Chapter7uavbook.byu.edu/lib/exe/fetch.php?media=lecture:chap7.pdfBeard & McLain, “Small Unmanned Aircraft,” Princeton University Press,2012, Chapter 7, Slide 17 Altitude Measurement

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 1

Chapter 7

Sensors


Architecture

Path planner

Path manager

Path following

Autopilot

Unmanned Vehicle

Waypoints

On-board sensors

Position error

Tracking error

Status

Destination, obstacles

Servo commands State estimator

Wind

Path Definition

Airspeed, Altitude, Heading,

commands

Map


Sensors for MAVs

• The following types of sensors are

commonly used for guidance and control

of MAVs

– accelerometers

– rate gyros

– pressure sensors

– magnetometers (digital compasses)

– GPS


MEMS Accelerometer

Newtons 2

ndlaw gives

mx = k(y � x)

Note that the acceleration of the proof mass pro-

portional to deflection of the suspension

Taking the Laplace transform gives

X(s)

Y (s)

=

1

mk s

2+ 1

Also, since X(s)/Y (s) = s

2X(s)/s

2Y (s), we can

also write

AX(s)

AY (s)=

1

mk s

2+ 1

Accelerometers also have bias and zero mean Gaus-

sian noise. Therefore, the sensor model is

⌥accel = kaccela+ �accel + ⌘

0accel.


MEMS Accelerometer


Acceleration Measurement

Tricky concept: Measured acceleration is the total acceleration of the accelerometer

casing minus the acceleration of gravity

Example: Set the accelerometer on a table top.

What does it measure?

Accels measure components of linear, coriolis, and externally applied acceleration.

They do not measure gravity, since both the proof mass and the casing are acted

on by gravity in exactly the same way

a =1

m(F

total

� Fgravity

)


Acceleration Measurement

Said another way, accelerometers measure specific force, which is defined as the

sum of the non-gravitational forces divided by the mass

Example: Set the accelerometer on a table top.

What does it measure?

Hint: (Draw FBD of accel housing)

ameasured

=1

m

⇣XF

non-gravitational

⌘

=1

m

⇣XF� F

gravitational

⌘


Acceleration on Fixed-Wing Aircraft

Fdrag

Flift

Fthrust

Fgravity

ameasured

=1

m(F

total

� Fgravity

)

=1

m((F

lift

+ Fdrag

+ Fthrust

+ Fgravity

)� Fgravity

)

=1

m(F

lift

+ Fdrag

+ Fthrust

)


Acceleration on Fixed-Wing Aircraft

Recall from Chapter 3, that

m

✓dv

dtb

+ !b/i

⇥ v

◆= F

total

.

Using the expression

ameasured

=

1

m(F

total

� Fgravity

) ,

the output of the accelerometer can be expressed as

ameasured

=

dv

dtb

+ !b/i

⇥ v � 1

mF

gravity

.

Expressing this relationship in the body frame gives

ax

= u+ qw � rv + g sin ✓

ay

= v + ru� pw � g cos ✓ sin�

az

= w + pv � qu� g cos ✓ cos�


Accelerometer Models

yaccel,x

= u+ qw � rv + g sin ✓ + ⌘accel,x

yaccel,y

= v + ru� pw � g cos ✓ sin�+ ⌘accel,y

yaccel,z

= w + pv � qu� g cos ✓ cos�+ ⌘accel,z

or

yaccel,x

=

⇢V 2

a

S

2m

C

X

(↵) + CXq (↵)

cq

2Va

+ CX�e

(↵)�e

�

+

⇢Sprop

Cprop

2m[(k

motor

�t

)

2 � V 2

a

] + ⌘accel,x

yaccel,y

=

⇢V 2

a

S

2m

C

Y0 + CY�� + C

Yp

bp

2Va

+ CYr

br

2Va

+ CY�a

�a

+ CY�r

�r

�+ ⌘

accel,y

yaccel,z

=

⇢V 2

a

S

2m

C

Z

(↵) + CZq (↵)

cq

2Va

+ CZ�e

(↵)�e

�+ ⌘

accel,z


Accelerometer Models

or

yaccel,x =

fx

m+ g sin ✓ + ⌘accel,x

yaccel,y =

fy

m� g cos ✓ sin�+ ⌘accel,y

yaccel,z =

fz

m� g cos ✓ cos�+ ⌘accel,z


MEMS Rate Gyro

Sensor measures deflection of proof mass due to coriolis acceleration

Vgyro

= kC |aC |= 2kC |⌦⇥ v|= ⌦|v|= 2kC⌦|A!n sin(!nt)|= 2kCA!n⌦

= KC⌦

Point translating on a rotating rigid

body experiences a coriolis acceleration:

aC = 2⌦⇥ v

MEMS rate gyro – resonating proof

mass:

|v| = A!n sin(!nt)


MEMS Rate Gyro


Rate Gyro Model

The manufacturing process implies that rate gyros will have a drift term, as

well as zero mean Gaussian noise:

⌥

gyro

= kgyro

⌦+ �gyro

+ ⌘0gyro

,

where ⌥

gyro

is in units of voltage. Calibrating to units of rad/s gives

ygyro,x

= p+ �gyro,x

+ ⌘gyro,x

ygyro,y

= q + �gyro,y

+ ⌘gyro,y

ygyro,z

= r + �gyro,z

+ ⌘gyro,z


Pressure Measurement


Pressure Measurement


Altitude MeasurementThe basic equation of hydrostatics is

P2

� P1

= ⇢g(z2

� z1

)

Using the ground as reference, and assuming constant air density

gives

P � Pground

= �⇢g(h� hground

)

= �⇢ghAGL

Below 11,000 m, can use barometric formula:

P = P0

T0

T0

+ L0

hASL

� gMRL0

,

where P0

: standard pressure at sea level

T0

: standard temperature at sea level

L0

: rate of temperature decrease

g: gravitational constantR: universal gas constant for air

M : standard molar mass of atmospheric air,

which takes into account change in density with altitude and tem-

perature.


Altitude Measurement

0 2000 4000 6000 8000 10000

−2

0

2

4

6

8

10

x 104

pres

sure

(N/m

2 )

altitude (m)0 100 200 300 400 500

9.5

9.6

9.7

9.8

9.9

10

10.1

10.2x 104

pres

sure

(N/m

2 )

altitude (m)

constant densityideal gas law

We usually assume density is constant:

yabs pres

= (Pground

� P ) + �abs pres

+ ⌘abs pres

= ⇢ghAGL

+ �abs pres

+ ⌘abs pres

Is this valid?


Airspeed Measurement

From Bernoulli’s equation:

Pt = Ps +⇢V 2

a

2

or

⇢V 2a

2

= Pt � Ps

Pitot-static pressure sensor measures dynamic pressure:

ydi↵ pres =⇢V 2

a

2

+ �di↵ pres + ⌘di↵ pres


north magneticnorth

declination

localmagneticfield

iv1

m�

iv m0

Magnetometers & Digital Compasses

Heading is sum of magnetic declination angle andmagnetic heading

= � + m

Magnetic heading determined from measurementsof body-frame components of magnetic field pro-jected onto horizontal plane

mv10 =

0

@mv1

0x

mv10y

mv10z

1

A = Rv1b

(�, ✓)mb

0

= Rv1v2(✓)Rv2

b

(�)mb

00

@mv1

0x

mv10y

mv10z

1

A =

0

@c✓

s✓

s�

s✓

c�

0 c�

�s�

�s✓

c✓

s�

c✓

c�

1

Amb

0

Solving for heading gives

m

= �atan2(mv10y,m

v10x).


0° 30° 60° 90° 120° 150° 180°330°300°270°240°210°180°

0° 30° 60° 90° 120° 150° 180°330°300°270°240°210°180°

0°

30°

60°

-30°

-60°

0°

30°

60°

-30°

-60°

10

20

30

Magnetic Declination Variation

World Magnetic Model, National Geophysical Data Center


Magnetic Inclination

From Wikipedia: "Magnetic dip or magnetic inclination is the angle made by

a compass needle with the horizontal at any point on the Earth's surface.

Positive values of inclination indicate that the field is pointing downward,

into the Earth, at the point of measurement.

hg

60

60

40

20

0

80

40

20

0

20

80

60

40

0

-80

-20

-40

-60

-20

-80

-40

-60

-20

-40

-60

-60

-60

70°N 70°N

70°S 70°S180°

180°

180° 135°E

135°E

90°E

90°E

45°E

45°E

0°

0°

45°W

45°W

90°W

90°W

135°W

135°W

60°N 60°N

45°N 45°N

30°N 30°N

15°N 15°N

0° 0°

15°S 15°S

30°S 30°S

45°S 45°S

60°S 60°S

180°

US/UK World Magnetic Model -- Epoch 2010.0Main Field Inclination (I)

Map developed by NOAA/NGDC & CIREShttp://ngdc.noaa.gov/geomag/WMM/Map reviewed by NGA/BGSPublished January 2010

Main field inclination (I)Contour interval: 2 degrees, red contours positive (down); blue negative (up); green zero line.Mercator Projection. : Position of dip polesg


Global Positioning System

• 24 satellites orbiting the earth

• Altitude 20,180 km

• Any point on Earth’s surface can

be seen by at least 4 satellites at

all times

• Time of flight of radio signal from

4 satellites to receiver used to

trilaterate location of receiver in 3

dimensions

• 4 range measurements needed to

account for clock offset error

• 4 nonlinear equations in 4

unknowns results:

– latitude

– longitude

– altitude

– receiver clock time offset


GPS Error Sources

• Time of flight of radio signal from satellite to receiver used to calculate

pseudorange

– Called pseudorange to distinguish it from true range

– Inaccuracies in timing information result in pseudorange ~= range

• Numerous sources of error in time of flight measurement:

– Ephemeris Data – errors in satellite location

– Satellite Clock – due to clock drift

– Ionosphere – upper atmosphere, free electrons slow transmission of GPS

signal

– Troposphere – lower atmosphere, weather (temperature and density)

affect speed of light, GPS signal transmission

– Multipath Reception – signals not following direct path

– Receiver Measurement – limitations in accuracy of receiver timing

• Small timing errors can result in large position errors

– 10 ns timing error à 3 m pseudorange error


GPS Error Characterization

• Cumulative effect of GPS pseudorange errors is

described by user equivalent range error (UERE)

• UERE has two components

– Bias

– Random

138 SENSORS FOR MAVS

Table 7.1: Standard pseudorange error model (1-�, in meters) [34].

Error source Bias Random TotalEphemeris data 2.1 0.0 2.1Satellite clock 2.0 0.7 2.1Ionosphere 4.0 0.5 4.0Troposphere monitoring 0.5 0.5 0.7Multipath 1.0 1.0 1.4Receiver measurement 0.5 0.2 0.5UERE, rms 5.1 1.4 5.3Filtered UERE, rms 5.1 0.4 5.1

RECEIVER MEASUREMENT

Receiver measurement errors stem from the inherent limits with which thetiming of the satellite signal can be resolved. Improvements in signal track-ing and processing have resulted in modern receivers that can compute thesignal timing with sufficient accuracy to keep ranging errors due to the re-ceiver less than 0.5 m.

Pseudorange errors from the sources described above are treated as sta-tistically uncorrelated and can be added using the root sum of squares. Thecumulative effect of each of these error sources on the pseudorange measur-ment is called the user-equivalent range error (UERE). Parkinson, et al. [34]characterized these errors as a combination of slowly varying biases andrandom noise. The magnitudes of these errors are tabulated in Table 7.1.Recent publications indicate that measurement accuracies have improved inrecent years due to improvements in error modeling and receiver technologywith total UERE being estimated as approximately 4.0 m (1-�) [33].

The pseudorange error sources described above contribute to the UEREin the range estimates for individual satellites. An additional source of po-sition error in the GPS system comes from the geometric configuration ofthe satellites used to compute the position of the receiver. This satellite ge-ometry error is expressed in terms of a single factor called the dilution ofprecision (DOP) [33]. The DOP value describes the increase in positioningerror attributed to the positioning of the satellites in the constellation. Ingeneral, a GPS position estimate from a group of visible satellites that arepositioned close to one another will result in a higher DOP value, while aposition estimate from a group of visible satellites that are spread apart willresult in a lower DOP value.

There are a variety of DOP terms defined in the literature. The two DOPterms of greatest interest to us are the horizontal DOP (HDOP) and the ver-

Forthcoming publication of Princeton University Press

138 SENSORS FOR MAVS

Table 7.1: Standard pseudorange error model (1-�, in meters) [34].

Error source Bias Random TotalEphemeris data 2.1 0.0 2.1Satellite clock 2.0 0.7 2.1Ionosphere 4.0 0.5 4.0Troposphere monitoring 0.5 0.5 0.7Multipath 1.0 1.0 1.4Receiver measurement 0.5 0.2 0.5UERE, rms 5.1 1.4 5.3Filtered UERE, rms 5.1 0.4 5.1

RECEIVER MEASUREMENT

Receiver measurement errors stem from the inherent limits with which thetiming of the satellite signal can be resolved. Improvements in signal track-ing and processing have resulted in modern receivers that can compute thesignal timing with sufficient accuracy to keep ranging errors due to the re-ceiver less than 0.5 m.

Pseudorange errors from the sources described above are treated as sta-tistically uncorrelated and can be added using the root sum of squares. Thecumulative effect of each of these error sources on the pseudorange measur-ment is called the user-equivalent range error (UERE). Parkinson, et al. [34]characterized these errors as a combination of slowly varying biases andrandom noise. The magnitudes of these errors are tabulated in Table 7.1.Recent publications indicate that measurement accuracies have improved inrecent years due to improvements in error modeling and receiver technologywith total UERE being estimated as approximately 4.0 m (1-�) [33].

The pseudorange error sources described above contribute to the UEREin the range estimates for individual satellites. An additional source of po-sition error in the GPS system comes from the geometric configuration ofthe satellites used to compute the position of the receiver. This satellite ge-ometry error is expressed in terms of a single factor called the dilution ofprecision (DOP) [33]. The DOP value describes the increase in positioningerror attributed to the positioning of the satellites in the constellation. Ingeneral, a GPS position estimate from a group of visible satellites that arepositioned close to one another will result in a higher DOP value, while aposition estimate from a group of visible satellites that are spread apart willresult in a lower DOP value.

There are a variety of DOP terms defined in the literature. The two DOPterms of greatest interest to us are the horizontal DOP (HDOP) and the ver-

Forthcoming publication of Princeton University Press


GPS Error Characterization

• Effect of satellite geometry on position calculation

is expressed by dilution of precision (DOP)

• Satellites close together à high DOP

• Satellites far apart à low DOP

• DOP varies with time

• Horizontal DOP is smaller than vertical DOP

• Nominal HDOP = 1.3

• Nominal VDOP = 1.8


Total GPS Error (RMS)

Standard deviation of RMS error in the north-east plane:

En-e,rms = HDOP⇥UERErms

= (1.3)(5.1 m)

= 6.6 m

Standard deviation of RMS altitude error:

Eh,rms = VDOP⇥UERErms

= (1.8)(5.1 m)

= 9.2 m


GPS Error Model• Interested in transient behavior of errors – how does GPS error change

with time

• We use Gauss-Markov error model proposed by Rankin

⌫[n+ 1] = e�kGPSTs⌫[n] + ⌘GPS[n]

Nominal 1-� error (m) Model Parameters

Direction Bias Random Std. Dev. ⌘GPS (m) 1/kGPS (s) Ts (s)

North 4.7 0.4 0.21 1100 1.0

East 4.7 0.4 0.21 1100 1.0

Altitude 9.2 0.7 0.40 1100 1.0

yGPS,n[n] = pn[n] + ⌫n[n]

yGPS,e[n] = pe[n] + ⌫e[n]

yGPS,h[n] = �pd[n] + ⌫h[n]


GPS Gauss Markov Process Error Model

0 2 4 6 8 10 12−30

−20

−10

0

10

20

time (hours)

altit

ude

erro

r (m

)

100 110 120 130 140 150 160 170 180 190−2

−1

0

1

time (s)

altit

ude

erro

r (m

)


Project

• Add sensor models to the simulation

Documents

Chapter7uavbook.byu.edu/lib/exe/fetch.php?media=lecture:chap7.pdfBeard & McLain, “Small Unmanned Aircraft,” Princeton University Press,2012, Chapter 7, Slide 17 Altitude Measurement