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%%\title{CH.4 Reducing differential motion of aLIGO seismic platforms}
%\date{}
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%\begin{document}

%\begin{figure}
%\centering
%\includegraphics[scale=0.5]{images/LIGO_LOGO.PNG}
%\end{figure}

\chapter{Control of seismic platforms motion and LSC offloading}
\label{CPSdiff}
During 2019, I spent some months working at the LIGO Hanford site (Washington, USA). This experience allowed me to be critically involved in the complicated life of a gravitational-wave interferometer. In particular, I was given the opportunity to study how to improve LIGO performance at low-frequency, focussing on the reduction of seismic motion of the platforms where the optics are located.\\
In this chapter I will demonstrate how we can modify seismic control configuration of LIGO: in particular, this study should help reduce the differential motion between the chambers, making them move in sync, and help reduce and stabilize the rms motion of the auxiliary sensors, through an LSC offload. The final goal is to obtain different and possibly better performance for seismic motion stabilization, faster and longer locking mode and, ultimately, more gravitational waves detections.  The detailed computations included in this chapter are original and partially presented to the LIGO community and stored in LIGO DCC \cite{proposal} \cite{technote1} .\\
This work has been developed in collaboration with LIGO Hanford and LIGO Livingston laboratories, Stanford University, MIT and UoB and completed at UoB during 2020.\\
This chapter is partially including some technical notes I shared with LIGO collaboration and the contents of this study have been presented at conferences and workshops \cite{chiatalk}.\\
Essential information about the sections of LIGO involved in this study has been exposed in detail in Chapter \ref{LIGO}.

\section{Motivation: Duty cycle on LIGO}
Lock loss events are the main sources of preventing continuous observations for long periods of time: when light loses resonance in the cavities, a lock loss happens and the control systems of the optical cavities are under effort to restore stabilization. This means that during lock loss the interferometer is no longer able to be stable and the observing time is interrupted \cite{biscans}.\\
Duty cycle is a major focus for commissioners before starting an observing run \cite{biscans} \cite{kisseltalk1}. It is needed not only to observe more gravitational waves, but also to identify noise sources and improve sensitivity \cite{biscanstalk}.\\
Since the number of detected events over a time period N(t) is proportional to the volume of Universe under observation V, the observing time t and the rate R of astrophysical sources that can occur in a certain volume:

\begin{equation}
\centering
N(t) = R\cdot V\cdot t,
\end{equation}\\
it can be seen that increasing the observing time towards a given direction, will increase the number of detected events.\\
\noindent
Other ways to increase the number of detectable events is to increase the observable volume: this can be achieved by spending time on hardware to improve sensitivity on a given frequency bandwidth \cite{kisseltalk1}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.35]{images/duty_cycle.png} \includegraphics[scale=0.55]{images/dutypele.jpg}
\caption[LHO duty cycle during O3b]{\textbf{Up:} Example of duty cycle for Hanford Observatory, during O3b \cite{kisseltalk1}. For almost 20\% of the running time the detector was not locked, which means that it was not observing \footnotemark . Of this 20\%, the \textbf{bottom} chart shows the causes of the lockloss: the main ones are seismic and "unknown" The study presented here could possibly reduce both.}
\label{duty}
\end{figure}


\newpage
\subsection{Differential motion between chambers}
We have seen that among the noise sources which contribute to lock loss events there is the ground motion, including earthquakes and microseismic events. \\
In particular, during O3 run, it was observed that the chambers in the corner station (CS) show differential seismic motion with respect to each other \cite{technote1}, because they move independently from each other with respect to ground. It is reasonable to think that if the chambers could have a synchronized motion, the whole interferometer would move following the ground motion, without being affected by it. This would in principle help the cavities to be stable and to maintain the resonance. In case of lock losses due to large earthquakes or high wind, stable resonance could be achieved in shorter times \cite{biswas}.\\
On another side, reducing the differential motion between the chambers means to reduce a source of noise at low frequency (5-30 Hz), as we will show in the next section: this would improve the sensitivity of the interferometer.
\footnotetext{For a full understanding of the legend, refer to Appendix C.}

\subsection{ISI stabilization}
Differential motion affects the ISI of the HAM and BSC chambers in the CS: these are the platforms that we want to stabilize. Several sensors are responsible for sensing the seismic motion, in all degrees of freedom of each stage. We have already introduced the seismic sensors in Section \ref{ligosei} of Chapter \ref{LIGO}: they are T240, L4C, GS13 and CPS \cite{kisselthesis}.\\
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.9]{images/isi.png}
%\caption[Example of ISI scheme]{Example of ISI inertial sensor scheme (figure taken from \cite{adl2}). All the main inertial sensors involved in the seismic isolation are shown in their locations. The CPSs are position sensors located between stages to measure the relative position.}
%\label{isi}
%\end{figure}
\noindent
In particular, CPS sensors are placed on every stage of every chamber: it is easy to compare the motion between HAM and BSC chambers through the signal of a device sensing the same motion on every chamber \cite{kisseltalk2}.\\
The idea which should stabilize ISIs to follow the ground motion is to lock the chambers to each other, in order to make them move on a synchronized way, following a common motion given by a driver chamber (or block of chambers).

\paragraph*{Role of the mode cleaner}
We started our design on the chambers on the X arm. Along this direction, the Input Mode Cleaner (IMC) lies totally on HAM2 and HAM3 platforms: it can be used as a reference, or witness, of the motion between chambers, once they are locked together.\\
%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.8]{images/IMC.png}
%\caption[Optical layout of the HAM2 and HAM3 chambers]{Optical layout of the HAM2 and HAM3 chambers.}
%\label{imc}
%\end{figure}\\

\noindent
In the next section we will demonstrate that CPS are good witnesses to sense differential motion and that they also can be used to lock the chambers with each other.

\section{Sensing differential motion via CPS}
The Capacitive Position Sensors (CPS) measure the relative motion between two stages of the isolation system. On HAM chambers they are set between HEPI and ground, and between Stage 1 and HEPI. On BSC chambers they also measure the relative motion between Stage 1 and Stage 2.  The plots in Fig. \ref{diff} show the differential motion seen by the CPS between BSC and HAM chambers: the sensors put in evidence that the HAM chambers have a more synchronized motion with respect to the motion between HAM and BSC and BSCs only. This means that the block of HAM chambers on X arm is more stable relatively to the other blocks and can be used as driver for the other chambers, with the mode cleaner acting as witness. We then projected the CPS of the X axis chambers to the suspension point in order to obtain PRCL and Input Model Cleaner Length (IMCL) traces like as they would be sensed by the CPS. For BSCs, we decided to sum the contributions of the CPSs on stage 1 and stage 2 and to project this sum to the suspension point.\\
\noindent
One of the main differences between the behaviour of CPS IMCL and CPS PRCL is that the former is involving only the HAM chambers. Since HAM2 and HAM3 have a very good common motion, IMCL can be considered more stable with respect to PRCL, which instead involves also BSCs. Indeed, CPS PRCL is following only the BSCs, at frequencies below 0.03 Hz \ref{sus}.\\

\noindent
Fig. \ref{sus} shows the plots of PRCL and ICML as sensed by CPS projection to the suspension point. These projections indicate that reducing the differential motion as seen by the CPSs will help to reduce the residual motion seen by the optical cavities. We expect this to be effective in a range of frequencies between 0.1 and 0.5 Hz.\\

\begin{figure}[H]
\centering
\includegraphics[scale=0.25]{images/ham2_ham3.PNG}\\
\includegraphics[scale=0.25]{images/bs_itmx.PNG}\\
\includegraphics[scale=0.25]{images/bs_ham2.PNG}
\caption[CPS differential motion]{CPS differential motion between the HAM and BSC chambers along X axis. ISIs move in common, particularly in the same building. This can be confirmed by noting that the difference between two chambers is much lower than individual chambers.}
\label{diff}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/CPS_SUSPOINT_IMCL.PNG}\\
\includegraphics[scale=0.3]{images/CPS_SUSPOINT_PRCL.PNG}
\caption[CPS suspoint projections]{CPS suspension point (suspoint) projections: IMCL and PRCL. These plots show that the CPS sensors are good to monitor the motion of the optics at suspension points. In particular, the PRCL sensed by the CPS matches the calibrated PRCL trace between 0.6 and 0.2 Hz. Below this value we were expecting the traces to match: the un-match is due to PRCL calibration issues under solving. The IMCL is currently monitored by the GS13, and the plot shows that even CPS can be a good sensor for it. Hence we can use these sensors to monitor the motion of the optics, additionally to the relative motions between the platforms.}
\label{sus}
\end{figure}

\newpage
\section{Locking chambers via CPS}
In the previous section we demonstrated that the CPSs are good sensors for differential motion and that they can be used to monitor the chamber motion at lower frequencies. That said, and remembering the aim of stabilizing the motion of the chambers making them moving in sync, it is possible to use the CPSs to lock HAM2 and HAM3 together, HAM4 and HAM5 together, BSCs in the Corner Station together and BSCs hosting the ETMs together (refer to Chapter \ref{LIGO} for the location of these chambers). This will stabilize the ISI differential motion with respect to a driving chamber.\\
Since we saw that HAM2 and HAM3 show a very good common motion and that we can use the IMC as a witness of it, our first step is to lock the HAM2 and HAM3 chambers together by feeding HAM3 a calculated differential CPS signal. This is performed with an additive offset to the setpoint of the HAM3 isolation control loop \cite{technote2}.\\
The block diagram in Fig. \ref{ham2b}, shows the structure of HAM2, where the signals from $d_{2}$ and $i_{2}$ represent the offsets given by CPS and inertial sensors \footnote{For a summary on control loops, general design and feature and how to solve a block diagram, refer to Appendix \ref{B}.}.\\
At low frequency the CPS noise is negligible because its contribution is about 10$^3$ times lower than the microseismic peak.\\
General block diagrams notations used in the hereafter are listed in Tab. \ref{tab1}\\

\begin{table}[h!]
\centering
\begin{tabular}{|c|| c| }
\hline
P & Plant\\
\hline
S & Sensor correction\\
\hline
C & Control\\
\hline
L & Low pass filter \\
\hline
H & High pass filter\\
\hline
$N_{g}$ & ground noise\\
\hline
$N_{i}$ & inertial noise\\
\hline
$x_{g}$ & ground motion\\
\hline
$x_{p}$ & plant motion\\
\hline
\end{tabular}
\caption{Notations used in the block diagrams.}
\label{tab1}
\end{table}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/ham2B.PNG}
\caption[HAM2 simplified block diagram]{Simplified block diagram for HAM2 chamber as it is at present on LIGO \cite{technote2}.}
\label{ham2b}
\end{figure}

\noindent
We can compute the signal $\textit{$d_{2}$}$ which will be the CPS offset to send to HAM3 chamber. In this case, HAM2 will drive HAM3 to follow its motion. Defining K = PC:\\

\begin{equation}
\centering
\begin{split}
&d_{2} = x_{p_{2}} -  x_{g}\\
&= K[L_{2}(SN_{g} + Sx_{g} + d_2) + H_2N_{1_2} + H_2(x_g + d_2)] + Px_g - x_g\\
&= KL_{2}S(N_{g} + x_{g}) + KL_{2}d_2 + KH_2N_{i_2} + KH_2x_g + KH_2d_2 + Px_g - x_g.
\end{split}
\end{equation}\\
Since L$_2$ + H$_2$ = 1, we get:\\
\begin{equation*}
\centering
d_2(1-K) = KL_{2}S(N_{g} + x_{g}) + KH_2N_{i_2} + KH_2x_g + Px_g - x_g,
\end{equation*}\\
which will lead to:\\
\begin{equation}
\centering
d_2 = \frac{K}{1-K} [L_2S(N_g + x_g) + H_2N_{i_2} + H_2x_g] + P\frac{x_g}{1-K} - \frac{x_g}{1-K}.
\label{d2}
\end{equation}\\
The platform motion of HAM2 can be computed following the block diagram in a similar way:\\
\begin{equation}
\centering
x_{p_{2}} = K\{L_{2}[S(N_g + x_g) + x_{p_2} - x_g] + H_{2}N_{i_{2}} + H_2x_{p_2}\} + Px_g.
\end{equation}\\
After some manipulations, and remembering that L$_2$ + H$_2$ = 1, we obtain:\\
\begin{equation}
\centering
x_{p_{2}} = \frac{K}{1-K}[L_2S(N_g + x_g) - L_2x_g + H_2N_{i_2}] + P\frac{x_g}{1-K}.
\label{xxp2}
\end{equation}\\
The result in Eq. \ref{xxp2} is the signal to subtract to HAM3 in order to feed HAM3 a CPS differential motion that is added to HAM3 as shown in the block diagram in Fig. \ref{ham3b}. In the original configuration, without any feeding into HAM3, the block diagrams for both chambers would be identical. With the provided feeding, instead, there is no sensor correction and ground noise on HAM3 because they both come from the contribution from HAM2, which is the offset $d_{2}$ added to HAM3.\\
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/ham3B.PNG}
\caption[HAM3 simplified block diagram with HAM2 offset]{Simplified block diagram for HAM3 in the new configuration where this chamber is now connected to HAM2: d$_{2}$ is the offset coming from HAM2 \cite{technote2}.}
\label{ham3b}
\end{figure}

\noindent
We then want to know the reaction on HAM3 plant in this configuration, in order to compute the differential motion between plants on both chambers.\\
Following the usual notations and the block diagram of HAM3:\\
\begin{equation*}
\centering
x_{p_{3}} = K[L_3d_2 + L_3x_p{_3} - L_3x_g + H_3N_{i_3} + H_3x_{p_3}] + Px_g,
\end{equation*}

\begin{equation}
\centering
x_{p_{3}} = \frac{K}{1-K}[L_3(d_2 - x_g) + H_3N_{i_3}] + P\frac{x_g}{1-K}.
\end{equation}\\
The differential motion will be:\\
\begin{equation}
\centering
\begin{split}
&x_{p_{3}} - x_{p_{2}} = \frac{K}{1-K}[L_3(d_2 - x_g) + H_3N_{i_3}] + P\frac{x_g}{1-K}\\
&  -\frac{K}{1-K}[L_2S(N_g + x_g) - L_2x_g + H_2N_{i_2}] - P\frac{x_g}{1-K}.
\end{split}
\end{equation}\\
In the approximation where K$\rightarrow \infty$, i.e. in a condition of infinite gain, the terms we computed become:\\
\begin{equation}
\centering
x_{p_{2}} = H_{2}N_{i_{2}} + L_{2}S(N_{g} + x_{g}) - L_{2}x_{g},
\label{xp2}
\end{equation}

\begin{equation}
\centering
d_{2} = H_{2}N_{i_{2}} + L_{2}S(N_{g} + x_{g}) + H_2x_{g},
\label{d2}
\end{equation}

\begin{equation}
x_{p_{3}} = H_{3}N_{i_{3}} + L_{3}(d_{2}-x_g).
\label{xp3}
\end{equation}

\begin{equation}
\centering
\begin{split}
x_{p_{3}} - x_{p_{2}} &= L_3d_2 - L_3x_g + H_3N_{i_3} - L_2S(N_g + x_g) + L_2x_g - H_2N_{i_2}\\
& = L_3L_{2}S(N_{g} + x_{g}) + L_3H_{2}N_{i_{2}} + L_3H_2x_{g}-\\
&L_3x_g + H_3N_{i_3} - L_2S(N_g + x_g) + L_2x_g - H_2N_{i_2}.
\end{split}
\end{equation}\\
If both chambers are mainly driven by the low pass filters (H $\rightarrow 0$), the differential motion becomes:\\
\begin{equation*}
\centering
\begin{split}
x_{p_{3}} - x_{p_{2}} & = L_3d_2 - L_3x_g - L_2S(N_g + x_g) + L_2x_g\\
& = L_3\cdot L_2S(N_g + x_g) - L_3x_g - L_2S(N_g + x_g) + L_2x_g,
\end{split}
\end{equation*}

\begin{equation}
x_{p_{3}} - x_{p_{2}} = L_2S(N_g + x_g)(L_3 -1) + x_g(L_2 - L_3),
\label{xp3xp2}
\end{equation}

\noindent
which is what we expect to be the signal of the differential motion sensed by the CPSs. In order to see this signal, we need to implement the modifications of the filters involved in the loop, as shown in the following section.

\section{Analysis of feasibility}
The next step is to study how to modify the low and high pass filters in order to obtain the best performance from each one in the new configuration of the chambers \cite{technote3}. To do this, we are going to change the blending filters, i.e. those filters whose combination gives the best performance of the set low+high pass filters.\\
If by definition we have L+H=1\footnote{This definition arises from the need to account for unconditional loop stability and noise contributions. For details about blending filters, refer to \cite{kisselthesis}.}, we can write it as:

\begin{equation}
\centering
L_{lh}+H_{lh}=\frac{(\alpha + b)^{l+h-1}}{(\alpha + b)^{l+h-1}}.
\label{binomial}
\end{equation}\\
According to the values of \textit{l} and \textit{h}, we have different orders of magnitude of the binomials, which can be solved for the real part.\\
\noindent
In our case, we have two main contributions given by inertial sensors and the CPS. We will apply the high-pass filter to the inertial sensors and the low-pass one to the CPS.\\
To do this, we need the specific contributions for each chamber to be specified, with all the components well defined. For example, in the case of the CPS contribution, we need to define the tilt component, the CPS noise and the ground motion, which will take part into the platform motion as seen by the CPS sensor. This is because these components are independent from each other and will need to be summed in quadrature.\\
Besides, as we saw in the previous computations, we will need to apply filters: the Sensor Correction filter will be the one used on LIGO and shown in Fig. \ref{SC}; the high- and low- pass filters will  be evaluated through blending several possible filters across a certain number of \textit{l} and \textit{h} order of magnitudes, as introduced before. The best blended filter will be given by a combination of two \textit{l} and \textit{h} values at a specific blending frequency.\\
At the end of the analysis for each chamber (HAM and BSC) in isolation, we will connect the chambers via CPS and look at the results.\\
All this analysis has been performed through Matlab software.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/SC.png}
\caption[Sensor correction filter]{The sensor correction filter as it is at present installed on LIGO. This filter is one of the contributors to take into account for when computing the blending filters.}
\label{SC}
\end{figure}

\subsection{Contributions from CPS and inertial sensors}
To calculate the CPS signal contribution, we need the ground motion and we used the ITMY STS (Streckheisen Tri-axial Seismometer) signal on X direction. This is going to be the same motion for every chamber, since there is only one sensor in the Corner Station to measure it, because it has been found that the ground motion is the same everywhere in the Corner Station. This signal includes a contribution from the tilt motion, dominating in the STS spectrum in the microseismic frequency (< 0.15 Hz); we call it $\theta_g = \theta \cdot g/\omega^2 $ and we separate it from the ground motion $x_g$:\\
\begin{equation}
x_g = STS - \theta_g.
\end{equation}\\
The CPS signal has been then computed summing in quadrature the contributions given by tilt, ground motion and CPS noise ($N_{cps}$), and applying the sensor correction filter:\\
\begin{equation}
CPS_{inj}=\sqrt{(\theta_g\cdot SC)^2+(x_g\cdot (1-SC))^2+(N_{cps})^2}.
\end{equation}\\
Figure \ref{cps_inj} shows the CPS signal and all its contributions. Since $\theta_g$ effects one STS, this injected signal is the same at all CPS sensors.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/cpsinj.png}
\caption[CPS contributions]{Plot of all the single CPS contributions calculated. This computation is valid for all chambers, since CPSs are installed on all chambers and are subjected to the same working principle.}
\label{cps_inj}
\end{figure}

\noindent
To calculate the platform motion of the BSC, we used data from the ITMX ISI along X direction. This is the signal from the T240 sensor. We applied the same technique and we separate the tilt contribution ($BSC\theta_p$) from the signal, and to obtain the inertial sensor contribution for the BSC chambers we sum in quadrature the contributions from tilt and T240 noise:
\begin{equation}
T240_{inj} = \sqrt{{BSC\theta_p}^2+{N_{T240}}^2}.
\end{equation}\\
Figure \ref{t240_inj} shows the T240 signal and its contributors.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/t240inj.png}
\caption[BSC contributions]{Plot of all the single BSC contributions computed from the inertial sensor involved in this chamber. We assume that the T240 is dominated by tilt effects below 80 mHz, and by sensor-noise at higher frequencies: the $\theta_p$ contribution is the $T240_{inj}$ signal below 80 mHz. We interpolated these two bands together to determine an effective input disturbance as witnessed by the T240.}
\label{t240_inj}
\end{figure}

\noindent
The inertial contribution for HAM chambers is computed in a similar way: the sensor in this case is the GS13 and we used data from the ISI of HAM2 in x direction. Calling variables in the usual way:
\begin{equation}
GS13_{inj}= \sqrt{{HAM\theta_p}^2+{N_{GS13}}^2}.
\end{equation}\\
Figure \ref{gs13_inj} shows the GS13 signal and its contributors.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/gs13inj.png}
\caption[HAM contributions]{Plot of all the single HAM contributions calculated for the inertial sensor involved in this chamber. Similarly to the previous case, the $\theta_p$ contribution is the $GS13_{inj}$ signal below 80 mHz.}
\label{gs13_inj}
\end{figure}

%\newpage
\subsection{Blending filters}
In order to compute the platform motion for the single chambers in isolation and, later, locked together via CPS, we need the low- and high- pass filters. Many possible blended filters have been found for different combinations of order of magnitudes and blending frequency: the plots in Fig. \ref{blend} show the velocity rms for every combination.\\
\noindent
The best combination has been found computing the orders and the blending frequency which give the minimum of the cost. The optimized blending filter has been then built using the best values of \textit{l} and \textit{h} orders and blending frequency. The cost is given by:
\begin{equation}
cost = \sqrt{\int((cps_{inj}\cdot LP)^2 + (T240_{inj}\cdot HP)^2)df}
\end{equation}\\
with a similar equation for the HAM chamber.\\
Fig. \ref{cost} shows the cost and its rms obtained with the best blending filters for BSC and HAM chambers.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.45]{images/bscblend.png}
\includegraphics[scale=0.3]{images/hamblend.png}
\caption[Blending filters]{Plots of all possible costs built with different combinations of blending filters. The orders of magnitude are indicated by the low and high pass indices l and h of the binomial filter in \ref{binomial} and the are going between l =[1,4] and h = [1,4]. The plateau on both plots is given by the fact that the SC filter is dominating over those frequencies: this means that the chosen cost function has too little power to discriminate between blends. This makes a new way to evaluate blending filters in presence of the SC filter necessary, in particular re-weighting the blends as functions of frequency may help the optimisation process.}
\label{blend}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/bscrms.png}
\includegraphics[scale=0.3]{images/hamrms.png}
\caption[HAM and BSC cost and their contributions]{HAM and BSC cost and their contributions from. For BSC, the best blending filter is given by l = 1, h = 4 at a blending frequency of fb = 0.13 Hz. For Ham, the best indeces are l = 1, h = 4 and fb = 1.14 Hz. Te green trace is the row, uncontrolled motion of the ISI, for reference.}
\label{cost}
\end{figure}


\subsection{Locking the chambers}
With these elements, we can proceed with the analysis of the behaviour of the chambers when locked via CPS. We refer to HAM2 and HAM3 chambers, since in the previous sections we made the computations for them. We recall here that the equations we need are \ref{xp2}, \ref{d2}, \ref{xp3} and \ref{xp3xp2}, where $x_{p_{2}}$ is HAM2 platform motion, $d_{2}$ is the signal from HAM2 to send to HAM3 and $x_{p_{3}}$ is HAM3 motion when attached to HAM2 via CPS.

\noindent
What we need to know is which terms of these equations are coherent, in order to separate them from the incoherent ones, which will need to be summed in quadrature. We assume that the ground translation at low-frequencies is the same everywhere in the CS, and we already estimate the tilt separately, so the terms involving $x_g$ can be considered coherent. Noises are instead independent from each other. The previous equations then become:
\begin{equation}
\centering
x_{p_{2}} = \sqrt{(H_{2}N_{i_{2}})^2 + (\theta_g\cdot SC\cdot L_2)^2} + (x_g\cdot SC\cdot L_2) - (x_g\cdot L_2),
\end{equation}

\begin{equation}
\centering
d_{2} = \sqrt{(N_{i_{2}}{H_2})^2 +(\theta_g\cdot SC\cdot L_2)^2} + (x_g\cdot SC\cdot L_2) + (x_g\cdot H_2),
\end{equation}

\begin{equation}
\centering
\begin{split}
x_{p_{3}} = &\sqrt{(N_{i_{2}}{H_2})^2 + [L_3\sqrt{(N_{i_{2}}{H_2})^2 +(\theta_g\cdot SC\cdot L_2)^2}]^2}\\
&+ (x_g\cdot SC\cdot L_2 + x_g\cdot H_2)\cdot L_3 - (x_g\cdot L_3).
\end{split}
\end{equation}\\
Since L$_3$ = L$_2$ and H$_2$=H$_3$:

\begin{equation}
\centering
\begin{split}
x_{p_{3}} - x_{p_{2}} = &|\sqrt{(L_2\cdot SC\cdot \theta_g\cdot L_2)^2 - (L_2\cdot SC\cdot \theta_g)^2}\\
&+(x_g\cdot SC\cdot {L_2}^2)-(x_g\cdot SC\cdot L_2)|.
\end{split}
\end{equation}
The plot in Fig. \ref{cpsdiff} shows the differential motion of HAM2 and HAM3 in isolation and the motions when the chambers are locked to each other. The improvement of the differential motion is evident below 0.1 Hz, but it is not efficient above this frequency: further studies of the blending filters involved and a re-weight of the cost in a different part of the spectrum could help to find a compromise.

%\begin{figure}[h!]
%\centering
%\includegraphics[scale=0.3]{images/diffham.png}
%\caption[HAM chambers in isolation]{Ham chambers in isolation: motion of HAM2 as a reference. The purple trace is the differential motion between HAM2 and HAM3 that we are interested in reducing below $\sim$ 0.1 Hz.}
%\label{diffham}
%\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{images/cpsdiff.png}
\caption[HAM chambers in CPS locking condition]{HAM chambers in CPS locking condition: the plot shows the motion of each chamber, where HAM3 depends on HAM2, through CPS locking, and the differential motion between them. There is an improvement of the differential motion in the new configuration (purple trace) with respect to the situation in isolation (green dashed trace) by a factor of 3 in order of magnitude below 0.1 Hz.}
\label{cpsdiff}
\end{figure}

\section{Inertial sensors locking}

In the frame of ISI stabilization from ground motion, the inertial sensors could also be used for the same purpose to lock chambers, in addition to CPS locking \cite{technote3}. This means that HAM3 will be fed by the inertial sensors signal coming from HAM2.\\
Signals from inertial sensors are represented by i$_2$ and i$_3$ notations in Fig. \ref{ham2b} and \ref{ham3b}. However, adding the signal i$_2$ to HAM3 implies that the block diagram will be modified as in Fig. \ref{ham3bi}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{images/ham3Bi.png}
\caption[Inertial sensor locking setup]{New setup of HAM3 chamber fed by CPS and inertial sensor signals from HAM2.}
\label{ham3bi}
\end{figure}

\noindent
Following a similar path for the math for this configuration of i$_2$ signal from HAM2:\\
\begin{equation}
\centering
i_2 = x_{p_2} + N_{i_2} = \frac{K}{1-K}[L_2S(N_g+x_g) - L_2(N_{i_2}+x_g)] + \frac{Px_g}{1-K}+\frac{N_{i_2}}{1-K}.
\end{equation}\\
This is the signal from HAM2 inertial sensors to inject into HAM3 as shown in Fig. \ref{ham3bi}. Then, the platform motion of HAM3 becomes:\\
\begin{equation*}
\centering
x_{p_3} = K[L_2d_2 +L_3x_{p_3} - L_3x_g +H_3i_2 + H_3N_{i_3} + H_3x_{p_3}] + Px_g,
\end{equation*}

\begin{equation}
\centering
x_{p_3} = \frac{K}{1-K}[L_3(d_2-x_g) + H_3(i_2+N_{i_3})] + \frac{Px_g}{1-K}.
\end{equation}\\
In the approximation where K $\rightarrow$ $\infty$:\\
\begin{equation}
\centering
i_2 = L_2S(N_g+x_g) - L_2(n_{i_2}+x_g),
\end{equation}

\begin{equation}
\centering
x_{p_3} = L_3(d_2-x_g)+ H_3(i_2+N_{i_3}).
\end{equation}\\
Remembering Eqs. \ref{xp2} and \ref{d2} for x$_{p_2}$ and d$_2$, we can compute the differential motion:\\
\begin{equation*}
\centering
\begin{split}
x_{p_3} - x_{p_2} & = L_3d_2-L_3x_g+ H_3i_2+H_3N_{i_3} - H_{2}N_{i_{2}} - L_{2}S(N_{g} + x_{g}) + L_{2}x_{g}\\
 = & L_3L_2S(N_{g} + x_{g}) + L_3H_2N_{i_2} + L_3H_2x_g - L_3x_g + H_3L_2S(N_{g} + x_{g}) -\\
&H_3L_2(N_{i_2}+x_g) + H_3N_{i_3}- L_2S(N_{g} + x_{g}) + L_2x_g - H_2N_{i_2},
\end{split}
\end{equation*}

\begin{equation}
\centering
x_{p_3} - x_{p_2} = x_g(L_3H_2 - H_3L_2 + L_2 - L_3) + N_{i_2}(L_3H_2 - H_3L_2 -H_2) +N_{i_3}H_3.
\end{equation}\\
The computation of the differential motion between HAM2 and HAM3 in the conditions where the two ISIs are connected both via CPS and inertial sensors shows that there is no contribution from the sensor correction and from the ground noise.\\
Besides, it is worth notice that if L$_2$ = L$_3$, also H$_2$=H$_3$ by definition and then the differential motion is:\\
\begin{equation}
\centering
\begin{split}
x_{p_3} - x_{p_2} &= N_{i_3}H_3 - N_{i_2} H_2\\
&= H(N_{i_3}-N_{i_2}),
\end{split}
\end{equation}\\
which is exactly the solution that we would obtain if the differential motion was computed without any feeding: this means that in this configuration the differential motion would be the same as in isolation condition.

\section{Test on LIGO Hanford and LSC signals optimization}
During the 2019 commissioning break, in collaboration with LIGO Livingston Observatory, we tried to apply the new CPS configuration in order to obtain improvements in ISI motion and LSC signals at LIGO Hanford.\\
This test has been performed before the detailed analysis exposed previously and hence a more detailed and precise study for the choice of the blending filters involved is essential to get the expected enhancements. However the preliminary tests at LHO showed an improvement of a factor of 3 at 60 mHz (Fig. \ref{isitest}), as detected by the IMC sensors, and an encouraging result detected by the DARM signal below 0.1 Hz when all the chambers inside and outside the CS were locked (Fig. \ref{darmtest}).
\noindent
This is an interesting result that shows that with the implementation of the correct filters as shown in the analysis it is possible to reduce the differential motion of the platforms. Some effects of this configuration have been testified even by IMCL, during a measurement in on/off offloading, reported in the LHO logbook post 52690. Other tests looking at the effect on the LSC cavities are reported in post 52729 of the LHO logbook.\\

\begin{figure}[h!]
\centering
\includegraphics[scale=1.15]{images/isi_sup.png}
\caption[ISI motion suppression]{Screenshot from LHO CDS showing a quick measurement with the chambers in the CS were locked: the witness is the IMC and we monitored also the ground motion to make sure that no important variations were happening at the moment of the measurement. The green traces represents the motion before the locking, while we took two measurement after the locking (blue and pink) to validate the test. \textbf{Left:} motion of the suspension point of the M2 and M3 optics (lying on HAM3 and HAM2 respectively). \textbf{Right:} ground motion and IMC cavity motion before (green) and after the locking mode (blue and pink).}
\label{isitest}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=1.2]{images/darm.png}
\caption[DARM improvement with locked chambers]{Screenshot taken from LHO CDS when a quick measurement of DARM reaction to the lock of all the chambers has been taken. The result is encouraging because it shows an improvement of the signal below 0.1 Hz.}
\label{darmtest}
\end{figure}

\noindent
With this in mind, a positive consequence of this effect might be the improvement of the LSC signals from LIGO cavities. Among them, DARM is particularly important, because it represents the gravitational wave signal. It might be convenient to make the optics of the LSC cavities, lying on the platforms and subjected to the ISI motion, be controlled by the ISI itself. This ideas has been developed and tested at LHO and is exposed in the following section.

\subsection{LSC offloading}
We saw that the cavities and the optical signals in LIGO are affected by the ISI motion, simply because they lie on them. Given the work done with the CPSs to suppress the ISI motion, we should see an improvement on LSC signals. This is not immediate, though, nor trivial, because the optics are just set on the optical bench, without any communication with the ISI, and there is no active control between the ISI and these optics. Despite there is a sort of benefit as testified by Fig. \ref{darmtest}, the motion of the optics on the chambers due to other factors than seismic noise is not seen by the platforms: if we could connect this motion to the platform via software, this would make the optics and the platform more dependent on each other. This means that we can control the stabilization of the cavity lengths also with the ISIs.\\
What we expect is a faster reach of locking and a longer state of lock of the interferometer during observing runs. Another advantage would lie in a reduction of the forces already used to stabilize the cavities and in less tilt motion.\\
\noindent
This work has been performed on LIGO Hanford in October and November 2019, during the commissioning break between O3a and O3b observing runs. The reason of this choice is that we needed the interferometer to \textit{not} be observing, since we were going to modify some software structure of the instrument.\\
\noindent
To lock the LSC signals to ISIs, we need to do something similar to what we did with the HAM chambers: we need to connect via software two different setups which do not talk to each other. We decided to start from the Power Recycling Cavity Length (PRCL) because we locked HAM2 and HAM3 chambers, so it was natural to start to lock the cavities on the x axis.\\
Through CPSs locking, we reduced the differential motion of HAM2 and HAM3 chambers and made them move in sync. So they can be considered as a whole block. The IMC is entirely lying on HAM2 and HAM3, and it is straightforward to use it as a witness: to make this real, we need to feed the HAM2-HAM3 block to IMCL. This will lock the cavity signal to the HAM2-HAM3 block. The same feeding will be performed with PRCL, SRCL, DARM and MICH cavities, whose optics are suspended on the other chambers, in and out the corner station. Fig. \ref{chamb} illustrates the chambers and the locations of the cavities of interest in this study.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.3]{images/chambs.pdf}
\caption[Sketch of the blocks and the locations of the cavities]{Sketch of the blocks and the locations of the PRC and IMC cavities (not in scale). the suspensions of the mode cleaner and the power recycling cavity lie all on HAM2 and HAM3 chambers. the signal for PRCL come form the Corner Station, which can be grouped as a BSC block.}
\label{chamb}
\end{figure}
\noindent
The same work is foreseen to be done for the other cavities: the very short period of time available during the commissioning break allowed us to modify only the control loop for PRCL. Moreover, during the commissioning break, time is also used to work on the chambers, profiting of the out-of-lock mode. This means that, for every attempt of software modification, a locking trial was needed, to see if the new configuration of the instrument was giving better performance and, also, if it was affecting negatively other sides of the instrument. To try to lock LIGO, we needed people not to work besides the chambers. This was a huge and collaborative work, which involved many people on site, and their time. Despite these challenges, the results obtained are encouraging and validated the analysis of feasibility exposed.

\paragraph{The Power Recycling Cavity Length (PRCL)}
We need to connect the ISI to the cavity and to do it we need to know how the PR cavity is going to communicate with the ISI (refer to Chapter \ref{LIGO} for details on the PR cavity). The block diagram in Fig. \ref{prcl} illustrates the simplified concept of the PR cavity connected to the ISIs of the block of HAM2 and HAM3 chambers \footnote{Some insights about the shape of the transfer function of the suspensions are in Appendix C.}.\\
The work done in this case is similar to the one done for the HAM chambers, except for the fact that a new filter needs now to be built in order to control how the ISI affect the motion of the PRC optics.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{images/PRCLfeed.png}
\caption[Block diagram of PRCL locked to the ISI]{Block diagram of PRCL locked to the ISI. This drawing highlights the details of the PRCL cavity sections involved in active control. In the standard diagram, only the PRCL sections would be involved, while now the cavity is connected via software to the ISI. The LSCfilter block is the crossover filter between the cavity and the ISI (and the connection between them is enabled by a switcher) while the ISItoM3 block represents the plant block of the suspension point of M3 after the connection. The blocks in pink represents the 3 stages of optics involved in the HSTS suspension and the controller already existing on LIGO.}
\label{prcl}
\end{figure}
\noindent
This block diagram has been solved with Mathematica in order to find the correct crossover filters to add. The system was simulated via Matlab and includes information from calibration filter modules, PRM control filters, and HAM Small Triple Suspension (HSTS) models via the calibration filters. This is needed to simulate the addition of the ISI as a PRCL actuator. The aim is to offload low-frequencies to the ISI and hence we need to decide the best configuration of gains and offsets of the crossover filter.\\
After every simulation which could possibly work for the system, we locked the interferometer and took a measurement of the PRM suspension point. The plot in Fig. \ref{prcltest} shows a comparison between the simulation and the actual measured PRCL signal: the outcome is positive because the two traces differ by only a factor of 2, which says that the crossover filter should be adjusted by a factor of 2 to match the real signal. This result has been obtained implementing the filter in Fig. \ref{prclfilter}. The test shows that the offloading works as expected and that the PRCL signal can be driven (and hence controlled) by the ISI.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{images/PRM.png}
\caption[PRCL-ISI offloading test]{Best measurement of the PRCL signal with respect to the expected signal from the simulations: the two traces differ by a factor of 2.}
\label{prcltest}
\end{figure}

\begin{figure}[h!]
\centering
\includegraphics[scale=0.65]{images/prcl_filter.png}
\caption[PRCL-ISI crossover filter]{Open loop gain (OLG) crossover filter implemented at LHO for a measurement of PRCL signal in offloading conditions.}
\label{prclfilter}
\end{figure}

\section{A follow-up test at LHO}
The new configuration proposed and tested has changed a crucial section of the structure of LIGO. This change might have consequences on other sides of the instrument, for example affecting other noise sources. A test about the effect of this configuration has been performed at LHO in 2020 by the LHO, LLO and Seismic teams. This test studied the impact of CPS differential controls on scattered light glitches on O3b run. The teams investigated the reason of an increase in the rate of glitches. The synchronized motion of the chambers with the ground could in principle make the instrument more sensitive to glitches (and other noise sources) which were hidden by the seismic and sensor noises. The study is showing that the configuration is not responsible for this increase, but there is an effect on the sensitivity of LIGO to glitches, when the configuration is activated. The complete study is exposed in details in the LHO logbook post \cite{lhotest}.\\
This is an example of the impact of the CPS differential control on LIGO: it has been used to test the effect of wind in microseismic regions \cite{lhotest2} and further tests might help to understand the impact of less seismic motion on other noise sources.

\section*{Conclusions}
\addcontentsline{toc}{section}{Conclusions}
This study is promising to provide a significant contribution to the improvement of LIGO LSC signals and the detector stability when it is running in observing mode. The tests at LHO demonstrated that the experiment succeeded in lowering the seismic motion of the platforms by a factor of 3 at low frequencies and that also the DARM signal benefited from it. The simulations have shown that it is possible to reduce the differential motion of the chambers by a factor of 3 in order of magnitude below 0.1 Hz. The test on the Power Recycling Cavity Length highlighted that the signal can be controlled by the ISI according with the software simulations.\\
As we saw, the implications go straight to the basics of the instrument: a more stable detector produces a less noisy signal which can last longer into the cavities, assuring a longer observing time and giving the possibility to detect more gravitational waves and in lower ranges of frequency \cite{lantztalk} \cite{jenne}. 
LIGO Livingston site has also actuated a similar process, following the progression at LHO during the work on site in 2019 \cite{llo}. Due to the limited time of the commissioning break, it was not possible to take further measurements of ISI motion and LSC signals, especially with an accurate study of the blending filters. However, since the software skeleton of the new configuration has been built and installed on both LIGO CDSs, further studies and tests were due in 2020 to complete the last steps and test it fully on the interferometer. Meanwhile, the impact of the new configuration is under investigation on both sites: the reduced seismic motion could affect other noise sources that were previously hidden by the seismic and sensor noises, like glitches and wind. The results make then the idea worthy of future developments and improvements, and we are confident that these tests could be carried out in the near future.


