The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
878 
where: 
¿Po ~ position error at time to 
dv 0 = velocity error at time to 
At = t -1 0 = total elapsed time 
5boa = residual accelerometer bias at time to (uncompensated) 
Sbog = residual gyro bias at time to (uncompensated) 
S(Xq = Horizontal misalignment at time to 
SHoz = Azimuth misalignment multiplied by the approximate 
distance VAt 
SFoa and SFog = accelerometer and gyroscope residual scale 
factor errors 
F = sensed acceleration 
g = gravity acceleration (~ 9,81 m/sec 2 ) 
The equations above show the importance of bias, scale-factor 
and non-orthogonality errors. 
The appropriate use of the Kalman filter and the introduction of 
these parameters like additional states of a filtering algorithm, 
produce their valuation and the correction of velocity and 
position drift. But the filter may converge very slow, and may 
even diverge if inappropriate starting values are given. 
For MEMSs sensors, the bias, scale factor and misalignment 
instabilities are much large and it is important an appropriate 
procedure for the correct evaluation of the initial parameters of 
these factors. 
2. INS CALIBRATION METHODS AND THE MULTI 
POSITION CALIBRATION METHOD 
The calibration of the IMU is the process of comparing the 
instruments outputs with known reference information and the 
determination of the coefficients in the output equation, that 
agree the reference information (Chatfield, 1997). 
For the tactical grade IMUs the process of calibration requires 
the inertial system installed on a very levelled platform and the 
IMU oriented with one of the three axes x, y and z 
perpendicular to the levelled plane, first in up and then in down 
position (Titterton and Weston, 2004). 
These six positions allow the bias and scale factors 
determination by solving the equations: 
For the determination of the misalignment of the sensor axes, 
we need to evaluate, at the same time, all error factors by using 
the misalignment matrix. This matrix contains in the diagonal 
terms the bias and scale factors too (Niu, 2002). 
The system, in the case of the accelerometers, has the form: 
L 
m xx 
m 
xy 
m xz 
a x 
K 
U 
= 
m y X 
m 
yy 
m yz 
a y 
+ 
K 
(6) 
L. 
m z x 
m 
zy 
™zz_ 
_a z _ 
K 
where the diagonal elements m xx , niyy, m 2z , are the scale factors, 
the non-diagonal elements are the non-orthogonality factors and 
the b elements are the bias factors. 
The six positions of the IMU on the levelled plate produce 18 
equations (6x3) and allow, by the mean square procedure, the 
bias, scale and axis misalignment factors determination. 
This calibration procedure is very delicate and expensive and 
also very difficult to realise with an IMU made with MEMS 
sensors. 
Then they have been studied and proposed calibration 
procedures, that don’t require the precise alignment of IMU 
with given directions, and a perfectly levelled positioning plate. 
An effective system is the multi-position calibration method 
(Shin and El Sheimy, 2002), based on 18 different and 
independent positions of the sensors. The sensors are collocated 
on a locally levelled plane. The axes of the reference sensors 
and the reference levelled plan form three angles characterized 
by directors cosines a, P e y. 
The acceleration components on the group of three axis of the 
sensors, in the static case, are: 
gx = g cosa 
gy= g cos p (7) 
gz = g cosy 
b = 
The equations of the calibration model are: 
(4) 
gx 2 + gy 2 + gz 2 = I g\ 2 (cosci 2 + cosi? +COSJ?) = \g\ 2 (8) 
1? + if""-lxk 
S = -X L 
2 xk 
This model is valid for the calibration of the accelerometers; we 
can use the known terrestrial rotation velocity for the 
gyroscopes calibration equations. 
where , /^ OH ” are the sensor measures in the up and down 
positions, while K is the known reference value, equal, in the 
accelerometers case, to earth gravity, and in the gyroscopes 
case, to earth rotation vector. 
For the MEMS sensors the bias, scale and misalignment factors 
are very bigger than the same factors of the tactical grade 
sensors. Thus it is more effective the use of the modified multi 
position calibration method (Syed, et al. 2007). 
This method needs only the approssimate sensor positions, and 
we don’t need to know exactly the sensor attitude, respect to 
the three axes of the reference plane. Besides, the MEMS