Nearest Neighbor (CNN Policy At arrial rates below a critical alue, optimizing the location at which the ehicle waits for new targets to arrie assumes importance The CNN policy stated in Algorithm 1 is motiated by this idea Theorem 4: (Performance of CNN policy Gien λ c = 1/(ρ Te, the expected fraction of targets c sericed by the CNN policy satisfies ρ (, if λ λ c c = 1/(λTe, if λ λ c Theorem 43: (CNN policy lower bound The expected fraction of targets c sericed by the CNN policy satisfies the following conditions if x =, c ρ ( if λ 1/4 1/4λ(1 + x, if λ 1/4 if x 1 ρ ( if λ /4, c 1/4λ(1 + x, if λ /4 ρ ( if λ /4 /4λ, if λ /4 if x 1, c Remark 44: (Optimality of the CNN policy When λ λ c, the expected fraction of targets caught by the CNN policy from Theorem 4 is ρ ( and based on Theorem 41, this performance is optimal Algorithm : Sector Wise (SW policy Gien: a,b, θ 1 i = 1 initialization; θ 1 (i 1 θ modulo π, θ i θ modulo π ; 3 if A(a,b,θ 1,θ contains outstanding demands then 4 serice demand closest to (b,θ 1 and in A(a,b,θ 1,θ and moe to (b,θ ; 5 i = i + 1 ; 6 else 7 moe to (b,θ along the arc with radius b ; 8 i = i + 1 ; 9 end B Sector-wise policy (SW Policy The SW policy stated in Algorithm is designed for the regime where λ 48/π(1 3 The ehicle traerses an annular sector in eery iteration of the SW policy The parameters a, b and θ of the SW policy are optimized to maximize the capture fraction Definition 45: (Annular Sector The set A(a,b,θ 1,θ := (r,θ : a r b θ [θ 1,θ ]}, where b a Lemma 46: (Farthest capturable target Gien A(a, b,, θ with θ π, if a ehicle traeling with unit speed and starting from (b, can capture a target located at (b, θ and moing radially outward with speed then it can capture any target starting in A(a,b,, θ and moing radially outward with speed Theorem 47: (Routing in A(a,b,, θ Gien an unsericed annular sector A(a,b,, θ, with a < b < 1, the following holds true: (i If a = (b 3 6π/λ θ 1/3, then E[n] = 1 where n is the number of outstanding ( targets in A(a,b,, θ b (ii If θ = cos (1 b /, then starting b at (b,, a ehicle can capture any moing target in A(a,b,, θ before it escapes the disk (iii If πb (b a/, then the ehicle starting from (b, at time t and moing along the circle of radius b will not return back to A(a, b,, θ before all targets accumulated in A(a,b,, θ at time t escape A(a,b,, θ The constraints (i - (iii in Theorem 47 can be simplified and the maximum alue b that b can take subject to these constraints can be found by soling the following concae maximization problem: [ 1 maximize b (1,1 subject to b 3 θ 1π λ, ( b θ = cos (1 b / b b 3 6π θ λ(6π 1π + 8π 3 3, (

5 When λ 48/π(1 3, the concae maximization problem is feasible and can be soled to obtain b The other two parameters can then be set: θ = cos 1 ( b + 1 (1 b / b ( a = b 1π 1/3 λ θ, Theorem 48: (SW policy lower bound The expected fraction c of targets sericed by the SW policy satisfies 1 c λ (1 b b b θ Theorem 49: (SW policy upper bound The expected fraction c of targets sericed by the SW policy satisfies c 1 b λ θ C Stay Near Boundary (SNB Policy The SNB policy is now introduced for the high arrial regime In inoles using solution to the Euclidean Minimum Hamiltonian Path (EMHP problem which can be stated as follows: Gien a set D of n points, determine the length of the shortest path which isits each point exactly once Algorithm 3: Stay Near Boundary (SNB policy Gien: ã, b 1 if A(ã, b,, π contains outstanding demands then s 1 := set of locations of outstanding targets in A(ã, b,,π; 3 s := set of their locations if they moe radially outward by distance (1 b ; 4 x := order of the EMHP starting from (1,, isiting demands in s and ending at (1, ; 5 serice demands in s 1 in order x using constant bearing principle and return to ( b, ; 6 end The parameters ã and b of the SNB policy are solution of the following optimization problem: ( ( 1 b 3 a 3 maximize a,b λπ b a subject to 3β T SP δ 1 (b a (1 b, 1 δ (b + a, 3 (b 3 a 3, a < b < 1 where δ 1 = k/(1, δ = k/(1 + and k = β T SP λπ/ Theorem 41: (SNB policy lower bound The expected fraction c of targets sericed by the SNB policy as λ + satisfies (1 3 c β T SP πλ Proof: Consider the set D := (r,θ A(a,b,,π} of locations (r,θ of targets accumulated in A(a,b,,π The normalized probability density function of the radial locations of these targets is f (r = 3r /(b 3 a 3 for r [a,b] Let D be the set of locations (r,θ of these targets if they moe outwards by distance d and occupy A(a + d,b + d,,π The normalized probability density functions for the random ariables r and θ which denote locations of these targets are f (r = 3(r d b 3 a 3 and ḡ(θ = 1 π Using Theorem 36, ET SP( D 3 θ lim = β T SP n + n b 3 a 3 ( b a Here n = D and since the targets are accumulated in A(a,b,,π, E[n] = λ(b 3 a 3 /3 For n +, the length of the ETSP through static targets D conerges to the length of the EMHP starting from s = (1,, passes through each of the n points in D exactly once, and terminating at f = (1, This follows from the fact that as n +, the upper bound on the distance between two arbitrary points in D is negligible when compared to the length of the tour/path Using Theorem 34 and assuming that the ehicle traerses the path through moing targets in D before they escape the enironment, the expected length of the path through escaping targets accumulated in A(a, b,, π and located in the set D is upper bounded by t u := β T SP 1 The condition β T SP πλ 1 ( πλ b a ( b a 1 b ensures that the targets in D are sericed before they escape the enironment and Theorem 34 is applicable The condition ( β T SP πλ b a b a 1 + ensures that A(a, b,, π is unsericed when the ehicle begins its tour at (1, Finally, when the condition (b 3 a 3 3 is satisfied, E[n] + as λ + The lower bound on the expected rate at which targets are sericed using the SNB policy is E[n]/t u and ( 1 c 3β T SP λπ ( b 3 a 3 b a Subject to the constraints in the optimization problem, (b 3 a 3 /(b a 3(1 / and the result is obtained

6 c CNN, λ = experimental expected fraction lower bound c SW policy, λ = 1 simulation lower bound upper bound Fig 4 Performance of the (a CNN and (b SW policies for arrial rates λ = and λ = 1 respectiely fraction target speed 1 arrial rate Remark 411: (Necessary condition for SNB policy A necessary condition for existence of a solution to the optimization problem in Theorem 41 for the SNB policy is that 1/delta < so that the second constraint in the problem leads to alid configurations (a,b for the policy This is true when (1 + λ > πβt SP Numerical simulations suggest that this condition is also sufficient for the existence of a solution to the optimization problem in Theorem 41 Theorem 41: (SNB policy upper bound The expected fraction c of targets sericed by the SNB policy as λ + satisfies 1 + c β T SP πλ Remark 413: (Optimality of SNB policy In the limit as + and λ ((1 + πβt SP +, the capture fraction of the SNB policy satisfies c (1 3 /(1+ Since x = and ρ = (1 for +, the SNB policy is within a constant factor (1 /(1 + of optimal in this regime V NUMERICAL SIMULATIONS We now present numerical simulations for the CNN and SW policies The CNN policy gies optimal performance for λ λ c based on numerical simulations Its performance at λ = is examined The actual simulation results, shown in Fig 4(a comply with the performance bounds obtained in Theorem 4 and 43 Performance of the SW policy in numerical simulations is shown for λ = 1 in Fig 4(b Lastly, the lower bounds on the fraction of targets captured for both the CNN and SW policies are compared for range of parameters and λ and the larger of the two alues chosen, shown in Fig 5 For a particular target speed, the CNN policy is suitable for lower arrial rates for the RET problem The SW policy shows a better performance for higher arrial rates VI CONCLUSION We hae introduced a dynamic ehicle routing problem in which demands appear uniformly distributed on a disk according to temporal Poisson process We propose three policies for a single ehicle to serice the demands before they escape the disk The CNN policy is optimal for arrial rate below a critical alue for all target speeds (,1 The performance of the SNB policy is close to optimal in Fig 5 Lower bound on c from the CNN and SW policy is calculated numerically The maximum of the two is shown for RET(,λ the limiting regime of λ + and + The SW policy is suitable for other regimes of the RET problem Extensions of the RET problem like for instance, when the distribution of targets in the enironment is not uniform, or when the enironment is an arbitrary closed cure are open to exploration REFERENCES [1] Y Asahiro, E Miyano, and S Shimoirisa Grasp and deliery for moing objects on broken lines Theory of Computing Systems, 4(3:89 35, 8 [] N Bansal, A Blum, S Chawla, and A Meyerson Approximation algorithms for deadline-tsp and ehicle routing with time-windows In Proceedings of the thirty-sixth annual ACM symposium on Theory of Computing, pages , 4 [3] J Beardwood, J Halton, and J Hammersly The shortest path through many points In Proceedings of the Cambridge Philosophy Society, olume 55, pages 99 37, 1959 [4] S D Bopardikar, S L Smith, F Bullo, and J P Hespanha Dynamic ehicle routing for translating demands: Stability analysis and receding-horizon policies IEEE Transactions on Automatic Control, 55(11: , 1 [5] F Bullo, E Frazzoli, M Paone, K Sala, and S L Smith Dynamic ehicle routing for robotic systems Proceedings of the IEEE, 99(9: , 11 [6] P Chalasani and R Motwani Approximating capacitated routing and deliery problems SIAM Journal on Computing, 8(6: , 1999 [7] M Hammar and B J Nilsson Approximation results for kinetic ariants of TSP Discrete and Computational Geometry, 7(4: , [8] C S Helig, G Robins, and A Zelikosky The moing-target traeling salesman problem Journal of Algorithms, 49(1: , 3 [9] R Isaacs Differential Games Wiley, 1965 [1] M Paone, N Bisnik, E Frazzoli, and V Isler A stochastic and dynamic ehicle routing problem with time windows and customer impatience ACM/Springer Journal of Mobile Networks and Applications, 14(3:35 364, 9 [11] A G Percus and O C Martin Finite size and dimensional dependence of the Euclidean traeling salesman problem Physical Reiew Letters, 76(8: , 1996 [1] V Pillac, M Gendreau, C Guéret, and A L Medaglia A reiew of dynamic ehicle routing problems European Journal of Operational Research, 5(1:1 11, 1 [13] H N Psaraftis Dynamic ehicle routing: Status and prospects Annals of Operations Research, 61: , 1995 [14] S L Smith, S D Bopardikar, and F Bullo A dynamic boundary guarding problem with translating demands In IEEE Conf on Decision and Control and Chinese Control Conference, pages , Shanghai, China, December 9 [15] P Toth and D Vigo, editors The Vehicle Routing Problem Monographs on Discrete Mathematics and Applications SIAM, 1