Quarternion Models for Aircraft Loss-of-Control Analysis

Harry G. Kwatny∗

Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA

Loss-of-Control (LOC) is a major factor in fatal aircraft accidents. LOC is gen-

erally associated with flight outside of the normal flight envelope almost always

involving nonlinear behaviors and often including extreme attitude conditions. Re-

cent and ongoing work at the NASA Langley Research Center has focused on

building aerodynamic models adequate for simulation and analysis in these regimes.

Models that capture out-of-envelope behavior with extreme attitudes also require

quaternion rather than Euler attitude parametrization. This paper describes such

expanded models and examines how these nonlinearities affect the ability to control

the aircraft and how they may contribute to loss-of-control.

I. Introduction

Recent published data shows that during the ten year period 1997-2006, 59% of fatalaircraft accidents were associated with Loss-of-Control (LOC) [1]. Yet the notion of

loss-of-control is not well-defined in terms suitable for rigorous control systems analysis.The importance of LOC is emphasized in [2] where the inadequacy of current definitionsis also noted. On the other hand, flight trajectories have been successfully analyzed interms of a set of five two-parameter envelopes to classify aircraft incidents as LOC [3]. Asnoted in that work, LOC is ordinarily associated with flight outside of the normal flightenvelope, extreme attitude variation, with nonlinear behaviors, resulting in the inability ofthe pilot to control the aircraft. The results in [3] provide a means for analyzing accidentdata to establish whether or not the accident should be classified as LOC. Moreover, theyhelp identify when the initial upset occurred, and when control was lost. The analysisalso suggests which variables were involved, thereby providing clues as to the underlyingmechanism of upset. However, it does not provide direct links to the flight mechanicsof the aircraft, so it cannot be used proactively to identify weaknesses or limitations inthe aircraft or its control systems. Moreover, it does not explain how departures fromcontrolled flight occur. In particular, we would like to know how environmental conditions(like icing) or faults (like a jammed surface or structural damage) impact the vulnerabilityof the aircraft to LOC.

Investigating LOC requires the use of aircraft dynamical models that are accurateoutside of the normal flight envelope. In particular it is necessary to characterize poststall and spin behaviors that are often associated with LOC events. Until recently, suchmodels were not available for large transport aircraft. Recent and ongoing work at theNASA Langley Research Center has focused on building aerodynamic models adequatefor simulation and analysis in these regimes [4, 5]. A central element in this effort isNASA’s Generic Transport Model (GTM) [6] – a 5.5 % dynamically scaled commercialtransport model. The GTM will be used to provide analysis examples. Nonlinear modelsthat capture out-of-envelope behavior also require quaternion rather than Euler attitudeparametrization. These models are discussed below as are their implications for departure

∗S. Herbert Raynes Professor, hgk22@drexel.edu

Itzhack Y. Bar-Itzhack Memorial Symposium on Estimation, Navigation, and Spacecraft Control, Haifa,Israel, October 14–17, 2012

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from controlled flight. Bifurcation analysis is used to study aircraft control propertiesand how they change with the flight condition and parameters of the aircraft.

The organization of the paper is as follows. Section II provides a short discussion andliterature review of the LOC problem. In Section III the six degrees of freedom GTMmathematical model we use is described. Section IV addresses the bifurcation analysisof controlled dynamical systems. Control issues that arise near stall are specificallyaddressed. Uncontrolled departures of the GTM near stall are illustrated as well as somefirst illustrations of recovery from post departure states. Finally, Section VI containsconcluding remarks.

II. The Loss-of-Control Problem

Although the majority of fatal aircraft crashes over the past decade or so have beenattributed to LOC, its meaning is ambiguous. Generally, a pilot will report LOC if theaircraft does not respond as expected. Consequently, pilot experience can be a majorvariable in assessing LOC. What LOC is to one pilot may not be to another. Recently,Wilborn and Foster [3] have proposed quantitative measures of LOC. These QuantitativeLoss-of-Control (QLC) metrics consist of envelopes defined in two dimensional parameterspaces. Based on the analysis of 24 data sets compiled by the Commercial Aircraft SafetyTeam (CAST) Joint Safety Analysis Team (JSAT) for LOC [7] five envelopes have beendefined:

1. Adverse Aerodynamics Envelope: (normalized) angle of attack vs. sideslip angle

2. Unusual Attitude Envelope: bank angle vs. pitch angle

3. Structural Integrity Envelope: normal load factor vs. normalized air speed

4. Dynamic Pitch Control Envelope: dynamic pitch attitude (θ + θ∆t) vs. % pitchcontrol command

5. Dynamic Roll Control Envelope: dynamic roll attitude (φ + φ∆t) vs. % lateralcontrol command

The authors provide a compelling discussion of why these envelopes are appropriate anduseful. Flight trajectories from the 24 CAST data sets are plotted and the authorsconclude maneuvers that exceed three or more envelopes can be classified as LOC, thosethat exceed two are borderline LOC and normal maneuvers rarely exceed one. Accordingto Ref. [3], the precipitating events of the CAST LOC incidents were: stalls (45.8%),sideslip-induced rolls (25.0%), rolls from other causes (12.5%), pilot-induced oscillation(12.5%) , and yaw (4.2%).

These results are important. They provide a means for analyzing accident data toestablish whether or not the accident should be classified as LOC. Moreover, they helpidentify when the initial upset occurred, when control was lost and suggest which variableswere involved. However, because the approach does not directly connect to the flightmechanics of the aircraft, it does not identify weaknesses or limitations in the aircraft orits control systems. Moreover, it does not explain how departures from controlled flightoccur. In particular, we would like to know how environmental conditions or actuatorfailures or structural damage impact the vulnerability of the aircraft to LOC.

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Another important study [2] reviews 74 transport LOC accidents in the fifteen yearperiod 1993-2007. Of these the major underlying causes of LOC are identified as stalls,ice contaminated airfoils, spatial disorientation, and faulty recovery technique.

A recent study [8] goes further, investigating 126 LOC accidents that span threedecades from 1979-2009. Accident reports were used to identify fourteen causal andcontributing events. These fourteen events are grouped into three broad categories:

1. Adverse onboard conditions

2. External hazards and disturbances

3. Vehicle upsets

The analysis identifies six events as the most significant contributors to LOC incidentsin terms of the number of fatalities: Category 1 – System Faults/Failures/Errors, VehicleImpairment/Damage, Inappropriate Crew Response; Category 2 – Atmospheric Distur-bances related to Wind Shear/Gusts, and Snow/Icing; Category 3 – Stall/Departure fromcontrolled flight.

Most importantly, the authors note that almost all LOC incidents involve a sequenceof events. They observe that vehicle upsets are rarely the precipitating factor in theLOC sequence, but most LOC sequences include vehicle upset somewhere in the chain ofevents. This category includes five events:

1. Abnormal attitude

2. Abnormal airspeed, Abnormal angular rates, asymmetric forces

3. Abnormal flight trajectory

4. Uncontrolled descent

5. Stall, departure from controlled flight

Of these, Stall/departure is clearly the most prevalent. The prevalence of stalls in LOCincidents suggests the importance of bifurcation behavior as a factor in LOC. We discussthis further in Section IV.

An aircraft must typically operate in multiple modes that have significantly differentdynamics and control characteristics. For example, cruise and landing configurations.Within each mode there may be some parametric variation, such as weight or center ofmass location, that also affects aircraft behavior. Each mode has associated with it a flightenvelope restricting speed, attitude and other flight variables. Under normal conditionskeeping within the flight envelope provides sufficient maneuverability to perform the modemission while insuring structural integrity of the vehicle for all admissible parametervariations and all anticipated disturbances. Abnormal conditions, e.g., icing, faults ordamage, will alter aircraft dynamics and may require the definition of a new mode withits own flight envelope.

III. Rigid Aircraft Dynamics

In our studies of LOC [9–11], nonlinear symbolic models are used to perform bi-furcation analysis and nonlinear control analysis and design. The requisite models are

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generated in various coordinate systems automatically using Mathematica with tools de-scribed in [12]. Simulation models are also automatically assembled from the symbolicmodel in the form of optimized C-code that compiles as a MEX file for use as an S-function in Simulink. In this section these models are described with emphasis on thequaternion model.

A. Poincare’s Equations

The six degrees of freedom rigid aircraft model has 12 states when Euler angles are usedto parameterize attitude. In the this case the model can be generated in the form ofPoincare’s equations [12],

q = V (q)p (1)

M (q) p+ C (q,p)p+ F (p,q, u) = 0 (2)

where q = (φ, θ, ψ,X, Y, Z)T is the generalized coordinate vector, p = (p, q, r, u, v, w)T isthe (quasi-) velocity vector, u is a vector of control inputs.

The kinematic matrix, (1), is of the form

V = diag(V1, V2)

where

V1 =

1 sin (φ) tan (θ) cos (φ) tan (θ)

0 cos (φ) −sin (φ)

0 sec (θ) sin (φ) cos (φ) sec (θ)

V2 =

cos (ψ) cos (θ) − cos (φ) sin (ψ) + cos (ψ) sin (ψ) sin (θ) sin (φ) sin (ψ) + cos (φ) cos (ψ) sin (θ)

cos (θ) sin (ψ) cos (φ) cos (ψ) + sin (φ) sin (ψ) sin (θ) − cos (ψ) sin (φ) + cos (φ) sin (ψ) sin (θ)

− sin (θ) cos (θ) sin (φ) cos (φ) cos (θ)

The parameters in the dynamics, (2), are:

M =

Ix 0 −Ixz 0 0 0

0 Iy +mx2cg 0 0 0 −mxcg

−Ixz 0 Iz +mx2cg 0 mxcg 0

0 0 0 m 0 0

0 0 mxcg 0 m 0

0 −mxcg 0 0 0 m

C =

−Ixzq −mvxcg − r(

Iy +mx2cg

)

−mwxcg + q(

Iz +mx2cg

)

0 −mw +mqxcg mv +mrxcg

Ixzp+ Ixr muxcg −Ixzr − p(

Iz +mx2cg

)

mw −mpxcg −mu

−Ixq p(

Iy +mx2cg

)

Ixzq +muxcg −mv mu −mpxcg

0 −mqxcg −mrxcg 0 −mr mq

0 mpxcg 0 mr 0 −mp

0 0 mpxcg −mq mp 0

F (p, q) =

−L

−M − ltT + gmxcg cos (φ) cos (θ)

−N − gmxcg cos (θ) sin (φ)

−T −X + gm sin (θ)

−Y − gm cos (θ) sin (φ)

−Z − gm cos (φ) cos (θ)

705

where the aerodynamics forces, X, Y, Z, and moments, L,M,N , are defined in termsof the aerodynamic coefficients, CX , CY , CZ , CL, CM , CN :

[

X Y Z L M N]

=12ρV 2S

[

CX CY CZ bCL cCM bCN

]

We can combine the kinematics (1) and dynamics (2) to obtain the state equations

x = f(x, u, µ) (3)

where µ ∈ Rk is an explicitly identified vector of distinguished aircraft parameters such asmass or center of mass location (or even set points for regulated variables), x ∈ Rn is thestate vector, and u ∈ Rm is the control vector. In the Euler angle case, n = 12 and thestate is given by x = [φ, θ, ψ,X, Y, Z, p, q, r, u, v, w]T . There are 4 control inputs, givenby u = [T, δe, δa, δr]

T . The controls are limited as follows: thrust 0 ≤ T ≤ 40 lbf, elevator−40 ≤ δe ≤ 20, aileron −20 ≤ δa ≤ 20, and rudder −30 ≤ δr ≤ 30. Velocity is inft/s.

Because LOC can involve extreme attitude variations Euler angle parametrization isnot reliable. Consequently, we wish to replace the Euler representation by a quaternionrepresentation. To do this the central problem is to characterize the rotation matrix interms of the quaternion. This was done by Bar-Itzhak in [13]. The following discussionsummarizes the essential properties of the quaternion.

B. Quaternion Representation of a Rotating Frame

Consider a rigid body free to rotate in space. Suppose that a point, fixed in the body,is represented by vector r defined in a body fixed frame. Let s be its representationin a space fixed frame. The rotation matrix transforms r from body fixed coordinatesto space coordinates. By convention the matrix transforming space coordinates to bodycoordinates is designated L. Recall, that a unit quaternion is a four tuple of real numbersq = (q0, q1, q2, q3) with unit norm. It is convenient to envision a quaternion as composedof two parts, a scalar q0 and vector q = (q1, q2, q3).

In a unit quaternion parametrization the rotation matrix has the form:

LT (q) =

q20 + q21 − q22 − q23 2q1q2 − 2q0q3 2q0q2 + 2q1q32q1q2 + 2q0q3 q20 − q21 + q22 − q23 −2q1q2 + 2q0q3−2q0q2 + 2q1q3 2q0q1 + 2q2q3 q20 − q21 − q22 + q23

(4)

Specifically, if r is a vector represented in the coordinates of the rotated body frame, ands its coordinates in the fixed space frame, then

r = L (q) s, s = LT (q) r

It is necessary to relate the body angular velocity to the rate of change Euler parameters.To do this begin with time differentiation of the quaternion product s = q r q∗ (recall,q∗ denotes the conjugate of q):

s = q r q∗ + q r q∗

Now substituting r = q∗ s q

s = q q∗ s+ s q q∗

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Since q is a unit quaternion, the time rate of change of its norm is zero. This fact allowsus to easily establish that the scalar parts of qq∗ and q q∗ are both zero. Furthermore, ifq q∗ = (0,w), then q q∗ = (0,−w). It follows that the vectorial part of s is s = 2w× s.From this we identify Ω = 2w, where Ω is the angular velocity of the body in spacecoordinates. Thus,

d

dt

[

q0

q

]

=1

2

(

q0 −qT

q q0I3 − q

)[

0

Ω

]

=1

2

(

−qT

q0I3 − q

)

Ω (5)

It is generally more convenient to write these equations using angular velocity in bodycoordinates

d

dt

[

q0

q

]

=1

2

(

−qT

q0I3 + q

)

ω (6)

where ω is the body angular velocity in body coordinates. Expanding, with ω = (p, q, r)

d

dt

q0

q1

q2

q3

=1

2

−q1 −q2 −q3

q0 −q3 q2

q3 q0 −q1

−q2 q1 q0

p

q

r

(7)

Exact integration of these equations for any specified ω (t) and initial condition q (0)of unit length produce a solution q (t) with ‖q (t)‖ = 1 for all t. However, imperfectintegration requires a correction. One common remedy for digital computation is toimplement

d

dt

q0

q1

q2

q3

=1

2

q1 q2 q3

q0 −q3 q2

q3 q0 −q1

−q2 q1 q0

p

q

r

+ κλ

q0

q1

q2

q3

(8)

where λ = (1− q20 − q21 − q22 − q23) and the parameter κ is chosen so that κh ≤ 1 and h isthe step size.

C. Euler Angles from Quaternion

It is necessary to determine the Euler angles from the quaternion. For Euler angles inthe 3-2-1 or zyx convention, the rotation matrix, from body to space coordinates, is

L (φ, θ, ψ) =

cos θ cosψ cos θ sinψ − sin θ

sinφ sin θ cosψ − cosφ sinψ sinφ sin θ sinψ + cosφ cosψ sinφ cos θ

cosφ sin θ cosψ + sinφ sinψ cosφ sin θ sinψ − sinφ cosψ cosφ cos θ

(9)

The corresponding rotation matrix in terms of the quaternion is

L (q) =

q20 + q21 − q22 − q23 2 (q1q2 + q0q3) 2 (q1q3 − q0q2)

2 (q2q1 − q0q3) q20 − q21 + q22 − q23 2 (q2q3 + q0q1)

2 (q1q3 + q0q2) 2 (q3q2 − q0q1) q20 − q21 − q22 + q23

(10)

Note that both of these matrices convert vector in body fixed coordinates to space coor-dinates. Equation (10) is obtained by inverting (equivalently, transposing) the matrix in(4).

707

Now, compare elements of (9) and (10) to obtained the following

φ = tan−1 2 (q2q3 + q0q1)

q20 − q21 − q22 + q23(11)

θ = sin−12 (q0q2 − q1q3) (12)

ψ = tan−1 2 (q2q1 + q0q3)

q20 + q21 − q22 − q23(13)

D. Quaternion from Euler Angles

It is convenient and sometimes necessary to be able solve for the quaternion parametersgiven a the Euler angles. For instance when integrating equation (8), the initial attitudeis often specified in Euler angles which need to be converted to quaternion parameters.

Comparing (9) and (10) the following four equations are obtained.

tan θ (q20 − q21 − q22 + q23) = 2 (q2q3 + q0q1)

sin θ = 2 (q0q2 − q1q3)

tanψ (q20 + q21 − q22 − q23) = 2 (q2q1 + q0q3)

q20 + q21 + q22 + q23 = 1

(14)

It is easily verified that these equations are satisfied by

q0 = cos 12ψ cos 1

2θ cos 1

2φ+ sin 1

2ψ sin 1

2θ sin 1

2φ

q1 = cos 12ψ cos 1

2θ sin 1

2φ− sin 1

2ψ sin 1

2θ cos 1

2φ

q2 = cos 12ψ sin 1

2θ cos 1

2φ+ sin 1

2ψ cos 1

2θ sin 1

2φ

q3 = − cos 12ψ sin 1

2θ sin 1

2φ+ sin 1

2ψ cos 1

2θ cos 1

2φ

(15)

E. The Generic Transport Model

In the subsequent discussion examples based on NASA’s Generic Transport Model (GTM)are given. Data obtained from NASA have been used to develop symbolic and simula-tion models. Aerodynamic models were obtained from NASA. They are based on dataobtained with a 5.5 % scale model in the NASA Langley 14 ft × 22 ft wind tunnel asdescribed in [5]. Aerodynamic force and moment coefficients are generated using a multi-variate orthogonal function method as described in [14,15]. In the original NASA modelseveral regions of angle of attack were used to capture severe nonlinearity. These modelswere blended using Gaussian weighting. For simplicity we use only one of the models forthe analysis herein.

IV. Aircraft Trim and Bifurcation Analysis

Departures from controlled flight like stall and spin have concerned aircraft engineersfrom the earliest days of flight. In recent years departure has been analyzed by usinga combination of simulation and flight test (with manned aircraft or scale models - see,for example, the informative report [16]). An example, that may be considered ‘state-of-the-art’ for studies of this type concerns the falling leaf and related behaviors of theF-18 [17–19]. Only in the past two decades have formal methods of bifurcation analysisbeen applied to aircraft [20–30]. Bifurcation analysis has been employed to identify the

708

conditions for occurrence of undesirable behaviors, to investigate recovery methods fromdangerous post bifurcation modes and to formulate feedback control systems that modifybifurcation behavior.

In this paper the regulation of aircraft to a desired trim condition is considered. Ingeneral the equations of motion of a rigid aircraft involve six degrees of freedom involvingthe seven coordinates q0, q1, q2, q3, X, Y, Z and six (quasi-) velocities p, q, r, u, v, w. Inthe study of steady motions it is usual to ignore the inertial location (X, Y, Z) andconsider only the velocities and attitude of the vehicle. The reduced dynamics comprisea ten dimensional system of state equation that are nonlinear and may be parameterdependent. Of concern are steady motions that can be defined in terms of these variables.In particular, these motions are trajectories in the inertial (X, Y, Z) space, that can beassociated with equilibria of the ten state system. It is important to know whether ornot it is possible to regulate to and steer along these motions. This naturally leads tothe study of the existence and stabilizability of equilibria of the reduced state system.

It is important to emphasize that most applications of bifurcation analysis to air-craft view the aircraft as a dynamical system in which the control inputs are treated asthe bifurcation parameters. In this work, as in [25], the system is viewed as a controlsystem, that is, a dynamical system with an input-output structure as well as parame-ters. Consequently the bifurcation points are linked with control system properties likecontrollability and observability rather than stability. This gives a unique view of howoperation near bifurcation points impacts the ability to control the aircraft.

A. Trim conditions

The idea of aircraft trim is so broadly entrenched that one would assume it requires nofurther discussion. However, there are subtleties that need to be explored. A generalformulation is as follows. Assume the aircraft is described by the state equations in theform of (1) and (2), or (3).

Definition 1. Steady Motion A steady motion is one for which all 6 velocities areconstant, i.e., u = 0, v = 0, w = 0, (equivalently, V = 0, α = 0, β = 0) and p = 0, q =0, r = 0. From (2), with p = 0,

g (x, u, µ) :=C (q,p)p+ F (p,q, u) = 0 (16)

The steady motion requirement, (??), provides six equations to which we add a set of

n+m− 6 trim equations:h (x, u, µ) = 0 (17)

Equations (??) and (16) form a set of n+m equations in n+m+k variables. Ordinarilywe fix the k parameters, µ, and solve for the remaining n+m variables – the state x andthe control u.

Definition 2. Trim Point Given the steady motion equation (??), the trim condition(16), an envelope C and a control restraint set U , a viable trim point or simply a trimpoint, with respect to the fixed parameter µ is a pair (x, u) that satisfies (??) and (16)and also x ∈ C, u ∈ U . The set of viable trim points is called the trim set, T .

The important point is that for a prescribed trim condition (16) there are often mul-tiple viable trim points. The general study of the trim point structure as a function ofthe parameters is a problem of static bifurcation analysis [9,25,31–33]. When k = 1 this

709

can be carried out using a continuation method. Let us consider two examples of thetrim condition for the six degree of freedom model described in Section III. In this casethere are four controls. First consider straight wings-level flight:

1. speed, V = V ∗

2. constant roll, pitch and heading, φ = 0, θ = 0, ψ = 0

3. roll angle, φ = 0

4. flight path angle, sin γ∗ + cos θ (cos β cosφ sinα + sin β sinφ)− cosα cos β sin θ = 0

Second, consider a coordinated turn. In this case the aircraft rotates at constant angularvelocity, ω∗ about the inertial z-axis. Thus the attitude of the aircraft varies periodicallywith time.

1. speed, V = V ∗

2. coordinate turn condition, pV cos β sinα− rV cos β cosα + g cos θ sinφ = 0

3. angular velocity, p = −ω∗ sin θ, q = ω∗ cos θ sinφ, r = ω∗ cos θ cosφ

4. flight path angle, sin γ∗ + cos θ (cos β cosφ sinα + sin β sinφ)− cosα cos β sin θ = 0

B. Control issues at stall

Consider a parameter dependent, nonlinear control system given by

x = f(x, u, µ)

z = h(x, u, µ) (18)

where x ∈ Rn are the states, u ∈ Rp are the control inputs, z ∈ Rr are the regulatedvariables and µ ∈ R is any parameter. Assume that f , h are smooth (sufficiently dif-ferentiable). The parameter could be a physical variable like the weight of the aircraftor the center of gravity location; or a regulated variable like velocity, flight path angle,altitude or roll angle; or the position of a stuck control surface. The regulator problemis solvable only if p ≥ r. Since the number of controls can always be reduced, henceforthassume p = r.

A triple (x⋆, u⋆, µ⋆) is an equilibrium point (or trim point) of (17) if

F (x⋆, u⋆, µ⋆) :=(

f(x⋆, u⋆, µ⋆)h(x⋆, u⋆, µ⋆)

)

= 0 (19)

The equilibrium surface is the set E =

(x, u, µ) ∈ Rn+m+k |F (x, u, µ) = 0

.

Definition 3. (from [25]) An equilibrium point (x⋆, u⋆, µ⋆) is regular if there is aneighborhood of µ⋆ on which there exist unique, continuously differentiable functionsx(µ), u(µ) satisfying

F (x(µ), u(µ), µ) = 0 (20)

If an equilibrium point is not a regular point it is a static bifurcation point. TheImplicit Function Theorem implies that an equilibrium point is a bifurcation point onlyif det J = 0. The Jacobian J is given by

J = [DxF (x⋆, u⋆, µ⋆) DuF (x⋆, u⋆, µ⋆)] (21)

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Now, if A,B,C,D denotes the linearization at (x⋆, u⋆, µ⋆) of (17) with output z so that

J =

[

A B

C 0

]

then we have the following theorem for a static bifurcation point.

Theorem 1 (from [25]) An equilibrium point (x⋆, u⋆, µ⋆) is a static bifurcation point onlyif

Im

(

A B

C 0

)

= Rn+r (22)

Recall that the system matrix is

P (λ) =(

λI − A B

−C 0

)

From this observation, necessary conditions for a static bifurcation point can be obtainedas follows [34]:

Theorem 2 The equilibrium point (x⋆, u⋆, µ⋆) is a static bifurcation point of (17) only ifone of the following conditions is true for its linearization:

1. there is a transmission zero at the origin,

2. there is an uncontrollable mode with zero eigenvalue,

3. there is an unobservable mode with zero eigenvalue,

4. it has insufficient independent controls,

5. it has redundant regulated variables.

The key implication of this theorem is that the ability to locally regulate the systemdiminishes as the bifurcation point is approached [35, 36]. In fact a linear regulator(indeed a smooth feedback regulator) does not exist at the bifurcation point [25]. Atthe bifurcation point an arbitrarily small perturbation of parameters changes the zerostructure of the system thereby requiring a fundamental change in the controller [36].The point we wish to emphasize is that losing the capacity to regulate nonlinear flightdynamics is intimately connected to the bifurcation structure of the trim equations ofthe aircraft. Thus, at least some forms of LOC can be rigorously connected to the flightdynamics. This argument is also made very strongly in [27].

V. Control Behavior of the GTM Near Stall

In this section the goal is to illustrate the control behavior of the GTM around trimpoints near stall. First, uncontrolled departures from controlled flight near stall bifurca-tion points are examined. Then controllability change as stall is approached is illustrated.Finally, some brief remarks about recovery from departures near stall are given.

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A. Uncontrolled departures near stall: GTM example

Simulated GTM departures from a coordinated turn at various speeds near the stallspeed are considered. In each case the aircraft is trimmed very near a coordinated turnequilibrium condition. The controls are then fixed and the resultant trajectories areobserved. Below results for three speeds – 90 ft/sec, 87 ft/sec, and 85 ft/sec are shown.The last is very close to the stall speed. See Figure 1. The equilibrium surface wasgenerated as a function of airspeed using a continuation method as described in [33].One projection of this surface is shown in Figure 1 which shows elevator deflection δe asa function of speed, the parameter V . The points selected for simulation are identifiedin the figure.

85 90 95 100

60

50

40

30

20

10

0

V, ftsec

∆e,deg

Figure 1. A portion of the coordinated turn equilibrium surface shows elevator deflectionas a function of speed.

Figures 2 on the following page and 3 on page 13 present a snapshot of the results.The former show the ground tracks and the latter display the aircraft attitude in termsof the Euler angles.

-200 0 200 400 600 800 10000

200

400

600

800

1000

1200

1400

x (ft)

y (

ft)

(a) 90 ft/sec

0 500 1000 1500 2000 2500 3000 3500 4000-2500

-2000

-1500

-1000

-500

0

500

1000

x (ft)

y (

ft)

(b) 87 ft/sec

-200 0 200 400 600 800 1000 1200 14000

500

1000

1500

2000

2500

3000

x (ft)

y (

ft)

(c) 85 ft/sec

Figure 2. Coordinated turn ground track. (a) at 90 ft/sec the the aircraft stabilizes in acoordinated turn. (b) at 87 ft/sec, closer to the stall speed, the aircraft departs from thecoordinated turn and enters a periodic motion much like an extremely exaggerated Dutchroll. (c) at 85 ft/sec, approximately stall speed, the aircraft departs to an erratic motion,possibly chaotic.

At 90 ft/sec the aircraft cleanly enters the coordinated turn. At 87 ft/sec, closer

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to stall speed, it departs from trim and within 20 seconds enters a well-formed, slowlydescending, periodic motion with rather violent swings in attitude, particularly roll. Nearstall speed, at 85 ft/sec, the vehicle departs from trim and enters a steeply descending,apparently chaotic motion with increasingly violent attitude swings. The simulationmodel utilizes a quaternion representation of attitude from which the Euler angles arecomputed. These are displayed in Figure 3.

0 20 40 60 80 100 120 140 160-200

-150

-100

-50

0

50

100

150

200

Time (sec)

φ,θ,ψ

(d

eg

)

φ

θ

ψ

(a) 90 ft/sec

0 20 40 60 80 100 120 140 160-200

-150

-100

-50

0

50

100

150

200

Time (sec)

φ,θ,ψ

(d

eg

)

φ

θ

ψ

(b) 87 ft/sec

0 10 20 30 40 50 60 70-200

-150

-100

-50

0

50

100

150

200

Time (sec)

φ,θ,ψ

(d

eg

)

φ

θ

ψ

(c) 85 ft/sec

Figure 3. Coordinated turn attitude. (a) at 90 ft/sec the attitude stabilizes as expected.(b) at 87 ft/sec the aircraft enters a periodic trajectory. (c) at 85 ft/sec, approximatelystall speed, the aircraft departs to an erratic motion.

As described in [3], the angle of attack versus sideslip angle plot, i.e., the AdverseAerodynamics Envelope, is a useful indicator of LOC. The simulation results are shownin Figure 4. We see the well contained data set for 90 ft/sec, and even for 87 ft/sec.However, the stall departure case, 85 ft/sec, suggests a serious control problem althoughthis trajectory is obtained with fixed controls. This is consistent with the observationsin Section B.

4 6 8 10 12 14 16 18 20 22-30

-20

-10

0

10

20

30

40

β (deg)

α (

de

g)

(a) 90 ft/sec

4 6 8 10 12 14 16 18 20 22-30

-20

-10

0

10

20

30

40

β (deg)

α (

de

g)

(b) 87 ft/sec

4 6 8 10 12 14 16 18 20 22-30

-20

-10

0

10

20

30

40

β (deg)

α (

de

g)

(c) 85 ft/sec

Figure 4. The adverse aerodynamics plot of the simulated trajectories suggests a seriouscontrol problem for the stall departure case (c).

With reference to Figure 3 (c) the large roll angles and roll rates suggest that otherLOC indicators identified in [3] are also triggered – specifically, the Unusual Attitudeenvelope and the Dynamic Roll Control envelope.

B. Control properties around the bifurcation point: GTM example

In accordance with Theorem 2 some degeneracy in the linear system zero dynamics at thebifurcation point is anticipated. To illustrate this, again consider the GTM, this time ina straight, wings level climb with flight path angle of 0.1 rad (5.7 deg). The equilibriumpoints are plotted as a function of speed in Figure 5. Now, a two-parameter family oflinear systems (LPV model) is constructed using the method described in [37]. In thiscase the parameters are airspeed, V , and flight path angle, γ. The equilibrium surfaceapproximation (and hence the LPV model) is third order in the parameters and generated

713

at the bifurcation point shown in Figure 5 (a). The ‘other’ equilibria, shown in Figure 5(b) are obtained from the cubic approximation and displayed in the figure with the trueequilibrium curve.

83 84 85 86 87 88

50

40

30

20

10

V, ftsec

∆e,deg

(a) Bifurcation point.

83.0 83.5 84.0 84.5 85.0

30

25

20

15

10

V, ftsec

∆e,deg

(b) Other equilibria.

Figure 5. The equilibrium curve is shown for straight, wings level climb with speed as theparameter. The stall point is indicated in (a). A two parameter LPV model is derivedaround the bifurcation point with parameters speed and flight path angle. This involves de-riving a polynomial approximation to the equilibrium surface around the bifurcation point(see [37]). In (b) several points computed from the approximating surface are comparedto the actual surface.

Analysis shows that the LPV system is uncontrollable at the bifurcation point, butcontrollable at points arbitrarily near the bifurcation point. Furthermore, controllabilitydegrades as the bifurcation point is approached. To see this, the controllability matrixis evaluated and its minimum singular value is computed at each of the points shownin Figure 5. The results are shown in Table 1, with the bifurcation point shown inbold typeface. As explained in [37], parameterization of the equilibrium manifold aroundsingular points requires replacement of the physical parameters by suitable coordinates onthe manifold. The variables s1, s2 in Table 1 denote the coordinates used herein. Fixings2 = 0 and varying s1 produces a slice through the surface corresponding to fixed γ andvarying V .

Table 1. Degree of Controllability

s1 s2 V δe σmin

0.010000 0.00 83.7759 -16.4569 0.00186924

0.005000 0.00 83.3882 -18.3686 0.00098419

0.001000 0.00 83.2658 -20.1175 0.00022455

-0.000104 0.00 83.2599 -20.6466 0.00000000

-0.000500 0.00 83.2607 -20.8418 0.00008107

-0.006500 0.00 83.4491 -24.2201 0.00153969

-0.010000 0.00 83.7040 -26.6229 0.00276331

Even though the system fails to be linearly controllable at the bifurcation point itis locally (nonlinearly) controllable in the sense that the controllability distribution hasfull generic rank around the bifurcation point. The implication is that any stabilizingfeedback controller would certainly be nonlinear and probably nonsmooth.

714

C. Remarks on recovery from stall

The GTM appears to be remarkably stable. In the coordinated turn illustrated above,the stall speed is about 85 ft/sec. With the controls fixed at their stall equilibrium valuesthe aircraft enters a spin and dives. After several seconds into the spin the controls arereset to their stable 90 ft/sec values. Figure 6 shows the recovery when the controlsare reset after 10 seconds. This was intended as an experiment to determine if thevehicle dynamics would permit recovery. No assessment was made as to whether this isan acceptable strategy. In particular structural integrity was not evaluated although itappears that peak acceleration is less than 2 g’s. The vehicle drops about 1400 feet.

0 10 20 30 40 50 60 70 8080

100

120

140

160

180

200

220

Time (sec)

Sp

ee

d (

ft/s

)

(a) Speed

-200 0 200 400 600 800 1000-800

-600

-400

-200

0

200

400

600

800

x (ft)

y (

ft)

(b) Ground Track

-200 0 200 400 600 800 10001.86

1.88

1.9

1.92

1.94

1.96

1.98

2

2.02x 10

4

x (ft)

h (

ft)

(c) Altitude

Figure 6. Coordinated turn recovery from stall. The vehicle stalls at about 85 ft/sec duringa coordinated turn. After 10 seconds the controls are reset to the 90 ft/sec values. (a) Thespeed peaks at about 215 ft/sec, 13 sec after stall, before stabilizing at 90 ft/sec. (b) Theground track follows the pattern of Figure 2 until the controls are reset. Then it stabilizesinto the turn. (c) The aircraft drops about 1400 ft. during the recovery.

It is worth noting that the simulations show that recovery can take place even furtherinto the spin. Of course, with significantly greater loss of altitude.

VI. Conclusions

In conclusion, the study of an aircraft’s equilibrium point structure and the associatedcontrol and regulation properties provide direct links between pilot loss of control expe-rience and analytical flight mechanics. The concepts, methods and tools for performingsuch studies are presented in this paper. Their application is illustrated using NASA’sGeneric Transport Model. The trim conditions employed in the examples are straight,wings level flight with specified airspeed and flight path angle or a coordinated turn withspecified airspeed, turn rate and flight path angle. The analysis is developed using modelsthat extend the validity of aircraft models into regions of unusual attitudes.

Appendix I – The GTM 6 Degree-of-Freedom Model

The model used here for all of our examples is derived from an early model of theGTM simulation model (V09.03). Since we made certain simplifications and the GTMmodel is regularly revised and improved we summarize the details of the model usedherein.

715

CX = −0.0390905 + 0.35218α + 5.36708α2 − 23.1537α3 − 26.2264α4 + 109.938α5

+q(2.46995+24.4028α+58.4581α2)c

2V+ 0.125409αδe + 0.0857469α3δe − 0.00961977α5δe

−0.0811392δ2e + 0.0405696α2δ2e − 0.0033808α4δ2e − 0.389796αδ3e+0.064966α3δ3e − 0.0032483α5δ3e

CY = −1.0499β + 0.254159β3 +r(0765433+010909α+0.553414α2)b

2V

+p(1.22326α+1.26322α2

−39.4599α3)b2V

+ 0.175591δr

CZ = −0.0261857− 5.38662α + 0.339087α2 + 28.0138α3 − 23.0418α4 − 12.8899α5

+q(−28.2259−62.5918α−460.841α2)c

2V− 0.445354δe − 0.0972682α2δe + 0.0347678α4δe

−0.0811392αδ2e + 0.0135232α3δ2e − 0.00067616α5δ2e + 0.389796δ3e − 0.194898α2δ3e+0.0162415α4δ3e

CL = −0.126318β − 0.22119αβ + 0.255338β3 − 0.191268β5

+r(0.0608527+0.730792α+2.90179α2)b

2V

+p(−0.414849−0.325859α+6.67529α2+125.613α4)b

2V− 0.0247139δa + 0.0193176δr

CM = 0.181738− 1.10553α− 15.1134α4 +q(−47.6756+69.4945α+308.277α2)c

2V

−1.76253δe − 0.920542αδ2e + 1.35544δ3e+(

−0.0261857− 5.38662α+ 0.339087α2 + 28.0138α3 − 23.0418α4)

(−xcg + xcg)(

−12.8899α5 +q(−28.2259−62.5918α−460.841α2)c

2V− 0.445354δe − 0.0972682α2δe

)

(−xcg + xcg)(

+0.0347678α4δe − 0.0811392αδ2e + 0.0135232α3δ2e − 0.00067616α5δ2e)

(−xcg + xcg)(

+0.389796δ3e − 0.194898α2δ3e + 0.0162415α4δ3e)

(−xcg + xcg)

CN = 0.202546β − 0.143331β3 +r(−0.379639−0.205145α−0.937344α2)b

2V

+p(−0.00731187−0.45033α+0.724553α2+16.4433α3)b

2V− 0.112626δr − 0.000470559βδr

−1bc

(

−1.0499β + 0.254159β3 +r(0.765433+0.10909α+0.553414α2)b

2V

)

(−xcg + xcg)

−1bc

(

p(1.22326α+1.26322α2−39.4599α3)b

2V+ 0.175591δr

)

(−xcg + xcg)

Acknowledgement

This research was supported by the National Aeronautics and Space Administrationunder Contract numbers NNX09CE93P and NNX10CB28C.

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