From 519580d31c6f83ed9eff60808dcd059f56a839b4 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 17 Feb 2021 15:17:19 +0100 Subject: [PATCH] Rename files --- index.html | 4220 +--------------------------------- svd-control.html | 4207 +++++++++++++++++++++++++++++++++ index.org => svd-control.org | 2 +- index.pdf => svd-control.pdf | Bin 3863625 -> 3863375 bytes 4 files changed, 4209 insertions(+), 4220 deletions(-) mode change 100644 => 120000 index.html create mode 100644 svd-control.html rename index.org => svd-control.org (99%) rename index.pdf => svd-control.pdf (99%) diff --git a/index.html b/index.html deleted file mode 100644 index 3aa56ca..0000000 --- a/index.html +++ /dev/null @@ -1,4219 +0,0 @@ - - - - - - -Diagonal control using the SVD and the Jacobian Matrix - - - - - - - - -
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-

Diagonal control using the SVD and the Jacobian Matrix

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-

Table of Contents

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- -
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-

This report is also available as a pdf.

-
- -

-In this document, the use of the Jacobian matrix and the Singular Value Decomposition to render a physical plant diagonal dominant is studied. -Then, a diagonal controller is used. -

- -

-These two methods are tested on two plants: -

- - -
-

1 Gravimeter - Simscape Model

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-

- -

-
- -
-

1.1 Introduction

-
-

-In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model: -

-
    -
  • Section 1.2: the model is described and its parameters are defined.
    • -
  • Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
    • -
  • Section 1.4: the plant is decoupled using the Jacobian matrices.
    • -
  • Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
    • -
  • Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
    • -
  • Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
    • -
  • Section 1.8: the obtained decoupled plants are compared
    • -
  • Section 1.9: the diagonal controller is developed
    • -
  • Section 1.10: the obtained closed-loop performances for the two methods are compared
    • -
  • Section 1.11: the robustness to a change of actuator position is evaluated
    • -
  • Section 1.12: the choice of the reference frame for the evaluation of the Jacobian is discussed
    • -
  • Section 1.13: the decoupling performances of SVD is evaluated for a low damped and an highly damped system
    • -
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1.2 Gravimeter Model - Parameters

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-

- -

- -

-The model of the gravimeter is schematically shown in Figure 1. -

- - -
-

gravimeter_model.png -

-

Figure 1: Model of the gravimeter

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- - -
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leg_model.png -

-

Figure 2: Model of the struts

-
- -

-The parameters used for the simulation are the following: -

-
-
l  = 1.0; % Length of the mass [m]
-h  = 1.7; % Height of the mass [m]
-
-la = l/2; % Position of Act. [m]
-ha = h/2; % Position of Act. [m]
-
-m = 400; % Mass [kg]
-I = 115; % Inertia [kg m^2]
-
-k = 15e3; % Actuator Stiffness [N/m]
-c = 2e1; % Actuator Damping [N/(m/s)]
-
-deq = 0.2; % Length of the actuators [m]
-
-g = 0; % Gravity [m/s2]
-
-
-
-
- -
-

1.3 System Identification

-
-

- -

- -
-
%% Name of the Simulink File
-mdl = 'gravimeter';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-G = linearize(mdl, io);
-G.InputName  = {'F1', 'F2', 'F3'};
-G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-
-
- -

-The inputs and outputs of the plant are shown in Figure 3. -

- -

-More precisely there are three inputs (the three actuator forces): -

-\begin{equation} - \bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix} -\end{equation} -

-And 4 outputs (the two 2-DoF accelerometers): -

-\begin{equation} - \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix} -\end{equation} - - -
-

gravimeter_plant_schematic.png -

-

Figure 3: Schematic of the gravimeter plant

-
- -

-We can check the poles of the plant: -

- - - --- - - - - - - - - - - - - - - - - - - - - - - - - - -
-0.12243+13.551i
-0.12243-13.551i
-0.05+8.6601i
-0.05-8.6601i
-0.0088785+3.6493i
-0.0088785-3.6493i
- -

-As expected, the plant as 6 states (2 translations + 1 rotation) -

-
-
size(G)
-
-
- -
-State-space model with 4 outputs, 3 inputs, and 6 states.
-
- - -

-The bode plot of all elements of the plant are shown in Figure 4. -

- - -
-

open_loop_tf.png -

-

Figure 4: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

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- -
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1.4 Decoupling using the Jacobian

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-

- -

- -

-Consider the control architecture shown in Figure 5. -

- -

-The Jacobian matrix \(J_{\tau}\) is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass: -

-\begin{equation} - \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix} -\end{equation} - -

-The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass: -

-\begin{equation} - \begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix} -\end{equation} - -

-We thus define a new plant as defined in Figure 5. -\[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \] -

- -

-\(\bm{G}_x(s)\) correspond to the \(3 \times 3\) transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5). -

- - -
-

gravimeter_decouple_jacobian.png -

-

Figure 5: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

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- -

-The Jacobian corresponding to the sensors and actuators are defined below: -

-
-
Ja = [1 0 -h/2
-      0 1  l/2
-      1 0  h/2
-      0 1  0];
-
-Jt = [1 0 -ha
-      0 1  la
-      0 1 -la];
-
-
- -

-And the plant \(\bm{G}_x\) is computed: -

-
-
Gx = pinv(Ja)*G*pinv(Jt');
-Gx.InputName  = {'Fx', 'Fy', 'Mz'};
-Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
-
-
- -
-size(Gx)
-State-space model with 3 outputs, 3 inputs, and 6 states.
-
- - -

-The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6. -

- -

-It is shown at the system is: -

-
    -
  • decoupled at high frequency thanks to a diagonal mass matrix (the Jacobian being evaluated at the center of mass of the payload)
    • -
  • coupled at low frequency due to the non-diagonal terms in the stiffness matrix, especially the term corresponding to a coupling between a force in the x direction to a rotation around z (due to the torque applied by the stiffness 1).
    • -
- -

-The choice of the frame in this the Jacobian is evaluated is discussed in Section 1.12. -

- - -
-

gravimeter_jacobian_plant.png -

-

Figure 6: Diagonal and off-diagonal elements of \(G_x\)

-
-
-
- -
-

1.5 Decoupling using the SVD

-
-

- -

- -

-In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required. -

- -

-Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). -

-
-
wc = 2*pi*10; % Decoupling frequency [rad/s]
-
-H1 = evalfr(G, j*wc);
-
-
- -

-The real approximation is computed as follows: -

-
-
D = pinv(real(H1'*H1));
-H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 1: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
0.0092-0.00390.0039
-0.00390.00480.00028
-0.0040.0038-0.0038
8.4e-090.00250.0025
- - -

-Now, the Singular Value Decomposition of \(H_1\) is performed: -\[ H_1 = U \Sigma V^H \] -

- -
-
[U,S,V] = svd(H1);
-
-
- - - - --- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 2: \(U\) matrix
-0.780.26-0.53-0.2
0.40.61-0.04-0.68
0.48-0.14-0.850.2
0.030.730.060.68
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 3: \(V\) matrix
-0.790.11-0.6
0.510.67-0.54
-0.350.730.59
- -

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7. -

- - -
-

gravimeter_decouple_svd.png -

-

Figure 7: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

-
- -

-The decoupled plant is then: -\[ \bm{G}_{SVD}(s) = U^{-1} \bm{G}(s) V^{-H} \] -

- -
-
Gsvd = inv(U)*G*inv(V');
-
-
- -
-size(Gsvd)
-State-space model with 4 outputs, 3 inputs, and 6 states.
-
- - -

-The 4th output (corresponding to the null singular value) is discarded, and we only keep the \(3 \times 3\) plant: -

-
-
Gsvd = Gsvd(1:3, 1:3);
-
-
- -

-The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8. -

- -
-

gravimeter_svd_plant.png -

-

Figure 8: Diagonal and off-diagonal elements of \(G_{svd}\)

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-
-
- -
-

1.6 Verification of the decoupling using the “Gershgorin Radii”

-
-

- -

- -

-The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\): -

- -

-The “Gershgorin Radii” of a matrix \(S\) is defined by: -\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \] -

- - -
-

gravimeter_gershgorin_radii.png -

-

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

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-
-
- -
-

1.7 Verification of the decoupling using the “Relative Gain Array”

-
-

- -

- -

-The relative gain array (RGA) is defined as: -

-\begin{equation} - \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T -\end{equation} -

-where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix. -

- -

-The obtained RGA elements are shown in Figure 10. -

- - -
-

gravimeter_rga.png -

-

Figure 10: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

-
- -

-The RGA-number is also a measure of diagonal dominance: -

-\begin{equation} - \text{RGA-number} = \| \Lambda(G) - I \|_\text{sum} -\end{equation} - - -
-

gravimeter_rga_num.png -

-

Figure 11: RGA-Number for the Gravimeter

-
-
-
- -
-

1.8 Obtained Decoupled Plants

-
-

- -

- -

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12. -

- - -
-

gravimeter_decoupled_plant_svd.png -

-

Figure 12: Decoupled Plant using SVD

-
- -

-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13. -

- - -
-

gravimeter_decoupled_plant_jacobian.png -

-

Figure 13: Gravimeter Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

-
-
-
- -
-

1.9 Diagonal Controller

-
-

- -The control diagram for the centralized control is shown in Figure 14. -

- -

-The controller \(K_c\) is “working” in an cartesian frame. -The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators. -

- - -
-

centralized_control_gravimeter.png -

-

Figure 14: Control Diagram for the Centralized control

-
- -

-The SVD control architecture is shown in Figure 15. -The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). -

- - -
-

svd_control_gravimeter.png -

-

Figure 15: Control Diagram for the SVD control

-
- - -

-We choose the controller to be a low pass filter: -\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] -

- -

-\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) -

- -
-
wc = 2*pi*10;  % Crossover Frequency [rad/s]
-w0 = 2*pi*0.1; % Controller Pole [rad/s]
-
-
- -
-
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
-L_cen = K_cen*Gx;
-
-
- -
-
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
-L_svd = K_svd*Gsvd;
-U_inv = inv(U);
-
-
- -

-The obtained diagonal elements of the loop gains are shown in Figure 16. -

- - -
-

gravimeter_comp_loop_gain_diagonal.png -

-

Figure 16: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

-
-
-
- -
-

1.10 Closed-Loop system Performances

-
-

- -

- -

-Now the system is identified again with additional inputs and outputs: -

-
    -
  • \(x\), \(y\) and \(R_z\) ground motion
    • -
  • \(x\), \(y\) and \(R_z\) acceleration of the payload.
    • -
- -
-
%% Name of the Simulink File
-mdl = 'gravimeter';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-G = linearize(mdl, io);
-G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
-G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-
-
- -

-The loop is closed using the developed controllers. -

-
-
G_cen = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
-G_svd = lft(G, -inv(V')*K_svd*U_inv(1:3, :));
-
-
- -

-Let’s first verify the stability of the closed-loop systems: -

-
-
isstable(G_cen)
-
-
- -
-ans =
-  logical
-   1
-
- - -
-
isstable(G_svd)
-
-
- -
-ans =
-  logical
-   1
-
- - -

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. -

- - -
-

gravimeter_platform_simscape_cl_transmissibility.png -

-

Figure 17: Obtained Transmissibility

-
- - -
-

gravimeter_cl_transmissibility_coupling.png -

-

Figure 18: Obtain coupling terms of the transmissibility matrix

-
-
-
- - -
-

1.11 Robustness to a change of actuator position

-
-

- -

- -

-Let say we change the position of the actuators: -

-
-
la = l/2*0.7; % Position of Act. [m]
-ha = h/2*0.7; % Position of Act. [m]
-
-
- -
-
%% Name of the Simulink File
-mdl = 'gravimeter';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-G = linearize(mdl, io);
-G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
-G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-
-
- -

-The loop is closed using the developed controllers. -

-
-
G_cen_b = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
-G_svd_b = lft(G, -inv(V')*K_svd*U_inv(1:3, :));
-
-
- -

-The new plant is computed, and the centralized and SVD control architectures are applied using the previously computed Jacobian matrices and \(U\) and \(V\) matrices. -

- -

-The closed-loop system are still stable in both cases, and the obtained transmissibility are equivalent as shown in Figure 19. -

- - -
-

gravimeter_transmissibility_offset_act.png -

-

Figure 19: Transmissibility for the initial CL system and when the position of actuators are changed

-
-
-
- -
-

1.12 Choice of the reference frame for Jacobian decoupling

-
-

- -

-

-If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobian at a point where the stiffness matrix is diagonal. -A displacement (resp. rotation) of the mass at this particular point should induce a pure force (resp. torque) on the same point due to stiffnesses in the system. -This can be verified by geometrical computations. -

- - -

-If we want to decouple the system at high frequency (determined by the mass matrix), we have tot compute the Jacobians at the Center of Mass of the suspended solid. -Similarly to the stiffness analysis, when considering only the inertia effects (neglecting the stiffnesses), a force (resp. torque) applied at this point (the center of mass) should induce a pure acceleration (resp. angular acceleration). -

- - -

-Ideally, we would like to have a decoupled mass matrix and stiffness matrix at the same time. -To do so, the actuators (springs) should be positioned such that the stiffness matrix is diagonal when evaluated at the CoM of the solid. -

-
-
-

1.12.1 Decoupling of the mass matrix

-
- -
-

gravimeter_model_M.png -

-

Figure 20: Choice of {O} such that the Mass Matrix is Diagonal

-
- -
-
la = l/2; % Position of Act. [m]
-ha = h/2; % Position of Act. [m]
-
-
- -
-
%% Name of the Simulink File
-mdl = 'gravimeter';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-G = linearize(mdl, io);
-G.InputName  = {'F1', 'F2', 'F3'};
-G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-
-
- -

-Decoupling at the CoM (Mass decoupled) -

-
-
JMa = [1 0 -h/2
-       0 1  l/2
-       1 0  h/2
-       0 1  0];
-
-JMt = [1 0 -ha
-       0 1  la
-       0 1 -la];
-
-
- -
-
GM = pinv(JMa)*G*pinv(JMt');
-GM.InputName  = {'Fx', 'Fy', 'Mz'};
-GM.OutputName  = {'Dx', 'Dy', 'Rz'};
-
-
- - -
-

jac_decoupling_M.png -

-

Figure 21: Diagonal and off-diagonal elements of the decoupled plant

-
-
-
- -
-

1.12.2 Decoupling of the stiffness matrix

-
- -
-

gravimeter_model_K.png -

-

Figure 22: Choice of {O} such that the Stiffness Matrix is Diagonal

-
- -

-Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame): -

-
-
JKa = [1 0  0
-       0 1 -l/2
-       1 0 -h
-       0 1  0];
-
-JKt = [1 0  0
-       0 1 -la
-       0 1  la];
-
-
- -

-And the plant \(\bm{G}_x\) is computed: -

-
-
GK = pinv(JKa)*G*pinv(JKt');
-GK.InputName  = {'Fx', 'Fy', 'Mz'};
-GK.OutputName  = {'Dx', 'Dy', 'Rz'};
-
-
- - -
-

jac_decoupling_K.png -

-

Figure 23: Diagonal and off-diagonal elements of the decoupled plant

-
-
-
- -
-

1.12.3 Combined decoupling of the mass and stiffness matrices

-
- -
-

gravimeter_model_KM.png -

-

Figure 24: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal

-
- -

-To do so, the actuator position should be modified -

- -
-
la = l/2; % Position of Act. [m]
-ha = 0; % Position of Act. [m]
-
-
- -
-
%% Name of the Simulink File
-mdl = 'gravimeter';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
-
-G = linearize(mdl, io);
-G.InputName  = {'F1', 'F2', 'F3'};
-G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
-
-
- -
-
JMa = [1 0 -h/2
-       0 1  l/2
-       1 0  h/2
-       0 1  0];
-
-JMt = [1 0 -ha
-       0 1  la
-       0 1 -la];
-
-
- -
-
GKM = pinv(JMa)*G*pinv(JMt');
-GKM.InputName  = {'Fx', 'Fy', 'Mz'};
-GKM.OutputName  = {'Dx', 'Dy', 'Rz'};
-
-
- - -
-

jac_decoupling_KM.png -

-

Figure 25: Diagonal and off-diagonal elements of the decoupled plant

-
-
-
- -
-

1.12.4 Conclusion

-
-

-Ideally, the mechanical system should be designed in order to have a decoupled stiffness matrix at the CoM of the solid. -

- -

-If not the case, the system can either be decoupled as low frequency if the Jacobian are evaluated at a point where the stiffness matrix is decoupled. -Or it can be decoupled at high frequency if the Jacobians are evaluated at the CoM. -

-
-
-
- -
-

1.13 SVD decoupling performances

-
-

- -As the SVD is applied on a real approximation of the plant dynamics at a frequency \(\omega_0\), it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation. -

- -

-Let’s do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping). -

- -

-Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26. -

-
-
c = 2e1; % Actuator Damping [N/(m/s)]
-
-
- - -
-

gravimeter_svd_low_damping.png -

-

Figure 26: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping

-
- -

-Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27. -

-
-
c = 5e2; % Actuator Damping [N/(m/s)]
-
-
- - -
-

gravimeter_svd_high_damping.png -

-

Figure 27: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping

-
-
-
-
- -
-

2 Parallel Manipulator with Collocated actuator/sensor pairs

-
-

- -

-

-In this section, we will see how the Jacobian matrix can be used to decouple a specific set of mechanical systems (described in Section 2.1). -

- -

-The basic decoupling architecture is shown in Figure 29 where the Jacobian matrix is used to both compute the actuator forces from forces/torques that are to be applied in a specific defined frame, and to compute the displacement/rotation of the same mass from several sensors. -

- -

-This is rapidly explained in Section 2.2. -

- - -
-

block_diagram_jacobian_decoupling.png -

-
- -

-Depending on the chosen frame, the Stiffness matrix in that particular frame can be computed. -This is explained in Section 2.3. -

- -

-Then three decoupling in three specific frames is studied: -

-
    -
  • Section 2.4: control in the frame of the legs
    • -
  • Section 2.5: control in a frame whose origin is at the center of mass of the payload
    • -
  • Section 2.6: control in a frame whose origin is located at the “center of stiffness” of the system
    • -
- -

-Conclusions are drawn in Section 2.7. -

-
- -
-

2.1 Model

-
-

- -

- -

-Let’s consider a parallel manipulator with several collocated actuator/sensors pairs. -

- -

-System in Figure 29 will serve as an example. -

- -

-We will note: -

-
    -
  • \(b_i\): location of the joints on the top platform
    • -
  • \(\hat{s}_i\): unit vector corresponding to the struts direction
    • -
  • \(k_i\): stiffness of the struts
    • -
  • \(\tau_i\): actuator forces
    • -
  • \(O_M\): center of mass of the solid body
    • -
  • \(\mathcal{L}_i\): relative displacement of the struts
    • -
- - -
-

gravimeter_model_analytical.png -

-

Figure 29: Model of the gravimeter

-
- -

-The parameters are defined as follows: -

-
-
l  = 1.0; % Length of the mass [m]
-h  = 2*1.7; % Height of the mass [m]
-
-la = l/2; % Position of Act. [m]
-ha = h/2; % Position of Act. [m]
-
-m = 400; % Mass [kg]
-I = 115; % Inertia [kg m^2]
-
-c1 = 2e1; % Actuator Damping [N/(m/s)]
-c2 = 2e1; % Actuator Damping [N/(m/s)]
-c3 = 2e1; % Actuator Damping [N/(m/s)]
-
-k1 = 15e3; % Actuator Stiffness [N/m]
-k2 = 15e3; % Actuator Stiffness [N/m]
-k3 = 15e3; % Actuator Stiffness [N/m]
-
-
- -

-Let’s express \({}^Mb_i\) and \(\hat{s}_i\): -

-\begin{align} -{}^Mb_1 &= [-l/2,\ -h_a] \\ -{}^Mb_2 &= [-la, \ -h/2] \\ -{}^Mb_3 &= [ la, \ -h/2] -\end{align} - -\begin{align} -\hat{s}_1 &= [1,\ 0] \\ -\hat{s}_2 &= [0,\ 1] \\ -\hat{s}_3 &= [0,\ 1] -\end{align} - -
-
s1 = [1;0];
-s2 = [0;1];
-s3 = [0;1];
-
-Mb1 = [-l/2;-ha];
-Mb2 = [-la; -h/2];
-Mb3 = [ la; -h/2];
-
-
- -

-Frame \(\{K\}\) is chosen such that the stiffness matrix is diagonal (explained in Section 3). -

- -

-The positions \({}^Kb_i\) are then: -

-\begin{align} -{}^Kb_1 &= [-l/2,\ 0] \\ -{}^Kb_2 &= [-la, \ -h/2+h_a] \\ -{}^Kb_3 &= [ la, \ -h/2+h_a] -\end{align} - -
-
Kb1 = [-l/2; 0];
-Kb2 = [-la; -h/2+ha];
-Kb3 = [ la; -h/2+ha];
-
-
-
-
- -
-

2.2 The Jacobian Matrix

-
-

- -

- -

-Let’s note: -

-
    -

  • -\(\bm{\mathcal{L}}\) the vector of actuator displacement: -

    -\begin{equation} - \bm{\mathcal{L}} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix} -\end{equation}
    • -

  • -\(\bm{\tau}\) the vector of actuator forces: -

    -\begin{equation} - \bm{\tau} = \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} -\end{equation}
    • -

  • -\(\bm{\mathcal{F}}_{\{O\}}\) the vector of forces/torques applied on the payload on expressed in frame \(\{O\}\): -

    -\begin{equation} - \bm{\mathcal{F}}_{\{O\}} = \begin{bmatrix} \mathcal{F}_{\{O\},x} \\ \mathcal{F}_{\{O\},y} \\ \mathcal{M}_{\{O\},z} \end{bmatrix} -\end{equation}
    • -

  • -\(\bm{\mathcal{X}}_{\{O\}}\) the vector of displacement of the payload with respect to frame \(\{O\}\): -

    -\begin{equation} - \bm{\mathcal{X}}_{\{O\}} = \begin{bmatrix} \mathcal{X}_{\{O\},x} \\ \mathcal{X}_{\{O\},y} \\ \mathcal{X}_{\{O\},R_z} \end{bmatrix} -\end{equation}
    • -
- - - -

-The Jacobian matrix can be used to: -

-
    -
  • Convert joints velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\): -\[ \dot{\bm{\mathcal{X}}}_{\{O\}} = J_{\{O\}} \dot{\bm{\mathcal{L}}} \]
    • -
  • Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\): -\[ \bm{\mathcal{F}}_{\{O\}} = J_{\{O\}}^T \bm{\tau} \]
    • -
-

-with \(\{O\}\) any chosen frame. -

- -

-If we consider small displacements, we have an approximate relation that links the displacements (instead of velocities): -

-\begin{equation} -\bm{\mathcal{X}}_{\{M\}} = J_{\{M\}} \bm{\mathcal{L}} -\end{equation} - - -

-The Jacobian can be computed as follows: -

-\begin{equation} -J_{\{O\}} = \begin{bmatrix} - {}^O\hat{s}_1^T & {}^Ob_{1,x} {}^O\hat{s}_{1,y} - {}^Ob_{1,x} {}^O\hat{s}_{1,y} \\ - {}^O\hat{s}_2^T & {}^Ob_{2,x} {}^O\hat{s}_{2,y} - {}^Ob_{2,x} {}^O\hat{s}_{2,y} \\ - \vdots & \vdots \\ - {}^O\hat{s}_n^T & {}^Ob_{n,x} {}^O\hat{s}_{n,y} - {}^Ob_{n,x} {}^O\hat{s}_{n,y} \\ -\end{bmatrix} -\end{equation} - - -

-Let’s compute the Jacobian matrix in frame \(\{M\}\) and \(\{K\}\): -

-
-
Jm = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
-      s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
-      s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 4: Jacobian Matrix \(J_{\{M\}}\)
101.7
01-0.5
010.5
- -
-
Jk = [s1', Kb1(1)*s1(2)-Kb1(2)*s1(1);
-      s2', Kb2(1)*s2(2)-Kb2(2)*s2(1);
-      s3', Kb3(1)*s3(2)-Kb3(2)*s3(1)];
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 5: Jacobian Matrix \(J_{\{K\}}\)
100
01-0.5
010.5
- -

-In the frame \(\{M\}\), the Jacobian is: -

-\begin{equation} -J_{\{M\}} = \begin{bmatrix} 1 & 0 & h_a \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} -\end{equation} - -

-And in frame \(\{K\}\), the Jacobian is: -

-\begin{equation} -J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} -\end{equation} -
-
- -
-

2.3 The Stiffness Matrix

-
-

- -

- -

-For a parallel manipulator, the stiffness matrix expressed in a frame \(\{O\}\) is: -

-\begin{equation} - K_{\{O\}} = J_{\{O\}}^T \mathcal{K} J_{\{O\}} -\end{equation} -

-where: -

-
    -
  • \(J_{\{O\}}\) is the Jacobian matrix expressed in frame \(\{O\}\)
    • -

  • -\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal -

    -\begin{equation} -\mathcal{K} = \begin{bmatrix} - k_1 & & & 0 \\ - & k_2 & & \\ - & & \ddots & \\ - 0 & & & k_n -\end{bmatrix} -\end{equation}
    • -
- -

-We have the same thing for the damping matrix. -

- -
-
Kr = diag([k1,k2,k3]);
-Cr = diag([c1,c2,c3]);
-
-
-
-
- -
-

2.4 Equations of motion - Frame of the legs

-
-

- -

- -

-Applying the second Newton’s law on the system in Figure 29 at its center of mass \(O_M\), we obtain: -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}} -\end{equation} -

-with: -

-
    -
  • \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): -\[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
    • -
  • \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): -\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]
    • -
  • \(\bm{\mathcal{X}}_{\{M\}}\) are displacements/rotations of the mass \(x\), \(y\), \(R_z\) expressed in the frame \(\{M\}\)
    • -
  • \(\bm{\mathcal{F}}_{\{M\}}\) are forces/torques \(\mathcal{F}_x\), \(\mathcal{F}_y\), \(\mathcal{M}_z\) applied at the origin of \(\{M\}\)
    • -
- -

-Let’s use the Jacobian matrix to compute the equations in terms of actuator forces \(\bm{\tau}\) and strut displacement \(\bm{\mathcal{L}}\): -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \bm{\mathcal{L}} = J_{\{M\}}^T \bm{\tau} -\end{equation} - -

-And we obtain: -

-\begin{equation} -\left( J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} s^2 + \mathcal{K} \right) \bm{\mathcal{L}} = \bm{\tau} -\end{equation} - -

-The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is: -

-\begin{equation} -\boxed{\bm{G}(s) = {\left( J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} s^2 + \mathcal{K} \right)}^{-1}} -\end{equation} - - -
-

gravimeter_block_decentralized.png -

-

Figure 30: Block diagram of the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\)

-
- -
-
%% Mass Matrix in frame {M}
-Mm = diag([m,m,I]);
-
-
- -

-Let’s note the mass matrix in the frame of the legs: -

-\begin{equation} -M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} -\end{equation} - -
-
%% Mass Matrix in the frame of the struts
-Ml = inv(Jm')*Mm*inv(Jm);
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 6: \(M_{\{L\}}\)
400680-680
6801371-1171
-680-11711371
- -

-As we can see, the Stiffness matrix in the frame of the legs is diagonal. -This means the plant dynamics will be diagonal at low frequency. -

-
-
Kl = diag([k1, k2, k3]);
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 7: \(K_{\{L\}} = \mathcal{K}\)
1500000
0150000
0015000
- -
-
Cl = diag([c1, c2, c3]);
-
-
- -

-The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is defined below and its magnitude is shown in Figure 31. -

-
-
Gl = inv(Ml*s^2 + Cl*s + Kl);
-
-
- -

-We can indeed see that the system is well decoupled at low frequency. -

- - -
-

plant_frame_L.png -

-

Figure 31: Dynamics from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\)

-
-
-
- -
-

2.5 Equations of motion - “Center of mass” {M}

-
-

- -

- -

-The equations of motion expressed in frame \(\{M\}\) are: -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}} -\end{equation} - -

-And the plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is: -

-\begin{equation} -\boxed{\bm{G}_{\{X\}} = {\left( M_{\{M\}} s^2 + K_{\{M\}} \right)}^{-1}} -\end{equation} -

-with: -

-
    -
  • \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): -\[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
    • -
  • \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): -\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]
    • -
- - -
-

gravimeter_block_com.png -

-

Figure 32: Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\)

-
- - -
-
%% Mass Matrix in frame {M}
-Mm = diag([m,m,I]);
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 8: Mass matrix expressed in \(\{M\}\): \(M_{\{M\}}\)
40000
04000
00115
- -
-
%% Stiffness Matrix in frame {M}
-Km = Jm'*Kr*Jm;
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 9: Stiffness matrix expressed in \(\{M\}\): \(K_{\{M\}}\)
15000025500
0300000
25500050850
- -
-
%% Damping Matrix in frame {M}
-Cm = Jm'*Cr*Jm;
-
-
- -

-The plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is defined below and its magnitude is shown in Figure 33. -

-
-
%% Plant in frame {M}
-Gm = inv(Mm*s^2 + Cm*s + Km);
-
-
- -

-And the system is well decoupled at high frequency (above the suspension modes). -

- - -
-

plant_frame_M.png -

-

Figure 33: Dynamics from \(\bm{\mathcal{F}}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\)

-
-
-
- -
-

2.6 Equations of motion - “Center of stiffness” {K}

-
-

- -

- -

-Let’s now express the transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\). -We start from: -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \bm{\mathcal{L}} = J_{\{M\}}^T \bm{\tau} -\end{equation} - -

-And we make use of the Jacobian \(J_{\{K\}}\) to obtain: -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} J_{\{K\}} \bm{\mathcal{X}}_{\{K\}} = J_{\{M\}}^T J_{\{K\}}^{-T} \bm{\mathcal{F}}_{\{K\}} -\end{equation} - -

-And finally: -

-\begin{equation} -\left( J_{\{K\}}^T J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} J_{\{K\}} s^2 + J_{\{K\}}^T \mathcal{K} J_{\{K\}} \right) \bm{\mathcal{X}}_{\{K\}} = \bm{\mathcal{F}}_{\{K\}} -\end{equation} - -

-The transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is then: -

-\begin{equation} -\boxed{\bm{G}_{\{K\}} = {\left( J_{\{K\}}^T J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} J_{\{K\}} s^2 + J_{\{K\}}^T \mathcal{K} J_{\{K\}} \right)}^{-1}} -\end{equation} - -

-The frame \(\{K\}\) has been chosen such that \(J_{\{K\}}^T \mathcal{K} J_{\{K\}}\) is diagonal. -

- - -
-

gravimeter_block_cok.png -

-

Figure 34: Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\)

-
- -
-
Mk = Jk'*inv(Jm)'*Mm*inv(Jm)*Jk;
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 10: Mass matrix expressed in \(\{K\}\): \(M_{\{K\}}\)
4000-680
04000
-68001271
- -
-
Kk = Jk'*Kr*Jk;
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 11: Stiffness matrix expressed in \(\{K\}\): \(K_{\{K\}}\)
1500000
0300000
007500
- -

-The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined below and its magnitude is shown in Figure 35. -

-
-
Gk = inv(Mk*s^2 + Ck*s + Kk);
-
-
- - -
-

plant_frame_K.png -

-

Figure 35: Dynamics from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\)

-
-
-
- -
-

2.7 Conclusion

-
-

- -

- -

-Jacobian matrices can be used to decouple the presented system. -

- -

-Depending on the chosen frame used for the estimation of the Jacobian, different plant dynamics is obtained. -

-
-
-
- -
-

3 Diagonal Stiffness Matrix for a planar manipulator

-
-

- -

-
-
-

3.1 Model and Assumptions

-
-

-Consider a parallel manipulator with: -

-
    -
  • \(b_i\): location of the joints on the top platform are called \(b_i\)
    • -
  • \(\hat{s}_i\): unit vector corresponding to the struts
    • -
  • \(k_i\): stiffness of the struts
    • -
  • \(\tau_i\): actuator forces
    • -
  • \(O_M\): center of mass of the solid body
    • -
- -

-Consider two frames: -

-
    -
  • \(\{M\}\) with origin \(O_M\)
    • -
  • \(\{K\}\) with origin \(O_K\)
    • -
- -

-As an example, take the system shown in Figure 36. -

- - -
-

3dof_model_fully_parallel.png -

-

Figure 36: Example of 3DoF parallel platform

-
-
-
- -
-

3.2 Objective

-
-

-The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. -If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically. -

-
-
- -
-

3.3 Conditions for Diagonal Stiffness

-
-

-The stiffness matrix in the frame \(\{K\}\) can be expressed as: -

-\begin{equation} \label{eq:stiffness_formula_planar} - K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}} -\end{equation} -

-where: -

-
    -
  • \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)
    • -

  • -\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal -

    -\begin{equation} -\mathcal{K} = \begin{bmatrix} - k_1 & & & 0 \\ - & k_2 & & \\ - & & \ddots & \\ - 0 & & & k_n -\end{bmatrix} -\end{equation}
    • -
- -

-The Jacobian for a planar manipulator, evaluated in a frame \(\{K\}\), can be expressed as follows: -

-\begin{equation} \label{eq:jacobian_planar} -J_{\{K\}} = \begin{bmatrix} - {}^K\hat{s}_1^T & {}^Kb_{1,x} {}^K\hat{s}_{1,y} - {}^Kb_{1,x} {}^K\hat{s}_{1,y} \\ - {}^K\hat{s}_2^T & {}^Kb_{2,x} {}^K\hat{s}_{2,y} - {}^Kb_{2,x} {}^K\hat{s}_{2,y} \\ - \vdots & \vdots \\ - {}^K\hat{s}_n^T & {}^Kb_{n,x} {}^K\hat{s}_{n,y} - {}^Kb_{n,x} {}^K\hat{s}_{n,y} \\ -\end{bmatrix} -\end{equation} - -

-Let’s omit the mention of frame, it is assumed that vectors are expressed in frame \(\{K\}\). -It is specified otherwise. -

- -

-Injecting \eqref{eq:jacobian_planar} into \eqref{eq:stiffness_formula_planar} yields: -

-\begin{equation} -\boxed{ -K_{\{K\}} = \left[ \begin{array}{c|c} - k_i \hat{s}_i \hat{s}_i^T & k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) \cr - \hline - k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) & k_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x})^2 -\end{array} \right] -} -\end{equation} - -

-In order to have a decoupled stiffness matrix, we have the following two conditions: -

-\begin{align} -k_i \hat{s}_i \hat{s}_i^T &= \text{diag. matrix} \label{eq:diag_cond_2D_1} \\ -k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) &= 0 \label{eq:diag_cond_2D_2} -\end{align} - -

-Note that we don’t have any condition on the term \(k_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x})^2\) as it is only a scalar. -

- -

-Condition \eqref{eq:diag_cond_2D_1}: -

-
    -
  • represents the coupling between translations and forces
    • -
  • does only depends on the orientation of the struts and the stiffnesses and not on the choice of frame
    • -
  • it is therefore a intrinsic property of the chosen geometry
    • -
- -

-Condition \eqref{eq:diag_cond_2D_2}: -

-
    -
  • represents the coupling between forces/rotations and torques/translation
    • -
  • it does depend on the positions of the joints \(b_i\) in the frame \(\{K\}\)
    • -
- -

-Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\): -

-\begin{align} -{}^Kb_i &= {}^Mb_i - {}^MO_K \\ -{}^K\hat{s}_i &= {}^M\hat{s}_i -\end{align} - -

-And the goal is to find \({}^MO_K\) such that \eqref{eq:diag_cond_2D_2} is fulfilled: -

-\begin{equation} -k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x} - {}^MO_{K,x}\hat{s}_{i,y} + {}^MO_{K,y}\hat{s}_{i,x}) \hat{s}_i = 0 -\end{equation} -\begin{equation} -k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i = {}^MO_{K,x} k_i \hat{s}_{i,y} \hat{s}_i - {}^MO_{K,y} k_i \hat{s}_{i,x} \hat{s}_i -\end{equation} - -

-And we have two sets of linear equations of two unknowns. -

- -

-This can be easily solved by writing the equations in a matrix form: -

-\begin{equation} -\underbrace{k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i}_{2 \times 1} = -\underbrace{\begin{bmatrix} -& \\ -k_i \hat{s}_{i,y} \hat{s}_i & - k_i \hat{s}_{i,x} \hat{s}_i \\ -& \\ -\end{bmatrix}}_{2 \times 2} -\underbrace{\begin{bmatrix} -{}^MO_{K,x}\\ -{}^MO_{K,y} -\end{bmatrix}}_{2 \times 1} -\end{equation} - -

-And finally, if the matrix is invertible: -

-\begin{equation} -\boxed{ -{}^MO_K = {\begin{bmatrix} -& \\ -k_i \hat{s}_{i,y} \hat{s}_i & - k_i \hat{s}_{i,x} \hat{s}_i \\ -& \\ -\end{bmatrix}}^{-1} k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i -} -\end{equation} - -

-Note that a rotation of the frame \(\{K\}\) with respect to frame \(\{M\}\) would make not change on the “diagonality” of \(K_{\{K\}}\). -

-
-
- -
-

3.4 Example 1 - Planar manipulator with 3 actuators

-
-

-Consider system of Figure 37. -

- - -
-

3dof_model_fully_parallel.png -

-

Figure 37: Example of 3DoF parallel platform

-
- -

-The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: -

-
-
ki = [5,1,2]; % Stiffnesses [N/m]
-si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
-bi = [[-1;0.5],[-2;-1],[0;-1]]; % Joint's positions in frame {M}
-
-
- -

-Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true: -

- - - --- -- - - - - - - - - - - - -
50
02
- -

-Now, compute \({}^MO_K\): -

-
-
Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);
-
-
- - - - --- - - - - - - - - - -
-1
0.5
- -

-Let’s compute the new coordinates \({}^Kb_i\) after the change of frame: -

-
-
Kbi = bi - Ok;
-
-
- -

-In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\): -

-
-
Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
100
01-1
011
- -

-And the stiffness matrix: -

-
-
K = Jk'*diag(ki)*Jk
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
500
020
002
-
-
- -
-

3.5 Example 2 - Planar manipulator with 4 actuators

-
-

-Now consider the planar manipulator of Figure 38. -

- - -
-

model_planar_2.png -

-

Figure 38: Planar Manipulator

-
- -

-The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: -

-
-
ki = [1,2,1,1];
-si = [[1;0],[0;1],[-1;0],[0;1]];
-si = si./vecnorm(si);
-h = 0.2;
-L = 2;
-bi = [[-L/2;h],[-L/2;-h],[L/2;h],[L/2;h]];
-
-
- -

-Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true: -

-
-
ki.*si*si'
-
-
- - - - --- -- - - - - - - - - - - - -
20
03
- -

-Now, compute \({}^MO_K\): -

-
-
Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);
-
-
- - - - --- - - - - - - - - - -
-0.33333
0.2
- -

-Let’s compute the new coordinates \({}^Kb_i\) after the change of frame: -

-
-
Kbi = bi - Ok;
-
-
- -

-In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\): -

-
-
Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - -
100
01-0.66667
-100
011.3333
- -

-And the stiffness matrix: -

-
-
K = Jk'*diag(ki)*Jk
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
200
03-2.2204e-16
0-2.2204e-162.6667
-
-
-
- -
-

4 Diagonal Stiffness Matrix for a general parallel manipulator

-
-
-
-

4.1 Model and Assumptions

-
-

-Let’s consider a 6dof parallel manipulator with: -

-
    -
  • \(b_i\): location of the joints on the top platform are called \(b_i\)
    • -
  • \(\hat{s}_i\): unit vector corresponding to the struts
    • -
  • \(k_i\): stiffness of the struts
    • -
  • \(\tau_i\): actuator forces
    • -
  • \(O_M\): center of mass of the solid body
    • -
- -

-Consider two frames: -

-
    -
  • \(\{M\}\) with origin \(O_M\)
    • -
  • \(\{K\}\) with origin \(O_K\)
    • -
- -

-An example is shown in Figure 39. -

- - -
-

stewart_architecture_example.png -

-

Figure 39: Parallel manipulator Example

-
-
-
- -
-

4.2 Objective

-
-

-The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. -If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically. -

-
-
- -
-

4.3 Analytical formula of the stiffness matrix

-
-

-For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed in a frame \(\{K\}\) is: -

-\begin{equation} - K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}} -\end{equation} -

-where: -

-
    -
  • \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)
    • -

  • -\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal: -

    -\begin{equation} -\mathcal{K} = \begin{bmatrix} - k_1 & & & 0 \\ - & k_2 & & \\ - & & \ddots & \\ - 0 & & & k_n -\end{bmatrix} -\end{equation}
    • -
- - -

-The analytical expression of \(J_{\{K\}}\) is: -

-\begin{equation} -J_{\{K\}} = \begin{bmatrix} - {}^K\hat{s}_1^T & ({}^Kb_1 \times {}^K\hat{s}_1)^T \\ - {}^K\hat{s}_2^T & ({}^Kb_2 \times {}^K\hat{s}_2)^T \\ - \vdots & \vdots \\ - {}^K\hat{s}_n^T & ({}^Kb_n \times {}^K\hat{s}_n)^T -\end{bmatrix} -\end{equation} - -

-To simplify, we ignore the superscript \(K\) and we assume that all vectors / positions are expressed in this frame \(\{K\}\). -Otherwise, it is explicitly written. -

- -

-Let’s now write the analytical expressing of the stiffness matrix \(K_{\{K\}}\): -

-\begin{equation} -K_{\{K\}} = \begin{bmatrix} - \hat{s}_1 & \dots & \hat{s}_n \\ - (b_1 \times \hat{s}_1) & \dots & (b_n \times \hat{s}_n) -\end{bmatrix} -\begin{bmatrix} - k_1 & & \\ - & \ddots & \\ - & & k_n -\end{bmatrix} -\begin{bmatrix} - \hat{s}_1^T & (b_1 \times \hat{s}_1)^T \\ - \hat{s}_2^T & (b_2 \times \hat{s}_2)^T \\ - \vdots & \dots \\ - \hat{s}_n^T & (b_n \times \hat{s}_n)^T -\end{bmatrix} -\end{equation} - -

-And we finally obtain: -

-\begin{equation} -\boxed{ -K_{\{K\}} = \left[ \begin{array}{c|c} - k_i \hat{s}_i \hat{s}_i^T & k_i \hat{s}_i (b_i \times \hat{s}_i)^T \cr - \hline - k_i (b_i \times \hat{s}_i) \hat{s}_i^T & k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T -\end{array} \right] -} -\end{equation} - -

-We want the stiffness matrix to be diagonal, therefore, we have the following conditions: -

-\begin{align} -k_i \hat{s}_i \hat{s}_i^T &= \text{diag. matrix} \label{eq:diag_cond_1} \\ -k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T &= \text{diag. matrix} \label{eq:diag_cond_2} \\ -k_i \hat{s}_i (b_i \times \hat{s}_i)^T &= 0 \label{eq:diag_cond_3} -\end{align} - -

-Note that: -

-
    -
  • condition \eqref{eq:diag_cond_1} corresponds to coupling between forces applied on \(O_K\) to translations of the payload. -It does not depend on the choice of \(\{K\}\), it only depends on the orientation of the struts and the stiffnesses. -It is therefore an intrinsic property of the manipulator.
    • -
  • condition \eqref{eq:diag_cond_2} corresponds to the coupling between forces applied on \(O_K\) and rotation of the payload. -Similarly, it does also correspond to the coupling between torques applied on \(O_K\) to translations of the payload.
    • -
  • condition \eqref{eq:diag_cond_3} corresponds to the coupling between torques applied on \(O_K\) to rotation of the payload.
    • -
  • conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} do depend on the positions \({}^Kb_i\) and therefore depend on the choice of \(\{K\}\).
    • -
- -

-Note that if we find a frame \(\{K\}\) in which the stiffness matrix \(K_{\{K\}}\) is diagonal, it will still be diagonal for any rotation of the frame \(\{K\}\). -Therefore, we here suppose that the frame \(\{K\}\) is aligned with the initial frame \(\{M\}\). -

- -

-Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\): -

-\begin{align} -{}^Kb_i &= {}^Mb_i - {}^MO_K \\ -{}^K\hat{s}_i &= {}^M\hat{s}_i -\end{align} - -

-The goal is to find \({}^MO_K\) such that conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} are fulfilled. -

- -

-Let’s first solve equation \eqref{eq:diag_cond_3} that corresponds to the coupling between forces and rotations: -

-\begin{equation} -k_i \hat{s}_i (({}^Mb_i - {}^MO_K) \times \hat{s}_i)^T = 0 -\end{equation} - -

-Taking the transpose and re-arranging: -

-\begin{equation} -k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = k_i ({}^MO_K \times \hat{s}_i) \hat{s}_i^T -\end{equation} - -

-As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector, we obtain: -

-\begin{equation} -k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = {}^M\bm{O}_{K} ( k_i \hat{s}_i \hat{s}_i^T ) -\end{equation} - -

-with: -

-\begin{equation} \label{eq:skew_symmetric_cross_product} -{}^M\bm{O}_K = \begin{bmatrix} -0 & -{}^MO_{K,z} & {}^MO_{K,y} \\ -{}^MO_{K,z} & 0 & -{}^MO_{K,x} \\ --{}^MO_{K,y} & {}^MO_{K,x} & 0 -\end{bmatrix} -\end{equation} - -

-We suppose \(k_i \hat{s}_i \hat{s}_i^T\) invertible as it is diagonal from \eqref{eq:diag_cond_1}. -

- -

-And finally, we find: -

-\begin{equation} -\boxed{ -{}^M\bm{O}_{K} = \left( k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T\right) \cdot {\left( k_i \hat{s}_i \hat{s}_i^T \right)}^{-1} -} -\end{equation} - -

-If the obtained \({}^M\bm{O}_{K}\) is a skew-symmetric matrix, we can easily determine the corresponding vector \({}^MO_K\) from \eqref{eq:skew_symmetric_cross_product}. -

- -

-In such case, condition \eqref{eq:diag_cond_2} is fulfilled and there is no coupling between translations and rotations in the frame \(\{K\}\). -

- -

-Then, we can only verify if condition \eqref{eq:diag_cond_3} is verified or not. -

- -
-

-If there is no frame \(\{K\}\) such that conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} are valid, it would be interesting to be able to determine the frame \(\{K\}\) in which is coupling is minimal. -

- -
-
-
- -
-

4.4 Example 1 - 6DoF manipulator (3D)

-
-

-Let’s define the geometry of the manipulator (\({}^Mb_i\), \({}^Ms_i\) and \(k_i\)): -

-
-
ki = [2,2,1,1,3,3,1,1,1,1,2,2];
-si = [[-1;0;0],[-1;0;0],[-1;0;0],[-1;0;0],[0;0;1],[0;0;1],[0;0;1],[0;0;1],[0;-1;0],[0;-1;0],[0;-1;0],[0;-1;0]];
-bi = [[1;-1;1],[1;1;-1],[1;1;1],[1;-1;-1],[1;-1;-1],[-1;1;-1],[1;1;-1],[-1;-1;-1],[1;1;-1],[-1;1;1],[-1;1;-1],[1;1;1]]-[0;2;-1];
-
-
- -

-Cond 1: -

-
-
ki.*si*si'
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
600
060
008
- -

-Find Ok -

-
-
OkX = (ki.*cross(bi, si)*si')/(ki.*si*si');
-
-if all(diag(OkX) == 0) && all(all((OkX + OkX') == 0))
-    disp('OkX is skew symmetric')
-    Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
-else
-    error('OkX is *not* skew symmetric')
-end
-
-
- - - - --- - - - - - - - - - - - - - -
0
-2
1
- -
-
% Verification of second condition
-si*cross(bi-Ok, si)'
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
000
000
000
- -

-Verification of third condition -

-
-
ki.*cross(bi-Ok, si)*cross(bi-Ok, si)'
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
144-2
4142
-2212
- -

-Let’s compute the Jacobian: -

-
-
Jk = [si', cross(bi - Ok, si)'];
-
-
- -

-And the stiffness matrix: -

-
-
Jk'*diag(ki)*Jk
-
-
- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
600000
060000
008000
000144-2
0004142
000-2212
- -
-
figure;
-hold on;
-set(gca,'ColorOrderIndex',1)
-plot(b1(1), b1(2), 'o');
-set(gca,'ColorOrderIndex',2)
-plot(b2(1), b2(2), 'o');
-set(gca,'ColorOrderIndex',3)
-plot(b3(1), b3(2), 'o');
-set(gca,'ColorOrderIndex',1)
-quiver(b1(1),b1(2),0.1*s1(1),0.1*s1(2))
-set(gca,'ColorOrderIndex',2)
-quiver(b2(1),b2(2),0.1*s2(1),0.1*s2(2))
-set(gca,'ColorOrderIndex',3)
-quiver(b3(1),b3(2),0.1*s3(1),0.1*s3(2))
-
-plot(0, 0, 'ko');
-quiver([0,0],[0,0],[0.1,0],[0,0.1], 'k')
-
-plot(Ok(1), Ok(2), 'ro');
-quiver([Ok(1),Ok(1)],[Ok(2),Ok(2)],[0.1,0],[0,0.1], 'r')
-
-hold off;
-axis equal;
-
-
-
-
- -
-

4.5 Example 2 - Stewart Platform

-
-
-
-

5 Stiffness and Mass Matrices in the Leg’s frame

-
-
-
-

5.1 Equations

-
-

-Equations in the \(\{M\}\) frame: -

-\begin{equation} -\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \mathcal{X}_{\{M\}} = \mathcal{F}_{\{M\}} -\end{equation} - -

-Thank to the Jacobian, we can transform the equation of motion expressed in the \(\{M\}\) frame to the frame of the legs: -

-\begin{equation} -J_{\{M\}}^{-T} \left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \dot{\mathcal{L}} = \tau -\end{equation} - -

-And we have new stiffness and mass matrices: -

-\begin{equation} -\left( M_{\{L\}} s^2 + K_{\{L\}} \right) \dot{\mathcal{L}} = \tau -\end{equation} -

-with: -

-
    -
  • The local mass matrix: -\[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \]
    • -
  • The local stiffness matrix: -\[ K_{\{L\}} = J_{\{M\}}^{-T} K_{\{M\}} J_{\{M\}}^{-1} \]
    • -
-
-
- -
-

5.2 Stiffness matrix

-
-

-We have that: -\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \] -

- -

-Therefore, we find that \(K_{\{L\}}\) is a diagonal matrix: -

-\begin{equation} -K_{\{L\}} = \mathcal{K} = \begin{bmatrix} -k_1 & & 0 \\ - & \ddots & \\ -0 & & k_n -\end{bmatrix} -\end{equation} - -

-The dynamics from \(\tau\) to \(\mathcal{L}\) is therefore decoupled at low frequency. -

-
-
- -
-

5.3 Mass matrix

-
-

-The mass matrix in the frames of the legs is: -\[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \] -with \(M_{\{M\}}\) a diagonal matrix: -

-\begin{equation} -M_{\{M\}} = \begin{bmatrix} -m & & & & & \\ - & m & & & 0 & \\ - & & m & & & \\ - & & & I_x & & \\ - & 0 & & & I_y & \\ - & & & & & I_z -\end{bmatrix} -\end{equation} - -

-Let’s suppose \(M_{\{L\}} = \mathcal{M}\) diagonal and try to find what does this imply: -\[ M_{\{M\}} = J_{\{M\}}^{T} \mathcal{M} J_{\{M\}} \] -with: -

-\begin{equation} -\mathcal{M} = \begin{bmatrix} -m_1 & & 0 \\ - & \ddots & \\ -0 & & m_n -\end{bmatrix} -\end{equation} - -

-We obtain: -

-\begin{equation} -\boxed{ -M_{\{M\}} = \left[ \begin{array}{c|c} - m_i \hat{s}_i \hat{s}_i^T & m_i \hat{s}_i (b_i \times \hat{s}_i)^T \cr - \hline - k_i \hat{s}_i (b_i \times \hat{s}_i)^T & m_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T -\end{array} \right] -} -\end{equation} - -

-Therefore, we have the following conditions: -

-\begin{align} -m_i \hat{s}_i \hat{s}_i^T &= m \bm{I}_{3} \\ -m_i \hat{s}_i (b_i \times \hat{s}_i)^T &= \bm{O}_{3} \\ -m_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T &= \text{diag}(I_x, I_y, I_z) -\end{align} -
-
- - -
-

5.4 Planar Example

-
-

-The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: -

-
-
ki = [1,1,1]; % Stiffnesses [N/m]
-si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
-bi = [[-1; 0],[-10;-1],[0;-1]]; % Joint's positions in frame {M}
-
-
- -

-Jacobian in frame \(\{M\}\): -

-
-
Jm = [si', (bi(1,:).*si(2,:) - bi(2,:).*si(1,:))'];
-
-
- -

-And the stiffness matrix in frame \(\{K\}\): -

-
-
Km = Jm'*diag(ki)*Jm;
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
201
01-1
1-12
- -

-Mass matrix in the frame \(\{M\}\): -

-
-
m = 10; % [kg]
-I = 1; % [kg.m^2]
-
-Mm = diag([m, m, I]);
-
-
- - -

-Now compute \(K\) and \(M\) in the frame of the legs: -

-
-
ML = inv(Jm)'*Mm*inv(Jm)
-KL = inv(Jm)'*Km*inv(Jm)
-
-
- -
-
Gm = 1/(ML*s^2 + KL);
-
-
- -
-
freqs = logspace(-2, 1, 1000);
-figure;
-hold on;
-for i = 1:length(ki)
-    plot(freqs, abs(squeeze(freqresp(Gm(i,i), freqs, 'Hz'))), 'k-')
-end
-for i = 1:length(ki)
-    for j = i+1:length(ki)
-        plot(freqs, abs(squeeze(freqresp(Gm(i,j), freqs, 'Hz'))), 'r-')
-    end
-end
-hold off;
-xlabel('Frequency [Hz]');
-ylabel('Magnitude');
-set(gca, 'xscale', 'log');
-set(gca, 'yscale', 'log');
-
-
-
-
-
- -
-

6 Stewart Platform - Simscape Model

-
-

- -

-

-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 40. -

- -

-Some notes about the system: -

-
    -
  • 6 voice coils actuators are used to control the motion of the top platform.
    • -
  • the motion of the top platform is measured with a 6-axis inertial unit (3 acceleration + 3 angular accelerations)
    • -
  • the control objective is to isolate the top platform from vibrations coming from the bottom platform
    • -
- - -
-

SP_assembly.png -

-

Figure 40: Stewart Platform CAD View

-
- -

-The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections: -

-
    -
  • Section 6.1: The parameters of the Simscape model of the Stewart platform are defined
    • -
  • Section 6.2: The plant is identified from the Simscape model and the system coupling is shown
    • -
  • Section 6.3: The plant is first decoupled using the Jacobian
    • -
  • Section 6.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • -
  • Section 6.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • -
  • Section 6.6:
    • -
  • Section 6.7: The dynamics of the decoupled plants are shown
    • -
  • Section 6.8: A diagonal controller is defined to control the decoupled plant
    • -
  • Section 6.9: Finally, the closed loop system properties are studied
    • -
-
- -
-

6.1 Simscape Model - Parameters

-
-

- -

-
-
open('drone_platform.slx');
-
-
- -

-Definition of spring parameters: -

-
-
kx = 0.5*1e3/3; % [N/m]
-ky = 0.5*1e3/3;
-kz = 1e3/3;
-
-cx = 0.025; % [Nm/rad]
-cy = 0.025;
-cz = 0.025;
-
-
- -

-We suppose the sensor is perfectly positioned. -

-
-
sens_pos_error = zeros(3,1);
-
-
- -

-Gravity: -

-
-
g = 0;
-
-
- -

-We load the Jacobian (previously computed from the geometry): -

-
-
load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
-
-
- -

-We initialize other parameters: -

-
-
U = eye(6);
-V = eye(6);
-Kc = tf(zeros(6));
-
-
- - -
-

stewart_simscape.png -

-

Figure 41: General view of the Simscape Model

-
- - -
-

stewart_platform_details.png -

-

Figure 42: Simscape model of the Stewart platform

-
-
-
- -
-

6.2 Identification of the plant

-
-

- -

- -

-The plant shown in Figure 43 is identified from the Simscape model. -

- -

-The inputs are: -

-
    -
  • \(D_w\) translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
    • -
  • \(\tau\) the 6 forces applied by the voice coils
    • -
- -

-The outputs are the 6 accelerations measured by the inertial unit. -

- - -
-

stewart_platform_plant.png -

-

Figure 43: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

-
- -
-
%% Name of the Simulink File
-mdl = 'drone_platform';
-
-%% Input/Output definition
-clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1; % Ground Motion
-io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
-io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
-
-G = linearize(mdl, io);
-G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
-                'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
-G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
-
-% Plant
-Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
-% Disturbance dynamics
-Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
-
-
- -

-There are 24 states (6dof for the bottom platform + 6dof for the top platform). -

-
-
size(G)
-
-
- -
-State-space model with 6 outputs, 12 inputs, and 24 states.
-
- - -

-The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 44. -

- -

-One can easily see that the system is strongly coupled. -

- - -
-

stewart_platform_coupled_plant.png -

-

Figure 44: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

-
-
-
- -
-

6.3 Decoupling using the Jacobian

-
-

- -Consider the control architecture shown in Figure 45. -The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator. -

- -

-The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators). -

- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 12: Computed Jacobian Matrix
0.8110.00.584-0.018-0.0080.025
-0.406-0.7030.584-0.016-0.012-0.025
-0.4060.7030.5840.016-0.0120.025
0.8110.00.5840.018-0.008-0.025
-0.406-0.7030.5840.0020.0190.025
-0.4060.7030.584-0.0020.019-0.025
- - -
-

plant_decouple_jacobian.png -

-

Figure 45: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

-
- -

-We define a new plant: -\[ G_x(s) = G(s) J^{-T} \] -

- -

-\(G_x(s)\) correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform. -

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

6.4 Decoupling using the SVD

-
-

- -

- -

-In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required. -

- -

-Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). -

-
-
wc = 2*pi*30; % Decoupling frequency [rad/s]
-
-H1 = evalfr(Gu, j*wc);
-
-
- -

-The real approximation is computed as follows: -

-
-
D = pinv(real(H1'*H1));
-H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
-
-
- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 13: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
- - -

-Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix. -This can be seen on the Bode plots where the phase is close to 1. -This can be verified below where only the real value of \(G_u(\omega_c)\) is shown -

- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 14: Real part of \(G\) at the decoupling frequency \(\omega_c\)
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
- -

-Now, the Singular Value Decomposition of \(H_1\) is performed: -\[ H_1 = U \Sigma V^H \] -

- -
-
[U,~,V] = svd(H1);
-
-
- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 15: Obtained matrix \(U\)
-0.0057e-066e-11-3e-06-10.1
-7e-06-0.005-9e-09-5e-09-0.1-1
4e-08-2e-10-6e-11-13e-06-3e-07
-0.002-1-5e-062e-100.00060.005
1-0.002-1e-082e-08-0.0050.0006
-4e-095e-06-16e-11-2e-09-1e-08
- - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Table 16: Obtained matrix \(V\)
-0.20.5-0.4-0.4-0.6-0.2
-0.30.50.4-0.40.50.3
-0.3-0.5-0.4-0.40.4-0.4
-0.2-0.50.4-0.4-0.50.3
0.6-0.06-0.4-0.40.10.6
0.60.060.4-0.4-0.006-0.6
- -

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 46. -

- - -
-

plant_decouple_svd.png -

-

Figure 46: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

-
- -

-The decoupled plant is then: -\[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \] -

- -
-
Gsvd = inv(U)*Gu*inv(V');
-
-
-
-
- -
-

6.5 Verification of the decoupling using the “Gershgorin Radii”

-
-

- -

- -

-The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\): -

- -

-The “Gershgorin Radii” of a matrix \(S\) is defined by: -\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \] -

- -

-This is computed over the following frequencies. -

- -
-

simscape_model_gershgorin_radii.png -

-

Figure 47: Gershgorin Radii of the Coupled and Decoupled plants

-
-
-
- -
-

6.6 Verification of the decoupling using the “Relative Gain Array”

-
-

- -

- -

-The relative gain array (RGA) is defined as: -

-\begin{equation} - \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T -\end{equation} -

-where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix. -

- -

-The obtained RGA elements are shown in Figure 48. -

- - -
-

simscape_model_rga.png -

-

Figure 48: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

-
-
-
- -
-

6.7 Obtained Decoupled Plants

-
-

- -

- -

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 49. -

- - -
-

simscape_model_decoupled_plant_svd.png -

-

Figure 49: Decoupled Plant using SVD

-
- -

-Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 50. -

- - -
-

simscape_model_decoupled_plant_jacobian.png -

-

Figure 50: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

-
-
-
- -
-

6.8 Diagonal Controller

-
-

- -The control diagram for the centralized control is shown in Figure 51. -

- -

-The controller \(K_c\) is “working” in an cartesian frame. -The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators. -

- - -
-

centralized_control.png -

-

Figure 51: Control Diagram for the Centralized control

-
- -

-The SVD control architecture is shown in Figure 52. -The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). -

- - -
-

svd_control.png -

-

Figure 52: Control Diagram for the SVD control

-
- - -

-We choose the controller to be a low pass filter: -\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] -

- -

-\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) -

- -
-
wc = 2*pi*80;  % Crossover Frequency [rad/s]
-w0 = 2*pi*0.1; % Controller Pole [rad/s]
-
-
- -
-
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
-L_cen = K_cen*Gx;
-G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
-
-
- -
-
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
-L_svd = K_svd*Gsvd;
-G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
-
-
- -

-The obtained diagonal elements of the loop gains are shown in Figure 53. -

- - -
-

stewart_comp_loop_gain_diagonal.png -

-

Figure 53: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

-
-
-
- -
-

6.9 Closed-Loop system Performances

-
-

- -

- -

-Let’s first verify the stability of the closed-loop systems: -

-
-
isstable(G_cen)
-
-
- -
-ans =
-  logical
-   1
-
- - -
-
isstable(G_svd)
-
-
- -
-ans =
-  logical
-   1
-
- - -

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 54. -

- - -
-

stewart_platform_simscape_cl_transmissibility.png -

-

Figure 54: Obtained Transmissibility

-
-
-
-
-
-
-

Author: Dehaeze Thomas

-

Created: 2021-02-17 mer. 15:15

-
- - diff --git a/index.html b/index.html new file mode 120000 index 0000000..2c909f6 --- /dev/null +++ b/index.html @@ -0,0 +1 @@ +svd-control.html \ No newline at end of file diff --git a/svd-control.html b/svd-control.html new file mode 100644 index 0000000..8b926dd --- /dev/null +++ b/svd-control.html @@ -0,0 +1,4207 @@ + + + + + + +Diagonal control using the SVD and the Jacobian Matrix + + + + + + + + +
+ UP + | + HOME +
+

Diagonal control using the SVD and the Jacobian Matrix

+
+

Table of Contents

+
+ +
+
+
+

This report is also available as a pdf.

+
+ +

+In this document, the use of the Jacobian matrix and the Singular Value Decomposition to render a physical plant diagonal dominant is studied. +Then, a diagonal controller is used. +

+ +

+These two methods are tested on two plants: +

+ + +
+

1 Gravimeter - Simscape Model

+
+

+ +

+
+ +
+

1.1 Introduction

+
+

+In this part, diagonal control using both the SVD and the Jacobian matrices are applied on a gravimeter model: +

+
    +
  • Section 1.2: the model is described and its parameters are defined.
    • +
  • Section 1.3: the plant dynamics from the actuators to the sensors is computed from a Simscape model.
    • +
  • Section 1.4: the plant is decoupled using the Jacobian matrices.
    • +
  • Section 1.5: the Singular Value Decomposition is performed on a real approximation of the plant transfer matrix and further use to decouple the system.
    • +
  • Section 1.6: the effectiveness of the decoupling is computed using the Gershorin radii
    • +
  • Section 1.7: the effectiveness of the decoupling is computed using the Relative Gain Array
    • +
  • Section 1.8: the obtained decoupled plants are compared
    • +
  • Section 1.9: the diagonal controller is developed
    • +
  • Section 1.10: the obtained closed-loop performances for the two methods are compared
    • +
  • Section 1.11: the robustness to a change of actuator position is evaluated
    • +
  • Section 1.12: the choice of the reference frame for the evaluation of the Jacobian is discussed
    • +
  • Section 1.13: the decoupling performances of SVD is evaluated for a low damped and an highly damped system
    • +
+
+
+ +
+

1.2 Gravimeter Model - Parameters

+
+

+ +

+ +

+The model of the gravimeter is schematically shown in Figure 1. +

+ + +
+

gravimeter_model.png +

+

Figure 1: Model of the gravimeter

+
+ + +
+

leg_model.png +

+

Figure 2: Model of the struts

+
+ +

+The parameters used for the simulation are the following: +

+
+
l  = 1.0; % Length of the mass [m]
+h  = 1.7; % Height of the mass [m]
+
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
+
+m = 400; % Mass [kg]
+I = 115; % Inertia [kg m^2]
+
+k = 15e3; % Actuator Stiffness [N/m]
+c = 2e1; % Actuator Damping [N/(m/s)]
+
+deq = 0.2; % Length of the actuators [m]
+
+g = 0; % Gravity [m/s2]
+
+
+
+
+ +
+

1.3 System Identification

+
+

+ +

+ +
+
%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+ +

+The inputs and outputs of the plant are shown in Figure 3. +

+ +

+More precisely there are three inputs (the three actuator forces): +

+\begin{equation} + \bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix} +\end{equation} +

+And 4 outputs (the two 2-DoF accelerometers): +

+\begin{equation} + \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix} +\end{equation} + + +
+

gravimeter_plant_schematic.png +

+

Figure 3: Schematic of the gravimeter plant

+
+ +

+We can check the poles of the plant: +

+ + + +++ + + + + + + + + + + + + + + + + + + + + + + + + + +
-0.12243+13.551i
-0.12243-13.551i
-0.05+8.6601i
-0.05-8.6601i
-0.0088785+3.6493i
-0.0088785-3.6493i
+ +

+As expected, the plant as 6 states (2 translations + 1 rotation) +

+
+
size(G)
+
+
+ +
+State-space model with 4 outputs, 3 inputs, and 6 states.
+
+ + +

+The bode plot of all elements of the plant are shown in Figure 4. +

+ + +
+

open_loop_tf.png +

+

Figure 4: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers

+
+
+
+ +
+

1.4 Decoupling using the Jacobian

+
+

+ +

+ +

+Consider the control architecture shown in Figure 5. +

+ +

+The Jacobian matrix \(J_{\tau}\) is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass: +

+\begin{equation} + \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix} +\end{equation} + +

+The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass: +

+\begin{equation} + \begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix} +\end{equation} + +

+We thus define a new plant as defined in Figure 5. +\[ \bm{G}_x(s) = J_a^{-1} \bm{G}(s) J_{\tau}^{-T} \] +

+ +

+\(\bm{G}_x(s)\) correspond to the \(3 \times 3\) transfer function matrix from forces and torques applied to the gravimeter at its center of mass to the absolute acceleration of the gravimeter’s center of mass (Figure 5). +

+ + +
+

gravimeter_decouple_jacobian.png +

+

Figure 5: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

+
+ +

+The Jacobian corresponding to the sensors and actuators are defined below: +

+
+
Ja = [1 0 -h/2
+      0 1  l/2
+      1 0  h/2
+      0 1  0];
+
+Jt = [1 0 -ha
+      0 1  la
+      0 1 -la];
+
+
+ +

+And the plant \(\bm{G}_x\) is computed: +

+
+
Gx = pinv(Ja)*G*pinv(Jt');
+Gx.InputName  = {'Fx', 'Fy', 'Mz'};
+Gx.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+
+ +
+size(Gx)
+State-space model with 3 outputs, 3 inputs, and 6 states.
+
+ + +

+The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6. +

+ +

+It is shown at the system is: +

+
    +
  • decoupled at high frequency thanks to a diagonal mass matrix (the Jacobian being evaluated at the center of mass of the payload)
    • +
  • coupled at low frequency due to the non-diagonal terms in the stiffness matrix, especially the term corresponding to a coupling between a force in the x direction to a rotation around z (due to the torque applied by the stiffness 1).
    • +
+ +

+The choice of the frame in this the Jacobian is evaluated is discussed in Section 1.12. +

+ + +
+

gravimeter_jacobian_plant.png +

+

Figure 6: Diagonal and off-diagonal elements of \(G_x\)

+
+
+
+ +
+

1.5 Decoupling using the SVD

+
+

+ +

+ +

+In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required. +

+ +

+Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). +

+
+
wc = 2*pi*10; % Decoupling frequency [rad/s]
+
+H1 = evalfr(G, j*wc);
+
+
+ +

+The real approximation is computed as follows: +

+
+
D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
0.0092-0.00390.0039
-0.00390.00480.00028
-0.0040.0038-0.0038
8.4e-090.00250.0025
+ + +

+Now, the Singular Value Decomposition of \(H_1\) is performed: +\[ H_1 = U \Sigma V^H \] +

+ +
+
[U,S,V] = svd(H1);
+
+
+ + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 2: \(U\) matrix
-0.780.26-0.53-0.2
0.40.61-0.04-0.68
0.48-0.14-0.850.2
0.030.730.060.68
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 3: \(V\) matrix
-0.790.11-0.6
0.510.67-0.54
-0.350.730.59
+ +

+The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 7. +

+ + +
+

gravimeter_decouple_svd.png +

+

Figure 7: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

+
+ +

+The decoupled plant is then: +\[ \bm{G}_{SVD}(s) = U^{-1} \bm{G}(s) V^{-H} \] +

+ +
+
Gsvd = inv(U)*G*inv(V');
+
+
+ +
+size(Gsvd)
+State-space model with 4 outputs, 3 inputs, and 6 states.
+
+ + +

+The 4th output (corresponding to the null singular value) is discarded, and we only keep the \(3 \times 3\) plant: +

+
+
Gsvd = Gsvd(1:3, 1:3);
+
+
+ +

+The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8. +

+ +
+

gravimeter_svd_plant.png +

+

Figure 8: Diagonal and off-diagonal elements of \(G_{svd}\)

+
+
+
+ +
+

1.6 Verification of the decoupling using the “Gershgorin Radii”

+
+

+ +

+ +

+The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\): +

+ +

+The “Gershgorin Radii” of a matrix \(S\) is defined by: +\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \] +

+ + +
+

gravimeter_gershgorin_radii.png +

+

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

+
+
+
+ +
+

1.7 Verification of the decoupling using the “Relative Gain Array”

+
+

+ +

+ +

+The relative gain array (RGA) is defined as: +

+\begin{equation} + \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T +\end{equation} +

+where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix. +

+ +

+The obtained RGA elements are shown in Figure 10. +

+ + +
+

gravimeter_rga.png +

+

Figure 10: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

+
+ +

+The RGA-number is also a measure of diagonal dominance: +

+\begin{equation} + \text{RGA-number} = \| \Lambda(G) - I \|_\text{sum} +\end{equation} + + +
+

gravimeter_rga_num.png +

+

Figure 11: RGA-Number for the Gravimeter

+
+
+
+ +
+

1.8 Obtained Decoupled Plants

+
+

+ +

+ +

+The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 12. +

+ + +
+

gravimeter_decoupled_plant_svd.png +

+

Figure 12: Decoupled Plant using SVD

+
+ +

+Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 13. +

+ + +
+

gravimeter_decoupled_plant_jacobian.png +

+

Figure 13: Gravimeter Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

+
+
+
+ +
+

1.9 Diagonal Controller

+
+

+ +The control diagram for the centralized control is shown in Figure 14. +

+ +

+The controller \(K_c\) is “working” in an cartesian frame. +The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators. +

+ + +
+

centralized_control_gravimeter.png +

+

Figure 14: Control Diagram for the Centralized control

+
+ +

+The SVD control architecture is shown in Figure 15. +The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). +

+ + +
+

svd_control_gravimeter.png +

+

Figure 15: Control Diagram for the SVD control

+
+ + +

+We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] +

+ +

+\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) +

+ +
+
wc = 2*pi*10;  % Crossover Frequency [rad/s]
+w0 = 2*pi*0.1; % Controller Pole [rad/s]
+
+
+ +
+
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_cen = K_cen*Gx;
+
+
+ +
+
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_svd = K_svd*Gsvd;
+U_inv = inv(U);
+
+
+ +

+The obtained diagonal elements of the loop gains are shown in Figure 16. +

+ + +
+

gravimeter_comp_loop_gain_diagonal.png +

+

Figure 16: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

+
+
+
+ +
+

1.10 Closed-Loop system Performances

+
+

+ +

+ +

+Now the system is identified again with additional inputs and outputs: +

+
    +
  • \(x\), \(y\) and \(R_z\) ground motion
    • +
  • \(x\), \(y\) and \(R_z\) acceleration of the payload.
    • +
+ +
+
%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
+G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+ +

+The loop is closed using the developed controllers. +

+
+
G_cen = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
+G_svd = lft(G, -inv(V')*K_svd*U_inv(1:3, :));
+
+
+ +

+Let’s first verify the stability of the closed-loop systems: +

+
+
isstable(G_cen)
+
+
+ +
+ans =
+  logical
+   1
+
+ + +
+
isstable(G_svd)
+
+
+ +
+ans =
+  logical
+   1
+
+ + +

+The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 17. +

+ + +
+

gravimeter_platform_simscape_cl_transmissibility.png +

+

Figure 17: Obtained Transmissibility

+
+ + +
+

gravimeter_cl_transmissibility_coupling.png +

+

Figure 18: Obtain coupling terms of the transmissibility matrix

+
+
+
+ + +
+

1.11 Robustness to a change of actuator position

+
+

+ +

+ +

+Let say we change the position of the actuators: +

+
+
la = l/2*0.7; % Position of Act. [m]
+ha = h/2*0.7; % Position of Act. [m]
+
+
+ +
+
%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Dx'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Dy'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Rz'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Abs_Motion'], 3, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'Dx', 'Dy', 'Rz', 'F1', 'F2', 'F3'};
+G.OutputName = {'Ax', 'Ay', 'Arz', 'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+ +

+The loop is closed using the developed controllers. +

+
+
G_cen_b = lft(G, -pinv(Jt')*K_cen*pinv(Ja));
+G_svd_b = lft(G, -inv(V')*K_svd*U_inv(1:3, :));
+
+
+ +

+The new plant is computed, and the centralized and SVD control architectures are applied using the previously computed Jacobian matrices and \(U\) and \(V\) matrices. +

+ +

+The closed-loop system are still stable in both cases, and the obtained transmissibility are equivalent as shown in Figure 19. +

+ + +
+

gravimeter_transmissibility_offset_act.png +

+

Figure 19: Transmissibility for the initial CL system and when the position of actuators are changed

+
+
+
+ +
+

1.12 Choice of the reference frame for Jacobian decoupling

+
+

+ +

+

+If we want to decouple the system at low frequency (determined by the stiffness matrix), we have to compute the Jacobian at a point where the stiffness matrix is diagonal. +A displacement (resp. rotation) of the mass at this particular point should induce a pure force (resp. torque) on the same point due to stiffnesses in the system. +This can be verified by geometrical computations. +

+ + +

+If we want to decouple the system at high frequency (determined by the mass matrix), we have tot compute the Jacobians at the Center of Mass of the suspended solid. +Similarly to the stiffness analysis, when considering only the inertia effects (neglecting the stiffnesses), a force (resp. torque) applied at this point (the center of mass) should induce a pure acceleration (resp. angular acceleration). +

+ + +

+Ideally, we would like to have a decoupled mass matrix and stiffness matrix at the same time. +To do so, the actuators (springs) should be positioned such that the stiffness matrix is diagonal when evaluated at the CoM of the solid. +

+
+
+

1.12.1 Decoupling of the mass matrix

+
+ +
+

gravimeter_model_M.png +

+

Figure 20: Choice of {O} such that the Mass Matrix is Diagonal

+
+ +
+
la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
+
+
+ +
+
%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+ +

+Decoupling at the CoM (Mass decoupled) +

+
+
JMa = [1 0 -h/2
+       0 1  l/2
+       1 0  h/2
+       0 1  0];
+
+JMt = [1 0 -ha
+       0 1  la
+       0 1 -la];
+
+
+ +
+
GM = pinv(JMa)*G*pinv(JMt');
+GM.InputName  = {'Fx', 'Fy', 'Mz'};
+GM.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+
+ + +
+

jac_decoupling_M.png +

+

Figure 21: Diagonal and off-diagonal elements of the decoupled plant

+
+
+
+ +
+

1.12.2 Decoupling of the stiffness matrix

+
+ +
+

gravimeter_model_K.png +

+

Figure 22: Choice of {O} such that the Stiffness Matrix is Diagonal

+
+ +

+Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame): +

+
+
JKa = [1 0  0
+       0 1 -l/2
+       1 0 -h
+       0 1  0];
+
+JKt = [1 0  0
+       0 1 -la
+       0 1  la];
+
+
+ +

+And the plant \(\bm{G}_x\) is computed: +

+
+
GK = pinv(JKa)*G*pinv(JKt');
+GK.InputName  = {'Fx', 'Fy', 'Mz'};
+GK.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+
+ + +
+

jac_decoupling_K.png +

+

Figure 23: Diagonal and off-diagonal elements of the decoupled plant

+
+
+
+ +
+

1.12.3 Combined decoupling of the mass and stiffness matrices

+
+ +
+

gravimeter_model_KM.png +

+

Figure 24: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal

+
+ +

+To do so, the actuator position should be modified +

+ +
+
la = l/2; % Position of Act. [m]
+ha = 0; % Position of Act. [m]
+
+
+ +
+
%% Name of the Simulink File
+mdl = 'gravimeter';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/F1'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F2'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F3'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
+
+G = linearize(mdl, io);
+G.InputName  = {'F1', 'F2', 'F3'};
+G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
+
+
+ +
+
JMa = [1 0 -h/2
+       0 1  l/2
+       1 0  h/2
+       0 1  0];
+
+JMt = [1 0 -ha
+       0 1  la
+       0 1 -la];
+
+
+ +
+
GKM = pinv(JMa)*G*pinv(JMt');
+GKM.InputName  = {'Fx', 'Fy', 'Mz'};
+GKM.OutputName  = {'Dx', 'Dy', 'Rz'};
+
+
+ + +
+

jac_decoupling_KM.png +

+

Figure 25: Diagonal and off-diagonal elements of the decoupled plant

+
+
+
+ +
+

1.12.4 Conclusion

+
+

+Ideally, the mechanical system should be designed in order to have a decoupled stiffness matrix at the CoM of the solid. +

+ +

+If not the case, the system can either be decoupled as low frequency if the Jacobian are evaluated at a point where the stiffness matrix is decoupled. +Or it can be decoupled at high frequency if the Jacobians are evaluated at the CoM. +

+
+
+
+ +
+

1.13 SVD decoupling performances

+
+

+ +As the SVD is applied on a real approximation of the plant dynamics at a frequency \(\omega_0\), it is foreseen that the effectiveness of the decoupling depends on the validity of the real approximation. +

+ +

+Let’s do the SVD decoupling on a plant that is mostly real (low damping) and one with a large imaginary part (larger damping). +

+ +

+Start with small damping, the obtained diagonal and off-diagonal terms are shown in Figure 26. +

+
+
c = 2e1; % Actuator Damping [N/(m/s)]
+
+
+ + +
+

gravimeter_svd_low_damping.png +

+

Figure 26: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with small damping

+
+ +

+Now take a larger damping, the obtained diagonal and off-diagonal terms are shown in Figure 27. +

+
+
c = 5e2; % Actuator Damping [N/(m/s)]
+
+
+ + +
+

gravimeter_svd_high_damping.png +

+

Figure 27: Diagonal and off-diagonal term when decoupling with SVD on the gravimeter with high damping

+
+
+
+
+ +
+

2 Parallel Manipulator with Collocated actuator/sensor pairs

+
+

+ +

+

+In this section, we will see how the Jacobian matrix can be used to decouple a specific set of mechanical systems (described in Section 2.1). +

+ +

+The basic decoupling architecture is shown in Figure 29 where the Jacobian matrix is used to both compute the actuator forces from forces/torques that are to be applied in a specific defined frame, and to compute the displacement/rotation of the same mass from several sensors. +

+ +

+This is rapidly explained in Section 2.2. +

+ + +
+

block_diagram_jacobian_decoupling.png +

+
+ +

+Depending on the chosen frame, the Stiffness matrix in that particular frame can be computed. +This is explained in Section 2.3. +

+ +

+Then three decoupling in three specific frames is studied: +

+
    +
  • Section 2.4: control in the frame of the legs
    • +
  • Section 2.5: control in a frame whose origin is at the center of mass of the payload
    • +
  • Section 2.6: control in a frame whose origin is located at the “center of stiffness” of the system
    • +
+ +

+Conclusions are drawn in Section 2.7. +

+
+ +
+

2.1 Model

+
+

+ +

+ +

+Let’s consider a parallel manipulator with several collocated actuator/sensors pairs. +

+ +

+System in Figure 29 will serve as an example. +

+ +

+We will note: +

+
    +
  • \(b_i\): location of the joints on the top platform
    • +
  • \(\hat{s}_i\): unit vector corresponding to the struts direction
    • +
  • \(k_i\): stiffness of the struts
    • +
  • \(\tau_i\): actuator forces
    • +
  • \(O_M\): center of mass of the solid body
    • +
  • \(\mathcal{L}_i\): relative displacement of the struts
    • +
+ + +
+

gravimeter_model_analytical.png +

+

Figure 29: Model of the gravimeter

+
+ +

+The parameters are defined as follows: +

+
+
l  = 1.0; % Length of the mass [m]
+h  = 2*1.7; % Height of the mass [m]
+
+la = l/2; % Position of Act. [m]
+ha = h/2; % Position of Act. [m]
+
+m = 400; % Mass [kg]
+I = 115; % Inertia [kg m^2]
+
+c1 = 2e1; % Actuator Damping [N/(m/s)]
+c2 = 2e1; % Actuator Damping [N/(m/s)]
+c3 = 2e1; % Actuator Damping [N/(m/s)]
+
+k1 = 15e3; % Actuator Stiffness [N/m]
+k2 = 15e3; % Actuator Stiffness [N/m]
+k3 = 15e3; % Actuator Stiffness [N/m]
+
+
+ +

+Let’s express \({}^Mb_i\) and \(\hat{s}_i\): +

+\begin{align} +{}^Mb_1 &= [-l/2,\ -h_a] \\ +{}^Mb_2 &= [-la, \ -h/2] \\ +{}^Mb_3 &= [ la, \ -h/2] +\end{align} + +\begin{align} +\hat{s}_1 &= [1,\ 0] \\ +\hat{s}_2 &= [0,\ 1] \\ +\hat{s}_3 &= [0,\ 1] +\end{align} + +
+
s1 = [1;0];
+s2 = [0;1];
+s3 = [0;1];
+
+Mb1 = [-l/2;-ha];
+Mb2 = [-la; -h/2];
+Mb3 = [ la; -h/2];
+
+
+ +

+Frame \(\{K\}\) is chosen such that the stiffness matrix is diagonal (explained in Section 3). +

+ +

+The positions \({}^Kb_i\) are then: +

+\begin{align} +{}^Kb_1 &= [-l/2,\ 0] \\ +{}^Kb_2 &= [-la, \ -h/2+h_a] \\ +{}^Kb_3 &= [ la, \ -h/2+h_a] +\end{align} + +
+
Kb1 = [-l/2; 0];
+Kb2 = [-la; -h/2+ha];
+Kb3 = [ la; -h/2+ha];
+
+
+
+
+ +
+

2.2 The Jacobian Matrix

+
+

+ +

+ +

+Let’s note: +

+
    +

  • +\(\bm{\mathcal{L}}\) the vector of actuator displacement: +

    +\begin{equation} + \bm{\mathcal{L}} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix} +\end{equation}
    • +

  • +\(\bm{\tau}\) the vector of actuator forces: +

    +\begin{equation} + \bm{\tau} = \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} +\end{equation}
    • +

  • +\(\bm{\mathcal{F}}_{\{O\}}\) the vector of forces/torques applied on the payload on expressed in frame \(\{O\}\): +

    +\begin{equation} + \bm{\mathcal{F}}_{\{O\}} = \begin{bmatrix} \mathcal{F}_{\{O\},x} \\ \mathcal{F}_{\{O\},y} \\ \mathcal{M}_{\{O\},z} \end{bmatrix} +\end{equation}
    • +

  • +\(\bm{\mathcal{X}}_{\{O\}}\) the vector of displacement of the payload with respect to frame \(\{O\}\): +

    +\begin{equation} + \bm{\mathcal{X}}_{\{O\}} = \begin{bmatrix} \mathcal{X}_{\{O\},x} \\ \mathcal{X}_{\{O\},y} \\ \mathcal{X}_{\{O\},R_z} \end{bmatrix} +\end{equation}
    • +
+ + + +

+The Jacobian matrix can be used to: +

+
    +
  • Convert joints velocity \(\dot{\mathcal{L}}\) to payload velocity and angular velocity \(\dot{\bm{\mathcal{X}}}_{\{O\}}\): +\[ \dot{\bm{\mathcal{X}}}_{\{O\}} = J_{\{O\}} \dot{\bm{\mathcal{L}}} \]
    • +
  • Convert actuators forces \(\bm{\tau}\) to forces/torque applied on the payload \(\bm{\mathcal{F}}_{\{O\}}\): +\[ \bm{\mathcal{F}}_{\{O\}} = J_{\{O\}}^T \bm{\tau} \]
    • +
+

+with \(\{O\}\) any chosen frame. +

+ +

+If we consider small displacements, we have an approximate relation that links the displacements (instead of velocities): +

+\begin{equation} +\bm{\mathcal{X}}_{\{M\}} = J_{\{M\}} \bm{\mathcal{L}} +\end{equation} + + +

+The Jacobian can be computed as follows: +

+\begin{equation} +J_{\{O\}} = \begin{bmatrix} + {}^O\hat{s}_1^T & {}^Ob_{1,x} {}^O\hat{s}_{1,y} - {}^Ob_{1,x} {}^O\hat{s}_{1,y} \\ + {}^O\hat{s}_2^T & {}^Ob_{2,x} {}^O\hat{s}_{2,y} - {}^Ob_{2,x} {}^O\hat{s}_{2,y} \\ + \vdots & \vdots \\ + {}^O\hat{s}_n^T & {}^Ob_{n,x} {}^O\hat{s}_{n,y} - {}^Ob_{n,x} {}^O\hat{s}_{n,y} \\ +\end{bmatrix} +\end{equation} + + +

+Let’s compute the Jacobian matrix in frame \(\{M\}\) and \(\{K\}\): +

+
+
Jm = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
+      s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
+      s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 4: Jacobian Matrix \(J_{\{M\}}\)
101.7
01-0.5
010.5
+ +
+
Jk = [s1', Kb1(1)*s1(2)-Kb1(2)*s1(1);
+      s2', Kb2(1)*s2(2)-Kb2(2)*s2(1);
+      s3', Kb3(1)*s3(2)-Kb3(2)*s3(1)];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 5: Jacobian Matrix \(J_{\{K\}}\)
100
01-0.5
010.5
+ +

+In the frame \(\{M\}\), the Jacobian is: +

+\begin{equation} +J_{\{M\}} = \begin{bmatrix} 1 & 0 & h_a \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} +\end{equation} + +

+And in frame \(\{K\}\), the Jacobian is: +

+\begin{equation} +J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} +\end{equation} +
+
+ +
+

2.3 The Stiffness Matrix

+
+

+ +

+ +

+For a parallel manipulator, the stiffness matrix expressed in a frame \(\{O\}\) is: +

+\begin{equation} + K_{\{O\}} = J_{\{O\}}^T \mathcal{K} J_{\{O\}} +\end{equation} +

+where: +

+
    +
  • \(J_{\{O\}}\) is the Jacobian matrix expressed in frame \(\{O\}\)
    • +

  • +\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal +

    +\begin{equation} +\mathcal{K} = \begin{bmatrix} + k_1 & & & 0 \\ + & k_2 & & \\ + & & \ddots & \\ + 0 & & & k_n +\end{bmatrix} +\end{equation}
    • +
+ +

+We have the same thing for the damping matrix. +

+ +
+
Kr = diag([k1,k2,k3]);
+Cr = diag([c1,c2,c3]);
+
+
+
+
+ +
+

2.4 Equations of motion - Frame of the legs

+
+

+ +

+ +

+Applying the second Newton’s law on the system in Figure 29 at its center of mass \(O_M\), we obtain: +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}} +\end{equation} +

+with: +

+
    +
  • \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): +\[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
    • +
  • \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): +\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]
    • +
  • \(\bm{\mathcal{X}}_{\{M\}}\) are displacements/rotations of the mass \(x\), \(y\), \(R_z\) expressed in the frame \(\{M\}\)
    • +
  • \(\bm{\mathcal{F}}_{\{M\}}\) are forces/torques \(\mathcal{F}_x\), \(\mathcal{F}_y\), \(\mathcal{M}_z\) applied at the origin of \(\{M\}\)
    • +
+ +

+Let’s use the Jacobian matrix to compute the equations in terms of actuator forces \(\bm{\tau}\) and strut displacement \(\bm{\mathcal{L}}\): +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \bm{\mathcal{L}} = J_{\{M\}}^T \bm{\tau} +\end{equation} + +

+And we obtain: +

+\begin{equation} +\left( J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} s^2 + \mathcal{K} \right) \bm{\mathcal{L}} = \bm{\tau} +\end{equation} + +

+The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is: +

+\begin{equation} +\boxed{\bm{G}(s) = {\left( J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} s^2 + \mathcal{K} \right)}^{-1}} +\end{equation} + + +
+

gravimeter_block_decentralized.png +

+

Figure 30: Block diagram of the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\)

+
+ +
+
%% Mass Matrix in frame {M}
+Mm = diag([m,m,I]);
+
+
+ +

+Let’s note the mass matrix in the frame of the legs: +

+\begin{equation} +M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} +\end{equation} + +
+
%% Mass Matrix in the frame of the struts
+Ml = inv(Jm')*Mm*inv(Jm);
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 6: \(M_{\{L\}}\)
400680-680
6801371-1171
-680-11711371
+ +

+As we can see, the Stiffness matrix in the frame of the legs is diagonal. +This means the plant dynamics will be diagonal at low frequency. +

+
+
Kl = diag([k1, k2, k3]);
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 7: \(K_{\{L\}} = \mathcal{K}\)
1500000
0150000
0015000
+ +
+
Cl = diag([c1, c2, c3]);
+
+
+ +

+The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is defined below and its magnitude is shown in Figure 31. +

+
+
Gl = inv(Ml*s^2 + Cl*s + Kl);
+
+
+ +

+We can indeed see that the system is well decoupled at low frequency. +

+ + +
+

plant_frame_L.png +

+

Figure 31: Dynamics from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\)

+
+
+
+ +
+

2.5 Equations of motion - “Center of mass” {M}

+
+

+ +

+ +

+The equations of motion expressed in frame \(\{M\}\) are: +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}} +\end{equation} + +

+And the plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is: +

+\begin{equation} +\boxed{\bm{G}_{\{X\}} = {\left( M_{\{M\}} s^2 + K_{\{M\}} \right)}^{-1}} +\end{equation} +

+with: +

+
    +
  • \(M_{\{M\}}\) is the mass matrix expressed in \(\{M\}\): +\[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \]
    • +
  • \(K_{\{M\}}\) is the stiffness matrix expressed in \(\{M\}\): +\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \]
    • +
+ + +
+

gravimeter_block_com.png +

+

Figure 32: Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\)

+
+ + +
+
%% Mass Matrix in frame {M}
+Mm = diag([m,m,I]);
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 8: Mass matrix expressed in \(\{M\}\): \(M_{\{M\}}\)
40000
04000
00115
+ +
+
%% Stiffness Matrix in frame {M}
+Km = Jm'*Kr*Jm;
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 9: Stiffness matrix expressed in \(\{M\}\): \(K_{\{M\}}\)
15000025500
0300000
25500050850
+ +
+
%% Damping Matrix in frame {M}
+Cm = Jm'*Cr*Jm;
+
+
+ +

+The plant from \(\bm{F}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\) is defined below and its magnitude is shown in Figure 33. +

+
+
%% Plant in frame {M}
+Gm = inv(Mm*s^2 + Cm*s + Km);
+
+
+ + +
+

plant_frame_M.png +

+

Figure 33: Dynamics from \(\bm{\mathcal{F}}_{\{M\}}\) to \(\bm{\mathcal{X}}_{\{M\}}\)

+
+
+
+ +
+

2.6 Equations of motion - “Center of stiffness” {K}

+
+

+ +

+ +

+Let’s now express the transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\). +We start from: +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \bm{\mathcal{L}} = J_{\{M\}}^T \bm{\tau} +\end{equation} + +

+And we make use of the Jacobian \(J_{\{K\}}\) to obtain: +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} J_{\{K\}} \bm{\mathcal{X}}_{\{K\}} = J_{\{M\}}^T J_{\{K\}}^{-T} \bm{\mathcal{F}}_{\{K\}} +\end{equation} + +

+And finally: +

+\begin{equation} +\left( J_{\{K\}}^T J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} J_{\{K\}} s^2 + J_{\{K\}}^T \mathcal{K} J_{\{K\}} \right) \bm{\mathcal{X}}_{\{K\}} = \bm{\mathcal{F}}_{\{K\}} +\end{equation} + +

+The transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is then: +

+\begin{equation} +\boxed{\bm{G}_{\{K\}} = {\left( J_{\{K\}}^T J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} J_{\{K\}} s^2 + J_{\{K\}}^T \mathcal{K} J_{\{K\}} \right)}^{-1}} +\end{equation} + +

+The frame \(\{K\}\) has been chosen such that \(J_{\{K\}}^T \mathcal{K} J_{\{K\}}\) is diagonal. +

+ + +
+

gravimeter_block_cok.png +

+

Figure 34: Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\)

+
+ +
+
Mk = Jk'*inv(Jm)'*Mm*inv(Jm)*Jk;
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 10: Mass matrix expressed in \(\{K\}\): \(M_{\{K\}}\)
4000-680
04000
-68001271
+ +
+
Kk = Jk'*Kr*Jk;
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 11: Stiffness matrix expressed in \(\{K\}\): \(K_{\{K\}}\)
1500000
0300000
007500
+ +

+The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined below and its magnitude is shown in Figure 35. +

+
+
Gk = inv(Mk*s^2 + Ck*s + Kk);
+
+
+ + +
+

plant_frame_K.png +

+

Figure 35: Dynamics from \(\bm{\mathcal{F}}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\)

+
+
+
+ +
+

2.7 Conclusion

+
+

+ +

+
+
+
+ +
+

3 Diagonal Stiffness Matrix for a planar manipulator

+
+

+ +

+
+
+

3.1 Model and Assumptions

+
+

+Consider a parallel manipulator with: +

+
    +
  • \(b_i\): location of the joints on the top platform are called \(b_i\)
    • +
  • \(\hat{s}_i\): unit vector corresponding to the struts
    • +
  • \(k_i\): stiffness of the struts
    • +
  • \(\tau_i\): actuator forces
    • +
  • \(O_M\): center of mass of the solid body
    • +
+ +

+Consider two frames: +

+
    +
  • \(\{M\}\) with origin \(O_M\)
    • +
  • \(\{K\}\) with origin \(O_K\)
    • +
+ +

+As an example, take the system shown in Figure 36. +

+ + +
+

3dof_model_fully_parallel.png +

+

Figure 36: Example of 3DoF parallel platform

+
+
+
+ +
+

3.2 Objective

+
+

+The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. +If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically. +

+
+
+ +
+

3.3 Conditions for Diagonal Stiffness

+
+

+The stiffness matrix in the frame \(\{K\}\) can be expressed as: +

+\begin{equation} \label{eq:stiffness_formula_planar} + K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}} +\end{equation} +

+where: +

+
    +
  • \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)
    • +

  • +\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal +

    +\begin{equation} +\mathcal{K} = \begin{bmatrix} + k_1 & & & 0 \\ + & k_2 & & \\ + & & \ddots & \\ + 0 & & & k_n +\end{bmatrix} +\end{equation}
    • +
+ +

+The Jacobian for a planar manipulator, evaluated in a frame \(\{K\}\), can be expressed as follows: +

+\begin{equation} \label{eq:jacobian_planar} +J_{\{K\}} = \begin{bmatrix} + {}^K\hat{s}_1^T & {}^Kb_{1,x} {}^K\hat{s}_{1,y} - {}^Kb_{1,x} {}^K\hat{s}_{1,y} \\ + {}^K\hat{s}_2^T & {}^Kb_{2,x} {}^K\hat{s}_{2,y} - {}^Kb_{2,x} {}^K\hat{s}_{2,y} \\ + \vdots & \vdots \\ + {}^K\hat{s}_n^T & {}^Kb_{n,x} {}^K\hat{s}_{n,y} - {}^Kb_{n,x} {}^K\hat{s}_{n,y} \\ +\end{bmatrix} +\end{equation} + +

+Let’s omit the mention of frame, it is assumed that vectors are expressed in frame \(\{K\}\). +It is specified otherwise. +

+ +

+Injecting \eqref{eq:jacobian_planar} into \eqref{eq:stiffness_formula_planar} yields: +

+\begin{equation} +\boxed{ +K_{\{K\}} = \left[ \begin{array}{c|c} + k_i \hat{s}_i \hat{s}_i^T & k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) \cr + \hline + k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) & k_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x})^2 +\end{array} \right] +} +\end{equation} + +

+In order to have a decoupled stiffness matrix, we have the following two conditions: +

+\begin{align} +k_i \hat{s}_i \hat{s}_i^T &= \text{diag. matrix} \label{eq:diag_cond_2D_1} \\ +k_i \hat{s}_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x}) &= 0 \label{eq:diag_cond_2D_2} +\end{align} + +

+Note that we don’t have any condition on the term \(k_i (b_{i,x}\hat{s}_{i,y} - b_{i,y}\hat{s}_{i,x})^2\) as it is only a scalar. +

+ +

+Condition \eqref{eq:diag_cond_2D_1}: +

+
    +
  • represents the coupling between translations and forces
    • +
  • does only depends on the orientation of the struts and the stiffnesses and not on the choice of frame
    • +
  • it is therefore a intrinsic property of the chosen geometry
    • +
+ +

+Condition \eqref{eq:diag_cond_2D_2}: +

+
    +
  • represents the coupling between forces/rotations and torques/translation
    • +
  • it does depend on the positions of the joints \(b_i\) in the frame \(\{K\}\)
    • +
+ +

+Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\): +

+\begin{align} +{}^Kb_i &= {}^Mb_i - {}^MO_K \\ +{}^K\hat{s}_i &= {}^M\hat{s}_i +\end{align} + +

+And the goal is to find \({}^MO_K\) such that \eqref{eq:diag_cond_2D_2} is fulfilled: +

+\begin{equation} +k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x} - {}^MO_{K,x}\hat{s}_{i,y} + {}^MO_{K,y}\hat{s}_{i,x}) \hat{s}_i = 0 +\end{equation} +\begin{equation} +k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i = {}^MO_{K,x} k_i \hat{s}_{i,y} \hat{s}_i - {}^MO_{K,y} k_i \hat{s}_{i,x} \hat{s}_i +\end{equation} + +

+And we have two sets of linear equations of two unknowns. +

+ +

+This can be easily solved by writing the equations in a matrix form: +

+\begin{equation} +\underbrace{k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i}_{2 \times 1} = +\underbrace{\begin{bmatrix} +& \\ +k_i \hat{s}_{i,y} \hat{s}_i & - k_i \hat{s}_{i,x} \hat{s}_i \\ +& \\ +\end{bmatrix}}_{2 \times 2} +\underbrace{\begin{bmatrix} +{}^MO_{K,x}\\ +{}^MO_{K,y} +\end{bmatrix}}_{2 \times 1} +\end{equation} + +

+And finally, if the matrix is invertible: +

+\begin{equation} +\boxed{ +{}^MO_K = {\begin{bmatrix} +& \\ +k_i \hat{s}_{i,y} \hat{s}_i & - k_i \hat{s}_{i,x} \hat{s}_i \\ +& \\ +\end{bmatrix}}^{-1} k_i ({}^Mb_{i,x}\hat{s}_{i,y} - {}^Mb_{i,y}\hat{s}_{i,x}) \hat{s}_i +} +\end{equation} + +

+Note that a rotation of the frame \(\{K\}\) with respect to frame \(\{M\}\) would make not change on the “diagonality” of \(K_{\{K\}}\). +

+
+
+ +
+

3.4 Example 1 - Planar manipulator with 3 actuators

+
+

+Consider system of Figure 37. +

+ + +
+

3dof_model_fully_parallel.png +

+

Figure 37: Example of 3DoF parallel platform

+
+ +

+The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: +

+
+
ki = [5,1,2]; % Stiffnesses [N/m]
+si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
+bi = [[-1;0.5],[-2;-1],[0;-1]]; % Joint's positions in frame {M}
+
+
+ +

+Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true: +

+ + + +++ ++ + + + + + + + + + + + +
50
02
+ +

+Now, compute \({}^MO_K\): +

+
+
Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);
+
+
+ + + + +++ + + + + + + + + + +
-1
0.5
+ +

+Let’s compute the new coordinates \({}^Kb_i\) after the change of frame: +

+
+
Kbi = bi - Ok;
+
+
+ +

+In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\): +

+
+
Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
100
01-1
011
+ +

+And the stiffness matrix: +

+
+
K = Jk'*diag(ki)*Jk
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
500
020
002
+
+
+ +
+

3.5 Example 2 - Planar manipulator with 4 actuators

+
+

+Now consider the planar manipulator of Figure 38. +

+ + +
+

model_planar_2.png +

+

Figure 38: Planar Manipulator

+
+ +

+The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: +

+
+
ki = [1,2,1,1];
+si = [[1;0],[0;1],[-1;0],[0;1]];
+si = si./vecnorm(si);
+h = 0.2;
+L = 2;
+bi = [[-L/2;h],[-L/2;-h],[L/2;h],[L/2;h]];
+
+
+ +

+Let’s first verify that condition \eqref{eq:diag_cond_2D_1} is true: +

+
+
ki.*si*si'
+
+
+ + + + +++ ++ + + + + + + + + + + + +
20
03
+ +

+Now, compute \({}^MO_K\): +

+
+
Ok = inv([sum(ki.*si(2,:).*si, 2), -sum(ki.*si(1,:).*si, 2)])*sum(ki.*(bi(1,:).*si(2,:) - bi(2,:).*si(1,:)).*si, 2);
+
+
+ + + + +++ + + + + + + + + + +
-0.33333
0.2
+ +

+Let’s compute the new coordinates \({}^Kb_i\) after the change of frame: +

+
+
Kbi = bi - Ok;
+
+
+ +

+In order to verify that the new frame \(\{K\}\) indeed yields a diagonal stiffness matrix, we first compute the Jacobian \(J_{\{K\}}\): +

+
+
Jk = [si', (Kbi(1,:).*si(2,:) - Kbi(2,:).*si(1,:))'];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + +
100
01-0.66667
-100
011.3333
+ +

+And the stiffness matrix: +

+
+
K = Jk'*diag(ki)*Jk
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
200
03-2.2204e-16
0-2.2204e-162.6667
+
+
+
+ +
+

4 Diagonal Stiffness Matrix for a general parallel manipulator

+
+
+
+

4.1 Model and Assumptions

+
+

+Let’s consider a 6dof parallel manipulator with: +

+
    +
  • \(b_i\): location of the joints on the top platform are called \(b_i\)
    • +
  • \(\hat{s}_i\): unit vector corresponding to the struts
    • +
  • \(k_i\): stiffness of the struts
    • +
  • \(\tau_i\): actuator forces
    • +
  • \(O_M\): center of mass of the solid body
    • +
+ +

+Consider two frames: +

+
    +
  • \(\{M\}\) with origin \(O_M\)
    • +
  • \(\{K\}\) with origin \(O_K\)
    • +
+ +

+An example is shown in Figure 39. +

+ + +
+

stewart_architecture_example.png +

+

Figure 39: Parallel manipulator Example

+
+
+
+ +
+

4.2 Objective

+
+

+The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal. +If the conditions are fulfilled, a second objective is to fine the location of the frame \(\{K\}\) analytically. +

+
+
+ +
+

4.3 Analytical formula of the stiffness matrix

+
+

+For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed in a frame \(\{K\}\) is: +

+\begin{equation} + K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}} +\end{equation} +

+where: +

+
    +
  • \(J_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)
    • +

  • +\(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal: +

    +\begin{equation} +\mathcal{K} = \begin{bmatrix} + k_1 & & & 0 \\ + & k_2 & & \\ + & & \ddots & \\ + 0 & & & k_n +\end{bmatrix} +\end{equation}
    • +
+ + +

+The analytical expression of \(J_{\{K\}}\) is: +

+\begin{equation} +J_{\{K\}} = \begin{bmatrix} + {}^K\hat{s}_1^T & ({}^Kb_1 \times {}^K\hat{s}_1)^T \\ + {}^K\hat{s}_2^T & ({}^Kb_2 \times {}^K\hat{s}_2)^T \\ + \vdots & \vdots \\ + {}^K\hat{s}_n^T & ({}^Kb_n \times {}^K\hat{s}_n)^T +\end{bmatrix} +\end{equation} + +

+To simplify, we ignore the superscript \(K\) and we assume that all vectors / positions are expressed in this frame \(\{K\}\). +Otherwise, it is explicitly written. +

+ +

+Let’s now write the analytical expressing of the stiffness matrix \(K_{\{K\}}\): +

+\begin{equation} +K_{\{K\}} = \begin{bmatrix} + \hat{s}_1 & \dots & \hat{s}_n \\ + (b_1 \times \hat{s}_1) & \dots & (b_n \times \hat{s}_n) +\end{bmatrix} +\begin{bmatrix} + k_1 & & \\ + & \ddots & \\ + & & k_n +\end{bmatrix} +\begin{bmatrix} + \hat{s}_1^T & (b_1 \times \hat{s}_1)^T \\ + \hat{s}_2^T & (b_2 \times \hat{s}_2)^T \\ + \vdots & \dots \\ + \hat{s}_n^T & (b_n \times \hat{s}_n)^T +\end{bmatrix} +\end{equation} + +

+And we finally obtain: +

+\begin{equation} +\boxed{ +K_{\{K\}} = \left[ \begin{array}{c|c} + k_i \hat{s}_i \hat{s}_i^T & k_i \hat{s}_i (b_i \times \hat{s}_i)^T \cr + \hline + k_i (b_i \times \hat{s}_i) \hat{s}_i^T & k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T +\end{array} \right] +} +\end{equation} + +

+We want the stiffness matrix to be diagonal, therefore, we have the following conditions: +

+\begin{align} +k_i \hat{s}_i \hat{s}_i^T &= \text{diag. matrix} \label{eq:diag_cond_1} \\ +k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T &= \text{diag. matrix} \label{eq:diag_cond_2} \\ +k_i \hat{s}_i (b_i \times \hat{s}_i)^T &= 0 \label{eq:diag_cond_3} +\end{align} + +

+Note that: +

+
    +
  • condition \eqref{eq:diag_cond_1} corresponds to coupling between forces applied on \(O_K\) to translations of the payload. +It does not depend on the choice of \(\{K\}\), it only depends on the orientation of the struts and the stiffnesses. +It is therefore an intrinsic property of the manipulator.
    • +
  • condition \eqref{eq:diag_cond_2} corresponds to the coupling between forces applied on \(O_K\) and rotation of the payload. +Similarly, it does also correspond to the coupling between torques applied on \(O_K\) to translations of the payload.
    • +
  • condition \eqref{eq:diag_cond_3} corresponds to the coupling between torques applied on \(O_K\) to rotation of the payload.
    • +
  • conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} do depend on the positions \({}^Kb_i\) and therefore depend on the choice of \(\{K\}\).
    • +
+ +

+Note that if we find a frame \(\{K\}\) in which the stiffness matrix \(K_{\{K\}}\) is diagonal, it will still be diagonal for any rotation of the frame \(\{K\}\). +Therefore, we here suppose that the frame \(\{K\}\) is aligned with the initial frame \(\{M\}\). +

+ +

+Let’s make a change of frame from the initial frame \(\{M\}\) to the frame \(\{K\}\): +

+\begin{align} +{}^Kb_i &= {}^Mb_i - {}^MO_K \\ +{}^K\hat{s}_i &= {}^M\hat{s}_i +\end{align} + +

+The goal is to find \({}^MO_K\) such that conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} are fulfilled. +

+ +

+Let’s first solve equation \eqref{eq:diag_cond_3} that corresponds to the coupling between forces and rotations: +

+\begin{equation} +k_i \hat{s}_i (({}^Mb_i - {}^MO_K) \times \hat{s}_i)^T = 0 +\end{equation} + +

+Taking the transpose and re-arranging: +

+\begin{equation} +k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = k_i ({}^MO_K \times \hat{s}_i) \hat{s}_i^T +\end{equation} + +

+As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector, we obtain: +

+\begin{equation} +k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = {}^M\bm{O}_{K} ( k_i \hat{s}_i \hat{s}_i^T ) +\end{equation} + +

+with: +

+\begin{equation} \label{eq:skew_symmetric_cross_product} +{}^M\bm{O}_K = \begin{bmatrix} +0 & -{}^MO_{K,z} & {}^MO_{K,y} \\ +{}^MO_{K,z} & 0 & -{}^MO_{K,x} \\ +-{}^MO_{K,y} & {}^MO_{K,x} & 0 +\end{bmatrix} +\end{equation} + +

+We suppose \(k_i \hat{s}_i \hat{s}_i^T\) invertible as it is diagonal from \eqref{eq:diag_cond_1}. +

+ +

+And finally, we find: +

+\begin{equation} +\boxed{ +{}^M\bm{O}_{K} = \left( k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T\right) \cdot {\left( k_i \hat{s}_i \hat{s}_i^T \right)}^{-1} +} +\end{equation} + +

+If the obtained \({}^M\bm{O}_{K}\) is a skew-symmetric matrix, we can easily determine the corresponding vector \({}^MO_K\) from \eqref{eq:skew_symmetric_cross_product}. +

+ +

+In such case, condition \eqref{eq:diag_cond_2} is fulfilled and there is no coupling between translations and rotations in the frame \(\{K\}\). +

+ +

+Then, we can only verify if condition \eqref{eq:diag_cond_3} is verified or not. +

+ +
+

+If there is no frame \(\{K\}\) such that conditions \eqref{eq:diag_cond_2} and \eqref{eq:diag_cond_3} are valid, it would be interesting to be able to determine the frame \(\{K\}\) in which is coupling is minimal. +

+ +
+
+
+ +
+

4.4 Example 1 - 6DoF manipulator (3D)

+
+

+Let’s define the geometry of the manipulator (\({}^Mb_i\), \({}^Ms_i\) and \(k_i\)): +

+
+
ki = [2,2,1,1,3,3,1,1,1,1,2,2];
+si = [[-1;0;0],[-1;0;0],[-1;0;0],[-1;0;0],[0;0;1],[0;0;1],[0;0;1],[0;0;1],[0;-1;0],[0;-1;0],[0;-1;0],[0;-1;0]];
+bi = [[1;-1;1],[1;1;-1],[1;1;1],[1;-1;-1],[1;-1;-1],[-1;1;-1],[1;1;-1],[-1;-1;-1],[1;1;-1],[-1;1;1],[-1;1;-1],[1;1;1]]-[0;2;-1];
+
+
+ +

+Cond 1: +

+
+
ki.*si*si'
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
600
060
008
+ +

+Find Ok +

+
+
OkX = (ki.*cross(bi, si)*si')/(ki.*si*si');
+
+if all(diag(OkX) == 0) && all(all((OkX + OkX') == 0))
+    disp('OkX is skew symmetric')
+    Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
+else
+    error('OkX is *not* skew symmetric')
+end
+
+
+ + + + +++ + + + + + + + + + + + + + +
0
-2
1
+ +
+
% Verification of second condition
+si*cross(bi-Ok, si)'
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
000
000
000
+ +

+Verification of third condition +

+
+
ki.*cross(bi-Ok, si)*cross(bi-Ok, si)'
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
144-2
4142
-2212
+ +

+Let’s compute the Jacobian: +

+
+
Jk = [si', cross(bi - Ok, si)'];
+
+
+ +

+And the stiffness matrix: +

+
+
Jk'*diag(ki)*Jk
+
+
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
600000
060000
008000
000144-2
0004142
000-2212
+ +
+
figure;
+hold on;
+set(gca,'ColorOrderIndex',1)
+plot(b1(1), b1(2), 'o');
+set(gca,'ColorOrderIndex',2)
+plot(b2(1), b2(2), 'o');
+set(gca,'ColorOrderIndex',3)
+plot(b3(1), b3(2), 'o');
+set(gca,'ColorOrderIndex',1)
+quiver(b1(1),b1(2),0.1*s1(1),0.1*s1(2))
+set(gca,'ColorOrderIndex',2)
+quiver(b2(1),b2(2),0.1*s2(1),0.1*s2(2))
+set(gca,'ColorOrderIndex',3)
+quiver(b3(1),b3(2),0.1*s3(1),0.1*s3(2))
+
+plot(0, 0, 'ko');
+quiver([0,0],[0,0],[0.1,0],[0,0.1], 'k')
+
+plot(Ok(1), Ok(2), 'ro');
+quiver([Ok(1),Ok(1)],[Ok(2),Ok(2)],[0.1,0],[0,0.1], 'r')
+
+hold off;
+axis equal;
+
+
+
+
+ +
+

4.5 Example 2 - Stewart Platform

+
+
+
+

5 Stiffness and Mass Matrices in the Leg’s frame

+
+
+
+

5.1 Equations

+
+

+Equations in the \(\{M\}\) frame: +

+\begin{equation} +\left( M_{\{M\}} s^2 + K_{\{M\}} \right) \mathcal{X}_{\{M\}} = \mathcal{F}_{\{M\}} +\end{equation} + +

+Thank to the Jacobian, we can transform the equation of motion expressed in the \(\{M\}\) frame to the frame of the legs: +

+\begin{equation} +J_{\{M\}}^{-T} \left( M_{\{M\}} s^2 + K_{\{M\}} \right) J_{\{M\}}^{-1} \dot{\mathcal{L}} = \tau +\end{equation} + +

+And we have new stiffness and mass matrices: +

+\begin{equation} +\left( M_{\{L\}} s^2 + K_{\{L\}} \right) \dot{\mathcal{L}} = \tau +\end{equation} +

+with: +

+
    +
  • The local mass matrix: +\[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \]
    • +
  • The local stiffness matrix: +\[ K_{\{L\}} = J_{\{M\}}^{-T} K_{\{M\}} J_{\{M\}}^{-1} \]
    • +
+
+
+ +
+

5.2 Stiffness matrix

+
+

+We have that: +\[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \] +

+ +

+Therefore, we find that \(K_{\{L\}}\) is a diagonal matrix: +

+\begin{equation} +K_{\{L\}} = \mathcal{K} = \begin{bmatrix} +k_1 & & 0 \\ + & \ddots & \\ +0 & & k_n +\end{bmatrix} +\end{equation} + +

+The dynamics from \(\tau\) to \(\mathcal{L}\) is therefore decoupled at low frequency. +

+
+
+ +
+

5.3 Mass matrix

+
+

+The mass matrix in the frames of the legs is: +\[ M_{\{L\}} = J_{\{M\}}^{-T} M_{\{M\}} J_{\{M\}}^{-1} \] +with \(M_{\{M\}}\) a diagonal matrix: +

+\begin{equation} +M_{\{M\}} = \begin{bmatrix} +m & & & & & \\ + & m & & & 0 & \\ + & & m & & & \\ + & & & I_x & & \\ + & 0 & & & I_y & \\ + & & & & & I_z +\end{bmatrix} +\end{equation} + +

+Let’s suppose \(M_{\{L\}} = \mathcal{M}\) diagonal and try to find what does this imply: +\[ M_{\{M\}} = J_{\{M\}}^{T} \mathcal{M} J_{\{M\}} \] +with: +

+\begin{equation} +\mathcal{M} = \begin{bmatrix} +m_1 & & 0 \\ + & \ddots & \\ +0 & & m_n +\end{bmatrix} +\end{equation} + +

+We obtain: +

+\begin{equation} +\boxed{ +M_{\{M\}} = \left[ \begin{array}{c|c} + m_i \hat{s}_i \hat{s}_i^T & m_i \hat{s}_i (b_i \times \hat{s}_i)^T \cr + \hline + k_i \hat{s}_i (b_i \times \hat{s}_i)^T & m_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T +\end{array} \right] +} +\end{equation} + +

+Therefore, we have the following conditions: +

+\begin{align} +m_i \hat{s}_i \hat{s}_i^T &= m \bm{I}_{3} \\ +m_i \hat{s}_i (b_i \times \hat{s}_i)^T &= \bm{O}_{3} \\ +m_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T &= \text{diag}(I_x, I_y, I_z) +\end{align} +
+
+ + +
+

5.4 Planar Example

+
+

+The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below: +

+
+
ki = [1,1,1]; % Stiffnesses [N/m]
+si = [[1;0],[0;1],[0;1]]; si = si./vecnorm(si); % Unit Vectors
+bi = [[-1; 0],[-10;-1],[0;-1]]; % Joint's positions in frame {M}
+
+
+ +

+Jacobian in frame \(\{M\}\): +

+
+
Jm = [si', (bi(1,:).*si(2,:) - bi(2,:).*si(1,:))'];
+
+
+ +

+And the stiffness matrix in frame \(\{K\}\): +

+
+
Km = Jm'*diag(ki)*Jm;
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
201
01-1
1-12
+ +

+Mass matrix in the frame \(\{M\}\): +

+
+
m = 10; % [kg]
+I = 1; % [kg.m^2]
+
+Mm = diag([m, m, I]);
+
+
+ + +

+Now compute \(K\) and \(M\) in the frame of the legs: +

+
+
ML = inv(Jm)'*Mm*inv(Jm)
+KL = inv(Jm)'*Km*inv(Jm)
+
+
+ +
+
Gm = 1/(ML*s^2 + KL);
+
+
+ +
+
freqs = logspace(-2, 1, 1000);
+figure;
+hold on;
+for i = 1:length(ki)
+    plot(freqs, abs(squeeze(freqresp(Gm(i,i), freqs, 'Hz'))), 'k-')
+end
+for i = 1:length(ki)
+    for j = i+1:length(ki)
+        plot(freqs, abs(squeeze(freqresp(Gm(i,j), freqs, 'Hz'))), 'r-')
+    end
+end
+hold off;
+xlabel('Frequency [Hz]');
+ylabel('Magnitude');
+set(gca, 'xscale', 'log');
+set(gca, 'yscale', 'log');
+
+
+
+
+
+ +
+

6 Stewart Platform - Simscape Model

+
+

+ +

+

+In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 40. +

+ +

+Some notes about the system: +

+
    +
  • 6 voice coils actuators are used to control the motion of the top platform.
    • +
  • the motion of the top platform is measured with a 6-axis inertial unit (3 acceleration + 3 angular accelerations)
    • +
  • the control objective is to isolate the top platform from vibrations coming from the bottom platform
    • +
+ + +
+

SP_assembly.png +

+

Figure 40: Stewart Platform CAD View

+
+ +

+The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections: +

+
    +
  • Section 6.1: The parameters of the Simscape model of the Stewart platform are defined
    • +
  • Section 6.2: The plant is identified from the Simscape model and the system coupling is shown
    • +
  • Section 6.3: The plant is first decoupled using the Jacobian
    • +
  • Section 6.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • +
  • Section 6.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • +
  • Section 6.6:
    • +
  • Section 6.7: The dynamics of the decoupled plants are shown
    • +
  • Section 6.8: A diagonal controller is defined to control the decoupled plant
    • +
  • Section 6.9: Finally, the closed loop system properties are studied
    • +
+
+ +
+

6.1 Simscape Model - Parameters

+
+

+ +

+
+
open('drone_platform.slx');
+
+
+ +

+Definition of spring parameters: +

+
+
kx = 0.5*1e3/3; % [N/m]
+ky = 0.5*1e3/3;
+kz = 1e3/3;
+
+cx = 0.025; % [Nm/rad]
+cy = 0.025;
+cz = 0.025;
+
+
+ +

+We suppose the sensor is perfectly positioned. +

+
+
sens_pos_error = zeros(3,1);
+
+
+ +

+Gravity: +

+
+
g = 0;
+
+
+ +

+We load the Jacobian (previously computed from the geometry): +

+
+
load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
+
+
+ +

+We initialize other parameters: +

+
+
U = eye(6);
+V = eye(6);
+Kc = tf(zeros(6));
+
+
+ + +
+

stewart_simscape.png +

+

Figure 41: General view of the Simscape Model

+
+ + +
+

stewart_platform_details.png +

+

Figure 42: Simscape model of the Stewart platform

+
+
+
+ +
+

6.2 Identification of the plant

+
+

+ +

+ +

+The plant shown in Figure 43 is identified from the Simscape model. +

+ +

+The inputs are: +

+
    +
  • \(D_w\) translation and rotation of the bottom platform (with respect to the center of mass of the top platform)
    • +
  • \(\tau\) the 6 forces applied by the voice coils
    • +
+ +

+The outputs are the 6 accelerations measured by the inertial unit. +

+ + +
+

stewart_platform_plant.png +

+

Figure 43: Considered plant \(\bm{G} = \begin{bmatrix}G_d\\G_u\end{bmatrix}\). \(D_w\) is the translation/rotation of the support, \(\tau\) the actuator forces, \(a\) the acceleration/angular acceleration of the top platform

+
+ +
+
%% Name of the Simulink File
+mdl = 'drone_platform';
+
+%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Dw'],              1, 'openinput');  io_i = io_i + 1; % Ground Motion
+io(io_i) = linio([mdl, '/V-T'],             1, 'openinput');  io_i = io_i + 1; % Actuator Forces
+io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
+
+G = linearize(mdl, io);
+G.InputName  = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
+                'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
+G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
+
+% Plant
+Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
+% Disturbance dynamics
+Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
+
+
+ +

+There are 24 states (6dof for the bottom platform + 6dof for the top platform). +

+
+
size(G)
+
+
+ +
+State-space model with 6 outputs, 12 inputs, and 24 states.
+
+ + +

+The elements of the transfer matrix \(\bm{G}\) corresponding to the transfer function from actuator forces \(\tau\) to the measured acceleration \(a\) are shown in Figure 44. +

+ +

+One can easily see that the system is strongly coupled. +

+ + +
+

stewart_platform_coupled_plant.png +

+

Figure 44: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

+
+
+
+ +
+

6.3 Decoupling using the Jacobian

+
+

+ +Consider the control architecture shown in Figure 45. +The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator. +

+ +

+The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators). +

+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 12: Computed Jacobian Matrix
0.8110.00.584-0.018-0.0080.025
-0.406-0.7030.584-0.016-0.012-0.025
-0.4060.7030.5840.016-0.0120.025
0.8110.00.5840.018-0.008-0.025
-0.406-0.7030.5840.0020.0190.025
-0.4060.7030.584-0.0020.019-0.025
+ + +
+

plant_decouple_jacobian.png +

+

Figure 45: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

+
+ +

+We define a new plant: +\[ G_x(s) = G(s) J^{-T} \] +

+ +

+\(G_x(s)\) correspond to the transfer function from forces and torques applied to the top platform to the absolute acceleration of the top platform. +

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

6.4 Decoupling using the SVD

+
+

+ +

+ +

+In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required. +

+ +

+Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_u(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). +

+
+
wc = 2*pi*30; % Decoupling frequency [rad/s]
+
+H1 = evalfr(Gu, j*wc);
+
+
+ +

+The real approximation is computed as follows: +

+
+
D = pinv(real(H1'*H1));
+H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 13: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
+ + +

+Note that the plant \(G_u\) at \(\omega_c\) is already an almost real matrix. +This can be seen on the Bode plots where the phase is close to 1. +This can be verified below where only the real value of \(G_u(\omega_c)\) is shown +

+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 14: Real part of \(G\) at the decoupling frequency \(\omega_c\)
4.4-2.1-2.14.4-2.4-2.4
-0.2-3.93.90.2-3.83.8
3.43.43.43.43.43.4
-367.1-323.8323.8367.143.3-43.3
-162.0-237.0-237.0-162.0398.9398.9
220.6-220.6220.6-220.6220.6-220.6
+ +

+Now, the Singular Value Decomposition of \(H_1\) is performed: +\[ H_1 = U \Sigma V^H \] +

+ +
+
[U,~,V] = svd(H1);
+
+
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 15: Obtained matrix \(U\)
-0.0057e-066e-11-3e-06-10.1
-7e-06-0.005-9e-09-5e-09-0.1-1
4e-08-2e-10-6e-11-13e-06-3e-07
-0.002-1-5e-062e-100.00060.005
1-0.002-1e-082e-08-0.0050.0006
-4e-095e-06-16e-11-2e-09-1e-08
+ + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 16: Obtained matrix \(V\)
-0.20.5-0.4-0.4-0.6-0.2
-0.30.50.4-0.40.50.3
-0.3-0.5-0.4-0.40.4-0.4
-0.2-0.50.4-0.4-0.50.3
0.6-0.06-0.4-0.40.10.6
0.60.060.4-0.4-0.006-0.6
+ +

+The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 46. +

+ + +
+

plant_decouple_svd.png +

+

Figure 46: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

+
+ +

+The decoupled plant is then: +\[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \] +

+ +
+
Gsvd = inv(U)*Gu*inv(V');
+
+
+
+
+ +
+

6.5 Verification of the decoupling using the “Gershgorin Radii”

+
+

+ +

+ +

+The “Gershgorin Radii” is computed for the coupled plant \(G(s)\), for the “Jacobian plant” \(G_x(s)\) and the “SVD Decoupled Plant” \(G_{SVD}(s)\): +

+ +

+The “Gershgorin Radii” of a matrix \(S\) is defined by: +\[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \] +

+ +

+This is computed over the following frequencies. +

+ +
+

simscape_model_gershgorin_radii.png +

+

Figure 47: Gershgorin Radii of the Coupled and Decoupled plants

+
+
+
+ +
+

6.6 Verification of the decoupling using the “Relative Gain Array”

+
+

+ +

+ +

+The relative gain array (RGA) is defined as: +

+\begin{equation} + \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T +\end{equation} +

+where \(\times\) denotes an element by element multiplication and \(G(s)\) is an \(n \times n\) square transfer matrix. +

+ +

+The obtained RGA elements are shown in Figure 48. +

+ + +
+

simscape_model_rga.png +

+

Figure 48: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

+
+
+
+ +
+

6.7 Obtained Decoupled Plants

+
+

+ +

+ +

+The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 49. +

+ + +
+

simscape_model_decoupled_plant_svd.png +

+

Figure 49: Decoupled Plant using SVD

+
+ +

+Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant \(G_x(s)\) using the Jacobian are shown in Figure 50. +

+ + +
+

simscape_model_decoupled_plant_jacobian.png +

+

Figure 50: Stewart Platform Plant from forces (resp. torques) applied by the legs to the acceleration (resp. angular acceleration) of the platform as well as all the coupling terms between the two (non-diagonal terms of the transfer function matrix)

+
+
+
+ +
+

6.8 Diagonal Controller

+
+

+ +The control diagram for the centralized control is shown in Figure 51. +

+ +

+The controller \(K_c\) is “working” in an cartesian frame. +The Jacobian is used to convert forces in the cartesian frame to forces applied by the actuators. +

+ + +
+

centralized_control.png +

+

Figure 51: Control Diagram for the Centralized control

+
+ +

+The SVD control architecture is shown in Figure 52. +The matrices \(U\) and \(V\) are used to decoupled the plant \(G\). +

+ + +
+

svd_control.png +

+

Figure 52: Control Diagram for the SVD control

+
+ + +

+We choose the controller to be a low pass filter: +\[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \] +

+ +

+\(G_0\) is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to \(\omega_c\) +

+ +
+
wc = 2*pi*80;  % Crossover Frequency [rad/s]
+w0 = 2*pi*0.1; % Controller Pole [rad/s]
+
+
+ +
+
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_cen = K_cen*Gx;
+G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
+
+
+ +
+
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
+L_svd = K_svd*Gsvd;
+G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
+
+
+ +

+The obtained diagonal elements of the loop gains are shown in Figure 53. +

+ + +
+

stewart_comp_loop_gain_diagonal.png +

+

Figure 53: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

+
+
+
+ +
+

6.9 Closed-Loop system Performances

+
+

+ +

+ +

+Let’s first verify the stability of the closed-loop systems: +

+
+
isstable(G_cen)
+
+
+ +
+ans =
+  logical
+   1
+
+ + +
+
isstable(G_svd)
+
+
+ +
+ans =
+  logical
+   1
+
+ + +

+The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 54. +

+ + +
+

stewart_platform_simscape_cl_transmissibility.png +

+

Figure 54: Obtained Transmissibility

+
+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2021-02-17 mer. 15:17

+
+ + diff --git a/index.org b/svd-control.org similarity index 99% rename from index.org rename to svd-control.org index 23cba56..2098af4 100644 --- a/index.org +++ b/svd-control.org @@ -43,7 +43,7 @@ #+begin_export html
-

This report is also available as a pdf.

+

This report is also available as a pdf.

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