RECENTLY, there has been significant interest at NASA and

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1 JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 35, No., Noveber Deceber Robust Landing-Guidance Law for Ipaired Aircraft P. K. Menon, S. S. Vaddi, and P. Sengupta Optial Synthesis Inc., Los Altos, California 9 DOI:.5/.53 A guidance law for landing an ipaired aircraft is described. The approach involves a planning phase wherein the perforance liits of the aircraft are used to design the landing pattern and the reference speed, followed by the guidance coputations for generating attitude coands. Coanded attitude can be displayed for the pilot on a heads-up display for anual control, or it can be coupled to the autopilot. Although not detailed in the paper, the guidance law can also generate the flap-deployent schedules and the auto-brake syste-initiation trigger in equipped aircraft. Guidance-syste developent is based on a point-ass, nonlinear odel of the aircraft. The feedback-linearization technique is used to transfor the syste dynaics into a linear, tie-invariant for. Finite interval differential-gae forulation is then used to derive the robust optial guidance law. Inverse transforation of the guidance coands to the original coordinate syste produces the roll, pitch, and yaw attitude coands. Attitude coands can be tailored for both the crabbed or sideslip flight odes, and for the transitions between the. Siulation results are given to deonstrate the perforance and robustness of the proposed approach. I. Introduction RECENTLY, there has been significant interest at NASA and other aviation-research centers in aking aircraft resilient with respect to perforance ipairent. Operational experience in the ajor wars during the twentieth century, in civil air transportation, and in ilitary flight operations over the past three decades has repeatedly deonstrated the capacity of fixed-wing aircraft to survive substantial ipairent and continue on to safe landing [ ]. In ost cases, pilot skill has been the key factor in ensuring favorable outcoes. Inspired by such exaples, researchers at NASA and other aerospace organizations have been investing resources in developing flight-control systes that will allow ipaired aircraft to land successfully. Eventually, these ipairent-adaptive, flightcontrol systes are expected to help copensate for the variations in the huan pilot-skill level and contribute toward higher levels of flight safety. Landing is considered one of the ost hazardous phases of an aircraft flight. Even under noral conditions, landing is the highest workload phase of flight operations. During this phase, the pilot is required to reduce the aircraft speed while descending, deploy flaps, extend the landing gear, align the flight path with the runway centerline, and execute the flare or the roundout [5] aneuver. If crosswinds are present, an additional decrab aneuver ust be executed at touchdown. Aircraft ipairent adds another diension to this already coplex task. By introducing additional uncertainties in the flight envelope, aircraft perforance ipairent requires the pilot to odify the standard landing procedures under liited inforation. For instance, flap deployent ay need to be delayed or altogether eliinated, and the flare aneuver ay need to be steeper than noinal to accoodate for the loss of lift. Moreover, ipaired aircraft ay not have the option of going around for another landing attept in case the first attept fails. For the present research, it is assued that inner-loop adaptive stabilization systes such as those described in Refs. [,] will Presented as Paper 77 at the AIAA Guidance, Navigation, and Control Conference, Toronto Ontario, August 5, ; received March ; revision received 3 February ; accepted for publication February. Copyright by Optial Synthesis Inc.. Published by the Aerican Institute of Aeronautics and Astronautics, Inc., with perission. Copies of this paper ay be ade for personal or internal use, on condition that the copier pay the $. per-copy fee to the Copyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 93; include the code 73-59/ and $. in correspondence with the CCC. Chief Scientist, 95 First Street, Suite. Fellow AIAA. Senior Research Scientist, 95 First Street, Suite. Meber AIAA. Research Scientist, 95 First Street, Suite. Senior Meber AIAA. 85 ensure that the airfrae is returned to a stable flying condition after the ipairent. Moreover, it will be assued that the aircraft flight envelope and aneuver liits can be estiated within a certain error bound using algoriths such as those described in Ref. []. The estiated perforance odels can then be used as the basis for deriving the guidance laws for flare and touchdown. Ipairents considered in this paper include the loss of lift fro ajor flying surfaces such as wings and the vertical fin, and partial loss of engine power, including thrust asyetry. Typical landing sequence of an aircraft is illustrated in Fig.. The ajor phases of the landing sequences are indicated by waypoints indicated using black hollow circles. Aircraft generally approach the landing area in the downwind direction with an airspeed of approxiately kt above the reference speed V REF [5,7] at an altitude of 5 ft. Note that V REF is 3% higher than the stall speed V SO. This portion of the landing pattern is generally tered as the downwind leg. The flap deployent is initiated when the aircraft is abea with the runway at approxiately its idpoint. The landing gear is deployed when the aircraft is abea with the expected touchdown point, indicated by the star sybol in Fig.. Flap deployent is coplete as the aircraft turns into the base leg beginning at waypoint A. The airspeed in the base leg is approxiately kt above V REF. The final leg of the landing pattern begins when the flight path is aligned with the runway centerline, denoted by the waypoint B in the figure. The diensions of the approach pattern depend on aircraft perforance capabilities. For instance, the turn to base portion of the approach pattern ay be roughly one ile or less for a sall general aviation aircraft, but ay involve several iles of airspace for a heavy jet aircraft. During the final leg of the landing pattern, the aircraft flight path is stabilized along the glideslope terinating at a desired ai point on the runway, and its velocity vector is aligned along the runway centerline at approxiately 5 kt above V REF. Note that in soe cases, the airspace ay allow a straight-in landing trajectory. In that case, the downwind and base legs will be absent in the pattern, being replaced by a long final leg. The aircraft trajectory ust be stabilized by the tie it reaches 5 ft altitude, otherwise it is recoended that a go-around aneuver is initiated for another landing attept. Assuing that the trajectory is stabilized, the aircraft can initiate flare aneuver at approxiately 5-ft altitude over the runway threshold to touchdown at approxiately ft down the runway. The flare aneuver decreases the aircraft descent rate to between and ft per inute. The first phase of the landing guidance law is to deterine the diensions of the approach pattern based on a knowledge of the aircraft perforance capabilities, such as turn rate and stall speed.

2 8 MENON, VADDI, AND SENGUPTA A V REF + Kts Fig. B Landing Gear Down V REF Decision Height V REF + Kts 5 Typical landing sequence of an aircraft. decrab Maneuver and Touchdown These are known under noral conditions, but ethods such as discussed in Ref. [] ust be eployed in the case of aircraft ipairent. The proble of designing the approach pattern on board an ipaired aircraft will not be further elaborated in the present paper. All through the final phase, the aircraft flight path is required to be aligned along the runway centerline. If the crosswinds are inial, the trajectory can be aligned without difficulty, but if crosswinds are present, the aircraft ust adopt either a crabbed-approach path or a sideslip-approach path to continue tracking the runway centerline []. Pilots of large aircraft tend to prefer the crabbed approach because it preserves the visual reference with respect to the horizon until the aircraft is close to the runway surface. Note that this aneuver requires use of the rudder to aintain the sideslip while applying the ailerons to counter the roll. Very close to the touchdown point, a decrabbing aneuver is executed by rolling the aircraft to touchdown on one of the ain gears while yawing the aircraft nose to align with the runway centerline and copleting the touchdown with the other ain gear. The decrab aneuver is carried out siultaneously with the flare. Because both of these aneuvers involve large attitude gyrations close to the ground, extree care ust be exercised to ensure that the airfrae does not contact the ground at any point other than at the landing gear. Several exaple videos illustrating the difficulty of landing in severe crosswinds are available on the worldwide web. The flare aneuver can be challenging even under ideal conditions, and the decrab aneuver during the touchdown in crosswinds adds to the coplexity of the task, soeties requiring a go-around in severe crosswind situations. The coplexity of the flare and decrab aneuvers is largely due to the need to siultaneously satisfy ultiple requireents on the flight-control syste along the path and at the final touchdown point. For instance, the angle of attack ust be precisely controlled to ensure a sall negative vertical velocity at touchdown to prevent ballooning and bouncing. The forward speed ust be aintained to ensure that sufficient lift is available until the wheels touch down, and then rapidly decreased to iniize floating caused by the ground effect. The crab angle ust be aintained to align the aircraft velocity vector along the runway centerline until close to the ground, at which tie the aircraft ust be rolled to ensure that only one of the ain landing gears touches the runway surface before the aircraft nose is aligned along the runway centerline. Iediately after this has occurred, the aircraft ust be gently rolled back to allow the other landing gear to touch down while siultaneously yawing to align the aircraft nose with the runway centerline. If it appears that any of these actions ay not lead to a successful landing, the aircraft ust iediately execute the go-around procedure by increasing the thrust to takeoff setting, retracting the flaps and the landing gear during the clib out. Alternately, if the ain gears have touched down with the nose pointing in the right direction, the aircraft nose ust be aintained high enough to rapidly bleed off the airspeed to allow the nose gear to touch down. Reverse thrust and spoilers can then be applied to bring the speed down to a point where the aircraft wheel brakes can be safely applied, copleting the landing aneuver. Several books on flight-control syste design discuss the flare-control proble [8,9], but none have addressed the siultaneous execution of flare and decrab aneuvers in the presence of uncertainties. Interestingly, the coplexities of the flare aneuver otivated one of the early attepts at eploying neural networks [] for designing an adaptive control law for this aneuver. Schaefer [] developed functional requireents for a B757/B77 autoland syste. Shakarian [] used Monte Carlo techniques to assess the dispersion perforance of a B757/B77 autoland syste. Wagner et al. [3] designed digital control laws for autolanding using quantitative feedback-control theory. Shue et al. [] designed robust control laws for glideslope capture under windshear using H technique. Fuzzy logic-based techniques were used by Nho and Agarwal [5,] for design of glideslope capture and autoland systes. Heffley [7] conducted closed-loop analysis of the flare aneuver using linear odels that were derived fro anual landings of real and siulated aircraft. Arents et al. [8] and Ngoc et al. [9] studied the ipact of synthetic vision on landing guidance and the flare aneuver. Prasad and Pradeep [] used the technique of feedback linearization for developing an autolanding syste for a fighter aircraft. Kluver [,] developed landing-guidance laws for unpowered aircraft using trajectory planning. Ki et al. [3] developed a glidepath-tracking algorith for autolanding of an unanned aerial vehicle. The current work develops a nonlinear, closed-for analytic three-diensional (3-D) landing-guidance law suitable for onboard ipleentation. It considers all the key phases of landing, including stabilized approach, flare, and decrab aneuvers. The guidance law is deonstrated on a realistic coercial aircraft odel obtained fro NASA. Siulation scenarios include both landing-undercrosswind and syetric loss-of-lift conditions. Asyetric loss of lift is dynaically equivalent to the crabbed crosswind-landing scenario. Robustness of the guidance law is deonstrated in Monte Carlo siulations, and coparison with a nonlinear nuerical optiization-based guidance solution is given. The following sections will first discuss the aircraft odels used in the guidance-law developent, followed by the guidancelaw derivation and siulation assessent. Near optiality of the proposed approach will be illustrated by coparing its perforance with those fro a nonlinear prograing-based trajectory optiization. Finally, the robustness of the guidance law will be deonstrated through Monte Carlo siulations. II. Aircraft Models Rigid-body dynaics of the aircraft odel are often used for autopilot design, but point-ass odels are ore appropriate for use in guidance-law developent due to their close connection with aircraft perforance paraeters. The following subsections will discuss two odels used in the present work. The first odel is used for guidance-law derivation, whereas the second odel is used for siulation evaluation of the guidance law. A. Point-Mass Dynaics The point-ass odel is defined with respect to a runway-fixed coordinate syste shown in Fig.. The equations of otion with the assuption of thrust acting along the velocity vector are: _x V cos cos () _y V cos sin () _h V sin (3) _V T D g sin () _ g Lh cos V W (5)

3 MENON, VADDI, AND SENGUPTA 87 h _T k pt T T coand (9) y x The variable is the atospheric density; C x, C y, and C z are the body-axis-referenced aerodynaic force coefficients; S ref is the reference area;,, and _ are the body attitudes; and T is the thrust. The subscript coand denotes coanded values. The approxiate proportional gains in the autopilot-control loops are k p, k p, and k p3 ; and the derivative gains are denoted by k v, k v, and k v3. The engine proportional gain is denoted by k pt. V REF Fig. Coordinate syste for point-ass odel. L y _ V cos In these equations, T is the thrust, is the ass, V is the speed, is the flight-path angle, is the heading angle, L h and L y are the aerodynaic forces noral to the velocity vector, and D is the aerodynaic drag. The point-ass odel is derived with the assuption that the aircraft is continuously aintained in oent equilibriu using the control surfaces. Thus, the vehicle attitude dynaics are treated as algebraic, and the attitudes are considered to be the control variables. Note that the forces L h and L y can be expressed in ters of lift, and the bank angle can be expressed in bank-to-turn (zero-slip) aneuvers, or in ters of lift and side force for skid-to-turn aneuvers. The lift and side-force coponents can be expressed in ters of angle of attack and angle of sideslip. Given the orientation of the velocity vector in the inertial frae through the flight-path angle and the heading angle, the pitch-yaw-roll body attitudes necessary to generate a desired angle of attack and the angle of sideslip can be coputed. These angles can be used for pilot guidance or they can be directly coupled to the attitude-tracking autopilot. B. Siulation Model In addition to the point-ass dynaics, the siulation odel captures the autopilot-attitude dynaics through linear secondorder dynaic systes defined along each axis. Note that this representation of the attitude-tracking autopilot dynaics will be exact if its design is based on the feedback linearization ethodology. The point-ass equations of otion of the aircraft in an earth-fixed inertial frae are given by: 3 3 X c c s s c c s c s c s s Y 7 5 c s s s s c c c s s s c 7 5 Z s s c c c 3 3 T V S ref C x ; V S ref C y ; 7 5 V S ref C z ; g () 7 5 (7) Note that the sine and cosine functions in the transforation atrix in Eq. (7) have been abbreviated for the sake of copactness. The control variables in this odel are the aircraft attitude coponents acting through the angle-of-attack/angle-of-sideslip variables and thrust. The autopilot is odeled as three decoupled second-order dynaic systes, and the engine dynaics are odeled as a first-order syste given by following equations: k p coand k v _ k p coand k v _ k p3 coand k v3 _ (8) III. Landing-Guidance Law Because the point-ass odel is nonlinear, it is not readily aenable to the derivation of guidance laws in feedback for using the theory of optial control [] and differential gaes [5]. Following previous research efforts [ 9], the syste dynaics can be first transfored into linear, tie-invariant for using feedback linearization [3 3]. The guidance proble can then be solved using the transfored dynaics, and then inverse transfored in ters of the original variables. The resulting solution will be in nonlinear-feedback for, suitable for real-tie ipleentation as outlined in the following sections. A. Feedback Linearization of the Aircraft Point-Mass Dynaics The first step in the transforation involves the differentiation of the kineatic Eqs. ( 3), followed by the substitutions fro Eqs. ( ) to yield: x y T D T D h cos cos L h sin cos L y sin cos sin L h sin sin L y cos () () T D sin L h cos g () Expressions ( ) describe the aircraft point-ass dynaics in the inertial frae. The transcendental functions given in ters of the flight-path angle and the heading angle on the right-hand sides of these equations can be coputed fro the inertial velocity coponents as: _h sin q _h _x _y _y sin p _x _y p _x _y cos q (3) _h _x _y _x cos p () _x _y Because the aerodynaic forces on the aircraft are based on its relative speed with respect to the atosphere, the transforation of the horizontal- and vertical-lift coponents to the aircraft attitudes ust be carried out with respect to the relative velocity vector. For this purpose, the coponents of the wind-velocity vector are assued to be specified in the sae X-Y-H coordinate syste as the equations of otion. The airspeed is then given by: q V a _x _x w _y _y w _h _h w (5) The flight-path angle and the heading angle of the airspeed vector can be coputed as:

4 88 MENON, VADDI, AND SENGUPTA _h _h sin w q w _x _x w _y _y w _h _h w p _x _x cos w w _y _y w q () _x _x w _y _y w _h _h w _y _y sin w p w _x _x w _y _y w _x _x cos w p w (7) _x _x w _y _y w The equations of otion can then be recast in ters of the atosphere-relative velocity vector to yield: x T D cos w cos w L hw sin w cos w L yh sin w U x (8) T D y cos w sin w L hw sin w sin w L yw cos w U y (9) T D h sin w L hw cos w g U h () The right-hand sides of Eqs. (8 ) are next denoted by pseudocontrol variables U x, U y, and U z to define the transfored dynaics of the aircraft. Note that the syste is in linear, tie-invariant, and decoupled for in ters of the pseudo-control variables. The state variables in the transfored odel are the position and velocity coponents of the aircraft in the inertial frae. The pseudo-control variables are the acceleration coponents in the inertial frae. The pseudo-control variables can be inverse transfored to thrust, pitch, yaw, and roll attitude using algebraic transforations, as illustrated in Sec. III.E. An available degree of freedo in this transforation is the fact that the aircraft can be operated in bank-toturn or skid-to-turn odes. Pitch-yaw attitudes at arbitrary roll attitudes can be used to create skid-to-turn aneuvers such as the crabbed approach, whereas specific pitch-yaw-roll attitudes can be used to create bank-to-turn aneuvers useful for sideslip approaches. It is reasonable to assue that inor changes to aircraft thrust will be ade by the auto-throttle or the pilot to keep the speed essentially constant during the descent. Consequently, the thrust agnitude can be assued to be equal to drag. Because the transfored point-ass dynaics are linear and tie invariant with respect to the pseudo-control variables, linear optialcontrol and differential-gae theories can be directly applied to this guidance proble, as illustrated in the following section. B. Robust Finite Interval Optial Guidance Because the transfored point-ass dynaics of the aircraft for are in decoupled for, the guidance-law derivation will only be illustrated along one the vertical direction. The developent for the lateral direction can be carried out in an entirely analogous for. The coupling between the lateral and longitudinal dynaics occurs through the inverse feedback-linearizing transforation. The feedback-linearized odel of the aircraft in the vertical direction can be written as: Z a b () landing-guidance law is to transfer the aircraft fro its initial position and velocity to a desired waypoint with a specified velocity at a specified final tie while subjecting it to the least aount of acceleration. The guidance law should siultaneously iniize the ipact of external disturbances on the trajectories. This can be achieved by iniizing a perforance index of the for. J S Z f Z fdes S _Z f _Z fdes Z tf Ra "b dt () The first ter in the perforance index penalizes the deviations in the aircraft position fro a specified waypoint location at the specified final tie, and the second ter penalizes the velocity error. The first ter in the integrand iposes the objective of iniizing the acceleration. The second ter in the integrand arises fro the need to ake the guidance law robust with respect to external disturbance b. The negative sign of this ter arises fro the differential-gae view of the disturbances, which assues that the disturbances always act in such a way as to axiize the perforance index. The paraeters S, S, and R can be chosen to establish relative weighting between the terinal-position error, the velocity error, and the acceleration agnitudes. The variational Hailtonian for the optial-control proble, () and (), can be used to derive the co-state equations while enforcing the optiality condition. This process yields: t t f H Ra "b _Z a b (3) _ () _ t (5) Optiality condition [] can be used to derive the optial control and worst case disturbance as: a R t () R b " t (7) " Boundary conditions on the co-states at the specified final tie are given by: f S Z f Z fdes (8) f S _Z f _Z fdes (9) Substituting the expression for control and worst-case disturbance into the transfored equations of otion and integrating yields the following expressions for acceleration, velocity, and position: Z t t R " t R (3) " Z f Z _Z t f R t f " R t 3 f " (3) Here a is acceleration in the vertical direction, and b represents the external disturbance in the sae direction. A siilar equation can be written for the Y coordinate as well. The objective of the _Z f _Z t f R " t f R " (3)

5 MENON, VADDI, AND SENGUPTA 89 Substituting these expressions in the expressions for co-state boundary conditions (8) and (9) yields the following linear syste of equations: " # Det with S t f R " S t 3 f R " " # S _Z fdes _Z S Z fdes Z _Z t f Det R " R " R " 3 S t f 7 t 5 f S t f R S t f " R S t f " t f S t 3 f R S " t f (33) Given the waypoint locations and the desired speeds at those waypoints, the expression (33) can be solved to yield the initial values of the co-states. These can then be used to copute the control in the vertical direction at the initial tie as: a (3) R Following Ref. [], this control variable can be recast in ters of tie-to-go to define the control variable at the initial tie given by (3) as the control variable at the present tie t. C. Inverse Transforation The finite-interval guidance law developed in the previous section coputes the acceleration coponents in the vertical and horizontal directions for landing the aircraft in the presence of disturbances while satisfying waypoint constraints. These acceleration coponents can be transfored into the body attitude coands for use by the pilot in a heads-up display or for direct tracking by the autopilot. Because this transforation process is coplex, it will be carried out in ultiple steps in the following section. Given U y and U h, the coponents of the accelerations noral to the relative velocity vector required to execute the landing trajectory, along with the assuption that T D, the coponents of the aerodynaic forces noral to the relative velocity vector can be coputed fro: L yw U cos y L hw sin w sin w (35) w L hw U cos h g (3) w The next step in the transforation is that of converting the horizontal and vertical forces given by Eqs. (35) and (3) first into aerodynaic coefficients and then into aircraft attitudes. Note that the transfored odel involves only the two coponents of acceleration with respect to the inertial frae. The aircraft attitude, however, has three coponents: the pitch, yaw, and roll attitudes. Thus, the transforation fro the acceleration coponents into attitudes has a free-degree-of-freedo. This additional degree of freedo can be used to enforce an additional constraint. During conventional 3-D aneuvers involving aircraft with a vertical plane of syetry, this constraint is enforced through the coordinated flight requireent, which constrains the angle of sideslip to reain zero throughout aneuvers. A second approach is the skid-to-turn aneuver, which explicitly eploys an angle of sideslip to generate lateral acceleration. Such aneuvers ay occur in ipaired aircraft that ay not have a vertical plane of syetry. Skid-to-turn aneuvers ay also arise during crabbed flight under crosswind conditions. In fighter aircraft, the skid aneuver is often used for pointing body-fixed weapon systes in a desired direction. Notably, ost axisyetric issiles eploy skid-to-turn aneuvers during their noral operation. The following sections discuss transforations to enable both these aneuvers. D. Skid-to-Turn Mode The skid-to-turn ode ay be eployed during the stabilizedapproach phase of the landing. In the lateral direction, this ode is characterized by the existence of non-zero angles of sideslip. Pitch and yaw attitudes are used to realize the desired angle of attack and angle of sideslip, with the roll attitude aintained at zero. First, the desired lift and side-force coefficients are coputed using the forces given by Eqs. (35) and (3) to yield: L hw C L desired :5 VaS L yw C S desired :5 VaS (37) The angle of attack and angle of sideslip corresponding to the desired lift and side-force coefficients can be deterined fro the tables listing the relationship between angle of attack and lift coefficient, and the angle of sideslip and side force. If these tables are sybolically denoted by the functions f L and f S, these coputations can be suarized as: w f L C L desired w f S C S desired (38) At sall angles of attack and angles of sideslip, these functional relationships are linear, but they can be highly nonlinear if the aircraft is operating at an extree angle of attack and angle of sideslip. Once the angle of attack and angle of sideslip are coputed, the next step in the process is that of coputing the corresponding yaw-pitch attitudes or pitch-roll attitudes. For sall angles of attack and angles of sideslip, the roll-pitch-yaw attitudes can be expressed as: w w w w (39) E. Bank-to-Turn Mode The bank-to-turn ode is eployed by aircraft under noral aircraft operations. Angle of attack and bank angle are used to realize the desired aerodynaic forces in this ode. Because the angle of sideslip is required to be zero, the side-force coefficients are not used in the coputations. The aerodynaic lift and the desired lift coefficient are first coputed as: q L L hw L yw ) C L desired L :5 V as () Next, the angle of attack and bank angle are coputed as: Lyw Tw fl C L desired w tan () L hw These controls will yield bank-to-turn or coordinated aneuvers ( ). The roll-pitch attitudes for sall angles of attack and sall bank angles can then be recovered as: w Tw w () The yaw attitude is coputed such that ( ). Note that the expressions for attitude in ters of the angle of attack and bank angle assue sall-angle approxiations. The relationships for large angles are a little ore involved. Aircraft operate in either one of the odes during ost of their flight, and thus there is a specific aneuver that eploys a cobination of these odes. This is the terinal decrab aneuver wherein the aircraft nose is intentionally rotated away fro the wind direction to point along the runway centerline. This creates an angle of sideslip, requiring the aircraft to bank toward the wind to counter the side force to keep it aligned along the runway centerline.

6 87 MENON, VADDI, AND SENGUPTA IV. Closed-Loop Siulation Results The guidance law discussed in the previous section is evaluated in a siulation. The following paraeters extracted fro the NASA GTM aircraft odel [33] are used in the siulation. 9 lbs S ref 95 :75 slug=ft 3 V stall 7 knots V ref 5 knots The following autopilot tie-constants fro Ref. [9] are used in the roll, pitch, and yaw channels: t cr :3 s t cp 3:88 s t cy :3 s The daping ratios for all three axes are set as r p y :77. After several trials, the terinal state-weighting factors, controlweighting factor, and the disturbance-weighing factor for the optial guidance law are chosen as: S e S e8 R " A. Guidance Law Ipleentation Only the final leg of the landing pattern consisting of stabilized approach, flare, and decrab are presented in this paper, as they for the ost crucial portions of the landing trajectory. Three waypoints are chosen for the final landing phase based on code of federal regulations (CFR) Part [5]. These are ) stabilized-approach checkpoint, ) threshold, and 3) the touchdown point. CFR Part recoends the glideslope to be between.5 and 3 deg. The glideslope is set to.75 deg in the present siulations, with the duration of flare being set to six seconds and the vertical speed at touchdown required to be at ft= in. Desired states at the three waypoints are specified as: h 5 ft; h _ V ref sin :75 ; y ; 5 5 tan :75 L rwy (3) h 5 ft; _h V ref sin :75; y ; L rwy () h 3 ; _h 3 ft= in; y 3 ; _y 3 3 V ref cos :75 t flare L rwy (5) These serve as boundary conditions for optial-guidance law in the final phase. The tie-to-go for the guidance law is coputed using the equation: t go range waypo int x x i y y i z z i range rate x x i _x y y i _y z z i _z () The aircraft is expected to stay close to the reference speed throughout the final segent of the landing pattern. Thrust required for aintaining constant airspeed is coputed by setting the derivative of the airspeed to zero to yield: 3 _V a _x _x V w _y _y w _h _h w U y 5 (7) a U h Because U y and U h are known fro the guidance-law coputations, the value of U x that will ensure constant airspeed can be coputed. This value of acceleration, together with the lateral and vertical aerodynaic forces, and the drag can be used to copute the thrust setting that will achieve constant airspeed. This thrust setting is used until the aircraft reaches the stabilized approach check altitude of 5 ft. The thrust is fixed beyond that point. U x h (ft) h (ft) Fig x Y (ft) Final segent of the landing trajectory in three diensions. Trajectory Glideslope SA Checkpoint Runway Fig. Landing trajectory in the vertical plane. B. Landing Under Large Initial Condition Errors In this landing scenario, the aircraft starts with a lateral error of 5 ft with an initial altitude error of ft fro the glideslope at an initial flight path angle of. deg. Figures 3 5 show the landing trajectories in three diensions, vertical plane, and horizontal plane, y (ft) 5 3 Runway Trajectory SA Checkpoint Fig. 5 Landing trajectory in the horizontal plane.

7 MENON, VADDI, AND SENGUPTA 87 Descent Rate (ft/in) Glideslope DR Touch Down DR V (knots) Stall Speed Ref. Speed 5 h (ft) Fig x Descent rates during the stabilized approach and flare Fig. 7 Trajectory in the vertical plane during the flare aneuver. respectively. It ay be observed that the position errors in the lateral and vertical directions are corrected by the tie the aircraft reaches the stabilized approach-check altitude. Thereafter, the aircraft reains very close to the glideslope in the vertical direction and Descent Rate (ft/in) x Fig. 9 Airspeed. aligned with the runway centerline in the lateral direction. Figure gives the descent rate (DR) of the aircraft. Figures 7 and 8 provide the trajectory in the vertical plane and the descent rate as a function of height above the ground during the flare. Descent rate at touchdown is approxiately 97 ft= in, which is within the specified rate of ft= in to ft= in in CFR Part. Figure 9 shows that the reference airspeed reained within three knots of the specified value throughout the landing trajectory. The coanded and the actual thrust profiles are shown in Fig.. The observed thrust transients are due to the tight control tolerances used in the present study. Note that the agnitudes are well within the axiu thrust capacity of, lbf for the NASA GTM aircraft odel. Figures 3 give the coanded and the actual roll, pitch, and yaw attitudes. Coanded and actual values of the roll attitude are zero during the stabilized-approach phase. Following the current operating procedure of using the crabbed approach, the yaw attitude is used to aintain the aircraft on course until the flare aneuver, at which tie the roll angle is used to keep the aircraft aligned along the runway centerline. This can be observed in Fig., which shows inor roll gyrations toward the end. Pitch-attitude history shown in Fig. assues negative values before the aircraft approaches the stabilized approach-check altitude to correct for the -ft excess altitude. The pitch-control activity reains sall after this stage. The aircraft pitches up by deg to reduce the descent rate during the flare. The yaw-attitude history shown in Fig. 3 exhibits the ost activity before the stabilized approach-check altitude is reached. Thereafter, Thrust (Klbf) 8 Coand Fig. 8 Descent rate as a function of altitude above the runway during the flare aneuver x Fig. Coanded and actual thrust.

8 87 MENON, VADDI, AND SENGUPTA.5 Coand 5 φ (deg) -.5 Y (ft) 3 Runway Trajectory SA Checkpoint - θ (deg) x Fig. and coanded roll-attitude histories Coand x Fig. and coanded pitch-attitude histories. the aircraft trajectory reains aligned with the runway centerline with very little yaw control activity. The dashed lines in all three plots show the attitude coands generated by the guidance law, and the solid lines indicate the aircraft attitude. The jups in the coanded values at the waypoints are due to the fact that the control objective ψ (deg) - Coand x Fig. 3 and coanded yaw-attitude histories Fig. Landing trajectory in the horizontal plane under -knot crosswind. changes between each waypoint segent and no requireent has been placed on the soothness of control. If desired, such a requireent can be iposed by introducing additional dynaics at the input. Studies such as these will be of future interest. C. Landing Under a -Knot Crosswind A -kt steady crosswind along the positive Y axis is introduced into the siulation to study the response of the guidance law. The landing trajectory in the horizontal plane is shown in Fig.. It can be seen in this figure that the aircraft corrects a 5-ft lateral offset in the presence of crosswind by the tie it reaches the stabilized approachcheck altitude. The pronounced bulge in the trajectory between the stabilized-approach checkpoint and the threshold is due to the cobined effect of the crosswind and the airspeed-vector orientation requireent at the waypoint. The specified conditions at the runway threshold and the touchdown point are et by the guidance law. The aircraft lands with a yaw attitude of less than. deg, a lateral offset of.37 ft, and a heading angle of :5 deg. Descent rate at touchdown is 8 ft= in, which is slightly higher than the specified axiu rate, as seen in Fig. 5. Figures and 7 provide coparisons of the yaw attitude, roll attitude, and angle of sideslip histories under a -kt crosswind with those of the noinal landing scenario discussed in the foregoing section. The yaw attitude of the aircraft under crosswind assues larger values than the noinal landing scenario. This is ore Descent Rate (ft/in) Glideslope DR Touch Down DR x Fig. 5 Descent rate during the stabilized approach and flare under - knot crosswind.

9 MENON, VADDI, AND SENGUPTA 873 Noinal kt Crosswind - ψ (deg) φ (deg) - - Noinal kt Crosswind Fig x Yaw-attitude history during the stabilized approach and flare. pronounced in the segent between the stabilized approach-check altitude and the runway threshold, where the yaw attitude assues a value of approxiately 8 deg to tighten the lateral deviation between the aircraft trajectory and the runway centerline in the presence of crosswind. During the decrab aneuver, however, the yaw attitude jups back to align the nose of the aircraft with the runway, the angle of sideslip develops, and the aircraft banks in the direction of the crosswind, as can be observed in Fig. 8. It should be noted that the aircraft rolls by as uch as deg, as opposed to the near-zero values in the noinal-landing scenario. D. Landing Under % Syetric Reduction of Lift Coefficient This scenario siulates the loss of lift due to aircraft ipairent. The aircraft lift coefficient is reduced by % to siulate this case. The landings at 5, 7, and 75 kt are then investigated. Figure 9 gives the vertical plane trajectories at all three speeds, as well as the noinal trajectory at the -kt landing speed. All four trajectories are close to one another, indicating that the guidance law was able to accoodate for the reduction in the lift coefficient. This is also confired by the descent-rate plots in Fig., where the % loss-oflift coefficient resulted in a vertical speed of approxiately 85 ft=s when copared with the 97 ft=s in the noinal scenario. The difference between the two scenarios anifests itself in the angle-ofattack and pitch-attitude histories shown in Figs. and, respectively. The aircraft under % loss-of-lift coefficient pulls a β (deg) Noinal kt Crosswind x Fig. 8 Roll-attitude history during the stabilized approach and flare. h (ft) Glideslope % C L Loss, kt % C L Loss, 75kt % C L Loss, 7kt % C L Loss, 5kt Fig. 9 Vertical-plane trajectories for different landing speeds with % loss-of-lift coefficient. larger angle of attack and pitch attitude to generate siilar lift agnitudes during the flare aneuver. If higher landing speeds are perissible, the axiu angle of attack and pitch attitude will be observed to decrease due to the higher dynaic pressure. Descent Rate (ft/in) Touchdown DR % C L Loss, kt % C L Loss, 75kt % C L Loss, 7kt % C L Loss, 5kt - Fig. 7 flare x Angle of sideslip history during the stabilized approach and x Fig. Descent-rate histories for different landing speeds at % lossof-lift coefficient.

10 87 MENON, VADDI, AND SENGUPTA α (deg) % C L Loss, kt % C L Loss, 75kt % C L Loss, 7kt % C L Loss, 5kt Wind Speed (ft/s) W x W y W z x Fig. Angle-of-attack histories for different landing speeds at % loss-of-lift coefficient. The siulation results given in the foregoing sections deonstrate that the guidance syste is capable of safely landing noral or ipaired aircraft under widely varying operating conditions. The reaining investigations are those of guidance-law robustness and near-optiality, discussed in the following sections. E. Robustness Evaluation in Monte Carlo Siulations Because the closed-loop guidance law cobined with the aircraft dynaics fors a nonlinear-dynaic syste, Monte Carlo siulation ethodology can be used to assess the robustness of the syste to paraeter variations and disturbances. All uncertainties except the wind gust and ground effect were assued to be uniforly distributed rando nubers. The following uncertainties are included in these siulations: ) Initial condition errors in altitude easured up to ft. ) Initial lateral offset fro the runway easured up to 5 ft. 3) Initial heading angle errors of 5 deg occurred. ) Initial flight-path deviations fro the glideslope by 5 deg were easured. 5) Loss-of-lift coefficient ranged between 35 and 5%. ) Rando wind profiles were generated using the Dryden windgust odel. Figure 3 shows saple plots of wind coponents. The initial seed used in generating the band-liited unit variance whitenoise signal is set to the Monte Carlo trial nuber. 8 % C L Loss, kt % C L Loss, 75kt % C L Loss, 7kt % C L Loss, 5kt Tie (secs) Fig. 3 Wind speeds fro Dryden wind-gust odel. 7) A ground effect odel fro Ref. [33] is used in all the Monte Carlo siulation runs. The perforance paraeters of interest fro the Monte Carlo siulation trials are ) descent rate at touchdown, ) lateral offset at touchdown, 3) heading angle at touchdown, and ) yaw attitude at touchdown. Figure shows the distribution of the descent rates obtained fro the 5 Monte Carlo siulation trials. The aircraft akes a positive touchdown (> ft=s descent rate) in 93% of the trials. The reaining 7% of the trials result in the ballooning of the trajectory. It is noted that the descent rates at touchdown are lower than the intended ft= in due to the extra lift derived fro the ground-effect odel. The offset fro the centerline at touchdown is less than 5 ft in all the trials, as shown in Fig. 5. The heading angle at touchdown is within deg in ost trials, as seen in Fig.. Finally, Fig. 7 shows that the yaw attitude of the aircraft at touchdown is within :5 deg. The Monte Carlo siulation results given in this section deonstrate that the guidance syste is robust to syste uncertainties and external disturbances within the specified uncertainty bounds. F. Near Optiality of the Landing-Guidance Law To exaine the optiality of the closed-loop guidance law based on transfored dynaics, a separate nuerical trajectoryoptiization proble was forulated in the pitch plane. The perforance index is chosen as the iniization of the integrated values of the load factor along the vertical-plane trajectory. Note that the closed-loop guidance-law forulation used the integral of the 8 θ (deg) - Frequency x Fig. Pitch-attitude histories for different landing speeds at % loss-of-lift coefficient Descent Touchdown (ft/in) Fig. Distribution of descent rate at touchdown.

11 MENON, VADDI, AND SENGUPTA 875 Frequency 8 h (ft) Optial Guidance Nuerical Optiization GlideSlope Frequency y (ft) td χ (ft) td Fig. Distribution of heading angle at touchdown. vehicle acceleration in the inertial frae. The desired boundary conditions are iposed as constraints in the proble. The pitchattitude history is approxiated by piecewise sixth-order polynoials in tie in each of the three landing phases. The 5 Fig. 5 Distribution of lateral offset at touchdown Fig. 8 Coparison between altitude histories during stabilizedapproach and flare. perforance index is evaluated by siulating the vertical-plane trajectory using the paraeterized pitch-attitude history. The coefficients of the polynoials describing the pitch-attitude history were then chosen by a nonlinear prograing algorith. The MATLAB fincon [3] is used in the present nuerical optiization study. The optiization proble does not include the autopilot response to the guidance coands or the response of the engine to the thrust coands. Therefore, these dynaics are reoved fro the siulation odel for the coparisons. Trajectory and descent rates obtained fro nuerical optiization copare favorably with those obtained fro closed-loop siulations, as shown in Figs Both schees achieve the desired waypoint conditions, although the interediate trajectories see slightly different. Figures 3 and 3 give the pitch-attitude and the angle-of-attack histories. The plots are qualitatively siilar in the stabilized-approach segents, differing by less than deg throughout the landing phase. The axiu angle of attack in the flare segent is approxiately 3.75 deg, and the axiu pitch attitude is 3 deg, indicating a very sall flight-path angle at the tie of touchdown. These nuerical optiization results deonstrate the near optiality of the closed-loop guidance law. Minor differences observed ay be attributed to the differences in the perforance index and the paraeterization of the pitch attitude Optial Guidance Nuerical Optiization 3 Frequency 5 h (ft) ψ (deg) td Fig. 7 Distribution of yaw attitude at touchdown Fig. 9 Coparison between altitude histories during the flare aneuver.

12 87 MENON, VADDI, AND SENGUPTA Descent Rate (ft/in) Optial Guidance Nuerical Optiization Fig. 3 Coparison between descent rate during stabilized approach and flare. θ (deg) Optial Guidance Nuerical Optiization Fig. 3 Coparison between pitch-attitude histories during the stabilized approach and flare. α (deg) Optial Guidance Nuerical Optiization Fig. 3 Coparison between angle-of-attack histories during the stabilized approach and flare. V. Conclusions This paper described the developent of a landing-guidance law that can be used under noral and ipaired conditions. Guidancelaw derivation eployed a feedback, linearized nonlinear, pointass dynaic odel of the aircraft. The syste dynaics were transfored to a linear, tie-invariant, decoupled for using differential geoetric transforation. Differential-gae theory was then eployed to synthesize the guidance law. Inverse transforation of the guidance law then produced a nonlinear, tievarying, coupled guidance law. Specialization of the guidance law for bank-to-turn and skid-to-turn aneuvers was illustrated. Perforances of the guidance law under both noral and ipaired aircraft conditions were then deonstrated. Robustness of the guidance law to a class of disturbances such as crosswind, wind gust, aerodynaic odel uncertainties, and ground effect was deonstrated in Monte Carlo siulations. Finally, the guidancelaw perforance was copared with a nuerically optiized trajectory to verify the near optiality of the proposed approach. Evaluation of this guidance law in anned siulations will be of future interest. The following conclusions can be drawn based on the present research: ) The present work deonstrated that it is feasible to develop guidance laws for executing coplex flight aneuvers required for landing ipaired aircraft. ) The guidance law ust be ade robust with respect to aircraft odel and environental uncertainties. This paper showed that the required robustness can be achieved using the theory of differential gaes. 3) The proposed analytical approach has been shown to be near optial by coparing the results with a nonlinear prograingbased nuerical trajectory optiization. The coputations required for ipleenting the guidance law are odest enough to perit its integration with existing flight-control systes on board odern transport aircraft. Acknowledgents This research was supported under NASA Contract nuber NNX9CCP, with John Kaneshige as the Technical Monitor. The authors would like to thank John Kaneshige, K. Krishnakuar, and Nhan Nguyen of NASA Aes Research Center for their support and enthusias for this research. Vivian Lin of Optial Synthesis, Inc., developed the data interface with FlightGear siulation progra to create the aniations used in evaluating the siulation results. References [] Nguyen, N., Krishnakuar, K., Kaneshige, J., and Nespeca, P., Dynaics and Adaptive Control for Stability Recovery of Daaged Asyetric Aircraft, AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA Paper -9, Keystone, Colorado, Aug.,. [] Nguyen, N., Krishnakuar, K., Kaneshige, J., and Nespeca, P., Flight Dynaics and Hybrid Adaptive Control of Daaged Aircraft, Journal of Guidance, Control, and Dynaics, Vol. 3, No. 3, May June 8, pp doi:.5/.8 [3] Rysdyk, R. T., and Calise, A. J., Fault-Tolerant Flight Control via Adaptive Neural Network Augentation, AIAA Paper , 998. [] Tandale, M. D., and Valasek, J., Fault-Tolerant Structured Adaptive Model Inversion Control, Journal of Guidance, Control, and Dynaics, Vol. 9, No. 3, May June,, pp. 35. doi:.5/.5 [5] Anon., Airplane Flying Handbook, FAA Docuent No. H-883-3A, U.S. Departent of Transportation: Federal Aviation Adinistration, Washington, DC,. [] Ki, J., Palaniappan, K., and Menon, P. K., Rapid Estiation of Ipaired Aircraft Aerodynaic Paraeters Aircraft Perforance Models using Differential Vortex Panel Method and Extended Kalan Filter, Journal of Aircraft, Vol. 7, No., July Aug., pp. 8.

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