70 KiB
Decoupling Properties of the Cubic Architecture
- Introduction
- Static Properties
- Dynamical Decoupling
- Decentralized Control
- Cubic architecture with Cube's center above the top platform
- Conclusion
Introduction ignore
The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart cite:&stewart65_platf_with_six_degrees_freed. This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_example.
Typically, the struts have similar length to the cube's edges, as illustrated in Figure ref:fig:detail_kinematics_cubic_architecture_example. Practical implementations of such configurations can be observed in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp. It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure ref:fig:detail_kinematics_ulb_pz.
Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm: simplified kinematics relationships and dynamical analysis cite:&geng94_six_degree_of_freed_activ; uniform stiffness in all directions cite:&hanieh03_activ_stewar; uniform mobility cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2; and minimization of the cross coupling between actuators and sensors in different struts cite:&preumont07_six_axis_singl_stage_activ. This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system.
These properties are examined in this section to assess their relevance for the nano-hexapod. The mobility and stiffness properties of the cubic configuration are analyzed in Section ref:ssec:detail_kinematics_cubic_static. Dynamical decoupling is investigated in Section ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section ref:ssec:detail_kinematics_decentralized_control. Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design. The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
Static Properties
<<ssec:detail_kinematics_cubic_static>>
Stiffness matrix for the Cubic architecture
Consider the cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_schematic_full. The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s.
\begin{equation}\label{eq:detail_kinematics_cubic_s} \hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{equation}
Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation eqref:eq:detail_kinematics_cubic_vertices.
\begin{equation}\label{eq:detail_kinematics_cubic_vertices} ~{\bm{b}}_1 = ~{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad \tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}, \quad ~{\bm{b}}_3 = ~{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}, \quad ~{\bm{b}}_5 = ~{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix}
\end{equation}
In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation eqref:eq:detail_kinematics_cubic_stiffness. Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size $H_c$.
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness}
\bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix}
2 & 0 & 0 & 0 & 0 & 0
0 & 2 & 0 & 0 & 0 & 0
0 & 0 & 2 & 0 & 0 & 0
0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0
0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
\end{bmatrix}
\end{equation}
However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure ref:fig:detail_kinematics_cubic_schematic). In that case, the location of the top joints can be expressed by equation eqref:eq:detail_kinematics_cubic_edges, yet the computed stiffness matrix remains identical to Equation eqref:eq:detail_kinematics_cubic_stiffness.
\begin{equation}\label{eq:detail_kinematics_cubic_edges} \bm{b}_i = ~{\bm{b}}_i + α \hat{\bm{s}}_i
\end{equation}
The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is located at the center of the cube (shown by frame $\{C\}$). This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center). This specific location where the stiffness matrix is diagonal is referred to as the "Center of Stiffness" (analogous to the "Center of Mass" where the mass matrix is diagonal).
%% Analytical formula for Stiffness matrix of the Cubic Stewart platform
% Define symbolic variables
syms k Hc alpha H
assume(k > 0); % k is positive real
assume(Hc, 'real'); % Hc is real
assume(H, 'real'); % H is real
assume(alpha, 'real'); % alpha is real
% Define si matrix (edges of the cubes)
si = 1/sqrt(3)*[
[ sqrt(2), 0, 1]; ...
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
[-sqrt(2)/2, sqrt(3/2), 1]; ...
[ sqrt(2), 0, 1]; ...
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
[-sqrt(2)/2, sqrt(3/2), 1] ...
];
% Define ci matrix (vertices of the cubes)
ci = Hc * [
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
[-sqrt(2), 0, 0.5]; ...
[-sqrt(2), 0, 0.5] ...
];
% Apply vertical shift to ci
ci = ci + H * [0, 0, 1];
% Calculate bi vectors (Stewart platform top joints)
bi = ci + alpha * si;
% Initialize stiffness matrix
K = sym(zeros(6,6));
% Calculate elements of the stiffness matrix
for i = 1:6
% Extract vectors for each leg
s_i = si(i,:)';
b_i = bi(i,:)';
% Calculate cross product vector
cross_bs = cross(b_i, s_i);
% Build matrix blocks
K(1:3, 4:6) = K(1:3, 4:6) + s_i * cross_bs';
K(4:6, 1:3) = K(4:6, 1:3) + cross_bs * s_i';
K(1:3, 1:3) = K(1:3, 1:3) + s_i * s_i';
K(4:6, 4:6) = K(4:6, 4:6) + cross_bs * cross_bs';
end
% Scale by stiffness coefficient
K = k * K;
% Simplify the expressions
K = simplify(K);
% Display the analytical stiffness matrix
disp('Analytical Stiffness Matrix:');
pretty(K);Effect of having frame $\{B\}$ off-centered
When the reference frames $\{A\}$ and $\{B\}$ are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
Considering a vertical shift as shown in Figure ref:fig:detail_kinematics_cubic_schematic, the stiffness matrix transforms into that shown in Equation eqref:eq:detail_kinematics_cubic_stiffness_off_centered. Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered $\{B\}$ frame.
\begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered}
\bm{K}_{\{B\} ≠ \{C\}} = k \begin{bmatrix}
2 & 0 & 0 & 0 & -2 H & 0
0 & 2 & 0 & 2 H & 0 & 0
0 & 0 & 2 & 0 & 0 & 0
0 & 2 H & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0 & 0
-2 H & 0 & 0 & 0 & \frac{3}{2} H_c^2 + 2 H^2 & 0
0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\
\end{bmatrix}
\end{equation}
This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures. Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center. This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform. This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design.
Uniform Mobility
The translational mobility of the Stewart platform with constant orientation was analyzed. Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated. The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_translations.
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke. These findings suggest that the mobility pattern is more subtle than sometimes described in the literature cite:&mcinroy00_desig_contr_flexur_joint_hexap, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions. This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans.
The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibits greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction. Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
%% Cubic configuration
H = 100e-3; % Height of the Stewart platform [m]
Hc = 100e-3; % Size of the useful part of the cube [m]
FOc = 50e-3; % Center of the cube at the Stewart platform center
MO_B = -50e-3; % Position {B} with respect to {M} [m]
MHb = 0;
stewart_cubic = initializeStewartPlatform();
stewart_cubic = initializeFramesPositions(stewart_cubic, 'H', H, 'MO_B', MO_B);
stewart_cubic = generateCubicConfiguration(stewart_cubic, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', MHb);
stewart_cubic = computeJointsPose(stewart_cubic);
stewart_cubic = initializeStrutDynamics(stewart_cubic, 'k', 1);
stewart_cubic = computeJacobian(stewart_cubic);
stewart_cubic = initializeCylindricalPlatforms(stewart_cubic, 'Fpr', 150e-3, 'Mpr', 150e-3);
% Let's now define the actuator stroke.
L_max = 50e-6; % [m]
Dynamical Decoupling
<<ssec:detail_kinematics_cubic_dynamic>>
Introduction ignore
This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform. When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control.
Low frequency and High frequency coupling
As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation eqref:eq:detail_kinematics_transfer_function_cart. At low frequency, the behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j ω) \xrightarrow[ω → 0]{} \bm{K}-1
\end{equation}
In Section ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center. In this case, the "Cartesian" plant is decoupled at low frequency. At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$) eqref:eq:detail_kinematics_transfer_function_high_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j ω) \xrightarrow[ω → ∞]{} - ω^2 \bm{M}-1
\end{equation}
To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame.
To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure ref:fig:detail_kinematics_cubic_payload). Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames. When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com). Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok).
%% Input/Output definition of the Simscape model
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, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External metrology [m,rad]
% Prepare simulation
controller = initializeController('type', 'open-loop');
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'R', 100e-3);
%% Cubic Stewart platform with payload above the top platform - B frame at the CoM
H = 200e-3; % height of the Stewart platform [m]
MO_B = 50e-3; % Position {B} with respect to {M} [m]
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, ...
'Mpm', 1e-6, ... % Massless platform
'Fpm', 1e-6, ... % Massless platform
'Mph', 20e-3, ... % Thin platform
'Fph', 20e-3, ... % Thin platform
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', 1e-6, ... % Massless strut
'Msm', 1e-6, ... % Massless strut
'Fsh', stewart.geometry.l(1)/2, ...
'Msh', stewart.geometry.l(1)/2 ...
);
% Run the linearization
G_CoM = linearize(mdl, io)*inv(stewart.kinematics.J).';
G_CoM.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
G_CoM.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
%% Same geometry but B Frame at cube's center (CoK)
MO_B = -100e-3; % Position {B} with respect to {M} [m]
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, ...
'Mpm', 1e-6, ... % Massless platform
'Fpm', 1e-6, ... % Massless platform
'Mph', 20e-3, ... % Thin platform
'Fph', 20e-3, ... % Thin platform
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', 1e-6, ... % Massless strut
'Msm', 1e-6, ... % Massless strut
'Fsh', stewart.geometry.l(1)/2, ...
'Msh', stewart.geometry.l(1)/2 ...
);
% Run the linearization
G_CoK = linearize(mdl, io)*inv(stewart.kinematics.J.');
G_CoK.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
G_CoK.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
Payload's CoM at the cube's center
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components cite:&li01_simul_fault_vibrat_isolat_point. This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure ref:fig:detail_kinematics_cubic_centered_payload). This approach was physically implemented in several studies cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm, as shown in Figure ref:fig:detail_kinematics_uw_gsp. The resulting dynamics are indeed well-decoupled (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices. The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform. If a design similar to Figure ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
%% Cubic Stewart platform with payload above the top platform
H = 200e-3;
MO_B = -100e-3; % Position {B} with respect to {M} [m]
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, ...
'Mpm', 1e-6, ... % Massless platform
'Fpm', 1e-6, ... % Massless platform
'Mph', 20e-3, ... % Thin platform
'Fph', 20e-3, ... % Thin platform
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', 1e-6, ... % Massless strut
'Msm', 1e-6, ... % Massless strut
'Fsh', stewart.geometry.l(1)/2, ...
'Msh', stewart.geometry.l(1)/2 ...
);
% Sample at the Center of the cube
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'H_offset', -H/2-50e-3);
% Run the linearization
G_CoM_CoK = linearize(mdl, io)*inv(stewart.kinematics.J.');
G_CoM_CoK.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
G_CoM_CoK.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
Conclusion
The analysis of dynamical properties of the cubic architecture yields several important conclusions. Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames $\{A\}$ and $\{B\}$ are positioned at the cube's center. Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$. While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
Decentralized Control
<<ssec:detail_kinematics_decentralized_control>>
Introduction ignore
The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. Two sensor types integrated in the struts are considered: displacement sensors and force sensors. The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix.
The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms. The first employs a cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_payload. The second uses a non-cubic Stewart platform shown in Figure ref:fig:detail_kinematics_non_cubic_payload, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture.
%% Input/Output definition of the Simscape model
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, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Displacement sensors [m]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensor [N]
% Prepare simulation : Payload above the top platform
controller = initializeController('type', 'open-loop');
sample = initializeSample('type', 'cylindrical', 'm', 10, 'H', 100e-3, 'R', 100e-3);
%% Cubic Stewart platform
H = 200e-3; % height of the Stewart platform [m]
MO_B = 50e-3; % Position {B} with respect to {M} [m]
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, ...
'Mpm', 1e-6, ... % Massless platform
'Fpm', 1e-6, ... % Massless platform
'Mph', 20e-3, ... % Thin platform
'Fph', 20e-3, ... % Thin platform
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', 1e-6, ... % Massless strut
'Msm', 1e-6, ... % Massless strut
'Fsh', stewart.geometry.l(1)/2, ...
'Msh', stewart.geometry.l(1)/2 ...
);
% Run the linearization
G_cubic = linearize(mdl, io);
G_cubic.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
%% Non-Cubic Stewart platform
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
stewart = generateCubicConfiguration(stewart, 'Hc', H, 'FOc', H/2, 'FHa', 25e-3, 'MHb', 25e-3);
stewart = generateGeneralConfiguration(stewart, 'FH', 25e-3, 'FR', 250e-3, 'MH', 25e-3, 'MR', 250e-3, ...
'FTh', [-22, 22, 120-22, 120+22, 240-22, 240+22]*(pi/180), ...
'MTh', [-60+22, 60-22, 60+22, 180-22, 180+22, -60-22]*(pi/180));
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, 'k', 1e6, 'c', 1e1);
stewart = initializeJointDynamics(stewart, 'type_F', '2dof', 'type_M', '3dof');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeCylindricalPlatforms(stewart, ...
'Mpm', 1e-6, ... % Massless platform
'Fpm', 1e-6, ... % Massless platform
'Mph', 20e-3, ... % Thin platform
'Fph', 20e-3, ... % Thin platform
'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)), ...
'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)));
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', 1e-6, ... % Massless strut
'Msm', 1e-6, ... % Massless strut
'Fsh', stewart.geometry.l(1)/2, ...
'Msh', stewart.geometry.l(1)/2 ...
);
% Run the linearization
G_non_cubic = linearize(mdl, io);
G_non_cubic.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_non_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};Relative Displacement Sensors
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure ref:fig:detail_kinematics_decentralized_dL. As anticipated from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$ eqref:eq:detail_kinematics_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency. At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal.
No significant advantage is evident for the cubic architecture (Figure ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure ref:fig:detail_kinematics_non_cubic_decentralized_dL). The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes.
Force Sensors
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms. The results are presented in Figure ref:fig:detail_kinematics_decentralized_fn. The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
Conclusion
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature. Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
Cubic architecture with Cube's center above the top platform
<<ssec:detail_kinematics_cubic_design>>
Introduction ignore
As demonstrated in Section ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. As shown in Section ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center. However, the $\{B\}$ frame is typically positioned above the top platform where forces are applied and displacements are measured.
This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration.
Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the center of mass relative to the mobile platform (coincident with the cube's center).
Depending on the cube's size $H_c$ in relation to $H$ and $H_{CoM}$, different designs emerge. In the following examples, $H = 100\,mm$ and $H_{CoM} = 20\,mm$.
%% Cubic configurations with center of the cube above the top platform
H = 100e-3; % height of the Stewart platform [m]
MO_B = 20e-3; % Position {B} with respect to {M} [m]
FOc = H + MO_B; % Center of the cube with respect to {F}Small cube
When the cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$ eqref:eq:detail_kinematics_cube_small, the resulting design is shown in Figure ref:fig:detail_kinematics_cubic_above_small.
\begin{equation}\label{eq:detail_kinematics_cube_small} H_c < 2 HCoM
\end{equation}
This configuration is similar to that described in cite:&furutani04_nanom_cuttin_machin_using_stewar, although they do not explicitly identify it as a cubic configuration. Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other.
This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness.
%% Small cube
Hc = 2*MO_B; % Size of the useful part of the cube [m]
stewart_small = initializeStewartPlatform();
stewart_small = initializeFramesPositions(stewart_small, 'H', H, 'MO_B', MO_B);
stewart_small = generateCubicConfiguration(stewart_small, 'Hc', Hc, 'FOc', FOc, 'FHa', 5e-3, 'MHb', 5e-3);
stewart_small = computeJointsPose(stewart_small);
stewart_small = initializeStrutDynamics(stewart_small, 'k', 1);
stewart_small = computeJacobian(stewart_small);
stewart_small = initializeCylindricalPlatforms(stewart_small, 'Fpr', 1.1*max(vecnorm(stewart_small.platform_F.Fa)), 'Mpr', 1.1*max(vecnorm(stewart_small.platform_M.Mb)));
Medium sized cube
Increasing the cube's size such that eqref:eq:detail_kinematics_cube_medium is verified produces an architecture with intersecting struts (Figure ref:fig:detail_kinematics_cubic_above_medium).
\begin{equation}\label{eq:detail_kinematics_cube_medium} 2 HCoM < H_c < 2 (HCoM + H)
\end{equation}
This configuration resembles the design proposed in cite:&yang19_dynam_model_decoup_contr_flexib (Figure ref:fig:detail_kinematics_yang19), although their design is not strictly cubic.
Large cube
When the cube's height exceeds twice the sum of the platform height and CoM height eqref:eq:detail_kinematics_cube_large, the architecture shown in Figure ref:fig:detail_kinematics_cubic_above_large is obtained.
\begin{equation}\label{eq:detail_kinematics_cube_large} 2 (HCoM + H) < H_c
\end{equation}
Platform size
%% Get the analytical formula for the location of the top and bottom joints
% Define symbolic variables
syms k Hc Hcom alpha H
assume(k > 0); % k is positive real
assume(Hcom > 0); % k is positive real
assume(Hc > 0); % Hc is real
assume(H > 0); % H is real
assume(alpha, 'real'); % alpha is real
% Define si matrix (edges of the cubes)
si = 1/sqrt(3)*[
[ sqrt(2), 0, 1]; ...
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
[-sqrt(2)/2, sqrt(3/2), 1]; ...
[ sqrt(2), 0, 1]; ...
[-sqrt(2)/2, -sqrt(3/2), 1]; ...
[-sqrt(2)/2, sqrt(3/2), 1] ...
];
% Define ci matrix (vertices of the cubes)
ci = Hc * [
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), -sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
[1/sqrt(2), sqrt(3)/sqrt(2), 0.5]; ...
[-sqrt(2), 0, 0.5]; ...
[-sqrt(2), 0, 0.5] ...
];
% Apply vertical shift to ci
ci = ci + (H + Hcom) * [0, 0, 1];
% Calculate bi vectors (Stewart platform top joints)
bi = ci + alpha * si;
% Extract the z-component value from the first row of ci
% (all rows have the same z-component)
ci_z = ci(1, 3);
% The z-component of si is 1 for all rows
si_z = si(1, 3);
alpha_for_0 = solve(ci_z + alpha * si_z == 0, alpha);
alpha_for_H = solve(ci_z + alpha * si_z == H, alpha);
% Verify the results
% Substitute alpha values and check the resulting bi_z values
bi_z_0 = ci + alpha_for_0 * si;
disp('Radius for fixed base:');
simplify(sqrt(bi_z_0(1,1).^2 + bi_z_0(1,2).^2))
bi_z_H = ci + alpha_for_H * si;
disp('Radius for mobile platform:');
simplify(sqrt(bi_z_H(1,1).^2 + bi_z_H(1,2).^2))For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation eqref:eq:detail_kinematics_cube_joints.
\begin{subequations}\label{eq:detail_kinematics_cube_joints}
\begin{align} R_{b_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 H_{CoM}^2} \label{eq:detail_kinematics_cube_top_joints} \\ R_{a_i} &= \sqrt{\frac{3}{2} H_c^2 + 2 (H_{CoM} + H)^2} \label{eq:detail_kinematics_cube_bot_joints} \end{align}\end{subequations}
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined based on rotational stiffness requirements. Subsequently, using Equation eqref:eq:detail_kinematics_cube_joints, the dimensions of the top and bottom platforms can be calculated.
Conclusion
The analysis of the cubic architecture for Stewart platforms yielded several important findings. While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center. Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space. This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected. While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described. Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center. However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform. Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.