The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
291 
knowledge as planes, slopes. Some factors of there unknown 
planes are estimated together with the calibration parameters. 
The parameters of a plane j are described as 
S;=\S> 
(2) 
where S lf S 2 and 5 3 are the direction cosines of the plane's 
normal vector and S 4 is the negative orthogonal distance 
between the plane and the coordinate system origin. The 
observation equation for an point i expressed by its 
coordinates x , » T,, z t lying on plane j has the form 
SljX i +S 2j y l +S 3 jZ i +S 4 j=0 
(3) 
Figure 3. Navigator position RMS for X, Y, and Z direction 
Note that the direction cosines must satisfy the following unit 
length constraint 
3. ERROR REVOERY MODEL 
3.1 Geo-Referencing of LiDAR Measurements 
As already mentioned, it is necessary to compute the laser point 
geo-referencing that represent mathematical models. The 
geo-referencing of the laser points is viewed as a function of 
the observation from the above parameters estimated. The 
LiDAR geo-referencing equation in a local reference frame can 
be given in eq. 1: 
(1) 
Xi 
m 
X 
m / 
Ax 
■°T 
y t 
= 
Y 
+c 
Ay 
+ R m Rs 
m 
0 
3. 
Z 
Az 
Ip]) 
where: 
x r y r z i 
X, Y,Z 
=the laser footprint position in the mapping 
frame 
=position of the phase center of GPS antenna 
in the mapping frame 
(4) 
Combining equation (1) with equation (2) consist of the form 
that constraint conditions and laser points position as a function 
of the systematic errors: 
F(0,X) = 0 (5) 
where, O is the observations, X is the systematic errors. 
The geo-referencing of the laser points in the laser coordination 
system is viewed as a function of the GPS, INS, range, 
scan-angle, and the systematic biases. In section 2, the 
systematic biases are selected from bore-sight angles, ranging 
biases, and scanning angle biases. After adding the systematic 
errors to the equation (1), the geo-referencing of the laser 
points is changed by the following form: 
Ax, Ay, Az =ever arm vector from the phase center of 
X 
m 
X 
m / 
Ax 
GPS antenna to laser scanner center 
y 
= 
Y 
+RL 
Ay 
R" mu =the rotate matrix between the IMU frame 
z 
Z 
Az 
+ AR m R„ARR, 
(6) 
and the mapping frame described by roll, pitch 
and yaw observation 
=a priori known rotation matrix from the IMU 
frame to the LS coordinate frame that depends 
on the mounting situation. 
=laser scanner rotation 
cos# -sin# 
sin# cos# 
y =the LiDAR encoder angular value 
p =the LiDAR range at time t 
3.2 Recovery Function Model 
The following model is based on constraining the target objects 
to the surface extracted from the laser points and known 
AR„ = 
0),(p,K 
=the range bias 
=rotation matrix with alignment error a 
defined in section 2 
=bore-sight rotation matrix 
=bore-sight angles 
Since the equation (6) is non-linear and each laser point 
position is represented by more than one observation, the 
adjustment model must be used. Substituting the equation (6) to