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Course 556 Details

Course 556: Inertial Systems, Kalman Filtering and GPS / INS Integration. THIS COURSE HAS BEEN REPLACED BY COURSE 557. PLEASE REFER TO COURSE 557 FOR DETAILS  
Instructors: Dr. Alan Pue, Johns Hopkins University, APL and  Mr. Michael Vaujin, Consultant


This course on GNSS-aided navigation will thoroughly immerse the student in the fundamental concepts and practical implementations of the various types of Kalman filters that optimally fuse GPS receiver measurements with a strapdown inertial navigation solution. Course 556 was expanded from the former Course 546 based on student requests for more review material on the first day for for even more content depth and MATLAB computer demonstrations. The course includes the fundamentals of inertial navigation, inertial instrument technologies, technology surveys and trends, integration architectures, practical Kalman filter design techniques, case studies, and illustrative demonstrations using MATLAB.
Note that the morning of Day 1 is a review of concepts and mathematical principles associated with this course and may be skipped by those who already have a sufficient mastery of the concepts.


  • Familiarity with principles of engineering analysis, including matrix algebra and linear systems.
  • A basic understanding of probability, random variables, and stochastic processes.
  • An understanding of the GPS operational principles in Course 356, or equivalent experience.

Who Should Attend?

  • GPS/GNSS professionals who are engineers, scientists, systems analysts, program specialists and others concerned with the integration of inertial sensors and systems.
  • Those needing a working knowledge of Kalman filtering, or those who work in the fields of either navigation or target tracking.

Equipment You Should Bring

  • A laptop (PC or Mac) with full version of MATLAB 5.0 (or later) installed. This will allow you to work the problems in class and do the practice "homework" problems each evening. All of the problems will also be worked in class by the instructor, so this equipment is not required, but is recommended.
  • These course notes are searchable and you can take electronic notes with the Adobe Acrobat 9 Reader we will provide you.

Materials You Will Keep

  • A color electronic copy of all course notes will be provided on a USB Drive or CD-ROM. Bringing a laptop to this class is highly recommended; power access will be provided. 
  • A black and white hard copy of the course notes, printed 3 slides to a page. 
  • Public Venue Attendees: Introduction to Random Signals and Applied Kalman Filtering, 3rd edition, by R. Grover Brown and Patrick Hwang, John Wiley & Sons, Inc., 1996. (Note: This arrangement does not apply to on-site contracts. Any books for on-site group contracts are negotiated on a case by case basis.)

Day 1, Morning

Introduction to INS/GPS Integration
  • Inertial navigation
  • Integration architectures
  • Example applications
Vectors, Matrices, and State Space
  • Vectors and matrices
  • State-space description
  • Examples
Random Processes
  • Random variables
  • Covariance matrices
  • Random process descriptions

Day 1, Afternoon

Kalman Filter
  • Filtering principles
  • Least squares estimation
  • Kalman filter derivation
Filter Implementation
  • Filter processing example
  • Off-line analysis
  • Filter tuning
Navigation Coordinate Systems
  • Earth model
  • Navigation coordinates
  • Earth relative kinematics

Day 2, Morning

Inertial Navigation Mechanization
  • Gravity model
  • Navigation equations
  • Implementation options
Inertial Sensor Technologies
  • Accelerometer technologies
  • Optical gyros
  • MEMS technologies
  • Technology survey
Strapdown Systems
  • Quaternions
  • Orientation vector
  • Coning and sculling compensation

Day 2, Afternoon

Navigation System Errors
  • Tilt angle definitions
  • Navigation error dynamics
  • Simplified error characteristics
System Initialization
  • INS static alignment
  • Transfer alignment
  • Simplified error analysis
Loosely-Coupled INS/GPS Design
  • Measurement processing
  • Filter design and tuning
  • Navigation systems update

Day 3, Morning

INS Aiding of Receiver Tracking
  • Code and carrier tracking
  • Track look deign trades
  • Interference suppression
  • Autocorrelation, power spectral density
  • Deep integration
Tightly coupled INS/GPS Design
  • Measurement processing
  • Filter parameter selection
  • Pseudo-range and delta pseudo-range measurement models
Multi-Sensor Integration
  • Terrain aiding and relative GPS
  • Carrier phase differential integration
  • GPS interferometer/INS integration

Day 3, Afternoon

Building Extended Kalman Filters
  • Linearized and extended Kalman filters
  • Radar tracking of a vertical body motion with non-linear dynamics
  • Radar tracking of an  accelerating body with non-linear measurements
Implementation Considerations
  • Covariance matrix numerical stability
  • Sequential vs batch measurement processing
  • Measurement correlation and de-correlation
  • Matrix partitioning for computational efficiency
Aided Psi-Angle Navigator
  • Description and demonstration of a loosely coupled 15-state Psi-angle wander azimuth navigator flying an aircraft type trajectory
  • Homework problems

Day 4, Morning

Partials of Measurement Equations
  • Techniques for taking partials
  • Psi-angle and Phi-angle feedback to strapdown
  • Homework solutions
  • Integrated true velocity error
Initialization and Process Noise
  • Strapdown and covariance matrix initialization
  • Process noise for vertical deflections/anomaly & velocity/attitude random walk
  • Typical sensor error models, random constant, random walk & first order Gauss Markov
  • Process noise for un-modeled states
Aided Phi-Angle Navigator
  • Description and demonstration of a Phi-angle north-slaved navigator modeling position error as latitude/longitude error
  • Modeling position error as latitude/longitude error
  • Comparison of state dynamics matrix elements
  • Modeling position error as tilt error
  • Navigation in ECEF coordinates

Day 4, Afternoon

Measurement Editing and Adaptive Filters
  • Online and offline residual analysis
  • Advanced methods of outlier detection and rejection
  • Multiple model adaptive estimation
  • Application to carrier phase integer ambiguity resolution
Methods of Smoothing
  • Optimal prediction and fixed Interval smoothing
  • Fixed point and fixed lag smoothing
  • Application to navigation
Square Root Filtering
  • Square root covariance filtering and smoothing
  • Information filter derivation
  • Square root information filters
  • UD factorization and UD filtering

Day 5, Morning

Suboptimal Covariance Analysis
  • Effects of mis-modeling errors
  • Optimal and sub-optimal (two pass) covariance analysis
  • Error budget and reduced state analysis
Unscented Kalman Filters
  • Limitations of EKFs
  • Sigma points and the unscented transform
  • Augmentation and application to navigation
  • Performance vs EKF
Particle Filtering and Miscellaneous Topics
  • Small azimuth static alignment and leveling
  • Large azimuth static alignment
  • Gyro-compassing and magnetometer aiding
  • Attitude matching and boresight error states
  • Particle filter theory
  • Terrain matching and multi-model position solutions
  • Performance vs UKF and EKF

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