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IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 519 On the Information Flow Required for Tracking Control in Networks of Mobile Sensing Agents Liang Chen, Sandip Roy, and Ali Saberi Abstract—We design controllers that permit mobile agents with distributed or networked sensing capabilities to track (follow) desired trajectories, identify what trajectory information must be distributed to each agent for tracking, and develop methods to minimize the communication needed for the trajectory information distribution. Index Terms—Cooperative control, dynamical networks, tracking. Ç 1 INTRODUCTION I N several modern applications, teams of autonomous agents with distributed sensing and/or communication capabilities are required to cooperatively complete a other agents’ desired trajectories must be communicated for informed use of the sensed information. It is this coupling between sensing capabilities and trajectory-dissemination complex task (see, e.g., [1], [2], [3]). In many of these needs that makes tracking challenging. In this paper, we applications, the teams must complete tracking tasks—ones provide a methodology for tracking control in networks in which each agent’s state (e.g., position) must follow a whose agents depend on distributed sensing/communica- specified command signal (e.g., [3], [4], [5]). For instance, tion capabilities to complete desired tasks. We stress that we teams of autonomous vehicles, which sense relative posi- have encountered problems of this form in a wide family of tions, may need to follow a “lawn-mower” pattern to search applications, including in air-vehicle or land-vehicle control a minefield. Similarly, a bank of antennas may need to in hazardous environments (where agents must use follow a path in a coordinated fashion. relative-position measurements to achieve control tasks) At first glance, these tracking problems for networks of and in domains where a large number of cheap mobile communicating/sensing agents seem no more challenging elements with highly localized sensing capabilities are than automated tracking problems for single devices; needed. We will give a concrete example of such a network seemingly, we could distribute to each agent in the network tracking problem after the problem formulation. its desired path, which the agent could then independently Very broadly, we argue that these tracking problems follow. However, many of the tracking problems that our require two sorts of information flow: 1) local sensing for group has encountered—in applications ranging from air control, i.e., using which each agent can correctly actuate its traffic management to sensor fusion and vehicle control— dynamics (change its state) so as to follow a specified path; turn out to be much more challenging. Fundamentally, and 2) higher-level communication for trajectory distribution, what makes these problems challenging is that, due to cost i.e., for disseminating each agent’s desired path to it. Often, or security or complexity constraints, individual agents do in tracking applications for modern communicating-agent not have sophisticated enough observation capabilities to teams, the information flow must be highly limited, in that, independently know where they are, and so move as they it must be both, local in space and sparse in time (see, e.g., wish in their environment. Instead, each agent depends in [7] for motivation regarding AUVs). In this paper, we marry an essential way on sensing of or communication with other well-known techniques for servo control with techniques agents to be able to follow desired paths (or complete for decentralized control and distributed algorithm devel- various other tasks, for that matter). From another view- opment to develop low-information-flow tracking algo- point, we have found that each agent fundamentally needs rithms for networks of sensing/communicating agents. In to sense/receive information about other agents simply to doing so, we delineate the role played by sensing topology operate, i.e., coordination is needed for task completion. For of the network in our ability to achieve tracking. We also tracking tasks in particular, this fundamental need to use study what trajectory information must be distributed to the sensed information also implies that information about agents and explore means for distributing this information with sparse or no communication. Recently, several articles have sought to expose the role . L. Chen is with Apple Corporation, Santa Clara, CA, and the Washington played by the sensing topology for formation (convergence State University, Pullman, WA. E-mail: lifemarcher@yahoo.com. . S. Roy and A. Saberi are with the School of Electrical Engineering and to a fixed or constant-velocity pattern) in communicating- Computer Science, Washington State University, PO Box 642752, agent networks (e.g., [1], [2]). Our recent work on formation Pullman, WA 99164-2752. E-mail: sroy@eecs.wsu.edu. has exposed that linear decentralized static and dynamic Manuscript received 8 June 2006; revised 25 Nov. 2007; accepted 23 Jan. controllers can be used to stabilize a network of double- 2010; published online 27 Aug. 2010. integrator agents with sensing capabilities and actuator For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-0160-0606. saturation, under broad connectivity requirements on the Digital Object Identifier no. 10.1109/TMC.2010.165. sensing topology [2]. Our aim here is to extend the 1536-1233/11/$26.00 ß 2011 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS 520 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 controllers/algorithms for formation in double-integrator 2.1 The Double-Integrator Network networks to permit tracking of temporal signals rather than We consider a network of n mobile sensors or vehicles or simple approach to a fixed value for general communica- agents, labeled 1; . . .; n. These agents aim to cooperatively tion/sensing topologies. complete a dynamic task, by actuating their internal Our work connects with the classical literature on dynamics using sensed observations. Let us motivate and decentralized tracking (e.g., [6]), as well as more recent introduce our model for the internal dynamics for the studies on tracking control for (specifically) teams of agents, and then consider the sensing architecture. autonomous vehicles (e.g., [4], [5]). With respect to Increasingly, modern communicating-agent networks (net- the classical literature, our study differs significantly in works of mobile sensors, autonomous vehicle teams) are the following sense: Our class of decentralized control made up of agents with simple but highly constrained problems—which are appropriate for sensing-network dynamics. For many such networks, a plausible canonical applications—requires that trajectories from a set of model for an agent’s dynamics is a saturating double neighbors (or more specifically, a function of these integrator. We adopt this saturating double-integrator model trajectories) be distributed to each agent; sending only the desired trajectory for a particular agent to that agent is for our agents’ internal dynamics, i.e., we assume agent i is insufficient. With regard to the recent literature on governed by autonomous-vehicle control, our approach differs in that ri ¼ ðui Þ; € ð1Þ it permits us to decouple the stabilization, path-following, 1 and path-distribution aspects of the control task. Hence, where ri is the position of agent i, ui is agent i’s input or we are able to achieve tracking for a very general class of actuation, and ðÞ is the standard saturation function. We sensing topologies, and to track essentially arbitrary also find it convenient to refer to the derivative of agent i’s signals when sufficient communication for path-distribu- position with respect to time (i.e., vi ¼ r) as agent i’s velocity. _ tion is permitted. Our approach also permits us to achieve We note that for the above dynamics, each agent is assumed tracking control in the presence of actuator saturation, and to have a scalar position. In the case where an agent moving to take a design-based perspective on tracking (one in in an n-dimensional space is governed by double-integrator which a broad class of feasible controllers and distribution dynamics in n directions (as is representative for certain schemes are identified, and hence, a suitable controller can autonomous-vehicle teams), the internal dynamics, observa- be chosen to meet other requirements). tions, and controller of the multiple-dimensional agent can For the sake of clarification, we should stress that we are be straightforwardly represented using multiple scalar concerned with designing controllers for teams of mobile agents (see [2]). It is worth noting that agents with coupled agents with distributed sensing capabilities so that they kinematics and actuator saturation are not addressed in this follow (track) desired paths. The term tracking is also used to work. This double-integrator network model has been very describe efforts for detecting a temporal trajectory from noisy commonly used to represent the dynamics of autonomous sensor observations. The two notions for tracking are deeply vehicle teams, and also has found application for agreement related (see, e.g., the literature on agreement [8], [9]), and our or data fusion tasks in sensor networks. methods can be modified to address trajectory detection; For some applications (e.g., computational ones, where however, here we are focused solely on the first notion. agents’ states represent opinions/estimates rather than We take the following approach to study tracking control actual positions), actuator saturation may not be a significant in communicating-agent networks. Section 2 develops the constraint. Motivated by such applications, and so as to make double-integrator network model and pose the decentra- the presentation of results clearer, we often first consider the lized tracking problem. In Section 3, we show how decentralized tracking can be achieved in a double- case where agents are governed by pure double integrators, integrator network with a given communication/sensing r i ¼ ui ; € ð2Þ topology when the appropriate trajectory signals are distributed to each agent. In turn, we identify the function rather than saturating double integrators. of trajectory signals (from neighbors) that must be dis- Each agent in the network has certain sensing capabil- tributed to each agent in order to complete the tracking task. ities. Our model for sensing is quite general: Each agent is In Section 4, we introduce a set of motions—specifically, assumed to make multiple observations, each of which is a translation, rotation, and expansion—that are typical of linear combination of (in general, multiple) agents’ current multiagent formations. We show that, given a set of agents positions or velocities. We note that such a model permits that only need moving in formation (i.e., along these representation of absolute and relative position observa- trajectories), the path-distribution task can be achieved tions, among others. Specifically, we suppose agent i has through very sparse communication. Finally, Section 5 2mi observations (mi observations on positions, and mi explores how agents can infer trajectory information rather observations on velocities) that can be written in the form than being sent this information. api ¼ Gi r; ð3Þ 2 MODEL AND PROBLEM FORMULATION avi ¼ Gi v; In this section, we motivate and describe a model comprising a network of communicating/sensing agents with double- 1. We use the term position for an agent’s state with the vehicle-control application in mind (in which Newton’s law yields the double-integrator integrator internal dynamics, and then introduce the tracking model), but in fact, double-integrator models can capture various dynamics problem in the context of this double-integrator network. of interest in communicating-agent networks (see [2] for some examples). CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 521 4 4 w h e r e r ¼ ½ r1 . . . rn T ; v ¼ ½ v1 . . . vn T , a n d t h e loss of generality) that each desired trajectory is generated graph matrix Gi has dimension mi Â n. It should be noted, by the exosystem as can be seen from (3), that each agent is assumed to have identically structured observations on positions and velo- wi ¼ Si wi ; _ ð4Þ cities. This restriction is plausible in many applications (see ri ¼ " dT wi ; i ð5Þ [2] for motivation), and also can be eliminated by consider- qi ing more general controllers than pursued here; the where wi 2 R ; the system matrix Si is assumed, without methods developed here can be generalized to this case. loss of generality, to have eigenvalues in the closed right For convenience, we also define the full graph matrix as half plane (since tracking is an asymptotic task, see [10]); G ¼ ½GT . . . GT T . and di is a constant vector of dimension qi Â 1. The initial 1 n conditions wi ð0Þ are set so that the desired trajectory is Remark. We stress that, in general, agents in our model generated. The following example illustrates trajectory cannot guide their dynamics autonomously without the generation using an exosystem. consideration of network interactions through sensing; for instance, agents may only have relative-position Example 1. Nearly all desired trajectories of interest can be information, and hence must depend on the motion of represented piecewise as responses of null-controllable other agents with absolute-position information to linear systems (systems with eigenvalues on the jw-axis). achieve their desired absolute positions. Similarly, we That is, desired trajectories can be represented piecewise can model a circumstance where, for security purposes, as sums of polynomials, sinusoids, and their products. each agent only can measure one component of its Here, let us illustrate generation of such a trajectory (vector) position, and must depend on the other agents to using an exosystem. fully guide its dynamics. In particular, suppose the desired trajectory for an agent i is of the form ri ðtÞ ¼ t þ sinð2tÞ. Notice that the " We assume that agent i is available from its observations trajectory consists a ramp and a sinusoidal signal. The for computing its actuation (input). That is, the observations ramp signal has an initial value of 0 and a slope of 1, and api and avi of agent i are considered as information that is the sinusoidal signal has an amplitude of 1, a frequency available to the local controller. Our goal is to design a static of 2rad=s, and an initial phase of 0o . linear controller (a controller without memory, specifically This trajectory signal can be generated from a linear one that sets the input ui to a linear combination of the system, as follows: current observations) for each agent i, so as to globally 2 3 achieve a tracking task.2 0 1 0 0 60 0 0 07 We refer to the internal dynamics, communication 6 7 ri ¼ ½ 1 0 1 0 wi ; where wi ¼ 6 " _ 7wi topology, and decentralized control paradigm described 40 0 0 25 above together as a saturating double-integrator network 0 0 À2 0 (or, double-integrator network in the case where actuator 2 3 0 saturation is ignored). 617 6 7 2.2 Tracking Problem Formulation with wi ð0Þ ¼ 6 7: 415 Motivated by a range of applications, we aim to design 0 controllers for the double-integrator network for tracking, i.e., controllers using each agent i’s position can follow a Notice that this tracking signal is thus specified solely by desired trajectory ri ðtÞ; t ! 0. We find it convenient to refer to " the parameters (modes) of the linear exosystem, as well a set of desired trajectories r1 ðtÞ; . . . ; rn ðtÞ together as the " " as the initial conditions of this exosystem. tracking task ð"1 ; . . . ; rn Þ. Controllers for tracking are needed, r " Our aim is to design controllers for each agent, so that for instance, to permit a team of autonomous vehicles to their positions follow the desired trajectories in an surround and destroy a target or sweep a minefield. asymptotic sense. Let us define the achievement of the Typically, in the controls literature, desired trajectories tracking task formally. are assumed to be ones that can be generated by an autonomous linear system (which is termed an exosystem); Definition 1. A double-integrator network is said to achieve or almost all trajectories of interest (for instance, ramp, step, complete the tracking task ð"1 ; . . . ; rn Þ, if the error signals r " 4 or sinusoidal signals) can be represented in this way (see, ei ðtÞ ¼ ri ðtÞ À ri ðtÞ; i ¼ 1; . . . ; n, approach 0 as t ! 1. " e.g., [10]). For us, signals of this form are compelling with respect to their distribution to agents, since they can be We note that tracking is an asymptotic task; a settling time specified by the parameters (more specifically, eigenvalues describing how quickly tracking is achieved can be obtained or modes) of the autonomous systems and the initial (see [10] for details). condition of the autonomous system’s state. Thus, 2.3 Example: Four Autonomous Vehicles whenever convenient, we assume (without meaningful Let us conclude the model formulation with a simple 2. Our motivation for considering static controllers is that they are easily example that helps to conceptualize the class of problems implementable even in devices with limited complexity. Consideration of that we are pursuing. static control also clarifies the exposition of information flow for tracking; many results readily generalize to settings where more complicated Let us consider a set of four autonomous land vehicles controllers are used. following each other along a road. These vehicles are tasked 522 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 2 3 with following a desired time trajectory—for instance, kT 1 6 .. 7 traveling at constant speeds for periods of time and K¼4 . 5 stopping for designated intervals to permit completion of kT n other tasks—while maintaining a designated separation (say a separation of one unit between each vehicle). The (where each ki is a mi -component vector) such that all tracking task for this group of four vehicles is made more eigenvalues of KG are in the open left half plane (OLHP). challenging by cost and security constraints, which limit the In this case, the tracking task can be achieved by driving each agent i with the input sensing capabilities of the vehicles. In particular, we assume that only one of the vehicles, say the lead vehicle, has a ! ! Â Ã api Gi Dw global positioning system, and so has absolute position ui ¼ kT kT i i À 2 þ dT Si wi ; i ð7Þ avi Gi DSw information. We also assume that this vehicle has the ability to sense a signal sent by the fourth vehicle, from which it where is a sufficiently large positive number, can deduce its relative position to the fourth vehicle. 2 T 3 2 3 Meanwhile, vehicles 2-4 do not have the capability to d1 S1 6 .. 7 6 .. 7 measure their absolute positions, and instead can only sense D¼6 4 . 7; S ¼ 6 5 4 . 7; 5 and their position relative to the vehicle in front. T dn Sn In many instances, vehicle dynamics are well modeled 2 3 using double integrators, and in this example, we assume w1 that each vehicle’s movement along the road is well 6 . 7 6 . 7: w¼4 . 5 modeled as a double-integrator dynamics. Based on the sensing capabilities described above, we model the four wn agents’ sensing topologies as follows: ! Proof. We prove our theorem by verifying that each agent i G1 ¼ 1 0 0 0 ; actually follows the desired trajectory dT wi with the i 1 0 0 À1 above input signal. G2 ¼ ½ À1 1 0 0 ; ð6Þ Noting that api ¼ Gi r and avi ¼ Gi v, when the above G3 ¼ ½ 0 À1 1 0 ; input signal is applied, the dynamics of agent i is G4 ¼ ½ 0 0 À1 1 : given by Also, the trajectory signal r1 ðtÞ describes the desired path of & r i ¼ vi ; _ the leading vehicle, while in this example ri ðtÞ ¼ riÀ1 ðtÞ À 1 vi ¼ ui ¼ kT Gi ðr À DwÞ þ kT Gi ðv À DSwÞ þ dT Si wi : _ i i i 2 for i ¼ 2; 3; 4. We are interested in designing a static control scheme that permits completion of the task. ð8Þ This example clarifies that the vehicles must operate Making the state transform of zpi ¼ ri À and dT wi i together for all of them to complete the tracking task. Since T zvi ¼ vi À di Si wi , the dynamics for agent i becomes vehicles 2-4 critically require sensed information about & other vehicles to achieve tracking, we might expect that zpi ¼ zvi ; _ ð9Þ these vehicles in fact will need to know/use the trajectory zvi ¼ kT Gi zp þ kT Gi zv ; _ i i signals of other vehicles to properly use sensed informa- tion for tracking, and in fact, our analysis shows that this where zp ¼ ½zp1 . . . zpn T , and zv ¼ ½zv1 . . . zvn T . Assem- is the case. bling the dynamics of n agents in the double-integrator network, we get the following state equation: ! ! ! 3 TRACKING AND THE REQUIRED INFORMATION _ zp 0 I zp ¼ : ð10Þ FLOW _ zv KG KG zv In this section, we first give conditions under which a In our previous work, it has been proved that if the double-integrator network (with and without actuation eigenvalues of KG are in the OLHP, the eigenvalues of saturation) can achieve tracking and show how a controller ! can be designed to do so. Throughout this section, we 0 I assume full information flow regarding the tracking task, KG KG i.e., we assume that the agents are given all required are guaranteed to be in OLHP for sufficiently large information about the signals in the tracking task ð"1 ; . . . ; rn Þ. We also delineate carefully what information r " positive (see Theorem 6 in [2] and note that the proof about the tracking task must be given to each agent. naturally can be adapted to the generalized eigenvector First, we consider a double-integrator network that is not case). Hence, zp ; zv ! 0 as t ! 1, and then, it follows subject to actuation saturation, and show how to develop a that for each agent i, ei ¼ zpi ! 0 as t ! 1. t u controller for the tracking task, in the process giving broad A couple of notes about this theorem are worthwhile: conditions under which tracking is possible. 1) We stress that the controllers being used to achieve the Theorem 1. A double-integrator network can complete any tracking task are static (memoryless) linear ones; the input tracking task ð"1 ; . . . ; rn Þ if there exists a block diagonal r " ui ðtÞ at each time t is a linear function of the current matrix observations. CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 523 2) We have chosen to present the control inputs in terms In this case, the tracking task can be achieved by driving each of the exosystem parameters with the motivation that this agent i with the following family of input signals, parameterized form is often the most easily implemented one. We can in ": easily phrase the input in terms of the desired trajectories. In this notation, we find that ! ! Â Ã api Gi Dw ! ! ui ¼ "2 kT "kT i i À Â Ã api Gi r avi Gi DSw ð11Þ ui ¼ kT kT À € þ ri ; where T 2 i i avi _ Gi r þ di Si wi ; rT ¼ ½r1 . . . rn : where " 2 ð0; "Ã , and "Ã is a function of the radius of the initial condition ball for (1) and the double-integrator network According to Theorem 1, tracking is possible whenever there parameters; is a sufficiently large positive number; exists an appropriately-structured (block-diagonal) K such 2 T 3 2 3 that KG has all eigenvalues in the OLHP. Essentially, d1 S1 whenever this condition holds, the controller can be chosen 6 .. 7 6 .. 7 to make the trajectory stable.3 Thus, we refer to a double- D¼6 4 . 7; S ¼ 6 5 4 . 7; and 5 integrator network for which there is K such that KG has all dT Sn n eigenvalues in the OLHP as a stabilizable double-integrator 2 3 w1 network. We note that the linear algebra problem of whether 6 . 7 there exists block-diagonal K such that KG has eigenvalues in w ¼ 6 . 7: 4 . 5 the OLHP is well studied (e.g., [11], [12]), and there are wn several broad classes of full graph matrices for which such stabilizing K exists. We refer the reader to [2], [12] for details. Proof. We prove this theorem by showing that for each So far, we have developed conditions under which an given closed and bounded set of initial condtions X0 , unsaturating double-integrator network can achieve track- there exists an "Ã > 0 such that for all " 2 ð0; "Ã , the ing; we note that the amplitude of the input signal ui may be tracking task can be achieved using the proposed control arbitrarily large when the controller in Theorem 1 is used, law when the two conditions hold. and hence, the result does not necessarily carry through to The family of inputs for agent i are of the form the case where actuators may saturate. We next develop a condition for tracking in a saturating double-integrator ! ! Â 2 T Ã api Gi Dw network. Conceptually, tracking under saturation requires ui ¼ " ki "ki T À 2 þ dT Si wi : i avi Gi DSw the further condition that the actuator can provide enough acceleration at all times to move each agent along its desired Note that the constructed input signal takes the same trajectory, plus an arbitrarily small amount of further form as that given in Theorem 1, except the gain matrix is acceleration for convergence to the trajectory (stabilization). parameterized in ". With this input signal, the closed- Under these conditions, by making the convergence to the loop dynamics of each agent i is trajectory sufficiently slow, we can achieve tracking for an arbitrarily large set of initial conditions. That is, tracking is ri ¼ vi ; _ ð12Þ achieved in a semiglobal sense, i.e., given any closed and vi ¼ ðui Þ: _ ð13Þ bounded ball of initial conditions for the saturating double- integrator network, there exists a controller that achieves Making a state transform (identical to that in the proof of tracking for any initial condition in this ball. This notion is Theorem 1) zpi ¼ ri À dT wi and zvi ¼ vi À dT Si wi , the i i formalized in the following theorem. dynamics for agent i becomes Theorem 2. Consider the tracking problem of a saturating 8 double-integrator network with n agents. A tracking task < zpi ¼ zvi ; _ À Á ð"1 ; . . . ; rn Þ can be achieved for any given closed and bounded r " zvi ¼ "2 kT Gi zp þ "kT Gi zv þ dT Si2 wi _ i i i ð14Þ : set of initial conditions, say X0 , for (1), if the two following À dT Si2 wi ; i conditions hold: where zp ¼ ½zp1 . . . zpn T , and zv ¼ ½ zv1 . . . zvn T . (I) There exists a block-diagonal matrix We then show that by choosing the parameter " 2 T 3 sufficiently small, the input of each agent i remains in the k1 6 .. 7 linear region 8t ! T , and the tracking task is achieved. K¼4 . 5 Let us first consider the transformed dynamics of kT n agent i if the actuation were not subject to saturation: (where ki is an mi -component vector) such that all eigenvalues & zpi ¼ zvi ; _ of KG are in the OLHP. ð15Þ (II) There exists a > 0 and a T ! 0 such that zvi ¼ "2 kT Gi zp þ "kT Gi zv : _ i i kdT Si2 wi k1;T 1 À ; i Assembling the closed-loop dynamics for the n agents d2 ri " assuming no saturation, we have the state equation (or equivalently, k 2 k1;T 1 À ) for all i ¼ 1; . . . ; n. ! ! ! dt _ zp 0 I zp 3. That is, the desired trajectories are attractive and also stable in the ¼ 2 : ð16Þ _ zv " KG "KG zv sense of Lyapunov. 524 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 We note that the state matrix of this system is similar to the applications of the double-integrator network model, for the state matrix of the system instance, in autonomous-vehicle-team control. Let us give an ! example motivated by this autonomous-vehicle-control _ 0 I application. h¼ h; "KG "KG Example 2. Several typical tracking tasks are simulated, for through the similarity transformation a network of three agents (autonomous vehicles) moving ! in a plane. The agents’ sensing capabilities are distrib- 1 0 uted, i.e., each agent makes different observations that : 0 1 are combinations of multiple agents’ states. We assume However, this new system is simply a time scaling of the that the sensing topology of the agents is a grounded system 10, and so its response remains bounded regard- Laplacian one, in particular with less of . From the transformation, we thus automatically 2 3 2 À1 0 obtain that the maximum absolute value of the response G ¼ 4 À1 3 À1 5: zp is upper bounded by a linear function of 1 , while " 0 À1 3 the maximum value of z is bounded (in an order 1 sense). We also make the observation that if the eigenvalues of Fig. 1 illustrates tracking of three sets of desired ! trajectories, namely, circular, sinusoidal, and lawn- 0 I mower trajectories, using the control law proposed in KG KG Theorem 1 (with identical gains used by each agent). The simulation results are shown in Fig. 1. We notice that are denoted as 1 ; . . . ; 2n , then the eigenvalues of each agent has a transient motion, before converging to ! 0 I the desired trajectory. In the case of the lawn-mower "2 KG "KG pattern, which is generated piecewise from several exosystems, slight transients are evident at the times are "1 ; . . . ; "2n (since this matrix is a transformation of that the agents must change from a translational motion the scaled matrix). Thus, for every ", the closed-loop to a rotational one and vice versa. system of interest is stable, and further, the component of From the form of the inputs in Theorem 1, we notice that the input "2 KGzp þ "KGzv is bounded by a linear computation of the input requires knowledge of the desired function of . trajectories of multiple agents. In the subsequent sections, We then consider the original system in which agents we will develop methods for providing the agents with this are subject to actuation saturation. information using minimal communication when the agents We observe that at time T , the transformed state are known to move in formation, and will even explore ! zp ðT Þ whether agents can deduce the required trajectories when zv ðT Þ only a leader has been told the tracking task. Before considering these special cases, we first identify the belongs to a bounded set, since trajectory information that, in general, must be provided ! to agent i to achieve the tracking task. The required zp ð0Þ information is deeply connected to the structure of the zv ð0Þ communication/sensing topology, i.e., the graph matrices is bounded and G1 ; . . . ; Gn . To formalize the connection, let us define a ! graph for the network communication/sensing. zp ðT Þ zv ðT Þ Definition 2. The network graph is a directed graph with n nodes labeled 1; . . . ; n, which corresponds to the n agents. The is determined by a linear differential equation with network graph has an edge from vertex j to vertex i, if and only bounded inputs. if Gij 6¼ ~ where i 6¼ j and Gij is the jth column of the graph 0, Then, if we consider the dynamics of agent i from matrix Gi . time T onwards, there exists an "Ã such that for all t ! T , i "2 kT Gi zp þ "kT Gi zv < if " 2 ð0; "Ã . Hence, the input i i i We note that a directed edge from vertex j to vertex i of each agent remains in the linear (unsaturated) region indicates that the observations made by agent i depend on for all t ! T , when " 2 ð0; "Ã , where "Ã ¼ minf"Ã ; . . . ; "Ã g. 1 n the position/velocity of agent j. Finally, since zpi ! 0 as t ! 1, we have ei ! 0 as We also find it convenient to define the notion of t ! 1, and the tracking task is achieved. t u neighbors from the network graph. In particular, if the network graph has an edge from vertex j to vertex i, we We notice that the low-gain parameter " depends on the refer to vertex (equivalently, agent) j as an upstream size of the initial-condition ball, as detailed in the above neighbor of vertex (equivalently, agent) i. We use the proof. This selection of low gains in control of saturating notation U(i) for the set of upstream neighbors of agent i, systems is classical (see, e.g., [10]). and use the term upstream neighbor set for this set. The above two theorems for tracking in double-integrator Let us now give the general result on the trajectory networks provide a practical strategy for path-following for information required by each agent. CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 525 Fig. 1. Tracking of (a) circular, (b) sinusoidal, and (c) lawn-mower trajectories is illustrated. In each plot, a time trace of each agent’s horizontal and vertical positions is shown. Theorem 3. When the control laws proposed in Theorems 1 and 2 CANNOT be achieved by simply sending each agent its are used to achieve a tracking task, each agent i requires a own desired trajectory, even when other controllers are signal zi , which is a function of the trajectories rj , j 2 UðiÞ, as " " considered; a combined statistic is necessary for tracking. well as the trajectory ri , in order to achieve the tracking task. " This can be shown using the regulator equation, which is used Specifically, the agent i requires the signal for designing tracking controllers, see [10] for background. Since our focus here is on designing working controllers, we r d"j d2 ri " feel the details are outside the scope of this work. zi ð"UðiÞ ; ri Þ ¼ ÀkT Æj2UðiÞ Gij rj þ " r " i " þ 2; ð17Þ dt dt Let us put forth a couple of perspectives that give context where we have used the notation rUðiÞ for the set of trajectories " to the trajectory distribution requirement in our design. rj , j 2 UðiÞ. " A Decentralized-Control Perspective. Our development of a trajectory-distribution paradigm advances the study of Proof. Equation (17) follows directly from the control laws decentralized tracking. While there is a wide literature in this (7) and (11). u t area (e.g., [6]), our work differs from the existing literature in the following way: We do not assume that each agent’s Theorem 3 makes clear that each agent i requires a signal observations include the variable(s) that must follow the zi , which is a function of the trajectories that the agent and " desired trajectory. That is, although we expect for the its upstream neighbors must follow. We refer to this position of agent i to follow a desired trajectory, we do not function zi ð"UðiÞ ; ri Þ as the trajectory information distribu- " r " assume that the agent i necessarily has an observation of its tion function (TIDF) for agent i, and also refer to a signal zi" position in an absolute frame. Since the agents that we generated by this function for a particular set of trajectories consider have limited sensing capability, we believe this as a trajectory information signal (TIS) for agent i. We note generalization to be absolutely necessary. that agent i’s input (actuation), or equivalently its TIS, can The comparison of our work with the literature leads us be viewed as comprising two components. The first to consider the important notion of decentralization in 2 " component (d ri ) provides the agent with the power needed dt2 tracking problems. In our setting, trajectory distribution is to follow its desired trajectory, and the second component not completely decentralized, in that, each agent’s actuation r d" ðÀkT Æj2UðiÞ Gij ð"j þ dtj ÞÞ allows stabilization about the i r depends not only on its own desired trajectory but on the trajectory and depends on the desired trajectories of up- desired trajectories of other agents. However, this depen- stream neighbors. We stress here that the tracking task dence is sparse, in that, each agent’s actuation only depends 526 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 on the trajectories of upstream neighbors in addition to its We have thus given conditions on the communication/ own. Hence, we can view trajectory distribution as being sensing topology for tracking in double-integrator networks partially decentralized. and saturating double-integrator networks. In the process, We stress that understanding this structural dependence we have exposed the need for trajectory distribution, of the actuation signal on desired trajectories is important, identified the trajectory information signals needed by each in that, it clarifies the content/complexity of computations agent, and indicated the dependence of these signals on the and communications needed for trajectory distribution. As sensing/communication topology. a concrete example, let us consider using the tracking controller in autonomous-vehicle-team applications. In 4 INFORMATION DISTRIBUTION IN FORMATION some applications, it may be advantageous for a vehicle to specify its desired trajectories (for instance in reaction to an In the autonomous-vehicle-coordination literature, there observed stimulus). In such cases, tracking requires com- has been wide interest in recent years on formation and munication of the desired trajectory to some but not all the motion-in-formation tasks (e.g., [1], [3], [13], [14]). These other vehicles in the team; in particular, the trajectory signal are tracking tasks where the agents in a network maintain a must be distributed to upstream neighbors. In other geometric pattern as they move through space. Further, in applications, a central authority may select and commu- some cases, this formation moves through space according nicate the desired trajectories. In this case, the central to some simple rules (e.g., the formation has to be at certain authority needs to construct the TIS for each vehicle locations at given times rather than following an arbitrarily according to the specified sensing topology. set trajectory, or the formation only needs to rotate). The An Information-Communication Perspective. Let us tracking controller that we have developed in Section 3 now consider trajectory distribution from the perspective permits completion of a wide range of formation-tracking of how the required information can be communicated to the tasks. Here, we develop controllers for a family of agents. In general, trajectory distribution can be achieved by formation-tracking tasks. We note that the formation- sending each agent’s TIS (or the desired trajectories from tracking tasks considered here are a special subcase of the which the TIS can be computed) to it before or during a general tracking tasks studied in Section 3. We will show tracking task. Alternately, trajectory distribution can be how the trajectory distribution aspect of the tracking task achieved without any special communication if the agents can be greatly simplified for this special case, and hence show that formation tracking can be achieved with little can somehow infer the necessary desired trajectory informa- communication. tion through their sensed observations. Such detection of or To consider tracking in formation, we impose a geo- adaptation to the correct trajectory may be feasible when the metric (spatial) interpretation for the agents’ states. Speci- possible desired trajectories of the agents are limited to a fically, motivated by typical autonomous-vehicle-control small set (see Section 5). applications, we consider a network of n agents moving in In the typical case that trajectory information must be the plane.4 Each agent’s x-direction (horizontal) and y- sent to the agents, a trajectory-communication scheme direction (vertical) motions are governed by double inte- which overlays the sensing network (observation graph) grators, i.e., the agent’s controller sets accelerations in each may well be needed. The TIDF for each agent makes clear direction with the goal of controlling its position in the the minimal statistic that must be communicated to that plane. Each agent makes a set of position observations, agent to achieve the tracking task. which are linear combinations of the agents’ x- and y- In many applications, the desired trajectories may be direction positions. For each position observation, an agent decided on by a central authority. For instance, a set of is assumed to make a corresponding velocity observation, robots may be tasked to sweep a minefield in a specified which is the same combination of the agents’ velocities. We pattern. In such cases, each TIDF indicates the sparsity of refer to this model as a planar double integrator network the computation required for the authority to generate the (PDIN). In the case where actuators may saturate, we use corresponding TIS; specifically, the authority must combine the term saturating planar double integrator network. the desired trajectories of upstream neighbors (and their We are interested in achieving tracking in the PDIN. As time derivatives) to obtain the TIS for a particular agent, with the double-integrator network, we specify a tracking which is then sent to the agent using some overlayed task with a set of trajectories for the desired motions of each communication scheme. agent. For a PDIN, the agents move in the plane, so we In other applications, the desired trajectories may specify a desired x-trajectory (desired trajectory in the x- themselves be chosen in a distributed fashion; either each direction) rix ðtÞ and desired y-trajectory riy ðtÞ for each agent may choose its own trajectory, or the trajectories may agent i. We refer to the desired trajectories together as the be set by a group of leaders in the network. (Such tracking task for the double-integrator network. distributed trajectory generation is sensible, for instance, As briefly discussed in Section 2 and in [2], we can in networks with a large number of agents that are straightforwardly reformulate a PDIN as a double-integra- concurrently participating in several different tasks.) In tor network with 2n agents. From the graph matrices of the such fully distributed settings, the TIDF indicates that equivalent double-integrator network, we can decide communication of the trajectory signals from upstream whether the double-integrator network is stabilizable, and neighbors is sufficient for each agent to compute its TIS. hence, whether the PDIN can achieve the tracking task. If it Thus, in this case, the TIDFs explicitly illustrate the sparsity and required topology for trajectory communication. 4. The generalization to higher-dimensional motion is straightforward. CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 527 Fig. 2. (a) Translation in the x- and y-directions, (b) rotation around the reference point, and (c) expansion around the reference point. can, we also refer to it as stabilizable. For a stabilizable The three typical trajectories—translation in the x and PDIN, we can again find a minimum statistic about the y-directions, rotation around the reference point, and expan- desired trajectories that must be provided to each agent to sion around the reference point—are illustrated in Fig. 2. permit completion of the tracking task. In keeping with the We are interested in having a network of agents that general case, we refer to the (two) signals that must be given completes a tracking task while in formation. Let us thus to agent i (for x-direction and y-direction tracking) to permit formally define the notion of a formation-tracking task. tracking as the trajectory information signals (TIS). Definition 5. A formation-F0 tracking task is one in which the We are interested in trajectory distribution for formation- set of desired trajectories ðr1x ðtÞ; r1y ðtÞÞ; . . . ; ðrnx ðtÞ; rny ðtÞÞ is tracking tasks in stabilizable PDINs, i.e., tracking tasks in in the formation F0 , at each time t. which the desired trajectories maintain a fixed pattern in the plane. In particular, we claim that trajectory distribution for That is, a formation-F0 tracking task is one in which the formation tracking can be achieved with simpler/less desired or nominal trajectories of the agents are in the communication than for general tracking tasks. To expose formation F0 at all times t. We stress here that formation is this simplification, let us first define the notion of a enforced on the desired trajectories, not on the agents formation and formation tracking. To do so, we first define themselves. If the agents are able to complete the tracking the notion of a nominal formation, i.e., a set of points that task, however, they enter and remain in formation after describe a pattern in the plane. some time passes. This is a sensible assumption for many Definition 3. A nominal formation F0 is an ordered set of n pairs applications, in that, vehicles/agents must move into ðb1x ; b1y Þ; . . . ; ðbnx ; rny Þ, along with a reference b0x and b0y . r r r b r r formation at the commencement of a task, and then remain The nominal formation describes a pattern of points in the in formation. We note that our notion of formation tracking plane, together with a reference point for this pattern. is identical to the notion of convergence to formation developed in [3], except in that our notion permits rotation and expansion in addition to translation. We refer to a set of points as being in the formation F0 , if Since the desired trajectories are in the formation F0 at these points form the same pattern in space as the nominal each time t, it is automatic that the desired trajectories for formation F0 : each agent i, 1 i n, can be written in the form Definition 4. An ordered set of n points in the plane (i.e., n pairs) ðr1x ; r1y Þ; . . . ; ðrnx ; rny Þ is said to be in the formation F0 , if all rix ðtÞ ¼ aðtÞ½ðbix À b0x ÞcosððtÞÞ r r points ðbix ; biy Þ in the nominal formation can be placed on the r r b þ ðbiy À r0y ÞsinððtÞÞ þ px ðtÞ; r corresponding points ðrix ; riy Þ through expansion around the ð18Þ riy ðtÞ ¼ aðtÞ½Àðbix À b0x ÞsinððtÞÞ r r reference point, rotation around the reference point, and b þ ðbiy À r0y ÞcosððtÞÞ þ py ðtÞ; r translation in the x- and y-directions. That is, ðr1x ; r1y Þ; . . . ; ðrnx ; rny Þ is in formation F0 if there are parameters a, , px , and for four signals aðtÞ, ðtÞ, px ðtÞ, and py ðtÞ. We refer to the b py such that rix ¼ a½ðbix À r0x ÞcosðÞ þ ðbiy À b0y ÞsinðÞ þ r r r signals aðtÞ, ðtÞ, px ðtÞ, and py ðtÞ as the expansion, rotation, px and riy ¼ a½Àðbix À b0x ÞsinðÞ þ ðbiy À b0y ÞcosðÞ þ py , r r r r x-translation, and y-translation parameters, respectively. for all i. We notice that these formation-tracking signals together with the nominal formation specify completely the tracking 528 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 task. Equation (18) clarifies that the formation-tracking task (unsaturated) PDIN with the same observation topology is is an example of the more general tracking task discussed in stabilizable, and further, the formation-tracking signals Section 3, for which trajectories are constrained to the form change slowly enough. In the case where standard forma- given in the equation. tion tracking is required, we can always meet this Since a formation-tracking task is specified by its additional condition by providing a sufficient time interval formation-tracking signals and nominal formation, only between each desired formations. this information or some subset thereof is needed to be communicated to each agent. Let us formalize this notion. 5 INFORMATION FLOW THROUGH ADAPTATION Theorem 4. Consider a stabilizable PDIN performing a forma- So far, we have considered distribution of desired- tion-tracking task. The TIS for each agent i’s x-trajectory or y- trajectory information through explicit (though hopefully trajectory is in general a function of the formation-tracking sparse) overlayed communication. In some applications, signals ðaðtÞ; ðtÞ; px ðtÞ; py ðtÞÞ as well as the nominal-forma- such additional communication may be infeasible, or r b tion parameters ðbix ; biy Þ, ðbjx ; bjy Þ, and ðb0x ; r0y Þ, j 2 UðiÞ. r r r r tracking of an agent with hidden or unknown trajectory information may be desired. In this final section, we Theorem 4 makes clear the information distribution explore, in the case of formation tracking, how to achieve required for formation tracking. In many applications, we trajectory information flow distribution through adaptation, envision that agents know in advance (or can obtain with i.e., how agents can identify and follow trajectories based minimal communication) their required information about on signatures in the actuation signals rather than through the nominal formation, i.e., the location of the reference and explicit overlayed communication. the locations of upstream agents (and the agent itself) in Our contention is that adaptation is most feasible for formation. In this case, the agents only need to be given, or formation-tracking tasks where agents can only follow be able to obtain, the four formation-tracking signals. trajectories having a small number of unknown parameters. While formation tracking permits a significant simplifi- In such cases, agents without knowledge of desired trajectory cation in information distribution, the distribution task is information can infer this information from transients in their still taxing in the sense that (in general) the signals aðtÞ, ðtÞ, actuation signals, and hence adapt their TIS appropriately. px ðtÞ, and py ðtÞ must be distributed to the agents or Let us explore information flow through adaptation in computed by them. In most settings, the formation may the context of an example. Specifically, we consider the case only need to follow a few simple trajectories to achieve that agents aim to adapt their TIS to follow a turn action, desired tasks. In such cases, we can develop a paradigm for while maintaining formation. In particular, we assume that providing agents with parametric information about the the agents are initially moving with constant velocity. A desired trajectories, from which the agents can compute leader agent is given a command to turn left or right (in their inputs. Precisely, as is very common in tracking both cases, the new desired trajectory is a straight-line path applications (see, e.g., [10]), we can limit the number of and is perpendicular to the original path before the turn), modes (signal frequencies) contained in the desired trajec- and the other agents must infer their TIS, and hence follow tories, and hence, simply communicate the modes and this new path. In addition, we make two assumptions on trajectory-initial conditions. There are of course several the communication/sensing topology of the double-inte- plausible paradigms for limiting the motions of the agent. grator network: 1) the leader agent measures its own state The following is one such paradigm: in an absolute frame (and so can follow its new TIS), and 2) in the corresponding network graph (as defined in Section Definition 6. A standard formation-F0 tracking task is a 3), there exists at least one path from the leader agent to formation-F0 tracking task in which the expansion, rotation, and each other agents in the network. In this case, we will show translation signals aðtÞ, ðtÞ, px ðtÞ, and py ðtÞ are set to desired that the TIS can be adaptively determined by each agent, values at specified times t ¼ 0; t1 ; t2 ; . . . , and are interpolated and hence, the formation tracking can be achieved. linearly between these desired values at intermediate times. We stress that we are considering a special case with only one type of trajectory—translation with constant veloci- Consider a standard formation-F0 tracking task in a ty—and two possible new trajectories. It should also be noted stabilizable PDIN. We notice that we can specify the that such a limited set of trajectories means that there are a formation-tracking signals for this task simply by specifying small number of unknown parameters corresponding to that the signals at time 0; t1 ; t2 ; . . . . Thus, it is automatic that we set. We note, however, that if these trajectories are generated can distribute these formation-tracking signals by distribut- by exosystems as defined in Section 2, the set of initial ing the values of the signals only at the specified times. Let conditions for the new exosystem may be infinite because the us formalize this notion. turn (left or right) can happen at any time. Theorem 5. Consider a stabilizable PDIN that must complete a Before presenting our method for TIS adaptation, it is standard formation-F0 tracking task. If each agent i is provided useful to define the shortest path from the leader agent to with the formation-tracking signals ðaðtÞ; ðtÞ; px ðtÞ; py ðtÞÞ at another agent (i.e., an agent that has no measurement on its times 0; t1 ; t2 ; . . . as well as the nominal-formation parameters own state in an absolute frame), the weight of an edge, and the weight of a path. ðbix ; biy Þ, ðbjx ; bjy Þ, and ðb0x ; b0y Þ, j 2 UðiÞ, then, the agent can r r r r r r compute its TIS, and hence, the tracking task can be achieved. Definition 7. A shortest path from the leader agent to another agent i is a directed path from the leader agent to In the interest of space, we omit detailed discussion of agent i in the network graph, which has no more edges than formation tracking in saturating PDINs. Briefly, it is easy to any other path from the leader agent to agent i. show that tracking in saturating PDINs is possible, if a CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 529 We note that there may be more than one shortest path Therefore, we have the following expressions: from the leader agent to a nonleading agent. À ðmÞ Á ðmÞ ðmÞ ðmÞ ðmÞ ui jt¼tT ¼ f1 ðiÞ þ f2 ðiÞ þ f3 ðiÞ þ f4 ðiÞ jt¼tT ; Definition 8. The weight of a directed edge from vertex j to vertex i is defined to be kT Gij , where ki is agent i’s position ðmþ1Þ 1 ðmÞ ðmþ1Þ 1 ðmÞ i ui jt¼tT ¼ f ðiÞ þ f2 ðiÞ þ f4 ðiÞ control gain, and Gij is the jth column of the graph matrix Gi . 2 ðmþ1Þ We note that edge weights are well defined even when þ f4 ðiÞ jt¼tT : agents have multiple observations. From the above expressions, together with the fact that Definition 9. The weight of a directed path is the product of edge weights along that path. ð0Þ ð0Þ ð0Þ ð0Þ f1 ðiÞ jt¼tT ¼ f2 ðiÞ jt¼tT ¼ f3 ðiÞ jt¼tT ¼ f4 ðiÞ jt¼tT ¼ 0 We are ready to show how the nonleading agents can and adapt their TIS appropriately. It turns out that, after the ðmþ1Þ ðmÞ leader agent makes a turn, a transient appears in the f2 ðiÞ jt¼tT ¼ ki Gii ui jt¼tT ; actuation signal of each nonleading agent. The sign of this ðÞ ðÞ we can recursively show that ¼ f4 ðiÞ jt¼tT , if ui jt¼tT transient is closely related to the acceleration of the leader ðmÞ ui jt¼tT ¼ 0 for m ¼ 0; . . . ; À 1. agent and the weights of the shortest paths from the leader Hence, we can recursively express the th-order agent to that particular nonleading agent. Formally, we derivative of ui with respect to time as: have the following result: ðÞ ðÀ1Þ Theorem 6. Suppose that a group of n agents in a PDIN (with a ui jt¼tT ¼ Æj1 2UðiÞ ki Gij1 uj1 jt¼tT leader agent and a path from this leader agent to each other ... agent in the network graph) are moving in a straight line with ð1Þ ujÀ1 jt¼tT ¼ Æj 2UðjÀ1 Þ kjÀ1 GjÀ1 j uj jt¼tT : constant velocity, and a leader agent is given a command to make a left or right turn. When the leader agent makes the If < , then the leader agent is not in the set UðjÀ1 Þ. ðÞ turn, each of the nonleading agents can infer the direction and Therefore, ui jt¼tT ¼ 0. time of the turn from their actuation-signal transients, and If ¼ , then the leader agent is in the set UðjÀ1 Þ. ðÞ hence, the appropriate TIS can be adaptively determined by Therefore, ui jt¼tT ¼ pi ul jt¼tT 6¼ 0, where pi is the sum each nonleading agent. Specifically, immediately after the of weights of the shortest paths from the leader agent to leader agent turns, we have that sgnðui Þ ¼ sgnðpi ul Þ, where agent i, and ul is the actuation signal of the leader agent. ðÞ ðÞ ui is the acceleration of agent i (in the x or y-direction), ul is Since ui jt¼tT ¼ 0 for all < and ui jt¼tT 6¼ 0, the sign of agent i’s acceleration ui (in the x-direction) the acceleration of the leader agent (in the x or y-direction), immediately after the leader-agent turns is the same as and pi 6¼ 0 (which can always be done by adjusting the ðÞ the sign of ui jt¼tT , i.e., sgnðui Þ ¼ sgnðpi ul Þ. t u control gain) is the sum of weights of the shortest paths from the leader agent to agent i. In many applications, the nonleading agents have Proof. We prove this theorem by first explicitly computing observations on their positions relative to other agents. In the derivatives of nonleading agent i’s actuation signal such cases, a grounded Laplacian sensing architecture (see (in the x-direction) with respect to time, and then [2] for definition, and note that in our case, only the leader arguing that the sign of nonleading agent i’s acceleration agent has absolute position measurement) is used, and we (in the x-direction) immediately after the leader agent have the following corollary: turns is determined by the th-order derivative, where Corollary 1. Suppose that n agents in a PDIN with grounded is the length of the shortest path from the leader agent to Laplacian sensing topology are moving in a straight line with agent i. Since the proof for the y-direction is exactly the constant velocity, and a leader agent is given a command to same, we only need to consider the x-direction. make a left or right turn. Then, immediately after the leader From the control law described in Theorem 1, the actuation signal of a nonleading agent i (in the x- agent turns, the sign of each nonleading agent i’s acceleration direction) is (in the x and y-direction) is the same as that of the leader agent (in the x and y-direction). ui ¼ ki Gii ðri À ri Þ þ ki Gii ðvi À vi Þ " " Proof. From the definition of grounded Laplacian sensing þ ki Æj2UðiÞ Gij ðrj À rj Þ þ ki Æj2UðiÞ Gij ðvj À vj Þ: " " architecture, all nonzero off-diagonal entries of the full For convenience, let ui ¼ f1 ðiÞ þ f2 ðiÞ þ f3 ðiÞ þ f4 ðiÞ, graph matrix G are negative. In addition, it is known that where the eigenvalues of KG (where K is the block-diagonal control gain matrix) are in the OLHP if and only if the f1 ðiÞ ¼ ki Gii ðri À ri Þ; " nonzero entries of K are negative. Therefore, the weight f2 ðiÞ ¼ ki Gii ðvi À vi Þ; " of path from the leader agent to each nonleading agent i f3 ðiÞ ¼ ki Æj2UðiÞ Gij ðrj À rj Þ; " is always positive, i.e., pi > 0. According to Theorem 6, u we have sgnðui Þ ¼ sgnðul Þ, and the corollary results. t and f4 ðiÞ ¼ ki Æj2UðiÞ Gij ðvj À vj Þ: " 530 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 10, NO. 5, APRIL 2011 Fig. 3. Information flow through adaptation is demonstrated. (a) Transient signature in nonleading agent 2’s y-direction acceleration in response to leader agent 1’s change in trajectory. (b) Tracking through adaptation. In both (a) and (b), a time trace of the horizontal and vertical positions of the agent(s) is shown. We thus notice that the sign of the leader agent’s tasks and networks is worthwhile. The fundamental com- acceleration, in the x and y-direction, can be inferred from plexity in adaptation lies in allowing agents to distinguish the nonleading agents’ local actuation signals. Such between different possible trajectories, using only data information is sufficient for nonleading agents to determine sensed through the network and assuming no prior knowl- the trajectory of the leader agent, and hence, the appropriate edge of the desired trajectory. This task is thus greatly TIS can be adapted by each nonleading agent. Specifically, ameliorated if the set of possible trajectories is highly limited, we use the following mechanics to adapt the TIS for each and the trajectories are simple so that changes can be nonleading agent i. determined through, e.g., simple thresholding mechanisms. First, we set a reasonable threshold for the actuation Also, it is worth noting that the transient motions induced signal of each nonleading agent i. When agent i is moving in the nonleader agents by the leader agent’s change in with constant velocity as desired, the amplitude of its direction are similar in flavor to the fluctuations observed in actuation signal is close to zero and should be smaller than follower agents in the string stability literature (for example, the threshold. In addition, if a turn is made by the leader [15]), though our analysis is valid for general topologies agent, each threshold should be exceeded by the ampli- rather than only strings, and also, the purpose of the analysis tude of the transient in each nonleading agent i’s actuation is different (in that, we use these fluctuations for decision signal. Then, from the sign of ui , agent i is able to making rather than seeking to prove that they are small). determine the acceleration, and hence the new trajectory of Finally, let us present an example that illustrates the the leader agent. Specifically, agent i can adaptively above adaptation scheme. determine its own new TIS by using its knowledge of the new velocity (speed and direction) of motion. In doing Example 3. We consider a double-integrator network so, we use the old TIS value when the threshold is consisting of four agents moving in the plane, with exceeded as the initial condition for the new TIS. It should sensing specified by the full graph matrix be noted that the TIS of agent i is not switched to the 2 3 newly-adapted TIS until all nonleading agents’ threshold 1 0 0 0 6 À1 1 0 0 7 have been exceeded. That is, after agent i’s threshold is G¼6 4 0 À 1 1 À 1 5: 7 exceeded and its new TIS is determined, a reasonable time 2 2 delay is desired before the new TIS is used by agent i. If À1 0 0 1 such a time delay is absent, the new behaviors of some Before a turn is made, we assume that the agents move nonleading agent i may affect the TIS adaptation of some along the x-axis with constant velocity. When the x- nonleading agent j whose actuation signal transient has coordinate of the leader agent (agent 1) reaches x ¼ 11, not exceeded its threshold. In this case, agent j may not agent 1 receives a command signal to turn right. The detect the new trajectory, or even determine an incorrect new trajectory. It should also be mentioned that, since we nonleading agents can use a threshold of 0.02 to detect simply use the old TIS values when threshold is exceeded the new trajectory of the leader agent, and change their as initial conditions for the new TIS, some error may be TIS to follow the new trajectory. The simulation result is introduced to the formation after the turn is made (since shown in Fig. 3. there is a delay between the time of the turn and the time Connection to [14]: In [14], a platoon of vehicles are when the threshold is exceeded). However, we also note performance regulated (i.e., a performance statistic of the that this error can be made arbitrarily small by making the platoon tracks a desired trajectory), while individual agents’ threshold small. trajectories are not explicitly set and are unknown until the A brief discussion of the complexities involved in closed-loop system is simulated. In order to realize perfor- generalizing the above adaptation to more general tracking mance regulation, an exosystem is used to generate the CHEN ET AL.: ON THE INFORMATION FLOW REQUIRED FOR TRACKING CONTROL IN NETWORKS OF MOBILE SENSING AGENTS 531 desired platoon performance, and an external device ACKNOWLEDGMENTS measures the actual platoon performance. This information This work was partly supported by the US National is broadcasted to all the vehicles. This method involves Science Foundation under Grant ECS 0528882 (Sensors), relatively little communication. Our method of tracking and by the US Office of Naval Research under Grant through adaptation is different from [14], in that, each N000140310848. agent’s trajectory and the nominal formation are explicitly specified, and hence, the performances of both individual agents and the group are known. Also, our method of REFERENCES tracking through adaptation does not require an external [1] J.A. Fax and R.M. Murray, “Information Flow and Cooperative device as in [14]. Instead, each agent can obtain the needed Control of Vehicle Formations,” IEEE Trans. Automatic Control, information locally (and hence no explicit communication is vol. 49, no. 9, pp. 1465-1476, Sept. 2004. needed). However, our method has the disadvantage that it [2] S. Roy, A. Saberi, and K. Herlugson, “Formation and Alignment of Distributed Sensing Agents with Double-Integrator Dynamics,” only may work for a limited set of possible trajectories. Sensor Network Operations, May 2006. One further note about the methodology in this section is [3] A. Williams, G. Lafferriere, and J.J.P. Veerman, “Stable Motions of needed. For ease of presentation, we have phrased the Vehicle Formations,” Proc. IEEE Conf. Decision and Control, pp. 72- 77, 2005. theorems in the section in terms of the parameters in KG [4] P. Ogren, E. Fiorelli, and N. Leonard, “Cooperative Control of rather than those in G. However, this formulation does not Mobile Sensor Networks,” IEEE Trans. Automatic Control, vol. 49, complicate use of the results; application of Theorem 6 only no. 8, pp. 1292-1302, Aug. 2004. [5] D.J. Stilwell, B.E. Bishop, and C.A. Sylvester, “Redundant requires that the controller design does not disconnect the Manipulator Techniques for Partially Decentralized Path Planning network graph (a condition which must be satisfied to and Control of a Platoon of Autonomous Vehicles,” IEEE Trans. achieve stability anyway). While the result of Theorem 6 Systems, Man, and Cybernetics, vol. 35, no. 4, pp. 842-848, Aug. 2005. [6] E. Davison, “The Robust Decentralized Control of a General does depend on K and G, we can compute the sign of ui , Servomechanism Problem,” IEEE Trans. Automatic Control, vol. 21, and hence detect a trajectory change and achieve adaptation no. 1, pp. 14-24, Feb. 1976. for any controller that we choose to use. [7] D.J. Stilwell and B.E. Bishop, “Platoons of Underwater Vehicles: Communication, Feedback, and Decentralized Control,” IEEE Control Systems Magazine, vol. 20, no. 6, pp. 45-52, Dec. 2000. 6 CONCLUSION [8] S. Roy, K. Herlugson, and A. Saberi, “A Control-Theoretic Approach on Distributed Discrete-Valued Decision-Making in We have developed tracking controllers for a canonical Networks of Sensing Agents,” IEEE Trans. Mobile Computing, communicating-agent network model, in particular a net- vol. 5, no. 8, Aug. 2006. [9] R.O. Saber and R.M. Murray, “Consensus Problems in Networks work of double integrators with sensing and communica- of Agents with Switching Topology and Time-Delays,” IEEE tion capabilities. Specifically, we have identified controllers Trans. Automatic Control, vol. 49, no. 9, pp. 1520-1533, Sept. 2004. that permit each agent in the network to track an arbitrarily [10] A. Saberi, A. Stoorvogel, and P. Sannuti, Output Regulation and Control Problems with Regulation Constraints. Springer-Verlag, desired trajectory and have characterized the information 1999. on the desired trajectories that must be distributed to each [11] M.E. Fisher and A.T. Fuller, “On the Stabilization of Matrices and agent. We have also specifically studied formation-tracking the Convergence of Linear Iterative Processes,” Proc. Cambridge Philosophical Soc., pp. 417-425, 1956. tasks (ones that involve movement of agents in a formation [12] S. Roy, J. Minteer, and A. Saberi, “Some New Results on through translation, rotation, and expansion/contraction), Stabilization by Scaling,” Proc. Am. Control Conf., 2006. showing that little communication is needed for informa- [13] R. Olfati-Saber, J.A. Fax, and R.M. Murray, “Consensus and Cooperation in Networked Multi-Agent Systems,” Proc. IEEE, tion distribution in this case. Finally, for some simple vol. 95, no. 1, pp. 215-233, Jan. 2007. tracking tasks, we are able to develop controllers that [14] D.J. Stilwell and B.E. Bishop, “Redundant Manipulator Techni- require no information flow at all. ques for Path Planning and Control of a Platoon of Autonomous Vehicles,” Proc. 41st IEEE Conf. Decision and Control, pp. 2093-2098, From a broad perspective, our efforts here contribute to 2002. the development/design of algorithms and controllers for [15] D. Swaroop and J.K. Hedrick, “String Stability of Interconnected modern networks (e.g., [2], [8], [1], [13], [16], [17]). The Systems,” IEEE Trans. Automatic Control, vol. 41, no. 3, pp. 349-357, Mar. 1996. research on decentralized controllers and control-theoretic [16] Y. Wan, S. Roy, and A. Saberi, “A New Focus in the Science of algorithms for networks can be viewed as having two main Networks: Toward Methods for Design,” Proc. Royal Soc. A, thrusts, namely toward 1) developing controllers/algorithms vol. 464, pp. 513-535, Mar. 2008. for varied and complex network tasks, and 2) designing high- [17] Y. Wan, S. Roy, A. Saberi, and A. Stoorvogel, “A Multiple Derivative and Multiple Delay Paradigm for Decentralized performance (fast, robust) algorithms and controllers. Our Controller Design,” Proc. 48th IEEE Conf. Decision and Control results here are aligned with the first of these two thrusts: (CDC 09), 2009. tracking algorithms/controllers are needed in numerous applications, and so our results contribute to the growing . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib. suite of algorithms/controllers for networks. We expect to address design of high-performance tracking controllers in future work; see, e.g., [16], [17] for recent work on designing high-performance decentralized controllers. We also note that the trajectory distribution through adaptation can be extended in several ways, including toward adaptation among a continuous set of possible trajectories and adapta- tion from a nonzero-acceleration trajectory.