Modelo ABCDE

The Kalman Filter provides an estimator which is optimal in the sense of minimal variance if the prior distribution is Gaussian and the measurement model is linear. The distributions which are derived from the initial orbit determination methods described above are not, however, Gaussian, and thus using a Kalman filter naively can provide invalid results.

Gaussian distributions are completely characterized by their first two moments, but the uniform distributions on the admissible regions described before need more

3.6 Experiments

parameters to be characterized. The Gaussian approximation that is done, then, truncates some of the available information and can generate samples that are not admissible. However, approximating this distribution as a Gaussian yields big per- formance gains since the number of required particles is relatively low, as discussed in section 3.5. Fortunately, the generated distributions are not multimodal as they are composed of Gaussian and uniform components, and so the Gaussian approxi- mation is reasonable. The uniform components could be better approximated with distributions with fatter tails, such as Student’s t distribution, and so filters for these such as the one in [77] could reasonably be used here. Alternatively, Gaussian mixtures such as the ones described in chapter2could be used to increase the faith- fulness of the approximation, as in [16]. The Kalman filter, however, is both simple and efficient compared to these two approaches so it will be used in this work.

Since the filtering distribution is approximated as a Gaussian when there is an update step, it is important to see how robust the filter is to these reparametrizations as they can entail some information loss. To do this, a Gaussian distribution in the sensor state space will be generated and non-linear transformations of two types will be applied to it:

• First, when the relative position of the object in the sensor frame of reference S∗ evolves;

• Second, when the spatial distribution of the object in the Earth frame of reference X is corrupted with noise.

In order to evaluate if a distribution is still Gaussian after each of these transforma- tions, Henze and Zirkler’s BHEP test [35] is used. This test compares the theoretical characteristic function of a Gaussian distribution with the empirical characteristic function of the available samples, and develops a test statistic which can be used to test whether it is plausible for the samples to have come from a Gaussian distribu- tion. The p-values of the test that are presented below indicate the probability of the test statistic (or a more extreme result) having been produce under the hypoth- esis of the distribution being Gaussian. The hypothesis is rejected if the p-value falls under a specified threshold probability, e.g., 0.05.

In the first experiment a Gaussian distribution in spherical co-ordinates is ini- tialized, corresponding to an object directly above a sensor and at a distance of 4 times the radius of the Earth, with covariance

3.6 Experiments Angle (degrees) 2 4 6 8 10 12 14 P-value 0 0.1 0.2 0.3 0.4 0.5

Figure 3.5: Results of the BHEP test averaged over 100 Monte Carlo runs for the first sensibility experiment. The red line indicates the 0.05 confidence interval, under which the hypothesis of the distribution being Gaussian is rejected.

where diag(·) denotes the diagonal matrix with elements in the diagonal given by its argument. The sensor frame of reference is then modified, corresponding to the sensor motion induced by the rotation of the Earth during 20 seconds, and the initial distribution is mapped to this new co-ordinate frame. The BHEP test is used to analyze the probability of the distribution being Gaussian in this new frame, and the process is iterated for a duration of approximately 1 hour. This was repeated for 100 Monte Carlo runs, and the average values of the p-values yielded by the test can be seen in Figure 3.5 graphed against the relative angle to the initial sensor frame of reference.

The results of the first experiment suggest that the validity of the Gaussian assumption in a sensor frame of reference is robust to moderate alterations of the sensor frame of reference. While out of the scope of this chapter, it opens the possibility for the exploitation of the sensor co-ordinate parameterization when two sensors observe simultaneously the same orbiting object.

In the second experiment a Gaussian distribution in spherical co-ordinates is initialized as in the first one. The spatial distribution is then mapped to the Carte- sian frame of reference, corrupted with Gaussian noise in this parametrization with covariance

diag([10002, 10002, 10002, 1002, 1002, 1002]),

before being mapped back to sensor frame of reference. The BHEP test is then used to assert whether the resulting distribution is Gaussian. This procedure is then repeated for growing levels of noise. The experiment was performed on 100 Monte Carlo runs, and the averaged p-values can be seen in Figure 3.6.

3.6 Experiments Noise level 20 40 60 80 100 P-value 0.1 0.2 0.3 0.4 0.5

Figure 3.6: Results of the BHEP test averaged over 100 Monte Carlo runs for the second sensibility experiment. The noise level indicates how many times Gaussian noise was added to the distribution in Cartesian co-ordinates. The red line indicates the 0.05 confidence interval.

T. a e i Ω ω

1 16495.0 km 0.01 2.0◦ 20.0◦ 311.0◦ 2 26352.5 km 0.6 11.3◦ 60.0◦ 351.0◦

Table 3.1: Target index, semi-major axis (a), eccentricity (e), inclination (i), right ascension of the ascending node (Ω), and argument of perigee (ω).

The results of the first experiment suggest that the validity of the Gaussian assumption in a sensor frame of reference is robust to the corruption of the distribu- tion in the Cartesian frame of reference with significant noise levels. In particular, since the level of the process noise Qk in the prediction step (3.12) is significantly lower than the threshold over which the BHEP test fails in Figure3.6, these results suggest that the validity of the Gaussian approximation hold for successive filtering steps while an object is in the sensor’s field of view and thus frequently observed and updated.