Rework the presentation of results + analysis
This commit is contained in:
@@ -1,64 +0,0 @@
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%% Remove first second
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Ts = t(2)-t(1);
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t = t(ceil(1/Ts):end);
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d = d(ceil(1/Ts):end);
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dp = dp(ceil(1/Ts):end);
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F = F(ceil(1/Ts):end);
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t = t - t(1);
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F = F - F(1);
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%% LPf
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s = tf('s');
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G_lpf = 1/(1+s/2/pi/10);
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F = lsim(G_lpf, F, t);
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d = lsim(G_lpf, d, t);
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dp = lsim(G_lpf, dp, t);
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%%
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figure;
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plot(t, F);
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figure;
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hold on;
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plot(t, d, 'DisplayName', 'Encoder');
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plot(t, dp, 'DisplayName', 'Mahr')
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hold off;
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legend;
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%%
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t_start = 0.05;
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t_end = 0.12;
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d = d(t < t_end & t > t_start);
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dp = dp(t < t_end & t > t_start);
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F = F(t < t_end & t > t_start);
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d = d - d(1);
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dp = dp - dp(1);
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F = F - F(1);
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%%
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figure;
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plot(F, dp);
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xlabel('Measured Force [N]');
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ylabel('Measured Displacement [m]');
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%%
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sprintf('Stiffness is %.3f [N/mm]', abs(1e-3*(d\F)))
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%%
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figure;
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hold on;
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plot(F, d);
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plot(F, F/(d\F));
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hold off;
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xlabel('Measured Force [N]');
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ylabel('Measured Displacement [m]');
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%%
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figure;
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hold on;
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plot(dp, d);
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hold off;
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xlabel('Probe [m]');
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ylabel('Encoder [m]');
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53
matlab/bench_dimensioning.m
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53
matlab/bench_dimensioning.m
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@@ -0,0 +1,53 @@
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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% Flexible joint Geometry
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% The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]].
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% Its bending stiffness is foreseen to be $k_{R_y}\approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 25\,mrad$.
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% #+name: fig:flexible_joint_geometry
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% #+caption: Geometry of the flexible joint
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% [[file:figs/flexible_joint_geometry.png]]
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% The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$.
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% Let's define the parameters on Matlab.
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kRx = 5; % Bending Stiffness [Nm/rad]
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Rxmax = 25e-3; % Bending Stroke [rad]
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h = 20e-3; % Height [m]
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% Required external applied force
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% The bending $\theta_y$ of the flexible joint due to the force $F_x$ is:
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% \begin{equation}
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% \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}}
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% \end{equation}
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% Therefore, the applied force to test the full range of the flexible joint is:
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% \begin{equation}
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% F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h}
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% \end{equation}
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Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
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% And we obtain:
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sprintf('\\begin{equation} F_{x,max} = %.1f\\, [N] \\end{equation}', Fxmax)
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% Required actuator stroke and sensors range
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% The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$.
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% The corresponding stroke at the location of the force sensor is:
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% \[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]
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dxmax = h*tan(Rxmax);
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sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax)
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67
matlab/error_budget.m
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67
matlab/error_budget.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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% Finite Element Model
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% From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables [[tab:axial_shear_characteristics]] and [[tab:bending_torsion_characteristics]].
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%% Stiffness
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ka = 94e6; % Axial Stiffness [N/m]
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ks = 13e6; % Shear Stiffness [N/m]
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kb = 5; % Bending Stiffness [Nm/rad]
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kt = 260; % Torsional Stiffness [Nm/rad]
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%% Maximum force
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Fa = 469; % Axial Force before yield [N]
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Fs = 242; % Shear Force before yield [N]
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Fb = 0.118; % Bending Force before yield [Nm]
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Ft = 1.508; % Torsional Force before yield [Nm]
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%% Compute the corresponding stroke
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Xa = Fa/ka; % Axial Stroke before yield [m]
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Xs = Fs/ks; % Shear Stroke before yield [m]
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Xb = Fb/kb; % Bending Stroke before yield [rad]
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Xt = Ft/kt; % Torsional Stroke before yield [rad]
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% Setup
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% The setup is schematically represented in Figure [[fig:test_bench_flex_side_bis]].
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% The force is applied on top of the flexible joint with a distance $h$ with the joint's center.
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% The displacement of the flexible joint is also measured at the same height.
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% The height between the joint's center and the force application point is:
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h = 25e-3; % Height [m]
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% Estimation error due to force sensor compression
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% The measured displacement is not done directly at the joint's location.
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% The force sensor compression will then induce an error on the joint's stiffness.
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% The force sensor stiffness $k_F$ is estimated to be around:
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kF = 50/0.05e-3; % [N/m]
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sprintf('k_F = %.1e [N/m]', kF)
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% Estimation error due to height estimation error
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% Let's consider an error in the estimation of the height from the application of the force to the joint's center:
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% \begin{equation}
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% h_{\text{est}} = h (1 + \epsilon)
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% \end{equation}
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% The computed bending stiffness will be:
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% \begin{equation}
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% k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x}
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% \end{equation}
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% And the stiffness estimation error is:
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% \begin{equation}
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% \frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2
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% \end{equation}
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h_err = 0.2e-3; % Height estimation error [m]
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62
matlab/probe_stiffness.m
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62
matlab/probe_stiffness.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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addpath('./mat/');
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% Results :ignore:
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% Let's load the measurement results.
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load('meas_stiff_probe.mat', 't', 'd', 'dp', 'F')
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%% Sampling time [s]
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Ts = (t(end) - t(1))/(length(t)-1);
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%% Remove first second
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t = t(ceil(1/Ts):end);
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d = d(ceil(1/Ts):end);
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dp = dp(ceil(1/Ts):end);
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F = F(ceil(1/Ts):end);
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%% Remove Offset
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t = t - t(1);
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F = F - mean(F(1:ceil(1/Ts)));
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% The time domain measured force and displacement are shown in Figure [[fig:mahr_time_domain]].
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%% Time Domain plots
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figure;
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tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile;
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plot(t, F);
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ylabel('Force [N]'); set(gca, 'XTickLabel',[]);
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ax2 = nexttile;
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plot(t, d);
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xlabel('Time [s]'); ylabel('Displacement [m]');
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% #+RESULTS:
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% : Stiffness is 0.039 [N/mm]
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% This is very close to the 0.04 [N/mm] written in the [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe datasheet]].
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% And compare the linear fit with the raw measurement data (Figure [[fig:mahr_stiffness_f_d_plot]]).
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figure;
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hold on;
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plot(F, d, 'DisplayName', 'Raw data');
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plot(F, F/(d\F), 'DisplayName', 'Linear fit');
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hold off;
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xlabel('Measured Force [N]');
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ylabel('Measured Displacement [m]');
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legend('location', 'southeast');
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86
matlab/probe_vs_encoder.m
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86
matlab/probe_vs_encoder.m
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@@ -0,0 +1,86 @@
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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addpath('./mat/');
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% Results :ignore:
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% Let's load the measurements.
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load('meas_probe_against_encoder.mat', 't', 'd', 'dp', 'F')
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%% Sampling time [s]
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Ts = (t(end) - t(1))/(length(t)-1);
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%% Remove first second
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t = t(ceil(1/Ts):end);
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d = -d(ceil(1/Ts):end);
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dp = -dp(ceil(1/Ts):end);
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F = F(ceil(1/Ts):end);
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% The time domain measured displacement by the probe and by the encoder is shown in Figure [[fig:comp_encoder_probe_time]].
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%% Time Domain plots
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figure;
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hold on;
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plot(t, d, 'DisplayName', 'Encoder');
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plot(t, dp, 'DisplayName', 'Probe');
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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% #+name: fig:comp_encoder_probe_time
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% #+caption: Time domain measurement
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% #+RESULTS:
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% [[file:figs/comp_encoder_probe_time.png]]
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% If we zoom, we see that there is some delay between the encoder and the probe (Figure [[fig:comp_encoder_probe_time_zoom]]).
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%% Zoom
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figure;
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hold on;
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plot(t, d, 'DisplayName', 'Encoder');
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plot(t, dp, 'DisplayName', 'Probe');
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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xlim([7.7, 7.9])
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% #+RESULTS:
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% : The time delay is approximately 15.8 [ms]
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% The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure [[fig:comp_encoder_probe_mismatch]].
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figure;
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hold on;
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plot(t, d-dp, 'DisplayName', 'Raw Mismatch');
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plot(t(1:end-finddelay(d, dp)), d(1:end-finddelay(d, dp))-dp(finddelay(d, dp)+1:end), 'DisplayName', 'Removed Delay');
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hold off;
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xlabel('Time [s]'); ylabel('Measurement Missmatch [m]');
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% #+name: fig:comp_encoder_probe_mismatch
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% #+caption: Measurement mismatch, with and without delay compensation
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% #+RESULTS:
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% [[file:figs/comp_encoder_probe_mismatch.png]]
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% Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure [[fig:comp_encoder_probe_linear_fit]]).
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figure;
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hold on;
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plot(1e3*d, 1e3*dp, 'DisplayName', 'Raw data');
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plot(1e3*d, 1e3*d*(d\dp), 'DisplayName', sprintf('Linear fit: $\\alpha = %.5f$', (d\dp)));
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hold on;
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xlabel('Encoder [mm]'); ylabel('Probe [mm]');
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legend('location', 'southeast')
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