diff --git a/index.html b/index.html index 2829521..8ddaf6c 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Diagonal control using the SVD and the Jacobian Matrix @@ -39,76 +39,76 @@

Table of Contents

@@ -127,61 +127,61 @@ 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

@@ -211,11 +211,11 @@ g = 0; % Gravity [m/s2]
-
-

1.3 System Identification

+
+

1.3 System Identification

- +

@@ -239,7 +239,7 @@ 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.

@@ -256,7 +256,7 @@ And 4 outputs (the two 2-DoF accelerometers): \end{equation} -

+

gravimeter_plant_schematic.png

Figure 3: Schematic of the gravimeter plant

@@ -312,11 +312,11 @@ 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

@@ -324,15 +324,15 @@ 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.

@@ -350,16 +350,16 @@ 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 \(J\)

@@ -397,7 +397,7 @@ 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.

@@ -409,11 +409,11 @@ 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 \(G_x\)

@@ -421,11 +421,11 @@ 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

- +

@@ -574,11 +574,11 @@ 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

@@ -609,10 +609,10 @@ 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 \(G_{svd}\)

@@ -620,11 +620,11 @@ 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”

- +

@@ -637,7 +637,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by:

-
+

gravimeter_gershgorin_radii.png

Figure 9: Gershgorin Radii of the Coupled and Decoupled plants

@@ -645,11 +645,11 @@ 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”

- +

@@ -663,11 +663,11 @@ 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

@@ -681,7 +681,7 @@ The RGA-number is also a measure of diagonal dominance: \end{equation} -
+

gravimeter_rga_num.png

Figure 11: RGA-Number for the Gravimeter

@@ -689,30 +689,30 @@ 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)

@@ -720,12 +720,12 @@ 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.

@@ -734,19 +734,19 @@ 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

@@ -782,11 +782,11 @@ 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

@@ -794,11 +794,11 @@ The obtained diagonal elements of the loop gains are shown in Figure -

1.10 Closed-Loop system Performances

+
+

1.10 Closed-Loop system Performances

- +

@@ -872,18 +872,18 @@ 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

@@ -892,11 +892,11 @@ 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

- +

@@ -948,11 +948,11 @@ 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

@@ -960,11 +960,11 @@ 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. @@ -984,11 +984,11 @@ 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

@@ -1043,7 +1043,7 @@ GM.OutputName = {'Dx', -
+

jac_decoupling_M.png

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

@@ -1051,11 +1051,11 @@ 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

@@ -1087,7 +1087,7 @@ GK.OutputName = {'Dx', -
+

jac_decoupling_K.png

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

@@ -1095,11 +1095,11 @@ 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

@@ -1155,7 +1155,7 @@ GKM.OutputName = {'Dx', +

jac_decoupling_KM.png

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

@@ -1163,8 +1163,8 @@ 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. @@ -1178,11 +1178,11 @@ 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.

@@ -1191,7 +1191,7 @@ 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)]
@@ -1199,14 +1199,14 @@ 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)]
@@ -1214,7 +1214,7 @@ 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

@@ -1223,15 +1223,15 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
-

2 Analytical Model

+
+

2 Analytical Model

-
-

2.1 Model

+
+

2.1 Model

-
+

gravimeter_model_analytical.png

Figure 28: Model of the gravimeter

@@ -1243,8 +1243,8 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
-

2.2 Stiffness and Mass matrices

+
+

2.2 Stiffness and Mass matrices

Stiffness matrix: @@ -1285,8 +1285,8 @@ Needs two Jacobians => complicated matrix

-
-

2.3 Equations

+
+

2.3 Equations

  • [ ] Ideally write the equation from \(\tau\) to \(\mathcal{L}\)
    • @@ -1302,8 +1302,8 @@ Needs two Jacobians => complicated matrix
-
-

2.4 Jacobians

+
+

2.4 Jacobians

Usefulness of Jacobians: @@ -1333,8 +1333,8 @@ J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatri

-
-

2.5 Parameters

+
+

2.5 Parameters

l  = 1.0; % Length of the mass [m]
@@ -1358,8 +1358,8 @@ k3 = 15e3; % Actuator Stiffness [N/m]
-
-

2.6 Transfer function from \(\tau\) to \(\delta \mathcal{L}\)

+
+

2.6 Transfer function from \(\tau\) to \(\delta \mathcal{L}\)

Mass, Damping and Stiffness matrices expressed in \(\{M\}\): @@ -1466,8 +1466,8 @@ Kt = inv(Jm')*Km

-
-

2.7 Transfer function from \(\mathcal{F}_{\{M\}}\) to \(\mathcal{X}_{\{M\}}\)

+
+

2.7 Transfer function from \(\mathcal{F}_{\{M\}}\) to \(\mathcal{X}_{\{M\}}\)

Gm = inv(Jm)*Gt*inv(Jm');
@@ -1538,8 +1538,8 @@ Kt = inv(Jm')*Km
-
-

2.8 Transfer function from \(\mathcal{F}_{\{K\}}\) to \(\mathcal{X}_{\{K\}}\)

+
+

2.8 Transfer function from \(\mathcal{F}_{\{K\}}\) to \(\mathcal{X}_{\{K\}}\)

Jacobian: @@ -1632,12 +1632,12 @@ Gk = inv(Jk)*Gt*inv(

-
-

2.9 Analytical

+
+

2.9 Analytical

-
-

2.9.1 Parameters

+
+

2.9.1 Parameters

syms la ha m I c k positive
@@ -1701,12 +1701,12 @@ Kk = Jk'*Kt*Jk;
-
-

3 Diagonal Stiffness Matrix for a planar manipulator

+
+

3 Diagonal Stiffness Matrix for a planar manipulator

-
-

3.1 Model and Assumptions

+
+

3.1 Model and Assumptions

Consider a parallel manipulator with: @@ -1728,11 +1728,11 @@ Consider two frames:

-As an example, take the system shown in Figure 29. +As an example, take the system shown in Figure 29.

-
+

3dof_model_fully_parallel.png

Figure 29: Example of 3DoF parallel platform

@@ -1740,8 +1740,8 @@ As an example, take the system shown in Figure 29.
-
-

3.2 Objective

+
+

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. @@ -1750,8 +1750,8 @@ If the conditions are fulfilled, a second objective is to fine the location of t

-
-

3.3 Conditions for Diagonal Stiffness

+
+

3.3 Conditions for Diagonal Stiffness

The stiffness matrix in the frame \(\{K\}\) can be expressed as: @@ -1764,7 +1764,17 @@ 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
    • +

  • +\(\mathcal{K}\) 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}

@@ -1883,15 +1893,15 @@ Note that a rotation of the frame \(\{K\}\) with respect to frame \(\{M\}\) woul

-
-

3.4 Example 1 - Planar manipulator with 3 actuators

+
+

3.4 Example 1 - Planar manipulator with 3 actuators

-Consider system of Figure 30. +Consider system of Figure 30.

-
+

3dof_model_fully_parallel.png

Figure 30: Example of 3DoF parallel platform

@@ -1970,15 +1980,15 @@ And the stiffness matrix:
-
-

3.5 Example 2 - Planar manipulator with 4 actuators

+
+

3.5 Example 2 - Planar manipulator with 4 actuators

-Now consider the planar manipulator of Figure 31. +Now consider the planar manipulator of Figure 31.

-
+

model_planar_2.png

Figure 31: Planar Manipulator

@@ -2061,12 +2071,12 @@ And the stiffness matrix:
-
-

4 Diagonal Stiffness Matrix for a general parallel manipulator

+
+

4 Diagonal Stiffness Matrix for a general parallel manipulator

-
-

4.1 Model and Assumptions

+
+

4.1 Model and Assumptions

Let’s consider a 6dof parallel manipulator with: @@ -2088,11 +2098,11 @@ Consider two frames:

-An example is shown in Figure 32. +An example is shown in Figure 32.

-
+

stewart_architecture_example.png

Figure 32: Parallel manipulator Example

@@ -2100,8 +2110,8 @@ An example is shown in Figure 32.
-
-

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. @@ -2110,8 +2120,8 @@ If the conditions are fulfilled, a second objective is to fine the location of t

-
-

4.3 Analytical formula of the stiffness matrix

+
+

4.3 Analytical formula of the stiffness matrix

For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed in a frame \(\{K\}\) is: @@ -2123,10 +2133,21 @@ For a fully parallel manipulator, the stiffness matrix \(K_{\{K\}}\) expressed i where:

    -
  • \(K_{\{K\}}\) is the Jacobian transformation from the struts to the frame \(\{K\}\)
    • -
  • \(\mathcal{K}\) is a diagonal matrix with the strut stiffnesses on the diagonal
    • +
  • \(J_{\{K\}}\) 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:

@@ -2232,7 +2253,7 @@ k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = k_i ({}^MO_K \times \hat{s}_i) \hat \end{equation}

-As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector, we obtain: +As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vehttps://rwth.zoom.us/j/92311133102?pwd=UTAzS21YYkUwT2pMZDBLazlGNzdvdz09tor, 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 ) @@ -2274,7 +2295,7 @@ 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.

@@ -2283,8 +2304,8 @@ If there is no frame \(\{K\}\) such that conditions \eqref{eq:diag_cond_2} and \
-
-

4.4 Example 1 - 6DoF manipulator (3D)

+
+

4.4 Example 1 - 6DoF manipulator (3D)

Let’s define the geometry of the manipulator (\({}^Mb_i\), \({}^Ms_i\) and \(k_i\)): @@ -2565,18 +2586,18 @@ hold off;

-
-

4.5 Example 2 - Stewart Platform

+
+

4.5 Example 2 - Stewart Platform

-
-

5 Stewart Platform - Simscape Model

+
+

5 Stewart Platform - Simscape Model

- +

-In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 33. +In this analysis, we wish to applied SVD control to the Stewart Platform shown in Figure 33.

@@ -2589,7 +2610,7 @@ Some notes about the system: -

+

SP_assembly.png

Figure 33: Stewart Platform CAD View

@@ -2599,23 +2620,23 @@ Some notes about the system: The analysis of the SVD/Jacobian control applied to the Stewart platform is performed in the following sections:

    -
  • Section 5.1: The parameters of the Simscape model of the Stewart platform are defined
    • -
  • Section 5.2: The plant is identified from the Simscape model and the system coupling is shown
    • -
  • Section 5.3: The plant is first decoupled using the Jacobian
    • -
  • Section 5.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • -
  • Section 5.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • -
  • Section 5.6:
    • -
  • Section 5.7: The dynamics of the decoupled plants are shown
    • -
  • Section 5.8: A diagonal controller is defined to control the decoupled plant
    • -
  • Section 5.9: Finally, the closed loop system properties are studied
    • +
  • Section 5.1: The parameters of the Simscape model of the Stewart platform are defined
    • +
  • Section 5.2: The plant is identified from the Simscape model and the system coupling is shown
    • +
  • Section 5.3: The plant is first decoupled using the Jacobian
    • +
  • Section 5.4: The decoupling is performed thanks to the SVD. To do so a real approximation of the plant is computed.
    • +
  • Section 5.5: The effectiveness of the decoupling with the Jacobian and SVD are compared using the Gershorin Radii
    • +
  • Section 5.6:
    • +
  • Section 5.7: The dynamics of the decoupled plants are shown
    • +
  • Section 5.8: A diagonal controller is defined to control the decoupled plant
    • +
  • Section 5.9: Finally, the closed loop system properties are studied
-
-

5.1 Simscape Model - Parameters

+
+

5.1 Simscape Model - Parameters

- + 

open('drone_platform.slx');
@@ -2671,14 +2692,14 @@ Kc = tf(zeros(6));
-
+

stewart_simscape.png

Figure 34: General view of the Simscape Model

-
+

stewart_platform_details.png

Figure 35: Simscape model of the Stewart platform

@@ -2686,15 +2707,15 @@ Kc = tf(zeros(6));
-
-

5.2 Identification of the plant

+
+

5.2 Identification of the plant

- +

-The plant shown in Figure 36 is identified from the Simscape model. +The plant shown in Figure 36 is identified from the Simscape model.

@@ -2710,7 +2731,7 @@ The outputs are the 6 accelerations measured by the inertial unit.

-
+

stewart_platform_plant.png

Figure 36: 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

@@ -2752,7 +2773,7 @@ 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 37. +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 37.

@@ -2760,7 +2781,7 @@ One can easily see that the system is strongly coupled.

-
+

stewart_platform_coupled_plant.png

Figure 37: Magnitude of all 36 elements of the transfer function matrix \(G_u\)

@@ -2768,12 +2789,12 @@ One can easily see that the system is strongly coupled.
-
-

5.3 Decoupling using the Jacobian

+
+

5.3 Decoupling using the Jacobian

- -Consider the control architecture shown in Figure 38. + +Consider the control architecture shown in Figure 38. The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.

@@ -2855,7 +2876,7 @@ The Jacobian matrix is computed from the geometry of the platform (position and -
+

plant_decouple_jacobian.png

Figure 38: Decoupled plant \(\bm{G}_x\) using the Jacobian matrix \(J\)

@@ -2878,11 +2899,11 @@ Gx.InputName = {'Fx',
-
-

5.4 Decoupling using the SVD

+
+

5.4 Decoupling using the SVD

- +

@@ -3218,11 +3239,11 @@ 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 39. +The obtained matrices \(U\) and \(V\) are used to decouple the system as shown in Figure 39.

-
+

plant_decouple_svd.png

Figure 39: Decoupled plant \(\bm{G}_{SVD}\) using the Singular Value Decomposition

@@ -3240,11 +3261,11 @@ The decoupled plant is then:
-
-

5.5 Verification of the decoupling using the “Gershgorin Radii”

+
+

5.5 Verification of the decoupling using the “Gershgorin Radii”

- +

@@ -3260,7 +3281,7 @@ The “Gershgorin Radii” of a matrix \(S\) is defined by: This is computed over the following frequencies.

-
+

simscape_model_gershgorin_radii.png

Figure 40: Gershgorin Radii of the Coupled and Decoupled plants

@@ -3268,11 +3289,11 @@ This is computed over the following frequencies.
-
-

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

+
+

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

- +

@@ -3286,11 +3307,11 @@ where \(\times\) denotes an element by element multiplication and \(G(s)\) is an

-The obtained RGA elements are shown in Figure 41. +The obtained RGA elements are shown in Figure 41.

-
+

simscape_model_rga.png

Figure 41: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant

@@ -3298,30 +3319,30 @@ The obtained RGA elements are shown in Figure 41.
-
-

5.7 Obtained Decoupled Plants

+
+

5.7 Obtained Decoupled Plants

- +

-The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 42. +The bode plot of the diagonal and off-diagonal elements of \(G_{SVD}\) are shown in Figure 42.

-
+

simscape_model_decoupled_plant_svd.png

Figure 42: 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 43. +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 43.

-
+

simscape_model_decoupled_plant_jacobian.png

Figure 43: 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)

@@ -3329,12 +3350,12 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
-
-

5.8 Diagonal Controller

+
+

5.8 Diagonal Controller

- -The control diagram for the centralized control is shown in Figure 44. + +The control diagram for the centralized control is shown in Figure 44.

@@ -3343,19 +3364,19 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied

-
+

centralized_control.png

Figure 44: Control Diagram for the Centralized control

-The SVD control architecture is shown in Figure 45. +The SVD control architecture is shown in Figure 45. The matrices \(U\) and \(V\) are used to decoupled the plant \(G\).

-
+

svd_control.png

Figure 45: Control Diagram for the SVD control

@@ -3392,11 +3413,11 @@ G_svd = feedback(G, inv(V')

-The obtained diagonal elements of the loop gains are shown in Figure 46. +The obtained diagonal elements of the loop gains are shown in Figure 46.

-
+

stewart_comp_loop_gain_diagonal.png

Figure 46: Comparison of the diagonal elements of the loop gains for the SVD control architecture and the Jacobian one

@@ -3404,11 +3425,11 @@ The obtained diagonal elements of the loop gains are shown in Figure -

5.9 Closed-Loop system Performances

+
+

5.9 Closed-Loop system Performances

- +

@@ -3439,11 +3460,11 @@ ans =

-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 47. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 47.

-
+

stewart_platform_simscape_cl_transmissibility.png

Figure 47: Obtained Transmissibility

@@ -3454,7 +3475,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well

Author: Dehaeze Thomas

-

Created: 2021-02-05 ven. 13:58

+

Created: 2021-02-05 ven. 15:45

diff --git a/index.org b/index.org index f52b601..459fa0e 100644 --- a/index.org +++ b/index.org @@ -1929,6 +1929,14 @@ The stiffness matrix in the frame $\{K\}$ can be expressed as: 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} @@ -2203,8 +2211,17 @@ For a fully parallel manipulator, the stiffness matrix $K_{\{K\}}$ expressed in K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}} \end{equation} where: -- $K_{\{K\}}$ is the Jacobian transformation from the struts to the frame $\{K\}$ -- $\mathcal{K}$ is a diagonal matrix with the strut stiffnesses on the diagonal +- $J_{\{K\}}$ 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} @@ -2286,7 +2303,7 @@ Taking the transpose and re-arranging: 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: +As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vehttps://rwth.zoom.us/j/92311133102?pwd=UTAzS21YYkUwT2pMZDBLazlGNzdvdz09tor, 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}