stewart-simscape/org/control-tracking.org

Stewart Platform - Tracking Control

• Introduction
• Decentralized Control Architecture using Strut Length
  • Control Schematic
  • Initialize the Stewart platform
  • Identification of the plant
  • Plant Analysis
  • Controller Design
  • Simulation
  • Results
  • Conclusion
• Centralized Control Architecture using Pose Measurement
  • Control Schematic
  • Initialize the Stewart platform
  • Identification of the plant
  • Diagonal Control - Leg's Frame
    • Control Architecture
    • Plant Analysis
    • Controller Design
    • Simulation
  • Diagonal Control - Cartesian Frame
    • Control Architecture
    • Plant Analysis
    • Controller Design
    • Simulation
  • Diagonal Control - Steady State Decoupling
    • Control Architecture
    • Plant Analysis
    • Controller Design
  • Comparison
    • Obtained MIMO Controllers
    • Simulation Results
  • Conclusion
• Hybrid Control Architecture - HAC-LAC with relative DVF
  • Introduction
  • Control Schematic
  • Initialize the Stewart platform
  • First Control Loop - $\bm{K}_\mathcal{L}$
    • Identification
    • Obtained Plant
    • Controller Design
  • Second Control Loop - $\bm{K}_\mathcal{X}$
    • Identification
    • Obtained Plant
    • Controller Design
  • Simulations
  • Conclusion
• Comparison of all the methods
• Compute the pose error of the Stewart Platform

Introduction   ignore

Let's suppose the control objective is to position $\bm{\mathcal{X}}$ of the mobile platform of the Stewart platform such that it is following some reference position $\bm{r}_\mathcal{X}$.

In order to compute the pose error $\bm{\epsilon}_\mathcal{X}$ from the actual pose $\bm{\mathcal{X}}$ and the reference position $\bm{r}_\mathcal{X}$, some computation is required and explained in section sec:compute_pose_error.

Depending of the measured quantity, different control architectures can be used:

• If the struts length $\bm{\mathcal{L}}$ is measured, a decentralized control architecture can be used (Section sec:decentralized)
• If the pose of the mobile platform $\bm{\mathcal{X}}$ is directly measured, a centralized control architecture can be used (Section sec:centralized)
• If both $\bm{\mathcal{L}}$ and $\bm{\mathcal{X}}$ are measured, we can use an hybrid control architecture (Section sec:hybrid)

The control configuration are compare in section sec:comparison.

Decentralized Control Architecture using Strut Length

<<sec:decentralized>>

Control Schematic

The control architecture is shown in Figure fig:decentralized_reference_tracking_L.

The required leg length $\bm{r}_\mathcal{L}$ is computed from the reference path $\bm{r}_\mathcal{X}$ using the inverse kinematics.

Then, a diagonal (decentralized) controller $\bm{K}_\mathcal{L}$ is used such that each leg lengths stays close to its required length.

  \begin{tikzpicture}
    % Blocs
    \node[block, align=center] (J) at (0, 0) {Inverse\\Kinematics};
    \node[addb={+}{}{}{}{-}, right=1 of J] (subr) {};
    \node[block, right=0.8 of subr] (K) {$\bm{K}_\mathcal{L}$};
    \node[block, right=1 of K] (G) {$\bm{G}_\mathcal{L}$};

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

Decentralized control for reference tracking

Initialize the Stewart platform

  <<init-sim>>

Identification of the plant

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

  %% 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);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};

Plant Analysis

The diagonal and off-diagonal terms of the plant are shown in Figure fig:plant_decentralized_L.

All the diagonal terms are equal. We see that the plant is decoupled at low frequency which indicate that decentralized control may be a good idea.

<<plt-matlab>>

Obtain Diagonal and off diagonal dynamics (png, pdf)

Controller Design

The controller consists of:

• A pure integrator
• A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin

The obtained loop gains corresponding to the diagonal elements are shown in Figure fig:loop_gain_decentralized_L.

  wc = 2*pi*30;
  Kl = diag(1./diag(abs(freqresp(G, wc)))) * wc/s * 1/(1 + s/3/wc);

<<plt-matlab>>

Loop Gain of the diagonal elements (png, pdf)

Simulation

Let's define some reference path to follow.

  <<init-ref>>

The reference path is shown in Figure fig:tracking_control_reference_path.

  figure;
  plot3(squeeze(references.r.Data(1,1,:)), squeeze(references.r.Data(2,1,:)), squeeze(references.r.Data(3,1,:)));
  xlabel('X [m]');
  ylabel('Y [m]');
  zlabel('Z [m]');

<<plt-matlab>>

Reference path used for Tracking control (png, pdf)

  controller = initializeController('type', 'ref-track-L');

  set_param('stewart_platform_model', 'StopTime', '10')
  sim('stewart_platform_model');
  simout_D = simout;

  save('./mat/control_tracking.mat', 'simout_D', '-append');

Results

The reference path and the position of the mobile platform are shown in Figure fig:decentralized_control_3d_pos.

  figure;
  hold on;
  plot3(simout_D.x.Xr.Data(:, 1), simout_D.x.Xr.Data(:, 2), simout_D.x.Xr.Data(:, 3), 'k-');
  plot3(squeeze(references.r.Data(1,1,:)), squeeze(references.r.Data(2,1,:)), squeeze(references.r.Data(3,1,:)), '--');
  hold off;
  xlabel('X [m]'); ylabel('Y [m]'); zlabel('Z [m]');
  view([1,1,1])
  zlim([0, inf]);

<<plt-matlab>>

Reference path and Obtained Position (png, pdf)

<<plt-matlab>>

Position error $\bm{\epsilon}_\mathcal{X}$ (png, pdf)

<<plt-matlab>>

Position error $\bm{\epsilon}_\mathcal{L}$ (png, pdf)

Conclusion

Such control architecture is easy to implement and give good results. The inverse kinematics is easy to compute.

However, as $\mathcal{X}$ is not directly measured, it is possible that important positioning errors are due to finite stiffness of the joints and other imperfections.

Centralized Control Architecture using Pose Measurement

<<sec:centralized>>

Control Schematic

The centralized controller takes the position error $\bm{\epsilon}_\mathcal{X}$ as an inputs and generate actuator forces $\bm{\tau}$ (see Figure fig:centralized_reference_tracking). The signals are:

• reference path $\bm{r}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}$
• tracking error $\bm{\epsilon}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}$
• actuator forces $\bm{\tau} = \begin{bmatrix} \tau_1 & \tau_2 & \tau_3 & \tau_4 & \tau_5 & \tau_6 \end{bmatrix}$
• payload pose $\bm{\mathcal{X}} = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}$

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

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

Centralized Controller

Instead of designing a full MIMO controller $K$, we first try to make the plant more diagonal by pre- or post-multiplying some constant matrix, then we design a diagonal controller.

We can think of two ways to make the plant more diagonal that are described in sections sec:diagonal_control_L and sec:diagonal_control_X.

Note here that the subtraction shown in Figure fig:centralized_reference_tracking is not a real subtraction. It is indeed a more complex computation explained in section sec:compute_pose_error.

Initialize the Stewart platform

  <<init-sim>>

Identification of the plant

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

  %% 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; % Relative Displacement Outputs [m]

  %% Run the linearization
  G = linearize(mdl, io);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

Diagonal Control - Leg's Frame

<<sec:diagonal_control_L>>

Control Architecture

The pose error $\bm{\epsilon}_\mathcal{X}$ is first converted in the frame of the leg by using the Jacobian matrix. Then a diagonal controller $\bm{K}_\mathcal{L}$ is designed. The final implemented controller is $\bm{K} = \bm{K}_\mathcal{L} \cdot \bm{J}$ as shown in Figure fig:centralized_reference_tracking_L

Note here that the transformation from the pose error $\bm{\epsilon}_\mathcal{X}$ to the leg's length errors by using the Jacobian matrix is only valid for small errors.

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

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

    \begin{scope}[on background layer]
      \node[fit={(J.south west) (K.north east)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
      \node[above] at (Ktot.north) {$\bm{K}$};
    \end{scope}
  \end{tikzpicture}

Controller in the frame of the legs

Plant Analysis

We now multiply the plant by the Jacobian matrix as shown in Figure fig:centralized_reference_tracking_L to obtain a more diagonal plant.

  Gl = stewart.kinematics.J*G;
  Gl.OutputName  = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};

The bode plot of the plant is shown in Figure fig:plant_centralized_L. We can see that the diagonal elements are identical. This will simplify the design of the controller as all the elements of the diagonal controller can be made identical.

<<plt-matlab>>

Diagonal and off-diagonal elements of the plant $\bm{J}\bm{G}$ (png, pdf)

We can see that this totally decouples the system at low frequency.

This was expected since: \[ \bm{G}(\omega = 0) = \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0) = \bm{J}^{-1} \frac{\delta\bm{\mathcal{L}}}{\delta\bm{\tau}}(\omega = 0) = \bm{J}^{-1} \text{diag}(\mathcal{K}_1^{-1} \ \dots \ \mathcal{K}_6^{-1}) \]

Thus $J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0)$ is a diagonal matrix containing the inverse of the joint's stiffness.

Controller Design

The controller consists of:

• A pure integrator
• A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin

The obtained loop gains corresponding to the diagonal elements are shown in Figure fig:loop_gain_centralized_L.

  wc = 2*pi*30;
  Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);

<<plt-matlab>>

Loop Gain of the diagonal elements (png, pdf)

The controller $\bm{K} = \bm{K}_\mathcal{L} \bm{J}$ is computed.

  K = Kl*stewart.kinematics.J;

Simulation

We specify the reference path to follow.

  <<init-ref>>

  controller = initializeController('type', 'ref-track-X');

We run the simulation and we save the results.

  set_param('stewart_platform_model', 'StopTime', '10')
  sim('stewart_platform_model')
  simout_L = simout;
  save('./mat/control_tracking.mat', 'simout_L', '-append');

Diagonal Control - Cartesian Frame

<<sec:diagonal_control_X>>

Control Architecture

A diagonal controller $\bm{K}_\mathcal{X}$ take the pose error $\bm{\epsilon}_\mathcal{X}$ and generate cartesian forces $\bm{\mathcal{F}}$ that are then converted to actuators forces using the Jacobian as shown in Figure e fig:centralized_reference_tracking_X.

The final implemented controller is $\bm{K} = \bm{J}^{-T} \cdot \bm{K}_\mathcal{X}$.

  \begin{tikzpicture}
    % Blocs
    \node[addb={+}{}{}{}{-}] (subr) at (0, 0) {};
    \node[block, right= of subr] (K) {$\bm{K}_{\mathcal{X}}$};
    \node[block, right= of K] (J) {$\bm{J}^{-T}$};
    \node[block, right= of J] (G) {$\bm{G}$};

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

    \begin{scope}[on background layer]
      \node[fit={(K.south west) (J.north east)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
      \node[above] at (Ktot.north) {$\bm{K}$};
    \end{scope}
  \end{tikzpicture}

Controller in the cartesian frame

Plant Analysis

We now multiply the plant by the Jacobian matrix as shown in Figure fig:centralized_reference_tracking_X to obtain a more diagonal plant.

  Gx = G*inv(stewart.kinematics.J');
  Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

<<plt-matlab>>

Diagonal and off-diagonal elements of the plant $\bm{G} \bm{J}^{-T}$ (png, pdf)

The diagonal terms are not the same. The resonances of the system are "decoupled". For instance, the vertical resonance of the system is only present on the diagonal term corresponding to $D_z/\mathcal{F}_z$.

Here the system is almost decoupled at all frequencies except for the transfer functions $\frac{R_y}{\mathcal{F}_x}$ and $\frac{R_x}{\mathcal{F}_y}$.

This is due to the fact that the compliance matrix of the Stewart platform is not diagonal.

4.75e-08	-1.9751e-24	7.3536e-25	5.915e-23	3.2093e-07	5.8696e-24
-7.1302e-25	4.75e-08	2.8866e-25	-3.2093e-07	-5.38e-24	-3.2725e-23
7.9012e-26	-6.3991e-25	2.099e-08	1.9073e-23	5.3384e-25	-6.4867e-40
1.3724e-23	-3.2093e-07	1.2799e-23	5.1863e-06	4.9412e-22	-3.8269e-24
3.2093e-07	7.6013e-24	1.2239e-23	6.8886e-22	5.1863e-06	-2.6025e-22
7.337e-24	-3.2395e-23	-1.571e-39	9.927e-23	-3.2531e-22	1.7073e-06

One way to have this compliance matrix diagonal (and thus having a decoupled plant at DC) is to use a cubic architecture with the center of the cube's coincident with frame $\{A\}$.

This control architecture can also give a dynamically decoupled plant if the Center of mass of the payload is also coincident with frame $\{A\}$.

Controller Design

The controller consists of:

• A pure integrator
• A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin

The obtained loop gains corresponding to the diagonal elements are shown in Figure fig:loop_gain_centralized_X.

  wc = 2*pi*30;
  Kx = diag(1./diag(abs(freqresp(Gx, wc)))) * wc/s * 1/(1 + s/3/wc);

<<plt-matlab>>

Loop Gain of the diagonal elements (png, pdf)

The controller $\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}$ is computed.

  K = inv(stewart.kinematics.J')*Kx;

Simulation

We specify the reference path to follow.

  <<init-ref>>

  controller = initializeController('type', 'ref-track-X');

We run the simulation and we save the results.

  set_param('stewart_platform_model', 'StopTime', '10')
  sim('stewart_platform_model')
  simout_X = simout;
  save('./mat/control_tracking.mat', 'simout_X', '-append');

Diagonal Control - Steady State Decoupling

<<sec:diagonal_control_SD>>

Control Architecture

The plant $\bm{G}$ is pre-multiply by $\bm{G}^{-1}(\omega = 0)$ such that the "shaped plant" $\bm{G}_0 = \bm{G} \bm{G}^{-1}(\omega = 0)$ is diagonal at low frequency.

Then a diagonal controller $\bm{K}_0$ is designed.

The control architecture is shown in Figure fig:centralized_reference_tracking_S.

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

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

    \begin{scope}[on background layer]
      \node[fit={(K.south west) (J.north east)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
      \node[above] at (Ktot.north) {$\bm{K}$};
    \end{scope}
  \end{tikzpicture}

Static Decoupling of the Plant

Plant Analysis

The plant is pre-multiplied by $\bm{G}^{-1}(\omega = 0)$. The diagonal and off-diagonal elements of the shaped plant are shown in Figure fig:plant_centralized_SD.

  G0 = G*inv(freqresp(G, 0));

<<plt-matlab>>

Diagonal and off-diagonal elements of the plant $\bm{G} \bm{G}^{-1}(\omega = 0)$ (png, pdf)

Controller Design

We have that: \[ \bm{G}^{-1}(\omega = 0) = \left(\frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0)\right)^{-1} = \left( \bm{J}^{-1} \frac{\delta\bm{\mathcal{L}}}{\delta\bm{\tau}}(\omega = 0) \right)^{-1} = \text{diag}(\mathcal{K}_1^{-1} \ \dots \ \mathcal{K}_6^{-1}) \bm{J} \]

And because:

• all the leg stiffness are equal
• the controller equal to a $\bm{K}_0(s) = k(s) \bm{I}_6$

We have that $\bm{K}_0(s)$ commutes with $\bm{G}^{-1}(\omega = 0)$ and thus the overall controller $\bm{K}$ is the same as the one obtain in section sec:diagonal_control_L.

Comparison

Obtained MIMO Controllers

<<plt-matlab>>

Comparison of the MIMO controller $\bm{K}$ for both centralized architectures (png, pdf)

Simulation Results

The position error $\bm{\epsilon}_\mathcal{X}$ for both centralized architecture are shown in Figure fig:centralized_control_comp_Ex.

Based on Figure fig:centralized_control_comp_Ex, we can see that:

• There is some tracking error $\epsilon_x$
• The errors $\epsilon_y$, $\epsilon_{R_x}$ and $\epsilon_{R_z}$ are quite negligible
• There is some error in the vertical position $\epsilon_z$. The frequency of the error $\epsilon_z$ is twice the frequency of the reference path $r_x$.
• There is some error $\epsilon_{R_y}$. This error is much lower when using the diagonal control in the frame of the leg instead of the cartesian frame.

<<plt-matlab>>

Comparison of the position error $\bm{\epsilon}_\mathcal{X}$ for both centralized controllers (png, pdf)

Conclusion

Both control architecture gives similar results even tough the control in the Leg's frame gives slightly better results.

The main differences between the control architectures used in sections sec:diagonal_control_L and sec:diagonal_control_X are summarized in Table tab:centralized_control_comp.

	Leg's Frame	Cartesian Frame	Static Decoupling
Plant Meaning	$\delta\mathcal{L}_i/\tau_i$	$\delta\mathcal{X}_i/\mathcal{F}_i$	No physical meaning
Obtained Decoupling	Decoupled at DC	Dynamical decoupling except few terms	Decoupled at DC
Diagonal Elements	Identical with all the Resonances	Different, resonances are cancel out	No Alternating poles and zeros
Mechanical Architecture	Architecture Independent	Better with Cubic Architecture
Robustness to Uncertainty	Good (only depends on $J$)	Good (only depends on $J$)	Bad (depends on the mass)

Comparison of the two centralized control architectures

These decoupling methods only uses the Jacobian matrix which only depends on the Stewart platform geometry. Thus, this method should be quite robust against parameter variation (e.g. the payload mass).

Hybrid Control Architecture - HAC-LAC with relative DVF

<<sec:hybrid>>

Introduction   ignore

Control Schematic

Let's consider the control schematic shown in Figure fig:hybrid_reference_tracking.

The first loop containing $\bm{K}_\mathcal{L}$ is a Decentralized Direct (Relative) Velocity Feedback.

A reference $\bm{r}_\mathcal{L}$ is computed using the inverse kinematics and corresponds to the wanted motion of each leg. The actual length of each leg $\bm{\mathcal{L}}$ is subtracted and then passed trough the controller $\bm{K}_\mathcal{L}$.

The controller is a diagonal controller with pure derivative action on the diagonal.

The effect of this loop is:

• it adds damping to the system (the force applied in each actuator is proportional to the relative velocity of the strut)
• it however does not go "against" the reference path $\bm{r}_\mathcal{X}$ thanks to the use of the inverse kinematics

Then, the second loop containing $\bm{K}_\mathcal{X}$ is designed based on the already damped plant (represented by the gray area). This second loop is responsible for the reference tracking.

  \begin{tikzpicture}
    \node[block={3.0cm}{3.0cm}] (G) {$G$};

    % Input and outputs coordinates
    \coordinate[] (outputX) at ($(G.south east)!0.25!(G.north east)$);
    \coordinate[] (outputL) at ($(G.south east)!0.75!(G.north east)$);

    \draw[->] (outputX) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{X}}$};
    \draw[->] (outputL) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{L}}$};

    % Blocs
    \node[addb, left= of G] (addF) {};
    \node[block, left=1.2 of addF] (Kx) {$\bm{K}_\mathcal{X}$};
    \node[addb={+}{}{}{}{-}, left= of Kx] (subx) {};

    \node[block, above= of addF] (Kl) {$\bm{K}_\mathcal{L}$};
    \node[addb={+}{}{}{-}{}, above= of Kl] (subl) {};

    \node[block, align=center, left= of subl] (invK) {Inverse\\Kinematics};

    % Connections and labels
    \draw[<-] (subx.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1.2, 0);
    \draw[->] ($(subx.west) + (-0.8, 0)$)node[branch]{} |- (invK.west);
    \draw[->] (invK.east) -- (subl.west) node[above left]{$\bm{r}_\mathcal{L}$};
    \draw[->] (subl.south) -- (Kl.north) node[above right]{$\bm{\epsilon}_\mathcal{L}$};
    \draw[->] (Kl.south) -- (addF.north);

    \draw[->] (subx.east) -- (Kx.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
    \draw[->] (Kx.east) node[above right]{$\bm{\tau}_\mathcal{X}$} -- (addF.west);
    \draw[->] (addF.east) -- (G.west) node[above left]{$\bm{\tau}$};

    \draw[->] ($(outputL.east) + (0.4, 0)$)node[branch](L){} |- (subl.east);
    \draw[->] ($(outputX.east) + (1.2, 0)$)node[branch]{} -- ++(0, -1.6) -| (subx.south);

    \begin{scope}[on background layer]
      \node[fit={(G.south-|Kl.west) (L|-subl.north)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
    \end{scope}
  \end{tikzpicture}

Hybrid Control Architecture

Initialize the Stewart platform

  <<init-sim>>

First Control Loop - $\bm{K}_\mathcal{L}$

Identification

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

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

Obtained Plant

The obtained plant is shown in Figure fig:hybrid_control_Kl_plant.

<<plt-matlab>>

Diagonal and off-diagonal elements of the plant for the design of $\bm{K}_\mathcal{L}$ (png, pdf)

Controller Design

We apply a decentralized (diagonal) direct velocity feedback. Thus, we apply a pure derivative action. In order to make the controller realizable, we add a low pass filter at high frequency. The gain of the controller is chosen such that the resonances are critically damped.

The obtain loop gain is shown in Figure fig:hybrid_control_Kl_loop_gain.

  Kl = 1e4 * s / (1 + s/2/pi/1e4) * eye(6);

<<plt-matlab>>

Obtain Loop Gain for the DVF control loop (png, pdf)

Second Control Loop - $\bm{K}_\mathcal{X}$

Identification

  Kx = tf(zeros(6));

  controller = initializeController('type', 'ref-track-hac-dvf');

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

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'input');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]

  %% Run the linearization
  G = linearize(mdl, io);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

Obtained Plant

We use the Jacobian matrix to apply forces in the cartesian frame.

  Gx = G*inv(stewart.kinematics.J');
  Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};

The obtained plant is shown in Figure fig:hybrid_control_Kx_plant.

<<plt-matlab>>

Diagonal and Off-diagonal elements of the plant for the design of $\bm{K}_\mathcal{L}$ (png, pdf)

Controller Design

The controller consists of:

• A pure integrator
• A Second integrator up to half the wanted bandwidth
• A Lead around the cross-over frequency
• A low pass filter with a cut-off equal to three times the wanted bandwidth

  wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]

  h = 3; % Lead parameter

  Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/3));

  % Normalization of the gain of have a loop gain of 1 at frequency wc
  Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));

<<plt-matlab>>

Obtained Loop Gain for the controller $\bm{K}_\mathcal{X}$ (png, pdf)

Then we include the Jacobian in the controller matrix.

  Kx = inv(stewart.kinematics.J')*Kx;

Simulations

We specify the reference path to follow.

  <<init-ref>>

We run the simulation and we save the results.

  set_param('stewart_platform_model', 'StopTime', '10')
  sim('stewart_platform_model')
  simout_H = simout;
  save('./mat/control_tracking.mat', 'simout_H', '-append');

The obtained position error is shown in Figure fig:hybrid_control_Ex.

<<plt-matlab>>

Obtained position error $\bm{\epsilon}_\mathcal{X}$ (png, pdf)

Conclusion

Comparison of all the methods

<<sec:comparison>>

Let's load the simulation results and compare them in Figure fig:reference_tracking_performance_comparison.

  load('./mat/control_tracking.mat', 'simout_D', 'simout_L', 'simout_X', 'simout_H');

<<plt-matlab>>

Comparison of the position errors for all the Control architecture used (png, pdf)

Compute the pose error of the Stewart Platform

<<sec:compute_pose_error>>

Let's note:

• $\{W\}$ the fixed measurement frame (corresponding to the metrology frame / the frame where the wanted displacement are expressed). The center of the frame if $O_W$
• $\{M\}$ is the frame fixed to the measured elements. $O_M$ is the point where the pose of the element is measured
• $\{R\}$ is a virtual frame corresponding to the wanted pose of the element. $O_R$ is the origin of this frame where the we want to position the point $O_M$ of the element.
• $\{V\}$ is a frame which its axes are aligned with $\{W\}$ and its origin $O_V$ is coincident with the $O_M$

Reference Position with respect to fixed frame {W}: ${}^WT_R$

  Dxr = 0;
  Dyr = 0;
  Dzr = 0.1;
  Rxr = pi;
  Ryr = 0;
  Rzr = 0;

Measured Position with respect to fixed frame {W}: ${}^WT_M$

  Dxm = 0;
  Dym = 0;
  Dzm = 0;
  Rxm = pi;
  Rym = 0;
  Rzm = 0;

We measure the position and orientation (pose) of the element represented by the frame $\{M\}$ with respect to frame $\{W\}$. Thus we can compute the Homogeneous transformation matrix ${}^WT_M$.

    %% Measured Pose
    WTm = zeros(4,4);

    WTm(1:3, 1:3) = [cos(Rzm) -sin(Rzm) 0;
         sin(Rzm)  cos(Rzm) 0;
         0        0         1] * ...
        [cos(Rym)  0        sin(Rym);
         0        1        0;
         -sin(Rym)  0        cos(Rym)] * ...
        [1        0        0;
         0        cos(Rxm) -sin(Rxm);
         0        sin(Rxm)  cos(Rxm)];
    WTm(1:4, 4) = [Dxm ; Dym ; Dzm; 1];

We can also compute the Homogeneous transformation matrix ${}^WT_R$ corresponding to the transformation required to go from fixed frame $\{W\}$ to the wanted frame $\{R\}$.

    %% Reference Pose
    WTr = zeros(4,4);

    WTr(1:3, 1:3) = [cos(Rzr) -sin(Rzr) 0;
         sin(Rzr)  cos(Rzr) 0;
         0        0         1] * ...
        [cos(Ryr)  0        sin(Ryr);
         0        1        0;
         -sin(Ryr)  0        cos(Ryr)] * ...
        [1        0        0;
         0        cos(Rxr) -sin(Rxr);
         0        sin(Rxr)  cos(Rxr)];
    WTr(1:4, 4) = [Dxr ; Dyr ; Dzr; 1];

We can also compute ${}^WT_V$.

  WTv = eye(4);
  WTv(1:3, 4) = WTm(1:3, 4);

Now we want to express ${}^MT_R$ which corresponds to the transformation required to go to wanted position expressed in the frame of the measured element. This homogeneous transformation can be computed from the previously computed matrices: \[ {}^MT_R = ({{}^WT_M}^{-1}) {}^WT_R \]

    %% Wanted pose expressed in a frame corresponding to the actual measured pose
    MTr = [WTm(1:3,1:3)', -WTm(1:3,1:3)'*WTm(1:3,4) ; 0 0 0 1]*WTr;

Now we want to express ${}^VT_R$: \[ {}^VT_R = ({{}^WT_V}^{-1}) {}^WT_R \]

    %% Wanted pose expressed in a frame coincident with the actual position but with no rotation
    VTr = [WTv(1:3,1:3)', -WTv(1:3,1:3)'*WTv(1:3,4) ; 0 0 0 1] * WTr;

    %% Extract Translations and Rotations from the Homogeneous Matrix
    T = MTr;
    Edx = T(1, 4);
    Edy = T(2, 4);
    Edz = T(3, 4);

    % The angles obtained are u-v-w Euler angles (rotations in the moving frame)
    Ery = atan2( T(1, 3),          sqrt(T(1, 1)^2 + T(1, 2)^2));
    Erx = atan2(-T(2, 3)/cos(Ery), T(3, 3)/cos(Ery));
    Erz = atan2(-T(1, 2)/cos(Ery), T(1, 1)/cos(Ery));