stewart-simscape/org/control-study.org

Stewart Platform - Control Study

• First Control Architecture
  • Control Schematic
  • Initialize the Stewart platform
  • Identification of the plant
  • Plant Analysis
  • Controller Design
• Transmissibility Analysis
  • Initialize the Stewart platform
  • Transmissibility
• Compliance Analysis
  • Initialize the Stewart platform
  • Compliance
• Functions
  • Compute the Transmissibility
    • Function description
    • Optional Parameters
    • Identification of the Transmissibility Matrix
    • Computation of the Frobenius norm
  • Compute the Compliance
    • Function description
    • Optional Parameters
    • Identification of the Compliance Matrix
    • Computation of the Frobenius norm

First Control Architecture

Control Schematic

  \begin{tikzpicture}
    % Blocs
    \node[block] (J) at (0, 0) {$J$};
    \node[addb={+}{}{}{}{-}, right=1 of J] (subr) {};
    \node[block, right=0.8 of subr] (K) {$K_{L}$};
    \node[block, right=1 of K] (G) {$G_{L}$};

    % Connections and labels
    \draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0);
    \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$};
    \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$};
    \draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$};
    \draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south);
  \end{tikzpicture}

Initialize the Stewart platform

  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);

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');

Identification of the plant

Let's identify the transfer function from $\bm{\tau}}$ to $\bm{L}$.

  %% 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, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};

Plant Analysis

Diagonal terms

Compare to off-diagonal terms

Controller Design

One integrator should be present in the controller.

A lead is added around the crossover frequency which is set to be around 500Hz.

  % wint = 2*pi*100; % Integrate until [rad]
  % wlead = 2*pi*500; % Location of the lead [rad]
  % hlead = 2; % Lead strengh

  % Kl = 1e6 * ... % Gain
  %      (s + wint)/(s) * ... % Integrator until 100Hz
  %      (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead

  wc = 2*pi*100;
  Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc));
  Kl = Kl * eye(6);

Transmissibility Analysis

Initialize the Stewart platform

  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);

We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.

  ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
  payload = initializePayload('type', 'rigid');

Transmissibility

  %% 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'};

  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

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*}

  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

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

  Gamma = T_norm/sqrt(6);

  figure;
  plot(freqs, Gamma)
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');

Compliance Analysis

Initialize the Stewart platform

  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);

We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid');

Compliance

  %% 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'};

  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

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

  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

  figure;
  plot(freqs, C_norm)
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');

Functions

Compute the Transmissibility

<<sec:computeTransmissibility>>

Function description

  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]

Optional Parameters

    arguments
      args.plots logical {mustBeNumericOrLogical} = false
      args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end

  freqs = args.freqs;

Identification of the Transmissibility Matrix

  %% 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'};

If wanted, the 6x6 transmissibility matrix is plotted.

  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';
    ylabel(han, 'Frequency [Hz]');
    xlabel(han, 'Transmissibility [m/m]');
  end

Computation of the Frobenius norm

  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

  T_norm = T_norm/sqrt(6);

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

Compute the Compliance

<<sec:computeCompliance>>

Function description

  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]

Optional Parameters

    arguments
      args.plots logical {mustBeNumericOrLogical} = false
      args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
    end

  freqs = args.freqs;

Identification of the Compliance Matrix

  %% 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, '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'};

If wanted, the 6x6 transmissibility matrix is plotted.

  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

Computation of the Frobenius norm

  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

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