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

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);