A Framework for Analysis of Aircraft Maneuverability

95–3448

A FRAMEWORK FOR ANALYSIS OF AIRCRAFT

MANEUVERABILITY

Jeffrey A. Beck∗

Air Force Institute of Technology, Wright-Patterson AFB, OH 45433

Thomas J. Cord†

Wright Laboratory, Wright-Patterson AFB, OH 45433

We present a framework of definitions and relationships which represent the prelimi-nary development of a generalized theory of aircraft maneuverability. Maneuverability isdefined as a measure of the ability to achieve and transition between steady maneuvers. Wefurther categorize maneuverability measures as maneuver performance or agility measureswhere maneuver performance is a measure of steady maneuver capability and agility is ameasure of the ability to transition between steady maneuvers. The motion vectors (veloc-ity, acceleration, etc.) are decomposed in the Frenet axis system and a new categorizationof components is identified according to the above definitions. We highlight importantparameters appearing in the components from which we develop maneuver performanceand agility envelopes. This framework provides an analytical basis on which to considermaneuverability in the aircraft development process, establishing a bridge between designand evaluation. It provides insight into the use of maneuverability in close-in air com-bat situations through ties to (and extensions of) energy-maneuverability theory. We alsoconsider the relationship between maneuverability and handling qualities, emphasizing theneed to treat them as two separate design goals.

I. Introduction

I.A. Problem Definition

The maneuverability of an aircraft is a composite of the ability to achieve rates, accelerations, and all higherderivatives in translation and rotation. For relatively steady maneuvers of long duration, the higher-orderderivatives do not significantly influence the time to perform a specified task. The time required to transitionto the desired speed and rotation rate is small compared to the total time of the maneuver. As the durationof the maneuver is reduced or as the maneuver involves more transitioning, the importance of higher-orderderivative capability increases. We wish to describe in simple terms the application of aircraft motionto accomplish tasks which require a high degree of maneuvering. Our discussion centers on establishingrequirements for air-to-air combat. However, such a description would be equally useful for other tasks suchas lateral offsets on approach, terrain following, or surface-to-air weapon evasion.

Formulating a suitable description has proven difficult due to the interplay of various factors and con-straints. Parameters in the areas of performance, maneuverability, and handling qualities have, for certainsystems, correlated well with success in the air combat arena. However, improvements in technology canlead to new capabilities which require new descriptors to properly account for the impact on aircraft ef-fectiveness. Historic examples of this can be seen in propulsion (more speed, altitude, endurance, range),structures (increased load factor, payload), avionics (better situational awareness), and weapons (decreasedtime to kill). Energy-maneuverability was created in the 1960s to account for the state of technology andtactics that existed at that time. It was necessary because the individual technical area metrics were nolonger sufficient to describe capabilities and predict the outcome of air battles.

We may once again need a new, integrating parameter to describe the capability which exists in today’saircraft. This situation arises because of a combination of environmental issues and technical capabilities.

∗Doctoral Candidate, Department of Aeronautics and Astronautics. Senior Member AIAA.†Flying Qualities Engineer, Flight Dynamics Directorate.This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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American Institute of Aeronautics and Astronautics Paper 95–3448

We anticipate future air battles where our forces are at a numerical disadvantage, relying on high technologyto provide a “force multiplier.” Weapons have evolved to the point where they outperform the aircraft andhave eliminated the possibility of a safe disengagement. Expanded flight envelopes in terms of angle of attackand increased control power through thrust vectoring have given the pilot a whole new set of dynamics thathe can choose to use in air combat. Based on the assumption that close-in combat will continue to benecessary, whether because of strategy or just sheer numbers, the above factors result in a need to minimizethe time to kill one adversary and target the next. This line of reasoning led to the search for the elusiveelement known as agility.

I.B. Previous Developments

Aircraft agility has been a subject of considerable discussion and research in the recent past. In some of theearliest agility research, Stutz and Price? compared aircraft by performing standard maneuvers and thenanalyzing trajectories and time histories of parameters such as load factor and acceleration. Some more recentinvestigators such as Lawless and Butts? and Wilson, et al?,? have continued in this direction. There hasbeen increasing emphasis on the development of new metrics to quantify agility. Herbst, et al? and Mazza?

have sought to define agility in a strict mathematical sense, while Kalviste,? Skow, et al?,?,? and others havetaken a more empirical approach to developing metrics. Here we are interested in mathematical derivations.However, such an approach should not conflict with the intuition used in developing the empirical metrics.

The development of agility has also evolved into two primary views with regard to the dynamics ofinterest: flight-path agility, which is derived from analysis of the flight path or velocity vector and pointingagility, which arises from analysis of the vehicle attitude dynamics. The relative importance of these twoconcepts depends upon the specific task being performed. Flight-path analysis is the basis of the parametersand relationships presented in this report. However, it should not be difficult to establish a direct relationshipbetween the two views (at least for conventional fixed-wing aircraft).

The flight-path approach has been taken by several other researchers using the Frenet axis system and theidea that agility is the derivative of maneuverability. In Ref. [?], four different agility definitions are presented.Each derivation starts from the premise that maneuverability is embodied in the Frenet components of theacceleration vector and agility is the derivative of maneuverability. The derivation attributed to Hornfollows these ideas to their logical conclusion that agility is embodied in the components of the jerk vector.The derivation by Mazza? is equivalent to this rigorous derivation. The remaining derivations in Ref. [?]give variations on Horn’s result, trading rigor for intuition. The principal departure occurs in the torsioncomponent where the roll rate term is dropped in favor of roll acceleration. Herbst justifies this by notingthat Horn’s torsion component is nonzero during a steady barrel roll. This discrepancy was the principalfactor leading to the new grouping of parameters which we present below.

I.C. Approach

Because maneuverability is often thought of as “the ability to change the velocity vector,” earlier attemptsto mathematically define agility have started by equating maneuverability with the acceleration vector asdescribed above. This restricted interpretation of maneuverability essentially forces a mathematical definitionof agility which conflicts conceptually with some proposed empirical measures. Furthermore, it leads tothe less than satisfying conclusion that maneuverability and agility are little more than new titles for theacceleration and jerk vectors.

In order to derive a more useful result, we begin by restating the definition of maneuverability in broaderterms: Maneuverability is a measure of the ability to achieve and transition between steady maneuvers. Thisdefinition, like previous ones, suggests that maneuverability includes accelerations and all higher derivatives.A subtle but important difference arises in that our definition of maneuverability includes velocity, or, moreprecisely, speed capabilities. We introduce a new term, maneuver performance, as the measure of first-order (or steady) maneuver capability. We then define agility as the measure of second-order (or transition)maneuver capability. The Venn diagrams in Figure 1 highlight the difference between our approach andprevious mathematical derivations based on a restricted definition of maneuverability.

The presentation of this report follows three steps. In the next section, we present the derivation of themotion vectors (i.e., the velocity, acceleration, etc.) in the Frenet axis system. This reference frame is anoninertial system tied to the trajectory which decomposes motion along the flight path into its fundamental

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(a)

(b)

Figure 1. Maneuverability and agility as viewed in (a) previous derivations and (b) the current approach.

components. We then examine these components and determine how they are related to maneuver perfor-mance and agility as defined here. This process leads to a logical division of components which appearsto be more appropriate and in better agreement with proposed empirical measures. Finally, we provide apreliminary look at the utility of this new “framework.”

II. Motion Vector Components

We derive the components of the motion vectors (velocity, acceleration, etc.) in the Frenet coordinateframe. A similar development is given by Horn,? however, our derivation includes an additional derivative(for reasons which should become obvious). The Frenet frame is tied to the trajectory traversed by thevehicle center of mass and rotates relative to an inertial coordinate frame. Let t, n, b be the orthogonalset of basis vectors in the Frenet frame as shown in Figure 2. The orientation of the Frenet system withrespect to a North-East-Down inertial systema can be described by a 3–2–1 set of Euler angles χ, γ, σ.

Figure 2. The Frenet axis system.

The first two angles denote the standard heading and flight-path angles, respectively. The third we will referto as the force inclination angle. It measures the inclination of the normal force from the horizontal in thenormal plane (see Figure 5). To describe the rotation of the Frenet coordinate frame, define the rotationvector

~Ωf = pf t + qf n + rf b. (1)

aA flat Earth fixed in space is assumed for this development.

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The components of the rotation vector are then related to the rate of change of the Euler angles by

pf = σ − χ sin γ (2a)

qf = γ cosσ + χ cos γ sinσ (2b)

rf = χ cos γ cosσ − γ sinσ. (2c)

Certain characteristics of the Frenet coordinate system permit these expressions to be simplified. The totalforce acting on the vehicle consists of a component in the tangential direction and a component normal to theflight path, which by definition is in the direction of the normal axis. Therefore, the force is constrained tobe in the maneuver plane at all times. Since there is no component of force in the binormal direction, therecan be no rotation about the normal axis. Hence, qf is identically equal to zero.b Also, since the normalforce component is always in the direction of the positive normal axis, the rotation about the binormal axisis always positive (i.e., rf is greater than or equal to zero). The rotation terms then become

pf = σ − χ sin γ (3a)

qf = 0 (3b)

rf =√γ2 + χ2 cos2 γ =

χ cos γ

cosσ(3c)

Note that rf represents the turn rate in the maneuver plane and pf represents the maneuver-plane roll rate(about the velocity vector). In the following, we adopt the more common notation for turn rate, ω, but weemphasize that turn rate is measured in the maneuver plane which does not necessarily coincide with thehorizontal plane.

To express the velocity of a vehicle in the Frenet system, we recognize that the tangential unit vector, t,is defined to be in the direction of travel at any given instant. Therefore, if a particle moves with speed v,the velocity is given by

V = vt (4)

Note that the Frenet axis system unit vectors, t, n, and b, are time-varying with respect to an inertial frame.Taking the derivative of the velocity vector with respect to time in the inertial frame gives the accelerationvector:

A =idV

dt

=fdV

dt+ ~Ωf × V

= vt + vωn

(5)

where the left superscript on the derivative indicates the frame in which the derivative is defined. Differen-tiating again gives the jerk vector

J =idA

dt

=fdA

dt+ ~Ωf × A

= (v − vω2)t + (2vω + vω)n + (vωpf )b

(6)

bWe can show this more rigorously. Let F be the total force vector. Since F is constrained to be in the maneuver plane bydefinition, we have F · b ≡ 0. By Newton’s 2nd Law, this implies A · b ≡ 0 where A is the acceleration. (This is true regardless

of whether or not mass is assumed constant.) But it is easy to show that A = vt + vrf n− vqf b from which we must concludethat qf ≡ 0.

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Finally, differentiating a third time gives a quantity we will refer to as the burst vector:

B =idJ

dt

=fdJ

dt+ ~Ωf × J

= (...v − 3vω2 − 3vωω)t

+ (3vω + 3vω + vω − vω3 − vωp2f )n

+ (3vωpf + 2vωpf + vωpf )b

(7)

The complete set of motion vector components is presented in Table 1.

Table 1. Frenet Components of Motion Vectors

Axial Curvature Torsion

t n b

Velocity v

Acceleration v vω

Jerk v − vω2 2vω + vω vωpf

Burst...v − 3vω2 − 3vωω

3vω + 3vω + vω

− vω3 − vωp2f3vωpf + 2vωpf

+ vωpf

The turning or bending of the path in the maneuver plane is referred to as the curvature of the trajectory.It is defined mathematically by

κ(s)4=

∣∣∣∣dtds∣∣∣∣ (8)

where s is a measure of distance along the flight path. The curvature, κ, is the inverse of the instantaneousradius of curvature and is related to turn rate by

ω = vκ (9)

where v = ds/dt is the speed. Similarly, the rotation of the maneuver plane about the velocity vector isreferred to as the torsion of the flight path and it is defined mathematically by

τ(s)4= −n · db

ds(10)

The torsion, τ , is related to the maneuver-plane roll rate by

pf = vτ (11)

Thus τ and κ are the spatial equivalents of the temporal rotation rates pf and ω. With these relationshipsit is simple to show that the equations derived by Horn? and Mazza? are equivalent. In this analysis, weuse the temporal notation. However, some terminology is taken from the spatial development. We referto the components in the tangential, normal, and binormal directions as the axial, curvature, and torsioncomponents, respectively (see Table 1).

III. Maneuverability

III.A. Maneuverability Components

Steady maneuvers are idealized vehicle motions involving constant speed and constant rotation rates. Transi-tion maneuvers are motions involving changes in speed or rotation rates. In the most general form, a steady

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maneuver follows a helical path. Barrel rolls and spiraling climbs are examples of steady maneuvers in threedimensions. Straight-line flight and steady turns represent one- and two-dimensional steady maneuvers.

In Figure 3, we determine the reduced form of the motion vector components for steady maneuvers withone-, two-, and three-degrees-of-freedom. For steady helical flight we find that in each axis the first termis complete (as compared to Table 1) and the second term is identically zero. When considered with thedefinitions presented above, this suggests that the first component in each axis should be identified withmaneuver performance while the second term should be identified with agility. This breakdown is shownschematically in Table 2 with regard to the components identified in Table 1. As a basis for comparison,Tables 3 and 4 show the component categorizations given in [?] and [?].

Figure 3. Motion vector components during steady flight.

III.B. Maneuverability Parameters and Envelopes

Steady maneuvers and the maneuver performance components are completely specified by three quantities:v, ω, and pf . We refer to these as the maneuver performance parameters and we refer to their values at aparticular instant as the maneuver state. We can associate each parameter with the motion vector componentin which it first appears as we increase degrees of freedom. That is, we consider v the axial parameter, ωthe curvature parameter, and pf the torsion parameter. Each maneuver performance parameter measures a“rate” — v is a translation rate along the trajectory while ω and pf are rotation rates.

Similarly, the agility components are completely specified by the maneuver performance parameters andtheir first derivatives, v, ω, and pf (the translational and rotational accelerations). We may associate each of

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Table 2. Maneuver Performance and Agility Components

Table 3. Horn/Mazza Selection of Components

Table 4. Zehner/Eberle/Kiefer Selection of Components

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these accelerations with the corresponding agility components to get the axial, curvature, and torsion agilityparameters.

To represent maneuver performance and agility graphically, the rate and acceleration parameters are usedto define axes in three dimensions. For maneuver performance, the limits on speed, turn rate, and torsionrate are determined by structural, lift, and control limitations. Also, these limits are a function of altitudeand are distorted by the instantaneous orientation of the flight-path axes with respect to the gravity vector.The maneuver performance envelope bounds the set of possible maneuver states and is shown conceptuallyin Figure 4. For agility, the limits on the acceleration parameters are not only functions of altitude andinstantaneous orientation, they are also functions of the maneuver state. In effect this means that there isa different agility envelope for each point in the maneuver performance envelope. The limits on linear, turn,and torsional acceleration are primarily determined by control power and actuator rates.

Figure 4. Maneuver performance envelope.

III.C. Maneuverability vs. Handling Qualities

Handling qualities is an area of research which looks at the characteristics of an aircraft that influence howwell the pilot can perform a specified task. Some of the criteria involved relate directly to rotational andtranslational accelerations. Therefore, handling qualities researchers have shown considerable interest inaircraft agility. Several research efforts have specifically looked at the extent to which handling qualitiesrequirements determine agility.?,?,?,?

Maneuverability, as defined here, is a measure of the open-loop capabilities of an aircraft. The objectiveis quickness and the more available the better. In contrast, handling qualities involves the specification ofopen-loop parameters which have been correlated with desirable closed-loop performance. The objective isa proper combination of quickness and precision. There is usually an upper and lower bound on quicknessassociated with a given degree of desired precision. Too slow and the pilot considers the aircraft “sluggish”and may have a tendency to overcontrol. Too quick and the pilot finds the aircraft “jumpy” or “oversensitive.”This has led some researchers to conclude that handling qualities requirements determine agility.

In fact, some of the specifications in the handling qualities standard incorporate maneuverability require-ments. However, several problems arise from this approach. First, this assumes that achieving good handlingqualities is the primary objective, with maneuverability being a secondary consideration. Thus, an aircraftwith Level I handling qualities would always appear better than a more maneuverable aircraft with LevelII handling qualities. Secondly, this does not allow for the possibility that different control strategies mayenable improvements in maneuverability without degrading handling qualities. That is, without separatemaneuverability requirements, little impetus exists to seek improved maneuverability. For this reason, abetter approach might be to consider maneuverability (maneuver performance and agility) and handlingqualities as separate design objectives which must be traded off to achieve the best design.

IV. Applications

Given the above framework of maneuverability components and parameters, we next consider how itmight be utilized in the processes of aircraft development and deployment. In the development process,

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we are interested in (1) how to specify requirements for maneuverability in new aircraft, (2) how to designfor better maneuverability in new aircraft, and (3) how to evaluate the maneuverability of existing aircraft.For deployment, we would like to determine how to use a maneuverability advantage over an adversary (orcounter a disadvantage) in tactical situations.

IV.A. Development

IV.A.1. Specification:

As stated above, it is our view that the handling qualities standard is inappropriate as a sole means ofspecifying maneuverability requirements. Of the maneuver performance parameters, v, vω, and vωpf , thefirst two are normally addressed as performance requirements on maximum speed and load factor. The thirdis expressed in the handling qualities area as the time to roll through ninety degrees of bank angle with theaircraft loaded up to eighty percent of limit load factor. This is often ignored in aircraft design because of thedifficulty in quantifying the benefit, thus the ability will usually be lost in favor of reduced aircraft weight orincreased range. The agility parameters, with the exception of v, bear little resemblance to any specificationparameters. v is usually specified in terms of time to accelerate from one Mach number to another. Thecurvature and torsion components of agility have yet to be tied to aircraft design requirements, althoughthere may be a possibility of relating dynamic elements of the torsion component to the roll requirementsof the handling quality specification. The curvature component would likely be based on pilot limitation interms of the rate of change of load factor.

An alternate method would be to establish task-oriented performance criteria for a standard set ofevaluation maneuvers. The particulars of the maneuvers would be tailored to the mission needs of the aircraftbeing procured. The criteria would be specified in terms of the maneuver performance and agility parametersdemonstrated by or projected from existing aircraft capability to correlate with desired performance. Theaircraft designed to these values would be capable of performing the trajectories necessary to accomplishthe specified tasks. Handling qualities would still have to be satisfied in order to assure the pilot’s ability toemploy the weapon system with acceptable workload.

IV.A.2. Design:

For design, we would like to determine how conventional parameters (wing loading, thrust-to-weight ratio,etc.) relate to maneuverability and whether any additional design parameters are necessary to establish themaneuverability of a proposed aircraft. To this end, we examine the interplay of forces and moments in themeans by which the pilot controls the force vector.

The motion of a (rigid) aircraft is described by a twelfth-order system of nonlinear differential equations.These may be expressed as six second-order equations — three associated with translation of the center ofmass (the “force equations”) and three associated with rotation about the center of mass (the “momentequations”). Most previous attempts to develop mathematical expressions for agility have been based solelyon analysis of the force equations. Some of these researchers have argued that the attitude dynamics whichare described by the moment equations are a separate issue adequately handled by the flying qualitiesdiscipline. On the other hand, some researchers have concentrated strictly on moments, arguing that forcesdetermine maneuverability while moments determine agility. Virtually none (with the notable exception ofBurgess and Ilgenfritz?) have acknowledged the importance of understanding how the forces and momentsinteract. To measure maneuverability, analysis of the forces is sufficient. But to design for maneuverability,both the forces and the moments must be considered.

In order to demonstrate this, it is beneficial to consider how the pilot of a conventional aircraft controlseach axis specifically. In the tangential direction, the pilot controls the forces directly. He accelerates bythe application of thrust and he decelerates by the reduction of thrust and by the use of drag devices. Incurvature, however, the pilot does not control the force directly. Instead, he controls pitching moment whichhe uses to increase or decrease angle of attack, and subsequently increase or decrease normal force. Thepitching moment and angle of attack are also a secondary means of controlling the forces in the tangentialdirection. Torsion is controlled directly by the application of coordinated body-axis rolling and yawingmoments. Hence, the pilot uses both forces and moments to control the aircraft trajectory and both mustbe considered to understand the relationship between maneuverability and aircraft control.

For a conventional aircraft, holding the controls in a fixed position corresponds to a more or less “steady”maneuver (i.e., constant velocity and rotation rates) although some variations do occur due to gravitational

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and altitude effects. Constant rates imply that the tangential forces are balanced, as are the moments.The normal forces are not necessarily balanced, but are relatively constant so that the pilot senses a steadyload factor. Movement of the controls unbalances these forces and moments and results in accelerations ordecelerations toward a steady state corresponding to the new controller position. Thus, stick and throttleposition relate to the maneuver state (maneuver performance) and stick and throttle movement relate tochange of the maneuver state (agility). This same idea was the basis for one of the approaches to agilitydiscussed by Herbst? but did not correlate well with their definitions of agility due to their initial assumptionthat maneuverability is acceleration. It does, however, correlate with the mathematical definition presentedabove.

Ultimately, we would like to derive the precise relationships between design parameters and maneuver-ability parameters. This would a involve a transformation of the Frenet motion vector components to thebody frame. The wind axis system could be used as an intermediate system, so that the transformationwould be a function of the force inclination angle, σ, the wind axes roll angle, µ, the angle of attack, andthe sideslip angle. Figure 5 shows the relationship between the Frenet and wind axis systems. They differonly in the third Euler angle. We hope to pursue this topic in a future effort.

Figure 5. A comparison of the third Euler angle for the Frenet and Wind axis systems.

IV.A.3. Evaluation:

We wish to determine how completely and accurately existing flight-test maneuvers and metrics measure ma-neuverability and whether additional measures are required. Maneuverability is a measure of the open-loopcapability of the aircraft. As such, any evaluation maneuver which involves a full pilot input with essentiallyno attempt at precise control evaluates maneuverability. For instance, maximum level accelerations and de-celerations, time to turn through a given change in heading angle at various speeds, and time to roll througha given angle at various load factors are all maneuverability evaluation maneuvers. By turning or rollingthrough an angle large enough to ensure the maximum rate is achieved, we should be able to evaluate boththe acceleration and rate limits from analysis of the time history. Thus maneuverability evaluation is presentin current standard flight test maneuvers. Likewise, by repeating these with a capture requirement on theend of the maneuver, we can evaluate the closed-loop, handling qualities aspects. Thus, the only questionwhich may bear further investigation is whether we currently analyze the data from existing maneuvers in amanner which properly portrays the maneuverability of an aircraft. We do not pursue this issue here.

IV.B. Deployment

Development of tactical maneuvers is a complex matter which must account for many factors in additionto differences in aircraft maneuver capabilities. This subject goes well beyond the scope of the presentdiscussion. However, we can provide some insights into this problem which may aid in the development ofmaneuvers.

In the mid 1960s, Boyd and Christie developed a method of approximating the maneuvering capabilityof an aircraft based upon the energy concepts of Rutowski. This concept, known as energy-maneuverability,has proven itself useful to military pilots by providing a better means of understanding the capabilities ofan aircraft and comparing those capabilities to threat aircraft. In this section, energy and rate of changeof energy are considered along with maneuver performance and agility to show how energy-maneuverabilitytheory relates to the aircraft maneuverability framework presented in this report.

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The total energy of a vehicle, consisting of both the kinetic and potential energy, can be approximatedby

E =1

2mv2 +mgh (12)

where m is the mass of the vehicle, g is the acceleration due to the gravitational attraction of the Earth, h isthe height above some reference height, and v is the speed. This can be divided by the weight of the vehicleto give the specific energy

Es =v2

2g+ h. (13)

Taking the derivative of this quantity with respect to time gives

Ps =v

gv + h (14)

which is referred to as the specific excess power. This quantity identifies the rate at which the vehicle islosing or gaining energy.

Considering the force diagram shown in Figure 6, the equation for the tangential component of force is

Ft = Tt −D −mg sin γ = mv (15)

where D is the drag and Tt is the thrust along the flight path. Tt is a function of throttle setting, altitude,Mach number, angle of attack, and thrust vectoring deflections. Then

Tt −D = mv +mg sin γ (16)

Multiplying Equation (16) by v/(mg) and recognizing that h = v sin γ, the expression on the right-hand sidebecomes the right-hand side of Equation (14). Then we have

Ps =v

mg(Tt −D) (17)

which is the standard expression for specific excess power.

Figure 6. Force diagram in the vertical plane.

If the tangential thrust and drag forces are expressed as functions of v, ω, and pf for a specific altitude andthrottle setting, then surfaces of constant Ps can be generated in the maneuver space. The Ps = 0 surfacefor maximum thrust bounds the region of sustainable maneuvers. Along with the maneuver performanceenvelope, this determines which maneuver states are sustainable, which are achievable instantaneously, andwhich are unattainable. This is shown conceptually in Figure 7. Figure 7(a) shows the cross-section in thev–ω plane which is essentially the standard “doghouse” plot. Figure 7(b) shows the cross-section in the ω–pfplane at some fixed speed, vo. The maneuver performance envelope contour resembles the Eidetics lateralagility diagram.?c However, the diagram includes additional information regarding increases in energy bleedrate due to roll or torsion. The degree to which the Ps = 0 line falls away with increasing torsion rate shouldnot be significant in a conventional aircraft using purely aerodynamic controls. However, significant drop-offmay occur for aircraft utilizing thrust vectoring to augment yaw or to counter inertial pitch-up during a highangle-of-attack roll. Figure 7(c) shows the v–pf cross-section at ω = ωo. It identifies what combinations ofspeed and torsion are sustainable at a given turn rate (or load factor).

cThe Eidetics lateral agility parameter incorporates onset and capture transients effects so that contributions due to torsionalmaneuver performance, torsional agility, and handling qualities are present. For larger roll angles, the torsional maneuverperformance typically dominates.

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Figure 7. Energy-maneuverability diagram.

We may plot similar diagrams at various aircraft configurations and throttle settings. We refer to thesediagrams as energy-maneuverability (E-M) diagrams. The E-M diagram presents the rate of change of theenergy state at each point in the maneuver performance envelope. The rate of change of the energy state isclosely related to axial agility. From Equation (14),

v =g

v

(Ps − h

)(18)

or at constant altitude,

v =g

vPs (19)

Therefore, E-M diagrams with Ps contours for maximum thrust would give a first-order approximation to theupper axial agility (acceleration) bound and E-M diagrams for idle and speedbrakes give an approximationto the lower axial agility (deceleration) bound.

Referring once again to the maneuver performance and agility components of the motion vectors asidentified in Table 1, classical energy-maneuverability addresses the relationship between axial and curvaturemaneuver performance and axial agility. Consideration of the three-dimensional maneuver performanceenvelope adds torsional maneuver performance as well. The above ideas might be extended to consider Ps

and the agility envelope.

V. Conclusions

The original goal of the effort reported here was to develop a definition of agility and a set of metricswith which to measure agility. In the process, we were forced to reassess our conception of maneuverability.The result is a mathematical definition of maneuverability which incorporates agility and which provides alogical decomposition of motion and associated categorization of parameters and metrics.

Our ultimate objective is to apply those metrics to develop aircraft that have a heretofore undescribedadvantage over their adversary. Before this can occur, the parameters presented must be correlated withmission effectiveness and linked to aircraft design parameters. The framework presented in this reportprovides a foundation upon which that work should proceed.

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VI. Acknowledgments

The authors wish to thank Carmen Mazza for several lively discussions on agility. We also wish to thankMark Burgess and Doug Ilgenfritz of Boeing, along with Bob Shaw of Fighter Command International, fortheir helpful insights. Finally, we would like to express our appreciation to the members of AGARD WorkingGroup 19 whose discussions provided a stimulus for the development presented.

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Documents

A Framework for Analysis of Aircraft Maneuverability