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-

4.1.1 Without Encoder

+
+

4.1.1 Without Encoder

- +

-
-
4.1.1.1 FRF Identification - Setup
+
+
4.1.1.1 FRF Identification - Setup

The identification is performed in three steps: @@ -2367,8 +2368,8 @@ We get the frequency vector that will be the same for all the frequency domain a

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-
4.1.1.2 FRF Identification - Displacement
+
+
4.1.1.2 FRF Identification - Displacement

In this section, the dynamics from the excitation voltage \(V_a\) to the interferometer \(d_a\) is identified. @@ -2384,14 +2385,14 @@ We compute the coherence for 2nd and 3rd identification:

-
+

strut_1_frf_dvf_plant_coh.png

Figure 53: Obtained coherence for the plant from \(V_a\) to \(d_a\)

-The transfer function from \(V_a\) to the interferometer measured displacement \(d_a\) is estimated and shown in Figure 54. +The transfer function from \(V_a\) to the interferometer measured displacement \(d_a\) is estimated and shown in Figure 54.

[dvf_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
@@ -2400,7 +2401,7 @@ The transfer function from \(V_a\) to the interferometer measured displacement \
 
-
+

strut_1_frf_dvf_plant_tf.png

Figure 54: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the interferometer \(d_a\))

@@ -2408,15 +2409,15 @@ The transfer function from \(V_a\) to the interferometer measured displacement \
-
-
4.1.1.3 FRF Identification - IFF
+
+
4.1.1.3 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 55. +First the coherence is computed and shown in Figure 55. 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).

@@ -2428,14 +2429,14 @@ It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the ex
-
+

strut_1_frf_iff_plant_coh.png

Figure 55: Obtained coherence for the IFF plant

-Then the FRF are estimated and shown in Figure 56 +Then the FRF are estimated and shown in Figure 56

[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
@@ -2444,7 +2445,7 @@ Then the FRF are estimated and shown in Figure 56
 
-
+

strut_1_frf_iff_plant_tf.png

Figure 56: Identified IFF Plant for the Strut 1

@@ -2453,15 +2454,15 @@ Then the FRF are estimated and shown in Figure 56
-
-

4.1.2 With Encoder

+
+

4.1.2 With Encoder

- +

-
-
4.1.2.1 Measurement Data
+
+
4.1.2.1 Measurement Data
leg_enc_sweep    = load(sprintf('frf_data_leg_coder_badly_align_%i_noise.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
@@ -2471,8 +2472,8 @@ leg_enc_noise_hf = load(sprintf('frf_data_leg_coder_bad
 
-
-
4.1.2.2 FRF Identification - DVF
+
+
4.1.2.2 FRF Identification - DVF

In this section, the dynamics from \(V_a\) to \(d_e\) is identified. @@ -2488,7 +2489,7 @@ We compute the coherence for 2nd and 3rd identification:

-
+

strut_1_enc_frf_dvf_plant_coh.png

Figure 57: Obtained coherence for the plant from \(V_a\) to \(d_e\)

@@ -2507,14 +2508,14 @@ We compute the coherence for 2nd and 3rd identification:

-The obtained transfer functions are shown in Figure 58. +The obtained transfer functions are shown in Figure 58.

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. @@ -2528,7 +2529,7 @@ Maybe it could be due to the amplifier?

-
+

Why is there a double resonance at around 94Hz?

@@ -2536,7 +2537,7 @@ Why is there a double resonance at around 94Hz?
-
+

strut_1_enc_frf_dvf_plant_tf.png

Figure 58: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

@@ -2544,25 +2545,25 @@ Why is there a double resonance at around 94Hz?
-
-
4.1.2.3 Comparison of the Encoder and Interferometer
+
+
4.1.2.3 Comparison of the Encoder and Interferometer

The interferometer could here represent the case where the encoders are fixed to the plates and not the APA.

-The dynamics from \(V_a\) to \(d_e\) and from \(V_a\) to \(d_a\) are compared in Figure 59. +The dynamics from \(V_a\) to \(d_e\) and from \(V_a\) to \(d_a\) are compared in Figure 59.

-
+

strut_1_comp_enc_int.png

Figure 59: Comparison of the transfer functions from excitation voltage \(V_a\) to either the encoder \(d_e\) or the interferometer \(d_a\)

-
+

It will clearly be difficult to do something (except some low frequency positioning) with the encoders fixed to the APA.

@@ -2571,15 +2572,15 @@ It will clearly be difficult to do something (except some low frequency position
-
-
4.1.2.4 APA Resonances Frequency
+
+
4.1.2.4 APA Resonances Frequency

-As shown in Figure 60, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz. +As shown in Figure 60, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.

-
+

strut_1_spurious_resonances.png

Figure 60: Magnitude of the transfer function from excitation voltage \(V_a\) to encoder measurement \(d_e\). The frequency of the resonances are noted.

@@ -2591,33 +2592,33 @@ These resonances correspond to parasitic resonances of the APA itself. They are very close to what was estimated using the FEM:

    -
  • X-bending mode at around 190Hz (Figure 61)
  • -
  • Y-bending mode at around 290Hz (Figure 62)
  • -
  • Z-torsion mode at around 400Hz (Figure 63)
  • +
  • X-bending mode at around 190Hz (Figure 61)
  • +
  • Y-bending mode at around 290Hz (Figure 62)
  • +
  • Z-torsion mode at around 400Hz (Figure 63)
-
+

mode_bending_x.gif

Figure 61: X-bending mode (189Hz)

-
+

mode_bending_y.gif

Figure 62: Y-bending mode (285Hz)

-
+

mode_torsion_z.gif

Figure 63: Z-torsion mode (400Hz)

-
+

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

@@ -2626,18 +2627,81 @@ The resonances are indeed due to limited stiffness of the APA.
-
-
4.1.2.5 Estimated Flexible Joint axial stiffness
+
+
4.1.2.5 Estimated Flexible Joint axial stiffness
+
+
+
load(sprintf('frf_data_leg_coder_%i_add_mass_closed_circuit.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
+de = de - de(1);
+da = da - da(1);
+
-
-
4.1.2.6 FRF Identification - IFF
+ +
+
figure;
+hold on;
+plot(t, de, 'DisplayName', 'Encoder')
+plot(t, da, 'DisplayName', 'Interferometer')
+hold off;
+xlabel('Time [s]'); ylabel('Displacement [m]');
+legend('location', 'northeast');
+
+
+ +
+
de_steps = [mean(de(t > 13 & t < 14));
+            mean(de(t > 19 & t < 20));
+            mean(de(t > 24 & t < 25));
+            mean(de(t > 28.5 & t < 29.5));
+            mean(de(t > 49 & t < 50))] - mean(de(t > 13 & t < 14));
+
+da_steps = [mean(da(t > 13 & t < 14));
+            mean(da(t > 19 & t < 20));
+            mean(da(t > 24 & t < 25));
+            mean(da(t > 28.5 & t < 29.5));
+            mean(da(t > 49 & t < 50))] - mean(da(t > 13 & t < 14));
+
+
+ +
+
added_mass = [0; 1; 2; 3; 4];
+
+
+ +
+
lin_fit = (added_mass\abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)));
+
+
+ +
+
figure;
+hold on;
+plot(added_mass, abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)), 'ok')
+plot(added_mass, added_mass*lin_fit, '-r')
+hold off;
+
+
+ +
+

+What is strange is that the encoder is measuring a larger displacement than the interferometer. +That should be the opposite. +Maybe is is caused by the fact that the APA is experiencing some bending and therefore a larger motion is measured on the encoder. +

+ +
+
+
+ +
+
4.1.2.6 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 55. +First the coherence is computed and shown in Figure 55. 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).

@@ -2649,14 +2713,14 @@ It is only dropping near a zeros at 40Hz, and near the resonance at 95Hz (the ex
-
+

strut_1_frf_iff_plant_coh.png

Figure 64: Obtained coherence for the IFF plant

-Then the FRF are estimated and shown in Figure 65 +Then the FRF are estimated and shown in Figure 65

[iff_enc_sweep,    ~] = tfestimate(leg_enc_sweep.Va,    leg_enc_sweep.Vs,    win, [], [], 1/Ts);
@@ -2665,23 +2729,23 @@ Then the FRF are estimated and shown in Figure 65
 
-
+

strut_1_enc_frf_iff_plant_tf.png

Figure 65: Identified IFF Plant

-Let’s now compare the IFF plants whether the encoders are fixed to the APA or not (Figure 66). +Let’s now compare the IFF plants whether the encoders are fixed to the APA or not (Figure 66).

-
+

strut_1_frf_iff_effect_enc.png

Figure 66: Effect of the encoder on the IFF plant

-
+

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

@@ -2692,18 +2756,18 @@ We can see that the IFF does not change whether of not the encoder are fixed to
-
-

4.2 Comparison of all the Struts

+
+

4.2 Comparison of all the Struts

- +

Now all struts are measured using the same procedure and test bench.

-
-

4.2.1 FRF Identification - Setup

+
+

4.2.1 FRF Identification - Setup

The identification is performed in two steps: @@ -2777,8 +2841,8 @@ We get the frequency vector that will be the same for all the frequency domain a

-
-

4.2.2 FRF Identification - DVF

+
+

4.2.2 FRF Identification - DVF

In this section, the dynamics from \(V_a\) to \(d_e\) is identified. @@ -2804,12 +2868,12 @@ coh_noise_hf = zeros(length(f), length(leg_nums));

-The coherence is shown in Figure 67. +The coherence is shown in Figure 67. It is clear that the Noise 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.

-
+

struts_frf_dvf_plant_coh.png

Figure 67: Obtained coherence for the plant from \(V_a\) to \(d_e\)

@@ -2836,22 +2900,22 @@ dvf_noise_hf = zeros(length(f), length(leg_nums));

-The obtained transfer functions are shown in Figure 68. +The obtained transfer functions are shown in Figure 68.

They are all superimposed except for the LEG7.

-
+

struts_frf_dvf_plant_tf.png

Figure 68: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

-
+

-Depending on how the APA are mounted with the flexible joints, the dynamics can change a lot as shown in Figure 68. +Depending on how the APA are mounted with the flexible joints, the dynamics can change a lot as shown in Figure 68. In the future, a “pin” will be used to better align the APA with the flexible joints. We can expect the amplitude of the spurious resonances to decrease.

@@ -2860,8 +2924,8 @@ We can expect the amplitude of the spurious resonances to decrease.
-
-

4.2.3 FRF Identification - DVF with interferometer

+
+

4.2.3 FRF Identification - DVF with interferometer

In this section, the dynamics from \(V_a\) to \(d_a\) is identified. @@ -2887,11 +2951,11 @@ coh_noise_hf = zeros(length(f), length(leg_nums));

-The coherence is shown in Figure 69. +The coherence is shown in Figure 69. It is clear that the Noise 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.

-
+

struts_frf_int_plant_coh.png

Figure 69: Obtained coherence for the plant from \(V_a\) to \(d_e\)

@@ -2917,7 +2981,7 @@ dvf_a_noise_hf = zeros(length(f), length(leg_nums));

-The obtained transfer functions are shown in Figure 70. +The obtained transfer functions are shown in Figure 70.

@@ -2925,7 +2989,7 @@ They are all superimposed except for the LEG7.

-
+

struts_frf_int_plant_tf.png

Figure 70: Estimated FRF for the DVF plant (transfer function from \(V_a\) to the encoder \(d_e\))

@@ -2933,15 +2997,15 @@ They are all superimposed except for the LEG7.
-
-

4.2.4 FRF Identification - IFF

+
+

4.2.4 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 71. +First the coherence is computed and shown in Figure 71. 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).

@@ -2963,14 +3027,14 @@ coh_noise_hf = zeros(length(f), length(leg_nums));
-
+

struts_frf_iff_plant_coh.png

Figure 71: Obtained coherence for the IFF plant

-Then the FRF are estimated and shown in Figure 72 +Then the FRF are estimated and shown in Figure 72

%% FRF estimation of the transfer function from Va to Vs
@@ -2989,7 +3053,7 @@ iff_noise_hf = zeros(length(f), length(leg_nums));
 
-
+

struts_frf_iff_plant_tf.png

Figure 72: Identified IFF Plant

@@ -2999,19 +3063,19 @@ iff_noise_hf = zeros(length(f), length(leg_nums));
-
-

5 Test Bench APA300ML - Simscape Model

+
+

5 Test Bench APA300ML - Simscape Model

-
-

5.1 Introduction

+
+

5.1 Introduction

-A simscape model (Figure 73) of the measurement bench is used. +A simscape model (Figure 73) of the measurement bench is used.

-
+

model_bench_apa.png

Figure 73: Screenshot of the Simscape model

@@ -3019,27 +3083,21 @@ A simscape model (Figure 73) of the measurement bench
-
-

5.2 First Identification

+
+

5.2 First Identification

The APA is first initialized with default parameters and the transfer function from excitation voltage \(V_a\) (before the amplification of 20 due to the PD200 amplifier) to the sensor stack voltage \(V_s\), encoder \(d_L\) and interferometer \(z\) is identified.

-
n_hexapod = struct();
+
%% Initialize the structure containing the data used in the model
+n_hexapod = struct();
 n_hexapod.actuator = initializeAPA('type', '2dof');
 
-
%% Options for Linearized
-options = linearizeOptions;
-options.SampleTime = 0;
-
-%% Name of the Simulink File
-mdl = 'test_bench_apa300ml';
-
-%% Input/Output definition
+
%% Input/Output definition
 clear io; io_i = 1;
 io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
 io(io_i) = linio([mdl, '/Vs'],  1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
@@ -3054,18 +3112,18 @@ Ga.OutputName = {'Vs', 
 

-The obtain dynamics are shown in Figure 74 and 75. +The obtain dynamics are shown in Figure 74 and 75.

-
+

apa_model_bench_bode_vs.png

Figure 74: Bode plot of the transfer function from \(V_a\) to \(V_s\)

-
+

apa_model_bench_bode_dl_z.png

Figure 75: Bode plot of the transfer function from \(V_a\) to \(d_L\) and to \(z\)

@@ -3073,17 +3131,16 @@ The obtain dynamics are shown in Figure 74 and
- -
-

5.3 Identify Sensor/Actuator constants and compare with measured FRF

+
+

5.3 Identify Sensor/Actuator constants and compare with measured FRF

-
-

5.3.1 How to identify these constants?

+
+

5.3.1 How to identify these constants?

-
-
5.3.1.1 Piezoelectric Actuator Constant
+
+
5.3.1.1 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 \(d\). @@ -3108,8 +3165,8 @@ From the two gains, it is then easy to determine \(g_a\):

-
-
5.3.1.2 Piezoelectric Sensor Constant
+
+
5.3.1.2 Piezoelectric Sensor Constant

Similarly, it is easy to determine the gain from the excitation voltage \(V_a\) to the voltage generated by the sensor stack \(V_s\): @@ -3143,11 +3200,12 @@ Then, the “sensor” constant is:

-
-

5.3.2 Identification Data

+
+

5.3.2 Identification Data

-
leg_sweep    = load(sprintf('frf_data_%i_sweep.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
+
%% Load the identification Data
+leg_sweep    = load(sprintf('frf_data_%i_sweep.mat',    1), 't', 'Va', 'Vs', 'de', 'da');
 leg_noise_hf = load(sprintf('frf_data_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da');
 
@@ -3159,7 +3217,7 @@ The time is the same for all measurements.
%% Time vector
 t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s]
 
-%% Sampling
+%% Sampling Time / Frequency
 Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s]
 Fs = 1/Ts; % Sampling Frequency [Hz]
 
@@ -3169,7 +3227,8 @@ Fs = 1/Ts; % Sampling Fr Then we defined a “Hanning” windows that will be used for the spectral analysis:

-
win = hanning(ceil(0.5*Fs)); % Hannning Windows
+
%% Window used for spectral analysis
+win = hanning(ceil(0.5*Fs)); % Hannning Windows
 
@@ -3177,46 +3236,44 @@ Then we defined a “Hanning” windows that will be used for the spectr We get the frequency vector that will be the same for all the frequency domain analysis.

-
% Only used to have the frequency vector "f"
+
%% Get the frequency vector "f" common to all further spectral data
 [~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts);
 
-
[dvf_sweep, ~]    = tfestimate(leg_sweep.Va,    leg_sweep.da,    win, [], [], 1/Ts);
+
%% Estimate the transfer function from u to dLm
+[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);
 
-
[iff_sweep, ~] = tfestimate(leg_sweep.Va, leg_sweep.Vs, win, [], [], 1/Ts);
+
%% Estimate the transfer function from u to taum
+[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);
 
- -
-
freqs = 2*logspace(0, 3, 1000);
-
-
-
-

5.3.3 2DoF APA

+
+

5.3.3 2DoF APA

-
-
5.3.3.1 2DoF APA
+
+
5.3.3.1 2DoF APA
-
n_hexapod.actuator = initializeAPA('type', '2dof', 'ga', 1, 'gs', 1);
+
%% Initialize a 2DoF APA with Ga=Gs=1
+n_hexapod.actuator = initializeAPA('type', '2dof', 'ga', 1, 'gs', 1);
 
-
-
5.3.3.2 Identification without actuator or sensor constants
+
+
5.3.3.2 Identification without actuator or sensor constants
%% Input/Output definition
@@ -3235,16 +3292,17 @@ Gs.OutputName = {'Vs', 
 
-
-
5.3.3.3 Actuator Constant
+
+
5.3.3.3 Actuator Constant
-
ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
+
%% Estimated Actuator Constant
+ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
 
-ga = -46.4 [N/V]
+ga = -32.1 [N/V]
 
@@ -3255,16 +3313,17 @@ ga = -46.4 [N/V]
-
-
5.3.3.4 Sensor Constant
+
+
5.3.3.4 Sensor Constant
-
gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
+
%% Estimated Sensor Constant
+gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
 
-gs = 0.098 [V/m]
+gs = 0.085 [V/m]
 
@@ -3275,14 +3334,15 @@ gs = 0.098 [V/m]
-
-
5.3.3.5 Comparison
+
+
5.3.3.5 Comparison

Identify the dynamics with included constants.

-
Gs = linearize(mdl, io, 0.0, options);
+
%% Identify again the dynamics with correct Ga,Gs
+Gs = linearize(mdl, io, 0.0, options);
 Gs = Gs*exp(-Ts*s);
 Gs.InputName  = {'Va'};
 Gs.OutputName = {'Vs', 'dL', 'z'};
@@ -3290,14 +3350,14 @@ Gs.OutputName = {'Vs', 
 
-
+

apa_act_constant_comp.png

-

Figure 76: Comparison of the experimental data and Simscape model (\(V_a\) to \(d_e\))

+

Figure 76: Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))

-
+

apa_sens_constant_comp.png

Figure 77: Comparison of the experimental data and Simscape model (\(V_a\) to \(V_s\))

@@ -3306,25 +3366,27 @@ Gs.OutputName = {'Vs',
-
-

5.3.4 Flexible APA

+
+

5.3.4 Flexible APA

-
-
5.3.4.1 Flexible APA
+
+
5.3.4.1 Flexible APA
-
n_hexapod.actuator = initializeAPA('type', 'flexible', 'ga', 1, 'gs', 1);
+
%% Initialize the APA as a flexible body
+n_hexapod.actuator = initializeAPA('type', 'flexible', 'ga', 1, 'gs', 1);
 
-
-
5.3.4.2 Identification without actuator or sensor constants
+
+
5.3.4.2 Identification without actuator or sensor constants
-
Gs = linearize(mdl, io, 0.0, options);
+
%% Identify the dynamics
+Gs = linearize(mdl, io, 0.0, options);
 Gs.InputName  = {'Va'};
 Gs.OutputName = {'Vs', 'dL', 'z'};
 
@@ -3332,11 +3394,12 @@ Gs.OutputName = {'Vs',
-
-
5.3.4.3 Actuator Constant
+
+
5.3.4.3 Actuator Constant
-
ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
+
%% Actuator Constant
+ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V]
 
@@ -3352,16 +3415,17 @@ ga = 23.4 [N/V]
-
-
5.3.4.4 Sensor Constant
+
+
5.3.4.4 Sensor Constant
-
gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
+
%% Sensor Constant
+gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m]
 
-gs = -4674824.519 [V/m]
+gs = -4674826.805 [V/m]
 
@@ -3372,11 +3436,12 @@ gs = -4674824.519 [V/m]
-
-
5.3.4.5 Comparison
+
+
5.3.4.5 Comparison
-
Gs = linearize(mdl, io, 0.0, options);
+
%% Identify with updated constants
+Gs = linearize(mdl, io, 0.0, options);
 Gs = Gs*exp(-Ts*s);
 Gs.InputName  = {'Va'};
 Gs.OutputName = {'Vs', 'dL', 'z'};
@@ -3384,92 +3449,67 @@ Gs.OutputName = {'Vs', 
 
-
+

apa_act_constant_comp_flex.png

-

Figure 78: Comparison of the experimental data and Simscape model (\(V_a\) to \(d_e\))

+

Figure 78: Comparison of the experimental data and Simscape model (\(u\) to \(d\mathcal{L}_m\))

-
+

apa_sens_constant_comp_flex.png

-

Figure 79: Comparison of the experimental data and Simscape model (\(V_a\) to \(V_s\))

+

Figure 79: Comparison of the experimental data and Simscape model (\(u\) to \(\tau_m\))

-
-

5.4 Compare 2-DoF with flexible

+
+

5.4 Optimize 2-DoF model to fit the experimental Data

-

-APA - 2 DoF -

-
n_hexapod.actuator = initializeAPA('type', '2dof');
-
-G_2dof = linearize(mdl, io, 0.0, options);
-G_2dof.InputName  = {'Va'};
-G_2dof.OutputName = {'Vs', 'dL', 'z'};
+
%% Optimized parameters
+n_hexapod.actuator = initializeAPA('type', '2dof', ...
+                                   'Ga', -30.0, ...
+                                   'Gs', 0.098, ...
+                                   'k',  ones(6,1)*0.38e6, ...
+                                   'ke', ones(6,1)*1.75e6, ...
+                                   'ka', ones(6,1)*3e7, ...
+                                   'c',  ones(6,1)*1.3e2, ...
+                                   'ce', ones(6,1)*1e1, ...
+                                   'ca', ones(6,1)*1e1 ...
+                                   );
 
-

-APA - Fully Flexible + +

+

comp_apa_plant_after_opt.png

-
-
n_hexapod.actuator = initializeAPA('type', 'flexible');
-
-G_flex = linearize(mdl, io, 0.0, options);
-G_flex.InputName  = {'Va'};
-G_flex.OutputName = {'Vs', 'dL', 'z'};
-
-
- -

-Comparison -

- -
-

apa_effect_joint_comp_vs.png -

-

Figure 80: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(V_s\)

-
- - -
-

apa_effect_joint_comp_dl.png -

-

Figure 81: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(d_L\) (encoder)

-
- - -
-

apa_effect_joint_comp_z.png -

-

Figure 82: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(z\) (interferometer)

+

Figure 80: Comparison of the measured FRF and the optimized model

-
-

6 Test Bench Struts - Simscape Model

+
+

6 Test Bench Struts - Simscape Model

-
-

6.1 Introduction

+
+

6.1 Introduction

-
-

6.2 First Identification

+
+

6.2 First Identification

The object containing all the data is created.

-
n_hexapod = struct();
+
%% Initialize structure containing data for the Simscape model
+n_hexapod = struct();
 n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '2dof');
 n_hexapod.actuator = initializeAPA('type', '2dof');
@@ -3495,29 +3535,30 @@ Gs.OutputName = {'Vs', 
 
-
+

strut_bench_model_iff_bode.png

-

Figure 83: Identified transfer function from \(V_a\) to \(V_s\)

+

Figure 81: Identified transfer function from \(V_a\) to \(V_s\)

-
+

strut_bench_model_dvf_bode.png

-

Figure 84: Identified transfer function from \(V_a\) to \(d_L\)

+

Figure 82: Identified transfer function from \(V_a\) to \(d_L\)

-
-

6.3 Effect of flexible joints

+
+

6.3 Effect of flexible joints

Perfect

-
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
+
%% Perfect Joints
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
 
 Gp = linearize(mdl, io, 0.0, options);
@@ -3530,7 +3571,8 @@ Gp.OutputName = {'Vs', 
 Top Flexible
 

-
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
+
%% Top joint with Axial stiffness
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
 
 Gt = linearize(mdl, io, 0.0, options);
@@ -3543,7 +3585,8 @@ Gt.OutputName = {'Vs', 
 Bottom Flexible
 

-
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+
%% Bottom joint with Axial stiffness
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
 
 Gb = linearize(mdl, io, 0.0, options);
@@ -3556,7 +3599,8 @@ Gb.OutputName = {'Vs', 
 Both Flexible
 

-
n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+
%% Both joints with Axial stiffness
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
 
 Gf = linearize(mdl, io, 0.0, options);
@@ -3566,31 +3610,31 @@ Gf.OutputName = {'Vs', 
 

-Comparison in Figures 85, 86 and 87. +Comparison in Figures 83, 84 and 85.

-
+

strut_effect_joint_comp_vs.png

-

Figure 85: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(V_s\)

+

Figure 83: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(V_s\)

-
+

strut_effect_joint_comp_dl.png

-

Figure 86: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(d_L\) (encoder)

+

Figure 84: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(d_L\) (encoder)

-
+

strut_effect_joint_comp_z.png

-

Figure 87: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(z\) (interferometer)

+

Figure 85: Effect of the joint’s flexibility on the transfer function from \(V_a\) to \(z\) (interferometer)

-
+

Imperfection of the flexible joints has the largest impact on the transfer function from \(V_a\) to \(d_L\). However, it has relatively small impact on the transfer functions from \(V_a\) to \(V_s\) and to \(z\). @@ -3600,15 +3644,16 @@ However, it has relatively small impact on the transfer functions from \(V_a\) t

-
-

6.4 Integral Force Feedback

+
+

6.4 Integral Force Feedback

-
-

6.4.1 Initialize the system

+
+

6.4.1 Initialize the system

-
n_hexapod = struct();
+
%% Initialize typical system
+n_hexapod = struct();
 n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof');
 n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof');
 n_hexapod.actuator = initializeAPA('type', '2dof');
@@ -3617,54 +3662,91 @@ n_hexapod.actuator = initializeAPA('type', 
 
-
-

6.4.2 Plant Identification

+
+

6.4.2 Plant Identification

-
G = linearize(mdl, io, 0.0, options);
+
%% Identify th edynamics
+G = linearize(mdl, io, 0.0, options);
 G.InputName  = {'Va'};
 G.OutputName = {'Vs', 'dL', 'z'};
 
-
Giff = G('Vs', 'Va');
+
%% IFF Plant (transfer function from u to taum)
+Giff = G('Vs', 'Va');
 
-
-

6.4.3 Root Locus

+
+

6.4.3 Root Locus

\begin{equation} K_{\text{IFF}} = \frac{g}{s + \omega_c} \end{equation}
-
wc = 2*pi*20;
+
%% Cut-off frequency of the LPF
+wc = 2*pi*20;
 
+ +
+

6.5 Comparison with the experimental Data

+
+
+
%% Initialize Simscape data
+n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof');
+n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof');
+n_hexapod.actuator = initializeAPA('type', '2dof');
+
-
-

7 Function

+
+
%% Input/Output definition
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/Va'],  1, 'openinput');  io_i = io_i + 1; % Actuator Voltage
+io(io_i) = linio([mdl, '/Vs'],  1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/z'],   1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
+
+%% Identification
+Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
+Gs.InputName  = {'Va'};
+Gs.OutputName = {'Vs', 'z'};
+
+
+ + +
+

comp_strut_plant_after_opt.png +

+

Figure 86: Comparison of the measured FRF and the optimized model

+
+
+
+
+ +
+

7 Function

-
-

7.1 initializeBotFlexibleJoint - Initialize Flexible Joint

+
+

7.1 initializeBotFlexibleJoint - Initialize Flexible Joint

- +

-
-

Function description

-
+
+

Function description

+
function [flex_bot] = initializeBotFlexibleJoint(args)
 % initializeBotFlexibleJoint -
@@ -3681,9 +3763,9 @@ K_{\text{IFF}} = \frac{g}{s + \omega_c}
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
@@ -3703,9 +3785,9 @@ K_{\text{IFF}} = \frac{g}{s + \omega_c}
 
-
-

Initialize the structure

-
+
+

Initialize the structure

+
flex_bot = struct();
 
@@ -3713,9 +3795,9 @@ K_{\text{IFF}} = \frac{g}{s + \omega_c}
-
-

Set the Joint’s type

-
+
+

Set the Joint’s type

+
switch args.type
   case '2dof'
@@ -3730,9 +3812,9 @@ K_{\text{IFF}} = \frac{g}{s + \omega_c}
 
-
-

Set parameters

-
+
+

Set parameters

+
flex_bot.kRx = args.kRx;
 flex_bot.kRy = args.kRy;
@@ -3752,17 +3834,17 @@ flex_bot.cz  = args.cz;
 
-
-

7.2 initializeTopFlexibleJoint - Initialize Flexible Joint

+
+

7.2 initializeTopFlexibleJoint - Initialize Flexible Joint

- +

-
-

Function description

-
+
+

Function description

+
function [flex_top] = initializeTopFlexibleJoint(args)
 % initializeTopFlexibleJoint -
@@ -3779,9 +3861,9 @@ flex_bot.cz  = args.cz;
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.type      char   {mustBeMember(args.type,{'2dof', '3dof', '4dof'})} = '2dof'
@@ -3801,9 +3883,9 @@ flex_bot.cz  = args.cz;
 
-
-

Initialize the structure

-
+
+

Initialize the structure

+
flex_top = struct();
 
@@ -3811,9 +3893,9 @@ flex_bot.cz = args.cz;
-
-

Set the Joint’s type

-
+
+

Set the Joint’s type

+
switch args.type
   case '2dof'
@@ -3828,9 +3910,9 @@ flex_bot.cz  = args.cz;
 
-
-

Set parameters

-
+
+

Set parameters

+
flex_top.kRx = args.kRx;
 flex_top.kRy = args.kRy;
@@ -3850,17 +3932,17 @@ flex_top.cz  = args.cz;
 
-
-

7.3 initializeAPA - Initialize APA

+
+

7.3 initializeAPA - Initialize APA

- +

-
-

Function description

-
+
+

Function description

+
function [actuator] = initializeAPA(args)
 % initializeAPA -
@@ -3877,9 +3959,9 @@ flex_top.cz  = args.cz;
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.type      char   {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'
@@ -3888,9 +3970,9 @@ flex_top.cz  = args.cz;
     args.Gs  (1,1) double {mustBeNumeric} = 0
 
     % For 2DoF
-    args.k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.35e6
-    args.ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.5e6
-    args.ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*43e6
+    args.k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.38e6
+    args.ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.75e6
+    args.ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e7
 
     args.c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e1
     args.ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1
@@ -3910,9 +3992,9 @@ flex_top.cz  = args.cz;
 
-
-

Initialize Structure

-
+
+

Initialize Structure

+
actuator = struct();
 
@@ -3920,9 +4002,9 @@ flex_top.cz = args.cz;
-
-

Type

-
+
+

Type

+
switch args.type
   case '2dof'
@@ -3937,14 +4019,14 @@ flex_top.cz  = args.cz;
 
-
-

Actuator/Sensor Constants

-
+
+

Actuator/Sensor Constants

+
if args.Ga == 0
     switch args.type
       case '2dof'
-        actuator.Ga = -46.4;
+        actuator.Ga = -30.0;
       case 'flexible frame'
         actuator.Ga = 1; % TODO
       case 'flexible'
@@ -3974,9 +4056,9 @@ flex_top.cz  = args.cz;
 
-
-

2DoF parameters

-
+
+

2DoF parameters

+
actuator.k  = args.k;  % [N/m]
 actuator.ke = args.ke; % [N/m]
@@ -3992,9 +4074,9 @@ actuator.Leq = args.Leq; % [m]
 
-
-

Flexible frame and fully flexible

-
+
+

Flexible frame and fully flexible

+
switch args.type
   case 'flexible frame'
@@ -4017,17 +4099,17 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
-
-

7.4 generateSweepExc: Generate sweep sinus excitation

+
+

7.4 generateSweepExc: Generate sweep sinus excitation

- +

-
-

Function description

-
+
+

Function description

+
function [U_exc] = generateSweepExc(args)
 % generateSweepExc - Generate a Sweep Sine excitation signal
@@ -4050,9 +4132,9 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
@@ -4070,9 +4152,9 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
-
-

Sweep Sine part

-
+
+

Sweep Sine part

+
t_sweep = 0:args.Ts:args.exc_duration;
 
@@ -4101,9 +4183,9 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
-
-

Smooth Ends

-
+
+

Smooth Ends

+
if args.t_start > 0
     t_smooth_start = args.Ts:args.Ts:args.t_start;
@@ -4132,9 +4214,9 @@ actuator.cs = args.cs; % Damping of one stack [N/m]
 
-
-

Combine Excitation signals

-
+
+

Combine Excitation signals

+
V_exc = [V_smooth_start, V_sweep, V_smooth_end];
 t_exc = args.Ts*[0:1:length(V_exc)-1];
@@ -4149,17 +4231,17 @@ t_exc = args.Ts*[0:1
 
-
-

7.5 generateShapedNoise: Generate Shaped Noise excitation

+
+

7.5 generateShapedNoise: Generate Shaped Noise excitation

- +

-
-

Function description

-
+
+

Function description

+
function [U_exc] = generateShapedNoise(args)
 % generateShapedNoise - Generate a Shaped Noise excitation signal
@@ -4179,9 +4261,9 @@ t_exc = args.Ts*[0:1
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
@@ -4196,9 +4278,9 @@ t_exc = args.Ts*[0:1
 
-
-

Shaped Noise

-
+
+

Shaped Noise

+
t_noise = 0:args.Ts:args.exc_duration;
 
@@ -4216,9 +4298,9 @@ t_exc = args.Ts*[0:1
 
-
-

Smooth Ends

-
+
+

Smooth Ends

+
t_smooth_start = args.Ts:args.Ts:args.t_start;
 
@@ -4242,9 +4324,9 @@ V_smooth_end   = zeros(size(t_smooth_start));
 
-
-

Combine Excitation signals

-
+
+

Combine Excitation signals

+
V_exc = [V_smooth_start, V_noise, V_smooth_end];
 t_exc = args.Ts*[0:1:length(V_exc)-1];
@@ -4259,17 +4341,17 @@ t_exc = args.Ts*[0:1
 
-
-

7.6 generateSinIncreasingAmpl: Generate Sinus with increasing amplitude

+
+

7.6 generateSinIncreasingAmpl: Generate Sinus with increasing amplitude

- +

-
-

Function description

-
+
+

Function description

+
function [U_exc] = generateSinIncreasingAmpl(args)
 % generateSinIncreasingAmpl - Generate Sinus with increasing amplitude
@@ -4290,9 +4372,9 @@ t_exc = args.Ts*[0:1
 
-
-

Optional Parameters

-
+
+

Optional Parameters

+
arguments
     args.Ts              (1,1) double  {mustBeNumeric, mustBePositive} = 1e-4
@@ -4308,9 +4390,9 @@ t_exc = args.Ts*[0:1
 
-
-

Sinus excitation

-
+
+

Sinus excitation

+
t_noise = 0:args.Ts:args.sin_period*args.sin_num;
 sin_exc = [];
@@ -4326,9 +4408,9 @@ sin_exc = [];
 
-
-

Smooth Ends

-
+
+

Smooth Ends

+
t_smooth_start = args.Ts:args.Ts:args.t_start;
 
@@ -4352,9 +4434,9 @@ V_smooth_end   = zeros(size(t_smooth_start));
 
-
-

Combine Excitation signals

-
+
+

Combine Excitation signals

+
V_exc = [V_smooth_start, sin_exc, V_smooth_end];
 t_exc = args.Ts*[0:1:length(V_exc)-1];
@@ -4377,7 +4459,7 @@ t_exc = args.Ts*[0:1
 

Author: Dehaeze Thomas

-

Created: 2021-06-07 lun. 18:42

+

Created: 2021-06-09 mer. 19:01

diff --git a/test-bench-apa300ml.org b/test-bench-apa300ml.org index eb7e2f6..0ab6a97 100644 --- a/test-bench-apa300ml.org +++ b/test-bench-apa300ml.org @@ -49,6 +49,8 @@
#+end_export +#+latex: \clearpage + * Introduction :ignore: The goal of this test bench is to extract all the important parameters of the Amplified Piezoelectric Actuator APA300ML. @@ -2555,6 +2557,58 @@ The resonances are indeed due to limited stiffness of the APA. #+end_important **** TODO Estimated Flexible Joint axial stiffness +#+begin_src matlab +load(sprintf('frf_data_leg_coder_%i_add_mass_closed_circuit.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); +de = de - de(1); +da = da - da(1); +#+end_src + +#+begin_src matlab +figure; +hold on; +plot(t, de, 'DisplayName', 'Encoder') +plot(t, da, 'DisplayName', 'Interferometer') +hold off; +xlabel('Time [s]'); ylabel('Displacement [m]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab +de_steps = [mean(de(t > 13 & t < 14)); + mean(de(t > 19 & t < 20)); + mean(de(t > 24 & t < 25)); + mean(de(t > 28.5 & t < 29.5)); + mean(de(t > 49 & t < 50))] - mean(de(t > 13 & t < 14)); + +da_steps = [mean(da(t > 13 & t < 14)); + mean(da(t > 19 & t < 20)); + mean(da(t > 24 & t < 25)); + mean(da(t > 28.5 & t < 29.5)); + mean(da(t > 49 & t < 50))] - mean(da(t > 13 & t < 14)); +#+end_src + +#+begin_src matlab +added_mass = [0; 1; 2; 3; 4]; +#+end_src + +#+begin_src matlab +lin_fit = (added_mass\abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1))); +#+end_src + +#+begin_src matlab +figure; +hold on; +plot(added_mass, abs(de_steps-da_steps) - abs(de_steps(1)-da_steps(1)), 'ok') +plot(added_mass, added_mass*lin_fit, '-r') +hold off; +#+end_src + +#+begin_question +What is strange is that the encoder is measuring a larger displacement than the interferometer. +That should be the opposite. +Maybe is is caused by the fact that the APA is experiencing some bending and therefore a larger motion is measured on the encoder. +#+end_question + **** FRF Identification - IFF In this section, the dynamics from $V_a$ to $V_s$ is identified. @@ -4050,6 +4104,7 @@ A simscape model (Figure [[fig:model_bench_apa]]) of the measurement bench is us #+end_src #+begin_src matlab :tangle no +%% Add useful folders to the path addpath('matlab/'); addpath('matlab/test_bench_apa300ml/'); addpath('matlab/mat/'); @@ -4058,6 +4113,7 @@ addpath('matlab/png/'); #+end_src #+begin_src matlab :eval no +%% Add useful folders to the path addpath('test_bench_apa300ml/'); addpath('png/'); addpath('mat/'); @@ -4065,32 +4121,30 @@ addpath('src/'); #+end_src #+begin_src matlab -%% Options for Linearized +%% Frequency vector used for many plots +freqs = 2*logspace(0, 3, 1000); +#+end_src + +#+begin_src matlab +%% Open Simscape Model options = linearizeOptions; options.SampleTime = 0; -%% Name of the Simulink File +% Name of the Simulink File mdl = 'test_bench_apa300ml'; -%% Open the Simulink File -open([mdl, '.slx']) +open(mdl) #+end_src ** First Identification The APA is first initialized with default parameters and the transfer function from excitation voltage $V_a$ (before the amplification of 20 due to the PD200 amplifier) to the sensor stack voltage $V_s$, encoder $d_L$ and interferometer $z$ is identified. #+begin_src matlab +%% Initialize the structure containing the data used in the model n_hexapod = struct(); n_hexapod.actuator = initializeAPA('type', '2dof'); #+end_src #+begin_src matlab -%% Options for Linearized -options = linearizeOptions; -options.SampleTime = 0; - -%% Name of the Simulink File -mdl = 'test_bench_apa300ml'; - %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage @@ -4107,6 +4161,7 @@ Ga.OutputName = {'Vs', 'dL', 'z'}; The obtain dynamics are shown in Figure [[fig:apa_model_bench_bode_vs]] and [[fig:apa_model_bench_bode_dl_z]]. #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to taum freqs = logspace(1, 3, 1000); figure; @@ -4142,6 +4197,7 @@ exportFig('figs/apa_model_bench_bode_vs.pdf', 'width', 'wide', 'height', 'normal [[file:figs/apa_model_bench_bode_vs.png]] #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm and dZm freqs = logspace(1, 3, 1000); figure; @@ -4179,7 +4235,6 @@ exportFig('figs/apa_model_bench_bode_dl_z.pdf', 'width', 'wide', 'height', 'tall #+RESULTS: [[file:figs/apa_model_bench_bode_dl_z.png]] - ** Identify Sensor/Actuator constants and compare with measured FRF *** How to identify these constants? **** Piezoelectric Actuator Constant @@ -4222,6 +4277,7 @@ Then, the "sensor" constant is: *** Identification Data #+begin_src matlab +%% Load the identification Data leg_sweep = load(sprintf('frf_data_%i_sweep.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); leg_noise_hf = load(sprintf('frf_data_%i_noise_hf.mat', 1), 't', 'Va', 'Vs', 'de', 'da'); #+end_src @@ -4231,39 +4287,39 @@ The time is the same for all measurements. %% Time vector t = leg_sweep.t - leg_sweep.t(1) ; % Time vector [s] -%% Sampling +%% Sampling Time / Frequency 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 +%% Window used for spectral analysis 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" +%% Get the frequency vector "f" common to all further spectral data [~, f] = tfestimate(leg_sweep.Va, leg_sweep.de, win, [], [], 1/Ts); #+end_src #+begin_src matlab +%% Estimate the transfer function from u to dLm [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 #+begin_src matlab +%% Estimate the transfer function from u to taum [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 -freqs = 2*logspace(0, 3, 1000); -#+end_src - *** 2DoF APA **** 2DoF APA #+begin_src matlab +%% Initialize a 2DoF APA with Ga=Gs=1 n_hexapod.actuator = initializeAPA('type', '2dof', 'ga', 1, 'gs', 1); #+end_src @@ -4284,6 +4340,7 @@ Gs.OutputName = {'Vs', 'dL', 'z'}; **** Actuator Constant #+begin_src matlab +%% Estimated Actuator Constant ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V] #+end_src @@ -4292,7 +4349,7 @@ sprintf('ga = %.1f [N/V]', ga); #+end_src #+RESULTS: -: ga = -46.4 [N/V] +: ga = -32.1 [N/V] #+begin_src matlab n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V] @@ -4300,6 +4357,7 @@ n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V] **** Sensor Constant #+begin_src matlab +%% Estimated Sensor Constant gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m] #+end_src @@ -4308,7 +4366,7 @@ sprintf('gs = %.3f [V/m]', gs); #+end_src #+RESULTS: -: gs = 0.098 [V/m] +: gs = 0.085 [V/m] #+begin_src matlab n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m] @@ -4317,6 +4375,7 @@ n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m] **** Comparison Identify the dynamics with included constants. #+begin_src matlab +%% Identify again the dynamics with correct Ga,Gs Gs = linearize(mdl, io, 0.0, options); Gs = Gs*exp(-Ts*s); Gs.InputName = {'Va'}; @@ -4324,6 +4383,7 @@ Gs.OutputName = {'Vs', 'dL', 'z'}; #+end_src #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm (both Simscape and measured FRF) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4334,9 +4394,9 @@ plot(f(f<=350), abs(dvf_sweep( f<=350)), 'k-'); plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz')))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; -ylim([1e-9, 1e-3]); +ylim([1e-8, 1e-3]); ax2 = nexttile; hold on; @@ -4358,11 +4418,12 @@ exportFig('figs/apa_act_constant_comp.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:apa_act_constant_comp -#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $d_e$) +#+caption: Comparison of the experimental data and Simscape model ($u$ to $d\mathcal{L}_m$) #+RESULTS: [[file:figs/apa_act_constant_comp.png]] #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to taum (both Simscape and measured FRF) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4373,7 +4434,7 @@ plot(f(f<=350), abs(iff_sweep( f<=350)), 'k-'); plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); @@ -4404,11 +4465,13 @@ exportFig('figs/apa_sens_constant_comp.pdf', 'width', 'wide', 'height', 'tall'); *** Flexible APA **** Flexible APA #+begin_src matlab +%% Initialize the APA as a flexible body n_hexapod.actuator = initializeAPA('type', 'flexible', 'ga', 1, 'gs', 1); #+end_src **** Identification without actuator or sensor constants #+begin_src matlab +%% Identify the dynamics Gs = linearize(mdl, io, 0.0, options); Gs.InputName = {'Va'}; Gs.OutputName = {'Vs', 'dL', 'z'}; @@ -4416,6 +4479,7 @@ Gs.OutputName = {'Vs', 'dL', 'z'}; **** Actuator Constant #+begin_src matlab +%% Actuator Constant ga = -mean(abs(dvf_sweep(f>10 & f<20)))./dcgain(Gs('dL', 'Va')); % [N/V] #+end_src @@ -4432,6 +4496,7 @@ n_hexapod.actuator.Ga = ones(6,1)*ga; % Actuator gain [N/V] **** Sensor Constant #+begin_src matlab +%% Sensor Constant gs = -mean(abs(iff_sweep(f>400 & f<500)))./(ga*abs(squeeze(freqresp(Gs('Vs', 'Va'), 1e3, 'Hz')))); % [V/m] #+end_src @@ -4440,7 +4505,7 @@ sprintf('gs = %.3f [V/m]', gs); #+end_src #+RESULTS: -: gs = -4674824.519 [V/m] +: gs = -4674826.805 [V/m] #+begin_src matlab n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m] @@ -4448,6 +4513,7 @@ n_hexapod.actuator.Gs = ones(6,1)*gs; % Sensor gain [V/m] **** Comparison #+begin_src matlab +%% Identify with updated constants Gs = linearize(mdl, io, 0.0, options); Gs = Gs*exp(-Ts*s); Gs.InputName = {'Va'}; @@ -4455,6 +4521,7 @@ Gs.OutputName = {'Vs', 'dL', 'z'}; #+end_src #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to dLm (both Simscape and measured FRF) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4465,7 +4532,7 @@ plot(f(f<=350), abs(dvf_sweep( f<=350)), 'k-'); plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz')))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-9, 1e-3]); @@ -4489,11 +4556,12 @@ exportFig('figs/apa_act_constant_comp_flex.pdf', 'width', 'wide', 'height', 'tal #+end_src #+name: fig:apa_act_constant_comp_flex -#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $d_e$) +#+caption: Comparison of the experimental data and Simscape model ($u$ to $d\mathcal{L}_m$) #+RESULTS: [[file:figs/apa_act_constant_comp_flex.png]] #+begin_src matlab :exports none +%% Bode plot of the transfer function from u to taum (both Simscape and measured FRF) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4504,7 +4572,7 @@ plot(f(f<=350), abs(iff_sweep( f<=350)), 'k-'); plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude $\tau_m/u$ [V/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-2, 1e2]); @@ -4528,11 +4596,167 @@ exportFig('figs/apa_sens_constant_comp_flex.pdf', 'width', 'wide', 'height', 'ta #+end_src #+name: fig:apa_sens_constant_comp_flex -#+caption: Comparison of the experimental data and Simscape model ($V_a$ to $V_s$) +#+caption: Comparison of the experimental data and Simscape model ($u$ to $\tau_m$) #+RESULTS: [[file:figs/apa_sens_constant_comp_flex.png]] -** Compare 2-DoF with flexible +** Optimize 2-DoF model to fit the experimental Data +#+begin_src matlab +%% Optimized parameters +n_hexapod.actuator = initializeAPA('type', '2dof', ... + 'Ga', -30.0, ... + 'Gs', 0.098, ... + 'k', ones(6,1)*0.38e6, ... + 'ke', ones(6,1)*1.75e6, ... + 'ka', ones(6,1)*3e7, ... + 'c', ones(6,1)*1.3e2, ... + 'ce', ones(6,1)*1e1, ... + 'ca', ones(6,1)*1e1 ... + ); +#+end_src + +#+begin_src matlab :exports none +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs + +%% Identification with optimized parameters +Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options); +Gs.InputName = {'Va'}; +Gs.OutputName = {'Vs', 'dL'}; +#+end_src + +#+begin_src matlab :exports none +apa_nums = [1 2 4 5 6 8]; + +%% 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 + +%% 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] + +win = hanning(ceil(0.5*Fs)); % Hannning Windows +% Only used to have the frequency vector "f" +[~, f] = tfestimate(apa_sweep{1}.Va, apa_sweep{1}.de, win, [], [], 1/Ts); + +%% 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 + +%% 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 +%% Comparison of the experimental data and Simscape Model +freqs = 5*logspace(0, 3, 1000); +figure; +tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), abs(dvf_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), abs(dvf_sweep( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz')))) +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-8, 1e-3]); + +ax1b = nexttile([2,1]); +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), abs(iff_sweep( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) +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]); + +ax2 = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), 180/pi*angle(dvf_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), 180/pi*angle(dvf_sweep( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('dL', 'Va'), freqs, 'Hz')))) +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]); + +ax2b = nexttile; +hold on; +for i = 1:length(apa_nums) + plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), 180/pi*angle(iff_sweep( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz')))) +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]); + +linkaxes([ax1,ax2,ax1b,ax2b],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_apa_plant_after_opt.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:comp_apa_plant_after_opt +#+caption: Comparison of the measured FRF and the optimized model +#+RESULTS: +[[file:figs/comp_apa_plant_after_opt.png]] + +** Compare 2-DoF with flexible :noexport: APA - 2 DoF #+begin_src matlab n_hexapod.actuator = initializeAPA('type', '2dof'); @@ -4678,6 +4902,7 @@ exportFig('figs/apa_effect_joint_comp_z.pdf', 'width', 'wide', 'height', 'tall') #+end_src #+begin_src matlab :tangle no +%% Add useful folders to the path addpath('matlab/'); addpath('matlab/test_bench_struts/'); addpath('matlab/mat/'); @@ -4686,12 +4911,18 @@ addpath('matlab/png/'); #+end_src #+begin_src matlab :eval no +%% Add useful folders to the path addpath('test_bench_struts/'); addpath('png/'); addpath('mat/'); addpath('src/'); #+end_src +#+begin_src matlab +%% Frequency vector used for many plots +freqs = 2*logspace(0, 3, 1000); +#+end_src + #+begin_src matlab %% Options for Linearized options = linearizeOptions; @@ -4701,12 +4932,13 @@ options.SampleTime = 0; mdl = 'test_bench_struts'; %% Open the Simulink File -open([mdl, '.slx']) +open(mdl) #+end_src ** First Identification The object containing all the data is created. #+begin_src matlab +%% Initialize structure containing data for the Simscape model n_hexapod = struct(); n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '2dof'); @@ -4730,8 +4962,7 @@ Gs.OutputName = {'Vs', 'dL', 'z'}; #+end_src #+begin_src matlab :exports none -freqs = logspace(1, 3, 1000); - +%% Bode plot of the transfer function from u to taum figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4763,8 +4994,7 @@ exportFig('figs/strut_bench_model_iff_bode.pdf', 'width', 'wide', 'height', 'nor [[file:figs/strut_bench_model_iff_bode.png]] #+begin_src matlab :exports none -freqs = logspace(1, 4, 1000); - +%% Bode plot of the transfer function from u to dLm and dZm figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4803,6 +5033,7 @@ exportFig('figs/strut_bench_model_dvf_bode.pdf', 'width', 'wide', 'height', 'tal ** Effect of flexible joints Perfect #+begin_src matlab +%% Perfect Joints n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof'); @@ -4813,6 +5044,7 @@ Gp.OutputName = {'Vs', 'dL', 'z'}; Top Flexible #+begin_src matlab +%% Top joint with Axial stiffness n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof'); @@ -4823,6 +5055,7 @@ Gt.OutputName = {'Vs', 'dL', 'z'}; Bottom Flexible #+begin_src matlab +%% Bottom joint with Axial stiffness n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof'); @@ -4833,6 +5066,7 @@ Gb.OutputName = {'Vs', 'dL', 'z'}; Both Flexible #+begin_src matlab +%% Both joints with Axial stiffness n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof'); @@ -4844,8 +5078,7 @@ Gf.OutputName = {'Vs', 'dL', 'z'}; Comparison in Figures [[fig:strut_effect_joint_comp_vs]], [[fig:strut_effect_joint_comp_dl]] and [[fig:strut_effect_joint_comp_z]]. #+begin_src matlab :exports none -freqs = logspace(1, 4, 1000); - +%% Transfer function from u to taum - Effect of flexible joints figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4886,8 +5119,7 @@ exportFig('figs/strut_effect_joint_comp_vs.pdf', 'width', 'wide', 'height', 'tal [[file:figs/strut_effect_joint_comp_vs.png]] #+begin_src matlab :exports none -freqs = logspace(1, 4, 1000); - +%% Transfer function from u to dLm - Effect of flexible joints figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4928,8 +5160,7 @@ exportFig('figs/strut_effect_joint_comp_dl.pdf', 'width', 'wide', 'height', 'tal [[file:figs/strut_effect_joint_comp_dl.png]] #+begin_src matlab :exports none -freqs = logspace(1, 4, 1000); - +%% Transfer function from u to dZm - Effect of flexible joints figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); @@ -4977,6 +5208,7 @@ However, it has relatively small impact on the transfer functions from $V_a$ to ** Integral Force Feedback *** Initialize the system #+begin_src matlab +%% Initialize typical system n_hexapod = struct(); n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '2dof'); n_hexapod.flex_top = initializeTopFlexibleJoint('type', '3dof'); @@ -4985,12 +5217,14 @@ n_hexapod.actuator = initializeAPA('type', '2dof'); *** Plant Identification #+begin_src matlab +%% Identify th edynamics G = linearize(mdl, io, 0.0, options); G.InputName = {'Va'}; G.OutputName = {'Vs', 'dL', 'z'}; #+end_src #+begin_src matlab +%% IFF Plant (transfer function from u to taum) Giff = G('Vs', 'Va'); #+end_src @@ -5000,12 +5234,15 @@ K_{\text{IFF}} = \frac{g}{s + \omega_c} \end{equation} #+begin_src matlab +%% Cut-off frequency of the LPF wc = 2*pi*20; #+end_src #+begin_src matlab :exports none +% Gains used for the Root Locus plot gains = logspace(1, 5, 300); +%% Root Locus for the IFF figure; hold on; @@ -5020,16 +5257,164 @@ for g = gains plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); end -g = 8e2; +g = 2e2; clpoles = pole(feedback(Giff, (g/(s + wc)))); -set(gca,'ColorOrderIndex',1); -plot(real(clpoles), imag(clpoles), 'rx', 'DisplayName', '$g = 1.5 \cdot 10^4$'); +set(gca,'ColorOrderIndex',2); +plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$g=%.0f$', g)); axis square; -xlim([-600, 5]); ylim([-5, 600]); +xlim([-700, 0]); ylim([-0, 700]); xlabel('Real Part'); ylabel('Imaginary Part'); legend('location', 'northwest'); #+end_src +** Comparison with the experimental Data +#+begin_src matlab :exports none +%% Measured dynamics +leg_nums = [1 2 3 4 5]; + +% Second identification +leg_noise = {}; +for i = 1:length(leg_nums) + leg_noise(i) = {load(sprintf('frf_data_leg_coder_%i_noise.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +end + +% Third identification +leg_noise_hf = {}; +for i = 1:length(leg_nums) + leg_noise_hf(i) = {load(sprintf('frf_data_leg_coder_%i_noise_hf.mat', leg_nums(i)), 't', 'Va', 'Vs', 'de', 'da')}; +end + +t = leg_noise{1}.t - leg_noise{1}.t(1) ; % Time vector [s] +Ts = (t(end) - t(1))/(length(t)-1); % Sampling Time [s] +Fs = 1/Ts; % Sampling Frequency [Hz] + +win = hanning(ceil(0.5*Fs)); % Hannning Windows + +% Only used to have the frequency vector "f" +[~, f] = tfestimate(leg_noise{1}.Va, leg_noise{1}.de, win, [], [], 1/Ts); + +% Transfer function estimation +dvf_a_noise = zeros(length(f), length(leg_nums)); +for i = 1:length(leg_nums) + [frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.da, win, [], [], 1/Ts); + dvf_a_noise(:, i) = frf; +end + +dvf_a_noise_hf = zeros(length(f), length(leg_nums)); +for i = 1:length(leg_nums) + [frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.da, win, [], [], 1/Ts); + dvf_a_noise_hf(:, i) = frf; +end + +% FRF estimation of the transfer function from Va to Vs +iff_noise = zeros(length(f), length(leg_nums)); +for i = 1:length(leg_nums) + [frf, ~] = tfestimate(leg_noise{i}.Va, leg_noise{i}.Vs, win, [], [], 1/Ts); + iff_noise(:, i) = frf; +end + +iff_noise_hf = zeros(length(f), length(leg_nums)); +for i = 1:length(leg_nums) + [frf, ~] = tfestimate(leg_noise_hf{i}.Va, leg_noise_hf{i}.Vs, win, [], [], 1/Ts); + iff_noise_hf(:, i) = frf; +end +#+end_src + +#+begin_src matlab +%% Initialize Simscape data +n_hexapod.flex_bot = initializeBotFlexibleJoint('type', '4dof'); +n_hexapod.flex_top = initializeTopFlexibleJoint('type', '4dof'); +n_hexapod.actuator = initializeAPA('type', '2dof'); +#+end_src + +#+begin_src matlab +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage +io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage +io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs + +%% Identification +Gs = exp(-s*Ts)*linearize(mdl, io, 0.0, options); +Gs.InputName = {'Va'}; +Gs.OutputName = {'Vs', 'z'}; +#+end_src + +#+begin_src matlab :exports none +freqs = 5*logspace(0, 3, 1000); + +figure; +tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:length(leg_nums) + plot(f(f> 350), abs(dvf_a_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), abs(dvf_a_noise( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), '-') +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-8, 1e-3]); + +ax1b = nexttile([2,1]); +hold on; +for i = 1:length(leg_nums) + plot(f(f> 350), abs(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), abs(iff_noise( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '-') +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]); + +ax2 = nexttile; +hold on; +for i = 1:length(leg_nums) + plot(f(f> 350), 180/pi*angle(dvf_a_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), 180/pi*angle(dvf_a_noise( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('z', 'Va'), freqs, 'Hz'))), '-') +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]); + +ax2b = nexttile; +hold on; +for i = 1:length(leg_nums) + plot(f(f> 350), 180/pi*angle(iff_noise_hf(f> 350, i)), 'color', [0,0,0,0.2]); + plot(f(f<=350), 180/pi*angle(iff_noise( f<=350, i)), 'color', [0,0,0,0.2]); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, 180/pi*angle(squeeze(freqresp(Gs('Vs', 'Va'), freqs, 'Hz'))), '-') +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]); + +linkaxes([ax1,ax2,ax1b,ax2b],'x'); +xlim([10, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_strut_plant_after_opt.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:comp_strut_plant_after_opt +#+caption: Comparison of the measured FRF and the optimized model +#+RESULTS: +[[file:figs/comp_strut_plant_after_opt.png]] + * Function ** =initializeBotFlexibleJoint= - Initialize Flexible Joint :PROPERTIES: @@ -5249,9 +5634,9 @@ arguments args.Gs (1,1) double {mustBeNumeric} = 0 % For 2DoF - args.k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.35e6 - args.ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.5e6 - args.ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*43e6 + args.k (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*0.38e6 + args.ke (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*1.75e6 + args.ka (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e7 args.c (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*3e1 args.ce (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2e1 @@ -5302,7 +5687,7 @@ end if args.Ga == 0 switch args.type case '2dof' - actuator.Ga = -46.4; + actuator.Ga = -30.0; case 'flexible frame' actuator.Ga = 1; % TODO case 'flexible' diff --git a/test-bench-apa300ml.pdf b/test-bench-apa300ml.pdf index 5ec4c08..781917c 100644 Binary files a/test-bench-apa300ml.pdf and b/test-bench-apa300ml.pdf differ