Mathematical Introduction to Integrated Navigation Systems, with Applications


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:

• Navigation Overview - From Dead-Reckoning to Inertial Navigation.
• Coordinate Systems - How we relate information so that others can use it.
• Navigation Equations - Position, velocity and attitude onboard solution.
• Navigation Aides - Redundant information to correct navigation 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

I. Navigation Overview - From Dead-Reckoning to Inertial Navigation.
A. Dead-Reckoning - early sensors
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.
C. Integrated Navigation Systems
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*.

IV. Navigation Aides - Redundant information to correct navigation data.
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*,…

*MATLAB/Simulink examples.


Students will have the opportunity to purchase the recommended textbook “Applied Mathematics in Integrated Navigation Systems”, authored by Robert M. Rogers. The textbook includes software with examples implemented in MATLAB/Simulink ™. This commercially available product provides a graphical user interface that promotes the visualization of 1) the information flow between subsystems and functional elements of a navigation system, and 2) the outputs at various stages in the computational process including the outputs of the navigation filter. These software examples provide the student a direct hands-on experience in the design and evaluation of Kalman filters.

To realize the full benefit of the course, attendees should have a laptop computer and release 6.0 or later version of MATLAB, including Simulink, software.


Robert M. Rogers received B.S., M.S., and PhD degrees in Aerospace Engineering from the University of Florida and is an Associate Fellow of the American Institute of Aeronautics and Astronautics. As sole proprietor of Rogers Engineering & Associates, Dr. Rogers performs research, prototype development, systems analysis, test and evaluation for DoD and NASA in guidance, navigation and control aircraft, helicopters, sea-craft and land vehicles.


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