stewart-simscape/org/cubic-configuration.org

Cubic configuration for the Stewart Platform

• Introduction
• Stiffness Matrix for the Cubic configuration
  • Introduction
  • Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center
  • Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
  • Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
  • Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
  • Conclusion
• Configuration with the Cube's center above the mobile platform
  • Introduction
  • Having Cube's center above the top platform
  • Size of the platforms
  • Conclusion
• Cubic size analysis
  • Introduction
  • Analysis
  • Conclusion
• Dynamic Coupling in the Cartesian Frame
  • Introduction
  • Cube's center at the Center of Mass of the mobile platform
  • Cube's center not coincident with the Mass of the Mobile platform
  • Conclusion
• Dynamic Coupling between actuators and sensors of each strut
  • Introduction
  • Coupling between the actuators and sensors - Cubic Architecture
  • Coupling between the actuators and sensors - Non-Cubic Architecture
  • Conclusion
• Functions
  • generateCubicConfiguration: Generate a Cubic Configuration
    • Function description
    • Documentation
    • Optional Parameters
    • Check the stewart structure elements
    • Position of the Cube
    • Compute the pose
    • Populate the stewart structure
• Bibliography

Introduction   ignore

The Cubic configuration for the Stewart platform was first proposed in cite:geng94_six_degree_of_freed_activ. This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube. This configuration is now widely used (cite:preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).

According to cite:preumont07_six_axis_singl_stage_activ, the cubic configuration offers the following advantages:

This topology provides a uniform control capability and a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other).

In this document, the cubic architecture is analyzed:

• In section sec:cubic_conf_stiffness, we study the uniform stiffness of such configuration and we find the conditions to obtain a diagonal stiffness matrix
• In section sec:cubic_conf_above_platform, we find cubic configurations where the cube's center is located above the mobile platform
• In section sec:cubic_conf_size_analysis, we study the effect of the cube's size on the Stewart platform properties
• In section sec:cubic_conf_coupling_cartesian, we study the dynamics of the cubic configuration in the cartesian frame
• In section sec:cubic_conf_coupling_struts, we study the dynamic cross-coupling of the cubic configuration from actuators to sensors of each strut
• In section sec:functions, function related to the cubic configuration are defined. To generate and study the Stewart platform with a Cubic configuration, the Matlab function generateCubicConfiguration is used (described here).

Stiffness Matrix for the Cubic configuration

<<sec:cubic_conf_stiffness>>

The Matlab script corresponding to this section is accessible here.

To run the script, open the Simulink Project, and type run cubic_conf_stiffness.m.

Introduction   ignore

First, we have to understand what is the physical meaning of the Stiffness matrix $\bm{K}$.

The Stiffness matrix links forces $\bm{f}$ and torques $\bm{n}$ applied on the mobile platform at $\{B\}$ to the displacement $\Delta\bm{\mathcal{X}}$ of the mobile platform represented by $\{B\}$ with respect to $\{A\}$: \[ \bm{\mathcal{F}} = \bm{K} \Delta\bm{\mathcal{X}} \]

with:

• $\bm{\mathcal{F}} = [\bm{f}\ \bm{n}]^{T}$
• $\Delta\bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_{x}, \delta \theta_{y}, \delta \theta_{z}]^{T}$

If the stiffness matrix is inversible, its inverse is the compliance matrix: $\bm{C} = \bm{K}^{-1$ and: \[ \Delta \bm{\mathcal{X}} = C \bm{\mathcal{F}} \]

Thus, if the stiffness matrix is diagonal, the compliance matrix is also diagonal and a force (resp. torque) $\bm{\mathcal{F}}_i$ applied on the mobile platform at $\{B\}$ will induce a pure translation (resp. rotation) of the mobile platform represented by $\{B\}$ with respect to $\{A\}$.

One has to note that this is only valid in a static way.

We here study what makes the Stiffness matrix diagonal when using a cubic configuration.

Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center

We create a cubic Stewart platform (figure fig:cubic_conf_centered_J_center) in such a way that the center of the cube (black star) is located at the center of the Stewart platform (blue dot). The Jacobian matrix is estimated at the location of the center of the cube.

  H = 100e-3;     % height of the Stewart platform [m]
  MO_B = -H/2;     % Position {B} with respect to {M} [m]
  Hc = H;         % Size of the useful part of the cube [m]
  FOc = H + MO_B; % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
  stewart = computeJacobian(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);

<<plt-matlab>>

Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (png, pdf)

2	0	-2.5e-16	0	2.1e-17	0
0	2	0	-7.8e-19	0	0
-2.5e-16	0	2	-2.4e-18	-1.4e-17	0
0	-7.8e-19	-2.4e-18	0.015	-4.3e-19	1.7e-18
1.8e-17	0	-1.1e-17	0	0.015	0
6.6e-18	-3.3e-18	0	1.7e-18	0	0.06

Stiffness Matrix

Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center

We create a cubic Stewart platform with center of the cube located at the center of the Stewart platform (figure fig:cubic_conf_centered_J_not_center). The Jacobian matrix is not estimated at the location of the center of the cube.

  H    = 100e-3; % height of the Stewart platform [m]
  MO_B = 20e-3;  % Position {B} with respect to {M} [m]
  Hc   = H;      % Size of the useful part of the cube [m]
  FOc  = H/2;    % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
  stewart = computeJacobian(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);

<<plt-matlab>>

Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (png, pdf)

2	0	-2.5e-16	0	-0.14	0
0	2	0	0.14	0	0
-2.5e-16	0	2	-5.3e-19	0	0
0	0.14	-5.3e-19	0.025	0	8.7e-19
-0.14	0	2.6e-18	1.6e-19	0.025	0
6.6e-18	-3.3e-18	0	8.9e-19	0	0.06

Stiffness Matrix

Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center

Here, the "center" of the Stewart platform is not at the cube center (figure fig:cubic_conf_not_centered_J_center). The Jacobian is estimated at the cube center.

  H    = 80e-3; % height of the Stewart platform [m]
  MO_B = -30e-3;  % Position {B} with respect to {M} [m]
  Hc   = 100e-3;      % Size of the useful part of the cube [m]
  FOc  = H + MO_B;    % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
  stewart = computeJacobian(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 175e-3, 'Mpr', 150e-3);

<<plt-matlab>>

Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (png, pdf)

2	0	-1.7e-16	0	4.9e-17	0
0	2	0	-2.2e-17	0	2.8e-17
-1.7e-16	0	2	1.1e-18	-1.4e-17	1.4e-17
0	-2.2e-17	1.1e-18	0.015	0	3.5e-18
4.4e-17	0	-1.4e-17	-5.7e-20	0.015	-8.7e-19
6.6e-18	2.5e-17	0	3.5e-18	-8.7e-19	0.06

Stiffness Matrix

We obtain $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$, but the Stiffness matrix is not diagonal.

Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center

Here, the "center" of the Stewart platform is not at the cube center. The Jacobian is estimated at the center of the Stewart platform.

The center of the cube is at $z = 110$. The Stewart platform is from $z = H_0 = 75$ to $z = H_0 + H_{tot} = 175$. The center height of the Stewart platform is then at $z = \frac{175-75}{2} = 50$. The center of the cube from the top platform is at $z = 110 - 175 = -65$.

  H    = 100e-3; % height of the Stewart platform [m]
  MO_B = -H/2;  % Position {B} with respect to {M} [m]
  Hc   = 1.5*H;      % Size of the useful part of the cube [m]
  FOc  = H/2 + 10e-3;    % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 0, 'MHb', 0);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', ones(6,1));
  stewart = computeJacobian(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 215e-3, 'Mpr', 195e-3);

<<plt-matlab>>

Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (png, pdf)

2	0	1.5e-16	0	0.02	0
0	2	0	-0.02	0	0
1.5e-16	0	2	-3e-18	-2.8e-17	0
0	-0.02	-3e-18	0.034	-8.7e-19	5.2e-18
0.02	0	-2.2e-17	-4.4e-19	0.034	0
5.9e-18	-7.5e-18	0	3.5e-18	0	0.14

Stiffness Matrix

Conclusion

Here are the conclusion about the Stiffness matrix for the Cubic configuration:

• The cubic configuration permits to have $k_x = k_y = k_z$ and $k_{\theta_x} = k_{\theta_y}$
• The stiffness matrix $K$ is diagonal for the cubic configuration if the Jacobian is estimated at the cube center.

Configuration with the Cube's center above the mobile platform

<<sec:cubic_conf_above_platform>>

The Matlab script corresponding to this section is accessible here.

To run the script, open the Simulink Project, and type run cubic_conf_above_platform.m.

Introduction   ignore

We saw in section sec:cubic_conf_stiffness that in order to have a diagonal stiffness matrix, we need the cube's center to be located at frames $\{A\}$ and $\{B\}$. Or, we usually want to have $\{A\}$ and $\{B\}$ located above the top platform where forces are applied and where displacements are expressed.

We here see if the cubic configuration can provide a diagonal stiffness matrix when $\{A\}$ and $\{B\}$ are above the mobile platform.

Having Cube's center above the top platform

Let's say we want to have a diagonal stiffness matrix when $\{A\}$ and $\{B\}$ are located above the top platform. Thus, we want the cube's center to be located above the top center.

Let's fix the Height of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$:

  H    = 100e-3; % height of the Stewart platform [m]
  MO_B = 20e-3;  % Position {B} with respect to {M} [m]

We find the several Cubic configuration for the Stewart platform where the center of the cube is located at frame $\{A\}$. The differences between the configuration are the cube's size:

• Small Cube Size in Figure fig:stewart_cubic_conf_type_1
• Medium Cube Size in Figure fig:stewart_cubic_conf_type_2
• Large Cube Size in Figure fig:stewart_cubic_conf_type_3

For each of the configuration, the Stiffness matrix is diagonal with $k_x = k_y = k_y = 2k$ with $k$ is the stiffness of each strut. However, the rotational stiffnesses are increasing with the cube's size but the required size of the platform is also increasing, so there is a trade-off here.

  Hc   = 0.4*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

<<plt-matlab>>

Cubic Configuration for the Stewart Platform - Small Cube Size (png, pdf)

2	0	-2.8e-16	0	2.4e-17	0
0	2	0	-2.3e-17	0	0
-2.8e-16	0	2	-2.1e-19	0	0
0	-2.3e-17	-2.1e-19	0.0024	-5.4e-20	6.5e-19
2.4e-17	0	4.9e-19	-2.3e-20	0.0024	0
-1.2e-18	1.1e-18	0	6.2e-19	0	0.0096

Stiffness Matrix

  Hc   = 1.5*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

<<plt-matlab>>

Cubic Configuration for the Stewart Platform - Medium Cube Size (png, pdf)

2	0	-1.9e-16	0	5.6e-17	0
0	2	0	-7.6e-17	0	0
-1.9e-16	0	2	2.5e-18	2.8e-17	0
0	-7.6e-17	2.5e-18	0.034	8.7e-19	8.7e-18
5.7e-17	0	3.2e-17	2.9e-19	0.034	0
-1e-18	-1.3e-17	5.6e-17	8.4e-18	0	0.14

Stiffness Matrix

  Hc   = 2.5*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

<<plt-matlab>>

Cubic Configuration for the Stewart Platform - Large Cube Size (png, pdf)

2	0	-3e-16	0	-8.3e-17	0
0	2	0	-2.2e-17	0	5.6e-17
-3e-16	0	2	-9.3e-19	-2.8e-17	0
0	-2.2e-17	-9.3e-19	0.094	0	2.1e-17
-8e-17	0	-3e-17	-6.1e-19	0.094	0
-6.2e-18	7.2e-17	5.6e-17	2.3e-17	0	0.37

Stiffness Matrix

Size of the platforms

The minimum size of the platforms depends on the cube's size and the height between the platform and the cube's center.

Let's denote:

• $H$ the height between the cube's center and the considered platform
• $D$ the size of the cube's edges

Let's denote by $a$ and $b$ the points of both ends of one of the cube's edge.

Initially, we have:

\begin{align} a &= \frac{D}{2} \begin{bmatrix}-1 \\ -1 \\ 1\end{bmatrix} \\ b &= \frac{D}{2} \begin{bmatrix} 1 \\ -1 \\ 1\end{bmatrix} \end{align}

We rotate the cube around its center (origin of the rotated frame) such that one of its diagonal is vertical. \[ R = \begin{bmatrix} \frac{2}{\sqrt{6}} & 0 & \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{6}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{6}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{bmatrix} \]

After rotation, the points $a$ and $b$ become:

\begin{align} a &= \frac{D}{2} \begin{bmatrix}-\frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ -\frac{1}{\sqrt{3}}\end{bmatrix} \\ b &= \frac{D}{2} \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ -\sqrt{2} \\ \frac{1}{\sqrt{3}}\end{bmatrix} \end{align}

Points $a$ and $b$ define a vector $u = b - a$ that gives the orientation of one of the Stewart platform strut: \[ u = \frac{D}{\sqrt{3}} \begin{bmatrix} -\sqrt{2} \\ 0 \\ -1\end{bmatrix} \]

Then we want to find the intersection between the line that defines the strut with the plane defined by the height $H$ from the cube's center. To do so, we first find $g$ such that: \[ a_z + g u_z = -H \] We obtain:

\begin{align} g &= - \frac{H + a_z}{u_z} \\ &= \sqrt{3} \frac{H}{D} - \frac{1}{2} \end{align}

Then, the intersection point $P$ is given by:

\begin{align} P &= a + g u \\ &= \begin{bmatrix} H \sqrt{2} \\ D \frac{1}{\sqrt{2}} \\ H \end{bmatrix} \end{align}

Finally, the circle can contains the intersection point has a radius $r$:

\begin{align} r &= \sqrt{P_x^2 + P_y^2} \\ &= \sqrt{2 H^2 + \frac{1}{2}D^2} \end{align}

By symmetry, we can show that all the other intersection points will also be on the circle with a radius $r$.

For a small cube: \[ r \approx \sqrt{2} H \]

Conclusion

We found that we can have a diagonal stiffness matrix using the cubic architecture when $\{A\}$ and $\{B\}$ are located above the top platform. Depending on the cube's size, we obtain 3 different configurations.

Cube's Size	Paper with the corresponding cubic architecture
Small	cite:furutani04_nanom_cuttin_machin_using_stewar
Medium	cite:yang19_dynam_model_decoup_contr_flexib
Large

Cubic size analysis

<<sec:cubic_conf_size_analysis>>

The Matlab script corresponding to this section is accessible here.

To run the script, open the Simulink Project, and type run cubic_conf_size_analysis.m.

Introduction   ignore

We here study the effect of the size of the cube used for the Stewart Cubic configuration.

We fix the height of the Stewart platform, the center of the cube is at the center of the Stewart platform and the frames $\{A\}$ and $\{B\}$ are also taken at the center of the cube.

We only vary the size of the cube.

Analysis

We initialize the wanted cube's size.

  Hcs = 1e-3*[250:20:350]; % Heights for the Cube [m]
  Ks = zeros(6, 6, length(Hcs));

The height of the Stewart platform is fixed:

  H    = 100e-3; % height of the Stewart platform [m]

The frames $\{A\}$ and $\{B\}$ are positioned at the Stewart platform center as well as the cube's center:

  MO_B = -50e-3;  % Position {B} with respect to {M} [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

We find that for all the cube's size, $k_x = k_y = k_z = k$ where $k$ is the strut stiffness. We also find that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ are varying with the cube's size (figure fig:stiffness_cube_size).

  <<plt-matlab>>

$k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ function of the size of the cube

Conclusion

We observe that $k_{\theta_x} = k_{\theta_y}$ and $k_{\theta_z}$ increase linearly with the cube size.

In order to maximize the rotational stiffness of the Stewart platform, the size of the cube should be the highest possible.

Dynamic Coupling in the Cartesian Frame

<<sec:cubic_conf_coupling_cartesian>>

The Matlab script corresponding to this section is accessible here.

To run the script, open the Simulink Project, and type run cubic_conf_coupling_cartesian.m.

Introduction   ignore

In this section, we study the dynamics of the platform in the cartesian frame.

We here suppose that there is one relative motion sensor in each strut ($\delta\bm{\mathcal{L}}$ is measured) and we would like to control the position of the top platform pose $\delta \bm{\mathcal{X}}$.

Thanks to the Jacobian matrix, we can use the "architecture" shown in Figure fig:local_to_cartesian_coordinates to obtain the dynamics of the system from forces/torques applied by the actuators on the top platform to translations/rotations of the top platform.

  \begin{tikzpicture}
    \node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$};
    \node[block, right= of Jt] (G) {$\bm{G}$};
    \node[block, right= of G] (J) {$\bm{J}^{-1}$};

    \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}$};
    \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
    \draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
    \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
  \end{tikzpicture}

From Strut coordinate to Cartesian coordinate using the Jacobian matrix

We here study the dynamics from $\bm{\mathcal{F}}$ to $\delta\bm{\mathcal{X}}$.

One has to note that when considering the static behavior: \[ \bm{G}(s = 0) = \begin{bmatrix} 1/k_1 & & 0 \\ & \ddots & 0 \\ 0 & & 1/k_6 \end{bmatrix}\]

And thus: \[ \frac{\delta\bm{\mathcal{X}}}{\bm{\mathcal{F}}}(s = 0) = \bm{J}^{-1} \bm{G}(s = 0) \bm{J}^{-T} = \bm{K}^{-1} = \bm{C} \]

We conclude that the static behavior of the platform depends on the stiffness matrix. For the cubic configuration, we have a diagonal stiffness matrix is the frames $\{A\}$ and $\{B\}$ are coincident with the cube's center.

Cube's center at the Center of Mass of the mobile platform

Let's create a Cubic Stewart Platform where the Center of Mass of the mobile platform is located at the center of the cube.

We define the size of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$.

  H    = 200e-3; % height of the Stewart platform [m]
  MO_B = -10e-3;  % Position {B} with respect to {M} [m]

Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.

  Hc   = 2.5*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with $\{A\}$ and $\{B\}$.

  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
                                                    'Mpm', 10, ...
                                                    'Mph', 20e-3, ...
                                                    'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));

And we set small mass for the struts.

  stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
  stewart = initializeInertialSensor(stewart);

No flexibility below the Stewart platform and no payload.

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');
  controller = initializeController('type', 'open-loop');

The obtain geometry is shown in figure fig:stewart_cubic_conf_decouple_dynamics.

<<plt-matlab>>

Geometry used for the simulations - The cube's center, the frames $\{A\}$ and $\{B\}$ and the Center of mass of the mobile platform are coincident (png, pdf)

We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.

  open('stewart_platform_model.slx')

  %% 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 = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};

Now, thanks to the Jacobian (Figure fig:local_to_cartesian_coordinates), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.

  Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
  Gc = inv(stewart.kinematics.J)*G*stewart.kinematics.J;
  Gc.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure fig:stewart_cubic_decoupled_dynamics_cartesian.

<<plt-matlab>>

Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ (png, pdf)

It is interesting to note here that the system shown in Figure fig:local_to_cartesian_coordinates_bis also yield a decoupled system (explained in section 1.3.3 in cite:li01_simul_fault_vibrat_isolat_point).

  \begin{tikzpicture}
    \node[block] (Jt) at (0, 0) {$\bm{J}$};
    \node[block, right= of Jt] (G) {$\bm{G}$};
    \node[block, right= of G] (J) {$\bm{J}^{-1}$};

    \draw[->] ($(Jt.west)+(-0.8, 0)$) -- (Jt.west);
    \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
    \draw[->] (G.east) -- (J.west) node[above left]{$\delta\bm{\mathcal{L}}$};
    \draw[->] (J.east) -- ++(0.8, 0) node[above left]{$\delta\bm{\mathcal{X}}$};
  \end{tikzpicture}

Alternative way to decouple the system

The dynamics is well decoupled at all frequencies.

We have the same dynamics for:

• $D_x/F_x$, $D_y/F_y$ and $D_z/F_z$
• $R_x/M_x$ and $D_y/F_y$

The Dynamics from $F_i$ to $D_i$ is just a 1-dof mass-spring-damper system.

This is because the Mass, Damping and Stiffness matrices are all diagonal.

Cube's center not coincident with the Mass of the Mobile platform

Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform.

  H    = 200e-3; % height of the Stewart platform [m]
  MO_B = -100e-3;  % Position {B} with respect to {M} [m]

Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.

  Hc   = 2.5*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);

However, the Center of Mass of the mobile platform is not located at the cube's center.

  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
                                                    'Mpm', 10, ...
                                                    'Mph', 20e-3, ...
                                                    'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));

And we set small mass for the struts.

  stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
  stewart = initializeInertialSensor(stewart);

No flexibility below the Stewart platform and no payload.

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');
  controller = initializeController('type', 'open-loop');

The obtain geometry is shown in figure fig:stewart_cubic_conf_mass_above.

<<plt-matlab>>

Geometry used for the simulations - The cube's center is coincident with the frames $\{A\}$ and $\{B\}$ but not with the Center of mass of the mobile platform (png, pdf)

We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.

  open('stewart_platform_model.slx')

  %% 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 = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};

And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.

  Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
  Gc.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
  Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure fig:stewart_conf_coupling_mass_matrix.

<<plt-matlab>>

Obtained Dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ (png, pdf)

The system is decoupled at low frequency (the Stiffness matrix being diagonal), but it is not decoupled at all frequencies.

This was expected as the mass matrix is not diagonal (the Center of Mass of the mobile platform not being coincident with the frame $\{B\}$).

Conclusion

Some conclusions can be drawn from the above analysis:

• Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube's center
• Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}.

Dynamic Coupling between actuators and sensors of each strut

<<sec:cubic_conf_coupling_struts>>

The Matlab script corresponding to this section is accessible here.

To run the script, open the Simulink Project, and type run cubic_conf_coupling_struts.m.

Introduction   ignore

From cite:preumont07_six_axis_singl_stage_activ, the cubic configuration "minimizes the cross-coupling amongst actuators and sensors of different legs (being orthogonal to each other)".

In this section, we wish to study such properties of the cubic architecture.

We will compare the transfer function from sensors to actuators in each strut for a cubic architecture and for a non-cubic architecture (where the struts are not orthogonal with each other).

Coupling between the actuators and sensors - Cubic Architecture

Let's generate a Cubic architecture where the cube's center and the frames $\{A\}$ and $\{B\}$ are coincident.

  H    = 200e-3; % height of the Stewart platform [m]
  MO_B = -10e-3;  % Position {B} with respect to {M} [m]
  Hc   = 2.5*H;    % Size of the useful part of the cube [m]
  FOc  = H + MO_B; % Center of the cube with respect to {F}

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
                                                    'Mpm', 10, ...
                                                    'Mph', 20e-3, ...
                                                    'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
  stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
  stewart = initializeInertialSensor(stewart);

No flexibility below the Stewart platform and no payload.

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');
  controller = initializeController('type', 'open-loop');

  disturbances = initializeDisturbances();
  references = initializeReferences(stewart);

<<plt-matlab>>

Geometry of the generated Stewart platform (png, pdf)

And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure fig:coupling_struts_relative_sensor_cubic) and to the force sensors $\tau_{m,i}$ (Figure fig:coupling_struts_force_sensor_cubic).

<<plt-matlab>>

Dynamics from the force actuators to the relative motion sensors (png, pdf)

<<plt-matlab>>

Dynamics from the force actuators to the force sensors (png, pdf)

Coupling between the actuators and sensors - Non-Cubic Architecture

Now we generate a Stewart platform which is not cubic but with approximately the same size as the previous cubic architecture.

  H    = 200e-3; % height of the Stewart platform [m]
  MO_B = -10e-3;  % Position {B} with respect to {M} [m]

  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
  stewart = generateGeneralConfiguration(stewart, 'FR', 250e-3, 'MR', 150e-3);
  stewart = computeJointsPose(stewart);
  stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
                                                    'Mpm', 10, ...
                                                    'Mph', 20e-3, ...
                                                    'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
  stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
  stewart = initializeInertialSensor(stewart);

No flexibility below the Stewart platform and no payload.

  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'none');
  controller = initializeController('type', 'open-loop');

<<plt-matlab>>

Geometry of the generated Stewart platform (png, pdf)

And we identify the dynamics from the actuator forces $\tau_{i}$ to the relative motion sensors $\delta \mathcal{L}_{i}$ (Figure fig:coupling_struts_relative_sensor_non_cubic) and to the force sensors $\tau_{m,i}$ (Figure fig:coupling_struts_force_sensor_non_cubic).

<<plt-matlab>>

Dynamics from the force actuators to the relative motion sensors (png, pdf)

<<plt-matlab>>

Dynamics from the force actuators to the force sensors (png, pdf)

Conclusion

The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control.

Functions

<<sec:functions>>

generateCubicConfiguration: Generate a Cubic Configuration

<<sec:generateCubicConfiguration>>

This Matlab function is accessible here.

Function description

  function [stewart] = generateCubicConfiguration(stewart, args)
  % generateCubicConfiguration - Generate a Cubic Configuration
  %
  % Syntax: [stewart] = generateCubicConfiguration(stewart, args)
  %
  % Inputs:
  %    - stewart - A structure with the following fields
  %        - geometry.H [1x1] - Total height of the platform [m]
  %    - args - Can have the following fields:
  %        - Hc  [1x1] - Height of the "useful" part of the cube [m]
  %        - FOc [1x1] - Height of the center of the cube with respect to {F} [m]
  %        - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m]
  %        - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
  %        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}

Documentation

Cubic Configuration

Optional Parameters

  arguments
      stewart
      args.Hc  (1,1) double {mustBeNumeric, mustBePositive} = 60e-3
      args.FOc (1,1) double {mustBeNumeric} = 50e-3
      args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
      args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3
  end

Check the stewart structure elements

  assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
  H = stewart.geometry.H;

Position of the Cube

We define the useful points of the cube with respect to the Cube's center. ${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}.

  sx = [ 2; -1; -1];
  sy = [ 0;  1; -1];
  sz = [ 1;  1;  1];

  R = [sx, sy, sz]./vecnorm([sx, sy, sz]);

  L = args.Hc*sqrt(3);

  Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc];

  CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg
  CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg

Compute the pose

We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$).

  CSi = (CCm - CCf)./vecnorm(CCm - CCf);

We now which to compute the position of the joints $a_{i}$ and $b_{i}$.

  Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;
  Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi;

Populate the stewart structure

  stewart.platform_F.Fa = Fa;
  stewart.platform_M.Mb = Mb;

Bibliography   ignore

bibliographystyle:unsrtnat bibliography:ref.bib