JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 19, No. 2, March-April 1996

Unified Development of Lateral-Directional Departure Criteria

R H. Lutze,* W. C. Durham,* and W. H. Mason*Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0203

Several frequently used departure prediction indicators for both open- and closed-loop control of flight aredeveloped using a unified, rigorous analytical approach applied to a linear version of the aircraft model. Thesecriteria are for departure caused by aerodynamic disturbances only. It is shown that these indicators are limited intheir accuracy because of restrictive assumptions and terms omitted. A second approach is presented that leads tothe same results as the first, but is more applicable to the nonlinear problem. Some ideas concerning the applicationof the linear methods to the nonlinear problem are presented.

a, b, c, d, eBbCj , Cn , Cy

"Ap

g

I

In

LN

sVY

A(-)9X

Subscript

x

Superscripts

s

Nomenclature= system matrix, Eq. (1)= /, jth element of system matrix= coefficients of characteristic equation= control matrix, Eq. (1)= wing span, ft= roll, yaw moment, and sideforce nondimensional

^coefficients= lateral departure parameter, Eq. (17)= closed-loop lateral departure parameter, Eq. (33)= acceleration because of gravity, ft/s2

= /,/z-(/«)2,slllg-ft2

= n x n identity matrix= moments of inertia about the x, y, z axes, xz

product of inertia, slug-ft2

= dimensional roll moment, ft-lb= dimensional yaw moment, ft-lb= roll, pitch, and yaw rate about generic body x,y,

and z axes, rad/s= dynamic pressure, psf= reference wing planform area, ft2

= airspeed, ft/s= dimensional side force, Ib= angle-of-attack measured with respect to generic

body jc axis= sideslip angle= perturbation value of (•)= pitch-attitude angle= eigenvalue associated with system matrix= bank-attitude angle

: partial derivative of (•) with respect to x, where jccan be ft, /?, or r

• quantity (•) evaluated in stability axes= dimensional or nondimensional stabilityderivatives including moment of inertia terms,Eqs. (7) and (14)

= quantities with no superscript are evaluated ingeneric body axes

Received Dec. 5, 1994; revision received June 20, 1995; accepted forpublication Sept. 5, 1995. Copyright © 1995 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

* Professor, Department of Aerospace and Ocean Engineering. AssociateFellow AIAA.

^Assistant Professor, Department of Aerospace and Ocean Engineering.Member AIAA.

* Professor, Department of Aerospace and Ocean Engineering. MemberAIAA.

Introduction

A LTHOUGH current fighter aircraft are generally limited tolow angles of attack (AOA), future aircraft designed for an air

superiority role will be required to have high AOA maneuveringcapabilities.1"5 It is, therefore, necessary that the aircraft have suf-ficient inherent stability or that it provide sufficient control .powerto prevent departure from controlled flight. Sufficient control poweris required so that departure resistance can be maintained by ap-propriate pilot inputs or by using feedback stability augmentationmethods. In either case, some design criteria must be available tocharacterize departure behavior.

Although aircraft can depart in many different ways, most of theexisting departure criteria are related to lateral-directional modescharacterized by a nose-slice, a wing-drop, or a wing-rock motion.These departure modes can be caused by several different mecha-nisms, asymmetric wing stall or unstall (wing drop),6 aerodynamicyaw moments exceeding the control authority of the rudder (noseslice),7-8 and aerodynamic rate-damping moments becoming nega-tive (wing rock).

Attempts to develop criteria to be used early in the design processhave been made over the past several years.9"12 The early philoso-phies were developed prior to the computer age and thus aimed atsimple back-of-the-envelope calculations that lead to criteria easilyevaluated in the early design stages. Until recently, this philosophyhas not changed. One of the reasons for this lack of change is thesuccess of these early criteria when applied to rather conventionalaircraft in predicting departure resistance. With the recent interest inhigh AOA maneuvering and the new unconventional fighter aircraftconfigurations, however, these simple methods have become lessaccurate in predicting departure susceptibility.13 A recent survey ofmany of the currently used and proposed criteria is presented bySeltzer.14

Although the early criteria were based on heuristic argumentscoupled with analyzing a considerable amount of data, these, alongwith more recent criteria, can be developed rigorously from basicprinciples and put on a sound theoretical foundation.15'16 Basicallythere are two approaches that are used to develop such criteria,both of which start with the vehicle equations of motion. The firstapproach is to linearize these equations of motion about some ref-erence flight condition, and then do some linear system analysison the resulting equations,9'11'13 whereas the second approach dealsdirectly with the equations of motion with particular attention givento angular accelerations about appropriate axes.15'16 The latter ap-proach has been used recently to include coupling effects observedin dynamic maneuvers in the derived departure criteria.17'18 Un-fortunately in many of the earlier developments and subsequentapplications, rigor is not retained throughout and confusion arisesas to what reference systems are used and how the aerodynamicstability derivatives are defined. This paper shows that the most fre-quently used criteria for both open- and closed-loop flight can bedeveloped from a unified approach. It also shows that most currentcriteria are subject to severe limiting assumptions with regard toterms ignored.

489

490 LUTZE, DURHAM, AND MASON

Open-Loop Stability CriteriaIn general, the conditions for departure resistance are based on

one of two considerations, static stability or dynamic stability. Ineach case we must start with a reference flight condition. Typicallyfor departure considerations, this condition is symmetric or nearsymmetric flight at a high AOA. The static stability criterion asksthe question, if disturbed from this reference flight condition, doesthe disturbance cause forces and/or moments that oppose the dis-turbance? On the other hand, the dynamic stability criterion asksthe question, does the system return to the reference flight condi-tion in time? Hence, static stability requires the forces and momentsresulting from disturbances to be in the correct direction, whereasdynamic stability requires additionally that they be in the correctphase. Static stability considerations can be developed from knowl-edge of only aerodynamic characteristics, whereas dynamic stabilityconsiderations require knowledge of the complete dynamic system.

Static stability requirements can be obtained by determining thechange in a force or moment resulting from a change in some dis-turbance variable. The result is usually in the form of a conditionon some stability derivative such as Np > 0. However, care must betaken in interpreting the result.

Dynamic stability requirements are well defined for linear sys-tems of the form

= A A*+ 5 Aw (1)where AJC is the perturbed state vector and An the perturbed controlvector. For the lateral-directional dynamics of an aircraft, the lineardynamic system has four states of interest,

Ajcr = [A/J, Ap, Ar, A(/>] (2)

A necessary and sufficient condition for dynamic stability is that theroots (eigenvalues) of the characteristic equation

|X / 4 -A |=0 (3)

have negative real parts. This characteristic equation has the genericform

aX4 + Z?X3 + cA,2 + d\ + e = 0 (4)

Although an analytic solution to Eq. (4) is available, it is not easilywritten in a useful literal form. An early mathematician by the nameof Routh, however, determined criteria that must be satisfied if theroots to a polynomial are to have negative real parts.19 For a fourth-order equation these criteria are

a, b, c, d, e > 0; be — ad > 0

d(bc - ad) -b2e>0(5)

For a third-order system it is required that all of the coefficients bepositive, as Eq. (5), and that (be — ad) > 0, and for a second-ordersystem the only requirement is that all of the coefficients be positive.In all cases the condition that the last coefficient be greater than zero(for the quartic, e > 0) is sometimes considered as the generalizedstatic stability criterion for the complete system because when itchanges sign a real root with a positive real part will appear. Hence,this criterion defines more precise requirements for static stabilitythan the simple derivative idea mentioned previously.19 It is this lastresult that is used to derive the various departure criteria of interest.

Lateral-Directional Linear StabilityIt is of interest to examine the generalized static stability term

for the lateral-directional motion to give additional insight into thedeparture problem. We linearize the equations of motion about asymmetric (ft = 0), steady-state flight condition. Under these cir-cumstances the lateral-directional system matrix is given by13'19

A =L'PN'P1

N>tan#

gcos 0/V000

The subscript (ref) indicates that the quantities in the matrix areevaluated at the reference flight condition of interest. The primedquantities are defined as

m 30

(7)

Note that the aerodynamic moments, stability derivatives, and inertiaproperties are dependent on the axis system selected.

The characteristic equation of the matrix in Eq. (6) has its coef-ficients given in terms of the matrix elements a/7- by

a =

C = 0H022

b = - (an + a22 + 033)

+ 022033 ~ 012021 — 013031 ~ 032023

d = 0n (032023 ~ ^22033) + 012(021033 ~ 031023)

— 021032) — 014 (021 + 031043)

(8)

e = 014[(021033 ~ 031023) + 043(031022 ~ 021032)]

The generalized static stability criterion for lateral-directional mo-tion is given by e > 0, or

(g/ V) [(L'pN'r - N'ftL'r (N'ftL'p - L'ftN'p) 0 (9)

Also note that the condition e > 0 holds for all coordinate systemsprovided that all of the aerodynamic stability derivatives and inertiaterms are calculated in the same coordinate system.

It can be observed from the condition given by Eq. (9) that grav-ity is an important contributor. If gravity were neglected, this termwould vanish. We will assert that gravity should not directly causea lateral-directional aerodynamic instability. Hence, if we restrictour interest to departure resulting from aerodynamic causes, we canneglect the gravity term and, thus, any results obtained will be thoseresulting solely from aerodynamic disturbances.

The basic static stability conditions require that the perturbationsin the state variables develop forces or moments that oppose theperturbations. The objective here is to show that if we include onlyforce and moment terms resulting from displacements in the systemmatrix, Eq. (6), the generalized static stability requirement will leadto the classic Cn/,d n > 0 requirement. The approach taken here issimilar to that use3 by Calico.11 The fundamental idea is that theaerodynamic forces and moments resulting from angular displace-ments must provide the stability for departure prevention. Hence, ifwe keep only these terms and neglect all others (including gravity),we have

A =

sin a.h — cos oti,0 00 01 tan# ref

The characteristic equation becomes

X2[X2 - YpX + Np cosah - Up sine*/,] = 0

(10)

(11)

Here we have lost two roots, one resulting from the assumption of nogravity and the other resulting from the assumption of no dampingterms (the spiral and rolling convergence modes, respectively). Therequirements due to Routh for stability of the roots of the remainingquadratic equation are

Yp < 0, Np cos ab - Lfp sin ab > 0 (12)

As already observed, Eq. (12) holds for all axis systems. Hence,we can select the stability axes system (ah = 0) and write the lastcondition as

(6) 0 (13)

LUTZE, DURHAM, AND MASON 491

where the superscript s refers to the body-fixed stability axes. If wedefine the terms

[cn,

Equations (7), (12), and (13) can be reduced to the equivalent con-dition

l sine*, > 0

/;)c - C cosa, - (IJIx)C'lj si (15)

= CnK > 0"Alyn

where the absence of a superscript refers to an arbitrarily selectedbody-fixed axes system. In terms of the basic stability derivativeswe have

C"Alyn ~COSCfy,

sin a/,

-\'J

(16)

(17)

For the special case of principal axes (/„ = 0), these results re-duce to those originally proposed by Moul and Paulson.9 For the re-duced system given by Eq. (10), Cnp& n is directly related to thegeneralized static stability requirement. It also indicates that thestandard lateral-directional stability parameter criterion Cnft > 0 isvalid only if evaluated in stability axes!

An alternative approach similar to that suggested by Pelikan16 andrelated to the approach of Kalviste and Eller18 that can be applied tothe complete nonlinear system is presented next. It will be appliedto the reduced linear system of equations for comparative purposes.If we select generic body-fixed axes, we can write the followingrelations from Eqs. (1), (2), and (10):

Ar = (18)

Transformation to stability axes and the introduction of Eq. (18)leads to the result

Ar'v = —A/? sinah -f Arcosa&

(19)

lier. Under these circumstances the system reduces to a system witha third-order characteristic equation. The system matrix becomes

A =

o

L'PN'Pi tan#

(21)

The associated characteristic equation is a quartic with one zeroroot,

+ bX2 + cX + d] = 0 (22)

where

- [(Y'p/V) +

d = [(Yfp/V)

^ - [(7,7 V) - co*ab]N'ft (23)

jjtf; - N'ftL'r) + Y'p(N'pL'r - L'pN'r)

The conditions for stability are that b, c, and d > 0 and thatbd — c > 0. The requirement that c > 0 leads to terms that includeCn/?d n. In particular, we have

AT; cos** - L'p sin«> > -^(L; + N'r) - L'pN'r

(24)

Hence, a lower bound on Cnft is determined that is related to theearlier ignored rate terms. Similar results are presented in Ref. 13.Additional criteria can be determined from the other conditions thatb and d > 0 and be — d > 0. In each case the equations can be ex-panded in terms of the basic stability derivatives Cn/J, Cifj, Cnp, Cnr,etc., and the moments and products of inertia. The forms of the re-sulting equations are quite complicated but are easily implementedon a computer.

Closed-Loop Departure CriteriaThe techniques of the preceding section can be extended to the

closed-loop controlled case in a straightforward manner. Considerthe case of full state feedback with only aileron and rudder as con-trols. In this case the feedback control law takes the form

1AM =l A « r JApAr

(25)

Using Eq. (19) along with the static stability requirement of havingan angular acceleration (moment) to oppose a disturbance in thesideslip angle, and noting that the sideslip angle is measured aboutthe stability z axis, we have

(20)Is.

This result is the same as that developed earlier in Eq. (15) andvalidates this approach as a method to predict lateral-directionaldeparture. It is suitable for the complete nonlinear dynamic systembut a discussion of that digresses from the objectives of this paper.

From the preceding remarks one can see that Cnp > 0 is really astatic stability criterion and is obtained in its pure form only by mak-ing several restrictive assumptions. The information lost because ofassumptions, however, can cause inaccurate results. Attempts havebeen made to reduce these assumptions but still retain tractableresults.13 We will eliminate only the gravity term, as indicated ear-

AM = KAx (26)

With this control law substituted into Eq. (1), the closed-loop con-trolled system is given by

where

AJC = Aci A*

Acl = A + BK

(27)

(28)

Yaw StabilityWe can parallel the development in the preceding section and

consider first the case where the rate terms (and gravity) are ignoredor assumed zero. Also we will assume only a rudder-sideslip anglefeedback gain, all others being zero. We can, therefore, designate

492 LUTZE, DURHAM, AND MASON

the gain k2\ — k. Under these conditions the augmented systemmatrix becomes

0

sin ofh — cos ab 0"0 0 00 0 01 tan# 0

(29)

With suitable definitions of primed control derivatives [see Eqs. (7)and (14)], Routh's criteria for the reduced system become

<0 (30)

and

- (L'P + kL'Sr) sinojz, + (N'ft + kN'Sr) cose*,, > 0 (31)

This last equation leads to the result

- (It/I,)(C'lf + kC'hr ) sin ab > 0

Define

(32)

r 0 (33)

Equation (33) is the closed-loop departure criterion for the casewhere k is the sideslip-rudder gain for the aircraft.

The results are determined from the linearized equations of mo-tion and all of the derivatives are evaluated at the reference flight con-dition. Since aerodynamic terms become highly nonlinear at highAOA and local slopes required for the derivatives may not be repre-sentative of the global behavior, alternative approximations to thesederivatives can be used. Pelikan16 suggests using sideslip relatedderivatives evaluated using the secant method. Here, the derivativeis determined as the value of the force or moment coefficient dividedby the sideslip angle. Using this definition, we can now show thatthe result in Eq. (33) leads to the same result obtained by Pelikan.16

First observe that since the reference flight condition is at zerosideslip angle the perturbation value of the sideslip angle is equalto the sideslip angle and they can be used interchangeably, that is,A/3 = ft. We note that the gain term is

k =

Substituting this value into Eq. (33) yields the typical term

(34)

A/*

A/*(35)

The last line is the nonlinear equivalent of the linear results onthe previous line. C'n(ft) is the total yaw-moment coefficient for agiven sideslip angle with no rudder control, C'n(8r) is the total yawmoment coefficient resulting from rudder control deflection, and ftis the total sideslip angle. Hence, the linear result is converted backto a form useful for nonlinear analysis with the two terms being thesecant derivatives. If these results are substituted into Eq. (33), theresulting expression for the Cn/?apparent term is given by

Cn'^apparent

[C'n(fi) + C'n(S,)-\COSCfy,

(36)

Equation (36) is the equation developed by Pelikan16 using the angu-lar acceleration requirement, (3 Ar*/9 A/3 > 0), described earlier.

Roll Stability (Lateral Control Departure Parameter)Here we will consider the case where all of the elements in the

gain matrix, Eq. (25), except ku = k are equal to zero. Under thesecircumstances the reduced augmented system (no rate or gravityterms) becomes

(37)

~Y'pL'pN'p_ 0

sin oLb

001

— cos ah

00

tan#

"qkLikNio"_-Iref

The characteristic equation is a quartic with the coefficients

= -Y'

c = -kN' tan0

d = -kY8a(Nfptane + L'

cos ah)

tan0 +

+ cosa/;)

(38)

For stability we require all of the coefficients to be positive. Thegeneralized static stability requirements is that e > 0. Assumingthe magnitude of the pitch attitude and AOA are less than 90 deg,this requirement reduces to

-k(N'ftL',a-L'ftNla)>0 (39)

Equation (39) is the required stability criterion and can be shownto reduce to the lateral control departure parameter (LCDP), whichis currently used. If it is assumed that the product of kL'Sa is negative,so that a positive roll angle would cause a negative rolling momentresulting from aileron control, Eq. (39) can be rearranged in theform

In terms of the nondimensional stability derivatives, Eq. (40)becomes

rc^+(w/.)c/3 1[c!8a+(ixz/iz)cnsa\ > (41)\ + (W4)C^

Equations (40) and (41) are versions of the LCDP.14'15 Note that thesame results are obtained if we assume only roll-rate feedback toaileron, k\2 ^ 0.

Aileron Rudder InterconnectThe stability of a system that includes an aileron-rudder inter-

connect (ARI) can be determined in a similar manner by includingthe gains ku and £24 in the gain matrix Eq. (26). Here we have

= k\ (42)

(43)

where k is ARI gearing ratio. Under these conditions the extra feed-back control terms enter into the fourth column of the system matrix,with these terms becoming

k2L28r

044=0(44)

All of the coefficients of the characteristic equation are changedsomewhat because of the addition of the bank-angle feedback torudder. The term of interest, however, is the last coefficient, e. With

LUTZE, DURHAM, AND MASON 493

the same assumptions as in the preceding section, the interestingpart of this term becomes

0 (45)

Again if we assume the coefficient of the N'ft is negative, we canrearrange the terms in Eq. (45) to put the equation in the desiredform,

N' - L'i. -kL'ir

>0

or

[N'f - L'p(N'r/L'Sr)]

(46)

> 0 (47)

where k = A8r/A8a, as defined earlier in Eq. (43). Note that thesame results are obtained assuming only roll-rate feedback to aileronand rudder.

These results are the same as those found in Refs. 9 and 15, wherethey were found using an approach completely different from thatused here.

Summary and ConclusionsAs originally stated, the purpose of this paper is to show that the

most frequently used departure predicting parameters can be derivedfrom a unified approach. This approach consists of linearizing theequations of motion, making suitable assumptions, and examiningthe stability of the resulting dynamic system using some of Routh'scriteria. In particular, the criterion used most frequently was thegeneralized static stability criterion: that the coefficient of the zerothpower of the eigenvalue in the characteristic equation be greaterthan zero. In general, this criterion leads to Cnf) for the open-loopsystem and Q^ arent for the closed-loop systelS, as well as to theLCPD and ARI requirements for selected feedback gain structures.

In obtaining these results, several assumptions were made and, inaddition, a considerable amount of information was not used that isprovided by the remaining criteria of Routh. Included in this cate-gory are that the remaining coefficients of the characteristic equationmust be greater than zero and that each of the Routh discriminantsmust be greater than zero. These additional criteria provide infor-mation on dynamic stability requirements, as well as static stabilityrequirements.

The idea of Pelikan,16 which uses secant derivatives, can be usedto extend results obtained from linear system analysis to be appli-cable to nonlinear systems. In addition, the direct application of thebasic idea of restoring moments or accelerations generated by per-turbations in the variables was shown to lead to the same resultsas the linear theory provided the moment of interest was measuredabout the axis along which the angular perturbation displacementwas taken.

AcknowledgmentsThe authors would like to acknowledge partial support for this

work from NASA Langley Research Center, Grant NCC1-158, withJohn Foster as Technical Monitor.

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