# MICROSCOPE instrument description and validation

Françoise Liorzou1, Pierre Touboul1, Manuel Rodrigues1, Gilles Métris2, Yves
André3, Joel Bergé1, Damien Boulanger1, Stefanie Bremer4111Current adress:
DLR, Institute of Space Systems, Robert-Hooke-Str. 7, D-28359 Bremen, Germany,
Ratana Chhun1, Bruno Christophe1, Pascale Danto3, Bernard Foulon1, Daniel
Hagedorn5, Emilie Hardy1, Phuong-Anh Huynh1, Claus Lämmerzahl4, Vincent
Lebat1, Meike List4222Current adress: DLR Institute for Satellite Geodesy and
Inertial Sensing, Am Fallturm 9, D-28359 Bremen, Germany, Frank Löffler5,
Benny Rievers4, Alain Robert3, Hanns Selig4333Current adress: GERADTS GMBH,
Kleiner Ort 8, D-28357 Bremen, Germany 1 DPHY, ONERA, Université Paris Saclay,
F-92322 Châtillon, France  2 Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, IRD, Géoazur, 250 avenue Albert Einstein, F-06560 Valbonne,
France  3 CNES, 18 avenue E Belin, F-31401 Toulouse, France  4 ZARM, Center of
Applied Space Technology and Microgravity, University of Bremen, Am Fallturm,
D-28359 Bremen, Germany  5 PTB, Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Brunswick, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Dedicated accelerometers have been developed for the MICROSCOPE mission taking
into account the specific range of acceleration to be measured on board the
satellite. Considering one micro-g and even less as the full range of the
instrument, leads to a customized concept and a high performance electronics
for the sensing and servo-actuations of the accelerometer test-masses. In
addition to a very accurate geometrical sensor core, a high performance
electronics architecture provides the measurement of the weak electrostatic
forces and torques applied to the test-masses. A set of capacitive sensors
delivers the position and the attitude of the test-mass with respect to a very
steady gold coated cage made in silica. The voltages applied on the electrodes
surrounding each test-mass are finely controlled to generate the adequate
electrical field and so the electrostatic pressures on the test-mass. This
field maintains the test-mass motionless with respect to the instrument
structure. Digital control laws are implemented in order to enable instrument
operation flexibility and a weak position sensor noise. These electronics
provide both the scientific data for MICROSCOPE’s test of General Relativity
and the data for the satellite drag-free and attitude control system (DFACS).

††: Class. Quantum Grav.

  * December 2020

Keywords: General Relativity, Experimental Gravitation, Equivalence Principle,
Space accelerometers, Micro-satellite, Accelerometer, capacitive sensing,
electrostatic actuators, space mission, gravitation test.

##  1 Introduction

The MICROSCOPE (Micro-Satellite à traînée Compensée pour l’Observation du
Principe d’Equivalence) mission was fully dedicated to the in-orbit test of
the Universality of free fall, the so-called Weak Equivalence Principle (WEP)
[1, 2]. The accuracy objective was expected to be better than $10^{-15}$ [3].
The test principle consists in comparing the accelerations of two test-masses
of different composition in the Earth gravitational field. The payload of the
MICROSCOPE satellite is a double electrostatic accelerometer composed of two
cylindrical and concentric test-masses made of different materials. The
accelerometer measures the difference of accelerations needed to be applied on
the two test-bodies to keep their motion identical with respect to each other
while undergoing the same gravity. A non-null difference indicates a violation
of the WEP.

Based on a long legacy [4], MICROSCOPE payload developments re-use some very
well-known and mastered processes and technologies concerning the electronics,
as well as the accelerometer mechanical cores. The instrument is called T-SAGE
which stands for “Twin Space Accelerometer for Gravity Experiment’. The target
value at $10^{-15}$ requires space accelerometers with resolution in the
femto-g range. This means stringent specifications on both the instrument and
its environment, and extensive pre-flight testing to verify that this
performance level can be reached once in space. The instrument must be then
carefully characterized to identify the effects of any imperfections and to
measure its sensitivity to space environment.

This paper provides a more detailed description of the instrument T-SAGE than
the first result papers [5, 6]. It shows some steps among the most critical,
of verifications, validations and tests performed on ground and when possible,
confirmation of the characterized performance during the mission in space.

##  2 T-SAGE instrument description

###  2.1 Instrument sub-system and data production

The MICROSCOPE payload is actually made up of two independent “Space
Accelerometer for Gravity Experiment” (SAGE) sensors. The first one, designed
for the WEP test, embarks two test-masses made of different composition, one
in platinum rhodium and one in titanium alloys. The other sensor is identical
except for the test-masses composition, both in platinum alloy, and is used to
check the experiment (from measurement to the data process). Each SAGE is
composed of three units described hereafter. The Sensor Unit (SU) comprises
two concentric cylindrical test-masses each surrounded by a set of electrodes.
The sensor with platinum/titanium masses is called SU-EP and the sensor with
platinum/platinum masses is called SU-REF. The material composition and main
physical parameters of the test-masses are summarised in Tab. 1.

The payload is organised in three units as shown in Fig. 1. The Front End
Electronic Unit (FEEU) includes the capacitive sensing of test-mass motion,
the reference voltage sources and the analog electronics to generate the
electrical voltages applied onto the electrodes. The Interface Control Unit
(ICU) hosts the satellite power converters and the digital electronics
associated to the FEEU for the servo-control of the test-masses and the
interfaces to the data and power bus of the satellite.

The SU and FEEU have been integrated inside a highly stabilised thermal case
in the centre of the satellite while the ICU less thermally demanding is
accommodated in one of the satellite wall equipment.

The two-times-six acceleration measurements provided by the accelerometers of
one SAGE along and around its three perpendicular sensitive axes are also used
in real time on board the satellite by the Drag-Free and Attitude Control
System (DFACS)[7]. The “drag-free point” is defined by a selected combination
of the four test-mass acceleration measurements and was regularly changed to
the benefit of the experiment validation. The satellite has been in inertial
pointing or rotating (called also satellite spin) about the normal axis to the
orbital plane during the mission. The angular acceleration measured by the
accelerometers was associated to the satellite star-sensor in order to
estimate and then to finely control the attitude of the satellite.

The experiment was performed during two years in different configurations in
order to assess the instrument sensitivities to the environment. The data
delivered by T-SAGE contain scientific measurements for the equivalence
principle test (accelerations) and housekeeping measurements to monitor the
instrument (electronics modes, reference voltages, temperature, error status,
test-mass position and attitude) with precise time. The science data is used
in real time by the DFACS. All data are stored on board by the satellite
computer before being transmitted to the ground control station.

Table 1:  Test-masses main physical parameters measured in laboratory before
flight. (∗) Deduced by density comparison between slices of material extracted
at each end of the test-mass during manufacturing.

  * Parameter | IS1-SUREF | IS2-SUREF | IS1-SUEP | IS2-SUEP
---|---|---|---|---
Inner radius [mm] | 30.801 | 60.799 | 30.801 | 60.802
Outer radius [mm] | 39.390 | 69.397 | 39.390 | 69.401
Length [mm] | 43.331 | 79.821 | 43.330 | 79.831
Inertia about $X$ [kg mm2] | 125.0206 | 1442.454 | 125.0775 | 319.0266
Inertia about $Y$ [kg mm2] | 125.0021 | 1442.139 | 125.0524 | 318.9978
Inertia about $Z$ [kg mm2] | 125.0070 | 1442.214 | 125.0549 | 318.9867
Maximum relative difference | 0.0004 | 0.0007 | 0.001 | 0.0001
in moment of inertia |  |  |  |
Mass [kg] | 0.401533 | 1.359813 | 0.401706 | 0.300939
Density @ 20oC [g cm-3] | 19.967 | 19.980 | 19.972 | 4.420
Density homogeneity along $X^{(*)}$ | 0.04% | 0.05% | 0.1% | 0.001%

Figure 1: View of the T-SAGE elements with their characteristics (volume,
mass, consumption).

###  2.2 Operation principle

####  2.2.1 Servo-loop overview.

The measurement principle of the T-SAGE instrument relies on the levitation of
a cylindrical test-mass maintained centred with respect to surrounding
electrodes by means of electrostatic forces. The test-mass position with
respect to the surrounding electrodes is finely measured by capacitive
position sensors. The electrostatic forces are generated from a digital servo-
loop controller (Fig. 2). The resultant motions of the test-masses arround the
Earth follow the same gravitational orbit.

The set of control electrodes is placed on two cylinders inside and outside
each test-mass. The configuration of the electrodes is shown in Fig. 3 and 4.
The translation axes, $Y$ and $Z$, and the rotation axes $\theta$ and $\psi$
respectively about $Y$ and $Z$, are controlled by the eight electrodes of the
inner cylinder, two associated pairs for $Y$ and $\theta$ two associated pairs
for $Z$ and $\psi$. The position is obtained by averaging the capacitive
measurement of each associated pair while the rotation is calculated by their
half difference. The axis of the cylinder, $X$, and the associated rotation,
$\phi$, are controlled by the electrodes on the outer cylinder. The
measurement of the electrodes $\phi$, four pairs wired together as a single
pair, is possible thanks to flat areas on the outside surface of the test-mass
which create a variation of the distance between the mass and the electrode
when the mass rotates.

The $X$ axis provides the most sensitive measurements which are used for the
equivalence principle test.

This set of electrodes is used for both measuring the position of the test-
mass and applying the control voltage calculated by the servo command laws
running at about 1027 Hz (period of 973$\,\mu$s). This is made possible by the
use of a sinusoidal reference voltage $V_{d}$ (detection) applied on the test-
mass as schematised in Fig. 2. This is typically a 100 kHz signal of a few
volts; the specific amplitude is chosen in conjunction with the desired
detector gain, while the 100 kHz detection frequency is set much higher than
the control loop bandwidth to reduce the sensor noise.

Figure 2: Schematic of one degree of freedom servo-loop control.

By properly combining the electrodes, six servo-loops channels are controlling
independently the translations and attitudes of the test-mass. The operation
of the accelerometer is detailed hereafter in section 2.2.2. The signals are
digitised in the loop. The selected control law for each mass (each degree of
freedom) is a PID (Proportional, Integral, Derivative) configuration. The
optimisation of these controllers taking into account the test conditions are
made possible by the use of parameters patchable inside the digital
electronics of the ICU.

The range of the inertial sensor is limited by the voltage range that can be
applied to the electrodes, but must in particular exceed the weak residual
accelerations managed by the satellite control. The voltage applied on the
electrodes depends on the geometrical and electrical configuration and on the
mass of the test-mass (see section 2.2.2).

The value of the test-mass DC voltage, $V_{p}$, can be set to modify the
electrostatic acceleration behaviour: the higher $V_{p}$, the easier the
acquisition of the test-mass but the higher the noise. In order to facilitate
the acquisition of the test-mass during the free-fall tests [8] in ZARM Bremen
tower, the $V_{p}$ voltage and the maximum electrode range voltage have been
specifically increased to 90 V with the Engineering Model Electronics only :
component derating rules don’t allow this increase on flight models.

Both $V_{d}$ and $V_{p}$ voltages are fed to the test-mass through a very thin
gold wire described in section 2.3.3.

Scientific measurements are produced by T-SAGE at a frequency of 4Hz, they are
of two kinds. The first one is devoted to the satellite Drag-Free and Attitude
Control System (DFACS): linear and angular accelerations are deduced from the
forces and torques that the controller commands. The second one was originally
devoted to the scientific measurement: it is directly picked up on the two $X$
electrodes, in order to reduce the defects of the actuator electronics by the
gain of the loop (Readout circuit in Fig. 2). The PID corrector is adjusted to
fix the electrostatic levitation bandwidth to a few Hz, which is compatible
with the measurement frequency needs and with the DFACS frequency bandwidth.

####  2.2.2 Forces and torques.

As shown in Fig. 3 the set of $n$ electrodes surrounding the hollow test-mass,
inside or outside, exerts a total force along or around any degree of freedom
on the test-mass that can be expressed by:

$F=\sum_{n}{\frac{1}{2}\nabla(C_{n})(V_{n}-V_{p}-v_{d})^{2}}$ (1)

Where $V_{p}$ is the DC voltage applied on the test-mass, $V_{n}$ the voltage
applied on the $n$th electrode and $C_{n}$ the capacitance between the $n$th
electrode and the test-mass. $v_{d}=\sqrt{2}\,V_{d}\,\sin(\omega_{\rm
100\,kHz}t)$ is the 100 kHz voltage applied on the test-mass superimposed to
$V_{p}$: $V_{d}$ is the root mean value (rms).

For $X$, the expression of the capacitance is established by considering that
the electrode is long enough with respect to the front end of the test-mass. A
2D axisymmetric simulation of electrostatic field showed that the field lines
are not disturbed anymore when the length of the electrode is greater than 2
times the thickness of the hollow test-mass cylinder. In Fig. 3, there are two
electrodes to be considered, $X+$ and $X-$. The capacitance of the two
electrodes with respect to the test-mass is given by:

$\begin{split}C_{x_{+}}=\frac{2\pi\epsilon_{0}(h+x)}{ln(R_{x}/R_{p})}\\\
C_{x_{-}}=\frac{2\pi\epsilon_{0}(h-x)}{ln(R_{x}/R_{p})}\end{split}$ (2)

with $R_{x}$ the electrode $X$ radius, $R_{p}$ the test-mass outer radius, $h$
the electrode length covering the test-mass when it is centred. The difference
$(C_{x_{-}}-C_{x_{+}})$ is representative of the displacement $x$:
$(C_{x_{-}}-C_{x_{+}})=-\frac{4\pi\epsilon_{0}}{ln(R_{x}/R_{p})}x$. A
capacitive sensor transforms the difference of capacitance into a proportional
voltage which enters at the input of the digital control loop.

Figure 3: Electrode arrangement for $X$ (left), $Y$, $Z$, $\Theta$ and $\Psi$
(right).

Due to the electrical configuration, the force control along $X$ is obtained
by:

$F_{x}=\frac{1}{2}[\nabla(C_{{x_{+}}})(V_{x+}-V_{p}-v_{d})^{2}+\nabla(C_{{x_{-}}})(V_{x-}-V_{p}-v_{d})^{2}]$
(3)

When the servo-loop operates, $V_{x+}=v_{x}+V^{\prime}_{p_{x}}$ and
$V_{x-}=-v_{x}+V^{\prime}_{p_{x}}$, Eq. (3) simplifies into:
$F_{x}=-\frac{4\pi\epsilon_{0}}{ln(R_{x}/R_{p})}(V_{p}-V^{\prime}_{p_{x}})v_{x}$
in the control bandwidth lower than $10\,$Hz,

Indeed, the mean value at low frequency of the AC voltage $v_{d}$ is null. In
this expression $v_{x}$ is the voltage calculated by the servo-loop to control
the test-mass motionless. $V^{\prime}_{p_{x}}$ is a DC bias voltage applied to
each $X$ electrode. The force divided by the mass $m$ of the test-mass
represents the electrostatic acceleration that compensates the other
accelerations applied to the test-mass in order to maintain the relative
motion null [9]. The acceleration measurement
$\Gamma_{x}=\frac{1}{m}F_{x}=\hat{G_{ex}}v_{x}$ along $X$ is determined by
$v_{x}$ and computed into an acceleration with the definition of the physical
gain :
$\hat{G_{ex}}=-\frac{4\pi\epsilon_{0}}{m\,ln(R_{x}/R_{p})}(V_{p}-V^{\prime}_{p_{x}})$
where $(V_{p}-V^{\prime}_{p_{x}})$ fixes the range of control for the
considered geometry.

Figure 4: Electrode arrangement for $\phi$ measurement.

The same principle can be applied to each degree of freedom $w$, the voltage
applied to the electrode is: $V_{w\pm{}}=\pm{}v_{w}+V^{\prime}_{p_{w}}$. As
shown below, the physical gain becomes proportional to
$(V_{p}-V^{\prime}_{p_{w}})$. When the bias $V^{\prime}_{p_{w}}$ is null,
$V_{p}$ defines the range of the control for the $w$ axis, otherwise the range
of the control is reduced or increased depending on the sign of
$V^{\prime}_{p_{w}}$. The $V_{p}$ voltage is a very stable voltage reference
applied on the test-mass. $V^{\prime}_{p_{w}}$ is produced by a digital
controller and converted by a Digital-to-Analog Converter (DAC) to an analog
DC voltage.

For all degrees of freedom, the electrodes are associated by pair: one set in
the positive direction and the other one in the negative one. For $Y$
(respectively $Z$), two electrodes $Y1+/-$, $Y2+/-$ (resp. $Z1+/-$, $Z2+/-$)
are combined for the position measurement with the help of the capacitive
sensing. The capacitance is given by:

$C_{Y_{k}s}=\frac{\epsilon_{0}S}{e_{\rm in}+s\,y+s(2k-3)\,\psi\,d}$ (4)

where $S$ is the area of one $Y$ or $Z$ electrode, $e_{\rm in}$ the gap
between inner electrodes and the test-mass, $d$ is the mean distance of the
electrode centre from the $Y$ axis (the lever arm of the rotation torque), $k$
denotates the electrode 1 ($k=1$) or 2 ($k=2$) in Fig. 3 and $s=+1$ for the
electrodes in the positive side of the axis and $s=-1$ for the ones on the
negative side.

When the test-mass is controlled by the accelerometer servo-loop to be
motionless, $y\ll{}e_{\rm in}$ and $\psi d\ll{}e_{\rm in}$, the difference
$\sum_{k}(C_{Y_{k-}}-C_{Y_{k+}})$ is representative of the displacement $y$ at
first order and simplifies to
$\sum_{k}(C_{Y_{k-}}-C_{Y_{k+}})\approx\frac{4\epsilon_{0}S}{e_{\rm
in}^{2}}y$.

In the same way, the difference $\sum_{s}{(C_{Y_{1s}}-C_{Y_{2s})}}$ is
representative of the angular motion $\psi$ and at first order simplifies to
$\sum_{s}{(C_{Y_{1s}}-C_{Y_{2s})}}\approx\frac{4\epsilon_{0}S\,d}{e_{\rm
in}^{2}}\psi$.

The total force exerted by the four $Y$ electrodes along $Y$ is evaluated with
the applied voltage on each electrode “$ks$”
$V_{yks}=-s\,v_{y}+s(2k-3)v_{\psi}+V^{\prime}_{p_{y}}$ :

$F_{y}\sim-\frac{4\epsilon_{0}S}{e_{\rm
in}^{2}}(V^{\prime}_{p_{y}}-V_{p})v_{y}+\frac{4\epsilon_{0}S}{e_{\rm
in}^{3}}((V^{\prime}_{p_{y}}-V_{p})^{2}+v_{y}^{2}+v_{\psi}^{2}+V_{d}^{2})y$
(5)

The linear physical gain along $Y$ of the accelerometer is
$\hat{G_{ey}}=-\frac{4\epsilon_{0}S}{me_{\rm
in}^{2}}(V^{\prime}_{p_{y}}-V_{p})$.

The term $\frac{4\epsilon_{0}S}{e_{\rm
in}^{3}}((V^{\prime}_{p_{y}}-V_{p})^{2}+V_{d}^{2})$ represents the
electrostatic stiffness. The term $\frac{4\epsilon_{0}S}{e_{\rm
in}^{3}}(v_{y}^{2})y_{0}$ represents the quadratic term when the position
measurement is biased by $y_{0}$. Then the servo-loop controlling the output
of the PID will apply a force to displace the test-mass to compensate the
bias. Finally the term $v_{\psi}^{2}$ expresses the coupling between the
angular control and the linear control.

It has to be noted that at first order with respect to the displacement of the
test-mass, there is no stiffness and no non-linear term along $X$. Actually
the back action of the electrodes $Z$, $X$ and $\Phi$ also has an effect on
the $Y$ control. The force along $Y$ can be detailed as following:

$\displaystyle F_{y}\sim$ $\displaystyle-\frac{4\epsilon_{0}S}{e_{\rm
in}^{2}}\frac{\sin(\alpha_{y}/2)}{(\alpha_{y}/2)}(V^{\prime}_{p_{y}}-V_{p})v_{y}$
(6) $\displaystyle+\frac{2\epsilon_{0}S}{e_{\rm
in}^{3}}(1+\frac{\sin(\alpha_{y})}{\alpha_{y}})((V^{\prime}_{p_{y}}-V_{p})^{2}+v_{y}^{2}+v_{\psi}^{2}+V_{d}^{2})y$
$\displaystyle+\frac{2\epsilon_{0}S}{e_{\rm
in}^{3}}(1+\frac{\sin(\alpha_{y})}{\alpha_{y}})((V^{\prime}_{p_{z}}-V_{p})^{2}+v_{z}^{2}+v_{\theta}^{2}+V_{d}^{2})y$
$\displaystyle+\frac{2\pi\epsilon_{0}R_{x}h}{e_{\rm
out}^{3}}((V^{\prime}_{p_{x}}-V_{p})^{2}+v_{x}^{2}+V_{d}^{2})y$
$\displaystyle+\frac{\pi\epsilon_{0}R_{\phi}L_{\phi}}{e_{\rm
out}^{3}}((V^{\prime}_{p_{\phi}}-V_{p})^{2}+v_{\phi}^{2}+V_{d}^{2})y$

where $e_{\rm out}$ is the gap between the outer electrode cylinder and the
test-mass. If an acceleration is applied along $X$, or a torque applied about
$X$, $v_{x}$ or $v_{\phi}$ is not null and it couples to the measurement along
$Y$. This point is discussed in Ref. [10]. The total electrostatic stiffness
$K_{Y}$ depends on the geometry of the inner and outer electrode cylinders :

$\displaystyle K_{Y}$ $\displaystyle=\frac{4\epsilon_{0}S}{e_{\rm
in}^{3}}((V^{\prime}_{p_{y}}-V_{p})^{2}+V_{d}^{2})$ (7)
$\displaystyle+\frac{4\epsilon_{0}S}{e_{\rm
in}^{3}}((V^{\prime}_{p_{z}}-V_{p})^{2}+V_{d}^{2})$
$\displaystyle+\frac{2\pi\epsilon_{0}R_{x}h}{e_{\rm
out}^{3}}((V^{\prime}_{p_{x}}-V_{p})^{2}+V_{d}^{2})$
$\displaystyle+\frac{\pi\epsilon_{0}R_{\phi}L_{\phi}}{e_{\rm
out}^{3}}((V^{\prime}_{p_{\phi}}-V_{p})^{2}+V_{d}^{2})$

A similar analysis can be performed on $Z$ axis in which $z$ plays the same
role as $y$. We can switch the role of $z$ and $y$ and the role of the
associated rotations $\theta$ and $\psi$.

For the angular torque about $Y$ ($\Theta$) or about $Z$ ($\Psi$), the
equations are quite similar with $V^{\prime}_{p_{\psi}}=V^{\prime}_{p_{y}}$,
respectively $V^{\prime}_{p_{\theta}}=V^{\prime}_{p_{z}}$ because $\psi$,
respectively $\theta$, is controlled by the same set of electrodes as $Y$,
respectively $Z$. The impact of $X$ and $\Phi$ on the rotation has to be
considered through the force multiplied by the lever arms defined by the
centre of electrodes. For $X$, if a force is exerted by the electrode along
$Y$ or $Z$ at a distance $\frac{1}{2}(L_{p}-h)$ (being the distance of the $X$
electrode centre from the $Y$ or $Z$ axis), a torque shall appear. In the same
way for $\Phi$ electrodes, the lever arm is $\frac{1}{4}L_{\phi}$, where
$L_{p}$ is the test-mass length and $L_{\phi}$ the $\Phi$ electrode length.
The total torque for $\Psi$ degree of freedom is thus given by:

$\displaystyle T_{\psi}\sim$ $\displaystyle-\frac{4\epsilon_{0}S\,d}{e_{\rm
in}^{2}}(V^{\prime}_{p_{y}}-V_{p})v_{\psi}$ (8)
$\displaystyle+\frac{4\epsilon_{0}S\,d}{e_{\rm
in}^{3}}\left((V^{\prime}_{p_{y}}-V_{p})^{2}+v_{\psi}^{2}+v_{y}^{2}+V_{d}^{2}\right)\psi$
$\displaystyle+\frac{4\epsilon_{0}S\,d}{e_{\rm
in}^{3}}\left((V^{\prime}_{p_{z}}-V_{p})^{2}+v_{z}^{2}+v_{\theta}^{2}+V_{d}^{2}\right)\psi$
$\displaystyle+\frac{2\pi\epsilon_{0}R_{x}h}{e_{\rm
out}^{3}}\frac{(L_{p}-h)}{2}\left((V^{\prime}_{p_{x}}-V_{p})^{2}+v_{x}^{2}+V_{d}^{2}\right)\psi$
$\displaystyle+\frac{\pi\epsilon_{0}R_{x}L_{\phi}}{e_{\rm
out}^{3}}\frac{L_{\phi}}{4}\left((V^{\prime}_{p_{\phi}}-V_{p})^{2}+v_{\phi}^{2}+V_{d}^{2}\right)\psi$

The physical gain of the angular acceleration $T_{\psi}/J_{z}$ is given by :
$\hat{G_{e\psi}}=-\frac{4\epsilon_{0}S\,d}{J_{z}e_{\rm
in}^{2}}(V^{\prime}_{p_{y}}-V_{p})$ where $J_{z}$ is the mechanical inertia
moment of the test-mass about $Z$.

For the control in rotation about $\Phi$, the angular motion is defined by the
sum of the difference of capacitance of the 4 pairs $\Phi$ electrodes (Fig. 4)
evaluated at first order to : $\Delta C_{\phi}\approx
16\epsilon_{0}\sin(\frac{\alpha_{e}}{2})R_{x}\frac{L_{\phi}}{e_{\rm
out}^{2}}\phi$. $\alpha_{e}$ is the angle defining the arc of the electrode in
Fig. 4 ($\sim 2\pi/8$ if we neglect the spaces between the electrodes). The
torque is expressed by:

$T_{\phi}\sim\frac{8\epsilon_{0}L_{\phi}\,R_{x}}{e_{\rm
out}}\left[1-\frac{1+\cos(\alpha_{f}/2)}{1-\cos(\alpha_{f}/2)}\right](V^{\prime}_{p_{\phi}}-V_{p})v_{\phi}$
(9)

where $\alpha_{f}$ is the angle defining the flat area on the test-mass. In
this particular case of $\Phi$ degree of freedom, the physical gain converting
the applied voltage to an angular acceleration becomes:
$\hat{G_{\phi}}=\frac{8\epsilon_{0}L_{\phi}\,R_{x}}{J_{x}e_{\rm
out}}\left[1-\frac{1+\cos(\alpha_{f}/2)}{1-\cos(\alpha_{f}/2)}\right](V^{\prime}_{p_{\phi}}-V_{p})$.
The test-mass has been defined with a particular relationship between its
length and its radiuses in order to have all moment of inertia identical about
all axes $J_{x}\sim J_{y}\sim J_{z}$. This equality is questioned by the
accuracy of the machined parts and the homogeneity of the material given in
Tab. 1. In the following, the scale factor is defined as the ratio of the
estimated physical gain $\hat{G_{ew}}$ to the real one $G_{ew}$, and should be
close to 1: $K_{1w}=\frac{\hat{G}_{ew}}{G_{ew}}$ ($w=x,y,z,\phi,\theta$ or
$\psi$).

Table 2:  Theoretical physical gain and position sensitivity for each degree
of freedom used in flight for the DFACS and for the ground segment.

  * Physical gain | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
---|---|---|---|---|---|---
|  $\mu$m s-2/V |  $\mu$rad s-2/V
IS1 SU-REF | 0.0688 | 0.161 | 0.161 | 2.09 | 4.65 | 4.65
IS2 SU-REF | 0.0538 | 0.438 | 0.438 | 1.29 | 7.03 | 7.03
IS1 SU-EP | 0.0689 | 0.161 | 0.161 | 2.09 | 4.65 | 4.65
IS2 SU-EP | 0.0805 | 0.992 | 0.991 | 5.79 | 1.57 | 1.58
Position sensor gain | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
|  $\mu$m/V |  $\mu$rad/V
IS1 SU-REF | 3.33 | 4.35 | 4.35 | 708 | 493 | 493
IS2 SU-REF | 3.85 | 3.23 | 3.23 | 100 | 191 | 191
IS1 SU-EP | 3.33 | 4.35 | 4.35 | 708 | 493 | 493
IS2 SU-EP | 3.85 | 3.23 | 3.23 | 100 | 191 | 191

###  2.3 Sensor Unit (SU) descriptions

The two Sensor Units SU-REF and SU-EP are fixed on a mechanical interface with
the satellite, named SUMI (Fig. 6). In order to allow the sensor alignments
during the integration on the satellite, an optical cube is implemented on the
SUMI as well.

Each Sensor Unit is a mechanical assembly of the following elements (Fig. 5):

  * •

Two test-masses: they serve as inertial reference for the drag-free and
attitude control system and as mass probes for the test of the equivalence
principle;

  * •

The electrode set for the capacitive sensing and for the electrostatic control
of the test-masses; they are made of gold coated silica cylinders;

  * •

A mounting base-plate made of gold coated silica for a precise positioning of
the electrode sets;

  * •

A sole plate made of Invar for the positioning of the mounting base plate, and
which supports the titanium inner housing and the vacuum tight housing;

  * •

Mechanical stops that prevent the test-mass to be electrically in contact with
the electrodes and that take the shock of any mechanical contact to preserve
the silica electrode set;

  * •

One 7$\mu$m gold wire connecting each test-mass used to apply a very stable
voltage on the test-masses;

  * •

A test-mass blocking mechanism in order to prevent any damage from the heavy
test-masses (0.4 kg to 1.4 kg) because of launch phase vibrations and micro-
satellite in-orbit release shocks. The test-masses are unlocked before the
first in-orbit switch on;

  * •

A vacuum system to maintain a sufficiently low vacuum and cleanliness of the
electrode set and to ensure the performance of the instrument;

  * •

A vacuum Invar tight housing encloser that closes hermetically the SU;

  * •

Hermetic electrical connectors.

Figure 5: Cross-cut view of the differential accelerometer sensor unit,
comprising two concentric accelerometers in a tight housing with the blocking
mechanism on the bottom of the SU and the vacuum system at the top of the SU.

####  2.3.1 Test-masses.

A particular effort has been paid on the definition of the test-mass shape.
This shape guarantees a certain low level of sensitivity to gravity
disturbances in the acceleration differential measurement. The gravity
acceleration module $g^{(i)}$ undergone by the $i$th test-mass of mass
$m^{(i)}$, considered perfect, from a source mass $M_{s}$ at distance $R_{s}$
is expressed at order 2 of the mechanical moments in [11]:

$g^{(i)}\approx\frac{GM_{s}}{R_{s}^{2}}\left[1+\frac{3}{2m^{(i)}R_{s}^{2}}(J_{y}^{(i)}+J_{z}^{(i)}-2J_{x}^{(i)})+...\right]$
(10)

where $J_{x}^{(i)}$ , $J_{y}^{(i)}$ and $J_{z}^{(i)}$ are the mechanical
moment of inertia around the respective test-mass axes $X$, $Y$ and $Z$. When
measuring the differential acceleration between two perfect concentric test-
masses, the term $g^{(1)}-g^{(2)}$ does not totally disappear [9]. A term
depending on the geometry remains at first of order:

$g^{(1)}-g^{(2)}\approx\frac{GM_{s}}{R_{s}^{2}}\frac{3}{2R_{s}^{2}}\left[\frac{1}{m^{(1)}}\left(J_{y}^{(1)}+J_{z}^{(1)}-2J_{x}^{(1)}\right)-\frac{1}{m^{(2)}}\left(J_{y}^{(2)}+J_{z}^{(2)}-2J_{x}^{(2)}-2J_{x}^{(1)}\right)\right]$
(11)

Equation (11) shows that for a sphere, with all mechanical moments being
identical, the gravity field should not depend any more on the test-mass
geometry. For cylindrical test-masses, by adjusting the length for a giving
radius to get $\left(J_{y}^{(i)}+J_{z}^{(i)}-2J_{x}^{(i)}\right)=0$, it is
possible to cancel the effect of local gravity at first order.

####  2.3.2 Sensor core: electrode cylinders and sole plate.

Each test-mass is surrounded by two cylinders (one inner, one outer) made of
silica (Suprasil 2®). The cylinders are engraved by ultrasonic machining
(ONERA patent) and gold coated in order to provide the necessary electrodes to
control the six degrees of freedom of each test-mass. They are integrated on
the Silica base-plate, also accurately machined and gold coated, see Fig. 6.
The ultrasonic machining ensures an accuracy up to $5\,\mu$m. The control of
the silica parts is performed at $1\,\mu$m accuracy with a 3D metrology
system. All silica parts are accurately mounted on an Invar® sole plate which
serves as reference for all the core centring and alignment. The sensor core
is closed by an inner housing and a clamp. The sole plate also hosts the
mobile parts of the blocking system.

Figure 6: Left: step of integration of the inner SUREF test-mass laying on the
stops and surrounding the first silica gold coated cylinder laying on the
silica reference base-plate. This base-plate is mounted on the polished Invar
sole plate. Middle: all gold coated cylinders integrated around the two test-
masses of the SUEP. Right: SUEP and SUREF fixed on the SUMI titanium gold
coated plate. SUREF is the SU closer to to the optical cube. Coaxial shielded
connectors deliver all the capacitive sensing information.

Internal circuit boards complete the mechanical core of the double
accelerometer. This core is closed by a hermetic tight housing composed of the
outer housing, the hermetic connector support and the vacuum management
system.

The vacuum is ensured passively in flight thanks to a getter material ST172®.
Its porous structure allows the gases to diffuse inside the getter where a
sorption mechanism acts irreversibly for the molecules of CO, CO2, O2, N2 and
in a reversible way for those of H2. An ionic pump is used on ground until
delivery. Through a resistive wire embedded in the material, the material is
heated to a temperature of 900 °C during 10 minutes. This is the activation
process that is applied when integrating the SU. Then during the life of the
instrument, the surface of the getter is exposed to a gradual saturation due
firstly to the residual degassing of the internal parts to the sealed
enclosure (first months after closing) and secondly to the porosity of the
materials of the enclosure and especially ceramics of the various sealed
crossings (storage on the ground until launch). Even if the leakage rate level
of the hermetic housing and connectors is very low (better than $10^{-10}$
mbar l s-1) and ensures the low vacuum level inside the sensor unit, the
getter has been periodically reactivated by heating to restart the process of
diffusion into the material and clean the surface.

In orbit, the vacuum has been specified to be lower than $10^{-5}$ Pa and the
outgassing rate inside the housing to be lower than $3\times{}10^{-10}$ Pa
m3s-1. A test, performed on a representative model from January 2007 to
September 2015, showed that the pressure was maintained under the
specification during 600 days over 930 days of test, only limited by the time
between two reactivations of the getter. This test duration largely covers the
ground test and pre-launch storage period. The hermiticity of the mechanical
interfaces is ensured by indium seals laid on the polished surfaces.

The external housing supports the electrical connection of the electrodes, the
test-masses and the temperature probes. This housing has been tested with
$10^{-9}$ mbar l s-1 ($10^{-13}$ Pa m-3s-1) tightness. The connectors are
D-subminiature to enable the transmission of more than 40 shielded signals
assembled in a specially developed circuit board.

####  2.3.3 Gold wire.

A 7$\mu$m diameter gold wire is connected to each test-mass. The goal of such
a mechanical link is to control the test-mass charge in order to avoid any
parasitic forces fluctuation due to the flux of charged particles in orbit and
to keep as stable as possible the voltage on the test-mass and thus the scale
factor. The gold wire is thin and as long as possible to minimise the
resulting stiffness and damping. The wires are made of gold (rust free) with
the main characteristics in Tab. 3.

Table 3:  Gold wire characteristics.

  * Material | pure gold stabilised with 5 ppm of beryllium
---|---
Diameter |  $7\,\mu$m
Length |  $\sim 20$ mm
Elongation |  $0.5$%
Tensile strength |  $1$ cN
Electrical conductivity |  $870\,\Omega$ m-1

This wire is implemented along the sensitive $X$-axis: the first end is glued
with electrically conductive adhesive on the bottom of the test mass, and the
other end is glued on a pin fixed on the base plate for the outer test-mass
and on a specific electrode for the inner one.

In orbit, before the release of the test-masses, the conduction between the
masses and the sources of references $V_{p}$ and $V_{d}$ is done both through
the gold wire and through contact with the mechanical stops. In order to
verify the integrity of the gold wire and thus their resistance to the launch
vibrations, an electrical test is performed. A remote control command opens
the connection to the stops through a relay. In presence of the gold wire, the
capacitive sensing keeps on giving the same signal representative of the test-
mass position since $V_{d}$ is still coming through the gold wire. In case of
gold wire failure, when the relay of the stops is open, $V_{d}$ is not applied
on the test-mass anymore neither through the gold wire nor through the stops:
the detector signal drops to zero. This simple test was successfully performed
in orbit at the beginning of the mission to green light moving towards the
next step, the commissioning phase.

####  2.3.4 Blocking system and mechanical stops.

The blocking system limits the test-mass motion during the launch vibrations.
It is retracted once in orbit in order to release the test-masses and start
the instrument operation. As shown in Fig. 5, it consists of three fixed stops
at the top end of each mass and three mobile stops at the bottom end of each
mass. The rounded end of each stop fits into a corresponding conical hole in
the test-mass. The shapes of both stop and hole are designed to prevent the
mass from sticking to each other after a prolonged contact under the locking
force. The retractation system consists of a pressure vessel which applies a
total $2500$ N locking force on the stops. Once in orbit, an electro-valve is
activated to open the vessel to the space vacuum, which allows a compressed
spring to push away the mobile stops. They are retracted by $150\,\mu$m, which
allows for a theoretical free motion of the test-mass of $\pm{}75\,\mu$m. As
the stops are not fully retracted from the holes the radial motion is also
limited to the same $\pm{}75\,\mu$m value thanks to the conic shape of the
holes. The limited free motion ensures an easier electrostatic levitation of
the test-mass in orbit. At the same time, the gap between the stops and the
mass must be as large as possible to limit the patch effect disturbances.

###  2.4 Electronics description

The payload electronics puts up together different functional sub-systems
distributed over the FEEU or the ICU boxes.

The Interface Control Unit (ICU) contains:

  * •

The Power conversion units (PCU) delivering 6 secondary voltage buses (5 V,
3.3 V, $\pm{}$15 V, $\pm{}$48 V) with line regulation and low noise and
temperature sensitivity. The PCU function is the only one to have a cold
redundancy;

  * •

The Digital Signal Processing (DSP) board for both the computation of the
servo-control signals and the interface with the satellite;

  * •

A serial high-speed (RS422 bi-directional) link that transmits back and forth
between FEEU and ICU the position data and actuation control signals at a
baud-rate of 1.25 Mbps;

  * •

The telemetry and command control with the satellite.

The Front End Electronics Unit (FEEU) comprises:

  * •

The Front-end capacitive position sensors and the amplifiers for the
electrostatic actuation;

  * •

The voltage references ($V_{p}$, $V_{d}$) for the test-masses, and the ones
for the Analog to Digital Converters (ADC).

The electrode signals, the test-mass voltages and the temperature probe
signals are carried out between the SU and the FEEU through three harnesses of
more than 40 pico-coaxial cables. The harnesses exhibit low thermal
conductivity and stray capacitance compatible with the tuning of the
capacitive bridge and the space outgassing requirements.

####  2.4.1 Position detector: Capacitive sensing.

Capacitive detectors are contact-less, high-resolution position sensors [12].
They consist in an electromechanical transducer and a conditioning electronic
circuit. The electromechanical transducer is constituted by a couple of
electrodes facing the test-mass opposite surfaces. The displacements of the
test-mass modulate the voltage difference between the electrodes and the test-
mass set to a $100$ kHz sine voltage $V_{d}$. The displacements induce thus a
variation of charge that is sensed by the detector conditioning electronics.
The conditioning electronics is based on a 100 kHz synchronous detection
scheme whose pumping frequency is far beyond the displacement bandwidth.
Capacitive detectors can thus be implemented on the same set of electrodes as
that used for actuation. The first stage of the electronics is a differential
transformer linked to the sensor electrodes. The differential transformer and
the capacitances of the test-mass/electrodes pair form a capacitive bridge. It
is terminated by the feedback capacitance of a charge preamplifier. The
preamplifier stage carries out a charge to voltage transformation resulting in
a scale factor from 16 to 80 V/pF after amplification by one or two band-pass
filters. The detector front-end is followed by a synchronous detection stage
(demodulation of the 100 kHz signal) and a DC amplifier that implements anti-
aliasing filtering. Finally, the output stage of the capacitive position
sensor is based on an analog-to-digital converter (24-bit $\Sigma\Delta$ ADCs)
which delivers position data to the control laws.

####  2.4.2 Actuation voltage.

The drive voltage amplifiers deliver two opposite voltages to opposite
electrodes through a superimposition network and from the outputs of the
control laws, delivered by a DAC. These amplifiers are specified to be very
symmetric and are mainly composed of a first stage providing the open gain of
the amplifier and a push-pull circuit providing the level of the necessary
high voltage ($\pm{}40$ V).

####  2.4.3 $V_{p}$ reference voltage.

The role of the $V_{p}$ voltage is to linearise the electrostatic actuation as
seen in section 2.2.2. In High-Resolution Mode (HRM), the $V_{p}$ voltage is
set to 5 V to minimise the back-action of the position sensor noise through
the electrostatic stiffness phenomenon while in Full-Range Mode (FRM), the
$V_{p}$ voltage is set to 42 V to increase the accelerometer full-scale range.
In FRM, all $V_{p}^{\prime}$ are set to 0 V. In addition to that, the $V_{p}$
voltage can be set to 7.5 V for characterisation purposes.

####  2.4.4 $V_{d}$ detection voltage.

The $V_{d}$ generator is an RC-oscillator at 100 kHz whose amplitude is servo-
controlled. The amplitude is set to 5 Vrms in HRM measurement phase and to 1
Vrms in all other modes in particular in FRM for the test-mass acquisition. In
FRM, the gain of the detectors is reduced to increase the range of the
position sensor.

####  2.4.5 Analog-to-digital conversion.

The $2\times{}6$ position signals delivered by the capacitive detectors are
digitised before being fed to the servo-control. Twelve analog-to-digital
converters (24-bit $\Sigma\Delta$ ADC7712) simultaneously sample the position
signals to ensure synchronisation of the measurements along the 6 degrees of
freedom. The sampling frequency, 1027 Hz, is consistent with the control
bandwidth. As a $\Sigma\Delta$ converter, the AD7712 relies upon a master
clock at 10 MHz that is distributed to all the FEEU ADCs. Each chip sets its
Data Ready output to low when the acquisition is completed, i.e.
simultaneously after a fixed number of master clock periods. A Field
Programmable Gate Arrays (FPGA) polls all the ADCs simultaneously,
deserialises the data and forms the packet to be transmitted to the ICU. For
the digital to analog conversion, the MAX542 (Serial link, 16-bit
quantisation, “latch” input for synchronisation, low power) is used. At
reception of the ICU packet, the FEEU FPGA dispatches the data to the 24 DACs,
the conversion is latched until the ADCs have completed their acquisition. The
delay between the acquisition of position and the actuation is therefore
deterministic without skew. The total delay, comprising the FEEU-ICU
transmission, the ICU processing, the ICU-FEEU transmission equals 2 sampling
periods (2/1027 sec).

####  2.4.6 ICU DSP board

The architecture of the DSP board is based on the radiation-hardened Atmel®
version of the SHARC ADSP 21020, the TSC21020F model. The DSP board is mounted
with the typical peripherals PROM, EEPROM, and SRAM. The input-output
constraints of the DSP (separate bus for program and data memory) made it
necessary to implement a companion FPGA which in addition to the handling of
the memory, takes in charge the interfaces with the satellite computer and the
FEEU. The FPGA is an RT54SX72-S from Actel®, hardened component insensitive to
latch-ups and upsets (triplication voted internally on an anti-fuse base). The
DSP has a 20 MHz cycle and operates the servo loop at a submultiple of this
frequency: 1027 Hz.

####  2.4.7 ICU Software

The DSP software of the payload is split into two pieces of code:

  * •

The Initialisation Software, which is stored and executed in PROM. It enables
the interface with the on board computer, loads the Application software and
if ordered by a specific telecommand (TC) completes the patch of the
Application software in EEPROM.

  * •

The Application software stored in EEPROM and executed in SRAM. It handles the
digital tasks of the accelerometers in a deterministic and cyclic way as:

    * –

receiving data from position sensors;

    * –

applying the control laws;

    * –

transmitting the actuation signals;

    * –

processing telemetries;

    * –

handling asynchronous or synchronous TC that come from the satellite on-board
computer (OBC).

####  2.4.8 On-board computer interface and telemetry package.

The data produced by the instrument are transmitted and synchronised to the
on-board computer by a serial link. They are transmitted in two telemetry
packets at 1 Hz for the “housekeeping data” and at 4 Hz for the science data.

The housekeeping or maintenance data are processed to track the good operation
of the instrument. Integrity checks are performed and reported to ensure the
consistency of the transmitted TM data. They contain the position and attitude
of the test-masses, the $V_{p}$ and $V_{d}$ voltage monitoring, the SU, the
FEEU and ICU temperatures, the synchronisation data, the FEEU and ICU status
(configuration, mode, flags), the secondary supply voltage monitoring, the SU
force sensor used before test-mass release, the Single Event Upset (SEU)
counter detected in SRAM and the Overload counter.

The 4 Hz sampled science telemetry, called $TM_{\rm 4Hz}$, contains the
acceleration measurement data:

  * •

12 degrees of freedom data are taken at the output of the PID controls (at
1027 Hz), filtered through a low pass infinite impulse response filter and
averaged through a simple averaging filter. A priori scale factor is applied
to generate an acceleration data directly usable on board by the DFACS;

  * •

2 specific out-of-the-loop measurements for the $X$ axis. An additional ADC
per test-mass measures the actual voltage applied to the $X$ electrodes. This
ADC operates synchronously with the PID control output ones. Therefore, the
processing chain is executed in parallel with those of the other axes. The
signal path is basically the same; however, the requirements for anti-aliasing
are different. There is a separate processing chain for this out of the loop
measurement which is transmitted in $TM_{\rm 4Hz}$ packets after the PID
output data.

The $TM_{\rm 4Hz}$ packets are formed four times per second with a phasing
parameterised with respect to the 1 Hz clock signal delivered to the ICU by
the On board Computer. The real-time cycle of the ICU is set by the ADCs the
FEEU. Moreover as the integration time of ADCs $T_{\rm
ADC}=512\times{}19/f_{clock}\approx 973\mu s$, with $f_{clock}=10\,$MHz the
ADC frequency clock signal, the theoretical number of samples in a $TM_{\rm
4Hz}$ packet is close to 257. Indeed, the number of samples is an integer
closest to the ratio $250ms/T_{\rm ADC}=256.99013$. The number of samples is
thus 257 most of the time. But as the ratio of clocks is not an integer, this
means that the 257${}^{t}h$ sample of the FEEU arrives for each $TM_{\rm 4Hz}$
packet closer and closer to when the $TM_{\rm 4Hz}$ packet is formed. There
comes a time when this 257${}^{t}h$ sample arrives ”too late” to be
integrated. This is when the ICU ”purges” the delay by considering only 256
samples for its packet. The delay is thus reset to zero and the next packet
has 257 samples. With perfect setting of the clocks the occurrence of 256
samples in a $TM_{\rm 4Hz}$ is 1 over 102 $TM_{\rm 4Hz}$. In flight due to the
FEEU and ICU clock tuning, the occurrence is one over 150 for the SU EP and 1
over 128 for the SU REF. The specification was set to an occurrence greater
lower than one over 100, and the measurements show some margins with the
needs.

##  3 T-SAGE instrument sub-systems test and performance assessment

###  3.1 SU on-ground verification of T-SAGE instrument

####  3.1.1 Test mass metrology.

The cylindrical test-masses have been finely produced and characterised in the
laboratory of Physikalisch-Technische-Bundesanstalt (PTB), in Brunswick, with
accuracy better than $2\,\mu$m [13]. The masses have been measured with a
maximum error of 0.025 mg. The density was evaluated by cutting two slices at
the ends of the real test-masses (giving also an estimation of the homogeneity
of the obtained material) and measuring the density of the slices at accuracy
better than 0.001 g cm-3.

The test-masses have 4 small flat areas along $X$ to break the cylindrical
symmetry and to manage the angular control of the test-mass about $X$ ($\Phi$
axis) while the length of the test-mass is optimised to keep an inertia
momentum identical about the 3 axes. The inertia momentums have been
calculated taking into account the measured dimensions and densities. The
dispersion of the measured geometrical characteristics, in addition to the
measured density, leads to a total relative dispersion of the inertia with
respect to the ideal ‘test-mass’ behaving like a homogeneous sphere: the worst
case of relative deviation has been evaluated to $10^{-3}$ and is sufficient
to minimise the effect of local gravity gradients [9]. A synthesis of the
test-mass metrology data is summarised in Tab. 1, in good agreement with the
geometry specifications up to $2\,\mu$m tolerance deviations.

####  3.1.2 Core metrology: test-mass centring.

The centring of the two concentric test-masses once levited in orbit is
specified to $20\,\mu$m [9]. This centring is achieved by the accuracy of
realisation of the different parts and their integration accuracy that leads
to produce a symmetric electrostatic field of one test-mass with respect to
the other. The integration of all parts has been realised with several
critical steps of 3D machine control of the partially assembled parts with up
to $1\,\mu$m accuracy. The capacitive measurement during the integration
process completes the information on the geometry metrology.

By positioning the $X$ axis vertically up or down under the Earth’s gravity
field, the test-masses rest on the stops at a distance of $\pm{}75\,\mu$m for
the SUEP and at $+90/-75\,\mu$m for the SUREF with respect to the theoretical
geometric centre. The capacitances of $X$ electrodes are estimated by
$C_{+}(x)=\frac{2\pi\epsilon_{0}}{ln(R_{x}/R_{p})}x+C_{0_{+}}$ and
$C_{-}(x)=-\frac{2\pi\epsilon_{0}}{ln(R_{x}/R_{p})}x+C_{0_{-}}$ and compared
to actual measurements for different test-mass displacements $x$ under
gravity. $C_{0_{+}}$ and $C_{0_{-}}$ are the capacitance values when the test-
mass is placed at the geometrical center ($x=0$). It comprises also border
effects mainly due to chamfers which bias the capacitance centre value. The
measurement of the capacitances in the two positions associated to the
knowledge of the geometry gives the position where the capacitances should be
balanced between $C_{+}(x)$ and $C_{-}(x)$: this is at this position that the
servo-loop controls the test-mass once in orbit. The balance is not the
geometrical centre but a position deduced by the capacitance measurements in
different positions. For each test-mass, the position $x_{0}$ verifying
$C_{+}(x_{0})=C_{-}(x_{0})$ is given by:

$x_{0}=\frac{1}{2\alpha}[\frac{1}{2}(C_{-}(d_{1})+C_{-}(d_{2}))-\frac{1}{2}(C_{+}(d_{1})+C_{+}(d_{2})]-(d2+d1)$
(12)

where $\alpha=\frac{2\pi\epsilon_{0}}{ln(R_{x}/R_{p})}$, $d_{1}$ is the actual
position corresponding to the theoretical displacement of $+75\,\mu$m for the
SUEP and $+90\,\mu$m for the SUREF. In the same way, $d_{2}$ is the actual
position corresponding to the theoretical displacement of $-75\,\mu$m for the
SUEP and the SUREF. Because of the accuracy of the mechanical parts (the stops
in particular) and of the integration of the stops, $d_{1}$ and $d_{2}$ are
deduced from the metrology with a few micrometers accuracy. Eq. (12) assumes
that the equation of perfect concentric infinite cylinders applies. The term
$\alpha$ can also be deduced from the measurements:

$\alpha=\frac{1}{2(d_{1}-d_{2})}[(C_{+}(d_{1})-C_{+}(d_{2}))-(C_{-}(d_{1})-C_{-}(d_{2})]$
(13)

When replacing Eq. (13) in Eq. (12), the offcentering is deduced without
strong assumption on the capacitance formulas except linearity and reported in
Tab. 4.

A similar approach was performed with $Y$ and $Z$ but due to the distribution
of the 3 stops about $X$ (every $\frac{2\pi}{3}$ rad) at each end of the test-
mass, the position under gravity when relying on $Y$ or $Z$ is not as precise
as for $X$. That’s why in Tab. 4 the figures for $Y$ and $Z$ can be biased by
a few microns. In that case $d_{1}$ and $d_{2}$ are determined as upper or
lower limit values to verify that the free motion is sufficient to enable a
good operation of the servo control with margins. The offcentering along $Y$
and $Z$ in Table 4 are mainly determined by the metrology of the cylinders
once integrated by considering capacitive effects as negligible. On ground and
in flight measurements are compatible at 3$\sigma$ and shows the difficulty to
predict the flight centring to better than 8$\mu$m with ground metrology alone
mainly because it is not possible to levitate the test-mass (see section 4).
Error bars are also difficult to determine with metrology from different
sources and methods.

The centring $\vv{O_{1}O_{2}}$ estimated in flight are also presented in Table
4 by extracting from the accelerometric measurement the effect of the Earth’s
gravity gradient [14]. On ground several errors cumulate: 4$\mu$m error on the
test-mass position lying on the stops (deduced from 18 mechanical measurements
at 1$\mu$m accuracy each) and 2$\mu$m metrology error on the geometry. In
flight, the error comes from the dispersion of the different estimations made
with the Earth’s gravity field knowledge and the accelerometric measurements
[14], without any temperature dependency correction.

Table 4:  Evaluation of the SU test-masses relative centring (∗errors do not
take into account capacitive border effects).

  * | X | Y | Z
---|---|---|---
On-ground estimation |  |  |
SU-EP |  $24\pm{}6\,\mu$m |  $0\pm{}4\,\mu$m∗ |  $-6\pm{}4\,\mu$m∗
SU-REF |  $-28\pm{}6\,\mu$m |  $4\pm{}4\,\mu$m∗ |  $0\pm{}4\,\mu$m∗
In-flight estimation |  |  |
SU-EP |  $20.1\pm{}0.1\,\mu$m |  $-8.0\pm{}0.8\,\mu$m |  $-5.7\pm{}0.1\,\mu$m
SU-REF |  $-35.5\pm{}0.3\,\mu$m |  $5.6\pm{}0.4\,\mu$m |  $5.7\pm{}0.3\,\mu$m

####  3.1.3 Core metrology: test-mass alignment.

The test-mass alignments limit the projection of undesired signals on the X
axis. There are specifications on the value and its knowledge for a potential
correction. The mean alignment of the test-mass with the star sensor has to be
known to better than $10^{-3}$ rad to limit the error of the Earth’s gravity
gradient projection on the $X$ axis during the data process. The orthogonality
of the test-mass axes has also to be set to better than $10^{-4}$ rad to limit
cross-axis projection. This requirement is completed by the test-mass relative
alignment of $1.5\times{}10^{-3}$ rad calibrated in-orbit to an accuracy of
$10^{-4}$ rad to reject the common mode accelerations. This last requirement
was verified in flight and showed some good margins [14].

The sensors alignment with respect to the SUMI reference frame is obtained on
the three faces of an optical cube glued on the SUMI (Fig. 6). The budget is
established considering defects such as flatness of the SUMI, parallelism of
the base plate and presented in Tab. 5 and shows good agreement with the
requirement of $1.5\times{}10^{-3}$ rad.

Table 5:  On ground budget error and estimation accuracy of the SU alignment
with respect to the SUMI Optical Cube reference.

  * | Around $Y$ or $Z$ | Around $X$
---|---|---
| rad | rad
SU-EP |  |
Assembly budget | $10^{-4}$ | $4.6\times{}10^{-4}$
Accuracy | $0.8\times{}10^{-4}$ | $4.5\times{}10^{-4}$
SU-REF |  |
Assembly budget | $1.2\times{}10^{-4}$ | $7.1\times{}10^{-4}$
Accuracy | $10^{-4}$ | $7.1\times{}10^{-4}$
Requirement | $15\times{}10^{-4}$ | $15\times{}10^{-4}$

###  3.2 T-SAGE instrument electronics on-ground verification

####  3.2.1 Capacitive sensor.

As a sensor, the capacitive position detector is characterised by an
equivalent input noise spectrum, an equivalent input bias, a gain, a dynamic
range and non-linearity parameters. Both gain and bias parameters have their
own thermal sensitivity. The main sources of bias are the dissymmetries
originating from the differential transformers, the stray capacitance
difference of the coaxial cables connecting each electrode pair to the primary
winding and the two superimposed capacitors allowing for the action feedback
on each opposite electrode. A defect of symmetry of the differential
transformers, thus of the inductance $\delta L$ introduces a detector output
voltage proportional to $V_{d}$ and $\delta L$. The differential transformers
are manually wound and finely tuned in order to get the primary coils as
symmetric as possible. The relative symmetry $\delta L/L$ achievement is
better than $10^{-5}$.

The superimposed capacitors associated to each electrode pair has to exhibit
the smallest current leakage by choosing high capacitance values and getting
the same values. Each superimposed capacitor is carefully sorted in order to
reach the best matching values before implementation.

In order to minimise the bias, at the last stage of the FEEU production, the
detector channel bias is adjusted by the addition of a stray capacitance of a
few picofarads in one of the two branches. By doing so, the maximum bias
obtained was limited to 50 mV, corresponding to a capacitance misbalance of
$6\times{}10^{-4}$ pF.

The bias detector thermal sensitivity error source is found to be in the
differential transformer and extensive tests have been conducted in order to
select the best techniques in term of coil shielding, coil potting and coil
packaging to minimise the thermal effects.

The measurement of the electromechanical transducer characteristics is not
straightforward [15]. Indeed the mechanical core cannot operate under ground
gravity. Thus a mechanical core simulator has been used as a specific item of
the test equipment that allows to operate and to test the capacitive detector
individually. It simulates the test-mass / electrodes capacitance thanks to
ceramic capacitors. A schematic is provided in Fig. 7. The motion of the test-
mass is simulated by modifying the balance of the test-mass / electrodes
capacitors. This bi-level command is performed thanks to a switch. The
amplitude of the so-called motion is controlled by the input 100 kHz voltage
$V_{d_{inj}}$. Amplitude modulation with $V_{d}$ as carrier permits to
simulate a low-frequency displacement signal. The capacitances are measured
with an accurate capacitance bridge (Andeen-Hagerling® AH 2500) in order to
fully characterise the SU simulator with femtofarad accuracy.

Figure 7: A baseline for the capacitances of the test bench set to :
$C_{inj}=1$ pF, $C_{\sigma}=1.2$ pF and $C_{\delta}=27$ pF. The misbalance
equals to $0.005$ pF when the switch is set on position 1 or 2, short-
circuiting one of the 2 $C_{\delta}$. The “0 position” shall lead to a
capacitance balance. Since it is $C_{\sigma}$ that drives the misbalance
figure, the tolerance has been set to a maximum of $7$ fF between the two
capacitances $C_{\sigma}$. 1% accuracy capacitances are implemented for
$C_{inj}$ and $C_{\delta}$.

To obtain the sensitivity measurement, the detector output is recorded for the
three positions of the SU simulator, and then redone with permutation of the
electrode cables. This provided six measurements used to determine the
detector and simulator properties simultaneously from the equation: $V=G_{\rm
det}(\Delta C_{i}+b_{s})+b_{\rm det}$, where $\Delta C_{i=0,1,2}$ is the
approximately 0, +1, or -1 pF capacitance difference of the simulator,
depending on the switch position $(i=0,1,2)$, $b_{s}$ is its bias, and the
entire term between the parentheses becomes negative when the simulator is
connected in the opposite side. This method eliminates the bias induced by the
simulator itself. Hence the detector gain $G_{det}$ and bias $b_{det}$ are
determined.

To demonstrate the detector linearity a change in proof mass position is
simulated by changing the amplitude of the reference voltage input
$V_{d_{inj}}$ to the SU simulator while maintaining the demodulator reference
at $5$ Vrms. The amplitude is always positive to stay in phase with the
demodulator reference ($V_{d0}$) signal and the sign change is obtained by
using the two positions of the simulator. The input variation created by
changing the $V_{d_{inj}}$ amplitude is converted to a corresponding variation
of capacitance $\Delta C=\Delta C_{0}V_{d_{inj}}/V_{d0}$, where $V_{d0}$ is
the 5 Vrms level input to the detector demodulator.

The detector noise is measured by grounding the input of the SU simulator. The
major noise sources are the transformer and the first amplifier. The frequency
domain of interest is from very low frequency down to $10^{-4}$ Hz to
approximately $100$ Hz, corresponding to the control loop bandwidth. This
requires long time measure in a stable thermal enclosure. Fig. 8 shows a
typical noise spectrum after 7 hours of data recording. The data analysis
required a down-sampling from 1027 Hz to 10 Hz associated with a type
averaging FIR filter, which explains the decrease of the spectrum with respect
to frequency above 5 Hz.

Figure 8: Typical detector spectrum noise with an example of channel X1i (X
channel, inner test-mass).

The thermal sensitivity measurement especially on the detector bias requires a
particular procedure due to the small amount of variation expected. The
classical method consisting in measuring $N$ values of bias at $N$
temperatures (the FEEU box placed in a thermal chamber) is not efficient here
since the measurement points are under the noise level. To measure such a
small fluctuation, a different method has been applied for MICROSCOPE
electronics consisting in the application of low frequency sinusoidal thermal
fluctuation to the electronic box at $2\times{}10^{-4}$ Hz and to retrieve the
thermal bias sensitivity by frequency analysis.

Tab. 6 summarises the main measurement results obtained on MICROSCOPE’s FEEU
flight models capacitive sensors. Taking into account the theoretical physical
gains in Tab. 2, the measurements show a position sensor resolution better
than $4\times{}10^{-11}$m Hz-1/2 for any linear test-mass degree of freedom
and better than $8.5\times{}10^{-9}$rad Hz-1/2 for any angular test-mass
degree of freedom. Concerning the bias, the impact in terms of position
sensing is lower than 0.08$\mu$m and in terms of angular sensing lower than
35$\mu$rad.

Table 6:  Laboratory measured resolution, bias, thermal sensitivity and gain,
of the capacitive sensors at 20 °C.

  * Capacitive sensor’s resolution | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
---|---|---|---|---|---|---
at $10^{-2}$ Hz [$\mu$V Hz-1/2] |  |  |  |  |  |
IS1 SU-REF | 8.1 | 3.8 | 3.7 | 9.3 | 3.8 | 3.7
IS2 SU-REF | 5.4 | 2.1 | 2.1 | 9.6 | 2.1 | 2.1
IS1 SU-EP | 12 | 3.1 | 3.1 | 12 | 3.1 | 3.1
IS2 SU-EP | 5.6 | 1.9 | 1.9 | 5.5 | 1.9 | 1.9
Bias [mV] | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
IS1 SU-REF | -6 | -7 | -1 | 14 | 2 | -3
IS2 SU-REF | 14 | -1 | -1 | -46 | 0 | 1
IS1 SU-EP | 14 | -1 | -9 | 49 | 1 | -4
IS2 SU-EP | 21 | -2 | -5 | 30 | 1 | 1
Thermal sensitivity [$\mu$V/K] | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
IS1 SU-REF | 64.7 | 10.1 | 14.3 | 36 | 1.7 | -2.1
IS2 SU-REF | 16.7 | 12.0 | 25.4 | 130.9 | 2.7 | 12.2
IS1 SU-EP | 65.9 | 23.3 | 10.7 | 37.7 | 10 | -0.6
IS2 SU-EP | 16.3 | 8.9 | 14 | 193.3 | 1.2 | -0.5
Capacitive sensor’s gain [V/pF] | $X$ | $Y$ | $Z$ | $\phi$ | $\theta$ | $\psi$
IS1 SU-REF | 82.5 | 16.9 | 17.3 | 82.2 | 16.9 | 17.3
IS2 SU-REF | 40.6 | 5.0 | 5.2 | 84.5 | 5.0 | 5.2
IS1 SU-EP | 81.2 | 16.0 | 16.0 | 81.0 | 16.0 | 16.1
IS2 SU-EP | 39.3 | 5 | 5 | 85.0 | 5.0 | 5.0

####  3.2.2 Electrostatic actuation channel test.

An actuation channel is composed of a pair of electrodes with, associated to
each one, a Drive Voltage Amplifier (DVA) and a DAC that commands the DVA.

The DVA is a low noise feedback amplifier capable of delivering voltages in
the range of about $-40$ V to $+40$ V when powered with $\pm{}48$ V power
lines. It consists of 4 stages:

  * •

A 16-bit DAC that feeds the return branch of the accelerometer loop with the
control voltage provided by the PID in the ICU. Its performance in terms of
noise and linearity is a critical item for the resolution of the instrument;

  * •

A second stage based on an operational amplifier which matches impedance and
isolates the amplifier input from noise; the internal feedback, although not
theoretically necessary, prevents the DVA from auto-oscillations;

  * •

A third stage which delivers the output voltage across a capacitive load.

As part of the feedback branch of the accelerometer, the DVAs are one of the
major contributors to the accelerometer performance: they impose noise and
linearity limits. Since the measurement bandwidth starts at $0.1$ mHz, the
main criterion for the selection of an operational amplifier technology for
this function is its behaviour at this bound. While previous models using OP97
exhibit a typical slope of $1/f$ noise in the measurement bandwidth, chopper-
stabilized amplifiers are free of this effect. Then the equivalent noise of
the amplifier is that of the first stage.

The linearity and thermal sensitivity of DVAs have been measured to assess the
stability of the scale factor. The resistors involved in the transfer function
have a thermal stability better than $0.05$ ppm/°C.

Tab. 7 gathers the main results obtained on the DVA electronics of the X axis.
All gains of Electrode + and Electrode - have been matched to better than
$0.015$% for X axis and $0.02$% for all the other axes which have very similar
gains. For what concern the bias, all others axes exhibit a maximum bias of
1.7 mV and a maximum bias sensitivity of 3.5$\mu$V/°C.

Table 7:  Bias and thermal sensitivity of the actuator electronics (DVA) at
electrodes $X_{+}$ and $X_{-}$ (which control the translation along the $X$
axis) as measured in the laboratory before flight at 20°C.

  * | Electrode $X_{+}$ | Electrode $X_{-}$
---|---|---
Gain of DVA :
IS1-SUREF | $-15.93472$ | $-15.93264$
IS2-SUREF | $-15.93419$ | $-15.93323$
IS1-SUEP | $-15.93285$ | $-15.93053$
IS2-SUEP | $-15.93227$ | $-15.93926$
Noise (any test-mass) : $0.4\mu$V Hz-1/2
Bias of DVA ($\mu$V) :
IS1-SUREF | $157$ | $168$
IS2-SUREF | $118$ | $146$
IS1-SUEP | $147$ | $225$
IS2-SUEP | $256$ | $254$
Thermal sensitivity $\mu$V/K :
IS1-SUREF | $0.3$ | $1.0$
IS2-SUREF | $-0.8$ | $0.9$
IS1-SUEP | $-2.0$ | $-1.1$
IS2-SUEP | $0.4$ | $-0.9$

####  3.2.3 Reference voltage test.

The $V_{p}$ voltage is obtained by the average of two very stable voltage
references in parallel. In order to guarantee a very stable scale factor, the
$V_{p}$ function thermal stability has been specified to $3$ ppm/°C. $V_{p}$
production and characteristics relies mainly on the quality of the DC source
used. As the requirement is more stringent than what can be found on the data
sheet for space qualified components (ceramic case), the same reference in
plastic case with much better performance has been sorted to select the best
pair of components for each $V_{p}$ function. $V_{p}$ has been measured in
flight as $V_{p}=5.003\pm{}0.013$V with respect to the FEEU zero voltage
(through the 1 Hz housekeeping) and can be compared to ground
characterisations in Table 8.

The 100 kHz $V_{d}$ detection voltage amplitude modulation noise is measured
through a rectified diode circuit in order to characterise the low frequencies
behaviour.

Tab. 8 gathers the main results obtained on ground.

Table 8: On-ground characterisation of reference voltages: $V_{p}$ and $V_{d}$
functions at 20 °C.

  * | IS1-SUREF | IS2-SUREF | IS1-SUEP | IS2-SUEP
---|---|---|---|---
$V_{p}$
Value [V] | 4.9973 | 4.9983 | 4.9997 | 4.9990
Broadband noise [$\mu$V Hz-1/2] | 0.16 | 0.16 | 0.22 | 0.22
Corner frequency [Hz] | 2 | 2 | 2 | 2
Thermal sensitivity [ppm/°C] | -2.1 | -1.2 | -1.0 | -0.9
$V_{d}$
Value [V${{}_{r}ms}$] | 5.0282 | 5.0152 | 5.0288 | 5.0188
Frequency [kHz] | 99.91 | 99.92 | 99.86 | 99.86
Broadband noise [$\mu$V Hz-1/2] | 1.5 | 1.5 | 1.5 | 1.5
Corner frequency [Hz] | 2 | 2 | 2 | 2
Thermal sensitivity [ppm/°C] | 50 | 50 | 49 | 59

####  3.2.4 Read-out channel test.

The “off-the-loop” or read-out channel takes a measurement at electrode level
by doing a difference between electrode $X_{+}$ and electrode $X_{-}$. This
design was meant to help reject the DVA noise present in the loop. This
measurement is captured by an ADC. As for the DVA function, a very accurate
attenuation factor conditioning the signal before the digital conversion, has
been obtained by the use of ultra-high precision and stability resistor in
order to use this channel for science purpose. The obtained thermal
sensitivity of the bias was lower than $1.8\,\mu$V/°C and of the gain lower
than 7.9 ppm/°C. The four channels show a bias lower than 1mV and a noise
lower than $1.6\,\mu$V Hz-1/2.

In spite of this outstanding performance on ground, the SUEP-IS2 $X$ output
exhibited in flight an acceleration higher than expected, identified as a bias
due to a stronger gold wire stiffness. This led to the saturation of the read-
out channel on this test-mass. The read-out channel was afterward only used
for technology purposes. Furthermore the noise is dominated by this gold wire
parasitic disturbance and thus minimises the interest of using this channel
eventually exhibiting the same performance as the “in-the-loop” channel.

####  3.2.5 Digital control validation.

The servo-loop correctors are implemented in the rad-hard 32/40-bit IEEE
floating point DSP TSC21020F. The control loop operation cycle was defined at
1027 Hz leading to a CPU load of 77% typical and 95% maximum. Calculation
accuracy was a key point throughout the development, computation errors
introduced by the digitisation inevitably appears during the operations of
quantification and rounding. Both are difficult to detect and therefore had
needed careful tests and measurements.

Moreover in such a DSP, the variables which intervene in the different
computations are memorised using the IEEE 754 floating point standard with 32
bits and the computations (additions or multiplications) are performed using a
40-bit format. For some critical operations, the error analysis showed that
the variables have to be stored in 40 bits for the rounding errors to be
negligible. For the read-out channel, the global measurement error due to the
32/40-bit conversion format is dominated by the amplification of error during
the different filtering processes. This issue has been solved by the use of a
new computation method which minimises the error propagation. It had been
shown that replacing the regular filter by a null gain filter improves the
results significantly.

For on-ground test, the digital control has been very convenient to validate
the “in-the-loop” electronic functions by implementing appropriate control law
or by simply implementing unitary control law.

In flight, the digital control has permitted to optimise the servo control by
patching new control law parameters, and even to implement a new
functionality: the possibility to store 8 seconds of actuation data at 1027 Hz
(instead of 4 Hz in the regular telemetry) for investigation purpose.

##  4 T-SAGE instrument validation

The development of the payload has benefited from the experience acquired with
four previous space accelerometer missions: ASTRE with COLUMBIA shuttle, STAR
[16] with CHAMP mission, SuperSTAR with GRACE mission [17], and the 6
accelerometers of GOCE mission [18]. Nevertheless drastic modifications are
considered because of a new constraint: the operating ranges of the
accelerometer in its Full or High Resolution Modes (FRM and HRM) are not
compatible with ground operations because of the intensity of the gravity. The
needed electrostatic forces are out of access. Nevertheless these ranges
remain reachable with the accelerometric environment in the Bremen free-fall
drop tower [19] in certain conditions.

We have followed the classical development flow of engineering model (EM),
qualification model (QM) and flight models (FM) for both the SU and the
electronics FEEU and ICU. However, the SU development was more complex due to
operational constraints explained above.

First, SU prototype and pre-qualification models have been developed to assess
the vibration resistance and thermal behaviour, then an SU engineering model
has been realised and tested with light silica test-masses in order to be
levitated on ground. This step allowed us to operate the digital control of
the loops and to establish the control laws for the future qualification model
tested in the free-fall tower.

Then two particular vibration test models, representative of the flight model
for the mechanical aspects only, have been produced, assembled and vibrated.
The first test model was dedicated to the validation of the test-mass blocking
system when submitted to the vibration environments. It also allowed us to
check the free motion of test-masses after vibration and switch on of the
release system. The second test model was fully representative of the flight
model from a mechanical point of view, with fewer engravings and less cabling
since its purpose was to establish the integration processes and to test the
general behaviour of the core mechanics under vibration conditions. It does
not feature the vacuum pump and was not functional (left on Fig. 9).

Later, the Qualification Model (QM) (middle on Fig. 9) has been integrated as
fully identical to the flight model except for the test-masses in wolfram
alloy test-masses (Inermet®). Inermet® density is 17.6, 10% lower than the
PtRh density. The QM has been used for final vibrations, shocks, thermal and
EMC testing and was operated during free-fall tests in the ZARM drop tower.

Finally, the two Flight Models have undergone campaigns with limited vibration
and thermal tests (Fig. 9). The development of the instrument concluded on a
final drop test in the ZARM free-fall tower prior to the integration into the
satellite and the launch.

Figure 9: Left: P5.6 vibration model for the gold wire resistance. Middle: QM
model under shock qualification tests. Right: FM model under thermal
acceptance tests.

###  4.1 SU vibration tests

The payload was delivered to CNES in October 2014. The first qualification
process specified a 9 grms load for the random vibrations in the frequency
band [20 Hz-2000 Hz]. Unlike previous accelerometers developed by ONERA,
T-SAGE possesses a blocking device in order to handle the 0.4 kg to 1.4 kg
test-masses safely during the launch phase (test-masses are surrounded by
silica cylinders and might have been damaged by moving masses). However, this
blocking device has a dynamic motion at eigen frequencies (400 Hz, 600 Hz and
800 Hz) of a few hundreds of microns that led to a fatigue of the $7\,\mu$m
gold wire. Two years of development were necessary to understand the fatigue
process and to correct it.

In cooperation with CNES, a specific damping device was implemented at payload
interface with the satellite that reduces to 3.5 grms the levels of vibration
and thus the amplitude of the motion of the gold wire. The integration process
was also improved to guarantee a repeatability of the gold wire gluing and a
better resistance. Before delivery, the qualification model successfully
sustained 5 times the vibration duration that would be applied to the flight
model from acceptance tests to in-orbit operation. This endurance test, quite
out of the generic qualification process, helped to get confidence on the
resistance of such a sensitive payload.

The levels of vibrations were less critical to sustain for the electronics,
and no endurance tests were necessary.

###  4.2 Thermal assessment

The two Sensor Units on their interface plate (SUMI) and the two FEEU are
integrated in the payload case with highly stabilised thermal environment. The
temperature variation is the main error source. Six temperature probes (Fig.
10) have been integrated into each SU, and five temperature probes have been
integrated into each FEEU in order to carefully monitor the evolution of the
thermal environment within the payload case. Particular tests have been
performed at satellite level, to evaluate the thermal performance of the
payload case for which a passive stabilisation was chosen. The SU temperature
variations were specified at $f_{\rm{EP}}$ to better than 1 mK at SU level and
5 mK at FEEU level. A dedicated model was developed with representative
thermal characteristics and power dissipation in order to evaluate the way the
different thermal power sources can affect the payload at $f_{\rm{EP}}$. The
test was done in CNES Toulouse premises. The test result analysis has allowed
readjusting the numerical thermal model of the payload case with accuracy
better than 15% [14].

Figure 10: Position of the SU internal thermal probes.

###  4.3 Functional assessment: drop tower test

####  4.3.1 Facility and instrument configuration.

A unique facility in Europe, the drop tower in ZARM Institute provides
conditions for experiments under microgravity. The height of the tower limits
the “free-fall” experiment period to 4.74 s for drops, extended to 9.5 s since
December 2012 with catapult shots [20]. For the MICROSCOPE mission, 25
campaigns with a total of 71 falls have been conducted in order to assess the
dynamic behaviour of T-SAGE twelve digital servo-loops: 49 drops with the
“free-flyer” [8] and 22 drops with the catapult. Previously the drop capsule
in “free-flyer” configuration was settled with a SuperSTAR accelerometer
(engineering model of GRACE mission) with some 100 drops. The flight models
were tested with only 8 drops to limit the stress. QM campaign drops are
presented and discussed in Ref. [8]. In the following, the catapult shot
results obtained with the QM model are shown and the very last test, performed
in June 2014 of the flight model is presented.

Due to the very short free-fall duration and the small range of the
accelerometer, the electronics configuration had to be boosted to 100 V
instead of the nominal flight supply of 48 V in order to increase the applied
voltage capabilities and thus the electrostatic forces for the test-mass
acquisition. The DC voltage applied to the mass is also changed from 41.4 V to
89.2 V. With this drop tower configuration, the full scale range of $X$
measurement for the smallest of the 4 test-masses is of $2.7\times{}10^{-5}$ m
s-2 versus $1.6\times{}10^{-6}$ m s-2 in flight HRM mode.

The accelerometers and their electronics are accommodated in a cylindrical
capsule (2 m high and 80 cm in diameter). Its weight is about 300 kg with its
electronics, its on-board data acquisition computer and the experiment
payload. At the bottom of the capsule, a 50 cm long cone stabilises it during
the deceleration inside the 8 meters high braking tank filled with fine graded
polystyrol. The deceleration process lasts about 0.2 s and the maximum
deceleration rate does not exceed 400 m s-2 ($\approx$40 g) and can be applied
without damage to the instrument. For the free-fall, the $X$ axis is always
horizontal, the $Y$ axis is aligned along gravity (the same axis as the
cylindrical capsule) and the $Z$ axis is also horizontal. The residual drag of
the capsule is thus measured along the $Y$ axis. Optimisation of the capsule
balance is a key point in the experiment preparation: without any prior
balancing of the capsule, the centre of gravity is not located on the capsule
axis of symmetry anymore, also aligned with the axis joining the two sensor
units along $Y$. The test-masses positions versus the centre of gravity of the
capsule are better than 1 mm along the radial axes.

A final pressure of about 10 Pa is achieved inside the drop tube in order to
minimise the drag effect at 200 $\mu$m s-2. A particular care is also taken to
any motion inside the capsule that may generate small vibrations which may be
measured by the accelerometers: for instance, cables are carefully fixed and
the on-board computer fans are switched off.

####  4.3.2 Drop results.

Since the instrument is configured in a coarse acquisition mode because of the
short free fall time, its ultimate performances are out of reach. However,
such drops are the only way to test the servo control dynamics. Their
objective is to demonstrate that the sensor operates as expected, especially
that the test-mass is controlled at the zero position with desaturated
acceleration measurements along all degrees of freedom.

Fig. 11 illustrates a typical catapult test with the flight models. Before the
capsule release, under 1 $g$, the test-masses lay on the stops. Along the $Y$
axis (vertical, direction of the fall), they are resting at a distance of
about $+90\,\mu$m from the centre of the electrode set (see section 3.1.2)
with $+Y$ direction pointing to the ground. The instrument is switched on
before the release of the capsule, so when under microgravity, the acquisition
of the mass electrostatic suspension is automatically performed. The capsule
is catapulted to the top of the “free-fall” tower. As soon as the level of the
capsule acceleration is within the accelerometer full range, the mass position
is controlled to the centre of the electrode set and then the acceleration
measurement is desaturated (Fig. 11).

In a standard drop, the drag due to the residual air in the tower is
continuously increasing from the time of the capsule release to the end of the
drop. During a catapult shot the drag decreases to zero at the apex of the
trajectory i.e. at the very precise moment when the velocity is also zero. In
Fig. 11, the bias of the instrument can be estimated at the apex, however this
determination is not precise because the test-masses are not yet controlled
along and about all directions particularly along $X$. With the radial
electrodes, the control of the test-masses is sufficiently stiff to get the
convergence of the PID. The measured acceleration provided by each
accelerometer can be considered as the sum of the external acceleration and
the accelerometer bias. Thus the simple difference of the measurements
provides an estimate of the differential bias in a coarse mode
($V_{p}=100\,V$).

Figure 11: Measurements during the last catapult test of SU-REF and SU-EP
flight models. On the left, the $X$ axis measurements. On the right, the $Y$
axis measurements. The upper graphs show the acceleration of inner (red) and
outer (blue) test-masses. The lower graphs show the position of the test-
masses.

###  4.4 Last instrument “good health test”

As the accelerometers cannot operate on ground, good health tests have been
established in order to assess the integrity of the instrument during the
integration on the satellite, the satellite environmental test and after
transportation to the launch pad site. It consists in verifying as
exhaustively as possible the functionality of the SU, FEEU and ICUME knowing
that the instrument is in launch configuration: test-masses locked on their
stops.

The main functionalities verifications are obtained through dedicated
housekeeping data and through the scientific data themselves contained in the
telemetries:

  * •

At SU level: the gold wire health is checked thanks to a command electrically
disconnecting the stops. The capacitive detectors must show then the position
of the test-masses locked on the stops if the $V_{d}$ voltage is well applied
by the wire.

  * •

A force sensor measures the blocking mechanism efficiency: a total $1600$ N is
measured during locked phases and less than $200$ N (the sensor bias) when the
test-masses are free.

  * •

At FEEU level: the capacitive detector channels provide repeatable
measurements with different “expected” values with respect to the two values
of $V_{d}$. This validates the electronics, the SU/FEEU harness and the SU
core integrity at once.

  * •

At ICU level: the digital control loop through the open loop response is
verified as well as the data production, the command control, and the power
supply.

##  5 Conclusion

T-SAGE inherits from previous developments of electrostatic accelerometers
[21, 4, 16]. The major improvement is the use of a digital controller for the
test-mass servo loop with a large capability of configuration change or
parameterisation in order to make this instrument as user friendly as a
laboratory experiment. Another difference with previous developments was the
impossibility to levitate the test-masses on ground which led to a development
strongly relying on free-fall tests for the servo loop adjusting phase and for
the qualification verifications.

The other key points are the test-mass shape, the cylindrical test-masses and
the gold-coated electrode sets needed new procedures of machining and
metrology. Finally it was the first time, a locking device system is used for
the test-masses. Previously, the test-masses did not weigh more than 0.3 kg
and mechanical studies and experience showed that they could be free even
during the launch vibration phase. With the 1.3 kg test-mass, a locking device
was mandatory. New developments were necessary to define a locking device that
could be safely released in orbit, with no damage on the surface of the test-
mass and that could not induce sticking forces at the release or disturbing
forces during flight. The electrodes around the test-mass and all others
surfaces that could load an electrical charge have been carefully designed to
limit any electrostatic disturbances on the test-mass motion. In particular
the mechanical stops have multilayer coatings to differentiate areas of
different voltages in front of electrodes or of the test-mass.

The confidence in the final performance of the instrument has relied on the
verification of each individual specification in the mechanics (dimension,
position, mass…), in the environment (level of vacuum, temperature, magnetic
characteristics…) and in the electronics (noise, offset, gain, thermal
stability…). The performance cannot be measured on ground which is why a
particular care has been taken to establish each contributor to the detailed
error budget that contributes to the mission performance budget as detailed in
[9].

The authors express their gratitude to all the different services involved in
the mission partners and in particular CNES, the French space agency in charge
of the satellite. This work is based on observations made with the T-SAGE
instrument, installed on the CNES-ESA-ONERA-CNRS-OCA-DLR-ZARM MICROSCOPE
mission. ONERA authors’ work is financially supported by CNES and ONERA
fundings. Authors from OCA, Observatoire de la Côte d’Azur, have been
supported by OCA, CNRS, the French National Center for Scientific Research,
and CNES. ZARM authors’ work is supported by the German Space Agency of DLR
with funds of the BMWi (FKZ 50 OY 1305) and by the Deutsche
Forschungsgemeinschaft DFG (LA 905/12-1). The authors would like to thank the
Physikalisch-Technische Bundesanstalt (PTB), in Brunswick, Germany, for their
contribution to the development of the test-masses with funds of CNES and DLR.

## References

## References

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