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A second generation micro-vehicle testbed for cooperative control andsensing strategiesKevin K. Leung1, Chung H. Hsieh2, Yuan R. Huang2, Abhijeet Joshi2, Vlad Voroninski3, andAndrea L. Bertozzi3AbstractThis paper describes the second generation ofan economical cooperative control testbed described in C.H. Hsieh et al Proc. Am. Cont. Conf., 2006. The originalcar-based vehicle is improved with on-board range sensing,limited on board computing, and wireless communication, whilemaintaining economic feasibility and scale. A second, tank-based platform, uses a flexible caterpillar-belt drive and thesame modular sensing and communication components. Wedemonstrate practical use of the testbed for algorithm validationby implementing a recently proposed cooperative steering lawinvolving obstacle avoidance.I. INTRODUCTIONComputer simulation, as a tool to validate cooperativecontrol algorithms, is only as good as the accuracy of themodel as a reflection of real-world parameters. A real vehicletestbed is an important step forward to verify algorithmeffectiveness. As the interest in autonomous multi-vehiclemotion continues to rise, the testbed remains as an invaluablelearning tool to observe theory in real world action. A1:1 scale multi-vehicle testbed is impractical for both costand space constraints. Even with a scaled-down approach,many testbeds involve 10-50cm size vehicles that can notmaneuver well in an indoor environment. To this end, aminiaturized, economical micro-car testbed was developedin 1, using 1/64 size vehicles in an integrated system withoverhead camera positioning and off-board motion planning.The platform demonstrated the possibility of a multi-functioncooperative testbed arena in a cost-effective design (less than$4,000 for all material and computing costs). This paperdescribes the second generation of this facility, which hasmany new features while preserving the overall cost andscale of the original design. The first generation vehicle isbased on a microsizer car chassis, which has three discretesteering states, a single speed, no on-board processing, anda slow one-way wireless communication rate (13Hz). Thesecond generation vehicles communicate two-way at 30Hzand possess on-board processing and on-board range sensing.Two different chassis designs are implemented, one basedon a car platform and a second based on a tank witha caterpillar-style drive, allowing for a negligible turningradius. Hardware is divided into multiple sub-modules which1Dept. of Electrical and Computer Engineering, University of California,Davis 95616kevinlg2Dept. of Electrical Engineering, University of California Los An-geles,LosAngeles,CA90095 chung hsieh,yuanh,abhijeet3DepartmentofMathematics,UniversityofCal-iforniaLosAngeles,LosAngeles,CA90095vladvoro,bertozziease future expansion and upgrade. Updated camera posi-tioning software allows for better overhead tracking. Thepath planning software is now implemented both on andoff-board depending on the application. In this paper wedemonstrate an application utilizing the dynamic coordinatedcontrol laws in 14 and a cumulative sum algorithm forbarrier detection inspired from 19. All the motion planningis done on-board and the off-board computer is only usedfor communication of overhead camera information andsensor data from the vehicles. Our new ground vehicles havecomparable functionality to those in 2-11 , while keepingmaterial cost at the order of $160 per vehicle, on a sub-palm-sized chassis. A pocket size transmitter connects to a laptopwith a serial cable, thereby making the entire platform veryportable. In some scenarios we are also interested in real timeobstacle detection in which the on-board IR sensors play arole. The position tracking system update rate and accuracyare improved.This paper is organized as follows. In Section II we presentthe overall system structure, tracking system, the vehiclehardware, local motion control, and the physics model. Sec-tion III presents mathematical models for the motion of thevehicles. Section IV describes a suite of steering control lawsfor different tasks. Section V presents the implementation ofthe control laws for the tasks of circle following, splitting andmerging of a group, and point to point motion of a groupwith dynamic barrier avoidance.II. MULTIPLE MICRO-VEHICLE TESTBEDA. Vehicle PlatformFigure 1 shows a system diagram of the platform.Fig. 1.Testbed system diagramProceedings of the 2007 American Control ConferenceMarriott Marquis Hotel at Times SquareNew York City, USA, July 11-13, 2007WeC14.61-4244-0989-6/07/$25.00 2007 IEEE.1900Fig. 2.(left) Car based vehicle. (right) Tank based vehicle.Fig. 3.Diagram of vehicle on-board system. The tank system diagram isthe same as above, except that it has another motor instead of the servo.B. Vehicle InformationWe design two vehicles: a car and a tank. The car isbuilt from a ZipZaps micro RC Special Edition car witha 21,500 RPM motor and 12:1 gearing. It has rear wheeldrive in either a forward or backward direction. It uses apotentiometer steering gauge that provides feedback to thesteering controller. The tank is built from an Ecoman R/Cmini tank with 96:1 gearing and the ability to climb a 38slope. It has two motors, one controlling the left belt andthe other the right. Their electronic system infrastructure isidentical (see figure 3). For ease of installation and flexibility,the vehicles have four major hardware modules, a processingH-Bridge board (Megabitty), an upper circuit board, a lowerdeck, and the vehicle chassis. They are interconnected byscrews and/or socket pins (see figure 2). This configurationallows future expansions and upgrades. One will only needto modify the lower deck, if a different chassis or type ofvehicle is needed. Furthermore, the processing power couldalso be increased by replacing the Megabitty. Table I showsthe physical dimension of the new vehicles compared withthe first generation cars in 1. Note that vehicles equippedwith long range IR sensors are 1 cm longer and 2g heavier.TABLE IPHYSICALDIMENSIONS OF THEVEHICLE.Dimensions (L x W x H)Weight ( with batteries)Old Car(5.8 3 3.2)cm50gNew Car(7 3.4 4.3)cm68gTank(7 3.8 4.6)cm65gFig. 4.(left) Wireless communication module. (right) Sample tag forpositioning.C. Vehicle HardwareThe Processing Board. We use the pre-assembled pro-cessor board from J, Megabitty. It has an 8 - bitRISC AVR 16MHz microprocessor and two 500mA ratedH-bridges. The module is slightly modified to accommodatethe overall hardware structure. This board sits on the upperdeck and connects to the wireless module through a 3.3V to5V level shifter.The Upper Deck holds the main component of the robot,the communication module, power structure, the infraredsensor, and connectors. The main power reservoir is a singlecell 3.7V 740mAh Lithium polymer battery. We utilize thestep up regulator to provide an 8V power rail such that alow drop out 5V and 3.3V regulator can maintain stabilityfor the communication unit and the processors module.The Lower Deck. Screws are bolted tight through the boardto the chassis. It holds the mechanics mount between thechassis and the upper deck. It also provides the space toinsert the Li - poly battery and as well the on / off switch.The IR Sensors are proximity model numbers GP2Y0A02YK and GP 2Y0A21 YK from Sharp. They havea range of 20 to 150 cm and 10 to 80 cm, respectively. Theoutput voltage is higher if an object is closer in their effectiverange. The sensors were manually calibrated and the detailsare discussed in 12.Wireless Communication Module. To aid the cost of de-velopment we choose the pre-assembled Radiotronix Wi.232DTS transceiver module 13 as the wireless bridge betweenthe micro controller and the station. The module costs$27, is 2cm 2.5cm, and operates at 16mA. As a low-power UART-to-antenna serial interface, this module can beeasily integrated with both the Megabitty and the trackingcomputer. Operating on the 902-928MHz ISM band, themodule has the flexibility to transmit and receive on separatechannels, thus allow us to achieve a full duplex system. Thewireless communication module is shown in figure 4. Withthe current setup, we achieve a maximum data rate of 57.6kbit/sec, which is sufficient for the 30Hz positioning update.D. Software ArchitectureThe software architecture includes a low level controllayer and a user application layer. The control layer consistsof four parts: a task scheduler, a basic motion (steeringand speed) controller, sensor measurement acquisition, andWeC14.61901Fig. 5.Steering response, from far left (50) to center (25), as measuredby the munication. Application software can access the controllayer to alter the vehicle motion, communicate with trackingsystem, and retrieve range sensor data.Scheduler. A simple task scheduler regulates the updaterate of the steering, motor drive control, and sensor readings.On start up, each the task registers with the schedulerwith its priority, update rate, and a callback function. Sincethe primary use of the scheduler is to update the variouslocal control systems, none of the tasks are allowed to runfor longer than the scheduler resolution of one ms. Thescheduled task cannot perform any blocking calls; if the taskis waiting for additional resources, it is required to rescheduleitself to run again later. Scheduled task conflicts are resolvedvia priorities, if two tasks have the same priority, the taskswill be executed in the order of initial registration. One usertask can be scheduled, but it has the lowest priority and canbe preempted by any of the controller tasks to ensure correctoperation of the vehicles.Basic Motion Software On The Car. Both the steering andspeed motor are controlled by two pulse width modulation(PWM) channels via two H-Bridges. The speed motor issimply controlled by altering the pulse width, while thesteering wheel control requires task scheduling for closed-loop feedback control. A potentiometer feeds the analogvoltage to the analog-to-digital converter (ADC) on themicro-processor. Task scheduling allows the ADC call torelease processing time to other jobs, while waiting for theconversion to be completed. The returned ADC value feedsinto the classical proportional derivative controller operatingat 250Hz. The steering angle has a total of 51 degrees, 50for far left, 25for center and 0for right, with an accuracyof one degree. The settling time for a 25offset is 0.18s.Figure 5 illustrates the performance of the steering controller.Basic Motion Software On The Tank. The tank drives twobelts independently, resulting in turns of arbitrary radius,while moving forward and backward. In practice we findit simpler to construct paths composed of either straight linemotion or turning in place. A state machine is responsible forthe execution sequence of the two maneuvers. Both the speedand heading direction are the input parameters. We assumethe heading direction has higher priority than speed. Thecomplete motion state sequence is described in table II. BothTABLE IITANKMOTIONCurrent heading directionNew heading direction RequestedSpeed = 0No motion1) Turn to desired direction2) StandSpeed 0Move Forward1) First Turn to new direction2) Move forwardthe straight line and turning maneuvers rely on the trackingsystem heading angle feedback. We use a simple proportionalderivative feedback control scheme to stabilize the tank at thedesire heading direction. Note that the left and right driver arenot identical, some electrical and mechanical discrepanciesexist. A proportional closed-loop controller alters the left andright drive strength to maintain a straight line motion.Infrared Sensor Measurement Acquisition.The infraredsensor uses another ADC channel to provide an instantaneousdigitized measurement. The execution rate is specified in thetask scheduling, typically at 25Hz. Various filters can becoded to fit application needs. For the application describedin this paper, we implement a cumulative sum algorithm forobstacle detection (see below). Another application, that ofdynamic visibility 12, uses an ENO scheme method forprocessing spatial point cloud data obtained by the rangesensors.Communications Software. Vehicles tracking data arestacked together and sent through the serial port as onepackage. The receipt of the tracking data is interrupt-driven.Upon receiving the whole package, the vehicle can extractits own and other vehicles tracking information. We utilizethe Windows API to interface with the serial port on thetracking computer.E. Tracking SystemThe tracking of the vehicles is accomplished via ananalysis of images from two overhead cameras. The physicalsetup is the same as in 1. The tags used to identify the carsare enlarged by 15% and the pattern of the rectangular bitson the tags has been changed from linear to checkered toavoid misidentification. Figure 4 (right) shows a sample tagpattern for the new vehicles.Additionally, the performance of the tracking algorithmhas been enhanced by identifying contours in a thresholdedimage as those formed by the car tags. Specifically, insteadof building a minimum area rectangle around each contour(both sides of the rectangle being parallel to the imageplane), a bounding rectangle is built such that it encloses acontour without the restriction of being parallel to the image.Thereby, restrictions can be made on the length and widthof the rectangle, instead of its area, the former being a morediscriminative selection process.The revised video tracking algorithm achieves a maximumheading error of 3, where the old algorithm has a maximum9error, and also maintain an average position error of1 pixel. Additionally, without the presence of noise orocclusion (i.e. for clean image input files), the probabilityof misidentifying a vehicle has been reduced.WeC14.61902Fig. 6.Open-loop test of vehicle motion laws (1-2). The stars representthe motion of the car on the testbed, while the dots represent a computersimulation of the motion.III. VEHICLEMOTIONMODELSA. The Simple Car ModelThe following system of equations, adapted from 1,model the motion of the cars. x = v cos(), y = v sin(),(1)M v = (q1/q1max)F v, =vLcartan?q2180?, (2)where x and y represent the position coordinates of thevehicle in the laboratory frame, v is the vehicle speed(positive if going forward and negative if going backwards), indicates the angle of the vehicle heading, M is the vehiclemass, F is the maximum driving force of the vehicle, and is the friction coefficient with the floor. The parameterLcaris the length of the car. The input control parameterq1 255,255 corresponds to the strength of the throttle,-255 being full backward, 255 being full forward, andq1max= max | q1|. The parameter q2 0,50 indicatesthe 51 possible steering angles of the wheels. Figure 6compares a computer simulation of vehicle motion to thephysical realization of the commands on the testbed. HereF = 631.8M cm/sec2where M is the mass of the vehicle.Lcaris the length from the front wheels to the back wheelsand = 3.0 M/sec.B. Differential Drive ModelWe adapted the model in 18 to formulate a first ordersystem for the tank, x = r/2(l+ r)cos(),(3) y = r/2(l+ r)sin(),(4) = r/Ltank(l r),(5)where land rare the angular velocities (rad/sec) of theleft and right tank belts, r is the radius of the circle thathas the same circumference as a tank belt, and Ltankisthe width of the tank minus the width of one of the belts.In practice, we restrict the tanks motion such thatddt 0only whendxdt=dydt= 0. Given this restriction, |l| =3.3357c/255 0.9656, where c 0,160 is the controlparameter to the left belt. When rotating counterclockwise,l= |l| = r. When rotating clockwise r= |r| =l. When going forward (backward), l= |l| = r.IV. ACOOPERATIVE STEERING CONTROL LAWA. Basic TheoryWe consider a recent Frenet-Serret frame based algorithmfor cooperative steering control in the presence of obstaclesdue to Morgan and Schwartz 14, inspired by the originalwork of Justh and Krishnaprasad 15. The motion of avehicle is described by a differentiable curve z(s) IR2parameterized by arc length. Let x denote the unit vectorfrom the position of the vehicle, in the direction of thetangent vector dz/ds and y = xis positively oriented withrespect to x. The motion of each vehicle is modeled by zk= xk, xk= ukyk, yk= ukxk(6)where k is the vehicle index. The vehicles move at unit speedand the curvature of the kthvehicle path is the scalar uk.The control law is specified by dynamically changing uktocreate pairwise interactions between vehicles. Define uk=Pj6=kujkwhereujk= ?rjk|rjk| xk?rjk|rjk| yk?f (|rjk|)?rjk|rjk| yk?+ xj yk(7)where rjk zk zj, f (|rjk|) = 1 ?r0|rjk|?2, and = (|r|), = (|r|), = (|r|) are specified functions.The term ?rjk|rjk| xk?rjk|rjk| yk?aligns the vehiclesperpendicular to their common baseline. The potential func-tion f (|rjk|) regulates the spacing between the vehicles andthe term xjyksteers the vehicles to a common orientation.This control law requires position information of other agentswithin a neighborhood as described below.B. Local Coupling and Leader Following Control LawLocal coupling is manifested by limiting the visible dis-tance range of each vehicle to a neighborhood around thatvehicle. To ensure swarming, any two vehicles have to bewithin a specified distance of each other. The local couplingcontrol law isuGLk=Xj6=kc(|rjk|,0,w)ujk(8)where ujkis (7) andc(|rjk|,q,w) =(1if|rjk| w,qotherwise.(9)Such local coupling is advantageous for large numbers ofagents due to the scalability of the communication step.A designated leader vehicle steers a swarm in a particulardirection. The rest of the vehicles (follower vehicles) followaccordingly, by using the local coupling control law, withstronger coupling between follower and leader vehicles.ufollowerk= c?|rl(k)k|,0,w?lcul(k)k(10)+Xj6=k,l(k)c(|rjk|,0,w)ujk,(11)WeC14.61903where lcis a leader coupling constant and l(k) is the index ofthe leader vehicle closest to the kthvehicle. The control lawfor the leader vehicles depends on the particular application.C. Homotopy Control LawIn order to transition smoothly from a global control lawto a local coupling leader following control law, a homotopyparameter is introduced. The homotopy control law uk(),0 1 satisfiesuk( = 0) = uGk,uk( = 1) = uLk(12)where uGkis the global control law and uLkis the local controllaw. If there are m leader vehicles, the homotopy control lawfor the n m follower agents isufollowerk() =c?|rl(k)k|,1 ,w?(lc 1) + 1ul(k)k(13)+Xj6=k,l(k)c(|rjk|,1 ,w)ujk,where ujkis given by eqn. (7) and lc 1 is a couplingconstant which strongly attracts the followers strongly totheir leaders. The leader agent homotopy control law isuleaderk() =Xj6=kujk(1 ) +skn 1.(14)When more than one leader is present, local coupling can beexploited to separate a swarm into two sub-swarms, as leadervehicles drive in different directions and follower vehiclesfollow their respective closest leaders. Such an example isdemonstrated on the testbed in the next section.D. Target SeekingTo move towards a specified target, the kth vehicle usesuk=Xj6=kc(|rjk|,0,w) 1 ?r0|rjk|?2!(rjk yk)?+ 1 ?r0|rk|?2!(rk yk),(15)where rkis the vector from the location of the kth agentto the target, and is a weighting constant. Only the firstterm involves interaction between the vehicles in order toavoid collisions. The second term directs each vehicle tomove toward the target. This control law does not guaranteeswarming, but if the agents start out in a swarm-like orien-tation, it is likely that they will stay together.E. Barrier AvoidanceConsider a fixed convex object in the plane, the exterior ofwhich is defined by a set of m points bi IR2. The averagebarrier direction vector is computed asvk=Pmi=1(biyk)c(|biyk|,0,w)|Pmi=1(biyk)c(|biyk|,0,w)|,if|Pmi=1(bi yk)c(|bi yk|,0,w)| 6= 0,0,otherwise,(16)where c() is a step cutoff function that is zero outside of aspecified radius. The control law for barrier avoidance thenconsists of adding the term?sign?vk yk?(vk yk) to con-trol law in (10). This term orients the vehicle perpendicularto vkand the sign steers the vehicle away from the averagebarrier direction.V. IMPLEMENTATIONIn this section we consider several path planning strategiesthat build on the above control laws. To date there havenot been many published experimental studies using thisclass of control laws. We mention a related paper 16that implements a curvature-based steering control law forfollowing boundaries of walls and is demonstrated with asingle agent. A companion paper 17 to ours develops acontrol law for motion camouflage using the frameworkof 15 with implementation on our new testbed. In theexamples below we consider both single agent and multi-agent tasks, including ones that rely on the range sensors toidentify obstacles for avoidance.A. Steering AngleTo map the curvature control ukto a cars desired steeringangle k, we use the formula = Lcar/tank18, where is the car turning radius. Thus, the steering angle of a carcan be calculated as:k= tan1(Lcar/) = tan1(Lcaruk).(17)Lcaris measured to be 4 cm. The steering law assumes thecars to have unit speed thus we scale Lcaraccording to theactual vehicular speed.B. Implementation of a Basic Circle TrackerThis example has the car follow the circumference of acircle of given radius about a particular point on the testbed. We use a two-car model in which one of the cars isfixed at the center of the circle. We set = 0, = = 1,r0equal to the circle radius r and |r| equal to the distancefrom the car to the center of the circle. The basic controllaw reduces tou = (|r|)(r|r|x)(r|r|y) f(|r|)(r|r| y)where r = r2r1, the distance between the cars. Figure 7shows an implementation of this basic circle followingcontrol law, on the testbed platform. A single car followsthree different circular paths in sequence with a change in rupon reaching a particular heading.C. Implementation of Homotopy Control LawDue to the testbed physical size constraints, the homotopycontrol law discussed above is modified to a sequence ofthree stages. In stage one, two leaders are designated, withsteering program uleaderk( = 0) given by eqn. (14) whichis reduced to global control law eqn. (7). The regular globalcontrol law, which in this case includes only the leaders.Four followers are given control laws ufollowerk( = 0) from(13). In stage two, the leaders are switched to uleaderk( = 1)WeC14.61904Fig. 7. A single car path (solid line) tracing, in sequence, concentric circles(dashed lines) of radii 65.5cm, 41.7cm, and 28.3cm.where skis an explicit non-interactive control law causingthe leaders to drive away from each other. The control law ofthe followers remains the same, causing a propagation of twoswarms led by separate leaders. In stage 3, the global controllaw eqn. (7) between the leaders is reinstated, causing thetwo swarms to merge back together. Note that the control law(7) does not define the direction of the swarm, the headingdirection is arbitrary and the outcome of the re-mergingof the group can result in a different overall heading aftermerging, as seen in the two experiments shown in Figures 8.D. Implementation of Target seeking, Dynamic Barrier De-tection and AvoidanceWe combine the target seeking IV-D, barrier avoidanceIV-E, and cumulative sum algorithm 19 for obstacledetection to generate a path dynamically. A box (W23 L6 H13) cm is placed along the path of a swarm of fourcars moving toward a common target. All cars have priorknowledge of the box dimensions and orientation, but notits location. The boxs widest surface is perpendicular tothe cars initial heading direction. We designate two frontcars as observers. They use the on board long range infraredsensor to estimate the box location. Once the obstacle islocated, data is sent to the computing station, which generatesa virtual barrier. This information is distributed to all fourcars. The virtual barrier is constructed as follows: let pibe the point on the obstacle detected by the ithobserver.A rectangle of length (2Lobstacle | p1 p2|) and width2Wobstacleis constructed with the center of the barrier side,closest to the swarm, located at ( p1+ p2)/2. Since we onlyhave two measures of distance to the obstacle, we extend thebarrier to ensure collision avoidance. To avoid crossing pathsand collision between cars after passing the obstacle, wereduce the barrier term weight when the car passes a certaindistance from the barrier. Figure 9 shows the trajectories andsnapshots of the implementation.Cumulative Sum Algorithm For Obstacle Detection. As theobserver approaches the obstacle, sensor readings increasefrom background noise to a level indicating the presenceof the object. Figure 10 shows an example of raw sensorreadings. To filter the signal, we use a particular version19 of a standard cumulative sum algorithm 20. Let Xn070140 210070140Y (cm)X (cm)070140 210070140Y (cm)X (cm)070140 210070140Y (cm)X (cm)070140 210070140Y (cm)X (cm)070140 210070140Y (cm)X (cm)070140 210070140Y (cm)X (cm)Fig. 8.The top four figures show testbed data of a time sequence of sixcars performing the maneuver described in section V-C. The trajectoriesare shown in the middle figure. The circular dots show the position of theleaders; the triangular dots show the position of the followers. The bottomfigure shows trajectories for second experiment in which one follower goeswith the top leader and three go with the bottom leader. In this secondrun, the overall heading of the group is different after the merger.In theimplementation, the tracking information is interpreted in pixel domain andthe corresponding parameters are = = = 1,and w = 600 for everyagents, r0= 50 for leaders and r0= 40,lc= 20 for followers.denote the raw sensor signal at time level n and denote themean of the background noise when no obstacle is present.Define Zn= Xnc where c is a fraction of the expectedchange in sensor reading due to the obstacle. Next, defineWn= max(0,Zn+Wn1). The calculated value Wnshouldremain around zero until the change of state occurs, at whichpoint it ramps up. An example is shown in Figure 10. OnceWnpasses a designated threshold (large enough to avoidfalse alarms with a high probability) the object is detected.Using the car chassis at 1/5 of the full throttle, we test thecumulative sum algorithm for different values of c rangingfrom 150 to 400. The results are well-reproduced in multipletrials. These values lie closely on a linear fit, therefore we useWeC14.61905Fig. 9.Target seeking with barrier avoidance. Top four panels showsnapshots, at different times, of a single demonstration of the maneuver.The time progresses from top left to bottom right. The bottom figuresshows trajectories of the cars compared to both the actual barrier (dark)and the larger virtual barrier (light) as computed from the range sensors ofthe observers. In the implementation, the tracking information is interpretedin pixel domain and the corresponding parameters are = 1, = 25,r0= 60, and w = 100 for the target seeking term. The weighting constantfor the barrier avoidance term is 85 and is reduced to 1 after passing thebarrier with w = 180.the c = 200 state in practice for the most advanced warning.VI. CONCLUSIONThe second generation testbed is a marked improvementover the the first version, with the addition of onboard pro-cessing and sensing, improved communication rate, vehiclesdiversity, and scalability. All upgrades are implemented whilemaintaining the low cost and micro-scale feature of theoriginal testbed. The availability of two different vehicleplatforms, enables a wider range of study in cooperativecontrol. In this paper, we validate the effectiveness of severaldynamic multi-agent cooperative control laws in which somemodification of the algorithms are required to work with thehardware architecture.ACKNOWLEDGMENTSWe thank Alexander Tartakovsky for suggesting the thecumulative sum algorithm and Dave Morgan for usefulcomments about the path planning algorithm. This researchis supported by ARO MURI grant 50363-MA-MUR, ONRFig. 10.Cumulative sum algorithm applied to sensor data from a singlecar approaching an obstacle. (top left) raw data and cumulative sum, (topright) cumulative sums for different choices of c, (bottom) sample car pathavoiding obstacle with cumulative sum sensor output.grant N000140610059, and ARO grant W911NF-05-1-0112.In addition, AJ and VV are supported by NSF researchtraining group grant DMS-0601395.REFERENCES1 C. H. Hsieh, Y. Chuang, Y. Huang, K. K. Leung, A. L. Bertozzi,and E. Frazzoli, ”An Economical Micro-Car Testbed for Validation ofCooperative Control Strategies”, Proc. of the 2006 American ControlConference, Minneap
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