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JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 31, No. 2, March–April 2008Flight-Test Results of Autonomous Airplane TransitionsBetween Steady-Level and Hovering FlightEric N. Johnson, Allen Wu,† James C. Neidhoefer,‡Suresh K. Kannan,§ and Michael A. Turbe†Georgia Institute of Technology, Atlanta, Georgia 30332-0150DOI: 10.2514/1.29261Linear systems can be used to adequately model and control an aircraft in either ideal steady-level ﬂight or in idealhovering ﬂight. However, constructing a single uniﬁed system capable of adequately modeling or controlling anairplane in steady-level ﬂight and in hovering ﬂight, as well as during the highly nonlinear transitions between thetwo, requires the use of more complex systems, such as scheduled-linear, nonlinear, or stable adaptive systems. Thispaper discusses the use of dynamic inversion with real-time neural network adaptation as a means to provide a singleadaptive controller capable of controlling a ﬁxed-wing unmanned aircraft system in all three ﬂight phases: steadylevel ﬂight, hovering ﬂight, and the transitions between them. Having a single controller that can achieve andtransition between steady-level and hovering ﬂight allows utilization of the entire low-speed ﬂight envelope, evenbeyond stall conditions. This method is applied to the GTEdge, an eight-foot wingspan, ﬁxed-wing unmannedaircraft system that has been fully instrumented for autonomous ﬂight. This paper presents data from actual ﬂighttest experiments in which the airplane transitions from high-speed, steady-level ﬂight into a hovering condition andthen back again. f mNomenclature 2, B 1, A Aa a , a bv , bwegK, Rn1n2n3p Q q q0 ; q1 ; q2 ; q3 q Ti!b u t u, v, wV, W, v, wvX thrx, xxin , x zj , v , w linearized vehicle dynamicsacceleration or activation potentialtranslational dynamics and estimateneural network biaseserror between reference model and plantacceleration due to gravityinner-loop, outer-loop gain matricesnumber of neural network inputsnumber of neural neuronsnumber of neural network outputsposition vectorattitude error angle functionattitude quaternionEuler rotation to quaternion transformtransformation from inertial frame tobody framecontrol vector at time tbody axis velocity componentsneural network input and output weightsvelocitythrottle control derivativestate variable, state vectorneural network inputinput to jth hidden-layer neuronangular acceleration or angle of attackattitude dynamics and estimateneural network learning rate matricestotal function approximation error , , v , w ad , ad r , 0!! attitude correction throttle actuator deﬂection elevator, aileron, and rudder actuatordeﬂection actuator deﬂection and estimate damping ratio neural network thresholds E-modiﬁcation parameter adaptive element signals robustifying signal neuron sigmoidal function, gradient angular velocity natural frequencyI. IntroductionFIXED-WING aircraft with the ability to hover have the potentialof providing a nearly stationary surveillance platform whilemaintaining the high-speed, maneuverable, long-endurance dashcapabilities associated with such vehicles. There is currentlysigniﬁcant commercial and military interest in developing suchsystems, which would be well suited for a variety of missions,especially in urban or other constrained environments. Militaryapplications include the provision of persistent intelligence,surveillance, and reconnaissance (ISR) with the ability to stare toenhance target identiﬁcation. In the commercial sector, a hoveringmode could enhance the utility of unmanned aircraft systems (UASs)used for border patrol, trafﬁc monitoring, and hazardous siteinspection. The potential for performing tail-sitting takeoffs andlandings also increases the operational domains of such aircraft.Although the ability to transition ﬁxed-wing UASs betweensteady-level and hovering ﬂight has been demonstrated withremotely piloted vehicles, performing such maneuvers autonomously presents unique challenges. One such challenge results fromthe highly nonlinear nature of the actual transition between the twoﬂight regimes. Thus, although linear systems could be used to modeland control the aircraft while in either steady-level ﬂight or ideal(nearly stationary) hovering ﬂight, transitioning the aircraft betweenthe two would require the use of a more complex system (such as anonlinear, scheduled-linear, or a stable adaptive system) capable ofmodeling (or controlling) the nonlinearities that occur duringtransition.Received 11 December 2006; revision received 25 May 2007; accepted forpublication 25 May 2007. Copyright 2007 by Eric N. Johnson, Allen Wu,James C. Neidhoefer, Suresh K. Kannan, and Michael A. Turbe. Published bythe American Institute of Aeronautics and Astronautics, Inc., withpermission. Copies of this paper may be made for personal or internal use, oncondition that the copier pay the 10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0731-5090/08 10.00 in correspondence with the CCC. Lockheed Martin Assistant Professor of Avionics Integration, AerospaceEngineering Department.†Graduate Research Assistant, Aerospace Engineering Department.‡Research Engineer, Aerospace Engineering Department.§Senior Research Scientist, Guided Systems Technologies, 75 Fifth Street,Northwest, Suite 336.358

359JOHNSON ET AL.The speciﬁc contributions of this work include 1) the use ofdynamic inversion with adaptation as a means to provide stableaircraft control in steady-level ﬂight and hovering ﬂight, as well asduring the transition between the two regimes, with a single uniﬁedcontroller, and 2) detailed hover-transition ﬂight-test resultsgenerated using a fully instrumented research unmanned air vehicle(UAV) ﬂying in windy conditions. In this approach, a neural networkis adapted in real time to account for errors in a single vehicle modelthat is linearized about a hover condition. Additionally,pseudocontrol hedging (PCH) allows the neural network to continueadapting when actuator nonlinearities, such as saturation, occur [1–3].Work on hovering ﬁxed-wing aircraft is currently being performedby several research groups. William E. Green and Paul Y. Oh atDrexel University¶ have performed autonomous ﬂight tests of aﬁxed-wing, micro-UAS hovering in an urban environment. Theseexperiments involved the use of a Microstrain inertial measurementunit (IMU), a PIC16F87 microcontroller, and a linear controller tomaintain hover; however, transitions to and from hover wereperformed manually. Aerovironment’s SkyTote UAS†† [4] is a UASwith potential capabilities to hover, take off, and land vertically, andalso transition into conventional horizontal ﬁxed-wing ﬂight. Aprototype SkyTote is complete and hover testing is currentlyunderway. Aurora Flight Sciences’ GoldenEye family of UASs‡‡§§have been used in numerous ﬂight tests throughout 2005 and 2006 inwhich autonomous transitions were made from vertical hoveringﬂight to horizontal ﬂight and back again. Although the GoldenEyehas the ability to ﬂy in a ﬁxed-wing conﬁguration, its wing is notactually “ﬁxed,” in that it has the ability to vary the angle of incidenceof its wings during ﬂight. The University of Sydney is developing aT-wing UAS [5,6] capable of vertical takeoffs and landings as well assustained forward ﬁxed-wing ﬂight. To date, the T-wing has beenﬂown in hover mode both manually and under automatic controlusing command augmentation system (CAS) controllers. Manyother widely varying methods for autonomously guiding andcontrolling UASs have been developed [7–10].Throughout this paper the term “hovering” is used to indicate aﬂight regime in which groundspeed is very small (or even zero).Thus, hovering in this context usually indicates a ﬂight regimebeyond wing-stall, in which engine thrust is the primary forcekeeping the aircraft aloft. It is recognized that it is possible for a ﬁxedwing airplane to ﬂy at zero groundspeed while its wings are notstalled; however, in most cases, the high-wind conditions required toachieve this would preclude safe ﬂight.This paper provides a description of the GTEdge UAS and theguidance and adaptive control architectures used to successfullytransition the aircraft between steady-level and hovering ﬂightduring actual ﬂight tests. A description of the GTEdge is given ﬁrst,followed by a detailed discussion of the guidance system and theneural adaptive controller. The next section gives a justiﬁcation forthe use of adaptive control and a description of the role of theadaptive element in this application. Finally, ﬂight-test results arepresented, followed by a discussion of limitations in the presentarchitecture and suggestions for improving performance.II.Flight-Test Hardware DescriptionThe GTEdge UAS (see Fig. 1) consists of four major subsystems:1) the baseline commercial off-the-shelf (COTS) airframe, 2) the¶Green, W. E., and Oh, P. Y., “Autonomous Hovering of a Fixed-WingMicro Air Vehicle,” http://prism2.mem.drexel.edu/ billgreen/Publications/ﬁnalGreenIcra2006.pdf, Drexel Univ., Philadelphia (cited Dec. 2006). Green, W. E., and Oh, P. Y., “Micro Air Vehicle to Fly in Caves,Tunnels, and Forests,” http://www.pp.drexel.edu/ weg22/fwHovering/ﬁxedWingHovering.html, Drexel Univ., Philadelphia (cited Dec. 2006).††U.S. Air Force Research Laboratory Web site, .asp (cited December 2006).‡‡Aurora Flight Sciences Web site http://www.aurora.aero/GE50/index.html (cited December 2006).§§International Online Defence Magazine, 50.htm (cited December 2006).Fig. 1 GTEdge research UAS is a modiﬁed 33% scale Edge 540T,selected for its ability to carry moderate payloads and performaggressive aerobatic maneuvers.avionics used for autonomous guidance, navigation, and control,3) the software that runs onboard the ﬂight control computer, and4) the ground control station used for issuing commands to thevehicle. This section describes the speciﬁcations and role that eachsubsystem plays in the context of the operation of the GTEdge.A.GTEdge AirframeThe GTEdge is a modiﬁed commercially available Aeroworks33% scale Edge 540T aircraft. This baseline airframe was selectedfor its off-the-shelf availability and for its aerobatic capabilities,including a thrust-to-weight ratio greater than one, which allowshovering and accelerated vertical climbs. Payload requirementsgoverned the selection of the airplane’s scale. The GTEdge has thefollowing physical characteristics: 1) a wing span of 8.75 ft, wingarea of 13 ft2 , and length of 7.8 ft; 2) engine type is gasoline, DesertAircraft DA100 100 cc engine, 9.8 hp; 3) dry weight without payloadis 35 lb; 4) throttle, elevator, aileron, rudder actuated by JR8611Aultratorque digital servos; 5) actuators powered by lithium-ionbatteries; 6) endurance is approximately 30 min. at steady ﬂight of70 ft s.B.FCS20-based Avionics SuiteA small, integrated guidance, navigation, and control (GN&C)hardware and software system, referred to as the Flight ControlSystem Version 20 (FCS20) [11,12], recently developed by theGeorgia Institute of Technology, is the cornerstone of the GTEdgeavionics suite. This miniature computer uses a ﬂoating point digitalsignal processor (DSP) for high-level serial processing and a ﬁeldprogrammable gate array (FPGA) for low-level parallel processing,along with microelectromechanical systems (MEMS) sensors.The basic modules of the FCS20 are the EC20 processor board andthe SB20 sensor/power board. The EC20 processor board (see Fig. 2)handles FCS20 processing and internal and external communications. Embedded ﬂash memory included in the processor boardallows for high data rate onboard data recording during ﬂight. Pulsewidth modulation signals used for driving vehicle actuators aregenerated by the FPGA.The SB20 sensor/power board was designed to be compatible withthe EC20 processor board and provides three main functions:supplying regulated and ﬁltered power to the system, supportingonboard or external navigation sensors, and serving as an interface toexternal components. The eight main sensor components of the SB20consist of three analog devices ADXR300 rate gyros, two analogdevices ADXL210E two-axis 10g accelerometers, a mBlox globalpositioning system (GPS) module, and Freescale absolute anddifferential pressure sensors.Other signiﬁcant components of the GTEdge avionics suiteinclude a Novatel OEM4 differential GPS and a Freewave spread

360JOHNSON ET AL.data recording to supplement the data recorded by the GTEdgeonboard computer. GCS operators can also issue commands such asﬂight plans or requests for speciﬁc data by sending the appropriatemessage type from the GCS to the GTEdge.III. Guidance and Control SystemsFig. 2FCS20 with EC20 processor board on top.spectrum transceiver for communications with the ground controlstation. A diagram of the complete system can be seen in Fig. 3.C.Extended-Kalman-Filter-Based Navigation SystemAn integral component of the FCS20 is the 16-state extendedKalman ﬁlter (EKF) which uses data from the sensors on the SB20processor board to generate a navigation solution that closelyestimates the state of the system [13,14]. The EKF serves severalimportant functions, including 1) estimating the orientation of thesystem from accelerations and angular rates, 2) removing processand measurement noise from the measurements, and 3) providingstate estimates at 100 Hz, even though the GPS updates at a slowerrate of 10 Hz. Similar EKF implementations can be found forcomparison [15].D.Ground Control StationThe ground control station (GCS) is a laptop that communicateswith the GTEdge over a wireless serial link. The GTEdge sends aprimary message packet containing information needed for operationdown to the GCS at 10 Hz, and another message packet containingless vital information once every second. These message packetsprovide the GCS with data used to display vehicle information to theGCS operator and are also recorded on the GCS to provide redundantFig. 3An adaptive neural net-based controller [2] developed and ﬂighttested on other UASs [16] is used to control the GTEdge in steadylevel ﬂight, hovering ﬂight, and the transition between. An overviewof the controller architecture and the motivation for its use in thisapplication are included here. A proof of the underlying theory is alsoavailable [17].In this implementation, the controller consists of an outer loop fortracking translational states and an inner loop responsible for vehicleattitude dynamics. Both the inner and outer loops are organized insimilar fashion into four primary components: a reference model, anapproximate inversion, a linear proportional plus derivative (PD)compensator, and a hedging block, as shown in Fig. 4. Each loop usesfeedback linearization and, more speciﬁcally, dynamic inversion, asthe control strategy. The reference model is nonlinear and selected toimpose a desired closed-loop response and also impose limits on theevolution of states. Any step changes in the external command thusappear as continuous signals to the linear PD compensator.Dynamic inversion, when used alone, requires accurate systemmodels for all ﬂight regimes and these models can be costly anddifﬁcult to obtain [3]. However, in this application, a simple model ofthe vehicle, linearized about a hover condition, is used in theapproximate dynamic inversion, and a neural network is trainedonline to correct for the modeling errors. In this approach, certainnonlinear effects (such as actuator saturation) can create difﬁcultiesfor the adaptive element [1,2]. Thus, a technique calledpseudocontrol hedging (PCH) has been implemented to keep thenetwork from continuously trying to adapt to these effects, byadjusting the reference model with a hedging signal.In the cascaded inner- and outer-loop architecture shown in Fig. 4,the inner loop appears to the outer loop as an actuator that generatestranslational accelerations by means of a commanded attitude. Thus,in the case of a ﬁxed-wing aircraft, the outer loop generates a throttlecommand and an attitude augmentation to be added to the externalattitude command for the inner loop to achieve the desired vehicleposition and velocity. The inner loop combines the desired attitudefrom the outer loop with the commanded attitude from the trajectorygenerator to compute the appropriate moment actuator deﬂections. InFig. 4, a and are used to denote linear and angular acceleration,respectively.FCS20 data ﬂow diagram.

361JOHNSON ET AL.Fig. 4 Architecture used for control of a ﬁxed-wing UAS during steady-level and hovering ﬂight, as well as the transition between them.A.Figures 5 and 6 show the NN output during an actual ﬂight test(more data from the same test is presented in Sec. IV). Because thisdata was taken in actual ﬂight conditions (not in an ideal simulation),the inﬂuence of process noise (wind, turbulence, etc.), measurementnoise, lag, and other real-world effects can clearly be seen in theoverall noise level of the data. In addition, these same real-worldeffects make it much more difﬁcult for the closed-loop aircraft toachieve perfect “steady-level” or “stationary hovering” ﬂight.However, the aforementioned trend is still clearly visible in the data,particularly in Figs. 5a, 6a, and 6c: namely, the NN output exhibitsrelatively constant characteristics (e.g., amplitude and frequency)during steady-level ﬂight (the nonshaded areas in the plots) which aredifferent than the relatively constant characteristics exhibited duringhovering ﬂight (the shaded areas in the plots) and the ﬂight phases;including the gray transitions, which approximately cover thetransitions between steady-level and hovering ﬂight, are readilyidentiﬁable in the data. It is also noted here that in each experimentthe neural network weights were initialized to zero so that, during theexperiments, signiﬁcant and meaningful adaptation of the weightsoccurred.Need for 0-4(1)0-5-10-15-200time (sec)a) x axisp v0-10Controller SynthesisThe aircraft dynamics are described with the following nonlinearequations [2]:13015B.ν ad for a z65ν ad for a yν ad for axAs mentioned earlier, although linear systems can be used toadequately model and control an aircraft either in steady-level ﬂightor in ideal (nearly stationary) hovering ﬂight, the same cannot be saidfor the highly nonlinear transition between these two regimes.Accurately modeling or controlling an aircraft during this transitionwould require the use of systems capable of dealing with thesenonlinearities, such as nonlinear systems, multiple scheduled-linearsystems, or stable adaptive systems.In this work, a stable adaptive controller was used to address thischallenge; the adaptive element within the controller was a singlehidden-layer perceptron neural network (NN) used to model the errorbetween the linear hover model and the actual nonlinear aircraftsystem. Using an adaptive controller offers potential advantages overboth nonlinear systems and multiple scheduled-linear systems. Forexample, traditional methods (such as Lyapunov synthesis [18]) fordeveloping a stable nonlinear (nonadaptive) controller can provedifﬁcult in nonideal applications; likewise, the cost of developing themultiple linear systems required for scheduling can preclude theusefulness of that approach.Based on the controller formulation used in this work, certaincharacteristics of the neural network output during ﬂight can bepredicted. In particular, because during both steady-level andhovering ﬂight, the aircraft can be adequately represented with alinear model; the NN output (which represents the error between alinear model and the actual model) should have relatively constantcharacteristics during each of these two ﬂight phases.1020304050607080-250time (sec)b) y axis1020304050607080time (sec)c) z axisFig. 5 Output from the neural network adaptive element for translational accelerations of the outer loop in the slow transition, expressed as componentsof the body-ﬁxed axes; vertical axis is in units of ft s2 .

362JOHNSON ET AL.8-0.26-0.3876-0.452-0.5ν ad for α zν ad for α yν ad for α e (sec)a) x axis01020304050607080-20time (sec)b) y axis1020304050607080time (sec)c) z axisFig. 6 Output from the neural network adaptive element for angular accelerations of the inner loop in the slow transition, expressed as components ofthe body-ﬁxed axes; vertical axis is in units of rad s2 .v a p; v; q; !; f ; m x; ; h! des (2)(11)in whichq q q;! (3) a x; a x; a x; ; x; ; x; x; x; ; p; v; q; !; f ; m !(4)represents the acceleration that is not cancelled out exactly in thefeedback linearization due to errors in the model inversion.Furthermore, actuator saturation can introduce limitations on theachievable pseudocontrol, and so the desired pseudocontrol may beunrealizable; this effect introduces the ah and h terms in the closedloop dynamical equations. System stabilization can be achieved bychoosing the pseudocontrols in the following manner [2]:in which quaternions are used to express vehicle attitude.The control architecture is governed by nested outer and innerloops that handle the translational dynamics [Eq. (2)] and the attitudedynamics [Eq. (4)], respectively. The state vector of the vehicle isx pT vT qT !T TThe vector of control signals is written as:hiT Tf Tm(5)(6)where f represents the propeller thrust, and m represents theelevator, aileron, and rudder aerodynamic moment actuators. Theactuator dynamics are unknown but are assumed to be asymptoticallystable.A transformation is introduced to provide approximate feedbacklinearization. This transformation is [2] v; q; !; qdes ; fdes ; m a p;ades des p;v; q; !; f ; mdes (7) and are mappings selected to approximate the actualwhere a translational and rotational accelerations a and of the vehicle.Here, ades and des are commonly referred to as the pseudocontrolsignals and are analogous to desired translational and angularaccelerations. The control inputs and attitude needed to produce thedesired pseudocontrols are fdes , mdes , and qdes , respectively, and fand m are estimated actuator positions. If the accelerationapproximations are chosen to be invertible, expressions for thedesired control inputs and attitude can be calculated as fdes a 1 p; v; q; !; ades ; m (8)qdes mdes 1 p; v; q; !; f ; des (9)This approximate inversion provides the following closed-loopdynamics: a x; ; ahv ades (10)a des acr apd a ad(12) des cr pd ad(13)in which acr and cr are output from the vehicle dynamic referencemodels, apd and pd are output from the PD compensator, and a adand ad are output from the adaptive element designed to cancel the modeling error .The following sections outline the determination of the signalsused in the computation of the pseudocontrols. The proper selectionof these signals guarantees the boundedness of the error between theplant output and the reference model output given by the vector23pr p6 vr v 77(14)e 6 r ; q 54 Q q!r ! is a function that, given two quaternions, results in an errorwhere Qangle vector with three components, as follows: Q p;q 2 q1 p1 q2 p2 q3 p3 q4 p4 23 q1 p2 q2 p1 q3 p4 q4 p34 q1 p3 q2 p4 q3 p1 q4 p2 5 q1 p4 q2 p3 q3 p2 q4 p1The tracking error dynamics can be found by directly differentiatingEq. (14) to get23vr v6 v r v 77(15)e 64 !r ! 5! r !Furthermore, the linear PD compensator has the following form:

363JOHNSON ET AL. apdRp 0 pdC.Rd00Kp 0eKd ; e Ae B ad x;2Reference Model and Pseudocontrol HedgingCare must be taken in designing the reference model to ensure thatthe effects which introduce ah and h in Eqs. (10) and (11) are notpresent in the tracking error dynamics. Otherwise, the referencemodel will continue to generate commands as if there were noactuator saturation, and the adaptive element would try to correct forthis discrepancy. This can be avoided through the use of PCH in thereference model dynamics as follows [2]:v r acr pr ; vr ; pc ; vc ah! r cr qr ; !r ; qc06 RpA 64 00000 Kp300 77;I 5 Kda ad ad2 30 0(20)6I 0767B 40 050 IThus, the linear gain matrices must be chosen such that A is Hurwitz, and ad needs to be chosen to cancel the effect of .In this implementation, the following reference model was used:a cr Rp pc pr Rd vc vr (21)v r acr ah(22)(16)qdes ; !c h(17) c cr Kp Q qwhere the subscript c refers to commands and r to the referencemodel, and the expression qc qdes represents the combination oftwo quaternion vectors through quaternion multiplication. Thesignals ah and h , which represent the error between the commandedand realized pseudocontrol, are given as follows: qdes ; fdes ; m a x; f ; m ades a x; f ; m a h a x;(18) f x; f ; m des x; f ; m h x;(19)As can be seen from the reference model dynamics, the hedgingsignals shift the reference models by an estimate of the amount theplant did not move due to the saturation of any actuators. This meansthat when the estimated actual position of the actuators is the same asthe desired positions of the actuator (i.e., the actuator is notsaturated), then the ah and h terms vanish, and the reference modelis unaffected. In the presence of saturated actuators, however, the ahand h terms are nonzero and therefore are subtracted from thereference model dynamics to effectively hide the error due tosaturation in the error dynamics. The pseudocontrol hedging conceptis relevant to this application as the actuators often saturate during thehover-transition maneuvers and during hovering ﬂight. The utility ofPCH for enabling valid adaptation in these types of conditions hasbeen demonstrated [19–21].Substituting Eqs. (16–19) into the error dynamics of Eq. (15), itcan be shown that the error dynamics become [2]Table 1I Rd00 ad qdes ; qr Kd !c !r ! r cr h(23)(24)The four gain matrices (Rp;d and Kp;d ) in the preceding equationswere chosen to result in desirable closed-loop pole placement and arethe same as those in the PD compensator (speciﬁc values of Rp;d andKp;d are given in Table 1). Formulas for the gain values are given as[2]Rp !2o !2i!2i 4 o !o i !i !2o(25)!o !i o !i !o i !2i 4 o !o i !i !2o(26)Kp !2i 4 o !o i !i !2o(27)Kd 2 i !i 2 o !o(28)Rd 2where !i;o represent the natural frequencies for the inner and outerloops, and i;o represent the damping ratios. These scalar gains areplaced in the diagonal gain matrices shown in Eqs. (21) and (23).Controller design parameters used in this zRdx;y;zKpx;y;zKdx;y;zKrn1n2n3bvbwaj w v 1A 2A BX thr ftrim1 s2 1 s1 s2 1 sN/AN/AN/AN/AN/AN/AN/AN/AN/AN/A1 srad ft srad s2 ft s2 %maximum 1000.49, 0.3623, 0.241.4, 1.0145, 0.86.25, 8.64, 17.255.0, 4.8, 7.00.018560.50.50.2, 0.4, 0.6, 0.8, 1.0 for j 1; . . . ; 5Diag(1.0, 1.0, 1.0, 1.0, 1.0, 1.0)5.00.1diag(0.0, 0.0, 0.0)diag(0.0, 0.0, 0.0)diag(50.0, 15.0, 50.0)45.00.2344inner-loop proportional gaininner-loop derivative gainouter-loop proportional gainouter-loop proportional gaingain for robustifying termnumber neural network inputsnumber of neuronsnumber of neural network outputsneural network inner biasneural network outer biasactivation potentialsneural network outer learning rateneural network inner learning rateweight for damping term in adaptation lawmatrix for inner-loop reference modelmatrix for inner-loop reference modelcontrol effectiveness; inner-loop reference modelthrottle control derivativethrottle trim value for hover

364JOHNSON ET AL. P P1 Fig. 7P2 Neural network with a single hidden layer.Speciﬁc values for parameters used in this implementation aregiven in Table 1.D.P100P2 1 b2w 1R R2 p d 0Rp 1K K2 p d 0Kp1n4 2R2p 12 Rp R2d1R R2 p dKp2 12 Kp Kd21K K2 p dIn the preceding equations, the NN weight matrices are updatedaccording to the adaptation laws 0 V T x r T kekW wW(32) v x r T W T 0 kekV V(33)Adaptive Neural NetworkA single-hidden-layer perceptron neural network (NN) is used toapproximate the modeling error. As shown in Fig. 7, the number ofinputs, neurons, and outputs are given by n1 , n2 , and n3 , respectively.In the NN, the input-output relationship is given by adk bw wk n2Xwjk j zj (29)j 1where k 1; . . . ; n3 , bw is the outer-layer bias, wk is the kththreshold, and wjk represents the weights of the outer layer. Thesigmoidal activation function is j zj 1 1 e azj where w and v are positive deﬁnite matrices, and is a positivescalar. The 0 matrix is the Jacobian of the vector, and the B matrixcomes from the error dynamics in Eq. (20). The Z value is a bound onthe norm of the Z matrix corresponding to the ideal NN weightingmatrices in accordance with the universal approximation property.It has been proven [2] that Lyapunov analysis carried over the errordynamics in Eq. (20) using these weight adaptation rules given inEqs. (32) and (33) will result in the ultimate boundedness of thetracking error e and the NN weights with the bounds being explicitlydeﬁned in terms of system dynamics, and the Lyapunov equationmatrices [2].(30)E.with the neuronal activation potential given as a. For the jth neuron,the input iszj bv vj n1Xvij xini(31)i 1where bv is the inner-layer bias, vj is the jth threshold, vij representsthe weights of the inner layer, and xini represents the NN inputs. Boththe inner- and outer-layer biases are chosen based on experie

dynamic inversion with adaptation as a means to provide stable aircraft control in steady-level ﬂight and hovering ﬂight, as well as during the transition between the two regimes, with a single uniﬁed controller, and 2) detailed hover-transition ﬂight-test results generated using a fully instrumented research unmanned air vehicle

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