So the linearized observable equation can be developed as 
4-4-5 
Curvature in 1/km 
v(x,) = K 2 (y 0 )-da + db + a a K[(y a )-x r dm b - 
~ VU-Vo)du b - (x,)-a 0 K 2 (y 0 ) + b 0 ; 
For each iteration start solutions have to be developed for the 
linearization point y 0 and for the curvature K 2 (y 0 ). Unfortu 
nately the simple interpolation of discrete curvatures from the 
digital map is not sufficient for an exact calculation. Very good 
estimations of the functions and the first differential have to be 
developed. Therefore an intermediate solution via polynomial 
approximation is chosen. This adjustment develops towards very 
good results if the unknown parameters can be calculated 
iteratively. 
Example 1: Gyro measurement on a railway vehicle using 
curvature matching 
A rail-bounded vehicle follows very strongly the predetermined 
track especially in comparison to a road vehicle as a car. This 
means that curvature of the rail track can be developed very 
precisely. 
The cross correlation between measured curvature and the 
curvature derived from the map can be shown in picture 6. This 
yields in a very good phase shift between the 2 different signals. 
Cftiss correlation in % 
Difference in m 
Picture 6: Cross correlation 
Based on this phase shift a stable solution by least square 
techniques can be derived as shown in picture 7, curvature 
matching of a railway trajectory. 
Picture 7: Curvature matching of a railway trajectory 
The accuracy of distance and curvature is in the order of 0.02% 
resp. 0.01% and is sufficient for quite precise positioning of rail 
vehicles for vehicle autonomous rail guidance systems. 
Example 2:Road 
Between the measurement accuracy of the used sensors the 
accuracy and reliability of the image matching between the shape 
of different tracks are of tremendous importance. This is shown 
by an example inside the area of Stuttgart where due to hilly areas 
similar curvature patterns occur for different roads. 
Picture 8. Map matching on digital road maps in Stuttgart