This document discusses aerodynamic modeling and simulation of aircraft at high angles of attack. It outlines some of the challenges, including developing accurate mathematical models from wind tunnel data and flight tests. Nonlinear effects like separated and unsteady flow require dynamic modeling approaches. Qualitative analysis of the nonlinear dynamics can reveal critical flight conditions like departures and spins. Flight simulations are used alongside flight tests to evaluate control laws and train pilots. Accurate modeling of phenomena like aerodynamic asymmetry is important for understanding spin behavior.

1. Aerodynamics Modeling and Dynamics Simulation at High Angles
of Attack
M.Goman, A.Khramtsovsky and M.Shapiro
Central Aerohydrodynamic Institute (TsAGI), Russia
Abstract
Flight simulation problems at high angles of attack
ranged from development of adequate mathemati-
cal model for aerodynamic characteristics and non-
linear dynamics analysis to piloted simulation for
research and training purposes are discussed. The
mathematical model for high angles of attack aero-
dynamics is formulated based on di erent types of
experimental wind tunnel data and nally corrected
using the ight test results. The piloted ight sim-
ulation is planned using results of qualitative non-
linear dynamics investigation providing valuable in-
formationabout high angles of attack departure and
spin behavior. The experience in application of the
mini desktop and full size movable simulators for
highincidence ight investigationsincluding dynam-
ics analysis, post design assessment of control laws
for departure prevention and spin recovery, safety
support of special ight tests and pilot training is
outlined.
Introduction
High angles of attack ight practically for all types
of aircraft is associated with critical and emergen-
cy conditions due to serious changes in an aircraft
dynamic responses and control. For example, about
30% of aircraft losses in ight accidents are origi-
nated from aircraft departures at high angle of at-
tack and development of spin. However, the possible
bene ts of the high incidence ight expanding ma-
neuverability boundaries are considered for future
generation of combat aircraft. The new types of ma-
neuvers such as the Cobra and the Herbst ones have
been not onlydemonstrated in ight, but thoroughly
investigated in terms of their e ciency and applica-
tion.
Copyright c 2001 by Central Aerohydrodynamic Insti-
tute (TsAGI).
During the last three decades there have been
made signi cante orts indevelopmentof theoretical
and experimental methods for investigation of high
angles of attack ight conditions, including dynam-
ics analysis, piloted simulation and ight tests. The
most typical feature of all investigations done during
these years was the extensive use of di erent types
of simulators for research and pilot training purpos-
es. The set of simulators available now in TsAGI
for solving high incidence ight dynamics and safety
problems is presented in Fig.1. They can be selec-
tively used for accompanying ight tests, performing
engineering research and pilot training. The com-
plex research/engineering simulator on the Stuart
platform providing six degrees-of-freedom motion is
more precise and expensive, it is mostly used for pi-
loted simulation during the special ight tests. The
simple and the most cheap desktop simulators with
simpli edvisualizationsystem andcontrol levers can
be widely used for the rank-and- le pilots training.
The structure and all components of an aircraft
mathematical model can be the same in all these
simulatorsthus supporting the continuous process of
research work and pilot training. The general struc-
ture ofthe mathematicalmodelis presented inFig.2.
One of the most signi cant problems in mathe-
matical model development is connected with for-
mulation of the aerodynamic forces and moments at
high incidence ight conditions. Due to separated
and vortical ow the aerodynamic dependencies be-
come essentially nonlinear and motion dependent.
Unfortunately, the mathematical model built only
on the wind tunnel data requires further corrections
for better agreement with ight test data. There-
fore, control law design, dynamics analysis and pi-
loted simulation compose the closed-loop research
and development cycle (see Fig.3).
In this paper some problems and experience asso-
ciated with modeling and simulation of high angles
of attack ight of combat and general aviation air-
craft are discussed.
1

2. Closed-loop research and development
cycle
There are some links and interconnections between
components of the cycle in Fig.3, which are rather
natural for ight dynamics in general, but more
strong for high angles of attack conditions.
Flight Tests at High Angles of Attack are
very hazardous and expensive. They take long time
and require high-skilled test pilot, special equipment
ensuring ight safety and extended ground based
theoretical and experimental support. The mainob-
jectives of such special ght tests are assessment of
stall/spin resistance, search for an adequate control
technique for spin prevention/recovery and testing
of automatic control system.
Aerodynamics modeling is mainlyrelied on the
experimental wind tunnel data from static, forced
oscillation and rotary balance tests. These experi-
mental data allow to obtain rather good agreement
with ight tests, when aircraft motion is stable and
not agitated. The test pilot comments help to ad-
just the wind tunnel data to real ight conditions
(Fig.4).
Dynamic instability and large amplitude oscilla-
tions require correct modeling of unsteady aerody-
namic e ects due to internal ow dynamics sepa-
rated and vortical ow. The methods of unsteady
aerodynamics modeling is now under development
6, 7, 9, 10, 11, 12]. To reconcile the conventional
mathematical model with ight test results in such
agitated ight conditions the set of unknown param-
eters in the mathematical model are corrected using
ight test data using identi cation techniques.
The current wind tunnel experimentalfacilities re-
quire further development and improvement to sim-
ulate an aircraft high angles of attack motion condi-
tions (large amplitude and multi-degree of freedom
oscillations).
Ground based simulation and pilot training
at high incidence will be e cient only if we have
an adequate mathematical model for nonlinear un-
steady aerodynamics, and this is an iterative process
connected with ight tests.
Spin recovery for modern aircraft is too compli-
cated for rank-and- le pilots, that is why the lessons
learnt during the special ight tests and later re-
produced in simulation are extremely important for
pilot training. The simple and a ordable desktop
simulators are the most appropriate tools for these
objectives.
Aircraft dynamics has a multi-attractor nature
and depend on the style of piloting. Results of
nonlinear dynamics analysis help to plan the pilot
training exhaustively. Simulationreveals the control
techniques for spin entry and recovery and highlights
the critical ight conditions.
Stability & Dynamics Analysis is based on bi-
furcation and nonlinear dynamic theory methods
and application of specially developed software for
such qualitative investigation (the KRIT Package
4]).
Multiple equilibrium and periodical dynamical
states are investigated using continuation technique
and Poincare mapping method. This helps to pre-
dict aircraft departures and possible critical attrac-
tors. The same qualitative methods of analysis are
applied for pre- and post-design assessment of con-
trol laws 5].
Nonlinear control laws design for high angles
of attack mainlysolve the ight safety issues such as
warning, prevention and eliminationof critical ight
conditions. Normally, special control laws are de-
signed for departure prevention and spin recovery.
Unfortunately, due to loss of aerodynamic e ciency
of control surfaces at high angles of attack there are
a lot of limitations to solve this problem. The level
of aerodynamic characteristics uncertainty at high
angles of attack is much higher than at normal ight
conditions, that is why the advanced robust control
design methods and innovative control e ectors such
as thrust vectoring, vortex ow generators, etc., are
of great importance for high angles of attack ight.
Anti-spin parachute mathematical
model
Experimental aircraft are often equipped with anti-
spin parachute for safety reasons (see Fig.5). Stat-
ic line of the parachute is attached behind the air-
craft center of gravity. Parachute's drag force is
transferred through static line and brings out pitch
and yaw moments. These moments tend to recover
the aircraft to normal ight conditions with near-
zero angle-of-attack and sideslip. The moments are
su cient for spin recovery provided the parachute
canopy area is large enough.
The static line and the canopy may be deployed
in di erent manners, for example, using special con-
tainer equipped with powder rocket engines. In a
short time (' 0:7 sec) the parachute is ejected out
ofthe airplanewake. Ittakes about 0:5 0:7 sec more
for fullcanopy deployment. After that the drag force
on the canopy appears.
2

3. Due to the air ow, the canopy is moving with re-
spect to the airplane. To know exact position of the
parachute with respect to the airplane is important
for the calculation of additional pitching and yawing
moments.
The mathematicalmodel of the parachute motion
is based on the following assumptions:
the parachute oats in the air inertialessly
static line is long enough, so the disturbances
of the velocity eld in the airplane wake can be
ignored
the velocity eld is practically uniform near the
canopy
aerodynamic force is normal to the canopy.
The orientation of the static line of the parachute
is described by unit vector ~p = (px py pz). Ori-
entation of the vector ~p with respect to body-axis
frame of reference is given by the angles p and p
(analogous to angle-of-attack and sideslip). The re-
lationship between them is
px = ;cos p cos p
py = ;sin p
pz = ;sin p cos p
The airspeed vector at the canopy location is a
sum of airplane ight velocity vector and transla-
tional velocity due to aircraft rotation
~Vp = ~Vc + ~! (~rp + lp ~p)]
where
~Vc - airplane velocity vector at c.g.,
~! - airplane rotation rate vector,
~rp - radius-vector from c.g.
to static line attachment point,
lp - length of the static line.
Airspeed vector component normal to ~p will cause
to move the parachute with respect to the airplane.
The static line orientation is governed by the equa-
tion
d~p
dt
= ;
hh
~p ~Vp
i
~p
i
lp
(1)
If orientation unit vector ~p is known, additional
forces and moments can be calculated as follows
~Fp = CDpSp
~Vp ~p
2
2 ~p = CDpSp
^~V p ~p
2
QS~p
~Mp =
h
~rp ~Fp
i
where
Sp - the canopy area,
Sp = Sp=S - nondimensional canopy area,
CDp - parachute drag coe cient,
- air density,
Q - dynamic air pressure,
^~V p = ~Vp=V - nondimensional
parachute velocity vector.
Due to the air ow retardation in airplane wake,
the drag coe cient CDp depends on the airplane's
angle of attack. This dependence can be obtained
from wind tunnel tests of the airplane with the de-
ployed parachute.
For the equation (1), it is necessary to set correct-
ly initial static line orientation at the moment of full
canopy deployment. The parachute container eject
the parachute in a certain direction (along X-axis of
the airplane, p = 0, p = 0). These angles may
change during deployment stage due to aircraft ro-
tation. The initial conditions p0, p0 = 0 for the
equation (1) can be calculated taking into account
the mechanism of deployment.
At steady state rotation in spin conditions the ori-
entation of the static line of the parachute coincides
with the direction of the local velocity of the air ow:
p = ;Vp
Vp
= ;Vc +! (rp + lpp)
Vp
(2)
Assuming that Vp Vc, than the equation (2)
becomes linear with respect to p
p+
lp
Vc
! p = ;Vc + ! rp
Vc
= p (3)
Computing both vector and scalar products of the
right and left-hand parts of the equation (3) with
vector ! (fromthe left), one can obtain the following
expression
! p = ! p +
lp
Vc
!2
p;(! p )!
which after substitution into equation (3) gives the
following nal expression for vector p
p =
p + lp
Vc
p ! + lp
Vc
!(! p )
1 + lp
Vc
2
!2
Using this formula it is possible to calculate the
steady-state airplane spin parameters taking into ac-
count the in uence of the anti-spin parachute. It is
also possible to evaluate the needed parameters of
anti-spin parachute for successful spin recovery.
3

4. Unsteady aerodynamics modeling
Signi cant contribution to aerodynamic loads at
high angles of attack is generated by separated and
vortical ow. Their in uence produces at high inci-
dence nonlinearanddynamicaerodynamicresponses
to changes in an aircraft attitude. As a result the
conventional form of aerodynamic coe cients based
on the aerodynamic derivative concept becomes in-
accurate 8]. The most reasonable way of modeling
of nonlinear unsteady aerodynamics e ects is in ap-
plication of ordinary di erential equations for vorti-
cal and separated ow contributions.
The mathematical model for any force and mo-
ment coe cients may be represented using load par-
titioningin the followingform(here the normalforce
coe cient is considered as an example):
CN(t) = CNpt( )+ CN_ pt ( )_ + CNdyn (4)
where inertialess terms CNpt( ), CN_ pt ( ) are equiv-
alent to the conventional representation form with
aerodynamic derivatives, and dynamic contribution
CNdyn is governed by nonlinear equation
dCNdyn
dt
=
3X
i=1
ki( )(CNvb0 ;CNdyn ( ))i (5)
where t = 2t0V1c is dimensionless time, = k;1
1 ( )
is the characteristic time constant, extracted from
small amplitude responses, and the right hand side
function CNvb0
is de ned as
CNvb0
( ) = CNst( ); CNpt( ):
The linearized dynamic equation (5) behaves very
well in case of small amplitude oscillations, during
large amplitude motion the nonlinear terms in (5)
become rather large to obtain good agreement with
experimental results 7, 10, 9].
Such dynamic representation of all aerodynamic
coe cients is important for adequate modeling of
high angles of attack aircraft oscillatory motionsuch
as wing rock or agitated spin.
Nonlinear dynamics qualitative analy-
sis
The whole mathematical model of an aircraft dy-
namics at high incidence is highly nonlinear, it dis-
plays various types of behavior depending on the pi-
lot control manner. Sometimes di erent pilots pro-
voke di erent aircraft dynamics and some of critical
ightregimes maybe avoided. That is why the qual-
itativemethod ofanalysisprovidingallpossiblecriti-
calstates, their stabilityandregions ofattraction are
used to perform thorough simulation of an aircraft
dynamics at high angles of attack. The example of
qualitative dynamics analysis for a hypothetical air-
craft is presented in Fig.6. Along with stable normal
ight solutions the critical solution branches such as
roll-coupling modes, wing rock and oscillatory at
spin modes are identi ed. These solutions provide
not only magnitudes of motion parameters but also
the character of motion stability.
Aerodynamic asymmetry and aircraft
spin
Modern maneuverable aircraft con gurations espe-
cially when they are statically unstable at low an-
gles of attack su er with the lack of pitch-down con-
trol at high angles of attack (see Fig.7). The con-
trol system in such cases provides stability at nor-
mal ight regimes, however there are stable trims at
high incidence, where an aircraft can be locked-in.
These ight conditions, which are called deep stall
regimes, may be unrecoverable using conventional
control technique.
Similar critical unrecoverable situations can arise
due to aerodynamic yaw asymmetry producing at
spin regimes, where an aircraft can be also locked-in
(see Fig.8).
The asymmetricalaerodynamicrollingandyawing
aerodynamic moments at high angles of attack are
result of the onset ofasymmetricalvortical ow. The
aerodynamic asymmetry is observed both in wind
tunnel andin ight. The onlydi erence that in ight
the level of yaw asymmetry may be higher than in a
wind tunnel. The possible reason ofsuch di erence is
in aeroelastic vibrations of scaled aircraft model and
in di erent interference e ects available in a wind
tunnel.
The aerodynamic asymmetry in yaw extracted
from high incidence ight tests are presented in
Figs.9 and 10 respectively for the Su-27 aircraft and
the experimental X-31 aircraft. Although the ampli-
tudes of yaw asymmetry in these cases are di erent,
the qualitative dependence on angle of attack is sim-
ilar. Yaw asymmetry changes its sign with angle of
attack and displays dynamic hysteresis during pitch
up and pitch down attitude variations.
Fig.11 illustrates how the stable equilibrium at
spin solution appears in the moment balance equa-
tions with the increase of the yaw asymmetry am-
plitude. The aerodynamic asymmetry may signi -
cantly exceed the e ciency of rudder and ailerons
so that the at spin regime can be unrecoverable by
means of simple counteracting control de ections.
The only e cient control technique in the cases of
deep stall and at spin regimes is the so called pitch
4

5. rocking control. Actually it means that available
constrained control authority is applied to destabi-
lize the critical ight regime in a self-agitating man-
ner (Fig.12). Typical variations of aircraft motion
parameters during spin recovery using pitch rocking
control technique is presented in Fig.13. It is inter-
esting to note that pitch rocking control (75 90
seconds) produces increase in amplitude of oscilla-
tion not only in pitch, but also in roll due to inertia
coupling of both these forms of motion.
The e ciency of pitch rocking control in compar-
ison with simple counteracting control can be seen
from Fig.14, where the time of recovery from at
spin conditions is given as a function of the level of
yaw asymmetry.
Development of the adequate mathematical mod-
el for aerodynamic characteristics during the initial
stage of ight tests allowed to design the spin pre-
vention and recovery control system, which later had
been also tested in ight. The general block diagram
of this system is presented in Fig.15.
The adequate mathematical model veri ed in
ight tests and in piloted simulationswith participa-
tion of experienced test pilots has been applied for
development of simple and cheap desktop simulator
for training of the rank-and- le pilots. Special work
has been done for creation of the database of repre-
sentative set of simulated ights illustrating possible
pilot's mistakes and correct recovery control.
Fig.16 presents two examples from this database.
The rst one illustrates the deep stall departure and
following recovery (the time histories for motion pa-
rametersandcontrolare presented inFig.17)andthe
second one illustrates the at spin departure and re-
covery (the timehistories for motionparameters and
control are presented in Fig.18). The Cobra maneu-
ver simulation is presented in Fig.19.
High incidence ight simulation of gen-
eral aviation aircraft
General aviation aircraft Molnia-1 (Fig.20) with a
canard and high horizontal tail provides another
example of successful application of simulation ap-
proach for high incidence ight (see Fig.21).
A small positive installation angle of the canard
leads to earlier onset of ow separation on a canard
with respect to stall conditions on a wing. This
produces the pitch down moment in static depen-
dency of the pitch moment coe cient. Because the
ow separation on a canard occurs with some de-
lay it generates the anti-dampinge ect in pitch (see
Fig.22).
The unsteady aerodynamic model for the pitch
moment coe cient has been developed in the form
(5) and applied in the mathematicaland piloted sim-
ulationon complexresearch simulatorwith 6 degree-
of -freedom (see Fig.1, top). This piloted simulation
has been performed before the ight tests and helped
the test pilot to study the peculiarities of such air-
craft con guration. At high angles of attack (for
this con guration sens 18 deg) due to canard
ow separation occur the self-sustained oscillations
in pitch, which serve as warning factor for pilot of
high incidence ight. This pitch oscillations is stable
in recoverable when pilot applies a pitch down con-
trol. The predicted behavior ofanaircraft athighin-
cidence ight has been con rmed later in ight tests.
The example of timehistories for motionparameters
are presented in Fig.23.
Concluding remarks
Piloted simulation of an aircraft dynamics at high
incidence ight is extremely important element of
aircraft development and serti cation processes. It
helps in mathematical model assessment, accompa-
nyingthe special ighttests andthus increasingtheir
safetyand e ciency, and nallycan be used forrank-
and- le pilots training beyond the normal ight con-
ditions.
References
1] Aerodynamics, stability and controllability of
supersonic aircraft. Editor G.S.Bushgens, Nau-
ka, Fizmatlit, Moscow, 1998, 816 pp.
2] Ahrameev, V., Goman, M., Kalugin, A., Klu-
mov, A., Merkulov, A., Milash, E., Syrovatsky,
V., Khramtsovsky, A., and A.Scherbakov. Au-
tomatic aircraft recovery from spin regimes,
Technika Vozdushogo Flota, No.3, 1991, pp.15-
24 (in russian).
3] Zagaynov, G.I., and M.G.Goman Bifurcation
analysis of critical ight regimes, ICAS Pro-
ceedings, Vil.1, 1984, pp.217-223.
4] Goman M.G., Zagainov G.I and A.V.Khram-
tsovsky Application of Bifurcation Methods
to Nonlinear Flight Dynamics Problems. {
Progress in Aerospace Sciences, Vol.33, pp.539-
586, 1997, Elsevier Science, Ltd.
5] Goman M.G. and A.V.Khramtsovsky Applica-
tion of Bifurcation and Continuation Methods
for an Aircraft Control Law Design. { Phil.
Trans. R. Soc. Lond. A (1998) 356, 1-19, In the
RoyalSociety Theme Issue "Flight Dynamicsof
High Performance Manoeuvrable Aircraft".
5

6. 6] Tobak, M. and Schi , L.B. On the Formulation
of the Aerodynamic Characteristics in Aircraft
Dynamics, NASA TR-R-456, 1976.
7] Goman, M.G., and A.N.Khrabrov. State-Space
Representation of Aerodynamic Characteristics
of an Aircraft at High Angles of Attack, Jour-
nal of Aircraft, Vol.31, No.5, Sept.-Oct. 1994,
pp.1109 - 1115.
8] Greenwell, D.I.Di culties in the Application of
Stability Derivatives to the Maneuvering Aero-
dynamics of Combat Aircraft, ICAS Paper 98-
1.7.1,the 21th Congress ofthe AeronauticalSci-
ences, Sept. 1998, Melbourne, Australia.
9] Goman,M.G.,Greenwell, D.I.,andA.N.Khrab-
rov. The Characteristic Time Constant Ap-
proach for Mathematical Modeling of High An-
gle of Attack Aerodynamics, ICAS Paper, 22nd
Congress of the Aeronautical Sciences, Sept.
2000, Harrogate, UK, pp. 223.1-223.14.
10] Abramov, N.B., Goman, M.G., Khrabrov,
A.N., and K.A.KolinkoSimple Wings Unsteady
Aerodynamics at High Angles of Attack: Ex-
perimental and Modeling Results, Paper 99-
4013, AIAA Atmospheric Flight Mechanics
Conference, August 1999, Portland, OR.
11] Klein, V., and Noderer, K.D. Modeling of Air-
craft Unsteady Aerodynamic Characteristics,
Part 1 - Postulated Models, NASA TM 109120,
May 1994 Part 2 - Parameters Estimated From
Wind Tunnel Data, NASA TM 110161, April
1995 Part 3 - Parameters Estimated From
Flight Data, NASA TM 110259, May 1996.
12] Mark S.Smith Analysis of Wind Tunnel Oscil-
latory Data of the X-31A Aircraft, NASA/CR-
1999-208725, Feb. 1999.
13] B.R.Cobleigh,M.A.Croom, B.F.Tormat Com-
parison of X-31 Flight, Wind -Tunnel, and Wa-
ter Tunnel Yawing Moment Asymmetries at
High Angles of Attack, High Alpha Conference
IV -Electronic Workshop,NASA Dryden Flight
Research Center, July 12-14, 1994
Figure 1: Research/training simulator on the Stuart
platform (top), midi size training simulator (mid-
dle), desktop training simulator (bottom).
6

7. Equations
of motion
Undercarriage
model
Aerodynamic
forces and moments
model
Aerodynamic
characteristics
database
Atmospheric
turbulence
model
Engine model
Altitude-velocity
engine characteristics
Cockpit
Control system
and actuator
models
Flight tests
safety equipment
Figure 2: General structure of mathematical model
used in piloted simulation.
Figure 3: Research and development cycle at high
angles of attack.
Figure 4: Aerodynamicmodeldevelopmentbased on
wind tunnel and ight tests data.
Figure 5: Anti-spin parachute mathematical model.
7

8. Figure 6: Qualitative analysis of nonlinear aircraft
dynamics at high angles of attack.
Figure 7: Deep stall regimes.
Figure 8: Unrecoverable at spin regimes.
8

9. Figure 9: Aerodynamic yaw asymmetry extracted
from ight tests 1].
Figure 10: Aerodynamic yaw asymmetry extracted
from ight tests of the X-31 aircraft 13].
Cn0
= 0 Cn0
= 0.035Cn0
= 0.02
- balance in pitch moments
- balance in roll and yaw moments
- stable spin regime
- aperiodically unstable spin regime
Figure 11: Flat spin generated by aerodynamic yaw
asymmetry.
Figure 12: Pitch rockingcontrol technique (potential
function analogy).
9

10. Figure 13: Spin recovery using pitch rocking control.
Timeofrecovery(sec)
0
10
20
30
40
50
0 0.05 0.10 0.15
with rocking
without
rocking
Yaw asymmetry Cn0
Figure 14: E ciency of pitch rocking control.
Figure 15: Spin prevention and recovery control sys-
tem.
10

11. Marker time step: 5 sec
−2000
0
2000
4000
−4000 −3500 −3000 −2500 −2000 −1500 −1000 −500 0 500
5500
6000
6500
7000
7500
8000
8500
9000
xe
−ye
h
Marker time step: 10 sec
−2000
0
2000
4000 −5000
−4000
−3000
−2000
−1000
0
1000
0
1000
2000
3000
4000
5000
6000
−yexe
h
Figure 16: Maneuvers performed by experienced test
pilots on desktop simulator. Deep stall departure
andrecovery (top). Flat spin departure and recovery
(bottom).
0 10 20 30 40 50 60
−100
0
100
α,deg
0 10 20 30 40 50 60
−10
0
10
β,deg
0 10 20 30 40 50 60
−5
0
5
p,1sec
0 10 20 30 40 50 60
−0.5
0
0.5
r,1sec
0 10 20 30 40 50 60
−1
0
1
q,1sec
Time, sec
0 10 20 30 40 50 60
−100
0
100
θ,deg 0 10 20 30 40 50 60
−200
0
200
ψ,deg
0 10 20 30 40 50 60
−200
0
200
φ,deg
0 10 20 30 40 50 60
−10
0
10
−az
0 10 20 30 40 50 60
−0.2
0
0.2
a
y
Time, sec
0 10 20 30 40 50 60
−200
0
200
Xθ
,mm
0 10 20 30 40 50 60
−200
0
200
Xψ
,mm
0 10 20 30 40 50 60
−200
0
200
Xφ
,mm
0 10 20 30 40 50 60
0
50
100
XT
r
,mm
0 10 20 30 40 50 60
0
50
100
XT
l
,mm
Time, sec
Figure 17: Deep stall departure and recovery during
aircraft spatial maneuvering.
11

12. 0 20 40 60 80 100 120
−100
0
100
α,deg
0 20 40 60 80 100 120
−20
0
20
β,deg
0 20 40 60 80 100 120
−2
0
2
p,1sec
0 20 40 60 80 100 120
−2
0
2
r,1sec
0 20 40 60 80 100 120
−0.5
0
0.5
q,1sec
Time, sec
0 20 40 60 80 100 120
−100
0
100
θ,deg
0 20 40 60 80 100 120
−200
0
200
ψ,deg
0 20 40 60 80 100 120
−200
0
200
φ,deg
0 20 40 60 80 100 120
−10
0
10
−a
z
0 20 40 60 80 100 120
−0.1
0
0.1
ay
Time, sec
0 20 40 60 80 100 120
−200
0
200
X
θ
,mm
0 20 40 60 80 100 120
−200
0
200
X
ψ
,mm
0 20 40 60 80 100 120
0
50
100
X
φ
,mm
0 20 40 60 80 100 120
0
50
100
X
T
r
,mm
0 20 40 60 80 100 120
0
50
100
X
T
l
,mm
Time, sec
Figure 18: Flat spin departure and recovery during
aircraft spatial maneuvering.
Marker time step: 1 sec
1000
1500
2000
2500
3000
3500
4000
−500
0
500
1400
1600
1800
2000
2200
2400
xe
H0
=1423 ft; Mach=0.49; γ0
=0; Throttle=0.25 (t=3:1:10 seconds)
−ye
h
3 4 5 6 7 8 9 10
−100
0
100
α,deg
H0
=1423 ft; Mach=0.49; γ0
=0; Throttle=0.25 (pitch, roll and yaw control)
3 4 5 6 7 8 9 10
−2
0
2
q,rad/s
3 4 5 6 7 8 9 10
200
400
600
V,ft/s
3 4 5 6 7 8 9 10
−100
0
100
θ,deg
3 4 5 6 7 8 9 10
−20
−10
0
δe
,deg
3 4 5 6 7 8 9 10
−20
0
20
β,deg
3 4 5 6 7 8 9 10
−0.5
0
0.5
r,rad/s
3 4 5 6 7 8 9 10
−2
0
2
p,rad/s
3 4 5 6 7 8 9 10
−200
0
200
φ,deg
3 4 5 6 7 8 9 10
−40
−20
0
δa
,deg
3 4 5 6 7 8 9 10
−20
0
20
δr
,deg
Time, sec
Figure 19: Cobra maneuver simulation.
12

13. Figure 20: General aviation aircraft Molnia 1.
Figure 21: Canard ow separation.
Figure 22: Anti-damping e ect due to canard ow
separation.
Figure 23: Pitch oscillations at high angles of at-
tack of a general aviation aircraft with canard ( ight
tests).
13