The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
sensors aren’t very accurate and the angular velocity error is 
bigger than the terrestrial velocity rotation. Then for the 
gyroscopes calibration we use a rotation plate. 
The described method (equations 7, 8) takes into account the 
bias, axes misalignment and scale factors. The non 
orthogonality of the sensor axes is given by three misalignment 
angles: given a fixed x axis, the misalignment of y axis is 
obtained by a rotation angle of y axis respect to z axis, and 
by the rotation angles of the z axis 0 ZX and 0 zy , respect to x and 
y axis. 
By using the rotation matrices, and assuming small values for 
the above defined rotation angles, we obtain the formulation of 
the problem: 
gx 1 
1 
0 
o" 
gx 
gx 2 
= 
1 
0 
gy 
(9) 
gx 3. 
-3 
gzy 
1 
_gz_ 
where gx 1? gx 2 , and gx 3 are the components of the acceleration 
along the misaligned axes. 
The equation (9) can be rewritten, taking into account both bias 
and scale factors; we obtain the equation: 
l + s gx 
0 
0 
gx 
= 
- 9gy* 
1 + ^ 
0 
gy 
+ 
b sy 
. 
1 + -V 
_gz_ 
K 
where l gx , Igy, l gz , are the sensor measures, Sg*, Sgy and s^ are the 
scale factors and b gx bgy bgz the bias factors along the axes x, y 
e z of the reference frame. 
By using the abovementioned model, the state equations have 
the form: 
gx 2 + gy 2 + gz 2 - \g\ 2 = 0 (11) 
ax 2 + cay 2 + ox 2 - \cd [ 2 - 0 (12) 
for accelerometers and gyroscopes respectively. 
The unknowns of these equations are the misalignment 
parameters, the scale factors and the bias factors, and can be 
solved by using a least squares procedure. 
The combined system of (11) and (12) uses parameters properly 
weighed (Krakiwsky, 1990) and must be linearized. For writing 
the state equations, the precise determination of the sensor axes 
is not required. To have a number of equation exceeding the 
unknowns number, it is sufficient to position the IMU in 
different orientations, as possible distinct and independent each 
other. 
3. THE STATIC CASE - THE ANALISYS OF THE 
ACCELEROMETER RESULTS WITH KALMAN 
FILTER 
In this section we describe the test performed on an IMU, for 
the determination of bias, scale and misalignment factors of the 
accelerometers. 
First we have analyzed the accelerometers measures of the tri 
axis sensors, obtained in the static case, by using a Kalman 
filter. 
The utilized instrumentation is a MEMS IMU made by Analog 
Devices: the ADIS 16350. 
The ADIS 16350 is a complete triple axis gyroscope and triple 
axis accelerometer inertial sensing system. For the tri-axis 
accelerometer the measurement range is ± 10 g, with 14 bit 
resolution, the axis non-orthogonality is ± 0,25 degrees at 25°C 
and the bias is 0,7 mg at 25 °C, with a 4 mg/°C temperature 
coefficient. 
In the model of static case, with constant acceleration, we can 
write the following state equations: 
*, + i =x,+v,M 
■ V ,+I =v, +a,M (13) 
where: 
x is the position, v is the velocity and a is the acceleration; t and 
t+1 indicate the time, At is the time rate of acquisition and % is 
the model stochastic error. 
For the tri-dimensional case, the matrix formulation is: 
V. = ®,*, + *, ( 14 > 
where x is the unknowns vector with nine components (three of 
position, three of velocity and three of acceleration), O is the 
state matrix having 9x9 dimension, and e is the model 
stochastic error vector with nine components. 
The observation equation is: 
z, =Hx ( +v, (15) 
where z is the vector of the observation of the tri-axial sensor 
with three components (the acceleration measures of the 
sensor), H is the measurements matrix (3x9) and v is the noise 
sensor vector. 
The rate acquisition was 1/200 sec.