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Cubic configuration for the Stewart Platform

Table of Contents

The Cubic configuration for the Stewart platform was first proposed in 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 (preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm).

According to 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:

1 Stiffness Matrix for the Cubic configuration

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.

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.

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

We create a cubic Stewart platform (figure 1) 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);

cubic_conf_centered_J_center.png

Figure 1: Cubic Stewart platform centered with the cube center - Jacobian estimated at the cube center (png, pdf)

Table 1: Stiffness Matrix
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

1.2 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 2). 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);

cubic_conf_centered_J_not_center.png

Figure 2: Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center (png, pdf)

Table 2: Stiffness Matrix
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

1.3 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 3). 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);

cubic_conf_not_centered_J_center.png

Figure 3: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center (png, pdf)

Table 3: Stiffness Matrix
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

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

1.4 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);

cubic_conf_not_centered_J_stewart_center.png

Figure 4: Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center (png, pdf)

Table 4: Stiffness Matrix
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

1.5 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.

2 Configuration with the Cube’s center above the mobile 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.

We saw in section 1 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.

2.1 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 5
  • Medium Cube Size in Figure 6
  • Large Cube Size in Figure 7

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}

stewart_cubic_conf_type_1.png

Figure 5: Cubic Configuration for the Stewart Platform - Small Cube Size (png, pdf)

Table 5: Stiffness Matrix
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 
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}

stewart_cubic_conf_type_2.png

Figure 6: Cubic Configuration for the Stewart Platform - Medium Cube Size (png, pdf)

Table 6: Stiffness Matrix
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 
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_cubic_conf_type_3.png

Figure 7: Cubic Configuration for the Stewart Platform - Large Cube Size (png, pdf)

Table 7: Stiffness Matrix
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

2.2 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: \[ r \approx \sqrt{2} H \]

2.3 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 furutani04_nanom_cuttin_machin_using_stewar
Medium yang19_dynam_model_decoup_contr_flexib
Large  

3 Cubic 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.

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.

3.1 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 8).

stiffness_cube_size.png

Figure 8: \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) function of the size of the cube

3.2 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.

4 Dynamic Coupling in the Cartesian Frame

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.

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 9 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.

local_to_cartesian_coordinates.png

Figure 9: 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.

4.1 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 10.

stewart_cubic_conf_decouple_dynamics.png

Figure 10: 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 9), 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 11.

stewart_cubic_decoupled_dynamics_cartesian.png

Figure 11: Dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (png, pdf)

It is interesting to note here that the system shown in Figure 12 also yield a decoupled system (explained in section 1.3.3 in li01_simul_fault_vibrat_isolat_point).

local_to_cartesian_coordinates_bis.png

Figure 12: 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.

4.2 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 13.

stewart_cubic_conf_mass_above.png

Figure 13: 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 14.

stewart_conf_coupling_mass_matrix.png

Figure 14: 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 \(\{B\}\)).

4.3 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}.

5 Dynamic Coupling between actuators and sensors of each strut

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.

From 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).

5.1 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);

stewart_architecture_coupling_struts_cubic.png

Figure 15: 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 16) and to the force sensors \(\tau_{m,i}\) (Figure 17).

coupling_struts_relative_sensor_cubic.png

Figure 16: Dynamics from the force actuators to the relative motion sensors (png, pdf)

coupling_struts_force_sensor_cubic.png

Figure 17: Dynamics from the force actuators to the force sensors (png, pdf)

5.2 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');

stewart_architecture_coupling_struts_non_cubic.png

Figure 18: 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 19) and to the force sensors \(\tau_{m,i}\) (Figure 20).

coupling_struts_relative_sensor_non_cubic.png

Figure 19: Dynamics from the force actuators to the relative motion sensors (png, pdf)

coupling_struts_force_sensor_non_cubic.png

Figure 20: Dynamics from the force actuators to the force sensors (png, pdf)

5.3 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.

6 Functions

6.1 generateCubicConfiguration: Generate a Cubic Configuration

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-definition.png

Figure 21: 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

Author: Dehaeze Thomas

Created: 2020-03-12 jeu. 18:06