zotero-db/storage/RI3GLAG7/.zotero-ft-cache

MTR150184 MITRE TECHNICAL REPORT

MITRE

Derivation of a Point-Mass Aircraft Model used for Fast-Time Simulation

MITRE Technical Report

Sponsor: Federal Aviation Administration (FAA)
Dept. No.: F084 Project No.: 0215BB02-AS Outcome No.: 2 PBWP Reference: 2-6.1-1 “Development of the FIM SPR and the FIM MOPS”
Approved for Public Release.
This technical data was produced for the U.S. Government under Contract No. DTFAWA-10-C-00080, and is subject to the Rights in Technical Data— Noncommercial Items clause (DFARS) 252.227-7013 (NOV 1995)
c 2015 The MITRE Corporation. All Rights Reserved.
McLean, VA

Dr. Lesley A. Weitz April 2015

Approved By:

Robert C. Strain, Department Head

Date

Elida C. Smith, Outcome Leader

Date

Abstract
This document describes a closed-loop aircraft model for testing the performance of Flight-deck Interval Management (FIM) avionics. The derivation of a point-mass aircraft model with and without winds is presented. Guidance control laws used to track a four-dimensional trajectory, described as a reference horizontal path, a vertical profile, and an indicated airspeed command, are also presented. In particular, two speed management modes, used to control the aircraft’s speed during descent while also managing the aircraft’s altitude relative to the reference vertical profile, are described. The data format of the reference horizontal path is described in detail, including an algorithm for mapping the aircraft’s position in the horizontal plane onto the reference horizontal path. Lastly, parameters for representing different aircraft types using the general equations are enumerated. Keywords: aircraft dynamics, aircraft control.
iii

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Table of Contents

1 Introduction

1

2 Equations of Motion

1

2.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Equations of Motion with Winds

4

3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Sign Convention for the Wind Gradients . . . . . . . . . . . . . . . . . . . . . . . 6

4 Guidance Control Laws

7

4.1 Lateral Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Speed Control Using Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Speed Control Using Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.4 Control Gains and Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Reference Horizontal Path

14

5.1 Data Format for the Horizontal Path . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.2 Mapping to Along-Path Position . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2.1 Cross-track Error Calculation . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.2 Along-Path Position Calculation . . . . . . . . . . . . . . . . . . . . . . . 19

6 Aircraft-Specific Modeling

20

7 References

23

Appendix A Abbreviations and Acronyms

24

v

List of Figures
1 Sequence of three simple rotations: 3-2-1 Euler angle orientation sequence. . . . . 2 2 Free-body diagram of forces acting on the point mass. . . . . . . . . . . . . . . . . 3 3 Example wind profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 TAS and wind vectors in the horizontal plane. . . . . . . . . . . . . . . . . . . . . 8 5 Data format for the reference horizontal path. . . . . . . . . . . . . . . . . . . . . 14 6 Horizontal path generated from the data in Figure 5. . . . . . . . . . . . . . . . . . 15 7 High-level process for determining the along-path position by projecting the air-
craft’s x, y position onto the reference horizontal path. . . . . . . . . . . . . . . . . 16 8 Illustration of quadrants around each HPT. . . . . . . . . . . . . . . . . . . . . . . 18 9 Projection of aircraft positions onto horizontal path to determine DTG to end point
(HPT1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
List of Tables
1 Control Gains and Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . 13
vi

1 Introduction
Equations of motion (EOM) are derived for a point-mass aircraft model. The EOM may be used for modeling aircraft motion in a fast-time simulation environment. In Section 2, the EOM are derived by first defining reference frames, determining the aircraft acceleration in an appropriate reference frame, and the dynamics are derived using Newton’s Second Law. Section 3 shows the derivation of the EOM for non-zero wind conditions.
Control laws for tracking a reference four-dimensional trajectory are presented in Section 4. A detailed description of the reference horizontal path and a mapping between the aircraft’s position in the horizontal plane and the reference path is provided in Section 5.
In Section 6, an approach to representing aircraft-specific performance is presented, including a mapping to parameters from EUROCONTROL’s Base of Aircraft Data (BADA) [2].

2 Equations of Motion

2.1 Reference Frames

Assuming a flat, non-rotating Earth, an inertial reference frame N is defined with the nˆ1 axis aligned with east, the nˆ2 axis aligned with north, and the nˆ3 axis pointing up from the Earth.
A sequence of three simple rotations relates the inertial reference frame N to a non-inertial, rotating reference frame C. Forces acting on the aircraft are most easily coordinatized in reference frame C, which is often referred to as a body-fixed reference frame when deriving the full six degreeof-freedom (DOF) EOM. The sequence of simple rotations is illustrated in Figure 1. The first rotation is through an angle ψ about the nˆ3 axis (a “3” rotation) to an intermediate reference frame A; the second rotation is through an angle γ about the aˆ 2 axis (a “2” rotation) to an intermediate reference frame B; and, the third rotation is through an angle φ about the bˆ 1 axis (a “1” rotation) to reference frame C. This rotation sequence is known as a 3-2-1 Euler angle rotation sequence, which represents the orientation of the aircraft through the yaw (ψ), flight-path angle (γ), and roll (φ) angles.

A vector represented in the inertial frame may also be coordinatized in the C reference frame. The vector is acted upon by the rotation matrix [C321(ψ, γ, φ)], which is the product of the three simple rotation matrices: [C1(φ)][C2(γ)][C3(ψ)].

[C321(ψ, γ, φ)] = 100

0 cos φ − sin φ

csoin0sφφ csoin0sγγ

0 1 0

−cos0isnγγ −cos0isnψψ

sin ψ cos ψ
0

100

(1)

= −sicnoφs φsisninψcψo+s+γcoscsionφsφψsisninγγccoossψψ

cos γ sin ψ cos φ cos ψ + sin φ sin γ sin ψ − sin φ cos ψ + cos φ sin γ sin ψ

− sin γ sin φ cos cos φ cos

 γγ

(2)

1

a2

n2

a3

b3

c3

b3

ψ

γ

a1

ψ

n1

γ

φ

c2

φ

a1

b2

b1

Figure 1. Sequence of three simple rotations: 3-2-1 Euler angle orientation sequence.

2.2 Kinematics

The inertial position of the aircraft is given by the following.

P = xnˆ1 + ynˆ2 + hnˆ3

(3)

The inertial velocity can be written in the N and C reference frames.

V = x˙nˆ1 + y˙nˆ2 + h˙ nˆ3 = V cˆ1

(4)

Here, V is the aircraft’s true airspeed (TAS), which is equal to the inertial velocity only when there are no winds.

To derive the EOM using Newton’s Second Law, the inertial acceleration, coordinatized in the C frame, is determined using transport theorem [1].

Nd dt

(V)

=

Cd dt

(V)

+

ωC/N

×

V

(5)

The angular-velocity vector relating the C frame to the N frame, ωC/N, is given by the addition of angular velocities. Here, the angular velocities are expressed in the C frame by pre-multiplying by the appropriate rotation matrices.

ωC/N = ωC/B + ωB/A + ωA/N

(6)

= [φ˙, 0, 0]T + [C1(φ)][0, γ˙ , 0]T + [C1(φ)][C2(γ)][0, 0, ψ˙ ]T

(7)

ωC/N = φ˙ − ψ˙ sin γ cˆ1 + γ˙ cos φ + ψ˙ cos γ sin φ cˆ2 + −γ˙ sin φ + ψ˙ cos γ cos φ cˆ3 (8)

Making the appropriate substitutions into equation (5) yields the following expression for the inertial acceleration.

V˙ = V˙ cˆ1 + V −γ˙ sin φ + ψ˙ cos γ cos φ cˆ2 − V γ˙ cos φ + ψ˙ cos γ sin φ cˆ3

(9)

2

The kinematic equations are determined by projecting V cˆ1 into the inertial frame using the transpose of the rotation matrix, [C321(ψ, γ, φ)]T ; x˙, y˙, and h˙ are the velocities along the inertial axes,
nˆ1, nˆ2, and nˆ3, respectively.

hyx˙˙˙ = ccoo−ssγγsicsnoinγsψψ

− cos φ sin ψ + sin φ sin γ cos ψ cos φ cos ψ + sin φ sin γ sin ψ
sin φ cos γ

−sisninφφsicnoψscψo+s+cφoccsooφssφsγisninγγcossinψψ V00

(10)

The inertial velocity of the aircraft can be expressed as shown.

V = V cos γ cos ψnˆ1 + V cos γ sin ψnˆ2 − V sin γnˆ3

(11)

2.3 Dynamics

Figure 2 shows free-body diagrams of the forces acting on the point mass (the depiction of the aircraft in the figure is used for illustrative purposes only). The thrust and drag forces, T and D, respectively, are shown to be aligned with the airspeed vector in the cˆ1 direction; the lift force, L, is in the cˆ3 direction; and, the weight of the aircraft is in the nˆ3 direction.

L

L

c3

c1

T

− γ

D

c3
c2 φ

mg

mg

Figure 2. Free-body diagram of forces acting on the point mass: side view (left) and front view (right). The lift vector and the cˆ3 unit vector are shown in gray in the side view to indicate that they are projections onto the vertical plane.

The summation of forces acting on the aircraft are given in equation (12).

F = (T − D + mg sin γ) cˆ1 − mg cos γ sin φcˆ2 + (L − mg cos γ cos φ) cˆ3

(12)

The dynamics equations, expressed in the C reference frame, are shown below.

mV˙ = T − D + mg sin γ

(13)

mV −γ˙ sin φ + ψ˙ cos γ cos φ = −mg cos γ sin φ

(14)

−mV γ˙ cos φ + ψ˙ cos γ sin φ = L − mg cos γ cos φ

(15)

3

To decouple equations (14) and (15), equation (14) is multiplied by sin φ and equation (15) is multiplied by cos φ and the resulting expressions are added to get an equation for γ˙.

mVγ˙ = −L cos φ + mg cos γ

(16)

In a similar manner, equation (14) is multiplied by cos φ and equation (15) is multiplied by sin φ and the second expression is subtracted from the first to get an equation for ψ˙ .

mV cos γψ˙ = −L sin φ

(17)

The aircraft EOM are the following six first-order ordinary differential equations (ODEs), comprised of three kinematic and three dynamic equations.

x˙ = V cos γ cos ψ

(18)

y˙ = V cos γ sin ψ

(19)

h˙ = −V sin γ

(20)

V˙ = T − D + g sin γ

(21)

m

γ˙ = − L cos φ + g cos γ

(22)

mV

V

ψ˙

=

− L sin φ mV cos γ

(23)

3 Equations of Motion with Winds

3.1 Derivation

Suppose there is a constant wind field with wind velocity expressed in the inertial reference frame.

Vw = Vwx nˆ1 + Vwy nˆ2

(24)

The inertial velocity of the aircraft can be expressed as

VC/N = VC/W + VW/N ,

(25)

where VC/W is the velocity of the aircraft relative to the wind frame W, and VW/N is the wind velocity relative to the inertial frame, which moves with velocity Vw relative to the inertial frame. The inertial velocity is found by adding equations (11) and (24).

V = (V cos γ cos ψ + Vwx) nˆ1 + V cos γ sin ψ + Vwy nˆ2 − V sin γnˆ3

(26)

Given a constant wind field, the inertial velocity in equation (26) and the dynamics equations (21)-(23) comprise the EOM.

However, given a non-constant wind field, the time derivative of the wind velocity must be considered in the dynamics. The inertial acceleration, expressed in the C frame, has the following form
4

when wind-velocity derivatives are included. V˙ = V˙ + V˙ wxcγcψ + V˙ wycγsψ cˆ1+ V −γ˙ sφ + ψ˙ cγcφ + V˙ wx (−cφsψ + sφsγcψ) + V˙ wy (cφcψ + sφsγsψ) cˆ2+ (27) −V γ˙ cφ + ψ˙ cγsφ + V˙ wx (sφsψ + cφsγcψ) + V˙ wy (−sφcψ + cφsγsψ) cˆ3
Here, s(·) ≡ sin(·) and c(·) ≡ cos(·).

Applying Newton’s Second Law for the inertial acceleration in equation (27) and decoupling the equations for γ˙ and ψ˙ (as described in Section 2.3) results in the following set of equations.

V˙

=

T

−D m

+

g sin γ

−

V˙ wx

cos γ cos ψ

−

V˙ wy

cos γ

sin ψ

(28)

γ˙ = −L cos φ + mg cos γ + 1

mV

V

V˙ wx sin γ cos ψ + V˙ wy sin γ sin ψ

(29)

ψ˙

=

− L sin φ mV cos γ

+

V

1 cos γ

V˙ wx sin ψ − V˙ wy cos ψ

(30)

Often wind velocities are provided in a grid format, where the wind speed and direction are given at a point in the horizontal plane and at an altitude for a given time. Assuming that Vwx and Vwy are functions of t, x, y, and h, the time derivative of Vwx and Vwy can be computed by the chain rule. For example:

V˙ wx(t,

x, y, h)

=

dVwx dt

=

∂Vwx ∂t

+

∂Vwx ∂x

∂x ∂t

+

∂Vwx ∂y

∂y ∂t

+

∂Vwx ∂h

∂h ∂t

(31)

=

∂Vwx ∂t

+

∂Vwx ∂x

x˙ +

∂Vwx ∂y

y˙ +

∂Vwx ∂h

h˙

(32)

Additionally, it can be assumed that the wind conditions change slowly with time (∂Vwx/∂t ≈ 0). If the wind gradients in the x and y directions are much less than the wind gradient with respect to altitude (∂Vwx/∂x, ∂Vwx/∂y << ∂Vwx/∂h), the time rate of change of the wind velocity can be simplified to the following.

V˙ wx(t, x, y, h) =

∂Vwx ∂h

h˙ = −

∂Vwx ∂h

V sin γ

(33)

Substituting equation (33) and a similar expression for the wind derivative in the nˆ2 direction into the dynamics equations in (28)-(30) results in the following.

V˙ = T − D + g sin γ + V m

∂Vwx ∂h

·

cos

ψ

+

∂Vwy ∂h

· sin ψ

sin γ cos γ

(34)

γ˙ = −L cos φ + mg cos γ − mV

∂Vwx ∂h

·

cos ψ

+

∂Vwy ∂h

·

sin ψ

sin2 γ

(35)

ψ˙

=

− L sin φ mV cos γ

−

∂Vwx ∂h

·

sin ψ

−

∂Vwy ∂h

·

cos ψ

tan γ

(36)

Equations (34)-(36) in combination with the kinematic equations for x˙, y˙, and h˙, which are determined from the nˆ1, nˆ2, and nˆ3 components in equation (26), respectively, form the EOM for the point-mass aircraft model with non-constant winds.

5

Increasing

3.2 Sign Convention for the Wind Gradients
Equation (33) shows how the time derivative of the wind velocity is determined using the wind gradient with respect to altitude, ∂Vw/∂h, and the vertical rate, h˙. If dVw/dt < 0, the wind magnitude is decreasing with time; and, if dVw/dt > 0, the wind magnitude is increasing with time. An example wind profile is shown in Figure 3. Because the nˆ3 direction has been defined as positive in the upward direction, the wind gradient is positive; i.e., ∂Vw/∂h > 0.
Altitude
Wind Magnitude, Vw
Figure 3. Example wind profile.
Given the wind profile in Figure 3, and if the aircraft is descending, ∂Vw/∂h > 0 and h˙ < 0; therefore, dVw/dt < 0 and the wind magnitude is decreasing with time, which matches the sign convention described above. If the aircraft is climbing, ∂Vw/∂h > 0 and h˙ > 0; therefore, dVw/dt > 0 and the wind magnitude is increasing with time.
6

4 Guidance Control Laws

The control inputs to the point-mass aircraft model are T , γ, and φ. Equation (29) is replaced with a first-order differential equation to track a commanded flight-path angle, γc, with gain kγ; wind-gradient effects on the flight-path angle tracking are still included.

γ˙ = kγ (γc − γ) −

∂Vwx ∂h

·

cos ψ

+

∂Vwy ∂h

·

sin ψ

sin2 γ

(37)

First-order differential equations for thrust and roll angle also augment the equations of motion to model non-instantaneous changes to T and φ, where the gains kT and kφ are chosen to realistically model the aircraft’s response.

T˙ = kT (Tc − T )

(38)

φ˙ = kφ (φc − φ)

(39)

Here, Tc and φc are the commanded thrust and roll angle, respectively.
The guidance control laws are designed to track a reference horizontal path, a reference calibrated airspeed (CAS), and a reference vertical profile. These reference inputs (and other inputs such as vertical rate) are determined from a pre-calculated four-dimensional trajectory that is generated by the aircraft’s Flight Management System and is designed for a specified navigation procedure, which may include altitude and speed constraints at waypoints along the procedure.

When the aircraft is in level flight, the airspeed is controlled using thrust. However, there are two speed-management modes during the descent:

• Speed control using thrust: speed is managed using thrust and the altitude is managed using pitch (or flight-path angle in the point-mass model described here); and,

• Speed control using pitch: speed is managed by changing the pitch of the aircraft and the altitude is managed by modulating the thrust.

A speed brake model is also included to increase drag in cases when the aircraft needs to decelerate

(in the speed control using thrust mode) or decrease its altitude (in the speed control using pitch

mode), but is limited by the current state of the aircraft and/or its control inputs. When the speed

brake is fully deployed, the drag coefficient is increased by 60%. To model the non-instantaneous

deployment of the speed brake, the speed brake deployment fraction, b, is modeled as a first-order

differential equation as shown.

b˙ = kspeed brake (bc − b)

(40)

Here, kspeed brake prescribes the rate of deployment and bc is the commanded speed brake deployment fraction (e.g., 1.0 indicates full deployment and 0.5 indicates that the speed brake is only deployed halfway).

The lateral control law and the control laws for airspeed and altitude for each of the speedmanagement modes are described in the following subsections. Logic for deploying the speed brakes is included in the descriptions of the speed-management modes.
7

4.1 Lateral Control

The lateral control law is a function of the heading error, eψ, and the cross-track error, extrk. The heading error is the error in the commanded heading and the current heading.

eψ = ψc − ψ

(41)

The commanded heading is the sum of the course-angle guidance, θ, defined by the reference horizontal path, and the angle β, which is the angle between the horizontal projection of the TAS vector and the course angle.

ψc = θ + β

(42)

The angle β is determined from the winds at the aircraft’s current x, y position, the horizontal projection of the TAS, and the ground speed as illustrated in Figure 4.

n2 W

Vcos(γ) Vw

Vw ||

β

(Vcos(γ))2 - (Vw )2

θcrs n1

Figure 4. TAS and wind vectors in the horizontal plane.

The wind components parallel and perpendicular to the reference horizontal path, Vw|| and Vw⊥, respectively, are found by rotating the x and y components of the wind through angle θ.

Vw|| = Vwx cos θ + Vwy sin θ

(43)

Vw⊥ = −Vwx sin θ + Vwy cos θ

(44)

8

The ground speed, Vgs, and the wind magnitude, W, are calculated as shown.

Vgs = (V cos γ)2 − Vw2⊥ + Vw||

(45)

W = Vw2x + Vw2y

(46)

The magnitude of β is found using the Law of Cosines.

|β|

=

cos−1

 

(V

cos γ)2 + Vg2s − 2V cos γ · Vgs

W2

 

(47)

The sign of β is opposite to the sign of Vw⊥.

β = −|β| · sign Vw⊥

(48)

The heading error should be limited to [−π, π) to prevent a turn that is greater than 180 degrees.

The cross-track error is the perpendicular distance between the aircraft’s current position and the reference horizontal path. Equations to calculate the cross-track error based on the aircraft’s x, y position and the reference horizontal path are presented in Section 5.2.1.

The commanded roll angle is computed as a linear function of the heading and cross-track errors.

φc = −kψeψ − kxtrkextrk

(49)

4.2 Speed Control Using Thrust

The airspeed control law is a simple proportional control law that determines the desired acceleration from which the commanded thrust may be computed using dynamic inversion. The commanded calibrated airspeed, VCAS c, is converted to a commanded TAS, Vc. The desired acceleration, V˙ c, is a function of the error between the current and the commanded TAS.

V˙ c = kV (Vc − V)

(50)

The commanded thrust, Tc, is found by solving equation (34) for T and substituting the desired acceleration for V˙ .

Tc = mV˙ c + D − mg sin γ − mV

∂Vwx ∂h

·

cos ψ

+

∂Vwy ∂h

·

sin ψ

sin γ cos γ

(51)

The commanded thrust is bounded by the maximum and minimum thrust values, Tmax and Tmin, respectively. Section 6 describes how Tmax and Tmin are calculated for a specific aircraft type.

The altitude control law commands a flight-path angle, γc, to manage the aircraft’s altitude relative

to the reference vertical profile defined by the reference altitude, hre f , and the reference vertical rate, h˙ re f .

γc

=

sin−1

 

−

h˙ re f

+ kaltealt V

  ,

where ealt = hre f

−h

(52)

9

If the commanded thrust, Tc, is less than Tmin, the current CAS is checked against the upper CAS limit for the next flaps configuration to determine if the flight crew can deploy the next flaps setting to increase drag.1 The speed brake is deployed if the minimum thrust is commanded for more than 15 seconds and if the current TAS is more than 5 kt faster than Vc. The speed brake is deployed for at least 30 seconds or until the commanded thrust is greater than the minimum thrust (i.e., Tc > Tmin).
The following algorithm describes the logic for checking the aircraft configuration in the case that Tc < Tmin. The function getConfigForDrag is detailed in Section 6. At the start of the simulation, minT hrustCounter is initialized to 0, and the current flaps configuration is initialized based on the aircraft’s current CAS per the function getCon f ig, which is also detailed in Section 6.
Algorithm 1 Configuration for Increased Drag (Speed on Thrust)
if(Tc < Tmin) then Tc = Tmin newFlapCon f ig = getCon f igForDrag (VCAS , h, hFAF, currentFlapCon f ig) minT hrustCounter = minT hrustCounter + 1
else minT hrustCounter = 0 if(Tc > Tmax) Tc = Tmax end if
end if
The use of speed brakes is detailed in the following algorithms: Algorithm 2 is used to determine when to deploy speed brakes, and Algorithm 3 is used to determine when to retract speed brakes.
Algorithm 2 Speed Brake Deployment Logic (Speed on Thrust)
if(minT hrustCounter > 15 AND (Vc − V) < −5.0 kt) then if(newFlapCon f ig=={CR,AP,LDG}) then if(speedBrakeOn== f alse) then speedBrakeCounter = 0 bc = 0 speedBrakeOn = true else speedBrakeCounter = speedBrakeCounter + 1 bc = 0.5 end if else speedBrakeCounter = 0 bc = 0 speedBrakeOn = f alse end if
end if
1The aircraft model is assumed to have four potential flaps configurations: cruise (CR), approach (AP), landing (LDG), and landing with gear (LDG+GEAR), which are described more fully in Section 6.
10

Algorithm 3 Speed Brake Retraction Logic
if(speedBrakeOn == true AND speedBrakeCounter ≤ 30) then speedBrakeCounter = speedBrakeCounter + 1 bc = 0
else if(speedBrakeOn == true AND speedBrakeCounter > 30) if(minT hrustCounter==0) then speedBrakeCounter = 0 bc = 0 speedBrakeOn = f alse else speedBrakeCounter = speedBrakeCounter + 1 bc = 0.5 end if
end if
4.3 Speed Control Using Pitch
This speed management mode is only used when the aircraft is in descent; otherwise, the speed is controlled using thrust. Therefore, additional logic is needed to determine when the aircraft is descending and when the aircraft is in level flight.2
The logic for determining the speed management mode is based on the altitude error, ealt = h−hre f , and the reference vertical rate, h˙re f . If one of the following conditions applies, the aircraft is assumed to be in level flight, and speed is managed using thrust:
• ealt < hthreshold;
• the aircraft’s current altitude is 200 feet (ft) below the lower bound on the next downstream altitude constraint in the navigation procedure; or
• h˙ re f = 0
Here, hthreshold is a tolerance on the altitude error measured relative to the reference altitude. If the aircraft is in level flight, the commanded thrust is determined using equation (51) and γc = 0.
If the aircraft’s current altitude is more than 200 ft above the lower bound on the next downstream altitude constraint, the aircraft is assumed to be descending, and speed is managed using pitch.
The commanded flight-path angle is computed based on the Energy Share Factor (ESF), which is the ratio of power allocated to changing altitude versus changing airspeed. The error in TAS is eV = Vc − V, and ES F0 is the Energy Share Factor needed to maintain the aircraft’s current airspeed while descending. The algorithm outlines the logic for determining the ES F needed for accelerating or decelerating in order to achieve the commanded speed, Vc.
2This paper only deals with managing the aircraft during level or descending flight; i.e., climbing flight is not considered.
11

Algorithm 4 Logic for Determining Energy Share Factor (Speed on Pitch)

if(eV ≤ −Vthreshold) then ES F = 0.3

else if(eV > −Vthreshold AND eV ≤ 0) then

ES F =

e + ES F ES F0−0.3

Vthreshold

V

0

else if(eV > 0 AND eV ≤ Vthreshold) then

ES F =

1.7−ES F0 Vthreshold

eV + ES F0

else if(eV ≥ Vthreshold)

ES F = 1.7

end if

Here, Vthreshold is a parameter used to determine when the ES F should be set to the minimum or maximum values. If the aircraft needs to decelerate (eV < 0), the ES F is decreased (towards a minimum value of 0.3), meaning that the vertical path should become shallower. If the aircraft needs to accelerate (eV > 0), the ES F is increased (towards a maximum value of 1.7), meaning that the vertical path should become steeper. The ES F values used in this logic are proposed in
EUROCONTROL’s BADA user’s manual [2].

The commanded flight-path angle is then calculated using the following equations.

dh = (T − D) · V · ES F

(53)

dt

mg

γc

=

sin−1

 

−

dh dt

V

 

(54)

The commanded thrust is based on ealt and is calculated using the following logic. Algorithm 5 Calculation of Commanded Thrust (Speed on Pitch)

if(ealt < −hthreshold) then

Tc = 0.5 · Tmax

else if(ealt > hthreshold) then

Tc = Tmin

else

T = e + Tmin−0.5·Tmax

T min +0.50·T ma x

c

2·hthreshold

alt

2

end if

Again, Tmin is the idle-thrust setting and Tmax is the maximum thrust. If the current altitude is below the reference altitude by more than hthreshold, the commanded thrust, Tc, is set to half of the maximum thrust, Tmax. If the current altitude is above the reference altitude by more than hthreshold, the commanded thrust is set to the minimum thrust, Tmin.

In this speed management mode, Algorithm 6 is used to determine whether the next flaps configuration or speed brakes should be deployed to increase drag. As mentioned previously, Algorithm 3 is used to determine when to retract speed brakes.

12

Algorithm 6 Configuration for Increased Drag (Speed on Pitch)
if(Tc == Tmin) then if(ealt > hthreshold) then newFlapCon f ig = getCon f igForDrag (VCAS , h, hFAF, currentFlapCon f ig) if(newFlapCon f ig == currentFlapCon f ig AND newFlapCon f ig == {CR, AP, LDG}) then speedBrakeCounter = speedBrakeCounter + 1 bc = 0.5 speedBrakeOn = true end if end if minT hrustCounter = minT hrustCounter + 1
else minT hrustCounter = 0
end if

4.4 Control Gains and Parameter Values
The control gains and thresholds used in the guidance control laws are presented in Table 1.

Table 1. Control Gains and Parameter Values

Control Gain kT kφ
kspeed brake kψ kxtrk kV kalt
hthreshold Vthreshold

Value 0.352 s−1 0.4 s−1 0.10 s−1
3.0 5 × 10−4 m−1 0.1136 s−1
0.20 s−1
500 ft
10 kt

13

5 Reference Horizontal Path
The aircraft is controlled to a reference horizontal path, a reference altitude, and a reference CAS, which are determined from a four-dimensional trajectory. The reference altitude and airspeed are provided as a function of along-path position (or distance-to-go [DTG]); therefore, the reference inputs are determined using the aircraft’s along-path position.
The reference horizontal path is described in detail in this section, including an algorithm for mapping the aircraft’s x, y position to an along-path position.
5.1 Data Format for the Horizontal Path
The reference horizontal path is comprised of a series of straight and turning segments that, when put together, describe a continuous path for the aircraft. The data format is provided in Figure 5.

HPT_j
1 2 3 4 5

A
x
0 5279.26 7127.86 10593.80 12250.50

B
y
0 -9.23 482.84 2476.82 3989.59

C

D

E

DTG

Segment Type

Course

0 5279.3 7214.3 11212.9 13474.2

straight turn
straight turn -

6.2814 1.00E+07 0.5221 1.00E+07
-

F Turn Center
x 0 5285.72 0 8007.20 -

G Turn Center
y 0 3684.91 0 6973.01 -

H Angle Start of Turn
0 -1.5725
0 -1.0487
-

I Angle End of Turn
0 -1.0487
0 -0.6128
-

J
R
0 3694.14
0 5187.14
-

Figure 5. Data format for the reference horizontal path.
The horizontal path example in Figure 5 describes a reference path with two straight segments and two turning segments, which is illustrated in Figure 6. The reference horizontal path is generated backwards from the last point at x = 0, y = 0 (which is the first line in Figure 5). Each x, y position in Columns A and B represents a Horizontal Path Transition (HPT) point, where the horizontal path transitions between a straight segment and a turn segment and vice versa or connects two straight segments. The DTG to the end point from each HPT j is given in Column C. The segment type (straight or turn) that follows HPT j is given in Column D.3 If the segment between two transition points is straight, the course, defined on [0, 2π), is provided in Column E4; if the segment is a turn, the course value is set to 1.0 × 107. For turn segments, columns F, G, H, I, and J include the x and y coordinates of the turn center, the angles of the start and end of the turn, and the turn radius. The angles at the start and end of the turn are used to determine the sweep angle of the turn, are defined on [−π, π), and are measured relative to the x-axis. All values in Figure 5 are given in meters or radians.

3It should be noted that when the aircraft is flying along the horizontal path, the segment in line j precedes HPT j. 4The course value given in Figure 5 is the course used for generating the horizontal path backwards from the last point; i.e., the flown course is the value in Figure 5 + π.
14

y (nmi)

5

4

3

2

1
HPT
1 0

HPT
2

−1

−2

0

1

2

3

4

5

6

7

x (nmi)

y (nmi)

5

4

3
2
1
HPT
1 0
−1

Turn Center x,y

R

HPT
2

HPT
3

−2

0

1

2

3

4

5

6

7

x (nmi)

y (nmi)

5

4

3

2
1
HPT
1 0
−1

HPT
4

HPT
2

HPT
3

−2

0

1

2

3

4

5

6

7

x (nmi)

y (nmi)

5

4

Turn Center x,y

3
2
1
HPT
1 0
−1

HPT
5
R

HPT
2

HPT
3

HPT
4

−2

0

1

2

3

4

5

6

7

x (nmi)

Figure 6. Horizontal path generated from the data in Figure 5. The horizontal path starts with the last point (x, y) = (0,0) and the path is generated backwards using straight and turn segments, as specified. The first straight segment is shown in the top-left, the first turn segment is appended to first straight segment in the top-right, the second straight segment is added in bottom-left, and the second turn segment is appended in the bottom-right.

15

5.2 Mapping to Along-Path Position
The along-path position of the aircraft is found by orthogonally projecting the aircraft’s x, y position onto the reference horizontal path. Figure 7 illustrates the high-level process for determining the along-path position.
Calculate Distance dj between x,y and HPTj, j = 1..M
Calculate cross-track error for HPTj with smallest distance dj

Is cross-track error < 2.5 nmi?

no Calculate cross-track error for HPTj (sorted by dj)

Is cross-track

yes

error < 2.5 nmi?

index = j

yes

no

index = index(min(dj))

j = j – 1

Determine next downstream HPT

Calculate Distance-ToGo to downstream HPT

Along-path Position = –1 x (Distance-To-Go to downstream HPT + Distance-To-Go from downstream HPT to PTP)

Figure 7. High-level process for determining the along-path position by projecting the aircraft’s x, y position onto the reference horizontal path.

The distance between the x, y position and HPT j for j = 1..M is calculated using the Euclidean distance. Starting with the HPT point that results in the smallest distance, d j, the cross-track error is calculated between the aircraft’s position and the horizontal path.
If the absolute value of the cross-track error is less than 2.5 nmi, then determine the next downstream HPT. Otherwise, check the cross-track error for the HPT with the next smallest value of d j. Continue checking the cross-track errors until finding an HPT that meets the 2.5-nmi threshold. The DTG and along-path position are then calculated using the next downstream HPT.
The cross-track error and along-path position calculations are detailed below.
16

5.2.1 Cross-track Error Calculation

The cross-track error is calculated by first identifying the whether the aircraft is on a straight or a turn segment. This is done by defining four equal quadrants around HPT j and determining which quadrant the aircraft’s x, y position falls within. Figure 8 shows a horizontal path defined by HPT j for j = 1..4, comprised of two straight segments and one turn segment. If j = 2, meaning that the aircraft is closest in distance to HPT2, the quadrants are defined using α = course(HPT2) + π. If j = 1 or j = 3, then α = course(HPT1) + π or α = course(HPT3) + π, respectively. The four quadrants are then defined as follows.
• quadrant 1: [α, α + π/2)
• quadrant 2: [α + π/2, α + π)
• quadrant 3: [α + π, α + 3π/2)
• quadrant 4: [α + 3π/2, α + 2π)

The angle between the aircraft’s position and HPT j is defined on [0, 2π) and is determined as:

θHPT j = atan2 y − y(HPT j), x − x(HPT j)

(55)

To determine the quadrant where the aircraft is located, θHPT j is checked against the bounds of each quadrant. For example, if θHPT j is within quadrant 1, then diff1 = |α − θHPT j| ≤ π/2 AND diff2 = |α + π/2 − θHPT j| ≤ π/2. It should be noted the differences in angles must be defined on [−π, π).

The following logic is then applied to determine the next downstream HPT based on the quadrant that contains the aircraft’s position and the segment type.

Algorithm 7 Logic for Determining Downstream HPT

if[segment(HPT j) == turn AND (quadrant == 1 OR quadrant == 4)] then nextHPT = j − 1
else if[segment(HPT j) == turn AND (quadrant == 2 OR quadrant == 3)] then nextHPT = j
else if[segment(HPT j) == straight AND (quadrant == 1 OR quadrant == 4)] then if j == 1 then nextHPT = 1
else nextHPT = j − 1
end if else if[segment(HPT j) == straight AND (quadrant == 2 OR quadrant == 3)] then
nextHPT = j
end if

nextHPT is then used to determine whether the aircraft is on a straight or a turn segment: segment(nextHPT). Figure 8 shows three examples of aircraft positions in the horizontal plane: a, b, and c. If the aircraft is located at position a, nextHPT = 1 and the aircraft is assumed to be on a straight segment.
17

HPT4

HPT1

2 HPT3
31 4
c

4

3b
2

1

a

HPT2

Figure 8. Illustration of quadrants around each HPT.

If the aircraft is located at positions b or c, nextHPT = 2 and the aircraft is assumed to be on a turn segment.

For straight segments, the cross-track error, extrk, is calculated using the following equations.

θ = course(nextHPT), θ : [0, 2π)

(56)

extrk = − [x − x(nextHPT)] sin θ + y − y(nextHPT) cos θ

(57)

If the aircraft is determined to be on a turn segment, the cross-track error is found using the following equations.

R1 = x − xturncenter (nextHPT) 2 + y − yturncenter (nextHPT) 2

(58)

extrk = |R1 − R(nextHPT)| · θdir

(59)

Here, θdir is the direction of the turn. If the course angle at the end of the turn is greater than at the start of the turn, θdir = 1; otherwise, θdir = −1.

18

5.2.2 Along-Path Position Calculation

The DTG to the end point is calculated as a function of the downstream HPT point and the segment type (straight or turn).

The DTG is the cumulative path length from the next HPT to the end point, L(nextHPT), plus the distance from the aircraft’s current position to nextHPT , DTG(nextHPT ).

DTG = L(nextHPT ) + DTG(nextHPT )

(60)

If the aircraft is on a straight segment, the DTG(nextHPT ) is given by the following.

DTG(nextHPT ) = [x − x(nextHPT)] cos θ + y − y(nextHPT) sin θ

(61)

If the aircraft is on a turn segment, the following equations are used to find DTG(nextHPT ).

θturn = atan2 y − yturncenter (nextHPT ), x − xturncenter (nextHPT )

(62)

∆θ = θturnstart − θturn, ∆θ : [−π, π)

(63)

DTG(nextHPT ) = R · |∆θ|

(64)

Figure 9 illustrates the quantities used to calculate the DTG for both a straight and a turn segment. From aircraft position a, the DTG is calculated to the downstream point HPT1. From aircraft position c on the turn segment, the distance to HPT2 is the arclength of the projection onto the horizontal path, and the DTG is the sum of that distance and the distance from HPT2 to HPT1.
HPT4

HPT3 c

Dq R

Dq
d HPT1

a HPT2

Figure 9. Projection of aircraft positions onto horizontal path to determine DTG to end point (HPT1).

It is assumed that the along-path positions are negative and decreasing along the path to zero at the final point. Therefore, the DTG is multiplied by −1 to get the along-path position.
19

6 Aircraft-Specific Modeling
EUROCONTROL’s Base of Aircraft Data (BADA) is used to represent aircraft-specific performance [2]. This section lists the BADA parameters and equations from the BADA user’s manual that are used in the aircraft model and the control laws presented here.
The aircraft mass parameters are taken from the Operations Performance File (OPF) for the specified aircraft type. Three mass parameters are needed:
• Reference Mass, mre f
• Minimum Mass, mmin
• Maximum Mass, mmax
The aircraft mass should be bounded by the minimum and maximum mass values.
The wing reference area, S , which is used in the computation of the lift coefficient, is also given in the OPF.
The drag parameters, CD0 and CD2, are given in the OPF by configuration. For modeling descent trajectories, the cruise (CD0,CR, CD2,CR), approach (CD0,AP, CD2,AP), and landing (CD0,LDG, CD2,LDG) configurations are used. The landing gear configuration is modeled by adding the drag coefficient for the landing gear, CD0,∆LDG, to the drag coefficients for the landing configuration. The drag coefficients, CD, in the cruise, approach, and landing configurations are given by equations (BADA:3.6-2)-(BADA3.6-4).
The minimum transition speeds for each configuration are described in BADA: Section 3.5 (b) as a function of the stall speed for each configuration (corrected using the mass of the aircraft and assuming a stall speed margin). The transition (CAS) speed limits for each configuration are denoted as VAP,trans, VLDG,trans, and VLDG+GEAR,trans. The following function, getCon f ig, is applied to determine whether the aircraft should transition to next flaps configuration based on the aircraft’s current CAS and the minimum speed for its current configuration.
Algorithm 8 Get Configuration Based on Transition Speeds
if(currentFlapCon f ig == {CR} AND VCAS > VAP,trans) then newFlapCon f ig = CR
else if(currentFlapCon f ig == {CR, AP} AND VCAS ≤ VAP,trans AND VCAS > VLDG,trans AND (h − hFAF) < 10,000 ft) then newFlapCon f ig = AP
else if(currentFlapCon f ig == {CR, AP, LDG} AND VCAS ≤ VLDG,trans AND VCAS > VLDG+GEAR,trans AND (h − hFAF) < 10,000 ft) then newFlapCon f ig = LDG
else if(currentFlapCon f ig == {CR, AP, LDG, LDG + GEAR} AND VCAS < VLDG+GEAR,trans AND (h − hFAF) < 10,000 ft) then newFlapCon f ig = LDG + GEAR
else
20

newFlapCon f ig = currentFlapCon f ig end if
Because the commanded CAS, VCAS , must be limited to the upper CAS speed limit for a given configuration, the upper speed limits for each configuration (VAP,max, VLDG,max, and VLDG+GEAR,max) will also be provided for each aircraft type. The function getCon f igForDrag, described below, is used to determine whether the next flaps configuration can be deployed to increase drag. Inputs to the function are the aircraft’s current CAS, VCAS , the aircraft’s current altitude, h, and the current configuration, currentFlapCon f ig.
Algorithm 9 Get Configuration for Increased Drag
if(currentFlapCon f ig == CR AND VCAS < VAP,max AND (h − hFAF) < 10,000 ft) then newFlapCon f ig = AP
else if(currentFlapCon f ig == AP AND VCAS < VLDG,max AND (h − hFAF) < 10,000 ft) then newFlapCon f ig = LDG
else if(currentFlapCon f ig == LDG AND VCAS < VLDG+GEAR,max AND (h − hFAF) < 10,000 ft) then
newFlapCon f ig = LDG + GEAR else
newFlapCon f ig = currentFlapCon f ig end if
The drag force, D, is calculated as a function of air density, wing reference area, and the TAS using equation (BADA:3.6-5). The lift force is calculated similarly.
L = CL · ρ · V2 · S 2
The thrust parameters are also given in the OPF and are as follows:
• CT c,1
• CT c,2
• CT c,3
• CT des,high
• CT des,low
• CT des,app
• CT des,ld
• Hp,des
The maximum engine thrust, Tmax, is modeled using equations (BADA:3.7-1) and (BADA:3.78), assuming no temperature deviations from the standard atmospheric model. The minimum
21

thrust, Tmin, is modeled as a function of altitude and configuration using equations (BADA:3.7-9)(BADA:3.7-12). The maximum operating calibrated airspeed (CAS) and Mach are given in the OPF as VMO and MMO, respectively. The standard atmospheric model is given in Section BADA:3.1; conversions between CAS and TAS are described in equations (BADA:3.1-23) and (BADA:3.1-24).
22

7 References
[1] J. E. Hurtado, Kinematic and Kinetic Principles. Hurtado - Lulu.com, 2007. [2] “User manual for the base of aircraft data (BADA) revision 3.11,” Tech. Rep., May 2013,
Eurocontrol Experimental Center Technical/Scientific Report No. 13/04/16-01.
23

Appendix A Abbreviations and Acronyms

3-DOF 4D
AP
b bc BADA
CAASD CAS CR
DTG D
e.g. EUROCONTROL
FAA FAF ft
g γ γc
h hre f h˙ re f HPT
i.e. IM
kalt kφ kψ k s peedbrake kT kV kxtrk kt

Three Degrees of Freedom Four Dimensional
Approach Flaps Configuration
Speed brake deployment fraction Commanded speed brake deployment fraction Base of Aircraft Data
Center for Advanced Aviation System Development Calibrated Airspeed Cruise Flaps Configuration
Distance to Go Drag
For Example European Organization for the Safety of Air Navigation
Federal Aviation Administration Final Approach Fix Feet
Acceleration due to gravity Flight-path angle Commanded flight-path angle
Altitude of the aircraft (+h is pointing up from the surface of the Earth) Reference altitude from pre-calculated trajectory Reference vertical rate from pre-calculated trajectory Horizontal Path Transition
That Is Interval Management
Altitude tracking gain Roll-angle tracking gain Heading-angle tracking gain Speed-brake-deployment tracking gain Thrust tracking gain True airspeed tracking gain Cross-track error gain Knots

24

L LDG LDG + GEAR
m
nmi
φ φc ψ ψc
sec
T Tc θ θnextHPT
θturn
TOD
V V˙ V VAP,max VAP,trans
Vc VCAS c VLDG,max VLDG,trans
VLDG+G E AR,ma x VLDG+G E AR,tran s
Vw Vw|| Vw⊥ Vwx Vwy
x

Lift force Landing Flaps Configuration Landing Flaps Configuration plus Landing Gear
Aircraft mass
Nautical Miles
Roll angle Commanded roll angle Heading angle Commanded heading angle
Seconds
Thrust force Commanded thrust force Course-angle guidance from the reference horizontal path Angle between the x, y position of the aircraft and the next downstream Horizontal Path Transition Angle between the x, y position of the aircraft and the center of the turn for the next downstream Horizontal Path Transition Top of Descent
True airspeed of the aircraft Time rate of change of the TAS Inertial velocity vector Maximum CAS in the Approach Flaps Configuration Minimum Speed to transition from Cruise Flaps Configuration to the Approach Flaps Configuration Commanded TAS Commanded calibrated airspeed Maximum CAS in the Landing Flaps Configuration Minimum Speed to transition from Approach Flaps Configuration to the Landing Flaps Configuration Maximum CAS in the Landing Flaps + Gear Configuration Minimum Speed to transition from Landing Flaps Configuration to the Landing Flaps + Gear Configuration Wind velocity vector Wind speed aligned with aircraft’s course Wind speed perpendicular to aircraft’s course Wind speed in the x direction Wind speed in the y direction
Position of the aircraft in the x direction in the horizontal plane (+x direction is aligned with East)
25

y

Position of the aircraft in the y direction in the horizontal plane (+y direc-

tion is aligned with North)

26

Disclaimer The contents of this material reflect the views of the author and/or the Director of the Center for Advanced Aviation System Development (CAASD), and do not necessarily reflect the views of the Federal Aviation Administration (FAA) or the Department of Transportation (DOT). Neither the FAA nor the DOT makes any warranty or guarantee, or promise, expressed or implied, concerning the content or accuracy of the views expressed herein. This is the copyright work of The MITRE Corporation and was produced for the U.S. Government under Contract Number DTFAWA-10-C-00080 and is subject to Federal Aviation Administration Acquisition Management System Clause 3.5-13, Rights in Data-General, Alt. III and Alt. IV (Oct. 1996). No other use other than that granted to the U.S. Government, or to those acting on behalf of the U.S. Government, under that Clause is authorized without the express written permission of The MITRE Corporation. For further information, please contact The MITRE Corporation, Contract Office, 7515 Colshire Drive, McLean, VA 22102 (703) 983-6000. c 2015 The MITRE Corporation. The Government retains a nonexclusive, royalty-free right to publish or reproduce this document, or to allow others to do so, for “Government Purposes Only.”
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