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First draft of the control section

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Thomas Dehaeze 2025-02-11 17:25:54 +01:00
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36 changed files with 8760 additions and 75 deletions
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@ -214,10 +214,11 @@ CLOSED: [2025-02-05 Wed 16:04]
** QUES [#A] Should I talk about APA here?
- 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
- [ ] Should the NASS be validated using 1 DoF actuators?
How to deal with IFF?
- Add parallel stiffness
- *Add parallel stiffness*
- High pass filter
** TODO [#A] Make sure the Simulink file for the Stewart platform is working well
@ -260,9 +261,6 @@ It should be the exact model reference that will be included in the NASS model (
- [ ] Make sure they are all defined in correct order
- [ ] Make sure all vectors and matrices are bold
** TODO [#A] Should I include the effect of rotation somewhere?
Similar to what was done with the 3DoF model?
** DONE [#C] Better understand principle of virtual work
CLOSED: [2025-02-10 Mon 15:51]
@ -272,6 +270,12 @@ Better understand this: https://en.wikipedia.org/wiki/Virtual_work
Also add link or explanation for this equation.
** TODO [#A] Should I include the effect of rotation somewhere?
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
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
CLOSED: [2025-02-06 Thu 18:56]
@ -1249,9 +1253,9 @@ exportFig('figs/nhexa_multi_body_plant_fm.pdf', 'width', 'half', 'height', 600);
#+end_figure
** Conclusion
:properties:
:unnumbered: t
:end:
:PROPERTIES:
:UNNUMBERED: t
:END:
The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system.
Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform.
@ -1268,14 +1272,12 @@ The validated multi-body model will serve as a valuable tool for predicting syst
<<sec:nhexa_control>>
** Introduction :ignore:
MIMO control: much more complex than SISO control because of interaction.
Possible to ignore interaction when good decoupling is achieved.
Important to have tools to study interaction
Different ways to try to decouple a MIMO plant.
Reference book: [[cite:&skogestad07_multiv_feedb_contr]]
Control will be more detailed in chapter 2.
- Contrary to what was done with the uniaxial model SISO control => MIMO control: much more complex than because of interaction
- Possible to ignore interaction when good decoupling is achieved: important to have tools to study interaction.
- 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
- Control will be optimized during the detailed design phase
- Reference book: [[cite:&skogestad07_multiv_feedb_contr]]
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -1308,71 +1310,743 @@ Control will be more detailed in chapter 2.
- 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)
#+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.
[[file:figs/nhexa_stewart_decentralized_control.png]]
** Choice of the control space
<<ssec:nhexa_control_space>>
- [ ] file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
- 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 $\bm{r}_\mathcal{X}$.
- Jacobian matrices, CoK, CoM, control in the frame of the struts, SVD, Modal, ...
- Combined CoM and CoK => Discussion of cubic architecture ? (quick, as it is going to be in detailed in chapter 2)
- Explain also the link with the setpoint: it is interesting to have the controller in the frame of the performance variables
Also speak about disturbances? (and how disturbances can be mixed to different outputs due to control and interaction)
- Table that summarizes the trade-off for each strategy
- Say that in this study, we will do the control in the frame of the struts for simplicity (even though control in the cartesian frame was also tested)
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 $\bm{\tau}$.
*Maybe all details about control should be in chapter 2, dedicated to control*
*Here, just say that using kinematics, we control in the frame of the struts*
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.
#+begin_src latex :file nhexa_control_strut.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {Plant};
\coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
\node[block={2.0cm}{2.0cm}, left=0.8 of inputF] (K) {\begin{matrix}K_1 & & 0 \\ & \ddots & \\ 0 & & K_6\end{matrix}};
\node[block, left=0.8 of K] (J) {$\bm{J}$};
\node[addb={+}{}{}{}{-}, left=0.8 of J] (subr) {};
% \node[block, align=center, left=0.6 of subr] (J) {Inverse\\Kinematics};
% Connections and labels
\draw[->] (outputX) -- ++(0.8, 0);
\draw[->] ($(outputX) + (0.3, 0)$)node[branch]{} node[above]{$\bm{\mathcal{X}}$} -- ++(0, -1.2) -| (subr.south);
\draw[->] (subr.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (J.west);
\draw[->] (J.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{L}}$} (K.west);
\draw[->] (K.east) -- node[midway, above]{$\bm{\tau}$} (inputF);
\draw[<-] (subr.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0);
\end{tikzpicture}
#+end_src
#+begin_src latex :file nhexa_control_cartesian.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {Plant};
\coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
\node[block, left=0.8 of inputF] (J) {$\bm{J}^{-T}$};
\node[block={2.0cm}{2.0cm}, left=0.8 of J] (K) {\begin{matrix}K_x & & 0 \\ & \ddots & \\ 0 & & K_{R_z}\end{matrix}};
\node[addb={+}{}{}{}{-}, left=0.8 of K] (subr) {};
% Connections and labels
\draw[->] (outputX) -- ++(0.8, 0);
\draw[->] ($(outputX) + (0.3, 0)$)node[branch]{} node[above]{$\bm{\mathcal{X}}$} -- ++(0, -1.2) -| (subr.south);
\draw[->] (subr.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (K.west);
\draw[->] (K.east) -- node[midway, above]{$\bm{\mathcal{F}}$} (J.west);
\draw[->] (J.east) -- node[midway, above]{$\bm{\tau}$} (inputF.west);
\draw[<-] (subr.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0);
\end{tikzpicture}
#+end_src
#+name: fig:nhexa_control_frame
#+caption: Two control strategies
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_control_strut}Control in the frame of the struts. $\bm{J}$ is used to project errors in the frame of the struts}
#+attr_latex: :options {0.98\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/nhexa_control_strut.png]]
#+end_subfigure
\bigskip
#+attr_latex: :caption \subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-T}$ is used to project force and torques on each strut}
#+attr_latex: :options {0.98\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/nhexa_control_cartesian.png]]
#+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
- Say that in order to validate the conceptual design, the control will be performed in the frame of the struts for simplicity
- There are much to discuss about controlling a Stewart platform, this will be done during the detail design phase.
#+begin_src matlab
%% Control at the CoM
stewart = initializeSimplifiedNanoHexapod('Mpm', 1e-3); % Massless top platform
%% Identify plant from actuator forces to external metrology
stewart = initializeSimplifiedNanoHexapod();
initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External Metrology [m, rad]
% With no payload
G = linearize(mdl, io);
G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
J = stewart.geometry.J;
Gm = inv(J)*G*inv(J');
%% Plant in the Cartesian Frame
G_cart = G*inv(stewart.geometry.J');
G_cart.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
%% Plant in the frame of the struts
G_struts = stewart.geometry.J*G;
G_struts.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/\tau_i$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/\tau_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:),0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_plant_frame_struts.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Bode plot of the plant projected in the Cartesian frame
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_cart(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_cart(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{D_x}/\mathcal{F}_x$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(2,2), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{D_y}/\mathcal{F}_y$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(3,3), freqs, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', '$\epsilon_{D_z}/\mathcal{F}_z$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(4,4), freqs, 'Hz'))), 'color', colors(4,:), ...
'DisplayName', '$\epsilon_{R_x}/\mathcal{M}_x$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(5,5), freqs, 'Hz'))), 'color', colors(5,:), ...
'DisplayName', '$\epsilon_{R_y}/\mathcal{M}_y$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(6,6), freqs, 'Hz'))), 'color', colors(6,:), ...
'DisplayName', '$\epsilon_{R_z}/\mathcal{M}_z$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(1,5), freqs, 'Hz'))), 'color', [0, 0, 0, 0.5], ...
'DisplayName', 'Coupling')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 4e-3]);
leg = legend('location', 'southwest', 'FontSize', 7, 'NumColumns', 3);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart(i,i), freqs, 'Hz'))), 'color', colors(i,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_plant_frame_cartesian.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nhexa_plant_frame
#+caption: Bode plot of the transfer functions computed from the nano-hexapod multi-body model
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/nhexa_plant_frame_struts.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width \linewidth
[[file:figs/nhexa_plant_frame_cartesian.png]]
#+end_subfigure
#+end_figure
** Active Damping with Decentralized IFF
<<ssec:nhexa_control_iff>>
Guaranteed stability: [[cite:&preumont08_trans_zeros_struc_contr_with]]
- [ ] I think there is another paper about that
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).
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}
#+begin_src latex :file nhexa_decentralized_iff_schematic.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {Plant};
\coordinate[] (input) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputH) at ($(P.south east)!0.2!(P.north east)$);
\coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);
\node[block, above=0.2 of P] (Klac) {$\bm{K}_\text{IFF}$};
\node[addb, left=0.8 of input] (addF) {};
% Connections and labels
\draw[->] (outputL) -- ++(0.7, 0) coordinate(eastlac) |- (Klac.east);
\node[above right] at (outputL){$\bm{f}_m$};
\draw[->] (Klac.west) -| (addF.north);
\draw[->] (addF.east) -- (input) node[above left]{$\bm{f}$};
\draw[->] (outputH) -- ++(1.7, 0) node[above left]{$\bm{d_L}$};
\draw[<-] (addF.west) -- ++(-0.8, 0) node[above right]{$\bm{f}^{\prime}$};
\begin{scope}[on background layer]
\node[fit={(Klac.north-|eastlac) (addF.west|-P.south)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Pi) {};
\node[anchor={north west}] at (Pi.north west){\footnotesize{Damped Plant}};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:nhexa_decentralized_iff_schematic
#+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$
#+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 $\infty$.
- [ ] 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
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
How to optimize the added damping to all modes?
- [ ] Add some papers citations
Show effect of changed payload mass? (no maybe NASS section)
Compute:
- [ ] Plant dynamics
- [ ] Root Locus
- [ ] 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)?
#+begin_src matlab
%% Identify the IFF Plant
initializeSimplifiedNanoHexapod('actuator_kp', 0); % Ignoring parallel stiffness for now
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
% With no payload
G_iff = linearize(mdl, io);
G_iff.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff.OutputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
#+end_src
#+begin_src matlab
%% IFF Controller Design
Kiff = -500/s * ... % Gain
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Root Locus plot of the Decentralized IFF Control
gains = logspace(-2, 1, 200);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
plot(real(pole(G_iff)), imag(pole(G_iff)), 'x', 'color', colors(1,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_iff)), imag(tzero(G_iff)), 'o', 'color', colors(1,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_iff, g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
clpoles = pole(feedback(G_iff, Kiff, +1));
plot(real(clpoles), imag(clpoles), 'kx', ...
'DisplayName', '$g_{opt}$');
hold off;
axis equal;
xlim([-600, 50]); ylim([-50, 600]);
xticks([-600:100:0]);
yticks([0:100:600]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_iff_root_locus.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Loop gain for the Decentralized IFF
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(-G_iff(1,1)*Kiff(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e2]);
% leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
% leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(-G_iff(1,1)*Kiff(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
xlim([1, 1e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_iff_loop_gain.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nhexa_decentralized_iff_results
#+caption: Decentralized IFF
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.85
[[file:figs/nhexa_decentralized_iff_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.85
[[file:figs/nhexa_decentralized_iff_root_locus.png]]
#+end_subfigure
#+end_figure
** MIMO High-Authority Control - Low-Authority Control
<<ssec:nhexa_control_hac_lac>>
Compute:
- [ ] compare open-loop and damped plant (outputs are the encoders)
- [ ] Implement decentralized control?
- [ ] Check stability:
- Characteristic Loci: Eigenvalues of $G(j\omega)$ plotted in the complex plane
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.
#+begin_src latex :file nhexa_hac_iff_schematic.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {Plant};
\coordinate[] (input) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputH) at ($(P.south east)!0.2!(P.north east)$);
\coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);
\node[block, above=0.2 of P] (Klac) {$\bm{K}_\text{IFF}$};
\node[addb, left=0.8 of input] (addF) {};
\node[block, left=0.8 of addF] (Khac) {$\bm{K}_\text{HAC}$};
\node[block, left=0.8 of Khac] (inverseK) {$\bm{J}^{-1}$};
\node[addb={+}{}{}{}{-}, left=0.8 of inverseK] (subL) {};
% Connections and labels
\draw[->] (outputL) -- ++(0.7, 0) coordinate(eastlac) |- (Klac.east);
\node[above right] at (outputL){$\bm{f}_m$};
\draw[->] (Klac.west) -| (addF.north);
\draw[->] (addF.east) -- (input) node[above left]{$\bm{f}$};
\draw[->] (outputH) -- ++(1.7, 0) node[above left]{$\bm{\mathcal{X}}$};
\draw[->] (Khac.east) node[above right]{$\bm{f}^{\prime}$} -- (addF.west);
\draw[->] ($(outputH) + (1.2, 0)$)node[branch]{} |- ($(subL.south)+(0, -1.2)$) -- (subL.south);
\draw[->] (subL.east) -- (inverseK.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
\draw[->] (inverseK.east) -- (Khac.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$};
\draw[<-] (subL.west) -- ++(-0.8, 0) node[above right]{$\bm{r}_\mathcal{X}$};
\begin{scope}[on background layer]
\node[fit={(Klac.north-|eastlac) (addF.west|-P.south)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Pi) {};
\node[anchor={north west}] at (Pi.north west){\footnotesize{Damped Plant}};
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:nhexa_hac_iff_schematic
#+caption: HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts
#+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
#+begin_src matlab
%% Identify the IFF Plant
initializeController('type', 'iff');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External Metrology [m,rad]
% With no payload
G_hac = linearize(mdl, io);
G_hac.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_hac.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
%% Plant in the frame of the struts
G_hac_struts = stewart.geometry.J*G_hac;
G_hac_struts.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,j), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,2), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_hac_iff_plant_undamped.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_hac_struts(i,j), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$')
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i^\prime$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_struts(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,2), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j^\prime$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_struts(i,i), freqs, 'Hz'))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_hac_iff_plant_damped.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nhexa_decentralized_hac_iff_plant
#+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: :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: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nhexa_decentralized_hac_iff_plant_damped.png]]
#+end_subfigure
#+end_figure
From the obtained damped plant, the High Authority Controller is developed.
\begin{equation}\label{eq:nhexa_khac}
\bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
K_{\text{HAC}}(s) & & 0 \\
& \ddots & \\
0 & & K_{\text{HAC}}(s)
\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
#+begin_src matlab :exports none
%% High Authority Controller - Mid Stiffness Nano-Hexapod
% Wanted crossover
wc = 2*pi*20; % [rad/s]
% Integrator
H_int = wc/s;
% Lead to increase phase margin
a = 2; % Amount of phase lead / width of the phase lead / high frequency gain
H_lead = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
% Low Pass filter to increase robustness
H_lpf = 1/(1 + s/2/pi/200);
% Gain to have unitary crossover at 5Hz
H_gain = 1./abs(evalfr(G_hac_struts(1, 1), 1j*wc));
% Decentralized HAC
Khac = H_gain * ... % Gain
H_int * ... % Integrator
H_lpf * ... % Low Pass filter
eye(6); % 6x6 Diagonal
#+end_src
#+begin_src matlab :exports none
%% Plot of the eigenvalues of L in the complex plane
Ldet = zeros(6, length(freqs));
Lmimo = squeeze(freqresp(G_hac_struts*Khac, freqs, 'Hz'));
for i_f = 2:length(freqs)
Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f)));
end
figure;
hold on;
for i = 1:6
plot(real(squeeze(Ldet(i,:))), imag(squeeze(Ldet(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
end
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Real'); ylabel('Imag');
axis square
xlim([-1.8, 0.2]); ylim([-1, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_hac_iff_root_locus.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Loop gain for the Decentralized HAC_IFF
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
xlim([1, 1e3]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nhexa_decentralized_hac_iff_loop_gain.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nhexa_decentralized_hac_iff_results
#+caption: Decentralized HAC-IFF
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+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: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.85
[[file:figs/nhexa_decentralized_hac_iff_root_locus.png]]
#+end_subfigure
#+end_figure
- [ ] Show some performance metric? For instance compliance?
** Conclusion
@ -1380,7 +2054,6 @@ Compute:
:UNNUMBERED: t
:END:
* Conclusion
:PROPERTIES:
:UNNUMBERED: t

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@ -1,4 +1,4 @@
% Created 2025-02-11 Tue 09:54
% Created 2025-02-11 Tue 17:18
% Intended LaTeX compiler: pdflatex
\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}
\label{sec:nhexa_control}
MIMO control: much more complex than SISO control because of interaction.
Possible to ignore interaction when good decoupling is achieved.
Important to have tools to study interaction
Different ways to try to decouple a MIMO plant.
Reference book: \cite{skogestad07_multiv_feedb_contr}
Control will be more detailed in chapter 2.
\begin{itemize}
\item Contrary to what was done with the uniaxial model SISO control => MIMO control: much more complex than because of interaction
\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
\item Here, just basic strategy, similar to what was done with the uniaxial model is used to validate the concept
\item Control will be optimized during the detailed design phase
\item Reference book: \cite{skogestad07_multiv_feedb_contr}
\end{itemize}
\section{Centralized and Decentralized Control}
\label{ssec:nhexa_control_centralized_decentralized}
@ -643,62 +643,236 @@ Control will be more detailed in chapter 2.
\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})
\end{itemize}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_decentralized_control.png}
\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}
\section{Choice of the control space}
\label{ssec:nhexa_control_space}
\begin{itemize}
\item[{$\square$}] \url{file:///home/thomas/Cloud/research/matlab/decoupling-strategies/svd-control.org}
\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 \(\bm{r}_\mathcal{X}\).
\item Jacobian matrices, CoK, CoM, control in the frame of the struts, SVD, Modal, \ldots{}
\item Combined CoM and CoK => Discussion of cubic architecture ? (quick, as it is going to be in detailed in chapter 2)
\item Explain also the link with the setpoint: it is interesting to have the controller in the frame of the performance variables
Also speak about disturbances? (and how disturbances can be mixed to different outputs due to control and interaction)
\item Table that summarizes the trade-off for each strategy
\item Say that in this study, we will do the control in the frame of the struts for simplicity (even though control in the cartesian frame was also tested)
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}
\textbf{Maybe all details about control should be in chapter 2, dedicated to control}
\textbf{Here, just say that using kinematics, we control in the frame of the struts}
\begin{figure}[htbp]
\begin{subfigure}{0.98\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_control_strut.png}
\end{center}
\subcaption{\label{fig:nhexa_control_strut}Control in the frame of the struts. $\bm{J}$ is used to project errors in the frame of the struts}
\end{subfigure}
\bigskip
\begin{subfigure}{0.98\textwidth}
\begin{center}
\includegraphics[scale=1,scale=1]{figs/nhexa_control_cartesian.png}
\end{center}
\subcaption{\label{fig:nhexa_control_cartesian}Control in the Cartesian frame. $\bm{J}^{-T}$ is used to project force and torques on each strut}
\end{subfigure}
\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}
\item Say that in order to validate the conceptual design, the control will be performed in the frame of the struts for simplicity
\item There are much to discuss about controlling a Stewart platform, this will be done during the detail design phase.
\end{itemize}
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_struts.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_struts}Plant in the frame of the struts}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,width=\linewidth]{figs/nhexa_plant_frame_cartesian.png}
\end{center}
\subcaption{\label{fig:nhexa_plant_frame_cartesian}Plant in the Cartesian Frame}
\end{subfigure}
\caption{\label{fig:nhexa_plant_frame}Bode plot of the transfer functions computed from the nano-hexapod multi-body model}
\end{figure}
\section{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff}
Guaranteed stability: \cite{preumont08_trans_zeros_struc_contr_with}
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).
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}
\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).
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{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
How to optimize the added damping to all modes?
\begin{itemize}
\item[{$\square$}] Add some papers citations
\end{itemize}
Show effect of changed payload mass? (no maybe NASS section)
Compute:
\begin{itemize}
\item[{$\square$}] Plant dynamics
\item[{$\square$}] Root Locus
\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}
\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}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_root_locus.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_iff_root_locus}Root Locus}
\end{subfigure}
\caption{\label{fig:nhexa_decentralized_iff_results}Decentralized IFF}
\end{figure}
\section{MIMO High-Authority Control - Low-Authority Control}
\label{ssec:nhexa_control_hac_lac}
Compute:
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.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_hac_iff_schematic.png}
\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[{$\square$}] compare open-loop and damped plant (outputs are the encoders)
\item[{$\square$}] Implement decentralized control?
\item[{$\square$}] Check stability:
\item[{$\square$}] Maybe two subfigures for undamped and damped
\end{itemize}
\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}
\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}
\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.
\begin{equation}\label{eq:nhexa_khac}
\bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
K_{\text{HAC}}(s) & & 0 \\
& \ddots & \\
0 & & K_{\text{HAC}}(s)
\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 Characteristic Loci: Eigenvalues of \(G(j\omega)\) plotted in the complex plane
\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}
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_hac_iff_loop_gain.png}
\end{center}
\subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\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}
\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}