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AIAA 2016-1666 AIAA SciTech 4-8 January 2016, San Diego, California, USA AIAA Modeling and Simulation Technologies Conference Aerodynamic Model Update Using Parameter Identification Supporting a Cessna Grand Caravan Engineering Simulation Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Alfonso Paris1 and Omeed Alaverdi2 AMERICAN SYSTEMS, Engineering & Analytical Solutions Directorate, Lexington Park, MD 20653 This paper outlines the initial nonlinear aerodynamic model development, and focuses on the subsequent model updates based on flight test data, for an engineering simulation of the Cessna Grand Caravan (C208B) aircraft. This modeling effort was in support of creating a set of flight control laws and navigation algorithms capable of flying the aircraft in autonomous mode. An initial simulation model was developed based on first principles and empirical techniques. The aerodynamic model contained both airframe characteristics and control surface hinge moments. A flight test program was carried out to gather data upon which to further update and validate the simulation model. The resulting model was instrumental during support of control system design, system flight test risk reduction efforts, and overall system flight test preparation. Nomenclature ax,y,z b S SH,V TR,S u,v,w = = = = = = = = = = = = = = VH ,V = horizontal and vertical tail volume coefficients Vrw x,y XH,V X,Y,Z , E,A,R Φ,θ,ψ ηH,V τ ωP,SP,DR ζP,SP,DR Σ = = = = = = = = = = = = c CL,D,Y Cm,n,l f Ixx,yy,zz L,M,N p,q,r q 1 2 body axis linear accelerations along x, y, and z axes reference wing span reference planform mean aerodynamic chord non-dimensional lift, drag, and sideforce aerodynamic coefficients non-dimensional pitching, yawing, and rolling moment aerodynamic coefficients generic function body axis roll, pitch, and yaw moments of inertia dimensional roll, pitch, and yaw aerodynamic coefficients body axis roll, pitch, and yaw angular rates dynamic pressure wing reference planform area horizontal and vertical tail reference planform area roll mode and spiral mode time constants body axis air relative velocities in x, y, and z axes true airspeed linear model state and output vectors longitudinal distance from aircraft cg to horizontal and vertical tail aerodynamic centers dimensional body axis x, y, and z aerodynamic coefficients flow angle of attack and sideslip elevator, differential aileron, and rudder aerodynamic control deflections roll, pitch, and yaw Euler attitudes horizontal and vertical tail dynamic pressure ratio aerodynamic control surface effectiveness modeling term phugoid, short period, and dutch roll system natural frequency phugoid, short period, and dutch roll system damping summation Senior Aerospace Engineer, Engineering & Analytical Solutions Directorate, Senior Member AIAA. Senior Aerospace Engineer, Engineering & Analytical Solutions Directorate, Senior Member AIAA. 1 American Institute of Aeronautics and Astronautics Copyright © 2015 by AMERICAN SYSTEMS. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. I. Introduction Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 T HE C208B is a high wing, single engine turboprop aircraft capable of carrying both passengers and/or cargo. A fully nonlinear simulation of the aircraft was created to support development of autonomous flight controls, navigation software and ground station software in the loop (SIL) testing. The simulation was used to support flight test risk reduction and aided in flight test preparation efforts. The model included airframe mass characteristics, ground reaction, airframe aerodynamics, control surface rigging and aerodynamic hinge moments, propulsion, and system control actuators. The focus of this paper is on the airframe aerodynamic model. An initial simulation model was developed based on first principles and empirical techniques. A flight test program was then designed to collect data targeted at updating and validating the nonlinear simulation model through application of Parameter Identification (PID) techniques. A series of ground and test flights were performed in late 2006 to early 2007 resulting in the collection of flight test data suitable to support further model refinements as well as support PID efforts. Overall, data was collected in Cruise (CR) Flaps 0, CR Flaps 10, Takeoff (TO) Flaps 20, and Landing (LD) Flaps 30 configurations. This paper focuses on results obtained in the CR Flaps 0 and Flaps 10 configurations. Data archival, preprocessing, and PID analysis were performed within the Integrated Data Evaluation and Analysis System (IDEAS). IDEAS is a database management system and analysis software containing a full complement of flight data preprocessing, calibration, simulation, model estimation, verification, and validation tools.1,2 Flight test data collected for the purpose of aerodynamic parameter estimation typically consists of inertial and air-relative sensor outputs. Prior to performing PID analyses, or traditional data reduction, it was important to evaluate, and if necessary correct, the measured data to ensure kinematic consistency amongst them. Consequently, a rigorous post-flight data calibration study was performed within IDEAS using the Navigation Identification (NAVIDNT) tool. This tool couples an adaptive nonlinear Least-Squares Identification algorithm (LSIDNT) with a set of rigid body navigation equations that model aircraft motion over an oblate, rotating Earth. 1,3,4 Sensor biases and/or scale factors, as well as time invariant atmospheric winds, may be identified using this tool. Basic airframe aerodynamic forces and moments were extracted from the collected flight test data, in addition to control surface hinge moments, to support aerodynamic model update efforts. Aerodynamic model structures were developed through analysis of these data using an equation error technique in IDEAS known as Athena employing principal component axis regression. Athena expresses the overall aerodynamic forces and moments as linear combinations of parameters such as stability derivatives and/or increments. In addition, to develop parameter nonlinearities Athena supports the use of linear or cubic basis splines. Model refinements were installed in the engineering simulation for the airframe and validation completed. II. Aerodynamic Model Overview Detailed engineering drawings and airfoil type definitions from the aircraft manufacturer provided the needed data upon which to build a basis for an initial aerodynamic model. 5 Overall, the aerodynamic forces and moments for the simulation are derived using a component buildup method. The forces and moments for each individual aerodynamic component (wing, vertical tail, horizontal tail, and fuselage) are calculated and summed to produce the overall aerodynamic forces and moments on the airframe. The aircraft body axis velocities and angular rates are used to determine local velocities at the velocity reference center defined for each specific component of the wing, fuselage, and tail sections. The wing and horizontal tail sections are broken into two separate semi-span sections while the vertical tail is a single section. Each section has a defined velocity reference center out along its span where local velocities and flow angles are determined to produce a baseline local lift, drag, and pitching moment for the planform without aerodynamic controls deflected. Additional sub-sections along each semi-span define control surface regions where local velocities and flow angles are used, along with defined control deflections, to produce an incremental lift, drag, and pitching moment effect due to a deflected surface. The sum of forces and moments for each major planform section, and its associated control surface sub-sections, is made at an overall defined center of pressure for each planform section. The aerodynamic contribution of each major planform component, and control surface sub-component, in the model has a locally defined dihedral, sweep, and/or incidence relative to the aircraft body axis. A local lift, drag, and pitching moment model is defined for each component. The basic airframe lift, drag, and pitching moment for the wing, horizontal tail, and vertical tail planforms were determined using XFOIL and AVL to obtain characteristics without control surfaces deflected. Aerodynamic contributions resulting from deflected control surfaces were determined empirically using techniques for plain flap 2 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 and single slotted flap surfaces as outlined by McCormick.6 Control surface aerodynamic hinge moment models were determined using empirical techniques for plain flaps as outlined by Perkins and Hage.7 The Caravan wing includes ailerons, aileron trim tabs, fowler flaps, and spoilers that are modeled in the simulation. The wing is divided at the centerline into overall left and right sections. The wing sections, and each sub-section (left and right ailerons, flaps, spoilers, and trim tabs), are individually analyzed as noted above. The left and right horizontal tail sections contain both an elevator and elevator tab surface. The vertical tail contains only a rudder control surface. The individual lift, drag and pitching moment contribution of each of these components are calculated, transferred to the body axis, and then summed to form total forces and moments generated by the wing about the overall defined aerodynamic reference center for the aircraft. Main wing planform downwash is computed and affects the local flow angle at the horizontal tail. Thrust is produced given a separate propeller model while momentum theory is employed to determine slipstream effects on downstream surfaces such as the vertical and horizontal tail.8 The component buildup technique provides a resulting simulation with predictive capability based on physics. It allows for a simulation that offers a wider range of application over those defined strictly on aircraft stability derivatives. As an example, control surface off-nominal conditions can easily be modeled in such a system by simply forcing the affected surface to the desired location. Propulsion wake effects on local components can also be modeled readily. In addition, the requirement to initially define dynamic stability derivatives, such as C mq, Clp, and Cnr, is removed as they now become a direct fall-out of the model as defined above. Classical stability derivatives may be determined for the model using linear model extraction (LME) techniques as was often applied in this work. The simulation was hosted within the IDEAS environment. This allowed the simulation model to be easily accessible by the analysis tools available within IDEAS including LME analysis for providing linear models to the control law development team. This also provided easy access to flight test data for validation and model refinement purposes. III. Available Flight Test Maneuvers A flight test program was carried out to gather the data required to refine and validate the overall model. This testing consisted of two ground tests and eleven flight tests. Ground testing collected propulsion static and dynamic characteristics while flight testing collected aerodynamic flying qualities, hinge moment, and additional propulsion static and dynamic performance. Data was gathered at representative operational configurations as follows: • • • • CR 0 / 125 KIAS CR 10 / 125 KIAS TO 20 / 90 KIAS LD 30 / 90 KIAS Over three hundred individual test events were completed across the program within four weeks. Data from various maneuver types were collected during this program to support model refinement and validation. The two ground testing events supplied propulsion static and dynamic characteristics with the following maneuvers: • • • Ng maps Engine dynamic response Propeller governor dynamics The eleven flight test events provided data related to aerodynamic flying qualities, aerodynamic hinge moments, and propulsion system performance with the following maneuvers: • • • • • • • • • All axis doublets All axis frequency sweeps Pitch attitude captures Wheel bank-to-bank roll attitude captures Pedal bank-to-bank roll attitude captures Longitudinal static stability Lateral-directional static stability Maneuvering stability 1-G wings level stalls 3 American Institute of Aeronautics and Astronautics • • • • Flap transitions Power lever bodes Level flight Acceleration-Deceleration Ground effect Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Throughout this testing the data gathered was inspected daily for quality and any necessary repeats were scheduled into the subsequent flights. Having the data consumers, in this case the model and control law developers, involved in the test data collection process proved beneficial in gathering the necessary data as efficiently as possible, thus minimizing test event repeats. The quality of the test maneuvers, in addition to the data itself, is of utmost importance when updating and validating a simulation model. Figure 1 presents an overview of gathering data through flight test, data quality checking, and applying that data in support of model updates using PID and validation. At any point in the process the requirement may arise to return to flight test to gather additional data. The quality of the final model is only as good as the quality of the source data and its information content. In this test program the aircraft was fully instrumented, including a test boom. Data of interest for this study included, but was not limited to, the following: • • • • • • • • • • Inertial position and velocities Euler Attitudes Body axis angular rates Body axis linear accelerations Flow angles (angle-of-attack and flank angle) Air relative speed Pressure Altitude Ambient atmospheric conditions (OAT and Static Pressure) Pilot inputs via control cable throw Control surface deflections via control cable throw IV. Dynamic Flight Test Data Pre-processing Analysis within IDEAS examined pertinent channels from each maneuver for data dropouts and/or signal wrapping. Appropriate utilities within the IDEAS Data Pre-processing And Reconstruction (DATPAR) toolbox were employed to correct such anomalies when they occurred. 1,2 Specifically, the DATPAR UNWRAP utility was used to generate continuous signals from discontinuous, wrapped, sensor outputs such as heading when necessary. In addition, the DATPAR WILD_EDIT utility was used to discard “wild points” from the flight test data when encountered. Kinematic consistency evaluations were performed on the flight test data using NAVIDNT within IDEAS. Recall that NAVIDNT couples the nonlinear least-squares algorithm LSIDNT with a set of navigation equations of motion to form an output error sensor model identification scheme. 1,3,4 The navigation equations integrate flight measured linear accelerations and angular rates to generate rigid body airframe responses. The goal of this process, of course, is to ensure all accelerations, rates, angular orientations, flow angles, velocities (air-relative and inertial), and inertial positions are consistent. Recall that NAVIDNT is used to estimate sensor biases and/or scale factors, as well as time invariant atmospheric winds, to generate a kinematically consistent dataset. LSIDNT, the optimizer, is a robust, adaptive nonlinear least-squares scheme based on the N2F family of algorithms encompassing the Newton minimization scheme with an augmented version of the Gauss-Newton approximation.4 Details on the kinematic consistency process, and algorithms within IDEAS, are outlined in detail in prior work by the authors. 3 NAVIDNT studies within IDEAS indicated the need for adjustments to be made to the locally measured flow angles and air-relative velocity collected by the installed flight test boom. Note this boom was located on the starboard wing tip. Both flow angles and velocity measurements needed to have the effect of body axis rates removed in addition to accounting for necessary calibration adjustments. Figure 2 presents an example where test boom true airspeed is transferred from its local measurement location to the actual aircraft CG. Body yaw rate clearly is seen to affect the local airspeed measurement as expected during this pedal frequency sweep as the starboard wing advances and retreats. The angle of attack measurement had to be adjusted for a boom installation angle bias as well as account for local flow up-wash. Figure 3 presents an example of local angle of attack measurement transferred to the aircraft CG and adjusted for installation bias and up-wash effects for a column frequency sweep. Figure 4 presents a similar comparison but for a wheel frequency sweep. The removal of body axis rate effects during the transfer to the CG is clear. The flank angle measurement had to be adjusted for the boom 4 American Institute of Aeronautics and Astronautics installation angle as shown in Figure 5. Boom recorded air-relative speed required a calibration adjustment. All axis doublets, frequency sweeps, and wind estimation turns were used to estimate the sensor calibration adjustments mentioned using NAVIDNT within IDEAS. The linear accelerations and inertial velocities were transferred from their local measurement location to the CG in support of model development and evaluation efforts. Overall, DATPAR tools were used to compute a variety of important histories for each maneuver including body axis air relative velocities, sign convention adjustment, units book-keeping, and the transfer of locally measured quantities to a common reference location to support the model update and validation processes. 1,2 Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 V. Total Force and Moment Reconstruction A variety of data is necessary to successfully extract total force and moment coefficients from flight data. These data consist of body axis angular rates and accelerations, body axis linear accelerations, dynamic pressure, as well as aircraft mass and inertia data. In addition, to facilitate transferring the overall moments to a specific reference point, about which the aerodynamic model is to be developed, the center of gravity location for each maneuver must be determined. Ideally, inflight thrust estimates, or those computed based on a validated propulsion deck model, would be used to remove thrust effects from the overall total coefficients. In this case flight thrust was not available so the total force and moment coefficients included both aerodynamic and thrust effects. Note during PID maneuvers the power lever is maintained constant, and small amplitude airframe responses are targeted, thus resulting in thrust contributing to an overall bias in the force and moment total coefficients. Here the slope of the coefficients with respect to regressors such as angle of attack or sideslip are of greater significance and will be due primarily to the basic airframe aerodynamic effects. Angular rate, angular acceleration, linear acceleration, and dynamic pressure data were available in the flight test data and subject to DATPAR processing as outlined in the previous section. Suitable information regarding aircraft weight and balance characteristics such as mass, CG position, and moments of inertia were readily available from a post-processing calculator developed for this aircraft making use of known starting fuel state and flight test recorded fuel flow. All PID maneuvers were flown with throttles maintained at their trim position throughout each maneuver. A variety of IDEAS DATPAR tools were employed to extract the total force and moment coefficients from the flight data.1,2 This resulted in the overall body axis X, Y, and Z forces, as well as pitching moment, rolling moment, and yawing moment coefficient histories with respect to the aircraft CG per PID maneuver. Overall, IDEAS DATPAR utilities were used to back-calculate the aircraft body axis total forces and moments given pertinent measured airframe linear and angular accelerations, airframe states, and weight and balance information. All body axis total coefficients were reconstructed assuming a rigid body aircraft. An additional IDEAS DATPAR utility was used to transfer all reconstructed body axis total moments to the desired reference point about which the aerodynamic model would be developed.1,2 This transference involves the incremental effects due to the reconstructed body axis aerodynamic forces about the aircraft CG being offset from the desired aerodynamic reference center location. VI. Output Error PID Technique Initial control law design was being performed using the baseline nonlinear simulation model of the aircraft. This included linear models extracted using LME techniques directly from the nonlinear simulation within IDEAS. The longer term goal was to update the fidelity of the full nonlinear model given the available flight test data at hand. However, in the short term it was desired to develop linear models from flight test data to provide directly to the control law developers. This would allow for next step adjustments in the control law design while the full nonlinear model was being updated and validated. Both longitudinal and lateral-directional linear models were developed based on flight test data using nonlinear least squares tools within MATLAB®. The linear model structures were written in C and compiled into a MATLAB® mex-file format to reduce computational time during the iterative output error technique. The GaussNewton nonlinear least squares optimization algorithm was used to identify the linear model parameters of interest.16 The iterative nature of the output error process is shown in Figure 6. In this analysis multiple longitudinal or lateral-directional maneuvers were binned together at a given flight condition and analyzed. This allowed for the identification of a single linear model given data from multiple test events. In this process, the flight test data are used to define initial conditions for the linear models. Recorded flight test control inputs are applied during propagation of the linear models and outputs then compared to those from the flight test event. The GaussNewton optimization algorithm works iteratively to update the model parameters to minimize this error until the final parameters are determined. 5 American Institute of Aeronautics and Astronautics The general state space model format was applied in both longitudinal and lateral-directional linear models for all flap settings for which data was available: x Ax Bu (1) y Cx Du The longitudinal models were formulated as follows: Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 u X u w Z u q M u 0 Xw Zw X Z Mw 0 u 1 w 0 q 0 0 0 a z Z u a x X U W0 g cos 0 u X E U 0 g sin 0 w Z E q q M E E Mq 0 1 0 0 q 0 0 1 0 0 1 0 1 U0 Zw 0 Zq Xw Xq 0 0 0 0 0 u 0 0 1 w 0 E 0 q 0 Z E 0 X 0 E (2) (3) The lateral-directional models were formulated as follows: v Yv p L v r N v 0 0 Y p W0 Lp Np 1 0 v 1 p 0 r 0 0 0 1 U a 0 y Yv Yr U 0 g cos 0 0 v Y A 0 0 p L A Nr 0 0 r N A tan 0 0 0 0 sec 0 0 0 0 Lr 0 0 1 0 0 1 0 0 0 0 0 0 Yp Yr 0 0 0 0 0 v 0 0 0 p 0 1 0 r 0 0 1 0 0 0 0 Y 0 0 A Y R L R A N R R 0 0 0 0 0 0 A 0 R 0 Y R (4) (5) In each flight condition, and flap configuration of interest, a group of pilot applied frequency sweeps was presented to the optimization scheme simultaneously to generate a single linear model accounting for all the data available at that condition. For example, in the CR Flaps 0 case the longitudinal model for 125 KIAS at 10k ft was developed using a binned group of four longitudinal pilot applied frequency sweeps. The lateral-directional model at the same flight condition was developed using a binned group of three wheel and two pedal frequency sweeps. Sample linear model comparison plots are shown in Figure 7 for a longitudinal frequency sweep for CR Flaps 0. The resulting linear model output is shown along with the recorded flight test data. Note the steady state values of the states, controls, and responses have been removed as these are the perturbation linear model responses. Linear model comparison to flight test data for a pedal frequency sweep at CR Flaps 0 configuration is shown in Figure 8. Sample linear model comparison plots for CR Flaps 10 are shown for a longitudinal frequency sweep and directional frequency sweep in Figure 9 and 10. In all cases the linear models perform well in emulating the aircraft 6 American Institute of Aeronautics and Astronautics flight responses. The binned frequency sweeps provide good information content for the output error model estimation process. The longitudinal linear models for CR Flaps 0 and CR Flaps 10 configuration are shown below in Table 1. The corresponding lateral-directional linear models are shown in Table 2. These models are representative of flight at 125 KIAS and 10k ft. Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Table 1. Longitudinal linear model parameter estimates using output error. Model Parameter Xu Xw Xq Xe Zu Zw Zq Ze Mu Mw Mq Me Uo Wo o Parameter Units 1/s 1/s ft/(s rad) ft/(s2 rad) 1/s 1/s ft/(s rad) ft/(s2 rad) rad/(ft s) rad/(ft s) 1/(s rad) 1/(s2 rad) ft/s ft/s rad CR Flaps 0 -0.0191 0.166 0 0 -0.265 -1.412 -8.0897 -37.236 0 -0.0541 -2.999 -11.950 239.8 14.8 0.1141 CR Flaps 10 -0.0444 0.126 0 0 -0.237 -1.518 -8.591 -37.482 0 -0.0373 -3.0751 -10.757 248.1 4.5 0.0092 Table 2. Lateral-directional linear model parameter estimates using output error. Model Parameter CR CR Parameter Units Flaps 0 Flaps 10 Yv 1/s -0.228 -0.220 Yp ft/(s rad) 0.0748 0.276 Yr ft/(s rad) 2.861 3.157 2 ft/(s rad) 0 0 Ya 2 ft/(s rad) 12.943 13.253 Yr Lv rad/(ft s) -0.0201 -0.0236 Lp 1/(s rad) -2.671 -3.691 Lr 1/(s rad) 0.283 0.355 1/(s2 rad) 2.207 2.648 La 1/(s2 rad) 0.115 0.306 Lr Nv rad/(ft s) 0.0106 0.0115 Np 1/(s rad) -0.394 -0.333 Nr 1/(s rad) -0.650 -0.621 1/(s2 rad) 0.171 0.163 Na 2 1/(s rad) -2.862 -2.990 Nr Uo ft/s 233.5 244.8 Wo ft/s 14.8 4.4 rad 0.0732 0.0157 o In this linear model structure the aileron control is an effective overall roll control in that it is modeled as total differential aileron (aL – aR). In addition, the upper wing surface roll control spoilers are geared to move with the ailerons. As a result, their power is included in the effective aileron control power model terms. 7 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 The style of maneuvers used to develop the linear models were targeted at exciting the phugoid, short period, dutch-roll, and roll modes. The natural frequencies, damping, roll mode, and spiral mode time constant were determined for the output error identified linear models. The initial full nonlinear model for this aircraft was examined using an LME facility to develop longitudinal and lateral-directional linear models at similar operating conditions and configurations as tested in flight. The same mode characteristics were determined from those models and are presented in Table 3 and Table 4 for comparison to those obtained from flight. Table 3. Comparison of linear model characteristic modes for CR Flaps 0 configuration. Mode CR Flaps 0 CR Flaps 0 Percent Output Initial Difference Error Nonlinear Model Model LME SP 4.12 6.63 60.92 (rad/s) 0.534 0.451 -15.54 SP P 0.177 0.141 -20.34 (rad/s) 0.0789 0.179 126.87 P DR 1.85 3.06 65.41 (rad/s) 0.18 0.155 -13.89 DR TR 0.351 0.220 -37.30 (s) TS 35.34 106.0 199.98 (s) Table 4. Comparison of linear model characteristic modes for CR Flaps 10 configuration. Mode CR Flaps 10 CR Flaps 10 Percent Output Initial Difference Error Model Nonlinear Model LME SP 3.69 5.98 62.06 (rad/s) 0.623 0.469 -24.72 SP P 0.154 0.151 -1.95 (rad/s) 0.148 0.168 13.51 P DR 1.87 2.91 55.61 (rad/s) 0.187 0.154 -17.65 DR TR 0.262 0.21 -19.99 (s) TS 44.74 198.0 342.53 (s) The initial non-linear model, based on first principles and empirical modeling, provided a starting point for overall analysis. Based on first look results of Table 3 and Table 4 the model clearly required updates to better match the aircraft mode characteristics. Best overall results with the original non-linear model were seen in the short period (SP) and dutch-roll damping (DR) terms. The horizontal and vertical tail lift forces play a significant role in these damping characteristics. The non-linear simulation horizontal and vertical tail planform lift curve slopes were generated using CFD as previously mentioned. These results indicate a close first estimate for these lift curve slopes. In the case of CR Flaps 0, SP was ~15% low while DR was ~ 14% low. The CR Flaps 10 configuration shows SP was ~25% low while DR was ~ 18% low. 8 American Institute of Aeronautics and Astronautics Given the linear model estimates for CR Flaps 0 of M q and Nr, a rapid update could be determined for the lift curve slope linear region of the full model horizontal and vertical tails. For example, consider the relation for pitch damping in the linear model: M q Cmq q0 Sc 2 2 I yy 0U 0 (6) This relation can be solved to determine Cmq given the known flight conditions, aircraft geometry, mass characteristics, and overall linear model Mq estimate from flight data. The non-dimensional pitch damping coefficient (Cmq) can also be defined as follows given Ref. 14: Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Cmq 2CL H HVH XH c (7) Where the horizontal tail lift curve slope can be determined as follows: CL H Cmq (8) X 2 HVH H c This relation provides a new overall estimate for the linear region of the horizontal tail planform lift curve slope that can be blended directly into the full model supporting rapid turnaround. Where: H VH qH q SH X AC H X CG S (9) X H X AC H X CG (10) (11) The dynamic pressure ratio H models any increase in dynamic pressure experienced at the horizontal tail versus free stream. In the full nonlinear simulation the term is defined given the modeled prop wake enveloping the tail section. Once overall airframe drag was adjusted to get the proper throttle setting to match flight, this value could be referenced for this particular flight condition and configuration. The dynamic pressure ratio could then be used to support calculation of a new linear lift curve slope value for the horizontal tail. Given the available flight test maneuvers in the CR Flaps 0 configuration it was determined that on overall increase in fuselage model drag of ~30% was necessary to capture the power setting matching flight. This is not unexpected as the fuselage initial aerodynamic model was not based on CFD but was built upon empirical component representative shapes to model the fuselage and strut aerodynamics. An identical technique was employed for the vertical tail planform linear region lift curve slope. The relation for yaw damping in the estimated linear model is as follows: N r Cnr q0 Sb2 2 I zz 0U 0 (12) This relation can be solved to determine Cnr given the known flight conditions, aircraft geometry, mass characteristics, and overall linear model Nr estimate from flight data. The non-dimensional yaw damping coefficient (Cnr) can also be defined as follows given Ref. 14: 9 American Institute of Aeronautics and Astronautics 2 X V2 S V V 2 b S Cnr CL V (13) Where the vertical tail lift curve slope can be determined as follows: CL V b2 S Cnr 2 X V2V SV (14) This relation provides a new overall estimate for the linear region of the vertical tail planform lift curve slope that can be blended directly into the full model supporting rapid turnaround. Where: qV q (15) XV X ACV X CG (16) Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 V Similarly, the dynamic pressure ratio V models any increase in dynamic pressure experienced at the vertical tail versus free stream. Also analyzed for rapid model update for CR Flaps 0 configuration was the elevator and rudder control power effectiveness parameters. Given the linear model structure identified, the elevator and rudder power force coefficients are defined as follows: Z E CL E Y R CY R qo S m qo S m (17) (18) The non-dimensional stability derivatives for lift due to elevator and side force due to rudder deflection are then determined as follows: CL Z E E CY Y R R m qo S m qo S (19) (20) Given the linear model estimates from flight, the known aircraft mass, geometry, and flight conditions, they can be directly calculated. The non-dimensional elevator and rudder control power coefficients can also be defined as follows given Ref. 14: CL E CL H H SH S E CY R CLV V SV S R 10 American Institute of Aeronautics and Astronautics (21) (22) The elevator and rudder effectiveness model terms can then be calculated for direct blending into the non-linear simulation model: E Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 R CL S E CL H H S H CY S R CLV V SV (23) (24) From these examples it is clear that other linear slope sections could be blended in a similar manner. However, only the tail planform lift curve slope linear sections were updated in this fashion along with the elevator and rudder control power. With this update the non-linear model overall pitch and yaw damping, along with pitch and yaw control power, would provide representative flight characteristics. The mode comparison results noted in Table 3 and 4 indicate the short period and dutch-roll frequencies of the initial nonlinear model did not compare as well as the damping. In the case of CR Flaps 0, SP was ~61% high while DR was ~ 65% high. The CR Flaps 10 configuration shows SP was ~62% high while DR was ~ 56% high. Both the damping and static coefficient terms are known to play a significant role in the mode frequencies. For example, Cmq and Cm shape the short period natural frequency together while Cnr and Cn shape the dutch-roll frequency. The aerodynamic model terms with the greatest uncertainty in the full non-linear model were those developed for the fuselage components. The best path forward was to blend in any necessary C m and Cn updates through the fuselage model itself. The overall rate damping terms would be captured by the tail planform lift slope corrections previously discussed. Any updates necessary for Cy and Clwould also be applied to the fuselage section aerodynamics. If additional updates were necessary for C L it was decided to apply those to the main wing planform. Any updates to the main wing planform lift curve slope would also effect the overall simulation roll damping as a natural fall-out which in turn would alter the roll mode time constant (T R). Main wing lift curve slope updates would also naturally alter the roll response with sideslip (Cl). Note Tables 3 and 4 indicate the non-linear simulation, for both CR Flaps 0 and CR Flaps 10, have a roll mode time constant lower than what was identified in flight by ~37% and ~20% respectively. This indicates the non-linear model overall roll damping is likely too high. In both flap configurations the initial spiral mode time constant was high by ~200% and ~343% for CR Flaps 0 and Flaps 10 respectively. Note the spiral mode root location is typically near zero on the root locus x-axis. Small changes in the root location can make a notable difference in the resulting spiral time constant. The spiral mode depends heavily on aircraft directional static stability (Cn) and dihedral effect (Cl). The prior discussion notes the dutch-roll frequency of the original model needs adjustment and, as such, it is expected the spiral mode match will improve as a direct fall-out. In summary the updates required to the non-linear simulation in this phase of analysis are shown for CR Flaps 0: Fuselage drag increase (~30%) Horizontal tail lift curve slope (CLH) Elevator control power (CLE) Vertical tail lift curve slope (CLV) Rudder control power (CYR) At this phase in the analysis the only update applied for CR Flaps 10 was an adjustment in drag due to wing flap deflection. The following section describes the equation error PID technique used to perform additional aerodynamic model updates. VII. Equation Error PID Technique The next stage of analysis involved an equation error PID technique to extract aerodynamic model terms. The equation error tool in IDEAS, known as Athena, is capable of determining the overall aerodynamic force and moment models as linear combinations of parameters (typically stability derivatives and/or incremental coefficients) using a principal component regression algorithm.1,2,9,10 This technique has been used with success in prior work by 11 American Institute of Aeronautics and Astronautics the authors.11,12 Athena allows these terms to be modeled with nonlinear dependencies as well. As an example, consider a simple pitching moment buildup: Cm Cmbasic Cme e (25) In this example the overall model structure is a nonlinear function of angle-of-attack. To model the nonlinearities, Athena supports the use of linear or cubic basis splines to estimate coefficients of the splines at the specified knot locations. Eq. (25) is rewritten as: K1 K2 i 1 i 1 Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Cm Cmbasic,i f1,i Cme,i f 2,i e (26) Cmbasic,i and Cme,i are the parameters to be estimated; K1 and K2 represent the number of angle-of-attack knot locations defined for Cmbasic and Cme respectively; and f1,i() and f2,i() are basis functions. The basis functions are defined such that each takes on a value of 1.0 at one knot location and a value of 0.0 at all other knot locations. Athena assumes the model may be represented as a set of linearly combined, time-independent parameters with the following structure: y Ap v (27) Vector y (size nx1) represents the total non-dimensional force or moment coefficient history vector under investigation. The parameter vector p (size mx1) represents the stability and control derivatives under estimation, and the regressor matrix A (size nxm) contains the independent variables. Vector v (size nx1) represents unmodeled aerodynamic responses and/or phenomena such as due to system/sensor noise. Consider the example outlined by Eq. (25) and Eq. (26) above assuming two angle-of-attack knot locations are chosen for both C mbasic and Cme. In this case, Eq. (27) becomes: C m 1 f 1,1 1 C m n f 1,1 n f 1, 2 1 f 2,1 1 e 1 f 1, 2 n f 2,1 n e n C mbasic,1 f 2, 2 1 e 1 v1 C mbasic, 2 C me ,1 f 2, 2 n e n vn C me, 2 (28) Athena uses a numerically robust singular value decomposition method to solve Eq. (27) and estimate the parameter vector p. In general, Athena first determines an incremental response vector ys by removing the prior model contributions (po): y s y - A po (29) Vector po consists of initial estimates of parameters as well as those that have been fixed and are not to be estimated. The regressor matrix A is then thinned such that it contains only those columns that correspond to parameters that are to be estimated. The thinned matrix As allows the identification statement of Eq. (27) to be reformulated as follows: ys A s p s v (30) The thinned regressor matrix As is then decomposed into the following form: A s U SV T 12 American Institute of Aeronautics and Astronautics (31) U and V are orthogonal and S is a diagonal matrix containing the singular values. Using Eq. (30) and Eq. (31) together the free parameters are then estimated in principal component axes: p s VS UT y s (32) Note that vectors po and ps are combined to yield the final parameter estimates: p po p s (33) Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Output statistics for this technique are provided in the form of a fit percentage based upon the Theil inequality coefficient statistic (U) defined as:14 1 N U 1 N N yˆ i 1 N yˆ i 1 2 i yi 2 i (34) N y 1 N i 1 2 i N is the total number of points in the residual vector. This coefficient represents the ratio of the root mean square fit error and the root mean square values of the estimated and actual signal summed together. The value of U always falls between 0 and 1, with 0 indicating a perfect fit and 1 the worst fit. The Athena fit percentage (F), a measure of signal fit quality, is defined as follows: F 1001 U (35) A 100% fit represents a perfect match with the measured data. Additionally, Athena breaks the fit error into bias (UB), variance (UV), and covariance (UC) proportions as follows:13 yˆ y 2 UB 1 N yˆi yi 2 N i1 1 yˆ y N (36) 2 UV yˆ y N i i 1 UC (37) 2 i 21 yˆ y 1 N yˆi yi 2 N i1 (38) Where and represent the correlation coefficient and standard deviation respectively. 1 y yˆ N yˆ N i 1 i yˆ yi y 13 American Institute of Aeronautics and Astronautics (39) Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 x 1 N N x i 1 i x 2 (40) The bias proportion presents the deviation of the average values of the simulated and measured data acting as a measure of model systematic error. The variance proportion acts as a measure of the model’s ability to duplicate the variability in the true system. The covariance proportion is a measure of non-systematic error (e.g. due to unmodelled random sensor/system noise). Note that these three proportions sum to 1, with the ideal fit having U B and UV close to zero, with UC close to 1. These fit statistics act as a measure of accuracy and/or certainty in the proposed model formulation under investigation and provide clues into the effectiveness, or lack thereof, of adjustments introduced in the model structure. An additional strength of this algorithm is its ability to analyze multiple segments of information, in this case PID maneuvers, at once. This allows for the extraction of global models from analysis groups of multiple PID maneuvers. Overall, this procedure is a fast, single pass algorithm that results in good base model structure determination. This technique was used to analyze groups of flight test data, spanning angle of attack for example, in order to develop both linear and nonlinear model coefficient updates. These updates were blended into the final aerodynamic model. Recall each maneuver to be evaluated in the PID process had total forces and moments backcalculated from the flight test data and presented with respect to a common aerodynamic model reference center matching that of the non-linear simulation model database. A summary of updates required to the non-linear simulation using this technique is as shown for CR Flaps 0: Main wing lift curve slope (CL) Fuselage wing-body adjustment for Cm Fuselage wing-body adjustment for Cy Fuselage wing-body adjustment for Cn Fuselage wing-body adjustment for Cl Roll control power (Cla and Cna) Updates required to the non-linear simulation for CR Flaps 10 were applied to the same terms as those above with corrections implemented as a function of flap deflection. As an example of this analysis consider the CR Flaps 0 configuration lift curve slope update. In this phase a group of maneuvers were analyzed using this equation error technique to develop model updates for the main wing lift curve slope (CL). This data was binned together and included the following maneuvers analyzed simultaneously: Four column frequency sweep events Three 1-g wings level stall events The angle-of-attack for these maneuvers collectively ranged from 1 o to 16o. The column frequency sweeps remained in the linear region of the lift curve slope while the 1-g stalls spanned from the linear into the non-linear region and stall break. The model structure considered in Athena was the following: qc CL E CL CL CLq E 2Vrw (41) Due to the flight data coverage for angle-of-attack the model knot locations for Athena regarding the function C L() were anchored at [1.0 5.0 9.0 11.0 13.0 16.0]. This instructs the algorithm to estimate six model entries for C L anchored at the defined angle-of-attack values specified. Recall the prior analysis had already updated the tail planform lift curve slopes as well as the elevator control power based on the linear model analysis. With those updates having been blended into the non-linear simulation, a sweep utility was used to determine values of C Lq and CLE from the non-linear simulation. The simulation sweep facility allows for configuring, and trimming, the aircraft at a specified flight condition. Desired variables, such as pitch rate and elevator deflection for example, can 14 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 then be individually perturbed over a specified range with the resulting change in overall aerodynamic coefficients saved out. In this case the simulation based values for C Lq and CLE could then be set constant in the Athena model. Athena was used to estimate the parameters for C L(). Lift coefficient model matches versus flight test data are shown in Figure 11 for a sample column frequency sweep and 1-g stall maneuver. The angle-of-attack from flight is also presented for each test case for reference. The resulting model indicates a good match against flight test data with all seven maneuvers binned together resulting in an overall Theil fit percentage of 97.7% with U B = 0, UV = 0.004, and UC = 0.996. A comparison of the Athena identified lift curve slope with that obtained from the nonlinear model using the simulation sweep facility is shown in Figure 12. Overall it presents a close match below 9 o with the simulation slope being slightly higher. Above 9 o the flight identified lift curve slope begins to develop a non-linear, slope reducing, trend while the initial simulation predicted higher lift. Comparisons such as this were used to re-shape simulation database coefficient characteristics such that they fair through the trends identified from flight data. This equation error technique was used to analyze the other parameters noted above with the same blending technique employed to update the final model for both CR Flaps 0 and Flaps 10 configurations. VIII. Simulation Validation Having finalized and blended the aerodynamic model updates into the non-linear simulation a new set of linear models were extracted. The simulation was configured to match the flight conditions of the linear models identified previously from flight. The LME facility was then used again to generate corresponding linear models from the simulation. The aerodynamic model characteristic comparisons are shown below in Table 5 and 6 for CR Flaps 0 and CR Flaps 10 respectively. Table 5. Comparison of linear model characteristic modes for CR Flaps 0 after simulation update. Mode CR Flaps 0 CR Flaps 0 Percent Output Updated Difference Error Nonlinear Model Model LME SP 4.12 4.42 7.28 (rad/s) 0.534 0.54 1.12 SP P 0.177 0.187 5.65 (rad/s) 0.0789 0.0965 22.31 P DR 1.85 1.85 0.00 (rad/s) 0.18 0.217 20.56 DR TR 0.351 0.308 -12.25 (s) TS 35.34 26.7 -24.45 (s) 15 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Table 6. Comparison of linear model characteristic modes for CR Flaps 10 after simulation update. Mode CR Flaps 10 CR Flaps 10 Percent Output Updated Difference Error Model Nonlinear Model LME SP 3.69 4.01 8.67 (rad/s) 0.623 0.628 0.80 SP P 0.154 0.148 -3.90 (rad/s) 0.148 0.0968 -34.59 P DR 1.87 1.76 -5.88 (rad/s) 0.187 0.232 24.06 DR TR 0.262 0.267 1.91 (s) TS 44.74 45.3 1.25 (s) Overall, the non-linear simulation mode characteristics show significant improvement following the PID model updates. The initial non-linear simulation mode characteristics can be referenced in Table 3 and 4 for comparison. Note the spiral mode time constants show vast improvement. In addition, the roll mode time constant has improved. Another test of model fidelity entails the use of simulation propagation with outputs compared to flight data. Flight test measurements for pilot column, wheel, pedal, and throttle inputs were used to override pilot control inputs in the simulation. The atmospheric model within the simulation was initialized to flight test conditions. Aircraft weight and balance information were initialized to data describing each flight test event examined. Prior to , p , q, r ) in the test propagation, the simulation was trimmed to match any non-steady initial conditions ( u, v, w data. In addition, the simulation was initialized with the appropriate airspeed, altitude, body axis angular rates, and Euler attitudes as determined from the flight data. In support of validation efforts a set of tolerance bounds were selected based upon information available in FAA Advisory Circular 120-40B.15 Bounds are applied to the flight test data to display a region of acceptable match in the validation plots. A simulation output propagation compared against flight test data can be seen in Figure 11. This maneuver is a piloted longitudinal frequency sweep at CR Flaps 0, 125 KIAS, 10k ft. The flight test data was used to create the bounds shown after applying tolerances from FAA AC-120-40B. Figure 11 presents a set of output histories for this maneuver that indicate an accurate match in the longitudinal axis regarding both high and low frequency portions of the event. The elevator control power, pitch damping, and region of short period mode natural frequency excitation followed by the response attenuation are well emulated by the model for this frequency sweep. Figure 14 presents a sample roll rate validation match for a lateral frequency sweep at CR Flaps 0, 125 KIAS, 10k ft. Overall the model predicts the roll rate response well across the frequency ranges covered by this maneuver. Roll axis control power and subsequent roll damping are captured by the model. Figure 15 presents a sample yaw rate validation match for a directional frequency sweep at CR Flaps 0, 125 KIAS, 10k ft. Similarly, the updated model predicts the yaw rate response well across the frequency ranges covered by this maneuver. The yaw axis control power, and subsequent rate damping, are captured by the model. In addition, the model captures the excitation of the dutch-roll mode similarly to that of the flight test aircraft. Yaw rate response attenuation as the pedal input frequency increases beyond the dutch-roll excitation frequency is also emulated well by the model. IX. Conclusion An initial aerodynamic non-linear model of a Cessna Grand Caravan (C208B) aircraft was developed to support engineering analysis and flight test studies. A flight test program was developed to collect the required data to support the update, and subsequent validation, of the model. Both linear and non-linear model extraction techniques were successfully applied to provide a more accurate flight model. The resulting product was instrumental in 16 American Institute of Aeronautics and Astronautics control system design, engineering analysis, off-nominal condition studies, and flight test preparation. Further detail regarding the flight test build-up and final system demonstration for this aircraft can be found in Ref. 17. References 1 Linse, D. J., "Aircraft System Identification Using Integrated Software Tools", RTA-SCI Symposium, Madrid, Spain, May 1998. 2 Linse, D. J., "Improving Simulator Accuracy With Integrated Analysis of Flight Data", Interservice/Industry Training, Simulation, and Education Conference, I/ITSEC Paper EC-046, November 2000. 3 Paris, A. C., and Alaverdi, O., "Post-Flight Inertial And Air-Data Sensor Calibration", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 98-4450, Boston 1998. Dennis, J. E. Jr., Gay, D. M., and Welsch, R. E., “An Adaptive Nonlinear Least-Squares Algorithm,” ACM Transactions on Mathematical Software, Vol. 7, No. 3, September 1981, pp 348-368. Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 4 5 Cessna Aircraft Company, “Substantiation and Critical Loads Summary Report S-208B-33”, Jan 2006. 6 McCormick, B. W., "Aerodynamics, Aeronautics, and Flight Mechanics", 1979. 7 Perkins, C. D., Hage, R. E., “Airplane Performance Stability and Control”, 1949. 8 McCormick, B. W., "Aerodynamics of VSTOL Flight", Academic Press, 1967. 9 Anderson, Laurence, C., "Robust Parameter Identification for Nonlinear Systems Using a Principal Components Regression Algorithm", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 85-1766, August 1985. Linse, D. J., “System Identification Software Design Document and User’s Manual for the Integrated Data Evaluation and Analysis System (IDEAS)”, Science Applications International Corporation, SAIC Report No. 011393-2990-A005/A006, California, MD, November 1997. 10 11 Paris, A. C., and Alaverdi, O., "Nonlinear Aerodynamic Model Extraction From Flight Test Data for the S-3B Viking", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2001-4015, Montreal, Canada 2001. 12 Paris, A. C., and Bonner, M., "Nonlinear Model Development From Flight Test Data for the F/A-18E Super Hornet", AIAA Atmospheric Flight Mechanics Conference, AIAA Paper 2003-5535, Austin, Texas 2003. 13 Pindyck, R. S., and Rubinfeld, D. L., "Econometric Models and Economic Forecasts", 3 rd Edition, McGraw-Hill, Inc., 1991, pp 336-342. Roskam, J., "Airplane Flight Dynamics and Automatic Flight Controls – Part I", Roskam Aviation and Engineering Corporation, Ottawa, Kansas, 1982, pp 195-197, pp 203 - 205. 14 15 Federal Aviation Administration, "Airplane Simulator Qualification", U.S. Department of Transportation, AC-12040B, July 1991. Maine, R.E., Iliff, K.W., "Application of Parameter Estimation to Aircraft Stability and Control – The Output Error Approach", NASA-RP-1168, 1986. 16 17 Haider, M., Alaverdi, O., "Manned but Unpiloted: A Discussion of the Flight Testing Leading to a Demonstration of the Caravan Optionally Piloted Aircraft (COPA)", Society of Experimental Test Pilots, 2008. 17 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Flight Test Instrumentation Correction Onboard or Telemetry Data Kinematic Consistency and Sensor Calibration System Identification Model Structure Determination and Parameter Estimation Model Update and Validation Simulation Update and Validation Analysis New Simulation Model Supporting Engineering Analysis, Flight Control Design, Training, and Testing Data Problems Insufficient Identification Data Insufficient Validation Data Figure 1. Overview of the flight test process supporting System Identification based model updates and final model validation. Figure 2. Body axis rate effects removed during transfer of test boom local true airspeed to actual aircraft CG location for a pedal frequency sweep. Figure 4. Local angle of attack measurement transferred to the aircraft CG and adjusted for installation bias and up-wash effects for a wheel frequency sweep. Figure 5. Flank angle measurement adjusted for boom installation angle as shown in this pedal frequency sweep. Initial Parameter Estimates Flight Test Database Binned Flight Test Events Similar Flight Condition & IC’s Linear Model Structure y Updated Parameter Estimates + Figure 3. Local angle of attack measurement transferred to the aircraft CG and adjusted for installation bias and up-wash effects for a column frequency sweep. - Nonlinear Least Squares Optimizer Figure 6. Output error parameter identification technique. 18 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 7. Longitudinal frequency sweep linear model response comparison against flight test data for CR Flaps 0, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output] 19 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 8. Directional frequency sweep linear model response comparison against flight test data for CR Flaps 0, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output] 20 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 9. Longitudinal frequency sweep linear model response comparison against flight test data for CR Flaps 10, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output] 21 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 10. Directional frequency sweep linear model response comparison against flight test data for CR Flaps 10, 125 KIAS, 10k ft. [Solid Blue Line = Flight Data, Red Dash Line = Model Output] 22 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Column Frequency Sweep 1-g Stall Figure 11. Equation error CL model response comparison against flight test data and corresponding angleof-attack envelope for CR Flaps 10, 125 KIAS, 10k ft maneuvers. [Solid Blue Line = Flight Data, Red Dash Line = Model Output] Figure 12. Equation error lift coefficient match from flight compared to initial simulation database for the CR Flaps 0 configuration. 23 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 13. Nonlinear simulation output compared to flight test data for a piloted longitudinal frequency sweep at 125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied. 24 American Institute of Aeronautics and Astronautics Downloaded by Alfonso Paris on January 5, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-1666 Figure 14. Nonlinear simulation output compared to flight test data for a piloted lateral frequency sweep at 125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied. Figure 15. Nonlinear simulation output compared to flight test data for a piloted directional frequency sweep at 125 KIAS, 10k ft, in the CR Flaps 0 configuration. FAA AC-120-40B tolerance bounds applied. 25 American Institute of Aeronautics and Astronautics