APPENDIX A
APPENDIX A Project Publications and Contributors A. PROJECT PUBLICATIONS AND CONTRIBUTORS A.1 Project Publications (Supported Wholly or in Part by NASA Grant) 1. R.R. Mohler, Nonlinear Systems: 1Iol. 2 Applicatz'ons to Bilinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.
2. R.R. Zakrzewski and R.R. Mohler, "On Nonlinear Model Algorithm Controller Design," Proceedings IFIP Conf. Sys. Modeling and Optimiz., Zurich, 1991.
3. R.R. Mohler, V. Rajkumar, and R.R. Zakrzewski, "Nonlinear Time-Series Based Adaptive Control Applications," Proceedings IEEE Conf. Decision & Control, Brighton, 1991.
4. R.R. Mohler, V. Rajlmmar, and R.R. Zakrzewski, "On Discrete Nonlinear Self-Tuning Control," Proceedings Korean Control Conf., Seoul, 1991.
5. R.R. Mohler, R. Zakrzewski, S. Cho, and C. Koo, "New Results on Nonlinear Adaptive High Alpha Control," Comcon 3, Victoria, 1991.
6. J.E. Kurek, "Analysis of Nonlinear Stability Using Robust Stability Analysis for Linear Systems," submitted to IEEE Trans. Aurora. Control, 1993.
7. R.R. Mohler and R.R. Zakrzewski, "Suboptimal Intelligent Control with High Alpha Aircraft Application," Proceedings IFIP Conf. Sys. Modeling & Optimiz., Compiegne, France, 1993.
8. R.R. Mohler, "Nonlinear Control of High Performance Aircraft," Proceedings American Control Conf., San Francisco, 1993.
9. A. Khapalov and R. Mohler, "Reachable Sets and Controllability of Bilinear Time-Invariant Systems," submitted to IEEE Trans. Aurora. Control, 1993.
Three other journal papers are being prepared on adaptive/intelligent aircraft control.
A.2 Project Contributors Staff Senior 1. R.R. Mohler, P.I., Professor 2.
A. Khapalov, Visiting Professor 3. J. Kurek, Visiting Professor 4. M. Boasson, Visiting Professor 5.
A. Yagen, Visiting Research Associate 6.
J. Dory, Visiting Research Associate A-1
Students
R. Zakrzewski*, Graduate Research Assistant (GRA) I.
2.
D. Collins, NASA Fellow/NSF REU 3. S. Cho, GRA 4.
C. Koo, GRA 5.
J. Young**, NSF (REU) 6. D. Aaberg**, NSF (REU) 7. H. Travis**, NSF (REU) 8. S. Bloom**, NSF (REU) *NASA-NSF support **NSF support A-2
APPENDIX B
APPENDIX B Equations of Motion and Aerodynamic Model EQUATIONS OF MOTION AND AERODYNAMIC MODEL Introduction and Notation In the following the equations of motion of an airplane in the longitudinal mode will be derived The curve fitting technique for the from the basic six degrees of freedom equations of a rigid body.
stability derivatives will also be presented.
Coordinate System Figure B. 1 depicts the body axes coordinate system used in this work.
Figure B. 1. Coordinate System (body axes) XYZ: Aerodynamic forces in Xb, Yb, Zb directions FR, FQ, FP: Aerodynamic moments in X b, Yb, Zb The aerodynamic lift and drag forces are defined in Fig. B. 1. Also, the flight path angle (3'), pitch angle (0), and angle of attack (c0 are related, as shown in Fig. B. 1.
The side slip angle/_ is the angle between V T (the velocity vector) and the XbZ b plane. Angle of attack, _, is the angle between the projection of V t on the XbZ b plane and X b.
B-1
Thefollowingrelations aredef'med
(B.1) V 2 = u 2 ÷ v 2 ÷ w 2 Nomenclature reference length for lateral derivative (span, ft) b,b' reference length for stability derivatives (MAC, ft) center of mass (gravity) c.g.
coefficients, functions of moments of inertia, Ixx, Iyy, Izz,Ixz Cij pitching moment coefficient Cm drag coefficient CD lift coefficient eL D drag force (lb) external force components 0b) Fx, Fy, F z angular acceleration components 0b-ft) FP, FQ, FR gravitational acceleration (ft/sec 2) g h altitude (ft) principal moments of inertia (slugs-ft a) cross product of inertia (slugs-ft 2) L lift force (lb) position vector from e.g. to engine thrust center (ft) ,exg, lye, ezg position vector from e.g. to aerodynamic center (ft) ex, ey, t z M math number external moment components 0b-ft) m aircraft mass (slug) roll rate (rad/sex:) P pitch rate (rad/sec) q dynamic pressure 0b/ft 2)
-4
r yaw rate (rad/sec) reference area ((wing area) ft2) S engine thrust components (lb) Ty, B-2 U velocity component in x b direction (It/see) V velocity component in Yb direction (It/see) W velocity component in z b direction (it/see) X,Y,Z force components in Xb, Yb, Zb directions (Ib) Ot angle of attack (tad or degree) angle of side slip (rad or degree) flight path angle (rad or degree) 3' Euler angles (pitch, yaw, roll) (rad)* O, ¢+, + P air density (slug/it 3) 6h, 6a, _r deflection angles of stabilator, ailerons, and rudder (-) standard atmosphere density ratio throttle setting (.)
Sr
B.2 General Equations of Motion (GDOF) The general, GDOF equations of motion derived from Newtons laws are given in many references. The form used in [B. 1] is shown here for the force and moment equations and in []3.2] for the Euler equations.
Force Equations F x = m(ti-vr +wq) (B.2) E Fy = m(9-wp +ur) E F z = m (g,- uq + vp) Moment Equations M x = p Ixx - i" Ixz + (Izz - Iyy) qr - pq Ixz (B.3) E My : _lIyy + rP_xx-Izz) + _32-r2)Ixz E Mz -- tIzz - I_ Ixz + (Iyy - Ixx ) pq + Ixz qr *Get order of rotations: yaw, pitch, roll B-3 Euler Equations = qcos_ - rsin_ 03.4) = p + qsin_tgO+ rcos_tgO = (qsin_ +rcos _)secO Here the aircraft is treated as a rigid, symmetrical body (so only the Ixz cross product exits); also (j mean d/(dt) the time derivative.
The GDOF equations can be written in terms of u, v, w and Dyr [B.3].
Force Equations (body axes) T X fl =rv - qw - gsin0 + X + m T_ 03.5) v = pw - ru + gcosOsin$ + Y + m T Z w = qu - pv + gcos0costk + Z + m m Moment Equations (body axes) C43 tl T - Tx ) + C * I_ = C41 pg + C42 qr + C43 FR + C * FP + -_=_ zt y gyt I---_- (eye Tz - gze Ty) + 1 03.6) q = CslPr + C52(r2-p 2) + FQ .-_(eztTx-extTz) f = C61 pq + C62 qr + C63 FP + C * FR + C63 C * Tx ) I-'-_"(gy' Tz - exe Ty) + I--'_ (exe Ty - ey e The three Euler equations 03.4) remain unchanged.
B-4
In these equations
X = [-D cos _x + L sin ot]/m
(normal aerodynamic acceleration) Z = [-D sin o_ - L cos a]/m (axial aerodynamic acceleration) (aerodynamic side acceleration) (B.7) Y = qS Cy/m
and
L = qSC L (lift) D = qSC D (drag) The aerodynamic accelerations are (rolling) FP = [qSbC' + m(eyZ-ezY)l/Ixx (B.8) (pitching) FQ = [qScCm +m(ezX- ,xZ)]/Iyy (yawing) Note that reference length for roll and yaw is b (wing span) and for pitch is c, mean aerodynamic chord (MAC).
The inertia moments are assumed to be constant and their coefficients are defined as follows C . __ C4o = Ixx" I=/(IxxI=-I 2) C41 : C* Ixz_zz+Ixx-Iyy)/IxxIzz C42 = C * [Izz Qyy - I=) - I2] / Ixx Ixz C43 -- C * Ixz I Ixx 03.9) C51 = (Izz - Ixx ) / Iyy C52 = It. z/Iyy C61 = C * [Ixx Oxx - Iyy)"I2] / IxxI= C62 = C * Ixzgyy-lzz-Ixx)/Ixxlzz C63 = C "1=/122 B-5 B.3 Form of Aerodynamic Coefficients (Stability Derivatives) The aerodynamic coefficients, also called stability derivatives, are written in the following form: Drag Coefficient C D = CDo (ot, M,h,6h) Lift Coefficient (B.9) C L = CLo(a,M,h, 6h) + _-_[CLq(a,M,h) q +CL_ (a,M,h)&] Pitching Moment Cm = Cmo(_,M,h,dih ) + ___[Cmq(a,M,h)q +Cma (_,M,h)S] The aerodynamic coefficients CDo , CLo , Cmo depend on angle of attack, c_, mach, M, altitude, h, and control surface angles, (5, (which are the controlling factor). The damping coefficients CLq, CL, _, Cmq , Cma are dependent on o_, M, h but not functions of 6.
B.4 Equations of Motion for the Longitudinal Mode The case in which the airplane moves without side slip and rolling motion is called the longitudinal mode. In this the motion is restricted to a plane containing the XbZ b plane. For this case we may make the following assumptions.
03.10) 3, 3, v, p, r, _, _b will all be zero identically The equation for longitudinal motion, based on (B.5) and (B.6) are T x fi = -qw-gsinO +X + m Tz (B.11) _¢ =qu +gcosO +Z+-- m qScC m + (m(ezX-exZ)+ezeTx-exeTz)/I.
Iyy and the remaining Euler equation (B.4) (B.12) _=q It is more convenient to select state variables or, V, q, 0, instead of u, w, q, 0 (B.4).
B-6 Using the relationship (Fig. B-4, fl = 0) U = V cos ot W = Vsint_ and V 2 = u 2 + w 2 By taking derivatives with respect to time, we get u_' - _w V & + sin a ft.
& - =,. #¢ = (B.13) V 2 cos a Using the expression for _, in (B. 11), after some algebraic manipulation, we get the equation for &: & = q + (cos O cos a + sin O sincO - L _ sina + _cosa
g rx Tz
mV mV mV In a similar fashion we derive expressions for _r, q, and 0: Tx Tz Q = -g sin O cos a + g sino_ cos 0 _ __D+ _ cos ct + sin ot m m m L (e zsina+t xCOsot) + 1 T (B 13)
q = qScCmlyy + D(-ezCOs +exsin )l. + T- (ezerx-exe z) "
0=q By substitution of the form of aerodynamic COefficients 03.9) and defining qll = -tz COs cz + t x Sin ot 03.14) q12 = ez sin o_ + e x cos c_ and some more simple algebra, we get SCL o g COs7-q_ +
1 /[1 s cL,]
q +V mV 1 +p Sc 4m CLa 03.15") _ sino_ + _ cos ot Tx Tz } mV mV B-7 _/ = -g sin3, - _ S Tx Tz m -- CDo + m coscz + -- sin r_ m m 2Iyy tyy pSCL° V - Tx sinot (B.15) COS 7 - + P Sc(cCm_ + q12 CL_) { pS_ 2m mV 4I-_ [i :_CLa _ [1- 4_ CLq} g tztTx - txeTz + Tz coso t V 4- mV Iyy _)=q These are the four equations of motion used in this work.
B.5 Values of Constants and Aerodynamic Coefficients The values of the various constants for this airplane were taken from [13.4].
Constants: aircraft -- McDonnel-Douglas F-18 Fighter mass m = 1035.308 slugs w = 33,310 lb weight moments of inertia: Ixx = 23,000 slug-ft 2 Iyy = 151,293 slug'R 2 Izz = 169,945 slug'R 2 Ixz = -2,971 slugs-ff 2 s = 4oo ft2 wing area = 11.52 R MAC b = 37.42 ft wing span geometry e x = -0.297R ey = 0 g z = 0.233 ft B-8 txt = -19.37 ft ey e = 0 eze = 0.233 ft = I12 p V 2 dynamic pressure air density p = 0.0023709 a Aerodynamic Coefficients Reference [I3.4] gives the aerodynamic coefficients for this aircraft in the form of curve fits to wind tunnel data. Values are given for the following range of parameters: angle of attack -100 < ot < 90 ° side slip -20 ° < /_ < 20 ° math number 0.2 < M < 2 altitude 0 < h < 60,000 ft aileron deflection -25 ° < 6a < 25 rudder deflection -30 ° < f < 30 ° stabilator deflection -24 ° < _h < 10.5 ° throttle setting 30 ° (idle) < b r < 131 ° (full afterburner) The coefficients are given in the form of piecewise arc tangent functions for discrete values of the parameters. Interim values are then interpolated.
Example: CMo - pitching moment in Eq. (B.9): CMo(a,M = 0.6, cSh = 10.5 °, h = 15,000 ft) = CMo X 6 (c0 CMo X 6 (or) = (.26/2.75) tan -1 (-(a°-5) 1/10) + (-.39/2.75) tan -1 ((a°-l) 1/8) + (.8/2.75)tan 1 ((a*-5) 1/13) + (.70/2.75)tan 1 (-(a*-10) 1/65) + (1.2/2.75)tan "1 (ot*-49) 1/15) + (2.1/2.75)tan -1 (-(-a°-69) 1/15) + (-.45/2.75) tan -1 (-(a0-77) 1/2) - 3.98 Similar expressions are given for all the other relevant aerodynamic coefficients. This data is sufficient for the longitudinal case and also for small lateral angles. Ref. [13.4] also shows the accuracy of the curve filled data by comparing it to the actual wind tunnel results. In most cases the agreement between the curve filled and actual data is excellent. Small deviations can be expected in the high angle of attack range (_ > w-700).
B-9 The values of the aerodynamic coefficients are given for the following range of parameters: CDo 6h = 10.5 °, 0 °, -5 °, -24 ° ] CLo h= 15k' 5h= 10.5 ° M=0.6,0.9 ] -10° < c_ < 90 ° _Sh = -24 ° M = 0.6, 0.9 CLq M = 0.6 h = 15 k' CL a M = 0.6 h = 15k' h = 15 k' Cmo M = 0.6, 0.9 _Sh = 10.5 °, 5 °, 2*, 0", -5 °, -12.5 °, -24 ° Cm,t M = 0.6 h = 15 k' Cma M = 0.6 h = 15 k' Further aerodynamic data can be found in Ref. [13.5] and [B.6] (manufacturers data) and several more reports which are in our hands.
B.6 Future Developments The method of derivation for the equations of motion shown here can be used for the lateral mode also. In order to check the longitudinal controllers which were developed and to develop lateral controllers (if necessary) for combined pitch, yaw, and roll maneuvers which are needed in modern air combat, it will be necessary to develop a GDOF aerodynamic model, including GDOF equations of motion. This will be done gradually, by first constraining the lateral movements to small side slip angles, as was done by several references -- for example, Safanov [B.7] discussing the Herbst lB.5] maneuver and Ostroff [B.9]. One can do this in two ways -- either develop the full GDOF equations, from [B.5] and [B.6], and add a small/_ constraint to the result, or use the small constraint from the beginning to get a somewhat simpler equation system, and move on to the full case later on. For the full GDOF case the data given in [B.4] must be widened by using the full data available in [B.5] and [B.6]. The group has already done preliminary work in this direction. A convenient method for showing results on real time simulations is given in Ref. [B. 10]. This method uses spherical mapping to transform the various angular relations to a two-dimensional plane. Future lateral results may be displayed in this form.
References B.7 Perkins, C.D. and Haye, R.E., Airplane Performance, Stability, and Control, J. Wiley & Sons, lB.1] 1949.
Atkin, B., Dynamics of Flight -- Stability and Control (2nd Ed.), J. Wiley & Sons, 1982.
Cho, S., Ph.D. Dissertation, to be published in 1993.
B-10 03.4] Cuo, J., F. Garret Jr., E. Hoffman, and H. Stalford, "Analytical Aerodynamic Model of a High Alpha Research Vehicle Wind Tunnel Model," Georgia Institute of Technology, Atlanta, GA, 1990 (Also NASA Grant NAG10959).
03.5] MDC Rep. No. A7247- "FIA 18 Stability and Control Data Report", Vol. I -- Low Angle of Attack; Vol. II -- High Angle of Attack; August 31, 1981.
lB.6] MDC Report No. A8575 -- "FIA 18 Basic Aerodynamic Data," McDonnel Aircraft Comp., St.
Louis, MO, March 31, 1984.
['13.7] Chiang, R.Y., M.G. Safonov, K.P. Madden, and J.A. Tekway, "A Fixed H o* Controller for a Supermaneuverable Fighter Performing a Herbst Maneuver," Proc. IEEE Conf. on Decision and Control, Honolulu, December 5-7, 1990.
113.8] Herbst, W.B., "Future Fighter Technology," Jr. of Aircraft, Vol. 17, No. 8, August 1980.
[B.91 Ostroff, A.J., "Application of Variable Gain for Output Feedback for High Alpha Control.
[B.10] Kalviste, J., "Spherical Mapping and Analysis of Aircraft Angles for Maneuvering Flight," NIAH paper 86-2283.
B-II