3. NUMERICAL EXPERIMENTS 
In our situation the complexity of testing of the developed 
method was connected with that for real videos of spacecraft 
docking there are no exact measurements of trajectory 
coordinates and angles. For testing of algorithm the modeling 
data were used. These data include the trajectory, simulating 
real trajectory of spacecraft docking and synthesized by 
specialized software modeling high realistic video film 
corresponding to given trajectory. Known values of coordinates 
and angles allow to make an assessment of accuracy of 
algorithm at this stage. Then the algorithm was tested on a real 
video with only visual quality assessment. The 3D model of the 
ISS used for the experiments presented in this work consisted of 
about 16000 vertices. 
The algorithm for iterative model-based pose estimation has 
been used to track the International Space Station by first 
analyzing a synthetic video sequence depicting the approach 
procedure. Although, in principle, the ISS contains moving 
parts such as solar panels, we assume that its exact 
configuration is known for each particular docking mission. The 
initial estimate of SCS position (three angles and three 
coordinates) relative to the ISS is known also. The algorithm 
was initialized by the approximate pose parameters. In Figure 3 
we demonstrate the initial estimate superimposed over the video 
image, and in Figure 4 the final solution obtained by running 
the iterative contour-based algorithm is shown. Visual quality 
assessment from Figure 5 shows that contour visually matches 
the real image edges. The tracking of the model was performed 
over a period of several minutes during which the SCS travelled 
along a spiral-like trajectory for a total displacement of about 
600 meters. Figures 6-8 demonstrate the orientation angles 
corresponding to the modeled trajectory (solid lines) as well as 
the orientation angles recovered by our contour-based tracking 
algorithm (dashed lines). The mean squared errors (MSE) for 
angles under estimation are: 
MSE( o) ^ 0.1627, 
MSE( g ) — 0.6571, 
MSE( « ) ^ 0.4397, 
all values are satisfactory. 
The distance threshold for the search of closest edges was fixed 
at R — 50, the Tukey function parameter was set to c — 10. 
These parameters can be changed depending on the translation 
value between two consecutive video frames and the standard 
deviation of distances from the projected contour to the nearest 
edges. Regions of the image were treated as potential edges if 
the absolute value of the directional derivative exceeded a 
threshold of 5. The threshold value depends on the image 
contrast, and in principle it should be updated dynamically 
depending on photometric conditions. 
Five hundred random uniformly distributed contour points were 
used for computing the misfit function on each iteration. Most 
computation time was spent on computing the projected contour 
and calculation of the Jacobian matrix. These operations 
involve simple 3x3 or 4x4 matrix-vector multiplications and are 
easily parallelizable. It was observed that the Levenberg- 
Marquardt method usually converges in about eight iterations, 
and the misfit function is computed, on average, 15-17 times 
per video frame. 
10 
  
  
Figure 4. Model contour fitted to image using found pose 
parameters 
  
, degrees 
  
  
  
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