#+TITLE: Identification of the Stewart Platform using Simscape
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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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

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#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:

* Introduction                                                        :ignore:
In this document, we discuss the various methods to identify the behavior of the Stewart platform.

- [[sec:modal_analysis]]
- [[sec:transmissibility]]
- [[sec:compliance]]

* Modal Analysis of the Stewart Platform
<<sec:modal_analysis>>
** 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 :results none :exports none
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open('stewart_platform_model.slx')
#+end_src

** Initialize the Stewart Platform
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');
  controller = initializeController('type', 'open-loop');
#+end_src

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}

  %% Run the linearization
  G = linearize(mdl, io);
  % G.InputName  = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
  % G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
#+end_src

Let's check the size of =G=:
#+begin_src matlab :results replace output
  size(G)
#+end_src

#+RESULTS:
: size(G)
: State-space model with 12 outputs, 6 inputs, and 18 states.
: 'org_babel_eoe'
: ans =
:     'org_babel_eoe'

We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
#+begin_src matlab :results replace output
  Gm = minreal(G);
#+end_src

#+RESULTS:
: Gm = minreal(G);
: 6 states removed.

And indeed, we obtain 12 states.

** Coordinate transformation
We can perform the following transformation using the =ss2ss= command.
#+begin_src matlab
  Gt = ss2ss(Gm, Gm.C);
#+end_src

Then, the =C= matrix of =Gt= is the unity matrix which means that the states of the state space model are equal to the measurements $\bm{Y}$.

The measurements are the 6 displacement and 6 velocities of mobile platform with respect to $\{B\}$.

We could perform the transformation by hand:
#+begin_src matlab
  At = Gm.C*Gm.A*pinv(Gm.C);

  Bt = Gm.C*Gm.B;

  Ct = eye(12);
  Dt = zeros(12, 6);

  Gt = ss(At, Bt, Ct, Dt);
#+end_src

** Analysis
#+begin_src matlab
  [V,D] = eig(Gt.A);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  ws = imag(diag(D))/2/pi;
  [ws,I] = sort(ws)

  xi = 100*real(diag(D))./imag(diag(D));
  xi = xi(I);

  data2orgtable([[1:length(ws(ws>0))]', ws(ws>0), xi(xi>0)], {}, {'Mode Number', 'Resonance Frequency [Hz]', 'Damping Ratio [%]'}, ' %.1f ');
#+end_src

#+RESULTS:
| Mode Number | Resonance Frequency [Hz] | Damping Ratio [%] |
|-------------+--------------------------+-------------------|
|         1.0 |                    780.6 |               0.4 |
|         2.0 |                    780.6 |               0.3 |
|         3.0 |                    903.9 |               0.3 |
|         4.0 |                   1061.4 |               0.3 |
|         5.0 |                   1061.4 |               0.2 |
|         6.0 |                   1269.6 |               0.2 |

** Visualizing the modes
To visualize the i'th mode, we may excite the system using the inputs $U_i$ such that $B U_i$ is co-linear to $\xi_i$ (the mode we want to excite).

\[ U(t) = e^{\alpha t} (  ) \]

Let's first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
#+begin_src matlab
  ws = imag(diag(D));
  [ws,I] = sort(ws)
  ws = ws(7:end); I = I(7:end);
#+end_src

#+begin_src matlab
  for i = 1:length(ws)
#+end_src

#+begin_src matlab
  i_mode = I(i); % the argument is the i'th mode
#+end_src

#+begin_src matlab
  lambda_i = D(i_mode, i_mode);
  xi_i = V(:,i_mode);

  a_i = real(lambda_i);
  b_i = imag(lambda_i);
#+end_src

Let do 10 periods of the mode.
#+begin_src matlab
  t = linspace(0, 10/(imag(lambda_i)/2/pi), 1000);
  U_i = pinv(Gt.B) * real(xi_i * lambda_i * (cos(b_i * t) + 1i*sin(b_i * t)));
#+end_src

#+begin_src matlab
  U = timeseries(U_i, t);
#+end_src

Simulation:
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', num2str(t(end)));
  sim(mdl);
#+end_src

Save the movie of the mode shape.
#+begin_src matlab
  smwritevideo(mdl, sprintf('figs/mode%i', i), ...
               'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
               'FrameRate', 30, ...
               'FrameSize', [800, 400]);
#+end_src

#+begin_src matlab
  end
#+end_src

#+name: fig:mode1
#+caption: Identified mode - 1
[[file:figs/mode1.gif]]

#+name: fig:mode3
#+caption: Identified mode - 3
[[file:figs/mode3.gif]]

#+name: fig:mode5
#+caption: Identified mode - 5
[[file:figs/mode5.gif]]

* Transmissibility Analysis
<<sec:transmissibility>>
** Introduction                                                      :ignore:
** Matlab Init                                                     :noexport:
#+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('stewart_platform_model.slx')
#+end_src

** Initialize the Stewart platform
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src

We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
  ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
  payload = initializePayload('type', 'rigid');
  controller = initializeController('type', 'open-loop');
#+end_src

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances/D_w'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
  io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]

  %% Run the linearization
  T = linearize(mdl, io, options);
  T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
  T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_src matlab
  freqs = logspace(1, 4, 1000);

  figure;
  for ix = 1:6
    for iy = 1:6
      subplot(6, 6, (ix-1)*6 + iy);
      hold on;
      plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylim([1e-5, 10]);
      xlim([freqs(1), freqs(end)]);
      if ix < 6
        xticklabels({});
      end
      if iy > 1
        yticklabels({});
      end
    end
  end
#+end_src

From cite:preumont07_six_axis_singl_stage_activ, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
\begin{align*}
  \| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
                       &= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}

#+begin_src matlab
  freqs = logspace(1, 4, 1000);

  T_norm = zeros(length(freqs), 1);

  for i = 1:length(freqs)
    T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
  end
#+end_src

And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]

#+begin_src matlab
  Gamma = T_norm/sqrt(6);
#+end_src

#+begin_src matlab
  figure;
  plot(freqs, Gamma)
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src

* Compliance Analysis
<<sec:compliance>>
** Introduction                                                      :ignore:
** Matlab Init                                                     :noexport:
#+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('stewart_platform_model.slx')
#+end_src

** Initialize the Stewart platform
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart);
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src

We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid');
  controller = initializeController('type', 'open-loop');
#+end_src

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

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances/F_ext'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
  io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]

  %% Run the linearization
  C = linearize(mdl, io, options);
  C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
  C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_src matlab
  freqs = logspace(1, 4, 1000);

  figure;
  for ix = 1:6
    for iy = 1:6
      subplot(6, 6, (ix-1)*6 + iy);
      hold on;
      plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylim([1e-10, 1e-3]);
      xlim([freqs(1), freqs(end)]);
      if ix < 6
        xticklabels({});
      end
      if iy > 1
        yticklabels({});
      end
    end
  end
#+end_src

We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.

#+begin_src matlab
  freqs = logspace(1, 4, 1000);

  C_norm = zeros(length(freqs), 1);

  for i = 1:length(freqs)
    C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
  end
#+end_src

#+begin_src matlab
  figure;
  plot(freqs, C_norm)
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src

* Functions
** Compute the Transmissibility
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeTransmissibility>>

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [T, T_norm, freqs] = computeTransmissibility(args)
  % computeTransmissibility -
  %
  % Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
  %        - freqs [] - Frequency vector to estimate the Frobenius norm
  %
  % Outputs:
  %    - T      [6x6 ss] - Transmissibility matrix
  %    - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
  %    - freqs  [length(freqs)x1] - Frequency vector in [Hz]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
    arguments
      args.plots logical {mustBeNumericOrLogical} = false
      args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end
#+end_src

#+begin_src matlab
  freqs = args.freqs;
#+end_src

*** Identification of the Transmissibility Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances/D_w'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
  io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]

  %% Run the linearization
  T = linearize(mdl, io, options);
  T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
  T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
  p_handle = zeros(6*6,1);

  if args.plots
    fig = figure;
    for ix = 1:6
      for iy = 1:6
        p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
        hold on;
        plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        if ix < 6
            xticklabels({});
        end
        if iy > 1
            yticklabels({});
        end
      end
    end

    linkaxes(p_handle, 'xy')
    xlim([freqs(1), freqs(end)]);
    ylim([1e-5, 1e2]);

    han = axes(fig, 'visible', 'off');
    han.XLabel.Visible = 'on';
    han.YLabel.Visible = 'on';
    xlabel(han, 'Frequency [Hz]');
    ylabel(han, 'Transmissibility [m/m]');
  end
#+end_src

*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  T_norm = zeros(length(freqs), 1);

  for i = 1:length(freqs)
    T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
  end
#+end_src

#+begin_src matlab
  T_norm = T_norm/sqrt(6);
#+end_src

#+begin_src matlab
  if args.plots
    figure;
    plot(freqs, T_norm)
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]');
    ylabel('Transmissibility - Frobenius Norm');
  end
#+end_src

** Compute the Compliance
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeCompliance.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeCompliance>>

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [C, C_norm, freqs] = computeCompliance(args)
  % computeCompliance -
  %
  % Syntax: [C, C_norm, freqs] = computeCompliance(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
  %        - freqs [] - Frequency vector to estimate the Frobenius norm
  %
  % Outputs:
  %    - C      [6x6 ss] - Compliance matrix
  %    - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
  %    - freqs  [length(freqs)x1] - Frequency vector in [Hz]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
    arguments
      args.plots logical {mustBeNumericOrLogical} = false
      args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end
#+end_src

#+begin_src matlab
  freqs = args.freqs;
#+end_src

*** Identification of the Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances/F_ext'],      1, 'openinput');  io_i = io_i + 1; % External forces [N, N*m]
  io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]

  %% Run the linearization
  C = linearize(mdl, io, options);
  C.InputName  = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
  C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
  p_handle = zeros(6*6,1);

  if args.plots
    fig = figure;
    for ix = 1:6
      for iy = 1:6
        p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
        hold on;
        plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        if ix < 6
            xticklabels({});
        end
        if iy > 1
            yticklabels({});
        end
      end
    end

    linkaxes(p_handle, 'xy')
    xlim([freqs(1), freqs(end)]);

    han = axes(fig, 'visible', 'off');
    han.XLabel.Visible = 'on';
    han.YLabel.Visible = 'on';
    xlabel(han, 'Frequency [Hz]');
    ylabel(han, 'Compliance [m/N, rad/(N*m)]');
  end
#+end_src

*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  freqs = args.freqs;

  C_norm = zeros(length(freqs), 1);

  for i = 1:length(freqs)
    C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
  end
#+end_src

#+begin_src matlab
  if args.plots
    figure;
    plot(freqs, C_norm)
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    xlabel('Frequency [Hz]');
    ylabel('Compliance - Frobenius Norm');
  end
#+end_src