diff --git a/figs/apa300ml_comp_simpler_model.pdf b/figs/apa300ml_comp_simpler_model.pdf index 1141940..65f5566 100644 Binary files a/figs/apa300ml_comp_simpler_model.pdf and b/figs/apa300ml_comp_simpler_model.pdf differ diff --git a/figs/apa300ml_comp_simpler_model.png b/figs/apa300ml_comp_simpler_model.png index 493b5d5..725e098 100644 Binary files a/figs/apa300ml_comp_simpler_model.png and b/figs/apa300ml_comp_simpler_model.png differ diff --git a/figs/apa300ml_iff_plant.pdf b/figs/apa300ml_iff_plant.pdf index 2f6eafa..f2c981f 100644 Binary files a/figs/apa300ml_iff_plant.pdf and b/figs/apa300ml_iff_plant.pdf differ diff --git a/figs/apa300ml_iff_plant.png b/figs/apa300ml_iff_plant.png index 906a13b..2671c25 100644 Binary files a/figs/apa300ml_iff_plant.png and b/figs/apa300ml_iff_plant.png differ diff --git a/figs/apa300ml_iff_root_locus.pdf b/figs/apa300ml_iff_root_locus.pdf index b9a2d24..45f6b25 100644 Binary files a/figs/apa300ml_iff_root_locus.pdf and b/figs/apa300ml_iff_root_locus.pdf differ diff --git a/figs/apa300ml_iff_root_locus.png b/figs/apa300ml_iff_root_locus.png index 8bcb9f3..31065b7 100644 Binary files a/figs/apa300ml_iff_root_locus.png and b/figs/apa300ml_iff_root_locus.png differ diff --git a/figs/apa300ml_plant_dynamics.pdf b/figs/apa300ml_plant_dynamics.pdf index 8993e0c..eb09fea 100644 Binary files a/figs/apa300ml_plant_dynamics.pdf and b/figs/apa300ml_plant_dynamics.pdf differ diff --git a/figs/apa300ml_plant_dynamics.png b/figs/apa300ml_plant_dynamics.png index 5712c5f..31ef786 100644 Binary files a/figs/apa300ml_plant_dynamics.png and b/figs/apa300ml_plant_dynamics.png differ diff --git a/figs/apa300ml_resonance.pdf b/figs/apa300ml_resonance.pdf index 6a97872..a6f736c 100644 Binary files a/figs/apa300ml_resonance.pdf and b/figs/apa300ml_resonance.pdf differ diff --git a/figs/apa300ml_resonance.png b/figs/apa300ml_resonance.png index f4b8747..6e356a0 100644 Binary files a/figs/apa300ml_resonance.png and b/figs/apa300ml_resonance.png differ diff --git a/figs/flexor_ID16_compare_bushing_joint.pdf b/figs/flexor_ID16_compare_bushing_joint.pdf index deb3f9d..612d404 100644 --- a/figs/flexor_ID16_compare_bushing_joint.pdf +++ b/figs/flexor_ID16_compare_bushing_joint.pdf @@ -3,7 +3,7 @@ 1 0 obj << /Producer (Apache FOP Version 2.4.0-SNAPSHOT: PDFDocumentGraphics2D) -/CreationDate (D:20200804121346+02'00') +/CreationDate (D:20200831100334+02'00') >> endobj 2 0 obj @@ -1185,332 +1185,354 @@ endobj 198 0 obj << /Length 199 0 R /Filter /FlateDecode >> stream -xԽˮ,ɒ7X/I_A(h Q!z@hG%: w33ܾr =|OxsVŔ/j(|?p__W_O8#_o?_W_ }GH;8"xXachq-/}h`%9C6 ,?pʇmXa=19 9i9Q߾A>R){E_FVT1Ood_1O [Cb1v/}dhtZj\i-1 }Cü -){_6oehakX>2m-=B!5wϵ -shM{o~~#C?j>2Y#y#.,~ƚW}RnK,~.X_^оgs.Z奏 {?glxV}v?vy#CG Nmy#Cjz=奏 {AjY\xB>jpd>2fq[Khi m~#Cj=奏 {A‚yI|vqy#COG=| <./}dhߴ.8 {&kw>n}Oq2Kl}w>2ן"}>2<./}dh[ àKVx\>2Lma奏 {Auw>2чy\^о&9/}$ mk]<./}dh߳y\^оgC{qy#CX)f/}dh߳!=}wqy#CG=BOq2K쿇OO1u?wK)0-ZnG׸;KVE<ɮ-/}dh[ |}o59G;KV<./}dh[ <|y,߳8Ϸ%_оi]Oq6KAETG=EHD| l~#mW>2Yvq~#fP} 5@8}s~di8GѾ1\F]gFs%{ÿZ=ffCEobˏT{o.6!?ˏ{;5x>ڣ;g]~dH9Կvg[~dH]!v%>B̉oYNm~`@1 i? 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Finite Element Model with Simscape

Table of Contents

-
-

1 Amplified Piezoelectric Actuator - 3D elements

+
+

1 Amplified Piezoelectric Actuator - 3D elements

The idea here is to: @@ -103,15 +129,15 @@ The idea here is to:

-
-

1.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

+
+

1.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates

We first extract the stiffness and mass matrices.

-
K = extractMatrix('piezo_amplified_3d_K.txt');
-M = extractMatrix('piezo_amplified_3d_M.txt');
+
K = extractMatrix('piezo_amplified_3d_K.txt');
+M = extractMatrix('piezo_amplified_3d_M.txt');
@@ -119,22 +145,22 @@ M = extractMatrix('piezo_amplified_3d_M.txt'); Then, we extract the coordinates of the interface nodes.

-
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_3d.txt');
+
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_3d.txt');
-
save('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
save('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
-
-

1.2 Output parameters

+
+

1.2 Output parameters

-
load('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
load('./mat/piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
@@ -170,7 +196,7 @@ Then, we extract the coordinates of the interface nodes. -
+

amplified_piezo_interface_nodes.png

Figure 1: Interface Nodes for the Amplified Piezo Actuator

@@ -628,26 +654,26 @@ Using K, M and int_xyz, we can use the -
-

1.3 Piezoelectric parameters

+
+

1.3 Piezoelectric parameters

Parameters for the APA95ML:

-
d33 = 3e-10; % Strain constant [m/V]
-n = 80; % Number of layers per stack
-eT = 1.6e-7; % Permittivity under constant stress [F/m]
-sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
-ka = 235e6; % Stack stiffness [N/m]
-C = 5e-6; % Stack capactiance [F]
+
d33 = 3e-10; % Strain constant [m/V]
+n = 80; % Number of layers per stack
+eT = 1.6e-7; % Permittivity under constant stress [F/m]
+sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
+ka = 235e6; % Stack stiffness [N/m]
+C = 5e-6; % Stack capactiance [F]
-
na = 2; % Number of stacks used as actuator
-ns = 1; % Number of stacks used as force sensor
+
na = 2; % Number of stacks used as actuator
+ns = 1; % Number of stacks used as force sensor
@@ -657,7 +683,7 @@ We denote this constant by \(g_a\) and: \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]

-
d33*(na*n)*(ka/(na + ns)) % [N/V]
+
d33*(na*n)*(ka/(na + ns)) % [N/V]
@@ -680,7 +706,7 @@ where:
  • \(\Delta h\): relative displacement [m]
    • -
    1e-6*d33/(eT*sD*ns*n) % [V/um]
+
    1e-6*d33/(eT*sD*ns*n) % [V/um]
    @@ -690,8 +716,8 @@ where:
    -
    -

    1.4 Identification of the Dynamics

    +
    +

    1.4 Identification of the Dynamics

    The flexible element is imported using the Reduced Order Flexible Solid simscape block. @@ -718,15 +744,15 @@ We first set the mass to be zero. The dynamics is identified from the applied force to the measured relative displacement.

    -
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
 
-Gh = -linearize(mdl, io);
+Gh = -linearize(mdl, io);
    @@ -743,23 +769,23 @@ And the dynamics is identified.

    -The two identified dynamics are compared in Figure 2. +The two identified dynamics are compared in Figure 2.

    -
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
 
-Ghm = -linearize(mdl, io);
+Ghm = -linearize(mdl, io);
    -
    +

    dynamics_act_disp_comp_mass.png

    Figure 2: Dynamics from \(F\) to \(d\) without a payload and with a 10kg payload

    @@ -767,24 +793,24 @@ Ghm = -linearize(mdl, io);
    -
    -

    1.5 Comparison with Ansys

    +
    +

    1.5 Comparison with Ansys

    Let’s import the results from an Harmonic response analysis in Ansys.

    -
    Gresp0 = readtable('FEA_HarmResponse_00kg.txt');
-Gresp10 = readtable('FEA_HarmResponse_10kg.txt');
+
    Gresp0 = readtable('FEA_HarmResponse_00kg.txt');
+Gresp10 = readtable('FEA_HarmResponse_10kg.txt');

    -The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure 3. +The obtained dynamics from the Simscape model and from the Ansys analysis are compare in Figure 3.

    -
    +

    dynamics_force_disp_comp_anasys.png

    Figure 3: Comparison of the obtained dynamics using Simscape with the harmonic response analysis using Ansys

    @@ -792,15 +818,15 @@ The obtained dynamics from the Simscape model and from the Ansys analysis are co
    -
    -

    1.6 Force Sensor

    +
    +

    1.6 Force Sensor

    The dynamics is identified from internal forces applied between nodes 3 and 11 to the relative displacement of nodes 11 and 13.

    -The obtained dynamics is shown in Figure 4. +The obtained dynamics is shown in Figure 4.

    @@ -809,13 +835,13 @@ The obtained dynamics is shown in Figure 4.
    -
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gf = linearize(mdl, io);
 
    
@@ -827,20 +853,20 @@ Gf = linearize(mdl, io);
    -
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gfm = linearize(mdl, io);
    -
    +

    dynamics_force_force_sensor_comp_mass.png

    Figure 4: Dynamics from \(F\) to \(F_m\) for \(m=0\) and \(m = 10kg\)

    @@ -848,8 +874,8 @@ Gfm = linearize(mdl, io);
    -
    -

    1.7 Distributed Actuator

    +
    +

    1.7 Distributed Actuator

    m = 0;
@@ -860,13 +886,13 @@ Gfm = linearize(mdl, io);
 The dynamics is identified from the applied force to the measured relative displacement.
 
    
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d_distri';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d_distri';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gd = linearize(mdl, io);
 
    
@@ -884,13 +910,13 @@ Then, we add 10Kg of mass:
 And the dynamics is identified.
 
    
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d_distri';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d_distri';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gdm = linearize(mdl, io);
 
    
@@ -898,8 +924,8 @@ Gdm = linearize(mdl, io);
 
    
 
    
 
-
    
-
    1.8 Distributed Actuator and Force Sensor
    
+
    
+
    1.8 Distributed Actuator and Force Sensor
    
 
    
 
    
 
    m = 0;
@@ -907,13 +933,13 @@ Gdm = linearize(mdl, io);
 
    
 
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d_distri_act_sens';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d_distri_act_sens';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gfd = linearize(mdl, io);
 
    
@@ -925,13 +951,13 @@ Gfd = linearize(mdl, io);
 
    
 
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d_distri_act_sens';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d_distri_act_sens';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1;
 
 Gfdm = linearize(mdl, io);
 
    
@@ -939,8 +965,8 @@ Gfdm = linearize(mdl, io);
 
    
 
    
 
-
    
-
    1.9 Dynamics from input voltage to displacement
    
+
    
+
    1.9 Dynamics from input voltage to displacement
    
 
    
 
    
 
    m = 5;
@@ -952,30 +978,30 @@ And the dynamics is identified.
 
    
 
 
    
-The two identified dynamics are compared in Figure 2.
+The two identified dynamics are compared in Figure 2.
 
    
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/V'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/V'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
 
-G = -linearize(mdl, io);
+G = -linearize(mdl, io);
 
    
 
    
 
 
    
-
    save('../test-bench-apa/mat/fem_model_5kg.mat', 'G')
+
    save('../test-bench-apa/mat/fem_model_5kg.mat', 'G')
 
    
 
    
 
    
 
    
 
-
    
-
    1.10 Dynamics from input voltage to output voltage
    
+
    
+
    1.10 Dynamics from input voltage to output voltage
    
 
    
 
    
 
    m = 5;
@@ -983,776 +1009,23 @@ G = -linearize(mdl, io);
 
    
 
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_3d';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_3d';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1;
 
-G = -linearize(mdl, io);
-
    
-
    
-
    
-
    
-
    
-
-
    
-
    2 APA300ML
    
-
    
-
-
    
-
    apa300ml_ansys.jpg
-
    
-
    Figure 5: Ansys FEM of the APA300ML
    
-
    
-
    
-
-
    
-
    2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
-
    
-
    
-We first extract the stiffness and mass matrices.
-
    
-
    
-
    K = extractMatrix('mat_K-48modes-7MDoF.matrix');
-M = extractMatrix('mat_M-48modes-7MDoF.matrix');
-
    
-
    
-
-
    
-
    K = extractMatrix('mat_K-80modes-7MDoF.matrix');
-M = extractMatrix('mat_M-80modes-7MDoF.matrix');
-
    
-
    
-
-
    
-Then, we extract the coordinates of the interface nodes.
-
    
-
    
-
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('Nodes_MDoF_NLIST_MLIST.txt');
-
    
-
    
-
-
    
-
    save('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+G = -linearize(mdl, io);
 
    
 
    
 
    
 
    
 
-
    
-
    2.2 Output parameters
    
-
    
-
    
-
    load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
-
    
-
    
-
-
-
-
---
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
    



    Total number of Nodes7
    Number of interface Nodes7
    Number of Modes6
    Size of M and K matrices48
    
-
-
-
-
---
--
--
--
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
    Table 4: Coordinates of the interface nodes






    Node iNode Numberx [m]y [m]z [m]
    1.053917.00.0-0.0150.0
    2.053918.00.00.0150.0
    3.053919.0-0.03250.00.0
    4.053920.0-0.01250.00.0
    5.053921.0-0.00750.00.0
    6.053922.00.01250.00.0
    7.053923.00.03250.00.0
    
-
-
-
-
---
--
--
--
--
--
--
--
--
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
    Table 5: First 10x10 elements of the Stiffness matrix











    200000000.030000.050000.0200.0-100.0-300000.010000000.0-6000.020000.0-60.0
    30000.07000000.010000.030.0-30.0-70.07000.0-500000.03000.0-10.0
    50000.010000.030000000.0200000.0-200.0-100.020000.0-2000.02000000.0-9000.0
    200.030.0200000.01000.0-0.8-0.450.0-69000.0-30.0
    -100.0-30.0-200.0-0.810000.00.2-40.010.020.0-0.05
    -300000.0-70.0-100.0-0.40.2900.0-30000.010.0-40.00.1
    10000000.07000.020000.050.0-40.0-30000.0200000000.0-50000.030000.0-50.0
    -6000.0-500000.0-2000.0-610.010.0-50000.07000000.0-4000.08
    20000.03000.02000000.09000.020.0-40.030000.0-4000.030000000.0-200000.0
    -60.0-10.0-9000.0-30.0-0.050.1-50.08-200000.01000.0
    
-
-
-
-
-
---
--
--
--
--
--
--
--
--
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
    Table 6: First 10x10 elements of the Mass matrix











    0.017e-06-5e-06-6e-083e-09-5e-05-0.0005-2e-07-3e-061e-08
    7e-060.0094e-076e-09-4e-09-3e-08-2e-076e-055e-07-1e-09
    -5e-064e-070.012e-052e-083e-08-2e-06-1e-07-0.00029e-07
    -6e-086e-092e-053e-071e-103e-10-7e-092e-10-9e-073e-09
    3e-09-4e-092e-081e-101e-07-3e-126e-09-2e-10-3e-099e-12
    -5e-05-3e-083e-083e-10-3e-126e-071e-06-3e-092e-08-7e-11
    -0.0005-2e-07-2e-06-7e-096e-091e-060.01-8e-06-2e-069e-09
    -2e-076e-05-1e-072e-10-2e-10-3e-09-8e-060.0091e-072e-09
    -3e-065e-07-0.0002-9e-07-3e-092e-08-2e-061e-070.01-2e-05
    1e-08-1e-099e-073e-099e-12-7e-119e-092e-09-2e-053e-07
    
-
-
    
-Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.
-
    
-
    
-
    
-
-
    
-
    2.3 Piezoelectric parameters
    
-
    
-
    
-Parameters for the APA300ML:
-
    
-
-
    
-
    d33 = 3e-10; % Strain constant [m/V]
-n = 80; % Number of layers per stack
-eT = 1.6e-8; % Permittivity under constant stress [F/m]
-sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
-ka = 235e6; % Stack stiffness [N/m]
-C = 5e-6; % Stack capactiance [F]
-
    
-
    
-
-
    
-
    na = 2; % Number of stacks used as actuator
-ns = 1; % Number of stacks used as force sensor
-
    
-
    
-
-
    
-The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V].
-We denote this constant by \(g_a\) and:
-\[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
-
    
-
    
-
    d33*(na*n)*(ka/(na + ns)) % [N/V]
-
    
-
    
-
-
    -0.42941
-
    
-
-
-
    
-From (Fleming and Leang 2014) (page 123), the relation between relative displacement and generated voltage is:
-\[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
-where:
-
    
-
      
-
    • \(V_s\): measured voltage [V]
      • 
-
    • \(d_{33}\): strain constant [m/V]
      • 
-
    • \(\epsilon^T\): permittivity under constant stress [F/m]
      • 
-
    • \(s^D\): elastic compliance under constant electric displacement [m^2/N]
      • 
-
    • \(n\): number of layers
      • 
-
    • \(\Delta h\): relative displacement [m]
      • 
-
    
-
    
-
    1e-6*d33/(eT*sD*ns*n) % [V/um]
-
    
-
    
-
-
    -5.8594
-
    
-
    
-
    
-
-
    
-
    2.4 Identification of the APA Characteristics
    
-
    
-
    
-
    
-
    2.4.1 Stiffness
    
-
    
-
    
-The transfer function from vertical external force to the relative vertical displacement is identified.
-
    
-
-
    
-The inverse of its DC gain is the axial stiffness of the APA:
-
    
-
    
-
    1e-6/dcgain(G) % [N/um]
-
    
-
    
-
-
    -1.8634
-
    
-
-
-
    
-The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
-
    
-
    
-
    
-
-
    
-
    2.4.2 Resonance Frequency
    
-
    
-
    
-The resonance frequency is specified to be between 650Hz and 840Hz.
-This is also the case for the FEM model (Figure 6).
-
    
-
-
-
    
-
    apa300ml_resonance.png
-
    
-
    Figure 6: First resonance is around 800Hz
    
-
    
-
    
-
    
-
-
    
-
    2.4.3 Amplification factor
    
-
    
-
    
-The amplification factor is the ratio of the axial displacement to the stack displacement.
-
    
-
-
    
-The ratio of the two displacement is computed from the FEM model.
-
    
-
    
-
    -dcgain(G(1,1))./dcgain(G(2,1))
-
    
-
    
-
-
    -4.936
-
    
-
-
-
    
-If we take the ratio of the piezo height and length (approximation of the amplification factor):
-
    
-
    
-
    75/15
-
    
-
    
-
-
    -5
-
    
-
    
-
    
-
-
    
-
    2.4.4 Stroke
    
-
    
-
    
-Estimation of the actuator stroke:
-\[ \Delta H = A n \Delta L \]
-with:
-
    
-
      
-
    • \(\Delta H\) Axial Stroke of the APA
      • 
-
    • \(A\) Amplification factor (5 for the APA300ML)
      • 
-
    • \(n\) Number of stack used
      • 
-
    • \(\Delta L\) Stroke of the stack (0.1% of its length)
      • 
-
    
-
-
    
-
    1e6 * 5 * 3 * 20e-3 * 0.1e-2
-
    
-
    
-
-
    -300
-
    
-
-
-
    
-This is exactly the specified stroke in the data-sheet.
-
    
-
    
-
    
-
    
-
-
    
-
    2.5 Identification of the Dynamics
    
-
    
-
    
-The flexible element is imported using the Reduced Order Flexible Solid simscape block.
-
    
-
-
    
-To model the actuator, an Internal Force block is added between the nodes 3 and 12.
-A Relative Motion Sensor block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
-
    
-
-
    
-One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
-
    
-
-
    
-We first set the mass to be zero.
-The dynamics is identified from the applied force to the measured relative displacement.
-The same dynamics is identified for a payload mass of 10Kg.
-
    
-
    
-
    m = 10;
-
    
-
    
-
-
-
    
-
    apa300ml_plant_dynamics.png
-
    
-
    Figure 7: Transfer function from forces applied by the stack to the axial displacement of the APA
    
-
    
-
    
-
    
-
-
    
-
    2.6 IFF
    
-
    
-
    
-Let’s use 2 stacks as actuators and 1 stack as force sensor.
-
    
-
-
    
-The transfer function from actuator to sensors is identified and shown in Figure 8.
-
    
-
-
    
-
    apa300ml_iff_plant.png
-
    
-
    Figure 8: Transfer function from actuator to force sensor
    
-
    
-
-
    
-For root locus corresponding to IFF is shown in Figure 9.
-
    
-
-
-
    
-
    apa300ml_iff_root_locus.png
-
    
-
    Figure 9: Root Locus for IFF
    
-
    
-
    
-
    
-
-
    
-
    2.7 DVF
    
-
    
-
    
-Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure 10.
-
    
-
-
    
-
    apa300ml_dvf_plant.png
-
    
-
    Figure 10: Transfer function from stack actuator to relative motion sensor
    
-
    
-
-
    
-The root locus for DVF is shown in Figure 11.
-
    
-
-
-
    
-
    apa300ml_dvf_root_locus.png
-
    
-
    Figure 11: Root Locus for Direct Velocity Feedback
    
-
    
-
    
-
    
-
-
    
-
    2.8 Identification for a simpler model
    
-
    
+
    
+
    1.11 Identification for a simpler model
    
+
    
 
    
 The goal in this section is to identify the parameters of a simple APA model from the FEM.
 This can be useful is a lower order model is to be used for simulations.
@@ -1763,19 +1036,19 @@ The presented model is based on (Souleille et al.
 
    
 
 
    
-The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 12).
+The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 5).
 The parameters are shown in the table below.
 
    
 
 
-
    
+
    
 
    souleille18_model_piezo.png
 
    
-
    Figure 12: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
    
+
    Figure 5: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
    
 
    
 
 
-
+@@ -1830,8 +1103,8 @@ If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
 From the identified dynamics, compute \(\alpha\) and \(\beta\)
 
    
-
    alpha = abs(dcgain(G('y', 'Fa')));
-beta  = abs(dcgain(G('y', 'Fd')));
+
    alpha = abs(dcgain(G('y', 'Fa')));
+beta  = abs(dcgain(G('y', 'Fd')));
 
    
 
    
 
@@ -1839,7 +1112,7 @@ beta  = abs(dcgain(G('y', 'Fd')));
 \(k_a\) is estimated using the following formula:
 
    
-
    ka = 0.8/abs(dcgain(G('y', 'Fa')));
+
    ka = 0.9/abs(dcgain(G('y', 'Fa')));
 
    
 
    
@@ -1850,8 +1123,8 @@ The factor can be adjusted to better match the curves.
 Then \(k_e\) and \(k_1\) are computed.
 
    
-
    ke = ka/(beta/alpha - 1);
-k1 = 1/beta - ke*ka/(ke + ka);
+
    ke = ka/(beta/alpha - 1);
+k1 = 1/beta - ke*ka/(ke + ka);
 
    
 
    
 
@@ -1872,7 +1145,936 @@ k1 = 1/beta - ke*ka/(ke + ka);
 
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 7: Parameters used for the model of the APA 100MTable 4: Parameters used for the model of the APA 100M
 
 
 

 
 
 
 
 
    
 ka42.954.9
    ke25.1
    k14.3
    
+
+
    
+The damping in the system is adjusted to match the FEM model if necessary.
+
    
+
    
+
    c1 = 1e2;
+
    
+
    
+
+
    
+Analytical model of the simpler system:
+
    
+
    
+
    Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
+     [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
+      -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
+
+Ga.InputName = {'Fd', 'w', 'Fa'};
+Ga.OutputName = {'y', 'Fs'};
+
    
+
    
+
+
    
+Adjust the DC gain for the force sensor:
+
    
+
    
+
    F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
+
    
+
    
+
+
+
    
+
    apa95ml_comp_simpler_model.png
+
    
+
    Figure 6: Comparison of the Dynamics between the FEM model and the simplified one
    
+
    
+
+
    
+We save the parameters of the simplified model for the APA95ML:
+
    
+
    
+
    save('./mat/APA95ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');
+
    
+
    
+
    
+
    
+
    
+
+
    
+
    2 APA300ML
    
+
    
+
+
    
+
    apa300ml_ansys.jpg
+
    
+
    Figure 7: Ansys FEM of the APA300ML
    
+
    
+
    
+
+
    
+
    2.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
+
    
+
    
+We first extract the stiffness and mass matrices.
+
    
+
    
+
    K = readmatrix('mat_K.CSV');
+M = readmatrix('mat_M.CSV');
+
    
+
    
+
+
    
+Then, we extract the coordinates of the interface nodes.
+
    
+
    
+
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+
    
+
    
+
+
    
+
    save('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
    
+
    
+
+
    
+
    2.2 Output parameters
    
+
    
+
    
+
    load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    Total number of Nodes7
    Number of interface Nodes7
    Number of Modes120
    Size of M and K matrices162
    
+
+
+
+
+++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 5: Coordinates of the interface nodes






    Node iNode Numberx [m]y [m]z [m]
    1.0697783.00.00.0-0.015
    2.0697784.00.00.00.015
    3.0697785.0-0.03250.00.0
    4.0697786.0-0.01250.00.0
    5.0697787.0-0.00750.00.0
    6.0697788.00.01250.00.0
    7.0697789.00.03250.00.0
    
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 6: First 10x10 elements of the Stiffness matrix











    200000000.030000.0-20000.0-70.0300000.040.010000000.010000.0-6000.030.0
    30000.030000000.02000.0-200000.060.0-10.04000.02000000.0-500.09000.0
    -20000.02000.07000000.0-10.0-30.010.06000.0900.0-500000.03
    -70.0-200000.0-10.01000.0-0.10.08-20.0-9000.03-30.0
    300000.060.0-30.0-0.1900.00.130000.020.0-10.00.06
    40.0-10.010.00.080.110000.020.09-50.03
    10000000.04000.06000.0-20.030000.020.0200000000.010000.09000.050.0
    10000.02000000.0900.0-9000.020.0910000.030000000.0-500.0200000.0
    -6000.0-500.0-500000.03-10.0-59000.0-500.07000000.0-2
    30.09000.03-30.00.060.0350.0200000.0-21000.0
    
+
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 7: First 10x10 elements of the Mass matrix











    0.01-2e-061e-066e-095e-05-5e-09-0.0005-7e-076e-07-3e-09
    -2e-060.018e-07-2e-05-8e-092e-09-9e-07-0.00021e-08-9e-07
    1e-068e-070.0095e-101e-09-1e-09-5e-073e-086e-051e-10
    6e-09-2e-055e-103e-072e-11-3e-123e-099e-07-4e-103e-09
    5e-05-8e-091e-092e-116e-07-4e-11-1e-06-2e-091e-09-8e-12
    -5e-092e-09-1e-09-3e-12-4e-111e-07-2e-09-1e-09-4e-10-5e-12
    -0.0005-9e-07-5e-073e-09-1e-06-2e-090.011e-07-3e-07-2e-08
    -7e-07-0.00023e-089e-07-2e-09-1e-091e-070.01-4e-072e-05
    6e-071e-086e-05-4e-101e-09-4e-10-3e-07-4e-070.009-2e-10
    -3e-09-9e-071e-103e-09-8e-12-5e-12-2e-082e-05-2e-103e-07
    
+
+
    
+Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.
+
    
+
    
+
    
+
+
    
+
    2.3 Piezoelectric parameters
    
+
    
+
    
+Parameters for the APA300ML:
+
    
+
+
    
+
    d33 = 3e-10; % Strain constant [m/V]
+n = 80; % Number of layers per stack
+eT = 1.6e-8; % Permittivity under constant stress [F/m]
+sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
+ka = 235e6; % Stack stiffness [N/m]
+C = 5e-6; % Stack capactiance [F]
+
    
+
    
+
+
    
+
    na = 2; % Number of stacks used as actuator
+ns = 1; % Number of stacks used as force sensor
+
    
+
    
+
+
    
+The ratio of the developed force to applied voltage is \(d_{33} n k_a\) in [N/V].
+We denote this constant by \(g_a\) and:
+\[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
+
    
+
    
+
    d33*(na*n)*(ka/(na + ns)) % [N/V]
+
    
+
    
+
+
    +3.76
+
    
+
+
+
    
+From (Fleming and Leang 2014) (page 123), the relation between relative displacement and generated voltage is:
+\[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
+where:
+
    
+
      
+
    • \(V_s\): measured voltage [V]
      • 
+
    • \(d_{33}\): strain constant [m/V]
      • 
+
    • \(\epsilon^T\): permittivity under constant stress [F/m]
      • 
+
    • \(s^D\): elastic compliance under constant electric displacement [m^2/N]
      • 
+
    • \(n\): number of layers
      • 
+
    • \(\Delta h\): relative displacement [m]
      • 
+
    
+
    
+
    1e-6*d33/(eT*sD*ns*n) % [V/um]
+
    
+
    
+
+
    +11.719
+
    
+
    
+
    
+
+
    
+
    2.4 Identification of the APA Characteristics
    
+
    
+
    
+
    
+
    2.4.1 Stiffness
    
+
    
+
    
+The transfer function from vertical external force to the relative vertical displacement is identified.
+
    
+
+
    
+The inverse of its DC gain is the axial stiffness of the APA:
+
    
+
    
+
    1e-6/dcgain(G) % [N/um]
+
    
+
    
+
+
    +1.753
+
    
+
+
+
    
+The specified stiffness in the datasheet is \(k = 1.8\, [N/\mu m]\).
+
    
+
    
+
    
+
+
    
+
    2.4.2 Resonance Frequency
    
+
    
+
    
+The resonance frequency is specified to be between 650Hz and 840Hz.
+This is also the case for the FEM model (Figure 8).
+
    
+
+
+
    
+
    apa300ml_resonance.png
+
    
+
    Figure 8: First resonance is around 800Hz
    
+
    
+
    
+
    
+
+
    
+
    2.4.3 Amplification factor
    
+
    
+
    
+The amplification factor is the ratio of the axial displacement to the stack displacement.
+
    
+
+
    
+The ratio of the two displacement is computed from the FEM model.
+
    
+
    
+
    abs(dcgain(G(1,1))./dcgain(G(2,1)))
+
    
+
    
+
+
    +5.0749
+
    
+
+
+
    
+If we take the ratio of the piezo height and length (approximation of the amplification factor):
+
    
+
    
+
    75/15
+
    
+
    
+
+
    +5
+
    
+
    
+
    
+
+
    
+
    2.4.4 Stroke
    
+
    
+
    
+Estimation of the actuator stroke:
+\[ \Delta H = A n \Delta L \]
+with:
+
    
+
      
+
    • \(\Delta H\) Axial Stroke of the APA
      • 
+
    • \(A\) Amplification factor (5 for the APA300ML)
      • 
+
    • \(n\) Number of stack used
      • 
+
    • \(\Delta L\) Stroke of the stack (0.1% of its length)
      • 
+
    
+
+
    
+
    1e6 * 5 * 3 * 20e-3 * 0.1e-2
+
    
+
    
+
+
    +300
+
    
+
+
+
    
+This is exactly the specified stroke in the data-sheet.
+
    
+
    
+
    
+
    
+
+
    
+
    2.5 Identification of the Dynamics
    
+
    
+
    
+The flexible element is imported using the Reduced Order Flexible Solid simscape block.
+
    
+
+
    
+To model the actuator, an Internal Force block is added between the nodes 3 and 12.
+A Relative Motion Sensor block is added between the nodes 1 and 2 to measure the displacement and the amplified piezo.
+
    
+
+
    
+One mass is fixed at one end of the piezo-electric stack actuator, the other end is fixed to the world frame.
+
    
+
+
    
+We first set the mass to be zero.
+The dynamics is identified from the applied force to the measured relative displacement.
+The same dynamics is identified for a payload mass of 10Kg.
+
    
+
    
+
    m = 10;
+
    
+
    
+
+
+
    
+
    apa300ml_plant_dynamics.png
+
    
+
    Figure 9: Transfer function from forces applied by the stack to the axial displacement of the APA
    
+
    
+
    
+
    
+
+
    
+
    2.6 IFF
    
+
    
+
    
+Let’s use 2 stacks as actuators and 1 stack as force sensor.
+
    
+
+
    
+The transfer function from actuator to sensors is identified and shown in Figure 10.
+
    
+
+
    
+
    apa300ml_iff_plant.png
+
    
+
    Figure 10: Transfer function from actuator to force sensor
    
+
    
+
+
    
+For root locus corresponding to IFF is shown in Figure 11.
+
    
+
+
    
+
    apa300ml_iff_root_locus.png
+
    
+
    Figure 11: Root Locus for IFF
    
+
    
+
    
+
    
+
+
    
+
    2.7 DVF
    
+
    
+
    
+Now the dynamics from the stack actuator to the relative motion sensor is identified and shown in Figure 12.
+
    
+
+
    
+
    apa300ml_dvf_plant.png
+
    
+
    Figure 12: Transfer function from stack actuator to relative motion sensor
    
+
    
+
+
    
+The root locus for DVF is shown in Figure 13.
+
    
+
+
+
    
+
    apa300ml_dvf_root_locus.png
+
    
+
    Figure 13: Root Locus for Direct Velocity Feedback
    
+
    
+
    
+
    
+
+
    
+
    2.8 Identification for a simpler model
    
+
    
+
    
+The goal in this section is to identify the parameters of a simple APA model from the FEM.
+This can be useful is a lower order model is to be used for simulations.
+
    
+
+
    
+The presented model is based on (Souleille et al. 2018).
+
    
+
+
    
+The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure 5).
+The parameters are shown in the table below.
+
    
+
+
+
    
+
    souleille18_model_piezo.png
+
    
+
    Figure 14: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 8: Parameters used for the model of the APA 100M



     Meaning
    \(k_e\)Stiffness used to adjust the pole of the isolator
    \(k_1\)Stiffness of the metallic suspension when the stack is removed
    \(k_a\)Stiffness of the actuator
    \(c_1\)Added viscous damping
    
+
+
    
+The goal is to determine \(k_e\), \(k_a\) and \(k_1\) so that the simplified model fits the FEM model.
+
    
+
+
    
+\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+
    
+
+
    
+If we can fix \(k_a\), we can determine \(k_e\) and \(k_1\) with:
+\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
+\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
+
    
+
+
    
+From the identified dynamics, compute \(\alpha\) and \(\beta\)
+
    
+
    
+
    alpha = abs(dcgain(G('y', 'Fa')));
+beta  = abs(dcgain(G('y', 'Fd')));
+
    
+
    
+
+
    
+\(k_a\) is estimated using the following formula:
+
    
+
    
+
    ka = 0.8/abs(dcgain(G('y', 'Fa')));
+
    
+
    
+
    
+The factor can be adjusted to better match the curves.
+
    
+
+
    
+Then \(k_e\) and \(k_1\) are computed.
+
    
+
    
+
    ke = ka/(beta/alpha - 1);
+k1 = 1/beta - ke*ka/(ke + ka);
+
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
@@ -1899,12 +2101,12 @@ The damping in the system is adjusted to match the FEM model if necessary.
 Analytical model of the simpler system:
 
    
-
    Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
-     [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
-      -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
+
    Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
+     [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
+      -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
 
-Ga.InputName = {'Fd', 'w', 'Fa'};
-Ga.OutputName = {'y', 'Fs'};
+Ga.InputName = {'Fd', 'w', 'Fa'};
+Ga.OutputName = {'y', 'Fs'};
 
    
 
    
 
@@ -1912,44 +2114,277 @@ Ga.OutputName = {'y', 'Fs'};
 Adjust the DC gain for the force sensor:
 
    
-
    lambda = dcgain(Ga('Fs', 'Fd'))/dcgain(G('Fs', 'Fd'));
+
    F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
 
    
 
    
 
 
-
    
+
    
 
    apa300ml_comp_simpler_model.png
 
    
-
    Figure 13: Comparison of the Dynamics between the FEM model and the simplified one
    
+
    Figure 15: Comparison of the Dynamics between the FEM model and the simplified one
    
+
    
+
+
    
+We now compare the FEM model with the simplified simscape model.
+
    
+
+
    
+
    apa300ml_comp_simpler_simscape.png
+
    
+
    Figure 16: Comparison of the Dynamics between the FEM model and the simplified simscape model
    
+
    
+
+
    
+We save the parameters of the simplified model for the APA300ML:
+
    
+
    
+
    save('./mat/APA300ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');
+
    
+
    
+
    
+
+
+
    
+
    2.9 Identification of the stiffness properties
    
+
    
+
    
+
    
+
    2.9.1 APA Alone
    
+
    
+
    



     Value [N/um]
    ka40.5
 
    
 
 
    
 
 
    
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    CaracteristicsValue
    Kx [N/um]0.8
    Ky [N/um]1.6
    Kz [N/um]1.8
    Rx [Nm/rad]71.4
    Ry [Nm/rad]148.2
    Rz [Nm/rad]4241.8
    
+
    
+
    
+
+
    
+
    2.9.2 See how the global stiffness is changing with the flexible joints
    
+
    
+
    
+
    flex = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    CaracteristicValue
    Kx [N/um]0.0
    Ky [N/um]0.0
    Kz [N/um]1.8
    Rx [Nm/rad]722.9
    Ry [Nm/rad]129.6
    Rz [Nm/rad]115.3
    
+
+
+
    
+
    freqs = logspace(-2, 5, 1000);
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
+plot(freqs, abs(squeeze(freqresp(Gf(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
+hold off;
+legend('location', 'northeast');
+
    
+
    
+
+
    
+
    freqs = logspace(-2, 5, 1000);
+
+figure;
+hold on;
+plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
+plot(freqs, abs(squeeze(freqresp(Gf(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
+hold off;
+legend('location', 'northeast');
+
    
 
    
 
    
 
    
 
    
 
-
    
-
    3 Flexible Joint
    
+
    
+
    2.10 Effect of APA300ML in the flexibility of the leg
    
+
    
+
+
+
+++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    




    CaracteristicRigid APAFlexible APA
    Kx [N/um]0.0180.019
    Ky [N/um]0.0180.018
    Kz [N/um]60.02.647
    Rx [Nm/rad]16.705557.682
    Ry [Nm/rad]16.535185.939
    Rz [Nm/rad]118.0114.803
    
+
    
+
    
+
    
+
+
    
+
    3 Flexible Joint
    
 
    
 
    
-The flexor in Figure 14 is studied with a FEM.
+The studied flexor is shown in Figure 17.
+
    
+
+
    
+The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
+It is then imported into Simscape.
+
    
+
+
    
+A simplified model of the flexor is then developped.
 
    
 
 
-
    
+
    
 
    flexor_id16_screenshot.png
 
    
-
    Figure 14: Flexor studied
    
+
    Figure 17: Flexor studied
    
 
    
 
    
 
-
    
-
    3.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
+
    
+
    3.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
 
    
 
    
 We first extract the stiffness and mass matrices.
 
    
 
    
-
    K = extractMatrix('mat_K_6modes_2MDoF.matrix');
-M = extractMatrix('mat_M_6modes_2MDoF.matrix');
+
    K = extractMatrix('mat_K_6modes_2MDoF.matrix');
+M = extractMatrix('mat_M_6modes_2MDoF.matrix');
 
    
 
    
 
@@ -1957,22 +2392,22 @@ M = extractMatrix('mat_M_6modes_2MDoF.matrix');
 Then, we extract the coordinates of the interface nodes.
 
    
 
    
-
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
 
    
 
    
 
 
    
-
    save('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    save('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
 
    
 
    
 
    
 
    
 
-
    
-
    3.2 Output parameters
    
+
    
+
    3.2 Output parameters
    
 
    
 
    
-
    load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
 
    
 
    
 
@@ -2008,7 +2443,7 @@ Then, we extract the coordinates of the interface nodes.
 
 
 
-
+@@ -2050,7 +2485,7 @@ Then, we extract the coordinates of the interface nodes.
 
    Table 8: Coordinates of the interface nodesTable 9: Coordinates of the interface nodes
 
 
 

    
 
 
-
+@@ -2208,7 +2643,7 @@ Then, we extract the coordinates of the interface nodes.
 
 
 
    Table 9: First 10x10 elements of the Stiffness matrixTable 10: First 10x10 elements of the Stiffness matrix
 
 
 

    
-
+@@ -2370,8 +2805,8 @@ Using K, M and int_xyz, we can use the 
 
 
-
    
-
    3.3 Flexible Joint Characteristics
    
+
    
+
    3.3 Flexible Joint Characteristics
    
 
    
 
    
 The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
@@ -2401,6 +2836,12 @@ The most important parameters of the flexible joint can be directly estimated fr
 
    
 
+
+
+
+
+
+
 
@@ -2423,8 +2864,8 @@ The most important parameters of the flexible joint can be directly estimated fr
 
 
-
    
-
    3.4 Identification of the parameters using Simscape
    
+
    
+
    3.4 Identification of the parameters using Simscape
    
 
    
 
    
 The flexor is now imported into Simscape and its parameters are estimated using an identification.
@@ -2462,49 +2903,49 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
 
    
-
+
-
+
-
+
    Table 10: First 10x10 elements of the Mass matrixTable 11: First 10x10 elements of the Mass matrix
 
 
 

    60
 
    Shear Stiffness [N/um]110
    
 Bending Stiffness [Nm/rad]
 33
 
    
 Bending Stiffness Rx [Nm/rad]
 333334
 
    
 
 
    
 Bending Stiffness Ry [Nm/rad]
 3333126
 
    
 
 
    
 Torsion Stiffness Rz [Nm/rad]
 236236238
 
    
     
 
    
 
    
 
    
 
-
    
-
    3.5 Simpler Model
    
+
    
+
    3.5 Simpler Model
    
 
    
 
    
-Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 15.
+Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 18.
 
    
 
 
-
    
+
    
 
    flexible_joint_simscape.png
 
    
-
    Figure 15: Bushing Joint used to model the flexible joint
    
+
    Figure 18: Bushing Joint used to model the flexible joint
    
 
    
 
 
    
 The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
 
    
 
    
-
    Kx = K(1,1); % [N/m]
-Ky = K(2,2); % [N/m]
-Kz = K(3,3); % [N/m]
-Krx = K(4,4); % [Nm/rad]
-Kry = K(5,5); % [Nm/rad]
-Krz =  K(6,6); % [Nm/rad]
+
    Kx = K(1,1); % [N/m]
+Ky = K(2,2); % [N/m]
+Kz = K(3,3); % [N/m]
+Krx = K(4,4); % [Nm/rad]
+Kry = K(5,5); % [Nm/rad]
+Krz =  K(6,6); % [Nm/rad]
 
    
 
    
 
@@ -2514,32 +2955,35 @@ The two obtained dynamics are compared in Figure
 
    
 
 
-
    
+
    
 
    flexor_ID16_compare_bushing_joint.png
 
    
-
    Figure 16: Comparison of the Joint compliance between the FEM model and the simpler model
    
+
    Figure 19: Comparison of the Joint compliance between the FEM model and the simpler model
    
 
    
 
    
 
    
 
    
 
-
    
-
    4 Integral Force Feedback with Amplified Piezo
    
+
    
+
    4 Optimal Flexible Joint
    
 
    
-
    
-In this section, we try to replicate the results obtained in (Souleille et al. 2018).
+
+
    
+
    CS.jpg
 
    
+
    Figure 20: Flexor studied
    
+
    
 
    
 
-
    
-
    4.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
+
    
+
    4.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
 
    
 
    
 We first extract the stiffness and mass matrices.
 
    
 
    
-
    K = extractMatrix('piezo_amplified_IFF_K.txt');
-M = extractMatrix('piezo_amplified_IFF_M.txt');
+
    K = readmatrix('mat_K.CSV');
+M = readmatrix('mat_M.CSV');
 
    
 
    
 
@@ -2547,17 +2991,605 @@ M = extractMatrix('piezo_amplified_IFF_M.txt');
 Then, we extract the coordinates of the interface nodes.
 
    
 
    
-
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_IFF.txt');
+
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+
    
+
    
+
+
    
+
    save('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
 
    
 
    
 
    
 
    
 
-
    
-
    4.2 IFF Plant
    
+
    
+
    4.2 Output parameters
    
 
    
+
    
+
    load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    Total number of Nodes2
    Number of interface Nodes2
    Number of Modes6
    Size of M and K matrices18
    
+
+
+
+
+++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 12: Coordinates of the interface nodes






    Node iNode Numberx [m]y [m]z [m]
    1.0528875.00.00.00.0
    2.0528876.00.00.0-0.0
    
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 13: First 10x10 elements of the Stiffness matrix











    12700000.0-18.5-26.80.00162-4.6364.0-12700000.018.326.70.00234
    -18.512700000.0-499.0-132.00.00414-0.49518.4-12700000.0499.0132.0
    -26.8-499.094000000.0-470.00.00771-0.85526.8498.0-94000000.0470.0
    0.00162-132.0-470.04.832.61e-070.000123-0.00163132.0470.0-4.83
    -4.630.004140.007712.61e-074.834.43e-054.63-0.00413-0.00772-4.3e-07
    64.0-0.495-0.8550.0001234.43e-05260.0-64.00.4950.855-0.000124
    -12700000.018.426.8-0.001634.63-64.012700000.0-18.2-26.7-0.00234
    18.3-12700000.0498.0132.0-0.004130.495-18.212700000.0-498.0-132.0
    26.7499.0-94000000.0470.0-0.007720.855-26.7-498.094000000.0-470.0
    0.00234132.0470.0-4.83-4.3e-07-0.000124-0.00234-132.0-470.04.83
    
+
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 14: First 10x10 elements of the Mass matrix











    0.0068e-09-2e-08-1e-103e-053e-080.003-3e-099e-092e-12
    8e-090.021e-07-3e-051e-116e-101e-080.003-5e-083e-09
    -2e-081e-070.01-6e-08-6e-11-8e-12-1e-071e-080.003-1e-08
    -1e-10-3e-05-6e-081e-067e-146e-131e-101e-06-1e-083e-10
    3e-051e-11-6e-117e-142e-071e-103e-08-7e-126e-11-6e-16
    3e-086e-10-8e-126e-131e-105e-071e-08-5e-10-1e-111e-13
    0.0031e-08-1e-071e-103e-081e-080.02-2e-081e-07-4e-12
    -3e-090.0031e-081e-06-7e-12-5e-10-2e-080.006-8e-083e-05
    9e-09-5e-080.003-1e-086e-11-1e-111e-07-8e-080.01-6e-08
    2e-123e-09-1e-083e-10-6e-161e-13-4e-123e-05-6e-082e-07
    
+
 
    
-The transfer function from the force actuator to the force sensor is identified and shown in Figure 17.
+Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.
+
    
+
    
+
    
+
+
    
+
    4.3 Flexible Joint Characteristics
    
+
    
+
    
+The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    CaracteristicValue
    Axial Stiffness [N/um]94.0
    Shear Stiffness [N/um]12.7
    Bending Stiffness [Nm/rad]4.8
    Bending Stiffness [Nm/rad]4.8
    Torsion Stiffness [Nm/rad]260.2
    
+
    
+
    
+
+
    
+
    4.4 Identification of the parameters using Simscape
    
+
    
+
    
+The flexor is now imported into Simscape and its parameters are estimated using an identification.
+
    
+
+
    
+The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
+And we find the same parameters as the one estimated from the Stiffness matrix.
+
    
+
+
+
+
+++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    




    CaracteristicValueIdentification
    Axial Stiffness Dz [N/um]94.093.9
    Bending Stiffness Rx [Nm/rad]4.84.8
    Bending Stiffness Ry [Nm/rad]4.84.8
    Torsion Stiffness Rz [Nm/rad]260.2260.2
    
+
    
+
    
+
+
    
+
    4.5 Simpler Model
    
+
    
+
    
+Let’s now model the flexible joint with a “perfect” Bushing joint as shown in Figure 18.
+
    
+
+
+
    
+
    flexible_joint_simscape.png
+
    
+
    Figure 21: Bushing Joint used to model the flexible joint
    
+
    
+
+
    
+The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
+
    
+
    
+
    Kx = K(1,1); % [N/m]
+Ky = K(2,2); % [N/m]
+Kz = K(3,3); % [N/m]
+Krx = K(4,4); % [Nm/rad]
+Kry = K(5,5); % [Nm/rad]
+Krz =  K(6,6); % [Nm/rad]
+
    
+
    
+
+
    
+The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
+The two obtained dynamics are compared in Figure
+
    
+
+
+
    
+
    flexor_ID16_compare_bushing_joint.png
+
    
+
    Figure 22: Comparison of the Joint compliance between the FEM model and the simpler model
    
+
    
+
    
+
    
+
    
+
+
    
+
    5 Integral Force Feedback with Amplified Piezo
    
+
    
+
    
+In this section, we try to replicate the results obtained in (Souleille et al. 2018).
+
    
+
    
+
+
    
+
    5.1 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
+
    
+
    
+We first extract the stiffness and mass matrices.
+
    
+
    
+
    K = extractMatrix('piezo_amplified_IFF_K.txt');
+M = extractMatrix('piezo_amplified_IFF_M.txt');
+
    
+
    
+
+
    
+Then, we extract the coordinates of the interface nodes.
+
    
+
    
+
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_IFF.txt');
+
    
+
    
+
    
+
    
+
+
    
+
    5.2 IFF Plant
    
+
    
+
    
+The transfer function from the force actuator to the force sensor is identified and shown in Figure 23.
 
    
 
 
    
@@ -2571,13 +3603,13 @@ The transfer function from the force actuator to the force sensor is identified
 
    
 
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_IFF';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_IFF';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Kiff'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/G'],    1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Kiff'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/G'],    1, 'openoutput'); io_i = io_i + 1;
 
 Gf = linearize(mdl, io);
 
    
@@ -2594,62 +3626,62 @@ Gf = linearize(mdl, io);
 
    
 
 
-
    
+
    
 
    piezo_amplified_iff_plant.png
 
    
-
    Figure 17: IFF Plant
    
+
    Figure 23: IFF Plant
    
 
    
 
    
 
    
 
-
    
-
    4.3 IFF controller
    
-
    
+
    
+
    5.3 IFF controller
    
+
    
 
    
-The controller is defined and the loop gain is shown in Figure 18.
+The controller is defined and the loop gain is shown in Figure 24.
 
    
 
    
-
    Kiff = -1e12/s;
+
    Kiff = -1e12/s;
 
    
 
    
 
 
-
    
+
    
 
    piezo_amplified_iff_loop_gain.png
 
    
-
    Figure 18: IFF Loop Gain
    
+
    Figure 24: IFF Loop Gain
    
 
    
 
    
 
    
 
-
    
-
    4.4 Closed Loop System
    
-
    
+
    
+
    5.4 Closed Loop System
    
+
    
 
    
 
    m = 10;
 
    
 
    
 
 
    
-
    Kiff = -1e12/s;
+
    Kiff = -1e12/s;
 
    
 
    
 
 
    
-
    %% Name of the Simulink File
-mdl = 'piezo_amplified_IFF';
+
    %% Name of the Simulink File
+mdl = 'piezo_amplified_IFF';
 
-%% Input/Output definition
+%% Input/Output definition
 clear io; io_i = 1;
-io(io_i) = linio([mdl, '/Dw'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
-io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/G'],  1, 'output'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Dw'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/G'],  1, 'output'); io_i = io_i + 1;
 
 Giff = linearize(mdl, io);
-Giff.InputName  = {'w', 'f', 'F'};
-Giff.OutputName  = {'x1', 'Fs'};
+Giff.InputName  = {'w', 'f', 'F'};
+Giff.OutputName  = {'x1', 'Fs'};
 
    
 
    
 
@@ -2660,24 +3692,564 @@ Giff.OutputName  = {'x1', 'Fs'};
 
 
    
 
    G = linearize(mdl, io);
-G.InputName  = {'w', 'f', 'F'};
-G.OutputName  = {'x1', 'Fs'};
+G.InputName  = {'w', 'f', 'F'};
+G.OutputName  = {'x1', 'Fs'};
 
    
 
    
 
 
-
    
+
    
 
    piezo_amplified_iff_comp.png
 
    
-
    Figure 19: OL and CL transfer functions
    
+
    Figure 25: OL and CL transfer functions
    
 
    
 
 
 
-
    
+
    
 
    souleille18_results.png
 
    
-
    Figure 20: Results obtained in souleille18_concep_activ_mount_space_applic
    
+
    Figure 26: Results obtained in souleille18_concep_activ_mount_space_applic
    
+
    
+
    
+
    
+
    
+
+
    
+
    6 Complete Strut with Encoder
    
+
    
+
    
+
    
+
    6.1 Introduction
    
+
    
+
+
    
+
    points3.jpg
+
    
+
    Figure 27: Interface points
    
+
    
+
+
    
+Flexible joints have 0.25mm width.
+
    
+
    
+
    
+
+
    
+
    6.2 Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
    
+
    
+
    
+We first extract the stiffness and mass matrices.
+
    
+
    
+
    K = readmatrix('mat_K.CSV');
+M = readmatrix('mat_M.CSV');
+
    
+
    
+
+
    
+Then, we extract the coordinates of the interface nodes.
+
    
+
    
+
    [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+
    
+
    
+
+
    
+
    save('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
    
+
    
+
+
    
+
    6.3 Output parameters
    
+
    
+
    
+
    load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+
    
+
    
+
+
+
+
+++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    



    Total number of Nodes8
    Number of interface Nodes8
    Number of Modes6
    Size of M and K matrices54
    
+
+
+
+
+++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 15: Coordinates of the interface nodes






    Node iNode Numberx [m]y [m]z [m]
    1.0504411.00.00.00.0405
    2.0504412.00.00.0-0.0405
    3.0504413.0-0.03250.00.0
    4.0504414.0-0.01250.00.0
    5.0504415.0-0.00750.00.0
    6.0504416.00.03250.00.0
    7.0504417.00.0040.0145-0.00175
    8.0504418.00.0040.0166-0.00175
    
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 16: First 10x10 elements of the Stiffness matrix











    2000000.01000000.0-3000000.0-400.0300.0200.0-30.02000.0-10000.00.3
    1000000.04000000.0-8000000.0-900.0400.0-50.0-6000.010000.0-20000.03
    -3000000.0-8000000.020000000.02000.0-900.0200.0-10000.020000.0-300000.07
    -400.0-900.02000.05-0.10.051-36-0.0007
    300.0400.0-900.0-0.150.04-0.10.5-30.0001
    200.0-50.0200.00.050.04300.04-0.01-13e-05
    -30.0-6000.0-10000.01-0.143000000.0-1000000.0-2000000.0-300.0
    2000.010000.020000.0-30.5-0.01-1000000.06000000.07000000.01000.0
    -10000.0-20000.0-300000.06-3-1-2000000.07000000.020000000.02000.0
    0.337-0.00070.00013e-05-300.01000.02000.05
    
+
+
+
+
+
+++
++
++
++
++
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
    Table 17: First 10x10 elements of the Mass matrix











    0.04-0.0050.0072e-060.0001-5e-07-1e-05-9e-078e-05-5e-10
    -0.0050.030.02-0.00011e-06-3e-073e-05-0.00018e-05-3e-08
    0.0070.020.08-6e-06-5e-06-7e-074e-05-0.00010.0005-3e-08
    2e-06-0.0001-6e-062e-06-4e-102e-11-8e-093e-08-2e-086e-12
    0.00011e-06-5e-06-4e-103e-062e-10-3e-093e-09-7e-096e-13
    -5e-07-3e-07-7e-072e-112e-105e-07-2e-085e-09-5e-091e-12
    -1e-053e-054e-05-8e-09-3e-09-2e-080.040.0040.0031e-06
    -9e-07-0.0001-0.00013e-083e-095e-090.0040.02-0.020.0001
    8e-058e-050.0005-2e-08-7e-09-5e-090.003-0.020.08-5e-06
    -5e-10-3e-08-3e-086e-126e-131e-121e-060.0001-5e-062e-06
    
+
+
    
+Using K, M and int_xyz, we can use the Reduced Order Flexible Solid simscape block.
+
    
+
    
+
    
+
+
+
+
+
    
+
    6.4 Piezoelectric parameters
    
+
    
+
    
+Parameters for the APA300ML:
+
    
+
+
    
+
    d33 = 3e-10; % Strain constant [m/V]
+n = 80; % Number of layers per stack
+eT = 1.6e-8; % Permittivity under constant stress [F/m]
+sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
+ka = 235e6; % Stack stiffness [N/m]
+C = 5e-6; % Stack capactiance [F]
+
    
+
    
+
+
    
+
    na = 2; % Number of stacks used as actuator
+ns = 1; % Number of stacks used as force sensor
+
    
+
    
+
    
+
    
+
+
    
+
    6.5 Identification of the Dynamics
    
+
    
+
    
+The dynamics is identified from the applied force to the measured relative displacement.
+The same dynamics is identified for a payload mass of 10Kg.
+
    
+
    
+
    m = 10;
+
    
 
    
 
    
 
    
@@ -2695,7 +4267,7 @@ G.OutputName  = {'x1', 'Fs'};
 
    
 
    
 
    Author: Dehaeze Thomas
    
-
    Created: 2020-08-04 mar. 12:14
    
+
    Created: 2020-10-29 jeu. 10:08
    
 
    
 
 
diff --git a/index.org b/index.org
index 6e9cab5..d914e23 100644
--- a/index.org
+++ b/index.org
@@ -6,8 +6,8 @@
 #+EMAIL: dehaeze.thomas@gmail.com
 #+AUTHOR: Dehaeze Thomas
 
-#+HTML_LINK_HOME: ./index.html
-#+HTML_LINK_UP: ./index.html
+#+HTML_LINK_HOME: ../index.html
+#+HTML_LINK_UP:   ../index.html
 
 #+HTML_HEAD: 
 #+HTML_HEAD: 
@@ -180,7 +180,7 @@ Parameters for the APA95ML:
 The ratio of the developed force to applied voltage is $d_{33} n k_a$ in [N/V].
 We denote this constant by $g_a$ and:
 \[ F_a = g_a V_a, \quad g_a = d_{33} n k_a \]
-#+begin_src matlab :results replace value
+ #+begin_src matlab :results replace value
   d33*(na*n)*(ka/(na + ns)) % [N/V]
 #+end_src
 
@@ -633,8 +633,8 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
   %% Input/Output definition
   clear io; io_i = 1;
-  io(io_i) = linio([mdl, '/Va'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1;
 
   G = -linearize(mdl, io);
 #+end_src
@@ -646,7 +646,9 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
   ax1 = subplot(2,1,1);
   hold on;
-  plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G(2,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G(3,1), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G(3,1) + G(3,2), freqs, 'Hz'))));
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
@@ -654,13 +656,174 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
   ax2 = subplot(2,1,2);
   hold on;
-  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G(2,1), freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G(3,1), freqs, 'Hz')))));
+  plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G(3,1) + G(3,2), freqs, 'Hz')))));
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
   yticks(-360:90:360);
   xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
   hold off;
   linkaxes([ax1,ax2],'x');
-  xlim([10, 5e3]);
+  xlim([10, 1e4]);
+#+end_src
+
+** Identification for a simpler model
+The goal in this section is to identify the parameters of a simple APA model from the FEM.
+This can be useful is a lower order model is to be used for simulations.
+
+The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
+
+The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
+The parameters are shown in the table below.
+
+#+name: fig:souleille18_model_piezo
+#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
+[[file:./figs/souleille18_model_piezo.png]]
+
+#+caption: Parameters used for the model of the APA 100M
+|       | Meaning                                                        |
+|-------+----------------------------------------------------------------|
+| $k_e$ | Stiffness used to adjust the pole of the isolator              |
+| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
+| $k_a$ | Stiffness of the actuator                                      |
+| $c_1$ | Added viscous damping                                          |
+
+The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.
+
+\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
+
+If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
+\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
+\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
+
+#+begin_src matlab :exports none
+  m = 10;
+#+end_src
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'piezo_amplified_3d_identification';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
+  io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion [m]
+  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
+  io(io_i) = linio([mdl, '/z'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
+  io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
+
+  G = linearize(mdl, io);
+
+  G.InputName = {'Fd', 'w', 'Fa'};
+  G.OutputName = {'y', 'Fs'};
+#+end_src
+
+From the identified dynamics, compute $\alpha$ and $\beta$
+#+begin_src matlab
+  alpha = abs(dcgain(G('y', 'Fa')));
+  beta  = abs(dcgain(G('y', 'Fd')));
+#+end_src
+
+$k_a$ is estimated using the following formula:
+#+begin_src matlab
+  ka = 0.9/abs(dcgain(G('y', 'Fa')));
+#+end_src
+The factor can be adjusted to better match the curves.
+
+Then $k_e$ and $k_1$ are computed.
+#+begin_src matlab
+  ke = ka/(beta/alpha - 1);
+  k1 = 1/beta - ke*ka/(ke + ka);
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+|    | Value [N/um] |
+|----+--------------|
+| ka |         54.9 |
+| ke |         25.1 |
+| k1 |          4.3 |
+
+The damping in the system is adjusted to match the FEM model if necessary.
+#+begin_src matlab
+  c1 = 1e2;
+#+end_src
+
+Analytical model of the simpler system:
+#+begin_src matlab
+  Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
+       [ 1 ,              k1 + c1*s + ke*ka/(ke + ka)  , ke/(ke + ka) ;
+        -ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 ,       -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
+
+  Ga.InputName = {'Fd', 'w', 'Fa'};
+  Ga.OutputName = {'y', 'Fs'};
+#+end_src
+
+Adjust the DC gain for the force sensor:
+#+begin_src matlab
+  F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 5, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 2, 1);
+  title('$\displaystyle \frac{x_1}{f}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax2 = subplot(2, 2, 2);
+  title('$\displaystyle \frac{x_1}{F}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax3 = subplot(2, 2, 3);
+  title('$\displaystyle \frac{F_s}{f}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax4 = subplot(2, 2, 4);
+  title('$\displaystyle \frac{F_s}{F}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
+
+  linkaxes([ax1,ax2,ax3,ax4],'x');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa95ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full');
+#+end_src
+
+#+name: fig:apa95ml_comp_simpler_model
+#+caption: Comparison of the Dynamics between the FEM model and the simplified one
+#+RESULTS:
+[[file:figs/apa95ml_comp_simpler_model.png]]
+
+We save the parameters of the simplified model for the APA95ML:
+#+begin_src matlab
+  save('./mat/APA95ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');
 #+end_src
 
 * APA300ML
@@ -681,7 +844,7 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 
 #+begin_src matlab
   addpath('./src/');
-  addpath('./data/APA300ML/');
+  addpath('./data/APA300ML_new/');
 #+end_src
 
 #+begin_src matlab :exports none
@@ -691,18 +854,13 @@ The two identified dynamics are compared in Figure [[fig:dynamics_act_disp_comp_
 ** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
 We first extract the stiffness and mass matrices.
 #+begin_src matlab
-  K = extractMatrix('mat_K-48modes-7MDoF.matrix');
-  M = extractMatrix('mat_M-48modes-7MDoF.matrix');
-#+end_src
-
-#+begin_src matlab
-  K = extractMatrix('mat_K-80modes-7MDoF.matrix');
-  M = extractMatrix('mat_M-80modes-7MDoF.matrix');
+  K = readmatrix('mat_K.CSV');
+  M = readmatrix('mat_M.CSV');
 #+end_src
 
 Then, we extract the coordinates of the interface nodes.
 #+begin_src matlab
-  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('Nodes_MDoF_NLIST_MLIST.txt');
+  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
 #+end_src
 
 #+begin_src matlab
@@ -719,10 +877,10 @@ Then, we extract the coordinates of the interface nodes.
 #+end_src
 
 #+RESULTS:
-| Total number of Nodes     |  7 |
-| Number of interface Nodes |  7 |
-| Number of Modes           |  6 |
-| Size of M and K matrices  | 48 |
+| Total number of Nodes     |   7 |
+| Number of interface Nodes |   7 |
+| Number of Modes           | 120 |
+| Size of M and K matrices  | 162 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
   data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
@@ -730,15 +888,15 @@ Then, we extract the coordinates of the interface nodes.
 
 #+caption: Coordinates of the interface nodes
 #+RESULTS:
-| Node i | Node Number |   x [m] |  y [m] | z [m] |
-|--------+-------------+---------+--------+-------|
-|    1.0 |     53917.0 |     0.0 | -0.015 |   0.0 |
-|    2.0 |     53918.0 |     0.0 |  0.015 |   0.0 |
-|    3.0 |     53919.0 | -0.0325 |    0.0 |   0.0 |
-|    4.0 |     53920.0 | -0.0125 |    0.0 |   0.0 |
-|    5.0 |     53921.0 | -0.0075 |    0.0 |   0.0 |
-|    6.0 |     53922.0 |  0.0125 |    0.0 |   0.0 |
-|    7.0 |     53923.0 |  0.0325 |    0.0 |   0.0 |
+| Node i | Node Number |   x [m] | y [m] |  z [m] |
+|--------+-------------+---------+-------+--------|
+|    1.0 |    697783.0 |     0.0 |   0.0 | -0.015 |
+|    2.0 |    697784.0 |     0.0 |   0.0 |  0.015 |
+|    3.0 |    697785.0 | -0.0325 |   0.0 |    0.0 |
+|    4.0 |    697786.0 | -0.0125 |   0.0 |    0.0 |
+|    5.0 |    697787.0 | -0.0075 |   0.0 |    0.0 |
+|    6.0 |    697788.0 |  0.0125 |   0.0 |    0.0 |
+|    7.0 |    697789.0 |  0.0325 |   0.0 |    0.0 |
 
 #+begin_src matlab :exports results :results value table replace :tangle no
   data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
@@ -746,16 +904,16 @@ Then, we extract the coordinates of the interface nodes.
 
 #+caption: First 10x10 elements of the Stiffness matrix
 #+RESULTS:
-| 200000000.0 |   30000.0 |    50000.0 |    200.0 |  -100.0 | -300000.0 |  10000000.0 |   -6000.0 |    20000.0 |     -60.0 |
-|     30000.0 | 7000000.0 |    10000.0 |     30.0 |   -30.0 |     -70.0 |      7000.0 | -500000.0 |     3000.0 |     -10.0 |
-|     50000.0 |   10000.0 | 30000000.0 | 200000.0 |  -200.0 |    -100.0 |     20000.0 |   -2000.0 |  2000000.0 |   -9000.0 |
-|       200.0 |      30.0 |   200000.0 |   1000.0 |    -0.8 |      -0.4 |        50.0 |        -6 |     9000.0 |     -30.0 |
-|      -100.0 |     -30.0 |     -200.0 |     -0.8 | 10000.0 |       0.2 |       -40.0 |      10.0 |       20.0 |     -0.05 |
-|   -300000.0 |     -70.0 |     -100.0 |     -0.4 |     0.2 |     900.0 |    -30000.0 |      10.0 |      -40.0 |       0.1 |
-|  10000000.0 |    7000.0 |    20000.0 |     50.0 |   -40.0 |  -30000.0 | 200000000.0 |  -50000.0 |    30000.0 |     -50.0 |
-|     -6000.0 | -500000.0 |    -2000.0 |       -6 |    10.0 |      10.0 |    -50000.0 | 7000000.0 |    -4000.0 |         8 |
-|     20000.0 |    3000.0 |  2000000.0 |   9000.0 |    20.0 |     -40.0 |     30000.0 |   -4000.0 | 30000000.0 | -200000.0 |
-|       -60.0 |     -10.0 |    -9000.0 |    -30.0 |   -0.05 |       0.1 |       -50.0 |         8 |  -200000.0 |    1000.0 |
+| 200000000.0 |    30000.0 |  -20000.0 |     -70.0 | 300000.0 |    40.0 |  10000000.0 |    10000.0 |   -6000.0 |     30.0 |
+|     30000.0 | 30000000.0 |    2000.0 | -200000.0 |     60.0 |   -10.0 |      4000.0 |  2000000.0 |    -500.0 |   9000.0 |
+|    -20000.0 |     2000.0 | 7000000.0 |     -10.0 |    -30.0 |    10.0 |      6000.0 |      900.0 | -500000.0 |        3 |
+|       -70.0 |  -200000.0 |     -10.0 |    1000.0 |     -0.1 |    0.08 |       -20.0 |    -9000.0 |         3 |    -30.0 |
+|    300000.0 |       60.0 |     -30.0 |      -0.1 |    900.0 |     0.1 |     30000.0 |       20.0 |     -10.0 |     0.06 |
+|        40.0 |      -10.0 |      10.0 |      0.08 |      0.1 | 10000.0 |        20.0 |          9 |        -5 |     0.03 |
+|  10000000.0 |     4000.0 |    6000.0 |     -20.0 |  30000.0 |    20.0 | 200000000.0 |    10000.0 |    9000.0 |     50.0 |
+|     10000.0 |  2000000.0 |     900.0 |   -9000.0 |     20.0 |       9 |     10000.0 | 30000000.0 |    -500.0 | 200000.0 |
+|     -6000.0 |     -500.0 | -500000.0 |         3 |    -10.0 |      -5 |      9000.0 |     -500.0 | 7000000.0 |       -2 |
+|        30.0 |     9000.0 |         3 |     -30.0 |     0.06 |    0.03 |        50.0 |   200000.0 |        -2 |   1000.0 |
 
 
 #+begin_src matlab :exports results :results value table replace :tangle no
@@ -764,16 +922,16 @@ Then, we extract the coordinates of the interface nodes.
 
 #+caption: First 10x10 elements of the Mass matrix
 #+RESULTS:
-|    0.01 |  7e-06 |  -5e-06 | -6e-08 |  3e-09 | -5e-05 | -0.0005 | -2e-07 |  -3e-06 |  1e-08 |
-|   7e-06 |  0.009 |   4e-07 |  6e-09 | -4e-09 | -3e-08 |  -2e-07 |  6e-05 |   5e-07 | -1e-09 |
-|  -5e-06 |  4e-07 |    0.01 |  2e-05 |  2e-08 |  3e-08 |  -2e-06 | -1e-07 | -0.0002 |  9e-07 |
-|  -6e-08 |  6e-09 |   2e-05 |  3e-07 |  1e-10 |  3e-10 |  -7e-09 |  2e-10 |  -9e-07 |  3e-09 |
-|   3e-09 | -4e-09 |   2e-08 |  1e-10 |  1e-07 | -3e-12 |   6e-09 | -2e-10 |  -3e-09 |  9e-12 |
-|  -5e-05 | -3e-08 |   3e-08 |  3e-10 | -3e-12 |  6e-07 |   1e-06 | -3e-09 |   2e-08 | -7e-11 |
-| -0.0005 | -2e-07 |  -2e-06 | -7e-09 |  6e-09 |  1e-06 |    0.01 | -8e-06 |  -2e-06 |  9e-09 |
-|  -2e-07 |  6e-05 |  -1e-07 |  2e-10 | -2e-10 | -3e-09 |  -8e-06 |  0.009 |   1e-07 |  2e-09 |
-|  -3e-06 |  5e-07 | -0.0002 | -9e-07 | -3e-09 |  2e-08 |  -2e-06 |  1e-07 |    0.01 | -2e-05 |
-|   1e-08 | -1e-09 |   9e-07 |  3e-09 |  9e-12 | -7e-11 |   9e-09 |  2e-09 |  -2e-05 |  3e-07 |
+|    0.01 |  -2e-06 |  1e-06 |  6e-09 |  5e-05 | -5e-09 | -0.0005 |  -7e-07 |  6e-07 | -3e-09 |
+|  -2e-06 |    0.01 |  8e-07 | -2e-05 | -8e-09 |  2e-09 |  -9e-07 | -0.0002 |  1e-08 | -9e-07 |
+|   1e-06 |   8e-07 |  0.009 |  5e-10 |  1e-09 | -1e-09 |  -5e-07 |   3e-08 |  6e-05 |  1e-10 |
+|   6e-09 |  -2e-05 |  5e-10 |  3e-07 |  2e-11 | -3e-12 |   3e-09 |   9e-07 | -4e-10 |  3e-09 |
+|   5e-05 |  -8e-09 |  1e-09 |  2e-11 |  6e-07 | -4e-11 |  -1e-06 |  -2e-09 |  1e-09 | -8e-12 |
+|  -5e-09 |   2e-09 | -1e-09 | -3e-12 | -4e-11 |  1e-07 |  -2e-09 |  -1e-09 | -4e-10 | -5e-12 |
+| -0.0005 |  -9e-07 | -5e-07 |  3e-09 | -1e-06 | -2e-09 |    0.01 |   1e-07 | -3e-07 | -2e-08 |
+|  -7e-07 | -0.0002 |  3e-08 |  9e-07 | -2e-09 | -1e-09 |   1e-07 |    0.01 | -4e-07 |  2e-05 |
+|   6e-07 |   1e-08 |  6e-05 | -4e-10 |  1e-09 | -4e-10 |  -3e-07 |  -4e-07 |  0.009 | -2e-10 |
+|  -3e-09 |  -9e-07 |  1e-10 |  3e-09 | -8e-12 | -5e-12 |  -2e-08 |   2e-05 | -2e-10 |  3e-07 |
 
 Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
 
@@ -802,7 +960,7 @@ We denote this constant by $g_a$ and:
 #+end_src
 
 #+RESULTS:
-: 0.42941
+: 3.76
 
 From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement and generated voltage is:
 \[ V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h \]
@@ -818,7 +976,7 @@ where:
 #+end_src
 
 #+RESULTS:
-: 5.8594
+: 11.719
 
 ** Identification of the APA Characteristics
 *** Stiffness
@@ -846,7 +1004,7 @@ The inverse of its DC gain is the axial stiffness of the APA:
 #+end_src
 
 #+RESULTS:
-: 1.8634
+: 1.753
 
 The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$.
 
@@ -885,7 +1043,7 @@ The amplification factor is the ratio of the axial displacement to the stack dis
   %% Input/Output definition
   clear io; io_i = 1;
   io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
   io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
 
   G = linearize(mdl, io);
@@ -893,11 +1051,11 @@ The amplification factor is the ratio of the axial displacement to the stack dis
 
 The ratio of the two displacement is computed from the FEM model.
 #+begin_src matlab :results replace value
-  -dcgain(G(1,1))./dcgain(G(2,1))
+  abs(dcgain(G(1,1))./dcgain(G(2,1)))
 #+end_src
 
 #+RESULTS:
-: 4.936
+: 5.0749
 
 If we take the ratio of the piezo height and length (approximation of the amplification factor):
 #+begin_src matlab :results replace value
@@ -947,7 +1105,7 @@ The dynamics is identified from the applied force to the measured relative displ
   %% Input/Output definition
   clear io; io_i = 1;
   io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
 
   Gh = -linearize(mdl, io);
 #+end_src
@@ -964,7 +1122,7 @@ The same dynamics is identified for a payload mass of 10Kg.
   %% Input/Output definition
   clear io; io_i = 1;
   io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;
-  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
 
   Ghm = -linearize(mdl, io);
 #+end_src
@@ -1063,7 +1221,6 @@ The transfer function from actuator to sensors is identified and shown in Figure
 [[file:figs/apa300ml_iff_plant.png]]
 
 For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
-
 #+begin_src matlab :exports none
   figure;
 
@@ -1400,7 +1557,7 @@ If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
   io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
   io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion [m]
   io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
-  io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
+  io(io_i) = linio([mdl, '/z'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
   io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
   io(io_i) = linio([mdl, '/d'],  1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]
 
@@ -1435,7 +1592,7 @@ Then $k_e$ and $k_1$ are computed.
 #+RESULTS:
 |    | Value [N/um] |
 |----+--------------|
-| ka |         42.9 |
+| ka |         40.5 |
 | ke |          1.5 |
 | k1 |          0.4 |
 
@@ -1456,7 +1613,7 @@ Analytical model of the simpler system:
 
 Adjust the DC gain for the force sensor:
 #+begin_src matlab
-  lambda = dcgain(Ga('Fs', 'Fd'))/dcgain(G('Fs', 'Fd'));
+  F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
 #+end_src
 
 #+begin_src matlab :exports none
@@ -1464,7 +1621,7 @@ Adjust the DC gain for the force sensor:
 
   figure;
 
-  ay = subplot(2, 3, 1);
+  ax1 = subplot(2, 3, 1);
   title('$\displaystyle \frac{x_1}{w}$')
   hold on;
   plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
@@ -1494,8 +1651,8 @@ Adjust the DC gain for the force sensor:
   ax4 = subplot(2, 3, 4);
   title('$\displaystyle \frac{F_s}{w}$')
   hold on;
-  plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'w'), freqs, 'Hz'))));
-  plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'w'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz'))));
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
@@ -1503,8 +1660,8 @@ Adjust the DC gain for the force sensor:
   ax5 = subplot(2, 3, 5);
   title('$\displaystyle \frac{F_s}{f}$')
   hold on;
-  plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'Fa'), freqs, 'Hz'))));
-  plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
@@ -1512,8 +1669,8 @@ Adjust the DC gain for the force sensor:
   ax6 = subplot(2, 3, 6);
   title('$\displaystyle \frac{F_s}{F}$')
   hold on;
-  plot(freqs, abs(squeeze(freqresp(lambda*G( 'Fs', 'Fd'), freqs, 'Hz'))));
-  plot(freqs, abs(squeeze(freqresp(Ga('Fs', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
   hold off;
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
@@ -1530,10 +1687,239 @@ Adjust the DC gain for the force sensor:
 #+RESULTS:
 [[file:figs/apa300ml_comp_simpler_model.png]]
 
+We now compare the FEM model with the simplified simscape model.
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_test_bench_simplified';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
+  io(io_i) = linio([mdl, '/w'],  1, 'openinput');  io_i = io_i + 1; % Base Motion [m]
+  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force [N]
+  io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
+  io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
+
+  Gs = linearize(mdl, io);
+
+  Gs.InputName = {'Fd', 'w', 'Fa'};
+  Gs.OutputName = {'y', 'Fs'};
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 5, 1000);
+
+  figure;
+
+  ax1 = subplot(2, 3, 1);
+  title('$\displaystyle \frac{x_1}{w}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
+
+  ax2 = subplot(2, 3, 2);
+  title('$\displaystyle \frac{x_1}{f}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax3 = subplot(2, 3, 3);
+  title('$\displaystyle \frac{x_1}{F}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax4 = subplot(2, 3, 4);
+  title('$\displaystyle \frac{F_s}{w}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('Fs', 'w'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
+
+  ax5 = subplot(2, 3, 5);
+  title('$\displaystyle \frac{F_s}{f}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('Fs', 'Fa'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
+
+  ax6 = subplot(2, 3, 6);
+  title('$\displaystyle \frac{F_s}{F}$')
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
+  plot(freqs, abs(squeeze(freqresp(Gs('Fs', 'Fd'), freqs, 'Hz'))));
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
+
+  linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full');
+#+end_src
+
+#+name: fig:apa300ml_comp_simpler_simscape
+#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model
+#+RESULTS:
+[[file:figs/apa300ml_comp_simpler_simscape.png]]
+
+We save the parameters of the simplified model for the APA300ML:
+#+begin_src matlab
+  save('./mat/APA300ML_simplified_model.mat', 'ka', 'ke', 'k1', 'c1', 'F_gain');
+#+end_src
+
+** Identification of the stiffness properties
+*** APA Alone
+#+begin_src matlab :exports none
+  m = 10;
+#+end_src
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_characterisation';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
+  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
+
+  G = linearize(mdl, io);
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristics*', '*Value*'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+| *Caracteristics* | *Value* |
+|------------------+---------|
+| Kx [N/um]        |     0.8 |
+| Ky [N/um]        |     1.6 |
+| Kz [N/um]        |     1.8 |
+| Rx [Nm/rad]      |    71.4 |
+| Ry [Nm/rad]      |   148.2 |
+| Rz [Nm/rad]      |  4241.8 |
+
+*** See how the global stiffness is changing with the flexible joints
+#+begin_src matlab
+  flex = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_characterisation_with_joints';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
+  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
+
+  Gf = linearize(mdl, io);
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+| *Caracteristic* | *Value* |
+|-----------------+---------|
+| Kx [N/um]       |     0.0 |
+| Ky [N/um]       |     0.0 |
+| Kz [N/um]       |     1.8 |
+| Rx [Nm/rad]     |   722.9 |
+| Ry [Nm/rad]     |   129.6 |
+| Rz [Nm/rad]     |   115.3 |
+
+
+#+begin_src matlab
+  freqs = logspace(-2, 5, 1000);
+
+  figure;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
+  plot(freqs, abs(squeeze(freqresp(Gf(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
+  hold off;
+  legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab
+  freqs = logspace(-2, 5, 1000);
+
+  figure;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
+  plot(freqs, abs(squeeze(freqresp(Gf(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
+  hold off;
+  legend('location', 'northeast');
+#+end_src
+
+** Effect of APA300ML in the flexibility of the leg
+#+begin_src matlab :exports none
+  m = 10;
+#+end_src
+
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'APA300ML_flex_joints';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fd'], 1, 'openinput');  io_i = io_i + 1; % External Vertical Force [N]
+  io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
+#+end_src
+
+#+begin_src matlab :exports none
+  G = linearize(mdl, io);
+#+end_src
+
+#+begin_src matlab :exports none
+  Gf = linearize(mdl, io);
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([[1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', [1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]'], {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Rigid APA*', '*Flexible APA*'}, ' %.3f ');
+#+end_src
+
+
+#+RESULTS:
+| *Caracteristic* | *Rigid APA* | *Flexible APA* |
+|-----------------+-------------+----------------|
+| Kx [N/um]       |       0.018 |          0.019 |
+| Ky [N/um]       |       0.018 |          0.018 |
+| Kz [N/um]       |        60.0 |          2.647 |
+| Rx [Nm/rad]     |      16.705 |        557.682 |
+| Ry [Nm/rad]     |      16.535 |        185.939 |
+| Rz [Nm/rad]     |       118.0 |        114.803 |
+
 * Flexible Joint
 ** Introduction                                                      :ignore:
+The studied flexor is shown in Figure [[fig:flexor_id16_screenshot]].
 
-The flexor in Figure [[fig:flexor_id16_screenshot]] is studied with a FEM.
+The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
+It is then imported into Simscape.
+
+A simplified model of the flexor is then developped.
 
 #+name: fig:flexor_id16_screenshot
 #+caption: Flexor studied
@@ -1640,13 +2026,14 @@ Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= sims
 The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
 
 #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
-  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6); 60, 15, 15, 20]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Estimation by Francois*'}, ' %0.f ');
+  data2orgtable([1e-6*K(3,3), 1e-6*K(2,2), K(4,4), K(5,5), K(6,6); 60, 0, 15, 15, 20]', {'Axial Stiffness [N/um]', 'Shear Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Estimation by Francois*'}, ' %0.f ');
 #+end_src
 
 #+RESULTS:
 | *Caracteristic*            | *Value* | *Estimation by Francois* |
 |----------------------------+---------+--------------------------|
 | Axial Stiffness [N/um]     |     119 |                       60 |
+| Shear Stiffness [N/um]     |      11 |                        0 |
 | Bending Stiffness [Nm/rad] |      33 |                       15 |
 | Bending Stiffness [Nm/rad] |      33 |                       15 |
 | Torsion Stiffness [Nm/rad] |     236 |                       20 |
@@ -1681,9 +2068,9 @@ And we find the same parameters as the one estimated from the Stiffness matrix.
 | *Caracteristic*               | *Value* | *Identification* |
 |-------------------------------+---------+------------------|
 | Axial Stiffness Dz [N/um]     |     119 |              119 |
-| Bending Stiffness Rx [Nm/rad] |      33 |               33 |
-| Bending Stiffness Ry [Nm/rad] |      33 |               33 |
-| Torsion Stiffness Rz [Nm/rad] |     236 |              236 |
+| Bending Stiffness Rx [Nm/rad] |      33 |               34 |
+| Bending Stiffness Ry [Nm/rad] |      33 |              126 |
+| Torsion Stiffness Rz [Nm/rad] |     236 |              238 |
 
 ** Simpler Model
 
@@ -1773,6 +2160,247 @@ The two obtained dynamics are compared in Figure
 #+RESULTS:
 [[file:figs/flexor_ID16_compare_bushing_joint.png]]
 
+* Optimal Flexible Joint
+** Introduction                                                      :ignore:
+#+name: fig:optimal_flexor
+#+caption: Flexor studied
+[[file:data/flexor_circ_025/CS.jpg]]
+
+** 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
+
+#+begin_src matlab
+  addpath('./src/');
+  addpath('./data/flexor_circ_025/');
+#+end_src
+
+#+begin_src matlab :exports none
+  open('flexor_025.slx');
+#+end_src
+
+** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
+We first extract the stiffness and mass matrices.
+#+begin_src matlab
+  K = readmatrix('mat_K.CSV');
+  M = readmatrix('mat_M.CSV');
+#+end_src
+
+Then, we extract the coordinates of the interface nodes.
+#+begin_src matlab
+  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+#+end_src
+
+#+begin_src matlab
+  save('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+** Output parameters
+#+begin_src matlab
+  load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
+#+end_src
+
+#+RESULTS:
+| Total number of Nodes     |  2 |
+| Number of interface Nodes |  2 |
+| Number of Modes           |  6 |
+| Size of M and K matrices  | 18 |
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
+#+end_src
+
+#+caption: Coordinates of the interface nodes
+#+RESULTS:
+| Node i | Node Number | x [m] | y [m] | z [m] |
+|--------+-------------+-------+-------+-------|
+|    1.0 |    528875.0 |   0.0 |   0.0 |   0.0 |
+|    2.0 |    528876.0 |   0.0 |   0.0 |  -0.0 |
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
+#+end_src
+
+#+caption: First 10x10 elements of the Stiffness matrix
+#+RESULTS:
+|  12700000.0 |       -18.5 |       -26.8 |  0.00162 |    -4.63 |      64.0 | -12700000.0 |        18.3 |        26.7 |   0.00234 |
+|       -18.5 |  12700000.0 |      -499.0 |   -132.0 |  0.00414 |    -0.495 |        18.4 | -12700000.0 |       499.0 |     132.0 |
+|       -26.8 |      -499.0 |  94000000.0 |   -470.0 |  0.00771 |    -0.855 |        26.8 |       498.0 | -94000000.0 |     470.0 |
+|     0.00162 |      -132.0 |      -470.0 |     4.83 | 2.61e-07 |  0.000123 |    -0.00163 |       132.0 |       470.0 |     -4.83 |
+|       -4.63 |     0.00414 |     0.00771 | 2.61e-07 |     4.83 |  4.43e-05 |        4.63 |    -0.00413 |    -0.00772 |  -4.3e-07 |
+|        64.0 |      -0.495 |      -0.855 | 0.000123 | 4.43e-05 |     260.0 |       -64.0 |       0.495 |       0.855 | -0.000124 |
+| -12700000.0 |        18.4 |        26.8 | -0.00163 |     4.63 |     -64.0 |  12700000.0 |       -18.2 |       -26.7 |  -0.00234 |
+|        18.3 | -12700000.0 |       498.0 |    132.0 | -0.00413 |     0.495 |       -18.2 |  12700000.0 |      -498.0 |    -132.0 |
+|        26.7 |       499.0 | -94000000.0 |    470.0 | -0.00772 |     0.855 |       -26.7 |      -498.0 |  94000000.0 |    -470.0 |
+|     0.00234 |       132.0 |       470.0 |    -4.83 | -4.3e-07 | -0.000124 |    -0.00234 |      -132.0 |      -470.0 |      4.83 |
+
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
+#+end_src
+
+#+caption: First 10x10 elements of the Mass matrix
+#+RESULTS:
+|  0.006 |  8e-09 | -2e-08 | -1e-10 |  3e-05 |  3e-08 |  0.003 | -3e-09 |  9e-09 |  2e-12 |
+|  8e-09 |   0.02 |  1e-07 | -3e-05 |  1e-11 |  6e-10 |  1e-08 |  0.003 | -5e-08 |  3e-09 |
+| -2e-08 |  1e-07 |   0.01 | -6e-08 | -6e-11 | -8e-12 | -1e-07 |  1e-08 |  0.003 | -1e-08 |
+| -1e-10 | -3e-05 | -6e-08 |  1e-06 |  7e-14 |  6e-13 |  1e-10 |  1e-06 | -1e-08 |  3e-10 |
+|  3e-05 |  1e-11 | -6e-11 |  7e-14 |  2e-07 |  1e-10 |  3e-08 | -7e-12 |  6e-11 | -6e-16 |
+|  3e-08 |  6e-10 | -8e-12 |  6e-13 |  1e-10 |  5e-07 |  1e-08 | -5e-10 | -1e-11 |  1e-13 |
+|  0.003 |  1e-08 | -1e-07 |  1e-10 |  3e-08 |  1e-08 |   0.02 | -2e-08 |  1e-07 | -4e-12 |
+| -3e-09 |  0.003 |  1e-08 |  1e-06 | -7e-12 | -5e-10 | -2e-08 |  0.006 | -8e-08 |  3e-05 |
+|  9e-09 | -5e-08 |  0.003 | -1e-08 |  6e-11 | -1e-11 |  1e-07 | -8e-08 |   0.01 | -6e-08 |
+|  2e-12 |  3e-09 | -1e-08 |  3e-10 | -6e-16 |  1e-13 | -4e-12 |  3e-05 | -6e-08 |  2e-07 |
+
+Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
+
+** Flexible Joint Characteristics
+The most important parameters of the flexible joint can be directly estimated from the stiffness matrix.
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([1e-6*K(3,3), 1e-6*K(2,2), K(4,4), K(5,5), K(6,6)]', {'Axial Stiffness [N/um]', 'Shear Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+| *Caracteristic*            | *Value* |
+|----------------------------+---------|
+| Axial Stiffness [N/um]     |    94.0 |
+| Shear Stiffness [N/um]     |    12.7 |
+| Bending Stiffness [Nm/rad] |     4.8 |
+| Bending Stiffness [Nm/rad] |     4.8 |
+| Torsion Stiffness [Nm/rad] |   260.2 |
+
+** Identification of the parameters using Simscape
+The flexor is now imported into Simscape and its parameters are estimated using an identification.
+
+#+begin_src matlab :exports none
+  m = 1;
+#+end_src
+
+The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'flexor_025';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
+
+  G = linearize(mdl, io);
+#+end_src
+
+And we find the same parameters as the one estimated from the Stiffness matrix.
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %.1f ');
+#+end_src
+
+#+RESULTS:
+| *Caracteristic*               | *Value* | *Identification* |
+|-------------------------------+---------+------------------|
+| Axial Stiffness Dz [N/um]     |    94.0 |             93.9 |
+| Bending Stiffness Rx [Nm/rad] |     4.8 |              4.8 |
+| Bending Stiffness Ry [Nm/rad] |     4.8 |              4.8 |
+| Torsion Stiffness Rz [Nm/rad] |   260.2 |            260.2 |
+
+** Simpler Model
+
+Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].
+
+#+name: fig:flexible_joint_simscape
+#+caption: Bushing Joint used to model the flexible joint
+[[file:figs/flexible_joint_simscape.png]]
+
+The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
+#+begin_src matlab
+  Kx = K(1,1); % [N/m]
+  Ky = K(2,2); % [N/m]
+  Kz = K(3,3); % [N/m]
+  Krx = K(4,4); % [Nm/rad]
+  Kry = K(5,5); % [Nm/rad]
+  Krz =  K(6,6); % [Nm/rad]
+#+end_src
+
+The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'flexor_025_simplified';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/T'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
+
+  Gs = linearize(mdl, io);
+#+end_src
+
+The two obtained dynamics are compared in Figure
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 5, 1000);
+
+  figure;
+
+  ax1 = subplot(1,2,1);
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), '-', 'DisplayName', '$D_x/F_x$');
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(Gs(1,1), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', '$D_y/F_y$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(Gs(2,2), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', '$D_z/F_z$');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, abs(squeeze(freqresp(Gs(3,3), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
+  hold off;
+  legend('location', 'northeast');
+
+  ax2 = subplot(1,2,2);
+  hold on;
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), '-', 'DisplayName', '$R_x/M_x$');
+  set(gca,'ColorOrderIndex',1)
+  plot(freqs, abs(squeeze(freqresp(Gs(4,4), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), '-', 'DisplayName', '$R_y/M_y$');
+  set(gca,'ColorOrderIndex',2)
+  plot(freqs, abs(squeeze(freqresp(Gs(5,5), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), '-', 'DisplayName', '$R_z/M_z$');
+  set(gca,'ColorOrderIndex',3)
+  plot(freqs, abs(squeeze(freqresp(Gs(6,6), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
+  hold off;
+  legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:flexor_ID16_compare_bushing_joint
+#+caption: Comparison of the Joint compliance between the FEM model and the simpler model
+#+RESULTS:
+[[file:figs/flexor_ID16_compare_bushing_joint.png]]
+
 * Integral Force Feedback with Amplified Piezo
 ** Introduction                                                      :ignore:
 In this section, we try to replicate the results obtained in cite:souleille18_concep_activ_mount_space_applic.
@@ -2034,6 +2662,201 @@ The controller is defined and the loop gain is shown in Figure [[fig:piezo_ampli
 #+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
 [[file:figs/souleille18_results.png]]
 
+* Complete Strut with Encoder
+** Introduction
+
+#+name: fig:strut_encoder_points3
+#+caption: Interface points
+[[file:data/strut_encoder/points3.jpg]]
+
+Flexible joints have 0.25mm width.
+
+** 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
+
+#+begin_src matlab
+  addpath('./src/');
+  addpath('./data/strut_encoder/');
+#+end_src
+
+#+begin_src matlab
+  open('strut_encoder.slx');
+#+end_src
+
+** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
+We first extract the stiffness and mass matrices.
+#+begin_src matlab
+  K = readmatrix('mat_K.CSV');
+  M = readmatrix('mat_M.CSV');
+#+end_src
+
+Then, we extract the coordinates of the interface nodes.
+#+begin_src matlab
+  [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
+#+end_src
+
+#+begin_src matlab
+  save('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+** Output parameters
+#+begin_src matlab
+  load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
+#+end_src
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
+#+end_src
+
+#+RESULTS:
+| Total number of Nodes     |  8 |
+| Number of interface Nodes |  8 |
+| Number of Modes           |  6 |
+| Size of M and K matrices  | 54 |
+
+#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
+  data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
+#+end_src
+
+#+caption: Coordinates of the interface nodes
+#+RESULTS:
+| Node i | Node Number |   x [m] |  y [m] |    z [m] |
+|--------+-------------+---------+--------+----------|
+|    1.0 |    504411.0 |     0.0 |    0.0 |   0.0405 |
+|    2.0 |    504412.0 |     0.0 |    0.0 |  -0.0405 |
+|    3.0 |    504413.0 | -0.0325 |    0.0 |      0.0 |
+|    4.0 |    504414.0 | -0.0125 |    0.0 |      0.0 |
+|    5.0 |    504415.0 | -0.0075 |    0.0 |      0.0 |
+|    6.0 |    504416.0 |  0.0325 |    0.0 |      0.0 |
+|    7.0 |    504417.0 |   0.004 | 0.0145 | -0.00175 |
+|    8.0 |    504418.0 |   0.004 | 0.0166 | -0.00175 |
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
+#+end_src
+
+#+caption: First 10x10 elements of the Stiffness matrix
+#+RESULTS:
+|  2000000.0 |  1000000.0 | -3000000.0 |  -400.0 |  300.0 | 200.0 |      -30.0 |     2000.0 |   -10000.0 |     0.3 |
+|  1000000.0 |  4000000.0 | -8000000.0 |  -900.0 |  400.0 | -50.0 |    -6000.0 |    10000.0 |   -20000.0 |       3 |
+| -3000000.0 | -8000000.0 | 20000000.0 |  2000.0 | -900.0 | 200.0 |   -10000.0 |    20000.0 |  -300000.0 |       7 |
+|     -400.0 |     -900.0 |     2000.0 |       5 |   -0.1 |  0.05 |          1 |         -3 |          6 | -0.0007 |
+|      300.0 |      400.0 |     -900.0 |    -0.1 |      5 |  0.04 |       -0.1 |        0.5 |         -3 |  0.0001 |
+|      200.0 |      -50.0 |      200.0 |    0.05 |   0.04 | 300.0 |          4 |      -0.01 |         -1 |   3e-05 |
+|      -30.0 |    -6000.0 |   -10000.0 |       1 |   -0.1 |     4 |  3000000.0 | -1000000.0 | -2000000.0 |  -300.0 |
+|     2000.0 |    10000.0 |    20000.0 |      -3 |    0.5 | -0.01 | -1000000.0 |  6000000.0 |  7000000.0 |  1000.0 |
+|   -10000.0 |   -20000.0 |  -300000.0 |       6 |     -3 |    -1 | -2000000.0 |  7000000.0 | 20000000.0 |  2000.0 |
+|        0.3 |          3 |          7 | -0.0007 | 0.0001 | 3e-05 |     -300.0 |     1000.0 |     2000.0 |       5 |
+
+
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
+#+end_src
+
+#+caption: First 10x10 elements of the Mass matrix
+#+RESULTS:
+|   0.04 |  -0.005 |   0.007 |   2e-06 | 0.0001 | -5e-07 | -1e-05 |  -9e-07 |  8e-05 | -5e-10 |
+| -0.005 |    0.03 |    0.02 | -0.0001 |  1e-06 | -3e-07 |  3e-05 | -0.0001 |  8e-05 | -3e-08 |
+|  0.007 |    0.02 |    0.08 |  -6e-06 | -5e-06 | -7e-07 |  4e-05 | -0.0001 | 0.0005 | -3e-08 |
+|  2e-06 | -0.0001 |  -6e-06 |   2e-06 | -4e-10 |  2e-11 | -8e-09 |   3e-08 | -2e-08 |  6e-12 |
+| 0.0001 |   1e-06 |  -5e-06 |  -4e-10 |  3e-06 |  2e-10 | -3e-09 |   3e-09 | -7e-09 |  6e-13 |
+| -5e-07 |  -3e-07 |  -7e-07 |   2e-11 |  2e-10 |  5e-07 | -2e-08 |   5e-09 | -5e-09 |  1e-12 |
+| -1e-05 |   3e-05 |   4e-05 |  -8e-09 | -3e-09 | -2e-08 |   0.04 |   0.004 |  0.003 |  1e-06 |
+| -9e-07 | -0.0001 | -0.0001 |   3e-08 |  3e-09 |  5e-09 |  0.004 |    0.02 |  -0.02 | 0.0001 |
+|  8e-05 |   8e-05 |  0.0005 |  -2e-08 | -7e-09 | -5e-09 |  0.003 |   -0.02 |   0.08 | -5e-06 |
+| -5e-10 |  -3e-08 |  -3e-08 |   6e-12 |  6e-13 |  1e-12 |  1e-06 |  0.0001 | -5e-06 |  2e-06 |
+
+Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
+
+
+
+
+** Piezoelectric parameters
+Parameters for the APA300ML:
+
+#+begin_src matlab
+  d33 = 3e-10; % Strain constant [m/V]
+  n = 80; % Number of layers per stack
+  eT = 1.6e-8; % Permittivity under constant stress [F/m]
+  sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
+  ka = 235e6; % Stack stiffness [N/m]
+  C = 5e-6; % Stack capactiance [F]
+#+end_src
+
+#+begin_src matlab
+  na = 2; % Number of stacks used as actuator
+  ns = 1; % Number of stacks used as force sensor
+#+end_src
+
+** Identification of the Dynamics
+#+begin_src matlab :exports none
+  m = 0.01;
+#+end_src
+
+The dynamics is identified from the applied force to the measured relative displacement.
+#+begin_src matlab :exports none
+  %% Name of the Simulink File
+  mdl = 'strut_encoder';
+
+  %% Input/Output definition
+  clear io; io_i = 1;
+  io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1;
+  io(io_i) = linio([mdl, '/L'], 1, 'openoutput'); io_i = io_i + 1;
+
+  Gh = -linearize(mdl, io);
+#+end_src
+
+The same dynamics is identified for a payload mass of 10Kg.
+#+begin_src matlab
+  m = 10;
+#+end_src
+
+#+begin_src matlab :exports none
+  Ghm = -linearize(mdl, io);
+#+end_src
+
+#+begin_src matlab :exports none
+  freqs = logspace(0, 4, 5000);
+
+  figure;
+  tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
+
+  % ax1 = subplot(2,1,1);
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
+  plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+  axis padded 'auto x'
+  hold off;
+
+  ax2 = nexttile;
+  % ax2 = subplot(2,1,2);
+  hold on;
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Gh, freqs, 'Hz'))), '-', ...
+       'DisplayName', '$m = 0kg$');
+  plot(freqs, 180/pi*angle(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-', ...
+       'DisplayName', '$m = 10kg$');
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+  yticks(-360:90:360);
+  % ylim([-360 0]);
+  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+  hold off;
+  axis padded 'auto x'
+
+  % linkaxes([ax1,ax2],'x');
+  % xlim([freqs(1), freqs(end)]);
+  % legend('location', 'southwest');
+#+end_src
+
 * Bibliography                                                        :ignore:
 bibliographystyle:unsrt
 bibliography:ref.bib