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@ -781,14 +781,10 @@ Then, the high authority controller uses the computed errors in the frame of the
<<sec:nass_active_damping>> <<sec:nass_active_damping>>
** Introduction :ignore: ** Introduction :ignore:
- How to apply/optimize IFF on an hexapod? Building upon the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy.
- Robustness to payload mass Springs in parallel to the force sensors are used to guarantee the control robustness as was found using the 3DoF rotating model.
- Root Locus The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$.
- Damping optimization Used payloads have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg.
Explain which samples are tested:
- cylindrical, 250mm height
- mass of 1kg, 25kg and 50kg
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -816,16 +812,15 @@ Explain which samples are tested:
#+end_src #+end_src
** IFF Plant ** IFF Plant
<<ssec:nass_active_damping_plant>>
Using the multi-body model, the transfer functions from the six actuator forces $f_i$ to the six force sensors $f_{mi}$ are computed. Transfer functions from actuator forces $f_i$ to force sensor measurements $f_{mi}$ are computed using the multi-body model.
Figure ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects the plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg.
First the effect of added parallel stiffness on the plant dynamics is studied in Figure ref:fig:nass_iff_plant_effect_kp. Without parallel stiffness (Figure ref:fig:nass_iff_plant_no_kp), the dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
The plant is identified while the Spindle is rotating at a maximum velocity $\Omega_z = 360\,\text{deg/s}$. Adding parallel stiffness (Figure ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation.
The payload mass is 25kg.
The obtained dynamics without the parallel stiffness (Figure ref:fig:nass_iff_plant_no_kp) has non-minimum phase zeros at low frequency, as was predicted using the 3-DoF rotating model.
When the parallel stiffness is added (Figure ref:fig:nass_iff_plant_kp), a minimum phase complex conjugate zero is obtained instead, which permits to use decentralized IFF with unconditional stability.
In both cases, high coupling around resonances, but should have guaranteed stability thanks to the collocated nature of actuators and sensors. Though both cases show significant coupling around resonances, stability is guaranteed by the collocated arrangement of actuators and sensors [[cite:&preumont08_trans_zeros_struc_contr_with]].
#+begin_src matlab #+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model %% Identify the IFF plant dynamics using the Simscape model
@ -1004,11 +999,10 @@ exportFig('figs/nass_iff_plant_kp.pdf', 'width', 'half', 'height', 600);
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
The effect of rotation, 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).
- Effect of Rotation ref:fig:nass_iff_plant_effect_rotation: almost negligible thanks to sufficient actuator stiffness of $1\,\mu m/N$ (determined using the uniaxial model) Figure ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics.
coupling is increased at much lower frequency that the first mode to damp and should therefore do not impact the control performances While the poles and zeros are shifting with payload mass, the alternating pattern of poles and zeros is maintained, ensuring that the phase remains bounded between 0 and 180 degrees, and thus good robustness properties.
- Effect of payload's mass ref:fig:nass_iff_plant_effect_payload: still have alternating poles and zeros, and therefore bounded phase between 0 and 180 degrees.
Location of poles change with the payload's mass as expected.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Effect of spindle's rotation on the IFF Plant %% Effect of spindle's rotation on the IFF Plant
@ -1140,13 +1134,15 @@ exportFig('figs/nass_iff_plant_effect_payload.pdf', 'width', 'half', 'height', 6
#+end_figure #+end_figure
** Controller Design ** Controller Design
<<ssec:nass_active_damping_control>>
Using the 3DoF rotating model, it was shown that the decentralized IFF with pure integrators becomes unstable due to gyroscopic induced by the spindle's rotation. Previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to gyroscopic effects caused by spindle rotation.
It was shown that adding sufficient stiffness in parallel with the force sensor allows to make the decentralized IFF unconditional stable again. This finding is also confirmed with the multi-body model of the NASS: the system is unstable when using pure integrators and without parallel stiffness.
Such parallel stiffness are here added to the nano-hexapod, and using the multi-body model of the NASS, it is verified that without parallel stiffness, the system would be unstable when using decentralized IFF with pure integrators.
Even though pure integrators will give stable systems and guaranteed stability when parallel stiffness are added, it would lead to unnecessary gain at low frequency that would modify the damped plant dynamics at low frequency. This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
To avoid that, a second order low pass filter is added at low frequency eqref:eq:nass_kiff. However, as illustrated in Figure ref:fig:nass_iff_plant_kp, adding parallel stiffness increases the low frequency gain.
If using pure integrators, this would results in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
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.
\begin{equation}\label{eq:nass_kiff} \begin{equation}\label{eq:nass_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix} \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
@ -1156,6 +1152,9 @@ To avoid that, a second order low pass filter is added at low frequency eqref:eq
\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} \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}
\end{equation} \end{equation}
The cut-off frequency of the second-order high-pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at $5\,\text{Hz}$.
The overall gain is then increased to have large loop gain around resonances to be damped, as illustrated in Figure ref:fig:nass_iff_loop_gain.
#+begin_src matlab #+begin_src matlab
%% Verify that parallel stiffness permits to have a stable plant %% Verify that parallel stiffness permits to have a stable plant
Kiff_pure_int = -200/s*eye(6); Kiff_pure_int = -200/s*eye(6);
@ -1189,9 +1188,6 @@ save('./matlab/mat/nass_K_iff.mat', 'Kiff');
save('./mat/nass_K_iff.mat', 'Kiff'); save('./mat/nass_K_iff.mat', 'Kiff');
#+end_src #+end_src
The frequency of the second order high pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass (here at $4\,\text{Hz}$).
The overall gain is increased to have some authority on the nano-hexapod modes that we want to damp (Figure ref:fig:nass_iff_loop_gain).
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Loop gain for the decentralized IFF %% Loop gain for the decentralized IFF
figure; figure;
@ -1254,8 +1250,8 @@ exportFig('figs/nass_iff_loop_gain.pdf', 'width', 'wide', 'height', 'normal');
#+RESULTS: #+RESULTS:
[[file:figs/nass_iff_loop_gain.png]] [[file:figs/nass_iff_loop_gain.png]]
In order to check the stability, root loci for the three payload configurations are computed and shown in Figure ref:fig:nass_iff_root_locus. To verify stability, root loci for the three payload configurations are computed and shown in Figure ref:fig:nass_iff_root_locus.
It is shown that the closed-loop poles are bounded to the left-half plane indicating the good robustness properties of the applied decentralized IFF. The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability properties of the applied decentralized IFF.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Root Locus for the Decentralized IFF controller - 1kg Payload %% Root Locus for the Decentralized IFF controller - 1kg Payload
@ -1379,7 +1375,7 @@ exportFig('figs/nass_iff_root_locus_50kg.pdf', 'width', 'third', 'height', 'norm
#+end_src #+end_src
#+name: fig:nass_iff_root_locus #+name: fig:nass_iff_root_locus
#+caption: Measurement of strut flexible modes #+caption: Root Loci for Decentralized IFF for three payload masses. Closed-loop poles are shown by the black crosses.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$} #+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$}
@ -1402,11 +1398,6 @@ exportFig('figs/nass_iff_root_locus_50kg.pdf', 'width', 'third', 'height', 'norm
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
* Centralized Active Vibration Control * Centralized Active Vibration Control
:PROPERTIES: :PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nass_3_hac.m :HEADER-ARGS:matlab+: :tangle matlab/nass_3_hac.m

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@ -1,4 +1,4 @@
% Created 2025-02-17 Mon 21:45 % Created 2025-02-17 Mon 22:37
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -169,7 +169,7 @@ Finally, these errors are mapped to the strut space through the nano-hexapod Jac
\bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}} \bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}}
\end{equation} \end{equation}
\section{Control Architecture} \section{Control Architecture - Summary}
\label{ssec:nass_control_architecture} \label{ssec:nass_control_architecture}
The complete control architecture is summarized in Figure \ref{fig:nass_control_architecture}. The complete control architecture is summarized in Figure \ref{fig:nass_control_architecture}.
@ -189,29 +189,20 @@ Then, the high authority controller uses the computed errors in the frame of the
\chapter{Decentralized Active Damping} \chapter{Decentralized Active Damping}
\label{sec:nass_active_damping} \label{sec:nass_active_damping}
\begin{itemize} Building upon the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy.
\item How to apply/optimize IFF on an hexapod? Springs in parallel to the force sensors are used to guarantee the control robustness as was found using the 3DoF rotating model.
\item Robustness to payload mass The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from \(1\) to \(50\,\text{kg}\) and rotational velocity up to \(360\,\text{deg/s}\).
\item Root Locus Used payloads have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg.
\item Damping optimization
\end{itemize}
Explain which samples are tested:
\begin{itemize}
\item cylindrical, 250mm height
\item mass of 1kg, 25kg and 50kg
\end{itemize}
\section{IFF Plant} \section{IFF Plant}
\label{ssec:nass_active_damping_plant}
Using the multi-body model, the transfer functions from the six actuator forces \(f_i\) to the six force sensors \(f_{mi}\) are computed. Transfer functions from actuator forces \(f_i\) to force sensor measurements \(f_{mi}\) are computed using the multi-body model.
Figure \ref{fig:nass_iff_plant_effect_kp} examines how parallel stiffness affects the plant dynamics, with identification performed at maximum spindle velocity \(\Omega_z = 360\,\text{deg/s}\) and with a payload mass of 25 kg.
First the effect of added parallel stiffness on the plant dynamics is studied in Figure \ref{fig:nass_iff_plant_effect_kp}. Without parallel stiffness (Figure \ref{fig:nass_iff_plant_no_kp}), the dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
The plant is identified while the Spindle is rotating at a maximum velocity \(\Omega_z = 360\,\text{deg/s}\). Adding parallel stiffness (Figure \ref{fig:nass_iff_plant_kp}) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation.
The payload mass is 25kg.
The obtained dynamics without the parallel stiffness (Figure \ref{fig:nass_iff_plant_no_kp}) has non-minimum phase zeros at low frequency, as was predicted using the 3-DoF rotating model.
When the parallel stiffness is added (Figure \ref{fig:nass_iff_plant_kp}), a minimum phase complex conjugate zero is obtained instead, which permits to use decentralized IFF with unconditional stability.
In both cases, high coupling around resonances, but should have guaranteed stability thanks to the collocated nature of actuators and sensors. Though both cases show significant coupling around resonances, stability is guaranteed by the collocated arrangement of actuators and sensors \cite{preumont08_trans_zeros_struc_contr_with}.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
@ -229,13 +220,10 @@ In both cases, high coupling around resonances, but should have guaranteed stabi
\caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and 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 to complex conjugate zeros (\subref{fig:nass_iff_plant_kp})} \caption{\label{fig:nass_iff_plant_effect_kp}Effect of stiffness parallel to the force sensor on the IFF plant with \(\Omega_z = 360\,\text{deg/s}\) and 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 to complex conjugate zeros (\subref{fig:nass_iff_plant_kp})}
\end{figure} \end{figure}
The effect of rotation, 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).
\begin{itemize} Figure \ref{fig:nass_iff_plant_effect_payload} illustrate the effect of payload mass on the plant dynamics.
\item Effect of Rotation \ref{fig:nass_iff_plant_effect_rotation}: almost negligible thanks to sufficient actuator stiffness of \(1\,\mu m/N\) (determined using the uniaxial model) While the poles and zeros are shifting with payload mass, the alternating pattern of poles and zeros is maintained, ensuring that the phase remains bounded between 0 and 180 degrees, and thus good robustness properties.
coupling is increased at much lower frequency that the first mode to damp and should therefore do not impact the control performances
\item Effect of payload's mass \ref{fig:nass_iff_plant_effect_payload}: still have alternating poles and zeros, and therefore bounded phase between 0 and 180 degrees.
Location of poles change with the payload's mass as expected.
\end{itemize}
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
@ -254,13 +242,15 @@ Location of poles change with the payload's mass as expected.
\end{figure} \end{figure}
\section{Controller Design} \section{Controller Design}
\label{ssec:nass_active_damping_control}
Using the 3DoF rotating model, it was shown that the decentralized IFF with pure integrators becomes unstable due to gyroscopic induced by the spindle's rotation. Previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to gyroscopic effects caused by spindle rotation.
It was shown that adding sufficient stiffness in parallel with the force sensor allows to make the decentralized IFF unconditional stable again. This finding is also confirmed with the multi-body model of the NASS: the system is unstable when using pure integrators and without parallel stiffness.
Such parallel stiffness are here added to the nano-hexapod, and using the multi-body model of the NASS, it is verified that without parallel stiffness, the system would be unstable when using decentralized IFF with pure integrators.
Even though pure integrators will give stable systems and guaranteed stability when parallel stiffness are added, it would lead to unnecessary gain at low frequency that would modify the damped plant dynamics at low frequency. This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
To avoid that, a second order low pass filter is added at low frequency \eqref{eq:nass_kiff}. However, as illustrated in Figure \ref{fig:nass_iff_plant_kp}, adding parallel stiffness increases the low frequency gain.
If using pure integrators, this would results in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
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}.
\begin{equation}\label{eq:nass_kiff} \begin{equation}\label{eq:nass_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix} \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
@ -270,8 +260,8 @@ To avoid that, a second order low pass filter is added at low frequency \eqref{e
\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} \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}
\end{equation} \end{equation}
The frequency of the second order high pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass (here at \(4\,\text{Hz}\)). The cut-off frequency of the second-order high-pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at \(5\,\text{Hz}\).
The overall gain is increased to have some authority on the nano-hexapod modes that we want to damp (Figure \ref{fig:nass_iff_loop_gain}). The overall gain is then increased to have large loop gain around resonances to be damped, as illustrated in Figure \ref{fig:nass_iff_loop_gain}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -279,8 +269,8 @@ The overall gain is increased to have some authority on the nano-hexapod modes t
\caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)} \caption{\label{fig:nass_iff_loop_gain}Loop gain for the decentralized IFF: \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)}
\end{figure} \end{figure}
In order to check the stability, root loci for the three payload configurations are computed and shown in Figure \ref{fig:nass_iff_root_locus}. To verify stability, root loci for the three payload configurations are computed and shown in Figure \ref{fig:nass_iff_root_locus}.
It is shown that the closed-loop poles are bounded to the left-half plane indicating the good robustness properties of the applied decentralized IFF. The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability properties of the applied decentralized IFF.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth} \begin{subfigure}{0.33\textwidth}
@ -301,11 +291,9 @@ It is shown that the closed-loop poles are bounded to the left-half plane indica
\end{center} \end{center}
\subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$} \subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
\end{subfigure} \end{subfigure}
\caption{\label{fig:nass_iff_root_locus}Measurement of strut flexible modes} \caption{\label{fig:nass_iff_root_locus}Root Loci for Decentralized IFF for three payload masses. Closed-loop poles are shown by the black crosses.}
\end{figure} \end{figure}
\section*{Conclusion}
\chapter{Centralized Active Vibration Control} \chapter{Centralized Active Vibration Control}
\label{sec:nass_hac} \label{sec:nass_hac}
\begin{itemize} \begin{itemize}