Inertial Navigation
Academic Year 2008/09
Master of Science in Computer Engineering,
Environmental and Land Planning Engineering
Inertial navigation
• Reference systems
• Inertial sensors
• Navigation equations
• Error budget
Pseudo inertial system
- origin in the Earth barycentre;
- x3 axis: oriented towards a celestial North Pole;
- x1 axis: intersection between ecliptic and celestial equatorial plane;
- x2 axis: to complete the right-handed triad.
Earth-fixed system
- origin in the Earth barycentre;
- x3 axis: oriented towards a conventional North Pole;
- x1 axis: intersection between Greenwich meridian plane and
terrestrial equatorial plane;
- x2 axis: to complete the right-handed triad.
x3
x2
x1
Navigation system
- origin in a generic point P;
- x1 axis: oriented in the East direction;
- x2 axis: oriented in the North direction;
- x3 axis: oriented as the normal to reference ellipsoid, Up direction.
x2
x3
x1
Body system
- origin in a generic point P;
- x1 axis: oriented in the motion direction;
- x3 axis: perpendicular to the vehicle plane and in the up direction;
- x2 axis: to complete the right-handed triad.
x3
yaw
roll
x1
x2
pitch
Inertial navigation systems
Inertial navigation system (INS):
- 3 accelerometers,
- 3 gyroscopes,
- the hardware collecting data,
- the software for real time processing.
INS ref. system oriented as:
- the navigation system
(by means of servomotors)
- the body system (cheaper)
(called strap-down INS)
Each component is called IMU
(Inertial Measurement Unit)
Accelerometers
The basic principle of accelerometers is to measure the forces
acting on a proof mass.
Two types of accelerometers:
- open loop (e.g. spring based accelerometers)
measure the displacement of the proof mass resulting from
external forces acting on the sensor.
- closed loop (e.g. pendulous or electrostatic accelerometers)
keep the proof mass in a state of equilibrium by generating
a force that is opposite to the applied force.
Spring accelerometers
• The system dynamics is described by:
m&l& + k v &l − k e l = F
• In case of F=const, the following law holds:
(after the initial oscillations have ceased)
− kel = F
• Knowing the proof mass, it is possible to
derive the acceleration a along the spring axis: F = m a
F
m
l
Pendulous accelerometers
M
P
S
m
I
f
F
= proof mass
= hinge
= optical sensor
= magnet
= current through the coil
= extern force to the pendulous
= induced force to the magnet
feedback
system
Electrostatic accelerometers
Accelerometers with scientific purposes and with extremely higher
accuracies (of the order of 10-10 m/s2).
Mounted on board geodetic satellites (CHAMP, GRACE, GOCE)
for the measurement of the gravitational field.
Accelerometer error
The accelerometer error can be modeled as:
δa = b + λa + cT (T − T0 ) + ν
where b is a bias, λ is a scale factor, cT is a thermal constant
depending on the temperature T and v is the measurement noise.
Typical values for a commercial instrument can be:
b = 25 mGal
λ =5 10 −5
cT = 0.5 mGal / °C
σ v = 40 mGal / Hz
(namely about 0.6 Gal for each observation,
with a sampling frequency of about 200 Hz)
Gyroscopes
Gyroscopes (or briefly gyros) measure the angular rate of the
sensor rotation with respect to an inertial reference system.
Two main types of gyroscopes:
- mechanical
(more expensive, suitable
for gimbaled platforms)
- optical
(cheaper, miniaturizable,
only for strapdown systems)
Fiber Optic Gyroscopes
They are based on a the so-called Sagnac effect.
- An observer moving with the fiber sees the light to cover exactly
one revolution ϑ’ = 2π
- An inertial observer sees the light to cover an angle ϑ = 2π+∆ϑ
∆ϑ =
ωR
c
⋅ 2π
∆L = R ∆ϑ =
ωR 2
c
⋅ 2π =
2ωA
c
If the source E emits another light beam at the same initial time t0 but
covering the coil in the opposite direction, then ϑ =2π-∆ϑ.
∆ϕ =
∆L+ − ∆L− 4ωA
=
λ
λc
The measure of the phase shift allows to
derive a measure of the angular rate ω.
Gyroscope error
The accelerometer error can be modeled as:
δω = b + λω + cT (T − T0 ) + ν
where b is a bias, λ is a scale factor, cT is a thermal constant
depending on the temperature T and v is the measurement noise.
Typical values for a commercial instrument can be:
b = 10 −3 ° / ora
λ =2 10 −6
cT = 5 10 −5 ° / ora °C
σ v = 6 10 −7 rad / s Hz
(namely about 10-5 rad/s for observation
with a sampling frequency of ≈ 200 Hz)
Rotations
2D:
 x1 ' = + x1 cos α + x 2 sin α

 x 2 ' = − x1 sin α + x 2 cos α
x2
x1’
x2’
x1
 cos α
U AB = 
− sin α
x' = U AB x
sin α 
cos α 
3D:
0
1
U 1 = 0 cos α 1
0 − sin α 1
0 
sin α 1 
cos α 1 
cos α 2
U 2 =  0
 sin α 2
0 − sin α 2 
1
0 
0 cos α 2 
 cos α 3
U 3 = − sin α 3
 0
sin α 3
cos α 3
0
0
0
1
U AB = U 3 (α 1 ) U 2 (α 2 ) U 1 (α 3 ) =
 cos α 3 cos α 2
= − sin α 3 cos α 2

sin α 2
sin α 3 cos α 1 + cos α 3 sin α 2 sin α 1
cos α 3 cos α 1 − sin α 3 sin α 2 sin α 1
− cos α 2 sin α 1
sin α 3 sin α 1 − cos α 3 sin α 2 cos α 1 
cos α 3 sin α 1 + sin α 3 sin α 2 cos α 1 

cos α 2 cos α 1
Rotations
Considering the infinitesimal angles and neglecting the second order terms, we have:
0
1
dU 1 = 0
1
0 − dα 1
0 
dα 1 
1 
 1
dU 2 =  0
dα 2
0 − dα 2 
1
0 
0
1 
 1
dU 3 = − dα 3
 0
 0
dA =  dα 3
− dα 2
dU AB = dU 3 ( dα 1 ) dU 2 ( dα 2 ) dU 1 (dα 3 ) = I − dA
− dα 3
0
dα 1
dα 3
1
0
0
0
1
dα 2 
− dα 1 
0 
By defining:
U AB (t 0 ) = U AB
U AB (t 0 + dt ) = U AB (dt ) U AB (t 0 ) = ( I − dA) U AB
the time derivative of the rotation matrix is:
U AB (t 0 + dt ) − U AB (t 0 )
( I − dA) U AB − U AB
dA B
B
&
U A = lim
= lim
= − lim
U A = −Ω ABU AB
dt →0
dt
dt
→
0
→
0
dt
dt
dt

 0
 dα
dA
3
= lim 
lim
dt →0 dt
dt → 0 
dt
 dα 2
− dt

−
dα 3
dt
0
dα 1
dt
dα 2 
dt   0
dα 
− 1  =  ω 3
dt 
 − ω 2
0 

− ω3
0
ω1
ω2 
− ω1  = Ω AB
0 
matrix of angular velocities
Rotations
By recalling the rotation matrices properties, it holds:
U AB U BA = I
By computing the time derivative of this expression, it holds:
U& AB U BA + U AB U& BA = 0
U& AB = −Ω ABU AB
U& AB U BA = −Ω AB U AB U BA
U& AB U BA = −Ω AB
− U AB U& BA = −Ω AB
− U BA U AB U& BA = −U BA Ω AB
U& BA = U BA Ω AB
Navigation equations
Navigation equations establish a link between the unknowns
(namely position, velocity and attitude of the vehicle) and the
observations of the accelerometers and of the gyroscopes (and
in case of the GPS receivers).
Two cases:
- Navigation equations in an inertial reference system
(suitable to describe space navigation, for example the orbits
of an artificial satellite)
- Navigation equations in an Earth-fixed reference system
(more suitable to describe terrestrial navigation, sometimes
requiring a further step towards the local-level system).
Navigation equations (IRF)
• In an inertial reference system Newton’s second law of dynamics
can be written as:
i
i
F tot = m &x&
i
F + mg
i
i stays for inertial reference system
g is the acceleration produced by the gravitational field
• Knowing the mass m, the navigation equations result:
&x&i = f i + g i
or equivalently
d xi
i
&
x
=

 dt

 d x& i
i
i

= f +g
 dt
Navigation equations (IRF)
• In strapdown systems, the accelerometers measure the specific
force f b in the body reference system. Therefore:
i
f = U bi f
b
accelerometer measurements
where U bi is the rotation matrix from body to inertial system. This
matrix can be derived by integrating gyroscope measurements:
U& bi = −Ω biU bi
gyroscope measurements
Navigation equations (IRF)
• Example of rotation around a single axis of the reference system:
 cos α
U AB = − sin α
 0
U& bi = −Ω biU bi
sin α
cos α
0
−ω
0
0
0
Ω AB = ω
 0
0
0
1
0
0
0
dα

−
sin
α

dt

dα
− cos α
dt

0


dα
dt
dα
− sin α
dt
0

0
0

0 = − ω

 0
0

dα

−
sin
α

dt

dα
− cos α
dt

0


dα
dt
dα
− sin α
dt
0

0
− ω sin α
 

0 = − ω cos α

0
0 

dα
=ω
dt
cos α
cos α
−ω
0
0
0  cos α
0 − sin α
0  0
sin α
cos α
0
ω cos α 0
− ω sin α 0
0
1
t
α (t ) = ∫ ω (τ )dτ + α 0
t0
requiring the knowledge of the initial attitude!
0
0
1
Navigation equations (ERF)
• The navigation equations in an Earth-fixed reference system can
be derived from Coriolis’ theorem:
&x&e = U ie ( &x&i − &x&i0 ) − ( 2 Ω ie x& e + Ω ie Ω ie x e + Ω& ie x e )
=0
=0
centrifugal acceleration
Coriolis’ acceleration
&x&e = U ie &x&i − 2 Ω ie x& e − Ω ie Ω ie x e
Navigation equations (ERF)
i
i
i
&
&
x
=
f
+
g
• Recalling that for Newton’s second law
:
&x&e = U ie ( f i + g i ) − 2Ω ie x& e − Ω ie Ω ie x e = f e + g e − 2Ω ie x& e − Ω ie Ω ie x e
• Typically gravitational acceleration and centrifugal acceleration
are grouped together to form the gravity acceleration vector:
g = g − Ω ie Ω ie x
e
e
e
• Navigation equations result:
&x&e = f e + g e − 2 Ω ie x& e
or
d xe
e
= x&

 dt
 e
 d x& = U e U i f b + g e − 2Ω e x& e
i
b
i
 dt

 i
U& b = U bi Ω ib
Navigation equations (ERF)
• Navigation equations can be written component by component:
0
Ω ie = ω E
 0
− ωE
0
0
− ω E2

Ω ie Ω ie =  0
 0

0
0
0
+ x&2e 


e
− 2Ω ie x& = 2ω E  − x&1e 
 0 


− ω E2
0
0

0
0
 x1e 
 
e
− Ω ie Ω ie x = ω E2  x 2e 
0
 
Coriolis’ acceleration
dx1e
 e
dx 2
dx3e

dx&1e

 e
dx& 2

 e
dx& 3

0
dt = f1e −
[( x )
dt = f 2e −
[( x )
dt = f 3e −
[( x )
e 2
1
e 2
1
e 2
1
GM
e 2
1
e
g =−
GM
x
GM
+ ( x 2e ) 2 + ( x3e ) 2
e 3
]
3/ 2
gravitational acceleration
]
]
x 2e + ω E2 x 2e − 2ω E x&1e
]
x3e
e 2 3/ 2
3
+ (x ) + (x )
e 2
2
e 2
[( x )
x1e + ω E2 x1e + 2ω E x& 2e
e 2 3/ 2
3
+ (x ) + (x )
GM
e 2
2
x
e
e 2 3/ 2
3
+ ( x 2e ) 2 + ( x )
GM
GM
g =−
centrifugal acceleration
dt = x&1e
dt = x& 2e
dt = x& 3e
e
g =
x
e
 x1e 
 e
 x2 
 x3e 
 
Error budget
• Let us consider a very simplified scenario where the Earth is assumed to be
spherical and not rotating, and the vehicle is moving in the surroundings of the
North Pole (with a maximum diameter of some kilometers).
&x&i = f i + g i = f i −
f
i
0
= f +ν
i
GM
x
i 3
i
x
i
&x&i = −
GM
x
i 3
x + f 0 −ν
i
i
i
( )
The dynamics can be split into two terms:
i
&x&i = &~
x& + δ &x&
i
GM ~ i
i
&~
x& = −
x
+
f
0
i 3
~
x
i
linea
ri
i
i +
~
x ~
x
P=
i 2
~
x
zing
δ &x&i = −
GM
(~
x1i ) 2 + ( ~
x 2i ) 2 + ( ~
x3i ) 2
[
]
3/ 2
( I − 3P ) δ x − ν
i
i
Error budget
• In the case under study
~
x1
~ 10 −3
R
2
(~
x1i )
1 ~i ~i
P ≅ 2  x1 x 2
R  ~i
x R
 1
~
~
x2
x3
~ 10 −3
R
~
x1i ~
x 2i ~
x1i R  0
 
i 2
i
~
~
(x2 ) x2 R ≅ 0
~
x 2i R R 2  0

~R
0 0
0 0
0 1
thus obtaining:
δ&x&1i = −ω s2δx1i − ν 1i
 i
2
i
i
δ&x&2 = −ω s δx 2 − ν 2
δ&x&i = 2ω 2δx i − ν i
3
3
s
 1
ωs =
GM
−3
≅
1
.
24
10
rad / s
3
R
Schuler frequency corresponding
to a period T=84.46 minutes
Error budget
• It is possible to study how the
system dynamics evolves as a result
of an impulsive noise by solving the
homogeneous system of equations:
δ&x&1i + ω s2δx1i = 0
 i
2
i
δ&x&2 + ω s δx 2 = 0
δ&x&i − 2ω 2δx i = 0
3
s
 1
• The solution of the first two equations is a harmonic oscillator at the Schuler
frequency:
δx ki (t ) = δx 0 sin(ω s t )
k=1,2
the noise effect on the horizontal components does not tend to damp down
but it is bounded therefore it is controllable in time.
• The solution of the last equations is an inverse harmonic oscillator:
(
δx3i (t ) = δx0 exp 2ω s t
)
after ten minutes, the error is already 3 times larger than the initial error!
“stand-alone” inertial system can be used for the “horizontal” navigation, but
not for the “vertical” one, where the support of other techniques, such as GPS,
is required.