[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference and Exhibit - Keystone, Colorado ()] AIAA Atmospheric Flight Mechanics Conference and

Effect of Thrust Vectoring on Aircraft Post – Stall

Trims, Stability, and Maneuvers

Kunal Ghosh∗ , Aditya Paranjape† , and N. Ananthkrishnan‡

Indian Institute of Technology – IIT Bombay, Mumbai 400076, India

High performance airplanes require maneuvering at high rates and under conditions ofhigh incidence where the dynamics can be extremely nonlinear and is often accompanied bycontrol input saturation, thereby limiting the flight envelope. In recent times, the urge toexpand the flight envelope, beyond that achievable using aerodynamic control surfaces, hasbeen addressed using thrust vectoring due to its superior characteristics at high angles ofattack. In this paper, effectiveness of thrust vectoring as a control input has been discussedin detail. Improvement in the post-stall angle of attack capture using thrust vectoringon saturation of aerodynamic controls has been demonstrated by extended bifurcationanalysis. Furthermore, the effect of control input saturation on stability of closed-loopairplane dynamics has been illustrated. Key maneuvers, such as turns, have been analyzedin detail to demonstrate effectiveness of closed-loop in stabilizing otherwise unstable open-loop turn trims. Instantaneous turn maneuver has been remodeled to incorporate realisticeffects of throttle saturation and negative specific excess power on maximum achievableturn rate. Improvement in the instantaneous turn rate with thrust vectoring is qualified andquantified. Finally, using thrust vectoring, controlled departure and subsequent recoveryto level flight is simulated for flight instabilities as pitch bucking and oscillatory spin.

Nomenclature

p, q, r roll, pitch, and yaw rates, respectivelyM Pitch momentN Yaw momentQ Dynamic pressureS Reference AreaV Aircraft velocityAOA Angle of AttackHARV High Alpha Research VehicleHHIRM Hypothetical High Incidence Research ModelITR Instantaneous Turn RateNDI Nonlinear Dynamic InversionSEP Specific Excess PowerTV Thrust Vectoring

Subscripts

(.)0 indicates trim value(.)C indicates external command input to the particular NDI control loop(.)sp short period

Symbols

α Angle of attack

∗Masters(Dual Degree) Student, Department of Aerospace Engineering; [email protected]. Member AIAA†Masters(Dual Degree) Student, Department of Aerospace Engineering; [email protected]. Member AIAA‡Associate Professor, Department of Aerospace Engineering; [email protected]. Senior Member AIAA.

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American Institute of Aeronautics and Astronautics

AIAA Atmospheric Flight Mechanics Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6486

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

β Sideslip angleδa, δe, δr Aileron, Elevator and Rudder respectivelyδpv, δyv Pitch vectoring and Yaw vectoring deflection respectivelyη Throttle setting, i.e the ratio of actual thrust to maximum thrustλ 2ζspωspV0/QSµ Roll angle about velocity vectorω Bandwidthφ Bank angleθ Pitch angleζ Damping coeffcient

Superscripts

( ˙ ) differentiation with respect to time, t

I. Introduction

Ever since the first fighter airplane appeared during World War I, maneuverability has been the keyto survival in aerial combat, and has received a lot of emphasis in the design history of fighter airplanes.Examples of highly maneuverable fighters include the Sopwith Camel of World War I, the P-51 Mustangof World War II, the F-86 Sabrejet used in the Korean conflict, and the F-15 and F-16 that were designedaround the experience gained in Vietnam. The race for evolution of fighter airplanes more maneuverablethan their potential adversaries has led to the need for extreme maneuvering capability, also known assuper-maneuverability.

Super-maneuverability, coined by Herbst,1 is the ability to execute maneuvers with controlled sideslip atangles of attack well beyond those for maximum lift. It often implies capabilities beyond those achievablethrough more efficient wings, better performing engines, or more sophisticated flight control systems. Amongthe various key concepts which constitute the present-day definition of super-maneuverability, post-stallmaneuverability (PSM) is of prominent interest as it has the potential to dramatically enlarge the flightenvelope of an airplane in terms of parameters such airspeed, turn rate, climb rate, etc. PSM, whichrepresents the ability of an airplane to perform controlled tactical maneuvers beyond its stall limit, oftenrequires generation of high pitch and yaw rates, and thus is primarily governed by the nature and effectivenessof various control inputs available on an airplane.

Traditionally, airplane controls have been aerodynamic, such as elevator, aileron, and rudder, which areplaced at specific locations on the airplane, such as wings or tail. When deflected, these surfaces modify theexterior shape of the airplane, thus creating an imbalance in the aerodynamic forces acting on it, causinga rotation about the center of gravity, resulting in a maneuver. These conventional aerodynamic controlsare therefore limited by aerodynamic constraints, because control can be compromised at flight conditionswhere aerodynamic forces are small. Such conditions typically occur at high AOA, and low flight speeds. Anaerodynamic force for a given surface is proportional to the square of the velocity. Therefore, deflection of acontrol surface will be effective in bringing about a differential in aerodynamic force large enough to createrotation of the airplane only above a certain threshold speed. Further, aerodynamic forces increase upto acritical AOA , beyond which the flow separation leads to aerodynamic stall and the aerodynamic forces fallrapidly with a consequent loss of efficiency of the control surfaces. Unfortunately, in post-stall maneuvers,AOA is very high and velocity of the airplane is usually low, and consequently, aerodynamic controls showa pronounced reduction in their effectiveness and cannot be reliably used for PSM.3

The development of another source of control is necessary, especially for high-performance airplanes,whose survivability in within-visual-range combat depends mostly on their ability to accomplish post-stallmaneuvers.2 Attention is then drawn to the other forces acting on the vehicle besides the aerodynamicones, viz., weight and thrust. Though effect of weight has already been used, such as in the Concorde, bymanipulating the center of gravity location, however, this is a cumbersome process and cannot be used asan active control for fast PSM. Though neglected for maneuvering purposes in almost all airplanes, thrustcan also be used to achieve maneuverability. Traditionally, the direction of thrust is fixed, and only themagnitude changes, according to the flight regime requirements. However, with the advent of gas turbineengine with a directing nozzle, thrust vectoring (TV) has developed as a promising control input that cansignificantly reduce or even eliminate problems of loss of control effectiveness at high AOA. As TV is largely

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independent of external flow field around the airplane, it can be a suitable choice for airplane control inputfor PSM.

Manned flight-testing of TV-enabled X-29 airplane has supported the fact that TV implementationimproves flight characteristics significantly. A decrease of as much as 50% in relative takeoff ground rolldistance of TV-enabled X-29 airplane has been noticed.4 Furthermore, in sustained level turn maneuver,Lee and Lan5 have illustrated that usage of TV leads to 15% decrease in minimum turn radius along with3% increase in maximum turn rate. Their analysis also demonstrated that performance gain obtained usingTV improves with increase in thrust-to-weight ratio. Effect of TV on airplane maneuverability has beendiscussed by Raghavendra et al.6 They have demostrated, for an F-18/HARV airplane model, that recoveryfrom flight conditions such as spin to low AOA level trim is possible only with TV as a supplementarycontrol input to aerodynamic controls. Results on how TV modifies the stability of open-loop steady statesas compared to the case with aerodynamic controls alone are rare and often restricted to how TV can beused for spin recovery.7 Commonly, based on stability analysis of airplane equations, control strategies forTV are devised to recover from spin conditions to low AOA stable trims.

The aim of the present work is to demonstrate the effect of closed loop control power saturation and thrustvectoring on airplane stability and maneuvering ability. In particular, turning maneuver with maximumturn rate has been examined in detail. Instantaneous turns have been realistically modeled to includenegative specific excess power (SEP) and throttle saturation. The improvement in turning performance withthrust vectoring has been demonstrated. Further, simulation results for instantaneous turns are analyzedto demonstrate controlled departure to flight instabilities such as spin and pitch bucking, and subsequentrecovery from these conditions.

The structure of the paper is as follows. The second section contains brief notes on the theory of bifurca-tion analysis and the NDI algorithm employed in this work. Section 3 deals with the Extended BifurcationAnalysis (EBA) of the airplane model in order to compare the longitudinal trim solutions with and withoutTV. Effect of control saturation on stability has been quantified using closed-loop bifurcation analysis. Insection 4, effect of flight stability on turning maneuvers has been discussed in detail. Instantaneous turnmaneuver has been studied and the improvement in the instantaneous turn rate with thrust vectoring ispresented. Controlled departure and subsequent recovery of the airplane from flight instabilities is presentedin section 5. The final section concludes the paper.

II. Analysis Tools

This chapter gives brief information on the analysis tools which have been the basis of the present work:

1. Bifurcation Analysis has been used to study and map the dynamics of the F-18/HARV airplane.

2. Nonlinear Dynamic Inversion (NDI)-based controller airplane is used to carry out closed-loopsimulations.

A. Bifurcation Analysis

Bifurcation analysis provides a convenient and efficient methodology for a global analysis of the airplanedynamics. Continuation algorithms8 are used to trace out equilibrium solutions from a given starting solutionas a control parameter (for e.g. the elevator deflection) is varied. These algorithms also characterize airplanestability at these equilibrium (or trim) points. Bifurcation points can then be identified as the points instate-parameter space where changes occur in the number of equilibrium points and their stability.9 In thepresent work, Standard Bifurcation Analysis (SBA) is used to identify the spin limit cycles of the open-loopF-18/HARV airplane dynamics and Extended Bifurcation Analysis (EBA)10 is used to compute the levelflight trims of the F-18/HARV airplane model. For a more detailed account of bifurcation analysis usingcontinuation, the reader is referred to Refs.11, 12

1. Standard Bifurcation Analysis (SBA)

Standard bifurcation analysis requires the airplane equations of motion to be represented as follows:

x = f(x, u) (1)

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where x, the vector of states, and u, the vector of control parameters, are as follows:

x = [V, α, β, p, q, r, φ, θ]

u = [δe, η, δa, δr, δpv, δyv]

In SBA, one of the control inputs is chosen as the principal continuation parameter (p), whereas the othercontrol inputs are kept constant. An initial equilibrium solution is required in order to start the continuation.The continuation algorithm varies the principal continuation parameter and obtains the equilibrium solutionsx(p), their stability information, the various bifurcation points, and also the bifurcated equilibrium branches(also called departure branches).

Aerodynamic data for the F-18/HARV airplane13 has been used for this study. AUTO200014 is thecontinuation algorithm used to carry out the bifurcation analysis.

2. Extended Bifurcation Analysis (EBA)

The EBA procedure allows the computation of equilibrium solutions subject to constraints on the statevariables x. The airplane dynamics given by Eq. (1) along with the constraint equations can be representedin the following form:

x = f(x, u1, u2, ..., um+1, um+2, ..., ur)

gi(x) = 0, i = 1, ..., m (2)

where gi are the m constraint functions, u1 is the principal continuation parameter, u2, ..., um+1 are them control parameters that are to be varied as a function of u1 in order to satisfy the constraints, andum+2, ..., ur are the controls that are kept constant. EBA computations are carried out in two steps. In thefirst step, both the state and constraint equations in Eq. (2) are solved together to simultaneously obtain theconstrained equilibrium solutions x(u1) and the control parameter schedules u2(u1), ..., um+1(u1) requiredto satisfy the constraints gi. This first step is called Constrained Bifurcation Analysis (CBA). In the secondstep, which is a normal SBA, only the state equations in Eq. (2), with the parameter schedules computed inthe first step incorporated as follows:

x = f(x1, u1, u2(u1), ..., um+1(u1), um+2, ..., ur) (3)

are solved to obtain the equilibrium states, their stability, bifurcation points, and bifurcated equilibriumbranches. It must be noted that the equilibrium solutions on the bifurcated branches represent departuresfrom the constrained trim states; these are valid solutions for the control parameter schedules u2(u1), ..., um+1(u1),but do not satisfy the constraints imposed in Eq. (2).

B. Nonlinear Dynamic Inversion

In the present work, the controller for the F-18/HARV model is designed using a nonlinear dynamic inversionalgorithm proposed by Snell et al.15 Using the fact that control inputs affect angular rates much faster thanattitude angles, two loops of different timescales, as shown in Fig. 1, are implemented to generate the controlinputs for the airplane. The faster inner loop inverts the body axis angular rate dynamics and the slowerouter loop inverts the attitude angles with respect to the velocity vector. The commanded variables inthe outer loop are attitude variables: AOA αc, sideslip angle βc, and roll angle about velocity vector µc.Inversion of the outer loop dynamics generates the commanded variables for the inner loop, which are theangular rates, pc, qc, rc.

Using the fact that elevator has a dominant influence on pitch angular rate, the inversion of the inner loopdynamics is first carried out for qc to generate commanded values of elevator δec. This commanded elevatordeflection is then passed through the saturation and rate limiter block. The elevator deflection so obtainedis then used for inversion of inner loop dynamics for pc and rc to generate commanded values of aileronδac and rudder δrc. These lateral control inputs, δac and δrc, are again passed through the saturation andrate limiter blocks before being fed to the airplane dynamics block. Thus, a sequential inner loop inversionis obtained as compared to the one-step procedure of Raghavendra et al.6 The idea behind the sequentialinversion is to eliminate undesirable effects of elevator saturation on pc and rc loops. For the case when onlyaerodynamic controls are implemented, TV controls are not commanded, and both commanded pitch TV

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toφ

µ

4s + 4

ωµ

ωβ

ωα

ω r

ω q

ω p

1

2s + 1

ηc

δyvc

pvcδ

cδ r

cδ a

cδ e

δyv

pvδ

δ r

δ a

δ e

.p

d

.rd

.qd

µc

cβ

αc

.βd

.αd

.µd

−

Out

er L

oop

Inve

rse

Dyn

amic

s

−

−

−

−

−

η

Satu

ratio

n an

d R

ate

limits

x

Air

craf

t Dyn

amic

s

pc

qc

r c

Inne

r Loo

p In

vers

e D

ynam

ics

α

µ

β

p

q

r

Figure 1. Block diagram of closed-loop airplane dynamics with nonlinear dynamic inversion law implemented.6

δpvc, and commanded yaw TV δyvc, are kept unchanged at zero. The bandwidths ωp, ωq , and ωr, alongroll, pitch, and yaw rate paths, respectively, in the inner loop are taken to be 10 rad/s. In the outer loop,the bandwidths ωα and ωβ are set to 2 rad/s along AOA and sideslip paths, respectively, and ωµ is set to1.5 rad/s for the roll path. The roll command is furthermore passed through a filter 4

s+4to tune the roll

response of the airplane. These values of bandwidth have been chosen so as to maintain timescale differencebetween the inner and outer loops. The throttle command ηc is passed through a filter 1

2s+1which models

the lag in the throttle response.

.qd

.pd

.rd Ntv

Mtv

δyvc

pvcδMc

L c

rcδ

acδ

ecδ eδ

aδ

rδNc

McNc

aM

aN

L

eac

c

r c

δδδ

Inve

rse

Dyn

amic

s

Con

trol

Eff

ectiv

enes

sIn

vers

e T

hrus

t Vec

tor

Eff

ectiv

enes

sIn

vers

e C

ontr

ol

Con

trol

Eff

ectiv

enes

s

Lim

itsR

ate

and

Satu

ratio

n

−

−

Figure 2. Block diagram of the daisychaining algorithm to compute thrust vectoring commands.6

In the F-18/HARV model, TV is obtained by directing engine exhaust, which, in turn, is obtained

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by orienting the nozzle or using external paddles attached to the nozzle. As nozzle parts or paddles areexposed to hot exhausts, frequent TV usage decreases their life drastically. Hence, TV is considered as asupplementary control input to aerodynamic controls. In order to take care of this fact, TV is implementedusing daisychaining algorithm.16 The implementation procedure is identical to that of Raghavendra et al.6

It can be observed in Fig. 2 that commanded pitch (Mc), and yaw (Nc) control moments are inverted togenerate commanded elevator δec, and rudder δrc, deflections. Whenever the commanded control deflections,δec and δrc, are more than their rate and/or saturation limits, the excess uninverted control moments, Mtv

and Ntv, are inverted to generate commanded pitch TV δpvc, and yaw TV δyvc, deflections. Both theseTV control inputs are then passed through saturation limits before being fed to the airplane dynamics. Nocontribution of TV to roll moment is modeled.

It is important to note that during closed-loop bifurcation analysis, filters in the control law have notbeen considered as these filters always augment the closed-loop system with poles that are in the openleft half-plane, and hence do not affect the stability of the closed-loop airplane dynamics. Furthermore, thecontrol input rate limiters have also not been considered as bifurcation analysis only accounts for quasi-staticvariation of controls.

III. Bifurcation Analysis of Longitudinal Flight

The problem of trade-off between stability and maneuverability has often played a decisive role duringdesign of an airplane. For example, commercial transport airplanes which are primarily designed for stabilitylack extreme maneuvering capabilities, whereas highly maneuverable airplanes such as the F-22 have negativeor marginal static stability to improve stealth and agility features for combat effectiveness. These super-maneuverable airplanes usually require active control augmentation throughout their mission, and thus itis more appropriate to describe their flight envelope by closed-loop behavior. Furthermore, these highperformance airplanes require maneuvering under extreme conditions such as high rates and large angles ofattack. In these circumstances, the dynamics is extremely nonlinear and maneuvers are often accompaniedby control input saturation limiting the flight envelope. Thus, the important issues in flight dynamics ofhighly maneuverable airplanes are control law effectiveness and control saturation

Investigations on the first issue have already been addressed in the literature using closed-loop bifurca-tion analysis.17 Avanzini and de Matteis18 showed that for closed-loop F-16 airplane, deep stall equilibriumbranches from the open-loop were absent due to alpha limiter employed in the stability and control aug-mentation system, leading to a marked change in the bifurcation diagrams. A similar investigation usingbifurcation analysis of closed-loop Hypothetical High Incidence Research Model (HHIRM) has been discussedby Littleboy and Smith.19 First, they identified regions of lateral and longitudinal coupling in the flight enve-lope by open-loop bifurcation analysis of fifth-order HHIRM. Then, a control law, designed using NonlinearDynamic Inversion (NDI), was implemented to obtain a continuous stretch of stable equilibria without anylateral effects until one of the control inputs saturated. Closed-loop bifurcation analysis of Littleboy andSmith illustrates that control laws based on NDI modify the stability of steady states without significantlychanging their structure in the bifurcation diagram. Furthermore, they showed that with the control lawimplemented, the only factor that restricts flight envelope of an airplane is control input saturation. Thus,for high incidence airplanes, which are inherently unstable, the flight envelope is primarily constrained bycontrol saturation, as instability of basic airframe is eliminated by flying closed-loop. In their analysis it wasnoted that AOA stops increasing after control saturation and the steady states become oscillatory unstablefor any further command to increase the AOA. The reason for this instability of steady states beyond controlinput saturation was not addressed by Littleboy and Smith. Further, reported applications of bifurcationanalysis to airplanes show that limiters, dead bands, control input saturation, are considered as problems tobifurcation analysis application as they generate discontinuous derivatives, and hence the questions posedabove have not been answered satisfactorily.

In this section, the above issues have been addressed by open and closed-loop bifurcation analysis ofF-18/HARV airplane for longitudinal flight conditions. Initially, open-loop bifurcation analysis is carried outto identify the basic airframe instabilities in longitudinal flight. Effectiveness of TV as supplementary controlto aerodynamic controls has been quantified and qualified using EBA. Subsequently, closed-loop bifurcationanalysis is carried out with NDI control law to estimate the effectiveness of the control augmentation and toanalyze the effects of control saturation on closed-loop stability.

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Table 1. Summary of longitudinal constraints

Branch Continuation Constraints Freed Fixed

parameter parameter parameter

A-B δe γ = 0, β = 0, φ = 0 η, δa, δr δpv = 0, δyv = 0

B-C δe β = 0, φ = 0 δr, δa η = 1, δpv = 0, δyv = 0

C-D δpv β = 0, φ = 0 δr, δa η = 1, δe = −0.43 rad, δyv = 0

D-E δpv β = 0, φ = 0 δyv, δa η = 1, δe = −0.43 rad, δr = −0.6 rad

A. Open-loop Bifurcation Analysis

0.00 0.25 0.50 0.75 1.00 1.25

0.00

0.25

0.50

0.75

1.00

1.25

Ang

le o

f Atta

ck (r

ad)

Angle of Attack (rad)

A

B

CD E

Figure 3. Extended bifurcation analysis diagram of AOA for longitudinal flight trims (full lines: stable equi-libriums; dashed lines: unstable equilibriums; filled squares: Hopf bifurcation points).

This section presents extended bifurcation analysis of the 6-DOF open-loop F-18/HARV airplane modelusing AUTO.14 The complete EBA is summarized in Table 1. Figure 3 shows the longitudinal trims: branchA-C is generated using elevator δe as the continuation parameter, with aileron δa and rudder δr varyied tokeep sideslip angle β and roll angle φ fixed at zero. The control input variations are shown in Fig. 4. Afterpoint C in Fig. 3, elevator saturates at −0.43 rad and pitch TV is used as the next continuation parameteruntil point E. At point D, the rudder saturates at −0.6 rad and then, yaw TV δyv is also used until pointE to maintain sideslip angle β at zero. Thus, with implementation of TV, the maximum controllable AOAincreases from 55 deg (0.96 rad) at point C to 67 deg (1.17 rad) at point E. The value of throttle η isscheduled as a function of α, shown in Fig. 4(a), to maintain level flight trim conditions until throttle getssaturated at point B. From point B to point E, all trim states correspond to descending flight.

It is observed in Fig. 3 that there are four major stable flight trim branches. These branches are approx-imately in the range of very high α (1.1−1.17 rad), high α (0.7−0.8 rad), moderate α (0.47−0.65 rad), andlow α (0− 0.37 rad). The largest unstable stretch of trim points is approximately over a range of α between0.8 − 1.1 rad. The instability in this α range is due to loss of short period damping and results in buckingoscillations.6 Figures 4(c) and 4(d) show notable negative deflections of aileron and rudder, respectively, atlarge angles of attack required to overcome the asymmetric lateral forces and moments as a result of rightand left elevator deflection and maintain symmetric flight. Trim values for sideslip angle β, bank angle φ,and angular rates p, q, r, remain equal to zero, and hence have not plotted here.

B. Closed-loop Bifurcation Analysis

1. With Aerodynamic Control

This section presents bifurcation analysis of the closed-loop airplane dynamics model in Fig. 1. Figure 5 showsthe trims generated using commanded AOA αC as the continuation parameter while commanded sideslip

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0.00 0.25 0.50 0.75 1.00 1.25

0.00

0.25

0.50

0.75

1.00

1.25

Thro

ttle


A

B C D E

(a)

0.00 0.25 0.50 0.75 1.00 1.25

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

Ele

vato

r (ra

d)


ED

A

B

C

(b)

0.00 0.25 0.50 0.75 1.00 1.25

-0.20

-0.10

0.00

0.10

0.20

0.30

Ail

eron

(ra

d)


A B

C

E

D

(c)

0.00 0.25 0.50 0.75 1.00 1.25

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

Rud

der (

rad)


A B

C

D E

(d)

0.00 0.25 0.50 0.75 1.00 1.25

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00


Pitc

h T

V

E

CBA

D

(e)

0.00 0.25 0.50 0.75 1.00 1.25

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0080

E

C DBA

Yaw

TV

(rad

)


(f)

Figure 4. Variation of (a) throttle η, (b) elevator deflection δe, (c) aileron deflection δa, (d) rudder deflectionδr, (e) pitch TV δpv, and (f) yaw TV δyv, control inputs with AOA α for longitudinal flight trims.

βC and wind-axis roll angle µC are held fixed at zero. The primary aim of the analysis is to understand theeffects of saturation on closed-loop airplane dynamics, and therefore, complexities due to AOA limiters andpilot stick have not been modeled.

Figure 5(a) shows the equilibrium values of AOA as a function of αC . Since the airplane is operating withcontrol law implemented, all trims for AOA range of 0 to 30 deg (0.55 rad) are stable and correspond to levelflight. At α of 30 deg (0.55 rad), throttle saturates and all stable trims obtained henceforth are descendingflights. The commanded control inputs, elevator, aileron, and rudder, are also shown in Fig. 5. Littleboyand Smith’s conclusion that AOA cannot be increased beyond the elevator saturation point19 is confirmedin Fig. 5(a). At αC of 55 deg (0.96 rad), elevator saturates at −25 deg (−0.43 rad) and all trims obtainedwith saturated elevator have same AOA of 55 deg (0.96 rad). Thus, as long as demanded elevator input

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(a) (b)

(c) (d)

Figure 5. Bifurcation diagram of (a) AOA α, (b) commanded elevator deflection δec, (c) commanded ailerondeflection δac, and (d) commanded rudder deflection δrc, with commanded AOA αc as the continuation pa-rameter (full lines: stable equilibriums; dashed lines: unstable equilibriums; filled squares: Hopf bifurcationpoints).

is below saturation point, AOA achievable is same as commanded AOA, αC , and beyond saturation, AOAremains fixed at that value where elevator saturation first occurred. This shows that control saturation is aprominent factor which restricts the closed-loop flight envelope of airplane, especially those which operate athigh AOA. Furthermore, it can be noticed in Fig. 5 that after elevator saturation all the trims obtained areoscillatory unstable. This is due to the fact that on saturation all 8 closed-loop eigenvalues of the Jacobianmatrix suffer a discontinuous jump in their values as shown in Fig. 6(a) and eigenvalues 1 and 2 have crossedover to open right half complex plane as can be noticed in Fig. 6(b).

2. With Both Aerodynamic and Thrust Vectoring Control

This subsection discusses results of bifurcation analysis of closed-loop airplane dynamics with TV controlimplemented (as in Fig. 2) to supplement the aerodynamic controls. The bifurcation diagram in Fig. 7illustrates the trims of the closed-loop generated using commanded AOA αC as the continuation parameterwith other parameters kept identical to that in the previous subsection.

With implementation of TV, the maximum controllable AOA increases from 55 deg (0.96 rad) to 67 deg(1.17 rad), as can be readily observed in Fig. 7(a). The variation of commanded controls required to achievecommanded AOA αC is shown in Figs. 7(b) through Fig. 7(f). It can be noticed in Fig. 7(d) that rudderinput saturates at 63 deg (1.1 rad), however, yaw TV supplements the saturated rudder and further increasein AOA with zero sideslip can be achieved. At α of 67 deg (1.17 rad), pitch TV saturates and no furtherincrease in AOA is possible. As in the case of elevator saturation previously, pitch TV saturation also leadsto jump relocation of eigenvalues as can be observed in Fig. 8(a). However, there is no crossing of eigenvalues

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(a)

(b)

Figure 6. Variation of real part of (a) all 8 closed-loop eigenvalues and (b) eigenvalues 1, 2, 3 and 4, withcommanded angle of attack αc, with only aerodynamic controls implemented.

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(a) (b)

(c) (d)

(e) (f)

Figure 7. Bifurcation diagram of (a) AOA α, (b) commanded elevator deflection δec, (c) commanded ailerondeflection δac, (d) commanded rudder deflection δrc, (e) commanded pitch TV δpvc, and (f) commanded yawTV δyvc, with commanded AOA αc as the continuation parameter (full lines: stable equilibria).

to open right half plane as shown clearly in fig. 8(b).In order to reason out why control saturation with aerodynamic controls alone resulted in migration

of eigenvalues from left half plane to right half plane causing a Hopf bifurcation in Fig. 5, the open-loopextended bifurcation analysis diagram (Fig. 3) of longitudinal flight trims with elevator as continuationparameter is analyzed at elevator saturation point ‘C’. It is observed that the open-loop trim at the elevatorsaturation point ‘C’ is unstable. The instability is due to short period instability.6 Thus, it is concludedthat instability in closed-loop behavior is not due to elevator saturation per se, as saturation only reverts theclosed-loop behavior to its open-loop by repositioning the eigenvalues of the closed-loop Jacobian matrix,and the consequent behavior is governed by the nature of open-loop stability. In this case, the open-loop is

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oscillatory unstable, so is the closed-loop. Similarly, the open-loop bifurcation diagram (Fig. 3) of longitudinalflight trims with pitch TV as continuation parameter is analyzed at its saturation value ‘E’. It is found thatthe open-loop trim at pitch TV saturation point ‘E’ is stable which explains why closed-loop is stable aftercontrol saturation in Fig. 7.

In summary, it can be seen that thrust vectoring permits airplane trims at angles at attack higher thanthose permitted by elevator alone. This could constitute a significant advantage in air-to-air combat. Further,control input saturation relocates closed-loop airplane eigenvalues to those while in the open-loop, and hence,airplane stability after control input saturation is governed by the nature of its open-loop stability at thecorresponding angles of attack.

IV. Bifurcation Analysis of Turning Flight

Turn rate and turn radius are accepted parameters used to characterize fighter airplane performance. Forsuperior air combat capability, the airplane should have the maximum possible turn rate and the minimumpossible turn radius.20 The maximum turn rate achieved is typically constrained by the maximum coeffi-cient of lift that can be obtained (CLmax

) and the maximum load factor (nmax) that can be sustained bythe airplane. These constitute aerodynamic and structural/human constraints respectively, with maximumthrust further limiting the turn rate for sustained level maneuvers.

Using the fact that turn maneuver is a steady trim solution of the airplane dynamical equations, anestimate of the turn rate χ is obtained using V = 0 and γ = 0 equations21 as follows:

χ =g

V

L

Wsin µ

T = D

L cosµ = W (4)

Thus,

χ =g

Vtan µ (5)

n =L

W= sec µ (6)

(7)

(a) (b)

Figure 8. Variation of real part of (a) all 8 closed-loop eigenvalues and (b) eigenvalues 1, 2, 3 and 4, withcommanded AOA αc, with both aerodynamic and TV controls implemented.

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This turn rate equation can be reformulated in terms of load factor, n, using Eq. (6) as

χ =g

V

√

n2− 1 (8)

(9)

A typical turn rate variation with Mach number obtained for a supersonic fighter airplane is shown in Fig. 9.The branch ‘a’ shows maximum turn rate achievable while turning at CLmax

limit. This branch starts from

a b

d

CP

Figure 9. Typical turn rate vs Mach No. plot.22

level longitudinal trim flight with n = 1. As µ increases, the airplane turns at an increasing load factor untilthe load factor reaches the structural limit at the corner point CP. At this Mach number the airplane isturning at its maximum attainable turn rate, as any further increase in turn rate is not possible by increasingbank angle µ due to load factor limit. Furthermore, any increment in AOA also leads to decrease in lift, andhence the turn rate (using Eq. (8)). Starting from corner point CP, branch ‘d’ in Fig. 9 represents variation ofmaximum turn rate with given Mach number as the airplane turns at its maximum load factor. As the loadfactor (Eq. (6)) is only dependent on wind-axis roll angle µ, branch d represents turns at constant µ. Branch‘b’ in Fig. 9 represents maximum sustained turn rate variation with Mach number. Points on branches ‘a’and ‘d’ with turn rate more than that on branch ‘b’ at the same Mach number are not sustainable. In otherwords, all turns above branch ‘b’ are instantaneous in nature. Turning flight at turn rate above branch ‘b’in Fig. 9 has to occur at negative SEP. Under this flight condition, the airplane decelerates and/or descendsin altitude and thus, the maximum achievable turn rate cannot be “sustained.” An important considerationin generating the turn rate plot of Fig. 9 is that all control inputs are assumed to be within their saturationlimit.

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A. Open-loop Bifurcation Analysis

Turn, which is often an important combat performance assessment maneuver, may experience a variety offlight instabilities. Since turning maneuver can be represented as a steady trim solution of airplane flightdynamics equations, bifurcation analysis is used to analyze its stability. Ananthkrishnan and Sinha10 usedthis concept to search for departure tendencies on constant load factor branch ‘d’ of Fig. 9. Stability analysisof turn trims requires use of the Extended Bifurcation Analysis (EBA), where a control input such as elevatorδe is chosen as the continuation parameter and other control inputs, aileron δa and rudder δr, are variedas a pre-scheduled function of the continuation parameter. This pre-scheduling allows generation of thedesired turn trims with correct stability information. A complete EBA of the eighth-order equations for

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

0.

5.

10.

15.

20.

25.

Mach Number

Tur

n R

ate

(deg

/s)

1

2

3

4

5a

b

cd

A

BE

C

D

CP

FG

Figure 10. Open-loop airplane performance showing turn rate vs Mach number (full lines: stable equilib-ria; dashed lines: unstable equilibriums; filled squares: Hopf bifurcation points; open squares: pitchfork ortranscritical bifurcation points).23

open-loop airplane turn maneuver for all three branches ‘a’, ‘b’, and ‘d’ has been described by Paranjapeand Ananthkrishnan23 for F-18/HARV airplane. Figure 10 shows their turn rate plot. Their analysis showsthat open-loop F-18/HARV airplane experiences departures on all the three turning branches ‘a’, ‘b’, and ‘d’(‘c’ is a constant load factor branch similar to ‘d’). Departures are of oscillatory nature on branches ‘b’, and‘d’, whereas until point C on branch ‘a,’ the airplane seems to depart to a non-coordinated turning trim statehaving same control input. The instability on branch ‘a’ limits the lowest velocity at which instantaneousturns can be performed to that at point ‘C’ in Fig. 10. Furthermore, the maximum sustained turn rate atpoint ‘G’ on branch ‘b’ is found to be oscillatory unstable. Thus, the stable maximum sustained turn rateis lower than the otherwise accepted maximum sustained turn rate at point ‘G’. The stability analysis ofRef. 23 for turn maneuvers with maximum achievable turn rate for a given velocity is an important steptowards incorporating effect of stability on commonly assessed turn performance. Their investigation showsthat besides maximum coefficient of lift (CLmax

), maximum sustainable load factor (nmax), and maximumthrust available, flight stability during a turn maneuver is also a limiting factor on maximum achievable turnrate. As CLmax

, nmax, and maximum thrust available are basic design parameters in evaluating airplaneperformance, open-loop stability analysis of turn maneuver can serve as an important input for airplanecontrol law design.

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Table 2. Summary of turning constraints

Label Constraints Freed parameters Fixed parameters

a αc = 0.42 rad, γ = 0, βc = 0 η, µc —

b γ = 0, βc = 0 αc, µc η = 1.0

c µc = 1.3 rad, γ = 0, βc = 0 η, αc —

d µc = 1.37 rad, γ = 0, βc = 0 η, αc —

(Note: Constraints µc = 1.3 rad and µc = 1.37 rad correspond to L/W = 3.7 and L/W = 5 respectively.)

B. Closed-loop Bifurcation Analysis

In this subsection, we present closed-loop (with the previous NDI controller) bifurcation analysis of the F-18/HARV model in a turn maneuver with aerodynamic controls alone using EBA. As maximum achievablesustained turn rate for a given velocity is bounded by various constraints discussed above, EBA is required,even in closed-loop bifurcation analysis, to generate steady turn trims which satisfy these constraints.

1. Zero Specific Excess Power Closed-loop Turns

Table 2 summarizes the various EBA constraints used. The constraints have been chosen suitably so as tocompare stability with open-loop turn plot (Fig. 10) of Ref. 23 under similar flight conditions.

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.

5.

10.

15.

20.

25.

D

a bc d

CP

Mach Number

Tur

n R

ate(

deg/

s)

Figure 11. Closed-loop airplane performance showing turn rate versus Mach number (full lines: stable equi-libriums).

Figure 11 shows the closed-loop turn rate plot for F-18/HARV with NDI controller. It can be noticedthat all the unstable open-loop turn trims of Fig. 10 are stablilized with controller in the loop. Thus,with control law incorporated, the F-18/HARV has maximum achievable turn rate at the same point ‘G’of Fig. 10. Point G in the open loop is found under the assumption that it will be stabilized in the closedloop. Furthermore, the closed-loop branches ‘a’, ‘b’, ‘c’, and ‘d’ have no departures. Hence, NDI controllaw is able to stabilize all the turn trims without changing the trim states. Moreover, all the control inputsare found to be within their saturation limits.23 Hence, the NDI law is able to cancel the natural dynamicsof the open-loop F-18/HARV airplane completely, and simultaneously place the closed-loop eigenvalues atthe desired locations to achieve designer’s desired stability behavior. It is important to notice that all theturn trims of branch ‘a’, ‘c’, and ‘d’ having turn rate more than that of branch ‘b’ at given Mach numberhave been generated with throttle η value exceeding saturation at 1.0. Thus, the instantaneous turn regionin the turn rate plot of Fig. 11 represents unrealistic turn trims with zero SEP. It is expected, at least forthe short durations of time, when an airplane is performing an instantaneous turn maneuver, it will havestates very similar to those computed here as zero-SEP turn trims. Thus, the zero-SEP turn trim at the

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corner point CP is presently accepted as the best possible indicator of maximum instantaneous turn rate forairplane performance assessment.

2. Negative Specific Excess Power Closed-loop Turns

An airplane flying in the instantaneous region of the turn rate plot of Fig.9 performs a turning maneuverwhich is not sustained in nature due to thrust limitations. In these cases, because the airplane motioninvolves loss of altitude along with velocity, its energy height decreases. Time rate of change of energy heightis related to the SEP as

SEP = he =V

gV + h (10)

It is a well-known fact that in instantaneous turn maneuvers, SEP is negative, as seen in Fig. 9, and itselfvarying with time. The variation of SEP depends on the initial flight condition and the control input sequencefollowed while engaging the instantaneous turn maneuver. Presently, in literature23, 20, 21 an approximationof SEP = 0 is used to evaluate the maximum turn rate obtainable during instantaneous turn maneuvers, asdiscussed in the previous section.

In this section, we investigate instantaneous turn maneuvers in a more realistic way, with throttle limi-tation and negative SEP incorporated. For this, the equation, γ = 0, is revisited to estimate analytical turnrate χ for negative SEP in instantaneous turn maneuvers as follows:24

χ =g

V cos γ

L

Wsinµ

L cosµ = W cos γ (11)

Thus,

χ =g

Vtan µ (12)

n =L

W=

cos γ

cosµ(13)

This turn rate equation can be reformulated in terms of load factor L/W using Eq. (13) as

χ =g

V

√

( n

cos γ

)2

− 1 (14)

It can be noticed that turn rate χ in Eq. (12) is the same during an instantaneous turn maneuver as comparedto Eq. (5) for a sustained maneuver; however, the load factor is now defined as cos γ/cosµ compared to 1/cosµpreviously. As a first step towards realistic modeling of instantaneous turn maneuver, it is assumed that theairplane loses altitude, but not velocity. Thus

SEP = h.

This assumption allows closed-loop bifurcation analysis of the complete eighth-order model of the airplanewithout dropping the V = 0 equation. Furthermore, the conventional turn rate plot of maximum turn ratevs Mach number is also retained. Figure 12 shows the new descending, negative SEP closed-loop turn rateplot, generated with throttle saturation in effect, for the same constraints discussed in Table 4.1. Figure 13shows variation of γ and load factor on various branches of Table 4.1. It can be noticed in Fig. 13(a) thatinstantaneous turn trims in these maneuvers are descending in nature as flight path angle γ is negative, asexpected. Furthermore, the load factor (Fig. 13(b)) attained at the corner point(CP) is less than that of thecorresponding non-descending turns of Fig. 11. This decrease in load factor (n = cos γ/cosµ) for the sameroll angle φ is due to the steep negative flight path angle γ.24 Thus, descending turns are limited at theircorner point by roll angle rather than load factor. Moreover, the turn rate attained at the corner point isapproximately 3 deg/s more for descending turns due to lower trim velocity as compared to non-descendingturn trims, but at the cost of severe loss of altitude.

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0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

0.

5.

10.

15.

20.

25.

D

CP

Mach Number

ab

cd

Tur

n R

ate

(deg

/s)

Figure 12. Closed-loop turn rate vs Mach number (level and descending turns)(full lines: stable equilibriums).

C. Turning with Thrust Vectoring

Recently, usage of TV for improving turn performance of airplane has been conceptualized.5, 25 This conceptis based on the fact that TV can be used to assist the lift in creating the centripetal acceleration on theairplane, during a turn maneuver. This can be understood by revisiting the dynamical equations, includingTV, as:

χ =g

V

(L + T sin(α + δpv)

W

)

sin µ

T cos(α + δpv) = D

L + T sin(α + δpv) cos µ = W (15)

Thus,

χ =g

Vtan µ (16)

n =L + T sin(α + δpv)

W= secµ (17)

This turn rate equation can be reformulated in terms of load factor (L + T sin(α + δpv))/W using Eq. (17)as

χ =g

V

√

n2− 1 (18)

Thus, an increase in thrust to weight ratio T/W and/or positive (down) deflection of the pitch TV δpvwill increase load factor and hence the turn rate. Based on this, Lee and Lan,5 have illustrated that TVusage leads to a significant decrease in minimum turn radius along with increase in maximum turn rateand maximum load factor, as can be observed in Fig. 14. It can be noticed in Fig. 14 (generated withT/W = 0.77) that the minimum radius of turn is equal to 1670 ft with δj = 0 deg and 1420 ft with δj = 45deg. Thus, TV usage leads to a significant decrease in minimum turn radius by as much as 15 percent. Themaximum turn rate for δj = 0 deg occurs at M = 0.77 with a value of 13.7 deg/s, and with δj = 18 deg, themaximum turnrate is 14.1 deg/s at M = 0.73, leading to an increment of 3 percent in the maximum turnrate. Lee and Lan’s5 analysis also demonstrates that the performance gains obtained by using TV increasewith increase in the thrust-to-weight ratio. Their results plotted in Fig. 15, generated using T/W = 1.0,show 18 percent decrease in minimum radius of turn and 5 percent increase in maximum turn rate usingTV. Thus, there is a decrement of 3 percent in relative minimum radius of turn and 2 percent increment inrelative maximum turn rate with 42 percent increase in T/W . Though Lee and Lan’s analysis illustrates theeffect of TV on various turning performance measures, however their analysis uses constant values of pitch

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0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

-0.90

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

Flig

ht P

ath

Ang

le(r

ad)

Mach Number

a

c

b

d

D

CP

(a)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Mach Number

a

b

c

D

d

CP

Loa

d Fa

ctor

(b)

Figure 13. Closed-loop bifurcation diagram of (a) flight path angle γ, and (b) load factor vs Mach number(level and descending turns) (full lines: stable equilibria).

TV, which may lead to increase in load factor beyond the allowable limit of the airplane. Nevertheless, theiranalysis shows TV can be a very effective control for increasing the net lift on the airplane, especially whenthe airplane is using its maximum aerodynamic lift in order to obtain steeper turn below its structural loadfactor limit.

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Tur

n R

ate

(deg

/s)

Mach No.

Figure 14. Effect of TV on sustained level-turn performance for a fighter airplane (T/W = 0.77).5

The above analysis is a good starting point for investigating TV effects on turn maneuvers but is limitedby their use of a 3-dof model. This is because Eq. (18) is derived considering only the forces acting on theairplane. The moments are assumed to be balanced by suitable deflection of elevator, aileron, and ruddersurfaces. For a 6-dof model, the moment balance, especially in pitch axis, is obtained by the lift generatedby deflecting elevator. Thus, to incorporate the influence of elevator on the lift during a turn, the lift in theturn rate equation should be evaluated as:

χ =g

V

√

[L(α, δe)

W+

T

Wsin(α + δpv)

]2

− 1 (19)

It is important to note that on maximum CL branch ‘a’ in Fig. 9, obtained using only aerodynamic controls,if pitch TV is to be incorporated, then extra elevator deflection is required to balance the pitch axis momentgenerated by the thrust component perpendicular to velocity vector. For example, to increase the T sin(α +δpv) term in Eq. (19), if pitch TV δpv is deflected downwards, themn, to maintain angle of attack formaximum CL, the elevator should be deflected up. In combat airplanes such as F-18/HARV, the location ofthe elevator and engine nozzle are at similar distance from the center of gravity as can be noticed in Fig. 16.Thus, for moment balance, the lift generated by the elevator is approximately the same as the componentof thrust perpendicular to velocity vector. Hence, increment of net force perpendicular to velocity vectoris expected to be lesser than that assumed by Lee and Lan,5 and consequently turn rates lower than thiseestimated by them are likely during turn analysis with complete 6-dof airplane model.

Figure 17 shows descending, negative SEP closed-loop turn rate plot with thrust vectoring incorporatedto improve turn rate on the constant AOA branch of α = 0.42 rad of Table 4.1. The new constant AOAbranch obtained in Fig. 17 with thrust vectoring is ‘e.’ On branch ‘e’, the elevator is fixed at maximum updeflection and pitch TV is allowed to vary with wind axis roll angle µ. This allows maximum possible down

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Mach No.

Figure 15. Effect of TV on sustained level-turn performance for a fighter airplane (T/W = 1).5

deflection of pitch TV to get thrust component along the lift vector, while allowing pitching moment balanceto keep angle of attack fixed at 0.42 rad. The branches ‘a’ and ‘b’ are same as in Fig. 12. The improvementin the turn rate obtained using thrust vectoring between branches ‘a’ and ‘e’ is of the order of 1 − 2 deg/sfor a given Mach number. The improvement in the turn rate for a constant wind-axis roll angle µ can beattributed to a decrease in the velocity which is due to the fact that deflecting nozzle downwards (positiveδpv) increases the thrust component in the direction of the lift, and consequently for weight balance, the liftcan decrease as compared to the lift without TV. As angle of attack is fixed, for lift to decrease, velocitydecreases. This phenomenon happens even under descending flight conditions in Fig. 17.

V. Controlled Departure and Recovery from Flight Instabilities

Bifurcation theory, as an approach to investigate airplane departures, has been examined by many re-searchers over the last three decades. Numerous studies (see Goman et al12) have demonstrated the capabilityof the technique to pinpoint departure-prone regions of the flight envelope by determining all steady-stateconditions attainable by the airplane and their stability. However, bifurcation analysis cannot determine thetransient behavior leading to departure. Thus, theoretical predictions are supplemented by offline simulationand/or piloted simulation for complete departure analysis. Two essential, distinct stages of investigation arepredicting departures using bifurcation analysis, and validating and investigating the nature of the predicteddepartures in off-line or piloted simulation.27

Piloted simulation in departure analysis is often associated with problems such as uncontrollability ofunstable open-loop airplanes, and finding suitable methods to induce specific departure mode. The firstproblem is addressed using control law based on nonlinear dynamic inversion. Littleboy and Smith19 haveshown, using closed-loop bifurcation analysis, that control law based on NDI only modifies the stability of

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Pitch TV

Elevator surface

Control surface contour C.G. contour

Moment arm

Figure 16. Top view of F-18/HARV airplane.26

0.1500.175

0.2000.225

0.2500.275

0.3000.325

0.350

0.

5.

10.

15.

20.

25.

Mach Number

ab

Tur

n R

ate

(deg

/s)

e

Figure 17. Closed-loop turn rate vs Mach number with thrust vectoring (level and descending turns) (fulllines: stable equilibriums).

the steady states without changing their structure. Furthermore, the only factor that restricts the flightenvelope of an NDI augmented airplane is control input saturation. These facts can be verified by comparingopen-loop and closed-loop bifurcation plots of longitudinal flight in section 3. Thus, NDI control is usedin this work to enable the pilot to attain departure-prone flight conditions in a controlled manner. TheNDI controller is switched on to reach the departure flight condition and then switched off to investigate theopen-loop departure modes. The second problem of inducing a particular departure mode is airplane-specific.

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The two flight instabilities that have been investigated in this work are pitch bucking and spin. In presentanalysis, departure to pitch bucking is attained from a longitudinal dive trim state that has highest value ofcontrollable angle of attack, whereas departure to spin is initiated from a falling turn trim state.

A. Instability Prediction

Instability detection in the F-18/HARV has been accomplished using standard bifurcation analysis. Elevatordeflection δe is used as the continuation parameter for the standard bifurcation analysis (SBA), with aileronand rudder fixed at zero, and throttle at 0.38. Figure 18(a) shows the equilibrium values of angle of attack α

(a) (b)

(c) (d)

Figure 18. Bifurcation diagrams of (a) angle of attack α, (b) roll rate p, (c) yaw rate r, and (d) pitch angleθ, plotted against δe, with δa = δr = δpv = δyv = 0 and η = 0.38 (Dashed lines: unstable equilibria; solid lines:stable equilibria; open squares: pitchfork or transcritical bifurcation points; filled squares: Hopf bifurcationpoints; filled circles: stable limit cycles; open circles: unstable limit cycles).6

as a function of elevator deflection δe. Over the low-to-moderate α range between 0 and 43 deg (0.75 rad),the airplane dynamics consists mostly of stable equilibria with very short stretches of unstable equilibriumpoints bounded on either side by Hopf bifurcations (marked with filled squares). Figures 18(b) and 18(c)show, respectively, the roll rate and yaw rate for these equilibria to be zero, thereby indicating that theseequilibria correspond to symmetric flight. Equilibrium solutions for negative values of angle of attack areunstable due to loss of spiral stability. At the other end of the stable branch of equilibria, beyond the Hopfbifurcation marked ‘H1’ at 43 deg angle of attack, the equilibrium solutions are all unstable. Instead, abranch of unstable limit cycles is created at ‘H1’ which turns stable at a fold bifurcation marked ‘F1’ atα = 50 deg (0.875 rad). The limit cycle motion on this branch can be seen from Figs. 18(a)-(d) to consist

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predominantly of a pitching oscillation, accompanied by comparatively smaller roll and yaw rates, that issometimes called “pitch bucking.”

Figures 18(b) and 18(c) reveal that the unstable equilibrium solutions beyond the Hopf bifurcation‘H1’ deviate towards negative roll and yaw rates. This is caused by non-zero and asymmetric values ofthe lateral force and moment coefficients due to right and left elevator deflection at higher angles of attack.Figure 18(a) further shows an apparently unconnected branch of unstable equilibrium solutions at high anglesof attack between 65 deg (1.14 rad) and 72 deg (1.25 rad). These equilibria were obtained by continuing thecomputations beyond the up-elevator deflection limit to the left of Fig. 18(a) where the branch of unstablesolutions passing through ‘H1’ was seen to fold back and reappear in the figure as a high angle of attackequilibrium branch. Figures 18(b), 18(c), and 18(d) show these equilibriums to correspond to an unstableequilibrium spin solution having a large negative flight path angle and with high negative roll and yaw rates.Further, a stable oscillatory spin solution is seen to emerge from this unstable equilibrium branch at a Hopfbifurcation point marked ‘H2’ at an angle of attack of about 70 deg (1.22 rad). The peak roll and yaw ratesin this stable limit cycling spin solution can be observed from Figs. 18(b) and 18(c) to be quite large andnegative, indicating a rapid left spin with nose pitched below the horizon. This oscillatory spin predicted bythe bifurcation analysis appears to match fairly well with observations on a scaled F-18/HARV model in aspin tunnel.28

B. Controlled Pitch Bucking Departure and Recovery

In this section, controlled departure to pitch bucking flight instability and subsequent recovery to level trimstate is attempted. The complete maneuver is summarized in Table 3.

Table 3. Summary of pitch bucking maneuver

Time (s) Maneuver Control law Status Commanded parameters

0-75 Dive longitudinal trim on αc = 1.17 rad, η = 1

βc = 0, µc = 0

75-150 Pitch bucking off δe = −0.4 rad, η = 0.38

δa = 0, δr = 0, δpv = 0, δyv = 0

150-165 Recovery to level trim on αc = 0.3 rad, η = 0.54

βc = 0, µc = 0

The airplane is initially placed (Fig. 19) in the dive trim state having highest value of controllable angleof attack of 64 deg (1.17 rad) as described earlier in Fig. 7(a). The control law is kept on till 75 sec toachieve the departure point without losing stability and also to account for possible errors in specificationof the initial conditions. At 75 sec, the airplane is allowed to depart by shutting off the control law andsetting the controls to δe = −0.4 rad, δa = 0, δr = 0, and η = 0.38, which corresponds to the pitch buckingcondition as can be observed in Fig. 18(a). It can be noticed in Fig. 19 that the airplane departs to pitchbucking instability in 25 sec. The control law is then reinitiated at 150 sec for recovery from pitch buckingto a level trim state. The trim state chosen for the recovery is M = 0.2, α = 0.3 rad, η = 0.54, δe = −0.1rad, with values of other variables taken to be zero. The commanded values of angle of attack αc = 0.3 rad,sideslip βc = 0, and win-axis roll angle µc = 0 are given as step commands at t = 150 sec. Time historiesof the various states and control inputs required during the recovery can be observed in Figs. 19 and 20,respectively. It is seen in Figs. 19(a) and (b) that both lateral and longitudinal variables settle down totheir trim values in about 10 seconds from the point of application of recovery controls at t = 150 sec. Thephugoid mode, which is uncontrolled, has a larger timescale but is damped as seen in Fig. 19(c), which showsthat level flight path angle γ of 0 deg is also achieved; that is, the airplane is recovered from pitch buckingto level flight.

C. Controlled Spin Departure and Recovery

In this section, controlled departure to oscillatory spin and subsequent recovery to level trim state is at-tempted. The complete maneuver has been summarized in the Table 4. It can be noticed in Fig. 18(a) thatthe oscillatory spin branch starts from ‘H2’ at α = 1.2 rad and continues to higher α with increase in up

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(a) (b)

(c) (d)

Figure 19. Time history of (a) α (blue), β (black), φ (red), (b) p (red), q (black), r (blue), (c) γ, and (d) M ,states during entry and recovery from pitch bucking.

Table 4. Summary of spin maneuver

Time (s) Maneuver Control law Status Commanded parameters

0-60 Falling turn trim on αc = 0.61 rad, η = 1

βc = 0, µc = −1.3 rad

60-150 Steady Spin on αc = 1.25 rad, η = 1

βc = 0, µc = −1.3 rad

150-200 Oscillatory Spin off δe = −0.43 rad, η = 0.38

δa = 0, δr = 0, δpv = 0, δyv = 0

200-220 Recovery to level trim on αc = 0.3 rad, η = 0.54

βc = 0, µc = 0

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elevator deflection. Thus the spin branch always lies at higher α than the maximum controllable α of 64 deg(1.17 rad) in Fig. 7(a). Furthermore, departure from this dive trim state of α = 64 deg (1.17 rad) by turningoff the controller is always seen to lead to pitch bucking as in Fig. 19. In order to initiate the airplane intodeparture to spin, it is placed in a descending turn as described in section 4. The initial state chosen forthe airplane is a falling left turn trim with M = 0.25, α = 0.6 rad, µ = −1.3 rad, p = −0.2 rad/s, q = 0.25rad/s, r = −0.22 rad/s, η = 1, δe = −0.1 rad. At 60 sec, the airplane is commanded to a controllablesteady spin. In this state, the control law is kept on till 150 sec to achieve the departure point without losingstability. At 150 sec, the airplane is allowed to depart by shutting off the controller and setting the controlsto δe = −0.436 rad, δa = 0, δr = 0, and η = 0.38, which represents oscillatory left spin condition as can be

(a)

(b)

Figure 20. Time history of (a) δe (red), δa (black), δr (blue), and (b) η (blue), δpv (black), δyv (red), controlinputs during entry and recovery from pitch bucking.

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observed in Fig. 18(a). It can be noticed in Fig. 21 that the airplane successfully departs to oscillatory spininstability in 50 sec.

The controller is then re-initiated at 200 sec for recovery from spin to the level trim state. The trim statechosen for the recovery is the same as in the previous section (M = 0.2, α = 0.3 rad, η = 0.54, δe = −0.1 rad)with other variables taken as zero. The commanded values of angle of attack αc = 0.3 rad, sideslip βc = 0,and wind-axis roll angle µc = 0 are given as step commands at t = 200 sec. Time histories of the variousstates and control inputs required during the recovery can be observed in Figs. 21 and 22, respectively. Itis seen in Figs. 21(a) and (b) that both lateral and longitudinal variables settle down to the trim values inabout 10 sec from the point of application of recovery controls at t = 200 sec. The phugoid mode is alsodamped as seen in Figs. 21(c) and (d), thus the airplane is recovered from spin to level flight.

It is important to note that in both the examples considered above, as the aerodynamic controls approachsaturation, thrust vectoring has to be engaged to effect recovery to the desired trim state. Although thrustvectoring is used in short ‘bursts,’ the deflection of the nozzles, given by δpv and δyv, is seen to be quitesignificant. This clearly demonstrates the important role played by thrust vectoring in the recovery process.

(a) (b)

(c) (d)

Figure 21. Time history of (a) α (blue), β (black), φ (red), (b) p (red), q (black), r (blue), (c) γ, and (d) M ,states during entry and recovery from oscillatory left spin.

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(a)

(b)

Figure 22. Time history of (a) δe (red), δa (black), δr (blue), and (b) η (blue), δpv (black), δyv (red), controlinputs during entry and recovery from oscillatory left spin.

VI. Conclusions

The objective of this work has been to qualify and quantify effect of thrust vectoring on airplane trim,stability, and maneuvers using bifurcation analysis and use the information obtained from bifurcation dia-grams as an aid for controlled departure and subsequent recovery from departed flight states to a level flighttrim using a nonlinear dynamic inversion (NDI) scheme as the control strategy. Bifurcation analysis is anonlinear analysis technique, which is uniquely suited to study the high angle of attack (AOA) dynamics ofan airplane with and without a controller in the loop.

1. Bifurcation Analysis of Longitudinal Flight: Extended Bifurcation Analysis has been carried outfor the F-18/HARV airplane model to generate longitudinal flight trims. It is found with implementa-

27 of 29


tion of TV as a supplementary control to aerodynamic controls, that maximum controllable angle ofattack increases from 55 deg (0.96 rad) to 67 deg (1.17 rad).

Effect of control saturation on closed-loop stability was identified. It is found that control saturationleads to relocation of eigenvalues of the Jacobian matrix of the closed-loop nonlinear airplane dynamicalequations in a manner that, after saturation, stability characteristics of the closed-loop airplane areprimarily determined by the nature of the corresponding open-loop airplane dynamics.

2. Bifurcation Analysis of Turning Flight: Extended Bifurcation Analysis was carried out to findall the turn trims of the airplane. It was observed that the NDI control law stabilizes all unstable turntrims without altering the trim states. Instantaneous descending turn, which accounts for throttlesaturation and negative SEP, shows that turn rate at corner point is limited by roll angle, not theload factor which decreases with steeper descending flight path angles. Furthermore, for instantaneousdescending turn, thrust vectoring is found to increase the turn rate by 5% (1 − 2 deg/s) for a givenvelocity, on the maximum CL branch.

3. Controlled Departure and Recovery from Flight Instabilities: Controlled departure to pitchbucking and spin were simulated using the closed-loop F-18/HARV airplane model. High angle ofattack longitudinal dive trim state was chosen for departure to pitch bucking. Aircraft was foundto successfully depart to pitch bucking by switching the control law off. Subsequent recovery wascompleted within 10 sec, using the control law. For departure to oscillatory left spin, the departurestate chosen was a descending left turn. Aircraft was found to successfully enter developed spin statein 50 sec, then spin recovery was initiated.

References

1Herbst, W. B., “Dynamics of Air Combat,” Journal of Aircraft, Vol. 20, No. 7, 1983, pp. 594-598.2Alcorn, C. W., Croom, M. A., Francis, M. S., and Ross, H., “The X-31 Aircraft: Advances in Aircraft Agility and

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Perseus Books Group, Cambridge, MA, USA, 1994.10Ananthkrishnan, N., and Sinha, N. K., “Level Flight Trim and Stability Analysis using an Extended Bifurcation and

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and Dynamics, Vol. 5, No. 5, 1982, pp. 529-536.12Goman, M. G., Zagaynov, G. I., and Khramtsovsky, A. V., “Application of Bifurcation Methods to Nonlinear Flight

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[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference and Exhibit - Keystone, Colorado ()] AIAA Atmospheric Flight Mechanics Conference and