diff --git a/svd-control.html b/svd-control.html index 18e0609..3ca87b7 100644 --- a/svd-control.html +++ b/svd-control.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Diagonal control using the SVD and the Jacobian Matrix @@ -39,100 +39,100 @@

Table of Contents

@@ -151,61 +151,61 @@ Then, a diagonal controller is used. These two methods are tested on two plants:

-
-

1 Gravimeter - Simscape Model

+
+

1 Gravimeter - Simscape Model

- +

-
-

1.1 Introduction

+
+

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
    • +
  • 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

+
+

1.2 Gravimeter Model - Parameters

- +

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

@@ -235,11 +235,11 @@ g = 0; % Gravity [m/s2]
-
-

1.3 System Identification

+
+

1.3 System Identification

- +

@@ -263,7 +263,7 @@ G.OutputName = {'Ax1',

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

@@ -280,7 +280,7 @@ And 4 outputs (the two 2-DoF accelerometers): \end{equation} -

+

gravimeter_plant_schematic.png

Figure 3: Schematic of the gravimeter plant

@@ -336,11 +336,11 @@ 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. +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

@@ -348,15 +348,15 @@ The bode plot of all elements of the plant are shown in Figure -

1.4 Decoupling using the Jacobian

+
+

1.4 Decoupling using the Jacobian

- +

-Consider the control architecture shown in Figure 5. +Consider the control architecture shown in Figure 5.

@@ -374,16 +374,16 @@ The Jacobian matrix \(J_{a}\) is used to compute the vertical acceleration, hori \end{equation}

-We thus define a new plant as defined in Figure 5. +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). +\(\bm{G}_x(s)\) 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 \(J\)

@@ -421,7 +421,7 @@ 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. +The diagonal and off-diagonal elements of \(G_x\) are shown in Figure 6.

@@ -433,11 +433,11 @@ It is shown at the system is:

-The choice of the frame in this the Jacobian is evaluated is discussed in Section 1.12. +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 \(G_x\)

@@ -445,11 +445,11 @@ The choice of the frame in this the Jacobian is evaluated is discussed in Sectio
-
-

1.5 Decoupling using the SVD

+
+

1.5 Decoupling using the SVD

- +

@@ -598,11 +598,11 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:

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

@@ -633,10 +633,10 @@ The 4th output (corresponding to the null singular value) is discarded, and we o

-The diagonal and off-diagonal elements of the “SVD” plant are shown in Figure 8. +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 \(G_{svd}\)

@@ -644,11 +644,11 @@ The diagonal and off-diagonal elements of the “SVD” plant are shown
-
-

1.6 Verification of the decoupling using the “Gershgorin Radii”

+
+

1.6 Verification of the decoupling using the “Gershgorin Radii”

- +

@@ -661,7 +661,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:

-
+

gravimeter_gershgorin_radii.png

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

@@ -669,11 +669,11 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:
-
-

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

+
+

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

- +

@@ -687,11 +687,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an

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

@@ -705,7 +705,7 @@ The RGA-number is also a measure of diagonal dominance: \end{equation} -
+

gravimeter_rga_num.png

Figure 11: RGA-Number for the Gravimeter

@@ -713,30 +713,30 @@ The RGA-number is also a measure of diagonal dominance:
-
-

1.8 Obtained Decoupled Plants

+
+

1.8 Obtained Decoupled Plants

- +

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

@@ -744,12 +744,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
-

1.9 Diagonal Controller

+
+

1.9 Diagonal Controller

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

@@ -758,19 +758,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control_gravimeter.png

Figure 14: Control Diagram for the Centralized control

-The SVD control architecture is shown in Figure 15. +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

@@ -806,11 +806,11 @@ U_inv = inv(U);

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

@@ -818,11 +818,11 @@ The obtained diagonal elements of the loop gains are shown in Figure -

1.10 Closed-Loop system Performances

+
+

1.10 Closed-Loop system Performances

- +

@@ -896,18 +896,18 @@ ans =

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

@@ -916,11 +916,11 @@ The obtained transmissibility in Open-loop, for the centralized control as well
-
-

1.11 Robustness to a change of actuator position

+
+

1.11 Robustness to a change of actuator position

- +

@@ -972,11 +972,11 @@ The new plant is computed, and the centralized and SVD control architectures are

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

@@ -984,11 +984,11 @@ The closed-loop system are still stable in both cases, and the obtained transmis
-
-

1.12 Choice of the reference frame for Jacobian decoupling

+
+

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. @@ -1008,11 +1008,11 @@ Ideally, we would like to have a decoupled mass matrix and stiffness matrix at t 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

+
+

1.12.1 Decoupling of the mass matrix

-
+

gravimeter_model_M.png

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

@@ -1067,7 +1067,7 @@ GM.OutputName = {'Dx', -
+

jac_decoupling_M.png

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

@@ -1075,11 +1075,11 @@ GM.OutputName = {'Dx',
-
-

1.12.2 Decoupling of the stiffness matrix

+
+

1.12.2 Decoupling of the stiffness matrix

-
+

gravimeter_model_K.png

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

@@ -1111,7 +1111,7 @@ GK.OutputName = {'Dx', -
+

jac_decoupling_K.png

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

@@ -1119,11 +1119,11 @@ GK.OutputName = {'Dx',
-
-

1.12.3 Combined decoupling of the mass and stiffness matrices

+
+

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

@@ -1179,7 +1179,7 @@ GKM.OutputName = {'Dx', +

jac_decoupling_KM.png

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

@@ -1187,8 +1187,8 @@ GKM.OutputName = {'Dx', -

1.12.4 Conclusion

+
+

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. @@ -1202,11 +1202,11 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C

-
-

1.13 SVD decoupling performances

+
+

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.

@@ -1215,7 +1215,7 @@ Let’s do the SVD decoupling on a plant that is mostly real (low damping) a

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

c = 2e1; % Actuator Damping [N/(m/s)]
@@ -1223,14 +1223,14 @@ Start with small damping, the obtained diagonal and off-diagonal terms are shown
-
+

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

c = 5e2; % Actuator Damping [N/(m/s)]
@@ -1238,7 +1238,7 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
+

gravimeter_svd_high_damping.png

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

@@ -1247,54 +1247,54 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
-

2 Parallel Manipulator with Collocated actuator/sensor pairs

+
+

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). +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. +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. +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. +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
    • +
  • 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. +Conclusions are drawn in Section 2.7.

-
-

2.1 Model

+
+

2.1 Model

- +

@@ -1302,7 +1302,7 @@ Let’s consider a parallel manipulator with several collocated actuator/sen

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

@@ -1318,7 +1318,7 @@ We will note: -

+

gravimeter_model_analytical.png

Figure 29: Model of the gravimeter

@@ -1374,7 +1374,7 @@ Mb3 = [ la; -h/2];

-Frame \(\{K\}\) is chosen such that the stiffness matrix is diagonal (explained in Section 4). +Frame \(\{K\}\) is chosen such that the stiffness matrix is diagonal (explained in Section 4).

@@ -1395,11 +1395,11 @@ Kb3 = [ la; -h/2

-
-

2.2 The Jacobian Matrix

+
+

2.2 The Jacobian Matrix

- +

@@ -1563,11 +1563,11 @@ J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatri

-
-

2.3 The Stiffness Matrix

+
+

2.3 The Stiffness Matrix

- +

@@ -1606,15 +1606,15 @@ Cr = diag([c1,c2,c3]);

-
-

2.4 Equations of motion - Frame of the legs

+
+

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: +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\}} @@ -1653,7 +1653,7 @@ The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) i \end{equation} -
+

gravimeter_block_decentralized.png

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

@@ -1755,7 +1755,7 @@ This means the plant dynamics will be diagonal at low frequency.

-The transfer function \(\bm{G}(s)\) from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\) is defined below and its magnitude is shown in Figure 31. +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);
@@ -1767,7 +1767,7 @@ We can indeed see that the system is well decoupled at low frequency.
 

 
 
-

+

 
plant_frame_L.png
 

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

@@ -1775,11 +1775,11 @@ We can indeed see that the system is well decoupled at low frequency.
 

 

 
-

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

+

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

 

 

-
+
 

 
 

@@ -1806,7 +1806,7 @@ with:
 
 
 
-

+

 
gravimeter_block_com.png
 

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

@@ -1894,7 +1894,7 @@ 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.
+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}
@@ -1903,7 +1903,7 @@ Gm = inv(Mm*s^2 
 
 
-

+

 
plant_frame_M.png
 

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

@@ -1911,11 +1911,11 @@ Gm = inv(Mm*s^2 
 

 
-

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

+

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

 

 

-
+
 

 
 

@@ -1952,7 +1952,7 @@ The frame \(\{K\}\) has been chosen such that \(J_{\{K\}}^T \mathcal{K} J_{\{K\}
 

 
 
-

+

 
gravimeter_block_cok.png
 

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

@@ -2031,7 +2031,7 @@ The frame \(\{K\}\) has been chosen such that \(J_{\{K\}}^T \mathcal{K} J_{\{K\}
 
 
 

-The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined below and its magnitude is shown in Figure 35.
+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);
@@ -2039,7 +2039,7 @@ The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined bel
 

 
 
-

+

 
plant_frame_K.png
 

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

@@ -2047,18 +2047,18 @@ The plant from \(\bm{F}_{\{K\}}\) to \(\bm{\mathcal{X}}_{\{K\}}\) is defined bel
 

 

 
-

-
2.7 Conclusion

+

+
2.7 Conclusion

 

 

-
+
 

 

 

 

 
-

-
3 SVD / Jacobian / Model decoupling comparison

+

+
3 SVD / Jacobian / Model decoupling comparison

 

 

 The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators.
@@ -2068,25 +2068,25 @@ The goal of this section is to compare the use of several methods for the decoup
 It is structured as follow:
 

 
    
-
  • Section 3.1: the model used to compare/test decoupling strategies is presented
    • 
-
  • Section 3.2: decoupling using Jacobian matrices is presented
    • 
-
  • Section 3.3: modal decoupling is presented
    • 
-
  • Section 3.4: SVD decoupling is presented
    • 
-
  • Section 3.5: the three decoupling methods are applied on the test model and compared
    • 
-
  • Section 3.7: conclusions are drawn on the three decoupling methods
    • 
+
  • Section 3.1: the model used to compare/test decoupling strategies is presented
    • 
+
  • Section 3.2: decoupling using Jacobian matrices is presented
    • 
+
  • Section 3.3: modal decoupling is presented
    • 
+
  • Section 3.4: SVD decoupling is presented
    • 
+
  • Section 3.5: the three decoupling methods are applied on the test model and compared
    • 
+
  • Section 3.7: conclusions are drawn on the three decoupling methods
    • 
 

 

 
-

-
3.1 Test Model

+

+
3.1 Test Model

 

 

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

 
 

-System in Figure 36 will serve as an example.
+System in Figure 36 will serve as an example.
 

 
 

@@ -2102,7 +2102,7 @@ We will note:
 
 
 
-

+

 
model_test_decoupling.png
 

 
Figure 36: Model use to compare decoupling techniques

@@ -2176,11 +2176,11 @@ G = J*inv(M*s
 
 

-The magnitude of the coupled plant \(G\) is shown in Figure 37.
+The magnitude of the coupled plant \(G\) is shown in Figure 37.
 

 
 
-

+

 
coupled_plant_bode.png
 

 
Figure 37: Magnitude of the coupled plant.

@@ -2189,11 +2189,11 @@ The magnitude of the coupled plant \(G\) is shown in Figure 
-
3.2 Jacobian Decoupling

+

+
3.2 Jacobian Decoupling

 

 

-
+
 

 
 

@@ -2221,13 +2221,13 @@ The obtained plan corresponds to forces/torques applied on origin of frame \(\{O
 

 
 
-

+

 
jacobian_decoupling_arch.png
 

 
Figure 38: Block diagram of the transfer function from \(\bm{\mathcal{F}}_{\{O\}}\) to \(\bm{\mathcal{X}}_{\{O\}}\)

 

 
-

+

 

 The Jacobian matrix is only based on the geometry of the system and does not depend on the physical properties such as mass and stiffness.
 

@@ -2248,11 +2248,11 @@ It is then easy to include a reference tracking input that specify the wanted mo
 

 

 
-

-
3.3 Modal Decoupling

+

+
3.3 Modal Decoupling

 

 

-
+
 

 
 

@@ -2375,21 +2375,21 @@ We now expressed the transfer function from input \(\bm{\tau}\) to output \(\bm{
 \end{equation}
 
 

-By inverting \(B_m\) and \(C_m\) and using them as shown in Figure 39, we can see that we control the system in the “modal space” in which it is decoupled.
+By inverting \(B_m\) and \(C_m\) and using them as shown in Figure 39, we can see that we control the system in the “modal space” in which it is decoupled.
 

 
 
-

+

 
decoupling_modal.png
 

 
Figure 39: Modal Decoupling Architecture

 

 
 

-The system \(\bm{G}_m(s)\) shown in Figure 39 is diagonal \eqref{eq:modal_eq}.
+The system \(\bm{G}_m(s)\) shown in Figure 39 is diagonal \eqref{eq:modal_eq}.
 

 
-

+

 

 Modal decoupling requires to have the equations of motion of the system.
 From the equations of motion (and more precisely the mass and stiffness matrices), the mode shapes \(\Phi\) are computed.
@@ -2408,11 +2408,11 @@ Using this decoupling strategy, it is possible to control each mode individually
 

 

 
-

-
3.4 SVD Decoupling

+

+
3.4 SVD Decoupling

 

 

-
+
 

 
 

@@ -2447,7 +2447,7 @@ H1 = pinv(D*real(H1'*
 

 
    
-
  • Use the singular input and output matrices to decouple the system as shown in Figure 40
+
  • Use the singular input and output matrices to decouple the system as shown in Figure 40
 \[ G_{svd}(s) = U^{-1} G(s) V^{-T} \]
    • 
 

 

@@ -2456,13 +2456,13 @@ H1 = pinv(D*real(H1'*
 
 
-

+

 
decoupling_svd.png
 

 
Figure 40: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

 

 
-

+

 

 In order to apply the Singular Value Decomposition, we need to have the Frequency Response Function of the system, at least near the frequency where we wish to decouple the system.
 The FRF can be experimentally obtained or based from a model.
@@ -2490,15 +2490,15 @@ The inputs and outputs are ordered from higher gain to lower gain at the chosen
 

 

 
-

-
3.5 Comparison

+

+
3.5 Comparison

 

 

-
+
 

 

-

-
3.5.1 Jacobian Decoupling

+

+
3.5.1 Jacobian Decoupling

 

 

 Decoupling properties depends on the chosen frame \(\{O\}\).
@@ -2516,7 +2516,7 @@ Gx.OutputName  = {'Dx', 
 
 
-

+

 
jacobian_plant.png
 

 
Figure 41: Plant decoupled using the Jacobian matrices \(G_x(s)\)

@@ -2524,11 +2524,11 @@ Gx.OutputName  = {'Dx', 
 

 
-

-
3.5.2 Modal Decoupling

+

+
3.5.2 Modal Decoupling

 

 

-For the system in Figure 36, we have:
+For the system in Figure 36, we have:
 

 \begin{align}
 \bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\
@@ -2549,7 +2549,7 @@ c & 0 & 0 \\
 \end{align}
 
 

-In order to apply the architecture shown in Figure 39, we need to compute \(C_{ox}\), \(C_{ov}\), \(\Phi\), \(\mu\) and \(J\).
+In order to apply the architecture shown in Figure 39, we need to compute \(C_{ox}\), \(C_{ov}\), \(\Phi\), \(\mu\) and \(J\).
 

 
 

@@ -2567,7 +2567,7 @@ Bm = inv(mu)*V'*J
 

 
-
+
@@ -2598,7 +2598,7 @@ Bm = inv(mu)*V'*J
 

 Table 12: \(B_m\) matrix
 
 


 
-
+
@@ -2630,7 +2630,7 @@ Bm = inv(mu)*V'*J
 
 

-And the plant in the modal space is defined below and its magnitude is shown in Figure 42.
+And the plant in the modal space is defined below and its magnitude is shown in Figure 42.
 

 

 
Gm = inv(Cm)*G*inv(Bm);
@@ -2638,7 +2638,7 @@ And the plant in the modal space is defined below and its magnitude is shown in
 

 
 
-

+

 
modal_plant.png
 

 
Figure 42: Modal plant \(G_m(s)\)

@@ -2650,9 +2650,26 @@ Let’s now close one loop at a time and see how the transmissibility change
 

 

 
-

-
3.5.3 SVD Decoupling

+

+
3.5.3 SVD Decoupling

 

+

+
%% Decoupling frequency [rad/s]
+wc = 2*pi*10;
+
+%% System's response at the decoupling frequency
+H1 = evalfr(G, j*wc);
+
+%% Real approximation of G(j.wc)
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+[U,S,V] = svd(H1);
+
+Gsvd = inv(U)*G*inv(V');
+

+

+
 

 Table 13: \(C_m\) matrix
 
 


 
@@ -2684,8 +2701,13 @@ Let’s now close one loop at a time and see how the transmissibility change
 

 Table 14: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
 

 
+
    
+
  • [ ] Do we have something special when applying SVD to a collocated MIMO system?
    • 
+
  • When applying SVD on a non-collocated MIMO system, we obtained a decoupled plant looking like the one in Figure 8
    • 
+

 
-

+
+

 
svd_plant.png
 

 
Figure 43: Svd plant \(G_m(s)\)

@@ -2694,15 +2716,15 @@ Let’s now close one loop at a time and see how the transmissibility change
 

 

 
-

-
3.6 Further Notes

+

+
3.6 Further Notes

 

 

-

-
3.6.1 Robustness of the decoupling strategies?

+

+
3.6.1 Robustness of the decoupling strategies?

 

 

-What happens if we have an additional resonance in the system (Figure 44).
+What happens if we have an additional resonance in the system (Figure 44).
 

 
 

@@ -2715,7 +2737,7 @@ Less actuator than DoF:
 
 
 
-

+

 
model_test_decoupling_spurious_res.png
 

 
Figure 44: Plant with spurious resonance (additional DoF)

@@ -2723,8 +2745,8 @@ Less actuator than DoF:
 

 

 
-

-
3.6.2 Other decoupling strategies

+

+
3.6.2 Other decoupling strategies

 

 
    
 
  • DC decoupling: pre-multiply the plant by \(G(0)^{-1}\)
    • 
@@ -2734,19 +2756,19 @@ Less actuator than DoF:
 

 

 
-

-
3.7 Conclusion

+

+
3.7 Conclusion

 

 

-
+
 

 
 

 The three proposed methods clearly have a lot in common as they all tend to make system more decoupled by pre and/or post multiplying by a constant matrix
-However, the three methods also differs by a number of points which are summarized in Table 15.
+However, the three methods also differs by a number of points which are summarized in Table 15.
 

 
-
+
@@ -2884,7 +2906,7 @@ However, the three methods also differs by a number of points which are summariz
 
-
+
@@ -2908,15 +2930,15 @@ However, the three methods also differs by a number of points which are summariz
 
 
 
-

-
4 Diagonal Stiffness Matrix for a planar manipulator

+

+
4 Diagonal Stiffness Matrix for a planar manipulator

 

 

-
+
 

 

-

-
4.1 Model and Assumptions

+

+
4.1 Model and Assumptions

 

 

 Consider a parallel manipulator with:
@@ -2938,11 +2960,11 @@ Consider two frames:
 
 
 

-As an example, take the system shown in Figure 45.
+As an example, take the system shown in Figure 45.
 

 
 
-

+

 
3dof_model_fully_parallel.png
 

 
Figure 45: Example of 3DoF parallel platform

@@ -2950,8 +2972,8 @@ As an example, take the system shown in Figure 45.
 

 

 
-

-
4.2 Objective

+

+
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.
@@ -2960,8 +2982,8 @@ If the conditions are fulfilled, a second objective is to fine the location of t
 

 

 
-

-
4.3 Conditions for Diagonal Stiffness

+

+
4.3 Conditions for Diagonal Stiffness

 

 

 The stiffness matrix in the frame \(\{K\}\) can be expressed as:
@@ -3103,15 +3125,15 @@ Note that a rotation of the frame \(\{K\}\) with respect to frame \(\{M\}\) woul
 

 

 
-

-
4.4 Example 1 - Planar manipulator with 3 actuators

+

+
4.4 Example 1 - Planar manipulator with 3 actuators

 

 

-Consider system of Figure 46.
+Consider system of Figure 46.
 

 
 
-

+

 
3dof_model_fully_parallel.png
 

 
Figure 46: Example of 3DoF parallel platform

@@ -3264,15 +3286,15 @@ And the stiffness matrix:
 

 

 
-

-
4.5 Example 2 - Planar manipulator with 4 actuators

+

+
4.5 Example 2 - Planar manipulator with 4 actuators

 

 

-Now consider the planar manipulator of Figure 47.
+Now consider the planar manipulator of Figure 47.
 

 
 
-

+

 
model_planar_2.png
 

 
Figure 47: Planar Manipulator

@@ -3440,12 +3462,12 @@ And the stiffness matrix:
 

 

 
-

-
5 Diagonal Stiffness Matrix for a general parallel manipulator

+

+
5 Diagonal Stiffness Matrix for a general parallel manipulator

 

 

-

-
5.1 Model and Assumptions

+

+
5.1 Model and Assumptions

 

 

 Let’s consider a 6dof parallel manipulator with:
@@ -3467,11 +3489,11 @@ Consider two frames:
 
 
 

-An example is shown in Figure 48.
+An example is shown in Figure 48.
 

 
 
-

+

 
stewart_architecture_example.png
 

 
Figure 48: Parallel manipulator Example

@@ -3479,8 +3501,8 @@ An example is shown in Figure 48.
 

 

 
-

-
5.2 Objective

+

+
5.2 Objective

 

 

 The objective is to find conditions for the existence of a frame \(\{K\}\) in which the Stiffness matrix of the manipulator is diagonal.
@@ -3489,8 +3511,8 @@ If the conditions are fulfilled, a second objective is to fine the location of t
 

 

 
-

-
5.3 Analytical formula of the stiffness matrix

+

+
5.3 Analytical formula of the stiffness matrix

 

 

 For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed in a frame \(\{K\}\) is:
@@ -3664,7 +3686,7 @@ In such case, condition \eqref{eq:diag_cond_2} is fulfilled and there is no coup
 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.
 

@@ -3673,8 +3695,8 @@ If there is no frame \(\{K\}\) such that conditions \eqref{eq:diag_cond_2} and \
 

 

 
-

-
5.4 Example 1 - 6DoF manipulator (3D)

+

+
5.4 Example 1 - 6DoF manipulator (3D)

 

 

 Let’s define the geometry of the manipulator (\({}^Mb_i\), \({}^Ms_i\) and \(k_i\)):
@@ -3955,16 +3977,16 @@ hold off;
 

 

 
-

-
5.5 Example 2 - Stewart Platform

+

+
5.5 Example 2 - Stewart Platform

 

 

-

-
6 Stiffness and Mass Matrices in the Leg’s frame

+

+
6 Stiffness and Mass Matrices in the Leg’s frame

 

 

-

-
6.1 Equations

+

+
6.1 Equations

 

 

 Equations in the \(\{M\}\) frame:
@@ -3998,8 +4020,8 @@ with:
 

 

 
-

-
6.2 Stiffness matrix

+

+
6.2 Stiffness matrix

 

 

 We have that:
@@ -4023,8 +4045,8 @@ The dynamics from \(\tau\) to \(\mathcal{L}\) is therefore decoupled at low freq
 

 

 
-

-
6.3 Mass matrix

+

+
6.3 Mass matrix

 

 

 The mass matrix in the frames of the legs is:
@@ -4080,8 +4102,8 @@ m_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T &= \text{diag}(I_x, I_y, I_z
 

 
 
-

-
6.4 Planar Example

+

+
6.4 Planar Example

 

 

 The stiffnesses \(k_i\), the joint positions \({}^Mb_i\) and joint unit vectors \({}^M\hat{s}_i\) are defined below:
@@ -4189,14 +4211,14 @@ ylabel('Magnitude');
 

 

 
-

-
7 Stewart Platform - Simscape Model

+

+
7 Stewart Platform - Simscape Model

 

 

-
+
 

 

-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 49.
+In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 49.
 

 
 

@@ -4209,7 +4231,7 @@ Some notes about the system:
 
 
 
-

+

 
SP_assembly.png
 

 
Figure 49: Stewart Platform CAD View

@@ -4219,23 +4241,23 @@ Some notes about the system:
 The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections:
 

 
    
-
  • Section 7.1: The parameters of the Simscape model of the Stewart platform are defined
    • 
-
  • Section 7.2: The plant is identified from the Simscape model and the system coupling is shown
    • 
-
  • Section 7.3: The plant is first decoupled using the Jacobian
    • 
-
  • Section 7.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • 
-
  • Section 7.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • 
-
  • Section 7.6:
    • 
-
  • Section 7.7: The dynamics of the decoupled plants are shown
    • 
-
  • Section 7.8: A diagonal controller is defined to control the decoupled plant
    • 
-
  • Section 7.9: Finally, the closed loop system properties are studied
    • 
+
  • Section 7.1: The parameters of the Simscape model of the Stewart platform are defined
    • 
+
  • Section 7.2: The plant is identified from the Simscape model and the system coupling is shown
    • 
+
  • Section 7.3: The plant is first decoupled using the Jacobian
    • 
+
  • Section 7.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • 
+
  • Section 7.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • 
+
  • Section 7.6:
    • 
+
  • Section 7.7: The dynamics of the decoupled plants are shown
    • 
+
  • Section 7.8: A diagonal controller is defined to control the decoupled plant
    • 
+
  • Section 7.9: Finally, the closed loop system properties are studied
    • 
 

 

 
-

-
7.1 Simscape Model - Parameters

+

+
7.1 Simscape Model - Parameters

 

 

-
+
 

 

 
open('drone_platform.slx');
@@ -4291,14 +4313,14 @@ Kc = tf(zeros(6));
 

 
 
-

+

 
stewart_simscape.png
 

 
Figure 50: General view of the Simscape Model

 

 
 
-

+

 
stewart_platform_details.png
 

 
Figure 51: Simscape model of the Stewart platform

@@ -4306,15 +4328,15 @@ Kc = tf(zeros(6));
 

 

 
-

-
7.2 Identification of the plant

+

+
7.2 Identification of the plant

 

 

-
+
 

 
 

-The plant shown in Figure 52 is identified from the Simscape model.
+The plant shown in Figure 52 is identified from the Simscape model.
 

 
 

@@ -4330,7 +4352,7 @@ The outputs are the 6 accelerations measured by the inertial unit.
 

 
 
-

+

 
stewart_platform_plant.png
 

 
Figure 52: 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

@@ -4372,7 +4394,7 @@ 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 53.
+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 53.
 

 
 

@@ -4380,7 +4402,7 @@ One can easily see that the system is strongly coupled.
 

 
 
-

+

 
stewart_platform_coupled_plant.png
 

 
Figure 53: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

@@ -4388,12 +4410,12 @@ One can easily see that the system is strongly coupled.
 

 

 
-

-
7.3 Decoupling using the Jacobian

+

+
7.3 Decoupling using the Jacobian

 

 

-
-Consider the control architecture shown in Figure 54.
+
+Consider the control architecture shown in Figure 54.
 The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
 

 
@@ -4475,7 +4497,7 @@ The Jacobian matrix is computed from the geometry of the platform (position and
 

 Table 15: Comparison of decoupling strategies
 
 

 
 If good decoupling at all frequencies => requires specific mechanical architecture
   Diagonal plants may not be easy to control
 

 
 

 
 
-

+

 
plant_decouple_jacobian.png
 

 
Figure 54: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

@@ -4498,11 +4520,11 @@ Gx.InputName  = {'Fx', 
 

 

 
-

-
7.4 Decoupling using the SVD

+

+
7.4 Decoupling using the SVD

 

 

-
+
 

 
 

@@ -4838,11 +4860,11 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
 
 
 

-The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 55.
+The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 55.
 

 
 
-

+

 
plant_decouple_svd.png
 

 
Figure 55: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

@@ -4860,11 +4882,11 @@ The decoupled plant is then:
 

 

 
-

-
7.5 Verification of the decoupling using the “Gershgorin Radii”

+

+
7.5 Verification of the decoupling using the “Gershgorin Radii”

 

 

-
+
 

 
 

@@ -4880,7 +4902,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:
 This is computed over the following frequencies.
 

 
-

+

 
simscape_model_gershgorin_radii.png
 

 
Figure 56: Gershgorin Radii of the Coupled and Decoupled plants

@@ -4888,11 +4910,11 @@ This is computed over the following frequencies.
 

 

 
-

-
7.6 Verification of the decoupling using the “Relative Gain Array”

+

+
7.6 Verification of the decoupling using the “Relative Gain Array”

 

 

-
+
 

 
 

@@ -4906,11 +4928,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an
 

 
 

-The obtained RGA elements are shown in Figure 57.
+The obtained RGA elements are shown in Figure 57.
 

 
 
-

+

 
simscape_model_rga.png
 

 
Figure 57: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

@@ -4918,30 +4940,30 @@ The obtained RGA elements are shown in Figure 57.
 

 

 
-

-
7.7 Obtained Decoupled Plants

+

+
7.7 Obtained Decoupled Plants

 

 

-
+
 

 
 

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 58.
+The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 58.
 

 
 
-

+

 
simscape_model_decoupled_plant_svd.png
 

 
Figure 58: 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 59.
+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 59.
 

 
 
-

+

 
simscape_model_decoupled_plant_jacobian.png
 

 
Figure 59: 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)

@@ -4949,12 +4971,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
 

 

 
-

-
7.8 Diagonal Controller

+

+
7.8 Diagonal Controller

 

 

-
-The control diagram for the centralized control is shown in Figure 60.
+
+The control diagram for the centralized control is shown in Figure 60.
 

 
 

@@ -4963,19 +4985,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
 

 
 
-

+

 
centralized_control.png
 

 
Figure 60: Control Diagram for the Centralized control

 

 
 

-The SVD control architecture is shown in Figure 61.
+The SVD control architecture is shown in Figure 61.
 The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).
 

 
 
-

+

 
svd_control.png
 

 
Figure 61: Control Diagram for the SVD control

@@ -5012,11 +5034,11 @@ G_svd = feedback(G, inv(V')
 

 
 

-The obtained diagonal elements of the loop gains are shown in Figure 62.
+The obtained diagonal elements of the loop gains are shown in Figure 62.
 

 
 
-

+

 
stewart_comp_loop_gain_diagonal.png
 

 
Figure 62: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

@@ -5024,11 +5046,11 @@ The obtained diagonal elements of the loop gains are shown in Figure 
-
7.9 Closed-Loop system Performances

+

+
7.9 Closed-Loop system Performances

 

 

-
+
 

 
 

@@ -5059,11 +5081,11 @@ ans =
 
 
 

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 63.
+The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 63.
 

 
 
-

+

 
stewart_platform_simscape_cl_transmissibility.png
 

 
Figure 63: Obtained Transmissibility

@@ -5074,7 +5096,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
 

 

 
Author: Dehaeze Thomas

-
Created: 2021-03-05 ven. 11:47

+
Created: 2021-03-05 ven. 12:02

 

 
 
diff --git a/svd-control.org b/svd-control.org
index c9c94c5..67f05bd 100644
--- a/svd-control.org
+++ b/svd-control.org
@@ -2474,7 +2474,7 @@ Using this decoupling strategy, it is possible to control each mode individually
 Procedure:
 - Identify the dynamics of the system from inputs to outputs (can be obtained experimentally)
 - Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice)
-#+begin_src matlab
+#+begin_src matlab :eval no
 %% Decoupling frequency [rad/s]
 wc = 2*pi*10;
 
@@ -2482,18 +2482,18 @@ wc = 2*pi*10;
 H1 = evalfr(G, j*wc);
 #+end_src
 - Compute a real approximation of the system's response at that frequency
-#+begin_src matlab
+#+begin_src matlab :eval no
 %% Real approximation of G(j.wc)
 D = pinv(real(H1'*H1));
 H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
 #+end_src
 - Perform a Singular Value Decomposition of the real approximation
-#+begin_src matlab
+#+begin_src matlab :eval no
 [U,S,V] = svd(H1);
 #+end_src
 - Use the singular input and output matrices to decouple the system as shown in Figure [[fig:decoupling_svd]]
   \[ G_{svd}(s) = U^{-1} G(s) V^{-T} \]
-#+begin_src matlab
+#+begin_src matlab :eval no
 Gsvd = inv(U)*G*inv(V');
 #+end_src
 
@@ -2688,6 +2688,21 @@ exportFig('figs/modal_plant.pdf', 'width', 'wide', 'height', 'normal');
 Let's now close one loop at a time and see how the transmissibility changes.
 
 *** SVD Decoupling
+#+begin_src matlab
+%% Decoupling frequency [rad/s]
+wc = 2*pi*10;
+
+%% System's response at the decoupling frequency
+H1 = evalfr(G, j*wc);
+
+%% Real approximation of G(j.wc)
+D = pinv(real(H1'*H1));
+H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
+
+[U,S,V] = svd(H1);
+
+Gsvd = inv(U)*G*inv(V');
+#+end_src
 
 #+begin_src matlab :exports results :results value table replace :tangle no
 data2orgtable(H1, {}, {}, ' %.2g ');
@@ -2701,6 +2716,9 @@ data2orgtable(H1, {}, {}, ' %.2g ');
 |  2.1e-06 | -1.3e-06 | -2.5e-08 |
 | -2.1e-06 | -2.5e-08 | -1.3e-06 |
 
+- [ ] Do we have something special when applying SVD to a collocated MIMO system?
+- When applying SVD on a non-collocated MIMO system, we obtained a decoupled plant looking like the one in Figure [[fig:gravimeter_svd_plant]]
+
 #+begin_src matlab :exports none
 freqs = logspace(-1, 2, 1000);
 figure;
@@ -2722,7 +2740,7 @@ end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Magnitude');
-ylim([1e-8, 1e-2]);
+% ylim([1e-8, 1e-2]);
 legend('location', 'northeast');
 #+end_src
 
@@ -2789,7 +2807,7 @@ However, the three methods also differs by a number of points which are summariz
 |---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
 | *Cons*                    | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations                                          | Loose the physical meaning of inputs /outputs          |
 |                           | Limited to parallel mechanisms (?)                                                     |                                                                    | Decoupling depends on the real approximation validity  |
-|                           | If good decoupling at all frequencies => requires specific mechanical architecture     |                                                                    |                                                        |
+|                           | If good decoupling at all frequencies => requires specific mechanical architecture     |                                                                    | Diagonal plants may not be easy to control             |
 |---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------|
 | *Applicability*           | Parallel Mechanisms                                                                    | Systems whose dynamics that can be expressed with M and K matrices | Very general                                           |
 |                           | Only small motion for the Jacobian matrix to stay constant                             |                                                                    | Need FRF data (either experimentally or analytically)  |
diff --git a/svd-control.pdf b/svd-control.pdf
index fffa121..fa7a091 100644
Binary files a/svd-control.pdf and b/svd-control.pdf differ