diff --git a/figs/block_diagram_jacobian_decoupling.pdf b/figs/block_diagram_jacobian_decoupling.pdf new file mode 100644 index 0000000..675598e Binary files /dev/null and b/figs/block_diagram_jacobian_decoupling.pdf differ diff --git a/figs/block_diagram_jacobian_decoupling.png b/figs/block_diagram_jacobian_decoupling.png new file mode 100644 index 0000000..f263dcd Binary files /dev/null and b/figs/block_diagram_jacobian_decoupling.png differ diff --git a/figs/gravimeter_block_cok.pdf b/figs/gravimeter_block_cok.pdf new file mode 100644 index 0000000..1db7763 Binary files /dev/null and b/figs/gravimeter_block_cok.pdf differ diff --git a/figs/gravimeter_block_cok.png b/figs/gravimeter_block_cok.png new file mode 100644 index 0000000..d820027 Binary files /dev/null and b/figs/gravimeter_block_cok.png differ diff --git a/figs/gravimeter_block_com.pdf b/figs/gravimeter_block_com.pdf new file mode 100644 index 0000000..eeb67dc Binary files /dev/null and b/figs/gravimeter_block_com.pdf differ diff --git a/figs/gravimeter_block_com.png b/figs/gravimeter_block_com.png new file mode 100644 index 0000000..f263dcd Binary files /dev/null and b/figs/gravimeter_block_com.png differ diff --git a/figs/gravimeter_block_decentralized.pdf b/figs/gravimeter_block_decentralized.pdf new file mode 100644 index 0000000..d4ec660 Binary files /dev/null and b/figs/gravimeter_block_decentralized.pdf differ diff --git a/figs/gravimeter_block_decentralized.png b/figs/gravimeter_block_decentralized.png new file mode 100644 index 0000000..019cfe8 Binary files /dev/null and b/figs/gravimeter_block_decentralized.png differ diff --git a/figs/gravimeter_model_analytical.pdf b/figs/gravimeter_model_analytical.pdf index b807c2a..c07bc78 100644 Binary files a/figs/gravimeter_model_analytical.pdf and b/figs/gravimeter_model_analytical.pdf differ diff --git a/figs/gravimeter_model_analytical.png b/figs/gravimeter_model_analytical.png index 98eabe0..dd46ae2 100644 Binary files a/figs/gravimeter_model_analytical.png and b/figs/gravimeter_model_analytical.png differ diff --git a/figs/plant_frame_K.pdf b/figs/plant_frame_K.pdf new file mode 100644 index 0000000..49ed89d Binary files /dev/null and b/figs/plant_frame_K.pdf differ diff --git a/figs/plant_frame_K.png b/figs/plant_frame_K.png new file mode 100644 index 0000000..dbb40bc Binary files /dev/null and b/figs/plant_frame_K.png differ diff --git a/figs/plant_frame_L.pdf b/figs/plant_frame_L.pdf new file mode 100644 index 0000000..e8df407 Binary files /dev/null and b/figs/plant_frame_L.pdf differ diff --git a/figs/plant_frame_L.png b/figs/plant_frame_L.png new file mode 100644 index 0000000..6da9b38 Binary files /dev/null and b/figs/plant_frame_L.png differ diff --git a/figs/plant_frame_M.pdf b/figs/plant_frame_M.pdf new file mode 100644 index 0000000..bc0239c Binary files /dev/null and b/figs/plant_frame_M.pdf differ diff --git a/figs/plant_frame_M.png b/figs/plant_frame_M.png new file mode 100644 index 0000000..55b800c Binary files /dev/null and b/figs/plant_frame_M.png differ diff --git a/index.html b/index.html index d1dff5d..3aa56ca 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 +39,78 @@

Table of Contents

-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 +423,11 @@ 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 +576,11 @@ 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 +611,10 @@ 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 +622,11 @@ 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 +639,7 @@ 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 +647,11 @@ 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 +665,11 @@ 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 +683,7 @@ 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 +691,30 @@ 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 +722,12 @@ 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 +736,19 @@ 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 +784,11 @@ 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 +796,11 @@ 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 +874,18 @@ 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 +894,11 @@ 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 +950,11 @@ 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 +962,11 @@ 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 +986,11 @@ 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 +1045,7 @@ GM.OutputName = {'Dx', -
+

jac_decoupling_M.png

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

@@ -1051,11 +1053,11 @@ 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 +1089,7 @@ GK.OutputName = {'Dx', -
+

jac_decoupling_K.png

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

@@ -1095,11 +1097,11 @@ 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 +1157,7 @@ GKM.OutputName = {'Dx', +

jac_decoupling_KM.png

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

@@ -1163,8 +1165,8 @@ 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 +1180,11 @@ 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 +1193,7 @@ 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 +1201,14 @@ 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 +1216,7 @@ 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

@@ -1223,119 +1225,86 @@ Now take a larger damping, the obtained diagonal and off-diagonal terms are show
-
-

2 Analytical Model

+
+

2 Parallel Manipulator with Collocated actuator/sensor pairs

-
-
-

2.1 Model

-
+

+ +

+

+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 28: Model of the gravimeter

+

Figure 29: Model of the gravimeter

-
    -
  • collocated actuators and sensors
    • -
-
-
- -
-

2.2 Stiffness and Mass matrices

-
-

-Stiffness matrix: -

-\begin{equation} - \mathcal{F}_{\{O\}} = -K_{\{O\}} \mathcal{X}_{\{O\}} -\end{equation} -

-with: -

-
    -
  • \(\mathcal{X}_{\{O\}}\) are displacements/rotations of the mass \(x\), \(y\), \(R_z\) expressed in the frame \(\{O\}\)
    • -
  • \(\mathcal{F}_{\{O\}}\) are forces/torques \(\mathcal{F}_x\), \(\mathcal{F}_y\), \(\mathcal{M}_z\) applied at the origin of \(\{O\}\)
    • -
-

-Mass matrix: +The parameters are defined as follows:

-\begin{equation} - \mathcal{F}_{\{O\}} = M_{\{O\}} \ddot{\mathcal{X}}_{\{O\}} -\end{equation} - - -

-Consider the two following frames: -

-
    -
  • \(\{M\}\): Center of mass => diagonal mass matrix -\[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \] -\[ K_{\{M\}} = \begin{bmatrix}k_1 & 0 & k_1 h_a \\ 0 & k_2 + k_3 & 0 \\ k_1 h_a & 0 & k_1 h_a + (k_2 + k_3)l_a\end{bmatrix} \]
    • -
  • \(\{K\}\): Diagonal stiffness matrix -\[ K_{\{K\}} = \begin{bmatrix}k_1 & 0 & 0 \\ 0 & k_2 + k_3 & 0 \\ 0 & 0 & (k_2 + k_3)l_a\end{bmatrix} \] -
      -
    • [ ] Compute the mass matrix \(M_{\{K\}}\) -Needs two Jacobians => complicated matrix
      • -
    • -
-
-
- -
-

2.3 Equations

-
-
    -
  • [ ] Ideally write the equation from \(\tau\) to \(\mathcal{L}\)
    • -
- -\begin{equation} - \mathcal{L} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix} -\end{equation} - -\begin{equation} - \tau = \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} -\end{equation} -
-
- -
-

2.4 Jacobians

-
-

-Usefulness of Jacobians: -

-
    -
  • \(J_{\{M\}}\) converts \(\dot{\mathcal{L}}\) to \(\dot{\mathcal{X}}_{\{M\}}\): -\[ \dot{\mathcal{X}}_{\{M\}} = J_{\{M\}} \dot{\mathcal{L}} \]
    • -
  • \(J_{\{M\}}^T\) converts \(\tau\) to \(\mathcal{F}_{\{M\}}\): -\[ \mathcal{F}_{\{M\}} = J_{\{M\}}^T \tau \]
    • -
  • \(J_{\{K\}}\) converts $\dot{\mathcal{K}}$to \(\dot{\mathcal{X}}_{\{K\}}\): -\[ \dot{\mathcal{X}}_{\{K\}} = J_{\{K\}} \dot{\mathcal{K}} \]
    • -
  • \(J_{\{K\}}^T\) converts \(\tau\) to \(\mathcal{F}_{\{K\}}\): -\[ \mathcal{F}_{\{K\}} = J_{\{K\}}^T \tau \]
    • -
- - -

-Let’s compute the Jacobians: -

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

2.5 Parameters

-
l  = 1.0; % Length of the mass [m]
 h  = 2*1.7; % Height of the mass [m]
@@ -1355,358 +1324,738 @@ 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];
+
+
-
-

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

+
+

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}

-Mass, Damping and Stiffness matrices expressed in \(\{M\}\): +

-
-
Mm = [m 0 0 ;
-      0 m 0 ;
-      0 0 I];
-
-Cm = [c1    0     c1*ha ;
-      0     c2+c3 0 ;
-      c1*ha 0     c1*ha + (c2+c3)*la];
-
-Km = [k1    0     k1*ha ;
-      0     k2+k3 0 ;
-      k1*ha 0     k1*ha + (k2+k3)*la];
-
-

-Jacobian \(J_{\{M\}}\): +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.

-
Jm = [1 0  ha ;
-      0 1 -la ;
-      0 1  la];
+
Gk = inv(Mk*s^2 + Ck*s + Kk);
-
-
Mt = inv(Jm')*Mm*inv(Jm);
-Ct = inv(Jm')*Cm*inv(Jm);
-Kt = inv(Jm')*Km*inv(Jm);
-
-
- - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 4: \(M_t\)
400.0340.0-340.0
340.0504.0-304.0
-340.0-304.0504.0
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 5: \(K_t\)
15000.00.00.0
0.024412.5-9412.5
0.0-9412.524412.5
- -
-
Gt = s^2*inv(Mt*s^2 + Ct*s + Kt);
-% Gt = JM*s^2*inv(MM*s^2 + CM*s + KM)*JM';
-
+
+

plant_frame_K.png +

+

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

-
-

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

+
+

2.7 Conclusion

-
-
Gm = inv(Jm)*Gt*inv(Jm');
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 6: \(M_{\{M\}}\)
400.00.00.0
0.0400.00.0
0.00.0115.0
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 7: \(K_{\{M\}}\)
15000.00.012750.0
0.030000.00.0
12750.00.027750.0
-
-
- -
-

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

-

-Jacobian: +

-
-
Jk = [1 0  0
-      0 1 -la
-      0 1  la];
-
-

-Mass, Damping and Stiffness matrices expressed in \(\{K\}\): +Jacobian matrices can be used to decouple the presented system.

-
-
Mk = Jk'*Mt*Jk;
-Ck = Jk'*Ct*Jk;
-Kk = Jk'*Kt*Jk;
-
-
- - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 8: \(M_{\{K\}}\)
400.00.0-340.0
0.0400.00.0
-340.00.0404.0
- - - - - --- -- -- - - - - - - - - - - - - - - - - - - - -
Table 9: \(K_{\{K\}}\)
15000.00.00.0
0.030000.00.0
0.00.016912.5
- -
-
% Gk = s^2*inv(Mk*s^2 + Ck*s + Kk);
-Gk = inv(Jk)*Gt*inv(Jk');
-
-
-
-
- -
-

2.9 Analytical

-
-
-
-

2.9.1 Parameters

-
-
-
syms la ha m I c k positive
-
-
- -
-
Mm = [m 0 0 ;
-      0 m 0 ;
-      0 0 I];
-
-Cm = [c    0   c*ha ;
-      0    2*c 0 ;
-      c*ha 0   c*(ha+2*la)];
-
-Km = [k    0   k*ha ;
-      0    2*k 0 ;
-      k*ha 0   k*(ha+2*la)];
-
-
- -
-
Jm = [1 0  ha ;
-      0 1 -la ;
-      0 1  la];
-
-
- -
-
Mt = inv(Jm')*Mm*inv(Jm);
-Ct = inv(Jm')*Cm*inv(Jm);
-Kt = inv(Jm')*Km*inv(Jm);
-
-
- -
-
Jk = [1 0  0
-      0 1 -la
-      0 1  la];
-
-

-Mass, Damping and Stiffness matrices expressed in \(\{K\}\): +Depending on the chosen frame used for the estimation of the Jacobian, different plant dynamics is obtained.

-
-
Mk = Jk'*Mt*Jk;
-Ck = Jk'*Ct*Jk;
-Kk = Jk'*Kt*Jk;
-
-
- -
-
['\begin{equation} M_{\{K\}} = ', latex(simplify(Kk)), '\end{equation}']
-
-
- -\begin{equation} M_{\{K\}} = \left(\begin{array}{ccc} k & 0 & 0\\ 0 & 2\,k & 0\\ 0 & 0 & k\,\left(-{\mathrm{ha}}^2+\mathrm{ha}+2\,\mathrm{la}\right) \end{array}\right)\end{equation} -
-
-

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: @@ -1728,20 +2077,20 @@ Consider two frames:

-As an example, take the system shown in Figure 29. +As an example, take the system shown in Figure 36.

-
+

3dof_model_fully_parallel.png

-

Figure 29: Example of 3DoF parallel platform

+

Figure 36: Example of 3DoF parallel platform

-
-

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 +2099,8 @@ 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: @@ -1893,18 +2242,18 @@ 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 37.

-
+

3dof_model_fully_parallel.png

-

Figure 30: Example of 3DoF parallel platform

+

Figure 37: Example of 3DoF parallel platform

@@ -2054,18 +2403,18 @@ 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 38.

-
+

model_planar_2.png

-

Figure 31: Planar Manipulator

+

Figure 38: Planar Manipulator

@@ -2230,12 +2579,12 @@ 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: @@ -2257,20 +2606,20 @@ Consider two frames:

-An example is shown in Figure 32. +An example is shown in Figure 39.

-
+

stewart_architecture_example.png

-

Figure 32: Parallel manipulator Example

+

Figure 39: Parallel manipulator Example

-
-

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. @@ -2279,8 +2628,8 @@ 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: @@ -2353,7 +2702,7 @@ And we finally obtain: 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 \hat{s}_i (b_i \times \hat{s}_i)^T & k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T + 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} @@ -2412,7 +2761,7 @@ 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 vehttps://rwth.zoom.us/j/92311133102?pwd=UTAzS21YYkUwT2pMZDBLazlGNzdvdz09tor, we obtain: +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 ) @@ -2454,7 +2803,7 @@ 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.

@@ -2463,8 +2812,8 @@ 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 \(k_i\)): @@ -2745,18 +3094,248 @@ hold off;

-
-

4.5 Example 2 - Stewart Platform

+
+

4.5 Example 2 - Stewart Platform

-
-

5 Stewart Platform - Simscape Model

+
+

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

@@ -2769,33 +3348,33 @@ Some notes about the system: -

+

SP_assembly.png

-

Figure 33: Stewart Platform CAD View

+

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 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 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
-
-

5.1 Simscape Model - Parameters

-
+
+

6.1 Simscape Model - Parameters

+

- + 

open('drone_platform.slx');
@@ -2851,30 +3430,30 @@ Kc = tf(zeros(6));
-
+

stewart_simscape.png

-

Figure 34: General view of the Simscape Model

+

Figure 41: General view of the Simscape Model

-
+

stewart_platform_details.png

-

Figure 35: Simscape model of the Stewart platform

+

Figure 42: Simscape model of the Stewart platform

-
-

5.2 Identification of the plant

-
+
+

6.2 Identification of the plant

+

- +

-The plant shown in Figure 36 is identified from the Simscape model. +The plant shown in Figure 43 is identified from the Simscape model.

@@ -2890,10 +3469,10 @@ 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

+

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

@@ -2932,7 +3511,7 @@ 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 44.

@@ -2940,20 +3519,20 @@ 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 \(G_u\)

+

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

-
-

5.3 Decoupling using the Jacobian

-
+
+

6.3 Decoupling using the Jacobian

+

- -Consider the control architecture shown in Figure 38. + +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.

@@ -2962,7 +3541,7 @@ The Jacobian matrix is computed from the geometry of the platform (position and

- +@@ -3035,10 +3614,10 @@ The Jacobian matrix is computed from the geometry of the platform (position and
Table 10: Computed Jacobian MatrixTable 12: Computed Jacobian Matrix
-
+

plant_decouple_jacobian.png

-

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

+

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

@@ -3058,11 +3637,11 @@ Gx.InputName = {'Fx',

-
-

5.4 Decoupling using the SVD

-
+
+

6.4 Decoupling using the SVD

+

- +

@@ -3089,7 +3668,7 @@ H1 = inv(D*real(H1'*

- +@@ -3169,7 +3748,7 @@ This can be verified below where only the real value of \(G_u(\omega_c)\) is sho

Table 11: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)Table 13: Real approximate of \(G\) at the decoupling frequency \(\omega_c\)
- +@@ -3252,7 +3831,7 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
Table 12: Real part of \(G\) at the decoupling frequency \(\omega_c\)Table 14: Real part of \(G\) at the decoupling frequency \(\omega_c\)
- +@@ -3325,7 +3904,7 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
Table 13: Obtained matrix \(U\)Table 15: Obtained matrix \(U\)
- +@@ -3398,14 +3977,14 @@ Now, the Singular Value Decomposition of \(H_1\) is performed:
Table 14: Obtained matrix \(V\)Table 16: Obtained matrix \(V\)

-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 46.

-
+

plant_decouple_svd.png

-

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

+

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

@@ -3420,11 +3999,11 @@ The decoupled plant is then:

-
-

5.5 Verification of the decoupling using the “Gershgorin Radii”

-
+
+

6.5 Verification of the decoupling using the “Gershgorin Radii”

+

- +

@@ -3440,19 +4019,19 @@ 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

+

Figure 47: Gershgorin Radii of the Coupled and Decoupled plants

-
-

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

-
+
+

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

+

- +

@@ -3466,55 +4045,55 @@ 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 48.

-
+

simscape_model_rga.png

-

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

+

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

-
-

5.7 Obtained Decoupled Plants

-
+
+

6.7 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 49.

-
+

simscape_model_decoupled_plant_svd.png

-

Figure 42: Decoupled Plant using SVD

+

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 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 50.

-
+

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)

+

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)

-
-

5.8 Diagonal Controller

-
+
+

6.8 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 51.

@@ -3523,22 +4102,22 @@ 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

+

Figure 51: Control Diagram for the Centralized control

-The SVD control architecture is shown in Figure 45. +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 45: Control Diagram for the SVD control

+

Figure 52: Control Diagram for the SVD control

@@ -3572,23 +4151,23 @@ 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 53.

-
+

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

+

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

-
-

5.9 Closed-Loop system Performances

-
+
+

6.9 Closed-Loop system Performances

+

- +

@@ -3619,14 +4198,14 @@ 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 54.

-
+

stewart_platform_simscape_cl_transmissibility.png

-

Figure 47: Obtained Transmissibility

+

Figure 54: Obtained Transmissibility

@@ -3634,7 +4213,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well

Author: Dehaeze Thomas

-

Created: 2021-02-05 ven. 16:05

+

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

diff --git a/index.org b/index.org index d404ab4..23cba56 100644 --- a/index.org +++ b/index.org @@ -1542,73 +1542,48 @@ exportFig('figs/gravimeter_svd_high_damping.pdf', 'width', 'wide', 'height', 'no #+RESULTS: [[file:figs/gravimeter_svd_high_damping.png]] -* Analytical Model -** Model +* Parallel Manipulator with Collocated actuator/sensor pairs +<> -#+name: fig:gravimeter_model_analytical -#+caption: Model of the gravimeter -[[file:figs/gravimeter_model_analytical.png]] +** Introduction :ignore: -- collocated actuators and sensors +In this section, we will see how the Jacobian matrix can be used to decouple a specific set of mechanical systems (described in Section [[sec:jac_decoupl_model]]). -** Stiffness and Mass matrices +The basic decoupling architecture is shown in Figure [[fig:gravimeter_model_analytical]] 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. -*Stiffness matrix*: -\begin{equation} - \mathcal{F}_{\{O\}} = -K_{\{O\}} \mathcal{X}_{\{O\}} -\end{equation} -with: -- $\mathcal{X}_{\{O\}}$ are displacements/rotations of the mass $x$, $y$, $R_z$ expressed in the frame $\{O\}$ -- $\mathcal{F}_{\{O\}}$ are forces/torques $\mathcal{F}_x$, $\mathcal{F}_y$, $\mathcal{M}_z$ applied at the origin of $\{O\}$ +This is rapidly explained in Section [[sec:jac_decoupl_jacobian]]. -*Mass matrix*: -\begin{equation} - \mathcal{F}_{\{O\}} = M_{\{O\}} \ddot{\mathcal{X}}_{\{O\}} -\end{equation} +#+begin_src latex :file block_diagram_jacobian_decoupling.pdf :tangle no :exports results +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.6 of G] (Jt) {$J_{\{M\}}^{-T}$}; + \node[block, right=0.6 of G] (Ja) {$J_{\{M\}}^{-1}$}; + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; -Consider the two following frames: -- $\{M\}$: Center of mass => diagonal mass matrix - \[ M_{\{M\}} = \begin{bmatrix}m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I\end{bmatrix} \] - \[ K_{\{M\}} = \begin{bmatrix}k_1 & 0 & k_1 h_a \\ 0 & k_2 + k_3 & 0 \\ k_1 h_a & 0 & k_1 h_a + (k_2 + k_3)l_a\end{bmatrix} \] -- $\{K\}$: Diagonal stiffness matrix - \[ K_{\{K\}} = \begin{bmatrix}k_1 & 0 & 0 \\ 0 & k_2 + k_3 & 0 \\ 0 & 0 & (k_2 + k_3)l_a\end{bmatrix} \] - - [ ] Compute the mass matrix $M_{\{K\}}$ - Needs two Jacobians => complicated matrix + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src -** Equations +#+RESULTS: +[[file:figs/block_diagram_jacobian_decoupling.png]] -- [ ] Ideally write the equation from $\tau$ to $\mathcal{L}$ +Depending on the chosen frame, the Stiffness matrix in that particular frame can be computed. +This is explained in Section [[sec:jac_decoupl_stiffness]]. -\begin{equation} - \mathcal{L} = \begin{bmatrix} \mathcal{L}_1 \\ \mathcal{L}_2 \\ \mathcal{L}_3 \end{bmatrix} -\end{equation} +Then three decoupling in three specific frames is studied: +- Section [[sec:jac_decoupl_legs]]: control in the frame of the legs +- Section [[sec:jac_decoupl_com]]: control in a frame whose origin is at the center of mass of the payload +- Section [[sec:jac_decoupl_cok]]: control in a frame whose origin is located at the "center of stiffness" of the system -\begin{equation} - \tau = \begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} -\end{equation} - -** Jacobians - -Usefulness of Jacobians: -- $J_{\{M\}}$ converts $\dot{\mathcal{L}}$ to $\dot{\mathcal{X}}_{\{M\}}$: - \[ \dot{\mathcal{X}}_{\{M\}} = J_{\{M\}} \dot{\mathcal{L}} \] -- $J_{\{M\}}^T$ converts $\tau$ to $\mathcal{F}_{\{M\}}$: - \[ \mathcal{F}_{\{M\}} = J_{\{M\}}^T \tau \] -- $J_{\{K\}}$ converts $\dot{\mathcal{K}}$ to $\dot{\mathcal{X}}_{\{K\}}$: - \[ \dot{\mathcal{X}}_{\{K\}} = J_{\{K\}} \dot{\mathcal{K}} \] -- $J_{\{K\}}^T$ converts $\tau$ to $\mathcal{F}_{\{K\}}$: - \[ \mathcal{F}_{\{K\}} = J_{\{K\}}^T \tau \] - - -Let's compute the Jacobians: -\begin{equation} -J_{\{M\}} = \begin{bmatrix} 1 & 0 & h_a \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} -\end{equation} - -\begin{equation} -J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatrix} -\end{equation} +Conclusions are drawn in Section [[sec:jac_decoupl_conclusion]]. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1619,7 +1594,26 @@ J_{\{K\}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -l_a \\ 0 & 1 & l_a \end{bmatri <> #+end_src -** Parameters +** Model +<> + +Let's consider a parallel manipulator with several collocated actuator/sensors pairs. + +System in Figure [[fig:gravimeter_model_analytical]] 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 + +#+name: fig:gravimeter_model_analytical +#+caption: Model of the gravimeter +[[file:figs/gravimeter_model_analytical.png]] + +The parameters are defined as follows: #+begin_src matlab l = 1.0; % Length of the mass [m] h = 2*1.7; % Height of the mass [m] @@ -1639,113 +1633,377 @@ k2 = 15e3; % Actuator Stiffness [N/m] k3 = 15e3; % Actuator Stiffness [N/m] #+end_src -** Transfer function from $\tau$ to $\delta \mathcal{L}$ -Mass, Damping and Stiffness matrices expressed in $\{M\}$: -#+begin_src matlab -Mm = [m 0 0 ; - 0 m 0 ; - 0 0 I]; +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} -Cm = [c1 0 c1*ha ; - 0 c2+c3 0 ; - c1*ha 0 c1*ha + (c2+c3)*la]; - -Km = [k1 0 k1*ha ; - 0 k2+k3 0 ; - k1*ha 0 k1*ha + (k2+k3)*la]; -#+end_src - -Jacobian $J_{\{M\}}$: -#+begin_src matlab -Jm = [1 0 ha ; - 0 1 -la ; - 0 1 la]; -#+end_src +\begin{align} +\hat{s}_1 &= [1,\ 0] \\ +\hat{s}_2 &= [0,\ 1] \\ +\hat{s}_3 &= [0,\ 1] +\end{align} #+begin_src matlab -Mt = inv(Jm')*Mm*inv(Jm); -Ct = inv(Jm')*Cm*inv(Jm); -Kt = inv(Jm')*Km*inv(Jm); +s1 = [1;0]; +s2 = [0;1]; +s3 = [0;1]; + +Mb1 = [-l/2;-ha]; +Mb2 = [-la; -h/2]; +Mb3 = [ la; -h/2]; #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Mt, {}, {}, ' %.1f '); +Frame $\{K\}$ is chosen such that the stiffness matrix is diagonal (explained in Section [[sec:diagonal_stiffness_planar]]). + +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} + +#+begin_src matlab +Kb1 = [-l/2; 0]; +Kb2 = [-la; -h/2+ha]; +Kb3 = [ la; -h/2+ha]; #+end_src -#+caption: $M_t$ +** 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 $\{K\}$: +#+begin_src matlab +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)]; +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Jm +#+end_src + +#+caption: Jacobian Matrix $J_{\{M\}}$ #+RESULTS: -| 400.0 | 340.0 | -340.0 | -| 340.0 | 504.0 | -304.0 | -| -340.0 | -304.0 | 504.0 | - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Kt, {}, {}, ' %.1f '); -#+end_src - -#+caption: $K_t$ -#+RESULTS: -| 15000.0 | 0.0 | 0.0 | -| 0.0 | 24412.5 | -9412.5 | -| 0.0 | -9412.5 | 24412.5 | +| 1 | 0 | 1.7 | +| 0 | 1 | -0.5 | +| 0 | 1 | 0.5 | #+begin_src matlab -Gt = s^2*inv(Mt*s^2 + Ct*s + Kt); -% Gt = JM*s^2*inv(MM*s^2 + CM*s + KM)*JM'; +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)]; #+end_src +#+begin_src matlab :results value replace :exports results :tangle no +ans = Jk +#+end_src + +#+caption: Jacobian Matrix $J_{\{K\}}$ +#+RESULTS: +| 1 | 0 | 0 | +| 0 | 1 | -0.5 | +| 0 | 1 | 0.5 | + +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} + +** 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. + +#+begin_src matlab +Kr = diag([k1,k2,k3]); +Cr = diag([c1,c2,c3]); +#+end_src + +** Equations of motion - Frame of the legs +<> + +Applying the second Newton's law on the system in Figure [[fig:gravimeter_model_analytical]] 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} + +#+begin_src latex :file gravimeter_block_decentralized.pdf :tangle no :exports results +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + + % Connections and labels + \draw[<-] (G.west) -- ++(-0.8, 0) node[above right]{$\bm{\tau}$};; + \draw[->] (G.east) -- ++( 0.8, 0) node[above left]{$\bm{\mathcal{L}}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:gravimeter_block_decentralized +#+caption: Block diagram of the transfer function from $\bm{\tau}$ to $\bm{\mathcal{L}}$ +#+RESULTS: +[[file:figs/gravimeter_block_decentralized.png]] + +#+begin_src matlab +%% Mass Matrix in frame {M} +Mm = diag([m,m,I]); +#+end_src + +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} + +#+begin_src matlab +%% Mass Matrix in the frame of the struts +Ml = inv(Jm')*Mm*inv(Jm); +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Ml +#+end_src + +#+caption: $M_{\{L\}}$ +#+RESULTS: +| 400 | 680 | -680 | +| 680 | 1371 | -1171 | +| -680 | -1171 | 1371 | + +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. +#+begin_src matlab +Kl = diag([k1, k2, k3]); +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Kl +#+end_src + +#+caption: $K_{\{L\}} = \mathcal{K}$ +#+RESULTS: +| 15000 | 0 | 0 | +| 0 | 15000 | 0 | +| 0 | 0 | 15000 | + +#+begin_src matlab +Cl = diag([c1, c2, c3]); +#+end_src + +The transfer function $\bm{G}(s)$ from $\bm{\tau}$ to $\bm{\mathcal{L}}$ is defined below and its magnitude is shown in Figure [[fig:plant_frame_L]]. +#+begin_src matlab +Gl = inv(Ml*s^2 + Cl*s + Kl); +#+end_src + +We can indeed see that the system is well decoupled at low frequency. + #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); +freqs = logspace(-2, 2, 1000); figure; % Magnitude hold on; for i_in = 1:3 for i_out = [1:i_in-1, i_in+1:3] - plot(freqs, abs(squeeze(freqresp(Gt(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + plot(freqs, abs(squeeze(freqresp(Gl(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end -plot(freqs, abs(squeeze(freqresp(Gt(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); +plot(freqs, abs(squeeze(freqresp(Gl(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'DisplayName', '$\mathcal{L}_i/\tau_j\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gt(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); + plot(freqs, abs(squeeze(freqresp(Gl(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', ['$\mathcal{L}_', int2str(i_in_out), '/\tau_', int2str(i_in_out), '$']); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'southeast'); -ylim([1e-8, 1e0]); +legend('location', 'northeast', 'FontSize', 8); +ylim([1e-8, 1e-2]); #+end_src -** Transfer function from $\mathcal{F}_{\{M\}}$ to $\mathcal{X}_{\{M\}}$ +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/plant_frame_L.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:plant_frame_L +#+caption: Dynamics from $\bm{\tau}$ to $\bm{\mathcal{L}}$ +#+RESULTS: +[[file:figs/plant_frame_L.png]] + +** 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 $\{M\}$: + \[ K_{\{M\}} = J_{\{M\}}^T \mathcal{K} J_{\{M\}} \] + +#+begin_src latex :file gravimeter_block_com.pdf :tangle no :exports results +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.6 of G] (Jt) {$J_{\{M\}}^{-T}$}; + \node[block, right=0.6 of G] (Ja) {$J_{\{M\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:gravimeter_block_com +#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{M\}}$ to $\bm{\mathcal{X}}_{\{M\}}$ +#+RESULTS: +[[file:figs/gravimeter_block_com.png]] + #+begin_src matlab -Gm = inv(Jm)*Gt*inv(Jm'); +%% Mass Matrix in frame {M} +Mm = diag([m,m,I]); #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Mm, {}, {}, ' %.1f '); +#+begin_src matlab :results value replace :exports results :tangle no +ans = Mm #+end_src -#+caption: $M_{\{M\}}$ +#+caption: Mass matrix expressed in $\{M\}$: $M_{\{M\}}$ #+RESULTS: -| 400.0 | 0.0 | 0.0 | -| 0.0 | 400.0 | 0.0 | -| 0.0 | 0.0 | 115.0 | +| 400 | 0 | 0 | +| 0 | 400 | 0 | +| 0 | 0 | 115 | -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Km, {}, {}, ' %.1f '); +#+begin_src matlab +%% Stiffness Matrix in frame {M} +Km = Jm'*Kr*Jm; #+end_src -#+caption: $K_{\{M\}}$ +#+begin_src matlab :results value replace :exports results :tangle no +ans = Km +#+end_src + +#+caption: Stiffness matrix expressed in $\{M\}$: $K_{\{M\}}$ #+RESULTS: -| 15000.0 | 0.0 | 12750.0 | -| 0.0 | 30000.0 | 0.0 | -| 12750.0 | 0.0 | 27750.0 | +| 15000 | 0 | 25500 | +| 0 | 30000 | 0 | +| 25500 | 0 | 50850 | + +#+begin_src matlab +%% Damping Matrix in frame {M} +Cm = Jm'*Cr*Jm; +#+end_src + +The plant from $\bm{F}_{\{M\}}$ to $\bm{\mathcal{X}}_{\{M\}}$ is defined below and its magnitude is shown in Figure [[fig:plant_frame_M]]. +#+begin_src matlab +%% Plant in frame {M} +Gm = inv(Mm*s^2 + Cm*s + Km); +#+end_src #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); +freqs = logspace(-2, 2, 1000); figure; % Magnitude @@ -1757,62 +2015,112 @@ for i_in = 1:3 end end plot(freqs, abs(squeeze(freqresp(Gm(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); + 'DisplayName', '$G_{\\\{M\\\}}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gm(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); + plot(freqs, abs(squeeze(freqresp(Gm(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', ['$G_{\\\{M\\\}}(', int2str(i_in_out), ',', int2str(i_in_out), ')$']); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'southeast'); -ylim([1e-8, 1e0]); +legend('location', 'southwest', 'FontSize', 8); +ylim([1e-8, 1e-2]); #+end_src -** Transfer function from $\mathcal{F}_{\{K\}}$ to $\mathcal{X}_{\{K\}}$ - -Jacobian: -#+begin_src matlab -Jk = [1 0 0 - 0 1 -la - 0 1 la]; +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/plant_frame_M.pdf', 'width', 'wide', 'height', 'normal'); #+end_src -Mass, Damping and Stiffness matrices expressed in $\{K\}$: -#+begin_src matlab -Mk = Jk'*Mt*Jk; -Ck = Jk'*Ct*Jk; -Kk = Jk'*Kt*Jk; -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Mk, {}, {}, ' %.1f '); -#+end_src - -#+caption: $M_{\{K\}}$ +#+name: fig:plant_frame_M +#+caption: Dynamics from $\bm{\mathcal{F}}_{\{M\}}$ to $\bm{\mathcal{X}}_{\{M\}}$ #+RESULTS: -| 400.0 | 0.0 | -340.0 | -| 0.0 | 400.0 | 0.0 | -| -340.0 | 0.0 | 404.0 | +[[file:figs/plant_frame_M.png]] +** Equations of motion - "Center of stiffness" {K} +<> -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Kk, {}, {}, ' %.1f '); +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. + +#+begin_src latex :file gravimeter_block_cok.pdf :tangle no :exports results +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.6 of G] (Jt) {$J_{\{K\}}^{-T}$}; + \node[block, right=0.6 of G] (Ja) {$J_{\{K\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; + \end{scope} +\end{tikzpicture} #+end_src -#+caption: $K_{\{K\}}$ +#+name: fig:gravimeter_block_cok +#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{K\}}$ to $\bm{\mathcal{X}}_{\{K\}}$ #+RESULTS: -| 15000.0 | 0.0 | 0.0 | -| 0.0 | 30000.0 | 0.0 | -| 0.0 | 0.0 | 16912.5 | +[[file:figs/gravimeter_block_cok.png]] #+begin_src matlab -% Gk = s^2*inv(Mk*s^2 + Ck*s + Kk); -Gk = inv(Jk)*Gt*inv(Jk'); +Mk = Jk'*inv(Jm)'*Mm*inv(Jm)*Jk; +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Mk +#+end_src + +#+caption: Mass matrix expressed in $\{K\}$: $M_{\{K\}}$ +#+RESULTS: +| 400 | 0 | -680 | +| 0 | 400 | 0 | +| -680 | 0 | 1271 | + +#+begin_src matlab +Kk = Jk'*Kr*Jk; +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Kk +#+end_src + +#+caption: Stiffness matrix expressed in $\{K\}$: $K_{\{K\}}$ +#+RESULTS: +| 15000 | 0 | 0 | +| 0 | 30000 | 0 | +| 0 | 0 | 7500 | + +The plant from $\bm{F}_{\{K\}}$ to $\bm{\mathcal{X}}_{\{K\}}$ is defined below and its magnitude is shown in Figure [[fig:plant_frame_K]]. +#+begin_src matlab +Gk = inv(Mk*s^2 + Ck*s + Kk); #+end_src #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); +freqs = logspace(-2, 2, 1000); figure; % Magnitude @@ -1824,80 +2132,32 @@ for i_in = 1:3 end end plot(freqs, abs(squeeze(freqresp(Gk(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_x(i,j)\ i \neq j$'); + 'DisplayName', '$G_{\\\{K\\\}}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gk(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out)); + plot(freqs, abs(squeeze(freqresp(Gk(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', ['$G_{\\\{K\\\}}(', int2str(i_in_out), ',', int2str(i_in_out), ')$']); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); -legend('location', 'southeast'); -ylim([1e-8, 1e0]); +legend('location', 'southwest', 'FontSize', 8); +ylim([1e-8, 1e-2]); #+end_src -** Analytical -*** Matlab Init :noexport:ignore: -#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) -<> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -<> -#+end_src - -*** Parameters -#+begin_src matlab -syms la ha m I c k positive -#+end_src - -#+begin_src matlab -Mm = [m 0 0 ; - 0 m 0 ; - 0 0 I]; - -Cm = [c 0 c*ha ; - 0 2*c 0 ; - c*ha 0 c*(ha+2*la)]; - -Km = [k 0 k*ha ; - 0 2*k 0 ; - k*ha 0 k*(ha+2*la)]; -#+end_src - -#+begin_src matlab -Jm = [1 0 ha ; - 0 1 -la ; - 0 1 la]; -#+end_src - -#+begin_src matlab -Mt = inv(Jm')*Mm*inv(Jm); -Ct = inv(Jm')*Cm*inv(Jm); -Kt = inv(Jm')*Km*inv(Jm); -#+end_src - -#+begin_src matlab -Jk = [1 0 0 - 0 1 -la - 0 1 la]; -#+end_src - -Mass, Damping and Stiffness matrices expressed in $\{K\}$: -#+begin_src matlab -Mk = Jk'*Mt*Jk; -Ck = Jk'*Ct*Jk; -Kk = Jk'*Kt*Jk; -#+end_src - -#+begin_src matlab :results replace value raw -['\begin{equation} M_{\{K\}} = ', latex(simplify(Kk)), '\end{equation}'] +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/plant_frame_K.pdf', 'width', 'wide', 'height', 'normal'); #+end_src +#+name: fig:plant_frame_K +#+caption: Dynamics from $\bm{\mathcal{F}}_{\{K\}}$ to $\bm{\mathcal{X}}_{\{K\}}$ #+RESULTS: -\begin{equation} M_{\{K\}} = \left(\begin{array}{ccc} k & 0 & 0\\ 0 & 2\,k & 0\\ 0 & 0 & k\,\left(-{\mathrm{ha}}^2+\mathrm{ha}+2\,\mathrm{la}\right) \end{array}\right)\end{equation} +[[file:figs/plant_frame_K.png]] + +** Conclusion +<> * Diagonal Stiffness Matrix for a planar manipulator +<> ** Model and Assumptions Consider a parallel manipulator with: - $b_i$: location of the joints on the top platform are called $b_i$ @@ -2261,7 +2521,7 @@ And we finally obtain: 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 \hat{s}_i (b_i \times \hat{s}_i)^T & k_i (b_i \times \hat{s}_i) (b_i \times \hat{s}_i)^T + 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} @@ -2303,7 +2563,7 @@ 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 vehttps://rwth.zoom.us/j/92311133102?pwd=UTAzS21YYkUwT2pMZDBLazlGNzdvdz09tor, we obtain: +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} @@ -2448,6 +2708,162 @@ axis equal; #+end_src ** TODO Example 2 - Stewart Platform +* Stiffness and Mass Matrices in the Leg's frame + +** 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: + \[ K_{\{L\}} = J_{\{M\}}^{-T} K_{\{M\}} J_{\{M\}}^{-1} \] + +** 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. + +** 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} + + +** Planar Example +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +The stiffnesses $k_i$, the joint positions ${}^Mb_i$ and joint unit vectors ${}^M\hat{s}_i$ are defined below: +#+begin_src matlab +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} +#+end_src + +Jacobian in frame $\{M\}$: +#+begin_src matlab +Jm = [si', (bi(1,:).*si(2,:) - bi(2,:).*si(1,:))']; +#+end_src + +And the stiffness matrix in frame $\{K\}$: +#+begin_src matlab +Km = Jm'*diag(ki)*Jm; +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +ans = Km +#+end_src + +#+RESULTS: +| 2 | 0 | 1 | +| 0 | 1 | -1 | +| 1 | -1 | 2 | + +Mass matrix in the frame $\{M\}$: +#+begin_src matlab +m = 10; % [kg] +I = 1; % [kg.m^2] + +Mm = diag([m, m, I]); +#+end_src + + +Now compute $K$ and $M$ in the frame of the legs: +#+begin_src matlab +ML = inv(Jm)'*Mm*inv(Jm) +KL = inv(Jm)'*Km*inv(Jm) +#+end_src + +#+begin_src matlab +Gm = 1/(ML*s^2 + KL); +#+end_src + +#+begin_src matlab +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'); +#+end_src + * Stewart Platform - Simscape Model :PROPERTIES: :header-args:matlab+: :tangle stewart_platform/script.m diff --git a/index.pdf b/index.pdf index a62cb57..a76ca20 100644 Binary files a/index.pdf and b/index.pdf differ