1.1

Instrument

The payload consists of 24 normal cameras and two fast cameras mounted on a highly stable optical bench. The normal cameras with a cadence of 25 s are grouped in four subsets with six telescopes. As shown in fig. 1b the four groups are tilted to cover a larger FoV. The fast cameras feature have a cadence of 2.5 s for the observation of very bright targets. Furthermore, they feature spectral filters (blue/red) to provide colour information for stellar analysis.

Figure 1.

PLATO Instrument

All cameras have the same optical design and contain four CCDs to cover the FoV. Each CCD has a sensitive area of 81.18 mm × 81.18 mm, equivalent to 4510 pixel × 4510 pixel with a pixel size of 18 μm. With an iFoV of 15 arcsec, a single camera covers a FoV of ±18.9°. To achieve the high cadence of the fast cameras, the CCDs are operated in frame transfer mode. Due to a metallization layer, as seen in fig. 2a, the light sensitive area for a frame transfer CCD is 4490 pixel × 2245 pixel. Light collected on the sensitive area is shifted within 2 s to the read-out area. Here it is continuously read out by the Front End Electronics (FEE) and processed by the Digital Processing Unit (DPU). The FGS algorithm is running on a Leon2-FT processor at 80MHz and needs to be highly optimized to process 30 guide stars in less than 300 ms. This is important to reduce latency which directly impacts the AOCS.

Figure 2.

Frame Transfer CCDs

1.2

Camera Model

A geometric camera model, sometimes referred to as FoV model, is needed to transform sky coordinates given in the camera boresight⁵ reference frame to the CCD reference frame and vice versa. It contains information regarding the Telescope Optical Unit (TOU), namely the focal length and a distortion model. Furthermore the position and rotation of the particular CCDs on the Focal Plane Assembly (FPA) w.r.t. the optical axis is determined. The different CCD reference frames are shown in fig. 2b.

Each camera model will be calibrated on-ground by using a collimated light source in combination with a hexapod to realize different field angles. This geometric calibration will be repeated on-board using dedicated stars.

2.1

Centroid Calculation

The current state of the art algorithm to estimate the position or centroid of a guide star is the Centre of Mass (CoM) algorithm. It calculates the raw image moments

with the intensity of a particular pixel y (i, j). The centroid is defined as a weighted mean over all pixels. This algorithm is fast, robust, and easy to implement but also has some major drawbacks. It does not handle possible background signals and is not optimal when subjected to random noise. Supposed the size of the star is considerably smaller than the window, the local background can be estimated by averaging the faintest pixels in the window, e.g. using the 20 border pixels of a 6 × 6 pixel window.

The better choice in terms of random noise is to use a PSF fitting algorithm. Note that there is no substantial improvement in terms of non-linearity. An observation h (i, j) of a single pixel is modelled as a Gaussian PSF which we define as an integral over the area of a pixel of a Gaussian distribution.

with the centroid position (u_c, v_c)^T, PSF width σ, intensity I_m, background D, and random noise ξ. The intensity of each star is computed with

with the measured flux F_m in e⁻/s, the exposure time t and a gain factor g. The flux depends on the star magnitude m, the effective light-collecting area A, the transmission of the optical system T, the quantum efficiency of the detector Q, and the reference flux F₀ of a star with m = 0.

The centroid of the PSF is estimated by minimizing the following non-linear functional

α

with observations for each pixel y(i, j) and unknowns α = (u_c, v_c, σ, D, I_m)^T.

2.2

Attitude Estimation

After transforming the measured centroids to vector observations, the optimal attitude is estimated by means of the QUEST algorithm.⁶ With the star directions

given in the camera boresight reference frame and the corresponding reference unit vectors r_i, given in an inertial reference frame, the following optimization problem arises.

Where A is an orthogonal matrix (direction-cosine matrix) and a_i denoting non negative weights with where N is the number of measurements. For measurements with variance σ², with . This optimization problem is often referred to as Wahba’s problem. An efficient solution is the QUEST algorithm, which solves for the optimal quaternion minimizing eq. (7).

A possible drawback of the QUEST algorithm is a singularity which occurs if the scalar part of the quaternion equals zero. This can be avoided e.g. by applying an a priori quaternion known from the nominal pointing before estimation. Because an a priory reference frame rotation is assumed to be known, the QUEST method can be used without risk to run into singular cases. In case of an unknown orientation, the methods ESOQ (Estimator of the Optimal Quaternion) and ESOQ2 (Second Estimator of the Optimal Quaternion) provide build in mechanisms to avoid the singular case not needing explicit reference frame rotations. In terms of speed, QUEST is superior to Davenports q-method and about equal to ESOQ, ESOQ2, and FOAM.⁷

3.1

Simulation

The PLATO simulator (PlatoSim⁸) is a software tool designed to simulate image time series. It models a CCD and its read-out electronic including an exhaustive list of common noise sources, e.g. read-out noise, shot noise, dark noise, open shutter smearing, and cosmic rays to name just a few.

Figure 4 shows the used open loop test set-up, consisting of the simulated camera, S/C AOCS, and the FGS itself. For now the AOCS assumes a worst case FGS noise and simulates an 8 Hz time series of the predicted S/C movement, which is fed to the camera simulator. In a later stage of the development the loop will be closed by feeding the FGS measurement back to the AOCS.

Figure 4.

Open loop configuration

Furthermore, the telescope optics can be specified in terms of geometry, transmission, and PSF by means of a Zemax simulation. Figure 5a shows a typical input PSF for a particular field position. PlatoSim also takes CCD charge diffusion, a potentially diffusion of charge into neighbour pixels, into account to produce a system PSF as seen in fig. 5b. After placing the high resolution PSF to its projected position on the CCD it is sampled to its final resolution, shown in fig. 5c.

Figure 5.

PlatoSim PSF Evolution

3.2

Performance Evaluation

The overall performance of the FGS can be quantified with two error terms. First, the Noise Equivalent Angle (NEA), describing short term random noise and second, bias-stability which is long term systematic noise due to non-linearities within the centroid estimation. Bias-stability, caused by Thermo-Elastic Distortion (TED) and stellar aberration, has been studied but will not be the focus of this paper. Besides the number and distribution of the chosen guide stars, the centroid precision is the main factor for the expected performance in terms of NEA.

In order to get representative results 134 guide stars have been chosen for the first stellar field to be observed. Guide stars need to be in the right magnitude range (5 – 7), must have low contamination from neighbouring stars, and low stellar variability. Due to the use of frame transfer CCDs only 80 of the chosen guide stars fall on the sensitive area of the CCDs. The baseline is to use 30 stars for the FGS algorithm.

3.2.1

Centroid Precision

The residual centroid error is calculated as difference between nominal star position – including averaged S/C jitter – and the estimated centroid. Figure 6 shows the measured centroid noise σ_x,y as well as an uncertainty estimate derived from the expected read-out noise and shot noise. Since this estimate does not include the influence of S/C jitter, it can be seen as a lower bound, mainly dependent on the star magnitude which determines the Signal to Noise Ratio (SNR).

Figure 6.

Residual centroid noise for blue non-Gaussian PSF

Besides the random noise sources, read-out noise and shot noise, an important driver for the centroid precision is the PSF shape which determines the sensitivity against S/C jitter. As seen in fig. 5, the PSF is rather non-Gaussian and varies widely over the FoV. This effect is amplified if local PSF features, e.g. minima or maxima, are located near a pixel edge. Small movements induced by jitter will lead to a disproportionately strong change in pixel illumination which results in additional noise for the estimated centroid position. In case of a Gaussian PSF shape this would be covered by the observation model given in eq. (2).

Table 2.

Non-Gaussian PSF: mean residual noise [mas]

	CoM	Gauss-fit
<σx>	<σy>	<σx>	<σy>
blue	31.4	31.9	15.9	16.7
red	31.4	31.9	16.6	17.5

Table 2 shows the mean residual noise for the centroid positions using the CoM method as well as the introduced Gaussian fit for a window size of 6 × 6 pixel. Here, the performance of the Gaussian fit is improved by a factor 2 compared to the traditional CoM algorithm. This means a factor 4 less guide stars are needed to achieve a similar FGS performance. Since we are dealing with a limited amount of very bright targets, this already is a significant improvement. Note that the CoM precision is reduced with larger windows, e.g. 44 mas for 7 × 7 pixel windows. This is not the case for the Gaussian fit algorithm which further increases its advantage.

3.2.2

Attitude Precision

The NEA does not only depend on the individual noise at centroids level but also on the number and distribution of the used guide stars. A rough estimation is made by averaging the mean centroid noise centroid with the number of stars N_stars. ⁹ The directional error is given as follows:

Assuming uniformly distributed stars across a rectangular FoV, the roll-axis precision can be estimated with

where β_FoV = 36 deg is the FoV angle. For 30 stars with a mean directional centroid precision of 17 mas as given in table 2, the directional FGS precision is 3.1 mas, respectively 12.1 mas for the roll axis. Due to the geometry of the frame transfer CCDs yielding a reduced FoV, the roll axis performance will be decreased.

The influence regarding selection and distribution of guide stars has been studied by setting up a Monte Carlo simulation. The outcome is a mean error with an uncertainty rather than a single result holding only for a particular set of guide stars. As shown in section 3.2, 80 guide stars are available for the given pointing direction in the simulation. Randomly chosen 30 guide stars are used to calculate the attitude as shown in section 2.2. With the simulation input jitter serving as a ground truth the residual FGS error can be calculated. Because the selection of guide stars is the same as in table 2, the FGS performance shown in table 3 nicely follows the estimation given in eqs. (8) and (9).

Table 3.

Mean FGS NEA for different PSFs [mas]

	σx	σy	σz
blue	3.1± 0.3	2.9± 0.3	16.5± 1.8
red	3.0± 0.3	3.2± 0.4	14.7± 1.4

To take possible guide star selections and distributions into account, the associated requirements are given with a 95 % Confidence Interval (CI) with ensemble statistical interpretation. The FGS performance is therefore calculated as given in table 4. It shows that the NEA requirement of 25 mas for transverse axes and 100 mas for the rotational axis is fulfilled in all cases including maturity margin.

Table 4.

Mean FGS NEA at 95 % CI [mas]

	σx	σy	σz
blue	7.2	6.8	40.1
red	7.1	8.0	34.8