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Dynamic Vehicle Routing for Translating Demands: Stability Analysis and Receding-Horizon Policies The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bopardikar, S.D. et al. “Dynamic Vehicle Routing for Translating Demands: Stability Analysis and Receding-Horizon Policies.” Automatic Control, IEEE Transactions On 55.11 (2010) : 25542569. © 2011 IEEE. As Published http://dx.doi.org/10.1109/tac.2010.2049278 Publisher Institute of Electrical and Electronics Engineers Version Final published version Accessed Thu May 26 19:14:28 EDT 2016 Citable Link http://hdl.handle.net/1721.1/64987 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms 2554 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 Dynamic Vehicle Routing for Translating Demands: Stability Analysis and Receding-Horizon Policies Shaunak D. Bopardikar, Member, IEEE, Stephen L. Smith, Member, IEEE, Francesco Bullo, Fellow, IEEE, and João P. Hespanha, Fellow, IEEE Abstract—We introduce a problem in which demands arrive stochastically on a line segment, and upon arrival, move with a fixed velocity perpendicular to the segment. We design a receding horizon service policy for a vehicle with speed greater than that of the demands, based on the translational minimum Hamiltonian path (TMHP). We consider Poisson demand arrivals, uniformly distributed along the segment. For a fixed segment width and fixed vehicle speed, the problem is governed by two parameters; the demand speed and the arrival rate. We establish a necessary condition on the arrival rate in terms of the demand speed for the existence of any stabilizing policy. We derive a sufficient condition on the arrival rate in terms of the demand speed that ensures stability of the TMHP-based policy. When the demand speed tends to the vehicle speed, every stabilizing policy must service the demands in the first-come-first-served (FCFS) order; and the TMHP-based policy becomes equivalent to the FCFS policy which minimizes the expected time before a demand is serviced. When the demand speed tends to zero, the sufficient condition on the arrival rate for stability of the TMHP-based policy is within a constant factor of the necessary condition for stability of any policy. Finally, when the arrival rate tends to zero for a fixed demand speed, the TMHP-based policy becomes equivalent to the FCFS policy which minimizes the expected time before a demand is serviced. We numerically validate our analysis and empirically characterize the region in the parameter space for which the TMHP-based policy is stable. Index Terms—Autonomous vehicles, dynamic vehicle routing, minimum Hamiltonian path, queueing theory. I. INTRODUCTION EHICLE routing problems are concerned with planning optimal vehicle routes for providing service to a given set of customers. The routes are planned with complete information of the customers, and thus the optimization is static, and typically combinatorial [3]. In contrast, dynamic vehicle routing (DVR) considers scenarios in which not all customer information is known a priori, and thus routes must be re-planned as V Manuscript received March 17, 2009; revised November 17, 2009; accepted February 27, 2010. Date of publication April 29, 2010; date of current version November 03, 2010. This work was supported in part by ARO-MURI Award W911NF-05-1-0219, ONR Award N00014-07-1-0721 and by the Institute for Collaborative Biotechnologies through grant DAAD19-03-D-0004 from the U.S. Army Research Office. Recommended by Associate Editor I. Paschalidis. S. D. Bopardikar, F. Bullo, and J. P. Hespanha are with the Center for Control, Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106 USA (e-mail: [email protected]; [email protected]; [email protected]). S. L. Smith is with the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge MA 02139 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2049278 new customer information becomes available. DVR problems naturally occur in scenarios where autonomous vehicles are deployed in complex and uncertain environments. Examples include search and reconnaissance missions, and environmental monitoring. An early DVR problem is the dynamic traveling repairperson problem (DTRP) [4], in which customers, or demands arrive sequentially in a region and a service vehicle seeks to serve them by reaching each demand location. In this paper, we introduce a dynamic vehicle routing problem in which the demands move with a specified velocity upon arrival, and we design a novel receding horizon control policy for a single vehicle to service them. This problem has applications in areas such as surveillance and perimeter defense, wherein the demands could be visualized as moving targets trying to cross a region under surveillance by an Unmanned Air Vehicle [5], [6]. Another application is in the automation industry where the demands are objects that arrive continuously on a conveyor belt and a robotic arm performs a pick-and-place operation on them [7]. A. Contributions We introduce a dynamic vehicle routing problem in which demands arrive via a stochastic process on a line segment of fixed length, and upon arrival, translate with a fixed velocity perpendicular to the segment. A service vehicle, modeled as a first-order integrator having speed greater than that of the demands, seeks to serve these mobile demands. The goal is to design stable service policies for the vehicle, i.e., the expected time spent by a demand in the environment is finite. We propose a novel receding horizon control policy for the vehicle that services the translating demands as per a translational minimum Hamiltonian path (TMHP). In this paper, we analyze the problem when the demands are uniformly distributed along the segment and the demand arrival of the segprocess is Poisson with rate . For a fixed length ment and the vehicle speed normalized to unity, the problem is governed by two parameters; the demand speed and the arrival rate . Our results are as follows. First, we derive a necessary condition on in terms of for the existence of a stable service policy. Second, we analyze our novel TMHP-based policy and derive a sufficient condition for in terms of that ensures stability of the policy. With respect to stability of the problem, we identify two asymptotic regimes: a) High speed regime: when the demands move as fast as the vehicle, i.e., (and therefore for stability, ); and b) High arrival regime: when the arrival rate tends to infinity, i.e., (and therefore for stability, ). In the high speed regime, we show that: i) for existence of a stabilizing policy, must converge to zero as , ii) every stabilizing policy must service the demands in the first-come-first-served (FCFS) order, 0018-9286/$26.00 © 2010 IEEE BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS Fig. 1. Summary of stability regions for the TMHP-based policy. Stable service policies exist only for the region under the solid black curve. In the top figure, the solid black curve is due to part (i) of Theorem III.1 and the dashed blue curve is due to part (i) of Theorem III.2. In the asymptotic regime shown in the bottom left, the dashed blue curve is described in part (ii) of Theorem III.2, and is different than the one in the top figure. In the asymptotic regime shown in the bottom right, the solid black curve is due to part (ii) of Theorem III.1, and is different from the solid black curve in the top figure. and iii) the TMHP-based policy becomes equivalent to the FCFS policy which we establish is optimal in terms of minimizing the expected time to service a demand. In the high arrival regime, we show that the sufficient condition on for the stability of the TMHP-based policy is within a constant factor of the necessary condition on for stability of any policy. Third, we identify another asymptotic regime, termed as the low arrival regime, in which the arrival rate for a fixed demand speed. In this low arrival regime, we establish that the TMHP-based policy becomes equivalent to the FCFS policy which we establish is optimal in terms of minimizing the expected time to service a demand. Fourth, for the analysis of the TMHP-based policy, we study the classic FCFS policy in which demands are served in the order in which they arrive. We determine necessary and sufficient conditions on for the stability of the FCFS policy. Fifth and finally, we validate our analysis with extensive simulations and provide an empirically accurate characterization of the region in the parameter space of demand speed and arrival rate for which the TMHP-based policy is stable. Our numerical results show that the theoretically established sufficient stability condition for the TMHP-based policy in the high arrival asymptotic regime serves as a very good approximation to the stability boundary, for nearly the entire range of demand speeds. A plot of the theoretically established necessary and sufficient conditions for stability in the - parameter space is shown in Fig. 1. The bottom figures are for the asymptotic regimes of , and , respectively. B. Related Work Dynamic vehicle routing (DVR) is an area of research that integrates fundamentals of combinatorics, probability and queueing theories. One of the early versions of a DVR problem is the Dynamic Traveling Repairperson Problem [4] in which 2555 the goal is to minimize the expected time spent by each demand before being served. In [4], the authors propose a policy that is optimal in the case of low arrival rate, and several policies within a constant factor of the optimal in the case of high arrival rate. In [8], they consider multiple vehicles, and vehicles with finite service capacity. In [9], a single policy is proposed which is optimal for the case of low arrival rate and performs within a constant factor of the best known policy for the case of high arrival rate. Recently, there has been an upswing in versions of DVR such as in [10] where different classes of demands have been considered; and in [11] which addresses the case of demand impatience. DVR problems addressing motion constraints on the vehicle have been presented in [12], while limited sensing range for the vehicle has been considered in [13]. Problems involving multiple vehicles and with minimal communication have been considered in [14]. In [15], adaptive and decentralized policies are developed for the multiple service vehicle versions, and in [16], a dynamic team-forming variation of the Dynamic Traveling Repairperson problem is addressed. Related dynamic problems include [17], wherein a receding horizon control has been proposed for multiple vehicles to visit target points in uncertain environments; and [18], wherein pickup and delivery problems have been considered. Another related area of research is the version of the Euclidean Traveling Salesperson Problem (ETSP) with moving points or objects. The static version of the ETSP consists of determining the minimum length tour through a given set of static points in a region [19]. Vehicle routing with objects moving on straight lines was introduced in [7], in which a fixed number of objects move in the negative direction with fixed speed, and the motion of the service vehicle is constrained to be parallel to either the or the axis. For a version of this problem wherein the vehicle has arbitrary motion, termed as the translational Traveling Salesperson Problem, a polynomial-time approximation scheme has been proposed in [20] to catch all objects in minimum time. Another variation of this problem with object motion on piece-wise straight line paths, and with different but finite object speeds has been addressed in [21]. Other variants of the ETSP in which the points are allowed to move in different directions have been addressed in [22]. A third area of research related to this work is a geometric location problem such as [23], and [24], where given a set of static demands, the goal is to find supply locations that minimize a cost function of the distance from each demand to its nearest supply location. In our problem, in the asymptotic regime of low arrival, when the arrival rate tends to zero for a fixed demand speed , the problem becomes one of providing optimal coverage. In this regime, the demands are served in a first-come-first-served order; such policies are common in classical queueing theory [25]. Another related work is [26], in which the authors study the problem of deploying robots into a region so as to provide optimal coverage. Other relevant literature include [27] in which multiple mobile targets are to be kept under surveillance by multiple mobile sensor agents. The authors in [28] propose mixed-integer-linearprogram approach to assigning multiple agents to time-dependent cooperative tasks such as target tracking. In an early version of this work [1], [2], we analyzed a preliminary version of the TMHP-based policy and the FCFS policy respectively. In this paper, we present and analyze a unified 2556 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 minimum Hamiltonian path (EMHP) is a Euclidean Hamiltonian path that has minimum length. In this paper, we also consider the problem of determining a constrained EMHP which starts at a specified start point, visits a set of points and terminates at a specified end point. More specifically, the EMHP problem is as follows. Fig. 2. Constant bearing control. The vehicle moves towards the point C := x; y + vT ), where x, y , v and T are as per Definition II.1, to reach the demand. ( policy for the problem. We also present new simulation results to numerically determine the region of stability for the policy proposed in this paper. As in [17] and in [18], the problem addressed in this paper applies several control theoretic concepts to a relevant robotic application. In particular, we design policies or control laws in order to stabilize a stochastic and dynamic system. The solution that we propose is a version of receding horizon control, in which one computes the optimal solution based on all of the present information and then repeatedly recomputes as new information becomes available. Our stability analysis relies on writing the closed-loop dynamical system as a recursion in a state of the system, and then ensuring that this state has bounded evolution. C. Organization This paper is organized as follows. A short review of results on optimal motion and combinatorics is presented in Section II. The problem formulation, the TMHP-based policy, and the main results are presented in Section III. The FCFS policy is presented and analyzed in Section IV. Utilizing the results of Section IV, the main results are proven in Section V. Finally, simulation results are presented in Section VI. Given static points placed in , determine the length of the shortest path which visits each point exactly once. An upper bound on the length of such a path for points in a unit square was given by Few [30]. Here we extend Few’s bound to the case of points in a rectangular region. For completeness, we have included the proof in Appendix. Lemma II.3 (EMHP Length): Given points in a rectangle in the plane, where , there exists a path that starts from a unit length edge of the rectangle, passes through each of the points exactly once, and terminates on the opposite unit length edge, with length upper bounded by Given a set of points in , the ETSP is to determine the shortest tour, i.e., a closed path that visits each point exactly denote the length of the ETSP tour through once. Let . The following is a classic result from [31]. Theorem II.4 (Asymptotic ETSP Length, [31]): If a set of points are distributed independently and uniformly in a compact region of area , then there exists a constant such that, with probability one (2) II. PRELIMINARY RESULTS In this section, we provide some useful background results. A. Constant Bearing Control The following control is used to reach a moving demand. Definition II.1 (Constant Bearing Control): Given initial locations and of the vehicle and a demand, respectively, with the demand moving in the positive -direction with constant speed , the motion of the service vehicle towards the point , where (1) with unit speed is defined as the constant bearing control. Shown in Fig. 2, the constant bearing control is known to reach a demand in minimum time as is characterized in the following proposition. Proposition II.2 (Minimum Time Control, [29]): The constant bearing control is the minimum time control for the service vehicle to reach the moving demand. B. Euclidean and Translational Minimum Hamiltonian Path (EMHP/TMHP) Problems Given a set of points in the plane, a Euclidean Hamiltonian path is a path that visits each point exactly once. A Euclidean The constant has been estimated numerically as , [32]. Next, we describe the TMHP problem which was proposed and solved in [20]. This problem is posed as follows. Given initial coordinates; of a start point, of a set of points, and of a finish point, all moving with the same constant speed and in the same direction, determine a path that starts at time zero from point , visits all points in the set exactly once and ends at the finish point, and the length of which is minimum. We wish to determine the TMHP through points which translate in the positive direction. We also assume the speed of the service vehicle to be normalized to unity, and hence consider the speed of the points . A solution for the TMHP problem is the Convert-to-EMHP method: i) For , define the conversion map by ii) Compute the EMHP that starts at , passes through the set of points given by and ends at . iii) Move between any two demands using the constant bearing control. The following result is established. BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS 2557 Fig. 4. An iteration of the TMHP-based policy. The vehicle shown as a square serves all outstanding demands shown as black dots as per the TMHP that begins at ( ) and terminates at which is the demand with the least -coordinate. The first figure shows a TMHP at the beginning of an iteration. The second figure shows the vehicle servicing the demands through which the TMHP has been computed while new demands arrive in the environment. The third figure shows the vehicle repeating the policy for the set of new demands when it has completed service of the demands present at the previous iteration. X;Y Fig. 3. Problem set-up. The thick line segment is the generator of mobile demands. A dark circle denotes a demand and the square denotes the vehicle. Lemma II.5 (TMHP Length, [20]): Let the initial coordinates and , and the speed of the points . The length of the TMHP is where denotes the length of the EMHP with starting point , passing through , and ending at . the set of points This lemma implies the following result: given a start point, a set of points and an end point all of whom translate in the positive vertical direction at speed , the order of the points followed by the optimal TMHP solution is the same as the order of the points followed by the optimal EMHP solution through a set of static locations equal to the locations of the moving points at initial time converted via the map . III. PROBLEM STATEMENT AND TMHP-BASED POLICY In this section, we pose the dynamic vehicle routing problem with translating demands and present the TMHP-based policy along with the main results. A. Problem Statement We consider a single service vehicle that seeks to service mobile demands that arrive via a spatio-temporal process on a line segment with length along the -axis, termed the generator. The vehicle is modeled as a first-order integrator with speed upper bounded by one. The demands arrive uniformly distributed on the generator via a temporal Poisson process with intensity , and move with constant speed along the positive -axis, as shown in Fig. 3. We assume that once the vehicle reaches a demand, the demand is served instantaneously. The vehicle is assumed to have unlimited fuel and demand servicing capacity. We define the environment as , and let denote the position of the service vehicle at time . Let denote the set of all demand locations at time , and the cardinality of . Servicing of a demand and removing it from the set occurs when the service vehicle reaches the location of the demand. A static feedback control policy for the system is a map , where is the set of finite subsets of , q y assigning a commanded velocity to the service vehicle as a func. tion of the current state of the system: Let denote the time that the th demand spends inside the set , that is, the delay between the generation of the th demand and the time it is serviced. The policy is stable if under its action that is, the steady state expected delay is finite. Equivalently, the policy is stable if under its action that is, if the vehicle is able to service demands at a rate that is—on average—at least as fast as the rate at which new demands arrive. In what follows, our goal is to design stable control policies for the system. To obtain further intuition into stability of a policy, consider the - parameter space. In the asymptotic regime of high speed, where , the arrival rate must tend to zero for stability, otherwise the service vehicle would have to move successively further away from the generator in expected value, thus making the system unstable. Similarly, stability in the asymptotic region of high arrival, where implies that the demand speed must tend to zero. Thus, our primary goals are: i) to characterize regions in the - parameter space in which one can never design any stable policy, and ii) design a novel policy and determine its stability region in the - parameter space, with additional emphasis in the above two asymptotic regimes. In addition, for the asymptotic regime of low arrival, where for a , the arrival rate , stability is intuitive fixed speed as demands arrive very rarely. Hence, in this regime, we seek to minimize the steady state expected delay for a demand. B. The TMHP-Based Policy We now present a novel receding horizon service policy for the vehicle that is based on the repeated computation of a translational minimum Hamiltonian path through successive groups of outstanding demands. For a given arrival rate and demand speed , let denote the vehicle location in the environment that minimizes the expected time to service a demand once it appears on the generator. The expression for is postponed to Section IV-A. The TMHP-based policy is summarized in Algorithm 1, and an iteration of the policy is illustrated in Fig. 4. 2558 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 i) For a general Algorithm 1: The TMHP-based policy Assumes: The optimal location 1 if no outstanding demands are present in is given. then ii) In the asymptotic regime of high arrival where (and so ), the policy is stable if 2 3 else 4 5 A plot of the necessary and sufficient conditions is shown in Fig. 1. In the asymptotic regime of high speed, the sufficient condition from part (i) of Theorem III.2 simplifies to Repeat. C. Main Results The following is a summary of our main results and the locations of their proofs within the paper. We begin with a necessary condition on the problem parameters for stability of any policy, the proof of which is presented in Section V-A. The condition is policy independent and especially in the asymptotic regime (and therefore ), we charof high speed, i.e., acterize the way in which . Theorem III.1 (Necessary Condition for Stability): The following are necessary conditions for the existence of a stabilizing policy: i) For a general ii) For the asymptotic regime of high speed, where , every stabilizing policy must serve the demands in the order in which they arrive. Further, the arrival rate must tend to zero as per Next, we present a sufficient condition on the problem parameters that ensures stability of the TMHP-based policy, the proof of which is presented in Section V-B. We present a condition for a general value of the demand speed and especially in the (and thereasymptotic regime of high arrival, i.e., fore ), we characterize the way in which and , in order to ensure stability of the TMHP-based policy. We first introduce the following notation. Let for , otherwise where , and is the solution to , and is approximately equal to 2/3. Theorem III.2 (Sufficient Condition for Stability): The following are sufficient conditions for stability of the TMHP-based policy. and the necessary condition established in part (ii) of Theorem III.1 simplifies to In the asymptotic regime of high arrival, the sufficient condition , from part (ii) of Theorem III.2 is and the necessary condition established in part (i) of Theorem . III.1 is Theorems III.1 and III.2 lead to the following corollary. Corollary III.3 (Constant Factor Sufficient Condition): In the asymptotic regime of i) high speed, which is the limit as , the ratio . ii) high arrival, which is the limit as , the ratio . The third and final result shows optimality of the TMHPbased policy with respect to minimizing the expected delay in the asymptotic regimes of low arrival and high speed. Theorem III.4 (Optimality of TMHP-Based Policy): In the asymptotic regimes of i) low arrival, where for a fixed ; and (and therefore ); ii) high speed, where the TMHP-based policy serves the demands in the order in which they arrive, and also minimizes the expected time to service a demand. In other words, the TMHP-based policy becomes equivalent to a first-come-first-served (FCFS) policy in the above two asymptotic regimes. Theorem III.4 is proven for the FCFS policy in Section V-B. By the equivalence between FCFS and the TMHP-based policies in the above two asymptotic regimes, the result also holds for the TMHP-based policy. To study the stability of the TMHP-based policy, we introduce and analyze the FCFS policy in Section IV. IV. THE FIRST-COME-FIRST-SERVED (FCFS) POLICY In this section, we present the FCFS policy and establish some of its properties, and use these properties as tools to analyze the TMHP-based policy. In the FCFS policy, the service vehicle uses constant bearing control and services the demands in the BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS Fig. 5. The FCFS policy. The vehicle services the demands in the order of their arrival in the environment, using the constant bearing control. order in which they arrive. If the environment contains no demands, the vehicle moves to the location which minimizes the expected time to catch the next demand to arrive. This policy is summarized in Algorithm 2. Algorithm 2: The FCFS policy Assumes: The optimal location 1 if no outstanding demands are present in is given. then 2559 We seek to place the vehicle at location that minimizes the expected time to service a demand once it appears on the generator. Demands appear at uniformly random positions on the generator and the vehicle uses the constant bearing control to to reach reach the demand. Thus, the expected time a demand generated at position from vehicle position is given by (3) The following lemma characterizes the way in which this expectation varies with the position . Lemma IV.2 (Properties of the Expected Time): The expected is convex in , for all . time Additionally, there exists a unique point that minimizes . Proof: Regarding the first statement, it suffices to show that is convex for all . To do this, the integrand in (3), with respect to and consider the Hessian of given by 2 3 else 4 5 Repeat. Fig. 5 illustrates an instance of the FCFS policy. The following lemma establishes the relationship between the FCFS policy and the TMHP-based policy. Lemma IV.1 (Relationship Between TMHP and FCFS): Given an arrival rate and a demand speed , if the FCFS policy is stable, then the TMHP-based policy is stable. Proof: Consider an initial vehicle position and a set of outstanding demands, all of which have lower -coordinates than the vehicle. Let us compare the amount of time required to service the outstanding demands using the TMHP-based policy with the amount of time required for the FCFS policy. Both policies generate paths through all outstanding demands, starting at the initial vehicle location, and terminating at the outstanding demand with the lowest -coordinate. By definition, the TMHPbased policy generates the shortest such path. Thus, the TMHPbased policy will require no more time to service all outstanding demands than the FCFS policy. Since this holds at every iteration of the policy, the region of stability of TMHP-based policy contains the region of stability for the FCFS policy. In Sections IV-A–C, we analyze the FCFS policy. We then combine these results with the above lemma to establish analogous results for the TMHP-based policy. The first question is, how do we compute the optimal position ? This is answered in Section IV-A. A. Optimal Vehicle Placement In this subsection, we study the FCFS policy when is fixed and . In this regime, stability is not an issue as demands arrive very rarely, and the problem becomes one of optimally placing the service vehicle (i.e., determining in the statement of the FCFS policy). . For , the Hessian is positive semi-definite because its determinant is zero and its trace is non-negative. This implies is convex in for each , from which the that first statement follows. Regarding the second statement, since demands are uniformly randomly generated on the interval , the optimal horizontal position is . Thus, it suffices to show that is strictly convex. From the term of the Hessian we see that is strictly convex for all . But, letting and we can write The integrand is strictly convex for all , implying that is strictly convex on the line , and that the point is the unique minimizer. Lemma IV.2 tells us that there exists a unique point which minimizes the expected travel time. In addition, we know that . Obtaining a closed form expression for does not appear to be possible. Computing the integral in (3), with , one can obtain where . For each value of and , this convex expression can be easily numerically minimized over , to obtain . A plot of as a function of for is shown in Fig. 6. For the optimal position , the expected delay between a demand’s arrival and its service completion is 2560 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 note that the only policy that ensures that a demand’s -coordinate never exceeds that of the service vehicle (i.e., that all demands remain below the service vehicle at all time) is the FCFS policy. In what follows, we prove the proposition statement by computing the expected time to travel between demands using the FCFS policy. First, consider a vehicle location and a demand location with initial location , the minimum time in which the vehicle can reach the demand is given by (4) Fig. 6. Optimal position Y of the vehicle that minimizes the expected distance . to a demand, as a function of v . The generator length W =1 Thus, a lower bound on the steady-state expected delay of any . We now characterize the steady-state expected policy is , as tends to zero. delay of the FCFS policy Lemma IV.3 (FCFS Optimality): Fix any . Then in the limit as , the FCFS policy minimizes the expected time to service a demand, i.e., . Proof: We have shown how to compute the position which minimizes (3). Thus, if the vehicle is located at , then the expected time to service the demand is minimized. , the probability that demand arrives before But, as the vehicle completes service of demand and returns to tends to zero. Thus, the FCFS policy is optimal as . Remark IV.4 (Minimizing the Worst-Case Time): Another metric that can be used to determine the optimal placement is the worst-case time to service a demand. Using an argument identical to that in the proof of Lemma IV.3, it is possible to show that for fixed , and as , the FCFS policy, with , minimizes the worst-case time to service a demand. . Second, assume there are many and is undefined if outstanding demands below the service vehicle, and none above. Suppose the service vehicle completed the service of . Let us comdemand at time and position pute the expected time to reach demand , with location . Since arrivals are Poisson, it follows that , with probability one. To simplify notation, and . we define Then, from (4) Taking expectation and noting that and Now, we note that , that constant independent of , and that Thus are independent is a positive , and for every B. A Necessary Condition for FCFS Stability In the previous subsection, we studied the case of fixed and . In this subsection, we analyze the problem when , and determine necessary conditions on the magnitude of that ensure the FCFS policy remains stable. To establish these conditions we utilize a standard result in queueing theory (cf. [25]) which states that a necessary condition for the existence of a stabilizing policy is that , where is the expected time to service a demand (i.e., the travel time between demands). We begin with the following result. Proposition IV.5 (Special Case of Equal Speeds): For , there does not exist a stabilizing policy. Proof: When , each demand and the service vehicle move at the same speed. If a demand has a higher vertical position than the service vehicle, then clearly the service vehicle cannot reach it. The same impossibility result holds if the demand has the same vertical position and a distinct horizontal position as the service vehicle. In summary, a demand can be reached only if the service vehicle is above the demand. Next, This means that the necessary condition for stability, i.e., , is violated. Thus, there does not exist a stabilizing policy. Next we look at the FCFS policy and give a necessary condition for its stability. Lemma IV.6 (Necessary Stability Condition for FCFS): A necessary condition for the stability of the FCFS policy is for otherwise , , where is the Euler constant; is the solution to the equation , and is approximately equal to . Proof: Suppose the service vehicle completed the service of demand at time at position , and demand is located at . Define where and BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS and time between demands is given by . For , the travel (5) Observe that the function inequality leads to is convex in and 2561 where is the 1st order Struve function and is 1st order Bessel function of the 2nd kind [33]. Using about a Taylor series expansion of the function , followed by a subsequent analysis of the higher order terms, one can show that . Jensen’s where , and is the Euler constant. This inequality implies that (8) can be written as Substituting the expressions for the expected values, we obtain where we have used the fact that From the necessary condition for stability, we must have To obtain a stability condition on we wish to remove the term. To do this, note that from (6) we have and thus from , By simplifying, we obtain (6) This provides a good necessary condition for low . Next, we obtain a much better necessary condition for large . Since is convex in , we apply Jensen’s inequality to (5) to obtain The necessary stability condition is necessary condition for stability is , from which a (7) . Now, the random variable is diswhere tributed exponentially with parameter and probability density function Solving for when , we obtain that (9) Un-conditioning (7) on , we obtain (8) The right hand side can be evaluated using the software Maple and equals The condition , implies that the above . We now have two bound holds for all , and (9) which holds bounds; (6) which holds for all for . The final step is to determine the values of for which each bound is active. To do this, we set the right-hand side of (6) equal to the right-hand side of (9) and denote the solution . Thus, the necessary stability condition is given by (6) by when , and by (9) when . C. A Sufficient Condition for FCFS Stability In Section IV-B, we determined a necessary condition for stability of the FCFS policy. In this subsection, we will derive the following sufficient condition on the arrival rate that ensures stability for the FCFS policy. To establish this condition, we utilize a standard result in queueing theory (cf. [25]) which states that a sufficient condition for the existence of a stabilizing policy is 2562 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 that , where is the expected time to service a demand (i.e., the travel time between demands). Lemma IV.7 (Sufficient Stability Condition for FCFS): The FCFS policy is stable if for where , and denotes the Bessel function of the first kind. To obtain a lower bound on , we observe that if , then due to the term , we can substitute in in the above lower bound for place of . Thus, we obtain , otherwise (13) where , and is the solution to , and is approximately equal to 2/3. Proof: We begin with the expression for the travel time between two consecutive demands using the constant bearing control (cf. Definition II.1) (10) where we used the inequality . Thus since the demands are distributed uniformly in the -direction and Poisson in the -direction. A sufficient condition for stability is Substituting into (12), we obtain which yields (14) Now, let be the least upper bound on for which the FCFS , the FCFS policy is policy is unstable, i.e., for every stable. To obtain , we need to solve . Using (14), we can obtain a lower bound on by simplifying (11) The upper bound on given by (10) is very conservative except for the case when is very small. Alternatively, taking expected value of conditioned on , and applying Jensen’s inequality to the square-root part, we obtain From the condition given by (11), the second term in the parentheses satisfies Thus, we obtain since . Following steps which are similar to those between (7) and (8), we obtain (12) In [33], polynomial approximations have been provided for the Struve and Bessel functions in the intervals and . We seek an upper bound for the right-hand side of (12) when is sufficiently large, i.e., when the argument of and is small. From [33], we know that where . Since stability, a sufficient condition for stability is implies (15) To determine the value of the speed beyond which this is a less conservative condition than (11), we solve For , one can verify that the numerical value of the argument of the Struve and Bessel functions is less than 2, and BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS 2563 that at time , contains points in (that is, the probability that ). We have Since the generation process is temporally Poisson and spatially uniform, the above equation can be rewritten as (16) Fig. 7. Necessary and sufficient conditions for the stability for the FCFS policy. The dashed curve is the necessary condition for stability as established in Lemma IV.6; while the solid curve is the sufficient condition for stability as established in Lemma IV.7. so the bounds given by (13) used in this analysis are valid. Thus, a sufficient condition for stability is given by equation(11) for , and by (15) for . Remark IV.8 (Tightness in Limiting Regimes): As , the sufficient condition for FCFS stability becomes , which is exactly equal to the necessary condition given in Lemma IV.6. Thus, the condition for stability is asymptotically tight in this limiting regime. Fig. 7 shows a comparison of the necessary and sufficient stability conditions for the FCFS policy. It should be noted that converges to extremely slowly as tends to , and still satisfy the sufficient stability condition in Lemma IV.7. For ex, the FCFS policy can stabilize the ample, with . system for an arrival rate of V. PROOFS OF THE MAIN RESULTS Now we compute and so that, substituting these expressions and adopting the shortand , (16) becomes hands Rewriting of binomial as (17) , and using the definition , (17) reads In this section, we present the proofs of the main results which were presented in Section III-C. A. Proof of Theorem III.1 We first present the proof of part (i). We begin by looking at the distribution of demands in the service region. Lemma V.1 (Distribution of Outstanding Demands): Suppose the generation of demands commences at time 0 and no demands are serviced in the interval . Let denote the set at time . Then, given a Borel of all demands in measurable set of area contained in Proof: We first establish the result for a rectangle. Let be a rectangle contained in with area . Let us calculate the probability Finally, since , we obtain where . Thus, the result is established for a closed rectangle. But notice that the result also holds for the case when a rectangle is open. Further, the result holds for a countable union of disjoint rectangles due to the demand generation process being uniform along the generator and Poisson in time. can be generated Now, a Borel measurable set on 2564 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 by countable unions and complements of rectangles. But both, the unions and the complements of rectangles can be written as a disjoint union of a countable number of rectangles. Hence, the . result holds for any Borel measurable set in Remark V.2 (Uniformly Distributed Demands): Lemma V.1 shows that the number of demands in an unserviced region , with area is Poisson distributed with parameter and conditioned on this number, the demands are distributed uniformly. Lemma V.3 (Travel Time Bound): Consider the set of demands that are uniformly distributed in at time . Let be a random variable giving the minimum amount of time required to . travel to a demand in from a vehicle position selected Then Proof: Let denote the vehicle location selected a priori. To obtain a lower bound on the minimum travel time, we consider the best-case scenario, when no demands have . Consider a demand in been serviced in the time interval with position at time . Using Proposition II.2, we can write the travel time from to implicitly as (18) as the collection of demands that can be Next, define the set reached from in or fewer time units. From (18) we see that when , the set is a disk of radius centered at . That is where we have omitted the dependence of on and . The area of the set , denoted , is upper bounded by , and the area is equal to if the does not intersect a boundary of . Now, by Lemma V.1 the demands in an unserviced region are uniformly randomly distributed with den. Let us compute the distribution of sity . For every vehicle position chosen before the generation of demands, the probability that is given by where the second equality is by Lemma V.1, and the last inequality comes from the fact that (Note that this is equivalent to assuming that the entire plane contains demands with density ). Hence, we have We can now prove part (i) of Theorem III.1. Proof of part (i) of Theorem III.1: A necessary condition for the stability of any policy is where demands is the steady-state expected travel time between and . For every policy . Thus, a necessary condition for stability is that Remark V.4 (Constant Fraction Service): A necessary condition for the existence of a policy which services a fraction of the demands is that Thus, for a fixed no policy can service a constant . This follows because in fraction of the demands as . order to service a fraction we require that In order to service a fraction of the demands, we consider a , with the arrival rate subset of the generator having length on that subset being equal to . The use of the TMHP-based policy on this subset and with the arrival rate gives a sufficient condition for stability analogous to Theorem III.2, but with an extra term of in the denominator. For the proof of part (ii) of Theorem III.1, we first recall from Lemma IV.6 that for stability of the FCFS policy, although must go to zero as , it can go very slowly to 0. Specifically, goes to zero as This quantity goes to zero more slowly than any polynomial in . We are now ready to prove Theorem III.1. Proof of part (ii) of Theorem III.1: The central idea of this , if the vehicle skips a deproof is that in the limit as mand and services the next one at an instance, then to reach the demand it skipped, the vehicle has to travel infinitely far from the generator which leads to instability. Observe that the condition on in the statement of part (ii) is the expression given by the necessary condition for FCFS , from Lemma stability in the asymptotic regime as IV.6. Therefore, suppose there is a policy that does not serve demands FCFS, but can stabilize the system with for some , and . Let be the first instant at which policy deviates from FCFS. Then, the demand served immediately after is demand for some . When the vehicle reaches demand at time , demand has moved above the vehicle. To ensure stability, demand must eventually be served. The time to travel to demand from any demand , where , is BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS where and are now the minimum of the and distances from to the . The random variable is Erlang distributed with shape and rate , implying for each , and in particular, for . Now, since as , , with probability one. Thus with probability one, as demands that arrive during 2565 where is the -coordinate of and is the time taken for , serves all the vehicle as per the TMHP that begins at and ends at . demands in other than of the TMHP for We seek an upper bound for the length which we use the Convert-to-EMHP method (cf. Section II-B). , and writing Invoking Lemma II.5 for for convenience, we have . Thus, the expected number of is as . This implies that with probability one, the policy becomes unstable when it deviates from FCFS and that any . Thus, deviation must occur with probability zero as a necessary condition for a policy to be stabilizing with is that, as , the policy must serve demands in the order in which they arrive. But this needs to hold for every and, by letting go to infinity, converges to zero for all . Thus, a non-FCFS policy cannot stabilize the system as . Hence, as , no matter how quickly every stabilizing policy must serve the demands in the order in which they arrive. Additionally, notice that the definition of the FCFS policy is that it uses the minimum time control (i.e., constant bearing control) to move between demands, thus the expression in part (ii) of Theorem III.1 is a necessary condition . for all stabilizing policies as B. Proofs of Theorem III.2 and Theorem III.4 We first present the proof of Theorem III.2. Proof of part (i) of Theorem III.2: The central idea is to derive a recurrence relation between the expected coordinate of the vehicle at the next iteration and the expected coordinate at the present iteration by using the background results on the computation of the TMHP in the iteration. The stability condition follows by ensuring that the resulting evolution of the expected coordinate with the number of iterations remains bounded. Note that if there are any demands “above” the vehicle initially, at the end of the first iteration of the TMHP-based policy, all outstanding demands have their y-coordinates less than or equal to that of the vehicle, and hence would be located “below” the vehicle as shown in the first of Fig. 4. Hence at the end of every iteration of the TMHP-based policy, all outstanding demands would be located “below” the vehicle. and Let the vehicle be located at denote the demand with the least y-coordinate at time instant . Let denote the number of demands in the set . If there exists a non-empty set of unserviced demands below the ve, with probability hicle at time , then for each where the first inequality is due to Lemma II.3, and the second . Thus, when is non-empty at time is due to If is empty at time , then the vehicle moves towards the . When a new demand arrives, the optimal location , then in the worst-case, vehicle moves towards it. If the vehicle is very close to an endpoint of the generator and the next demand arrives at the other endpoint. In this case, the vehicle moves with a vertical velocity component equal to and horizontal component equal to . So in the worst-case, at the beginning of the the vehicle is at a height . In this case, next iteration. The other possibility is if to get an upper bound on the height of the vehicle at the next iteration, we consider the vehicle motion when it first moves horizontally so that the -coordinate equals that of the demand, and then moves vertically down to meet the demand. This gives an upper bound on the height of the vehicle at the next iteration . Thus, if is empty, then the sum of as these two upper bounds is trivially an upper bound on the height of the vehicle at the beginning of the next iteration. Thus, if is empty, then Conditioned on , we have 2566 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 It can be shown that . Collecting the terms together and on further simplifying, with . Moreover, we may upper bound the bound for expected delay for a demand by where the equality follows since the arrival process is Poisson with rate and for a time interval . Substituting into (19), we obtain Proof of part (ii) of Theorem III.2: As the arrival rate , the necessary condition in Theorem III.1 states that for stability, must tend to zero. Now, for this part, we make use , the length of the of the following two facts. First, as TMHP constrained to start at the vehicle location and end at the lowest demand, is equal to the length of the EMHP in the corresponding static instance under the map , as described in of points which are Lemma II.5. Second, consider a set uniformly distributed in a region with finite area. Then, in the , the length of a constrained EMHP through limit as tends to the length of the ETSP tour through . More specifically, consider the th iteration of the TMHPbased policy, and let be the position of the service , the number of outstanding vehicle. In the limit as , and in addition, condidemands in that iteration tioned on , the demands are uniformly distributed in the re(cf. Remark V.2). Now using the above gion two facts, we can apply Theorem II.4 to obtain an expression for the length of the TMHP constrained to start at the vehicle , the location and ending at the lowest demand. As position of the vehicle at the end of the th iteration satisfies Using the law of iterated expectation, we have with probability one, where is the area of the region below the vehicle at the th iteration. Thus, conditioned on being bounded away from 0, we have (19) Applying Jensen’s inequality to the conditional expectation in the second term in the right hand side of (19) with probability one, where we have applied Jensen’s in. Thus, equality. Using Lemma V.1, with probability one (20) Thus, is finite if Thus, if satisfies the above condition, then expected number of demands in the environment is finite and the TMHP-based policy is stable. Finally, from Lemma IV.1, the region of stability for the FCFS policy is contained in the region of stability for the TMHP-based policy. Thus, the TMHP-based policy is stable for all arrival rates for which the FCFS policy is stable. Thus, the TMHP-based policy is stable for all arrival rates satisfying the bound in Lemma IV.7. This gives us the desired result. Remark V.5 (Upper Bound on Expected Delay): Since (20) , we can easily obtain an upper is a linear recurrence in Thus, the sufficient condition for stability of the TMHP-based (and thus ) is policy as Finally, we present the proof of Theorem III.4. Proof of Theorem III.4: It follows from Lemma IV.3 and Lemma IV.1 that the FCFS minimizes the steady state expected delay in the asymptotic regime of low arrival. Part (i) of Theorem III.4 follows since in the asymptotic regime of low arrival, the TMHP-based policy becomes equivalent to the FCFS policy. The proof of part (ii) follows from part (ii) of Theorem III.1 and Lemma IV.1 along with the fact that the TMHP-based policy spends the minimum amount of time to travel between demands. BOPARDIKAR et al.: DYNAMIC VEHICLE ROUTING FOR TRANSLATING DEMANDS Fig. 8. Numerically determined region of stability for the TMHP-based policy. A lightly shaded (green-colored) dot represents stability while a darkly shaded (blue-colored) dot represents instability. The uppermost (thick solid) curve is the necessary condition for stability for any policy as derived in Theorem III.1. The lowest (dashed) curve is the sufficient condition for stability of the TMHPbased policy as established by Theorem III.2. The broken curve between the two curves is the sufficient stability condition of the TMHP-based policy in the high arrival regime as derived in part (ii) of Theorem III.2. The environment width . is W =1 VI. SIMULATIONS In this section, we numerically determine the region of stability of the TMHP-based policy, and compare it with the theoretical results from the previous sections. In the actual implementation of the TMHP-based policy, the computational complexity increases undesirably as the values of approach the instability region. the problem parameters Therefore, we adopt a different procedure to characterize the stable/unstable region boundary, based upon the following central idea. For a given value of and a sufficiently high value of the height of the vehicle, if the policy is stable, then after one iteration of the policy, the vehicle’s height must decrease. In particular, the following procedure was adopted. i) For a collection of instructive pairs of the demand speed and in the region of interest, we set the generator width and we set the initial height of the environment of interest so that the expected number of demands in the environment are 1000. Thus, . ii) We repeated 10 times the following procedure. The vehicle is placed at the height and at a uniformly random location in the horizontal direction. A Poisson distributed number with parameter , of outstanding demands are uniformly randomly placed in the environment (cf. Lemma V.1). The vehicle uses the TMHP-based policy to serve all outstanding demands and we store the height of the vehicle at the end of the single iteration of the policy. Finally, we compute the average height of the 10 iterations. iii) If , then the policy is deemed to be stable for the chosen value of . Otherwise the policy is deemed to be unstable. 1 solver was used to generate approximations The to the TMHP at every iteration of the policy. The linkern solver takes as an input an instance of the ETSP. To transform the constrained EMHP problem into an ETSP, we replace the distance 1The TSP solver linkern is freely available for academic research use at http://www.tsp.gatech.edu/concorde.html. 2567 v Fig. 9. Variation of the steady state vehicle height with the product 1 at different demand speeds. The error bars correspond to 6 standard deviation about the mean. 1 between the start and end points with a large negative number, ensuring that this edge is included in the linkern output. The results of this numerical experiment are presented in Fig. 8. For the purpose of comparison, we overlay the plots for the theoretical curves, from Fig. 1. We observe that the numerically obtained stability boundary for the TMHP-based policy falls between the necessary and the sufficient conditions which were established in parts (i) of Theorems III.1 and III.2 respectively. Further, although the sufficient condition characterized in part (ii) of Theorem III.2, is theoretically an approximation of the stability boundary in the asymptotic regime of high arrival, our numerical results show that the condition serves as a very good approximation to the stability boundary, for nearly the entire range of demand speeds. in the stable region, we present For different values of simulations of the TMHP-based policy that shed light on the steady state height of the vehicle. For three different values of , we empirically determine the steady state vehicle height at various arrival rates. The results are shown in Fig. 9. We observe that the steady state height increases i) at higher arrival rates for a fixed demand speed, and ii) at higher demand speeds at fixed arrival rates, which is fairly intuitive to expect for a stable policy. VII. CONCLUSION We introduced a dynamic vehicle routing problem with translating demands. We determined a necessary condition on the arrival rate of the demands for the existence of a stabilizing policy. In the limit when the demands move as fast as the vehicle, we showed that every stabilizing policy must service the demands in the FCFS order. We proposed a novel receding horizon policy that services the moving demands as per a translational minimum Hamiltonian path. In the asymptotic regime when the demands move as fast as the vehicle, we showed that the TMHP-based policy minimizes the expected time to service a demand. We derived a sufficient condition for stability of the TMHP-based policy, and showed that in the asymptotic regime of low demand speed, the sufficient condition is within a constant factor of the necessary condition for stability. In a third asymptotic regime when arrival rate tends to zero for a fixed demand speed, we showed that the TMHP-based policy is optimal in terms of minimizing the expected time to service a demand. Finally, we presented an implementation of the TMHP-based 2568 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 where the notation denotes the shortest distance of point from the nearest of the lines. The second path is constructed similarly using the lines . This path also commences on , passes through the above lines (visiting the points whenever they are at the shortest distance from these lines) and ends on . The length of this path is where the notation denotes the shortest distance of point from the nearest of the new lines. Observe that . Hence Fig. 10. Illustration of the proof of Lemma II.3. The dots indicate the locations of the points inside a rectangle of size 1 h. The first of the two paths considered in the proof through the points begins at (1; h) and follows the direction of the arrows, visiting a point whenever it is within a distance of h=2m for a specific integer m from the solid horizontal lines. 2 policy to numerically determine its region of stability. Our numerical results show that the theoretically established sufficient condition in the asymptotic regime of low demand speeds also serves as a good approximation to the boundary of the stability region for a significantly large interval of values of demand speed. As future work, it would be of interest to extend the present analysis or to design an improved policy that is optimal also in the asymptotic regime of high arrival, in addition to the two other asymptotic regimes. Another interesting direction is to have on-site service times for the demands. For the case in which the on-site service times are independent and identically distributed with a known expected value, and the vehicle is permitted to move with the demand upon reaching it, the results in this paper could be extended using similar analysis. Recently, in [34], we addressed a version of the present problem in which the goal for the vehicle is to maximize the fraction of demands served before they reach a deadline, which is at a given distance from the generator. In [35], we addressed a vehicle placement problem that involves non-uniform spatial arrival density and also multiple service vehicles. APPENDIX In this Appendix, we present the proof of Lemma II.3. Proof: Suppose the rectangular region is given by , . Let be a positive integer (to be chosen later) and let the points be denoted by . We now construct two paths through the points. The first consists of (a) the lines ; (b) the shortest distances from each of the points to the nearest such line, each traveled twice, and (c) suitable portions of the lines , , and , . This is illustrated in Fig. 10. The length of this path is Now choose to be the integer nearest to , where . Thus , so that Thus, at least one of the two paths must have length upper . bounded by REFERENCES [1] S. D. Bopardikar, S. L. Smith, F. Bullo, and J. P. Hespanha, “Dynamic vehicle routing with moving demands—Part I: Low speed demands and high arrival rates,” in Proc. Amer. Control Conf., St. Louis, MO, Jun. 2009, pp. 1454–1459. [2] S. L. Smith, S. D. Bopardikar, F. Bullo, and J. P. Hespanha, “Dynamic vehicle routing with moving demands—Part II: High speed demands or low arrival rates,” in Proc. Amer. Control Conf., St. Louis, MO, Jun. 2009, pp. 1466–1471. [3] P. Toth and D. Vigo, The Vehicle Routing Problem, ser. Monographs on Discrete Mathematics and Applications. Philadelphia, PA: SIAM, 2001. 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Tang and Ü.Özgüner, “Motion planning for multi-target surveillance with mobile sensor agents,” IEEE Trans. Robotics, vol. 21, no. 5, pp. 898–908, Oct. 2005. [28] D. B. Kingston and C. J. Schumacher, “Time-dependent cooperative assignment,” in Proc. Amer. Control Conf., Portland, OR, Jun. 2005, pp. 4084–4089. [29] R. Isaacs, Differential Games. New York: Wiley, 1965. [30] L. Few, “The shortest path and the shortest road through n points in a region,” Mathematika, vol. 2, pp. 141–144, 1955. [31] J. Beardwood, J. Halton, and J. Hammersly, “The shortest path through many points,” in Proc. Cambridge Philosophy Soc., 1959, vol. 55, pp. 299–327. [32] A. G. Percus and O. C. Martin, “Finite size and dimensional dependence of the Euclidean traveling salesman problem,” Phys. Rev. Lett., vol. 76, no. 8, pp. 1188–1191, 1996. [33] J. N. Newman, “Approximations for the Bessel and Struve functions,” Math. Computation, vol. 43, no. 168, pp. 551–556, 1984. [34] S. L. Smith, S. D. Bopardikar, and F. Bullo, “A dynamic boundary guarding problem with translating demands,” in Proc. IEEE Conf. Decision Control, Shanghai, China, Dec. 2009, pp. 8543–8548. [35] S. D. Bopardikar, S. L. Smith, and F. Bullo, “Vehicle placement to intercept moving targets,” in Proc. Amer. Control Conf., Baltimore, MD, Jun. 2010, pp. 5538–5543. Shaunak D. Bopardikar (S’08–M’10) received the B.Tech. and M.Tech. degrees in mechanical engineering from Indian Institute of Technology, Bombay, in 2004, and the Ph.D. degree in mechanical engineering from the University of California at Santa Barbara, in 2010. From 2004 to 2005, he was an Engineer with General Electric India Technology Center, Bangalore, India. He is currently a Postdoctoral Researcher with the Center for Control Dynamical Systems and Computation (CCDC), University of California at Santa Barbara. His research interests include large-dimensional dynamic optimization, game theory, pursuit-evasion, and motion coordination for robotic systems. 2569 Stephen L. Smith (S’05–M’09) received the B.Sc. degree in engineering physics from Queen’s University, Kingston, ON, Canada, in 2003, the M.A.Sc. degree in electrical engineering from the University of Toronto, Toronto, ON, Canada in 2005, and the Ph.D. degree in mechanical engineering from the University of California, Santa Barbara, in 2009. He is currently a Postdoctoral Associate in the Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology (MIT), Cambridge. His research focuses on the distributed control of autonomous systems. Particular interests include persistent monitoring, vehicle routing, and task allocation in dynamic environments. Dr. Smith was a finalist for the Student Best Paper Award at the IEEE Conference on Decision and Control in 2007. Francesco Bullo (S’95–M’99–SM’03–F’10) received the Laurea degree (with highest honors) in electrical engineering from the University of Padova, Padova, Italy, in 1994, and the Ph.D. degree in control and dynamical systems from the California Institute of Technology, Pasadena, in 1999. From 1998 to 2004, he was an Assistant Professor with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign. He is currently a Professor with the Mechanical Engineering Department and the Center for Control, Dynamical Systems and Computation, University of California, Santa Barbara. He is the coauthor of Geometric Control of Mechanical Systems (Springer, 2004) and Distributed Control of Robotic Networks (Princeton Univ. Press, 2009). He has published more than 150 papers in international journals, books, and refereed conferences. He has served, or is serving, on the editorial boards of the ESAIM: Control, Optimization, and the Calculus of Variations and the SIAM Journal of Control and Optimization. His main research interest is multi-agent networks with application to robotic coordination, distributed computing and power networks; he has worked on problems in vehicle routing, geometric control, and motion planning. Dr. Bullo received the 2003 ONR Young Investigator Award and the 2008 Outstanding Paper Award for the IEEE Control Systems Magazine. He served on the editorial board of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. João P. Hespanha (M’94–F’08) was born in Coimbra, Portugal, in 1968. He received the Licenciatura in electrical and computer engineering from the Instituto Superior Técnico, Lisbon, Portugal in 1991 and the Ph.D. degree in electrical engineering and applied science from Yale University, New Haven, CT, in 1998. From 1999 to 2001, he was Assistant Professor at the University of Southern California, Los Angeles. He moved to the University of California, Santa Barbara in 2002, where he currently holds a Professor position with the Department of Electrical and Computer Engineering. He is Associate Director for the Center for Control, Dynamical-systems, and Computation (CCDC), Vice-Chair of the Department of Electrical and Computer Engineering, and a member of the Executive Committee for the Institute for Collaborative Biotechnologies (ICB). His current research interests include hybrid and switched systems; the modeling and control of communication networks; distributed control over communication networks (also known as networked control systems); the use of vision in feedback control; and stochastic modeling in biology. Dr. Hespanha received the Yale University’s Henry Prentiss Becton Graduate Prize for exceptional achievement in research in Engineering and Applied Science, a National Science Foundation CAREER Award, the 2005 best paper award at the 2nd International Conference on Intelligent Sensing and Information Processing, the 2005 Automatica Theory/Methodology best paper prize, the 2006 George S. Axelby Outstanding Paper Award, and the 2009 Ruberti Young Researcher Prize. From 2004 to 2007, he was an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. He is an IEEE Distinguished Lecturer since 2007.