A Low-Cost GPS Aided Inertial Navigation
System for Vehicular Applications
ISAAC SKOG
Master of Science Thesis
Stockholm, Sweden 2005-03-09
IR-SB-EX-0506
1
Abstract
In this report an approach for integration between GPS and inertial navigation
systems (INS) is described. The continuous-time navigation and error equations
for an earth-centered earth-fixed INS system are derived. Using zero order hold
sampling, the set of equations is discretized. An extended Kalman filter for
closed loop integration between the GPS and INS is derived. The filter propa-
gates and estimates the error states, which are fed back to the INS for correction
of the internal navigation states. The integration algorithm is implemented on
a host PC, which receives the GPS and inertial measurements via the serial port
from a tailor made hardware platform, which is briefly discussed. Simulation
results of the system are presented.
Contents
1 Introduction 4
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Thesis Outline and Contributes . . . . . . . . . . . . . . . . . . . 6
2 Coordinate Systems 8
2.1 Coordinate Frame Definition . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Earth-centered earth-fixed frame (ECEF, e-frame) . . . . 8
2.1.2 Local geodetic frame (t-frame) . . . . . . . . . . . . . . . 8
2.1.3 Inertial frame (i-frame) . . . . . . . . . . . . . . . . . . . 10
2.1.4 Body frame (b-frame) . . . . . . . . . . . . . . . . . . . . 10
2.2 Coordinate Frame Transformation . . . . . . . . . . . . . . . . . 10
2.2.1 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Plane Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Small angle rotation . . . . . . . . . . . . . . . . . . . . . 14
2.4 Rotating Coordinate Frame . . . . . . . . . . . . . . . . . . . . . 16
3 Inertial Navigation Equation 18
3.1 Fundamental Equations of Inertial Navigation . . . . . . . . . . . 18
3.1.1 Gravity force . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Inertial force . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . 19
3.2 Navigation Equations . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 General Navigation Equations . . . . . . . . . . . . . . . . 20
3.2.2 ECEF-frame navigation equations . . . . . . . . . . . . . 21
3.3 ECEF Navigation Error Model . . . . . . . . . . . . . . . . . . . 22
3.3.1 ECEF navigation error equations . . . . . . . . . . . . . . 22
4 Discretization 27
4.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 Discrete time navigation equations . . . . . . . . . . . . . 27
4.1.2 Discrete time error equations . . . . . . . . . . . . . . . . 29
5 Integration of GPS and INS 30
5.1 The Extend Kalman Filter . . . . . . . . . . . . . . . . . . . . . . 30
CONTENTS 3
6 Hardware 33
6.1 The Inertial Measurement Unit . . . . . . . . . . . . . . . . . . . 33
6.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 Simulation Results 35
7.1 Driving Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Conclusions and Further Work 39
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A The Kalman Filter 42
A.1 Derivation of the Kalman Filter Equations . . . . . . . . . . . . . 42
