From 9eaaa827b1506f2cc5d9892bfe7cf4ed98671726 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 29 Oct 2020 13:37:06 +0100 Subject: [PATCH] Update main page / figures --- figs/apa300ml_comp_simpler_model.pdf | Bin 145630 -> 148855 bytes figs/apa300ml_comp_simpler_model.png | Bin 151494 -> 154903 bytes figs/apa300ml_iff_plant.pdf | Bin 132385 -> 133586 bytes figs/apa300ml_iff_plant.png | Bin 108050 -> 116001 bytes figs/apa300ml_iff_root_locus.pdf | Bin 21254 -> 21323 bytes figs/apa300ml_iff_root_locus.png | Bin 26276 -> 26465 bytes figs/apa300ml_plant_dynamics.pdf | Bin 237225 -> 237589 bytes figs/apa300ml_plant_dynamics.png | Bin 149110 -> 150191 bytes figs/apa300ml_resonance.pdf | Bin 74086 -> 74836 bytes figs/apa300ml_resonance.png | Bin 64344 -> 66707 bytes figs/flexor_ID16_compare_bushing_joint.pdf | 738 ++-- figs/flexor_ID16_compare_bushing_joint.png | Bin 164009 -> 164189 bytes index.html | 3570 ++++++++++++++------ index.org | 971 +++++- 14 files changed, 3848 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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:
  1. \(\Delta h\): relative displacement [m]
  2. -
    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

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