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ANALYSIS OF THE KALMAN FILTER WITH DIFFERENT INS ERROR MODELS 
FOR GPS/INS INTEGRATION IN AERIAL REMOTE SENSING APPLICATIONS 
Hongxing Sun a , Jianhong Fu a , Xiuxiao Yuan 3 , Weiming Tang b 
a School of Remote Sensing & Info. Eng., Wuhan University, P. R. China 430079 
b GNSS Eng. Technology Research Center, Wuhan University, P. R. China 430079 
Commission V, ICWG V/I 
KEY WORDS: Direct Georeferencing, Aerial Triangulation, GPS/INS Integration, Kalman Filter 
ABSTRACT: 
In the Kalman filter used for the integration of GPS/INS, the inertial sensor error model is usually considered as a random constant or 
random walk for both gyroscopes and accelerometers. However, the Inertial Measurement Unit (IMU) used in aerial remote sensing 
applications for sensor positioning and orientation is typically of tactical grade, i.e., the gyroscope drifts are on the order of 0.1 deg/h 
and the accelerometer biases are 1 OOug respectively. In this case, there is the room to improve the system performance by developing 
more complicated error models for the inertial sensors. In this paper, 6-state, 12-state and 15-state error models for the inertial 
sensors are implemented, and their performance of each in the Kalman filter is compared and analyzed. Firstly, the commonly used 
6-state error model that includes three random walks for gyroscopes and three random walks for accelerometers is implemented. 
Then, a 12-state error model is formed by augmenting the 6-state model with three scale factors for the gyroscopes and three scale 
factors for the accelerometers. Thirdly, three first-order Markov procedures are considered for the gyroscopes in addition to the 
random walks and scale factors, thus resulting in a 15-state error model. Aerial GPS/INS data collected in the field with a tactical 
grade IMU and dual frequency GPS receivers is processed with these three error models. In the data processing, the loosely-coupled 
Kalman filter, which is the common coupling method for the aerial GPS/INS integration, is used. The 12-state and 15-state error 
models show obvious advantages over the 6-state error model in the test results. The accuracies of the integrated position (5cm), 
velocity (3cm/s) and attitude (0.002 degree for pitch and roll, 0.008 degree for heading) in the 12-state model are all better than that 
of the 6-state error model. However, the improvement of the 15-state error model relative to the 12-state error model is limited and 
insignificant. 
1. INTRODUCTION 
Direct georeferencing, also referred to as direct platform 
orientation (DPO), is defined as direct measurement of the 
imaging sensor external orientation parameters (EOP), using 
positioning and orientation sensors, typically the Global 
Positioning System (GPS) and Inertial Navigation System (INS) 
or Inertial Measurement Unit (IMU). With the increasing use of 
multi-sensor mapping, the DPO of the integrated GPS/IMU 
systems has become a crucial component of spatial data 
processing algorithms, and substantial research effort has been 
devoted to extensive algorithmic developments, performance 
analysis and practical implementations of GPS/IMU systems 
(Skaloud et al, 1996; Abdullah, 1997; Grejner-Brzezinska, 
1997; Toth and Grejner-Brzezinska, 1998; Grejner-Brzezinska, 
1999; Grejner-Brzezinska, 2001; Mostafa et al., 2001; Cramer 
et al., 2000; Cramer, 2001). However, investigation of the 
GPS/INS integration itself, especially for the inertial sensor 
error model, is not focused on as much. Grejner-Brzezinska et 
al., (2005) attempted to improve the performance of GPS/IMU 
integration by using a precise gravity model, signal de-noising 
and parameter refinement of the inertial sensor stochastic model, 
nevertheless the sensor stochastic model was still of 12 states. 
The commonly used IMU sensor stochastic model in the 
Kalman Filter (KF) supports 6 states (i.e., gyroscope drift and 
accelerometer bias) to 12 states (for which the scale factors of 
both of gyroscope and accelerometer are also included) (Cramer, 
2001; Grejner-Brzezinska et al., 2005). In aerial 
photogrammetric mapping or remote sensing, the IMU 
hardware is typically classified as high-end tactical grade sensor, 
i.e., the gyroscope drifts are on the order of 0.1 deg/h and the 
accelerometer biases are lOOug respectively. In this case, there 
is the room to improve the system performance by developing 
more complicated error models for the inertial sensors. In this 
paper, the 6-state, 12-state and 15-state inertial sensor error 
models are implemented, and the KF performance of each is 
compared and analyzed. 
2. STOCHASTIC ERROR MODEL OF THE INERTIAL 
SENEORS 
The performance characteristic of a gyroscope (or 
accelerometer) is determined by the dynamic model, which 
involves a scale factor, bias and random, random environmental 
sensitivity and misalignment (IEEE std. 952-1997). The 
situation is similar for accelerometers (IEEE std. 1293-1998). 
The environmental sensitivity and misalignment are generally 
ignored in the stochastic error model, so the focus in this paper 
is mainly on the first two items. The scale factor of the sensor is 
calibrated by the manufacturers in the factory before the sale. 
But post-factory calibration of the instrument can still influence 
the navigation performance significantly, therefore it can also 
be considered in the stochastic error model. The random 
component of the gyroscope and accelerometer data mainly 
include: (a) the gyro rate ramp (trend) defined as a gyro 
behavior characterized by quadratic growth within a certain 
range of time, (b) gyro rate (acceleration) random walk due to 
white noise in the angular acceleration (jerk) which is defined as 
the drift rate error (acceleration) build-up with time, (c) flicker 
noise (bias instability), defined as a random variation in bias, 
computed over a specific finite sample time and averaging time 
interval; (d) angle (velocity) random walk due to the white noise 
of gyroscope angular rate (acceleration) data, (e) quantization 
noise, defined as a random variation in the digitized output