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@ -1307,15 +1307,21 @@ The validated multi-body model will serve as a valuable tool for predicting syst
** Centralized and Decentralized Control
<<ssec:nhexa_control_centralized_decentralized>>
- Explain what is centralized and decentralized:
- linked to the sensor position relative to the actuators
- linked to the fact that sensors and actuators pairs are "independent" or each other (related to the control architecture, not because there is no coupling)
This does not mean there is no coupling
Decentralized = The controller state depends on one sensor only and will impact one actuator signal only
- When can decentralized control be used and when centralized control is necessary?
Study of interaction: RGA
- IFF: Decentralized (Section ref:ssec:nhexa_control_iff)
- HAC: Centralized (Section ref:ssec:nhexa_control_hac_lac)
In the control of MIMO systems and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies.
In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops as illustrated in Figure ref:fig:nhexa_stewart_decentralized_control.
While mechanical coupling between the struts exists, the control decisions are made locally, with each controller processing information from a single sensor-actuator pair.
This approach offers simplicity in implementation and reduced computational requirements.
Conversely, centralized control utilizes information from all sensors to determine the control action for each actuator.
This strategy potentially enables better performance by explicitly accounting for the mechanical coupling between the struts, though at the cost of increased complexity in both design and implementation.
The choice between these approaches depends significantly on the degree of interaction between the different control channels, but also on the available sensors and actuators.
For instance, when using external metrology systems that measure the platform's global position, centralized control becomes necessary as each sensor measurement depends on all actuator inputs.
In the context of the nano-hexapod, two distinct control strategies will be examined during the conceptual phase:
- Decentralized Integral Force Feedback (IFF), which utilizes collocated force sensors to implement independent control loops for each strut (Section ref:ssec:nhexa_control_iff)
- High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section ref:ssec:nhexa_control_hac_lac)
#+name: fig:nhexa_stewart_decentralized_control
#+caption: Decentralized control strategy using the encoders. The two controllers for the struts on the back are not shown for simplicity.
@ -1324,13 +1330,21 @@ The validated multi-body model will serve as a valuable tool for predicting syst
** Choice of the control space
<<ssec:nhexa_control_space>>
- Suppose an external metrology measures the pose of frame $\{B\}$ with respect to $\{A\}$, noted $\bm{\mathcal{X}}$.
The goal is to position the top platform to follow some reference signal $\bm{r}_\mathcal{X}$.
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.
This choice impacts both the control design and performance characteristics.
One strategy is to use the Jacobian matrix to perform an approximate inverse kinematics in real time to map the error in the frame of the struts $\bm{\epsilon}_\mathcal{L}$, and then a diagonal controller is used to control the position of each strut by output forces to be applied on each strut $\bm{\tau}$.
Another strategy is to have the controller get the cartesian errors as input $\bm{\epsilon}_{\mathcal{L}}$ and output forces and torques to apply to the top platform $\bm{\mathcal{F}}$.
The Jacobian is then used to map these forces and torque to force to be applied by each strut.
**** 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.
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 $\bm{\epsilon}_{\mathcal{L}}$.
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 diagonal terms of the plant (transfer functions from force to displacement of the same strut, as measured by the external metrology) are identical due to the system's symmetry.
This simplifies the control design as only one controller needs to be tuned.
Furthermore, at low frequencies, the plant exhibits good decoupling between struts, allowing for effective independent control of each axis.
#+begin_src latex :file nhexa_control_strut.pdf
\begin{tikzpicture}
@ -1397,17 +1411,20 @@ The validated multi-body model will serve as a valuable tool for predicting syst
#+end_subfigure
#+end_figure
- Trade-off for both strategies from looking at the obtained plant.
- The plant in the frame of the struts is shown in Figure ref:fig:nhexa_control_strut
equal diagonal plant elements: just one controller to design, well decoupled at low frequency
- The plant in Cartesian frame is shown in Figure ref:fig:nhexa_control_cartesian
less visible modes in some directions: vertical plant: second order plant, same for Rz
But Coupling: $\epsilon_{R_x}/\mathcal{F}_y$, $\epsilon_{R_y}/\mathcal{F}_x$, $\epsilon_{D_x}/\mathcal{M}_y$, $\epsilon_{D_y}/\mathcal{M}_x$
can choose the bandwidth for different DoF, but coupling may be present at low frequency
**** Control in Cartesian Space
- Say that in order to validate the conceptual design, the control will be performed in the frame of the struts for simplicity
Alternatively, control can be implemented directly in Cartesian space, as shown in Figure ref:fig:nhexa_control_cartesian.
Here, the controller processes Cartesian errors $\bm{\epsilon}_{\mathcal{X}}$ to generate forces and torques $\bm{\mathcal{F}}$, which are then mapped to actuator forces through the transpose of the inverse Jacobian matrix.
- There are much to discuss about controlling a Stewart platform, this will be done during the detail design phase.
The plant behavior in Cartesian space, illustrated in Figure ref:fig:nhexa_plant_frame_cartesian, reveals interesting characteristics.
Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics.
A key advantage of this approach is that control performance can be individually tuned for each direction.
This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others.
However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$).
For the conceptual validation of the nano-hexapod, control in the strut space has been selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies.
More sophisticated control strategies will be explored during the detailed design phase.
#+begin_src matlab
%% Identify plant from actuator forces to external metrology
@ -1557,17 +1574,9 @@ exportFig('figs/nhexa_plant_frame_cartesian.pdf', 'width', 'half', 'height', 600
** Active Damping with Decentralized IFF
<<ssec:nhexa_control_iff>>
Integral Force Feedback is implemented in a decentralized way (i.e. similarly to what is shown in Figure ref:fig:nhexa_stewart_decentralized_control, but using force sensors instead of relative motion sensors).
The decentralized Integral Force Feedback (IFF) control strategy is implemented using independent control loops for each strut, similarly to what is shown in Figure ref:fig:nhexa_stewart_decentralized_control, but using force sensors instead of relative motion sensors.
Block diagram is shown in Figure ref:fig:nhexa_decentralized_iff_schematic, with controller $\bm{K}_{\text{IFF}}(s)$ being a diagonal controller eqref:eq:nhexa_kiff (i.e. one independent controller for each strut) with pure integrators on the diagonal.
\begin{equation}\label{eq:nhexa_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
K_{\text{IFF}}(s) & & 0 \\
& \ddots & \\
0 & & K_{\text{IFF}}(s)
\end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s}
\end{equation}
The corresponding block diagram of the control loop is shown in Figure ref:fig:nhexa_decentralized_iff_schematic, in which the controller $\bm{K}_{\text{IFF}}(s)$ is a diagonal matrix where each diagonal element is a pure integrator eqref:eq:nhexa_kiff.
#+begin_src latex :file nhexa_decentralized_iff_schematic.pdf
\begin{tikzpicture}
@ -1601,28 +1610,28 @@ Block diagram is shown in Figure ref:fig:nhexa_decentralized_iff_schematic, with
#+RESULTS:
[[file:figs/nhexa_decentralized_iff_schematic.png]]
Note that here, we are not considering stiffness in parallel with the force sensors are the Stewart platform is not rotating (this will be studied in the next section when the Stewart platform will be located on top of the micro-station).
Similarly to what was done with the 3DoF model, the Root Locus plot is computed by estimating the poles of the closed-loop system as the controller gain $g$ is varied from $0$ to $\infty$.
\begin{equation}\label{eq:nhexa_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
K_{\text{IFF}}(s) & & 0 \\
& \ddots & \\
0 & & K_{\text{IFF}}(s)
\end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s}
\end{equation}
- [ ] Interaction around resonances is very high: show that with RGA (encoder outputs)
- [ ] But guaranteed stability with decentralized IFF [[cite:&preumont08_trans_zeros_struc_contr_with]]
- [ ] I think there is another paper about that
- [ ] nice way to have some control authority around that frequency, which would be impossible with positioning sensors
In this section, the stiffness in parallel with the force sensor has been omitted since the Stewart platform is not subjected to rotation.
The effect of this parallel stiffness will be examined in the next section when the platform is integrated into the complete NASS system.
For decentralized control: "MIMO root locus" can be used to estimate the damping / optimal gain
Poles and converging towards /transmission zeros/
=> Already explain in 3DoF model
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]].
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.
Show effect of changed payload mass? (no maybe NASS section)
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.
Compute:
- [ ] Plant dynamics (already shown earlier)
- [ ] Root Locus with decentralized IFF (only pure integrator?)
- [ ] show the poles for one value of the gain => How to optimize the added damping to all modes?
- [ ] Add some papers citations
- [ ] Effect of rotation and added parallel stiffness? Or maybe in next section (NASS + Spindle)?
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.
This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability.
#+begin_src matlab
%% Identify the IFF Plant
@ -1721,7 +1730,7 @@ exportFig('figs/nhexa_decentralized_iff_loop_gain.pdf', 'width', 'half', 'height
#+caption: Decentralized IFF
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain}
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{m1}}{f_1}(s)$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.85
@ -1738,11 +1747,12 @@ exportFig('figs/nhexa_decentralized_iff_loop_gain.pdf', 'width', 'half', 'height
** MIMO High-Authority Control - Low-Authority Control
<<ssec:nhexa_control_hac_lac>>
The HAC-IFF architecture is shown in Figure ref:fig:nhexa_hac_iff_schematic.
The reference signal $\bm{r}_{\mathcal{X}}$ is compared with the measured pose $\bm{\mathcal{X}}$.
The Jacobian matrix is used to solve the approximate inverse kinematics in real time.
Finally, the (diagonal) High Authority Controller $\bm{K}_{\text{HAC}}$ is doing the doing in the frame of the struts.
The design of the High Authority Control positioning loop is now examined.
The complete HAC-IFF control architecture is illustrated in Figure ref:fig:nhexa_hac_iff_schematic, where the reference signal $\bm{r}_{\mathcal{X}}$ represents the desired pose, and $\bm{\mathcal{X}}$ is the measured pose by the external metrology system.
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 $\bm{\epsilon}_{\mathcal{L}}$.
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 $\bm{f}^{\prime}$.
#+begin_src latex :file nhexa_hac_iff_schematic.pdf
\begin{tikzpicture}
@ -1787,9 +1797,10 @@ Finally, the (diagonal) High Authority Controller $\bm{K}_{\text{HAC}}$ is doing
#+RESULTS:
[[file:figs/nhexa_hac_iff_schematic.png]]
The transfer functions from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ (i.e. without the Decentralized IFF being implemented) are compared with the transfer functions from $\bm{f}^{\prime}$ to $\bm{\epsilon}_{\mathcal{L}}$ (i.e. with the Decentralized IFF being implemented).
- [ ] Maybe two subfigures for undamped and damped
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.
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.
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.
#+begin_src matlab
%% Identify the IFF Plant
@ -1915,13 +1926,13 @@ exportFig('figs/nhexa_decentralized_hac_iff_plant_damped.pdf', 'width', 'half',
#+caption: Plant in the frame of the strut for the High Authority Controller.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped}
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped plant in the frame of the struts}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nhexa_decentralized_hac_iff_plant_undamped.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped with Decentralized IFF}
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped plant with Decentralized IFF}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
@ -1929,7 +1940,9 @@ exportFig('figs/nhexa_decentralized_hac_iff_plant_damped.pdf', 'width', 'half',
#+end_subfigure
#+end_figure
From the obtained damped plant, the High Authority Controller is developed.
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.
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 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}
\bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
@ -1939,9 +1952,10 @@ From the obtained damped plant, the High Authority Controller is developed.
\end{bmatrix}, \quad 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}}
\end{equation}
- In order to check the stability of the feedback MIMO loop:
- Characteristic Loci: Eigenvalues of $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane
- Generalized Nyquist Criterion: If $G(s)$ has $p_0$ unstable poles, then the closed-loop system with return ratio $kG(s)$ is stable if and only if the characteristic loci of $kG(s)$, taken together, encircle the point $-1$, $p_0$ times anti-clockwise, assuming there are no hidden modes
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 $\infty$.
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.
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
%% High Authority Controller - Mid Stiffness Nano-Hexapod
@ -2039,7 +2053,7 @@ exportFig('figs/nhexa_decentralized_hac_iff_loop_gain.pdf', 'width', 'half', 'he
#+attr_latex: :scale 0.85
[[file:figs/nhexa_decentralized_hac_iff_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Root Locus}
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.85
@ -2047,13 +2061,13 @@ exportFig('figs/nhexa_decentralized_hac_iff_loop_gain.pdf', 'width', 'half', 'he
#+end_subfigure
#+end_figure
- [ ] Show some performance metric? For instance compliance?
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
* Conclusion
:PROPERTIES:
:UNNUMBERED: t

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@ -1,4 +1,4 @@
% Created 2025-02-11 Tue 17:18
% Created 2025-02-11 Tue 18:35
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -638,18 +638,22 @@ The validated multi-body model will serve as a valuable tool for predicting syst
\section{Centralized and Decentralized Control}
\label{ssec:nhexa_control_centralized_decentralized}
In the control of MIMO systems and more specifically of Stewart platforms, a fundamental architectural decision lies in the choice between centralized and decentralized control strategies.
In decentralized control, each actuator operates based on feedback from its associated sensor only, creating independent control loops as illustrated in Figure \ref{fig:nhexa_stewart_decentralized_control}.
While mechanical coupling between the struts exists, the control decisions are made locally, with each controller processing information from a single sensor-actuator pair.
This approach offers simplicity in implementation and reduced computational requirements.
Conversely, centralized control utilizes information from all sensors to determine the control action for each actuator.
This strategy potentially enables better performance by explicitly accounting for the mechanical coupling between the struts, though at the cost of increased complexity in both design and implementation.
The choice between these approaches depends significantly on the degree of interaction between the different control channels, but also on the available sensors and actuators.
For instance, when using external metrology systems that measure the platform's global position, centralized control becomes necessary as each sensor measurement depends on all actuator inputs.
In the context of the nano-hexapod, two distinct control strategies will be examined during the conceptual phase:
\begin{itemize}
\item Explain what is centralized and decentralized:
\begin{itemize}
\item linked to the sensor position relative to the actuators
\item linked to the fact that sensors and actuators pairs are ``independent'' or each other (related to the control architecture, not because there is no coupling)
This does not mean there is no coupling
Decentralized = The controller state depends on one sensor only and will impact one actuator signal only
\end{itemize}
\item When can decentralized control be used and when centralized control is necessary?
Study of interaction: RGA
\item IFF: Decentralized (Section \ref{ssec:nhexa_control_iff})
\item HAC: Centralized (Section \ref{ssec:nhexa_control_hac_lac})
\item Decentralized Integral Force Feedback (IFF), which utilizes collocated force sensors to implement independent control loops for each strut (Section \ref{ssec:nhexa_control_iff})
\item High-Authority Control (HAC), which employs a centralized approach to achieve precise positioning based on external metrology measurements (Section \ref{ssec:nhexa_control_hac_lac})
\end{itemize}
\begin{figure}[htbp]
@ -661,15 +665,21 @@ Study of interaction: RGA
\section{Choice of the control space}
\label{ssec:nhexa_control_space}
\begin{itemize}
\item Suppose an external metrology measures the pose of frame \(\{B\}\) with respect to \(\{A\}\), noted \(\bm{\mathcal{X}}\).
The goal is to position the top platform to follow some reference signal \(\bm{r}_\mathcal{X}\).
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.
This choice impacts both the control design and performance characteristics.
One strategy is to use the Jacobian matrix to perform an approximate inverse kinematics in real time to map the error in the frame of the struts \(\bm{\epsilon}_\mathcal{L}\), and then a diagonal controller is used to control the position of each strut by output forces to be applied on each strut \(\bm{\tau}\).
Another strategy is to have the controller get the cartesian errors as input \(\bm{\epsilon}_{\mathcal{L}}\) and output forces and torques to apply to the top platform \(\bm{\mathcal{F}}\).
The Jacobian is then used to map these forces and torque to force to be applied by each strut.
\end{itemize}
\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.
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 \(\bm{\epsilon}_{\mathcal{L}}\).
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 diagonal terms of the plant (transfer functions from force to displacement of the same strut, as measured by the external metrology) are identical due to the system's symmetry.
This simplifies the control design as only one controller needs to be tuned.
Furthermore, at low frequencies, the plant exhibits good decoupling between struts, allowing for effective independent control of each axis.
\begin{figure}[htbp]
\begin{subfigure}{0.98\textwidth}
@ -689,21 +699,20 @@ The Jacobian is then used to map these forces and torque to force to be applied
\caption{\label{fig:nhexa_control_frame}Two control strategies}
\end{figure}
\begin{itemize}
\item Trade-off for both strategies from looking at the obtained plant.
\begin{itemize}
\item The plant in the frame of the struts is shown in Figure \ref{fig:nhexa_control_strut}
equal diagonal plant elements: just one controller to design, well decoupled at low frequency
\item The plant in Cartesian frame is shown in Figure \ref{fig:nhexa_control_cartesian}
less visible modes in some directions: vertical plant: second order plant, same for Rz
But Coupling: \(\epsilon_{R_x}/\mathcal{F}_y\), \(\epsilon_{R_y}/\mathcal{F}_x\), \(\epsilon_{D_x}/\mathcal{M}_y\), \(\epsilon_{D_y}/\mathcal{M}_x\)
can choose the bandwidth for different DoF, but coupling may be present at low frequency
\end{itemize}
\paragraph{Control in Cartesian Space}
\item Say that in order to validate the conceptual design, the control will be performed in the frame of the struts for simplicity
Alternatively, control can be implemented directly in Cartesian space, as shown in Figure \ref{fig:nhexa_control_cartesian}.
Here, the controller processes Cartesian errors \(\bm{\epsilon}_{\mathcal{X}}\) to generate forces and torques \(\bm{\mathcal{F}}\), which are then mapped to actuator forces through the transpose of the inverse Jacobian matrix.
\item There are much to discuss about controlling a Stewart platform, this will be done during the detail design phase.
\end{itemize}
The plant behavior in Cartesian space, illustrated in Figure \ref{fig:nhexa_plant_frame_cartesian}, reveals interesting characteristics.
Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics.
A key advantage of this approach is that control performance can be individually tuned for each direction.
This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others.
However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., \(\epsilon_{R_x}/\mathcal{F}_y\) or \(\epsilon_{D_y}/\bm\mathcal{M}_x\)).
For the conceptual validation of the nano-hexapod, control in the strut space has been selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies.
More sophisticated control strategies will be explored during the detailed design phase.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -724,9 +733,16 @@ can choose the bandwidth for different DoF, but coupling may be present at low f
\section{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff}
Integral Force Feedback is implemented in a decentralized way (i.e. similarly to what is shown in Figure \ref{fig:nhexa_stewart_decentralized_control}, but using force sensors instead of relative motion sensors).
The decentralized Integral Force Feedback (IFF) control strategy is implemented using independent control loops for each strut, similarly to what is shown in Figure \ref{fig:nhexa_stewart_decentralized_control}, but using force sensors instead of relative motion sensors.
The corresponding block diagram of the control loop is shown in Figure \ref{fig:nhexa_decentralized_iff_schematic}, in which the controller \(\bm{K}_{\text{IFF}}(s)\) is a diagonal matrix where each diagonal element is a pure integrator \eqref{eq:nhexa_kiff}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_decentralized_iff_schematic.png}
\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs \(\bm{f}^{\prime}\)}
\end{figure}
Block diagram is shown in Figure \ref{fig:nhexa_decentralized_iff_schematic}, with controller \(\bm{K}_{\text{IFF}}(s)\) being a diagonal controller \eqref{eq:nhexa_kiff} (i.e. one independent controller for each strut) with pure integrators on the diagonal.
\begin{equation}\label{eq:nhexa_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
@ -736,49 +752,26 @@ Block diagram is shown in Figure \ref{fig:nhexa_decentralized_iff_schematic}, wi
\end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s}
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_decentralized_iff_schematic.png}
\caption{\label{fig:nhexa_decentralized_iff_schematic}Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs \(\bm{f}^{\prime}\)}
\end{figure}
Note that here, we are not considering stiffness in parallel with the force sensors are the Stewart platform is not rotating (this will be studied in the next section when the Stewart platform will be located on top of the micro-station).
In this section, the stiffness in parallel with the force sensor has been omitted since the Stewart platform is not subjected to rotation.
The effect of this parallel stiffness will be examined in the next section when the platform is integrated into the complete NASS system.
Similarly to what was done with the 3DoF model, the Root Locus plot is computed by estimating the poles of the closed-loop system as the controller gain \(g\) is varied 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}.
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 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.
\begin{itemize}
\item[{$\square$}] Interaction around resonances is very high: show that with RGA (encoder outputs)
\item[{$\square$}] But guaranteed stability with decentralized IFF \cite{preumont08_trans_zeros_struc_contr_with}
\begin{itemize}
\item[{$\square$}] I think there is another paper about that
\end{itemize}
\item[{$\square$}] nice way to have some control authority around that frequency, which would be impossible with positioning sensors
\end{itemize}
For decentralized control: ``MIMO root locus'' can be used to estimate the damping / optimal gain
Poles and converging towards \emph{transmission zeros}
=> Already explain in 3DoF model
Show effect of changed payload mass? (no maybe NASS section)
Compute:
\begin{itemize}
\item[{$\square$}] Plant dynamics (already shown earlier)
\item[{$\square$}] Root Locus with decentralized IFF (only pure integrator?)
\item[{$\square$}] show the poles for one value of the gain => How to optimize the added damping to all modes?
\begin{itemize}
\item[{$\square$}] Add some papers citations
\end{itemize}
\item[{$\square$}] Effect of rotation and added parallel stiffness? Or maybe in next section (NASS + Spindle)?
\end{itemize}
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.
This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain}
\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{m1}}{f_1}(s)$}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
@ -792,11 +785,12 @@ Compute:
\section{MIMO High-Authority Control - Low-Authority Control}
\label{ssec:nhexa_control_hac_lac}
The HAC-IFF architecture is shown in Figure \ref{fig:nhexa_hac_iff_schematic}.
The reference signal \(\bm{r}_{\mathcal{X}}\) is compared with the measured pose \(\bm{\mathcal{X}}\).
The Jacobian matrix is used to solve the approximate inverse kinematics in real time.
Finally, the (diagonal) High Authority Controller \(\bm{K}_{\text{HAC}}\) is doing the doing in the frame of the struts.
The design of the High Authority Control positioning loop is now examined.
The complete HAC-IFF control architecture is illustrated in Figure \ref{fig:nhexa_hac_iff_schematic}, where the reference signal \(\bm{r}_{\mathcal{X}}\) represents the desired pose, and \(\bm{\mathcal{X}}\) is the measured pose by the external metrology system.
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 \(\bm{\epsilon}_{\mathcal{L}}\).
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 \(\bm{f}^{\prime}\).
\begin{figure}[htbp]
\centering
@ -804,29 +798,30 @@ Finally, the (diagonal) High Authority Controller \(\bm{K}_{\text{HAC}}\) is doi
\caption{\label{fig:nhexa_hac_iff_schematic}HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts}
\end{figure}
The transfer functions from \(\bm{f}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (i.e. without the Decentralized IFF being implemented) are compared with the transfer functions from \(\bm{f}^{\prime}\) to \(\bm{\epsilon}_{\mathcal{L}}\) (i.e. with the Decentralized IFF being implemented).
\begin{itemize}
\item[{$\square$}] Maybe two subfigures for undamped and damped
\end{itemize}
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.
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.
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.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_undamped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped plant in the frame of the struts}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/nhexa_decentralized_hac_iff_plant_damped.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped with Decentralized IFF}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped plant with Decentralized IFF}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_hac_iff_plant}Plant in the frame of the strut for the High Authority Controller.}
\end{figure}
From the obtained damped plant, the High Authority Controller is developed.
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}.
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 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}
\bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
@ -836,13 +831,10 @@ From the obtained damped plant, the High Authority Controller is developed.
\end{bmatrix}, \quad 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}}
\end{equation}
\begin{itemize}
\item In order to check the stability of the feedback MIMO loop:
\begin{itemize}
\item Characteristic Loci: Eigenvalues of \(\bm{G}(j\omega)\bm{K}(j\omega)\) plotted in the complex plane
\item Generalized Nyquist Criterion: If \(G(s)\) has \(p_0\) unstable poles, then the closed-loop system with return ratio \(kG(s)\) is stable if and only if the characteristic loci of \(kG(s)\), taken together, encircle the point \(-1\), \(p_0\) times anti-clockwise, assuming there are no hidden modes
\end{itemize}
\end{itemize}
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 \(\infty\).
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.
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{subfigure}{0.48\textwidth}
@ -855,30 +847,15 @@ From the obtained damped plant, the High Authority Controller is developed.
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Root Locus}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF}
\end{figure}
\begin{itemize}
\item[{$\square$}] Show some performance metric? For instance compliance?
\end{itemize}
\section*{Conclusion}
\chapter*{Conclusion}
\label{sec:nhexa_conclusion}