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

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

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

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

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

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

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

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

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

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

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

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

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

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