From 9d1d089369490bf6b0990628ee267b13d7fce8ee Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 2 Jun 2021 23:58:32 +0200 Subject: [PATCH] Analysis all measurements --- figs/apa_meas_k_time.pdf | Bin 0 -> 189043 bytes figs/apa_meas_k_time.png | Bin 0 -> 53263 bytes figs/apa_meas_k_time_oc_cc.pdf | Bin 0 -> 64636 bytes figs/apa_meas_k_time_oc_cc.png | Bin 0 -> 32561 bytes figs/frf_dvf_plant_coh.pdf | Bin 0 -> 619651 bytes figs/frf_dvf_plant_coh.png | Bin 0 -> 97233 bytes figs/frf_dvf_plant_tf.pdf | Bin 0 -> 291597 bytes figs/frf_dvf_plant_tf.png | Bin 0 -> 144819 bytes figs/frf_iff_effect_R.pdf | Bin 0 -> 56122 bytes figs/frf_iff_effect_R.png | Bin 0 -> 100369 bytes figs/frf_iff_plant_coh.pdf | Bin 0 -> 601494 bytes figs/frf_iff_plant_coh.png | Bin 0 -> 77102 bytes figs/frf_iff_plant_tf.pdf | Bin 0 -> 275724 bytes figs/frf_iff_plant_tf.png | Bin 0 -> 118372 bytes figs/hyst_exc_signal_time.pdf | Bin 0 -> 37987 bytes figs/hyst_exc_signal_time.png | Bin 0 -> 61257 bytes figs/hyst_results_multi_ampl.pdf | Bin 0 -> 827557 bytes figs/hyst_results_multi_ampl.png | Bin 0 -> 120101 bytes ...=> frf_data_1_add_mass_closed_circuit.mat} | Bin ...t => frf_data_1_add_mass_open_circuit.mat} | Bin ..._high_freq.mat => frf_data_1_noise_hf.mat} | Bin ..._high_freq.mat => frf_data_2_noise_hf.mat} | Bin test-bench-apa300ml.html | 2687 ++++++++++++++--- test-bench-apa300ml.org | 1297 +++++++- 24 files changed, 3461 insertions(+), 523 deletions(-) create mode 100644 figs/apa_meas_k_time.pdf create mode 100644 figs/apa_meas_k_time.png create mode 100644 figs/apa_meas_k_time_oc_cc.pdf create mode 100644 figs/apa_meas_k_time_oc_cc.png create mode 100644 figs/frf_dvf_plant_coh.pdf create mode 100644 figs/frf_dvf_plant_coh.png create mode 100644 figs/frf_dvf_plant_tf.pdf create mode 100644 figs/frf_dvf_plant_tf.png create mode 100644 figs/frf_iff_effect_R.pdf create mode 100644 figs/frf_iff_effect_R.png create mode 100644 figs/frf_iff_plant_coh.pdf create mode 100644 figs/frf_iff_plant_coh.png create mode 100644 figs/frf_iff_plant_tf.pdf create mode 100644 figs/frf_iff_plant_tf.png create mode 100644 figs/hyst_exc_signal_time.pdf create mode 100644 figs/hyst_exc_signal_time.png create mode 100644 figs/hyst_results_multi_ampl.pdf create mode 100644 figs/hyst_results_multi_ampl.png rename matlab/mat/{frf_data_1_add_mass_closed_ciruit.mat => frf_data_1_add_mass_closed_circuit.mat} (100%) rename matlab/mat/{frf_data_1_add_mass_open_ciruit.mat => frf_data_1_add_mass_open_circuit.mat} (100%) rename matlab/mat/{frf_data_1_noise_high_freq.mat => frf_data_1_noise_hf.mat} (100%) rename matlab/mat/{frf_data_2_noise_high_freq.mat => frf_data_2_noise_hf.mat} (100%) diff --git a/figs/apa_meas_k_time.pdf b/figs/apa_meas_k_time.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aaf83cd5fb085f73716b08d01d5d767dd0e0c0b9 GIT binary patch literal 189043 zcmbTcRa9I}v}lb5hv4pRfo|O0Jvbq_Htz0j!KI-I5-hkwa1Rc_rGvY>+s*m^hkG9H z_+HoEb5+%@Sz}bK+VtwuvY&wL+^F>B6_r(}Kq>&0lbJ25urMlzx~r3ghqt0RYF};UQJ7o>R%dZCvy)6OGkG(S5s#jkhvSD zG&3rPl&htw`#(V1)ZLPbNm`H-zzGEK0ysH2xw(M+tN>0%0DzGhRa6w!($V67TIT#; zJ<#%Twxr^a@phNfbpOZy%O*huU*fL5Z`}+`2Pze z@LwPx;6G9STd@D<=>`5LDS-dU3gCaz0{DNupyE)rbhLK2p#uKTv};gt$b#(M|E(ep zS^IwuOIw;dSy)p2^WDw;-_meE_0AdS9GkQnOTpuxDt$1{58Mh&R#(EnpiduwDYyD5 zOJzennLh__)V(?`ywIXEP_CK&eDZyt%EbChAnO?g#hK&T`wea2hi9>%uy>Axr`z?n z8(RM%t;d$U_xkR)Qp!WW1PxpAh@kx-`+}9J*%u1SNR7zXg20v~bH|3wsLSJ^H)P|W zXWHLy+FLJAvx(TO@5e#c-L5U~FvIT;K|wExl<$u%e0NT7cjr}|?;pdMW=Sv>9)e!w zo_m5?-d}`{$KIc?yWLRA5^LVCX(Mai-H$T`FUc>ug@4@coxk^W73G~MWnZ@YI^C>y zI|W^--IX@(b@m3mb_FntEobWV@$fyJueXTZGCntWUgr4|zUKyjx?4Hy_}+H|4!R3I zwx*w_MDk=F3##9DKaKhE1S-{p2Grl_`6^cE{{N96sTz%{bzTmUxs^`c#Q%B8Qsp($ zG2+7EI^`)|x8*G8R>=L>nx4B2U#Wp|X=menzG}$$6?64T-}IZKge-Ip=E7+8eb4J) zZc6dQkl^*K<%{Hs+v_hw3p2X5>$HtEK@XG*9R4qP=LWIYLi=gIlz`DeTRjVwQmBpZWnD^UnZ-cC0 zSNV0zqM4Z&wBNe#VBYt^TQ?W3Q@3{(mu)}moi(MW*jBlA=YFt2qj|@CHfBVSGpSAm%4Ng!B@&OVuAf6Q#3(x_NIi8ygLt;O zHkne`#EfdZC7Z>#%bfEiY2mpfd;6&1&O_Sf9zmb}9mD@Q-~L$Aj6SLsYoO=zc5 zcj?3_V_Rd^@|nmJ`Kd$qE6?i^G)M)72+;w}<3$kyjoW=+OYRPR@u_9`i1z81)S#JAr+q9Gh zG_ClYUt5295l~v{98Br+nrKMm{9QwAUjEvC`}g$KAZVgFi@_;Wq*?B<4Dqyj_vi$9*N=5S2L_W9Le##flXU*^P! 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a/matlab/mat/frf_data_1_add_mass_closed_ciruit.mat b/matlab/mat/frf_data_1_add_mass_closed_circuit.mat similarity index 100% rename from matlab/mat/frf_data_1_add_mass_closed_ciruit.mat rename to matlab/mat/frf_data_1_add_mass_closed_circuit.mat diff --git a/matlab/mat/frf_data_1_add_mass_open_ciruit.mat b/matlab/mat/frf_data_1_add_mass_open_circuit.mat similarity index 100% rename from matlab/mat/frf_data_1_add_mass_open_ciruit.mat rename to matlab/mat/frf_data_1_add_mass_open_circuit.mat diff --git a/matlab/mat/frf_data_1_noise_high_freq.mat b/matlab/mat/frf_data_1_noise_hf.mat similarity index 100% rename from matlab/mat/frf_data_1_noise_high_freq.mat rename to matlab/mat/frf_data_1_noise_hf.mat diff --git a/matlab/mat/frf_data_2_noise_high_freq.mat b/matlab/mat/frf_data_2_noise_hf.mat similarity index 100% rename from matlab/mat/frf_data_2_noise_high_freq.mat rename to matlab/mat/frf_data_2_noise_hf.mat diff --git a/test-bench-apa300ml.html b/test-bench-apa300ml.html index 8fb9b45..8c4a83c 100644 --- a/test-bench-apa300ml.html +++ b/test-bench-apa300ml.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Amplifier Piezoelectric Actuator APA300ML - Test Bench @@ -39,93 +39,170 @@

      Table of Contents

        -
      • 1. Model of an Amplified Piezoelectric Actuator and Sensor
      • -
      • 2. Geometrical Measurements +
      • 1. Model of an Amplified Piezoelectric Actuator and Sensor
      • +
      • 2. Geometrical Measurements
      • -
      • 3. Electrical Measurements
      • -
      • 4. Stiffness measurement +
      • 3. Electrical Measurements
      • +
      • 4. Stroke measurement
      • -
      • 5. Stroke measurement +
      • 5. Stiffness measurement
      • -
      • 6. Test-Bench Description
      • -
      • 7. Measurement Procedure +
      • 6. Test-Bench Description
      • +
      • 7. Measurement Procedure
      • -
      • 8. Measurement Results
      • -
      • 9. Test Bench APA300ML - Simscape Model +
      • 8. FRF measurement -
        +

        apa300ML.png

        Figure 1: Picture of the APA300ML

        -
        -

        1 Model of an Amplified Piezoelectric Actuator and Sensor

        +
        +

        1 Model of an Amplified Piezoelectric Actuator and Sensor

        -Consider a schematic of the Amplified Piezoelectric Actuator in Figure 2. +Consider a schematic of the Amplified Piezoelectric Actuator in Figure 2.

        -
        +

        apa_model_schematic.png

        Figure 2: Amplified Piezoelectric Actuator Schematic

        @@ -191,11 +268,11 @@ We wish here to experimental measure \(g_a\) and \(g_s\).

        -The block-diagram model of the piezoelectric actuator is then as shown in Figure 3. +The block-diagram model of the piezoelectric actuator is then as shown in Figure 3.

        -
        +

        apa-model-simscape-schematic.png

        Figure 3: Model of the APA with Simscape/Simulink

        @@ -203,30 +280,30 @@ The block-diagram model of the piezoelectric actuator is then as shown in Figure
        -
        -

        2 Geometrical Measurements

        +
        +

        2 Geometrical Measurements

        -The received APA are shown in Figure 4. +The received APA are shown in Figure 4.

        -
        +

        IMG_20210224_143500.jpg

        Figure 4: Received APA

        -
        -

        2.1 Measurement Setup

        +
        +

        2.1 Measurement Setup

        -The flatness corresponding to the two interface planes are measured as shown in Figure 5. +The flatness corresponding to the two interface planes are measured as shown in Figure 5.

        -
        +

        IMG_20210224_143809.jpg

        Figure 5: Measurement Setup

        @@ -234,8 +311,8 @@ The flatness corresponding to the two interface planes are measured as shown in
        -
        -

        2.2 Measurement Results

        +
        +

        2.2 Measurement Results

        The height (Z) measurements at the 8 locations (4 points by plane) are defined below. @@ -281,10 +358,10 @@ Finally, the flatness is estimated by fitting a plane through the 8 points using

        -The obtained flatness are shown in Table 1. +The obtained flatness are shown in Table 1.

        - +
        @@ -339,10 +416,10 @@ The obtained flatness are shown in Table 1. -
        -

        3 Electrical Measurements

        +
        +

        3 Electrical Measurements

        -
        +

        The capacitance of the stacks is measure with the LCR-800 Meter (doc)

        @@ -350,7 +427,7 @@ The capacitance of the stacks is measure with the +
        Table 1: Estimated flatness
        @@ -422,7 +499,7 @@ The excitation frequency is set to be 1kHz.
        Table 2: Capacitance measured with the LCR meter. The excitation signal is a sinus at 1kHz
        -
        +

        There is clearly a problem with APA300ML number 3

        @@ -431,13 +508,207 @@ There is clearly a problem with APA300ML number 3
        -
        -

        4 Stiffness measurement

        +
        +

        4 Stroke measurement

        +

        +We here wish to estimate the stroke of the APA. +

        + +

        +To do so, one side of the APA is fixed, and a displacement probe is located on the other side as shown in Figure 7. +

        + +

        +Then, a voltage is applied on either one or two stacks using a DAC and a voltage amplifier. +

        + +
        +

        +Here are the documentation of the equipment used for this test bench: +

        + +
        -
        -

        4.1 APA test

        + + +
        +

        CE0EF55E-07B7-461B-8CDB-98590F68D15B.jpeg +

        +

        Figure 7: Bench to measured the APA stroke

        +
        +
        + +
        +

        4.1 Voltage applied on one stack

        +

        +Let’s first look at the relation between the voltage applied to one stack to the displacement of the APA as measured by the displacement probe. +

        + +

        +The applied voltage is shown in Figure 8. +

        + + +
        +

        apa_stroke_voltage_time.png +

        +

        Figure 8: Applied voltage as a function of time

        +
        + +

        +The obtained displacement is shown in Figure 9. +The displacement is set to zero at initial time when the voltage applied is -20V. +

        + + +
        +

        apa_stroke_time_1s.png +

        +

        Figure 9: Displacement as a function of time for all the APA300ML

        +
        + +

        +Finally, the displacement is shown as a function of the applied voltage in Figure 10. +We can clearly see that there is a problem with the APA 3. +Also, there is a large hysteresis. +

        + + +
        +

        apa_d_vs_V_1s.png +

        +

        Figure 10: Displacement as a function of the applied voltage

        +
        + +
        +

        +We can clearly see from Figure 10 that there is a problem with the APA number 3. +

        + +
        +
        +
        + +
        +

        4.2 Voltage applied on two stacks

        +
        +

        +Now look at the relation between the voltage applied to the two other stacks to the displacement of the APA as measured by the displacement probe. +

        + +

        +The obtained displacement is shown in Figure 11. +The displacement is set to zero at initial time when the voltage applied is -20V. +

        + + +
        +

        apa_stroke_time_2s.png +

        +

        Figure 11: Displacement as a function of time for all the APA300ML

        +
        + +

        +Finally, the displacement is shown as a function of the applied voltage in Figure 12. +We can clearly see that there is a problem with the APA 3. +Also, there is a large hysteresis. +

        + + +
        +

        apa_d_vs_V_2s.png +

        +

        Figure 12: Displacement as a function of the applied voltage

        +
        +
        +
        + +
        +

        4.3 Voltage applied on all three stacks

        +
        +

        +Finally, we can combine the two measurements to estimate the relation between the displacement and the voltage applied to the three stacks (Figure 13). +

        + + +
        +

        apa_d_vs_V_3s.png +

        +

        Figure 13: Displacement as a function of the applied voltage

        +
        + +

        +The obtained maximum stroke for all the APA are summarized in Table 3. +

        + + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
        Table 3: Measured maximum stroke
         Stroke \([\mu m]\)
        APA 1373.2
        APA 2365.5
        APA 3181.7
        APA 4359.7
        APA 5361.5
        APA 6363.9
        APA 7358.4
        +
        +
        +
        + +
        +

        5 Stiffness measurement

        +
        +
        +
        +

        5.1 APA test

        +
        load('meas_stiff_apa_1_x.mat', 't', 'F', 'd');
         
        @@ -502,204 +773,10 @@ plot(F_l, F_l*fit_l(1) +
        -
        -

        5 Stroke measurement

        -
        -

        -We here wish to estimate the stroke of the APA. -

        - -

        -To do so, one side of the APA is fixed, and a displacement probe is located on the other side as shown in Figure 7. -

        - -

        -Then, a voltage is applied on either one or two stacks using a DAC and a voltage amplifier. -

        - -
        -

        -Here are the documentation of the equipment used for this test bench: -

        - - -
        - - -
        -

        CE0EF55E-07B7-461B-8CDB-98590F68D15B.jpeg -

        -

        Figure 7: Bench to measured the APA stroke

        -
        -
        - -
        -

        5.1 Voltage applied on one stack

        -
        -

        -Let’s first look at the relation between the voltage applied to one stack to the displacement of the APA as measured by the displacement probe. -

        - -

        -The applied voltage is shown in Figure 8. -

        - - -
        -

        apa_stroke_voltage_time.png -

        -

        Figure 8: Applied voltage as a function of time

        -
        - -

        -The obtained displacement is shown in Figure 9. -The displacement is set to zero at initial time when the voltage applied is -20V. -

        - - -
        -

        apa_stroke_time_1s.png -

        -

        Figure 9: Displacement as a function of time for all the APA300ML

        -
        - -

        -Finally, the displacement is shown as a function of the applied voltage in Figure 10. -We can clearly see that there is a problem with the APA 3. -Also, there is a large hysteresis. -

        - - -
        -

        apa_d_vs_V_1s.png -

        -

        Figure 10: Displacement as a function of the applied voltage

        -
        - -
        -

        -We can clearly see from Figure 10 that there is a problem with the APA number 3. -

        - -
        -
        -
        - -
        -

        5.2 Voltage applied on two stacks

        -
        -

        -Now look at the relation between the voltage applied to the two other stacks to the displacement of the APA as measured by the displacement probe. -

        - -

        -The obtained displacement is shown in Figure 11. -The displacement is set to zero at initial time when the voltage applied is -20V. -

        - - -
        -

        apa_stroke_time_2s.png -

        -

        Figure 11: Displacement as a function of time for all the APA300ML

        -
        - -

        -Finally, the displacement is shown as a function of the applied voltage in Figure 12. -We can clearly see that there is a problem with the APA 3. -Also, there is a large hysteresis. -

        - - -
        -

        apa_d_vs_V_2s.png -

        -

        Figure 12: Displacement as a function of the applied voltage

        -
        -
        -
        - -
        -

        5.3 Voltage applied on all three stacks

        -
        -

        -Finally, we can combine the two measurements to estimate the relation between the displacement and the voltage applied to the three stacks (Figure 13). -

        - - -
        -

        apa_d_vs_V_3s.png -

        -

        Figure 13: Displacement as a function of the applied voltage

        -
        - -

        -The obtained maximum stroke for all the APA are summarized in Table 3. -

        - - - - --- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
        Table 3: Measured maximum stroke
         Stroke \([\mu m]\)
        APA 1373.2
        APA 2365.5
        APA 3181.7
        APA 4359.7
        APA 5361.5
        APA 6363.9
        APA 7358.4
        -
        -
        -
        - -
        -

        6 Test-Bench Description

        +
        +

        6 Test-Bench Description

        -
        +

        Here are the documentation of the equipment used for this test bench:

        @@ -714,7 +791,7 @@ Here are the documentation of the equipment used for this test bench:
        -
        +

        test_bench_apa_alone.png

        Figure 14: Schematic of the Test Bench

        @@ -722,12 +799,15 @@ Here are the documentation of the equipment used for this test bench:
        -
        -

        7 Measurement Procedure

        +
        +

        7 Measurement Procedure

        +

        + +

        -
        -

        7.1 Stroke Measurement

        +
        +

        7.1 Stroke Measurement

        Using the PD200 amplifier, output a voltage: @@ -755,8 +835,8 @@ Conclude on the obtained stroke.

        -
        -

        7.2 Stiffness Measurement

        +
        +

        7.2 Stiffness Measurement

        Add some (known) weight \(\delta m g\) on the suspended mass and measure the deflection \(\delta d\). @@ -776,8 +856,8 @@ Then the obtained stiffness is:

        -
        -

        7.3 Hysteresis measurement

        +
        +

        7.3 Hysteresis measurement

        Supply a quasi static sinusoidal excitation \(V_a\) at different voltages. @@ -796,16 +876,16 @@ Then, \(d\) is plotted as a function of \(V_a\) for all the amplitudes.

        -
        +

        expected_hysteresis.png

        -

        Figure 15: Expected Hysteresis (poel10_explor_activ_hard_mount_vibrat)

        +

        Figure 15: Expected Hysteresis poel10_explor_activ_hard_mount_vibrat

        -
        -

        7.4 Piezoelectric Actuator Constant

        +
        +

        7.4 Piezoelectric Actuator Constant

        Using the measurement test-bench, it is rather easy the determine the static gain between the applied voltage \(V_a\) to the induced displacement \(d\). @@ -832,8 +912,8 @@ From the two gains, it is then easy to determine \(g_a\):

        -
        -

        7.5 Piezoelectric Sensor Constant

        +
        +

        7.5 Piezoelectric Sensor Constant

        From a quasi static excitation of the piezoelectric stack, measure the gain from \(V_a\) to \(V_s\): @@ -872,8 +952,8 @@ This external force can be some weight added, or a piezo in parallel.

        -
        -

        7.6 Capacitance Measurement

        +
        +

        7.6 Capacitance Measurement

        Measure the capacitance of the 3 stacks individually using a precise multi-meter. @@ -881,8 +961,8 @@ Measure the capacitance of the 3 stacks individually using a precise multi-meter

        -
        -

        7.7 Dynamical Behavior

        +
        +

        7.7 Dynamical Behavior

        Perform a system identification from \(V_a\) to the measured displacement \(d\) by the interferometer and by the encoder, and to the generated voltage \(V_s\). @@ -898,8 +978,8 @@ This can also be performed with and without the encoder fixed to the APA.

        -
        -

        7.8 Compare the results obtained for all 7 APA300ML

        +
        +

        7.8 Compare the results obtained for all 7 APA300ML

        Compare all the obtained parameters for all the test APA. @@ -908,29 +988,1379 @@ Compare all the obtained parameters for all the test APA.

        -
        -

        8 Measurement Results

        +
        +

        8 FRF measurement

        +
        +
          +
        • [ ] Schematic of the measurement
        • +
        + + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
        Variable UnitHardware
        VaOutput DAC voltage[V]DAC - Ch. 1 => PD200 => APA
        VsMeasured stack voltage (ADC)[V]APA => ADC - Ch. 1
        deEncoder Measurement[m]PEPU Ch. 1 - IO318(1) - Ch. 1
        daAttocube Measurement[m]PEPU Ch. 2 - IO318(1) - Ch. 2
        tTime[s] 
        -
        -

        9 Test Bench APA300ML - Simscape Model

        -
        +
        +

        8.1 frf_setup.m - Measurement Setup

        +
        +

        +First is defined the sampling frequency: +

        +
        +
        %% Simulation configuration
        +Fs   = 10e3; % Sampling Frequency [Hz]
        +Ts   = 1/Fs; % Sampling Time [s]
        +
        -
        -

        9.1 Introduction

        + +
        +
        %% Data record configuration
        +Trec_start = 5;  % Start time for Recording [s]
        +Trec_dur   = 100; % Recording Duration [s]
        +
        -
        -

        9.2 Nano Hexapod object

        -
        + +
        +
        Tsim = 2*Trec_start + Trec_dur;  % Simulation Time [s]
        +
        +
        + +

        +The maximum excitation voltage at resonance is 9Vrms, therefore corresponding to 0.6V of output DAC voltage. +

        +
        +
        %% Sweep Sine
        +gc = 0.1;
        +xi = 0.5;
        +wn = 2*pi*94.3;
        +
        +% Notch filter at the resonance of the APA
        +G_sweep = 0.2*(s^2 + 2*gc*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
        +
        +V_sweep = generateSweepExc('Ts',      Ts, ...
        +                           'f_start', 10, ...
        +                           'f_end',   1e3, ...
        +                           'V_mean',  3.25, ...
        +                           't_start', Trec_start, ...
        +                           'exc_duration', Trec_dur, ...
        +                           'sweep_type',   'log', ...
        +                           'V_exc', G_sweep*1/(1 + s/2/pi/500));
        +
        +
        + + +
        +

        frf_meas_sweep_excitation.png +

        +

        Figure 16: Example of Sweep Sin excitation signal

        +
        + + +

        +A white noise excitation signal can be very useful in order to obtain a first idea of the plant FRF. +The gain can be gradually increased until satisfactory output is obtained. +

        +
        +
        %% Shaped Noise
        +V_noise = generateShapedNoise('Ts', 1/Fs, ...
        +                              'V_mean', 3.25, ...
        +                              't_start', Trec_start, ...
        +                              'exc_duration', Trec_dur, ...
        +                              'smooth_ends', true, ...
        +                              'V_exc', 0.05/(1 + s/2/pi/10));
        +
        +
        + + +
        +

        frf_meas_noise_excitation.png +

        +

        Figure 17: Example of Shaped noise excitation signal

        +
        + +
        +
        %% Sinus excitation with increasing amplitude
        +V_sin = generateSinIncreasingAmpl('Ts', 1/Fs, ...
        +                                    'V_mean', 3.25, ...
        +                                    'sin_ampls', [0.1, 0.2, 0.4, 1, 2, 4], ...
        +                                    'sin_period', 1, ...
        +                                    'sin_num', 5, ...
        +                                    't_start', 10, ...
        +                                    'smooth_ends', true);
        +
        +
        + + +
        +

        frf_meas_sin_excitation.png +

        +

        Figure 18: Example of Shaped noise excitation signal

        +
        + +

        +Then, we select the wanted excitation signal. +

        +
        +
        %% Select the excitation signal
        +V_exc = timeseries(V_noise(2,:), V_noise(1,:));
        +
        +
        + +
        +
        %% Save data that will be loaded in the Simulink file
        +save('./frf_data.mat', 'Fs', 'Ts', 'Tsim', 'Trec_start', 'Trec_dur', 'V_exc');
        +
        +
        +
        +
        + +
        +

        8.2 frf_save.m - Save Data

        +
        +

        +First, we get data from the Speedgoat: +

        +
        +
        tg = slrt;
        +
        +f = SimulinkRealTime.openFTP(tg);
        +mget(f, 'data/data.dat');
        +close(f);
        +
        +
        + +

        +And we load the data on the Workspace: +

        +
        +
        data = SimulinkRealTime.utils.getFileScopeData('data/data.dat').data;
        +
        +da = data(:, 1); % Excitation Voltage (input of PD200) [V]
        +de = data(:, 2); % Measured voltage (force sensor) [V]
        +Vs = data(:, 3); % Measurment displacement (encoder) [m]
        +Va = data(:, 4); % Measurement displacement (attocube) [m]
        +t  = data(:, end); % Time [s]
        +
        +
        + +

        +And we save this to a mat file: +

        +
        +
        apa_number = 1;
        +
        +save(sprintf('mat/frf_data_%i_huddle.mat', apa_number), 't', 'Va', 'Vs', 'de', 'da');
        +
        +
        +
        +
        + +
        +

        8.3 Measurements on APA 1

        +
        +

        +Measurements are first performed on the APA number 1. +

        +
        + +
        +

        8.3.1 Huddle Test

        +
        +
        +
        load(sprintf('frf_data_%i_huddle.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
        +
        +
        + +
        +
        %% Parameter for Spectral analysis
        +Ts = (t(end) - t(1))/(length(t)-1);
        +Fs = 1/Ts;
        +
        +win = hanning(ceil(10*Fs)); % Hannning Windows
        +
        +
        + +
        +
        [phh_da, f] = pwelch(da - mean(da), win, [], [], 1/Ts);
        +[phh_de, ~] = pwelch(de - mean(de), win, [], [], 1/Ts);
        +[phh_Vs, ~] = pwelch(Vs - mean(Vs), win, [], [], 1/Ts);
        +
        +
        +
        +
        + +
        +

        8.3.2 First identification with Noise

        +
        +
        +
        load(sprintf('mat/frf_data_%i_noise.mat', 1), 't', 'Va', 'Vs', 'da', 'de')
        +
        +
        + +
        +
        [pxx_da, f] = pwelch(da, win, [], [], 1/Ts);
        +[pxx_de, ~] = pwelch(de, win, [], [], 1/Ts);
        +[pxx_Vs, ~] = pwelch(Vs, win, [], [], 1/Ts);
        +
        +
        + +
        +
        [G_dvf, f] = tfestimate(Va, de, win, [], [], 1/Ts);
        +[G_d,   ~] = tfestimate(Va, da, win, [], [], 1/Ts);
        +[G_iff, ~] = tfestimate(Va, Vs, win, [], [], 1/Ts);
        +
        +[coh_dvf, ~] = mscohere(Va, de, win, [], [], 1/Ts);
        +[coh_d,   ~] = mscohere(Va, da, win, [], [], 1/Ts);
        +[coh_iff, ~] = mscohere(Va, Vs, win, [], [], 1/Ts);
        +
        +
        +
        +
        + +
        +

        8.3.3 Second identification with Sweep and high frequency noise

        +
        +
        +
        load(sprintf('mat/frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'da', 'de')
        +
        +
        + +
        +
        [pxx_da, f] = pwelch(da, win, [], [], 1/Ts);
        +[pxx_de, ~] = pwelch(de, win, [], [], 1/Ts);
        +[pxx_Vs, ~] = pwelch(Vs, win, [], [], 1/Ts);
        +
        +
        + +
        +
        [G_dvf, f] = tfestimate(Va, de, win, [], [], 1/Ts);
        +[G_d,   ~] = tfestimate(Va, da, win, [], [], 1/Ts);
        +[G_iff, ~] = tfestimate(Va, Vs, win, [], [], 1/Ts);
        +
        +[coh_dvf, ~] = mscohere(Va, de, win, [], [], 1/Ts);
        +[coh_d,   ~] = mscohere(Va, da, win, [], [], 1/Ts);
        +[coh_iff, ~] = mscohere(Va, Vs, win, [], [], 1/Ts);
        +
        +
        + +
        +
        load(sprintf('mat/frf_data_%i_noise_high_freq.mat', 1), 't', 'Va', 'Vs', 'da', 'de')
        +
        +
        + +
        +
        [phf_da, f] = pwelch(da, win, [], [], 1/Ts);
        +[phf_de, ~] = pwelch(de, win, [], [], 1/Ts);
        +[phf_Vs, ~] = pwelch(Vs, win, [], [], 1/Ts);
        +
        +
        + +
        +
        [cohhf_dvf, ~] = mscohere(Va, de, win, [], [], 1/Ts);
        +[cohhf_d,   ~] = mscohere(Va, da, win, [], [], 1/Ts);
        +[cohhf_iff, ~] = mscohere(Va, Vs, win, [], [], 1/Ts);
        +
        +[Ghf_dvf, f] = tfestimate(Va, de, win, [], [], 1/Ts);
        +[Ghf_d,   ~] = tfestimate(Va, da, win, [], [], 1/Ts);
        +[Ghf_iff, ~] = tfestimate(Va, Vs, win, [], [], 1/Ts);
        +
        +
        +
        +
        + +
        +

        8.3.4 Extract Parameters (Actuator/Sensor constants)

        +
        +

        +Quasi static gain between \(d\) and \(V_a\): +

        +
        +
        g_d_Va = mean(abs(G_dvf(f > 10 & f < 15)));
        +
        +
        + +
        +g_d_Va = 1.7e-05 [m/V]
        +
        + + +

        +Quasi static gain between \(V_s\) and \(V_a\): +

        +
        +
        g_Vs_Va = mean(abs(G_iff(f > 10 & f < 15)));
        +
        +
        + +
        +g_Vs_Va = 5.7e-01 [V/V]
        +
        +
        +
        +
        + +
        +

        8.4 Comparison of all the APA

        +
        +
        +
        +

        8.4.1 Stiffness - Comparison of the APA

        +
        +

        +In order to estimate the stiffness of the APA, a weight with known mass \(m_a\) is added on top of the suspended granite and the deflection \(d_e\) is measured using an encoder. +The APA stiffness is then: +

        +\begin{equation} + k_{\text{apa}} = \frac{m_a g}{d} +\end{equation} + +

        +Here are the number of the APA that have been measured: +

        +
        +
        apa_nums = [1 2 4 5 6 7 8];
        +
        +
        + +

        +The data are loaded. +

        +
        +
        apa_mass = {};
        +for i = 1:length(apa_nums)
        +    apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')};
        +    % The initial displacement is set to zero
        +    apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11));
        +end
        +
        +
        + +

        +The raw measurements are shown in Figure 19. +All the APA seems to have similar stiffness except the APA 7 which should have an higher stiffness. +

        + +
        +

        +It is however strange that the displacement \(d_e\) when the mass is removed is higher for the APA 7 than for the other APA. +What could cause that? +

        + +
        + + +
        +

        apa_meas_k_time.png +

        +

        Figure 19: Raw measurements for all the APA. A mass of 6.4kg is added at arround 15s and removed at arround 22s

        +
        + +
        +
        added_mass = 6.4; % Added mass [kg]
        +
        +
        + +

        +The stiffnesses are computed for all the APA and are summarized in Table 4. +

        + + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
        Table 4: Measured stiffnesses
        APA Num\(k [N/\mu m]\)
        11.68
        21.69
        41.7
        51.7
        61.7
        71.93
        81.73
        + +
        +

        +The APA300ML manual specifies the nominal stiffness to be \(1.8\,[N/\mu m]\) which is very close to what have been measured. +Only the APA number 7 is a little bit off. +

        + +
        +
        +
        + +
        +

        8.4.2 Stiffness - Effect of connecting the actuator stack to the amplifier and the sensor stack to the ADC

        +
        +

        +We wish here to see if the stiffness changes when the actuator stacks are not connected to the amplifier and the sensor stacks are not connected to the ADC. +

        + +

        +Note here that the resistor in parallel to the sensor stack is present in both cases. +

        + +

        +First, the data are loaded. +

        +
        +
        add_mass_oc = load(sprintf('frf_data_%i_add_mass_open_circuit.mat',   1), 't', 'de');
        +add_mass_cc = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de');
        +
        +
        + +

        +And the initial displacement is set to zero. +

        +
        +
        add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11));
        +add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11));
        +
        +
        + + +
        +

        apa_meas_k_time_oc_cc.png +

        +

        Figure 20: Measured displacement

        +
        + + +

        +And the stiffness is estimated in both case. +The results are shown in Table 5. +

        + +
        +
        apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5)));
        +apa_k_cc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5)));
        +
        +
        + + + + +++ ++ + + + + + + + + + + + + + + + + + +
        Table 5: Measured stiffnesses on “open” and “closed” circuits
         \(k [N/\mu m]\)
        Not connected2.3
        Connected1.7
        + +
        +

        +Clearly, connecting the actuator stacks to the amplified (basically equivalent as to short circuiting them) lowers the stiffness. +

        + +
        +
        +
        + +
        +

        8.4.3 Hysteresis

        +
        +

        +We here wish to visually see the amount of hysteresis present in the APA. +

        + +

        +The data is loaded. +

        +
        +
        apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'Va', 'de');
        +% Initial time set to zero
        +apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
        +
        +
        + +

        +The excitation voltage amplitudes are: +

        +
        +
        ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
        +
        +
        + +

        +The excitation voltage and the measured displacement are shown in Figure 21. +

        + +
        +

        hyst_exc_signal_time.png +

        +

        Figure 21: Excitation voltage and measured displacement

        +
        + +

        +For each amplitude, we only take the last sinus in order to reduce possible transients. +Also, it is centered on zero. +

        + +

        +The measured displacement at a function of the output voltage are shown in Figure 22. +

        + +
        +

        hyst_results_multi_ampl.png +

        +

        Figure 22: Obtained hysteresis for multiple excitation amplitudes

        +
        + +
        +

        +It is quite clear that hysteresis is increasing with the excitation amplitude. +

        + +

        +Also, no hysteresis is found on the sensor stack voltage. +

        + +
        +
        +
        + +
        +

        8.4.4 FRF Identification - Setup

        +
        +

        +The identification is performed in three steps: +

        +
          +
        1. White noise excitation with small amplitude. +This is used to determine the main resonance of the system.
        2. +
        3. Sweep sine excitation with the amplitude lowered around the resonance. +The sweep sine is from 10Hz to 400Hz.
        4. +
        5. High frequency noise. +The noise is band-passed between 300Hz and 2kHz.
        6. +
        + +

        +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. +

        + +

        +Here are the APA numbers that have been measured. +

        +
        +
        apa_nums = [1 2 4 5 6 7 8];
        +
        +
        + +

        +The data are loaded for both the second and third identification: +

        +
        +
        %% Second identification
        +apa_sweep = {};
        +for i = 1:length(apa_nums)
        +    apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
        +end
        +
        +%% Third identification
        +apa_noise_hf = {};
        +for i = 1:length(apa_nums)
        +    apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')};
        +end
        +
        +
        + +

        +The time is the same for all measurements. +

        +
        +
        %% Time vector
        +t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s]
        +
        +%% Sampling
        +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
        +Fs = 1/Ts; % Sampling Frequency [Hz]
        +
        +
        + +

        +Then we defined a “Hanning” windows that will be used for the spectral analysis: +

        +
        +
        win = hanning(ceil(0.5*Fs)); % Hannning Windows
        +
        +
        + +

        +We get the frequency vector that will be the same for all the frequency domain analysis. +

        +
        +
        % Only used to have the frequency vector "f"
        +[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts);
        +
        +
        +
        +
        + +
        +

        8.4.5 FRF Identification - DVF

        +
        +

        +In this section, the dynamics from \(V_a\) to \(d_e\) is identified. +

        + +

        +We compute the coherence for 2nd and 3rd identification: +

        +
        +
        %% Coherence computation
        +coh_sweep = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
        +    coh_sweep(:, i) = coh;
        +end
        +
        +coh_noise_hf = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
        +    coh_noise_hf(:, i) = coh;
        +end
        +
        +
        + +

        +The coherence is shown in Figure 23. +It is clear that the Sweep sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency. +

        + + +
        +

        frf_dvf_plant_coh.png +

        +

        Figure 23: Obtained coherence for the plant from \(V_a\) to \(d_e\)

        +
        + + +

        +Then, the transfer function from the DAC output voltage \(V_a\) to the measured displacement by the encoders is computed: +

        +
        +
        %% Transfer function estimation
        +dvf_sweep = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts);
        +    dvf_sweep(:, i) = frf;
        +end
        +
        +dvf_noise_hf = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts);
        +    dvf_noise_hf(:, i) = frf;
        +end
        +
        +
        + +

        +The obtained transfer functions are shown in Figure 24. +

        + +

        +They are all superimposed except for the APA7. +

        + +
        +

        +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. +

        + +

        +It seems that there is a “gain” problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +

        + +
        + +
        +

        +Why is there a double resonance at around 94Hz? +

        + +
        + + +
        +

        frf_dvf_plant_tf.png +

        +

        Figure 24: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

        +
        +
        +
        + + +
        +

        8.4.6 FRF Identification - IFF

        +
        +

        +In this section, the dynamics from \(V_a\) to \(V_s\) is identified. +

        + +

        +First the coherence is computed and shown in Figure 25. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). +

        + +
        +
        %% Coherence
        +coh_sweep = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
        +    coh_sweep(:, i) = coh;
        +end
        +
        +coh_noise_hf = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
        +    coh_noise_hf(:, i) = coh;
        +end
        +
        +
        + + +
        +

        frf_iff_plant_coh.png +

        +

        Figure 25: Obtained coherence for the IFF plant

        +
        + +

        +Then the FRF are estimated and shown in Figure 26 +

        +
        +
        %% FRF estimation of the transfer function from Va to Vs
        +iff_sweep = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts);
        +    iff_sweep(:, i) = frf;
        +end
        +
        +iff_noise_hf = zeros(length(f), length(apa_nums));
        +for i = 1:length(apa_nums)
        +    [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts);
        +    iff_noise_hf(:, i) = frf;
        +end
        +
        +
        + + +
        +

        frf_iff_plant_tf.png +

        +

        Figure 26: Identified IFF Plant

        +
        +
        +
        + +
        +

        8.4.7 Effect of the resistor on the IFF Plant

        +
        +

        +A resistor is added in parallel with the sensor stack. +This has the effect to form a high pass filter with the capacitance of the stack. +

        + +

        +We here measured the low frequency transfer function from \(V_a\) to \(V_s\) with and without this resistor. +

        + +
        +
        % With the resistor
        +wi_k = load('frf_data_2_sweep_lf_with_R.mat', 't', 'Vs', 'Va');
        +
        +% Without the resistor
        +wo_k = load('frf_data_2_sweep_lf.mat',        't', 'Vs', 'Va');
        +
        +
        + +
        +
        t = wo_k.t; % Time vector [s]
        +
        +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
        +Fs = 1/Ts; % Sampling Frequency [Hz]
        +
        +
        + +

        +We use a very long “Hanning” window for the spectral analysis in order to estimate the low frequency behavior. +

        +
        +
        win = hanning(ceil(50*Fs)); % Hannning Windows
        +
        +
        + +

        +And we estimate the transfer function from \(V_a\) to \(V_s\) in both cases: +

        +
        +
        [frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, [], [], 1/Ts);
        +[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, [], [], 1/Ts);
        +
        +
        + + +
        +
        f0 = 0.35;
        +G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0);
        +
        +
        + + +
        +

        frf_iff_effect_R.png +

        +

        Figure 27: Transfer function from \(V_a\) to \(V_s\) with and without the resistor \(k\)

        +
        +
        +
        +
        + + +
        +

        8.5 Measurement on Strut 1

        +
        +

        +Measurements are first performed on the strut number 1 with: +

        +
          +
        • APA 1
        • +
        • flex 1 and flex 2
        • +
        +
        + +
        +

        8.5.1 Without Encoder

        +
        +
        +
        +
        8.5.1.1 FRF Identification - Setup
        +
        +

        +The identification is performed in three steps: +

        +
          +
        1. White noise excitation with small amplitude. +This is used to determine the main resonance of the system.
        2. +
        3. Sweep sine excitation with the amplitude lowered around the resonance. +The sweep sine is from 10Hz to 400Hz.
        4. +
        5. High frequency noise. +The noise is band-passed between 300Hz and 2kHz.
        6. +
        + +

        +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. +

        + +
        +
        leg_sweep    = load(sprintf('frf_data_leg_%i_sweep.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
        +leg_noise_hf = load(sprintf('frf_data_leg_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
        +
        +
        + +

        +The time is the same for all measurements. +

        +
        +
        %% Time vector
        +t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]
        +
        +%% Sampling
        +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
        +Fs = 1/Ts; % Sampling Frequency [Hz]
        +
        +
        + +

        +Then we defined a “Hanning” windows that will be used for the spectral analysis: +

        +
        +
        win = hanning(ceil(0.5*Fs)); % Hannning Windows
        +
        +
        + +

        +We get the frequency vector that will be the same for all the frequency domain analysis. +

        +
        +
        % Only used to have the frequency vector "f"
        +[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);
        +
        +
        +
        +
        + +
        +
        8.5.1.2 FRF Identification - DVF
        +
        +

        +In this section, the dynamics from \(V_a\) to \(d_e\) is identified. +

        + +

        +We compute the coherence for 2nd and 3rd identification: +

        +
        +
        [coh_sweep, ~]    = mscohere(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
        +[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_dvf_plant_coh.png +

        +

        Figure 28: Obtained coherence for the plant from \(V_a\) to \(d_e\)

        +
        + + +
        +
        [dvf_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
        +[dvf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts);
        +
        +
        + +

        +The obtained transfer functions are shown in Figure 24. +

        + +

        +They are all superimposed except for the APA7. +

        + +
        +

        +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. +

        + +

        +It seems that there is a “gain” problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +

        + +
        + +
        +

        +Why is there a double resonance at around 94Hz? +

        + +
        + + +
        +

        frf_dvf_plant_tf.png +

        +

        Figure 29: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

        +
        +
        +
        + +
        +
        8.5.1.3 FRF Identification - IFF
        +
        +

        +In this section, the dynamics from \(V_a\) to \(V_s\) is identified. +

        + +

        +First the coherence is computed and shown in Figure 25. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). +

        + +
        +
        [coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
        +[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_iff_plant_coh.png +

        +

        Figure 30: Obtained coherence for the IFF plant

        +
        + +

        +Then the FRF are estimated and shown in Figure 26 +

        +
        +
        [iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
        +[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_iff_plant_tf.png +

        +

        Figure 31: Identified IFF Plant

        +
        +
        +
        +
        + +
        +

        8.5.2 With Encoder

        +
        +
        +
        +
        8.5.2.1 FRF Identification - Setup
        +
        +

        +The identification is performed in three steps: +

        +
          +
        1. White noise excitation with small amplitude. +This is used to determine the main resonance of the system.
        2. +
        3. Sweep sine excitation with the amplitude lowered around the resonance. +The sweep sine is from 10Hz to 400Hz.
        4. +
        5. High frequency noise. +The noise is band-passed between 300Hz and 2kHz.
        6. +
        + +

        +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. +

        + +
        +
        leg_enc_sweep    = load(sprintf('frf_data_leg_coder_%i_noise.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
        +leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
        +
        +
        +
        +
        + +
        +
        8.5.2.2 FRF Identification - DVF
        +
        +

        +In this section, the dynamics from \(V_a\) to \(d_e\) is identified. +

        + +

        +We compute the coherence for 2nd and 3rd identification: +

        +
        +
        [coh_enc_sweep, ~]    = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
        +[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_dvf_plant_coh.png +

        +

        Figure 32: Obtained coherence for the plant from \(V_a\) to \(d_e\)

        +
        + + +
        +
        [dvf_enc_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.de,    win, [], [], 1/Ts);
        +[dvf_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts);
        +
        +
        + +

        +The obtained transfer functions are shown in Figure 24. +

        + +

        +They are all superimposed except for the APA7. +

        + +
        +

        +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. +

        + +

        +It seems that there is a “gain” problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +

        + +
        + +
        +

        +Why is there a double resonance at around 94Hz? +

        + +
        + + +
        +

        frf_dvf_plant_tf.png +

        +

        Figure 33: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

        +
        +
        +
        + +
        +
        8.5.2.3 Comparison with Interferometer
        +
        +
        +
        [dvf_int_sweep, ~]    = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.da,    win, [], [], 1/Ts);
        +[dvf_int_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts);
        +
        +
        + +
        +

        +Clearly using the encoder like this will not be useful. +

        + +

        +Probably, the encoders will have to be fixed on the plates so that the resonances of the APA are not a problem anymore. +

        + +
        +
        +
        + +
        +
        8.5.2.4 APA Resonances Frequency
        +
        +

        +This is very close to what was estimated using the FEM. +

        + + +
        +

        mode_bending_x.gif +

        +

        Figure 34: X-bending mode (189Hz)

        +
        + + +
        +

        mode_bending_y.gif +

        +

        Figure 35: Y-bending mode (285Hz)

        +
        + + +
        +

        mode_torsion_z.gif +

        +

        Figure 36: Z-torsion mode (400Hz)

        +
        + +
        +

        +The resonances are indeed due to limited stiffness of the APA. +

        + +
        +
        +
        + +
        +
        8.5.2.5 FRF Identification - IFF
        +
        +

        +In this section, the dynamics from \(V_a\) to \(V_s\) is identified. +

        + +

        +First the coherence is computed and shown in Figure 25. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). +

        + +
        +
        [coh_enc_sweep,    ~] = mscohere(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
        +[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_iff_plant_coh.png +

        +

        Figure 37: Obtained coherence for the IFF plant

        +
        + +

        +Then the FRF are estimated and shown in Figure 26 +

        +
        +
        [iff_enc_sweep,    ~] = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
        +[iff_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts);
        +
        +
        + + +
        +

        frf_iff_plant_tf.png +

        +

        Figure 38: Identified IFF Plant

        +
        +
        +
        + +
        +
        8.5.2.6 Comparison to when the encoder is not fixed
        +
        +
        +
        [iff_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.Vs,    win, [], [], 1/Ts);
        +[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts);
        +
        +
        + +
        +

        +We can see that the IFF does not change whether of not the encoder are fixed to the struts. +

        + +
        +
        +
        +
        +
        +
        + +
        +

        9 Measurement Results

        +
        + +
        +

        10 Test Bench APA300ML - Simscape Model

        +
        +
        +
        +

        10.1 Introduction

        +
        +
        +

        10.2 Nano Hexapod object

        +
        n_hexapod = struct();
         
        -
        -

        9.2.1 APA - 2 DoF

        -
        +
        +

        10.2.1 APA - 2 DoF

        +
        n_hexapod.actuator = struct();
         
        @@ -953,9 +2383,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        9.2.2 APA - Flexible Frame

        -
        +
        +

        10.2.2 APA - Flexible Frame

        +
        n_hexapod.actuator.type = 2;
         
        @@ -974,9 +2404,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        9.2.3 APA - Fully Flexible

        -
        +
        +

        10.2.3 APA - Fully Flexible

        +
        n_hexapod.actuator.type = 3;
         
        @@ -993,9 +2423,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        9.3 Identification

        -
        +
        +

        10.3 Identification

        +
        %% Options for Linearized
         options = linearizeOptions;
        @@ -1020,13 +2450,13 @@ Ga.OutputName = {'Vs', 
         
        -
        -

        9.4 Compare 2-DoF with flexible

        -
        +
        +

        10.4 Compare 2-DoF with flexible

        +
        -
        -

        9.4.1 APA - 2 DoF

        -
        +
        +

        10.4.1 APA - 2 DoF

        +
        n_hexapod = struct();
         
        @@ -1058,9 +2488,9 @@ G_2dof.OutputName = {'Vs', 
        -

        9.4.2 APA - Fully Flexible

        -
        +
        +

        10.4.2 APA - Fully Flexible

        +
        n_hexapod = struct();
         
        @@ -1085,31 +2515,31 @@ G_flex.OutputName = {'Vs', 
        -

        9.4.3 Comparison

        +
        +

        10.4.3 Comparison

        -
        -

        10 Test Bench Struts - Simscape Model

        -
        +
        +

        11 Test Bench Struts - Simscape Model

        +
        -
        -

        10.1 Introduction

        +
        +

        11.1 Introduction

        -
        -

        10.2 Nano Hexapod object

        -
        +
        +

        11.2 Nano Hexapod object

        +
        n_hexapod = struct();
         
        -
        -

        10.2.1 Flexible Joint - Bot

        -
        +
        +

        11.2.1 Flexible Joint - Bot

        +
        n_hexapod.flex_bot = struct();
         
        @@ -1129,9 +2559,9 @@ n_hexapod.flex_bot.cz  = ones(6,1)*1e2; 
         
        -
        -

        10.2.2 Flexible Joint - Top

        -
        +
        +

        11.2.2 Flexible Joint - Top

        +
        n_hexapod.flex_top = struct();
         
        @@ -1151,9 +2581,9 @@ n_hexapod.flex_top.cz  = ones(6,1)*1e2; 
         
        -
        -

        10.2.3 APA - 2 DoF

        -
        +
        +

        11.2.3 APA - 2 DoF

        +
        n_hexapod.actuator = struct();
         
        @@ -1176,9 +2606,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        10.2.4 APA - Flexible Frame

        -
        +
        +

        11.2.4 APA - Flexible Frame

        +
        n_hexapod.actuator.type = 2;
         
        @@ -1197,9 +2627,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        10.2.5 APA - Fully Flexible

        -
        +
        +

        11.2.5 APA - Fully Flexible

        +
        n_hexapod.actuator.type = 3;
         
        @@ -1217,9 +2647,9 @@ n_hexapod.actuator.Gs = ones(6,1)*1; 
        -

        10.3 Identification

        -
        +
        +

        11.3 Identification

        +
        %% Options for Linearized
         options = linearizeOptions;
        @@ -1244,13 +2674,13 @@ Gs.OutputName = {'Vs', 
         
        -
        -

        10.4 Compare flexible joints

        -
        +
        +

        11.4 Compare flexible joints

        +
        -
        -

        10.4.1 Perfect

        -
        +
        +

        11.4.1 Perfect

        +
        n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
         n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
        @@ -1266,9 +2696,9 @@ Gp.OutputName = {'Vs', 
         
        -
        -

        10.4.2 Top Flexible

        -
        +
        +

        11.4.2 Top Flexible

        +
        n_hexapod.flex_bot.type = 1; % 1: 2dof / 2: 3dof / 3: 4dof
         n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
        @@ -1284,9 +2714,9 @@ Gt.OutputName = {'Vs', 
         
        -
        -

        10.4.3 Bottom Flexible

        -
        +
        +

        11.4.3 Bottom Flexible

        +
        n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
         n_hexapod.flex_top.type = 2; % 1: 2dof / 2: 3dof / 3: 4dof
        @@ -1302,9 +2732,9 @@ Gb.OutputName = {'Vs', 
         
        -
        -

        10.4.4 Both Flexible

        -
        +
        +

        11.4.4 Both Flexible

        +
        n_hexapod.flex_bot.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
         n_hexapod.flex_top.type = 3; % 1: 2dof / 2: 3dof / 3: 4dof
        @@ -1320,46 +2750,46 @@ Gf.OutputName = {'Vs', 
         
        -
        -

        10.4.5 Comparison

        +
        +

        11.4.5 Comparison

        -
        -

        11 Resonance frequencies - APA300ML

        -
        +
        +

        12 Resonance frequencies - APA300ML

        +
        -
        -

        11.1 Introduction

        -
        +
        +

        12.1 Introduction

        +

        Three main resonances are foreseen to be problematic for the control of the APA300ML:

          -
        • Mode in X-bending at 189Hz (Figure 16)
        • -
        • Mode in Y-bending at 285Hz (Figure 17)
        • -
        • Mode in Z-torsion at 400Hz (Figure 18)
        • +
        • Mode in X-bending at 189Hz (Figure 34)
        • +
        • Mode in Y-bending at 285Hz (Figure 35)
        • +
        • Mode in Z-torsion at 400Hz (Figure 36)
        -
        +

        mode_bending_x.gif

        -

        Figure 16: X-bending mode (189Hz)

        +

        Figure 39: X-bending mode (189Hz)

        -
        +

        mode_bending_y.gif

        -

        Figure 17: Y-bending mode (285Hz)

        +

        Figure 40: Y-bending mode (285Hz)

        -
        +

        mode_torsion_z.gif

        -

        Figure 18: Z-torsion mode (400Hz)

        +

        Figure 41: Z-torsion mode (400Hz)

        @@ -1372,16 +2802,16 @@ In this section, we try to find the resonance frequency of these modes when one

        -
        -

        11.2 Setup

        -
        +
        +

        12.2 Setup

        +

        -The measurement setup is shown in Figure 19. +The measurement setup is shown in Figure 42. A Laser vibrometer is measuring the difference of motion of two points. The APA is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.

        -
        +
        • Laser Doppler Vibrometer Polytec OFV512
        • Instrumented hammer
        • @@ -1390,28 +2820,28 @@ The APA is excited with an instrumented hammer and the transfer function from th
        -
        -

        measurement_setup_torsion.png +

        +

        measurement_setup_torsion.jpg

        -

        Figure 19: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer

        +

        Figure 42: Measurement setup with a Laser Doppler Vibrometer and one instrumental hammer

        -
        -

        11.3 Bending - X

        -
        +
        +

        12.3 Bending - X

        +

        -The setup to measure the X-bending motion is shown in Figure 20. +The setup to measure the X-bending motion is shown in Figure 43. The APA is excited with an instrumented hammer having a solid metallic tip. The impact point is on the back-side of the APA aligned with the top measurement point.

        -
        -

        measurement_setup_X_bending.png +

        +

        measurement_setup_X_bending.jpg

        -

        Figure 20: X-Bending measurement setup

        +

        Figure 43: X-Bending measurement setup

        @@ -1440,26 +2870,26 @@ The transfer function from the input force to the output “rotation”

        -The result is shown in Figure 21. +The result is shown in Figure 44.

        The can clearly observe a nice peak at 280Hz, and then peaks at the odd “harmonics” (third “harmonic” at 840Hz, and fifth “harmonic” at 1400Hz).

        -
        +

        apa300ml_meas_freq_bending_x.png

        -

        Figure 21: Obtained FRF for the X-bending

        +

        Figure 44: Obtained FRF for the X-bending

        -
        -

        11.4 Bending - Y

        -
        +
        +

        12.4 Bending - Y

        +

        -The setup to measure the Y-bending is shown in Figure 22. +The setup to measure the Y-bending is shown in Figure 45.

        @@ -1467,10 +2897,10 @@ The impact point of the instrumented hammer is located on the back surface of th

        -
        -

        measurement_setup_Y_bending.png +

        +

        measurement_setup_Y_bending.jpg

        -

        Figure 22: Y-Bending measurement setup

        +

        Figure 45: Y-Bending measurement setup

        @@ -1483,24 +2913,24 @@ The data is loaded, and the transfer function from the force to the measured rot

        -The results are shown in Figure 23. +The results are shown in Figure 46. The main resonance is at 412Hz, and we also see the third “harmonic” at 1220Hz.

        -
        +

        apa300ml_meas_freq_bending_y.png

        -

        Figure 23: Obtained FRF for the Y-bending

        +

        Figure 46: Obtained FRF for the Y-bending

        -
        -

        11.5 Torsion - Z

        -
        +
        +

        12.5 Torsion - Z

        +

        -Finally, we measure the Z-torsion resonance as shown in Figure 24. +Finally, we measure the Z-torsion resonance as shown in Figure 47.

        @@ -1508,10 +2938,10 @@ The excitation is shown on the other side of the APA, on the side to excite the

        -
        -

        measurement_setup_torsion_bis.png +

        +

        measurement_setup_torsion_bis.jpg

        -

        Figure 24: Z-Torsion measurement setup

        +

        Figure 47: Z-Torsion measurement setup

        @@ -1524,16 +2954,16 @@ The data is loaded, and the transfer function computed.

        -The results are shown in Figure 25. +The results are shown in Figure 48. We observe a first peak at 267Hz, which corresponds to the X-bending mode that was measured at 280Hz. And then a second peak at 415Hz, which corresponds to the X-bending mode that was measured at 412Hz. The mode in pure torsion is probably at higher frequency (peak around 1kHz?).

        -
        +

        apa300ml_meas_freq_torsion_z.png

        -

        Figure 25: Obtained FRF for the Z-torsion

        +

        Figure 48: Obtained FRF for the Z-torsion

        @@ -1546,37 +2976,37 @@ In order to verify that, the APA is excited on the top part such that the torsio

        -The two FRF are compared in Figure 26. +The two FRF are compared in Figure 49. It is clear that the first two modes does not correspond to the torsional mode. Maybe the resonance at 800Hz, or even higher resonances. It is difficult to conclude here.

        -
        +

        apa300ml_meas_freq_torsion_z_comp.png

        -

        Figure 26: Obtained FRF for the Z-torsion

        +

        Figure 49: Obtained FRF for the Z-torsion

        -
        -

        11.6 Compare

        -
        +
        +

        12.6 Compare

        +

        -The three measurements are shown in Figure 27. +The three measurements are shown in Figure 50.

        -
        +

        apa300ml_meas_freq_compare.png

        -

        Figure 27: Obtained FRF - Comparison

        +

        Figure 50: Obtained FRF - Comparison

        -
        -

        11.7 Conclusion

        -
        +
        +

        12.7 Conclusion

        +

        When two flexible joints are fixed at each ends of the APA, the APA is mostly in a free/free condition in terms of bending/torsion (the bending/torsional stiffness of the joints being very small).

        @@ -1587,8 +3017,8 @@ Therefore, it is quite obvious that we measured higher resonance frequencies tha It is however quite interesting that there is a factor \(\approx \sqrt{2}\) between the two (increased of the stiffness by a factor 2?).

        - - +
        Table 4: Measured frequency of the modes
        +@@ -1628,6 +3058,363 @@ It is however quite interesting that there is a factor \(\approx \sqrt{2}\) betw +
        +

        13 Function

        +
        +
        +
        +

        13.1 generateSweepExc: Generate sweep sinus excitation

        +
        +

        + +

        +
        + +
        +

        Function description

        +
        +
        +
        function [U_exc] = generateSweepExc(args)
        +% generateSweepExc - Generate a Sweep Sine excitation signal
        +%
        +% Syntax: [U_exc] = generateSweepExc(args)
        +%
        +% Inputs:
        +%    - args - Optinal arguments:
        +%        - Ts              - Sampling Time                                              - [s]
        +%        - f_start         - Start frequency of the sweep                               - [Hz]
        +%        - f_end           - End frequency of the sweep                                 - [Hz]
        +%        - V_mean          - Mean value of the excitation voltage                       - [V]
        +%        - V_exc           - Excitation Amplitude for the Sweep, could be numeric or TF - [V]
        +%        - t_start         - Time at which the sweep begins                             - [s]
        +%        - exc_duration    - Duration of the sweep                                      - [s]
        +%        - sweep_type      - 'logarithmic' or 'linear'                                  - [-]
        +%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]
        +
        +
        +
        +
        + +
        +

        Optional Parameters

        +
        +
        +
        arguments
        +    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
        +    args.f_start         (1,1) double  {mustBeNumeric, mustBePositive} = 1
        +    args.f_end           (1,1) double  {mustBeNumeric, mustBePositive} = 1e3
        +    args.V_mean          (1,1) double  {mustBeNumeric} = 0
        +    args.V_exc                                         = 1
        +    args.t_start         (1,1) double  {mustBeNumeric, mustBeNonnegative} = 5
        +    args.exc_duration    (1,1) double  {mustBeNumeric, mustBePositive} = 10
        +    args.sweep_type            char    {mustBeMember(args.sweep_type,{'log', 'lin'})} = 'lin'
        +    args.smooth_ends           logical {mustBeNumericOrLogical} = true
        +end
        +
        +
        +
        +
        + +
        +

        Sweep Sine part

        +
        +
        +
        t_sweep = 0:args.Ts:args.exc_duration;
        +
        +if strcmp(args.sweep_type, 'log')
        +    V_exc = sin(2*pi*args.f_start * args.exc_duration/log(args.f_end/args.f_start) * (exp(log(args.f_end/args.f_start)*t_sweep/args.exc_duration) - 1));
        +elseif strcmp(args.sweep_type, 'lin')
        +    V_exc = sin(2*pi*(args.f_start + (args.f_end - args.f_start)/2/args.exc_duration*t_sweep).*t_sweep);
        +else
        +    error('sweep_type should either be equal to "log" or to "lin"');
        +end
        +
        +
        + +
        +
        if isnumeric(args.V_exc)
        +    V_sweep = args.V_mean + args.V_exc*V_exc;
        +elseif isct(args.V_exc)
        +    if strcmp(args.sweep_type, 'log')
        +        V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start*(args.f_end/args.f_start).^(t_sweep/args.exc_duration), 'Hz')))'.*V_exc;
        +    elseif strcmp(args.sweep_type, 'lin')
        +        V_sweep = args.V_mean + abs(squeeze(freqresp(args.V_exc, args.f_start+(args.f_end-args.f_start)/args.exc_duration*t_sweep, 'Hz')))'.*V_exc;
        +    end
        +end
        +
        +
        +
        +
        + +
        +

        Smooth Ends

        +
        +
        +
        if args.t_start > 0
        +    t_smooth_start = args.Ts:args.Ts:args.t_start;
        +
        +    V_smooth_start = zeros(size(t_smooth_start));
        +    V_smooth_end   = zeros(size(t_smooth_start));
        +
        +    if args.smooth_ends
        +        Vd_max = args.V_mean/(0.7*args.t_start);
        +
        +        V_d = zeros(size(t_smooth_start));
        +        V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
        +        V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
        +        V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
        +
        +        V_smooth_start = cumtrapz(V_d)*args.Ts;
        +
        +        V_smooth_end = args.V_mean - V_smooth_start;
        +    end
        +else
        +    V_smooth_start = [];
        +    V_smooth_end = [];
        +end
        +
        +
        +
        +
        + +
        +

        Combine Excitation signals

        +
        +
        +
        V_exc = [V_smooth_start, V_sweep, V_smooth_end];
        +t_exc = args.Ts*[0:1:length(V_exc)-1];
        +
        +
        + +
        +
        U_exc = [t_exc; V_exc];
        +
        +
        +
        +
        +
        + +
        +

        13.2 generateShapedNoise: Generate Shaped Noise excitation

        +
        +

        + +

        +
        + +
        +

        Function description

        +
        +
        +
        function [U_exc] = generateShapedNoise(args)
        +% generateShapedNoise - Generate a Shaped Noise excitation signal
        +%
        +% Syntax: [U_exc] = generateShapedNoise(args)
        +%
        +% Inputs:
        +%    - args - Optinal arguments:
        +%        - Ts              - Sampling Time                                              - [s]
        +%        - V_mean          - Mean value of the excitation voltage                       - [V]
        +%        - V_exc           - Excitation Amplitude, could be numeric or TF               - [V rms]
        +%        - t_start         - Time at which the noise begins                             - [s]
        +%        - exc_duration    - Duration of the noise                                      - [s]
        +%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]
        +
        +
        +
        +
        + +
        +

        Optional Parameters

        +
        +
        +
        arguments
        +    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
        +    args.V_mean          (1,1) double  {mustBeNumeric} = 0
        +    args.V_exc                                         = 1
        +    args.t_start         (1,1) double  {mustBeNumeric, mustBePositive} = 5
        +    args.exc_duration    (1,1) double  {mustBeNumeric, mustBePositive} = 10
        +    args.smooth_ends           logical {mustBeNumericOrLogical} = true
        +end
        +
        +
        +
        +
        + +
        +

        Shaped Noise

        +
        +
        +
        t_noise = 0:args.Ts:args.exc_duration;
        +
        +
        +
        + +
        +
        if isnumeric(args.V_exc)
        +    V_noise = args.V_mean + args.V_exc*sqrt(1/args.Ts/2)*randn(length(t_noise), 1)';
        +elseif isct(args.V_exc)
        +    V_noise = args.V_mean + lsim(args.V_exc, sqrt(1/args.Ts/2)*randn(length(t_noise), 1), t_noise)';
        +end
        +
        +
        +
        +
        + +
        +

        Smooth Ends

        +
        +
        +
        t_smooth_start = args.Ts:args.Ts:args.t_start;
        +
        +V_smooth_start = zeros(size(t_smooth_start));
        +V_smooth_end   = zeros(size(t_smooth_start));
        +
        +if args.smooth_ends
        +    Vd_max = args.V_mean/(0.7*args.t_start);
        +
        +    V_d = zeros(size(t_smooth_start));
        +    V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
        +    V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
        +    V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
        +
        +    V_smooth_start = cumtrapz(V_d)*args.Ts;
        +
        +    V_smooth_end = args.V_mean - V_smooth_start;
        +end
        +
        +
        +
        +
        + +
        +

        Combine Excitation signals

        +
        +
        +
        V_exc = [V_smooth_start, V_noise, V_smooth_end];
        +t_exc = args.Ts*[0:1:length(V_exc)-1];
        +
        +
        + +
        +
        U_exc = [t_exc; V_exc];
        +
        +
        +
        +
        +
        + +
        +

        13.3 generateSinIncreasingAmpl: Generate Sinus with increasing amplitude

        +
        +

        + +

        +
        + +
        +

        Function description

        +
        +
        +
        function [U_exc] = generateSinIncreasingAmpl(args)
        +% generateSinIncreasingAmpl - Generate Sinus with increasing amplitude
        +%
        +% Syntax: [U_exc] = generateSinIncreasingAmpl(args)
        +%
        +% Inputs:
        +%    - args - Optinal arguments:
        +%        - Ts              - Sampling Time                                              - [s]
        +%        - V_mean          - Mean value of the excitation voltage                       - [V]
        +%        - sin_ampls       - Excitation Amplitudes                                      - [V]
        +%        - sin_freq        - Excitation Frequency                                       - [Hz]
        +%        - sin_num         - Number of period for each amplitude                        - [-]
        +%        - t_start         - Time at which the excitation begins                        - [s]
        +%        - smooth_ends     - 'true' or 'false': smooth transition between 0 and V_mean  - [-]
        +
        +
        +
        +
        + +
        +

        Optional Parameters

        +
        +
        +
        arguments
        +    args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
        +    args.V_mean          (1,1) double  {mustBeNumeric} = 0
        +    args.sin_ampls             double  {mustBeNumeric, mustBePositive} = [0.1, 0.2, 0.3]
        +    args.sin_period      (1,1) double  {mustBeNumeric, mustBePositive} = 1
        +    args.sin_num         (1,1) double  {mustBeNumeric, mustBePositive, mustBeInteger} = 3
        +    args.t_start         (1,1) double  {mustBeNumeric, mustBePositive} = 5
        +    args.smooth_ends           logical {mustBeNumericOrLogical} = true
        +end
        +
        +
        +
        +
        + +
        +

        Sinus excitation

        +
        +
        +
        t_noise = 0:args.Ts:args.sin_period*args.sin_num;
        +sin_exc = [];
        +
        +
        + +
        +
        for sin_ampl = args.sin_ampls
        +    sin_exc = [sin_exc, args.V_mean + sin_ampl*sin(2*pi/args.sin_period*t_noise)];
        +end
        +
        +
        +
        +
        + +
        +

        Smooth Ends

        +
        +
        +
        t_smooth_start = args.Ts:args.Ts:args.t_start;
        +
        +V_smooth_start = zeros(size(t_smooth_start));
        +V_smooth_end   = zeros(size(t_smooth_start));
        +
        +if args.smooth_ends
        +    Vd_max = args.V_mean/(0.7*args.t_start);
        +
        +    V_d = zeros(size(t_smooth_start));
        +    V_d(t_smooth_start < 0.2*args.t_start) = t_smooth_start(t_smooth_start < 0.2*args.t_start)*Vd_max/(0.2*args.t_start);
        +    V_d(t_smooth_start > 0.2*args.t_start & t_smooth_start < 0.7*args.t_start) = Vd_max;
        +    V_d(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) = Vd_max - (t_smooth_start(t_smooth_start > 0.7*args.t_start & t_smooth_start < 0.9*args.t_start) - 0.7*args.t_start)*Vd_max/(0.2*args.t_start);
        +
        +    V_smooth_start = cumtrapz(V_d)*args.Ts;
        +
        +    V_smooth_end = args.V_mean - V_smooth_start;
        +end
        +
        +
        +
        +
        + +
        +

        Combine Excitation signals

        +
        +
        +
        V_exc = [V_smooth_start, sin_exc, V_smooth_end];
        +t_exc = args.Ts*[0:1:length(V_exc)-1];
        +
        +
        + +
        +
        U_exc = [t_exc; V_exc];
        +
        +
        +
        +
        +
        +
        +

        Bibliography

        Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” CEAS Space Journal 10 (2). Springer:157–65.
        @@ -1635,7 +3422,7 @@ It is however quite interesting that there is a factor \(\approx \sqrt{2}\) betw

        Author: Dehaeze Thomas

        -

        Created: 2021-05-06 jeu. 21:55

        +

        Created: 2021-06-02 mer. 23:58

        diff --git a/test-bench-apa300ml.org b/test-bench-apa300ml.org index c227ce6..1652ac5 100644 --- a/test-bench-apa300ml.org +++ b/test-bench-apa300ml.org @@ -732,7 +732,6 @@ addpath('./src/'); #+end_src #+begin_src matlab :eval no :exports none - addpath('./src/'); #+end_src @@ -895,6 +894,7 @@ save('./frf_data.mat', 'Fs', 'Ts', 'Tsim', 'Trec_start', 'Trec_dur', 'V_exc'); ** =frf_save.m= - Save Data :PROPERTIES: :header-args: :tangle matlab/frf_save.m +:header-args:matlab+: :comments no :END: First, we get data from the Speedgoat: @@ -1124,7 +1124,7 @@ linkaxes([ax1,ax2],'x'); xlim([1, 2e3]); #+end_src -*** Second identification with Sweep +*** Second identification with Sweep and high frequency noise #+begin_src matlab load(sprintf('mat/frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'da', 'de') #+end_src @@ -1171,6 +1171,68 @@ xlim([1e1, 2e3]); [coh_iff, ~] = mscohere(Va, Vs, win, [], [], 1/Ts); #+end_src +#+begin_src matlab +load(sprintf('mat/frf_data_%i_noise_high_freq.mat', 1), 't', 'Va', 'Vs', 'da', 'de') +#+end_src + +#+begin_src matlab +[phf_da, f] = pwelch(da, win, [], [], 1/Ts); +[phf_de, ~] = pwelch(de, win, [], [], 1/Ts); +[phf_Vs, ~] = pwelch(Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab +[cohhf_dvf, ~] = mscohere(Va, de, win, [], [], 1/Ts); +[cohhf_d, ~] = mscohere(Va, da, win, [], [], 1/Ts); +[cohhf_iff, ~] = mscohere(Va, Vs, win, [], [], 1/Ts); + +[Ghf_dvf, f] = tfestimate(Va, de, win, [], [], 1/Ts); +[Ghf_d, ~] = tfestimate(Va, da, win, [], [], 1/Ts); +[Ghf_iff, ~] = tfestimate(Va, Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +%% +figure; +hold on; +plot(f, coh_dvf); +plot(f, cohhf_dvf); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlim([10, 2e3]); ylim([0, 1]); +#+end_src + + +#+begin_src matlab :exports none +%% +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f<350), abs(G_dvf( f<350)), 'DisplayName', 'Attocube'); +plot(f(f>350), abs(Ghf_dvf(f>350)), 'DisplayName', 'Encoder'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-8, 1e-3]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +plot(f(f<350), 180/pi*angle(G_dvf( f<350))); +plot(f(f>350), 180/pi*angle(Ghf_dvf(f>350))); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + #+begin_src matlab :exports none %% figure; @@ -1235,7 +1297,7 @@ linkaxes([ax1,ax2],'x'); xlim([10, 2e3]); #+end_src -*** Extract Parameters +*** Extract Parameters (Actuator/Sensor constants) Quasi static gain between $d$ and $V_a$: #+begin_src matlab @@ -1261,10 +1323,8 @@ sprintf('g_Vs_Va = %.1e [V/V]', g_Vs_Va) #+RESULTS: : g_Vs_Va = 5.7e-01 [V/V] -** Comparison of all APA -:PROPERTIES: -:header-args: :tangle matlab/frf_analyze.m -:END: +** Comparison of all the APA +*** Introduction :ignore: *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1276,99 +1336,1190 @@ sprintf('g_Vs_Va = %.1e [V/V]', g_Vs_Va) #+end_src #+begin_src matlab :tangle no +addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no +addpath('./mat/'); addpath('./src/'); #+end_src -*** Test with one APA +*** Stiffness - Comparison of the APA +In order to estimate the stiffness of the APA, a weight with known mass $m_a$ is added on top of the suspended granite and the deflection $d_e$ is measured using an encoder. +The APA stiffness is then: +\begin{equation} + k_{\text{apa}} = \frac{m_a g}{d} +\end{equation} + +Here are the number of the APA that have been measured: #+begin_src matlab -%% Load measurement data for APA number 1 -load(sprintf('mat/frf_data_%i.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +apa_nums = [1 2 4 5 6 7 8]; #+end_src -Time domain data: +The data are loaded. #+begin_src matlab +apa_mass = {}; +for i = 1:length(apa_nums) + apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')}; + % The initial displacement is set to zero + apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11)); +end +#+end_src + +The raw measurements are shown in Figure [[fig:apa_meas_k_time]]. +All the APA seems to have similar stiffness except the APA 7 which should have an higher stiffness. + +#+begin_question +It is however strange that the displacement $d_e$ when the mass is removed is higher for the APA 7 than for the other APA. +What could cause that? +#+end_question + +#+begin_src matlab :exports none figure; -plot(t, de); +hold on; +for i = 1:length(apa_nums) + plot(apa_mass{i}.t, apa_mass{i}.de, 'DisplayName', sprintf('APA %i', apa_nums(i))); +end +hold off; +xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]'); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); #+end_src -Compute transfer functions: -#+begin_src matlab -Ts = (t(end) - t(1))/(length(t)-1); -Fs = 1/Ts; +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/apa_meas_k_time.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src +#+name: fig:apa_meas_k_time +#+caption: Raw measurements for all the APA. A mass of 6.4kg is added at arround 15s and removed at arround 22s +#+RESULTS: +[[file:figs/apa_meas_k_time.png]] + +#+begin_src matlab +added_mass = 6.4; % Added mass [kg] +#+end_src + +The stiffnesses are computed for all the APA and are summarized in Table [[tab:APA_measured_k]]. + +#+begin_src matlab :exports none +apa_k = zeros(length(apa_nums), 1); +for i = 1:length(apa_nums) + apa_k(i) = 9.8 * added_mass / (mean(apa_mass{i}.de(apa_mass{i}.t > 12 & apa_mass{i}.t < 12.5)) - mean(apa_mass{i}.de(apa_mass{i}.t > 20 & apa_mass{i}.t < 20.5))); +end +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable(1e-6*apa_k, cellstr(num2str(apa_nums')), {'APA Num', '$k [N/\mu m]$'}, ' %.2f '); +#+end_src + +#+name: tab:APA_measured_k +#+caption: Measured stiffnesses +#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| APA Num | $k [N/\mu m]$ | +|---------+---------------| +| 1 | 1.68 | +| 2 | 1.69 | +| 4 | 1.7 | +| 5 | 1.7 | +| 6 | 1.7 | +| 7 | 1.93 | +| 8 | 1.73 | + +#+begin_important +The APA300ML manual specifies the nominal stiffness to be $1.8\,[N/\mu m]$ which is very close to what have been measured. +Only the APA number 7 is a little bit off. +#+end_important + +*** Stiffness - Effect of connecting the actuator stack to the amplifier and the sensor stack to the ADC +We wish here to see if the stiffness changes when the actuator stacks are not connected to the amplifier and the sensor stacks are not connected to the ADC. + +Note here that the resistor in parallel to the sensor stack is present in both cases. + +First, the data are loaded. +#+begin_src matlab +add_mass_oc = load(sprintf('frf_data_%i_add_mass_open_circuit.mat', 1), 't', 'de'); +add_mass_cc = load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', 1), 't', 'de'); +#+end_src + +And the initial displacement is set to zero. +#+begin_src matlab +add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11)); +add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11)); +#+end_src + +#+begin_src matlab :exports none +figure; +hold on; +plot(add_mass_oc.t, add_mass_oc.de, 'DisplayName', 'Not connected'); +plot(add_mass_cc.t, add_mass_cc.de, 'DisplayName', 'Connected'); +hold off; +xlabel('Time [s]'); ylabel('Displacement $d_e$ [m]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/apa_meas_k_time_oc_cc.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:apa_meas_k_time_oc_cc +#+caption: Measured displacement +#+RESULTS: +[[file:figs/apa_meas_k_time_oc_cc.png]] + + +And the stiffness is estimated in both case. +The results are shown in Table [[tab:APA_measured_k_oc_cc]]. + +#+begin_src matlab +apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5))); +apa_k_cc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5))); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(1e-6*[apa_k_oc; apa_k_cc], {'Not connected', 'Connected'}, {'$k [N/\mu m]$'}, ' %.1f '); +#+end_src + +#+name: tab:APA_measured_k_oc_cc +#+caption: Measured stiffnesses on "open" and "closed" circuits +#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | $k [N/\mu m]$ | +|---------------+---------------| +| Not connected | 2.3 | +| Connected | 1.7 | + +#+begin_important +Clearly, connecting the actuator stacks to the amplified (basically equivalent as to short circuiting them) lowers the stiffness. +#+end_important + +*** Hysteresis +We here wish to visually see the amount of hysteresis present in the APA. + +The data is loaded. +#+begin_src matlab +apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'Va', 'de'); +% Initial time set to zero +apa_hyst.t = apa_hyst.t - apa_hyst.t(1); +#+end_src + +The excitation voltage amplitudes are: +#+begin_src matlab +ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes +#+end_src + +The excitation voltage and the measured displacement are shown in Figure [[fig:hyst_exc_signal_time]]. +#+begin_src matlab :exports none +figure; +tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +plot(apa_hyst.t, apa_hyst.Va) +xlabel('Time [s]'); ylabel('Output Voltage [V]'); + +ax2 = nexttile; +plot(apa_hyst.t, apa_hyst.de) +xlabel('Time [s]'); ylabel('Measured Displacement [m]'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/hyst_exc_signal_time.pdf', 'width', 'full', 'height', 'normal'); +#+end_src + +#+name: fig:hyst_exc_signal_time +#+caption: Excitation voltage and measured displacement +#+RESULTS: +[[file:figs/hyst_exc_signal_time.png]] + +For each amplitude, we only take the last sinus in order to reduce possible transients. +Also, it is centered on zero. + +The measured displacement at a function of the output voltage are shown in Figure [[fig:hyst_results_multi_ampl]]. +#+begin_src matlab :exports none +figure; +tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([1,2]); +hold on; +for i = flip(1:6) + i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5; + plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim)), ... + 'DisplayName', sprintf('$V_a = %.1f [V]$', ampls(i))) +end +hold off; +xlabel('Output Voltage [V]'); ylabel('Measured Displacement [m]'); +legend('location', 'northeast'); +xlim([-4, 4]); ylim([-1.2e-4, 1.2e-4]); + +ax2 = nexttile; +hold on; +for i = flip(1:6) + i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5; + plot(apa_hyst.Va(i_lim) - mean(apa_hyst.Va(i_lim)), apa_hyst.de(i_lim) - mean(apa_hyst.de(i_lim))) +end +hold off; +xlim([-0.4, 0.4]); ylim([-0.8e-5, 0.8e-5]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/hyst_results_multi_ampl.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:hyst_results_multi_ampl +#+caption: Obtained hysteresis for multiple excitation amplitudes +#+RESULTS: +[[file:figs/hyst_results_multi_ampl.png]] + +#+begin_important +It is quite clear that hysteresis is increasing with the excitation amplitude. + +Also, no hysteresis is found on the sensor stack voltage. +#+end_important + +*** FRF Identification - Setup +The identification is performed in three steps: +1. White noise excitation with small amplitude. + This is used to determine the main resonance of the system. +2. Sweep sine excitation with the amplitude lowered around the resonance. + The sweep sine is from 10Hz to 400Hz. +3. High frequency noise. + The noise is band-passed between 300Hz and 2kHz. + +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. + +Here are the APA numbers that have been measured. +#+begin_src matlab +apa_nums = [1 2 4 5 6 7 8]; +#+end_src + +The data are loaded for both the second and third identification: +#+begin_src matlab +%% Second identification +apa_sweep = {}; +for i = 1:length(apa_nums) + apa_sweep(i) = {load(sprintf('frf_data_%i_sweep.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +end + +%% Third identification +apa_noise_hf = {}; +for i = 1:length(apa_nums) + apa_noise_hf(i) = {load(sprintf('frf_data_%i_noise_hf.mat', apa_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +end +#+end_src + +The time is the same for all measurements. +#+begin_src matlab +%% Time vector +t = apa_sweep{1}.t - apa_sweep{1}.t(1) ; % Time vector [s] + +%% Sampling +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] +Fs = 1/Ts; % Sampling Frequency [Hz] +#+end_src + +Then we defined a "Hanning" windows that will be used for the spectral analysis: +#+begin_src matlab win = hanning(ceil(0.5*Fs)); % Hannning Windows #+end_src +We get the frequency vector that will be the same for all the frequency domain analysis. #+begin_src matlab -[G_dvf, f] = tfestimate(Va, de, win, [], [], 1/Ts); -[G_d, ~] = tfestimate(Va, da, win, [], [], 1/Ts); -[G_iff, ~] = tfestimate(Va, Vs, win, [], [], 1/Ts); +% Only used to have the frequency vector "f" +[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts); #+end_src -#+begin_src matlab :exports none -figure; -tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); +*** FRF Identification - DVF +In this section, the dynamics from $V_a$ to $d_e$ is identified. -ax1 = nexttile; -hold on; -plot(f, abs(G_dvf)); -plot(f, abs(G_d)); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([10, 30]); +We compute the coherence for 2nd and 3rd identification: +#+begin_src matlab +%% Coherence computation +coh_sweep = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts); + coh_sweep(:, i) = coh; +end -ax2 = nexttile; -hold on; -plot(f, 180/pi*angle(G_dvf)); -plot(f, 180/pi*angle(G_d)); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); - -linkaxes([ax1,ax2],'x'); -xlim([5, 5e3]); -#+end_src - -#+begin_src matlab :exports none -figure; -tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile; -plot(f, abs(G_iff)); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]); -hold off; -ylim([10, 30]); - -ax2 = nexttile; -plot(f, 180/pi*angle(G_iff)); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); - -linkaxes([ax1,ax2],'x'); -xlim([5, 5e3]); -#+end_src - -*** Comparison of all APA -#+begin_src matlab :exports none -%% Load all the measurements -meas_data = {}; -for i = 1:7 - meas_data(i) = {load(sprintf('mat/frf_data_%i.mat', i), 't', 'Va', 'Vs', 'de', 'da')}; +coh_noise_hf = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts); + coh_noise_hf(:, i) = coh; end #+end_src +The coherence is shown in Figure [[fig:frf_dvf_plant_coh]]. +It is clear that the Sweep sine gives good coherence up to 400Hz and that the high frequency noise excitation signal helps increasing a little bit the coherence at high frequency. + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ... + 'DisplayName', 'HF Noise'); +plot(f, coh_sweep(:, 1), 'color', [colors(2, :), 0.5], ... + 'DisplayName', 'Sweep'); +for i = 2:length(apa_nums) + plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ... + 'HandleVisibility', 'off'); + plot(f, coh_sweep(:, i), 'color', [colors(2, :), 0.5], ... + 'HandleVisibility', 'off'); +end; +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_dvf_plant_coh +#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ +#+RESULTS: +[[file:figs/frf_dvf_plant_coh.png]] + + +Then, the transfer function from the DAC output voltage $V_a$ to the measured displacement by the encoders is computed: +#+begin_src matlab +%% Transfer function estimation +dvf_sweep = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.de, win, [], [], 1/Ts); + dvf_sweep(:, i) = frf; +end + +dvf_noise_hf = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.de, win, [], [], 1/Ts); + dvf_noise_hf(:, i) = frf; +end +#+end_src + +The obtained transfer functions are shown in Figure [[fig:frf_dvf_plant_tf]]. + +They are all superimposed except for the APA7. + +#+begin_question +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. + +It seems that there is a "gain" problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +#+end_question + +#+begin_question +Why is there a double resonance at around 94Hz? +#+end_question + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), abs(dvf_noise_hf(f> 350, i)), 'color', colors(i, :), ... + 'DisplayName', sprintf('APA %i', apa_nums(i))); + plot(f(f<=350), abs(dvf_sweep( f<=350, i)), 'color', colors(i, :), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350, i)), 'color', colors(i, :)); + plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350, i)), 'color', colors(i, :)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_dvf_plant_tf +#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+RESULTS: +[[file:figs/frf_dvf_plant_tf.png]] + + +*** FRF Identification - IFF +In this section, the dynamics from $V_a$ to $V_s$ is identified. + +First the coherence is computed and shown in Figure [[fig:frf_iff_plant_coh]]. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). + +#+begin_src matlab +%% Coherence +coh_sweep = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [coh, ~] = mscohere(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts); + coh_sweep(:, i) = coh; +end + +coh_noise_hf = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [coh, ~] = mscohere(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts); + coh_noise_hf(:, i) = coh; +end +#+end_src + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], 'DisplayName', 'HF Noise'); +plot(f, coh_sweep( :, 1), 'color', [colors(2, :), 0.5], 'DisplayName', 'Sweep'); +for i = 2:length(apa_nums) + plot(f, coh_noise_hf(:, i), 'color', [colors(1, :), 0.5], ... + 'HandleVisibility', 'off'); + plot(f, coh_sweep( :, i), 'color', [colors(2, :), 0.5], ... + 'HandleVisibility', 'off'); +end; +hold off; +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_iff_plant_coh +#+caption: Obtained coherence for the IFF plant +#+RESULTS: +[[file:figs/frf_iff_plant_coh.png]] + +Then the FRF are estimated and shown in Figure [[fig:frf_iff_plant_tf]] +#+begin_src matlab +%% FRF estimation of the transfer function from Va to Vs +iff_sweep = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [frf, ~] = tfestimate(apa_sweep{i}.Va, apa_sweep{i}.Vs, win, [], [], 1/Ts); + iff_sweep(:, i) = frf; +end + +iff_noise_hf = zeros(length(f), length(apa_nums)); +for i = 1:length(apa_nums) + [frf, ~] = tfestimate(apa_noise_hf{i}.Va, apa_noise_hf{i}.Vs, win, [], [], 1/Ts); + iff_noise_hf(:, i) = frf; +end +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', colors(i, :), ... + 'DisplayName', sprintf('APA %i', apa_nums(i))); + plot(f(f<=350), abs(iff_sweep( f<=350, i)), 'color', colors(i, :), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', colors(i, :)); + plot(ff<=350), 180/pi*angle(iff_sweep( f<=350, i)), 'color', colors(i, :)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_iff_plant_tf +#+caption:Identified IFF Plant +#+RESULTS: +[[file:figs/frf_iff_plant_tf.png]] + +*** Effect of the resistor on the IFF Plant +A resistor is added in parallel with the sensor stack. +This has the effect to form a high pass filter with the capacitance of the stack. + +We here measured the low frequency transfer function from $V_a$ to $V_s$ with and without this resistor. + +#+begin_src matlab +% With the resistor +wi_k = load('frf_data_2_sweep_lf_with_R.mat', 't', 'Vs', 'Va'); + +% Without the resistor +wo_k = load('frf_data_2_sweep_lf.mat', 't', 'Vs', 'Va'); +#+end_src + +#+begin_src matlab +t = wo_k.t; % Time vector [s] + +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] +Fs = 1/Ts; % Sampling Frequency [Hz] +#+end_src + +We use a very long "Hanning" window for the spectral analysis in order to estimate the low frequency behavior. +#+begin_src matlab +win = hanning(ceil(50*Fs)); % Hannning Windows +#+end_src + +And we estimate the transfer function from $V_a$ to $V_s$ in both cases: +#+begin_src matlab +[frf_wo_k, f] = tfestimate(wo_k.Va, wo_k.Vs, win, [], [], 1/Ts); +[frf_wi_k, ~] = tfestimate(wi_k.Va, wi_k.Vs, win, [], [], 1/Ts); +#+end_src + + +#+begin_src matlab +f0 = 0.35; +G_hpf = 0.6*(s/2*pi*f0)/(1 + s/2*pi*f0); +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f, abs(frf_wo_k)); +plot(f, abs(frf_wi_k)); +plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_{out}/V_{in}$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-3, 1e1]); + +ax2 = nexttile; +hold on; +plot(f, 180/pi*angle(frf_wo_k), 'DisplayName', 'Without $k$'); +plot(f, 180/pi*angle(frf_wi_k), 'DisplayName', 'With $k$'); +plot(f, 180/pi*angle(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', sprintf('HPF $f_o = %.2f [Hz]$', f0)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); ylim([-180, 180]); +legend('location', 'northeast') + +linkaxes([ax1,ax2],'x'); +xlim([0.1, 10]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_effect_R.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_iff_effect_R +#+caption: Transfer function from $V_a$ to $V_s$ with and without the resistor $k$ +#+RESULTS: +[[file:figs/frf_iff_effect_R.png]] + + +** TODO Measurement on Strut 1 +*** Introduction :ignore: +Measurements are first performed on the strut number 1 with: +- APA 1 +- flex 1 and flex 2 + +*** 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 :tangle no +addpath('./matlab/mat/'); +addpath('./matlab/src/'); +addpath('./matlab/'); +#+end_src + +#+begin_src matlab :eval no +addpath('./mat/'); +addpath('./src/'); +#+end_src + +*** Without Encoder +**** FRF Identification - Setup +The identification is performed in three steps: +1. White noise excitation with small amplitude. + This is used to determine the main resonance of the system. +2. Sweep sine excitation with the amplitude lowered around the resonance. + The sweep sine is from 10Hz to 400Hz. +3. High frequency noise. + The noise is band-passed between 300Hz and 2kHz. + +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. + +#+begin_src matlab +leg_sweep = load(sprintf('frf_data_leg_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +leg_noise_hf = load(sprintf('frf_data_leg_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +#+end_src + +The time is the same for all measurements. +#+begin_src matlab +%% Time vector +t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s] + +%% Sampling +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] +Fs = 1/Ts; % Sampling Frequency [Hz] +#+end_src + +Then we defined a "Hanning" windows that will be used for the spectral analysis: +#+begin_src matlab +win = hanning(ceil(0.5*Fs)); % Hannning Windows +#+end_src + +We get the frequency vector that will be the same for all the frequency domain analysis. +#+begin_src matlab +% Only used to have the frequency vector "f" +[~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts); +#+end_src + +**** TODO FRF Identification - DVF +In this section, the dynamics from $V_a$ to $d_e$ is identified. + +We compute the coherence for 2nd and 3rd identification: +#+begin_src matlab +[coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts); +[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ... + 'DisplayName', 'HF Noise'); +plot(f, coh_sweep(:, 1), 'color', [colors(2, :), 0.5], ... + 'DisplayName', 'Sweep'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +% xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_dvf_plant_coh +#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ +#+RESULTS: +[[file:figs/frf_dvf_plant_coh.png]] + + +#+begin_src matlab +[dvf_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.da, win, [], [], 1/Ts); +[dvf_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.da, win, [], [], 1/Ts); +#+end_src + +The obtained transfer functions are shown in Figure [[fig:frf_dvf_plant_tf]]. + +They are all superimposed except for the APA7. + +#+begin_question +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. + +It seems that there is a "gain" problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +#+end_question + +#+begin_question +Why is there a double resonance at around 94Hz? +#+end_question + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(dvf_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), abs(dvf_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_dvf_plant_tf +#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+RESULTS: +[[file:figs/frf_dvf_plant_tf.png]] + +**** TODO FRF Identification - IFF +In this section, the dynamics from $V_a$ to $V_s$ is identified. + +First the coherence is computed and shown in Figure [[fig:frf_iff_plant_coh]]. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). + +#+begin_src matlab +[coh_sweep, ~] = mscohere(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts); +[coh_noise_hf, ~] = mscohere(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_noise_hf, 'color', [colors(1, :), 0.5], 'DisplayName', 'HF Noise'); +plot(f, coh_sweep, 'color', [colors(2, :), 0.5], 'DisplayName', 'Sweep'); +hold off; +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_iff_plant_coh +#+caption: Obtained coherence for the IFF plant +#+RESULTS: +[[file:figs/frf_iff_plant_coh.png]] + +Then the FRF are estimated and shown in Figure [[fig:frf_iff_plant_tf]] +#+begin_src matlab +[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts); +[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), abs(iff_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_iff_plant_tf +#+caption:Identified IFF Plant +#+RESULTS: +[[file:figs/frf_iff_plant_tf.png]] + +*** With Encoder +**** FRF Identification - Setup +The identification is performed in three steps: +1. White noise excitation with small amplitude. + This is used to determine the main resonance of the system. +2. Sweep sine excitation with the amplitude lowered around the resonance. + The sweep sine is from 10Hz to 400Hz. +3. High frequency noise. + The noise is band-passed between 300Hz and 2kHz. + +Then, the result of the second identification is used between 10Hz and 350Hz and the result of the third identification if used between 350Hz and 2kHz. + +#+begin_src matlab +leg_enc_sweep = load(sprintf('frf_data_leg_coder_%i_noise.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +#+end_src + +**** TODO FRF Identification - DVF +In this section, the dynamics from $V_a$ to $d_e$ is identified. + +We compute the coherence for 2nd and 3rd identification: +#+begin_src matlab +[coh_enc_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts); +[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_enc_noise_hf(:, 1), 'color', [colors(1, :), 0.5], ... + 'DisplayName', 'HF Noise'); +plot(f, coh_enc_sweep(:, 1), 'color', [colors(2, :), 0.5], ... + 'DisplayName', 'Sweep'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +% xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_dvf_plant_coh +#+caption: Obtained coherence for the plant from $V_a$ to $d_e$ +#+RESULTS: +[[file:figs/frf_dvf_plant_coh.png]] + + +#+begin_src matlab +[dvf_enc_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.de, win, [], [], 1/Ts); +[dvf_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.de, win, [], [], 1/Ts); +#+end_src + +The obtained transfer functions are shown in Figure [[fig:frf_dvf_plant_tf]]. + +They are all superimposed except for the APA7. + +#+begin_question +Why is the APA7 off? +We could think that the APA7 is stiffer, but also the mass line is off. + +It seems that there is a "gain" problem. +The encoder seems fine (it measured the same as the Interferometer). +Maybe it could be due to the amplifier? +#+end_question + +#+begin_question +Why is there a double resonance at around 94Hz? +#+end_question + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(dvf_enc_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_dvf_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_dvf_plant_tf +#+caption: Estimated FRF for the DVF plant (transfer function from $V_a$ to the encoder $d_e$) +#+RESULTS: +[[file:figs/frf_dvf_plant_tf.png]] + +**** Comparison with Interferometer + +#+begin_src matlab +[dvf_int_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.da, win, [], [], 1/Ts); +[dvf_int_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.da, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :), ... + 'DisplayName', 'Encoder'); +plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'color', colors(1, :), ... + 'HandleVisibility', 'off'); +plot(f(f> 350), abs(dvf_int_noise_hf(f> 350)), 'color', colors(2, :), ... + 'DisplayName', 'Interferometer'); +plot(f(f<=350), abs(dvf_int_sweep( f<=350)), 'color', colors(2, :), ... + 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-9, 1e-3]); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(dvf_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(dvf_enc_sweep( f<=350)), 'color', colors(1, :)); +plot(f(f> 350), 180/pi*angle(dvf_int_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(dvf_int_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_important +Clearly using the encoder like this will not be useful. + +Probably, the encoders will have to be fixed on the plates so that the resonances of the APA are not a problem anymore. +#+end_important + +**** APA Resonances Frequency + +#+begin_src matlab :exports none +figure; +hold on; +plot(f(f> 350), abs(dvf_enc_noise_hf(f> 350)), 'k-'); +plot(f(f<=350), abs(dvf_enc_sweep( f<=350)), 'k-'); + +text(93, 4e-4, {'93Hz'}, 'VerticalAlignment','bottom','HorizontalAlignment','center') +text(197, 1.3e-4,{'197Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(290, 4e-6, {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +text(376, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-7, 1e-3]); xlim([10, 2e3]); +#+end_src + +This is very close to what was estimated using the FEM. + +#+name: fig:mode_bending_x +#+caption: X-bending mode (189Hz) +#+attr_latex: :width 0.9\linewidth +[[file:figs/mode_bending_x.gif]] + +#+name: fig:mode_bending_y +#+caption: Y-bending mode (285Hz) +#+attr_latex: :width 0.9\linewidth +[[file:figs/mode_bending_y.gif]] + +#+name: fig:mode_torsion_z +#+caption: Z-torsion mode (400Hz) +#+attr_latex: :width 0.9\linewidth +[[file:figs/mode_torsion_z.gif]] + +#+begin_important +The resonances are indeed due to limited stiffness of the APA. +#+end_important + +**** FRF Identification - IFF +In this section, the dynamics from $V_a$ to $V_s$ is identified. + +First the coherence is computed and shown in Figure [[fig:frf_iff_plant_coh]]. +The coherence is very nice from 10Hz to 2kHz. +It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the excitation amplitude being lowered). + +#+begin_src matlab +[coh_enc_sweep, ~] = mscohere(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts); +[coh_enc_noise_hf, ~] = mscohere(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +colors = get(gca,'colororder'); + +figure; +hold on; +plot(f, coh_enc_noise_hf, 'color', [colors(1, :), 0.5], 'DisplayName', 'HF Noise'); +plot(f, coh_enc_sweep, 'color', [colors(2, :), 0.5], 'DisplayName', 'Sweep'); +hold off; +xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlim([5, 5e3]); ylim([0, 1]); +legend('location', 'southeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_coh.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:frf_iff_plant_coh +#+caption: Obtained coherence for the IFF plant +#+RESULTS: +[[file:figs/frf_iff_plant_coh.png]] + +Then the FRF are estimated and shown in Figure [[fig:frf_iff_plant_tf]] +#+begin_src matlab +[iff_enc_sweep, ~] = tfestimate(leg_enc_sweep.Va, leg_enc_sweep.Vs, win, [], [], 1/Ts); +[iff_enc_noise_hf, ~] = tfestimate(leg_enc_noise_hf.Va, leg_enc_noise_hf.Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(iff_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), abs(iff_enc_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +hold off; +ylim([1e-2, 1e2]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(iff_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(iff_enc_sweep( f<=350)), 'color', colors(1, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/frf_iff_plant_tf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:frf_iff_plant_tf +#+caption:Identified IFF Plant +#+RESULTS: +[[file:figs/frf_iff_plant_tf.png]] + +**** Comparison to when the encoder is not fixed + +#+begin_src matlab +[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts); +[iff_noise_hf, ~] = tfestimate(leg_noise_hf.Va, leg_noise_hf.Vs, win, [], [], 1/Ts); +#+end_src + +#+begin_src matlab :exports none +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(f(f> 350), abs(iff_enc_noise_hf(f> 350)), 'color', colors(1, :), ... + 'DisplayName', 'Encoder'); +plot(f(f<=350), abs(iff_enc_sweep( f<=350)), 'color', colors(1, :), ... + 'HandleVisibility', 'off'); +plot(f(f> 350), abs(iff_noise_hf(f> 350)), 'color', colors(2, :), ... + 'DisplayName', 'Withotu Encoder'); +plot(f(f<=350), abs(iff_sweep( f<=350)), 'color', colors(2, :), ... + 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +hold off; +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-2, 1e2]); + +ax2 = nexttile; +hold on; +plot(f(f> 350), 180/pi*angle(iff_enc_noise_hf(f> 350)), 'color', colors(1, :)); +plot(f(f<=350), 180/pi*angle(iff_enc_sweep( f<=350)), 'color', colors(1, :)); +plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350)), 'color', colors(2, :)); +plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350)), 'color', colors(2, :)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_important +We can see that the IFF does not change whether of not the encoder are fixed to the struts. +#+end_important + * Measurement Results * TODO Compare with the FEM/Simscape Model :noexport:
        Table 6: Measured frequency of the modes

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