A General Solution to the Aircraft Trim Problem

The air raft trim problem General trim equations Examples A General Solution to the Aircraft Trim Problem Agostino De Marco1 Eugene L. Duke2 Jon ...
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The air raft trim problem

General trim equations

Examples

A General Solution to the Aircraft Trim Problem Agostino De Marco1

Eugene L. Duke2

Jon S. Berndt3

1 University of Naples “Federico II” Department of Aerospace Engineering (DIAS) Via Claudio, 21 – 80125 Naples, Italy 2 Rain

Mountain Systems Incorporated Glasgow, Virginia, 24555-2509, USA 3 JSBSim

Project Team Lead

League City, TX, USA AIAA Modelling & Simulation Technology Conference Hilton Head, South Carolina, 2007

Summary

The air raft trim problem

General trim equations

Examples

Outline The aircraft trim problem Definitions Considerations

General trim equations 6DOF aircraft equations of motion Trim conditions Implementation of the trim algorithm in JSBSim

Examples Trim for straight flight Trim for pull-up

Summary

The air raft trim problem

General trim equations

Examples

Outline The aircraft trim problem Definitions Considerations

General trim equations 6DOF aircraft equations of motion Trim conditions Implementation of the trim algorithm in JSBSim

Examples Trim for straight flight Trim for pull-up

Summary

The air raft trim problem

General trim equations

Examples

The aircraft trim problem: our approach

• We define what we mean by trim. • Examine a variety of trim conditions. • Derive the equations defining those trim conditions. • Present a general approach to the solution of trim problems:

constrained minimization of a multivariate, scalar cost function. • Provide an example of how trim algorithms are applied in

JSBSim, an open source flight dynamics model.

Summary

The air raft trim problem

General trim equations

Examples

Summary

Definitions: frames, aerodynamic angles, forces φW L

θW ≡ γ

flight path α ground track, ψW = 0

β

C

β

D

T

α

α β

yV

~ V

horizon

yW

xB

w v

xW γ

yB

xV

zW zB aircraft symmetry plane

u

mg

zV

The air raft trim problem

General trim equations

Examples

Summary

Definitions: state space concepts • Generic aircraft state equations



˙ x, u = 0 implicit form: g x,

explicit form: x˙ = f x, u

x = aircraft state vector, u = vector of inputs.



• Dynamic & kinematic parts

  T T x = xT d , xk

• Body-fixed frame components, Earth-frame c.g. coordinates

and A/C Euler angles  T  T xd = u , v , w , p , q , r , xk = xC , yC , xC , φ , θ , ψ

The air raft trim problem

General trim equations

Examples

Summary

Definitions: inputs

• Vector u of control inputs depends by the type of aircraft. • For a conventional configuration aircraft the minimum

arrangement of the inputs is usually given by T  u = δT , δe , δa , δr

• Inputs uj have standard signs. Their ranges depend on the

particular aircraft. Might be mapped to [−1, +1] or to [0, +1]. • We will always refer to the inputs as some required

combination of aerosurface deflections and thrust output.

The air raft trim problem

General trim equations

Examples

Summary

Steady-state conditions. Trim points. Equilibrium. • A classical concept in the theory of nonlinear systems is the

equilibrium point or trim point. • For an autonomous, time-invariant system the equilibrium

point is denoted as xeq and is defined as the particular x vector which satisfies  g 0 , xeq , ueq = 0 with x˙ ≡ 0 and ueq = constant

• Equilibrium condition above also defines a set of control

settings ueq that make the steady state possible. • Generalized idea of rest: the condition when all the

derivatives are identically zero. In our case this applies to the dynamic part xd , i.e. to those independent variables ruled by Newton’s laws: x˙ d ≡ 0.

The air raft trim problem

General trim equations

Examples

Summary

General conditions for steady-state flight The general steady-state flight condition resulting from the above discussion is given as follows:   zero accelerations ⇒ u, ˙ v, ˙ w˙ or V˙ , α, ˙ β˙ ≡ 0   p, ˙ q, ˙ r˙ or p˙W , q˙W , r˙W ≡ 0 , linear velocities ⇒ u, v, w ( or V, α, β)

= constant values , angular velocities ⇒ p, q, r ( or pW , qW , rW ) = prescribed/constrained constant values aircraft controls ⇒ δT , δe , δa , δr = appropriate constant values

The air raft trim problem

General trim equations

Examples

Summary

Equations of aircraft motion: flat-Earth hypothesis

• Round-Earth equations can be relaxed to flat-Earth

equations. • Usually the flat-Earth equations are considered satisfactory

for all control system design purposes. Consequently those equations are satisfactory also for the derivation of trim conditions. • Allowed steady-state conditions: wings-level horizontal flight,

in any direction, and constant altitude turning flight. • If the change in atmospheric density with altitude is

neglected, also wings-level climb and climbing turn are permitted as steady state flight conditions.

The air raft trim problem

General trim equations

Examples

Outline The aircraft trim problem Definitions Considerations

General trim equations 6DOF aircraft equations of motion Trim conditions Implementation of the trim algorithm in JSBSim

Examples Trim for straight flight Trim for pull-up

Summary

The air raft trim problem

General trim equations

Examples

Summary

System of nonlinear algebraic equations. Multiple solutions.

• An actual pilot may not find it very difficult to put an aircraft

into a steady-state flight condition. • Trimming an aircraft mathematical model requires the

solution of simultaneous nonlinear equations: g(0, xeq , ueq ) = 0. • A steady-state solution can only be found by using a

numerical method. Multiple feasible trim solutions for a given trim condition are possible.

The air raft trim problem

General trim equations

Examples

Summary

A general approach. Aerodynamic database required. • The treatment of aircraft trim proposed here starts from the

standard 6DOF EOM of an airplane in atmospheric flight but does not make any limiting assumptions on the geometrical properties of the aircraft nor on the curves of aerodynamic coefficient • All that one really expects is an aerodynamic model that

provides non-dimensional aerodynamic coefficients, no matter where those parameters come from or how they are derived. • In practice, a convenient aerodynamic model should be

available in the form of tabulated data for the widest possible ranges of aerodynamic angles and velocities and for all possible aircraft configurations.

The air raft trim problem

General trim equations

Examples

Outline The aircraft trim problem Definitions Considerations

General trim equations 6DOF aircraft equations of motion Trim conditions Implementation of the trim algorithm in JSBSim

Examples Trim for straight flight Trim for pull-up

Summary

The air raft trim problem

General trim equations

Examples

Aircraft equations of motion – Translation (u, v, w) The classical system reference  1   u˙ =   m   1 v˙ =  m      w˙ = 1 m

of scalar force equations in the body-axis

 XA + XT − wq + vr − g sin θ

 YA + YT − ur + wp + g cos θ sin φ

 ZA + ZT − vp + uq + g cos θ cos φ V =

u = V cos β cos α ,

p

u2 + v 2 + w2

v = V sin β ,

w = V cos β sin α

Summary

The air raft trim problem

General trim equations

Examples

Summary

Aircraft equations of motion – Translation (V, α, β)  h ˙ = 1 − D cos β + C sin β + XT cos α cos β + YT sin β + ZT sin α cos β  V   m    − mg sin θ cos α cos β − cos θ sin φ sin β    i    − cos θ cos φ sin α cos β       α˙ = q − tan β p cos α + r sin α   h  1   + − L + ZT cos α − XT sin α  V m cos β i

+ mg cos θ cos φ cos α + sin θ sin α       β˙ = + p sin α − r cos α     1 h   + D sin β + C cos β − XT cos α sin β + YT cos β − ZT sin α sin β   Vm     + m g sin θ cos α sin β + cos θ sin φ cos β    i   − cos θ cos φ sin α sin β

The air raft trim problem

General trim equations

Examples

Summary

Aircraft equations of motion – Rotation (p, q, r)    I1  p˙  1    I2 q˙ =   det I B r˙ I3 −

X

Ikr

I2 I4 I5

ω˙ kr

k



0 − r −q



 r   ix  ir  yr  iy k

−r 0 p

X k



  I3   LA + LT   MA + MT I5  ·    NA + NT I6 

  q Ix −p  ·  −Ixy 0 −Ixz 

Ikr ωkr 

0 r −q

−r 0 p

−Ixy Iy −Iyz

   −Ixz  p  −Iyz  · q   Iz r

   r  q  ix    −p  · ir   yr   0 iz k

The air raft trim problem

General trim equations

Examples

Summary

Auxiliary kinematic equations Navigation equations:  cos θ cos ψ ( )  x˙ E  y˙ E =  cos θ sin ψ z˙E 

sin φ sin θ cos ψ  − cos φ sin ψ

sin φ sin θ sin ψ  + cos φ cos ψ

− sin θ

sin φ cos θ

cos φ sin θ cos ψ  + sin φ sin ψ



( u )  cos φ sin θ sin ψ  · v   w  − sin φ cos ψ cos φ cos θ

Gimbal equations: 

 1  0 = θ˙  ψ˙       φ˙ 

0

sin φ sin θ cos θ cos φ

cos φ sin θ cos θ − sin φ

sin φ cos θ

cos φ cos θ



 ( p )  · q  r 

The air raft trim problem

General trim equations

Examples

Summary

Constraints   pW qW  rW     p     pW q qW = CB←W    r rW

  1 0 − sin γ  φ˙ W  = 0 cos φW cos γ sin φW  γ˙   ˙  0 − sin φW cos γ cos φW ψW      pW cos α cos β − qW cos α sin β − rW sin α  pW sin β + qW cos β =    pW sin α cos β − qW sin α sin β + rW cos α  



Example of constraint equations for a steady-state turn: " 1 # n2 − cos2 γ 2 g g ˙ ˙ γ˙ = φW = 0, ψW = tan φW = tan V V cos γ   2 g cos γ sin φW cos α sin β p=− sin γ tan φW cos α cos β + + cos γ sin φW sin α V cos φW   g cos γ sin2 φW cos β q=− sin γ tan φW sin β − V cos φW   g cos γ sin2 φW sin α sin β r=− sin γ tan φW sin α cos β + − cos γ sin φW cos α V cos φW

The air raft trim problem

General trim equations

Examples

Summary

Aerodynamics models Table 1: Dependen y of for e and moment omponents upon air raft state and

ontrol variables Component

State Variables

h

V

1 X A + XT

d/ e

2 YA + YT

d

/

d

/

When When When d When e When b

q

r

α˙

β˙













d

/

β is not zero.

∼ ∼

e

a

∼ ∼

e

symbols: a

p

a,b

d,e

5 MA + MT 6 NA + NT

β

d,e

3 ZA + ZT 4 LA + LT

e

α



∼ /

δr















= dependent on = could depend on, may vary with configuration ∼ = almost always not dependent on

thrusters work in nonaxial ow. a nonsymmetri thrust is applied. ground ee t is modelled. engine state is altitude dependent.

δT



∼ ∼

δa





∼ ∼

/

δe

/

The air raft trim problem

General trim equations

Examples

Outline The aircraft trim problem Definitions Considerations

General trim equations 6DOF aircraft equations of motion Trim conditions Implementation of the trim algorithm in JSBSim

Examples Trim for straight flight Trim for pull-up

Summary

The air raft trim problem

General trim equations

Examples

Summary

Trim cost function • Aircraft model equations: x˙ = f x, u



Steady-state ⇒ x˙ d ≡ 0 = fd xd , xk , u • Possible definitions of a cost function:



J = u˙ 2 + v˙ 2 + w˙ 2 + p˙ 2 + q˙ 2 + r˙ 2 J = V˙ 2 + α˙ 2 + β˙ 2 + p˙ 2 + q˙ 2 + r˙ 2 • More generally J = {x˙ d }T [W ]{x˙ d }, with [W ] a matrix of

weights. • Aircraft steady-state flight yields a J = 0. • For a given, desired trim condition: Who are the variables

upon which J depends?

The air raft trim problem

General trim equations

Examples

Summary

Stating a trim problem • Formally: J = J xd , xk , u



i.e. J = J u, v, w, p, q, r, h, φ, θ, ψ, δT , δe , δa , δr • A trim problem is stated by:



(i) declaring the desired center of mass trajectory, e.g. assigning V , γ, ψGT , h, ψ˙ GT , γ˙ ; (ii) when needed, requiring a given normal load factor e.g. turn ⇒ φW , or pull-up/push-over ⇒ q ; (iii) and possibly assigning aircraft control e.g. aileron failure, δa = δa,fail ; (iv) constraining the values of some other state variables according to the kinematic equations (e.g. deriving the values of p, q, r); and (v) finding the minimum of J as a function of the remaining non-frozen variables.

The air raft trim problem

General trim equations

Examples

Summary

Example: trim for straight flight. ~ V straight flight path

YA

yB

ψ φ

horizontal

G

xB NA

zB zV

β

C

yV

LA θ

γ ψLoadModel(aircraftPath, enginePath, aircraftName); //

initial

conditions

cfg

file

JSBSim::FGInitialCondition *ic = fdmExec->GetIC(); ic->Load(initFileName); //

the

trim

object

JSBSim::FGTrimAnalysis fgta(fdmExec, (JSBSim::TrimAnalysisMode)1); // // // //

the

0: Longitudinal , 1: Full , 2: Full , Wings - Level , 3: Coordinated Turn , 4: Turn , 5: Pull - up / Push - over Low level coding of new trim targets possible ic

cfg

file

contains

trim

directives

fgta.Load(initFileName); //

optimize

J

if ( !fgta.DoTrim() ) cout