# Mathematical Introduction to Integrated Navigation Systems, with Applications

#### Synopsis:

The subject of Integrated Navigation Systems is presented. Integrated Navigation Systems is the combination of an on-board navigation system solution for: position, velocity and attitude as derived from accelerometer and/or gyro inertial sensors, and navigation aids providing independent/redundant data to update or correct this on-board navigation solution. In this course, and described in the accompanying textbook, this combination is accomplished with the use of the Kalman filter algorithm.

This course is segmented into two parts. In the first part, elements of the basic mathematics, kinematics, equations describing navigation systems and their error models, aides to navigation, and Kalman filtering are reviewed. Detailed derivations are provided. The accompanying textbook provides exercises to expand the application of the materials presented.

Applications of the course material, presented in the first part, are presented in the second part for actual Integrated Navigation Systems. Examples of these systems are implemented in the MATLAB/Simulink ™ commercial product, and are provided for a hands-on experience in the use of the mathematical techniques developed

#### Key Topics:

• Coordinate Systems - How we relate information so that others can use it.
• Navigation Equations - Position, velocity and attitude onboard solution.
• Kalman Filtering - Optimal combination of navigation solution and aiding data.
• Applications - Calibration, Alignment, Integrated INS/GPS…
• MATLAB/Simulink examples in textbook’s second edition

#### Who Should Attend:

This course is designed for those directly involved with the design, integration, and test and evaluation of Integrated Navigation Systems. It is assumed that the attendees have a background in mathematics including calculus.

#### Course Information:

Type of Course: Instructor-Led Short Course
Course Level: Intermediate
Course Length: 2 days
AIAA CEU's available: Yes
Outline

1. Heading reference - magnetic needle, magnetic compass, …
2. Speed/distance traveled - velocity log, air speed, odometer, …
B. Inertial Navigation - current sensor technology
1. Accelerometer - …, MEMS,
2. Gyro - RLG, Fiber Optic,…, MEMS.
1. Kalman Filtering - recursive estimator,
2. Navigation Aides - Doppler, GPS, Line-of-Sight, …

II. Coordinate Systems - How we relate information so that others can use it.
A. Earth-Centered Earth-Fixed,
B. Local Level - geographic, Wander Azimuth,
C. Line-of-Sight - range, azimuth, elevation.

III. Navigation Equations - Position, velocity and attitude data for onboard use.
A. Terrestrial, Space Referenced Inertial, …
B. Linearization for Kalman filter implementation - importance of error model*.

A. Doppler - Radar, acoustic, odometer, …, speed/distance traveled,
B. GPS - orbital position, psuedo range, delta range, deterministic solution,
C. Line-of-Sight - azimuth/elevation recursive least squares.

V. Kalman Filtering - Optimal combination of navigation and aiding data.
A. Recursive Weighted Least Squares Estimator - starting from least-squares,
B. Minimum Variance - starting with and assumed linear estimator form,
C. U-D Factored Form - enhanced numerical precision,
D. Combination of Two (or more) Kalman Filters' Estimates - multiple systems.

VI. Applications - calibration, alignment, INS/GPS, …
A. Laboratory Sensor Calibration*,
B. Alignment/Initialization - ground*, in-motion, …
C. Integrated Navigation: INS/GPS*, Attitude Determination/ Estimation*,…