#+TITLE: Stewart Platform - Decentralized Active Damping
:DRAWER:
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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs
:END:

* Introduction                                                        :ignore:

The following decentralized active damping techniques are briefly studied:
- Inertial Control (proportional feedback of the absolute velocity): Section [[sec:active_damping_inertial]]
- Integral Force Feedback: Section [[sec:active_damping_iff]]
- Direct feedback of the relative velocity of each strut: Section [[sec:active_damping_dvf]]

* Inertial Control
:PROPERTIES:
:header-args:matlab+: :tangle matlab/active_damping_inertial.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
<<sec:active_damping_inertial>>

** Introduction                                                      :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)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab
  simulinkproject('./');
#+end_src

#+begin_src matlab
  open('simulink/stewart_active_damping.slx')
#+end_src

** Identification of the Dynamics
#+begin_src matlab
  stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'disable', true);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'],   1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Vm'],  1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
#+end_src

The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
  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]);
  legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inertial_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inertial_plant_coupling
#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity of the same leg $v_{m,i}$ and to the absolute velocity of the other legs $v_{m,j}$ with $i \neq j$ in grey ([[./figs/inertial_plant_coupling.png][png]], [[./figs/inertial_plant_coupling.pdf][pdf]])
[[file:figs/inertial_plant_coupling.png]]

** Effect of the Flexible Joint stiffness on the Dynamics
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
#+begin_src matlab
  stewart = initializeJointDynamics(stewart);
  Gf = linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
#+end_src

The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
  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]);
  legend('location', 'southwest')

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/inertial_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:inertial_plant_flexible_joint_decentralized
#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity sensor $v_{m,i}$ ([[./figs/inertial_plant_flexible_joint_decentralized.png][png]], [[./figs/inertial_plant_flexible_joint_decentralized.pdf][pdf]])
[[file:figs/inertial_plant_flexible_joint_decentralized.png]]

** Obtained Damping
The control is a performed in a decentralized manner.
The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
\[ K(s) = g
  \begin{bmatrix}
    1 & & 0 \\
    & \ddots & \\
    0 & & 1
  \end{bmatrix} \]

The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  plot(real(pole(G)),  imag(pole(G)),  'x');
  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
  set(gca,'ColorOrderIndex',1);
  plot(real(tzero(G)),  imag(tzero(G)),  'o');
  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
  for i = 1:length(gains)
    cl_poles = pole(feedback(G, gains(i)*eye(6)));
    set(gca,'ColorOrderIndex',1);
    plot(real(cl_poles), imag(cl_poles), '.');
    cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
    set(gca,'ColorOrderIndex',2);
    plot(real(cl_poles), imag(cl_poles), '.');
  end
  ylim([0,2000]);
  xlim([-2000,0]);
  xlabel('Real Part')
  ylabel('Imaginary Part')
  axis square
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/root_locus_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:root_locus_inertial_rot_stiffness
#+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
[[file:figs/root_locus_inertial_rot_stiffness.png]]

#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  for i = 1:length(gains)
    set(gca,'ColorOrderIndex',1);
    cl_poles = pole(feedback(G, gains(i)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
    set(gca,'ColorOrderIndex',2);
    cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/pole_damping_gain_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:pole_damping_gain_inertial_rot_stiffness
#+caption: Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_inertial_rot_stiffness.png][png]], [[./figs/pole_damping_gain_inertial_rot_stiffness.pdf][pdf]])
[[file:figs/pole_damping_gain_inertial_rot_stiffness.png]]

** Conclusion
#+begin_important
  Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
#+end_important

* Integral Force Feedback
:PROPERTIES:
:header-args:matlab+: :tangle matlab/active_damping_iff.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
<<sec:active_damping_iff>>

** Introduction                                                      :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)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab
  simulinkproject('./');
#+end_src

#+begin_src matlab
  open('simulink/stewart_active_damping.slx')
#+end_src

** Identification of the Dynamics with perfect Joints
We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
  stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'disable', true);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
  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]);
  legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/iff_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:iff_plant_coupling
#+caption: Transfer function from the Actuator force $F_{i}$ to the Force sensor of the same leg $F_{m,i}$ and to the force sensor of the other legs $F_{m,j}$ with $i \neq j$ in grey ([[./figs/iff_plant_coupling.png][png]], [[./figs/iff_plant_coupling.pdf][pdf]])
[[file:figs/iff_plant_coupling.png]]

** Effect of the Flexible Joint stiffness on the Dynamics
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
#+begin_src matlab
  stewart = initializeJointDynamics(stewart);
  Gf = linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
  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]);
  legend('location', 'southwest')

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/iff_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:iff_plant_flexible_joint_decentralized
#+caption: Transfer function from the Actuator force $F_{i}$ to the force sensor $F_{m,i}$ ([[./figs/iff_plant_flexible_joint_decentralized.png][png]], [[./figs/iff_plant_flexible_joint_decentralized.pdf][pdf]])
[[file:figs/iff_plant_flexible_joint_decentralized.png]]

** Obtained Damping
The control is a performed in a decentralized manner.
The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
\[ K(s) = g
  \begin{bmatrix}
    \frac{1}{s} & & 0 \\
    & \ddots & \\
    0 & & \frac{1}{s}
  \end{bmatrix} \]

The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  plot(real(pole(G)),  imag(pole(G)),  'x');
  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
  set(gca,'ColorOrderIndex',1);
  plot(real(tzero(G)),  imag(tzero(G)),  'o');
  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
  for i = 1:length(gains)
    cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
    set(gca,'ColorOrderIndex',1);
    plot(real(cl_poles), imag(cl_poles), '.');
    cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
    set(gca,'ColorOrderIndex',2);
    plot(real(cl_poles), imag(cl_poles), '.');
  end
  ylim([0,inf]);
  xlim([-3000,0]);
  xlabel('Real Part')
  ylabel('Imaginary Part')
  axis square
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/root_locus_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:root_locus_iff_rot_stiffness
#+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
[[file:figs/root_locus_iff_rot_stiffness.png]]

#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  for i = 1:length(gains)
    set(gca,'ColorOrderIndex',1);
    cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
    set(gca,'ColorOrderIndex',2);
    cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/pole_damping_gain_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:pole_damping_gain_iff_rot_stiffness
#+caption: Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_iff_rot_stiffness.png][png]], [[./figs/pole_damping_gain_iff_rot_stiffness.pdf][pdf]])
[[file:figs/pole_damping_gain_iff_rot_stiffness.png]]

** Conclusion
#+begin_important
  The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
  Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
#+end_important

* Direct Velocity Feedback
:PROPERTIES:
:header-args:matlab+: :tangle matlab/active_damping_dvf.m
:header-args:matlab+: :comments org :mkdirp yes
:END:
<<sec:active_damping_dvf>>

** Introduction                                                      :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)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab
  simulinkproject('./');
#+end_src

#+begin_src matlab
  open('simulink/stewart_active_damping.slx')
#+end_src

** Identification of the Dynamics with perfect Joints
We first initialize the Stewart platform without joint stiffness.
#+begin_src matlab
  stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'disable', true);
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
#+end_src

And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/F'],   1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src

The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 2:6
    set(gca,'ColorOrderIndex',2);
    p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
  end
  set(gca,'ColorOrderIndex',1);
  p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
  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]);
  legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dvf_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:dvf_plant_coupling
#+caption: Transfer function from the Actuator force $F_{i}$ to the Relative Motion Sensor $D_{m,j}$ with $i \neq j$ ([[./figs/dvf_plant_coupling.png][png]], [[./figs/dvf_plant_coupling.pdf][pdf]])
[[file:figs/dvf_plant_coupling.png]]


** Effect of the Flexible Joint stiffness on the Dynamics
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
#+begin_src matlab
  stewart = initializeJointDynamics(stewart);
  Gf = linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src

The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
  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]);
  legend('location', 'northeast');

  linkaxes([ax1,ax2],'x');
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dvf_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:dvf_plant_flexible_joint_decentralized
#+caption: Transfer function from the Actuator force $F_{i}$ to the relative displacement sensor $D_{m,i}$ ([[./figs/dvf_plant_flexible_joint_decentralized.png][png]], [[./figs/dvf_plant_flexible_joint_decentralized.pdf][pdf]])
[[file:figs/dvf_plant_flexible_joint_decentralized.png]]

** Obtained Damping
The control is a performed in a decentralized manner.
The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
\[ K(s) = g
  \begin{bmatrix}
    s & & \\
    & \ddots & \\
    & & s
  \end{bmatrix} \]

The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  plot(real(pole(G)),  imag(pole(G)),  'x');
  plot(real(pole(Gf)), imag(pole(Gf)), 'x');
  set(gca,'ColorOrderIndex',1);
  plot(real(tzero(G)),  imag(tzero(G)),  'o');
  plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
  for i = 1:length(gains)
    cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
    set(gca,'ColorOrderIndex',1);
    plot(real(cl_poles), imag(cl_poles), '.');
    cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
    set(gca,'ColorOrderIndex',2);
    plot(real(cl_poles), imag(cl_poles), '.');
  end
  ylim([0,inf]);
  xlim([-3000,0]);
  xlabel('Real Part')
  ylabel('Imaginary Part')
  axis square
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/root_locus_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:root_locus_dvf_rot_stiffness
#+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
[[file:figs/root_locus_dvf_rot_stiffness.png]]

#+begin_src matlab :exports none
  gains = logspace(0, 5, 1000);

  figure;
  hold on;
  for i = 1:length(gains)
    set(gca,'ColorOrderIndex',1);
    cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
    set(gca,'ColorOrderIndex',2);
    cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
    poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
    plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
  end
  xlabel('Control Gain');
  ylabel('Damping of the Poles');
  set(gca, 'XScale', 'log');
  ylim([0,pi/2]);
#+end_src

#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/pole_damping_gain_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+name: fig:pole_damping_gain_dvf_rot_stiffness
#+caption: Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/pole_damping_gain_dvf_rot_stiffness.png][png]], [[./figs/pole_damping_gain_dvf_rot_stiffness.pdf][pdf]])
[[file:figs/pole_damping_gain_dvf_rot_stiffness.png]]

** Conclusion
#+begin_important
  Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
#+end_important