The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
The state vector at time step k (s k ) ensambles four variables 
related to the inertial frame: position, orientation, translational 
velocity and angular velocity. 
For brevity sake we report here just the equations we used for 
the translational dynamic (eq. 8), the rotational dynamic (eq. 9) 
and for the state measurements (eq. 10-13). 
x k 
rvO 
' 1 ' 
' 'fa-1 ' 
'y, 
'.V,., 
1 
l h-i 
I - 
¿ k 
! - 
¿ k-l 
I A 
!» 
~k-1 
* x 
di + -- 
2 
0 
'h 
/ 4 
— 
+.-i 
!* 
¿ k-l 
+ 
'V 
!» 
¿k~1 
0 
0 
0 
0 
l h 
0 
0 
/fa. 
V. ¿ k-ij 
, 0 , 
, 0 , 
'fa' 
ÍH 
íá-.) 
"O' 
A 
¿«i 
0 
¥i 
= 
. !+ 
*+-• 
0 
fa 
k-i • 
0 j 2 
0 
A 
A . 
M i 
0 , 
0 
l 0 i 
,0, 
(9) 
\ 
/ ! .. \ 
acc Kt 
acc rt 
-i*' 
l ÿt 
B“», 
tu 
v 7 
(10) 
where ' B R denotes the rotation matrix converting attitude and 
position data from the Body frame to the Inertial frame. 
r *™hk) 
'1 0 -s3 k - 
*™l,k 
= 
0 C$ k s ^k C ^k 
r v <J 
0 -sf k cS k cfa 
"GPS«' 
(i■■ \ 
** 
lGPS n 
= 
7 v 
>k 
[ lGPS n j 
v j 
(fa ' 
mags* 
= 
*9, 
(ID 
(12) 
(13) 
3.4 The reference trajectory 
In the previous subsections the set of Simulink blocks we used 
to model different aspects of our helicopter fligth (dynamics, 
state measurements and Kalman filetring) has been presented. 
All related information have been then employed to get the 
helicopter guidance control trough step-by-step comparison 
with a reference trajectory. To properly design this flight path 
we developed an algorithm in Matlab based on following 
assumptions: 
1) The mapping area should have a rectangular shape; 
2) Vertex coordinates of the rectangle were defined in the 
inertial frame; 
3) The trajectory should be automatically generated once the 
four vertex of the rectangle were known; 
4) Trajectory had to stop at the same helicopter starting 
position; 
5) Digital images had to be captured in hovering mode, i.e. 
while the helicopter is kept stationary; 
6) Stop points should be chosen in such a way to ensure 
enough side and along path image overlap, according to 
user requirements; 
7) Trajectory should be designed taking into account the use of 
a pair of digital cameras acquiring simoultaneously, their 
Field of View and their tilting with respect to the vertical. 
In order to meet all these requirements, our trajectory 
generating algorithm requires a set of input parameters 
describing the size of the mapping area, the helicopter’s 
geometry and the digital camera pair mounting. Here we recall 
just the most important ones: helicopter starting position, 
operating height (20 m), stop time (10 s), average translational 
velocity (0.5 m/s), vertex coordinates of the rectangular area, 
along-path overlap (2 m), side overlap (3 m) and FOV. 
Figure 6 shows an example of a reference trajectory generated 
with the implemented algorithm. Red points represent the stop 
points for image capture in hovering mode. 
4. TEST & RESULTS 
For the test we set an average velocity of 0.5 m per second, 
though this not very high speed when compared with the typical 
performance of acrobatic helicopters, however it fits well in the 
case of automatically controlled flight. The designed trajectory 
has been used as a reference for the control loop (see figure 4), 
which we implemented as a MIMO (multiple input multiple 
output) system generating the four servos needed to control the 
flight path according to the values of the six variables 
describing helicopter’s position and attitude in 3D space. 
Figures 7 shows the helicopter position on the reference 
trajectory along the x axis: horizontal steps correspond to the 
red points in figure 6 (stop positions), while vertical steps 
denote helicopter motion in between. As shown in figure 8, the 
range of the yaw angle computed for the same trajectory lies 
between -90 and +90 degrees. These wide angle variations are