Figure 6.1 Planar 6-body model of the GG system with springs, gimbals and the related connections. The 6 bodies are: the spacecraft, the PGB laboratory, 2 test masses and 2 gimballed arms. The spin angular velocity of the spacecraft is around the Z-axis.

The sensors in DCAP are a way to monitor the dynamic state of the system, performing an ideal measurement. A set of sensors has been defined both for active control and post-processing purposes.

 
6.1.4 Actuators and Function Generators. Definition of External Disturbance

For the purposes of the present analysis the orbital motion of the satellite can be neglected, and also the gravity gradients components; the orbiting reference system is therefore approximated by an inertial reference frame. The following scaling has been applied to the disturbances.

The actuators in DCAP provide a way of exerting forces or torques on the system. A "jet" actuator is characterized by an application point and a direction for an external force.

6.1.5 Passive Devices

k_b : rotational (bending) local stiffness

k_t : translational stiffness

With respect to mid-point:

• De41, De42. bending connection of inner mass (TMi). The bending stiffness act on Hi4 rotational freedoms (x,y) between No20-40 (k_b).

Table 6.3 Passive Devices Numerical Values (with Q_PGB=90, Q_TM=500). Note that we assume Q_TM=500, which is about 1/4 of the measured value, in order to make the whirl instability of the test masses grow faster so as to speed up the numerical simulations. The problem is easier if instabilities are slower, so our analysis is carried out under conservative assumptions.

All the set of devices described above define the passive elastic connections of the GG model.

The implementation of active damping of whirling motion can be introduced in DCAP in different ways. We have chosen to add special devices ("user-defined" via Fortran routines), whose command is computed in the subroutines that implement the stabilization scheme will described in the following sections. These devices act in parallel to the passive ones for the supercritical dofs. A mnemonic identification number has been set, according to the following example.

• De241, De251. Stabilizer between S/C and PGB.
• De341, De351. Stabilizer between PGB and TMe.
• De441, De451. Stabilizer between S/C and TMi.

6.1.7 Controller

A User-Defined Discrete Controller (USDC) with sampling rate of T_s /10 sec (with T_s the GG spin period) is defined to implement the stabilization logic and the related filtering processes. The controller adds errors to the measurements, performs filtering and computes the magnitude and phasing of the stabilization forces in the rotating frames; actuation of forces is executed by user-devices described above.

USDC has 3 inputs (relative displacements in x, see 6.1.8) and 22 outputs. The outputs are used mainly for checking purposes and for Drag Free Control. Details on USDC definition can be found in Section 6.2.

6.1.8 Interconnets

Besides the interconnects used to activate external disturbance (see Sec. 6.1.4), three additional interconnects are used to feed the controller with the relative displacement signals. In this first implementation, only one plate is used, assuming it to be aligned with x direction of inboard body. The nominal sensors measurements are fed to USDC with:

• In201, In301, In401. The ideal measurements of sensors Se201, Se301, Se401 (see 6.1.3) are connected to the three inputs of user discrete controller.

6.1.9 Constant Data (CNTDTA)

The easiest way to pass data to user defined routines is through "constant data" blocks. Each CNTDTA has an ID number and a value (double precision) associated to it. The following sets of data have been defined:

• Cn 1. Spin Period
• Cn 2¸ 4. Coefficient of non rotating damping for PGB, TMe and TMi.
• Cn 5¸ 7. Whirling modes frequency, in rad/s, for PGB and common mode TMe and Tmi. These values are used for Fourier filter on the whirling period
• Cn 10. Standard deviation s w for noise in the angular rate measurement from Earth Elevation Sensor (E.E.S.), relative to spin rate (Cn10=s w /w _s).
• Cn 11. Sensor 201 RMS for the displacement measurement noise in the small read-out capacitance plates driving the stabilization loop of the PGB.
• Cn 12. P.S.D. for the rate measurement noise in the small read-out capacitance plates driving the stabilization loops. This value is obsolete and it was used only in old continuous stabilization schemes.
• Cn 13. P.S.D. for the noise in stabilization force introduced by the active capacitance plates.
• Cn 14. Bias in the displacement measurement.
• Cn 15. Angular misalignment between the measurement plates and the reference direction (x). This gives a systematic phase error, an angular bias, in the reconstructed signal.
• Cn 16. RMS of sensors 301 and 401 for the displacement measurement noise in the small read-out capacitance plates driving the stabilization loops of TMi and TMe with respect to the PGB.
• Cn 101. Time range, starting at t = 0 sec, during which the non rotating damping coefficient of the PGB is amplified to rapidly reduce the Pico Gravity Box whirling motion.
• Cn 102. Amplification factor of the non rotating damping coefficient of the PGB, utilized in the time range specified by Cn101.
• Cn 103. Time range during which the non rotating damping coefficient of the Test Masses is amplified to rapidly reduce their whirling motion. The initial instant of amplification is set by Cn 105.
• Cn 104. Amplification factor of the non rotating damping coefficient of the TMs, utilized in the time range specified by Cn103.
• Cn 105. Delay time of the amplification of the non rotating damping coefficient of the Test Masses (initial time of the amplification of TMs C_NR).
• Cn 106. Instant at which the PGB whirling control force is turned off.
• Cn 107. Instant at which the PGB whirling control force is turned on again (Cn 107 > Cn 106).
• Cn 108. Instant at which the TMs whirling control forces are turned off.
• Cn 109. Instant at which the TMs whirling control forces are turned on again (Cn 109 > Cn 108).
• Cn 110. Gain of the Notch filter (Drag Free Control).
• Cn 111. RMS of the Drag Free Control actuator (FEEP thrusters).
• Cn 111. Bias of the Drag Free Control actuator (FEEP thrusters).

  6.1.10 Eigenfrequencies

With the model set-up outlined in the previous sections, a linearization has been performed in order to determine the eigenfrequencies of the system. The eigenvalue problem has been solved for the zero-spin rate case (w _s=0), in order to determine the whirling frequencies; linearization at non zero spin cannot be performed in DCAP and has been done with the DYNROT software by Prof. Genta and Dr. Brusa at Politecnico di Torino. We have checked that the results reported in Table 6.4 are essentially unchanged, as expected.

The complete model has 6 rigid-body dofs plus 22 flexible dofs (18 for the PGB. TMe, TMi plus 4 for the gimbals conical modes). The results are shown in Table 6.4; it can be noticed that the different configurations of the gimballed arms (with and without rings) do not affect the other frequencies; they obviously affect the local modes of the arms. If the gimbals are supposed to be mounted on a rigid fixture, with equivalent bending stiffness k’q given by

k’q =2k_b+k× h²/2+kq @ k× h²/2+kq =0.01× 0.10603²/2+4.0× 10^-6=6.0212× 10^-5 Nm (6.4)

and with the values for J_t given in Table 6.1, the eigenfrequencies are 85.22 and 87.81 rad/sec, thus confirming the local nature of these modes.

The configuration of gimballed arms without the stabilizing ring is expected to affect the test-masses differential modes with larger contributions. Any misalignment of the J_p axis w.r.t. spin axis is amplified by inertial moments, for a given q ₀ initial out-of-phase angle if the configuration is unstable (J_p<J_t). Since the conical motion of the gimballed arms directly induces differential motion in the test masses, the two configurations have been analyzed in order to evaluate the effects on differential motion and define a baseline design. It is found that, due to the much smaller mass of the coupling arms compared to the mass of the test bodies which they couple, they are in fact stabilized once the test bodies themselves are stabilized by the active control of their whirl motions. Therefore, in the remainder of this Chapter, all the 6-body simulations refer to coupling arms without rings.

6.1.11  Whirling Stabilization

The stabilization of whirl motion (see Sec. 2.1.5 for a physical introduction) can be implemented by building in a rotating frame a damping command proportional to the relative velocity between bodies in the inertial (non-rotating) reference frame. A simple way to show this capability is by examining the problem of 2 bodies connected by a dissipative spring. Let us consider r_s the vector pointing to the Spacecraft center of mass, r_p the vector pointing to the PGB center of mass, e the vector locating the suspension point of the spring with respect to the Spacecraft center of mass, S the Suspension point of the spring (see Fig. 6.3). The spacecraft is rotating around the z-axis, which is perpendicular to the xy plane. The rotation is counter-clockwise. In the inertial reference frame the equations of motion read:

(6.5)

being e = e (t) a rotating vector in the inertial frame (fixed in the reference frame corotating with the spacecraft) and w_s the spin angular velocity. These equations show clearly the nature of rotating damping, namely damping between the rotating bodies. Using: x = r_p - r_s (relative position vector); m_r = m_s m_p /(m_s + m_p) (reduced mass), w ₀²= k/m_r (natural frequency of oscillation), c_r = k/(Qw _s) (coefficient of rotating damping, with Q the mechanical quality factor of the spring), we can write the equation for the relative motion of the two bodies:

(6.6)

The solution of the equation of relative motion is:

(6.7)

with:

(6.8)

the vector describing the displacement of the equilibrium position due to the action of the external drag and x_whirl (w _wt) the exponential term giving rise to the spirals shown in Fig. 6.4.

Figure 6.4 Simulation of the 2-body system with and without an external residual drag. It shows how the effect of an external force (e.g. air drag) is that of displacing the equilibrium position of the system. A small phase lag appears due to energy losses in the suspensions (finite Q), as expected. Due to these losses the whirl instability builds up in either case around the corresponding equilibrium position.

The 2-body system is unstable. The simplest way to stabilize it is adding some Non Rotating Damping (i.e. damping between non rotating parts of the system), which is mathematically expressed by the terms containing C_nr (the equations refer again to the inertial reference frame):

(6.9)

(6.10)

These equations are useful to see the difference between Rotating Damping (term with c_r), and Non Rotating Damping (term with C_nr, which is the number defined by Cn 2 - Cn 4); see Fig. 2.16 for a physical explanation. In the space experiment everything is corotating with the spacecraft, so we need to recast these equations in the rotating reference frame:

(6.11)

(6.12)

with and the relative displacement and its rate of change in the rotating reference frame. The minimum C_nr for stabilizing the whirling motion is C_nr = c_r w /w ₀.

6.1.12  Whirling Stabilization Control Law

We have seen in the previous Section that the stabilization of whirl motion can be implemented by building in the rotating frame a damping command proportional to the relative velocity between the bodies in the inertial (non-rotating) reference frame

(6.13)

A more compact complex notation can be used by replacing the vector with a complex number (z =x +ih ) and the cross product operator w _s´ by iw _s (i=Ö -1), giving

, (6.14)

all in the rotating frame. Conversely, if is the displacement in the inertial frame, the relative velocity in rotating frame, on which rotating damping acts, can be defined as .

The main drawback of a continuous approach is that the whirl motion builds up with long time constants, and consequently the rate of change of the relative amplitude is highly dominated by the measurement noise. The tangential component of the stabilization signal instead must be built from the difference of two large terms, i.e., (w _s-w _n)t and w _st, affecting the phasing of the compensation command. Furthermore, any bias in the measurement is amplified by the pre-multiplication of w _s.. A modified approach based on signal filtering has therefore been introduced.

Let us see the solution of the equation motion (relative displacement of the bodies) in the rotating reference frame; in absence of non rotating damping it reads:

(6.15)

The first term is a fixed vector in the rotating reference frame which depends on the initial offset of the mounting point of the springs; the second term is a vector corotating with the drag force (with the angular velocity w _s as seen in the rotating frame) and with a phase lag f with respect to the drag depending on the mechanical quality factor of the spring, tan(f ) = 1/Q; we have assumed a drag defined by a DC component plus a term at orbital frequency in the inertial reference frame, so, z_drag(t) has w _s and (w _s-w _orb) angular frequencies (in the rotating reference frame). The third term, describing the growing spiral of the whirl motion, has (w _s-w _w) angular frequency. The idea is of using the precise signature of the growing whirling motion to build a control law capable of stabilizing the system. The starting point is sampling the relative displacement signal within every spin period T_s, with N points, and to analyze the harmonic at T_s in order to reconstruct the modulating signal amplitude and phase: note that in this way any bias is removed (it is a constant additional term of z(t)). Given N sampled points of the displacement z(t) in the spin period interval T_s=w _s/2p :

(6.16)

the fundamental harmonic can be obtained from Fourier integral as

(6.17)

From the difference between the corresponding values in two spin periods, the velocity of the modulating signal is then reconstructed. We obtain in fact a pair of values (A_k, j _k) for each spin period. The first harmonic filtered amplitude A_k and phase j _k can be used with A_k-1, j _k-1 at previous step (t_k-1=t_k-T_s) to determine the relative velocity in the inertial frame. The velocity signal V_k and its phase j _Vk are determined according to the following sketch.

(6.18)

The estimated amplitude (A_k, j _k) and velocity information (V_k, j _Vk) will be delayed by one revolution (T_s), since one T_s is needed to sample the starting data. Moreover, no velocity can be estimated in the first two spin periods of the simulation. The following formulas will apply:

(6.19)

and

(6.20)

A number of 11 points over T_s has been found to be a good compromise in treating noisy signals. The implemented discrete sampler is in fact operating at 10/ T_s Hz.

The simplified analysis for a two-body system shows that the reconstructed velocity is predominantly driven by the noise. This can be easily understood: the whirl motion grows so slowly that its velocity cannot possibly be estimated with good accuracy by analyzing variations in just one spin period of 0.5 sec. Therefore, in order to improve the resolution of the forces the natural solution is to perform a second Fourier harmonic analysis at the whirl period T_w. This is feasible thanks to the following fact.

The solution to the eigensystem for uncontrolled whirling of a system with high quality factor can be simplified to ± (w _n/2Q)± i× w _n; the ratio of the time constant for the divergent forward whirling mode to the whirling period is therefore Q/p , i.e., the time-to-double is 0.32× Q times larger than T_w. By using a Q=90 for the PGB, and its natural period of 296 sec, the signal doubles in » 28.6× Tw (2.4 hours). This suggests that there is enough time to collect samples over the whirl period and run the filter extracting an approximate sinusoidal signal that will be very close to the marginally divergent real signal. Even if the driving velocity signal will be applicable to previous period and the whirling will have increased, we must recall that we are handling a "stabilization" problem, where a minimum gain grants stability; therefore, a slightly larger margin on the gains completely overrides the problem. Note that Q=90 is the lowest Q in the GG system; hence this is the fastest growing whirl instability and therefore the most difficult to handle.

The formulas for this second harmonic analysis are the same of the first Fourier filter, after substituting T_s with T_w and z_k(t_k) with the discretised velocity along the x and y axes. An example of the capability of this double-filter control is shown in the following pictures. A sample of 1000 sec of simulated evolution of the 2-body system S/C-PGB has been considered first in the ideal condition (perfect reading and actuating sensors), then in a realistic situation (imperfect sensors). A residual drag of 5 10^-9 N DC plus 2× 10^-9 N at orbital frequency, with 10% noise on both components, a mechanical quality factor Q = 90, and a non Rotating damping coefficient have been used in both the simulations; the spinning frequency is n _s = 2 Hz. Figures 6.5¸ 6.8 show the results obtained in reconstructing the slow whirl velocity to be damped and demonstrate the validity of our strategy in the 2-body model case (both ideal and with realistic error sources; see Chap. 3 for experimental results on the accuracy of capacitance sensors).

Figure 6.5 Actual relative displacement S/C-PGB in the inertial frame as predicted with DCAP, to be compared with the relative displacement as reconstructed by the controller from measurements of the rotating sensors after the first Fourier filter (w_s filter, dashed curve); case of ideal, perfect, sensors.

Figure 6.6 Actual relative displacement S/C-PGB in the inertial frame as predicted with DCAP, to be compared with the relative displacement reconstructed by the controller from measurements of the rotating sensors after the first Fourier filter (w_s filter). The errors included were: errors on sensors RMS = 10^-2 m m, bias = 10 mm, angular bias = 1^o, RMS(w) = 10^-4 w. The bias has been removed from the sensor signal curve due to the scale (10 mm vs 10^-7 m). The improvement due to the Fourier filter is apparent: The bias is cancelled out, the RMS is partially filtered out.

Figure 6.7 Actual relative velocity S/C-PGB in the inertial frame, as predicted with DCAP, to be compared with the relative velocity as reconstructed by the controller after the second Fourier filter (w_w filter): case of ideal, perfect, sensors.

6.1.13   Non Rotating Damping in Pulsed Mode

The continuous compensation described in Sec. 6.1.12 requires a stabilization force to be operated by the active plates in the form of a sinusoidal signal . An equivalent scheme, based on pulsed operation, has been conceived and implemented.

(6.21)

or, in Laplace domain (operator s) and complex notation ,

(6.22)

In order to limit the noise amplification due to the derivative term and the amplification of displacement readout biases through the w _s term, the integral estimate of velocity and pulsed command mode are introduced. The attracting force is applied in the form of sinusoidal pulses of width ± p /4 across velocity phase (j _V+p ), with two pulses in one turn, one for each actuator as in the sketches shown next page. The pulses must be sized 2× C_NR× |V| in order to provide the equivalent stabilization of the continuous command. In the reference frame of the spacecraft, with the phase reference (t₀) given by sensor plate S1, the required control forces are:

(6.23)

In this Section we apply the control laws in order to damp whirl motions in the 2-Body model case made by the PGB laboratory and one test mass (PGB-TM) and in the full 6-body GG system. In both cases we first we assume using ideal, perfect sensors, and then introduce realist errors as follows:

Figure 6-9 One day simulation of the 2-body PGB-TM system showing their relative distance (ideal case). Oscillations at orbital period are due to the oscillating term of the drag; the offset from the X axis is due to the DC term of the drag which displaces the equilibrium position of a fixed vector along X. The short period oscillations (see Fig. 6.10 for a close up), represent the whirl motion and decay in few thousand seconds.

This test of the ideal case was an important check of the principles of the whirl control: the results are in perfect agreement with the theoretical predictions, confirming the validity of the control law.

Figure 6.10 Short sample from the one-day simulation of the ideal 2-body PGB-TM system shown in Fig. 6.9: the rapid decay (over a few thousand seconds) of the short periodic whirl oscillations is apparent.

Figure 6.11 Sample of 15000 sec from the one day simulation of the ideal 2-body PGB-TM system shown in Fig. 6.9. This polar plot shows the approaching of the test mass to its ideal "equilibrium circle" caused by the residual drag force, and corresponding to a non dissipating spring (no whirl motion, infinite quality factor). The constant displacement along the X axis is caused by the DC term of air drag.

Figure 6.12 One day simulation of the ideal 2-body PGB-TM system as in Fig.6.9. Here we plot the whirl radius, which reduces to one or two Angstrom after 6000-7000 sec only.

Figure 6.13 One day simulation of the ideal 2-body PGB-TM system as in Fig. 6.9. Here we plot the passive elastic force of the spring for comparison with the active force of the control. Note that in these simulations the active force was taken to be 11 times larger than the minimum value required to provide stability, so as to speed up the simulations. We have checked that a force only 2.5 times larger than the minimum does stabilize the system, obviously less rapidly.

Figure 6.14 One day simulation of the 2-body PGB-TM system with realistic error sources. Oscillations at orbital period are due to the oscillating term of the drag; the offset from the X axis is due to the DC term of the drag which displaces the equilibrium position by a constant amount along X. The short period oscillations (see Fig. 6.15 for a close up) represent the whirl motion and decay in few thousand seconds.

Figure 6.15 Sample from the one-day simulation of the realistic 2-body PGB-TM system as in Fig. 6.14. The short period oscillations are the whirl oscillations, which decay in a few thousand seconds.

Figure 6.16 Sample of 15000 sec from the one day simulation of the realistic (noisy) 2-body PGB-TM system as in Fig. 6.14. This polar plot shows the approaching of the test mass to the ideal "equilibrium circle" due to the variations of the residual drag force, and corresponding to a non dissipating spring (infinite quality factor, no whirl motion). Note that the scales on the two coordinate axes are different (the zero whirl "circle" is therefore distorted).

Figure 6.17 One day simulation of the realistic (noisy) 2-body PGB-TM system as in Fig. 6.14. Here we plot the whirl radius, which reduces to a few Angstrom after 7000-8000 sec only.

Figure 6.18 One day simulation of the realistic (noisy) 2-body PGB-TM system as in Fig. 6.14. Here we show the passive elastic force of the spring for comparison with the active force of the control. We recall that, in order to speed up the simulations, we apply an active force 11 times larger than the minimum needed for stabilization, and that an active force only 2.5 times larger than the minimum has been tested and found to be sufficient.

All plots are shown in the non rotating frame. The gimballed arms were modeled without rings (see Sec. 6.1.10).

Figure 6.19 Ideal 6-body GG system. Simulation of 15000 sec. The figure shows the distance between the PGB and the spacecraft, and the distance between the external test mass and the PGB. There are three types of oscillations: one is at the orbital period, due to the component of the drag at this frequency, another is at the whirl period of the PGB (controlled), and the one with the shortest oscillation period is the controlled whirl motion of the test mass. The offset of the distances with respect to the X axis is due to the DC component of the drag force.

Figure 6.20 Ideal 6-body GG system. Polar plot showing the motion of the PGB and of the external test mass with respect to the center of mass of the spacecraft (0,0) during the first orbit (6000 s). The whirl motions start to be damped, but still need some time. Note that the scales on the two coordinate axes are different (the zero whirl "circle" is therefore distorted)

Figure 6.21 Ideal 6-body GG system. Polar plot showing the motion of the PGB and of the external test mass with respect to the spacecraft center of mass (0,0) during 9000 sec after the first orbit. It is now apparent that the whirl motions are damped and that the bodies are already very close to their zero-whirl orbits (the circles in this polar plot)

Figure 6.22 Ideal 6-body system. Here we plot the whirl radii of the PGB and of the external test mass for the same time interval as in Fig. 6.21.

Figure 6.23 Ideal 6-body GG system. Here we plot the passive elastic force for comparison with the active control force in the case of the external test mass. The value of C_NR , the coefficient of non rotating damping applied to the test body, is 0.009 Ns/m, which is about 11 times larger than the minimum required value. A force 2.5 times larger than the minimum was found to be sufficient.

Figure 6.25 The 6-body GG system with noise as given in the text.. Simulation of 12000 sec. The figure shows the distance between the PGB and the spacecraft and the distance between the external test mass and the PGB. As usual, there are three types of oscillations: one at the orbital period, due to the orbital component of the drag (the longest period), another with the whirl period of the PGB (controlled), and an even shorter one at the whirl period of the TM (also controlled) .The offset of the distances with respect to the X axis is due to the DC component of the drag force.

Figure 6.26 The 6-body GG system with realistic noise (see text). This polar plot shows the motion of the PGB and of the external test mass with respect to the center of mass (0,0) of the spacecraft for 9500 sec after the first 2500 sec.

Figure 6.27 The 6-body system with realistic noise (see text). We plot the whirl radius of the PGB and of the external test mass after the first 4000 sec. These values are in very good agreement with the theoretical predictions and the expected self centering.

Figure 6.28 The 6-body GG system with realistic noise (see text). Here we plot the passive elastic force of the spring for comparison with the control force acting on the external test mass. The value of the C_NR coefficient of non rotating damping applied to the test body is 0.009 Ns/m, which is about 11 times larger than the minimum required value. We have checked that an active force only 2.5 times larger than the minimum could stabilize the system.

Figure 6.29 The 6-body GG system with realistic noise (see text). Here we plot the relative displacement between the inner and the outer test mass (differential displacement) obtained after applying the active control of their whirl motions as outlined in the previous sections (by double Fourier analysis). Note that in this simulation the whirl control is always on, i.e. this is a worst case simulation because whirl control can in fact be switched off during scientific data acquisition (see Sec. 2.1.5). Yet, this result is impressive in that it shows how active control by means of electrostatic sensors/actuators can be so accurate as to make the GG test bodies (macroscopic rotors of 10 kg each) self-center on one another as expected in supercritical rotation in absence of dissipation (infinite mechanical quality factor, zero-whirl). Achieving such a small separation of the test bodies is very important in order to reduce classical disturbing effects (such as Earth tides). Note that - although very small indeed- this relative displacement is still 160 times larger than the one expected in case of an EP violation to the level of 1 part in 10¹⁷ (see Eq. 2.2), which is the target of the GG mission. However, in the non-rotating reference frame (as in Fig. 6.26) the signal appears - after synchronous 2-phase demodulation at the 2 Hz spinning frequency- as a constant displacement always along the direction to the center of the Earth (indicated as the Y axis in Fig. 6.26 and as the X axis in Fig. 2.21); to the contrary, the whirl motion (shown in Fig. 6.26) which gives rise to the separation plotted here as function of time, has a distinct frequency, namely the natural frequency of the whirl (as it is apparent). This allows us to separate the relative displacement signal.

6.1.15   Drag Free Control

(6.24)

The results are shown in the following figures; all plots are in the inertial (not rotating) frame of reference. The whirl control is always on, although the time-span of the simulation is always too short - given the initial very high level of drag- for the whirl motions to be damped; here the relevant issue is whether the DFC is effective in reducing the amplitude of the effect of drag at the orbital frequency, which it is.

Figure 6.30 Simplified model of the drag force used in the simulations of the DFC: drag along the x axis

Figure 6.33 A detail from the previous plot showing how the drag free control is effective in reducing the amplitude of the relative displacement in just a few orbital periods of the satellite around the Earth.

Figs. 6.34 and 6.35 show the dependence of the DFC performances from the sensitivity of the Earth Sensor: this is due to the last operation of the DFC, which needs to transform the force from the inertial to the rotating frame where the actuators (FEEP) operate. The following figure shows the result of the simulation for the 6-body GG system, indicating that the DFC works just as well as in the 2-body case, save for the fact that it is much more demanding in terms of CPU time needed to run the simulations. Therefore, in the full system we have used the lower Earth sensor noise in order to show a significant reduction of drag already in first 2 orbital periods. Our purpose was to show that the current design of the DFC is adequate to reduce drag at the orbital frequency.