1. EE 495 Modern Navigation Systems
Inertial Navigation in the ECI Frame
Wed, Feb 18
EE 495 Modern Navigation Systems

2. EE 495 Modern Navigation Systems
Inertial Navigation in the ECI Frame Background– The Fundamental Problem
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:
Know where we started (initial PVA: 𝑟 ?𝑏 ? , 𝑣 ?𝑏 ? , & 𝐶 𝑏 ? )
Inertial sensors ( 𝜔 𝑖𝑏 𝑏 and 𝑓 𝑖𝑏 𝑏 ) are error free (relax later)
Have a gravity ( 𝑔 𝑏 ? ) and/or gravitational ( 𝛾 ?𝑏 ? ) model
QUESTION: Where am I ? – Current PVA ?
With respect to which frame?
Wed, Feb 18
EE 495 Modern Navigation Systems

3. Inertial Navigation in the ECI Frame Background – Inertial Navigation
The process of “integrating” angular velocity & acceleration to determine one’s position, velocity, and attitude (PVA)
Effectively “dead reckoning”
To measure the acceleration and angular velocity vectors we need at least 3-gyros and 3-accels
Typically configured in an orthogonal triad
The “mechanization” can be performed wrt:
The ECI frame,
The ECEF frame, or
The Nav frame.
Wed, Feb 18
EE 495 Modern Navigation Systems

4. Inertial Navigation in the ECI Frame Background – ISA, IMU, & INS
An Inertial Navigation System (INS)
ISA – Inertial Sensor Assembly
Typically, 3-gyros + 3-accels + basic electronics (power, ADCs, …)
IMU – Inertial Measurement Unit
ISA + Compensation algorithms (i.e., basic processing)
INS – Inertial Navigation System
IMU + gravity model + “mechanization” algorithms
INS
Mechanization Equations
Initialization
Gravity Model
Position
Velocity
Attitude
IMU
Compensation Algorithms
ISA
Accels
Gyros
Raw sensor signals
Wed, Feb 18
EE 495 Modern Navigation Systems

5. Mechanization Equations
Inertial Navigation in the ECI Frame Background – The Mechanization Process
Current IMU Measurements
gyro
accel
Mechanization
Equations
Prior Attitude
Prior Velocity
Prior Position
Gravity / Gravitational Model
Prior PVA
Updated Attitude
Updated Velocity
Updated Position
Updated PVA
Wed, Feb 18
EE 495 Modern Navigation Systems

6. EE 495 Modern Navigation Systems
Inertial Navigation in the ECI Frame Background – A Four Step Mechanization Process
Can be generically performed in four steps:
Attitude Update
Update the prior attitude (rotation matrix) using the current angular velocity measurement ( 𝐶 1 0 = 𝐶 1 0   𝛺 01 1 = 𝛺 01 0 𝐶 1 0  )
Transform the specific force measurement ( 𝑓 𝑖𝑏 ? = 𝐶 𝑏 ? 𝑓 𝑖𝑏 𝑏 )
Typically, using the attitude computed in step 1.
Update the velocity
Essentially, integrate the result from step 2. with the use of a gravity/gravitation model ( 𝑓 𝑖𝑏 = 𝑎 𝑖𝑏 − 𝛾 𝑖𝑏 )
Update the Position
Essentially, integrate the result from step 3.
Wed, Feb 18
EE 495 Modern Navigation Systems

7. EE 495 Modern Navigation Systems
Inertial Navigation in the ECI Frame Background – A Four Step Mechanization
IMU Measurements
Prior Attitude
1. Attitude Update
2. SF Transform
Prior Velocity
3. Velocity Update
Grav Model
Prior PVA
Prior Position
4. Position Update
Updated Attitude
Updated Velocity
Updated Position
Updated PVA
Wed, Feb 18
EE 495 Modern Navigation Systems

8. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
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 𝑖𝑏 𝑖
Velocity: Velocity of the body frame wrt the inertial frame resolved in the inertial frame: 𝑣 𝑖𝑏 𝑖
Attitude: Orientation of the body frame wrt the inertial frame 𝐶 𝑏 𝑖
Wed, Feb 18
EE 495 Modern Navigation Systems

9. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
1. Attitude Update: Method A
Body orientation frame at time “k” wrt time “k-1”
t = Timek – Timek-1
Body Frame at time “k”
Body Frame at time “k-1”
Wed, Feb 18
EE 495 Modern Navigation Systems

10. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
1. Attitude Update: Method B
Body orientation frame at time “k” wrt time “k-1”
t = Timek – Timek-1
Body Frame at time “k”
Body Frame at time “k-1”
Wed, Feb 18
EE 495 Modern Navigation Systems

11. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
1. Attitude Update:
High Fidelity
Lower Fidelity
Wed, Feb 18
EE 495 Modern Navigation Systems

12. Inertial Navigation in the ECI Frame 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
Wed, Feb 18
EE 495 Modern Navigation Systems

13. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
1. Attitude Update
2. SF Transform or
Grav Model
3. Velocity Update
4. Position Update
Wed, Feb 18
EE 495 Modern Navigation Systems

14. Inertial Navigation in the ECI Frame Case 1: ECI Mechanization
In continuous time notation:
Attitude: 𝐶 𝑏 𝑖 = 𝐶 𝑏 𝑖   𝛺 𝑖𝑏 𝑏
Velocity: 𝑣 𝑖𝑏 𝑖 = 𝐶 𝑏 𝑖   𝑓 𝑖𝑏 𝑏 + 𝛾 𝑖𝑏 𝑖
Position: 𝑟 𝑖𝑏 𝑖 = 𝑣 𝑖𝑏 𝑖
Combining into a state-space equation:
Wed, Feb 18
EE 495 Modern Navigation Systems