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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.



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:
. 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:                                 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:, 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,
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;
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).

              4                         4
w h e r e r ¼ ½ r1 . . . rn ŠT ; v ¼ ½ v1 . . . vn ŠT , a n d t h e               loss of generality) that each desired trajectory is generated
graph matrix Gi has dimension mi  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-
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 ŠT .                                                             and di is a constant vector of dimension qi  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 Šwi ; 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
saturation is ignored).
                                                                                                                           6 7
2.2 Tracking Problem Formulation                                                                             with wi ð0Þ ¼ 6 7:
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        "
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
                                                                                             6            ..            7
traveling at constant speeds for periods of time and                                       K¼4                 .        5
stopping for designated intervals to permit completion of                                                          kT
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         .
                                                                                                                        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
                             1 0 0 À1                                   above input signal.
                     G2 ¼ ½ À1 1 0 0 Š;                     ð6Þ              Noting that api ¼ Gi r and avi ¼ Gi v, when the above
                     G3 ¼ ½ 0 À1 1 0 Š;                                 input signal is applied, the dynamics of agent i is
                     G4 ¼ ½ 0 0 À1 1 Š:                                 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
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
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 ;
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 ŠT , and zv ¼ ½zv1 . . . zvn ŠT . 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
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.

   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 Š:                                                      where " 2 ð0; "à Š, and "à 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
integrator network for which there is K such that KG has all                                            dT                       Sn
eigenvalues in the OLHP as a stabilizable double-integrator                                2     3
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 "à > 0 such that for all " 2 ð0; "à Š, 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 :
                                                                                                          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 ;
  conditions hold:                                                                where zp ¼ ½zp1 . . . zpn ŠT , and zv ¼ ½ zv1 . . . zvn ŠT .
      (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
                                6       ..      7                                 linear region 8t ! T , and the tracking task is achieved.
                          K¼4              .    5
                                                                                     Let us first consider the transformed dynamics of
                                               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 ;
                                                                                  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-
                                 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
          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 ,
      "2 kT Gi zp þ "kT Gi zv <  if " 2 ð0; "à Š. 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; "à Š, where "à ¼ minf"à ; . . . ; "à 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.

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
                                              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
component (d ri ) provides the agent with the power needed
                                                                               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
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-
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.

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 ÞsinððtÞފ þ px ðtÞ;
  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
                                                                                                           þ ðbiy À r0y ÞcosððtÞފ þ py ðtÞ;
  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
  py such that rix ¼ a½ðbix À r0x ÞcosðÞ þ ðbiy À b0y Þsinðފ 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 Þcosðފ 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

   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                        
  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                                                            ðÞ                ðÞ
                                                                          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:
                                                                                   ðÞ                                 ðÀ1Þ
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
                                                                                 ujÀ1 jt¼tT    ¼ Æj 2UðjÀ1 Þ kjÀ1 GjÀ1 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 UðjÀ1 Þ.
                                                                                        ðÞ
  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.
  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.
                                                                                     ðÞ                                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
  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,
                                                                         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:
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

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
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
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on the desired trajectories that must be distributed to each      [11]   M.E. Fisher and A.T. Fuller, “On the Stabilization of Matrices and
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modern networks (e.g., [2], [8], [1], [13], [16], [17]). The             Systems,” IEEE Trans. Automatic Control, vol. 41, no. 3, pp. 349-357,
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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
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.

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