Configuration used for Monte Carlo simulations

This section presents the configuration which was used for the Monte Carlo simulations described in the following sections.

Since one of the goals of the Monte Carlo simulations was to measure the misidentification rate, and this misidentification rate is typically very small, to make it measurable without having to run the star-ID algorithm hundreds of million times, which would take a prohibitively long time, we have considered in our simulations a very coarse hypothetical star tracker, with standard deviation in centroid position in the order of minutes of arc. To do that, we modified a parameter in gen_sia_inputx (described in Chapter 4) that defines the imager width and height in pixels, from the default value of 1024 pixels to 128 pixels, while still keeping the simulated star tracker field of view at 25.457° × 25.457°.

This had the effect of increasing the angular width and height of the central pixel in the imager from 91 arc-seconds to 728 arc-seconds. Since the standard deviation in position in gen_sia_inputx (for x ≥ 6) are scaled by the pixel size, increasing the pixel size has the effect of increasing position noise in the simulated observed stars. With this noise level, the tolerance δθ used for selecting star pair candidates from the catalog of star pairs has to be increased proportionally to keep the success rate at acceptable levels.

Algorithms gen_sia_inputx (with x = 5,...,8 being the version number) have a configuration parameter moffset that can be used to simulate the star tracker sensitivity to light, as explained in Section 4.4. Positive values of moffset mean increased sensitivity to dim stars, whereas negative values of moffset mean decreased sensitivity. In an actual star tracker, this change in sensitivity is usually accomplished through a change in exposure time, since many other parameters that also affect sensitivity such as optics aperture and image sensor quality are usually set to their best design values and cannot be easily changed.

In an actual application, there’s always the desire to work with the shortest exposure time (or more precisely, integration time) that will enable detection of

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the minimum number of stars required for a reliable identification. This is justified by the fact that long exposure times lead to blurring in star images if the spacecraft is slowly rotating. Another strong reason is related to the time needed for attitude acquisition and to the rate of attitude update in attitude tracking mode.

In modern wide field of view star trackers, the limiting factor for attitude acquisition and update rate is no longer the processing power, but the minimum exposure time required to detect stars⁷. However, a too low exposure time will decrease the star tracker availability, since in that case there will be many scenes without sufficient stars for stellar identification and attitude determination. To select the value of moffset to be used in the simulations, it was assumed that the simulated star tracker had a requirement of an attitude acquisition probability from its first acquired image of at least 99%, in other words, provided that its field of view is not obstructed or blinded by bright sources (Sun, Moon or Earth), it should be able to acquire an attitude in more than 99% of the cases without having to acquire a new image, assuming a random and uniformly distributed initial attitude.

After many tests, the minimum value of moffset that was found to give an availability of at least 99% was moffset = −0.42. This was taken as the baseline configuration for our tests.

With this configuration, the value of the angular separation tolerance δθ in star pairs that provided the best star tracker availability (or success rate) was δθ = 170 arc-seconds. This value was selected for our baseline.

In the simulations presented here, the effects of stellar aberration (Section 2.5) were not included. Considering that the angular separation tolerance of 170 arc-seconds adopted here is much wider than the maximum, worst-case, apparent displacement in star positions of 28 arc-seconds due to stellar aberration for an

7 In a wide FOV star tracker employing an old generation processor running at 12 MHz, star identification takes, on average, less than 0.6 s using a slow star-ID algorithm and less than 40 ms using a fast star-ID algorithm (FIALHO, 2007). These are comparable to 400 ms of minimum exposure time needed to detect enough stars in all sky configurations in the same STR design.

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Earth orbiting spacecraft⁸, not including this effect in the simulations will have little or negligible impact in the results obtained. However, simulations using much tighter tolerances should include this effect.

Following is the configuration used in all tests, except where noted otherwise:

• Global parameters:

o Simulated star tracker FOV: square FOV of 25.4570° ⨯ 25.4570°

o Catalog used by gen_sia_input6 and gen_sia_input8 for

generating the list of observed stars: limited at visual magnitude 7.0 with 15,513 stars (files: cat_m7g.cat or cat_m7gc2.cat) o Catalog used for star identification: subset of catalog used by

gen_sia_input6 and gen_sia_input8 containing the brightest 1612 stars (limited to visual magnitude 5.0)

• Default parameters used by gen_sia_input6 / gen_sia_input8:

o moffset = −0.42 ==> 99% probability of star detection at mv = 3.37;

50% probability of star detection at mv = 4.466 and 1% probability of star detection at mv = 5.20.

o imager resolution (width and height): 128 pixels ==> pixels near the center of the array have an angular size of 728”.

o maximum number of stars in the output: unlimited.

• Default parameters for mfPyramid v01 and mfPyramid v03:

o angular separation tolerance (δθ): 170”

o min_sep: 0.03 * FOVdiagonal = 1.06295°

o max_tries: 15 (maximum number of kernels to use before giving up, introduced in mfPyramid v02)

o number of stars used for identification: 1612 ==> catalog of star pairs with 129,678 star pairs.

o kvector_size_ratio = 1.5 => k-vector with 194,517 elements.

8 Assuming that the master catalog has its origin at the Solar System’s barycenter.

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All tests were performed on a machine with the following configuration:

• processor: Intel® Core™ i7-4500U @ 1.80 GHz / 2.40GHz;

• memory: 8GB;

• operating system: Windows10 64-bit.