159 
P(* +11 k +1) = [I - Uk +1) • H] P(* +11 k) ■... 
[l-L(t + l) H] + L(k + 1) ■ R ■ L(k +1)' 
(16) 
where in (11) S(k + 1 I k) is the new estimate at step 
(k+1), as derived by update S(k I k) at time step k and 
discretized function f k (•) (see 4.1), obtained from 
nonlinear relationship (6) without error w(t); in (12) 
P(fc + 1 I k) is the covariance matrix of a priori predi 
ction error, O(klk) is the fundamental matrix related to 
linear system =F(s(i))-<^ and Q(t) is the cova 
riance matrix of discretized error w(t), both reported in 
subsection 4.2; in (14) L(k+1) is the gain matrix of the 
filter; in (15) S(k + 1 I k + 1) is the new update at time 
step (k+1); finally in (16) Y*(k + 1 I k + 1) is the a 
posteriori estimate error covariance. 
4.1 PREDICTOR DISCRETIZATION. 
In the prediction step, unlike the EKF theory, which 
states to solve the differential equation 
s(i)=f(s(i)) s(t k ) = s(k\k) (17) 
to compute the predictor 
s(k + \ \ k) = s(t k+] ) (18) 
we employ the discrete scheme below, which yields the 
discretized form in (11) of the predictor itself: 
s(k +1) = fj (s(k)) = s(k) + (t k+] -t k )-... 
(t -t ) 2 • ( 19 ) 
•f(s(^))+ u+, 2 
4.2 CALCULATION OF O(klk) AND Q(t). 
If we define the Jacobian matrix 
(20) 
J=s(t) 
for t=tk, assuming s(t k ) = s(k I k) , then F(klk)= 
F(S(k I k) ) and performing a discretization we obtain 
O(k I k) = exp{(^ +1 -t k )-F(k I k)} (21) 
As regards Q(t), the covariance matrix of error w(t) 
assumes following form: 
Q(*) = (i* +1 -t k )-Q>(k I k)-Q-®(k I k)' (22) 
5. COORDINATES TRANSFORMATION. 
Our main purpose for realizing such a mobile mapping 
system consists to carry out tipically road surveys on 
behalf of various italian public administrations or 
private agencies which are interested in road data 
collection for GIS applications. To this aim in most 
cases we will have to deal with cadastral maps that in 
Italy involve two different cartographic projections: 
Gauss-Boaga and Cassini-Soldner. The first is based on 
Hayford ellypsoid, oriented in Roma-Monte Mario, 
while the second refers to Bessel ellypsoid oriented in 
Genova: these are the two national geodetic datums, 
commonly used for mapping applications. Therefore 
prior to use the corrected GPS positions, as measure 
ments input for the Kalman filter, we perform a geode 
tic datum transformation between WGS-84 and one of 
our national datums, employing the Wingeo software 
developed on the basis of algorithms proposed by [1]. 
6. TEST AND RESULTS. 
In order to test the accuracy of the positioning module, 
i.e. the goodness of our kinematic model so as the 
effectiveness of our implementation of Kalman filter, 
we have simulated a survey employing random gene 
rated true position data with Gaussian noise with the 
same variance error as our GPS receiver (single fre 
quency Trimble Pro-XR) according to its technical data 
sheet (50 cm RMS, adopting Everest 1 M multipath 
rejection technology). 
Comparing these true positions with that computed by 
the filter, 86% of angle estimates move less than 2 
degree away from true value, while 50% present a 
discrepancy limited to 1 degree (Fig. 8). 
On the basis of these results and positioning accuracy 
of our receivers, as guaranteed by Trimble, we could 
conclude that our NavPos software module provides a 
positional error about 30-35 cm for object at distance 
of 10 m from the vehicle. 
Bearing error 
Fig. 8: Simulation results on bearing error 
F(s(i)) = 
ds