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
Navigation
Database
Navigation
Performance
Computations
Data Link
Lateral
Guidance
Performance
Database
Vertical
Guidance
Trajectory
Prediction
Flight Planning
Lateral &
Vertical Profile
Flight Plan
Buffer
3
Flight Management – Typical
Inertial
Reference
Data
Link
Position,
velocities,
vert speed,
pitch, roll,
heading,
accels
Init data, flight
plans, clearance,
weather
Data entry,
display data
Air
Data
Navigation
Receivers
Altitude,
speeds,
temperatures
Initial
position
Freq, range, bearing, LOC deviation,
GPS position, GPS GS, time
Tuning cmds
Flight
Management
Fuel weight,
engine thrust
Thrust limits
Engine and
Fuel Systems
Flight ID, aircraft
state, trajectory
MCDU
Map scale,
display
selections
Aircraft
Displays
Flight plan &
path, nav
data, route
data, HIS data
Roll axis cmds,
pitch axis cmds,
thrust axis cmds
Tactical cmds,
modes
Flight
Controls
Trajectory
conflicts
Surveillance
Systems
4
VNAV Flight Path
𝑡0
𝑡1
IDLE descent,
constant velocity
JAIKE
13,000ft
ILENE
13,000ft
280kts
250kts
𝑡2
WACKI
11,000ft
250kts
𝑡3
REGLE
7,000ft
Example of a VNAV Path
250kts
5
Flight Path
𝑡0
𝑡1
IDLE descent,
constant velocity
JAIKE
13,000ft
ILENE
13,000ft
280kts
250kts
𝑡2
WACKI
11,000ft
250kts
𝑡3
Can this work?
𝑉
0
0
0
0
𝜓
0
0
𝛾
𝐱=
=
=
= 𝐟 𝑉, 𝜓, 𝛾, ℎ, 𝑟, 𝑊
𝑉𝑠𝑖𝑛(γ)
−𝑉𝐷 𝑊
ℎ
𝑉𝑐𝑜𝑠(𝛾)
𝑉𝑐𝑜𝑠(𝛾)
𝑟
0
0
𝑊
REGLE
7,000ft
250kts
Example of a VNAV Path
6
Altitude Change versus Distance
Descent @ IDLE Thrust vs Time
4
1.3
x 10
Altitude [ft]
1.25
Range @ IDLE Thrust vs Time
7
6
1.2
1.15
Range [NM]
5
4
1.1
3
2
1.05
1
0
0
0
10
20
30
40
50
60
70
80
90
Time elapsed [sec]
10
20
30
40
50
60
Time elapsed [sec]
70
80
90
Answer: does not work at IDLE thrust!
7
Adjust the Thrust?
Not IDLE thrust
ILENE
13,000ft
𝑇−𝐷 𝑉
= 𝑉𝑠𝑖𝑛 𝛾
𝑊
𝑇−𝐷
sin 𝛾 =
𝑊
𝑇 = 𝐷 + 𝑊𝑠𝑖𝑛(𝛾)
ℎ=
𝛾
𝛾𝑔𝑙𝑖𝑑𝑒
Δℎ𝑑𝑒𝑠
WACKI
11,000ft
(e.g. 13000ft)
Δ𝑟𝑑𝑒𝑠
Range @ IDLE Thrust vs Time
6NM
9
(e.g. 8NM)
8
7
𝑉
𝜓
𝛾
𝐱=
=
ℎ
𝑟
𝑊
𝑔
𝑊
= 𝐟 𝑉, 𝜓, 𝛾, ℎ, 𝑟, 𝑊
Δℎ𝑑𝑒𝑠
Δ𝑟𝑑𝑒𝑠
Range [NM]
𝛾 = 𝑎𝑡𝑎𝑛
6
𝑇 − 𝐷 − 𝑊𝑠𝑖𝑛(𝛾)
0
0
𝑉𝑠𝑖𝑛(γ)
𝑉𝑐𝑜𝑠(𝛾)
0
=
0
0
0
5
4
3
2
𝑉(𝑇 − 𝐷) 𝑊
1
𝑉𝑐𝑜𝑠(𝛾)
0
0
0
20
40
60
80
100
120
Time elapsed [sec]
8
Flight Path – First Segment
𝑡0
Straight and level,
Speed change
𝑡1
26NM
JAIKE
13,000ft
ILENE
13,000ft
𝑑𝑉
𝑑𝑟
280kts
𝑡2
250kts
WACKI
11,000ft
250kts
𝑡3
𝑉
𝜓
𝛾
𝐱=
=
ℎ
𝑟
𝑊
𝑔
𝑊
𝑇𝑐𝑜𝑠(𝛼) − 𝐷 − 𝑊𝑠𝑖𝑛(𝛾)
𝑔
𝑊𝑉𝑐𝑜𝑠(𝛾)
𝑔
𝑊𝑉
𝑔
𝑊
𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇
𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) − 𝑊𝑐𝑜𝑠(𝛾)
𝑉𝑠𝑖𝑛(γ)
𝑉𝑐𝑜𝑠(γ)
−𝜂 𝑉 𝑇
=
𝑇−𝐷
0
0
0
𝑉
−𝜂 𝑉 𝑇
REGLE
7,000ft
𝑉=
𝑑𝑉 𝑑𝑉 𝑑𝑟 𝑑𝑉
=
=
𝑉
𝑑𝑡
𝑑𝑟 𝑑𝑡 𝑑𝑟
Example of a VNAV Path
9
Flight Path – First Segment
Non-steady Straight and Level vs Time
30
275
25
270
20
Range [NM]
Aispeed [kts]
Non-steady Straight and Level vs Time
280
265
15
260
10
255
5
250
0
50
100
150
200
250
Time elapsed [sec]
300
350
400
0
0
50
100
150
200
250
300
350
400
Time elapsed [sec]
10
Automation Modes – B787
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)
11
VNAV Mode – B787
• 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
12
Important Note
𝑡0
JAIKE
13,000ft
𝑡1
ILENE
13,000ft
𝑡2
WACKI
11,000ft
250kts
𝑡3
The path is defined in an Earthreferenced frame (navigation
frame, earth-frame)
REGLE
7,000ft
13
What happens when we have a tail-wind?
𝑡0
𝑡1
ILENE
13,000ft
JAIKE
13,000ft
𝑉
𝑉𝑤
𝑡2
WACKI
11,000ft
250kts
𝑉𝑔
So, this 1st segment would be completed
faster than expected.
𝑡3
REGLE
7,000ft
14
What happens when we have a tail-wind?
ILENE
13,000ft
ILENE
13,000ft
𝑟
ℎ
ℎ
𝑉 = 𝑉𝑒
𝑟
𝑉𝑤
𝑉𝑒
Too high w.r.t. path
No wind
WACKI
11,000ft
250kts
Tailwind
WACKI
11,000ft
250kts
VNAV may disconnect;
Airspeed must somehow be reduced
(reduce thrust, spoilers, etc.)
15
Cost Index (CI)
Time-related direct operating cost (minus cost of fuel):
• flight crew wages (hourly or fixed);
• lease of engines, auxiliary power units, airplanes;
• maintenance costs;
𝑇𝑖𝑚𝑒 𝑐𝑜𝑠𝑡 ($ ℎ𝑟)
𝐶𝐼 =
𝐹𝑢𝑒𝑙 𝑐𝑜𝑠𝑡 (𝑐𝑒𝑛𝑡𝑠 𝑙𝑏)
Cost of fuel, may be complex calculation due to:
• variation of fuel cost as a function of location;
• fuel tankering;
• fuel hedging.
16
Cost Index (CI)
𝑇𝑖𝑚𝑒 𝑐𝑜𝑠𝑡 ($ ℎ𝑟)
𝐶𝐼 =
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
17
CI Ranges for Boeing Aircraft
B787 as well
From: W. Roberson, et al., “Fuel Conservation Strategies: Cost Index Explained,” Boeing
18
CI Results for Phases of Flight
Minimum fuel flight
Minimum time flight
From: W. Roberson, et al., “Fuel Conservation Strategies: Cost Index Explained,” Boeing
19
CI Impact Example
20
Airbus CI Examples
Old fuel prices!!!
Crew cost is between 10-20 US$/min
Maintenance cost is between 7 and 17 US$/min,
Based on: “Getting to Grips with the Cost Index,” Airbus, May 1998.
21
CI Effect on Climb
22
CI Effect on Climb
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)
23
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.
Optimum fuel mileage
Based on: W. Roberson, et al., “Fuel Conservation Strategies: Cruise Flight,” Boeing
24
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)
Typically this 1% means a 3 to 5 % higher cruise speed
26
MRC versus LRC
From: W. Roberson, et al., “Fuel Conservation Strategies: Cruise Flight,” Boeing
27
Typical CI Values
28
Cost Simulations
•
•
•
•
•
•
Price fuel: $2.94/gallon
Crew and maintenance: $45/minute
Altitude: 20,000ft
Cruise for 200NM
Cruise: steady straight and level flight
Change from Vmo down to 0.65Vmo
29
Jet Fuel Priced (Commodity)
6.84 lbs/US gallon = 3.10kg/ US gallon
30
Speed at Altitude
Cruise
440
420
Airspeed [NM]
400
380
360
340
320
300
280
0
500
1000
1500
2000
2500
Time elapsed [sec]
Includes conversion from CAS to TAS for standard atmosphere!
31
Range and Speed
Cruise
Cruise
250
238
237
Mass [tonnes]
Range [NM]
200
150
100
236
235
234
50
233
0
0
500
1000
1500
Time elapsed [sec]
2000
2500
232
0
500
1000
1500
2000
2500
Time elapsed [sec]
32
Fuel Usage
Fuel usage versus Mach no
Fuel usage versus Mach no
10400
1500
10000
Fuel usage [gal]
Fuel usage [lbs]
10200
9800
9600
1450
1400
9400
9200
0.45
0.5
0.55
0.6
Mach no.
0.65
0.7
0.75
1350
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Mach no.
33
Costs
Time-related Cost versus Mach
Fuel-related Cost versus Mach no.
1900
4400
4350
4300
Fuel-related Cost (US$)
Time-related Cost (US$)
1800
1700
1600
1500
1400
4250
4200
4150
4100
4050
1300
4000
1200
0.45
0.5
0.55
0.6
Mach no.
0.65
0.7
0.75
3950
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Mach no.
34
Fuel Mileage versus Mach
Fuel Mileage versus Mach no.
48
MRC
Fuel mileage [NM/1000kg]
47.5
LRC
47
46.5
46
45.5
45
44.5
44
43.5
43
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Mach no.
35
Cost Index versus Mach
Computed Cost Index
34
32
Cost Index
30
28
26
24
22
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Mach no.
36
Now for Flying at Different Altitude
Fuel-related Cost versus Mach no.
Fuel-related Cost versus Mach no.
4600
4400
4350
4500
Fuel-related Cost (US$)
Fuel-related Cost (US$)
4300
4250
4200
4150
4100
4400
4300
4200
4100
4050
4000
4000
3950
0.45
0.5
0.55
0.6
0.65
0.7
Mach no.
@ 20,000ft
(standard atmosphere)
0.75
3900
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Mach no.
@ 30,000ft
(standard atmosphere)
37
Now for Flying at Different Altitude
Computed Cost Index
Computed Cost Index
34
32
32
30
28
Cost Index
Cost Index
30
28
26
24
26
22
24
20
22
0.45
0.5
0.55
0.6
0.65
0.7
Mach no.
@ 20,000ft
(standard atmosphere)
0.75
18
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Mach no.
@ 30,000ft
(standard atmosphere)
38
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
𝑉
𝑉
𝑀= =
=
𝑉𝑎
𝛾𝑅𝑇
𝑉𝑐𝑎𝑠
2 𝑝0
=
𝜇 𝜌0
𝑝
1+
𝑝0
2
𝛾−1
𝛾−1
𝛾
𝑝𝑡
𝑝
𝜇𝜌 2
1+
𝑉
2𝑝
1
𝜇
−1
1
2
𝜇
−1
−1
42
Crossover Altitude
Typical Climb Profile
43
Primary Flight Display
Climb display
Cruise display
44
Flat Earth Approximation
• Remember the 3DOF equations of
motion:
𝑔
𝑉=
𝑊
𝑇𝑐𝑜𝑠(𝛼) − 𝐷 − 𝑊𝑠𝑖𝑛(𝛾)
𝑔
𝜓=
𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇
𝑊𝑉𝑐𝑜𝑠(𝛾)
𝑔
𝛾=
𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) − 𝑊𝑐𝑜𝑠(𝛾)
𝑊𝑉
ℎ = 𝑉𝑠𝑖𝑛(γ)
𝑟 = 𝑉𝑐𝑜𝑠 γ
𝑊 = −𝜂 𝑉 𝑇
• These assume a “Flat earth”
45
Flat Earth Approximation
• First extend by breaking ‘r’ into a ‘x’
(North) and ‘y’ (East) direction:
𝑉=
𝑔
𝑊
𝑇𝑐𝑜𝑠(𝛼) − 𝐷 − 𝑊𝑠𝑖𝑛(𝛾)
𝑔
𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇
𝑊𝑉𝑐𝑜𝑠(𝛾)
𝑔
𝛾=
𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) − 𝑊𝑐𝑜𝑠(𝛾)
𝑊𝑉
ℎ = 𝑉𝑠𝑖𝑛(γ)
𝑥 = 𝑉𝑐𝑜𝑠 γ cos 𝜓 = 𝑉𝑁
Still a flat earth (ENU)
𝑦 = 𝑉𝑐𝑜𝑠 𝛾 sin 𝜓 = 𝑉𝐸
𝑊 = −𝜂 𝑉 𝑇
𝜓=
46
Earth-Referenced Equations
• Assume a spherical Earth
• Latitude and longitude rates are then:
𝑉𝑁
𝐿=
𝑅𝑒 + ℎ
𝑉𝐸
𝜆=
𝑅𝑒 + ℎ cos(𝐿)
• Compare to non-spherical Earth
𝑉𝑁
𝐿=
𝑅𝑁 + ℎ
𝑉𝐸
𝜆=
𝑅𝐸 + ℎ cos(𝐿)
47
Non-Spherical Earth (FYI)
Equatorial plane
North Pole
RN
RE
North
pole


Side view
𝑅(1 − 𝑒 2 )
𝑅𝑁 =
1 − 𝑒 2 sin2 𝐿
𝑅𝐸 =
3/2
Top view
𝑅
1 − 𝑒 2 sin2 𝐿
1/2
Meridian Radius
Transverse Radius
It is the radius of the best fitting curve to
a meridian section of the reference earth ellipsoid
It is the radius of the best fitting curve to
a vertical east-west section of the reference earth ellipsoid
Length of semi-major axis: 𝑅
Length of semi-minor axis: 𝑅(1 − 𝑓)
Flattening: f = (𝑅 − 𝑟)/𝑅
Major eccentricity: e = f 2 − f 1/2
R = 6378137.0
e = 0.0818191908426
Mean Radius of Curvature: 𝑅0 = 𝑅𝐺 =
𝑅𝑁 𝑅𝐸
48
Back to Spherical Coordinates
• 3DOF EOM:
𝑔
𝑉=
𝑊
𝑇𝑐𝑜𝑠(𝛼) − 𝐷 − 𝑊𝑠𝑖𝑛(𝛾)
𝑔
𝜓=
𝑇𝑠𝑖𝑛 𝛼 + 𝐿 𝑠𝑖𝑛 𝜇
𝑊𝑉𝑐𝑜𝑠(𝛾)
𝑔
𝛾=
𝑇𝑠𝑖𝑛(𝛼) + 𝐿 𝑐𝑜𝑠(𝜇) − 𝑊𝑐𝑜𝑠(𝛾)
𝑊𝑉
ℎ = 𝑉𝑠𝑖𝑛(γ)
𝑉𝑐𝑜𝑠 γ cos 𝜓
𝐿=
𝑅𝑒 + ℎ
𝑉𝑐𝑜𝑠 𝛾 sin 𝜓
𝜆=
Can solve these equations again
𝑅𝑒 + ℎ cos(𝐿)
using the ODE solvers, but now the
𝑊 = −𝜂 𝑉 𝑇
results are in the spherical Earth
49
Example: 1200NM Cruise @ 20,000ft
Track: 45 degrees
50
Remember from earlier notes …
WAYPOINT ‘i’
𝐫𝑖 = 𝑅𝑒
𝑐𝑜𝑠𝜆𝑖 𝑐𝑜𝑠𝐿𝑖
𝑠𝑖𝑛𝜆𝑖 𝑐𝑜𝑠𝐿𝑖 = 𝑅𝑒 𝐞𝑖
𝑠𝑖𝑛𝐿𝑖
E
𝐿𝑖
𝜆𝑖
Radius of a sphere
(approximate Earth by a sphere)
𝐿𝑖 = waypoint latitude
𝜆𝑖 = waypoint longitude
51
Lateral Guidance
Great-circle route:
Δ𝑟𝑎𝑝−𝑔𝑡
= 𝑅𝑒 𝑎𝑐𝑜𝑠 𝐞𝑎𝑝 ∙ 𝐞𝑔𝑡
𝐞𝑁,𝑠𝑡
𝑅𝑒
𝐞𝐸,𝑠𝑡
East-pointing local
level unit vector
Δ𝑟𝑠𝑡−𝑔𝑡
= 𝑅𝑒 cos−1 𝐞𝒔𝒕 ∙ 𝐞𝑔𝑡
𝑅𝑒
st: start point
gt: go to
ap: along path
𝐞𝐸,𝑠𝑡 = 𝐞𝑍 × 𝐞𝑠𝑡
𝐞𝑁,𝑠𝑡 = 𝐞𝑠𝑡 × 𝐞𝐸,𝑠𝑡
𝐞𝑎𝑝
North-pointing local
level unit vector
𝐞𝑠𝑡
𝐞𝑔𝑡
Normal vector to 𝐞𝑠𝑡 𝐨𝐞𝑔𝑡 plane:
𝐧 = 𝐞𝑠𝑡 × 𝐞𝑔𝑡
𝐨
𝐞𝑧 = 0
0
1
𝑇
52
Lateral Guidance
Top-view
Cross-track error:
𝑋𝑇𝑅𝐾 = −𝑅𝑒 cos −1 𝐞𝑎𝑝 ∙ 𝐞𝑝𝑜𝑠
𝑋𝑇𝑅𝐾 = −𝑅𝑒 𝐧 ∙ 𝐞𝑝𝑜𝑠
𝐷𝑇𝑅𝐾
𝐞𝑎𝑝
𝑋𝑇𝑅𝐾
𝐞𝑠𝑡
𝐞𝑔𝑡
Track error:
𝑇𝑅𝐾𝐸𝑅𝑅 = 𝐷𝑇𝑅𝐾 − 𝐶𝑇𝑅𝐾
𝐞𝑝𝑜𝑠
𝐶𝑇𝑅𝐾
Desired track:
−𝐧 ∙ 𝐞𝑁,𝑎𝑝
𝐷𝑇𝑅𝐾 = 𝑎𝑡𝑎𝑛
−𝐧 ∙ 𝐞𝐸,𝑎𝑝
53
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:
𝜑𝑐𝑜𝑚𝑚𝑎𝑛𝑑𝑒𝑑 = 𝐺𝑋𝑇𝑅𝐾 ∙ 𝑋𝑇𝑅𝐾 + 𝐺𝑇𝑅𝐾𝐸𝑅𝑅 ∙ 𝑇𝑅𝐾𝐸𝑅𝑅 + 𝜑𝑛𝑜𝑚𝑖𝑛𝑎𝑙
54
Lateral – Waypoint Changing
Nominal bank angle may exist due to a desired (nominal) course change
Centripetal force
𝑉 = 𝜔𝑅
𝛼
𝑉2
𝑚
𝑅
𝐿1
𝑅
𝐿2
𝑅
Radius of turn
𝑅
𝑉2
𝑅=
𝑔 ∙ 𝑡𝑎𝑛𝜑𝑛𝑜𝑚𝑖𝑛𝑎𝑙
55
Vertical Guidance
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)
• Vertical path changed
using pitch and
autothrottle (thrust)
• Vertical paths so far:
– Climb, descent, vertical
speed, take-off, etc.
ALT (engaged)
G/S engaged)
G/P (engaged)
FLARE (armed)
FLARE (engaged)
56
Vertical Guidance - Data
From: Avionics Handbook – Chapter 15 - Flight
Management Systems, Randy Walter
57
Vertical Guidance
ℎ𝑠𝑡
ℎ𝑝𝑜𝑠
𝐞𝑎𝑝 Path gradient: 𝑔ℎ
ℎ
=
ℎ𝑎𝑝
Δℎ
Δ𝑟
Δℎ
Δ𝑟
ℎ𝑔𝑡
(really: distance between two points)
Path altitude:
ℎ𝑎𝑝 = ℎ𝑔𝑡 + 𝑔ℎ Δ𝑟𝑎𝑝−𝑔𝑡
Vertical deviation:
Desired V/S:
𝛿ℎ = ℎ𝑝𝑜𝑠 − ℎ𝑎𝑝
𝑉/𝑆𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑 = 𝑔ℎ 𝑉𝐺𝑆
58
Transitions and switches
•
Auto Flight Phase Transitions:
Cruise
Climb
ℎ
ℎ𝑐𝑟𝑢𝑖𝑠𝑒
Switch from climb to cruise (level):
ℎ𝑐𝑟𝑢𝑖𝑠𝑒 − ℎ < 𝐺𝑐𝑎𝑝𝑡𝑢𝑟𝑒 ℎ
•
Vertical leg switches:
𝐺𝑐𝑎𝑝𝑡𝑢𝑟𝑒 ℎ𝑝𝑎𝑡ℎ,𝑛 − ℎ𝑝𝑎𝑡ℎ,𝑛+1 < ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑛 − ℎ𝑑𝑒𝑠𝑖𝑟𝑒𝑑,𝑛+1
59
Vertical Guidance
From: Avionics Handbook – Chapter 15 - Flight Management Systems, Randy Walter
60
Vertical Guidance
Vpath:
ℎ𝑓𝑖𝑥𝑒𝑑−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 −ℎ
Capture:
Δ𝜃 = 𝐺𝑝𝑎𝑡ℎ−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 sin−1
Track:
Δ𝜃 = 𝐺𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝛿ℎ + 𝐺𝑉𝑆 ℎ𝑓𝑖𝑥𝑒𝑑−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 − ℎ /𝑉𝑡𝑟𝑢𝑒
Altitude error
𝑉𝑡𝑟𝑢𝑒
V/S error
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
ℎ = 𝑉 𝑆 =vertical speed
61
Vertical Guidance
Vspd:
Capture:
Track:
Δ𝜃 = 𝐺𝑠𝑝𝑒𝑒𝑑−𝑟𝑎𝑡𝑒 𝑉 − 𝑉𝑐𝑎𝑝𝑡𝑢𝑟𝑒
Δ𝜃 = 𝐺𝑎𝑖𝑟𝑠𝑝𝑒𝑒𝑑 𝛿𝑉 + 𝐺𝑠𝑝𝑒𝑒𝑑−𝑟𝑎𝑡𝑒 𝑉 /𝑉𝑡𝑟𝑢𝑒
Airspeed error
Airspeed rate
𝑉
𝛿𝑉
𝑉𝑡𝑟𝑢𝑒
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
62
Vertical Guidance
Valt:
Capture:
ℎ𝑐𝑎𝑝𝑡𝑢𝑟𝑒 = 𝐺𝑎𝑙𝑡−𝑐𝑎𝑝𝑡𝑢𝑟𝑒 𝛿ℎ
Δ𝜃 = 𝐺𝑉/𝑆 sin−1
Track:
ℎ𝑐𝑎𝑝𝑡𝑢𝑟𝑒 −ℎ
𝑉𝑡𝑟𝑢𝑒
Δ𝜃 = 𝐺𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝛿ℎ + 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
63
Vertical Guidance
• Thrust is set based on the equations of motion:
– Taking into account the thrust limit and idle power
(see BADA discussion)
𝑊 ℎ𝑎𝑣𝑒
𝑉𝑎𝑣𝑒 𝑑𝑉𝑡𝑟𝑢𝑒
𝑇=
1+
+𝐷
𝑉𝑎𝑣𝑒
𝑔
𝑑ℎ
64
Instrument Approach using VNAV
1.
2.
3.
4.
5.
6.
Cruise - before the top of
descent, FMC is in cruise mode
and commands VNAV PTH and
ECON cruise speed.
Descent - nearing descent speed,
VNAV commands a descent in
VNAV PTH at ECON descent
speed.
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
Descent and Approach - when
at target speed, VNAV commands
a descent and starts approach in
VNAV PTH at commanded
speed.5
Missed Approach - when
selected during missed approach,
VNAV activates in VNAV SPD.6
Missed Approach Level Off - at
missed approach altitude, VNAV
SPD changes to VNAV
B787 FCOM pp. 1210 and on …
65
Takeoff and Climb
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.
66
ILS Approach
67