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@ -211,7 +211,8 @@ CLOSED: [2025-02-05 Wed 16:04]
- control is performed - control is performed
- simulations => validation of the concept - simulations => validation of the concept
** QUES [#A] Should I talk about APA here? ** ANSW [#A] Should I talk about APA here?
CLOSED: [2025-02-11 Tue 23:00]
- It seems APA is not mentioned in chapter 1 - It seems APA is not mentioned in chapter 1
- Could be nice to only talk about APA in *chapter 2* as in chapter 1 we don't care too much about actual implementation/design - Could be nice to only talk about APA in *chapter 2* as in chapter 1 we don't care too much about actual implementation/design
@ -221,6 +222,11 @@ CLOSED: [2025-02-05 Wed 16:04]
- *Add parallel stiffness* - *Add parallel stiffness*
- High pass filter - High pass filter
** TODO [#A] Check all figures
- [ ] Captions
- [ ] Legend, units, etc...
** TODO [#A] Make sure the Simulink file for the Stewart platform is working well ** TODO [#A] Make sure the Simulink file for the Stewart platform is working well
SCHEDULED: <2025-02-10 Mon> SCHEDULED: <2025-02-10 Mon>
@ -256,11 +262,19 @@ It should be the exact model reference that will be included in the NASS model (
- [X] Log configuration - [X] Log configuration
- [ ] *Do I want to be able to change each individual parameter value of each strut => no* - [ ] *Do I want to be able to change each individual parameter value of each strut => no*
** TODO [#A] For simplicity, maybe not talk at all about parallel stiffness with the force sensor
This could be the topic of the NASS section.
** TODO [#B] Check all notations ** TODO [#B] Check all notations
- [ ] Make sure they are all defined in correct order - [ ] Make sure they are all defined in correct order
- [ ] Make sure all vectors and matrices are bold - [ ] Make sure all vectors and matrices are bold
** TODO [#B] Remove all un-used control architecture
- [ ] Make sure HAC-IFF works as explained in the document
** TODO [#C] Mention the Toolbox (maybe make a DOI for that)
** DONE [#C] Better understand principle of virtual work ** DONE [#C] Better understand principle of virtual work
CLOSED: [2025-02-10 Mon 15:51] CLOSED: [2025-02-10 Mon 15:51]
@ -270,13 +284,12 @@ Better understand this: https://en.wikipedia.org/wiki/Virtual_work
Also add link or explanation for this equation. Also add link or explanation for this equation.
** TODO [#A] Should I include the effect of rotation somewhere? ** DONE [#A] Should I include the effect of rotation somewhere?
CLOSED: [2025-02-11 Tue 23:00]
Similar to what was done with the 3DoF model? Similar to what was done with the 3DoF model?
** TODO [#A] For simplicity, maybe not talk at all about parallel stiffness with the force sensor => *no, it will be done in the NASS chapter*
This could be the topic of the NASS section.
** TODO [#C] Mention the Toolbox (maybe make a DOI for that)
** DONE [#B] Define the geometry for the simplified nano-hexapod ** DONE [#B] Define the geometry for the simplified nano-hexapod
CLOSED: [2025-02-06 Thu 18:56] CLOSED: [2025-02-06 Thu 18:56]
@ -1272,12 +1285,14 @@ The validated multi-body model will serve as a valuable tool for predicting syst
<<sec:nhexa_control>> <<sec:nhexa_control>>
** Introduction :ignore: ** Introduction :ignore:
- Contrary to what was done with the uniaxial model SISO control => MIMO control: much more complex than because of interaction The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature.
- Possible to ignore interaction when good decoupling is achieved: important to have tools to study interaction. While the uniaxial model demonstrated the effectiveness of the HAC-LAC strategy, its extension to Stewart platforms requires careful consideration discussed in this section.
- Different ways to try to decouple a MIMO plant
- Here, just basic strategy, similar to what was done with the uniaxial model is used to validate the concept First, the distinction between centralized and decentralized control approaches is discussed in Section ref:ssec:nhexa_control_centralized_decentralized.
- Control will be optimized during the detailed design phase The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section ref:ssec:nhexa_control_space, highlighting the trade-offs between direction-specific tuning and implementation simplicity.
- Reference book: [[cite:&skogestad07_multiv_feedb_contr]]
Building upon these analyses, a decentralized active damping strategy using Integral Force Feedback is developed in Section ref:ssec:nhexa_control_iff, followed by the implementation of a centralized High Authority Control for positioning in Section ref:ssec:nhexa_control_hac_lac.
This architecture, while simple, will be used to demonstrate the feasibility of the NASS concept and will provide a foundation for more sophisticated control strategies to be developed during the detailed design phase.
** 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)
@ -1327,18 +1342,16 @@ In the context of the nano-hexapod, two distinct control strategies will be exam
#+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity. #+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.
[[file:figs/nhexa_stewart_decentralized_control.png]] [[file:figs/nhexa_stewart_decentralized_control.png]]
** Choice of the control space ** Choice of the Control Space
<<ssec:nhexa_control_space>> <<ssec:nhexa_control_space>>
When controlling a Stewart platform using external metrology that measures the pose of frame $\{B\}$ with respect to $\{A\}$, denoted as $\bm{\mathcal{X}}$, the control architecture can be implemented in either the Cartesian space or the strut space. When controlling a Stewart platform using external metrology that measures the pose of frame $\{B\}$ with respect to $\{A\}$, denoted as $\bm{\mathcal{X}}$, the control architecture can be implemented in either Cartesian space or strut space.
This choice impacts both the control design and performance characteristics. This choice impacts both the control design and the obtained performance.
**** Control in the Strut space **** Control in the Strut space
In this approach, illustrated in Figure ref:fig:nhexa_control_strut, the control is performed in the space of the struts. In this approach, illustrated in Figure ref:fig:nhexa_control_strut, the control is performed in the space of the struts.
The Jacobian matrix is used to perform real-time approximate inverse kinematics, mapping position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$. The Jacobian matrix is used to solve the inverse kinematics in real-time, mapping position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$.
A diagonal controller then processes these strut-space errors to generate force commands for each actuator. A diagonal controller then processes these strut-space errors to generate force commands for each actuator.
The main advantage of this approach emerges from the plant characteristics in strut space, as shown in Figure ref:fig:nhexa_plant_frame_struts. The main advantage of this approach emerges from the plant characteristics in strut space, as shown in Figure ref:fig:nhexa_plant_frame_struts.
@ -1625,9 +1638,7 @@ The effect of this parallel stiffness will be examined in the next section when
The Root Locus analysis, shown in Figure ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$. The Root Locus analysis, shown in Figure ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$.
A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability [[cite:&preumont08_trans_zeros_struc_contr_with]]. A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability [[cite:&preumont08_trans_zeros_struc_contr_with]].
This is particularly valuable as the coupling is very large around resonance frequencies, and without this guaranteed stability property, it would be very difficult to have these modes inside the control bandwidth. This property is particularly valuable as the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone.
This control strategy provides effective damping of the resonant modes while maintaining guaranteed stability - a property that would be difficult to achieve using position feedback alone.
The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$.
The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes.
@ -1752,7 +1763,7 @@ The complete HAC-IFF control architecture is illustrated in Figure ref:fig:nhexa
Following the analysis from Section ref:ssec:nhexa_control_space, the control is implemented in the strut space. Following the analysis from Section ref:ssec:nhexa_control_space, the control is implemented in the strut space.
The Jacobian matrix $\bm{J}^{-1}$ performs real-time approximate inverse kinematics to map position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$. The Jacobian matrix $\bm{J}^{-1}$ performs real-time approximate inverse kinematics to map position errors from Cartesian space $\bm{\epsilon}_{\mathcal{X}}$ to strut space $\bm{\epsilon}_{\mathcal{L}}$.
A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these errors in the frame of the struts and computed to forces to apply to the damp plant $\bm{f}^{\prime}$. A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these errors in the frame of the struts.
#+begin_src latex :file nhexa_hac_iff_schematic.pdf #+begin_src latex :file nhexa_hac_iff_schematic.pdf
\begin{tikzpicture} \begin{tikzpicture}
@ -1799,8 +1810,8 @@ A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these
The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions. The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions.
Figure ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, characterized by pronounced resonant peaks. Figure ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, characterized by pronounced resonant peaks.
When decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, exhibit significantly attenuated resonances while preserving the plant's decoupled behavior at low frequencies. When decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, exhibit significantly attenuated resonances.
This damping of structural resonances serves two purposes: it reduces vibrations in the mechanical structure and simplifies the design of the high authority controller by providing a simpler plant dynamics. This damping of structural resonances serves two purposes: it reduces vibrations in the vicinity of resonances and simplifies the design of the high authority controller by providing a simpler plant dynamics.
#+begin_src matlab #+begin_src matlab
%% Identify the IFF Plant %% Identify the IFF Plant
@ -1940,8 +1951,8 @@ exportFig('figs/nhexa_decentralized_hac_iff_plant_damped.pdf', 'width', 'half',
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
Building upon the damped plant dynamics shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller is designed with the structure given in equation eqref:eq:nhexa_khac. Building upon the damped plant dynamics shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller is designed with the structure given in eqref:eq:nhexa_khac.
The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter ensuring robustness by attenuating the controller's response to high-frequency dynamics. The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness to unmodeled high-frequency dynamics.
The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_decentralized_hac_iff_loop_gain. The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_decentralized_hac_iff_loop_gain.
\begin{equation}\label{eq:nhexa_khac} \begin{equation}\label{eq:nhexa_khac}
@ -1954,7 +1965,7 @@ The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_
The stability of the MIMO feedback loop is analyzed through the /characteristic loci/ method. The stability of the MIMO feedback loop is analyzed through the /characteristic loci/ method.
Such characteristic loci, shown in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as frequency varies from $0$ to $\infty$. Such characteristic loci, shown in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as frequency varies from $0$ to $\infty$.
For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point. For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point [[cite:&skogestad07_multiv_feedb_contr]].
As seen in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the -1 point, confirming the stability of the closed-loop system. Additionally, the distance of the loci from the -1 point provides information about stability margins for the coupled system. As seen in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the -1 point, confirming the stability of the closed-loop system. Additionally, the distance of the loci from the -1 point provides information about stability margins for the coupled system.
#+begin_src matlab :exports none #+begin_src matlab :exports none
@ -2003,7 +2014,7 @@ end
plot(-1, 0, 'kx', 'HandleVisibility', 'off'); plot(-1, 0, 'kx', 'HandleVisibility', 'off');
hold off; hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Real'); ylabel('Imag'); xlabel('Real Part'); ylabel('Imaginary Part');
axis square axis square
xlim([-1.8, 0.2]); ylim([-1, 1]); xlim([-1.8, 0.2]); ylim([-1, 1]);
#+end_src #+end_src
@ -2066,7 +2077,18 @@ exportFig('figs/nhexa_decentralized_hac_iff_loop_gain.pdf', 'width', 'half', 'he
:UNNUMBERED: t :UNNUMBERED: t
:END: :END:
The control architecture developed for the uniaxial and the rotating models has been adapted for the Stewart platform.
Two fundamental choices were first addressed: the selection between centralized and decentralized approaches, and the choice of control space.
While control in Cartesian space enables direction-specific performance tuning, the implementation in strut space was selected for the conceptual design phase due to two key advantages: good decoupling at low frequencies and identical diagonal terms in the plant transfer functions, allowing a single controller design to be replicated across all struts.
The HAC-LAC strategy was then implemented.
The inner loop implements decentralized Integral Force Feedback for active damping.
The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes.
The outer loop implements High Authority Control, enabling precise positioning of the platform.
This control architecture will then be used for the conceptual validation of the NASS.
More sophisticated control strategies will be investigated during the detailed design phase
* Conclusion * Conclusion
:PROPERTIES: :PROPERTIES:

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@ -1,4 +1,4 @@
% Created 2025-02-11 Tue 18:35 % Created 2025-02-11 Tue 23:02
% 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}
@ -627,14 +627,14 @@ The validated multi-body model will serve as a valuable tool for predicting syst
\chapter{Control of Stewart Platforms} \chapter{Control of Stewart Platforms}
\label{sec:nhexa_control} \label{sec:nhexa_control}
\begin{itemize} The control of Stewart platforms presents distinct challenges compared to the uniaxial model due to their multi-input multi-output nature.
\item Contrary to what was done with the uniaxial model SISO control => MIMO control: much more complex than because of interaction While the uniaxial model demonstrated the effectiveness of the HAC-LAC strategy, its extension to Stewart platforms requires careful consideration discussed in this section.
\item Possible to ignore interaction when good decoupling is achieved: important to have tools to study interaction.
\item Different ways to try to decouple a MIMO plant First, the distinction between centralized and decentralized control approaches is discussed in Section \ref{ssec:nhexa_control_centralized_decentralized}.
\item Here, just basic strategy, similar to what was done with the uniaxial model is used to validate the concept The impact of the control space selection - either Cartesian or strut space - is then analyzed in Section \ref{ssec:nhexa_control_space}, highlighting the trade-offs between direction-specific tuning and implementation simplicity.
\item Control will be optimized during the detailed design phase
\item Reference book: \cite{skogestad07_multiv_feedb_contr} Building upon these analyses, a decentralized active damping strategy using Integral Force Feedback is developed in Section \ref{ssec:nhexa_control_iff}, followed by the implementation of a centralized High Authority Control for positioning in Section \ref{ssec:nhexa_control_hac_lac}.
\end{itemize} This architecture, while simple, will be used to demonstrate the feasibility of the NASS concept and will provide a foundation for more sophisticated control strategies to be developed during the detailed design phase.
\section{Centralized and Decentralized Control} \section{Centralized and Decentralized Control}
\label{ssec:nhexa_control_centralized_decentralized} \label{ssec:nhexa_control_centralized_decentralized}
@ -662,18 +662,16 @@ In the context of the nano-hexapod, two distinct control strategies will be exam
\caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.} \caption{\label{fig:nhexa_stewart_decentralized_control}Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.}
\end{figure} \end{figure}
\section{Choice of the control space} \section{Choice of the Control Space}
\label{ssec:nhexa_control_space} \label{ssec:nhexa_control_space}
When controlling a Stewart platform using external metrology that measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), denoted as \(\bm{\mathcal{X}}\), the control architecture can be implemented in either the Cartesian space or the strut space. When controlling a Stewart platform using external metrology that measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), denoted as \(\bm{\mathcal{X}}\), the control architecture can be implemented in either Cartesian space or strut space.
This choice impacts both the control design and performance characteristics. This choice impacts both the control design and the obtained performance.
\paragraph{Control in the Strut space} \paragraph{Control in the Strut space}
In this approach, illustrated in Figure \ref{fig:nhexa_control_strut}, the control is performed in the space of the struts. In this approach, illustrated in Figure \ref{fig:nhexa_control_strut}, the control is performed in the space of the struts.
The Jacobian matrix is used to perform real-time approximate inverse kinematics, mapping position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\). The Jacobian matrix is used to solve the inverse kinematics in real-time, mapping position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\).
A diagonal controller then processes these strut-space errors to generate force commands for each actuator. A diagonal controller then processes these strut-space errors to generate force commands for each actuator.
The main advantage of this approach emerges from the plant characteristics in strut space, as shown in Figure \ref{fig:nhexa_plant_frame_struts}. The main advantage of this approach emerges from the plant characteristics in strut space, as shown in Figure \ref{fig:nhexa_plant_frame_struts}.
@ -758,9 +756,7 @@ The effect of this parallel stiffness will be examined in the next section when
The Root Locus analysis, shown in Figure \ref{fig:nhexa_decentralized_iff_root_locus}, reveals the evolution of the closed-loop poles as the controller gain \(g\) varies from \(0\) to \(\infty\). The Root Locus analysis, shown in Figure \ref{fig:nhexa_decentralized_iff_root_locus}, reveals the evolution of the closed-loop poles as the controller gain \(g\) varies from \(0\) to \(\infty\).
A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability \cite{preumont08_trans_zeros_struc_contr_with}. A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability \cite{preumont08_trans_zeros_struc_contr_with}.
This is particularly valuable as the coupling is very large around resonance frequencies, and without this guaranteed stability property, it would be very difficult to have these modes inside the control bandwidth. This property is particularly valuable as the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone.
This control strategy provides effective damping of the resonant modes while maintaining guaranteed stability - a property that would be difficult to achieve using position feedback alone.
The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)), presented in Figure \ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(-90^o\) and \(+90^o\). The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)), presented in Figure \ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(-90^o\) and \(+90^o\).
The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes.
@ -790,7 +786,7 @@ The complete HAC-IFF control architecture is illustrated in Figure \ref{fig:nhex
Following the analysis from Section \ref{ssec:nhexa_control_space}, the control is implemented in the strut space. Following the analysis from Section \ref{ssec:nhexa_control_space}, the control is implemented in the strut space.
The Jacobian matrix \(\bm{J}^{-1}\) performs real-time approximate inverse kinematics to map position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\). The Jacobian matrix \(\bm{J}^{-1}\) performs real-time approximate inverse kinematics to map position errors from Cartesian space \(\bm{\epsilon}_{\mathcal{X}}\) to strut space \(\bm{\epsilon}_{\mathcal{L}}\).
A diagonal High Authority Controller \(\bm{K}_{\text{HAC}}\) then processes these errors in the frame of the struts and computed to forces to apply to the damp plant \(\bm{f}^{\prime}\). A diagonal High Authority Controller \(\bm{K}_{\text{HAC}}\) then processes these errors in the frame of the struts.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -800,8 +796,8 @@ A diagonal High Authority Controller \(\bm{K}_{\text{HAC}}\) then processes thes
The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions. The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions.
Figure \ref{fig:nhexa_decentralized_hac_iff_plant_undamped} shows the original transfer functions from actuator forces \(\bm{f}\) to strut errors \(\bm{\epsilon}_{\mathcal{L}}\), characterized by pronounced resonant peaks. Figure \ref{fig:nhexa_decentralized_hac_iff_plant_undamped} shows the original transfer functions from actuator forces \(\bm{f}\) to strut errors \(\bm{\epsilon}_{\mathcal{L}}\), characterized by pronounced resonant peaks.
When decentralized IFF is implemented, the transfer functions from modified inputs \(\bm{f}^{\prime}\) to strut errors \(\bm{\epsilon}_{\mathcal{L}}\), shown in Figure \ref{fig:nhexa_decentralized_hac_iff_plant_damped}, exhibit significantly attenuated resonances while preserving the plant's decoupled behavior at low frequencies. When decentralized IFF is implemented, the transfer functions from modified inputs \(\bm{f}^{\prime}\) to strut errors \(\bm{\epsilon}_{\mathcal{L}}\), shown in Figure \ref{fig:nhexa_decentralized_hac_iff_plant_damped}, exhibit significantly attenuated resonances.
This damping of structural resonances serves two purposes: it reduces vibrations in the mechanical structure and simplifies the design of the high authority controller by providing a simpler plant dynamics. This damping of structural resonances serves two purposes: it reduces vibrations in the vicinity of resonances and simplifies the design of the high authority controller by providing a simpler plant dynamics.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
@ -819,8 +815,8 @@ This damping of structural resonances serves two purposes: it reduces vibrations
\caption{\label{fig:nhexa_decentralized_hac_iff_plant}Plant in the frame of the strut for the High Authority Controller.} \caption{\label{fig:nhexa_decentralized_hac_iff_plant}Plant in the frame of the strut for the High Authority Controller.}
\end{figure} \end{figure}
Building upon the damped plant dynamics shown in Figure \ref{fig:nhexa_decentralized_hac_iff_plant_damped}, a high authority controller is designed with the structure given in equation \eqref{eq:nhexa_khac}. Building upon the damped plant dynamics shown in Figure \ref{fig:nhexa_decentralized_hac_iff_plant_damped}, a high authority controller is designed with the structure given in \eqref{eq:nhexa_khac}.
The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter ensuring robustness by attenuating the controller's response to high-frequency dynamics. The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness to unmodeled high-frequency dynamics.
The loop gain of an individual control channel is shown in Figure \ref{fig:nhexa_decentralized_hac_iff_loop_gain}. The loop gain of an individual control channel is shown in Figure \ref{fig:nhexa_decentralized_hac_iff_loop_gain}.
\begin{equation}\label{eq:nhexa_khac} \begin{equation}\label{eq:nhexa_khac}
@ -833,7 +829,7 @@ The loop gain of an individual control channel is shown in Figure \ref{fig:nhexa
The stability of the MIMO feedback loop is analyzed through the \emph{characteristic loci} method. The stability of the MIMO feedback loop is analyzed through the \emph{characteristic loci} method.
Such characteristic loci, shown in Figure \ref{fig:nhexa_decentralized_hac_iff_root_locus}, represent the eigenvalues of the loop gain matrix \(\bm{G}(j\omega)\bm{K}(j\omega)\) plotted in the complex plane as frequency varies from \(0\) to \(\infty\). Such characteristic loci, shown in Figure \ref{fig:nhexa_decentralized_hac_iff_root_locus}, represent the eigenvalues of the loop gain matrix \(\bm{G}(j\omega)\bm{K}(j\omega)\) plotted in the complex plane as frequency varies from \(0\) to \(\infty\).
For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point. For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point \cite{skogestad07_multiv_feedb_contr}.
As seen in Figure \ref{fig:nhexa_decentralized_hac_iff_root_locus}, all loci remain to the right of the -1 point, confirming the stability of the closed-loop system. Additionally, the distance of the loci from the -1 point provides information about stability margins for the coupled system. As seen in Figure \ref{fig:nhexa_decentralized_hac_iff_root_locus}, all loci remain to the right of the -1 point, confirming the stability of the closed-loop system. Additionally, the distance of the loci from the -1 point provides information about stability margins for the coupled system.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -853,8 +849,18 @@ As seen in Figure \ref{fig:nhexa_decentralized_hac_iff_root_locus}, all loci rem
\end{figure} \end{figure}
\section*{Conclusion} \section*{Conclusion}
The control architecture developed for the uniaxial and the rotating models has been adapted for the Stewart platform.
Two fundamental choices were first addressed: the selection between centralized and decentralized approaches, and the choice of control space.
While control in Cartesian space enables direction-specific performance tuning, the implementation in strut space was selected for the conceptual design phase due to two key advantages: good decoupling at low frequencies and identical diagonal terms in the plant transfer functions, allowing a single controller design to be replicated across all struts.
The HAC-LAC strategy was then implemented.
The inner loop implements decentralized Integral Force Feedback for active damping.
The collocated nature of the force sensors ensures stability despite strong coupling between struts at resonance frequencies, enabling effective damping of structural modes.
The outer loop implements High Authority Control, enabling precise positioning of the platform.
This control architecture will then be used for the conceptual validation of the NASS.
More sophisticated control strategies will be investigated during the detailed design phase
\chapter*{Conclusion} \chapter*{Conclusion}
\label{sec:nhexa_conclusion} \label{sec:nhexa_conclusion}