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< title > Nano-Hexapod - Test Bench< / title >
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< h1 class = "title" > Nano-Hexapod - Test Bench< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orgd5a3ff2" > 1. Encoders fixed to the Struts< / a >
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< ul >
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< li > < a href = "#orgaaf36d1" > 1.1. Introduction< / a > < / li >
< li > < a href = "#org4eac0e4" > 1.2. Identification of the dynamics< / a >
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< ul >
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< li > < a href = "#orge7631cb" > 1.2.1. Load Data< / a > < / li >
< li > < a href = "#org3d8f0db" > 1.2.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#orgfe475e0" > 1.2.3. DVF Plant< / a > < / li >
< li > < a href = "#org9c55cb0" > 1.2.4. IFF Plant< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgb32a800" > 1.3. Comparison with the Simscape Model< / a >
< ul >
< li > < a href = "#org49d6b51" > 1.3.1. Dynamics from Actuator to Force Sensors< / a > < / li >
< li > < a href = "#org68f8e6c" > 1.3.2. Dynamics from Actuator to Encoder< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orge6221eb" > 1.4. Integral Force Feedback< / a >
< ul >
< li > < a href = "#org1ccd985" > 1.4.1. Root Locus and Decentralized Loop gain< / a > < / li >
< li > < a href = "#orgd6bc33c" > 1.4.2. Multiple Gains - Simulation< / a > < / li >
< li > < a href = "#orgcbdb9eb" > 1.4.3. Experimental Results< / a > < / li >
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< / li >
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< / li >
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< li > < a href = "#org16300e1" > 2. Encoders fixed to the plates< / a > < / li >
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< hr >
< p > This report is also available as a < a href = "./test-bench-nano-hexapod.pdf" > pdf< / a > .< / p >
< hr >
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< p >
In this document, the dynamics of the nano-hexapod shown in Figure < a href = "#orgcaac3cd" > 1< / a > is identified.
< / p >
< div class = "note" id = "org64d5e50" >
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< p >
Here are the documentation of the equipment used for this test bench:
< / p >
< ul class = "org-ul" >
< li > Voltage Amplifier: PiezoDrive < a href = "doc/PD200-V7-R1.pdf" > PD200< / a > < / li >
< li > Amplified Piezoelectric Actuator: Cedrat < a href = "doc/APA300ML.pdf" > APA300ML< / a > < / li >
< li > DAC/ADC: Speedgoat < a href = "doc/IO131-OEM-Datasheet.pdf" > IO313< / a > < / li >
< li > Encoder: Renishaw < a href = "doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf" > Vionic< / a > and used < a href = "doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf" > Ruler< / a > < / li >
< li > Interferometers: Attocube< / li >
< / ul >
< / div >
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< div id = "orgcaac3cd" class = "figure" >
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< p > < img src = "figs/IMG_20210608_152917.jpg" alt = "IMG_20210608_152917.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Nano-Hexapod< / p >
< / div >
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< div id = "org6004b44" class = "figure" >
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< p > < img src = "figs/IMG_20210608_154722.jpg" alt = "IMG_20210608_154722.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Nano-Hexapod and the control electronics< / p >
< / div >
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< div id = "orgc32dab5" class = "figure" >
< p > < img src = "figs/nano_hexapod_signals.png" alt = "nano_hexapod_signals.png" / >
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< p > < span class = "figure-number" > Figure 3: < / span > Block diagram of the system with named signals< / p >
< / div >
< table id = "orgcb52f65" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > List of signals< / caption >
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< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
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< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > < b > Unit< / b > < / th >
< th scope = "col" class = "org-left" > < b > Matlab< / b > < / th >
< th scope = "col" class = "org-left" > < b > Vector< / b > < / th >
< th scope = "col" class = "org-left" > < b > Elements< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > Control Input (wanted DAC voltage)< / td >
< td class = "org-left" > < code > [V]< / code > < / td >
< td class = "org-left" > < code > u< / code > < / td >
< td class = "org-left" > \(\bm{u}\)< / td >
< td class = "org-left" > \(u_i\)< / td >
< / tr >
< tr >
< td class = "org-left" > DAC Output Voltage< / td >
< td class = "org-left" > < code > [V]< / code > < / td >
< td class = "org-left" > < code > u< / code > < / td >
< td class = "org-left" > \(\tilde{\bm{u}}\)< / td >
< td class = "org-left" > \(\tilde{u}_i\)< / td >
< / tr >
< tr >
< td class = "org-left" > PD200 Output Voltage< / td >
< td class = "org-left" > < code > [V]< / code > < / td >
< td class = "org-left" > < code > ua< / code > < / td >
< td class = "org-left" > \(\bm{u}_a\)< / td >
< td class = "org-left" > \(u_{a,i}\)< / td >
< / tr >
< tr >
< td class = "org-left" > Actuator applied force< / td >
< td class = "org-left" > < code > [N]< / code > < / td >
< td class = "org-left" > < code > tau< / code > < / td >
< td class = "org-left" > \(\bm{\tau}\)< / td >
< td class = "org-left" > \(\tau_i\)< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > Strut motion< / td >
< td class = "org-left" > < code > [m]< / code > < / td >
< td class = "org-left" > < code > dL< / code > < / td >
< td class = "org-left" > \(d\bm{\mathcal{L}}\)< / td >
< td class = "org-left" > \(d\mathcal{L}_i\)< / td >
< / tr >
< tr >
< td class = "org-left" > Encoder measured displacement< / td >
< td class = "org-left" > < code > [m]< / code > < / td >
< td class = "org-left" > < code > dLm< / code > < / td >
< td class = "org-left" > \(d\bm{\mathcal{L}}_m\)< / td >
< td class = "org-left" > \(d\mathcal{L}_{m,i}\)< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > Force Sensor strain< / td >
< td class = "org-left" > < code > [m]< / code > < / td >
< td class = "org-left" > < code > epsilon< / code > < / td >
< td class = "org-left" > \(\bm{\epsilon}\)< / td >
< td class = "org-left" > \(\epsilon_i\)< / td >
< / tr >
< tr >
< td class = "org-left" > Force Sensor Generated Voltage< / td >
< td class = "org-left" > < code > [V]< / code > < / td >
< td class = "org-left" > < code > taum< / code > < / td >
< td class = "org-left" > \(\tilde{\bm{\tau}}_m\)< / td >
< td class = "org-left" > \(\tilde{\tau}_{m,i}\)< / td >
< / tr >
< tr >
< td class = "org-left" > Measured Generated Voltage< / td >
< td class = "org-left" > < code > [V]< / code > < / td >
< td class = "org-left" > < code > taum< / code > < / td >
< td class = "org-left" > \(\bm{\tau}_m\)< / td >
< td class = "org-left" > \(\tau_{m,i}\)< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > Motion of the top platform< / td >
< td class = "org-left" > < code > [m,rad]< / code > < / td >
< td class = "org-left" > < code > dX< / code > < / td >
< td class = "org-left" > \(d\bm{\mathcal{X}}\)< / td >
< td class = "org-left" > \(d\mathcal{X}_i\)< / td >
< / tr >
< tr >
< td class = "org-left" > Metrology measured displacement< / td >
< td class = "org-left" > < code > [m,rad]< / code > < / td >
< td class = "org-left" > < code > dXm< / code > < / td >
< td class = "org-left" > \(d\bm{\mathcal{X}}_m\)< / td >
< td class = "org-left" > \(d\mathcal{X}_{m,i}\)< / td >
< / tr >
< / tbody >
< / table >
< div id = "outline-container-orgd5a3ff2" class = "outline-2" >
< h2 id = "orgd5a3ff2" > < span class = "section-number-2" > 1< / span > Encoders fixed to the Struts< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< h3 id = "orgaaf36d1" > < span class = "section-number-3" > 1.1< / span > Introduction< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
In this section, the encoders are fixed to the struts.
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< / div >
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< div id = "outline-container-org4eac0e4" class = "outline-3" >
< h3 id = "org4eac0e4" > < span class = "section-number-3" > 1.2< / span > Identification of the dynamics< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< h4 id = "orge7631cb" > < span class = "section-number-4" > 1.2.1< / span > Load Data< / h4 >
< div class = "outline-text-4" id = "text-1-2-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_data_lf = {};
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
meas_data_lf(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/frf_data_exc_strut_%i_noise_lf.mat'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 't'< / span > , < span class = "org-string" > 'Va'< / span > , < span class = "org-string" > 'Vs'< / span > , < span class = "org-string" > 'de'< / span > )};
meas_data_hf(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/frf_data_exc_strut_%i_noise_hf.mat'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 't'< / span > , < span class = "org-string" > 'Va'< / span > , < span class = "org-string" > 'Vs'< / span > , < span class = "org-string" > 'de'< / span > )};
< span class = "org-keyword" > end< / span >
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< div id = "outline-container-org3d8f0db" class = "outline-4" >
< h4 id = "org3d8f0db" > < span class = "section-number-4" > 1.2.2< / span > Spectral Analysis - Setup< / h4 >
< div class = "outline-text-4" id = "text-1-2-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Setup useful variables< / span > < / span >
< span class = "org-comment" > % Sampling Time [s]< / span >
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Ts = (meas_data_lf{1}.t(end) < span class = "org-type" > -< / span > (meas_data_lf{1}.t(1)))< span class = "org-type" > /< / span > (length(meas_data_lf{1}.t)< span class = "org-type" > -< / span > 1);
< span class = "org-comment" > % Sampling Frequency [Hz]< / span >
Fs = 1< span class = "org-type" > /< / span > Ts;
< span class = "org-comment" > % Hannning Windows< / span >
win = hanning(ceil(1< span class = "org-type" > *< / span > Fs));
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< span class = "org-comment" > % And we get the frequency vector< / span >
[< span class = "org-type" > ~< / span > , f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
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i_lf = f < span class = "org-type" > < < / span > 250; < span class = "org-comment" > % Points for low frequency excitation< / span >
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i_hf = f < span class = "org-type" > > < / span > 250; < span class = "org-comment" > % Points for high frequency excitation< / span >
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< div id = "outline-container-orgfe475e0" class = "outline-4" >
< h4 id = "orgfe475e0" > < span class = "section-number-4" > 1.2.3< / span > DVF Plant< / h4 >
< div class = "outline-text-4" id = "text-1-2-3" >
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< p >
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First, let’ s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure < a href = "#org7027095" > 4< / a > ).
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Coherence< / span > < / span >
coh_dvf_lf = zeros(length(f), 6, 6);
coh_dvf_hf = zeros(length(f), 6, 6);
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
coh_dvf_lf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = mscohere(meas_data_lf{< span class = "org-constant" > i< / span > }.Va, meas_data_lf{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
coh_dvf_hf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = mscohere(meas_data_hf{< span class = "org-constant" > i< / span > }.Va, meas_data_hf{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org7027095" class = "figure" >
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< p > < img src = "figs/enc_struts_dvf_coh.png" alt = "enc_struts_dvf_coh.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Obtained coherence for the DVF plant< / p >
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< / div >
< p >
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Then the 6x6 transfer function matrix is estimated (Figure < a href = "#orgeda62ff" > 5< / a > ).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% DVF Plant (transfer function from u to dLm)< / span > < / span >
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G_dvf_lf = zeros(length(f), 6, 6);
G_dvf_hf = zeros(length(f), 6, 6);
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
G_dvf_lf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = tfestimate(meas_data_lf{< span class = "org-constant" > i< / span > }.Va, meas_data_lf{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
G_dvf_hf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = tfestimate(meas_data_hf{< span class = "org-constant" > i< / span > }.Va, meas_data_hf{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orgeda62ff" class = "figure" >
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< p > < img src = "figs/enc_struts_dvf_frf.png" alt = "enc_struts_dvf_frf.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Measured FRF for the DVF plant< / p >
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< / div >
< / div >
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< div id = "outline-container-org9c55cb0" class = "outline-4" >
< h4 id = "org9c55cb0" > < span class = "section-number-4" > 1.2.4< / span > IFF Plant< / h4 >
< div class = "outline-text-4" id = "text-1-2-4" >
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< p >
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First, let’ s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure < a href = "#orga958a00" > 6< / a > ).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Coherence for the IFF plant< / span > < / span >
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coh_iff_lf = zeros(length(f), 6, 6);
coh_iff_hf = zeros(length(f), 6, 6);
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
coh_iff_lf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = mscohere(meas_data_lf{< span class = "org-constant" > i< / span > }.Va, meas_data_lf{< span class = "org-constant" > i< / span > }.Vs, win, [], [], 1< span class = "org-type" > /< / span > Ts);
coh_iff_hf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = mscohere(meas_data_hf{< span class = "org-constant" > i< / span > }.Va, meas_data_hf{< span class = "org-constant" > i< / span > }.Vs, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orga958a00" class = "figure" >
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< p > < img src = "figs/enc_struts_iff_coh.png" alt = "enc_struts_iff_coh.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Obtained coherence for the IFF plant< / p >
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< / div >
< p >
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Then the 6x6 transfer function matrix is estimated (Figure < a href = "#orgaa3ad1c" > 7< / a > ).
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% IFF Plant< / span > < / span >
G_iff_lf = zeros(length(f), 6, 6);
G_iff_hf = zeros(length(f), 6, 6);
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:6< / span >
G_iff_lf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = tfestimate(meas_data_lf{< span class = "org-constant" > i< / span > }.Va, meas_data_lf{< span class = "org-constant" > i< / span > }.Vs, win, [], [], 1< span class = "org-type" > /< / span > Ts);
G_iff_hf(< span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ) = tfestimate(meas_data_hf{< span class = "org-constant" > i< / span > }.Va, meas_data_hf{< span class = "org-constant" > i< / span > }.Vs, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orgaa3ad1c" class = "figure" >
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< p > < img src = "figs/enc_struts_iff_frf.png" alt = "enc_struts_iff_frf.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Measured FRF for the IFF plant< / p >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgb32a800" class = "outline-3" >
< h3 id = "orgb32a800" > < span class = "section-number-3" > 1.3< / span > Comparison with the Simscape Model< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
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< p >
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In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
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< / p >
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< / div >
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< div id = "outline-container-org49d6b51" class = "outline-4" >
< h4 id = "org49d6b51" > < span class = "section-number-4" > 1.3.1< / span > Dynamics from Actuator to Force Sensors< / h4 >
< div class = "outline-text-4" id = "text-1-3-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Initialize Nano-Hexapod< / span > < / span >
n_hexapod = initializeNanoHexapodFinal(< span class = "org-string" > 'flex_bot_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'flex_top_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'motion_sensor_type'< / span > , < span class = "org-string" > 'struts'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > '2dof'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Identify the IFF Plant (transfer function from u to taum)< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/F'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Fm'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Force Sensors< / span >
Giff = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
< / pre >
< / div >
< div id = "orgb002d1f" class = "figure" >
< p > < img src = "figs/enc_struts_iff_comp_simscape.png" alt = "enc_struts_iff_comp_simscape.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Diagonal elements of the IFF Plant< / p >
< / div >
< div id = "orgef9afdd" class = "figure" >
< p > < img src = "figs/enc_struts_iff_comp_offdiag_simscape.png" alt = "enc_struts_iff_comp_offdiag_simscape.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Off diagonal elements of the IFF Plant< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org68f8e6c" class = "outline-4" >
< h4 id = "org68f8e6c" > < span class = "section-number-4" > 1.3.2< / span > Dynamics from Actuator to Encoder< / h4 >
< div class = "outline-text-4" id = "text-1-3-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Initialization of the Nano-Hexapod< / span > < / span >
n_hexapod = initializeNanoHexapodFinal(< span class = "org-string" > 'flex_bot_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'flex_top_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'motion_sensor_type'< / span > , < span class = "org-string" > 'struts'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > '2dof'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Identify the DVF Plant (transfer function from u to dLm)< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/F'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/D'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Encoders< / span >
Gdvf = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
< / pre >
< / div >
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< div id = "org8001ef8" class = "figure" >
< p > < img src = "figs/enc_struts_dvf_comp_simscape.png" alt = "enc_struts_dvf_comp_simscape.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Diagonal elements of the DVF Plant< / p >
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< / div >
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< div id = "org8a8dc6a" class = "figure" >
< p > < img src = "figs/enc_struts_dvf_comp_offdiag_simscape.png" alt = "enc_struts_dvf_comp_offdiag_simscape.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Off diagonal elements of the DVF Plant< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orge6221eb" class = "outline-3" >
< h3 id = "orge6221eb" > < span class = "section-number-3" > 1.4< / span > Integral Force Feedback< / h3 >
< div class = "outline-text-3" id = "text-1-4" >
< / div >
< div id = "outline-container-org1ccd985" class = "outline-4" >
< h4 id = "org1ccd985" > < span class = "section-number-4" > 1.4.1< / span > Root Locus and Decentralized Loop gain< / h4 >
< div class = "outline-text-4" id = "text-1-4-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% IFF Controller< / span > < / span >
Kiff_g1 = (1< span class = "org-type" > /< / span > (s < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 40))< span class = "org-type" > *< / span > ...< span class = "org-comment" > % Low pass filter (provides integral action above 40Hz)< / span >
(s< span class = "org-type" > /< / span > (s < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 30))< span class = "org-type" > *< / span > ...< span class = "org-comment" > % High pass filter to limit low frequency gain< / span >
(1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 500))< span class = "org-type" > *< / span > ...< span class = "org-comment" > % Low pass filter to be more robust to high frequency resonances< / span >
eye(6); < span class = "org-comment" > % Diagonal 6x6 controller< / span >
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< / pre >
< / div >
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< div id = "org9d7fb85" class = "figure" >
< p > < img src = "figs/enc_struts_iff_root_locus.png" alt = "enc_struts_iff_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Root Locus for the IFF control strategy< / p >
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< / div >
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< p >
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Then the “ optimal” IFF controller is:
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% IFF controller with Optimal gain< / span > < / span >
Kiff = g< span class = "org-type" > *< / span > Kiff_g1;
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< / pre >
< / div >
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< div id = "org879ceab" class = "figure" >
< p > < img src = "figs/enc_struts_iff_opt_loop_gain.png" alt = "enc_struts_iff_opt_loop_gain.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Bode plot of the “ decentralized loop gain” \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-orgd6bc33c" class = "outline-4" >
< h4 id = "orgd6bc33c" > < span class = "section-number-4" > 1.4.2< / span > Multiple Gains - Simulation< / h4 >
< div class = "outline-text-4" id = "text-1-4-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Tested IFF gains< / span > < / span >
iff_gains = [4, 10, 20, 40, 100, 200, 400, 1000];
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Initialize the Simscape model in closed loop< / span > < / span >
n_hexapod = initializeNanoHexapodFinal(< span class = "org-string" > 'flex_bot_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'flex_top_type'< / span > , < span class = "org-string" > '4dof'< / span > , ...
< span class = "org-string" > 'motion_sensor_type'< / span > , < span class = "org-string" > 'struts'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > '2dof'< / span > , ...
< span class = "org-string" > 'controller_type'< / span > , < span class = "org-string" > 'iff'< / span > );
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< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Identify the (damped) transfer function from u to dLm for different values of the IFF gain< / span > < / span >
Gd_iff = {zeros(1, length(iff_gains))};
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/F'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Inputs< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/D'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Strut Displacement (encoder)< / span >
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(iff_gains)< / span >
Kiff = iff_gains(< span class = "org-constant" > i< / span > )< span class = "org-type" > *< / span > Kiff_g1< span class = "org-type" > *< / span > eye(6); < span class = "org-comment" > % IFF Controller< / span >
Gd_iff(< span class = "org-constant" > i< / span > ) = {exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options)};
isstable(Gd_iff{< span class = "org-constant" > i< / span > })
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "orgb5b5f55" class = "figure" >
< p > < img src = "figs/enc_struts_iff_gains_effect_dvf_plant.png" alt = "enc_struts_iff_gains_effect_dvf_plant.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)< / p >
< / div >
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< / div >
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< / div >
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< div id = "outline-container-orgcbdb9eb" class = "outline-4" >
< h4 id = "orgcbdb9eb" > < span class = "section-number-4" > 1.4.3< / span > Experimental Results< / h4 >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org16300e1" class = "outline-2" >
< h2 id = "org16300e1" > < span class = "section-number-2" > 2< / span > Encoders fixed to the plates< / h2 >
< / div >
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< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-06-09 mer. 18:13< / p >
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< / div >
< / body >
< / html >
