diff --git a/test-bench-id31.org b/test-bench-id31.org index 9e26e6b..a1d6364 100644 --- a/test-bench-id31.org +++ b/test-bench-id31.org @@ -206,7 +206,8 @@ One big advantage of doing the control in the cartesian plane, is that we don't Maybe this should be done in A6 (simscape-nass). Here it can be reminded when doing the control in the cartesian frame. -** TODO [#A] Add same specifications for all the curves +** DONE [#A] Add same specifications for all the curves +CLOSED: [2025-02-01 Sat 10:22] Peak to peak errors: - Dz < +/- 50nm @@ -245,7 +246,9 @@ Fn => Vs - RPM - rpm -- Wz (deg/s) +- *Wz (deg/s)* + +Maybe deg/s is the most adequate Make a choice, and then adapt all notations. Also, change the =initializeReferences= to accept the chosen description instead of =period=. @@ -393,6 +396,11 @@ This means that height of nano-hexapod <=> beam is 800 - 530 - 95 = *175mm and n it seems 150mm was used for the metrology jacobian! - [X] If something is change, update the previous Simscape models +** TODO [#C] Verify notations + +$\bm{\epsilon\mathcal{L}}$ and not $\bm{e\mathcal{L}}$ +$\bm{\epsilon\mathcal{X}}$ and not $\bm{e\mathcal{L}}$ + ** CANC [#B] Should the micro-hexapod position be adjusted to match the experiment CLOSED: [2024-11-13 Wed 18:05] @@ -411,7 +419,7 @@ First identification: - New identification for all masses - Better match with Simscape model! -** QUES [#B] Why now we have minimum phase zero for IFF Plant? +** QUES [#C] Why now we have minimum phase zero for IFF Plant? ** CANC [#C] Find identification where Rz was not taken into account CLOSED: [2024-11-12 Tue 16:03] @@ -919,28 +927,26 @@ exportFig('figs/test_id31_xy_map_sphere.pdf', 'width', 'half', 'height', 'normal #+end_subfigure #+end_figure -* Identified Open Loop Plant +* Open Loop Plant :PROPERTIES: :header-args:matlab+: :tangle matlab/test_id31_2_open_loop_plant.m :END: <> ** Introduction :ignore: +The NASS plant is schematically shown in Figure ref:fig:test_id31_block_schematic_plant. +The input $\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]$ is the command signal and corresponds to the voltages generated for each piezoelectric actuator. +After amplification, the voltages across the piezoelectric stack actuators are $\bm{V}_a = [V_{a1},\ V_{a2},\ V_{a3},\ V_{a4},\ V_{a5},\ V_{a6}]$. +From the setpoint of micro-station stages ($r_{D_y}$ for the translation stage, $r_{R_y}$ for the tilt stage and $r_{R_z}$ for the spindle), the reference pose of the sample $\bm{r}_{\mathcal{X}}$ is computed using the micro-station's kinematics. +The sample's position $\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]$ is measured using multiple sensors. +First, the five interferometers $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$ are used to measure the $[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$ degrees of freedom of the sample. +The $R_z$ position of the sample is computed from the spindle's setpoint $r_{R_z}$ and from the 6 encoders $\bm{d}_e$ integrated in the nano-hexapod. -- Force sensors: $\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]$ -- Encoders: $\bm{d}_e = [d_{e1},\ d_{e2},\ d_{e3},\ d_{e4},\ d_{e5},\ d_{e6}]$ -- Interferometers: $\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]$ -- Command signal: $\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]$ -- Voltage across the piezoelectric stack actuator: $\bm{V}_a = [V_{a1},\ V_{a2},\ V_{a3},\ V_{a4},\ V_{a5},\ V_{a6}]$ -- Motion of the sample measured by external metrology: $\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]$ -# - Sample motion expressed in the nano-hexapod frame: $\bm{\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]$ -# - Motion of the struts measured by external metrology: $\bm{\mathcal{L}} = [\mathcal{L}_1,\,\mathcal{L}_2,\,\mathcal{L}_3,\,\mathcal{L}_4,\,\mathcal{L}_5,\,\mathcal{L}_6]$ -- Error of the sample measured by external metrology: $\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]$ -- Error of the struts measured by external metrology: $\bm{\epsilon\mathcal{L}} = [\epsilon_{\mathcal{L}_1},\,\epsilon_{\mathcal{L}_2},\,\epsilon_{\mathcal{L}_3},\,\epsilon_{\mathcal{L}_4},\,\epsilon_{\mathcal{L}_5},\,\epsilon_{\mathcal{L}_6}]$ -- Spindle angle setpoint (or encoder): $r_{R_z}$ -- Translation stage setpoint: $r_{D_y}$ -- Tilt stage setpoint: $r_{R_y}$ +The sample's position $\bm{y}_{\mathcal{X}}$ is compared to the reference position $\bm{r}_{\mathcal{X}}$ to compute the position error in the frame of the (rotating) nano-hexapod $\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]$. +Finally, the Jacobian matrix $\bm{J}$ of the nano-hexapod is used to map $\bm{\epsilon\mathcal{X}}$ in the frame of the nano-hexapod struts $\bm{\epsilon\mathcal{L}} = [\epsilon_{\mathcal{L}_1},\,\epsilon_{\mathcal{L}_2},\,\epsilon_{\mathcal{L}_3},\,\epsilon_{\mathcal{L}_4},\,\epsilon_{\mathcal{L}_5},\,\epsilon_{\mathcal{L}_6}]$. + +Voltages generated by the force sensor piezoelectric stacks $\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]$ will be used for active damping. #+begin_src latex :file test_id31_block_schematic_plant.pdf \begin{tikzpicture} @@ -994,7 +1000,7 @@ exportFig('figs/test_id31_xy_map_sphere.pdf', 'width', 'half', 'height', 'normal #+end_src #+name: fig:test_id31_block_schematic_plant -#+caption: Schematic of the +#+caption: Schematic of the NASS plant #+RESULTS: [[file:figs/test_id31_block_schematic_plant.png]] @@ -1027,7 +1033,7 @@ exportFig('figs/test_id31_xy_map_sphere.pdf', 'width', 'half', 'height', 'normal <> #+end_src -** First Open-Loop Plant Identification +** Open-Loop Plant Identification <> The plant dynamics is first identified for a fixed spindle angle (at $0\,\text{deg}$) and without any payload. @@ -1035,10 +1041,10 @@ The model dynamics is also identified in the same conditions. A first comparison between the model and the measured dynamics is done in Figure ref:fig:test_id31_first_id. A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled). -However, the coupling for the transfer function from command signals $\bm{u}$ to estimated strut motion from the external metrology $e\bm{\mathcal{L}}$ is larger than expected (Figure ref:fig:test_id31_first_id_int). +However, the coupling for the transfer function from command signals $\bm{u}$ to the estimated strut motion from the external metrology $\bm{\epsilon\mathcal{L}}$ is larger than expected (Figure ref:fig:test_id31_first_id_int). The experimental time delay estimated from the FRF (Figure ref:fig:test_id31_first_id_int) is larger than expected. -After investigation, it was found that the additional delay was due to digital processing unit[fn:3] that was used to read the interferometers in the Speedgoat. +After investigation, it was found that the additional delay was due to a digital processing unit[fn:3] that was used to get the interferometers' signals in the Speedgoat. This issue was later solved. #+begin_src matlab @@ -1220,7 +1226,7 @@ exportFig('figs/test_id31_first_id_iff.pdf', 'width', 'half', 'height', 600); #+end_src #+name: fig:test_id31_first_id -#+caption: Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}). +#+caption: Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are shown with shaded lines. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:test_id31_first_id_int}External Metrology} @@ -1241,12 +1247,12 @@ exportFig('figs/test_id31_first_id_iff.pdf', 'width', 'half', 'height', 600); <> One possible explanation of the increased coupling observed in Figure ref:fig:test_id31_first_id_int is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes. -To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders is implemented. -This allowed to perform two straight movements of the nano-hexapod along the $x$ and $y$ axes in the frame of the nano-hexapod. -During these two movements, the external metrology measurement is recorded and shown in Figure ref:fig:test_id31_Rz_align_error. +To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders was implemented. +This allowed to perform two straight movements of the nano-hexapod along its $x$ and $y$ axes. +During these two movements, the external metrology measurement was recorded and are shown in Figure ref:fig:test_id31_Rz_align_error. It was found that there is a misalignment of 2.7 degrees (rotation along the vertical axis) between the interferometer axes and nano-hexapod axes. This was corrected by adding an offset to the spindle angle. -To check that the alignment has improved, the same movement was performed using the nano-hexapod while recording the signal of the external metrology. +After alignment, the same movement was performed using the nano-hexapod while recording the signal of the external metrology. Results shown in Figure ref:fig:test_id31_Rz_align_correct are indeed indicating much better alignment. #+begin_src matlab @@ -1335,10 +1341,7 @@ exportFig('figs/test_id31_Rz_align_correct.pdf', 'width', 'half', 'height', 'nor #+end_subfigure #+end_figure -** Open-Loop Identification after alignment -<> - -The plant dynamics is identified after the fine alignment and is compared with the model dynamics in Figure ref:fig:test_id31_first_id_int_better_rz_align. +The plant dynamics was identified again after the fine alignment and is compared with the model dynamics in Figure ref:fig:test_id31_first_id_int_better_rz_align. Compared to the initial identification shown in Figure ref:fig:test_id31_first_id_int, the obtained coupling has decreased and is now close to the coupling obtained with the multi-body model. At low frequency (below $10\,\text{Hz}$) all the off-diagonal elements have an amplitude $\approx 100$ times lower compared to the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized way (i.e. $6$ SISO controllers). Between $650\,\text{Hz}$ and $1000\,\text{Hz}$, several modes can be observed that are due to flexible modes of the top platform and modes of the two spheres adjustment mechanism. @@ -1405,8 +1408,10 @@ exportFig('figs/test_id31_first_id_int_better_rz_align.pdf', 'width', 'wide', 'h ** Effect of Payload Mass <> -The system dynamics was identified with four payload conditions that are shown in Figure ref:fig:test_id31_picture_masses. +In order to see how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions that are shown in Figure ref:fig:test_id31_picture_masses. The obtained direct terms are compared with the model dynamics in Figure ref:fig:test_nhexa_comp_simscape_diag_masses. +It is shown that the model dynamics well predicts the measured dynamics for all payload conditions. +Therefore the model can be used for model-based control is necessary. It is interesting to note that the anti-resonances in the force sensor plant are now appearing as minimum-phase, as the model predicts (Figure ref:fig:test_id31_comp_simscape_iff_diag_masses). @@ -1754,8 +1759,7 @@ exportFig('figs/test_id31_comp_simscape_iff_diag_masses.pdf', 'width', 'half', ' ** Effect of Spindle Rotation <> -The dynamics was then identified while the Spindle was rotating at constant velocity. -Three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$. +To verify that all the kinematics in Figure ref:fig:test_id31_block_schematic_plant are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at $36\,\text{deg}/s$ and at $180\,\text{deg}/s$. The comparison of the obtained dynamics from command signal $u$ to estimated strut error $e\mathcal{L}$ is done in Figure ref:fig:test_id31_effect_rotation. Both direct terms (Figure ref:fig:test_id31_effect_rotation_direct) and coupling terms (Figure ref:fig:test_id31_effect_rotation_coupling) are unaffected by the rotation. @@ -1924,10 +1928,9 @@ exportFig('figs/test_id31_effect_rotation_coupling.pdf', 'width', 'half', 'heigh :UNNUMBERED: t :END: -Thanks to the model, poor alignment between the nano-hexapod axes and the external metrology axes could be identified. -After alignment, the identified dynamics is well matching with the multi-body model. - -Also, the observed effects of the payload mass and of the spindle rotation on the dynamics are well matching the model predictions. +The identified frequency response functions from command signals $\bm{u}$ to the force sensors $\bm{V}_s$ and to the estimated strut errors $\bm{\epsilon\mathcal{L}}$ are well matching the developed multi-body model. +Effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used. +The spindle rotation has no visible effect on the measured dynamics, indicating that controllers should be robust to the spindle rotation. * Decentralized Integral Force Feedback :PROPERTIES: diff --git a/test-bench-id31.pdf b/test-bench-id31.pdf index e8576c8..bfc82ae 100644 Binary files a/test-bench-id31.pdf and b/test-bench-id31.pdf differ diff --git a/test-bench-id31.tex b/test-bench-id31.tex index 3cbf43d..434e6ae 100644 --- a/test-bench-id31.tex +++ b/test-bench-id31.tex @@ -1,4 +1,4 @@ -% Created 2025-01-31 Fri 18:54 +% Created 2025-02-01 Sat 10:36 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -267,28 +267,28 @@ The effect of the noise on the translation and rotation measurements is estimate \caption{\label{fig:test_id31_metrology_errors}Estimated measurement errors of the metrology. Cross-coupling between lateral motion and vertical measurement is shown in (\subref{fig:test_id31_xy_map_sphere}). Effect of interferometer noise on the measured translations and rotations is shown in (\subref{fig:test_id31_interf_noise}).} \end{figure} -\chapter{Identified Open Loop Plant} +\chapter{Open Loop Plant} \label{sec:test_id31_open_loop_plant} -\begin{itemize} -\item Force sensors: \(\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]\) -\item Encoders: \(\bm{d}_e = [d_{e1},\ d_{e2},\ d_{e3},\ d_{e4},\ d_{e5},\ d_{e6}]\) -\item Interferometers: \(\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]\) -\item Command signal: \(\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]\) -\item Voltage across the piezoelectric stack actuator: \(\bm{V}_a = [V_{a1},\ V_{a2},\ V_{a3},\ V_{a4},\ V_{a5},\ V_{a6}]\) -\item Motion of the sample measured by external metrology: \(\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]\) -\item Error of the sample measured by external metrology: \(\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]\) -\item Error of the struts measured by external metrology: \(\bm{\epsilon\mathcal{L}} = [\epsilon_{\mathcal{L}_1},\,\epsilon_{\mathcal{L}_2},\,\epsilon_{\mathcal{L}_3},\,\epsilon_{\mathcal{L}_4},\,\epsilon_{\mathcal{L}_5},\,\epsilon_{\mathcal{L}_6}]\) -\item Spindle angle setpoint (or encoder): \(r_{R_z}\) -\item Translation stage setpoint: \(r_{D_y}\) -\item Tilt stage setpoint: \(r_{R_y}\) -\end{itemize} +The NASS plant is schematically shown in Figure \ref{fig:test_id31_block_schematic_plant}. +The input \(\bm{u} = [u_1,\ u_2,\ u_3,\ u_4,\ u_5,\ u_6]\) is the command signal and corresponds to the voltages generated for each piezoelectric actuator. +After amplification, the voltages across the piezoelectric stack actuators are \(\bm{V}_a = [V_{a1},\ V_{a2},\ V_{a3},\ V_{a4},\ V_{a5},\ V_{a6}]\). + +From the setpoint of micro-station stages (\(r_{D_y}\) for the translation stage, \(r_{R_y}\) for the tilt stage and \(r_{R_z}\) for the spindle), the reference pose of the sample \(\bm{r}_{\mathcal{X}}\) is computed using the micro-station's kinematics. +The sample's position \(\bm{y}_\mathcal{X} = [D_x,\,D_y,\,D_z,\,R_x,\,R_y,\,R_z]\) is measured using multiple sensors. +First, the five interferometers \(\bm{d} = [d_{1},\ d_{2},\ d_{3},\ d_{4},\ d_{5}]\) are used to measure the \([D_x,\,D_y,\,D_z,\,R_x,\,R_y]\) degrees of freedom of the sample. +The \(R_z\) position of the sample is computed from the spindle's setpoint \(r_{R_z}\) and from the 6 encoders \(\bm{d}_e\) integrated in the nano-hexapod. + +The sample's position \(\bm{y}_{\mathcal{X}}\) is compared to the reference position \(\bm{r}_{\mathcal{X}}\) to compute the position error in the frame of the (rotating) nano-hexapod \(\bm{\epsilon\mathcal{X}} = [\epsilon_{D_x},\,\epsilon_{D_y},\,\epsilon_{D_z},\,\epsilon_{R_x},\,\epsilon_{R_y},\,\epsilon_{R_z}]\). +Finally, the Jacobian matrix \(\bm{J}\) of the nano-hexapod is used to map \(\bm{\epsilon\mathcal{X}}\) in the frame of the nano-hexapod struts \(\bm{\epsilon\mathcal{L}} = [\epsilon_{\mathcal{L}_1},\,\epsilon_{\mathcal{L}_2},\,\epsilon_{\mathcal{L}_3},\,\epsilon_{\mathcal{L}_4},\,\epsilon_{\mathcal{L}_5},\,\epsilon_{\mathcal{L}_6}]\). + +Voltages generated by the force sensor piezoelectric stacks \(\bm{V}_s = [V_{s1},\ V_{s2},\ V_{s3},\ V_{s4},\ V_{s5},\ V_{s6}]\) will be used for active damping. \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/test_id31_block_schematic_plant.png} -\caption{\label{fig:test_id31_block_schematic_plant}Schematic of the} +\caption{\label{fig:test_id31_block_schematic_plant}Schematic of the NASS plant} \end{figure} -\section{First Open-Loop Plant Identification} +\section{Open-Loop Plant Identification} \label{ssec:test_id31_open_loop_plant_first_id} The plant dynamics is first identified for a fixed spindle angle (at \(0\,\text{deg}\)) and without any payload. @@ -296,10 +296,10 @@ The model dynamics is also identified in the same conditions. A first comparison between the model and the measured dynamics is done in Figure \ref{fig:test_id31_first_id}. A good match can be observed for the diagonal dynamics (except the high frequency modes which are not modeled). -However, the coupling for the transfer function from command signals \(\bm{u}\) to estimated strut motion from the external metrology \(e\bm{\mathcal{L}}\) is larger than expected (Figure \ref{fig:test_id31_first_id_int}). +However, the coupling for the transfer function from command signals \(\bm{u}\) to the estimated strut motion from the external metrology \(\bm{\epsilon\mathcal{L}}\) is larger than expected (Figure \ref{fig:test_id31_first_id_int}). The experimental time delay estimated from the FRF (Figure \ref{fig:test_id31_first_id_int}) is larger than expected. -After investigation, it was found that the additional delay was due to digital processing unit\footnote{The ``PEPU'' \cite{hino18_posit_encod_proces_unit} was used for digital protocol conversion between the interferometers and the Speedgoat} that was used to read the interferometers in the Speedgoat. +After investigation, it was found that the additional delay was due to a digital processing unit\footnote{The ``PEPU'' \cite{hino18_posit_encod_proces_unit} was used for digital protocol conversion between the interferometers and the Speedgoat} that was used to get the interferometers' signals in the Speedgoat. This issue was later solved. \begin{figure}[htbp] @@ -315,19 +315,19 @@ This issue was later solved. \end{center} \subcaption{\label{fig:test_id31_first_id_iff}Force Sensors} \end{subfigure} -\caption{\label{fig:test_id31_first_id}Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}).} +\caption{\label{fig:test_id31_first_id}Comparison between the measured dynamics and the multi-body model dynamics. Both for the external metrology (\subref{fig:test_id31_first_id_int}) and force sensors (\subref{fig:test_id31_first_id_iff}). Direct terms are displayed with solid lines while off-diagonal (i.e. coupling) terms are shown with shaded lines.} \end{figure} \section{Better Angular Alignment} \label{ssec:test_id31_open_loop_plant_rz_alignment} One possible explanation of the increased coupling observed in Figure \ref{fig:test_id31_first_id_int} is the poor alignment between the external metrology axes (i.e. the interferometer supports) and the nano-hexapod axes. -To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders is implemented. -This allowed to perform two straight movements of the nano-hexapod along the \(x\) and \(y\) axes in the frame of the nano-hexapod. -During these two movements, the external metrology measurement is recorded and shown in Figure \ref{fig:test_id31_Rz_align_error}. +To estimate this alignment, a decentralized low-bandwidth feedback controller based on the nano-hexapod encoders was implemented. +This allowed to perform two straight movements of the nano-hexapod along its \(x\) and \(y\) axes. +During these two movements, the external metrology measurement was recorded and are shown in Figure \ref{fig:test_id31_Rz_align_error}. It was found that there is a misalignment of 2.7 degrees (rotation along the vertical axis) between the interferometer axes and nano-hexapod axes. This was corrected by adding an offset to the spindle angle. -To check that the alignment has improved, the same movement was performed using the nano-hexapod while recording the signal of the external metrology. +After alignment, the same movement was performed using the nano-hexapod while recording the signal of the external metrology. Results shown in Figure \ref{fig:test_id31_Rz_align_correct} are indeed indicating much better alignment. \begin{figure}[htbp] @@ -346,10 +346,7 @@ Results shown in Figure \ref{fig:test_id31_Rz_align_correct} are indeed indicati \caption{\label{fig:test_id31_Rz_align_error}Measurement of the Nano-Hexapod axes in the frame of the external metrology. Before alignment (\subref{fig:test_id31_Rz_align_error}) and after alignment (\subref{fig:test_id31_Rz_align_correct}).} \end{figure} -\section{Open-Loop Identification after alignment} -\label{ssec:test_id31_open_loop_plant_after_alignment} - -The plant dynamics is identified after the fine alignment and is compared with the model dynamics in Figure \ref{fig:test_id31_first_id_int_better_rz_align}. +The plant dynamics was identified again after the fine alignment and is compared with the model dynamics in Figure \ref{fig:test_id31_first_id_int_better_rz_align}. Compared to the initial identification shown in Figure \ref{fig:test_id31_first_id_int}, the obtained coupling has decreased and is now close to the coupling obtained with the multi-body model. At low frequency (below \(10\,\text{Hz}\)) all the off-diagonal elements have an amplitude \(\approx 100\) times lower compared to the diagonal elements, indicating that a low bandwidth feedback controller can be implemented in a decentralized way (i.e. \(6\) SISO controllers). Between \(650\,\text{Hz}\) and \(1000\,\text{Hz}\), several modes can be observed that are due to flexible modes of the top platform and modes of the two spheres adjustment mechanism. @@ -364,8 +361,10 @@ The flexible modes of the top platform can be passively damped while the modes o \section{Effect of Payload Mass} \label{ssec:test_id31_open_loop_plant_mass} -The system dynamics was identified with four payload conditions that are shown in Figure \ref{fig:test_id31_picture_masses}. +In order to see how the system dynamics changes with the payload, open-loop identification was performed for four payload conditions that are shown in Figure \ref{fig:test_id31_picture_masses}. The obtained direct terms are compared with the model dynamics in Figure \ref{fig:test_nhexa_comp_simscape_diag_masses}. +It is shown that the model dynamics well predicts the measured dynamics for all payload conditions. +Therefore the model can be used for model-based control is necessary. It is interesting to note that the anti-resonances in the force sensor plant are now appearing as minimum-phase, as the model predicts (Figure \ref{fig:test_id31_comp_simscape_iff_diag_masses}). @@ -416,8 +415,7 @@ It is interesting to note that the anti-resonances in the force sensor plant are \section{Effect of Spindle Rotation} \label{ssec:test_id31_open_loop_plant_rotation} -The dynamics was then identified while the Spindle was rotating at constant velocity. -Three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\). +To verify that all the kinematics in Figure \ref{fig:test_id31_block_schematic_plant} are correct and to check whether the system dynamics is affected by Spindle rotation of not, three identification experiments were performed: no spindle rotation, spindle rotation at \(36\,\text{deg}/s\) and at \(180\,\text{deg}/s\). The comparison of the obtained dynamics from command signal \(u\) to estimated strut error \(e\mathcal{L}\) is done in Figure \ref{fig:test_id31_effect_rotation}. Both direct terms (Figure \ref{fig:test_id31_effect_rotation_direct}) and coupling terms (Figure \ref{fig:test_id31_effect_rotation_coupling}) are unaffected by the rotation. @@ -442,10 +440,9 @@ This also indicates that the metrology kinematics is correct and is working in r \end{figure} \section*{Conclusion} -Thanks to the model, poor alignment between the nano-hexapod axes and the external metrology axes could be identified. -After alignment, the identified dynamics is well matching with the multi-body model. - -Also, the observed effects of the payload mass and of the spindle rotation on the dynamics are well matching the model predictions. +The identified frequency response functions from command signals \(\bm{u}\) to the force sensors \(\bm{V}_s\) and to the estimated strut errors \(\bm{\epsilon\mathcal{L}}\) are well matching the developed multi-body model. +Effect of payload mass is shown to be well predicted by the model, which can be useful if robust model based control is to be used. +The spindle rotation has no visible effect on the measured dynamics, indicating that controllers should be robust to the spindle rotation. \chapter{Decentralized Integral Force Feedback} \label{sec:test_id31_iff}