EE6900 Flight Management Systems

“Flight Management System – Part 2”

Dr. Maarten Uijt de Haag

Ohio University

Flight Management System (FMS)• Basic FMS functions:

– Navigation

• responsible for determining the best estimate of

the current nav state of the aircraft.

– Flight planning

• allows the crew to establish a specific routing for

the aircraft

– Trajectory prediction

• responsible for computing the predicted aircraft

profile along the entire specified routing

– Performance computations

• provides the crew with aircraft unique

performance information such as takeoff speeds,

altitude capability, and profile optimization

advisories

– Guidance

• responsible for producing commands to guide

the aircraft along both the lateral and vertical

computed profiles

2

FMS- Functional Block Diagram

3

Navigation

Flight PlanningVertical

Guidance

Lateral

Guidance

Performance

Computations

Trajectory

Prediction

Navigation

Database

Performance

Database

Lateral &

Vertical Profile

Data Link

Flight Plan

Buffer

Flight Management – Typical

4

Flight

Management

Inertial

Reference

Air

DataNavigation

Receivers

Aircraft

DisplaysFlight

Controls

Surveillance

Systems

Engine and

Fuel Systems

Data

Link

MCDU

Altitude,

speeds,

temperaturesInitial

position

Position,

velocities,

vert speed,

pitch, roll,

heading,

accels

Init data, flight

plans, clearance,

weather

Data entry,

display data

Map scale,

display

selections

Trajectory

conflicts

Flight plan &

path, nav

data, route

data, HIS data

Roll axis cmds,

pitch axis cmds,

thrust axis cmds

Tactical cmds,

modes

Flight ID, aircraft

state, trajectory

Fuel weight,

engine thrust

Thrust limits

Tuning cmds

Freq, range, bearing, LOC deviation,

GPS position, GPS GS, time

VNAV Flight Path

5

JAIKE

13,000ft

ILENE

13,000ft

WACKI

11,000ft

250kts

REGLE

7,000ft

Example of a VNAV Path

280kts 250kts

250kts

𝑡0 𝑡1

𝑡2

𝑡3

IDLE descent,

constant velocity

Flight Path

6

JAIKE

13,000ft

ILENE

13,000ft

WACKI

11,000ft

250kts

REGLE

7,000ft

Example of a VNAV Path

280kts 250kts

250kts

𝑡0 𝑡1

𝑡2

𝑡3

IDLE descent,

constant velocity

Can this work?

𝐱 =

𝑉 𝜓 𝛾 ℎ 𝑟 𝑊

=

000

𝑉𝑠𝑖𝑛(γ)

𝑉𝑐𝑜𝑠(𝛾)0

=

000 −𝑉𝐷 𝑊

𝑉𝑐𝑜𝑠(𝛾)0

= 𝐟 𝑉, 𝜓, 𝛾, ℎ, 𝑟,𝑊

0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

Time elapsed [sec]

Ra

ng

e [

NM

]

Range @ IDLE Thrust vs Time

Altitude Change versus Distance

7

Answer: does not work at IDLE thrust!

0 10 20 30 40 50 60 70 80 901.05

1.1

1.15

1.2

1.25

1.3x 10

4

Time elapsed [sec]

Alt

itu

de

[ft

]

Descent @ IDLE Thrust vs Time

Adjust the Thrust?

8

ILENE

13,000ft

WACKI

11,000ft

Δ𝑟𝑑𝑒𝑠

Δℎ𝑑𝑒𝑠

(e.g. 8NM)

(e.g. 13000ft)

𝛾

𝛾 = 𝑎𝑡𝑎𝑛Δℎ𝑑𝑒𝑠Δ𝑟𝑑𝑒𝑠

6NM

𝐱 =

𝑉 𝜓 𝛾 ℎ 𝑟 𝑊

=

𝑔𝑊

𝑇 − 𝐷 −𝑊𝑠𝑖𝑛(𝛾)

00

𝑉𝑠𝑖𝑛(γ)𝑉𝑐𝑜𝑠(𝛾)

0

=

000

𝑉(𝑇 − 𝐷) 𝑊𝑉𝑐𝑜𝑠(𝛾)

0

= 𝐟 𝑉,𝜓, 𝛾, ℎ, 𝑟,𝑊

ℎ =𝑇 − 𝐷 𝑉

𝑊= 𝑉𝑠𝑖𝑛 𝛾

sin 𝛾 =𝑇 − 𝐷

𝑊

𝑇 = 𝐷 +𝑊𝑠𝑖𝑛(𝛾)

Not IDLE thrust

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

Time elapsed [sec]

Ra

ng

e [

NM

]

Range @ IDLE Thrust vs Time

𝛾𝑔𝑙𝑖𝑑𝑒

Flight Path – First Segment

9

JAIKE

13,000ft

ILENE

13,000ft

WACKI

11,000ft

250kts

REGLE

7,000ft

Example of a VNAV Path

280kts 250kts

𝑡0 𝑡1

𝑡2

𝑡3

Straight and level,

Speed change

𝐱 =

𝑉 𝜓 𝛾 ℎ 𝑟 𝑊

=

𝑔

𝑊𝑇𝑐𝑜𝑠(𝛼) − 𝐷 −𝑊𝑠𝑖𝑛(𝛾)

𝑔

𝑊𝑉𝑐𝑜𝑠(𝛾)𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇

𝑔

𝑊𝑉𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) −𝑊𝑐𝑜𝑠(𝛾)

𝑉𝑠𝑖𝑛(γ)𝑉𝑐𝑜𝑠(γ)

−𝜂 𝑉 𝑇

=

𝑔

𝑊𝑇 − 𝐷

000𝑉

−𝜂 𝑉 𝑇

𝑉 =𝑑𝑉

𝑑𝑡=𝑑𝑉

𝑑𝑟

𝑑𝑟

𝑑𝑡=𝑑𝑉

𝑑𝑟𝑉

26NM

𝑑𝑉

𝑑𝑟

Flight Path – First Segment

10

0 50 100 150 200 250 300 350 400250

255

260

265

270

275

280

Time elapsed [sec]

Ais

pe

ed

[k

ts]

Non-steady Straight and Level vs Time

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

30

Time elapsed [sec]

Ra

ng

e [

NM

]

Non-steady Straight and Level vs Time

Automation Modes – B787

1111

Autothrottle modes Roll modes Pitch modes

THR LNAV (armed) TO/GA

THR REF LNAV (engaged) VNAV (armed)

HOLD HDG SEL (engaged) VNAV SPD (engaged)

IDLE TRK SEL (engaged) VNAV PTH (engaged)

SPD TRK HOLD (engaged) VNAV ALT (engaged)

ATT (engaged) V/S (engaged)

LOC (armed) FPA (engaged)

LOC (engaged) FLCH SPD (engaged)

FAC (armed) ALT (engaged)

FAC (engaged) G/S engaged)

B/CRS (armed) G/P (engaged)

B/CRS (engaged) FLARE (armed)

TO/GA FLARE (engaged)

ROLLOUT (armed)

ROLLOUT (engaged)

VNAV Mode – B787

12

• VNAV engages at 400 feet AGL

• if VNAV is selected and the FMC has insufficient data to provide VNAV guidance (such as the gross

weight is invalid or there is no end–of–descent point in descent) displays PERF/VNAV UNAVAILABLE in

the CDU help window

• VNAV SPD, VNAV PTH or VNAV ALT pitch mode is displayed in green (engaged) on the PFD and HUD

pitch flight mode annunciator

• in the VNAV SPD pitch mode, the AFDS commands pitch to hold target airspeed. The autothrottle

operates in the THR REF, THR, IDLE or HOLD mode, as required by the phase of flight

• in the VNAV PTH pitch mode, the AFDS commands pitch to maintain FMC target altitude or the VNAV

path. The autothrottle maintains speed

• in the VNAV ALT pitch mode, the AFDS commands pitch to maintain the MCP selected altitude when that

altitude is lower than the VNAV commanded altitude in climb or higher than the VNAV commanded

altitude in descent

• if VNAV is selected and VNAV commands a descent with the MCP altitude window above the current

airplane altitude, the autopilot maintains the altitude at which VNAV was selected. When on an

instrument approach using VNAV, selecting the missed approach altitude does not interfere with the

VNAV descent

• if VNAV is selected and VNAV commands a climb with the MCP altitude window below the current

airplane altitude, the autopilot maintains the altitude at which VNAV is selected

• with the VNAV ALT pitch mode engaged, the autothrottle operates in the speed (SPD) mode

Important Note

13

JAIKE

13,000ft

ILENE

13,000ft

WACKI

11,000ft

250kts

REGLE

7,000ft

𝑡0 𝑡1

𝑡2

𝑡3

The path is defined in an Earth-

referenced frame (navigation

frame, earth-frame)

What happens when we have a tail-wind?

14

JAIKE

13,000ft

ILENE

13,000ft

WACKI

11,000ft

250kts

REGLE

7,000ft

𝑡0 𝑡1

𝑡2

𝑡3𝑉 𝑉𝑤

𝑉𝑔

So, this 1st segment would be completed

faster than expected.

What happens when we have a tail-wind?

15

ILENE

13,000ft

WACKI

11,000ft

250kts

ℎ

𝑟

No wind

ILENE

13,000ft

WACKI

11,000ft

250kts

ℎ

Tailwind

𝑉𝑤

𝑉 = 𝑉𝑒

𝑉𝑒Too high w.r.t. path

VNAV may disconnect;

Airspeed must somehow be reduced

(reduce thrust, spoilers, etc.)

𝑟

Cost Index (CI)

16

𝐶𝐼 =𝑇𝑖𝑚𝑒 𝑐𝑜𝑠𝑡 ( $ ℎ𝑟)

𝐹𝑢𝑒𝑙 𝑐𝑜𝑠𝑡 ( 𝑐𝑒𝑛𝑡𝑠 𝑙𝑏)

Time-related direct operating cost (minus cost of fuel):

• flight crew wages (hourly or fixed);

• lease of engines, auxiliary power units, airplanes;

• maintenance costs;

Cost of fuel, may be complex calculation due to:

• variation of fuel cost as a function of location;

• fuel tankering;

• fuel hedging.

Cost Index (CI)

17

𝐶𝐼 =𝑇𝑖𝑚𝑒 𝑐𝑜𝑠𝑡 ( $ ℎ𝑟)

𝐹𝑢𝑒𝑙 𝑐𝑜𝑠𝑡 ( 𝑐𝑒𝑛𝑡𝑠 𝑙𝑏)

Must be entered in the control

display unit (CDU) of the FMC.

Good if fuels costs are high

and time costs are low

Good if fuels costs are low

and time costs are high

CI Ranges for Boeing Aircraft

18

From: W. Roberson, et al., “Fuel Conservation Strategies: Cost Index Explained,” Boeing

B787 as well

CI Results for Phases of Flight

19

From: W. Roberson, et al., “Fuel Conservation Strategies: Cost Index Explained,” Boeing

Minimum fuel flight

Minimum time flight

CI Impact Example

20

Airbus CI Examples

21

Old fuel prices!!!

Based on: “Getting to Grips with the Cost Index,” Airbus, May 1998.

Crew cost is between 10-20 US$/min

Maintenance cost is between 7 and 17 US$/min,

CI Effect on Climb

22

CI Effect on Climb

23

The higher the cost index:

• the steeper the descent path (the higher the speed),

• the shorter the descent distance

• the later the top of descent (TOD)

Cruise Flight - Strategy

• Speed selection during cruise:– Maximize the distance traveled for a given amount of

fuel (i.e., maximum range).

– Minimize the fuel used for a given distance covered

(i.e., minimum trip fuel).

– Minimize total trip time (i.e., minimum time).

– Minimize total operating cost for the trip (i.e.,

minimum cost, or economy [ECON] speed).

– Maintain the flight schedule.

24

Optimum fuel mileage

Based on: W. Roberson, et al., “Fuel Conservation Strategies: Cruise Flight,” Boeing

Cruise Flight – Short Term Constraints

• Strategy may be temporarily abandoned during flight

due to:

– Flying a fixed speed that is compatible with other

traffic on a specified route segment.

– Flying a speed calculated to achieve a required time

of arrival (i.e., RTA) at a fix.

– Flying a speed calculated to achieve minimum fuel

flow while holding (i.e., maximum endurance).

– When directed to maintain a specific speed by air

traffic control.

25

Cruise Schemes

• Maximum-Range Speed (MRC)

– The speed that will provide the furthest distance for

a given amount of fuel burned and the minimum fuel

burned for a given cruise distance

• Long-range Cruise (LRC)

– Speed above MRC that will result in a 1 percent

decrease in fuel mileage (in NM/kg fuel burned)

26

Typically this 1% means a 3 to 5 % higher cruise speed

MRC versus LRC

27

From: W. Roberson, et al., “Fuel Conservation Strategies: Cruise Flight,” Boeing

Typical CI Values

28

Cost Simulations

• Price fuel: $2.94/gallon

• Crew and maintenance: $45/minute

• Altitude: 20,000ft

• Cruise for 200NM

• Cruise: steady straight and level flight

• Change from Vmo down to 0.65Vmo

29

Jet Fuel Priced (Commodity)

30

6.84 lbs/US gallon = 3.10kg/ US gallon

Speed at Altitude

31

0 500 1000 1500 2000 2500280

300

320

340

360

380

400

420

440

Time elapsed [sec]

Air

sp

ee

d [

NM

]

Cruise

Includes conversion from CAS to TAS for standard atmosphere!

Range and Speed

32

0 500 1000 1500 2000 25000

50

100

150

200

250

Time elapsed [sec]

Ra

ng

e [

NM

]

Cruise

0 500 1000 1500 2000 2500232

233

234

235

236

237

238

Time elapsed [sec]

Ma

ss

[to

nn

es

]

Cruise

Fuel Usage

33

0.45 0.5 0.55 0.6 0.65 0.7 0.751350

1400

1450

1500

Mach no.

Fu

el u

sa

ge

[g

al]

Fuel usage versus Mach no

0.45 0.5 0.55 0.6 0.65 0.7 0.759200

9400

9600

9800

10000

10200

10400

Mach no.

Fu

el u

sa

ge

[lb

s]

Fuel usage versus Mach no

Costs

34

0.45 0.5 0.55 0.6 0.65 0.7 0.751200

1300

1400

1500

1600

1700

1800

1900

Mach no.

Tim

e-r

ela

ted

Co

st

(US

$)

Time-related Cost versus Mach

0.45 0.5 0.55 0.6 0.65 0.7 0.753950

4000

4050

4100

4150

4200

4250

4300

4350

4400

Mach no.

Fu

el-

rela

ted

Co

st

(US

$)

Fuel-related Cost versus Mach no.

Fuel Mileage versus Mach

35

0.45 0.5 0.55 0.6 0.65 0.7 0.7543

43.5

44

44.5

45

45.5

46

46.5

47

47.5

48

Mach no.

Fu

el m

ile

ag

e [

NM

/10

00

kg

]

Fuel Mileage versus Mach no.

MRC

LRC

Cost Index versus Mach

36

0.45 0.5 0.55 0.6 0.65 0.7 0.7522

24

26

28

30

32

34

Mach no.

Co

st

Ind

ex

Computed Cost Index

Now for Flying at Different Altitude

37

@ 20,000ft

(standard atmosphere)

@ 30,000ft

(standard atmosphere)

0.45 0.5 0.55 0.6 0.65 0.7 0.753950

4000

4050

4100

4150

4200

4250

4300

4350

4400

Mach no.

Fu

el-

rela

ted

Co

st

(US

$)

Fuel-related Cost versus Mach no.

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.953900

4000

4100

4200

4300

4400

4500

4600

Mach no.

Fu

el-

rela

ted

Co

st

(US

$)

Fuel-related Cost versus Mach no.

Now for Flying at Different Altitude

38

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9518

20

22

24

26

28

30

32

Mach no.

Co

st

Ind

ex

Computed Cost Index

0.45 0.5 0.55 0.6 0.65 0.7 0.7522

24

26

28

30

32

34

Mach no.

Co

st

Ind

ex

Computed Cost Index

@ 20,000ft

(standard atmosphere)

@ 30,000ft

(standard atmosphere)

Speed Schedule

• Climb:

– Economy (based on cost index) –optimizes the

overall cost

– Maximum angle of climb – maximum climb rate w.r.t.

distance

– Maximum rate of climb – maximum climb rate w.r.t.

time

– Required time of arrival speed (RTA) – optimizes

cost of operation, but at the same time achieve the

arrival at a specific waypoint at a specific time

39

Speed Schedule

• Cruise:

– Economy (based on cost index) – optimizes the

overall cost

– Maximum endurance – produces the lowest fuel

burn rate (MRC)

– Long range cruise – see LRC discussion (good fuel

rate, good range)

– Required time of arrival speed (RTA) – optimizes

cost of operation, but at the same time achieve the

arrival at a specific waypoint at a specific time

40

Speed Schedule

• Descent:

– Economy (based on cost index) –optimizes the

overall cost

– Maximum descent rate – maximum descent rate

w.r.t. time

– Required time of arrival speed (RTA) – optimizes

cost of operation, but at the same time achieve the

arrival at a specific waypoint at a specific time

41

Crossover Altitude

• Crossover Altitude (or transition altitude) is the altitude

at which a specified CAS (Calibrated airspeed) and

Mach value represent the same TAS (True airspeed)

value. Above this altitude the Mach number is used to

reference speeds

42

𝑀 =𝑉

𝑉𝑎=

𝑉

𝛾𝑅𝑇=

2

𝛾 − 1

𝑝𝑡𝑝

𝛾−1𝛾

− 1

𝑉𝑐𝑎𝑠 =2

𝜇

𝑝0𝜌0

1 +𝑝

𝑝01 +

𝜇

2

𝜌

𝑝𝑉2

1𝜇

− 1

𝜇

− 1

12

Crossover Altitude

43

Typical Climb Profile

Primary Flight Display

44

Climb display Cruise display

Flat Earth Approximation

• Remember the 3DOF equations of

motion:

• These assume a “Flat earth”

45

𝑉 =𝑔

𝑊𝑇𝑐𝑜𝑠(𝛼) − 𝐷 −𝑊𝑠𝑖𝑛(𝛾)

𝜓 =𝑔

𝑊𝑉𝑐𝑜𝑠(𝛾)𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇

𝛾 =𝑔

𝑊𝑉𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) −𝑊𝑐𝑜𝑠(𝛾)

ℎ = 𝑉𝑠𝑖𝑛(γ) 𝑟 = 𝑉𝑐𝑜𝑠 γ 𝑊 = −𝜂 𝑉 𝑇

Flat Earth Approximation

• First extend by breaking ‘r’ into a ‘x’

(North) and ‘y’ (East) direction:

46

𝑉 =𝑔

𝑊𝑇𝑐𝑜𝑠(𝛼) − 𝐷 −𝑊𝑠𝑖𝑛(𝛾)

𝜓 =𝑔

𝑊𝑉𝑐𝑜𝑠(𝛾)𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇

𝛾 =𝑔

𝑊𝑉𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) −𝑊𝑐𝑜𝑠(𝛾)

ℎ = 𝑉𝑠𝑖𝑛(γ) 𝑥 = 𝑉𝑐𝑜𝑠 γ cos 𝜓 = 𝑉𝑁 𝑦 = 𝑉𝑐𝑜𝑠 𝛾 sin 𝜓 = 𝑉𝐸 𝑊 = −𝜂 𝑉 𝑇

Still a flat earth (ENU)

Earth-Referenced Equations

• Assume a spherical Earth

• Latitude and longitude rates are then:

• Compare to non-spherical Earth

47

𝐿 =𝑉𝑁

𝑅𝑒 + ℎ

𝜆 =𝑉𝐸

𝑅𝑒 + ℎ cos(𝐿)

𝐿 =𝑉𝑁

𝑅𝑁 + ℎ

𝜆 =𝑉𝐸

𝑅𝐸 + ℎ cos(𝐿)

Non-Spherical Earth (FYI)

48

North Pole

RN

Meridian RadiusIt is the radius of the best fitting curve to

a meridian section of the reference earth ellipsoid

Equatorial plane

RE

North

pole

Top viewSide view

Transverse RadiusIt is the radius of the best fitting curve to

a vertical east-west section of the reference earth ellipsoid

Mean Radius of Curvature:

𝑅𝑁 =𝑅(1 − 𝑒2)

1 − 𝑒2 sin2 𝐿 3/2𝑅𝐸 =

𝑅

1 − 𝑒2 sin2 𝐿 1/2

𝑅0 = 𝑅𝐺 = 𝑅𝑁𝑅𝐸

Length of semi-major axis: 𝑅Length of semi-minor axis: 𝑅(1 − 𝑓)Flattening: f = (𝑅 − 𝑟)/𝑅Major eccentricity: e = f 2 − f 1/2

R = 6378137.0

e = 0.0818191908426

Back to Spherical Coordinates

• 3DOF EOM:

49

𝑉 =𝑔

𝑊𝑇𝑐𝑜𝑠(𝛼) − 𝐷 −𝑊𝑠𝑖𝑛(𝛾)

𝜓 =𝑔

𝑊𝑉𝑐𝑜𝑠(𝛾)𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇

𝛾 =𝑔

𝑊𝑉𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) −𝑊𝑐𝑜𝑠(𝛾)

ℎ = 𝑉𝑠𝑖𝑛(γ)

𝐿 =𝑉𝑐𝑜𝑠 γ cos 𝜓

𝑅𝑒 + ℎ

𝜆 =𝑉𝑐𝑜𝑠 𝛾 sin 𝜓

𝑅𝑒 + ℎ cos(𝐿) 𝑊 = −𝜂 𝑉 𝑇

Can solve these equations again

using the ODE solvers, but now the

results are in the spherical Earth

Example: 1200NM Cruise @ 20,000ft

50

Track: 45 degrees

Remember from earlier notes …

51

E

WAYPOINT ‘i’ 𝐫𝑖 = 𝑅𝑒

𝑐𝑜𝑠𝜆𝑖𝑐𝑜𝑠𝐿𝑖𝑠𝑖𝑛𝜆𝑖𝑐𝑜𝑠𝐿𝑖

𝑠𝑖𝑛𝐿𝑖

= 𝑅𝑒𝐞𝑖

𝐿𝑖

𝜆𝑖𝐿𝑖 = waypoint latitude

𝜆𝑖 = waypoint longitude

Radius of a sphere

(approximate Earth by a sphere)

Lateral Guidance

52

Great-circle route:

𝐞𝑔𝑡

𝐞𝑠𝑡

𝐞𝑎𝑝

st: start point

gt: go to

ap: along path𝐨

Normal vector to 𝐞𝑠𝑡𝐨𝐞𝑔𝑡 plane:

𝐧 = 𝐞𝑠𝑡 × 𝐞𝑔𝑡𝑅𝑒

𝑅𝑒

Δ𝑟𝑎𝑝−𝑔𝑡= 𝑅𝑒𝑎𝑐𝑜𝑠 𝐞𝑎𝑝 ∙ 𝐞𝑔𝑡

Δ𝑟𝑠𝑡−𝑔𝑡= 𝑅𝑒 cos

−1 𝐞𝒔𝒕 ∙ 𝐞𝑔𝑡

𝐞𝐸,𝑠𝑡 = 𝐞𝑍 × 𝐞𝑠𝑡𝐞𝑁,𝑠𝑡 = 𝐞𝑠𝑡 × 𝐞𝐸,𝑠𝑡

North-pointing local

level unit vector

𝐞𝑁,𝑠𝑡

East-pointing local

level unit vector

𝐞𝐸,𝑠𝑡

𝐞𝑧 = 0 0 1 𝑇

Lateral Guidance

53

Top-view

𝐞𝑔𝑡

𝐞𝑠𝑡

𝐞𝑎𝑝

𝐞𝑝𝑜𝑠

Cross-track error:

𝑋𝑇𝑅𝐾 = −𝑅𝑒 cos−1 𝐞𝑎𝑝 ∙ 𝐞𝑝𝑜𝑠

𝑋𝑇𝑅𝐾 = −𝑅𝑒𝐧 ∙ 𝐞𝑝𝑜𝑠

Desired track:

𝐷𝑇𝑅𝐾 = 𝑎𝑡𝑎𝑛−𝐧 ∙ 𝐞𝑁,𝑎𝑝

−𝐧 ∙ 𝐞𝐸,𝑎𝑝

𝐷𝑇𝑅𝐾

𝐶𝑇𝑅𝐾𝑋𝑇𝑅𝐾

Track error:

𝑇𝑅𝐾𝐸𝑅𝑅 = 𝐷𝑇𝑅𝐾 − 𝐶𝑇𝑅𝐾

Lateral Guidance

• LNAV is a so-called roll mode: a roll must

be commanded so the XTRK error and

the TRKERR can be reduced to zero

• Over-simplified control strategy:

54

𝜑𝑐𝑜𝑚𝑚𝑎𝑛𝑑𝑒𝑑 = 𝐺𝑋𝑇𝑅𝐾 ∙ 𝑋𝑇𝑅𝐾 + 𝐺𝑇𝑅𝐾𝐸𝑅𝑅 ∙ 𝑇𝑅𝐾𝐸𝑅𝑅 + 𝜑𝑛𝑜𝑚𝑖𝑛𝑎𝑙

Lateral – Waypoint Changing

55

𝑅

𝑅

Radius of turn

Centripetal force

𝑚𝑉2

𝑅

𝑉 = 𝜔𝑅 𝛼

𝐿1

𝑅 =𝑉2

𝑔 ∙ 𝑡𝑎𝑛𝜑𝑛𝑜𝑚𝑖𝑛𝑎𝑙

𝐿2

𝑅

Nominal bank angle may exist due to a desired (nominal) course change

Vertical Guidance

• Vertical path changed

using pitch and

autothrottle (thrust)

• Vertical paths so far:

– Climb, descent, vertical

speed, take-off, etc.

56

Autothrottle modes Pitch modes

THR TO/GA

THR REF VNAV (armed)

HOLD VNAV SPD (engaged)

IDLE VNAV PTH (engaged)

SPD VNAV ALT (engaged)

V/S (engaged)

FPA (engaged)

FLCH SPD (engaged)

ALT (engaged)

G/S engaged)

G/P (engaged)

FLARE (armed)

FLARE (engaged)

Vertical Guidance - Data

57

From: Avionics Handbook – Chapter 15 - Flight

Management Systems, Randy Walter

Vertical Guidance

58

Δℎ

(really: distance between two points)

ℎ

Δ𝑟

Path gradient: 𝑔ℎ =Δℎ

Δ𝑟

Path altitude: ℎ𝑎𝑝 = ℎ𝑔𝑡 + 𝑔ℎΔ𝑟𝑎𝑝−𝑔𝑡

𝐞𝑎𝑝

Vertical deviation: 𝛿ℎ = ℎ𝑝𝑜𝑠 − ℎ𝑎𝑝

ℎ𝑝𝑜𝑠

ℎ𝑎𝑝

ℎ𝑔𝑡

ℎ𝑠𝑡

Desired V/S: 𝑉/𝑆𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = 𝑔ℎ 𝑉𝐺𝑆

Transitions and switches

• Auto Flight Phase Transitions:

• Vertical leg switches:

59

Climb

Cruise

ℎ𝑐𝑟𝑢𝑖𝑠𝑒 − ℎ < 𝐺𝑐𝑎𝑝𝑡𝑢𝑟𝑒 ℎ

ℎ𝑐𝑟𝑢𝑖𝑠𝑒

ℎSwitch from climb to cruise (level):

𝐺𝑐𝑎𝑝𝑡𝑢𝑟𝑒 ℎ𝑝𝑎𝑡ℎ,𝑛 − ℎ𝑝𝑎𝑡ℎ,𝑛+1 < ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑛 − ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑛+1

Vertical Guidance

60

From: Avionics Handbook – Chapter 15 - Flight Management Systems, Randy Walter

Vertical Guidance

61

Vpath:

Capture: Δ𝜃 = 𝐺𝑝𝑎𝑡ℎ−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 sin−1

ℎ𝑓𝑖𝑥𝑒𝑑−𝑐𝑎𝑝𝑡𝑢𝑟𝑒− ℎ

𝑉𝑡𝑟𝑢𝑒

Track: Δ𝜃 = 𝐺𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒𝛿ℎ + 𝐺𝑉𝑆 ℎ𝑓𝑖𝑥𝑒𝑑−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 − ℎ /𝑉𝑡𝑟𝑢𝑒

Altitude error V/S error

ℎ = 𝑉 𝑆 =vertical speed

For example VNAV PATH (B787 FCOM):

in the VNAV PTH pitch mode, the AFDS commands pitch to maintain FMC target altitude or the

VNAV path. The autothrottle maintains speed

Vertical Guidance

62

Vspd:

Capture: Δ𝜃 = 𝐺𝑠𝑝𝑒𝑒𝑑−𝑟𝑎𝑡𝑒 𝑉 − 𝑉𝑐𝑎𝑝𝑡𝑢𝑟𝑒Track: Δ𝜃 = 𝐺𝑎𝑖𝑟𝑠𝑝𝑒𝑒𝑑𝛿𝑉 + 𝐺𝑠𝑝𝑒𝑒𝑑−𝑟𝑎𝑡𝑒 𝑉 /𝑉𝑡𝑟𝑢𝑒

𝑉𝑡𝑟𝑢𝑒

𝛿𝑉𝑉

Airspeed error Airspeed rate

For example VNAV SPD (B787 FCOM):

in the VNAV SPD pitch mode, the AFDS commands pitch to hold target airspeed (“track”). The

autothrottle operates in the THR REF, THR, IDLE or HOLD mode, as required by the phase of flight

Vertical Guidance

63

Valt:

Capture: ℎ𝑐𝑎𝑝𝑡𝑢𝑟𝑒 = 𝐺𝑎𝑙𝑡−𝑐𝑎𝑝𝑡𝑢𝑟𝑒𝛿ℎ

Δ𝜃 = 𝐺𝑉/𝑆 sin−1

ℎ𝑐𝑎𝑝𝑡𝑢𝑟𝑒− ℎ

𝑉𝑡𝑟𝑢𝑒

Track: Δ𝜃 = 𝐺𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒𝛿ℎ + 𝐺𝑉𝑆 ℎ /𝑉𝑡𝑟𝑢𝑒

For example VNAV ALT (B787 FCOM):

in the VNAV ALT pitch mode, the AFDS commands pitch to maintain the MCP selected altitude

when that altitude is lower than the VNAV commanded altitude in climb or higher than the VNAV

commanded altitude in descent

Vertical Guidance

• Thrust is set based on the equations of motion:

– Taking into account the thrust limit and idle power

(see BADA discussion)

64

𝑇 =𝑊 ℎ𝑎𝑣𝑒𝑉𝑎𝑣𝑒

1 +𝑉𝑎𝑣𝑒𝑔

𝑑𝑉𝑡𝑟𝑢𝑒𝑑ℎ

+ 𝐷

Instrument Approach using VNAV

65

B787 FCOM pp. 1210 and on …

1. Cruise - before the top of

descent, FMC is in cruise mode

and commands VNAV PTH and

ECON cruise speed.

2. Descent - nearing descent speed,

VNAV commands a descent in

VNAV PTH at ECON descent

speed.

3. Descent Deceleration Phase -

before the speed restriction

altitude, the FMC commands the

target descent airspeed. The pitch

mode remains VNAV PTH and the

descent rate approximates 500

feet per minute.4

4. Descent and Approach - when

at target speed, VNAV commands

a descent and starts approach in

VNAV PTH at commanded

speed.5

5. Missed Approach - when

selected during missed approach,

VNAV activates in VNAV SPD.6

6. Missed Approach Level Off - at

missed approach altitude, VNAV

SPD changes to VNAV

Takeoff and Climb

66

1. Takeoff - if armed for takeoff, VNAV activates at 400 feet RA and

pitch guidance continues to maintain the target airspeed. During

takeoff, the FMC updates the target airspeed to the current airspeed

until VNAV activates. The target airspeed is between V2 + 15 and V2

+ 25 knots.

2. Acceleration Height - at acceleration height or flap retraction, VNAV

commands an airspeed increase to a speed 5 knots below the flap

placard speed for the existing flap setting. When flaps are retracted or

at an AFDS capture altitude, VNAV commands the greater of VREF +

80 knots or the speed transition associated with the origin airport,

limited by configuration. The FMC changes the thrust reference mode

to the selected climb thrust at the thrust reduction point.

3. VNAV Climb - the VNAV climb profile uses VNAV SPD or VNAV

PTH at the default climb speed or pilot selected climb speed to

remain within all airspeed and altitude constraints that are part of the

SID entered into the active route. Autothrottle uses selected climb

thrust limit.

4. Climb Constraints - VNAV enters the VNAV PTH mode to remain

within departure or waypoint constraints. Speed maintained during

this time can be: procedure based speed restriction, waypoint speed

restriction, default VNAV climb speed, manually entered climb speed.

If the FMC predicts the airplane will not reach an altitude constraint,

the FMS–CDU help window message UNABLE NEXT ALTITUDE

displays. Speed intervention can be used by pushing the IAS/MACH

selector and manually setting a lower airspeed to provide a steeper

climb; or, climb derates can be deleted on the THRUST LIMIT page.

5. Top Of Climb (T/C) - the point where the climb phase meets the

cruise altitude is called the top of climb. Approaching this point, the

FMC changes from the climb phase to the cruise phase. The T/C

displays any time the FMC calculates a change from a climb phase to

a cruise phase, such as a step climb.

ILS Approach

67