1. Inertial Navigation System Overview – Mechanization Equation
Huy-Tien, Bui
裴輝進
Department of Mechanical Engineering
National Cheng Kung University
Tainan, Taiwan
Fed 28, 2013

2. Outline
Inertial Navigation System Overview

3. Inertial Navigation System Overview
Inertial Navigation System (INS) provides position, velocity and heading information. A INS system is that in which a current state is evaluated by the relative increment from the previous known state.
INS is based on measurements obtained from an Inertial Measurement Unit (IMU).
A typical IMU consists of three accelerometers and three gyroscopes mounted in orthogonal triads.

4. Inertial Navigation System Overview
The measurements from these accelerometers and gyroscopes are combined using mechanization equations.
The principle of inertial navigation comes from Newton’s 2nd law of motion which states that ‘Acceleration is proportional to the resultant force and is in the same directions as this force’. This external force produces an acceleration which is observed by the accelerometers.

5. Inertial Navigation System Overview
This acceleration, when integrated, gives the velocity which if integrated again gives the change in position relative to the initial point. Through appropriate initialization, INS is capable of continuous determination of absolute values of position, velocity and attitude.
However, in an actual system, these integrations are accomplished after appropriate processing of the data which involves rotating the data in the desired frame. This drives the need for gyroscopes in the system.

6. Inertial Navigation System Overview
Integration of angular measurements from gyroscopes provides change in attitude of the system, which when combined by initial attitude provides the new attitude of the system. Attitude is defined by rotations along the three axes which are called roll, pitch and azimuth or yaw. In this work, roll is defined as rotations along the y axis, pitch is rotations along the x axis and azimuth (or yaw) is rotation along the z axis.

7. Inertial Navigation System Overview
Based on the attitude parameters, the rotational relation is established between the IMU body frame and the navigation frame, which is used for rotating the accelerations.
Errors in gyroscope measurements will therefore lead to errors in acceleration and finally position. For example, a gyroscope bias introduces a quadratic error in velocity and a cubic error in position. Therefore, the quality of an IMU and thus an INS is often judged by the quality of gyros contained in the sensor system.

8. Inertial Navigation System Overview
Coordinate Frames
There are four coordinate frames which are generally used when considering inertial navigation systems.
Inertial Frame (i-frame)
i-frame is considered to be non-rotating and non-accelerating frame relative to far-off galaxies. The origin of the inertial frame is arbitrary, and the coordinate axis may point in any three mutually perpendicular directions.

9. Inertial Navigation System Overview
But for practical purposes, an inertial frame is defined as follows
Origin: Earth’s center of mass
Zi –axis: parallel to spin axis of the earth
Xi –axis: pointing towards the mean vernal equinox
Yi –axis: orthogonal to X and Z completing the right hand system

10. Inertial Navigation System Overview
Earth Centered Earth Fixed Frame (ECEF or e- frame) is defined as follows:
Origin: earth’s center of mass
Ze –axis: parallel to mean spin axis of the earth
Xe –axis: pointing towards the meridian of GreenWich
Ye –axis: orthogonal to X and Z completing the right hand system.

11. Inertial Navigation System Overview
Local Level Frame (LLF or l-frame) is defined as follows:
Origin: coinciding with the center of the navigation system
Zl –axis: orthogonal to reference ellipsoid pointing Up
Xl –axis: pointing towards to geodetic East
Yl –axis: pointing towards to geodetic North
This forms a right hand East-North-Up (ENU) frame.

12. Inertial Navigation System Overview
ECEF, LLEF and ENU frames with the latitude () and longitude () angles

13. Inertial Navigation System Overview
Body Frame (b-frame) is an orthogonal frame that represents the orientation of the body to which it is connected. The body frame is assumed to be aligned with the pedestrian frame in such way so that:
Origin: center of the IMU or be body comprising multiple IMUs
Xb –axis: pointing towards the right to the direction of motion
Yb –axis: pointing towards the front (in the direction of motion)
Zb–axis: pointing up to complete the orthogonal right hand system

14. Inertial Navigation System Overview
The data integration is performed in one of above frames. This frame is called a navigation frame. The above mentioned coordinate frames can be transformed from one frame to another using rotation matrices or quaternions.

15. Inertial Navigation System Overview
INS Mechanization Equations
Mechanization equations are set of equations which are used to convert the specific force (fb) and angular velocity (b) measurements obtained from an IMU into position, velocity and attitude information. Users can select any coordinate frame as the navigation frame depending on the application requirements. Such a choice mainly depends on the system designer and/or system requirements.

16. Inertial Navigation System Overview
Mechanization equations in the ECEF frame is given by

17. Inertial Navigation System Overview
INS mechanization in ECEF frame block diagram

18. Inertial Navigation System Overview
The mechanization equation can be solved by the following basic steps
Sensor error correction
Attitude update
Transformation of specific force to navigation frame
Velocity and position calculation

19. Inertial Navigation System Overview
Sensor error correction
The raw measurements obtained from a low cost IMU are generally corrupted by errors such as constant bias, bias drift, scale factor errors and axis non-orthogonal. The values for these errors are estimated from lab-calibration.

20. Inertial Navigation System Overview
One the errors such as bias and scale factor have been estimated, the gyroscopes and accelerometers measurement can be corrected

21. Inertial Navigation System Overview
Attitude update
The body angular rates with respect to ECEF frame