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High Fidelity Mathematical Modeling of the
DA42 L360
PREPARED BY: Aditya, Ron
Contents
1. INTRODUCTION ..........................................................................................................................................1
1.1 General Parameters .....................................................................................................................................2 1.2 Wing Parameters .........................................................................................................................................2 1.3 Control Surface Parameters .........................................................................................................................3 1.4 Engine Thrust Vector ...................................................................................................................................3 1.5 Aerodynamic Center Approximation ............................................................................................................3 1.6 Aircraft Sign Convention ..............................................................................................................................5
2. LONGITUDINAL MODELING ......................................................................................................................6
2.1 Introduction ................................................................................................................................................6 2.2 Method .........................................................................................................................................................7 2.3 Model Equations ........................................................................................................................................9 2.4 Results ........................................................................................................................................................9
2.4.1 Gear Up Flaps Up ................................................................................................................................ 11 2.4.2 Gear Up Flaps Approach .................................................................................................................... 15 2.4.3 Gear Down Flaps Up ........................................................................................................................... 19 2.4.4 Gear Down Flaps Approach ............................................................................................................... 23 2.4.5 Gear Down Flaps Landing .................................................................................................................. 27
3. LATERAL MODELING ............................................................................................................................. 31
3.1 Introduction ................................................................................................................................................ 31 3.2 Method ...................................................................................................................................................... 31 3.3 Model Equations ....................................................................................................................................... 31 3.4 Results ...................................................................................................................................................... 31
3.4.1 Gear Up Flaps Up ................................................................................................................................. 33 3.4.2 Gear Up Flaps Approach ...................................................................................................................... 38 3.4.3 Gear Down Flaps Up ............................................................................................................................. 42 3.4.4 Gear Down Flaps Approach .................................................................................................................. 47 3.4.5 Gear Down Flaps Landing .................................................................................................................... 52
4. STATIC DERIVATIVE DETERMINATION ............................................................................................... 57
4.1 Method ...................................................................................................................................................... 57 4.2 Coefficient of Lift ....................................................................................................................................... 58 4.3 Coefficient of Drag .................................................................................................................................... 59 4.4 Coefficient of Pitching Moment ................................................................................................................. 60
5. NON-LINEAR AERODYNAMICS ............................................................................................................. 61
5.1 Coefficient of Lift ....................................................................................................................................... 61 5.2 Coefficient of Drag .................................................................................................................................... 64 5.3 Coefficient of Pitching Moment ................................................................................................................. 66
6. VALIDATION ............................................................................................................................................ 68
6.1 Method ...................................................................................................................................................... 68 6.2 Short Period Pitching Oscillations Validation ............................................................................................ 70
6.2.1 Gear Up Flaps Up ................................................................................................................................. 70 6.2.2 Gear Up Flaps Approach ...................................................................................................................... 71
6.3 Dutch Roll Validation ................................................................................................................................. 72 6.3.1 Gear Up Flaps Up ................................................................................................................................. 72 6.3.2 Gear Down Flaps Approach .................................................................................................................. 73
PAGE iii of viii
6.3.3 Gear Down Flaps Landing .................................................................................................................... 74 6.4 Stall ........................................................................................................................................................... 75
7. AIR DATA SYSTEMS AND FILTERING .................................................................................................. 76
8. SOURCES CITED: ................................................................................................................................... 79
9. APPENDICES ........................................................................................................................................... 80
Appendix A: Data Channel Definitions .................................................................................................................... 80 Appendix B: (PID of Longitudinal Coefficients) ....................................................................................................... 88
9.1.1 Linear Regression Method .................................................................................................................... 88 9.1.2 Output Error Method ............................................................................................................................. 89
Appendix C: (Navion 6 DOF Hand Trimming Code): .............................................................................................. 90
1
1. INTRODUCTION
Just as one can characterize a particular chemical by the constituent ingredients, one can also characterize
an aircraft’s ‘feel’ by its stability derivatives. Stability derivatives define how an aircraft responds to a stabilizing
or destabilizing perturbation which may include wind shear, control surface input, and angle of attack, just to
name a few. Within the last few decades, quantum leaps in the ability to obtain high-accuracy estimations of
these stability derivatives have become possible primarily due to increases in computing power,
instrumentation sensitivity and collective knowledge of the tools and techniques required. Classical, time-
consuming parameter identification techniques such as measurements of free oscillations and steady-state
error can be supplemented by time-effective mathematical modeling using computers. The rising cost of a
robust flight test program only further warrants the need to obtain high-fidelity aircraft mathematical modeling.
Parameter identification (PID) is the process of determining the stability and control derivatives from time-
history flight test data and comparing the response of a simulated math model to the measured aircraft
response.
The goal of this text is to provide a validated Simulink mathematical model of the Diamond DA42 L360 aircraft
and to present all results in tabulated and graphical forms. Stability derivatives of the following aircraft
configurations were obtained in order to establish a well-rounded mathematical model:
I. Flaps Retracted, Gear Up
II. Flaps Approach, Gear Up
III. Flaps Retracted, Gear Down
IV. Flaps Approach, Gear Down
V. Flaps Landing, Gear Down
The outputs of the mathematical model will quantitatively illustrate the impact of landing gear position and flap
deployment on the aircraft’s stability derivatives. Logic diagrams and lookup tables can then be inputted into
the Simulink model to alter aircraft configuration in flight. A standard computer joystick will interface with the
model to allow the user to fly the aircraft in a flight simulation program (FlightGear) in real-time.
Geometric data of the aircraft was obtained from released manufacturer technical publications and real flight
test data was used as inputs to a dynamic Simulink model that simulates aircraft flight characteristics as a
result of pilot input, aircraft configuration and aircraft states. Chapter 1 of this report provides basic DA42 L360
data that was used in the Simulink model. A figure is provided to illustrate the calculation of the Mean
Aerodynamic Center (MAC) and the engine thrust application vector.
3
1.3 Control Surface Parameters
TABLE 4: CONTROL SURFACE PARAMETERS
1.4 Engine Thrust Vector
TABLE 5: ENGINE THRUST VECTOR TABLE 6: THRUST APPLICATION VECTOR
1.5 Aerodynamic Center Approximation
The aerodynamic center of the DA42 L360 was geometrically determined using scaled drawings of the aircraft from
the manufacturer’s technical publications. The lengths of the tip chord and root chord were added to the wing root
and tip, respectively, to find the location of the mean aerodynamic chord (MAC). The aerodynamic center (AC) was
approximated to be 25% of the MAC, and, assuming a perfectly symmetrical structure, the aerodynamic center (AC)
was projected onto BL0. The location of the aerodynamic center was validated by ensuring it falls between the
manufacturer’s suggested forward and aft CG limits. The AC location was further confirmed during model validation
in chapter 6. Figure 1 below provides a high-level overview of the AC and CG locations:
Surface Area, Total (ft2) Deflection (°)
Aileron 7.1 TE up 25º, ± 2º TE down 15º,+ 2º/- 0 º
Elevator 7.1 TE up 15.5º, ± 0.5º TE down 13º, ± 1º
Elevator Trim Tab Included in
elevator
+ 17º, ± 5º (Nose up at elevator 10°
up)
- 35º, ± 5º (Nose down at elevator 10° up)
Rudder 8.4 Left 27º, ± 1º Right 29º, ± 1º
Rudder Trim Tab Included in
rudder + 34º, ± 5º
(Trim RH at rudder 20° LH) + 18º, ± 5º
(Trim LH at rudder 20° LH)
Flaps, Cruise
23.4
0°, + 2°/- 0°
Flaps, Approach 20º, + 4º/- 2°
Flaps, Landing 42º, +3º/- 1º
Left Engine ��� = (. ���, . ���,�)
Right Engine ��� = (.999, .044,0)
Left Engine ��� = (��. ��, ��.��,��. ��)��
Right Engine ��� = (53.92, 64.76, 10.06)��
6
2. LONGITUDINAL MODELING
2.1 Introduction
Longitudinal modeling of the DA42 aircraft was performed by analytically solving for the longitudinal stability
derivatives. Several large files of raw DA42 flight test data were obtained with 227 columns of data each
representing a data channel that corresponds to a specific parameter. An interface control document (ICD) is
provided in Appendix A: Data Channel Definitions for the reader to track the mapping of data channels within
the flight test data. It should be noted that each flight test data file represents a specific aircraft configuration,
i.e. flaps down gear up, flaps approach gear up, etc.
Perhaps one of the most important requirements for obtaining correct stability derivatives is to prevent the
state variables from tracking one another. In real life, angle of attack and angle of sideslip track well when the
flight path angle is close to zero. According to the Advisory Group for Aerospace Research & Development:
“The […] investigations have shown that an accurate identification of the stability and
control derivatives is guaranteed only if the aircraft is excited by input signals which fulfill
certain frequency requirements.”
Parameter Identification [3-10]
Tracking can be limited by stimulating the natural frequencies of the aircraft (phugoid and short period for
longitudinal motion) via control surface input. In lieu of experimentally determining the exact natural frequency
of the particular aircraft and precisely inputting controls to stimulate it, a 3-2-1-1 maneuver can be
implemented. The 3-2-1-1- maneuver consists of a stick input of three counts, followed by an opposite input
of two counts and two one-count inputs in the same fashion. A step, doublet or chirp input could have been
used to perform the maneuvers but the probability of stimulating ϣ�decreases. Preliminary examination of the
flight test data concluded that longitudinal 3-2-1-1 maneuvers were performed at various angles of attack
(AOA) once the aircraft was in a trimmed state. Either through manual pilot input or automated control systems,
performing 3-2-1-1 maneuvers provides a good compromise of practicality and precision. AGARD also
advocates the 3-2-1-1 maneuver for parameter identification:
“As the step contains energy only at lower frequencies it cannot excite the higher frequency
natural modes. On the other hand, the aircraft rapidly departs from the linear flight regime
when excited by the step. For these reasons the step is unfit for parameter identification.
The doublet excites a particular band at a higher frequency. By the choice of Δt the peak
of the power spectral density can be shifted to the range of the higher frequency natural
oscillations. However, the natural frequencies are not known exactly since they are
calculated from the a-priori-values of the derivatives. On the other hand they may vary due
to a change of the flight conditions. As the doublet is a relatively narrow-band signal it may
happen that the natural modes are not excited effectively […] these difficulties do not occur
when the aircraft is excited by a wide-band signal like the "3211"-input.”
Parameter Identification [3-3]
8
modeling and the residuals are provided below to illustrate the results obtained using the equation error solver
with only the most significant regressors:
FIGURE 4: CL∝ MATCH MODELING FIGURE 5: CL∝ RESIDUALS
Although CL∝ residuals do not reflect pure instrumentation noise towards the end of the data capture, adding
additional regressors did not improve the equation error results; large errors and expanded confidence
intervals were realized by doing so. Removing ��as a regressor of ��∝decreased the total error of all
regressors from 14.2% to 2.7%. Plots of the match modeling and residuals of ��∝with �� removed are provided
below:
FIGURE 6: CL∝ MATCH MODELING (REDUCED) FIGURE 7: CL∝ RESIDUALS (REDUCED)
The non-dimensional ��∝ coefficient changed from 4.861 to 5.320 by removing �� as a regressor, however, the
coefficient of ��� differed by only .002. ��∝ residuals more closely tracked the state response by removing ��
as a regressor: figures 8 and 9 illustrate the high residuals slightly before locations of high amplitude.
Regressors were individually added and removed within each configuration in effort to decrease error and
narrow the confidence interval of the coefficients. Once appropriate regressors were chosen and each
23
2.4.4 Gear Down Flaps Approach
The PID solution of the gear down flaps approach configuration appears similar to the gear down flaps up
configuration, however, the residuals plotted in FIGURE 39 show correlation to the CM matching. Derivative
estimation was found to improve by using the output error method.
FIGURE 38: CM MATCHING, EQUATION ERROR
FIGURE 39: CM RESIDUALS, EQUATION ERROR
FIGURE 40: CM MATCHING, OUTPUT ERROR
FIGURE 41: CM RESIDUALS, OUTPUT ERROR
31
3. LATERAL MODELING
3.1 Introduction
As was done in chapter 2 to determine the DA42 longitudinal stability derivatives, the lateral derivatives were
analytically determined using the same methods. Because each flight test data file corresponding to a specific
airplane configuration contained a longitudinal maneuver followed by a lateral maneuver, the lateral data was
cut using Matlab scripts. This method proved useful for group coordination to ensure the same data was being
analyzed each time the SIDPAC GUI was run. The succeeding sections of chapter 3 will highlight the method,
the reasoning justifying the method, and the ultimate results of performing lateral parameter identification on
the DA42 aircraft.
3.2 Method
The lateral stability derivatives were found using the same methods as were used for the longitudinal
derivatives. Raw flight test data files were loaded into Matlab and the SIDPAC toolbox was used to trim
individual lateral maneuvers performed at varying angles of attack. The derivatives were initially solved using
the equation error method and regressors were selected by comparing and contrasting the results obtained
from each iteration. In addition to equation error, the output error method was also used to obtain the lateral
stability derivatives. Although similar results were obtained, the math model ultimately received and validated
the output error PID results in accordance with the project requirements.
3.3 Model Equations
Whether the equation error or output error method is used within SIDPAC, only the individual stability
derivatives are provided; the stability coefficients must be manually calculated using standard equations. The
following equations were solved within the Simulink model to calculate the respective lateral stability
coefficients:
�� = ��� + ���� + �����
2�+ ����
�
2�+ ������
�� = ��� + ���� + �����
2�+ ����
�
2�+ ������
�� = ��� + ����
3.4 Results
The output error method provided exceptional matching of lateral stability derivatives as compared to equation
error. Matching and residuals plots are included below for each configuration to demonstrate the accuracy of
the solver. For the gear down flaps landing configuration, the average percent error of the lateral derivatives
using output error was roughly 9.5% whereas the average percent error of the longitudinal derivatives is 7.28%.
The increased average error of the lateral derivatives can be attributed to the greater number of lateral
42
3.4.3 Gear Down Flaps Up
Figures 90 and 91 represent �� matching and residuals using the equation error solver. The residuals represent
the derivative response, thus, the matching is insufficient. Figures 92 and 93 represent �� matching and
residuals of the same maneuver using the output error technique. It is evident that substantially better results
are obtained using the output error technique.
FIGURE 89: CL MATCHING, EQUATION ERROR
FIGURE 90: CL RESIDUALS, EQUATION ERROR
FIGURE 91: LATERAL MATCHING, OUTPUT ERROR
FIGURE 92: LATERAL RESIDUALS, OUTPUT ERROR
Table 31 provides the raw PID derivative values at various AOAs using the SIDPAC toolbox. Although the
values are naturally in the body reference frame, they were not converted to the wind reference frame as the
maneuvers were performed roughly at 0 degrees of sideslip and small angle approximation is acceptable.
57
4. STATIC DERIVATIVE DETERMINATION
4.1 Method
The coefficients for lift, drag, and pitching moment have been determined for each flap and gear configuration
for the DA42 aircraft. Most static coefficients have been provided directly from the flight test data set; however,
the lift, drag, and pitching moment variables presented here have been calculated based on other
supplementary information. Only the linear portions of the data are presented here. All data corresponding to
post stall can be found in Chapter 5. The following equations define the coefficients of lift and drag:
�� = C� cos(�) + ��sin(�)
�� = | �� sin(�) ��cos(�)|
Where CZ and CX are defined as:
�� = �
�����
�� =�� + ��2�����
Where T1 and T2 correspond to the average thrust levels of engines 1 and 2, respectively. CX and CZ can
also be extracted directly from the flight test data, however equating these static coefficients from flight
conditions was shown to produce more reasonable results for lift and drag. Pitching moment curves were
determined from CMα and CM0 values implemented in the DA42 model for each alpha. The following equation
details pitching moment:
�� = ���� + ���
Most flight test data is available for limited angles of attack. CL, CD, and CM were fit using these limited angles
and then extrapolated to the maximum angle of attack. In some cases, small modifications were made near
αmax to incorporate the nonlinear portion of each coefficient. The following figures show the lift, drag, and
pitching moment curves for each evaluated configuration as a function of angle of attack.
59
4.3 Coefficient of Drag
To extrapolate the remaining drag data from the limited flight test data, a parabola was fit to the known data
and the remaining linear regime was appended to create the full drag bucket. Figure 139 shows the drag
curves for each configuration vs. angle of attack. There are some nonlinearities that exist in the drag model
that occur around CDmin. These have been captured here.
FIGURE 138: LINEAR REGIME DRAG COEFFICIENT VS. ANGLE OF ATTACK FOR ALL CONFIGURATIONS
Drag data for the pre-stall portion of the flight regime have been tabulated and can be found in Table 38. The
data here only represents the drag curve up to the maximum angle of attack. This table is an abbreviation of
the final drag curves and may not contain all nonlinearities shown in the figure above.
TABLE 36: ABBREVIATED LOOKUP TABLE FOR LINEAR CD FOR ALL CONFIGURATIONS
Alpha Gear Up, Flaps
Up Gear Up, Flaps
Approach Gear Down,
Flaps Up
Gear Down, Flaps
Approach
Gear Down, Flaps Landing
-4 0.0611 0.0851 0.1425 0.0887 0.1195
-2 0.0284 0.0505 0.0700 0.0866 0.0927
0 0.0119 0.0321 0.0249 0.0862 0.0659
2 0.0117 0.0300 0.0071 0.0866 0.0391
4 0.0277 0.0441 0.0166 0.0887 0.0300
6 0.0599 0.0744 0.0533 0.0952 0.0784
8 0.1083 0.1210 0.1174 0.1107 0.1425
10 0.1730 0.1837 0.2087 0.1413 0.2338
61
5. NON-LINEAR AERODYNAMICS
Non-linear aerodynamic behavior has been determined for the Diamond DA-42 aircraft from flight test data for
the flaps up, gear up configuration. The provided flight test data for the stall regime for this configuration was
presented up to an angle of attack of 18 degrees. The data for each coefficient was fit to the end of the each
linear range for other configurations based on various factors including engineering judgment. Coefficient and
lift, drag, and pitching moment have been determined based on this nonlinear flight test data and the linear
ranges determined by static analysis.
5.1 Coefficient of Lift
The raw flight test data for the coefficient of lift was fitted to a third order polynomial using engineering judgment
and educated knowledge of general aviation stall curves; this includes accounting for low Reynold’s numbers
effects. In a low Reynold’s number aircraft such as a DA42, the stall break in lift will be gradual and the loss
of lift post stall will slowly increase. Hysteresis effects were also considered when choosing the manner in
which to fit the raw lift data. Figure 141 presents the raw stall flight test data overlaid with the fit stall data
curve.
FIGURE 140: RAW CL STALL DATA AND FIT CL CURVE
The determined stall data was then plotted against the original PID data for the gear up, flaps up configuration
to check for validity. The stall data joined cleanly with the original PID data indicating the fit was suitable for
further investigation and validation. Figure 142 shows the correlation between the chosen stall data curve and
the original flaps up gear up PID data
(deg)
4 6 8 10 12 14 16 18 200.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5CL vs.
Raw Stall Data
Reduced Stall Data
68
6. VALIDATION
Validation is a necessary step to quantitatively verify the simulation behaves like the physical aircraft it
represents. Because aircraft simulators are often used in a training environment, their accuracy is
fundamentally paramount. In addition, parameter identification can be used to facilitate the flight test process
and reduce the associated risk. The FAA provides general guidelines within Title 14, Part 60 to ensure
accuracy of these simulators is maintained. This chapter is devoted to demonstrating the DA42 L360 simulator
meets all requirements set forth by the instructor by visually confirming the aircraft responds properly to a
known input. Further validation will be demonstrated via manually flying the aircraft in FlightGear with a joystick
and observing the aircraft response. Table 44 below provides a high level overview of the Part 60 requirements
for simulator validation with respect to natural frequency response:
Part 60 Requirement
Natural Frequency
Numerical Requirement Flight Profile
Validation Requirement
2.c.9.a. Phugoid
Dynamics
±10% period, ±10% of time to 1/2 or double amplitude or ±.02
of damping ratio Cruise
The test must include whichever is less of the following: Three full cycles (six overshoots after the
input is completed), or the number of cycles sufficient to
determine time to 1/2 or double amplitude. CCA: Test in Non-
normal control state
2.c.9.b. Phugoid Dynamics
±10% period, Representative damping
Cruise
The test must include whichever is less of the following: Three full cycles (six overshoots after the
input is completed), or the number of cycles sufficient to
determine representative damping. CCA: Test in Non-
normal control state
2.c.10. Short Period
Dynamics
±1.5° pitch angle or ±2°/sec pitch rate, ±0.10g acceleration.
Cruise CCA: Test in Non-normal control
state.
2.d.7.
Dutch Roll, (Yaw
Damper OFF)
±0.5 sec or ±10% of period, ±10% of time to 1/2 or double amplitude or ±.02 of damping ratio. ±20% or ±1 sec of time difference between peaks of
bank and sideslip
Cruise, and Approach or Landing
Record results for at least 6 complete cycles with stability
augmentation OFF CCA: Test in Non-normal control
state.
TABLE 42: PART 60 VALIDATION PARAMETERS FOR PHUGOID, SHORT PERIOD & DUTCH ROLL
6.1 Method
Unfiltered validation data of maneuvers stimulating the dutch roll and short period frequencies were inputted
into the validation wrapper of the Simulink model. A second, identical set of this data was used as the base
model input to which the aircraft was trimmed using the JJ trim and JJ lin trimming functions. A +/- 5% upper
and lower limit was applied to the lateral and longitudinal states to visually display the allowable tolerance
70
6.2 Short Period Pitching Oscillations Validation
6.2.1 Gear Up Flaps Up
FIGURE 151: ELEVATOR INPUT TO MODEL FROM FLIGHT TEST DATA
FIGURE 152: ALPHA RESPONSE WITH BOUNDS
FIGURE 153: PITCH RATE RESPONSE WITH BOUNDS
FIGURE 154: THETA RESPONSE WITH BOUNDS
74
6.3.3 Gear Down Flaps Landing
FIGURE 167: RUDDER INPUT TO MODEL FROM FLIGHT TEST DATA
FIGURE 168: SIDESLIP RESPONSE WITH BOUNDS
FIGURE 169: ROLL RATE RESPONSE WITH BOUNDS
FIGURE 170: YAW RATE RESPONSE WITH BOUNDS
76
7. AIR DATA SYSTEMS AND FILTERING The filtering of data is used to simplify or correct the results produced from sensors. Often times, the data has
inaccuracies, or ‘noise’, that can throw off the measurement of a variable. In aircraft, spacecraft, and missile
design, a very common filtering method is the Kalman filter. It recursively uses streams of noisy/inaccurate
measurement data to produce an optimal estimate of the core system state. The Kalman filter is especially
useful for aerospace applications as it can run in real time using only the present input measurements and the
previously calculated state and its uncertainty matrix; no additional past information is required.
Using these fundamental ideas, the Kalman filter was applied to a maneuver from the DA42L’s approach (gear
down, flaps approach) flight test data in order to find the moment coefficient (CM) stability derivatives in real
time. The CM longitudinal model equation used is shown below:
�� = ��� + ���� + ����
�� + ������
The CM value, calculated using the moment equation for a rigid aircraft, is used as the measurement input.
Even with a high noise variance on 0.1 (mean at 0) in the measured values the Kalman Filter produces an
excellent estimate.
FIGURE 171: CM DERIVATIVES ESTIMATES WITH +/-20% BOUND OF THE ORIGINAL DERIVATIVE
Another important part of a Kalman filter is the variance or uncertainty in the predicted result. Because an initial
covariance is applied to the input measurements, the state variable results are initially very inaccurate. As the
88
Appendix B: (PID of Longitudinal Coefficients)
9.1.1 Linear Regression Method
Accepted Form: Linear Regression Result:
�� = ���(�) + ��� �� = 5.064(�) .742
�� = ���(�) + ����(��) + ��� ���̅
2�� +��� �� = .5058(�) + .5951(��) .00211 ��
�̅
2�� .05891
�� = ���(�) + ����(��) + ��� ���̅
2�� +��� �� = .0866(�) + .00616(��) .00225 ��
�̅
2�� + .7169
�� = ���(�) + ����(��) +��� �� = .0105(�) .00167(��) + .1287
�� = ����(��) + ��� �� = .0011(��) .1726
��:
PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX
--------- -------- --------- ------- ------------------------ ----- P( 1 ) -5.064E+00 5.296E-02 1.0 [ -5.170 , -4.959 ] 1 P( 2 ) -7.420E-01 2.751E-03 0.4 [ -0.748 , -0.737 ] 10
��:
PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX
--------- -------- --------- ------- ------------------------ -----
P( 1 ) -5.058E-001 7.369E-003 1.5 [ -0.521 , -0.491 ] 1
P( 2 ) 5.951E-001 1.246E-002 2.1 [ 0.570 , 0.620 ] 10
P( 3 ) -2.108E-003 7.866E-005 3.7 [ -0.002 , -0.002 ] 100
P( 4 ) -5.891E-002 9.958E-004 1.7 [ -0.061 , -0.057 ] 1000
��:
PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX
--------- -------- --------- ------- ------------------------ ----- P( 1 ) 8.655E-02 1.166E-03 1.3 [ 0.084 , 0.089 ] 1 P( 2 ) 6.159E-03 2.100E-03 34.1 [ 0.002 , 0.010 ] 10 P( 3 ) -2.252E-03 7.594E-04 33.7 [ -0.004 , -0.001 ] 100 P( 4 ) 7.169E-01 9.545E-03 1.3 [ 0.698 , 0.736 ] 1000
90
Appendix C: (Navion 6 DOF Hand Trimming Code):
clear all clc %Initial attitude initbank = 0; initpitch= .1425; inithead=0; %Trim Velocity components initu = 130; initv = 0; initw = initpitch*initu; %Initial rates initp=0; initq=0; initr=0; %Initial Position initnorth=0; initeast=0; initalt=10000; % NAVION stability derivatives % CL cla=4.44; cladot=0; clq=3.8; clde=0.355; clo=0.41; % CD cda2=0.33; cdo=0.05; % CM cma=-0.683; cmde=-0.923; cmadot=-4.36; cmq=-9.96; %CY cyb = -0.564; cydr = 0.157; %CL - Roll clb = -0.074; clp = -0.410; clr = 0.107; clda = -0.134;
