figure table

# Pyxis: A ground-based demonstrator for formation-flying optical
interferometry

Jonah T. Hansen  Research School of Astronomy and Astrophysics, College of
Science, Australian National University, ACT 2611, Australia  Samuel Wade
Advanced Instrumentation Technology Centre, Research School of Astronomy and
Astrophysics, College of Science, Australian National University, ACT 2611,
Australia  Michael J. Ireland  Research School of Astronomy and Astrophysics,
College of Science, Australian National University, ACT 2611, Australia  Tony
D. Travouillon  Advanced Instrumentation Technology Centre, Research School of
Astronomy and Astrophysics, College of Science, Australian National
University, ACT 2611, Australia  Tiphaine Lagadec  Research School of
Astronomy and Astrophysics, College of Science, Australian National
University, ACT 2611, Australia  Nicholas Herrald  Advanced Instrumentation
Technology Centre, Research School of Astronomy and Astrophysics, College of
Science, Australian National University, ACT 2611, Australia  Joice Mathew
Advanced Instrumentation Technology Centre, Research School of Astronomy and
Astrophysics, College of Science, Australian National University, ACT 2611,
Australia  Stephanie Monty  Research School of Astronomy and Astrophysics,
College of Science, Australian National University, ACT 2611, Australia
Institute of Astronomy, University of Cambridge, Madingley Rd, Cambridge, CB3
0HA, UK  Adam D. Rains  Research School of Astronomy and Astrophysics, College
of Science, Australian National University, ACT 2611, Australia  Division of
Astronomy and Space Physics, Department of Physics and Astronomy, Uppsala
University, Box 516, 75120 Uppsala, Sweden

###### Abstract

In the past few years, there has been a resurgence in studies towards space-
based optical/infrared interferometry, particularly with the vision to use the
technique to discover and characterise temperate Earth-like exoplanets around
solar analogues. One of the key technological leaps needed to make such a
mission feasible is demonstrating that formation flying precision at the level
needed for interferometry is possible. Here, we present Pyxis, a ground-based
demonstrator for a future small satellite mission with the aim to demonstrate
the precision metrology needed for space-based interferometry. We describe the
science potential of such a ground-based instrument, and detail the various
subsystems: three six-axis robots, a multi-stage metrology system, an
integrated optics beam combiner and the control systems required for the
necessary precision and stability. We end by looking towards the next stage of
Pyxis: a collection of small satellites in Earth orbit.

###### keywords:

Interferometry, Satellites, Metrology, Robotics, Telescopes

*Email<EMAIL_ADDRESS>
##  1 Introduction and Background

High angular resolution astrophysics, in particular optical/infrared (IR)
interferometry, is in a golden age. Instruments such as GRAVITY [1] and
MATISSE [2] at the VLTI and MIRC-X [3] on the CHARA array have produced
stunning scientific results, including imaging the starspots on a distant
giant star[4], the first astrometric confirmation of a planet[5], and of
course, the characterisation of a supermassive compact object at the centre of
our galaxy[6], leading to the 2020 Nobel Prize in physics. However, there are
still questions yet unexplored that only interferometry will be able to probe.

One such avenue is the direct imaging of exoplanets. One of the major goals of
exoplanet research is identifying potentially habitable worlds that may
harbour life, and to that end one needs access to the atmosphere of the planet
to look for biosignatures [7]. Transmission spectroscopy is promising for the
characterisation of hydrogen rich atmospheres, but is challenging for
terrestrial atmospheres[8]. Hence direct imaging is one of the only techniques
available to obtain atmospheric spectra from terrestrial planets. In order to
accomplish this, however, we require minimising the contrast between a planet
and it’s host star—which for terrestrial planets lies in the mid-infrared
(MIR). To obtain the sensitivity needed, as well as avoiding telluric
contamination, we also require these telescopes to be in space; producing
challenges in having large coronagraphic apertures with the required angular
resolution.

Hence space-based interferometry has long been recognised as the only way to
take MIR spectra of Earth-like planets[9, 10], and is simulated to have
similar or greater yield compared to the largest launchable $>$$10B
coronagraphic telescopes in detecting any habitable planet biosignatures [11,
12]. Once the required aperture diameter becomes too large to construct
mechanically or to launch, the only option is to launch parts of the mirror as
separate light collector spacecraft with light combined in a beam combiner
spacecraft—becoming a space interferometer.

Many previous studies have been made of space interferometer missions, in one
of two categories. These were either connected element interferometers[13, 14]
which are much more limited than formation flying, or formation flying
interferometers at the Sun-Earth L2 point[15, 9], which are vastly more
expensive than low earth orbit compatible designs, mostly due to launch costs.
In the late 2000s, both NASA and ESA shelved plans for large scale space-based
interferometers (TPF-I[16] and Darwin[9] respectively), primarily due to a
lack of understanding of the potential planet yield of such a mission, as well
as technical unreadiness. Since that time, however, missions such as Kepler
[17] and more recent exoplanet missions (e.g. TESS[18] and CHEOPS [19]) have
greatly increased our knowledge of planet demographics to the point where
simulations of a large MIR space interferometer have shown it would detect
approximately 20 Earth-like planets [12, 20]. This renewal of interest in
space interferometry has lead to the development of the Large Interferometer
For Exoplanets (LIFE) initiative; a revival of the TPF-I/Darwin concept to
detect and characterise planets in the MIR [12].

In late 2021, ESA released its Voyage 2050 plan, for which the
characterisation of planets in the MIR was one of the top priorities for a
large scale mission [21]. As a caveat to this recommendation, however, was the
requirement to prove that such a space interferometer mission would be
feasible technologically, as well as scientifically. Two critical technology
areas have not been at an adequate level to progress the formation flying
optical and infrared interferometry missions: compact, cryogenic compatible
nulling beam combiners (a target of the Nulling Interferometer Cryogenic
Experiment (NICE) [22]), and formation flying itself, including metrology
systems. It is this second technology that is the primary purpose of Pyxis,
the subject of this paper, though other investigations into formation flying
interferometry are currently ongoing [23, 24].

Pyxis is a multi-platform, linear-formation, robotic ground-based optical
interferometer in development at the Australian National University’s (ANU)
Research School of Astronomy and Astrophysics (RSAA) located at Mt Stromlo
Observatory. It will serve as a crucial technology demonstration of formation
control and metrology for future formation flying space-interferometry
missions and enable for more flexible ground-based stellar interferometry. A
schematic of the Pyxis interferometer is found in Figure 1, highlighting the
major components and subsystems. Here we distinguish between the side
collector platforms as “deputies”, and the central beam combining platform as
the “chief”.

Pyxis has a number of novel key features that will allow it to achieve its
goal of formation flying interferometry. Firstly, it utilises a frame of
reference tied to a precision star tracker on a movable platform, rather than
the Earth itself. This is made possible through the use of newly affordable
MEMS (microelectromechanical system) accelerometers and a fibre laser
gyroscope to define this frame of reference, and will be discussed more in
Section 5. Secondly, we implement a multi-stage metrology system, using camera
based coarse metrology supplemented by a time of flight (TOF) sensor; and a
sub-wavelength path-differential interferometric metrology sensor using laser
diodes. These systems allow us to vastly simplify our system architecture and
will be discussed in Section 3. Finally, Pyxis has a linear array architecture
that can be used in Low Earth Orbit (LEO) [25] and has a continuous range of
reconfigurable baselines, nominally between 1 and 60 m.

Figure 1: Schematic showing Pyxis’ two science telescope platforms (known as
the deputies) and the single central beam combining platform (known as the
chief), each of which are separate wheeled 6-axis platforms able to move to
position on a moderately flat surface and then track spatial and angular
positions. Starlight is focused and collimated by the telescope primary and
secondary mirrors, before reflecting off of 45∘ flat mirrors to the tip/tilt,
fibre injection, and beam combiner systems on the central platform. Spatial
locations are measured using accelerometers and a white-light fast metrology
system, with the frame of reference of the central beam combiner determined by
a combination of a fibre optic gyroscope and star tracker. In addition to the
electro-mechanical and control systems, the Pyxis concept can be broadly
broken into the following four systems: Coarse Positioning (Red): 1: 3 axis
wheels & base (X, Y, $\theta_{\rm Z}$); 2: 3 axis height adjustment (Z,
$\theta_{\rm X}$, $\theta_{\rm Y}$); 3: Shock absorbers; 4: Goniometer
($\theta_{\rm X}$). Angular Metrology (Blue): 5: Deputy star-tracker camera;
6: Chief star-tracker camera; 7: Fibre optic gyroscope; 8: Coarse metrology
camera; 9: Coarse metrology LEDs. Interplatform distance metrology (Orange):
10: Retroreflector; 11: Interferometric lasers & sensors, time-of-flight
sensor; 12: Metrology fibres, beam combiner, & spectrograph; Science
instruments (Green): 13: Science telescope; 14: Tip tilt, fibre injection; 15:
Science fibres, beam combiner, fringe tracking, & spectrograph.

This paper is structured as follows: the remainder of Section 1 details the
scientific potential and aims of Pyxis; Section 2 describes the mechanical
interface and architecture of the interferometer; Section 3 details the
metrology system; Section 4 discusses the beam combiner; and finally Section 5
will describe the control systems.

###  1.1 Scientific Aims

While the primary purpose of Pyxis is to act as a demonstrator for formation-
flying interferometry, it will be well placed to make important, unique
astrophysical measurements on its own. Pyxis is designed to nominally work in
the $R$ band with a wavelength range between $\sim$620 and 760 nm, where the
upper bound at 760 nm is due to the atmospheric telluric band corresponding to
the Fraunhofer A O2 band and the lower bound corresponding to the single-mode
cutoff of our 630 nm fibres. The science telescopes on each deputy platform
(see Figure 1) have an aperture diameter of 94 mm, and are expected to achieve
a 10% throughput including fibre coupling. We thus expect to be able to
achieve a limiting magnitude of approximately R$\sim$6 with 5 ms exposures,
corresponding to approximately 10 pixels per spectral channel. The aperture
size was chosen due to the only moderate sensitivity gains when increasing the
aperture beyond Fried’s parameter ($r_{0}=5-10$ cm) without adaptive optics.
Instead, the sensitivity of Pyxis is increased through having a simplistic
optical design (covered in Section 4).

With Pyxis working in the $R$ band, it is well placed to complement the
existing suite of interferometers globally; particularly as it will be the
only visible light interferometer in the Southern Hemisphere following the
decommissioning of the SUSI interferometer in the mid-2010s. Figure 2 shows
the current angular resolution capabilities of the VLTI in Chile, and NPOI and
the CHARA array in the USA as a function of wavelength, compared with Pyxis.
We also include the non-redundant baselines of the Keck Aperture Masking
experiment[26], as a comparison for short baseline capabilities. As can be
seen, Pyxis will span a unique combination of wavelength and angular
resolution within the southern hemisphere, and the ability for it to span a
continuous range of baselines and position angles (and therefore UV plane data
points) can allow it to probe optimal spatial frequencies for a given object’s
visibility curve.

Figure 2: Comparison of the wavelength-angular resolution space of Pyxis and
current interferometric facitilites. To create this plot, the most popular and
accessible baselines were chosen for each facility, noting that both NPOI and
VLTI have a much wider range of baselines that are not normally available. The
phase space of the Keck Aperture Masking experiment is also included. Angular
resolution is calculated as $0.5\lambda/B$. Of particular note is that Pyxis
spans a unique portion of this phase space for the Southern Hemisphere.

One of the key areas that Pyxis will provide insight into is the measurement
of fundamental parameters of stars. In recent years, obtaining the stellar
masses, ages and radii of stars with high precision is especially important
due to the burgeoning fields of Galactic archaeology and exoplanet research,
where the properties of the planet host star are critical in extracting the
exoplanet parameters [27, 28, 29]. Pyxis will build off of the success of PAVO
in measuring stellar diameters [30] utilising an even simpler architecture,
single mode spatial filtering, and polarisation control.

Precision stellar masses are another key parameter in the studies of stellar
evolution, as they determine stellar age and Galactic evolution timescales.
This parameter can be obtained through giant star asteroseismology, but
suffers from a lack of calibration and benchmarks [31, 32]. Simple stellar
diameter measurements have been shown to help calibrate these techniques [33],
and so Pyxis will be able to augment Gaia astrometry of the brightest
astrometric binaries with precise interferometric separation measurements,
thus providing a precise stellar mass with which we can calibrate other
measurements.

Arguably the more exciting and unique science case of Pyxis, however, is
utilising its polarimetry functionality. First pioneered by Ireland (2005)
[34], multi-wavelength interferometric polarimetry has resulted in exciting
observations in resolving spherically symmetric dust shells around very large
stars that would be otherwise undetectable with a non-interferometric
instrument [35]. Such studies have probed the dust grain size distribution
around giant stars, which in turn informs the processes of how dust is made
and stellar mass loss. However, these studies have been rare due to the lack
of available instrumentation to make these measurements, and questions remain
regarding when dust scattering provides the dominant mechanism for giant star
mass loss [36].

Pyxis requires a polarisation split in order to accurately calibrate the
visibilties, and due to its much simpler geometry than existing long baseline
interferometers, it is straightforward to split the polarisation for
scientific measurements; Pyxis should achieve an estimated calibrated
differential polarisation fringe visibility of 2% precision, comparable to
that of previous measurements [35]. Hence Pyxis should be able to make simple
time and wavelength-dependent interferometric polarimetry measurements around
these bright giant stars to resolve some of these questions.

##  2 Mechanical Design

###  2.1 Robotic Platforms

The Pyxis robotic platforms comprise a vibration-isolated upper platform,
where lasers, telescopes, cameras, and fibre injection systems are mounted,
and a lower platform housing the beam combiner, circuitry, and control
computers. The upper platform payload on each robot is mounted on a stepper-
motor-controlled goniometer to achieve precise elevation control at the level
of a few arcseconds. These goniometers are coupled to the upper platform
through a set of passive mechanical vibration isolators, intended to attenuate
vibrations from roughness in the surface the robot traverses, as well as from
the motors themselves.

The platforms are designed to allow control of all six degrees of freedom of
the upper platform payload as a ground based simulation of satellite control.
The three degrees of freedom in the “ground” plane (two horizontal
translations and rotation about the vertical axis) are controlled by stepper
motors, coupled to precision planetary gearboxes and bi-directional omni-
wheels in a “Kiwi drive” arrangement. The three remaining degrees of freedom
(vertical translation, tip and tilt) are controlled by a set of three linear
actuators (also stepper-motor driven) with a three way rotationally symmetric,
ball and V-groove kinematic interface. The V-grooves are created by pairs of
hardened dowel pins, creating a pair of point contacts with the truncated and
threaded balls attached to the ends of the linear actuators. A photograph of
one of the deputy platforms is shown in Figure 3.

Figure 3: A photograph of the deputy platform. The upper platform containing a
goniometer, telescope and star tracker is shown above vibration isolators. On
the chief platform, this upper platform also contains the fibre injection unit
(see Section 4) and fibre optic gyroscope.

Within the optical subsystems, fine tip/tilt control is achieved using piezo
actuators and optical path difference is controlled with a piezo stick/slip
stage. In order for the system to achieve stable fringes using these fine
control subsystems, we determined the following mechanical requirements for
the robotic platforms:

  * •

The RMS motion above 100 Hz for anything on the upper platform goniometer must
be below 50 nm, so that fringe visibility remains high in 5 ms exposures.

  * •

The RMS velocity must not exceed 10 µm/s at frequencies $<$100 Hz, so that
with a 5 ms servo lag, fringe tracking at this 50 nm level is possible, with
residual fringe motion being dominated by astronomical seeing.

  * •

The absolute positioning must be accurate to within 3 mm, in order to have a 1
part in 1,000 baseline knowledge for a 3 m science baseline, and in order to
keep the fringes within the range of the position actuator.

  * •

The attitude of the platforms must be accurate to within 30”, in order to
inject the light into the field of view of the injection unit.

  * •

The attitude velocity error cannot exceed 100”/s, so that with a 5 ms servo
lag, there is no more than a 0.5” angular error (required for single mode
fibre injection).

Figure 4: Theoretical frequency/resonance response of the platforms for
various axes of motion in terms of transmissibility, the ratio of input
amplitude to output amplitude. Generated through fitting a damped oscillator
to the parameters of the upper platform’s springs. Black dashed line indicates
a transmissibility of one.

Since precise positioning is essential for the successful operation of the
optics, the mechanical vibration isolators were characterised both through
simulation and testing. The simulated transmissibility plot of their frequency
response in the horizontal and vertical directions is displayed in Figure 4,
produced by fitting a damped oscillator function to the listed parameters of
the upper platform springs. The resonance peaks with over-unity gain in the
5-10 Hz range will cause fringe motions that are measurable and partly
controllable by the fringe tracker operating at a 200 Hz loop speed. From
these peaks, attenuation steadily improves over the 5-75 Hz range, and it is
expected that vibrations exceeding 75 Hz will not be of much concern. It is
also clear that the resonance in the vertical direction has greater gain, as
well as a higher frequency, and so is likely to be more of an issue than the
horizontal resonance. Simulations were also carried out on a two-dimensional
model of the system, and showed how horizontal motion would cause significant
angular perturbation of the platform given the high centre of mass, leading to
the installation of a 1kg counterweight suspended below the upper platform.

The system response between the lower and upper platform was physically
tested, by applying a sinusoidal frequency sweep between 0 and 75 Hz using the
motors in each axis (X, Y, Z), and recording the data from a set of three
3-axis accelerometers placed on the vibration-isolated platform. The
accelerometer readings were bias-corrected and transformed into body-frame
accelerations, before being Fourier transformed to find the peak amplitude for
the given test frequency. The transmissibility – the ratio of lower platform
(input) to upper platform (output) amplitudes – is plotted for all six axes of
motion in Figure 5 in both a linear and logarithmic scale. Between 1-15 Hz,
these results largely agree with the simulation, showing peaks close to where
they were expected for horizontal (X/Y) and vertical (Z) inputs, and
significantly higher gain in the vertical direction. Unlike the theoretical
plot, however, we do see some amplification of frequencies around 20-30 Hz in
all three axes. These resonances are likely due to the coupling of the full
system to the springs, and while the addition of a counterweight was able to
suppress these resonances they were not able to be removed completely. Beyond
30 Hz, once again the springs attenuate to below unity gain.

Figure 5: Measured frequency/resonance response of the platforms for various
axes of motion in terms of transmissibility, the ratio of input amplitude to
output amplitude. Black dashed line indicates a transmissibility of one.

It is clear from both model and test results that the vibration-isolators
alone are insufficient to attenuate vibration to the required levels over the
full range of frequencies. However, given their good performance at very high
($>$100 Hz) frequencies, it is expected that an active control system will be
able to handle the lower frequency vibrations, discussed further in Section 5.

###  2.2 Diamond-turned Telescope

The science telescopes are one of the subsystems that the project designed to
be “space ready”. The optics need to be significantly compressed to be
compatible with the CubeSat format without need for refocusing, while
sustaining the standard NASA GEVS vibration qualification profile. Current
demonstration units have passed this requirement. In addition, it requires a
wide enough field of view so that it could correct angular errors in the
deputy position without moving parts. Our solution was to design and prototype
the telescopes from diamond-turned aluminium. A photograph of one of the
telescopes is shown in Figure 6.

Figure 6: A photograph of one of the diamond-turned aluminium telescopes.

Each deputy telescope has a 5:1 magnification conjugated at infinity, in a
Cassegrain design with two paraboloids. The primary has a 94 mm clear aperture
diameter with a 200 mm radius of curvature, while the secondary is 25 mm
across with a 40 mm radius of curvature. This size allows the telescope to fit
in one end of a 3U CubeSat, and can be seen attached to the upper platform in
Figure 3. A 45 degree flat tertiary mirror reflects the beam at 90 degrees to
the optical axis. The telescopes were manufactured at Optofab-ANU, part of the
Australian National Fabrication Facility, using a diamond lathe turning
RSA-6061 aluminium. The complete telescope structure, including mirrors, is
formed from aluminium, so that the telescope is naturally resistant to optical
aberrations caused by thermal expansion.

The telescope manufacturing and assembling process went through several
prototypes, with the best iteration of the primary mirror exhibiting a level
of aberration of a 96% Strehl ratio. We have also minimised stress on the
primary mirror caused by the assembly of the two parts of the telescope: we
moved away from an original shrink fit design and have developed an assembly
that minimises distortions by using a 50 micron thick gap around the edge of
the primary mirror filled with a thermal-expansion matched vacuum compatible
adhesive. Coma and defocus are actively set to have negligible amplitudes
interferometrically during adhesive curing, by find adjustment of primary
mirror tilt and piston. While a residual amount of spherical aberration
remains, along with some minor astigmatism caused by the gluing and mounting
of the tertiary mirror, the final complete telescopes exhibit a $\sim$70%
Strehl ratio.

##  3 Metrology System

One of the primary goals of Pyxis is to demonstrate a metrology system capable
of sustaining satellite formation flight at an adequate level to make precise
interferometric measurements. Pyxis’s distance metrology approach is divided
into two broad parts: a coarse metrology system that measures the large
distances between the platforms, and a fine metrology system that complements
the coarse metrology using interferometry and fringe patterns to achieve the
necessary sub-wavelength precision. The fine metrology system concept was
already described in Lagadec et. al (2020)[37]. This system is not yet fully
commissioned, in particular with a decision point remaining as to whether
temperature stabilised Fabry-Perot laser diodes are enough, or if three single
frequency laser diodes are required. However, the combination of coarse
metrology described below, combined with a fringe search (see Section 5.4)
using starlight will be enough for initial on-sky fringes.

###  3.1 Coarse Metrology

The coarse metrology itself has two parts: measuring the parallax between two
LEDs, and a time-of-flight (TOF) laser system. The former consists of two LEDs
mounted 90 mm apart on the sides of each of the deputy platforms, and a camera
mounted on the chief robot. We measure the angular separation of these LEDs to
0.05 pixels or 0.7” using a 7.6 mm focal length F/2 lens attached to a FLIR
Firefly FFY-U3-16S2M-S camera. This in turn will enable a position measurement
of better than 1 cm for our maximum chief-deputy separation of 30 m.

However, for the purposes of designing a future space-based mission with a
baseline of $\sim$300 m, this camera-based metrology system is almost
certainly too challenging, even with an extended LED spacing of 300 mm. This
is due to the precision required for angular separation measurements, which is
proportional to the square of the spacecraft separation. Hence, while we
implement this system for Pyxis, the TOF system is more critical for
demonstrating a complete space compatible metrology system. The TOF system is
also required to obtain a sub-centimetre position estimate, in order to be
within the fine metrology system’s capture range.

The TOF distance ranging system utilises the time it takes for light to travel
from the chief platform to a retroreflector mounted on a deputy platform and
back. Pulses of light are generated by a laser diode and the distance the
light travel determines the delay before the reflected light returns. This is
equivalent to a phase shift in the pulses of the returning light relative to
the transmitted pulses. The distance to the reflecting object, $d$, is half
the total distance the light has travelled and is related to the delay in the
returning light pulse $t_{\rm d}$ by the speed of light, $c$. Hence this can
be expressed as $d=t_{\rm d}\cdot\frac{c}{2}$.

Implementing a TOF system with analogue electronics requires two capacitors
that act as timing elements to measure the delay and a clock source to
generate pulses of light. The same clock pulses are used to switch between the
two-timing elements so that the first capacitor is charged while the pulse is
high and the second capacitor charges while the pulse is low. When the
capacitor is connected, it is charged by a photo-diode which allows a current
to flow when it receives the returning pulses of light, charging the connected
capacitor. An FPGA with an integrated microcontroller and some additional
circuitry including the laser, photodiodes, timing capacitors, fast analogue
switches and an analogue-to-digital converter (ADC) is used to implement the
system.

A high frequency, square wave signal is produced by clock conditioning
circuity within the FPGA fabric. Two signals with the same frequency are
produced, one to drive the laser modulator and the other to drive the
switching between timing elements. The timing is implemented using a pair of
capacitors that are charged by the photocurrent from a photodiode detector and
a high-speed analogue 2:1 multiplexer (MUX) is used to direct the current
between the two charge-integrating capacitors. After an integrating cycle, the
capacitors are sequentially sampled by an ADC, which has a built-in MUX, and
the capacitors are reset by a command from the microcontroller.

In the full system, there are four pairs of capacitors and accompanying
photodiodes, so that the system can measure two distances to each of the
deputy platforms. The light from a single modulated laser diode is optically
split for the four light paths. All four channels are charged in parallel and
then be sampled sequentially. The laser modulator is controlled by the clock
signal and provides the required DC bias and modulation current to drive the
laser diode. The laser diodes used in this system are shared with the fine
metrology system (described next in Section 3.2), with the laser diodes and
TOF photodiodes shown in the metrology schematic in Figure 7.

###  3.2 Fine Interferometric Metrology

To achieve sub-wavelength precision metrology, we require the use of
interferometric fringe measurement. However, interferometric metrology itself
relies on the coherence of its beams to the sub-wavelength level. Such
required precision poses a challenge for a multiple platform system like Pyxis
for which fringe scanning - necessary to unambiguously determine the phase and
optical path length difference - is complex and time consuming. Fortunately,
it is possible to employ the paradigm of Multi-Wavelength Interferometry[38]
to broaden the wavelength range where the phase can be unambiguously measured;
using several wavelengths can broaden the range by three orders of magnitude
compared to the narrow $\pm\frac{\lambda}{2}$ possible with a single
wavelength. The details of this technique as applied to Pyxis are described in
Lagadec et. al (2020)[37], but in brief a ‘synthetic’ wavelength is created
between each combination of wavelengths, given by:

$\Lambda=\frac{\lambda_{1}\lambda_{2}}{(\lambda_{2}-\lambda_{1})},$ (1)

for two wavelengths $\lambda_{1}$ and $\lambda_{2}$. Hence, as long as the
coarse metrology can provide an estimate of the distance to within the range
of the longest synthetic wavelength, we can provide a precise measurement
using this system.

To implement this in Pyxis, we use a similar setup to Lagadec et. al
(2020)[37], which is shown in Figures 7 and 8. First describing Figure 7, the
light from each diode is collimated by a 4.5 mm focal length lens (L1) and
injected into an optical fibre by a 6.25 mm focal length lens (L2). The
injection unit involves an in-house constructed diode-to-fibre injection
system that couples into an APC fibre connector of a polarisation maintaining
fibre via a polarising element. This results with a high ($\sim$30 dB)
polarisation extinction ratio into one of the orthogonal fibre modes. Note the
schematic only shows one of the two laser injection units.

Figure 7: Schematic of the metrology system as described in Section 3.2 and
adapted from Lagadec et. al (2020) [37]. Note that this schematic only shows
one out of the two laser diodes, and one of the four beam paths through the
chip towards the deputy retro-reflectors. The laser diodes are also used for
the time-of-flight coarse metrology (see Section 3.1), and the photodiodes
displayed are specifically included for this system. Figure 8: Metrology
architecture adapted from Lagadec et. al (2020)[37]. Ultimately, the metrology
system is designed to measure the differential distance between the deputies
($d_{1}-d_{2}$) to sub-wavelength precision. The light from the metrology is
emitted from transceiver units located at positions A, B, C and D. The light
is then retro-reflected from the deputies back into the metrology system in
the chief, providing four distance measurements $d_{a}$, $d_{b}$, $d_{c}$ and
$d_{4}$.

The fibres are fed through a V-groove into a photonic chip, where the light is
mixed by a directional x-coupler. This chip is currently being manufactured
through Australian National Fabrication Facility OptoFab node through direct
write ultrafast laser inscription (ULI) [39, 40], and has the advantage of
being able to combine elements of the coarse and fine metrology systems in a
compact form factor ideal for a future space mission.

The light is split into four separate beams (only one beam path is shown in
Figure 7) and output into a series of optical fibres. These are then fed into
four transceiver units, which consist of a 25 mm focal length collimating lens
(L3) and a quarter-wave plate. The collimated light is sent out of the chief
platform at positions A, B, C and D shown in Figure 8, reflected off of a
retro-reflector mounted on each deputy platform (element 10 in Figure 1), and
back into the transceiver unit. The quarter-wave plate ensures that the
outgoing and incoming polarisations are orthogonal and will not interfere with
each other.

A test configuration using two Fabry-Perot laser diodes as injection units has
been constructed. Continued development of this system, including thermal
stabilisation of the Fabry-Perot diodes and investigation of single-frequency
diodes is ongoing.

##  4 Injection and Beam Combination

###  4.1 Fibre Injection

With Pyxis being a relatively simple, single baseline interferometer, we aimed
to produce a beam combiner that was as simple as possible to maximise
throughput. We also aimed to have limited moving parts, to maximise the amount
of coherent integration. This will also serve well in translating this beam
combiner design to a future space-based mission. In this vein, we chose to
base the beam combiner around an integrated optics (IO) photonic chip modelled
after the success of IO chips in the GRAVITY [1] and GLINT [41] instruments.

The $\sim$18 mm diameter beams from each of the two deputy collector platforms
are reflected towards the central platform and enter into the fibre injection
system; shown in Figure 9 for one of these light paths. This system is also
designed to be CubeSat compatible, measuring 200 by 100 mm. The beam is
transmitted through a $f=100$ mm focal length infinity corrected tube lens
assembly (L1), then a piezo-controlled translating $f=6$ mm, 4 mm diameter
collimating lens (L2), which acts as the tip/tilt actuator through X-Y
translation orthogonal to the optical axis. We used Piezosystem Jena PXY 200
D12 stages, which when accounting for PWM filtering and voltage conversions,
provides us with about 150 µm of stroke.

An 595 nm cutoff dichroic is used to split the shorter wavelengths used for
alignment from the longer science wavelengths, which are transmitted towards
the fibre injection unit. The $<$595 nm wavelengths are polarisation-split and
recombined so that a single FLIR Firefly FFY-U3-16S2M-S camera can coarsely
image both input pupils for pupil alignment with one polarisation while
simultaneously performing fast ($\sim$200 Hz sampling rate) tip/tilt with the
other polarisation. The control system is explained in Section 5.3.

The pupil viewing path is designed such that the light passes through a $f=48$
mm lens (L4), which is focused at a pupil imaging plane located before the
dichroic (seen in Figure 9(a)). The camera images both paths through a $f=12$
mm imaging lens (L5) focused at infinity with a right-angled prism glued in
front to ensure the camera fits within the tight footprint. It is also worth
mentioning that while the footprint is tight, space inside the unit has been
allocated to allow the insertion of a star tracking camera. This will be
utilised for the CubeSat version of Pyxis.

(a) Schematic of one of the two optical paths for the fibre injection system,
as explained in Section 4. The various coloured beams denote different light
paths of the subsystem: light redwards of 595 nm is sent towards the beam
combiner (red), whereas the shorter wavelengths are sent into an alignment
path, where one polarisation is used for tip/tilt correction (orange), and the
other polarisation images the pupil (yellow). The alignment paths are imaged
onto a camera within the assembly. Note that only one of the two fibre
injection units are mounted on a linear stage for fringe tracking.

(b) CAD model of the fibre injection and alignment system.

(c) Photograph of the assembled fibre injection and alignment system.

Figure 9: Schematic and model of the fibre injection system. The purpose of
this subsystem is to inject the fibres of the beam combiner with the light
from the deputy platforms, which it performs through the monitoring and
control of a piezo controlled tip/tilt actuator. The system also
simultaneously images the pupils of both incoming beams.

###  4.2 Science Beam Combiner

The longer wavelengths, used for science, are injected into polarisation
maintaining 630-HP fibres using two $f=4.5$ mm, 3 mm diameter achromat lenses
(L3). One of the arms is mounted onto a small SmarAct SLC-1720-L translation
stage, with 20 nm steps and 8 mm of stroke, for fine path control and fringe
tracking. The fibres transport the light from the upper platform towards the
bottom platform, where the beam combiner is located. The fibres are then fed
into a V-groove, which is attached to the photonic chip.

The chip features a “tricoupler” waveguide scheme, where three inputs are fed
towards each other in a equilateral triangle formation, and then fed back out
in three outputs. As described in Hansen et. al. (2022) [42], the tricoupler
results in the output beams having a phase shift of $\frac{2\pi}{3}$, and
allows for the full recovery of the complex coherence in a single frame
without modulation (see Sections 2.3 and 2.4 in that paper). The three outputs
that this chip provides is the minimum possible while retaining the above
qualities of full information without modulation, and thus maximises
throughput. The photonic chip was measured to have a mean throughput of 85
$\pm$ 7%, and a coupling ratio between 33:33:33% and 51:31:17% over the
science bandpass of approximately 620 nm to 760 nm. Because we only have two
inputs from the telescopes, the central input of the chip was not used. More
details regarding this photonic chip, including experimentally retrieved
visibilities and group delay, and a schematic of the chip and attached
V-groove, can be found in Hansen et. al. (2022) [42]. This paper also details
the reduction algorithms using a pixel to visibility matrix (PV2M), and group
delay extraction for fringe tracking based on Fourier transform numerical
integration. More details on the fringe tracking control loop can be found in
Section 5.4.

From the beam combiner chip, the three outputs are fed through a $f=12.5$ mm,
6.25 mm diameter collimating lens into a custom spectrograph. This
spectrograph includes a Wollaston prism for splitting linear polarisations
(and hence allowing for both polarisation calibration and polarimetry on
astrophysical sources) and a 45∘ BK7 dispersive prism. This results in the
spectrograph having a spectral resolution of $R\sim 50$, which was chosen as a
balance between throughput (i.e reducing the number of channels on the
detector) and scientific usefulness. The dispersed light is then fed through
an $f=15$ mm imaging lens onto the scientific camera. Due to the custom
optical components and small space limitations, a resin printed optical mount
was designed (shown in Figure 10) to hold the spectrograph lenses and prisms.
The mount contains a variety of screw holes and sprung inserts to ensure a
kinematic mount of all components.

Figure 10: 3D printed resin mount for the spectrograph. From left to right,
the optical components are: a $f=12.5$ mm collimating lens, a Wollaston prism,
a 45∘ BK7 dispersion prism and a $f=15$ mm imaging lens.

(a) CAD model of the beam combiner

(b) Assembly of the beam combiner.

Figure 11: Design and assembly of the science beam combiner, including the
photonic chip, spectrograph, camera and back illumination optics.

The full beam combiner design can be seen in Figure 11. The photonic chip is
mounted to a manual translation stage for focus adjustment; due to the limited
thermal expansion expected in the system, we do not anticipate needing to
constantly adjust this axis of motion. The chip is also mounted on top a piezo
stack to allow for small translations in the vertical direction; ensuring that
the centre of the output beam lines up with detector pixels.

The scientific detector chosen was a QHY2020 camera with the GPixel Gsense2020
BSI sensor. This was chosen for its low readout noise capabilities and high
frame rate. The camera is mounted with a set of telescope tube rings and is
connected to the spectrograph with a series of adapting rings. The whole
system fits on a 150x300 mm breadboard, and in principle the size footprint
could be made smaller through choosing a much smaller camera. The beam
combiner also features a small back-illumination setup, consisting of a 590 nm
LED that is reflected off of a miniature 45∘ mirror that is able to be flipped
in and out of the optical path onto the output waveguides of the photonic
chip. This will assist in fibre injection alignment, and show which pixel on
the pupil tracking camera corresponds to the science fibres.

###  4.3 Pupil Alignment Procedure

Before on-sky observations can be made, a variety of offsets and calibrations
are required to ensure that the system can be easily aligned. Notable among
these is a method to ensure an initial pupil alignment to within the capture
range of the pupil camera. This comes from the coarse metrology system, with
which we can identify a pixel offset from the two metrology LEDs corresponding
to the exit aperture.

Let us consider the two LEDs with coordinates $\mathbf{x_{1}}$ and
$\mathbf{x_{2}}$ that are measured from the mechanical design, as well as the
target exit aperture coordinate $\mathbf{x_{t}}$. These coordinates are a
projection of the absolute coordinates onto the plane perpendicular to the
baseline vector. We can then define a relationship between these three
positions with respect to two parameters, $\beta$ and $\gamma$:

$\displaystyle\mathbf{x_{t}}=\mathbf{x_{1}}+\beta(\mathbf{x_{2}}-\mathbf{x_{1}})+\gamma\mathbf{R}_{90}(\mathbf{x_{2}}-\mathbf{x_{1}}),$
(2)

where $\mathbf{R}_{90}$ is a 90 degree rotation matrix.

On the coarse metrology camera, we then measure the two dimensional angles
${\bf\it\alpha_{i}}$ corresponding to the two LEDs and can thus measure the
angle of the exit aperture:

$\displaystyle{\bf\it\alpha_{t}}=(\beta+\gamma\mathbf{R}_{90}){\bf\it\alpha_{2}}+(1-\beta-\gamma\mathbf{R}_{90}){\bf\it\alpha_{1}}.$
(3)

This angle can also be calculated when considering the frame of reference of
the chief:

$\displaystyle{\bf\it\alpha_{t}}=\frac{\mathbf{x_{0}}+{\bf\it\alpha_{c}}d}{d},$
(4)

where $\mathbf{x_{0}}$ is the aperture coordinate also taking into account the
offset from the chief metrology camera and $d$ is the distance between the two
platforms. The angle ${\bf\it\alpha_{c}}$ is calibrated in the lab using the
back-illuminated beam from the science camera, with the tip/tilt piezos at the
centre of their range. This reference pixel is identified ahead of time
through back illumination and set of retro-reflectors, and is the position on
the tip/tilt sensing camera where the light is injected into the fibre.

Hence, after calibrating the system, we simply need to adjust the deputy such
that the LED angles ${\bf\it\alpha_{1}}$ and ${\bf\it\alpha_{2}}$ satisfy:

$\displaystyle(\beta+\gamma\mathbf{R}_{90}){\bf\it\alpha_{2}}+(1-\beta-\gamma\mathbf{R}_{90}){\bf\it\alpha_{1}}=\frac{\mathbf{x_{0}}+{\bf\it\alpha_{c}}d}{d},$
(5)

while simultaneously measuring the distance $d$ through the approximation:

$\displaystyle
d\approx\frac{|\mathbf{x_{1}}-\mathbf{x_{2}}|}{|{\bf\it\alpha_{1}}-{\bf\it\alpha_{2}}|}.$
(6)

Using the small angle approximation here is adequate for all platform
separations exceeding 0.5m.

We mention here that while this calibration and alignment procedure will work
for Pyxis, it is insufficient for the space version; in a space environment,
we are not able to move the deputies in all directions. Instead, we change the
angle ${\bf\it\alpha_{c}}$ by changing the central target positions of the
tip/tilt piezo stages.

##  5 Control System

###  5.1 Sensors and Architecture

A key part of achieving the stability and positioning accuracy required for
the interferometry is the navigation and control system. This system is
comprised of a variety of sensors, actuators, and processing computers spread
across the three robotic platforms and interfaces with all Pyxis systems. Here
we describe an overview the physical elements that contribute to the control
of Pyxis, followed by the software architecture and the control system design.

It is clear from Figure 1 that many of the Pyxis subsystems are devoted to
metrology and navigation at varying levels of granularity in order to achieve
sub-wavelength precision, with coarse sensors providing sufficient accuracy
for unambiguous operation of higher resolution sensors. Absolute attitude
measurement is achieved using star tracker cameras on each deputy, described
further in Section 5.2. Attitude is also tracked via inertial measurements,
using six 3-axis accelerometers on each robot, and a fibre laser gyroscope
(FLG; VG035LND from Fizoptica) on the chief robot. The coarse metrology camera
also captures the position of the deputy satellites in 3 dimensions with
respect to the chief body frame. The central frame of reference, however, is
defined through a high precision star tracker on the chief robot. This star
tracker consists of a finite-conjugate version of the diamond-turned
telescopes described earlier in Section 2 combined with a FLIR Blackfly camera
containing an IMX178 sensor. This allows us to obtain a $\pm 40$ arcminutes
field of view, with 1.5” per pixel; ample enough to sample the FWHM of a guide
star under the seeing conditions of Mt Stromlo without pixel phase errors.
This, together with the attitude measurements supplied from the sensors listed
above, provides Pyxis with a reference frame independent from the Earth and
co-moving with the platforms themselves.

Critically, the FLG is used to measure the angle of rotation about the axis
orthogonal to the star and baseline vector. The FLG and star tracker together
enables the fine metrology measurements to to moved from the chief rigid body
frame to an intertial frame, in order to predict open-loop fringe motion. We
characterised the FLG to ensure it was within specifications, through driving
the gyroscope with a sinusoidal input at 0.1 Hz. A plot of the FLG voltage,
converted into angular velocity measurements through a gain of 0.152 rad/s/V,
against the integrated accelerations of the accelerometers, again in rad/s, is
shown in Figure 12. We see that the FLG does not drift substantially over a
long 100 s test. The RMS noise of the FLG was found to be 9.21$\times 10^{-6}$
rad/s, and the bias was calculated as $5.74\times 10^{-6}$ rad/s; indicating
that at a power bandwidth of 60 Hz, the FLG exhibits a random walk of 0.26”/s,
which is within the desired specification of 0.003 deg/hr1/2.

Figure 12: Plot of the fibre laser gyroscope (FLG) output angular velocity
against the integrated acceleration measured by accelerometers for a
sinusoidal input at 0.1 Hz.

Interfacing the array of sensors and actuators is a set of computers
comprising the physical elements of the control system. On the chief robot is
a PC with a 6-core i5-8500 processor, responsible for tracking the dynamic
state of all three robots, as well as coordinating the many systems mounted on
the chief robot (fibre optic gyroscope, beam combiner, time-of-flight
metrology, etc.). Multiple Teensy 4.1 microcontrollers are also connected to
the PC in order to manage all non-USB interfaces. The PC is connected to WiFi,
over which requests can be sent to its servers, and over which it can send
requests to servers running on the deputy robot computers. The deputy robots
mirror this configuration, but with a mini PC (Intel NUC with 4-core i5-10210U
processors) reflecting the fewer number of USB connections required and the
reduced computational demands. A schematic of this interface architecture for
the control system can be found in Figure 13.

Figure 13: Physical architecture of control elements and interfaces

In order to integrate this large number of different physical components, each
requiring different communication channels and protocols, a server-based
software architecture has been designed and implemented for a number of the
systems. In this architecture, each interface is given a server that can
respond to a variety of requests depending on the nature of the system it
interfaces. For example, a microcontroller on each deputy robot is used to
monitor voltage and current draw, as well as control the LEDs used by the
coarse camera metrology. This microcontroller connects via USB interface to
the control computer, where a server program runs managing it. The server has
a set of requests it can respond to by calling various functions, such as
reporting the latest voltage measurement, with requests coming either locally
from other servers running on the control computer, or over WiFi from the
chief or the user interface.

The control system itself is implemented through a PID
(proportional–integral–derivative) controller, that aims to minimise the error
in linear and angular position and velocity provided from the sensors through
a feedback loop. That is, for a given desired state $r(t)$, and a measured
state $y(t)$, the control function $u(t)$ in terms of the error value $e(t)$
is:

$\displaystyle e(t)$ $\displaystyle=r(t)-y(t)$ (7) $\displaystyle u(t)$
$\displaystyle=K_{1}e(t)+K_{2}\int^{t}_{0}e(x)dx+K_{3}\derivative{e(t)}{t}$
(8)

where $K_{1}$, $K_{2}$ and $K_{3}$ are tuning variables that are used to
maximise the performance of the control loop. A linear-quadratic-regulator
(LQG) controller was also considered, but was deemed unnecessary for the types
of error correction needed in Pyxis.

###  5.2 Pointing Control and Star Tracking

To ensure that Pyxis has sufficient attitude control, we rely on a pair of
star trackers on each deputy platform to provide an estimate of the position
and orientation of the robot. These star trackers take an image of the sky
using a 5∘ FOV, $f=50$ mm lens at a rate of approximately 3 Hz. This provides
a balance between attitude update speed and the ability to detect numerous
stars. An algorithm incorporating the Tetra3***GitHub:
https://github.com/esa/tetra3 and Astronomy.net[43]†††Website:
https://astrometry.net plate solvers then extracts the centroid positions of
the brightest stars in the image and matches them to a set of known stellar
positions located in an index file. These index files were compiled for
Astrometry.net utilising the Tycho-2 catalogue[44]. The matched positions are
then used to provide an estimate for the right ascension ($\alpha$) and
declination ($\delta$) of the centre of the image, as well as the position
angle of the image (angle of the top centre of the image from the North
Celestial Pole). These angles are converted into an altitude/azimuth/position
angle quaternion for use in correcting and adjusting the attitude of the
robot.

We tested the plate solving algorithm for two quantities: speed and accuracy.
The former is particularly important, as it cannot be slower than the frame
rate of the camera and, by extension, the attitude update speed. It is for
this reason that we adapted numerous plate solving algorithms to create a
high-speed version. In our tests, we found that the program could output an
attitude quaternion from an image in between 0.2 and 0.3 seconds ($\sim$ 4
Hz), which is sufficient for a frame rate of 3 Hz. We then tested the accuracy
by obtaining a number of random on-sky images using the same lens, running the
plate solver, and comparing the extracted positions with the positions located
in the index file. Each image was manually checked to ensure that the plate
solver matched with the correct stars. We converted the ($\alpha,\delta$)
coordinates of each star into polar coordinates to give us an estimate of the
radial position error and the azimuthal orientation error. We found that the
RMS error in the radial direction was about 2”, and the RMS error in
orientation was 30”. Our requirements are that the deputy platforms are able
to measure their angle to $\pm$100” in an angle about the star pointing vector
(that is, the position angle of the image) and $\pm 20$” in the other two
axes. Hence, our plate solver will be adequate in accuracy to function as an
attitude estimator for Pyxis.

The fine star tracker on the chief platform also utilises a plate solving
scheme, although it has a much tighter angular position requirement of 0.2”
along the optical path axis, and as such utilises a much smaller field of view
(approximately 1 degree). This leads to a much slower solve between 1.5 and 2
Hz. To augment this, we implement a centroiding algorithm that augments the
star tracker at 3-4 Hz and tracks the position of the target star, using a
sufficiently long exposure time so that the star is blurred through seeing
(avoiding the noise introduced through the movement of a star’s position by
the same seeing). Of course, the critical angle of alignment of the chief
platform is managed by the FLG, and so the star tracking system acts as a
fallback for the rest of the attitude determination.

###  5.3 Tip/Tilt Control

This tip/tilt control of the starlight injection into the chief platform,
described in Section 4.1, is measured though a weighted centre of gravity
(WCOG) centroiding algorithm, given by

$\displaystyle\mathbf{c}=\frac{\sum_{ij}{w(\mathbf{x}_{ij})F(\mathbf{x}_{ij})\mathbf{x}_{ij}}}{\sum_{ij}{w(\mathbf{x}_{ij})F(\mathbf{x}_{ij})}},$
(9)

where $F(\mathbf{x_{ij}})$ is the flux of a given pixel $x_{ij}$ and
$w(x_{ij})$ is a super-Gaussian weighting function of the form:

$\displaystyle
w(x,y)=e^{-\frac{1}{4\sigma^{4}}((x-x_{0})^{2}+(y-y_{0})^{2})^{2}}.$ (10)

The centroid is measured with respect to the tip/tilt reference position in
pixel coordinates, described in Section 4.3.

Now, the difference between the measured centroid and the reference position
is controlled to zero through two control systems with differing timescales.
On short timescales (around 200 Hz), the position error is directly sent to
the X-Y piezo stages and controlled through a proportional controller. Due to
the relatively small stroke of the actuators, however, we then implement a
second control loop on longer timescales (about 5 Hz). The positional error is
sent to the star tracker on the relevant deputy platform, and converted into a
motion to move the deputy angle to alleviate the reliance on the piezos.

This motion is calculated via applying an offset to the reference pixel of the
star tracker. That is, the plate solver solves for the location of the field
of view offset to the centre of the image; this is also required for
accounting for the offset between the star tracking camera and telescope. To
convert between the tip/tilt camera pixel frame ($\mathbf{x}$) and the star
tracker camera frame ($\mathbf{x^{\prime}}$), we use the following conversion
matrix:

$\displaystyle\mathbf{x}=\begin{bmatrix}\pm\cos^{2}{\epsilon}&\pm 5\\\
\cos{\epsilon}\sin{\epsilon}+5\cos^{2}{\epsilon}&1\end{bmatrix}\mathbf{x^{\prime}},$
(11)

where $\epsilon$ is the elevation pointing angle of the deputy, the factor of
5 comes from the telescope magnification and the relevant signs are flipped
for the two deputies. We also note that before the control loop is closed
during alignment, the server controlling the tip/tilt system can still send
alignment corrections to the star tracker, so that the centroid spot is
located well within the range of the tip/tilt piezos.

###  5.4 Fringe Tracking Control

The fringe tracker relies on a group delay estimator of the spectrally
dispersed fringes. This estimator comes from the production of an array of
$\mathbf{P2VM}$ matrices for each spectral channel and polarisation,
calculated using the output flux ratios of each input following the method
outlined in Hansen et. al. (2022)[42]. To begin with, a matrix of trial delay
phasors, $\mathbf{\tau}$ were calculated:

$\displaystyle\mathbf{x}$ $\displaystyle=[-a,-a+\delta,...,a-\delta,a]$ (12)
$\displaystyle{\bf\it\tau}$ $\displaystyle=e^{2\pi
i\left(\mathbf{x}\otimes\frac{1}{{\bf\it\lambda}}\right)},$ (13)

where $a$ is half the coherence length, $\delta$ is the group delay
resolution, $\mathbf{x}$ is the vector of trial delays and ${\bf\it\lambda}$
is the vector of wavelength channels.

As described in Hansen et. al. (2022)[42], we obtain the complex coherence of
each $i$th wavelength channel and polarisation by multiplying the
instantaneous intensity by the $\mathbf{P2VM}$ matrix:

$\displaystyle\begin{bmatrix}\mathfrak{R}(\gamma_{i})F_{0,i}\\\
\mathfrak{I}(\gamma_{i})F_{0,i}\\\ F_{0,i}\end{bmatrix}$
$\displaystyle=\mathbf{P2VM}_{i}\cdot\mathbf{I}_{i}$ (14)
$\displaystyle\gamma_{i}$
$\displaystyle=\mathfrak{R}(\gamma_{i})+i\mathfrak{I}(\gamma_{i}).$ (15)

The fringe group delay envelope, $H$ is simply the multiplication of the
complex coherence vector with the trial delay phasor matrix, effectively
sampling the Fourier transform of the coherence.

$\displaystyle H(x)={\bf\it\tau}\cdot{\bf\it\gamma}.$ (16)

The group delay is then the delay $x$ corresponding to the maximum of the
power spectrum $P(x)=|H(x)|^{2}$ of this envelope.

To reduce the effect of calibration error in the $\mathbf{P2VM}$ matrix, we
subtract a “foreground” power spectrum derived from the intensity of the
combined beams a long way away from the fringe envelope. Furthermore, to
mitigate some of the effects of scintillation and rapid variance in the group
delay estimator, we employ a fading memory controller, where each
instantaneous delay is combined with the average of the previous delays
$(\bar{P}(x))$ scaled by a factor $\alpha$. Combining these two effects, we
obtain:

$\displaystyle\bar{P}(x)_{i}=\alpha(P(x)_{i}-P(x)_{\text{foreground}})+(1-\alpha)\bar{P}(x)_{i-1},$
(17)

where the group delay associated with $\bar{P}(x)_{i}$ is sent to the
controller.

The fringe tracker utilises a proportional velocity controller, where the
speed of the delay line is proportional to the group delay estimator, scaled
by a gain factor $\beta$. Specifically, at each estimation of the group delay,
the SmarAct stage is given a command to move a set distance, clocking with
steps at a period given by:

$\displaystyle p=20\frac{\beta}{x_{\text{gd}}},$ (18)

where the prefactor is used to scale the units of group delay into step counts
and the gain into units of milliseconds. Each estimation overrides the
previous command, providing a smoother control than implementing a positional
controller. The performance of the servo loop can be seen in Figure 14, using
a gain of $\beta=50$ ms and a fading memory parameter of $\alpha=0.95$. The
external delay was purposely moved forwards and backwards by approximately 7
µm to be corrected by the fringe tracker. Note that the external delay was not
perfectly accurate due to the tolerances on the stage used, and the external
stage also exhibited oscillatory behaviour (as shown by the large oscillations
of the group delay estimate). Nevertheless, we see that the controller
responds well to external delay movements, and holds the group delay constant
at zero with an RMS of approximately 200 nm. Further optimisation of the
parameters $\alpha$ and $\beta$ will be done on-sky.

Figure 14: Response of the fringe tracking servo loop to a commanded top-hat
external delay modulation, and associated group delay estimate.

The fringe envelope can also be used to perform a signal to noise (SNR)
estimation for fringe searching in the case of a failing or absent fine
metrology system. The signal is given by the maximum of the foreground
subtracted power spectrum, and the background is provided by RMS of the median
of the real and imaginary components of the envelope:

$\displaystyle r$ $\displaystyle=\text{median}(|\mathfrak{R}(H(x))|^{2})$ (19)
$\displaystyle i$ $\displaystyle=\text{median}(|\mathfrak{I}(H(x))|^{2})$ (20)
SNR
$\displaystyle=\frac{\text{max}(P(x)_{i}-P(x)_{\text{foreground}})}{\sqrt{r^{2}+i^{2}}}$
(21)

The SmarAct stage, assuming that the coarse metrology system has equalised the
baselines up to the stroke of 8 mm, can then scan for the fringe envelope and
stop when the SNR reaches a predetermined threshold.

##  6 Summary

The Pyxis interferometer, when complete, will be a critical step towards
verifying the technological readiness of space-based interferometry in the
search for Earth-like exoplanets. Specifically, utilising its free-form
platform nature and multi-stage metrology system, it will provide a
demonstration of satellite-like formation-flying without the cost of space
qualification and launch. Furthermore, due to its relatively simple optical
design and beam combiner, it will be able to do unique visible-light
polarimetric interferometry; the only instrument of its kind in the Southern
Hemisphere.

Over the next few years, our goal will be to conduct on-sky demonstrations and
observations, while simultaneously preparing for the next phase in the
project: a satellite version of the same system (see Hansen and Ireland
(2020)[25]). To our knowledge, this would be the first demonstration of
optical interferometric fringe tracking in space, and furthermore that
utilising multiple separate spacecraft in formation. With these demonstrators,
it is the authors’ hope that the technological barriers will be sufficiently
eased such that the final goal of a large scale, mid-infrared space
interferometer such as LIFE will be achievable in the coming decades. For it
is only such a mission that will truly begin to probe one of the greatest
scientific questions of our time: “are there habitable worlds out there?”

### Disclosures

This paper was derived and adapted from two 2022 SPIE Astronomical Telescopes
and Instrumentation conference proceedings: Paper 12183-1B “The Pyxis
Interferometer (I): Scientific Context, Metrology System and Optical
Design”[45] and Paper 12183-1C “The Pyxis Interferometer (II): Control System,
Telescope and Mechanical Design” [46]. The fine metrology section (Section
3.2) also adapts a portion of the 2020 SPIE Astronomical Telescopes and
Instrumentation conference proceeding paper 11446-2F ”Compact unambiguous
differential path-length metrology with dispersed Fabry-Perot laser diodes for
a space interferometer array”[37].

### Acknowledgments

We acknowledge and celebrate the traditional custodians of the land on which
the Australian National University is based, the Ngunnawal and Ngambri
peoples, and pay our respects to elders past and present.

The authors also acknowledge the substantial work of the entire Pyxis team and
associates in progressing this project and its results: Julien Bernard,
Nicholas Bohlsen, Logan Corry, Michael Ellis, Steven Ellis, Alex Fan, Shanae
King, Weihao Luo, Stephen Madden, Joseph Mangos, Patrick Miller, Michael
Polkinghorne, Laura Schlueter, Thomas Scott, Hancheng Shao and Kunlun Yan.

This research was supported by funding from Australian Research Council grant
No. DP200102383. JH acknowledges support from the Australian Government
Research Training Program, and the College of Science’s Dean’s Merit HDR
supplementary scholarship.

### Code, Data, and Materials Availability

The data used in this paper is available upon reasonable request to the
authors.

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Jonah Hansen is a PhD candidate at the Australian National University,
researching into space interferometry. In particular, he is assisting to
design and build the Pyxis interferometer - a ground based pathfinder for an
eventual space interferometer mission, LIFE (Large Interferometer For
Exoplanets), designed to find and characterise exoplanets. He is also
assisting in modelling different array architectures and beam combiners for
this latter mission.

Samuel Wade is a mechatronics and project engineer at the Australian National
University, assisting in the engineering aspects of Pyxis \- a ground-based
pathfinder for an optical space interferometer. Samuel has had previous
experience in satellite sensors and ground-based space surveillance.

Michael Ireland is a Professor of Astrophysics and Instrumentation Science at
the Australian National University. He obtained his PhD from the University of
Sydney in 2006, and has since held positions at the California Institute of
Technology, the Australian Astronomical Observatory, Macquarie University and
the University of Sydney. His research focuses on stellar astrophysics,
exoplanet formation, the search for life on other worlds and technologies
needed to support these endeavours

Tony Travouillon is an associate professor at the Australian National
University and is responsible for the instrumentation and telescope
development of its school of astronomy. He received his PhD in 2005 from the
University of New South Wales.

Tiphaine Lagadec is an instrument scientist whose interest is the development
of interferometric techniques and technologies to investigate stellar systems
at high resolution. Tiphaine obtained a PhD from the University of Sydney in
2020 and has participated in the fields of intensity interferometry, nulling
interferometry, space interferometry and solar coronagraphy.

Joice Mathew is working as an instrumentation scientist at the Australian
National University (ANU). His research interests include electro-optical
payload development, instrument modelling, systems engineering, space
instrumentation, and qualification. Joice obtained his Ph.D. in astronomical
space instrumentation from the Indian Institute of Astrophysics, Bangalore.
Before joining ANU, he worked as a visiting instrument scientist at the Max
Planck Institute for Solar System Research, Germany on the Solar Orbiter
mission.

Stephanie Monty is a research associate at the University of Cambridge. She
completed her PhD at the Australian National University in 2022. Her research
focuses are Galactic Archaeology, stellar dynamics, fibre fed spectrographs
and adaptive optics.

Adam Rains completed his PhD in astronomy and astrophysics at the Australian
National University in 2021 and is now a postdoctoral researcher at Uppsala
University in Sweden. His research interests sit at the intersection of
stellar and exoplanetary astrophysics – in particular the spectroscopic
characterisation of low-mass stars and their planets.

Biographies of the other authors are not available at this time.