Tangle Matlab files without comments
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@ -22,18 +22,6 @@ colors = colororder;
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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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% IFF Plant
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% <<ssec:nass_active_damping_plant>>
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% Transfer functions from actuator forces $f_i$ to force sensor measurements $f_{mi}$ are computed using the multi-body model.
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% Figure ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg.
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% Without parallel stiffness (Figure ref:fig:nass_iff_plant_no_kp), the plant dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
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% Adding parallel stiffness (Figure ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation.
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% Although both cases show significant coupling around the resonances, stability is guaranteed by the collocated arrangement of the actuators and sensors [[cite:&preumont08_trans_zeros_struc_contr_with]].
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%% Identify the IFF plant dynamics using the Simscape model
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% Initialize each Simscape model elements
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@ -179,33 +167,6 @@ yticks([0:45:180]);
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linkaxes([ax1,ax2],'x');
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xlim([f(1), f(end)]);
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% #+name: fig:nass_iff_plant_effect_kp
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% #+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and a payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros into complex conjugate zeros (\subref{fig:nass_iff_plant_kp})
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% #+attr_latex: :options [h!tbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_iff_plant_no_kp.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_kp}with parallel stiffness}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_iff_plant_kp.png]]
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% #+end_subfigure
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% #+end_figure
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% The effect of rotation, as shown in Figure ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,N/\mu m$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model).
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% Figure ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics.
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% The poles and zeros shift in frequency as the payload mass varies.
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% However, their alternating pattern is preserved, which ensures the phase remains bounded between 0 and 180 degrees, thus maintaining robust stability properties.
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%% Effect of spindle's rotation on the IFF Plant
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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@ -305,29 +266,6 @@ yticks([0:45:180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Controller Design
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% <<ssec:nass_active_damping_control>>
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% The previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to the gyroscopic effects caused by spindle rotation.
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% This finding was also confirmed with the multi-body model of the NASS: the system was unstable when using pure integrators and without parallel stiffness.
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% This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
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% However, as illustrated in Figure ref:fig:nass_iff_plant_kp, adding parallel stiffness increases the low frequency gain.
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% Using pure integrators would result in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
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% To resolve this issue, a second-order high-pass filter is introduced to limit the low frequency gain, as shown in Equation eqref:eq:nass_kiff.
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% \begin{equation}\label{eq:nass_kiff}
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% \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
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% K_{\text{IFF}}(s) & & 0 \\
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% & \ddots & \\
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% 0 & & K_{\text{IFF}}(s)
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% \end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s} \cdot \frac{\frac{s^2}{\omega_z^2}}{\frac{s^2}{\omega_z^2} + 2 \xi_z \frac{s}{\omega_z} + 1}
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% \end{equation}
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% The cut-off frequency of the second-order high-pass filter was tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at $5\,\text{Hz}$.
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% The overall gain was then increased to obtain a large loop gain around the resonances to be damped, as illustrated in Figure ref:fig:nass_iff_loop_gain.
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%% Verify that parallel stiffness permits to have a stable plant
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Kiff_pure_int = -200/s*eye(6);
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isstable(feedback(G_iff_m25_Rz, Kiff_pure_int, 1))
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@ -401,18 +339,6 @@ yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:nass_iff_loop_gain
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% #+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$
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% #+attr_latex: :options [h!tbp]
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% #+RESULTS:
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% [[file:figs/nass_iff_loop_gain.png]]
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% To verify stability, the root loci for the three payload configurations were computed, as shown in Figure ref:fig:nass_iff_root_locus.
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% The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability of the applied decentralized IFF.
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%% Root Locus for the Decentralized IFF controller - 1kg Payload
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figure;
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gains = logspace(-2, 1, 200);
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@ -22,23 +22,6 @@ colors = colororder;
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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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% HAC Plant
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% <<ssec:nass_hac_plant>>
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% The plant dynamics from force inputs $\bm{f}$ to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were first extracted from the multi-body model without the implementation of the decentralized IFF.
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% The influence of spindle rotation on plant dynamics was investigated, and the results are presented in Figure ref:fig:nass_undamped_plant_effect_Wz.
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% While rotational motion introduces coupling effects at low frequencies, these effects remain minimal at operational velocities, owing to the high stiffness characteristics of the nano-hexapod assembly.
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% Payload mass emerged as a significant parameter affecting system behavior, as illustrated in Figure ref:fig:nass_undamped_plant_effect_mass.
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% As expected, increasing the payload mass decreased the resonance frequencies while amplifying coupling at low frequency.
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% These mass-dependent dynamic changes present considerable challenges for control system design, particularly for configurations with high payload masses.
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% Additional operational parameters were systematically evaluated, including the $R_y$ tilt angle, $R_z$ spindle position, and micro-hexapod position.
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% These factors were found to exert negligible influence on the plant dynamics, which can be attributed to the effective mechanical decoupling achieved between the plant and micro-station dynamics.
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% This decoupling characteristic ensures consistent performance across various operational configurations.
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% This also validates the developed control strategy.
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%% Identify the IFF plant dynamics using the Simscape model
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% Initialize each Simscape model elements
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@ -192,38 +175,6 @@ yticks([-180:45:180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:nass_undamped_plant_effect
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% #+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})
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% #+attr_latex: :options [h!tbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_undamped_plant_effect_Wz.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_mass}Effect of payload's mass}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_undamped_plant_effect_mass.png]]
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% #+end_subfigure
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% #+end_figure
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% The Decentralized Integral Force Feedback was implemented in the multi-body model, and transfer functions from force inputs $\bm{f}^\prime$ of the damped plant to the strut errors $\bm{\epsilon}_{\mathcal{L}}$ were extracted from this model.
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% The effectiveness of the IFF implementation was first evaluated with a $1\,\text{kg}$ payload, as demonstrated in Figure ref:fig:nass_comp_undamped_damped_plant_m1.
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% The results indicate successful damping of the nano-hexapod resonance modes, although a minor increase in low-frequency coupling was observed.
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% This trade-off was considered acceptable, given the overall improvement in system behavior.
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% The benefits of IFF implementation were further assessed across the full range of payload configurations, and the results are presented in Figure ref:fig:nass_hac_plants.
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% For all tested payloads ($1\,\text{kg}$, $25\,\text{kg}$ and $50\,\text{kg}$), the decentralized IFF significantly damped the nano-hexapod modes and therefore simplified the system dynamics.
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% More importantly, in the vicinity of the desired high authority control bandwidth (i.e. between $10\,\text{Hz}$ and $50\,\text{Hz}$), the damped dynamics (shown in red) exhibited minimal gain and phase variations with frequency.
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% For the undamped plants (shown in blue), achieving robust control with bandwidth above 10Hz while maintaining stability across different payload masses would be practically impossible.
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%% Identify HAC Plant with IFF
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initializeReferences(); % No Spindle Rotation
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initializeController('type', 'iff'); % Implemented IFF controller
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@ -345,34 +296,6 @@ yticks([-180:45:180]);
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linkaxes([ax1,ax2],'x');
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% xlim([1, 5e2]);
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% #+name: fig:nass_hac_plant
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% #+caption: Effect of Decentralized Integral Force Feedback on the positioning plant for a $1\,\text{kg}$ sample mass (\subref{fig:nass_undamped_plant_effect_Wz}). The direct terms of the positioning plants for all considered payloads are shown in (\subref{fig:nass_undamped_plant_effect_mass}).
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% #+attr_latex: :options [h!tbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_comp_undamped_damped_plant_m1.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :width 0.95\linewidth
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% [[file:figs/nass_hac_plants.png]]
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% #+end_subfigure
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% #+end_figure
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% The coupling between the nano-hexapod and the micro-station was evaluated through a comparative analysis of plant dynamics under two mounting conditions.
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% In the first configuration, the nano-hexapod was mounted on an ideally rigid support, while in the second configuration, it was installed on the micro-station with finite compliance.
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% As illustrated in Figure ref:fig:nass_effect_ustation_compliance, the complex dynamics of the micro-station were found to have little impact on the plant dynamics.
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% The only observable difference manifests as additional alternating poles and zeros above 100Hz, a frequency range sufficiently beyond the control bandwidth to avoid interference with the system performance.
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% This result confirms effective dynamic decoupling between the nano-hexapod and the supporting micro-station structure.
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%% Identify plant with "rigid" micro-station
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initializeGround('type', 'rigid');
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initializeGranite('type', 'rigid');
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@ -442,23 +365,6 @@ yticks([-180:45:180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Effect of Nano-Hexapod Stiffness on System Dynamics
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% <<ssec:nass_hac_stiffness>>
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% The influence of nano-hexapod stiffness was investigated to validate earlier findings from simplified uniaxial and three-degree-of-freedom (3DoF) models.
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% These models suggest that a moderate stiffness of approximately $1\,N/\mu m$ would provide better performance than either very stiff or very soft configurations.
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% For the stiff nano-hexapod analysis, a system with an actuator stiffness of $100\,N/\mu m$ was simulated with a $25\,\text{kg}$ payload.
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% The transfer function from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ was evaluated under two conditions: mounting on an infinitely rigid base and mounting on the micro-station.
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% As shown in Figure ref:fig:nass_stiff_nano_hexapod_coupling_ustation, significant coupling was observed between the nano-hexapod and micro-station dynamics.
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% This coupling introduces complex behavior that is difficult to model and predict accurately, thus corroborating the predictions of the simplified uniaxial model.
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% The soft nano-hexapod configuration was evaluated using a stiffness of $0.01\,N/\mu m$ with a $25\,\text{kg}$ payload.
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% The dynamic response was characterized at three rotational velocities: 0, 36, and 360 deg/s.
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% Figure ref:fig:nass_soft_nano_hexapod_effect_Wz demonstrates that rotation substantially affects system dynamics, manifesting as instability at high rotational velocities, increased coupling due to gyroscopic effects, and rotation-dependent resonance frequencies.
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% The current approach of controlling the position in the strut frame is inadequate for soft nano-hexapods; but even shifting control to a frame matching the payload's center of mass would not overcome the substantial coupling and dynamic variations induced by gyroscopic effects.
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%% Identify Dynamics with a Stiff nano-hexapod (100N/um)
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% Initialize each Simscape model elements
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@ -651,17 +557,6 @@ yticks([-180:90:180]);
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linkaxes([ax1,ax2],'x');
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xlim([f(1), f(end)]);
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% Controller design
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% <<ssec:nass_hac_controller>>
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% A high authority controller was designed to meet two key requirements: stability for all payload masses (i.e. for all the damped plants of Figure ref:fig:nass_hac_plants), and achievement of sufficient bandwidth (targeted at 10Hz) for high performance operation.
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% The controller structure is defined in Equation eqref:eq:nass_robust_hac, incorporating an integrator term for low frequency performance, a lead compensator for phase margin improvement, and a low-pass filter for robustness against high frequency modes.
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% \begin{equation}\label{eq:nass_robust_hac}
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% K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
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% \end{equation}
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%% HAC Design
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% Wanted crossover
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wc = 2*pi*10; % [rad/s]
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@ -689,13 +584,6 @@ Khac = -H_gain * ... % Gain
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% The designed HAC controller is saved
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save('./mat/nass_K_hac.mat', 'Khac');
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% The controller performance was evaluated through two complementary analyses.
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% First, the decentralized loop gain shown in Figure ref:fig:nass_hac_loop_gain, confirms the achievement of the desired 10Hz bandwidth.
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% Second, the characteristic loci analysis presented in Figure ref:fig:nass_hac_loci demonstrates robustness for all payload masses, with adequate stability margins maintained throughout the operating envelope.
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%% "Diagonal" loop gain for the High Authority Controller
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f = logspace(-1, 2, 1000);
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figure;
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@ -806,21 +694,6 @@ xlim([-1.8, 0.2]); ylim([-1, 1]);
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leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 15;
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% Tomography experiment
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% <<ssec:nass_hac_tomography>>
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% The Nano Active Stabilization System concept was validated through time-domain simulations of scientific experiments, with a particular focus on tomography scanning because of its demanding performance requirements.
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% Simulations were conducted at the maximum operational rotational velocity of $\Omega_z = 360\,\text{deg/s}$ to evaluate system performance under the most challenging conditions.
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% Performance metrics were established based on anticipated future beamline specifications, which specify a beam size of 200nm (horizontal) by 100nm (vertical).
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% The primary requirement stipulates that the point of interest must remain within beam dimensions throughout operation.
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% The simulation included two principal disturbance sources: ground motion and spindle vibrations.
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% Additional noise sources, including measurement noise and electrical noise from DAC and voltage amplifiers, were not included in this analysis, as these parameters will be optimized during the detailed design phase.
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% Figure ref:fig:nass_tomo_1kg_60rpm presents a comparative analysis of positioning errors under both open-loop and closed-loop conditions for a lightweight sample configuration (1kg).
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% The results demonstrate the system's capability to maintain the sample's position within the specified beam dimensions, thus validating the fundamental concept of the stabilization system.
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%% Simulation of tomography experiments
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% Sample is not centered with the rotation axis
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% This is done by offsetfing the micro-hexapod by 0.9um
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@ -927,34 +800,6 @@ yticks([-100:50:100]);
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leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 15;
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% #+name: fig:nass_tomo_1kg_60rpm
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% #+caption: Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at $360\,\text{deg/s}$. 1kg payload is placed on top of the nano-hexapod.
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% #+attr_latex: :options [h!tbp]
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% #+begin_figure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :scale 0.9
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% [[file:figs/nass_tomo_1kg_60rpm_xy.png]]
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% #+end_subfigure
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% #+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_yz}YZ plane}
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% #+attr_latex: :options {0.48\textwidth}
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% #+begin_subfigure
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% #+attr_latex: :scale 0.9
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% [[file:figs/nass_tomo_1kg_60rpm_yz.png]]
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% #+end_subfigure
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% #+end_figure
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% The robustness of the NASS to payload mass variation was evaluated through additional tomography scan simulations with 25 and 50kg payloads, complementing the initial 1kg test case.
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% As illustrated in Figure ref:fig:nass_tomography_hac_iff, system performance exhibits some degradation with increasing payload mass, which is consistent with predictions from the control analysis.
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% While the positioning accuracy for heavier payloads is outside the specified limits, it remains within acceptable bounds for typical operating conditions.
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% It should be noted that the maximum rotational velocity of 360deg/s is primarily intended for lightweight payload applications.
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% For higher mass configurations, rotational velocities are expected to be below 36deg/s.
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%% Simulation of tomography experiment - 1kg payload - 360deg/s - YZ errors
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figure;
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tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
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@ -22,7 +22,7 @@
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#+BIND: org-latex-bib-compiler "biber"
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :comments no
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#+PROPERTY: header-args:matlab+ :exports none
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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