Implementation and Flight Test Resultsof MILP-based UAV Guidance

Tom Schouwenaars∗ Mario Valenti∗ Eric Feron∗ Jonathan How∗∗∗Laboratory of Information and Decision Systems

∗∗Aerospace Controls LaboratoryMassachusetts Institute of Technology77 Massachusetts Ave, Room 32D-712

Cambridge, MA 02139, USA617-253-7446

{toms,valenti,feron,jhow}@mit.edu

Abstract—This paper discusses the implementation of a guid-ance system based on Mixed Integer Linear Programming(MILP) on a modified, autonomous T-33 aircraft equippedwith Boeing’s UCAV avionics package. A receding hori-zon MILP formulation is presented for safe, real-time trajec-tory generation in a partially-known, cluttered environment.Safety at all times is guaranteed by constraining the interme-diate trajectories to terminate in a loiter pattern that does notintersect with any no-fly zones and can always be used as asafe backup plan. Details about the real-time software im-plementation using CPLEX and Boeing’s OCP platform aregiven. A test scenario developed for the DARPA-sponsoredSoftware Enabled Control Capstone Demonstration is out-lined, and simulation and flight test results are presented.

TABLE OF CONTENTS

1 INTRODUCTION

2 EXPERIMENT OVERVIEW

3 SAFE TRAJECTORY PLANNING

4 MILP FORMULATION

5 SOFTWARE IMPLEMENTATION

6 RESULTS

7 CONCLUSION

1. INTRODUCTION

Unmanned aerial vehicles (UAVs) have been used by bothmilitary and civilian organizations in a number of applica-tions. Recent advances in guidance technologies have en-abled some UAVs to execute simple mission tasks withouthuman interaction. Many of these tasks are pre-planned us-ing reconnaissance or environment information. For exam-ple, air operations are executed according to an Air TaskingOrder (ATO), which may take up to 72 hours to plan, taskand execute [15]. In volatile situations, however, informationabout the vehicle’s operating environment may be limited: a

0-7803-8870-4/05/$20.00c©2005 IEEEIEEEAC paper # 1396

detailed map of the environment might not be available aheadof time, and obstacles might be detected while a mission iscarried out. In such situations, task planning flexibility andsafe trajectory solutions are essential to the survivability andsuccess of the autonomous system. Most unmanned vehicles,however, do not exhibit this level of performance. Intelligentguidance systems providing the required flexibility are there-fore a topic of active research.

An approach to adaptive autonomous trajectory planningbased on Mixed Integer Linear Programming (MILP) was re-cently introduced in [10]. MILP is a powerful mathematicalprogramming framework that extends continuous linear pro-gramming to include binary or integer decision variables [5].These variables can be used to model logical constraints suchas obstacle and collision avoidance rules, while the dynamicand kinematic properties of the vehicle are formulated as con-tinuous constraints. Thanks to the increase in computer speedand implementation of powerful state-of-the-art algorithms insoftware packages such as CPLEX [1], MILP has become afeasible option for real-time path planning, as demonstratedby the results discussed in this paper.

Under the DARPA-sponsored Software Enabled Control(SEC) program, a receding horizon MILP formulation wasdeveloped for safe, real-time trajectory generation in apartially-known, cluttered environment [11]. After each timeinterval of a certain duration, a new MILP problem is solvedthat incorporates updated information about the environment,the task and the state of the vehicle, and is constrained toterminate in a safe loiter pattern. The output of the MILPoptimization is a sequence of waypoints constituting a partialtrajectory to the goal. As such, an optimal reference trajec-tory achieving a particular task, such as to search for a targetin a partially unknown area, is computed in real-time, i.e. asthe mission unfolds.

This paper discusses the implementation of this MILP-basedguidance algorithm on a modified, autonomous T-33 aircraft,augmented with Boeing’s UCAV avionics package. The plan-ning software using CPLEX’s Concert Technologies [1] wasintegrated with Boeing’s real-time Open Control Platform

1

(OCP) [7] and flight tested during the capstone demonstrationof the SEC program in June 2004. A scenario was flown inwhich the T-33 acted as a UAV that was given mission levelcommands by an F-15 weapon systems officer (WSO). Thecommunication between the F-15 WSO and the T-33 UAVwas done using a natural language interface, that interpretedhuman language commands of the WSO and transformedthem in real-time into input data for the optimal guidanceproblem. These flight tests marked the first time an on-boardMixed Integer Linear Programming-based guidance systemwas used to control a UAV.

The paper is organized as follows. Section 2 gives anoverview of the SEC experiment and the associated technol-ogy development. Section 3 describes the basic trajectoryplanning problem and our approach to maintaining safety.The detailed MILP formulation of the latter is presented inSection 4. Next, Section 5 discusses the real-time softwareimplementation and some practical engineering decisions.Simulation and flight test results are presented in Section 6,and Section 7 concludes.

2. EXPERIMENT OVERVIEW

Mission Scenario

As originally discussed in [6], as part of the DARPA-sponsored Software Enabled Control Program, our team wastasked with developing a flight-test scenario that exhibitedUAV technology developed at MIT. For this demonstration,we had access to two flight assets: a Boeing F-15E fighterjet (similar to the aircraft shown in Figure 1) and a LockheedT-33 trainer fighter jet (similar to the aircraft shown in Fig-ure 2), equipped with Boeing’s UCAV avionics package. Theformer was to be flown by a pilot and will be referred to asthe Fixed-Wing (FW) vehicle. The latter was to be guidedby our technology and will be referred to as the UnmannedAerial Vehicle (UAV). Besides these aircraft, a ground sta-tion receiving state and user-defined information from bothvehicles was available to monitor the experiment.

To enable a hard real-time execution, our demonstration soft-ware needed to be integrated with Boeing’s Open ControlPlatform (OCP) [7] and loaded onto a laptop fitted in eachaircraft. The OCP software provided us with an aircraft inter-face including the following abilities:

• Send and receive state and user-defined data between bothaircraft using a Link-16 communications interface,• Receive the current vehicle state data,• Send a set of pre-defined commands to the aircraft avionicssystem which include Set and Hold Turn Rate, Set and HoldSpeed, Set and Hold Altitude, Set and Hold Heading, and• Memory storage and time frame execution.

Given these demonstration resources, we developed a mis-sion scenario (shown in Figure 3) in which the UAV performs

Figure 1. Boeing F-15E Strike Eagle

Figure 2. Lockheed T-33

tasks in support of the FW vehicle:

Mission—A manned fighter aircraft (FW) and a UAV willwork together on a mission to collect images of a possible sitein enemy territory. The FW Weapon Systems Officer (WSO)will communicate with the UAV using a Natural LanguageInterface, which allows the FW WSO to speak with the UAVusing normal sentence commands. The UAV will perform thereconnaissance for the mission in a partially-known environ-ment, and the FW WSO will decide how the UAV will beused to accomplish the mission goals. The UAV will possessthe ability to detect threats and collect images, whereas, ifapplicable, the FW vehicle will be able to deliver weapons.Since the environment is only partially-known, there may bethreats to both the manned and unmanned aircraft.

Starting Condition—The UAV will start in a pre-defined loi-ter pattern; the FW vehicle will be flying an air-patrol nearthe enemy territory. The environment is partially-known andupdated in real-time to both the UAV and the FW. A pop-upthreat may arise en route to the search site, which is currentlyunknown.

Mission Narrative—

1: The FW vehicle is commanded to gather information andpossibly destroy an enemy site located in unknown territory.Because of the mission risk, the FW vehicle assigns the UAV,stored in a nearby airspace volume, to gather information atthe designated site. The UAV leaves the loiter area and moves

2

Figure 3. MIT Flight Experiment Mission Overview

toward the designated task area. The F-15 follows behind ata higher altitude and a safe distance.

2: The UAV is informed of a pop-up threat en route to the taskarea. The UAV accounts for the threat dynamically, auto-matically generates a revised safe trajectory around the threatand other no-fly zones, while notifying the FW vehicle of thethreat’s position.

3: As the UAV moves within a few minutes of the task loca-tion, the UAV notifies the FW vehicle of its location. At thispoint, the FW vehicle will provide the UAV with the exactingress and egress conditions into and out of the search area.The UAV modifies its flight path to arrive at the site as com-manded.

4: The UAV enters the site, notifies the FW of its location andbegins its search for the target.

5: The UAV identifies the target and sends an image to the FWfor evaluation. The FW commands the UAV to return to itsoriginal loiter area, while the FW prosecutes the target.

Exit Conditions—The UAV safely returns to the original pre-defined loiter pattern location; the FW vehicle returns to fly-ing an air-patrol near the enemy territory.

Technology Development

The above mission scenario allowed us to demonstrate tech-nology developments in several areas, leading to three distinctsoftware components:

1: Natural Language Interface —This component interpretsand converts normal sentence commands (e.g., “Search thisregion for threats”) from the humans on-board the FW vehi-cle into data the UAV can use and understand and vice-versa.

It is aimed at minimizing the workload of the WSO when in-teracting with the computer-based UAV. Namely, the WSOcan give the UAV high level mission commands in Englishrather than low level guidance commands such as “Turn left”or “Speed up”.

2: Task Scheduling and Communications Interface —The pri-mary goal of this component is to interpret the command datafrom the Natural Language Interface and develop a series oftasks the vehicle can perform. The list of mission tasks thatwere developed includes: flying to a waypoint X, entering aloiter pattern, and performing a search pattern. For each ofthese tasks, the user must provide the task ingress/egress con-ditions, and the size and location of the task area. The com-ponent also contains the communications processing modulethat provides the FW WSO with the authority to send taskcommands and receive status updates, threat/obstacle avoid-ance information, and acknowledgement messages. In addi-tion, it provides the FW WSO with the ability to override theUAV’s navigation system in the event of an emergency or er-ror.

3: Trajectory Generation Using MILP —After the NaturalLanguage Interface and Task Scheduling component haveconverted the mission steps into a series of tasks for the vehi-cle to perform, the Trajectory Generation Module guides thevehicle from one task location to the next. Time-optimal safetrajectories that account for the current state of the vehicle andthe knowledge of the environment are computed using MixedInteger Linear Programming (MILP). Since we assume that inthe mission scenario the environment is only partially-knownand explored in real-time, the MILP guidance algorithm usesa receding horizon planning strategy, allowing for online tra-jectory computation.

In this paper, we will focus on the MILP Trajectory Genera-tion component. Further details about the Natural LanguageInterface can be found in [4] and [9] ; a description of theTask Scheduling and Communications Interface and detailsabout the integrated mission system are given in [14].

3. SAFE TRAJECTORY PLANNING

Basic Planning Problem

The basic problem tackled by the Trajectory Generation mod-ule is to guide the UAV from an initial state to a desired onethrough an obstacle field while optimizing a certain objective.The latter can be to minimize time, fuel or a more sophisti-cated cost criterion such as to minimize visibility or to max-imize the total area explored. We consider 2D scenarios inwhich no-fly zones or “obstacles” are detected while the mis-sion is carried out, but assume that the environment is fullycharacterized inside a certain detection regionD around theaircraft. For the SEC demonstration, we considered a circularregion of radius9 mi and assumed that all obstaclesO withinthat radius were static. The MILP formulation that we willpresent, however, can readily be generalized to account forany detection shape, such as a radar cone, and for unknown

3

areas within that shape.

Since trajectories must be dynamically feasible, the vehicledynamics and kinematics should be accounted for in the plan-ning problem. For optimization purposes, the vehicle is char-acterized by a discrete time, linear state space model (A,B)in an inertial 2D coordinate frame. As such, the state vectorxconsists of the position(x, y) and inertial velocity(ẋ, ẏ). De-pending on the particular model, the input vectoru is an in-ertial acceleration or reference velocity vector. In both cases,however, together with additional linear inequalities inx andu, the state space model must capture the closed-loop dynam-ics that result from augmenting the vehicle with a waypointtracking controller.

Since the environment is only partially-known and exploredin real-time, a receding horizon planning strategy is used toguide the vehicle towards the desired destination. The lat-ter is denoted byxf and is an ingress/egress state of a taskor some other waypoint with a corresponding inertial ve-locity vector. At each time step, a partial trajectory fromthe current state towards the goal is computed by solvingthe trajectory optimization problem over a limited horizonoflengthT . Because of the computation delay, the initial statex0 = (x0, y0, ẋ0, ẏ0) in the optimization problem should bean estimatexestim of the position and inertial velocity of theaircraft when the plan is actually implemented.

The solution to the optimization problem provides a sequenceof waypoints(xi, yi) and corresponding inertial reference ve-locities(ẋi, ẏi) to the aircraft for the nextT time steps. Typi-cally, however, only the first waypoint and reference velocityof this sequence are given to the waypoint follower, and theprocess is repeated at the next time step. As such, new infor-mation about the state of the vehicle and the environment canbe taken into account at each time step.

By introducing a cost functionJT over theT time steps, thegeneral trajectory optimization problem can be formulatedasfollows:

minxi,ui

JT =

T−1∑

i=0

ℓi(xi,ui,xf ) + f(xT ,xf ) (1)

subject to:

xi+1 = Axi + Bui, i = 0 . . . T − 1x0 = xestimxi ∈ X0ui ∈ U0

(xi, yi) ∈ D0(xi, yi) /∈ O0

(2)

The objective function (1) contains a terminal cost termf(xT ,xf ), which accounts for an estimate of the cost-to-gofrom the last statexT in the planning horizon to the goalstatexf . The setsX0 andU0 represent the (possibly non-convex) linear constraints on the vehicle dynamics and kine-matics, such as bounds on velocity, acceleration and turn rate.

Here, the 0-subscript denotes the fact that these constraintscan be dependent on the initial state. Lastly, the expressions(xi, yi) ∈ D0 and(xi, yi) /∈ O0 capture the requirement thatthe planned trajectory points should lieinsidethe known re-gionD0, but outsidethe obstaclesO0 as given at the currenttime stepi = 0.

Loiter Principle

Despite these constraints, however, the above receding hori-zon strategy has no safety guarantees regarding avoidance ofobstacles in the future. Namely, the algorithm may fail to pro-vide a solution in future time steps due to obstacles that arelocated beyond the surveillance and planning radius of thevehicle. For instance, when the planning horizon is too shortand the maximum turn rate relatively small, the aircraft mightapproach a no-fly zone too closely before accounting for it inthe trajectory planning problem. As a result, it might not beable to turn away in time, which translates into the optimiza-tion problem becoming infeasible at a certain time step.

In Ref. [11], we therefore proposed asafe receding hori-zon scheme based on maintaining a known feasible trajec-tory from the final statexT in the current planning horizontowards an obstacle-free loiter pattern. The latter must liein the region of the environment that is fully characterizedat the current time step and is computed and updated on-line. Assuming that the planned trajectories can be accuratelytracked, at each time step, the remaining part of the previousplan together with the loiter pattern can then always serve asa safe backup or “rescue” plan. Namely, if at a certain timestep, the guidance software fails to find a feasible solutionto the optimization problem within the timing constraints ofthe real-time system, the aircraft can just keep following theprevious trajectory. Eventually, that trajectory will endin aloiter pattern in which the aircraft can safely remain for anindefinite period of time,i.e., without flying into an obstacleor no-fly zone. At each time step, asafetrajectory that is con-sistent with the known environment can thus be determined.

More specifically, for the SEC demonstration, we explicitlyconstrained the final statexT to be an ingress state to a rightor left turning loiter circle, respectively denoted byCR andCL. As discussed in Ref. [11] and detailed in Section 4, thiscan be done by expressingN sample points along the loitercircles as affine functions ofxT , thereby accounting for thefact that the minimum turn radius scales inversely with thevelocity. This scaling property and the choice of turning di-rection give the aircraft more degrees of freedom to fit theloiter circles in the obstacle-free areas of the known environ-ment. Introducing an indexj for the sample points, the abovesafety constraint can be specified as follows:

CR(xT ) = {(xRj , yRj)} ∈ D0, j = 1 . . . NCR(xT ) = {(xRj , yRj)} /∈ O0

ORCL(xT ) = {(xLj , yLj)} ∈ D0, j = 1 . . . NCL(xT ) = {(xLj , yLj)} /∈ O0

(3)

4

4. MILP FORMULATION

Mixed Integer Linear Programming

The optimal safe trajectory planning problem outlined abovelends itself well to be formulated as amixed integer linearprogram(MILP). MILP is a powerful mathematical program-ming framework that allows inclusion of integer variablesand discrete logic in a continuous linear optimization prob-lem [5]. It is commonly used in Operations Research [13],and has more recently been introduced to the field of hybridsystems [3] and trajectory optimization [10]. In our case, thecontinuous optimization is done over the states and inputs;thediscrete logic is introduced by nonconvex constraints suchasobstacle avoidance and minimum speed requirements, and bythe binary selection between the right and left turning loitercircles.

As an illustration of how logical decisions can be incorpo-rated in an optimization problem, consider the following ab-stract example. Assume that a cost functionJ(x) needs to beminimized subject to either one of two constraintsℓ1(x) andℓ2(x) on the continuous decision vectorx:

minx

J(x)

subject to:ℓ1(x) ≤ 0

OR ℓ2(x) ≤ 0

(4)

By introducing a large, positive numberM and a binary vari-ableb, this optimization problem can equivalently be formu-lated as follows:

minx

J(x)

subject to:ℓ1(x) ≤ Mb

AND ℓ2(x) ≤ M(1 − b)b ∈ {0, 1}

(5)

When b = 0, constraintℓ1(x) must be satisfied, whereasℓ2(x) is relaxed. Namely, ifM is chosen sufficiently large,inequality ℓ2(x) ≤ M(1 − b) is always satisfied, indepen-dently of the value ofx. The situation is reversed whenb = 1.Sinceb can only take the binary values0 or 1, at least oneof the constraintsℓ1(x) andℓ2(x) will be satisfied, which isequivalent to the original “OR”-formulation in problem (4).In the special case whereJ(x), ℓ1(x) andℓ2(x) are (affine)linear expressions, problem (5) is a MILP.

The formulation can easily be extended to account for multi-ple constraintsℓk(x), k = 1 . . . K, out of which at leastNmust be satisfied simultaneously. This is done as follows:

minx

J(x)

subject to:ℓk(x) ≤ Mbk, k = 1 . . . K

∑

k bk ≤ K − Nbk ∈ {0, 1}

(6)

The additional summation constraint ensures that at leastNof the binary variablesbk are0, thus guaranteeing that at leastN of the inequalitiesℓk(x) ≤ 0 are satisfied simultaneously.

MILP Trajectory Planning Formulation

We now apply the above framework to the trajectory opti-mization problem of Section 3.

State Space Model—System identification experiments usingthe UAV DemoSim simulation software provided by Boeinggave us insight into the velocity response of the UAV. We de-cided that for guidance purposes a piece-wise linear first or-der approximation with an actuator delay was sufficient. Weidentified the time constant and DC gain of the transfer func-tion for a discrete set of forward velocities (from 350 fps to500 fps with a resolution of 10 fps) and stored these in a look-up table. At each iteration of the receding horizon strategy,the model corresponding to the velocity at that time step wasused, thus linearizing the nonlinear response into severalLTImodes scheduled around the initial velocity.

Taking the desired inertial velocity as input, we used the fol-lowing continuous state space model:

ẋ(t)ẏ(t)ẍ(t)ÿ(t)

=

0 0 1 00 0 0 10 0 − 1

τl0

0 0 0 − 1τl

x(t)y(t)ẋ(t)ẏ(t)

+

0 00 0

klτl

0

0 klτl

[

ẋcmd(t)ẏcmd(t)

]

(7)

Here,τl is the time constant andkl is the gain correspondingto thelth LTI mode. Note, however, that these dynamics arehomogeneous in thex- andy-coordinate and as such ignoredifferences in the lateral and longitudinal aircraft dynamics.As will be detailed further on, we introduced additional con-straints on the state and input vectors to correct for this.

Since the typical time constant was around 9.7s, using the bi-linear transform, we discretized the above model with a timestep of∆t = 10 s, thus obtaining:

[xi+1 yi+1 ẋi+1 ẏi+1]T

=

1 0 ∆t2

(

1 + 2τl−∆t2τl+∆t

)

0

0 1 0 ∆t2

(

1 + 2τl−∆t2τl+∆t

)

0 0 2τl−∆t2τl+∆t 0

0 0 0 2τl−∆t2τl+∆t

xiyiẋiẏi

+

kl(∆t)2

2τl+∆t0

0 kl(∆t)2

2τl+∆t2kl∆t

2τl+∆t0

0 2kl∆t2τl+∆t

[

ẋcmd,iẏcmd,i

]

(8)

Together with the extra constraints, we found this model to

5

produce good results for tasks requiring intensive waypointtracking and sharp turns (i.e. the loiter and search tasks).However, for straight trajectories with more or less constantspeed, we also found it to react too aggressively to perturba-tions along the nominal trajectory. To transition between twotask areas, we therefore used a more conservative double in-tegrator model that does not distinguish between the differentLTI modes:

xi+1yi+1ẋi+1ẏi+1

=

1 0 ∆t 00 1 0 ∆t0 0 1 00 0 0 1

xiyiẋiẏi

+

0 00 0

∆t 00 ∆t

[

ẍiÿi

]

. (9)

where we again used∆t = 10 s.

Initial State Estimate—Since it takes a certain time to com-pute the trajectory, the initial state should be an estimateofwhat the vehicle’s state will be when the plan is actually im-plemented. In our case, the computation delay was 1 s. Inaddition, we accounted for a1.2 s actuator delay. To ob-tain an estimate of the initial state at the next iteration, wethus propagated the dynamics following the previous plan for2.2 s.

Cost Function—Given the above models, we were interestedin guiding the UAV between waypoints in the fastest possibleway, while avoiding no-fly zones. The exact shortest time be-tween two states, however, can only be computed if the plan-ning horizon spans that arrival time, or if an exact cost-to-gois known. Since in the scenario of interest the environment isnot characterized beyond a certain detection radius aroundthevehicle, computing an exact time-to-go function as proposedin [2] is not possible. We therefore opted for the followingheuristic approach.

In case there are no known obstacles intersecting the straightline between the waypoint and the current location, that way-point is used as the desired state in the cost function. In casethere are obstacles blocking this direct line-of-sight, the short-est path (as far as the known obstacles are concerned) must gothrough one of the visible corner points of these no-fly zones.This point can then act as an intermediate waypoint en routeto the final destination. To determine this optimal interme-diate point, a grid is constructed between the corner pointsof all known obstacles interfering with the line of sight. Ashortest path algorithm is then run to compute the approxi-mate shortest time towards the goal from each visible cornerpoint, thereby assuming the UAV is flying at maximum speed.As such, the “best” intermediate waypoint is determined byminimizing the total time from the current location to one ofthe visible vertices and from that point to the destination asgiven by the approximate cost-to-go function.

Using this intermediate (or, in the obstacle-free case, theorig-inal) waypointpf = (xf , yf ), we used the following piece-wise linear cost function that aims at designing afast trajec-tory between the initial positionp0 = pestim = (x0, y0) inthe planning horizon andpf :

min J =

T∑

i=0

−qv′i(pf − pestim) + r|pi − pf | (10)

The first term tries to maximize the scalar product of the iner-tial velocityvi = (ẋi, ẏi) with the vector that is pointing fromthe initial position to the desired one. The effect is twofold:it will speed the aircraft up to its maximal velocity, whileturning it toward the right direction. In addition, the termr|pi − pf | tries to minimize the 1-norm distance towards thegoal. Both terms thus work towards the same objective, withq andr weighting the two contributions.

If, at a certain iteration, the planned trajectory passes throughor near waypointpf at a time stepTf ≤ T in the planninghorizon, the cost function for the next iteration is split intotwo parts:

min J̃ =

Tf−1∑

i=0

−qfv′i(pf − pestim) + rf |pi − pf |

+

T∑

i=Tf

−qnv′i(pn − pf ) + rn|pi − pn| (11)

in which pn is the next (intermediate) waypoint. The firstTf − 1 time steps are thus used to minimize the cost towardswaypointpf ; the remaining steps aim at minimizing the costtowards the next waypointpn. As a result, depending on therelative weighting, the MILP will produce a trajectory thatwill pass through or close bypf and aims forpn next. Theabsolute values in the cost function can be handled by in-troducing auxiliary variables and extra constraints, accordingto the following principle: using an auxiliary variables, theproblemmin |z| is equivalent to

min ss.t. z ≤ s−z ≤ s.

In the SEC demo, we usedT = 6, corresponding to an ef-fective planning length of 1 minute. All weights in the costfunctions (10) and (11) where set to 1. In case the double in-tegrator model was used, we also added a small input regular-ization term

∑T−1i=0 r|ui| with r = 10

−4 to produce smoothtrajectories.

Velocity Bounds—To ensure that the planned trajectory re-spects the velocity limits of the vehicle, we added the follow-ing maximum and minimum speed constraints:

6

∀i ∈ [0 . . . T − 1], ∀k ∈ [1 . . . K] :

ẋi sin(

2πkK

)

+ ẏi cos(

2πkK

)

≤ vmaxẋi sin

(

2πkK

)

+ ẏi cos(

2πkK

)

≥ vmin − Mcik∑K

k=1 cik ≤ K − 1cik ∈ {0, 1}

(12)

For the reference velocity model, these constraints were for-mulated in terms of the input commandsẋcmd,i and ẏcmd,iinstead. The maximum speed bound is thus approximated byconstraining the inertial velocity vector to lie inside aK-sidedpolygon. Using the binary variablescik and principle (6), theminimum speed requirement is captured by constraining thevelocity vector to lieoutsidea (smaller)K-sided polygon.

In our software, we usedK = 32. The minimum and max-imum bounds were respectively set tovmin = 400 fps andvmax = 450 fps. To avoid infeasible problems in the eventthe actual ground speed fell outside these bounds (e.g. be-cause of wind gusts), the latter were adapted accordingly.

Acceleration Bounds—As mentioned before, both models as-sume homogeneous dynamics in thex- and y-coordinates,thus ignoring differences in lateral and longitudinal dynam-ics. To correct for this, we added linear constraints that cap-ture limits on turn rate and on forward and lateral accelera-tion. When flying at a more or less constant speed, the fol-lowing acceleration constraints were sufficient [8]:

∀i ∈ [0 . . . T − 1], ∀k ∈ [1 . . . K] :

ẍi sin(

2πkK

)

+ ÿi cos(

2πkK

)

≤ alat (13)

for the double integrator model, and

∀i ∈ [1 . . . T − 1], ∀k ∈ [1 . . . K] :

(ẋcmd,i − ẋcmd,i−1) sin(

2πkK

)

+(ẋcmd,i − ẋcmd,i−1) cos

(

2πkK

)

≤ alat∆t(14)

for the reference velocity model. We set the lateral acceler-ation boundalat = 18.1 ft

2/s, corresponding to a maximumturn rate ofωmax = alat/vmin = 2.6 deg/s atvmin = 400 fps.

When the velocity is allowed to change, however, these in-equalities overestimate the available forward acceleration,which was limited toafwd = 5.0 ft

2/s. Therefore, to dis-tinguish between forward and lateral acceleration, we addedthe following constraints:

∀i ∈ [0 . . . T − 1], ∀k ∈ [1 . . . K] :

(ẍi + αv−10 ẏi) sin

(

2πkK

)

+(ÿi − αv

−10 ẋi) cos

(

2πkK

)

≤ βalat(ẍi − αv

−10 ẏi) sin

(

2πkK

)

+(ÿi + αv

−10 ẋi) cos

(

2πkK

)

≤ βalat

(15)

for the double integrator model, and

∀i ∈ [1 . . . T − 1], ∀k ∈ [1 . . . K] :

(ẋcmd,i − ẋcmd,i−1 + αv−10 ẏi∆t) sin

(

2πkK

)

+(ẏcmd,i − ẏcmd,i−1 − αv

−10 ẋi∆t) cos

(

2πkK

)

≤ βalat∆t(ẋcmd,i − ẋcmd,i−1 − αv

−10 ẏi∆t) sin

(

2πkK

)

+(ẏcmd,i − ẏcmd,i−1 + αv

−10 ẋi∆t) cos

(

2πkK

)

≤ βalat∆t(16)

for the reference velocity model. Here,v0 is the current abso-lute ground speed,α = (a2lat−a

2fwd)/(2afwd) = 30.4 fps

2/s,

andβ =√

α2 + a2lat = 35.4 fps2/s.

These inequalities describe the intersection of two circles inwhich the inertial acceleration vector has to lie. The shortaxisof this intersection has length2afwd and is aligned with thevelocity vector at the first time step. The long axis capturesthe larger lateral acceleration bound and has length2alat.Hence, this intersection approximates the dynamically fea-sible acceleration profile at the initial time step. More detailsabout the derivation of these constraints can be found in [12].

Obstacle Avoidance—In our demonstration, we only consid-ered rectangular no-fly zones aligned with the North-East co-ordinate frame. As mentioned before, only the zones lyinginside the current detection regionD0 of the aircraft neededto be accounted for. Denoting these obstacles byj = 1 . . . Jand describing them by their lower left (i.e. southwest) cor-ner (xmin,j , ymin,j) and upper right (i.e. northeast) corner(xmax,j , ymax,j), the avoidance constraints(xi, yi) /∈ O0were formulated as follows [10]:

∀i ∈ [1 . . . T ], ∀j ∈ [1 . . . J ] :

xi ≤ xmin,j + Mbi1−xi ≤ −xmax,j + Mbi2

yi ≤ ymin,j + Mbi3−yi ≤ −ymax,j + Mbi4

∑4k=1 bij ≤ 3

bij ∈ {0, 1}.

(17)

Applying principle (6) again, the last constraint ensures thatat least one of the coordinate inequalities is active, therebyguaranteeing that the trajectory point(xi, yi) lies outside therectanglesj = 1 . . . J . Because the resulting trajectory con-sists of discrete waypoints, to prevent the UAV from cuttingcorners we enlarged the actual obstacles with a safety bound-ary ofdsafe =

vmax∆t√2

≈ 3200 ft.

Loiter Constraints—As outlined in Section 3, safety can beguaranteed by ensuring that either a left or right loiter circlelying inside the detection region does not intersect with anyof the obstaclesj = 1 . . . J . As detailed in Ref. [11], we canexpress sample points along both circles as affine functionsof the last statexT in the planning horizon, and then intro-duce avoidance constraints similar to (17). As such,xT isconstrained to be ingress state to asafeloiter pattern.

Using an indexk to indicate the sample points along the cir-cles and introducing a binary variabled to select either the

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right or left circle, the safe loiter constraints at each recedinghorizon iteration can then explicitly be expressed as:

∀k ∈ [1 . . . N ], ∀j ∈ [1 . . . J ] :

xT − αc (cos kθs − 1) vyT − αc (sin kθs) vxT≤ xmin,j + Mbkj1 + Md

−xT + αc (cos kθs − 1) vyT + αc (sin kθs) vxT≤ −xmax,j + Mbkj2 + Md

yT − αc (sin kθs) vyT + αc (cos kθs − 1) vxT≤ ymin,j + Mbkj3 + Md

−yT + αc (sin kθs) vyT − αc (cos kθs − 1) vxT≤ −ymax,j + Mbkj4 + Md

AND

xT + αc (cos kθs − 1) vyT + αc (sin kθs) vxT≤ xmin,j + Mbkj1 + M(1 − d)

−xT − αc (cos kθs − 1) vyT − αc (sin kθs) vxT≤ −xmax,j + Mbkj2 + M(1 − d)

yT + αc (sin kθs) vyT − αc (cos kθs − 1) vxT≤ ymin,j + Mbkj3 + M(1 − d)

−yT − αc (sin kθs) vyT + αc (cos kθs − 1) vxT≤ −ymax,j + Mbkj4 + M(1 − d)

{

∑4l=1 bkjl ≤ 3bkjl, d ∈ {0, 1}

Hereθs = 2πN is the discretization angle around the circle.Again, because of the sampling procedure, the obstacle coor-dinates(xmin,j , ymin,j , xmax,j , ymax,j) are those obtained af-ter enlarging the obstacles on all sides by a thicknessdloiter.Given a maximum turn radius of1.9 mi, we usedN = 8 andsetdloiter = 0.6 mi.

5. SOFTWARE IMPLEMENTATION

Our guidance software module is implemented in C++ andILOG’s Concert Technologies. To interface with the UAVavionics and guarantee hard real-time execution, it is inte-grated with Boeing’s OCP software. The software runs ona Pentium 4 Linux laptop with 2.4 Ghz clock speed that ismounted in the aircraft, and interacts with the UAV avion-ics through a set of pre-defined command variables. Throughthe OCP interface the laptop receives GPS, ground speed andturn rate data, among other, at a rate of 20 Hz. The guidancemodule itself, however, only runs at 1 Hz. It consists of threesubmodules: a pre-processing step, an optimization step, anda post-processing step, which we will now discuss in moredetail.

Pre-Processing

The pre-processing routine is called every second and deter-mines all parameters of the MILP problem. It subsequently:

• Selects the correct LTI model,• Estimates the initial state for the current planning horizon,

• Determines the relevant obstacles,• Enlarges the obstacles with the appropriate safety band,• Determines the intermediate waypoint, and• Selects the appropriate cost function.

In addition, for numerical stability purposes and to speed upthe MILP optimization, all latitude/longitude position dataand obstacle coordinates are transformed to an East-Northaxis frame in kilometer units with the current position of theaircraft as the origin. Accordingly, the ground velocity oftheaircraft is scaled to km/s.

Optimization

The optimization step is implemented using the mathematicalprogramming package CPLEX from ILOG. CPLEX containsstate-of-the art routines for solving large MILPs and comeswith Concert Technologies, a C++ based modeling language.Using the latter, a MILP problem can be encoded in a com-pact form that is similar to the mathematical representation ofit.

An important feature of CPLEX is its optional limit on com-putation time, which is critical for a hard real-time system.We set this limit to 0.85 s and the optimality gap to10−4.After the allocated time has passed, CPLEX either returns afeasible solution within the optimality gap, a feasible solu-tion outside the optimality gap, or no solution at all. The lastsituation occurs when either the MILP itself is infeasible,orwhen no feasible solution can be found in time (e.g. becausethe problem is too complex).

Ideally, by definition of the safety constraints, the trajectoryplanning problem remains feasible at all times. However, be-cause of disturbances such as wind gusts, the initial velocitymight fall outside the constraint bounds, or the vehicle mightbe blown off course to a position from where an obstacle-freeMILP solution no longer exists. The first situation is easy tospot and can be resolved ahead of time by resetting the veloc-ity bounds in the pre-processing step. Infeasibilities causedby obstacles, however, are harder to predict and resolve. Inthat case, the UAV should resort to its backup plan, consistingof the previous plan minus the first time step and the previousloiter circle.

If the control authority used in the MILP problem (i.e. the ad-missible acceleration and turn rate limits) is somewhat con-servative w.r.t. the actual performance of the vehicle, robusttrajectories can be designed. Then, in case the UAV getsblown off course to a state from which no feasible solutionto the MILP exists, the aircraft can use its additional controlauthority to get back to feasibility within a few time steps.In our code we therefore use maximum acceleration and turnrate bounds that are smaller than the actual ones available tothe waypoint controller. As such, we never experienced in-feasibilities of more than 2 or 3 time steps.

Although the pre-processing step is repeated every second,

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the optimization function is nominally only executed every10 s. The reason for this is the∼10 s time constant of theT-33, which makes a higher planning rate unnecessary. Onlywhen a large disturbance or an additional obstacle is detected,or when the vehicle is in backup plan mode (i.e. when thelast MILP problem was infeasible), the optimization routineis executed at the next second. This way, the available timeslots can be occupied by computations required by the taskscheduling and natural language interface routines.

Post-Processing

The post-processing routine performs the feasibility check byinterpreting a CPLEX flag, and updates the current trajec-tory (i.e. the current waypoint list), the loiter directionand abackup plan counter accordingly. In the nominal case wherea feasible solution is found, the variables of interest are the6 new states of the planning horizon (i.e. the waypoint co-ordinates with corresponding velocity vectors) and the newloiter direction. The coordinates are first transformed back tothe original Greenwich-referenced longitude and latitudeaxisframe and the velocity is rescaled to fps. Next, the old plan isflushed, replaced by the new one, and the counter is set to 1,pointing to the first entry in the waypoint/state list. That en-try is then given to a waypoint controller that issues forwardvelocity and turn rate commands to the vehicle.

If no feasible solution is found, however, the remainder of theprevious trajectory is used as a backup plan. In this case, thebackup plan counter is increased by one, thus pointing to thenext waypoint of the existing plan. If the counter exceeds 6,depending on the value of the loiter direction binary, the leftor right loiter circle is initiated by issuing a “Set and HoldTurn Rate” command of 3 deg/s to the UAV. Notice that thisis slightly more aggressive than the maximum 2.6 deg/s turnrate accounted for in the plan. It results in a smaller loitercir-cle, which introduces some robustness to perturbations alongthe trajectory. As mentioned before, as long as the vehicleremains in the backup plan mode, the MILP optimization isexecuted every second (but the counter only updated every10s), until a new feasible plan is found.

6. RESULTS

Demonstration Scenario

Using the narrative outlined in Section 2, we designed a va-riety of sample scenarios that are depicted in Figure 4. Theflight area is approximately 40 mi across (east to west alongthe northern boundary) and 30 mi wide (north to south alongthe western boundary). There are two pre-determined no-fly zones (listed as “NFZ1” and “NFZ2”) and three potentialpop-up threats (denoted by “P Obs 1,” “P Obs 2,” and “P Obs3”), which can be activated during the demonstration. In ad-dition, there are two mission task areas (labeled “Search AreaAlpha” and “Search Area Bravo”). Each task area has threepotential ingress conditions which can be selected by the FW

Figure 4. Sample Scenario Map for the MIT SEC CapstoneDemonstration

WSO before the vehicle reaches the task area location. Eachtask area also includes a threat/target (denoted by “TrgFndA” and “TrgFnd B”) which the UAV searches for and locatesduring the mission. Finally, the UAV starts the mission fromthe UAV Base Loiter location in the southwest corner of theflight area, and the FW vehicle maintains a loiter pattern nearthe northern border until the target has been detected.

Simulation Results

To aid in the development of our guidance system, we built areal-time Simulation-In-the-Loop (SIL) test platform at MIT,which is shown in Figure 5. Besides the OCP, it includedBoeing’s DemoSim vehicle simulations for the UAV and FWaircraft, which ran on separate computers with a Link 16 com-munication interface.

Figure 5. MIT SEC Demonstration Simulation-In-the-Loop(SIL) laboratory setup

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Figure 6. SIL Test 1 - Initialization of the SECdemonstration: the UAV (in yellow) enters a loiter pattern.

Figure 6 shows one of the many initialization tests. As theFW aircraft and UAV approach the flight area, the demon-stration software is initialized: the UAV automatically fliesto the UAV Base Loiter Location, where it remains until it iscommanded another task. The loiter task itself is defined as aseries of six waypoints with a fixed ingress location and head-ing. As can be seen from the picture, the UAV successfullyavoids NFZ1 while flying to the loiter area.

Next, Figure 7 shows one of the pop-up obstacle avoidancetests used to verify the safety guarantees of the MILP trajec-tory planning algorithm. In this test, two pop-up obstacleswere placed into the demonstration area as the UAV was enroute to Search Area Alpha. The resulting trajectory high-lights the MILP algorithm’s ability to develop safe, dynami-

Figure 7. SIL Test 2 - Pop-up obstacle test: the UAV safelyavoids both pop-up threats.

Figure 8. SIL Test 3 - Simulated flight with two consecutivesearch missions.

cally feasible paths for the vehicle after unexpected changesto the environment right outside its detection radius. For ex-ample, when the UAV is south of the first pop-up obstacle(designated “P Obs 3”), the second one (designated “P Obs1”) is inserted, causing the UAV to immediately turn left andproceed northeast over it. After passing the pop-up obstacles,the vehicle levels out and flies at a safe distance from No-FlyZone 2 (NFZ 2) before turning north to enter Task Area Alphato perform a search task.

Figure 8 depicts a test where the UAV was commanded tofly two consecutive missions from the UAV Base Loiter Lo-cation. It shows the UAV’s coverage over both search areasduring the search task portions of the mission. Notice thatthe two search patterns are almost identical: the same taskdefining waypoint sequence (relative to the ingress positionof the task area) was used in both search tasks, showing thatthe MILP guidance approach can accurately track waypointplans. In addition, the UAV safely avoids two pop-up obsta-cles en route to each search area location.

Flight Test Setup

During the SEC Capstone Demonstration in late June 2004,multiple F-15/T-33 sorties were flown at NASA Dryden. Fig-ure 9 shows a system level diagram of the flight experimentsetup. During the test, the main role of the T-33’s two personcrew was to fly the vehicle to the demonstration area, activateour software and manage the vehicle in the event of failures.In addition, for technical reasons, the T-33 pilot executedtheforward velocity commands produced by the MILP guidancealgorithm. The turn rate, however, was directly commandedby the laptop.

Although the FW WSO was only able to select UAV-tasksfrom a pre-defined list of options, the actual mission wasnot pre-planned. The T-33’s rear-seat operator (nicknamed

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Figure 9. Flight experiment system level diagram

the “Guy-In-Back” or “GIB”) observed the progress of thedemonstration and a Ground Station Operator (GSO) addedpop-up obstacles via experiment key commands, introducinganother degree of uncertainty into the mission. Namely, al-though the possible locations of the mission task areas andpop-up obstacles were pre-defined, they were selected ran-domly in real-time during each flight experiment. The ex-periment Test Coordinator (TC) monitored the demonstrationfrom the ground station and communicated status informationabout the local airspace to the pilots.

Flight Test with Simulated F-15

In the first test, shown in Figure 10, the T-33/UAV flew a mis-sion with a simulated F-15 while the Ground Station Operatorissued the natural language commands. The flight took placein the morning of Thursday, June 17th, 2004, with a wind of5 to 10 knots blowing from the southwest corner of the flightarea (at a heading of 220 to 240 degrees).

Flight Account—The UAV started west of the ingress point(labelled “INPT / EGPT” in Figure 10) with a heading of 090.After the GIB initialized the mission software, the UAV be-gan turning south to avoid NFZ 1. As it turned south, thevehicle appeared to fluctuate between a heading of 125 and130. After it passed the lower left hand corner of NFZ 1, theTest Coordinator notified the T-33/UAV pilot and GIB thatthey had to change the flight card to avoid the southwesterncorner of the flight area (because of a real-time flight area re-striction). As such, when the vehicle was approximately twomiles SSW of NFZ 1, the GSO commanded the UAV to pro-ceed to Location Bravo 1. The UAV responded and beganturning left toward Task Area Bravo. Within two minutes ofthis command, the GSO inserted Pop-up Obstacle 3 into thetest area. At this point, the vehicle had a heading between070 and 080 and was approximately four miles from the ob-stacle. It began to turn right to avoid the obstacle and flewalong its southern boundary, successfully avoiding it. Afterpassing the obstacle, the UAV proceeded to turn north towardthe Bravo 1 location and initiated the search pattern. As thevehicle was facing SSE, it was near the target location and the

Figure 10. SEC Flight Test: UAV/T-33 (in yellow) withsimulated F-15

GSO inserted the target into the environment. The UAV sentthe proper status messages to the operator and the GSO com-manded the vehicle to return to base. It began to turn rightand safely flew southwest around Pop-up Obstacle 3. Sincethe southwest airspace was off-limits, however, the UAV wasnot permitted to return to its Base Loiter location and the mis-sion was ended.

Flight Evaluation—During this test flight, the vehicle wascommanded to fly to the Bravo Task Area before reachingthe UAV Base Loiter location. This verified the flexibility ofour mission software and the ability of an operator to easilychange the mission goals and needs in real-time. In addition,the pop-up obstacle was inserted when it was already locatedwithin the vehicle’s detection radius. Regardless, the MILPtrajectory planning software calculated a solution for theve-hicle, allowing it to safely avoid the obstacle. The flight testmarked the first time a MILP-based onboard guidance systemwas used to control a UAV.

Flight Test with actual F-15

In a second test, shown in Figure 11, the T-33/UAV flew asuccessful mission with the F-15 WSO issuing natural lan-guage commands. This flight took place in the afternoon onWednesday, June 23rd, 2004, with a SW wind (220-240 deg)between 10 and 15 knots.

Flight Account—The UAV again started west of the ingresspoint with a heading of 090, and began turning south to avoidNFZ 1 after the GIB started the experiment software. As itturned south, the vehicle steadily moved to the UAV BaseLocation. After reaching the loiter location, it started toturnright to a heading of 270 at which point the F-15 WSO com-manded it to fly to Task Area Bravo. The UAV respondedand began turning northward, thereby avoiding Pop-up Ob-stacle 3 and notifying the F-15 WSO that it detected a threat.As it approached the task area, the UAV notified the F-15

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Figure 11. SEC Flight Test - UAV/T-33 (in yellow) andactual F-15 providing natural language commands.

WSO that it was two minutes from the ingress point it wasinitially given. At that time, the WSO commanded the UAVto change the entrance location. It responded and proceededto the new ingress point.

After reaching the task area, the UAV notified the F-15 WSOand began its search pattern. As it turned left toward thesouthern boundary of the search area, the GSO inserted Tar-get B into the environment. The UAV notified the F-15 WSOand sent an image message to the F-15. The F-15 WSOthen commanded the UAV to return to base. The UAV ac-knowledged the command and flew back to its loiter loca-tion, thereby avoiding obstacles and providing status notifi-cations to the F-15 WSO along the way. The demonstrationwas ended when the UAV successfully returned to the UAVloiter area.

Flight Evaluation—During this flight, the MILP trajectoryplanning software again successfully managed to account forno-fly zones and for changing mission objectives. The pop-upobstacle was inserted well beyond the detection radius, suchthat the UAV had adequate time to plan around it. The flighttest marked the first time that a MILP-based guidance systemwas used to control a UAV in coordination with a mannedvehicle. In addition, it also marked the first time that a naturallanguage interface was used by a manned vehicle to commanda UAV in real-time.

7. CONCLUSION

This paper discussed the implementation of a UAV guidancesystem based on Mixed Integer Linear Programming that wasdeveloped for the Capstone Demonstration of DARPA’s Soft-ware Enabled Control Program. The guidance software waspart of a mission system allowing a fixed wing operator togive high level tasks to a UAV in natural language. Given task

waypoints, MILP was used to compute safe, dynamically fea-sible trajectories in real-time through a partially-knownenvi-ronment. The detailed MILP formulation using safety loitercircles was presented along with practical decisions abouttheimplementation in C++/CPLEX and the integration with Boe-ing’s OCP platform.

Simulation and June 2004 flight test results were presented,which mark the first time an on-board MILP-based guidancesystem was used to control a UAV. These results highlight theperformance, flexibility and robustness of the MILP approachto online trajectory planning in cluttered environments andthe feasibility of its implementation on future UAV systems.Currently we are extending this work to platforms with multi-ple unmanned vehicles for which real-time decentralized tra-jectory planning strategies are developed that exhibit similarsafety guarantees.

ACKNOWLEDGEMENTS

The authors would like to thank Timothy Espey, BrianMendel, Dr. Jim Paunicka, and Jared Rosson from BoeingPhantom Works, and Yoshiaki Kuwata from MIT for theirhelp with this project. The research was funded by DARPASEC contract F33615-01-C-1850.

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[3] A. Bemporad and M. Morari. Control of systems inte-grating logic, dynamics, and constraints.Automatica,35:407–427, 1999. Pergamon/Elsevier Science, NewYork NY.

[4] E. Craparo and E. Feron. Natural language processing inthe control of unmanned aerial vehicles. InProc. AIAAGuidance, Navigation, and Control Conference, Provi-dence, RI, August 2004.

[5] C. A. Floudas.Nonlinear and Mixed-Integer Program-ming - Fundamentals and Applications. Oxford Univer-sity Press, 1995.

[6] B. Mettler, M. Valenti, T. Schouwenaars, Y. Kuwata,J. How, J. Paunicka, and E. Feron. Autonomous uavguidance build-up: Flight-test demonstration and eval-uation plan. InProc. AIAA Guidance, Navigation, andControl Conference, Austin, TX, August 2003.

[7] J. Paunicka, B. Mendel, and D. Corman. The ocp - anopen middleware solution for embedded systems. InProc. 2001 American Control Conference, Arlington,VA, June 2001.

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[8] A. Richards and J. How. Aircraft trajectory planningwith collision avoidance using mixed integer linear pro-gramming. InProc. 2002 American Control Confer-ence, Anchorage, AK, May 2002.

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BIOGRAPHIES

Tom Schouwenaars is a Research As-sistant in the Laboratory for Informa-tion and Decision Systems at the Mas-sachusetts Institute of Technology. Heobtained a M.Sc. degree in electrical en-gineering from the Catholic Universityof Leuven, Belgium, in 2001, and is cur-rently a Ph.D. candidate in Aero- and

Astronautics at MIT. He is a Francqui Fellow of the Bel-gian American Educational Foundation and was awarded the2005 AIAA Unmanned Aerial Vehicles Graduate Award.

Mario Valenti is a Research Assistant inthe Laboratory for Information and De-cision Systems at MIT. He has B.Sc. de-grees in Mechanical and Electrical En-gineering from Temple University. Fortwo years, he worked as a Flight Con-trols Engineer in the Integrated DefenseSystems Division of the Boeing Com-

pany, where he is currently on a leave of absence. In June2003, he received a M.Sc. in Electrical Engineering and iscurrently pursuing his Ph.D. in Electrical Engineering atMIT.

Eric Feron is an Associate Professorof Aeronautics and Astronautics at MIT,where he is affiliated with the Labora-tory for Information and Decision Sys-tems. He is a graduate from EcolePolytechnique in Paris, and received hisPh.D. in Aero- and Astronautics fromStanford University in 1994.

Jonathan How is an Associate Pro-fessor of Aeronautics and Astronauticsat MIT, where he is affiliated with theAerospace Controls Laboratory. Hegraduated from the University of Torontowith a B.Sc. in Aerospace Engineering,and received the M.Sc. and Ph.D. de-grees in Aero- and Astronautics from

MIT in 1989 and 1993 respectively.

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