EE 570: Location and Navigation: Theory & Practice
Navigation Sensors and INS Mechanization
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 1 of 14
Navigation Sensors and INS Mechanization Navigation Equations – The Fundamental Problem
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• The Fundamental Inertial Navigation Problem:
Using inertial sensors (accels & gyros) and an initial position and orientation, determine the vehicle’s (i.e. body frame) current position, velocity, and attitude (PVA)
Assumptions:
1. Know where we started (initial PVA: 𝑟 ?𝑏? , 𝑣 ?𝑏
? , & 𝐶𝑏? )
2. Inertial sensors (𝜔𝑖𝑏𝑏 and 𝑓 𝑖𝑏
𝑏) are error free (relax later)
3. Have a gravity (𝑔 𝑏? ) and/or gravitational (𝛾 ?𝑏
? ) model
Where am I ? – Current PVA ?
o With respect to which frame?
Slide 2 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Inertial Navigation
• Inertial Navigation The process of “integrating” angular velocity & acceleration
to determine one’s position, velocity, and attitude (PVA) o Effectively “dead reckoning”
To measure the acceleration and angular velocity vectors we need at least 3-gyros and 3-accels o Typically configured in an orthogonal
triad
The “mechanization” can be performed wrt: o The ECI frame,
o The ECEF frame, or
o The Nav frame.
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 3 of 14
Navigation Sensors and INS Mechanization Navigation Equations – ISA, IMU, & INS
• An Inertial Navigation System (INS)
ISA – Inertial Sensor Assembly
o Typically, 3-gyros + 3-accels + basic electronics (power, …)
IMU – Inertial Measurement Unit
o ISA + Compensation algorithms (i.e. basic processing)
INS – Inertial Navigation System
o IMU + gravity model + “mechanization” algorithms
IMU
- Basic Processing
ISA Comp Algs
Raw sensor outputs
b
b
if
b
b
i
Mechanization Equations
INS
Initialization
Position, Velocity,
and Attitude (PVA)
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
Gravity Model
Slide 4 of 14
Mechanization Equations
Navigation Sensors and INS Mechanization Navigation Equations – Mechanization Process
Gravity / Gravitational
Model
b
ib b
ibf
Prior Attitude
Prior Velocity
Prior Position
Updated Attitude
Updated Velocity
Updated Position
Pri
or
PV
A
Updated PVA
IMU Measurements
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 5 of 14
Navigation Sensors and INS Mechanization Navigation Equations – A Four Step Mechanization
• Can be generically performed in four steps:
1. Attitude Update
o Update the prior attitude (rotation matrix) using the current
angular velocity measurement (𝐶 10 = 𝐶1
0 𝛺011 = 𝛺01
0 𝐶10 )
2. Transform the specific force measurement (𝑓 𝑖𝑏? = 𝐶𝑏
?𝑓 𝑖𝑏𝑏 )
o Typically, using the attitude computed in step 1.
3. Update the velocity
o Essentially integrate the result from step 2. with the use of
a gravity/gravitation model (𝑓 𝑖𝑏 = 𝑎 𝑖𝑏 − 𝛾 𝑖𝑏)
4. Update the Position
o Essentially integrate the result from step 3.
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 6 of 14
Navigation Sensors and INS Mechanization Navigation Equations – A Four Step Mechanization
2. SF Transform
b
ib
Prior Attitude
b
ibf
3. Position Update
Grav
Model Prior Velocity
Prior Position
3. Velocity Update
1. Attitude Update
Updated Attitude
Updated Velocity
Updated Position
Pri
or
PV
A
Updated PVA
IMU Measurements
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 7 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• CASE 1: ECI Frame Mechanization
Determine the Position, Velocity, and Attitude of the Body frame with respect to the Inertial Frame
• Determine our PVA wrt the ECI frame
Position: Vector from the origin of the inertial frame to the
origin of the body frame resolved in the inertial frame: 𝑟 𝑖𝑏𝑖
Velocity: Velocity of the body frame wrt the inertial frame
resolved in the inertial frame: 𝑣 𝑖𝑏𝑖
Attitude: Orientation of the body frame wrt the inertial
frame 𝐶𝑏𝑖
Slide 8 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
1. Attitude Update: Method A Body orientation frame at time “k” wrt time “k-1”
o t = Timek – Timek-1
( 1)b kx
( 1)b ky
( 1)b kz
( )b kx
( )b ky
( )b kz
Body Frame at time “k-1”
Body Frame at time “k”
i i b
b b ibC C
b
ib
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
lim ( ) ( 1)
0
i ii b bb
C k C kC
t t
( 1)i b
b ibC k
( ) ( ) ( )i i i b
b b b ibC C C t
( ) ( )i i b
b b ibC C I t
Slide 9 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
1. Attitude Update: Method B
Body orientation frame at time “k” wrt time “k-1”
o t = Timek – Timek-1
( 1)b kx
( 1)b ky
( 1)b kz
( )b kx
( )b ky
( )b kz
Body Frame at time “k-1”
Body Frame at time “k”
( 1)
( ) ( 1) ( )
i i b k
b k b k b kC C C
( 1)
( )
bib tb k
b kC e
( ) ( )bib ti i
b bC C e
b
ib
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
( )i b
b ibC I t
3 2
3 1
2 1
ˆ ˆ0
ˆ ˆ0
ˆ ˆ 0
ˆ( , )
k k
k k
k k
kR e e
K
2 2 3 3
2! 3!I
K KK
2sin( ) 1 cos( )I K K
e K
ˆb
ib t k
Slide 10 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
1. Attitude Update:
High Fidelity
Lower Fidelity
2( ) ( ) sin( ) 1 cos( )i i
b bC C I K K
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
( ) ( )i i b
b b ibC C I t
ˆb
ib t k
Slide 11 of 14
ˆSk kK
b b
ib ibSk
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
2. Specific Force Transformation
Simply coordinatize the specific force
3. Velocity Update
Assuming that we are in space (i.e. no centrifugal component)
Thus, by simple numerical integration
4. Position Update
By simple numerical integration
( )i i
ib i
b
b bf C f
a ib i
i i
ib
i
bf
( ) ( ) aib ib
i
ib
i iv v t
2
( ) ( ) ( ) a2
ib ib ib i
i
b
i i i tr r v t
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 12 of 14
aib i
i i
ib
i
bf
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
b
ib
1. Attitude Update
( )i
bC
( )i
bC
Grav
Model
3. Velocity Update
i
ib( )ib
iv
( )ib
iv
3. Position Update
( )ib
ir
( )ib
ir
2. SF Transform
b
ibf
i
ibf
( ) ( )i i b
b b ibC C I t
( )i i
ib i
b
b bf C f
( ) ( ) aib ib
i
ib
i iv v t
2
( ) ( ) ( ) a2
ib ib ib i
i
b
i i i tr r v t
a ib i
i i
ib
i
bf
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice Slide 13 of 14
Navigation Sensors and INS Mechanization Navigation Equations – Case 1: ECI Mechanization
Tuesday 26 Feb 2013 NMT EE 570: Location and Navigation: Theory & Practice
• In continuous time notation:
Attitude: 𝐶 𝑏𝑖 = 𝐶𝑏
𝑖 𝛺𝑖𝑏𝑏
Velocity: 𝑣 𝑖𝑏𝑖 = 𝐶𝑏
𝑖 𝑓 𝑖𝑏𝑏 + 𝛾 𝑖𝑏
𝑖
Position: 𝑟 𝑖𝑏𝑖 = 𝑣 𝑖𝑏
𝑖
• Combining into a state-space equation:
a ib i
i i
ib
i
bf
i i b
bib ibf C f
ii
i i b i
b
i i b
b
ibib
ib ib ib
b ib
vr
v C f
C C
ib
iv
Slide 14 of 14