Table of Contents
-
-
- 1. Encoders fixed to the Struts +
- 1. Encoders fixed to the Struts
-
-
- 1.1. Introduction -
- 1.2. Load Data -
- 1.3. Spectral Analysis - Setup -
- 1.4. DVF Plant -
- 1.5. IFF Plant -
- 1.6. Jacobian +
- 1.1. Introduction +
- 1.2. Load Data +
- 1.3. Spectral Analysis - Setup +
- 1.4. DVF Plant +
- 1.5. IFF Plant +
- 1.6. Jacobian
This report is also available as a pdf.
-+-Here are the documentation of the equipment used for this test bench:
@@ -59,29 +76,34 @@ Here are the documentation of the equipment used for this test bench:+-
Figure 1: Nano-Hexapod
+-
Figure 2: Nano-Hexapod and the control electronics
-1 Encoders fixed to the Struts
++1 Encoders fixed to the Struts
--1.1 Introduction
++-1.1 Introduction
+++In this section, the encoders are fixed to the struts. +
+-1.2 Load Data
++1.2 Load Data
-meas_data_lf = {}; @@ -95,8 +117,8 @@ Here are the documentation of the equipment used for this test bench:-1.3 Spectral Analysis - Setup
++1.3 Spectral Analysis - Setup
-% Sampling Time [s] @@ -126,11 +148,11 @@ i_hf = f > 250; % Poi
-1.4 DVF Plant
++1.4 DVF Plant
-First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 3). +First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 3).
@@ -147,14 +169,14 @@ coh_dvf_hf = zeros(length(f), 6, 6);-+
Figure 3: Obtained coherence for the DVF plant
-Then the 6x6 transfer function matrix is estimated (Figure 4). +Then the 6x6 transfer function matrix is estimated (Figure 4).
-%% DVF Plant @@ -169,7 +191,7 @@ G_dvf_hf = zeros(length(f), 6, 6);+-
Figure 4: Measured FRF for the DVF plant
@@ -178,11 +200,11 @@ G_dvf_hf = zeros(length(f), 6, 6);-1.5 IFF Plant
++1.5 IFF Plant
-First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 5). +First, let’s compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure 5).
@@ -199,14 +221,14 @@ coh_iff_hf = zeros(length(f), 6, 6);-+
Figure 5: Obtained coherence for the IFF plant
-Then the 6x6 transfer function matrix is estimated (Figure 6). +Then the 6x6 transfer function matrix is estimated (Figure 6).
-%% IFF Plant @@ -221,7 +243,7 @@ G_iff_hf = zeros(length(f), 6, 6);+-
Figure 6: Measured FRF for the IFF plant
@@ -229,33 +251,80 @@ G_iff_hf = zeros(length(f), 6, 6);-1.6 Jacobian
++1.6 Jacobian
+-+The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. +
+ ++Consider the plant shown in Figure 7 with: +
+-
+
- \(\tau\) the 6 input voltages (going to the PD200 amplifier and then to the APA) +
- \(d\mathcal{L}\) the relative motion sensor outputs (encoders) +
- \(\bm{\tau}_m\) the generated voltage of the force sensor stacks +
- \(J_a\) and \(J_s\) the Jacobians for the actuators and sensors +
++ +
+
+Figure 7: Plant in the cartesian Frame
++First, we load the Jacobian matrix (same for the actuators and sensors). +
load('jacobian.mat', 'J');
-1.6.1 DVF Plant
+ ++-1.6.1 DVF Plant
++The transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\) is computed and shown in Figure 8. +
++ + +G_dvf_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_lf, [2 3 1]), inv(J'))), [3 1 2]); G_dvf_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_hf, [2 3 1]), inv(J'))), [3 1 2]);
+
+
+Figure 8: Measured FRF for the DVF plant in the cartesian frame
+-@@ -263,7 +332,7 @@ G_iff_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_hf, [2 3 1]), i1.6.2 IFF Plant
++1.6.2 IFF Plant
++The transfer function from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{F}}_m\) is computed and shown in Figure 9. +
++ + +G_iff_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_lf, [2 3 1]), inv(J'))), [3 1 2]); G_iff_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_hf, [2 3 1]), inv(J'))), [3 1 2]);
+
+
+Figure 9: Measured FRF for the IFF plant in the cartesian frame
+-diff --git a/test-bench-nano-hexapod.org b/test-bench-nano-hexapod.org index 79856a6..e609e3a 100644 --- a/test-bench-nano-hexapod.org +++ b/test-bench-nano-hexapod.org @@ -47,8 +47,6 @@ #+end_export * Introduction :ignore: -* Test-Bench Description - #+begin_note Here are the documentation of the equipment used for this test bench: - Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]] @@ -60,14 +58,17 @@ Here are the documentation of the equipment used for this test bench: #+name: fig:picture_bench_granite_nano_hexapod #+caption: Nano-Hexapod +#+attr_latex: :width \linewidth [[file:figs/IMG_20210608_152917.jpg]] #+name: fig:picture_bench_granite_overview #+caption: Nano-Hexapod and the control electronics +#+attr_latex: :width \linewidth [[file:figs/IMG_20210608_154722.jpg]] * Encoders fixed to the Struts ** Introduction +In this section, the encoders are fixed to the struts. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -365,10 +366,50 @@ exportFig('figs/enc_struts_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); [[file:figs/enc_struts_iff_frf.png]] ** Jacobian +*** Introduction :ignore: +The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. + +Consider the plant shown in Figure [[fig:nano_hexapod_decentralized_schematic]] with: +- $\tau$ the 6 input voltages (going to the PD200 amplifier and then to the APA) +- $d\mathcal{L}$ the relative motion sensor outputs (encoders) +- $\bm{\tau}_m$ the generated voltage of the force sensor stacks +- $J_a$ and $J_s$ the Jacobians for the actuators and sensors + +#+begin_src latex :file schematic_jacobian_in_out.pdf +\begin{tikzpicture} + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {Plant}; + \coordinate[] (inputF) at (P.west); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + + \node[block, left= of inputF] (Ja) {$\bm{J}^{-T}_a$}; + \node[block, right= of outputL] (Js) {$\bm{J}^{-1}_s$}; + \node[block, right= of outputF] (Jf) {$\bm{J}^{-1}_s$}; + + % Connections and labels + \draw[->] ($(Ja.west)+(-1,0)$) -- (Ja.west) node[above left]{$\bm{\mathcal{F}}$}; + \draw[->] (Ja.east) -- (inputF) node[above left]{$\bm{\tau}$}; + \draw[->] (outputL) -- (Js.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Js.east) -- ++(1, 0) node[above left]{$d\bm{\mathcal{X}}$}; + \draw[->] (outputF) -- (Jf.west) node[above left]{$\bm{\tau}_m$}; + \draw[->] (Jf.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{F}}_m$}; +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_jacobian_in_out +#+caption: Plant in the cartesian Frame +#+RESULTS: +[[file:figs/schematic_jacobian_in_out.png]] + +First, we load the Jacobian matrix (same for the actuators and sensors). #+begin_src matlab load('jacobian.mat', 'J'); #+end_src + *** DVF Plant +The transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ is computed and shown in Figure [[fig:enc_struts_dvf_cart_frf]]. + #+begin_src matlab G_dvf_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_lf, [2 3 1]), inv(J'))), [3 1 2]); G_dvf_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_hf, [2 3 1]), inv(J'))), [3 1 2]); @@ -424,7 +465,18 @@ linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_dvf_cart_frf +#+caption: Measured FRF for the DVF plant in the cartesian frame +#+RESULTS: +[[file:figs/enc_struts_dvf_cart_frf.png]] + *** IFF Plant +The transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{F}}_m$ is computed and shown in Figure [[fig:enc_struts_iff_cart_frf]]. + #+begin_src matlab G_iff_J_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_lf, [2 3 1]), inv(J'))), [3 1 2]); G_iff_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_hf, [2 3 1]), inv(J'))), [3 1 2]); @@ -459,7 +511,7 @@ plot(f(i_lf), abs(G_iff_J_lf(i_lf, 1, 2)), 'color', [0, 0, 0, 0.2], ... hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-7, 1e-1]); +ylim([1e-3, 1e4]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; @@ -479,3 +531,12 @@ yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_cart_frf +#+caption: Measured FRF for the IFF plant in the cartesian frame +#+RESULTS: +[[file:figs/enc_struts_iff_cart_frf.png]] diff --git a/test-bench-nano-hexapod.pdf b/test-bench-nano-hexapod.pdf index 70e7c95..8b00a26 100644 Binary files a/test-bench-nano-hexapod.pdf and b/test-bench-nano-hexapod.pdf differCreated: 2021-06-08 mar. 22:15
+Created: 2021-06-08 mar. 22:38