The final goal of the work presented in this document is to have an accurate Simscape model of the struts that can then be included in the Simscape model of the nano-hexapod.
A mounting bench is used to greatly simply the mounting of the struts as well as ensuring the correct strut length and coaxiality of the flexible joint's interfaces.
This is very important in order to not loose any stroke when the struts will be mounted on the nano-hexapod.
The main part of the bench is here to ensure both the correct strut length and strut coaxiality as shown in Figure \ref{fig:test_struts_mounting_step_0}.
The tight tolerances of this element has been verified as shown in Figure \ref{fig:test_struts_mounting_bench_first_concept} and were found to comply with the requirements.
The flexible joints are rigidly fixed to cylindrical tools shown in Figure \ref{fig:cylindrical_mounting_part} which are then mounted on the mounting tool shown in Figure \ref{fig:test_struts_mounting_step_0}.
\item Screw flexible joints inside the cylindrical interface element shown in Figure \ref{fig:cylindrical_mounting_part} (Figure \ref{fig:test_struts_mounting_step_1})
After removing the strut from the mounting bench, we obtain a strut with ensured coaxiality between the two flexible joint's interfaces (Figure \ref{fig:test_struts_mounted_strut}).
Then, each end of the strut is fixed to a vertically guided stage as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_1_enc}.
From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:test_struts_mode_shapes}):
\caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model. a) X-bending mode at 189Hz. b) Y-bending mode at 285Hz. c) Z-torsion mode at 400Hz}
The ``X-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_1_enc}.
The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_2} with the encoder and in Figure \ref{fig:test_struts_meas_spur_res_struts_2_encoder} with the encoder.
Finally, the ``Z-torsion'' is measured as shown in Figure \ref{fig:test_struts_meas_spur_res_struts_3}.
The bench is shown in Figure \ref{fig:test_struts_bench_leg_overview}.
Measurements are performed either when no encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) or when one encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).
\subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder}
\end{subfigure}
\caption{\label{fig:test_struts_bench_leg}Experimental setup to measured the dynamics of the struts.}
\end{figure}
First, only one strut is measured in details (Section \ref{ssec:test_struts_meas_strut_1}), and then all the struts are measured and compared (Section \ref{ssec:test_struts_meas_all_struts}).
Measurements are first performed on one of the strut that contains:
\begin{itemize}
\item the Amplified Piezoelectric Actuator (APA) number 1
\item flexible joints 1 and 2
\end{itemize}
In Section \ref{sec:meas_strut_1_no_encoder}, the dynamics of the strut is measured without the encoder attached to it.
Then in Section \ref{sec:meas_strut_1_encoder}, the encoder is attached to the struts, and the dynamic is identified.
\subsection{Without Encoder}
\label{sec:meas_strut_1_no_encoder}
\paragraph{FRF Identification - Setup}
Similarly to what was done for the identification of the APA, the identification is performed in three steps:
\begin{enumerate}
\item White noise excitation with small amplitude.
This is used to determine the main resonance of the system.
\item Sweep sine excitation with the amplitude lowered around the resonance.
The sweep sine is from 10Hz to 400Hz.
\item High frequency noise.
The noise is band-passed between 300Hz and 2kHz.
\end{enumerate}
Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz.
The time is the same for all measurements.
We get the frequency vector that will be the same for all the frequency domain analysis.
\paragraph{FRF Identification - Interferometer}
In this section, the dynamics from the excitation voltage \(V_a\) to the interferometer \(d_a\) is identified.
We compute the coherence for 2nd and 3rd identification and combine them.
The combined coherence is shown in Figure \ref{fig:strut_1_frf_dvf_plant_coh}, and is found to be very good up to at least 1kHz.
\caption{\label{fig:strut_1_frf_dvf_plant_coh}Obtained coherence for the plant from \(V_a\) to \(d_a\)}
\end{figure}
The transfer function from \(V_a\) to the interferometer measured displacement \(d_a\) is estimated and shown in Figure \ref{fig:strut_1_frf_dvf_plant_tf}.
\caption{\label{fig:strut_1_int_with_enc_frf_dvf_plant_tf}Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))}
\end{figure}
The obtained FRF is very close to the one that was obtained when no encoder was fixed to the struts as shown in Figure \ref{fig:strut_leg_compare_int_frf}.
\caption{\label{fig:strut_leg_compare_int_frf}Comparison of the measured FRF from \(V_a\) to \(d_a\) with and without the encoders fixed to the struts}
\end{figure}
\paragraph{FRF Identification - Encoder}
In this section, the dynamics from \(V_a\) to \(d_e\) (encoder) is identified.
The coherence is computed and shown in Figure \ref{fig:strut_1_enc_frf_dvf_plant_coh}.
\caption{\label{fig:strut_1_comp_enc_int}Comparison of the transfer functions from excitation voltage \(V_a\) to either the encoder \(d_e\) or the interferometer \(d_a\)}
\end{figure}
\begin{important}
The dynamics from the excitation voltage \(V_a\) to the measured displacement by the encoder \(d_e\) presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure \ref{fig:strut_1_comp_enc_int}).
It will be further investigated why the two dynamics as so different and what are causing all these resonances.
\end{important}
\paragraph{APA Resonances Frequency}
As shown in Figure \ref{fig:strut_1_spurious_resonances}, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
\caption{\label{fig:strut_1_spurious_resonances}Magnitude of the transfer function from excitation voltage \(V_a\) to encoder measurement \(d_e\). The frequency of the resonances are noted.}
\end{figure}
These resonances correspond to parasitic resonances of the strut itself.
The resonances seen by the encoder in Figure \ref{fig:strut_1_spurious_resonances} are indeed corresponding to the modes of the strut as shown in Figure \ref{fig:test_struts_mode_shapes}.
\caption{\label{fig:strut_1_frf_iff_comp_enc}Effect of the encoder on the IFF plant}
\end{figure}
\begin{important}
The transfer function from the excitation voltage \(V_a\) to the generated voltage \(V_s\) by the sensor stack is not influence by the fixation of the encoder.
This means that the IFF control strategy should be as effective whether or not the encoders are fixed to the struts.
\end{important}
\paragraph{Non-Minimum phase zero?}
In order to determine if the complex conjugate zero of Figure \ref{fig:strut_1_enc_frf_iff_plant_tf} is minimum phase or non-minimum phase, longer measurements are performed.
Now all struts are measured using the same procedure and test bench as in Section \ref{sec:meas_strut_1}.
\subsection{FRF Identification - Setup}
The identification of the struts dynamics is performed in two steps:
\begin{enumerate}
\item The excitation signal is a white noise with small amplitude.
This is used to estimate the low frequency dynamics.
\item Then a high frequency noise band-passed between 300Hz and 2kHz is used to estimate the high frequency dynamics.
\end{enumerate}
Then, the result of the first identification is used between 10Hz and 350Hz and the result of the second identification if used between 350Hz and 2kHz.
Here are the leg numbers that have been measured.
The data are loaded for both the first and second identification:
The time is the same for all measurements.
Then we defined a ``Hanning'' windows that will be used for the spectral analysis:
We get the frequency vector that will be the same for all the frequency domain analysis.
\subsection{FRF Identification - Encoder}
In this section, the dynamics from \(V_a\) to \(d_e\) (encoder) is identified.
The coherence is computed and shown in Figure \ref{fig:struts_frf_dvf_plant_coh} for all the measured struts.
\caption{\label{fig:struts_frf_int_plant_coh}Obtained coherence for the plant from \(V_a\) to \(d_e\)}
\end{figure}
Then, the transfer function from the DAC output voltage \(V_a\) to the measured displacement by the Attocube is computed for all the struts and shown in Figure \ref{fig:struts_frf_int_plant_tf}.
\subsection{Misalignment of the APA and flexible joints}
The misalignment between the two flexible joints and the APA has been measured for all the struts:
\begin{itemize}
\item the strut is fixed to the mounting bench
\item using an indicator, the height difference from the flexible joints and the APA is measured both for the top and bottom joints and on both sides
\item then it is possible to obtain the misalignment for both flexible joints
\end{itemize}
The raw measurements are shown in Table \ref{tab:meas_misalignment_struts_raw}.
As the flexible joint's ``thickness'' is 1mm larger than the APA ``thickness'', ideally (i.e. if it were perfectly centered) we would measure \texttt{-0.50mm} each time.
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_raw}Measured misalignments of the struts (\texttt{R} means ``red'' side, and \texttt{B} means ``black side'') in [mm]}
Also, the sum of the measured distances on each side should be 1mm (equal to the thickness difference between the flexible joint and the APA).
This is verified in Table \ref{tab:meas_misalignment_struts_thickness}.
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_thickness}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.2\linewidth}{cll}
\toprule
\textbf{Strut}&\textbf{Top}&\textbf{Bot}\\
\midrule
1 & -1.0 & -0.98\\
2 & -0.97 & -0.97\\
3 & -0.95 & -0.95\\
4 & -0.94 & -0.93\\
5 & -0.97 & -1.0\\
\bottomrule
\end{tabularx}
\end{table}
The differences of the measured distances on each side corresponds to the misalignment on that same side (Table \ref{tab:meas_misalignment_struts_results}).
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_results}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.25\linewidth}{cll}
\toprule
\textbf{Strut}&\textbf{Top}&\textbf{Bot}\\
\midrule
1 & 0.1 & 0.33\\
2 & -0.185 & 0.145\\
3 & 0.405 & 0.315\\
4 & -0.01 & 0.535\\
5 & 0.155 & 0.02\\
\bottomrule
\end{tabularx}
\end{table}
\begin{important}
The misalignment of the APA and flexible joints is quite large and variable from one strut to the other.
\end{important}
\subsection{Conclusion}
\begin{important}
All the struts are giving very consistent behavior from the excitation voltage \(V_a\) to the force sensor generated voltage \(V_s\) and to the interferometer measured displacement \(d_a\).
However, the dynamics from \(V_a\) to the encoder measurement \(d_e\) is much more complex and variable from one strut to the other most likely due to poor alignment of the APA with respect to the flexible joints.
\end{important}
The measured FRF are now saved for further use.
\section{Comparison of all the (re-aligned) Struts}
The struts are re-aligned and measured using the same test bench.
\subsection{Measured misalignment of the APA and flexible joints}
The misalignment between the APA and the flexible joints are measured.
The results are defined below and summarized in Table \ref{tab:meas_misalignment_struts_new_raw}.
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_raw}Measured misalignments of the struts (\texttt{R} means ``red'' side, and \texttt{B} means ``black side'') in [mm]}
Also, the sum of the measured distances on each side should be 1mm (equal to the thickness difference between the flexible joint and the APA).
This is verified in Table \ref{tab:meas_misalignment_struts_new_thickness}.
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_thickness}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.2\linewidth}{cll}
\toprule
\textbf{APA}&\textbf{Top}&\textbf{Bot}\\
\midrule
1 & -1.04 & -1.02\\
2 & -0.99 & -0.98\\
4 & -0.98 & -0.96\\
5 & -0.96 & -0.96\\
6 & -1.0 & -1.0\\
8 & -0.99 & -0.97\\
\bottomrule
\end{tabularx}
\end{table}
The differences of the measured distances on each side corresponds to the misalignment on that same side (Table \ref{tab:meas_misalignment_struts_new_results}).
\begin{table}[htbp]
\caption{\label{tab:meas_misalignment_struts_new_results}Measured thickness difference between the flexible joints and the APA in [mm]}
\centering
\begin{tabularx}{0.25\linewidth}{cll}
\toprule
\textbf{APA}&\textbf{Top}&\textbf{Bot}\\
\midrule
1 & -0.02 & 0.01\\
2 & 0.055 & 0.0\\
4 & 0.01 & -0.02\\
5 & 0.03 & -0.03\\
6 & 0.0 & 0.0\\
8 & -0.005 & 0.055\\
\bottomrule
\end{tabularx}
\end{table}
\begin{important}
After using the alignment pins, the misalignment of the APA and flexible joints are much smaller (\(< 50\,\mu m\) for all the struts).
\end{important}
\subsection{FRF Identification - Setup}
The excitation signal is a low pass filtered white noise.
Both the encoder and the force sensor voltage are measured.
Here are the leg numbers that have been measured.
The time is the same for all measurements.
Then we defined a ``Hanning'' windows that will be used for the spectral analysis:
We get the frequency vector that will be the same for all the frequency domain analysis.
\subsection{FRF Identification - Encoder}
In this section, the dynamics from \(V_a\) to \(d_e\) (encoder) is identified.
The coherence is computed and shown in Figure \ref{fig:struts_align_frf_dvf_plant_coh} for all the measured struts.
\caption{\label{fig:struts_align_frf_dvf_plant_tf}Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))}
\end{figure}
\begin{important}
Even though the struts are much better aligned, we still observe high variability between the struts for the transfer function from \(V_a\) to \(d_e\).
\end{important}
\subsection{FRF Identification - Force Sensor}
In this section, the dynamics from \(V_a\) to \(V_s\) is identified.
First the coherence is computed and shown in Figure \ref{fig:struts_frf_iff_plant_coh}.
\item compare the measured FRF with the modelled FRF
\item help the correct understanding/interpretation of the results
\item tune the model of the struts (APA, flexible joints, encoder)
\end{itemize}
This study is structured as follow:
\begin{itemize}
\item Section \ref{sec:struts_comp_2dof}: the measured FRF are compared with the 2DoF APA model.
\item Section \ref{sec:struts_effect_misalignment}: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied.
It is found that the misalignment has a large impact on the dynamics from \(V_a\) to \(d_e\).
\item Section \ref{sec:struts_effect_joint_stiffness}: the effect of the flexible joint's stiffness on the dynamics is studied.
It is found that the axial stiffness of the joints has a large impact on the location of the zeros on the transfer function from \(V_s\) to \(d_e\).
\end{itemize}
\section{Comparison with the 2-DoF Model}
\label{sec:struts_comp_2dof}
\subsection{First Identification}
The strut is initialized with default parameters (optimized parameters identified from previous experiments).
The inputs and outputs of the model are defined.
The dynamics is identified and shown in Figure \ref{fig:strut_bench_model_bode}.
\caption{\label{fig:strut_bench_model_bode}Identified transfer function from \(V_a\) to \(V_s\) and from \(V_a\) to \(d_e,d_a\) using the simple 2DoF model for the APA}
\end{figure}
\subsection{Comparison with the experimental Data}
The experimentally measured FRF are loaded.
The FRF from \(V_a\) to \(d_a\) as well as from \(V_a\) to \(V_s\) are shown in Figure \ref{fig:comp_strut_plant_after_opt} and compared with the model.
They are both found to match quite well with the model.
\caption{\label{fig:comp_strut_plant_iff_after_opt}Comparison of the measured FRF and the optimized model}
\end{figure}
\begin{important}
The 2-DoF model is quite effective in modelling the transfer function from actuator to force sensor and from actuator to interferometer (Figure \ref{fig:comp_strut_plant_after_opt}).
But it is not effective in modeling the transfer function from actuator to encoder (Figure \ref{fig:comp_strut_plant_iff_after_opt}).
This is due to the fact that resonances greatly affecting the encoder reading are not modelled.
In the next section, flexible model of the APA will be used to model such resonances.
\end{important}
\section{Comparison with the Flexible Model}
\label{sec:struts_comp_flexible}
\subsection{First Identification}
The strut is initialized with default parameters (optimized parameters identified from previous experiments).
The inputs and outputs of the model are defined.
The dynamics is identified and shown in Figure \ref{fig:strut_bench_model_bode}.
\subsection{Comparison with the experimental Data}
The experimentally measured FRF are loaded.
The FRF from \(V_a\) to \(d_a\) as well as from \(V_a\) to \(V_s\) are shown in Figure \ref{fig:comp_strut_plant_after_opt} and compared with the model.
They are both found to match quite well with the model.
For instance, consider Figure \ref{fig:strut_misalign_schematic} where there is a misalignment in the \(y\) direction.
In such case, the mode at 200Hz is foreseen to be more excited as the misalignment \(d_y\) increases and therefore the dynamics from the actuator to the encoder should also change around 200Hz.
\caption{\label{fig:strut_misalign_schematic}Mis-alignement between the joints and the APA}
\end{figure}
If the misalignment is in the \(x\) direction, the mode at 285Hz should be more affected whereas a misalignment in the \(z\) direction should not affect these resonances.
Such statement is studied in this section.
But first, the measured FRF of the struts are loaded.
\subsection{Perfectly aligned APA}
Let's first consider that the strut is perfectly mounted such that the two flexible joints and the APA are aligned.
And define the inputs and outputs of the models:
\begin{itemize}
\item Input: voltage generated by the DAC
\item Output: measured displacement by the encoder
\end{itemize}
The transfer function is identified and shown in Figure \ref{fig:comp_enc_frf_align_perfect}.
From Figure \ref{fig:comp_enc_frf_align_perfect}, it is clear that:
\begin{enumerate}
\item The model with perfect alignment is not matching the measured FRF
\item The mode at 200Hz is not present in the identified dynamics of the Simscape model
\caption{\label{fig:comp_enc_frf_align_perfect}Comparison of the model with a perfectly aligned APA and flexible joints with the measured FRF from actuator to encoder}
\end{figure}
\begin{question}
Why is the flexible mode of the strut at 200Hz is not seen in the model in Figure \ref{fig:comp_enc_frf_align_perfect}?
Probably because the presence of this mode is not due because of the ``unbalanced'' mass of the encoder, but rather because of the misalignment of the APA with respect to the two flexible joints.
This will be verified in the next sections.
\end{question}
\subsection{Effect of a misalignment in y}
Let's compute the transfer function from output DAC voltage \(V_s\) to the measured displacement by the encoder \(d_e\) for several misalignment in the \(y\) direction:
The obtained dynamics are shown in Figure \ref{fig:effect_misalignment_y}.
\caption{\label{fig:effect_misalignment_y}Effect of a misalignement in the \(y\) direction}
\end{figure}
\begin{important}
The alignment of the APA with the flexible joints as a \textbf{huge} influence on the dynamics from actuator voltage to measured displacement by the encoder.
The misalignment in the \(y\) direction mostly influences:
\begin{itemize}
\item the presence of the flexible mode at 200Hz
\item the location of the complex conjugate zero between the first two resonances:
\begin{itemize}
\item if \(d_y < 0\): there is no zero between the two resonances and possibly not even between the second and third ones
\item if \(d_y > 0\): there is a complex conjugate zero between the first two resonances
\end{itemize}
\item the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
\end{itemize}
\end{important}
\subsection{Effect of a misalignment in x}
Let's compute the transfer function from output DAC voltage to the measured displacement by the encoder for several misalignment in the \(x\) direction:
The obtained dynamics are shown in Figure \ref{fig:effect_misalignment_x}.
\caption{\label{fig:effect_misalignment_x}Effect of a misalignement in the \(x\) direction}
\end{figure}
\begin{important}
The misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz.
\end{important}
\subsection{Find the misalignment of each strut}
From the previous analysis on the effect of a \(x\) and \(y\) misalignment, it is possible to estimate the \(x,y\) misalignment of the measured struts.
The misalignment that gives the best match for the FRF are defined below.
For each misalignment, the dynamics from the DAC voltage to the encoder measurement is identified.
The results are shown in Figure \ref{fig:comp_all_struts_corrected_misalign}.
\caption{\label{fig:comp_all_struts_corrected_misalign}Comparison (model and measurements) of the FRF from DAC voltage u to measured displacement by the encoders for all the struts}
\end{figure}
\begin{important}
By tuning the misalignment of the APA with respect to the flexible joints, it is possible to obtain a good fit between the model and the measurements (Figure \ref{fig:comp_all_struts_corrected_misalign}).
If encoders are to be used when fixed on the struts, it is therefore very important to properly align the APA and the flexible joints when mounting the struts.
In the future, a ``pin'' will be used to better align the APA with the flexible joints.
We can expect the amplitude of the spurious resonances to decrease.
\end{important}
\section{Effect of flexible joint's characteristics}
\label{sec:struts_effect_joint_stiffness}
As the struts are composed of one APA and two flexible joints, it is obvious that the flexible joint characteristics will change the dynamic behavior of the struts.
Using the Simscape model, the effect of the flexible joint's characteristics on the dynamics as measured on the test bench are studied:
\begin{itemize}
\item Section \ref{sec:struts_effect_bending_stiff_joints}: the effects of a change of bending stiffness is studied
\item Section \ref{sec:struts_effect_axial_stiff_joints}: the effects of a change of axial stiffness is studied
\item Section \ref{sec:struts_effect_bending_damping_joints}: the effects of a change of bending damping is studied
\end{itemize}
The studied dynamics is between \(V_a\) and the encoder displacement \(d_e\).
\subsection{Effect of bending stiffness of the flexible joints}
\label{sec:struts_effect_bending_stiff_joints}
Let's initialize an APA which is a little bit misaligned.
The bending stiffnesses for which the dynamics is identified are defined below.
Then the identification is performed for all the values of the bending stiffnesses.
The obtained dynamics from DAC voltage to encoder measurements are compared in Figure \ref{fig:effect_enc_bending_stiff}.
\caption{\label{fig:effect_enc_axial_stiff}Dynamics from DAC output to encoder for several axial stiffnesses}
\end{figure}
\begin{important}
The axial stiffness of the flexible joint has a large impact on the frequency of the complex conjugate zero.
Using the measured FRF on the test-bench, if is therefore possible to estimate the axial stiffness of the flexible joints from the location of the zero.
This method gives nice match between the measured FRF and the one extracted from the simscape model, however it could give not so accurate values of the joint's axial stiffness as other factors are also influencing the location of the zero.
Using this method, an axial stiffness of \(70 N/\mu m\) is found to give good results (and is reasonable based on the finite element models).
\end{important}
\subsection{Effect of bending damping}
\label{sec:struts_effect_bending_damping_joints}
Now let's study the effect of the bending damping of the flexible joints.
The tested bending damping are defined below:
Then the identification is performed for all the values of the bending damping.
The results are shown in Figure \ref{fig:effect_enc_bending_damp}.
