The Nano-Hexapod struts (shown in Figure \ref{fig:test_struts_picture_strut}) are composed of two flexible joints that are fixed at the two ends of the strut, one \acrfull{apa}\footnote{APA300ML from Cedrat Technologies} and one optical encoder\footnote{Vionic from Renishaw}.
The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section \ref{sec:test_struts_flexible_modes}).
Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section \ref{sec:test_struts_dynamical_meas}).
It was found that the dynamics from the \acrshort{dac} voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section \ref{sec:test_struts_flexible_modes}).
The strut models were then compared with the measured dynamics (Section \ref{sec:test_struts_simscape}).
The model dynamics from the \acrshort{dac} voltage to the axial motion of the strut (measured by an interferometer) and to the force sensor voltage well match the experimental results.
However, this is not the case for the dynamics from \acrshort{dac} voltage to the encoder displacement.
It is found that the complex dynamics is due to a misalignment between the flexible joints and the \acrshort{apa}.
It consists of a ``main frame'' (Figure \ref{fig:test_struts_mounting_step_0}) precisely machined to ensure both correct strut length and strut coaxiality.
The coaxiality is ensured by good flatness (specified at \(20\,\mu m\)) between surfaces A and B and between surfaces C and D.
Such flatness was checked using a FARO arm\footnote{FARO Arm Platinum 4ft, specified accuracy of \(\pm13\mu m\)} (see Figure \ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machined dowel holes.
\caption{\label{fig:test_struts_mounting_base_part}Main element of the mounting bench for the struts that ensure good coaxiality of the two flexible joints and correct struts length.}
The flexible joints were not directly fixed to the mounting bench but were fixed to a cylindrical ``sleeve'' shown in Figures \ref{fig:test_struts_cylindrical_mounting_part_top} and \ref{fig:test_struts_cylindrical_mounting_part_bot}.
The goal of these ``sleeves'' is to avoid mechanical stress that could damage the flexible joints during the mounting process.
These ``sleeves'' have one dowel groove (that are fitted to the dowel holes shown in Figure \ref{fig:test_struts_mounting_step_0}) that will determine the length of the mounted strut.
The ``sleeves'' were mounted to the main element as shown in Figure \ref{fig:test_struts_mounting_step_0}.
The left sleeve has a thigh fit such that its orientation is fixed (it is roughly aligned horizontally), while the right sleeve has a loose fit such that it can rotate (it will get the same orientation as the fixed one when tightening the screws).
The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure \ref{fig:test_struts_mounting_step_2} and screwed together using a torque screwdriver.
A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure \ref{fig:test_struts_mounting_joints}).
Thanks to this mounting procedure, the coaxiality and length between the two flexible joint's interfaces can be obtained within the desired tolerances.
A Finite Element Model\footnote{Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called \emph{17-4 PH}. Encoder and ruler support material is aluminium.} of the struts is developed and is used to estimate the flexible modes.
The inertia of the encoder (estimated at \(15\,g\)) is considered.
The mode shapes are displayed in Figure \ref{fig:test_struts_mode_shapes}: an ``X-bending'' mode at 189Hz, a ``Y-bending'' mode at 285Hz and a ``Z-torsion'' mode at 400Hz.
The setup used to measure the ``X-bending'' mode is shown in Figure \ref{fig:test_struts_meas_x_bending}.
The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_y_bending} and the ``Z-torsion'' measurement setup is shown in Figure \ref{fig:test_struts_meas_z_torsion}.
These tests were performed with and without the encoder being fixed to the strut.
The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure \ref{fig:test_struts_spur_res_frf_no_enc} when the encoder is not fixed to the strut and in Figure \ref{fig:test_struts_spur_res_frf_enc} when the encoder is fixed to the strut.
\subcaption{\label{fig:test_struts_spur_res_frf_enc}with the encoder}
\end{subfigure}
\caption{\label{fig:test_struts_spur_res_frf}Measured frequency response functions without the encoder \ref{fig:test_struts_spur_res_frf} and with the encoder \ref{fig:test_struts_spur_res_frf_enc}}
Table \ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies and the computed ones using the \acrfull{fem}.
The resonance frequencies of the 3 modes are only slightly decreased when the encoder is fixed to the strut.
In addition, the computed resonance frequencies from the \acrshort{fem} are very close to the measured frequencies when the encoder is fixed to the strut.
The strut mounted on the bench is shown in Figure \ref{fig:test_struts_bench_leg_overview}
A schematic of the bench and the associated signals are shown in Figure \ref{fig:test_struts_bench_schematic}.
A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools} is used to measure the motion of the granite (i.e. the axial motion of the strut).
System identification was performed without the encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) and with one encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).
\subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
\end{subfigure}
\caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})}
It was found that the encoder had very little effect on the transfer function from excitation voltage \(u\) to the axial motion of the strut \(d_a\) as measured by the interferometer (Figure \ref{fig:test_struts_effect_encoder_int}).
This means that the axial motion of the strut is unaffected by the presence of the encoder.
Similarly, it has little effect on the transfer function from \(u\) to the sensor stack voltage \(V_s\) (Figure \ref{fig:test_struts_effect_encoder_iff}).
This means that the encoder should have little effect on the effectiveness of the integral force feedback control strategy.
\subcaption{\label{fig:test_struts_comp_enc_int}$u$ to $d_e$, $d_a$}
\end{subfigure}
\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) (\subref{fig:test_struts_effect_encoder_int}) and from \(u\) to \(V_s\) (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int})}
The dynamics measured by the encoder (i.e. \(d_e/u\)) and interferometers (i.e. \(d_a/u\)) are compared in Figure \ref{fig:test_struts_comp_enc_int}.
The dynamics from the excitation voltage \(u\) to the displacement measured by the encoder \(d_e\) presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure \ref{fig:test_struts_comp_enc_int}).
The obtained dynamics from \(u\) to \(d_a\) are compared in Figure \ref{fig:test_struts_comp_interf_plants} while is dynamics from \(u\) to \(V_s\) are compared in Figure \ref{fig:test_struts_comp_iff_plants}.
All the struts exhibit very consistent behavior from the excitation voltage \(u\) to the force sensor generated voltage \(V_s\) and to the interferometer measured displacement \(d_a\).
However, the dynamics from \(u\) to the encoder measurement \(d_e\) is much more complex and vary from one strut to the another.
The reason for this variability will be studied in the next section thanks to the strut model.
The Simscape model of the strut was included in the Simscape model of the test bench (see Figure \ref{fig:test_struts_simscape_model}).
The obtained model was first used to compare the measured FRF with the existing model (Section \ref{ssec:test_struts_comp_model}).
Using a flexible APA model (extracted from a \acrshort{fem}), the effect of a misalignment of the APA with respect to flexible joints is studied (Section \ref{ssec:test_struts_effect_misalignment}).
It was found that misalignment has a large impact on the dynamics from \(u\) to \(d_e\).
This misalignment is estimated and measured in Section \ref{ssec:test_struts_meas_misalignment}.
The struts were then disassembled and reassemble a second time to optimize alignment (Section \ref{sec:test_struts_meas_all_aligned_struts}).
The model dynamics from DAC voltage \(u\) to the axial motion of the strut \(d_a\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_int}) and from DAC voltage \(u\) to the force sensor voltage \(V_s\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_iff}) are well matching the experimental identification.
For the 2DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis).
For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics.
\caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured frequency response functions, the Simscape model using the 2 DoF APA model, and using the ``flexible'' APA300ML model (Super-Element extracted from a Finite Element Model).}
As shown in Figure \ref{fig:test_struts_comp_enc_plants}, the identified dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\) are very different from one strut to the other.
In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the APA) can explain such dynamics.
For instance, consider Figure \ref{fig:test_struts_misalign_schematic} where there is a misalignment in the \(y\) direction between the two flexible joints (well aligned thanks to the mounting procedure in Section \ref{sec:test_struts_mounting}) and the APA300ML.
In this case, the ``x-bending'' mode at 200Hz (see Figure \ref{fig:test_struts_meas_x_bending}) can be expected to have greater impact on the dynamics from the actuator to the encoder.
To verify this assumption, the dynamics from the output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the flexible APA Simscape model for several misalignments in the \(y\) direction.
The obtained dynamics are shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
The alignment of the APA with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder.
\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)
The same can be done for misalignments in the \(x\) direction.
The obtained dynamics (Figure \ref{fig:test_struts_effect_misalignment_x}) are showing that misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure \ref{fig:test_struts_mode_shapes_2}).
A comparison of the experimental frequency response functions in Figure \ref{fig:test_struts_comp_enc_plants} with the model dynamics for several \(y\) misalignments in Figure \ref{fig:test_struts_effect_misalignment_y} indicates a clear similarity.
This similarity suggests that the identified differences in dynamics are caused by misalignment.
\subcaption{\label{fig:test_struts_effect_misalignment_x}Misalignment along $x$}
\end{subfigure}
\caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) direction (\subref{fig:test_struts_effect_misalignment_y}) and in the \(x\) direction (\subref{fig:test_struts_effect_misalignment_x})}
During the initial mounting of the struts, as presented in Section \ref{sec:test_struts_mounting}, the positioning pins that were used to position the APA with respect to the flexible joints in the \(y\) directions were not used (not received at the time).
Therefore, large \(y\) misalignments are expected.
\item the struts were fixed horizontally on the mounting bench, as shown in Figure \ref{fig:test_struts_mounting_step_3} but without the encoder
\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm0.5\,\mu m\)}, the height difference between the flexible joints surface and the APA shell surface was measured for both the top and bottom joints and for both sides
\item as the thickness of the flexible joint is \(21\,mm\) and the thickness of the APA shell is \(20\,mm\), \(0.5\,mm\) of height difference should be measured if the two are perfectly aligned
Large variations in the \(y\) misalignment are found from one strut to the other (results are summarized in Table \ref{tab:test_struts_meas_y_misalignment}).
To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is \(1\,mm\) (equal to the thickness difference between the flexible joint and the APA).
Thickness differences for all the struts were found to be between \(0.94\,mm\) and \(1.00\,mm\) which indicate low errors compared to the misalignments found in Table \ref{tab:test_struts_meas_y_misalignment}.
By using the measured \(y\) misalignment in the Simscape model with the flexible APA model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}.
A better match in the dynamics can be obtained by fine-tuning both the \(x\) and \(y\) misalignments (yellow curves in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}).
This confirms that misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\).
If encoders are fixed to the struts, the APA and flexible joints must be precisely aligned when mounting the struts.
With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
\caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measured displacement \(d_e\) by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the \(y\) misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned}
The alignment is then estimated using a length gauge, as described in the previous sections.
Measured \(y\) alignments are summarized in Table \ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu m\) for all the struts, which is much better than before (see Table \ref{tab:test_struts_meas_y_misalignment}).
\caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in \(mm\).}
The dynamics of the re-aligned struts were then measured on the same test bench (Figure \ref{fig:test_struts_bench_leg}).
A comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is presented in Figure \ref{fig:test_struts_comp_enc_frf_realign}.
\caption{\label{fig:test_struts_comp_enc_frf_realign}Comparison of the dynamics from \(u\) to \(d_e\) before and after proper alignment using the dowel pins}
The Hano-Hexapod struts are a key component of the developed \acrfull{nass}.
A mounting bench was used to obtain struts with good interface coaxiality, equal lengths, and ideally the same dynamics.
Using a test bench, it was found that while all the mounted struts had extremely similar dynamics when considering the axial motion and the integrated force sensor, the dynamics as seen by the encoder is much more complex and varied from one strut to the other.
