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< title > Nano-Hexapod - Test Bench< / title >
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< a accesskey = "H" href = "../index.html" > HOME < / a >
< / div > < div id = "content" >
< 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 = "#org850a381" > 1. Encoders fixed to the Struts< / a >
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< ul >
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< li > < a href = "#org9fdd0ef" > 1.1. Introduction< / a > < / li >
< li > < a href = "#org3af1b8c" > 1.2. Identification of the dynamics< / a >
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< ul >
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< li > < a href = "#org668e8ab" > 1.2.1. Load Measurement Data< / a > < / li >
< li > < a href = "#orgdd451e1" > 1.2.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#org502869a" > 1.2.3. DVF Plant< / a > < / li >
< li > < a href = "#org294fac1" > 1.2.4. IFF Plant< / a > < / li >
< li > < a href = "#org27bc005" > 1.2.5. Save Identified Plants< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org16dfab3" > 1.3. Comparison with the Simscape Model< / a >
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< ul >
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< li > < a href = "#orgbec3680" > 1.3.1. Load measured FRF< / a > < / li >
< li > < a href = "#org416f14a" > 1.3.2. Dynamics from Actuator to Force Sensors< / a > < / li >
< li > < a href = "#org7740929" > 1.3.3. Dynamics from Actuator to Encoder< / a > < / li >
< li > < a href = "#org07b9456" > 1.3.4. Effect of a change in bending damping of the joints< / a > < / li >
< li > < a href = "#org74cdaef" > 1.3.5. Effect of a change in damping factor of the APA< / a > < / li >
< li > < a href = "#org2265b20" > 1.3.6. Effect of a change in stiffness damping coef of the APA< / a > < / li >
< li > < a href = "#org81f3c3c" > 1.3.7. Effect of a change in mass damping coef of the APA< / a > < / li >
< li > < a href = "#org35c44c1" > 1.3.8. Using Flexible model< / a > < / li >
< li > < a href = "#org90aa71a" > 1.3.9. Flexible model + encoders fixed to the plates< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org3cf2cf1" > 1.4. Integral Force Feedback< / a >
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< ul >
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< li > < a href = "#orgfef1ee8" > 1.4.1. Identification of the IFF Plant< / a > < / li >
< li > < a href = "#orgabb73e5" > 1.4.2. Root Locus and Decentralized Loop gain< / a > < / li >
< li > < a href = "#orgbdac543" > 1.4.3. Multiple Gains - Simulation< / a > < / li >
< li > < a href = "#org3bd1e0b" > 1.4.4. Experimental Results - Gains< / a >
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< ul >
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< li > < a href = "#org69457dc" > 1.4.4.1. Load Data< / a > < / li >
< li > < a href = "#orge77daca" > 1.4.4.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#orga18329e" > 1.4.4.3. DVF Plant< / a > < / li >
< li > < a href = "#orgd7b3950" > 1.4.4.4. Experimental Results - Comparison of the un-damped and fully damped system< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org37f477e" > 1.4.5. Experimental Results - Damped Plant with Optimal gain< / a >
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< ul >
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< li > < a href = "#org2716b9a" > 1.4.5.1. Load Data< / a > < / li >
< li > < a href = "#org8de917f" > 1.4.5.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#orgfa9bce2" > 1.4.5.3. DVF Plant< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
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< li > < a href = "#org4284ac6" > 1.5. Modal Analysis< / a >
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< ul >
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< li > < a href = "#org2d0c26f" > 1.5.1. Effectiveness of the IFF Strategy - Compliance< / a > < / li >
< li > < a href = "#org3ff9878" > 1.5.2. Comparison with the Simscape Model< / a > < / li >
< li > < a href = "#orgd41f51c" > 1.5.3. Obtained Mode Shapes< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org2527b87" > 1.6. Accelerometers fixed on the top platform< / a >
< ul >
< li > < a href = "#org1fd49c3" > 1.6.1. Experimental Identification< / a > < / li >
< li > < a href = "#org4f8d7c7" > 1.6.2. Location and orientation of accelerometers< / a > < / li >
< li > < a href = "#org86ce650" > 1.6.3. COM< / a > < / li >
< li > < a href = "#org4309cac" > 1.6.4. COK< / a > < / li >
< li > < a href = "#org2577983" > 1.6.5. Comp with the Simscape Model< / a > < / li >
< / ul >
< / li >
< / ul >
< / li >
< li > < a href = "#org712faa3" > 2. Encoders fixed to the plates< / a >
< ul >
< li > < a href = "#orgd3ae77d" > 2.1. Identification of the dynamics< / a >
< ul >
< li > < a href = "#org7a53f0c" > 2.1.1. Load Measurement Data< / a > < / li >
< li > < a href = "#orgd3052ca" > 2.1.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#orga653b6d" > 2.1.3. DVF Plant< / a > < / li >
< li > < a href = "#orgfe3a401" > 2.1.4. IFF Plant< / a > < / li >
< li > < a href = "#orge8c6a42" > 2.1.5. Save Identified Plants< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgdfb05f4" > 2.2. Comparison with the Simscape Model< / a >
< ul >
< li > < a href = "#org37605d2" > 2.2.1. Load measured FRF< / a > < / li >
< li > < a href = "#org793238f" > 2.2.2. Dynamics from Actuator to Force Sensors< / a > < / li >
< li > < a href = "#org9c52c3e" > 2.2.3. Dynamics from Actuator to Encoder< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org3616410" > 2.3. Integral Force Feedback< / a >
< ul >
< li > < a href = "#org8cbdfcb" > 2.3.1. Identification of the IFF Plant< / a > < / li >
< li > < a href = "#orgab3fa5e" > 2.3.2. Effect of IFF on the plant - Simscape Model< / a > < / li >
< li > < a href = "#org1fbe8b3" > 2.3.3. Experimental Results - Damped Plant with Optimal gain< / a >
< ul >
< li > < a href = "#org456a248" > 2.3.3.1. Load Data< / a > < / li >
< li > < a href = "#org54fe798" > 2.3.3.2. Spectral Analysis - Setup< / a > < / li >
< li > < a href = "#orgf69ea08" > 2.3.3.3. Simscape Model< / a > < / li >
< li > < a href = "#orgf21ad91" > 2.3.3.4. DVF Plant< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org9090d9d" > 2.3.4. Effect of IFF on the plant - FRF< / a > < / li >
< li > < a href = "#org5a28e2e" > 2.3.5. Save Damped Plant< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org14e43ac" > 2.4. HAC Control - Frame of the struts< / a >
< ul >
< li > < a href = "#org991fc81" > 2.4.1. Simscape Model< / a > < / li >
< li > < a href = "#org457939e" > 2.4.2. HAC Controller< / a > < / li >
< li > < a href = "#org98d910a" > 2.4.3. Experimental Loop Gain< / a > < / li >
< li > < a href = "#org317850e" > 2.4.4. Verification of the Stability using the Simscape model< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org247dba8" > 2.5. Reference Tracking< / a >
< ul >
< li > < a href = "#orgd862f99" > 2.5.1. Load< / a > < / li >
< li > < a href = "#org2a92c68" > 2.5.2. Y-Z Scans< / a >
< ul >
< li > < a href = "#org3c5f01d" > 2.5.2.1. Generate the Scan< / a > < / li >
< li > < a href = "#org213a892" > 2.5.2.2. Reference Signal for the Strut lengths< / a > < / li >
< li > < a href = "#orge04e29b" > 2.5.2.3. Time domain simulation with 2DoF model< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org995b778" > 2.5.3. “ NASS” reference path< / a >
< ul >
< li > < a href = "#org065d872" > 2.5.3.1. Generate Path< / a > < / li >
< li > < a href = "#org57675af" > 2.5.3.2. Simscape Simulations< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org75465ae" > 2.5.4. Save Reference paths< / a > < / li >
< li > < a href = "#orge43ac25" > 2.5.5. Experimental Results< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org71bb087" > 2.6. Feedforward (Open-Loop) Control< / a >
< ul >
< li > < a href = "#org3ea802c" > 2.6.1. Introduction< / a > < / li >
< li > < a href = "#orgc7d508e" > 2.6.2. Simple Feedforward Controller< / a > < / li >
< li > < a href = "#org4c2c4ae" > 2.6.3. Test with Simscape Model< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgd127949" > 2.7. Feedback/Feedforward control in the frame of the struts< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org762064f" > 3. Functions< / a >
< ul >
< li > < a href = "#org9c82eb4" > 3.1. < code > generateXYZTrajectory< / code > < / a >
< ul >
< li > < a href = "#org6177d10" > Function description< / a > < / li >
< li > < a href = "#orgbf6b8b1" > Optional Parameters< / a > < / li >
< li > < a href = "#orgb818291" > Initialize Time Vectors< / a > < / li >
< li > < a href = "#org361222a" > XYZ Trajectory< / a > < / li >
< li > < a href = "#org84542e6" > Reference Signal< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org4433a1b" > 3.2. < code > generateYZScanTrajectory< / code > < / a >
< ul >
< li > < a href = "#org0479bfd" > Function description< / a > < / li >
< li > < a href = "#org7be834b" > Optional Parameters< / a > < / li >
< li > < a href = "#org81214f7" > Initialize Time Vectors< / a > < / li >
< li > < a href = "#org5ab49af" > Y and Z vectors< / a > < / li >
< li > < a href = "#org796db82" > Reference Signal< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org01056ff" > 3.3. < code > getTransformationMatrixAcc< / code > < / a >
< ul >
< li > < a href = "#org37f755d" > Function description< / a > < / li >
< li > < a href = "#orgf2435ad" > Transformation matrix from motion of the solid body to accelerometer measurements< / a > < / li >
< / ul >
< / li >
< li > < a href = "#orgab18063" > 3.4. < code > getJacobianNanoHexapod< / code > < / a >
< ul >
< li > < a href = "#org7e86fe2" > Function description< / a > < / li >
< li > < a href = "#orgb9e790e" > Transformation matrix from motion of the solid body to accelerometer measurements< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
< 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 >
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This document is dedicated to the experimental study of the nano-hexapod shown in Figure < a href = "#org1278ac9" > 1< / a > .
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< / p >
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< div id = "org1278ac9" 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 class = "note" id = "org19f78fa" >
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< p >
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Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure < a href = "#orga71ba57" > 2< / a > ):
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< / 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 = "orga71ba57" 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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< p >
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In Figure < a href = "#org5326b3a" > 3< / a > is shown a block diagram of the experimental setup.
When possible, the notations are consistent with this diagram and summarized in Table < a href = "#org0036e4f" > 1< / a > .
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< / p >
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< div id = "org5326b3a" class = "figure" >
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< p > < img src = "figs/nano_hexapod_signals.png" alt = "nano_hexapod_signals.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Block diagram of the system with named signals< / p >
< / div >
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< table id = "org0036e4f" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > List of signals< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< 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 >
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< p >
This document is divided in the following sections:
< / p >
< ul class = "org-ul" >
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< li > Section < a href = "#org261275f" > 1< / a > : the encoders are fixed to the struts< / li >
< li > Section < a href = "#orgcf084fc" > 2< / a > : the encoders are fixed to the plates< / li >
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< / ul >
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< div id = "outline-container-org850a381" class = "outline-2" >
< h2 id = "org850a381" > < 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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< p >
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< a id = "org261275f" > < / a >
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< / p >
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< / div >
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< div id = "outline-container-org9fdd0ef" class = "outline-3" >
< h3 id = "org9fdd0ef" > < 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.
< / p >
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< p >
It is divided in the following sections:
< / p >
< ul class = "org-ul" >
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< li > Section < a href = "#org0c70bc8" > 1.2< / a > : the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified.< / li >
< li > Section < a href = "#orgb7175f9" > 1.3< / a > : the obtained FRF matrix is compared with the dynamics of the simscape model< / li >
< li > Section < a href = "#orgffc7d0d" > 1.4< / a > : decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated.< / li >
< li > Section < a href = "#org10a48cf" > 1.5< / a > : a modal analysis of the nano-hexapod is performed< / li >
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< / ul >
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< / div >
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< / div >
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< div id = "outline-container-org3af1b8c" class = "outline-3" >
< h3 id = "org3af1b8c" > < 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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< p >
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< a id = "org0c70bc8" > < / a >
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< / p >
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< / div >
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< div id = "outline-container-org668e8ab" class = "outline-4" >
< h4 id = "org668e8ab" > < span class = "section-number-4" > 1.2.1< / span > Load Measurement Data< / h4 >
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< 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 >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgdd451e1" class = "outline-4" >
< h4 id = "orgdd451e1" > < span class = "section-number-4" > 1.2.2< / span > Spectral Analysis - Setup< / h4 >
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< 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 >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org502869a" class = "outline-4" >
< h4 id = "org502869a" > < span class = "section-number-4" > 1.2.3< / span > DVF Plant< / h4 >
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< 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 = "#orgca37e8c" > 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 >
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coh_dvf = zeros(length(f), 6, 6);
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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 >
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coh_dvf_lf = 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 = 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);
coh_dvf(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [coh_dvf_lf(i_lf, < span class = "org-type" > :< / span > ); coh_dvf_hf(i_hf, < span class = "org-type" > :< / span > )];
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orgca37e8c" 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 = "#org4233c6b" > 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 = zeros(length(f), 6, 6);
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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 >
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G_dvf_lf = 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 = 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);
G_dvf(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [G_dvf_lf(i_lf, < span class = "org-type" > :< / span > ); G_dvf_hf(i_hf, < span class = "org-type" > :< / span > )];
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org4233c6b" 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 >
< / div >
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< div id = "outline-container-org294fac1" class = "outline-4" >
< h4 id = "org294fac1" > < span class = "section-number-4" > 1.2.4< / span > IFF Plant< / h4 >
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< 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 = "#org50b0b59" > 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 = zeros(length(f), 6, 6);
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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 >
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coh_iff_lf = 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 = 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);
coh_iff(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [coh_iff_lf(i_lf, < span class = "org-type" > :< / span > ); coh_iff_hf(i_hf, < span class = "org-type" > :< / span > )];
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org50b0b59" 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 = "#orga852dc8" > 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 >
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G_iff = zeros(length(f), 6, 6);
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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 >
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G_iff_lf = 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 = 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);
G_iff(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = [G_iff_lf(i_lf, < span class = "org-type" > :< / span > ); G_iff_hf(i_hf, < span class = "org-type" > :< / span > )];
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "orga852dc8" 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 >
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< div id = "outline-container-org27bc005" class = "outline-4" >
< h4 id = "org27bc005" > < span class = "section-number-4" > 1.2.5< / span > Save Identified Plants< / h4 >
< div class = "outline-text-4" id = "text-1-2-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/identified_plants_enc_struts.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'Ts'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > )
< / pre >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org16dfab3" class = "outline-3" >
< h3 id = "org16dfab3" > < span class = "section-number-3" > 1.3< / span > Comparison with the Simscape Model< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
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< a id = "orgb7175f9" > < / a >
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< / p >
< 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-orgbec3680" class = "outline-4" >
< h4 id = "orgbec3680" > < span class = "section-number-4" > 1.3.1< / span > Load measured FRF< / h4 >
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< div class = "outline-text-4" id = "text-1-3-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load data< / span > < / span >
load(< span class = "org-string" > 'identified_plants_enc_struts.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'Ts'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > )
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org416f14a" class = "outline-4" >
< h4 id = "org416f14a" > < span class = "section-number-4" > 1.3.2< / span > Dynamics from Actuator to Force Sensors< / h4 >
< div class = "outline-text-4" id = "text-1-3-2" >
< div class = "org-src-container" >
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< 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;
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io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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" > '/dum'< / 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 >
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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 >
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< div id = "org7fc37a7" class = "figure" >
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< 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 >
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< div id = "orga701417" class = "figure" >
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< 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 >
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< div id = "outline-container-org7740929" class = "outline-4" >
< h4 id = "org7740929" > < span class = "section-number-4" > 1.3.3< / span > Dynamics from Actuator to Encoder< / h4 >
< div class = "outline-text-4" id = "text-1-3-3" >
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< 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 > , ...
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< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > );
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< / 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;
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io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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 >
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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 = "org82e0a0f" class = "figure" >
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< 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 = "org11c75ab" class = "figure" >
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< 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 >
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< div id = "outline-container-org07b9456" class = "outline-4" >
< h4 id = "org07b9456" > < span class = "section-number-4" > 1.3.4< / span > Effect of a change in bending damping of the joints< / h4 >
< div class = "outline-text-4" id = "text-1-3-4" >
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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" > %% Tested bending dampings [Nm/(rad/s)]< / span > < / span >
cRs = [1e< span class = "org-type" > -< / span > 3, 5e< span class = "org-type" > -< / span > 3, 1e< span class = "org-type" > -< / span > 2, 5e< span class = "org-type" > -< / span > 2, 1e< span class = "org-type" > -< / span > 1];
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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 DVF Plant (transfer function from u to dLm)< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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 >
< / pre >
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< / div >
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< p >
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Then the identification is performed for all the values of the bending damping.
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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" > %% Idenfity the transfer function from actuator to encoder for all bending dampins< / span > < / span >
Gs = {zeros(length(cRs), 1)};
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(cRs)< / 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" > 'flexible'< / span > , ...
< span class = "org-string" > 'flex_bot_cRx'< / span > , cRs(< span class = "org-constant" > i< / span > ), ...
< span class = "org-string" > 'flex_bot_cRy'< / span > , cRs(< span class = "org-constant" > i< / span > ), ...
< span class = "org-string" > 'flex_top_cRx'< / span > , cRs(< span class = "org-constant" > i< / span > ), ...
< span class = "org-string" > 'flex_top_cRy'< / span > , cRs(< span class = "org-constant" > i< / span > ));
G = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
G.InputName = {< span class = "org-string" > 'Va1'< / span > , < span class = "org-string" > 'Va2'< / span > , < span class = "org-string" > 'Va3'< / span > , < span class = "org-string" > 'Va4'< / span > , < span class = "org-string" > 'Va5'< / span > , < span class = "org-string" > 'Va6'< / span > };
G.OutputName = {< span class = "org-string" > 'dL1'< / span > , < span class = "org-string" > 'dL2'< / span > , < span class = "org-string" > 'dL3'< / span > , < span class = "org-string" > 'dL4'< / span > , < span class = "org-string" > 'dL5'< / span > , < span class = "org-string" > 'dL6'< / span > };
Gs(< span class = "org-constant" > i< / span > ) = {G};
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
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< ul class = "org-ul" >
< li > Could be nice< / li >
< li > Actual damping is very small< / li >
< / ul >
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< / div >
< / div >
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< div id = "outline-container-org74cdaef" class = "outline-4" >
< h4 id = "org74cdaef" > < span class = "section-number-4" > 1.3.5< / span > Effect of a change in damping factor of the APA< / h4 >
< div class = "outline-text-4" id = "text-1-3-5" >
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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" > %% Tested bending dampings [Nm/(rad/s)]< / span > < / span >
xis = [1e< span class = "org-type" > -< / span > 3, 5e< span class = "org-type" > -< / span > 3, 1e< span class = "org-type" > -< / span > 2, 5e< span class = "org-type" > -< / span > 2, 1e< span class = "org-type" > -< / span > 1];
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< / 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" > %% 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" > '/du'< / 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 >
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< / 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" > %% Idenfity the transfer function from actuator to encoder for all bending dampins< / span > < / span >
Gs = {zeros(length(xis), 1)};
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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(xis)< / 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" > 'flexible'< / span > , ...
< span class = "org-string" > 'actuator_xi'< / span > , xis(< span class = "org-constant" > i< / span > ));
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G = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
G.InputName = {< span class = "org-string" > 'Va1'< / span > , < span class = "org-string" > 'Va2'< / span > , < span class = "org-string" > 'Va3'< / span > , < span class = "org-string" > 'Va4'< / span > , < span class = "org-string" > 'Va5'< / span > , < span class = "org-string" > 'Va6'< / span > };
G.OutputName = {< span class = "org-string" > 'dL1'< / span > , < span class = "org-string" > 'dL2'< / span > , < span class = "org-string" > 'dL3'< / span > , < span class = "org-string" > 'dL4'< / span > , < span class = "org-string" > 'dL5'< / span > , < span class = "org-string" > 'dL6'< / span > };
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Gs(< span class = "org-constant" > i< / span > ) = {G};
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org9c60432" class = "figure" >
< p > < img src = "figs/bode_Va_dL_effect_xi_damp.png" alt = "bode_Va_dL_effect_xi_damp.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Effect of the APA damping factor \(\xi\) on the dynamics from \(u\) to \(d\mathcal{L}\)< / p >
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< / div >
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< div class = "important" id = "org9cb3638" >
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< p >
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Damping factor \(\xi\) has a large impact on the damping of the “ spurious resonances” at 200Hz and 300Hz.
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< / p >
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< / div >
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< div class = "question" id = "org3692ed5" >
< p >
Why is the damping factor does not change the damping of the first peak?
< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2265b20" class = "outline-4" >
< h4 id = "org2265b20" > < span class = "section-number-4" > 1.3.6< / span > Effect of a change in stiffness damping coef of the APA< / h4 >
< div class = "outline-text-4" id = "text-1-3-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > m_coef = 1e1;
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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" > %% Tested bending dampings [Nm/(rad/s)]< / span > < / span >
k_coefs = [1e< span class = "org-type" > -< / span > 6, 5e< span class = "org-type" > -< / span > 6, 1e< span class = "org-type" > -< / span > 5, 5e< span class = "org-type" > -< / span > 5, 1e< span class = "org-type" > -< / span > 4];
< / pre >
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< / 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 DVF Plant (transfer function from u to dLm)< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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 >
< / pre >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Idenfity the transfer function from actuator to encoder for all bending dampins< / span > < / span >
Gs = {zeros(length(k_coefs), 1)};
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" > 'flexible'< / 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(k_coefs)< / span >
k_coef = k_coefs(< span class = "org-constant" > i< / span > );
G = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
G.InputName = {< span class = "org-string" > 'Va1'< / span > , < span class = "org-string" > 'Va2'< / span > , < span class = "org-string" > 'Va3'< / span > , < span class = "org-string" > 'Va4'< / span > , < span class = "org-string" > 'Va5'< / span > , < span class = "org-string" > 'Va6'< / span > };
G.OutputName = {< span class = "org-string" > 'dL1'< / span > , < span class = "org-string" > 'dL2'< / span > , < span class = "org-string" > 'dL3'< / span > , < span class = "org-string" > 'dL4'< / span > , < span class = "org-string" > 'dL5'< / span > , < span class = "org-string" > 'dL6'< / span > };
Gs(< span class = "org-constant" > i< / span > ) = {G};
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< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div id = "org0191bb1" class = "figure" >
< p > < img src = "figs/bode_Va_dL_effect_k_coef.png" alt = "bode_Va_dL_effect_k_coef.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Effect of a change of the damping “ stiffness coeficient” on the transfer function from \(u\) to \(d\mathcal{L}\)< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org81f3c3c" class = "outline-4" >
< h4 id = "org81f3c3c" > < span class = "section-number-4" > 1.3.7< / span > Effect of a change in mass damping coef of the APA< / h4 >
< div class = "outline-text-4" id = "text-1-3-7" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > k_coef = 1e< span class = "org-type" > -< / span > 6;
< / 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" > %% Tested bending dampings [Nm/(rad/s)]< / span > < / span >
m_coefs = [1e1, 5e1, 1e2, 5e2, 1e3];
< / pre >
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< / 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 DVF Plant (transfer function from u to dLm)< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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 >
< / 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" > %% Idenfity the transfer function from actuator to encoder for all bending dampins< / span > < / span >
Gs = {zeros(length(m_coefs), 1)};
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" > 'flexible'< / 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(m_coefs)< / span >
m_coef = m_coefs(< span class = "org-constant" > i< / span > );
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G = exp(< span class = "org-type" > -< / span > s< span class = "org-type" > *< / span > Ts)< span class = "org-type" > *< / span > linearize(mdl, io, 0.0, options);
G.InputName = {< span class = "org-string" > 'Va1'< / span > , < span class = "org-string" > 'Va2'< / span > , < span class = "org-string" > 'Va3'< / span > , < span class = "org-string" > 'Va4'< / span > , < span class = "org-string" > 'Va5'< / span > , < span class = "org-string" > 'Va6'< / span > };
G.OutputName = {< span class = "org-string" > 'dL1'< / span > , < span class = "org-string" > 'dL2'< / span > , < span class = "org-string" > 'dL3'< / span > , < span class = "org-string" > 'dL4'< / span > , < span class = "org-string" > 'dL5'< / span > , < span class = "org-string" > 'dL6'< / span > };
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Gs(< span class = "org-constant" > i< / span > ) = {G};
< span class = "org-keyword" > end< / span >
< / pre >
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< / div >
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< div id = "orgeff421b" class = "figure" >
< p > < img src = "figs/bode_Va_dL_effect_m_coef.png" alt = "bode_Va_dL_effect_m_coef.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Effect of a change of the damping “ mass coeficient” on the transfer function from \(u\) to \(d\mathcal{L}\)< / p >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org35c44c1" class = "outline-4" >
< h4 id = "org35c44c1" > < span class = "section-number-4" > 1.3.8< / span > Using Flexible model< / h4 >
< div class = "outline-text-4" id = "text-1-3-8" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > d_aligns = [[< span class = "org-type" > -< / span > 0.05, < span class = "org-type" > -< / span > 0.3, 0];
[ 0, 0.5, 0];
[< span class = "org-type" > -< / span > 0.1, < span class = "org-type" > -< / span > 0.3, 0];
[ 0, 0.3, 0];
[< span class = "org-type" > -< / span > 0.05, 0.05, 0];
[0, 0, 0]]< span class = "org-type" > *< / span > 1e< span class = "org-type" > -< / span > 3;
< / pre >
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< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > d_aligns = zeros(6,3);
< span class = "org-comment" > % d_aligns(1,:) = [-0.05, -0.3, 0]*1e-3;< / span >
d_aligns< span class = "org-type" > (2,:) < / span > = [ 0, 0.3, 0]< span class = "org-type" > *< / span > 1e< span class = "org-type" > -< / span > 3;
< / pre >
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< / div >
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< 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" > 'flexible'< / span > , ...
< span class = "org-string" > 'actuator_d_align'< / span > , d_aligns);
< / pre >
< / div >
< div class = "question" id = "org45fc7d0" >
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< p >
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Why do we have smaller resonances when using flexible APA?
On the test bench we have the same resonance as the 2DoF model.
Could it be due to the compliance in other dof of the flexible model?
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< / p >
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< / div >
< div class = "org-src-container" >
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< 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" > '/du'< / 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 >
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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);
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< / pre >
< / div >
< div class = "org-src-container" >
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< 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" > '/du'< / 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" > '/dum'< / 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 >
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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 >
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< div class = "org-src-container" >
< pre class = "src src-matlab" >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org90aa71a" class = "outline-4" >
< h4 id = "org90aa71a" > < span class = "section-number-4" > 1.3.9< / span > Flexible model + encoders fixed to the plates< / h4 >
< div class = "outline-text-4" id = "text-1-3-9" >
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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" > %% 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" > '/du'< / 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" > % Force Sensors< / span >
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< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > d_aligns = [[< span class = "org-type" > -< / span > 0.05, < span class = "org-type" > -< / span > 0.3, 0];
[ 0, 0.5, 0];
[< span class = "org-type" > -< / span > 0.1, < span class = "org-type" > -< / span > 0.3, 0];
[ 0, 0.3, 0];
[< span class = "org-type" > -< / span > 0.05, 0.05, 0];
[0, 0, 0]]< span class = "org-type" > *< / span > 1e< span class = "org-type" > -< / span > 3;
< / 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" > %% 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" > 'flexible'< / span > , ...
< span class = "org-string" > 'actuator_d_align'< / span > , d_aligns);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gdvf_struts = 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 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > , ...
< span class = "org-string" > 'actuator_d_align'< / span > , d_aligns);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gdvf_plates = 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 = "org7c0b25c" class = "figure" >
< p > < img src = "figs/dvf_plant_comp_struts_plates.png" alt = "dvf_plant_comp_struts_plates.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / span > Comparison of the dynamics from \(V_a\) to \(d_L\) when the encoders are fixed to the struts (blue) and to the plates (red). APA are modeled as a flexible element.< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org3cf2cf1" class = "outline-3" >
< h3 id = "org3cf2cf1" > < span class = "section-number-3" > 1.4< / span > Integral Force Feedback< / h3 >
< div class = "outline-text-3" id = "text-1-4" >
< p >
< a id = "orgffc7d0d" > < / a >
< / p >
< / div >
< div id = "outline-container-orgfef1ee8" class = "outline-4" >
< h4 id = "orgfef1ee8" > < span class = "section-number-4" > 1.4.1< / span > Identification of the IFF Plant< / h4 >
< div class = "outline-text-4" id = "text-1-4-1" >
< 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" > '/du'< / 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" > '/dum'< / 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 >
< / div >
< div id = "outline-container-orgabb73e5" class = "outline-4" >
< h4 id = "orgabb73e5" > < span class = "section-number-4" > 1.4.2< / span > Root Locus and Decentralized Loop gain< / 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" > %% IFF Controller< / span > < / span >
Kiff_g1 = < 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 > 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 >
< / pre >
< / div >
< div id = "org345f502" 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 16: < / span > Root Locus for the IFF control strategy< / p >
< / div >
< p >
Then the “ optimal” IFF controller is:
< / p >
< div class = "org-src-container" >
< 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;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/Kiff.mat'< / span > , < span class = "org-string" > 'Kiff'< / span > )
< / pre >
< / div >
< div id = "orga311644" class = "figure" >
< p > < img src = "figs/enc_struts_iff_opt_loop_gain.png" alt = "enc_struts_iff_opt_loop_gain.png" / >
< / p >
< p > < span class = "figure-number" > Figure 17: < / span > Bode plot of the “ decentralized loop gain” \(G_\text{iff}(i,i) \times K_\text{iff}(i,i)\)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgbdac543" class = "outline-4" >
< h4 id = "orgbdac543" > < span class = "section-number-4" > 1.4.3< / span > Multiple Gains - Simulation< / h4 >
< div class = "outline-text-4" id = "text-1-4-3" >
< 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];
< / pre >
< / div >
< div class = "org-src-container" >
< 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 > );
< / pre >
< / div >
< 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" > '/du'< / 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 >
< 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 = "orga7ed5d5" class = "figure" >
< p > < img src = "figs/enc_struts_iff_gains_effect_dvf_plant.png" alt = "enc_struts_iff_gains_effect_dvf_plant.png" / >
< / p >
< p > < span class = "figure-number" > Figure 18: < / span > Effect of the IFF gain \(g\) on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org3bd1e0b" class = "outline-4" >
< h4 id = "org3bd1e0b" > < span class = "section-number-4" > 1.4.4< / span > Experimental Results - Gains< / h4 >
< div class = "outline-text-4" id = "text-1-4-4" >
< p >
Let’ s look at the damping introduced by IFF as a function of the IFF gain and compare that with the results obtained using the Simscape model.
< / p >
< / div >
< div id = "outline-container-org69457dc" class = "outline-5" >
< h5 id = "org69457dc" > < span class = "section-number-5" > 1.4.4.1< / span > Load Data< / h5 >
< div class = "outline-text-5" id = "text-1-4-4-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_iff_gains = {};
< 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 >
meas_iff_gains(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/iff_strut_1_noise_g_%i.mat'< / span > , iff_gains(< span class = "org-constant" > i< / span > )), < span class = "org-string" > 't'< / span > , < span class = "org-string" > 'Vexc'< / span > , < span class = "org-string" > 'Vs'< / span > , < span class = "org-string" > 'de'< / span > , < span class = "org-string" > 'u'< / span > )};
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge77daca" class = "outline-5" >
< h5 id = "orge77daca" > < span class = "section-number-5" > 1.4.4.2< / span > Spectral Analysis - Setup< / h5 >
< div class = "outline-text-5" id = "text-1-4-4-2" >
< div class = "org-src-container" >
< 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 >
Ts = (meas_iff_gains{1}.t(end) < span class = "org-type" > -< / span > (meas_iff_gains{1}.t(1)))< span class = "org-type" > /< / span > (length(meas_iff_gains{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));
< span class = "org-comment" > % And we get the frequency vector< / span >
[< span class = "org-type" > ~< / span > , f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orga18329e" class = "outline-5" >
< h5 id = "orga18329e" > < span class = "section-number-5" > 1.4.4.3< / span > DVF Plant< / h5 >
< div class = "outline-text-5" id = "text-1-4-4-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% DVF Plant (transfer function from u to dLm)< / span > < / span >
G_iff_gains = {};
< 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 >
G_iff_gains{< span class = "org-constant" > i< / span > } = tfestimate(meas_iff_gains{< span class = "org-constant" > i< / span > }.Vexc, meas_iff_gains{< span class = "org-constant" > i< / span > }.de(< span class = "org-type" > :< / span > ,1), win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org9939225" class = "figure" >
< p > < img src = "figs/comp_iff_gains_dvf_plant.png" alt = "comp_iff_gains_dvf_plant.png" / >
< / p >
< p > < span class = "figure-number" > Figure 19: < / span > Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain< / p >
< / div >
< div id = "org22bf508" class = "figure" >
< p > < img src = "figs/comp_iff_gains_dvf_plant_zoom.png" alt = "comp_iff_gains_dvf_plant_zoom.png" / >
< / p >
< p > < span class = "figure-number" > Figure 20: < / span > Transfer function from \(u\) to \(d\mathcal{L}_m\) for multiple values of the IFF gain (Zoom)< / p >
< / div >
< div class = "important" id = "org08ce868" >
< p >
The IFF control strategy is very effective for the damping of the suspension modes.
It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA).
This is very logical.
< / p >
< p >
Also, the experimental results and the models obtained from the Simscape model are in agreement.
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd7b3950" class = "outline-5" >
< h5 id = "orgd7b3950" > < span class = "section-number-5" > 1.4.4.4< / span > Experimental Results - Comparison of the un-damped and fully damped system< / h5 >
< div class = "outline-text-5" id = "text-1-4-4-4" >
< div id = "org5e4268d" class = "figure" >
< p > < img src = "figs/comp_undamped_opt_iff_gain_diagonal.png" alt = "comp_undamped_opt_iff_gain_diagonal.png" / >
< / p >
< p > < span class = "figure-number" > Figure 21: < / span > Comparison of the diagonal elements of the tranfer function from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) without active damping and with optimal IFF gain< / p >
< / div >
< div class = "question" id = "org63597e0" >
< p >
A series of modes at around 205Hz are also damped.
< / p >
< p >
Are these damped modes at 205Hz additional “ suspension” modes or flexible modes of the struts?
< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org37f477e" class = "outline-4" >
< h4 id = "org37f477e" > < span class = "section-number-4" > 1.4.5< / span > Experimental Results - Damped Plant with Optimal gain< / h4 >
< div class = "outline-text-4" id = "text-1-4-5" >
< p >
Let’ s now look at the \(6 \times 6\) damped plant with the optimal gain \(g = 400\).
< / p >
< / div >
< div id = "outline-container-org2716b9a" class = "outline-5" >
< h5 id = "org2716b9a" > < span class = "section-number-5" > 1.4.5.1< / span > Load Data< / h5 >
< div class = "outline-text-5" id = "text-1-4-5-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_iff_struts = {};
< 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_iff_struts(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/iff_strut_%i_noise_g_400.mat'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 't'< / span > , < span class = "org-string" > 'Vexc'< / span > , < span class = "org-string" > 'Vs'< / span > , < span class = "org-string" > 'de'< / span > , < span class = "org-string" > 'u'< / span > )};
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org8de917f" class = "outline-5" >
< h5 id = "org8de917f" > < span class = "section-number-5" > 1.4.5.2< / span > Spectral Analysis - Setup< / h5 >
< div class = "outline-text-5" id = "text-1-4-5-2" >
< div class = "org-src-container" >
< 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 >
Ts = (meas_iff_struts{1}.t(end) < span class = "org-type" > -< / span > (meas_iff_struts{1}.t(1)))< span class = "org-type" > /< / span > (length(meas_iff_struts{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));
< span class = "org-comment" > % And we get the frequency vector< / span >
[< span class = "org-type" > ~< / span > , f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgfa9bce2" class = "outline-5" >
< h5 id = "orgfa9bce2" > < span class = "section-number-5" > 1.4.5.3< / span > DVF Plant< / h5 >
< div class = "outline-text-5" id = "text-1-4-5-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% DVF Plant (transfer function from u to dLm)< / span > < / span >
G_iff_opt = {};
< 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_opt{< span class = "org-constant" > i< / span > } = tfestimate(meas_iff_struts{< span class = "org-constant" > i< / span > }.Vexc, meas_iff_struts{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "orgcc09028" class = "figure" >
2021-06-10 17:52:16 +02:00
< p > < img src = "figs/damped_iff_plant_comp_diagonal.png" alt = "damped_iff_plant_comp_diagonal.png" / >
< / p >
2021-06-30 22:45:20 +02:00
< p > < span class = "figure-number" > Figure 22: < / span > Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)< / p >
< / div >
< div id = "orgbd8c52a" class = "figure" >
< p > < img src = "figs/damped_iff_plant_comp_off_diagonal.png" alt = "damped_iff_plant_comp_off_diagonal.png" / >
< / p >
< p > < span class = "figure-number" > Figure 23: < / span > Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)< / p >
< / div >
< div class = "important" id = "orgcff805f" >
< p >
With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped.
Remains the undamped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz).
< / p >
< p >
The Simscape model and the experimental results are in very good agreement.
< / p >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org4284ac6" class = "outline-3" >
< h3 id = "org4284ac6" > < span class = "section-number-3" > 1.5< / span > Modal Analysis< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
< p >
< a id = "org10a48cf" > < / a >
< / p >
< p >
Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure < a href = "#orgd9eb98c" > 26< / a > .
< / p >
< div id = "orge611ee2" class = "figure" >
< p > < img src = "figs/accelerometers_nano_hexapod.jpg" alt = "accelerometers_nano_hexapod.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 24: < / span > Location of the accelerometers on top of the nano-hexapod< / p >
< / div >
< p >
The top platform is then excited using an instrumented hammer as shown in Figure < a href = "#org9e78cd0" > 25< / a > .
< / p >
< div id = "org9e78cd0" class = "figure" >
< p > < img src = "figs/hammer_excitation_compliance_meas.jpg" alt = "hammer_excitation_compliance_meas.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 25: < / span > Example of an excitation using an instrumented hammer< / p >
< / div >
< / div >
< div id = "outline-container-org2d0c26f" class = "outline-4" >
< h4 id = "org2d0c26f" > < span class = "section-number-4" > 1.5.1< / span > Effectiveness of the IFF Strategy - Compliance< / h4 >
< div class = "outline-text-4" id = "text-1-5-1" >
< p >
In this section, we wish to estimated the effectiveness of the IFF strategy concerning the compliance.
< / p >
< p >
The top plate is excited vertically using the instrumented hammer two times:
< / p >
< ol class = "org-ol" >
< li > no control loop is used< / li >
< li > decentralized IFF is used< / li >
< / ol >
< p >
The data is loaded.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > frf_ol = load(< span class = "org-string" > 'Measurement_Z_axis.mat'< / span > ); < span class = "org-comment" > % Open-Loop< / span >
frf_iff = load(< span class = "org-string" > 'Measurement_Z_axis_damped.mat'< / span > ); < span class = "org-comment" > % IFF< / span >
< / pre >
< / div >
< p >
The mean vertical motion of the top platform is computed by averaging all 5 accelerometers.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers)< / span > < / span >
d_frf_ol = 10< span class = "org-type" > /< / span > 5< span class = "org-type" > *< / span > (frf_ol.FFT1_H1_4_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_ol.FFT1_H1_7_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_ol.FFT1_H1_10_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_ol.FFT1_H1_13_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_ol.FFT1_H1_16_1_RMS_Y_Mod)< span class = "org-type" > ./< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > frf_ol.FFT1_H1_16_1_RMS_X_Val)< span class = "org-type" > .^< / span > 2;
d_frf_iff = 10< span class = "org-type" > /< / span > 5< span class = "org-type" > *< / span > (frf_iff.FFT1_H1_4_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_iff.FFT1_H1_7_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_iff.FFT1_H1_10_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_iff.FFT1_H1_13_1_RMS_Y_Mod < span class = "org-type" > +< / span > frf_iff.FFT1_H1_16_1_RMS_Y_Mod)< span class = "org-type" > ./< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > frf_iff.FFT1_H1_16_1_RMS_X_Val)< span class = "org-type" > .^< / span > 2;
< / pre >
< / div >
< p >
The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure < a href = "#orgd9eb98c" > 26< / a > .
< / p >
< div id = "orgd9eb98c" class = "figure" >
< p > < img src = "figs/compliance_vertical_comp_iff.png" alt = "compliance_vertical_comp_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 26: < / span > Measured vertical compliance with and without IFF< / p >
< / div >
< div class = "important" id = "orgbd002a6" >
< p >
From Figure < a href = "#orgd9eb98c" > 26< / a > , it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapode.
It also has the effect of degrading (slightly) the vertical compliance at low frequency.
< / p >
< p >
It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts.
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org3ff9878" class = "outline-4" >
< h4 id = "org3ff9878" > < span class = "section-number-4" > 1.5.2< / span > Comparison with the Simscape Model< / h4 >
< div class = "outline-text-4" id = "text-1-5-2" >
< p >
Let’ s now compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model.
< / p >
< p >
The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model.
The comparison is done in Figure < a href = "#orgfb4e6d0" > 27< / a > .
Again, the model is quite accurate!
< / p >
< div id = "orgfb4e6d0" class = "figure" >
< p > < img src = "figs/compliance_vertical_comp_model_iff.png" alt = "compliance_vertical_comp_model_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 27: < / span > Measured vertical compliance with and without IFF< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd41f51c" class = "outline-4" >
< h4 id = "orgd41f51c" > < span class = "section-number-4" > 1.5.3< / span > Obtained Mode Shapes< / h4 >
< div class = "outline-text-4" id = "text-1-5-3" >
< p >
Then, several excitation are performed using the instrumented Hammer and the mode shapes are extracted.
< / p >
< p >
We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure < a href = "#orgdee89e2" > 28< / a > .
< / p >
< div id = "orgdee89e2" class = "figure" >
< p > < img src = "figs/mode_shapes_annotated.gif" alt = "mode_shapes_annotated.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 28: < / span > Measured mode shapes for the first six modes< / p >
< / div >
< p >
Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure < a href = "#orgb2ad4e3" > 29< / a > ).
< / p >
< div id = "orgb2ad4e3" class = "figure" >
< p > < img src = "figs/ModeShapeFlex1_crop.gif" alt = "ModeShapeFlex1_crop.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 29: < / span > First flexible mode at 692Hz< / p >
< / div >
< p >
The obtained modes are summarized in Table < a href = "#org9394fd0" > 2< / a > .
< / p >
< table id = "org9394fd0" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 2:< / span > Description of the identified modes< / caption >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-right" > Mode< / th >
< th scope = "col" class = "org-right" > Freq. [Hz]< / th >
< th scope = "col" class = "org-left" > Description< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-right" > 1< / td >
< td class = "org-right" > 105< / td >
< td class = "org-left" > Suspension Mode: Y-translation< / td >
< / tr >
< tr >
< td class = "org-right" > 2< / td >
< td class = "org-right" > 107< / td >
< td class = "org-left" > Suspension Mode: X-translation< / td >
< / tr >
< tr >
< td class = "org-right" > 3< / td >
< td class = "org-right" > 131< / td >
< td class = "org-left" > Suspension Mode: Z-translation< / td >
< / tr >
< tr >
< td class = "org-right" > 4< / td >
< td class = "org-right" > 161< / td >
< td class = "org-left" > Suspension Mode: Y-tilt< / td >
< / tr >
< tr >
< td class = "org-right" > 5< / td >
< td class = "org-right" > 162< / td >
< td class = "org-left" > Suspension Mode: X-tilt< / td >
< / tr >
< tr >
< td class = "org-right" > 6< / td >
< td class = "org-right" > 180< / td >
< td class = "org-left" > Suspension Mode: Z-rotation< / td >
< / tr >
< tr >
< td class = "org-right" > 7< / td >
< td class = "org-right" > 692< / td >
< td class = "org-left" > (flexible) Membrane mode of the top platform< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
< / div >
< div id = "outline-container-org2527b87" class = "outline-3" >
< h3 id = "org2527b87" > < span class = "section-number-3" > 1.6< / span > Accelerometers fixed on the top platform< / h3 >
< div class = "outline-text-3" id = "text-1-6" >
< div id = "orgda28621" class = "figure" >
< p > < img src = "figs/acc_top_plat_top_view.jpg" alt = "acc_top_plat_top_view.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 30: < / span > Accelerometers fixed on the top platform< / p >
< / div >
< / div >
< div id = "outline-container-org1fd49c3" class = "outline-4" >
< h4 id = "org1fd49c3" > < span class = "section-number-4" > 1.6.1< / span > Experimental Identification< / h4 >
< div class = "outline-text-4" id = "text-1-6-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_acc = {};
< 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_acc(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/meas_acc_top_plat_strut_%i.mat'< / span > , < span class = "org-constant" > i< / span > ), < span class = "org-string" > 't'< / span > , < span class = "org-string" > 'Va'< / span > , < span class = "org-string" > 'de'< / span > , < span class = "org-string" > 'Am'< / span > )};
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< 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 >
Ts = (meas_acc{1}.t(end) < span class = "org-type" > -< / span > (meas_acc{1}.t(1)))< span class = "org-type" > /< / span > (length(meas_acc{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));
< span class = "org-comment" > % And we get the frequency vector< / span >
[< span class = "org-type" > ~< / span > , f] = tfestimate(meas_acc{1}.Va, meas_acc{1}.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< / pre >
< / div >
< p >
The sensibility of the accelerometers are \(0.1 V/g \approx 0.01 V/(m/s^2)\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Compute the 6x6 transfer function matrix< / span > < / span >
G_acc = 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_acc(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = tfestimate(meas_acc{< span class = "org-constant" > i< / span > }.Va, 1< span class = "org-type" > /< / span > 0.01< span class = "org-type" > *< / span > meas_acc{< span class = "org-constant" > i< / span > }.Am, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org4f8d7c7" class = "outline-4" >
< h4 id = "org4f8d7c7" > < span class = "section-number-4" > 1.6.2< / span > Location and orientation of accelerometers< / h4 >
< div class = "outline-text-4" id = "text-1-6-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Opm = [ 0.047, < span class = "org-type" > -< / span > 0.112, 10e< span class = "org-type" > -< / span > 3;
0.047, < span class = "org-type" > -< / span > 0.112, 10e< span class = "org-type" > -< / span > 3;
< span class = "org-type" > -< / span > 0.113, 0.011, 10e< span class = "org-type" > -< / span > 3;
< span class = "org-type" > -< / span > 0.113, 0.011, 10e< span class = "org-type" > -< / span > 3;
0.040, 0.113, 10e< span class = "org-type" > -< / span > 3;
0.040, 0.113, 10e< span class = "org-type" > -< / span > 3]< span class = "org-type" > '< / span > ;
Osm = [< span class = "org-type" > -< / span > 1, 0, 0;
0, 0, 1;
0, < span class = "org-type" > -< / span > 1, 0;
0, 0, 1;
< span class = "org-type" > -< / span > 1, 0, 0;
0, 0, 1]< span class = "org-type" > '< / span > ;
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org86ce650" class = "outline-4" >
< h4 id = "org86ce650" > < span class = "section-number-4" > 1.6.3< / span > COM< / h4 >
< div class = "outline-text-4" id = "text-1-6-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hbm = < span class = "org-type" > -< / span > 15e< span class = "org-type" > -< / span > 3;
M = getTransformationMatrixAcc(Opm< span class = "org-type" > -< / span > [0;0;Hbm], Osm);
J = getJacobianNanoHexapod(< span class = "org-variable-name" > Hbm< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_acc_CoM = zeros(size(G_acc));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(f)< / span >
G_acc_CoM(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > ) = inv(M)< span class = "org-type" > *< / span > squeeze(G_acc(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > ))< span class = "org-type" > *< / span > inv(J< span class = "org-type" > '< / span > );
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org4309cac" class = "outline-4" >
< h4 id = "org4309cac" > < span class = "section-number-4" > 1.6.4< / span > COK< / h4 >
< div class = "outline-text-4" id = "text-1-6-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Hbm = < span class = "org-type" > -< / span > 42.3e< span class = "org-type" > -< / span > 3;
M = getTransformationMatrixAcc(Opm< span class = "org-type" > -< / span > [0;0;Hbm], Osm);
J = getJacobianNanoHexapod(< span class = "org-variable-name" > Hbm< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G_acc_CoK = zeros(size(G_acc));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(f)< / span >
G_acc_CoK(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > ) = inv(M)< span class = "org-type" > *< / span > squeeze(G_acc(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > ))< span class = "org-type" > *< / span > inv(J< span class = "org-type" > '< / span > );
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org2577983" class = "outline-4" >
< h4 id = "org2577983" > < span class = "section-number-4" > 1.6.5< / span > Comp with the Simscape Model< / h4 >
< div class = "outline-text-4" id = "text-1-6-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > 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" > 'flexible'< / span > , ...
< span class = "org-string" > 'MO_B'< / span > , < span class = "org-type" > -< / span > 42.3e< span class = "org-type" > -< / span > 3);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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" > % Relative Motion Outputs< / span >
G = linearize(mdl, io, 0.0, options);
G.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G.OutputName = {< span class = "org-string" > 'D1'< / span > , < span class = "org-string" > 'D2'< / span > , < span class = "org-string" > 'D3'< / span > , < span class = "org-string" > 'D4'< / span > , < span class = "org-string" > 'D5'< / span > , < span class = "org-string" > 'D6'< / span > };
< / pre >
< / div >
< p >
Then use the Jacobian matrices to obtain the “ cartesian” centralized plant.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gc = inv(n_hexapod.geometry.J)< span class = "org-type" > *< / span > ...
G< span class = "org-type" > *< / span > ...
inv(n_hexapod.geometry.J< span class = "org-type" > '< / span > );
< / pre >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org712faa3" class = "outline-2" >
< h2 id = "org712faa3" > < span class = "section-number-2" > 2< / span > Encoders fixed to the plates< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
< a id = "orgcf084fc" > < / a >
< / p >
< p >
In this section, the encoders are fixed to the plates rather than to the struts as shown in Figure < a href = "#orgb246d13" > 31< / a > .
< / p >
< div id = "orgb246d13" class = "figure" >
< p > < img src = "figs/IMG_20210625_083801.jpg" alt = "IMG_20210625_083801.jpg" / >
< / p >
< p > < span class = "figure-number" > Figure 31: < / span > Nano-Hexapod with encoders fixed to the struts< / p >
< / div >
< p >
It is structured as follow:
< / p >
< ul class = "org-ul" >
< li > Section < a href = "#orgf841264" > 2.1< / a > : The dynamics of the nano-hexapod is identified< / li >
< li > Section < a href = "#org867bae7" > 2.2< / a > : The identified dynamics is compared with the Simscape model< / li >
< li > Section < a href = "#org65fc5ba" > 2.3< / a > : The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model< / li >
< li > Section < a href = "#org69c97ad" > 2.4< / a > : The High Authority Control (HAC) in the frame of the struts is developed< / li >
< li > Section < a href = "#org7f2d506" > 2.5< / a > : Some reference tracking tests are performed in order to experimentally validate the HAC-LAC control strategy.< / li >
< / ul >
< / div >
< div id = "outline-container-orgd3ae77d" class = "outline-3" >
< h3 id = "orgd3ae77d" > < span class = "section-number-3" > 2.1< / span > Identification of the dynamics< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
< p >
< a id = "orgf841264" > < / a >
< / p >
< / div >
< div id = "outline-container-org7a53f0c" class = "outline-4" >
< h4 id = "org7a53f0c" > < span class = "section-number-4" > 2.1.1< / span > Load Measurement Data< / h4 >
< div class = "outline-text-4" id = "text-2-1-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_data_lf = {};
< 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_exc_strut_%i_enc_plates_noise.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 >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgd3052ca" class = "outline-4" >
< h4 id = "orgd3052ca" > < span class = "section-number-4" > 2.1.2< / span > Spectral Analysis - Setup< / h4 >
< div class = "outline-text-4" id = "text-2-1-2" >
< div class = "org-src-container" >
< 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 >
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));
< 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);
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orga653b6d" class = "outline-4" >
< h4 id = "orga653b6d" > < span class = "section-number-4" > 2.1.3< / span > DVF Plant< / h4 >
< div class = "outline-text-4" id = "text-2-1-3" >
< p >
First, let’ s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure < a href = "#org00babe0" > 32< / a > ).
< / 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 = 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(< 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);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org00babe0" class = "figure" >
< p > < img src = "figs/enc_plates_dvf_coh.png" alt = "enc_plates_dvf_coh.png" / >
< / p >
< p > < span class = "figure-number" > Figure 32: < / span > Obtained coherence for the DVF plant< / p >
< / div >
< p >
Then the 6x6 transfer function matrix is estimated (Figure < a href = "#org7f5e25c" > 33< / a > ).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% DVF Plant (transfer function from u to dLm)< / span > < / span >
G_dvf = 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(< 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);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org7f5e25c" class = "figure" >
< p > < img src = "figs/enc_plates_dvf_frf.png" alt = "enc_plates_dvf_frf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 33: < / span > Measured FRF for the DVF plant< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orgfe3a401" class = "outline-4" >
< h4 id = "orgfe3a401" > < span class = "section-number-4" > 2.1.4< / span > IFF Plant< / h4 >
< div class = "outline-text-4" id = "text-2-1-4" >
< p >
First, let’ s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure < a href = "#org45ac66f" > 34< / a > ).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Coherence for the IFF plant< / span > < / span >
coh_iff = 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(< 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);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org45ac66f" class = "figure" >
< p > < img src = "figs/enc_plates_iff_coh.png" alt = "enc_plates_iff_coh.png" / >
< / p >
< p > < span class = "figure-number" > Figure 34: < / span > Obtained coherence for the IFF plant< / p >
< / div >
< p >
Then the 6x6 transfer function matrix is estimated (Figure < a href = "#org1630892" > 35< / a > ).
< / 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 = 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(< 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);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org1630892" class = "figure" >
< p > < img src = "figs/enc_plates_iff_frf.png" alt = "enc_plates_iff_frf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 35: < / span > Measured FRF for the IFF plant< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orge8c6a42" class = "outline-4" >
< h4 id = "orge8c6a42" > < span class = "section-number-4" > 2.1.5< / span > Save Identified Plants< / h4 >
< div class = "outline-text-4" id = "text-2-1-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/identified_plants_enc_plates.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'Ts'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > )
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgdfb05f4" class = "outline-3" >
< h3 id = "orgdfb05f4" > < span class = "section-number-3" > 2.2< / span > Comparison with the Simscape Model< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< p >
< a id = "org867bae7" > < / a >
< / p >
< p >
In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.
< / p >
< / div >
< div id = "outline-container-org37605d2" class = "outline-4" >
< h4 id = "org37605d2" > < span class = "section-number-4" > 2.2.1< / span > Load measured FRF< / h4 >
< div class = "outline-text-4" id = "text-2-2-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load data< / span > < / span >
load(< span class = "org-string" > 'identified_plants_enc_plates.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'Ts'< / span > , < span class = "org-string" > 'G_iff'< / span > , < span class = "org-string" > 'G_dvf'< / span > )
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org793238f" class = "outline-4" >
< h4 id = "org793238f" > < span class = "section-number-4" > 2.2.2< / span > Dynamics from Actuator to Force Sensors< / h4 >
< div class = "outline-text-4" id = "text-2-2-2" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / 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" > '/du'< / 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" > '/dum'< / 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 = "org2e32848" class = "figure" >
< p > < img src = "figs/enc_plates_iff_comp_simscape_all.png" alt = "enc_plates_iff_comp_simscape_all.png" / >
< / p >
< p > < span class = "figure-number" > Figure 36: < / span > IFF Plant for the first actuator input and all the force senosrs< / p >
< / div >
< div id = "orgc93c28d" class = "figure" >
< p > < img src = "figs/enc_plates_iff_comp_simscape.png" alt = "enc_plates_iff_comp_simscape.png" / >
< / p >
< p > < span class = "figure-number" > Figure 37: < / span > Diagonal elements of the IFF Plant< / p >
< / div >
< div id = "orga1ca50f" class = "figure" >
< p > < img src = "figs/enc_plates_iff_comp_offdiag_simscape.png" alt = "enc_plates_iff_comp_offdiag_simscape.png" / >
< / p >
< p > < span class = "figure-number" > Figure 38: < / span > Off diagonal elements of the IFF Plant< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org9c52c3e" class = "outline-4" >
< h4 id = "org9c52c3e" > < span class = "section-number-4" > 2.2.3< / span > Dynamics from Actuator to Encoder< / h4 >
< div class = "outline-text-4" id = "text-2-2-3" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / 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" > '/du'< / 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" > '/dL'< / 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 >
< div id = "org5781868" class = "figure" >
< p > < img src = "figs/enc_plates_dvf_comp_simscape_all.png" alt = "enc_plates_dvf_comp_simscape_all.png" / >
< / p >
< p > < span class = "figure-number" > Figure 39: < / span > DVF Plant for the first actuator input and all the encoders< / p >
< / div >
< div id = "org71185f6" class = "figure" >
< p > < img src = "figs/enc_plates_dvf_comp_simscape.png" alt = "enc_plates_dvf_comp_simscape.png" / >
< / p >
< p > < span class = "figure-number" > Figure 40: < / span > Diagonal elements of the DVF Plant< / p >
< / div >
< div id = "org9b58e6a" class = "figure" >
< p > < img src = "figs/enc_plates_dvf_comp_offdiag_simscape.png" alt = "enc_plates_dvf_comp_offdiag_simscape.png" / >
< / p >
< p > < span class = "figure-number" > Figure 41: < / span > Off diagonal elements of the DVF Plant< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org3616410" class = "outline-3" >
< h3 id = "org3616410" > < span class = "section-number-3" > 2.3< / span > Integral Force Feedback< / h3 >
< div class = "outline-text-3" id = "text-2-3" >
< p >
< a id = "org65fc5ba" > < / a >
< / p >
< div id = "org96b6cf3" class = "figure" >
< p > < img src = "figs/control_architecture_iff.png" alt = "control_architecture_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 42: < / span > Integral Force Feedback Strategy< / p >
< / div >
< / div >
< div id = "outline-container-org8cbdfcb" class = "outline-4" >
< h4 id = "org8cbdfcb" > < span class = "section-number-4" > 2.3.1< / span > Identification of the IFF Plant< / h4 >
< div class = "outline-text-4" id = "text-2-3-1" >
< 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" > 'plates'< / 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" > '/du'< / 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 >
< / div >
< div id = "outline-container-orgab3fa5e" class = "outline-4" >
< h4 id = "orgab3fa5e" > < span class = "section-number-4" > 2.3.2< / span > Effect of IFF on the plant - Simscape Model< / h4 >
< div class = "outline-text-4" id = "text-2-3-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'Kiff.mat'< / span > , < span class = "org-string" > 'Kiff'< / span > )
< / pre >
< / div >
< 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 >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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" > '/dL'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Plate Displacement (encoder)< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gd_ol = 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 class = "org-src-container" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > , ...
< span class = "org-string" > 'controller_type'< / span > , < span class = "org-string" > 'iff'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gd_iff = 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 >
< pre class = "example" >
1
< / pre >
< div id = "org6eb320c" class = "figure" >
< p > < img src = "figs/enc_plates_iff_gains_effect_dvf_plant.png" alt = "enc_plates_iff_gains_effect_dvf_plant.png" / >
< / p >
< p > < span class = "figure-number" > Figure 43: < / span > Effect of the IFF control strategy on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org1fbe8b3" class = "outline-4" >
< h4 id = "org1fbe8b3" > < span class = "section-number-4" > 2.3.3< / span > Experimental Results - Damped Plant with Optimal gain< / h4 >
< div class = "outline-text-4" id = "text-2-3-3" >
< p >
Let’ s now look at the \(6 \times 6\) damped plant with the optimal gain \(g = 400\).
< / p >
< / div >
< div id = "outline-container-org456a248" class = "outline-5" >
< h5 id = "org456a248" > < span class = "section-number-5" > 2.3.3.1< / span > Load Data< / h5 >
< div class = "outline-text-5" id = "text-2-3-3-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Load Identification Data< / span > < / span >
meas_iff_plates = {};
< 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_iff_plates(< span class = "org-constant" > i< / span > ) = {load(sprintf(< span class = "org-string" > 'mat/frf_exc_iff_strut_%i_enc_plates_noise.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-string" > 'u'< / span > )};
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org54fe798" class = "outline-5" >
< h5 id = "org54fe798" > < span class = "section-number-5" > 2.3.3.2< / span > Spectral Analysis - Setup< / h5 >
< div class = "outline-text-5" id = "text-2-3-3-2" >
< div class = "org-src-container" >
< 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 >
Ts = (meas_iff_plates{1}.t(end) < span class = "org-type" > -< / span > (meas_iff_plates{1}.t(1)))< span class = "org-type" > /< / span > (length(meas_iff_plates{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));
< span class = "org-comment" > % And we get the frequency vector< / span >
[< span class = "org-type" > ~< / span > , f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orgf69ea08" class = "outline-5" >
< h5 id = "orgf69ea08" > < span class = "section-number-5" > 2.3.3.3< / span > Simscape Model< / h5 >
< div class = "outline-text-5" id = "text-2-3-3-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'Kiff.mat'< / span > , < span class = "org-string" > 'Kiff'< / span > )
< / pre >
< / div >
< div class = "org-src-container" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > , ...
< span class = "org-string" > 'controller_type'< / span > , < span class = "org-string" > 'iff'< / span > );
< / pre >
< / div >
< 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 >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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" > '/dL'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Plate Displacement (encoder)< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gd_iff_opt = 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 >
< / div >
< div id = "outline-container-orgf21ad91" class = "outline-5" >
< h5 id = "orgf21ad91" > < span class = "section-number-5" > 2.3.3.4< / span > DVF Plant< / h5 >
< div class = "outline-text-5" id = "text-2-3-3-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% IFF Plant< / span > < / span >
G_enc_iff_opt = 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_enc_iff_opt(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = tfestimate(meas_iff_plates{< span class = "org-constant" > i< / span > }.Va, meas_iff_plates{< span class = "org-constant" > i< / span > }.de, win, [], [], 1< span class = "org-type" > /< / span > Ts);
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org1dd1189" class = "figure" >
< p > < img src = "figs/enc_plates_opt_iff_comp_simscape_all.png" alt = "enc_plates_opt_iff_comp_simscape_all.png" / >
< / p >
< p > < span class = "figure-number" > Figure 44: < / span > FRF from one actuator to all the encoders when the plant is damped using IFF< / p >
< / div >
< div id = "org579ee95" class = "figure" >
< p > < img src = "figs/damped_iff_plates_plant_comp_diagonal.png" alt = "damped_iff_plates_plant_comp_diagonal.png" / >
< / p >
< p > < span class = "figure-number" > Figure 45: < / span > Comparison of the diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)< / p >
< / div >
< div id = "org1ba23df" class = "figure" >
< p > < img src = "figs/damped_iff_plates_plant_comp_off_diagonal.png" alt = "damped_iff_plates_plant_comp_off_diagonal.png" / >
< / p >
< p > < span class = "figure-number" > Figure 46: < / span > Comparison of the off-diagonal elements of the transfer functions from \(\bm{u}\) to \(d\bm{\mathcal{L}}_m\) with active damping (IFF) applied with an optimal gain \(g = 400\)< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org9090d9d" class = "outline-4" >
< h4 id = "org9090d9d" > < span class = "section-number-4" > 2.3.4< / span > Effect of IFF on the plant - FRF< / h4 >
< div class = "outline-text-4" id = "text-2-3-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'identified_plants_enc_plates.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'G_dvf'< / span > );
< / pre >
< / div >
< div id = "org1fa27f9" class = "figure" >
< p > < img src = "figs/enc_plant_plates_effect_iff.png" alt = "enc_plant_plates_effect_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 47: < / span > Effect of the IFF control strategy on the transfer function from \(\bm{\tau}\) to \(d\bm{\mathcal{L}}_m\)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org5a28e2e" class = "outline-4" >
< h4 id = "org5a28e2e" > < span class = "section-number-4" > 2.3.5< / span > Save Damped Plant< / h4 >
< div class = "outline-text-4" id = "text-2-3-5" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/damped_plant_enc_plates.mat'< / span > , < span class = "org-string" > 'f'< / span > , < span class = "org-string" > 'Ts'< / span > , < span class = "org-string" > 'G_enc_iff_opt'< / span > )
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org14e43ac" class = "outline-3" >
< h3 id = "org14e43ac" > < span class = "section-number-3" > 2.4< / span > HAC Control - Frame of the struts< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< p >
< a id = "org69c97ad" > < / a >
< / p >
< p >
In a first approximation, the Jacobian matrix can be used instead of using the inverse kinematic equations.
< / p >
< div id = "org600fbb5" class = "figure" >
< p > < img src = "figs/control_architecture_hac_iff_L.png" alt = "control_architecture_hac_iff_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 48: < / span > HAC-LAC: IFF + Control in the frame of the legs< / p >
< / div >
< / div >
< div id = "outline-container-org991fc81" class = "outline-4" >
< h4 id = "org991fc81" > < span class = "section-number-4" > 2.4.1< / span > Simscape Model< / h4 >
< div class = "outline-text-4" id = "text-2-4-1" >
< p >
Let’ s start with the Simscape model and the damped plant.
< / p >
< p >
Apply HAC control and verify the system is stable.
< / p >
< p >
Then, try the control strategy on the real plant.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'Kiff.mat'< / span > , < span class = "org-string" > 'Kiff'< / span > )
< / pre >
< / div >
< div class = "org-src-container" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > , ...
< span class = "org-string" > 'controller_type'< / span > , < span class = "org-string" > 'iff'< / span > );
< / pre >
< / div >
< 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 >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/du'< / 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" > '/dL'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Plate Displacement (encoder)< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gd_iff_opt = 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 class = "org-src-container" >
< pre class = "src src-matlab" > isstable(Gd_iff_opt)
< / pre >
< / div >
< pre class = "example" >
1
< / pre >
< div id = "orgc147a1b" class = "figure" >
< p > < img src = "figs/hac_iff_struts_enc_plates_plant_bode.png" alt = "hac_iff_struts_enc_plates_plant_bode.png" / >
< / p >
< p > < span class = "figure-number" > Figure 49: < / span > Transfer function from \(u\) to \(d\mathcal{L}_m\) with IFF (diagonal elements)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org457939e" class = "outline-4" >
< h4 id = "org457939e" > < span class = "section-number-4" > 2.4.2< / span > HAC Controller< / h4 >
< div class = "outline-text-4" id = "text-2-4-2" >
< p >
Let’ s try to have 100Hz bandwidth:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Lead< / span > < / span >
a = 2; < span class = "org-comment" > % Amount of phase lead / width of the phase lead / high frequency gain< / span >
wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100; < span class = "org-comment" > % Frequency with the maximum phase lead [rad/s]< / span >
H_lead = (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > sqrt(a)))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > sqrt(a)));
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Low Pass filter< / span > < / span >
H_lpf = 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 > 200);
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Notch< / span > < / span >
gm = 0.02;
xi = 0.3;
wn = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 700;
H_notch = (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > gm< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > wn< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > wn< span class = "org-type" > ^< / span > 2)< span class = "org-type" > /< / span > (s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > 2< span class = "org-type" > *< / span > xi< span class = "org-type" > *< / span > wn< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > wn< span class = "org-type" > ^< / span > 2);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Khac_iff_struts = < span class = "org-type" > -< / span > (1< span class = "org-type" > /< / span > (2.87e< span class = "org-type" > -< / span > 5)) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Gain< / span >
H_lead < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Lead< / span >
H_notch < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Notch< / span >
(2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100< span class = "org-type" > /< / span > s) < span class = "org-type" > *< / span > ...< span class = "org-comment" > % Integrator< / span >
eye(6); < span class = "org-comment" > % 6x6 Diagonal< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/Khac_iff_struts.mat'< / span > , < span class = "org-string" > 'Khac_iff_struts'< / span > )
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Lhac_iff_struts = Khac_iff_struts< span class = "org-type" > *< / span > Gd_iff_opt;
< / pre >
< / div >
< div id = "org152d5ee" class = "figure" >
< p > < img src = "figs/loop_gain_hac_iff_struts.png" alt = "loop_gain_hac_iff_struts.png" / >
< / p >
< p > < span class = "figure-number" > Figure 50: < / span > Diagonal and off-diagonal elements of the Loop gain for “ HAC-IFF-Struts” < / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org98d910a" class = "outline-4" >
< h4 id = "org98d910a" > < span class = "section-number-4" > 2.4.3< / span > Experimental Loop Gain< / h4 >
< div class = "outline-text-4" id = "text-2-4-3" >
< ul class = "org-ul" >
< li class = "off" > < code > [  ]< / code > Find a more clever way to do the multiplication< / li >
< / ul >
< div class = "org-src-container" >
< pre class = "src src-matlab" > L_frf = zeros(size(G_enc_iff_opt));
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:size(G_enc_iff_opt, 1)< / span >
L_frf(< span class = "org-constant" > i< / span > , < span class = "org-type" > :< / span > , < span class = "org-type" > :< / span > ) = squeeze(G_enc_iff_opt(< span class = "org-constant" > i< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ))< span class = "org-type" > *< / span > freqresp(Khac_iff_struts, f(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > );
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< div id = "org5d41dbc" class = "figure" >
< p > < img src = "figs/hac_iff_plates_exp_loop_gain_diag.png" alt = "hac_iff_plates_exp_loop_gain_diag.png" / >
< / p >
< p > < span class = "figure-number" > Figure 51: < / span > Diagonal and Off-diagonal elements of the Loop gain (experimental data)< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org317850e" class = "outline-4" >
< h4 id = "org317850e" > < span class = "section-number-4" > 2.4.4< / span > Verification of the Stability using the Simscape model< / h4 >
< div class = "outline-text-4" id = "text-2-4-4" >
< div class = "org-src-container" >
< 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" > 'plates'< / span > , ...
< span class = "org-string" > 'actuator_type'< / span > , < span class = "org-string" > 'flexible'< / span > , ...
< span class = "org-string" > 'controller_type'< / span > , < span class = "org-string" > 'hac-iff-struts'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gd_iff_hac_opt = linearize(mdl, io, 0.0, options);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > isstable(Gd_iff_hac_opt)
< / pre >
< / div >
< pre class = "example" >
1
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org247dba8" class = "outline-3" >
< h3 id = "org247dba8" > < span class = "section-number-3" > 2.5< / span > Reference Tracking< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
< p >
< a id = "org7f2d506" > < / a >
< / p >
< / div >
< div id = "outline-container-orgd862f99" class = "outline-4" >
< h4 id = "orgd862f99" > < span class = "section-number-4" > 2.5.1< / span > Load< / h4 >
< div class = "outline-text-4" id = "text-2-5-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'Khac_iff_struts.mat'< / span > , < span class = "org-string" > 'Khac_iff_struts'< / span > )
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org2a92c68" class = "outline-4" >
< h4 id = "org2a92c68" > < span class = "section-number-4" > 2.5.2< / span > Y-Z Scans< / h4 >
< div class = "outline-text-4" id = "text-2-5-2" >
< / div >
< div id = "outline-container-org3c5f01d" class = "outline-5" >
< h5 id = "org3c5f01d" > < span class = "section-number-5" > 2.5.2.1< / span > Generate the Scan< / h5 >
< div class = "outline-text-5" id = "text-2-5-2-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Rx_yz = generateYZScanTrajectory(...
< span class = "org-string" > 'y_tot'< / span > , 4e< span class = "org-type" > -< / span > 6, ...
< span class = "org-string" > 'z_tot'< / span > , 8e< span class = "org-type" > -< / span > 6, ...
< span class = "org-string" > 'n'< / span > , 5, ...
< span class = "org-string" > 'Ts'< / span > , 1e< span class = "org-type" > -< / span > 3, ...
< span class = "org-string" > 'ti'< / span > , 2, ...
< span class = "org-string" > 'tw'< / span > , 0.5, ...
< span class = "org-string" > 'ty'< / span > , 2, ...
< span class = "org-string" > 'tz'< / span > , 1);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot(Rx_yz(< span class = "org-type" > :< / span > ,1), Rx_yz(< span class = "org-type" > :< / span > ,3), ...
< span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Y motion'< / span > )
plot(Rx_yz(< span class = "org-type" > :< / span > ,1), Rx_yz(< span class = "org-type" > :< / span > ,4), ...
< span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Z motion'< / span > )
hold off;
xlabel(< span class = "org-string" > 'Time [s]'< / span > );
ylabel(< span class = "org-string" > 'Displacement [m]'< / span > );
legend(< span class = "org-string" > 'location'< / span > , < span class = "org-string" > 'northeast'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot(Rx_yz(< span class = "org-type" > :< / span > ,3), Rx_yz(< span class = "org-type" > :< / span > ,4));
xlabel(< span class = "org-string" > 'y [m]'< / span > ); ylabel(< span class = "org-string" > 'z [m]'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org213a892" class = "outline-5" >
< h5 id = "org213a892" > < span class = "section-number-5" > 2.5.2.2< / span > Reference Signal for the Strut lengths< / h5 >
< div class = "outline-text-5" id = "text-2-5-2-2" >
< p >
Let’ s use the Jacobian to estimate the wanted strut length as a function of time.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > dL_ref = [n_hexapod.geometry.J< span class = "org-type" > *< / span > Rx_yz(< span class = "org-type" > :< / span > , 2< span class = "org-type" > :< / span > 7)< span class = "org-type" > '< / span > ]< span class = "org-type" > '< / span > ;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
< 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 >
plot(Rx_yz(< span class = "org-type" > :< / span > ,1), dL_ref(< span class = "org-type" > :< / span > , < span class = "org-constant" > i< / span > ))
< span class = "org-keyword" > end< / span >
xlabel(< span class = "org-string" > 'Time [s]'< / span > ); ylabel(< span class = "org-string" > 'Displacement [m]'< / span > );
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge04e29b" class = "outline-5" >
< h5 id = "orge04e29b" > < span class = "section-number-5" > 2.5.2.3< / span > Time domain simulation with 2DoF model< / h5 >
< div class = "outline-text-5" id = "text-2-5-2-3" >
< div class = "org-src-container" >
< 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" > '2dof'< / span > , ...
< span class = "org-string" > 'flex_top_type'< / span > , < span class = "org-string" > '3dof'< / span > , ...
< span class = "org-string" > 'motion_sensor_type'< / span > , < span class = "org-string" > 'plates'< / 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" > 'hac-iff-struts'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > mdl< / span > ,< span class = "org-string" > 'StopTime'< / span > , num2str(Rx_yz(< span class = "org-variable-name" > end< / span > ,1)))
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > mdl< / span > ,< span class = "org-string" > 'SimulationCommand'< / span > ,< span class = "org-string" > 'start'< / span > )
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > out.X.Data = out.X.Data < span class = "org-type" > -< / span > out.X.Data(1,< span class = "org-type" > :< / span > );
< / pre >
< / div >
< div id = "org7b6723e" class = "figure" >
< p > < img src = "figs/ref_track_hac_iff_struts_yz_plane.png" alt = "ref_track_hac_iff_struts_yz_plane.png" / >
< / p >
< p > < span class = "figure-number" > Figure 52: < / span > Simulated Y-Z motion< / p >
< / div >
< div id = "orgd3216d3" class = "figure" >
< p > < img src = "figs/ref_track_hac_iff_struts_yz_time.png" alt = "ref_track_hac_iff_struts_yz_time.png" / >
< / p >
< p > < span class = "figure-number" > Figure 53: < / span > Y and Z motion as a function of time as well as the reference signals< / p >
< / div >
< div id = "orgdf9d1bd" class = "figure" >
< p > < img src = "figs/ref_track_hac_iff_struts_pos_error.png" alt = "ref_track_hac_iff_struts_pos_error.png" / >
< / p >
< p > < span class = "figure-number" > Figure 54: < / span > Positioning errors as a function of time< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org995b778" class = "outline-4" >
< h4 id = "org995b778" > < span class = "section-number-4" > 2.5.3< / span > “ NASS” reference path< / h4 >
< div class = "outline-text-4" id = "text-2-5-3" >
< / div >
< div id = "outline-container-org065d872" class = "outline-5" >
< h5 id = "org065d872" > < span class = "section-number-5" > 2.5.3.1< / span > Generate Path< / h5 >
< div class = "outline-text-5" id = "text-2-5-3-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > ref_path = [ ...
0, 0, 0;
0, 0, 1; < span class = "org-comment" > % N< / span >
0, 4, 1;
3, 0, 1;
3, 4, 1;
3, 4, 0;
4, 0, 0;
4, 0, 1; < span class = "org-comment" > % A< / span >
4, 3, 1;
5, 4, 1;
6, 4, 1;
7, 3, 1;
7, 2, 1;
4, 2, 1;
4, 3, 1;
5, 4, 1;
6, 4, 1;
7, 3, 1;
7, 0, 1;
7, 0, 0;
8, 0, 0;
8, 0, 1; < span class = "org-comment" > % S< / span >
11, 0, 1;
11, 2, 1;
8, 2, 1;
8, 4, 1;
11, 4, 1;
11, 4, 0;
12, 0, 0;
12, 0, 1; < span class = "org-comment" > % S< / span >
15, 0, 1;
15, 2, 1;
12, 2, 1;
12, 4, 1;
15, 4, 1;
15, 4, 0;
];
< span class = "org-comment" > % Center the trajectory arround zero< / span >
ref_path = ref_path < span class = "org-type" > -< / span > (max(ref_path) < span class = "org-type" > -< / span > min(ref_path))< span class = "org-type" > /< / span > 2;
< span class = "org-comment" > % Define the X-Y-Z cuboid dimensions containing the trajectory< / span >
X_max = 10e< span class = "org-type" > -< / span > 6;
Y_max = 4e< span class = "org-type" > -< / span > 6;
Z_max = 2e< span class = "org-type" > -< / span > 6;
ref_path = ([X_max, Y_max, Z_max]< span class = "org-type" > ./< / span > max(ref_path))< span class = "org-type" > .*< / span > ref_path; < span class = "org-comment" > % [m]< / span >
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Rx_nass = generateXYZTrajectory(< span class = "org-string" > 'points'< / span > , ref_path);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot(1e6< span class = "org-type" > *< / span > Rx_nass(Rx_nass(< span class = "org-type" > :< / span > ,4)< span class = "org-type" > > < / span > 0, 2), 1e6< span class = "org-type" > *< / span > Rx_nass(Rx_nass(< span class = "org-type" > :< / span > ,4)< span class = "org-type" > > < / span > 0, 3), < span class = "org-string" > 'k.'< / span > )
xlabel(< span class = "org-string" > 'X [$\mu m$]'< / span > );
ylabel(< span class = "org-string" > 'Y [$\mu m$]'< / span > );
< span class = "org-type" > axis< / span > equal;
xlim(1e6< span class = "org-type" > *< / span > [min(Rx_nass(< span class = "org-type" > :< / span > ,2)), max(Rx_nass(< span class = "org-type" > :< / span > ,2))]);
ylim(1e6< span class = "org-type" > *< / span > [min(Rx_nass(< span class = "org-type" > :< / span > ,3)), max(Rx_nass(< span class = "org-type" > :< / span > ,3))]);
< / pre >
< / div >
< div id = "org73c947b" class = "figure" >
< p > < img src = "figs/ref_track_test_nass.png" alt = "ref_track_test_nass.png" / >
< / p >
< p > < span class = "figure-number" > Figure 55: < / span > Reference path corresponding to the “ NASS” acronym< / p >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot3(Rx_nass(< span class = "org-type" > :< / span > ,2), Rx_nass(< span class = "org-type" > :< / span > ,3), Rx_nass(< span class = "org-type" > :< / span > ,4), < span class = "org-string" > 'k-'< / span > );
xlabel(< span class = "org-string" > 'x'< / span > );
ylabel(< span class = "org-string" > 'y'< / span > );
zlabel(< span class = "org-string" > 'z'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
hold on;
plot(Rx_nass(< span class = "org-type" > :< / span > ,1), Rx_nass(< span class = "org-type" > :< / span > ,2));
plot(Rx_nass(< span class = "org-type" > :< / span > ,1), Rx_nass(< span class = "org-type" > :< / span > ,3));
plot(Rx_nass(< span class = "org-type" > :< / span > ,1), Rx_nass(< span class = "org-type" > :< / span > ,4));
hold off;
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
scatter(Rx_nass(< span class = "org-type" > :< / span > ,2), Rx_nass(< span class = "org-type" > :< / span > ,3), 1, Rx_nass(< span class = "org-type" > :< / span > ,4), < span class = "org-string" > 'filled'< / span > )
colormap winter
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org57675af" class = "outline-5" >
< h5 id = "org57675af" > < span class = "section-number-5" > 2.5.3.2< / span > Simscape Simulations< / h5 >
< div class = "outline-text-5" id = "text-2-5-3-2" >
< div class = "org-src-container" >
< 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" > '2dof'< / span > , ...
< span class = "org-string" > 'flex_top_type'< / span > , < span class = "org-string" > '3dof'< / span > , ...
< span class = "org-string" > 'motion_sensor_type'< / span > , < span class = "org-string" > 'plates'< / 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" > 'hac-iff-struts'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > mdl< / span > ,< span class = "org-string" > 'StopTime'< / span > , num2str(Rx_nass(< span class = "org-variable-name" > end< / span > ,1)))
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > mdl< / span > ,< span class = "org-string" > 'SimulationCommand'< / span > ,< span class = "org-string" > 'start'< / span > )
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > out.X.Data = out.X.Data < span class = "org-type" > -< / span > out.X.Data(1,< span class = "org-type" > :< / span > );
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org75465ae" class = "outline-4" >
< h4 id = "org75465ae" > < span class = "section-number-4" > 2.5.4< / span > Save Reference paths< / h4 >
< div class = "outline-text-4" id = "text-2-5-4" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/reference_path.mat'< / span > , < span class = "org-string" > 'Rx_yz'< / span > , < span class = "org-string" > 'Rx_nass'< / span > )
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-orge43ac25" class = "outline-4" >
< h4 id = "orge43ac25" > < span class = "section-number-4" > 2.5.5< / span > Experimental Results< / h4 >
< / div >
< / div >
< div id = "outline-container-org71bb087" class = "outline-3" >
< h3 id = "org71bb087" > < span class = "section-number-3" > 2.6< / span > Feedforward (Open-Loop) Control< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
< / div >
< div id = "outline-container-org3ea802c" class = "outline-4" >
< h4 id = "org3ea802c" > < span class = "section-number-4" > 2.6.1< / span > Introduction< / h4 >
< div class = "outline-text-4" id = "text-2-6-1" >
< div id = "org4a1bdc9" class = "figure" >
< p > < img src = "figs/control_architecture_iff_feedforward.png" alt = "control_architecture_iff_feedforward.png" / >
< / p >
< p > < span class = "figure-number" > Figure 56: < / span > Feedforward control in the frame of the legs< / p >
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< / div >
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< / div >
< / div >
< div id = "outline-container-orgc7d508e" class = "outline-4" >
< h4 id = "orgc7d508e" > < span class = "section-number-4" > 2.6.2< / span > Simple Feedforward Controller< / h4 >
< div class = "outline-text-4" id = "text-2-6-2" >
< p >
Let’ s estimate the mean DC gain for the damped plant (diagonal elements:)
< / p >
< pre class = "example" >
1.773e-05
< / pre >
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< p >
The feedforward controller is then taken as the inverse of this gain (the minus sign is there manually added as it is “ removed” by the < code > abs< / code > function):
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Kff_iff_L = < span class = "org-type" > -< / span > 1< span class = "org-type" > /< / span > mean(diag(abs(squeeze(mean(G_enc_iff_opt(f< span class = "org-type" > > < / span > 2 < span class = "org-type" > & < / span > f< span class = "org-type" > < < / span > 4,< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ))))));
< / pre >
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< / div >
< p >
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The open-loop gain (feedforward controller times the damped plant) is shown in Figure < a href = "#org2a759b4" > 57< / a > .
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< / p >
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< div id = "org2a759b4" class = "figure" >
< p > < img src = "figs/open_loop_gain_feedforward_iff_struts.png" alt = "open_loop_gain_feedforward_iff_struts.png" / >
< / p >
< p > < span class = "figure-number" > Figure 57: < / span > Diagonal elements of the “ open loop gain” < / p >
< / div >
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< p >
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And save the feedforward controller for further use:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Kff_iff_L = zpk(Kff_iff_L)< span class = "org-type" > *< / span > eye(6);
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > save(< span class = "org-string" > 'matlab/mat/feedforward_iff.mat'< / span > , < span class = "org-string" > 'Kff_iff_L'< / span > )
< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org4c2c4ae" class = "outline-4" >
< h4 id = "org4c2c4ae" > < span class = "section-number-4" > 2.6.3< / span > Test with Simscape Model< / h4 >
< div class = "outline-text-4" id = "text-2-6-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > load(< span class = "org-string" > 'reference_path.mat'< / span > , < span class = "org-string" > 'Rx_yz'< / span > );
< / pre >
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< / div >
< / div >
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< / div >
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< / div >
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< div id = "outline-container-orgd127949" class = "outline-3" >
< h3 id = "orgd127949" > < span class = "section-number-3" > 2.7< / span > Feedback/Feedforward control in the frame of the struts< / h3 >
< div class = "outline-text-3" id = "text-2-7" >
< div id = "org962f0fa" class = "figure" >
< p > < img src = "figs/control_architecture_hac_iff_L_feedforward.png" alt = "control_architecture_hac_iff_L_feedforward.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 58: < / span > Feedback/Feedforward control in the frame of the legs< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org762064f" class = "outline-2" >
< h2 id = "org762064f" > < span class = "section-number-2" > 3< / span > Functions< / h2 >
< div class = "outline-text-2" id = "text-3" >
< / div >
< div id = "outline-container-org9c82eb4" class = "outline-3" >
< h3 id = "org9c82eb4" > < span class = "section-number-3" > 3.1< / span > < code > generateXYZTrajectory< / code > < / h3 >
< div class = "outline-text-3" id = "text-3-1" >
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< p >
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< a id = "org4e35c09" > < / a >
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< / p >
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< / div >
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< div id = "outline-container-org6177d10" class = "outline-4" >
< h4 id = "org6177d10" > Function description< / h4 >
< div class = "outline-text-4" id = "text-org6177d10" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [ref]< / span > = < span class = "org-function-name" > generateXYZTrajectory< / span > (< span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateXYZTrajectory -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [ref] = generateXYZTrajectory(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - ref - Reference Signal< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbf6b8b1" class = "outline-4" >
< h4 id = "orgbf6b8b1" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-orgbf6b8b1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > args< / span > .points double {mustBeNumeric} = zeros(2, 3) < span class = "org-comment" > % [m]< / span >
< span class = "org-variable-name" > args< / span > .ti (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % Time to go to first point and after last point [s]< / span >
< span class = "org-variable-name" > args< / span > .tw (1,1) double {mustBeNumeric, mustBePositive} = 0.5 < span class = "org-comment" > % Time wait between each point [s]< / span >
< span class = "org-variable-name" > args< / span > .tm (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % Motion time between points [s]< / span >
< span class = "org-variable-name" > args< / span > .Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e< span class = "org-type" > -< / span > 3 < span class = "org-comment" > % Sampling Time [s]< / span >
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgb818291" class = "outline-4" >
< h4 id = "orgb818291" > Initialize Time Vectors< / h4 >
< div class = "outline-text-4" id = "text-orgb818291" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > time_i = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.ti;
time_w = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.tw;
time_m = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.tm;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org361222a" class = "outline-4" >
< h4 id = "org361222a" > XYZ Trajectory< / h4 >
< div class = "outline-text-4" id = "text-org361222a" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-comment" > % Go to initial position< / span >
xyz = (args.points(1,< span class = "org-type" > :< / span > ))< span class = "org-type" > '*< / span > (time_i< span class = "org-type" > /< / span > args.ti);
< span class = "org-comment" > % Wait< / span >
xyz = [xyz, xyz(< span class = "org-type" > :< / span > ,end)< span class = "org-type" > .*< / span > ones(size(time_w))];
< span class = "org-comment" > % Scans< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 2:size(args.points, 1)< / span >
< span class = "org-comment" > % Go to next point< / span >
xyz = [xyz, xyz(< span class = "org-type" > :< / span > ,end) < span class = "org-type" > +< / span > (args.points(< span class = "org-constant" > i< / span > ,< span class = "org-type" > :< / span > )< span class = "org-type" > '< / span > < span class = "org-type" > -< / span > xyz(< span class = "org-type" > :< / span > ,end))< span class = "org-type" > *< / span > (time_m< span class = "org-type" > /< / span > args.tm)];
< span class = "org-comment" > % Wait a litle bit< / span >
xyz = [xyz, xyz(< span class = "org-type" > :< / span > ,end)< span class = "org-type" > .*< / span > ones(size(time_w))];
< span class = "org-keyword" > end< / span >
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< span class = "org-comment" > % End motion< / span >
xyz = [xyz, xyz(< span class = "org-type" > :< / span > ,end) < span class = "org-type" > -< / span > xyz(< span class = "org-type" > :< / span > ,end)< span class = "org-type" > *< / span > (time_i< span class = "org-type" > /< / span > args.ti)];
< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org84542e6" class = "outline-4" >
< h4 id = "org84542e6" > Reference Signal< / h4 >
< div class = "outline-text-4" id = "text-org84542e6" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > t = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > *< / span > (length(xyz) < span class = "org-type" > -< / span > 1);
< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ref = zeros(length(xyz), 7);
ref(< span class = "org-type" > :< / span > , 1) = t;
ref(< span class = "org-type" > :< / span > , 2< span class = "org-type" > :< / span > 4) = xyz< span class = "org-type" > '< / span > ;
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< / pre >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org4433a1b" class = "outline-3" >
< h3 id = "org4433a1b" > < span class = "section-number-3" > 3.2< / span > < code > generateYZScanTrajectory< / code > < / h3 >
< div class = "outline-text-3" id = "text-3-2" >
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< p >
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< a id = "org5b64c20" > < / a >
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< / p >
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< / div >
< div id = "outline-container-org0479bfd" class = "outline-4" >
< h4 id = "org0479bfd" > Function description< / h4 >
< div class = "outline-text-4" id = "text-org0479bfd" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [ref]< / span > = < span class = "org-function-name" > generateYZScanTrajectory< / span > (< span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % generateYZScanTrajectory -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [ref] = generateYZScanTrajectory(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - ref - Reference Signal< / span >
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< / pre >
< / div >
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< / div >
< / div >
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< div id = "outline-container-org7be834b" class = "outline-4" >
< h4 id = "org7be834b" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-org7be834b" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > args< / span > .y_tot (1,1) double {mustBeNumeric} = 10e< span class = "org-type" > -< / span > 6 < span class = "org-comment" > % [m]< / span >
< span class = "org-variable-name" > args< / span > .z_tot (1,1) double {mustBeNumeric} = 10e< span class = "org-type" > -< / span > 6 < span class = "org-comment" > % [m]< / span >
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< span class = "org-variable-name" > args< / span > .n (1,1) double {mustBeInteger, mustBePositive} = 10 < span class = "org-comment" > % [-]< / span >
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< span class = "org-variable-name" > args< / span > .Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e< span class = "org-type" > -< / span > 4 < span class = "org-comment" > % [s]< / span >
< span class = "org-variable-name" > args< / span > .ti (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % [s]< / span >
< span class = "org-variable-name" > args< / span > .tw (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % [s]< / span >
< span class = "org-variable-name" > args< / span > .ty (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % [s]< / span >
< span class = "org-variable-name" > args< / span > .tz (1,1) double {mustBeNumeric, mustBePositive} = 1 < span class = "org-comment" > % [s]< / span >
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org81214f7" class = "outline-4" >
< h4 id = "org81214f7" > Initialize Time Vectors< / h4 >
< div class = "outline-text-4" id = "text-org81214f7" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > time_i = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.ti;
time_w = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.tw;
time_y = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.ty;
time_z = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.tz;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5ab49af" class = "outline-4" >
< h4 id = "org5ab49af" > Y and Z vectors< / h4 >
< div class = "outline-text-4" id = "text-org5ab49af" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-comment" > % Go to initial position< / span >
y = (time_i< span class = "org-type" > /< / span > args.ti)< span class = "org-type" > *< / span > (args.y_tot< span class = "org-type" > /< / span > 2);
< span class = "org-comment" > % Wait< / span >
y = [y, y(end)< span class = "org-type" > *< / span > ones(size(time_w))];
< span class = "org-comment" > % Scans< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:args.n< / span >
< span class = "org-keyword" > if< / span > mod(< span class = "org-constant" > i< / span > ,2) < span class = "org-type" > ==< / span > 0
y = [y, < span class = "org-type" > -< / span > (args.y_tot< span class = "org-type" > /< / span > 2) < span class = "org-type" > +< / span > (time_y< span class = "org-type" > /< / span > args.ty)< span class = "org-type" > *< / span > args.y_tot];
< span class = "org-keyword" > else< / span >
y = [y, (args.y_tot< span class = "org-type" > /< / span > 2) < span class = "org-type" > -< / span > (time_y< span class = "org-type" > /< / span > args.ty)< span class = "org-type" > *< / span > args.y_tot];
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > if< / span > < span class = "org-constant" > i< / span > < span class = "org-type" > < < / span > args.n
y = [y, y(end)< span class = "org-type" > *< / span > ones(size(time_z))];
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
< span class = "org-comment" > % Wait a litle bit< / span >
y = [y, y(end)< span class = "org-type" > *< / span > ones(size(time_w))];
< span class = "org-comment" > % End motion< / span >
y = [y, y(end) < span class = "org-type" > -< / span > y(end)< span class = "org-type" > *< / span > time_i< span class = "org-type" > /< / span > args.ti];
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-comment" > % Go to initial position< / span >
z = (time_i< span class = "org-type" > /< / span > args.ti)< span class = "org-type" > *< / span > (args.z_tot< span class = "org-type" > /< / span > 2);
< span class = "org-comment" > % Wait< / span >
z = [z, z(end)< span class = "org-type" > *< / span > ones(size(time_w))];
< span class = "org-comment" > % Scans< / span >
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:args.n< / span >
z = [z, z(end)< span class = "org-type" > *< / span > ones(size(time_y))];
< span class = "org-keyword" > if< / span > < span class = "org-constant" > i< / span > < span class = "org-type" > < < / span > args.n
z = [z, z(end) < span class = "org-type" > -< / span > (time_z< span class = "org-type" > /< / span > args.tz)< span class = "org-type" > *< / span > args.z_tot< span class = "org-type" > /< / span > (args.n< span class = "org-type" > -< / span > 1)];
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
< span class = "org-comment" > % Wait a litle bit< / span >
z = [z, z(end)< span class = "org-type" > *< / span > ones(size(time_w))];
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< span class = "org-comment" > % End motion< / span >
z = [z, z(end) < span class = "org-type" > -< / span > z(end)< span class = "org-type" > *< / span > time_i< span class = "org-type" > /< / span > args.ti];
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org796db82" class = "outline-4" >
< h4 id = "org796db82" > Reference Signal< / h4 >
< div class = "outline-text-4" id = "text-org796db82" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > t = 0< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > :< / span > args.Ts< span class = "org-type" > *< / span > (length(y) < span class = "org-type" > -< / span > 1);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > ref = zeros(length(y), 7);
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ref(< span class = "org-type" > :< / span > , 1) = t;
ref(< span class = "org-type" > :< / span > , 3) = y;
ref(< span class = "org-type" > :< / span > , 4) = z;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org01056ff" class = "outline-3" >
< h3 id = "org01056ff" > < span class = "section-number-3" > 3.3< / span > < code > getTransformationMatrixAcc< / code > < / h3 >
< div class = "outline-text-3" id = "text-3-3" >
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< p >
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< a id = "org39415ee" > < / a >
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< / p >
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< / div >
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< div id = "outline-container-org37f755d" class = "outline-4" >
< h4 id = "org37f755d" > Function description< / h4 >
< div class = "outline-text-4" id = "text-org37f755d" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [M]< / span > = < span class = "org-function-name" > getTransformationMatrixAcc< / span > (< span class = "org-variable-name" > Opm< / span > , < span class = "org-variable-name" > Osm< / span > )
< span class = "org-comment" > % getTransformationMatrixAcc -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [M] = getTransformationMatrixAcc(Opm, Osm)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - Opm - Nx3 (N = number of accelerometer measurements) X,Y,Z position of accelerometers< / span >
< span class = "org-comment" > % - Opm - Nx3 (N = number of accelerometer measurements) Unit vectors representing the accelerometer orientation< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - M - Transformation Matrix< / span >
< / pre >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgf2435ad" class = "outline-4" >
< h4 id = "orgf2435ad" > Transformation matrix from motion of the solid body to accelerometer measurements< / h4 >
< div class = "outline-text-4" id = "text-orgf2435ad" >
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< p >
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Let’ s try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in \(\{O\}\).
For any point \(p_i\) of the solid body (corresponding to an accelerometer), we can write:
< / p >
\begin{equation}
\begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + p_i \times \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}
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< p >
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We can write the cross product as a matrix product using the skew-symmetric transformation:
< / p >
\begin{equation}
\begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + \underbrace{\begin{bmatrix}
0 & p_{i,z} & -p_{i,y} \\
-p_{i,z} & 0 & p_{i,x} \\
p_{i,y} & -p_{i,x} & 0
\end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}
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< p >
If we now want to know the (scalar) acceleration \(a_i\) of the point \(p_i\) in the direction of the accelerometer direction \(\hat{s}_i\), we can just project the 3d acceleration on \(\hat{s}_i\):
< / p >
\begin{equation}
a_i = \hat{s}_i^T \cdot \begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}
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< p >
Which is equivalent as a simple vector multiplication:
< / p >
\begin{equation}
a_i = \begin{bmatrix}
\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]}
\end{bmatrix}
\begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix} = \begin{bmatrix}
\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]}
\end{bmatrix} {}^O\vec{x}
\end{equation}
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< p >
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And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form.
We obtain Eq. \eqref{eq:M_matrix}.
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< / p >
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< div class = "important" id = "org340212d" >
< p >
The transformation from solid body acceleration \({}^O\vec{x}\) from sensor measured acceleration \(\vec{a}\) is:
< / p >
\begin{equation} \label{eq:M_matrix}
\vec{a} = \underbrace{\begin{bmatrix}
\hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\
\vdots & \vdots \\
\hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]}
\end{bmatrix}}_{M} {}^O\vec{x}
\end{equation}
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< p >
with \(\hat{s}_i\) the unit vector representing the measured direction of the i’ th accelerometer expressed in frame \(\{O\}\) and \(P_{i,[\times]}\) the skew-symmetric matrix representing the cross product of the position of the i’ th accelerometer expressed in frame \(\{O\}\).
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< / p >
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< / div >
< p >
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Let’ s define such matrix using matlab:
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< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > M = zeros(length(Opm), 6);
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(Opm)< / span >
Ri = [0, Opm(3,< span class = "org-constant" > i< / span > ), < span class = "org-type" > -< / span > Opm(2,< span class = "org-constant" > i< / span > );
< span class = "org-type" > -< / span > Opm(3,< span class = "org-constant" > i< / span > ), 0, Opm(1,< span class = "org-constant" > i< / span > );
Opm(2,< span class = "org-constant" > i< / span > ), < span class = "org-type" > -< / span > Opm(1,< span class = "org-constant" > i< / span > ), 0];
M(< span class = "org-constant" > i< / span > , 1< span class = "org-type" > :< / span > 3) = Osm(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > '< / span > ;
M(< span class = "org-constant" > i< / span > , 4< span class = "org-type" > :< / span > 6) = Osm(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > )< span class = "org-type" > '*< / span > Ri;
< span class = "org-keyword" > end< / span >
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > end< / span >
< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgab18063" class = "outline-3" >
< h3 id = "orgab18063" > < span class = "section-number-3" > 3.4< / span > < code > getJacobianNanoHexapod< / code > < / h3 >
< div class = "outline-text-3" id = "text-3-4" >
< p >
< a id = "orgd78b26a" > < / a >
< / p >
< / div >
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< div id = "outline-container-org7e86fe2" class = "outline-4" >
< h4 id = "org7e86fe2" > Function description< / h4 >
< div class = "outline-text-4" id = "text-org7e86fe2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [J]< / span > = < span class = "org-function-name" > getJacobianNanoHexapod< / span > (< span class = "org-variable-name" > Hbm< / span > )
< span class = "org-comment" > % getJacobianNanoHexapod -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [J] = getJacobianNanoHexapod(Hbm)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - Hbm - Height of {B} w.r.t. {M} [m]< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - J - Jacobian Matrix< / span >
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb9e790e" class = "outline-4" >
< h4 id = "orgb9e790e" > Transformation matrix from motion of the solid body to accelerometer measurements< / h4 >
< div class = "outline-text-4" id = "text-orgb9e790e" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Fa = [[< span class = "org-type" > -< / span > 86.05, < span class = "org-type" > -< / span > 74.78, 22.49],
[ 86.05, < span class = "org-type" > -< / span > 74.78, 22.49],
[ 107.79, < span class = "org-type" > -< / span > 37.13, 22.49],
[ 21.74, 111.91, 22.49],
[< span class = "org-type" > -< / span > 21.74, 111.91, 22.49],
[< span class = "org-type" > -< / span > 107.79, < span class = "org-type" > -< / span > 37.13, 22.49]]< span class = "org-type" > '*< / span > 1e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Ai w.r.t. {F} [m]< / span >
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Mb = [[< span class = "org-type" > -< / span > 28.47, < span class = "org-type" > -< / span > 106.25, < span class = "org-type" > -< / span > 22.50],
[ 28.47, < span class = "org-type" > -< / span > 106.25, < span class = "org-type" > -< / span > 22.50],
[ 106.25, 28.47, < span class = "org-type" > -< / span > 22.50],
[ 77.78, 77.78, < span class = "org-type" > -< / span > 22.50],
[< span class = "org-type" > -< / span > 77.78, 77.78, < span class = "org-type" > -< / span > 22.50],
[< span class = "org-type" > -< / span > 106.25, 28.47, < span class = "org-type" > -< / span > 22.50]]< span class = "org-type" > '*< / span > 1e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Bi w.r.t. {M} [m]< / span >
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H = 95e< span class = "org-type" > -< / span > 3; < span class = "org-comment" > % Stewart platform height [m]< / span >
Fb = Mb < span class = "org-type" > +< / span > [0; 0; H]; < span class = "org-comment" > % Bi w.r.t. {F} [m]< / span >
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si = Fb < span class = "org-type" > -< / span > Fa;
si = si< span class = "org-type" > ./< / span > vecnorm(si); < span class = "org-comment" > % Normalize< / span >
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Bb = Mb < span class = "org-type" > -< / span > [0; 0; Hbm];
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J = [si< span class = "org-type" > '< / span > , cross(Bb, si)< span class = "org-type" > '< / span > ];
< / pre >
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< / div >
< / div >
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< / div >
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< / div >
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< / 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: 2021-06-30 mer. 22:41< / p >
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< / div >
< / body >
< / html >
