From 29ebdc27e02b3f9abab2aa3fccc5c0809db253f2 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 18 Feb 2021 11:38:56 +0100 Subject: [PATCH] Rework the presentation of results + analysis --- doc/flex_joints.pdf | Bin 0 -> 84684 bytes figs/flexible_joint_fem_geometry.pdf | Bin 0 -> 60052 bytes figs/flexible_joint_fem_geometry.png | Bin 0 -> 59972 bytes figs/nano_hexapod.pdf | Bin 0 -> 64663 bytes figs/nano_hexapod.png | Bin 0 -> 80043 bytes figs/test_bench_flex_overview.png | Bin 292357 -> 189393 bytes figs/test_bench_flex_side.pdf | Bin 79299 -> 79299 bytes figs/test_bench_flex_side.png | Bin 70963 -> 100566 bytes figs/test_bench_principle.pdf | Bin 0 -> 9494 bytes figs/test_bench_principle.png | Bin 0 -> 12306 bytes figs/test_bench_principle.svg | 2447 ++++++++++++++++++++++++++ matlab/analysis.m | 64 - matlab/bench_dimensioning.m | 53 + matlab/error_budget.m | 67 + matlab/probe_stiffness.m | 62 + matlab/probe_vs_encoder.m | 86 + test-bench-flexible-joints.html | 711 ++++++-- test-bench-flexible-joints.org | 436 ++++- test-bench-flexible-joints.pdf | Bin 3092747 -> 3267046 bytes 19 files changed, 3633 insertions(+), 293 deletions(-) create mode 100644 doc/flex_joints.pdf create mode 100644 figs/flexible_joint_fem_geometry.pdf create mode 100644 figs/flexible_joint_fem_geometry.png create mode 100644 figs/nano_hexapod.pdf create mode 100644 figs/nano_hexapod.png create mode 100644 figs/test_bench_principle.pdf create mode 100644 figs/test_bench_principle.png create mode 100644 figs/test_bench_principle.svg delete mode 100644 matlab/analysis.m create mode 100644 matlab/bench_dimensioning.m create mode 100644 matlab/error_budget.m create mode 100644 matlab/probe_stiffness.m create mode 100644 matlab/probe_vs_encoder.m diff --git a/doc/flex_joints.pdf b/doc/flex_joints.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9ab658e35910dd1df056494a8756a61ed483e2e9 GIT binary patch literal 84684 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F - F(1); - -%% LPf -s = tf('s'); -G_lpf = 1/(1+s/2/pi/10); -F = lsim(G_lpf, F, t); -d = lsim(G_lpf, d, t); -dp = lsim(G_lpf, dp, t); - -%% -figure; -plot(t, F); - -figure; -hold on; -plot(t, d, 'DisplayName', 'Encoder'); -plot(t, dp, 'DisplayName', 'Mahr') -hold off; -legend; - -%% -t_start = 0.05; -t_end = 0.12; -d = d(t < t_end & t > t_start); -dp = dp(t < t_end & t > t_start); -F = F(t < t_end & t > t_start); - -d = d - d(1); -dp = dp - dp(1); -F = F - F(1); - -%% -figure; -plot(F, dp); -xlabel('Measured Force [N]'); -ylabel('Measured Displacement [m]'); - -%% -sprintf('Stiffness is %.3f [N/mm]', abs(1e-3*(d\F))) - -%% -figure; -hold on; -plot(F, d); -plot(F, F/(d\F)); -hold off; -xlabel('Measured Force [N]'); -ylabel('Measured Displacement [m]'); - -%% -figure; -hold on; -plot(dp, d); -hold off; -xlabel('Probe [m]'); -ylabel('Encoder [m]'); diff --git a/matlab/bench_dimensioning.m b/matlab/bench_dimensioning.m new file mode 100644 index 0000000..8d66c74 --- /dev/null +++ b/matlab/bench_dimensioning.m @@ -0,0 +1,53 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +% Flexible joint Geometry +% The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]]. +% Its bending stiffness is foreseen to be $k_{R_y}\approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 25\,mrad$. + +% #+name: fig:flexible_joint_geometry +% #+caption: Geometry of the flexible joint +% [[file:figs/flexible_joint_geometry.png]] + +% The height between the flexible point (center of the joint) and the point where external forces are applied is $h = 20\,mm$. + +% Let's define the parameters on Matlab. + +kRx = 5; % Bending Stiffness [Nm/rad] +Rxmax = 25e-3; % Bending Stroke [rad] +h = 20e-3; % Height [m] + +% Required external applied force + +% The bending $\theta_y$ of the flexible joint due to the force $F_x$ is: +% \begin{equation} +% \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x h}{k_{R_y}} +% \end{equation} + +% Therefore, the applied force to test the full range of the flexible joint is: +% \begin{equation} +% F_{x,\text{max}} = \frac{k_{R_y} \theta_{y,\text{max}}}{h} +% \end{equation} + + +Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N] + + + +% And we obtain: + +sprintf('\\begin{equation} F_{x,max} = %.1f\\, [N] \\end{equation}', Fxmax) + +% Required actuator stroke and sensors range + +% The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$. +% The corresponding stroke at the location of the force sensor is: +% \[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \] + + +dxmax = h*tan(Rxmax); + +sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax) diff --git a/matlab/error_budget.m b/matlab/error_budget.m new file mode 100644 index 0000000..addc89b --- /dev/null +++ b/matlab/error_budget.m @@ -0,0 +1,67 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +% Finite Element Model +% 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]]. + + +%% Stiffness +ka = 94e6; % Axial Stiffness [N/m] +ks = 13e6; % Shear Stiffness [N/m] +kb = 5; % Bending Stiffness [Nm/rad] +kt = 260; % Torsional Stiffness [Nm/rad] + +%% Maximum force +Fa = 469; % Axial Force before yield [N] +Fs = 242; % Shear Force before yield [N] +Fb = 0.118; % Bending Force before yield [Nm] +Ft = 1.508; % Torsional Force before yield [Nm] + +%% Compute the corresponding stroke +Xa = Fa/ka; % Axial Stroke before yield [m] +Xs = Fs/ks; % Shear Stroke before yield [m] +Xb = Fb/kb; % Bending Stroke before yield [rad] +Xt = Ft/kt; % Torsional Stroke before yield [rad] + +% Setup + +% The setup is schematically represented in Figure [[fig:test_bench_flex_side_bis]]. + +% The force is applied on top of the flexible joint with a distance $h$ with the joint's center. +% The displacement of the flexible joint is also measured at the same height. + +% The height between the joint's center and the force application point is: + +h = 25e-3; % Height [m] + +% Estimation error due to force sensor compression +% The measured displacement is not done directly at the joint's location. +% The force sensor compression will then induce an error on the joint's stiffness. + +% The force sensor stiffness $k_F$ is estimated to be around: + +kF = 50/0.05e-3; % [N/m] + +sprintf('k_F = %.1e [N/m]', kF) + +% Estimation error due to height estimation error +% Let's consider an error in the estimation of the height from the application of the force to the joint's center: +% \begin{equation} +% h_{\text{est}} = h (1 + \epsilon) +% \end{equation} + +% The computed bending stiffness will be: +% \begin{equation} +% k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x} +% \end{equation} + +% And the stiffness estimation error is: +% \begin{equation} +% \frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2 +% \end{equation} + + +h_err = 0.2e-3; % Height estimation error [m] diff --git a/matlab/probe_stiffness.m b/matlab/probe_stiffness.m new file mode 100644 index 0000000..2075a6a --- /dev/null +++ b/matlab/probe_stiffness.m @@ -0,0 +1,62 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +addpath('./mat/'); + +% Results :ignore: +% Let's load the measurement results. + +load('meas_stiff_probe.mat', 't', 'd', 'dp', 'F') + +%% Sampling time [s] +Ts = (t(end) - t(1))/(length(t)-1); + +%% Remove first second +t = t(ceil(1/Ts):end); +d = d(ceil(1/Ts):end); +dp = dp(ceil(1/Ts):end); +F = F(ceil(1/Ts):end); + + +%% Remove Offset +t = t - t(1); +F = F - mean(F(1:ceil(1/Ts))); + + + +% The time domain measured force and displacement are shown in Figure [[fig:mahr_time_domain]]. + + +%% Time Domain plots +figure; +tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile; +plot(t, F); +ylabel('Force [N]'); set(gca, 'XTickLabel',[]); + +ax2 = nexttile; +plot(t, d); +xlabel('Time [s]'); ylabel('Displacement [m]'); + + + +% #+RESULTS: +% : Stiffness is 0.039 [N/mm] + +% 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]]. + +% And compare the linear fit with the raw measurement data (Figure [[fig:mahr_stiffness_f_d_plot]]). + + +figure; +hold on; +plot(F, d, 'DisplayName', 'Raw data'); +plot(F, F/(d\F), 'DisplayName', 'Linear fit'); +hold off; +xlabel('Measured Force [N]'); +ylabel('Measured Displacement [m]'); +legend('location', 'southeast'); diff --git a/matlab/probe_vs_encoder.m b/matlab/probe_vs_encoder.m new file mode 100644 index 0000000..0024d5f --- /dev/null +++ b/matlab/probe_vs_encoder.m @@ -0,0 +1,86 @@ +%% Clear Workspace and Close figures +clear; close all; clc; + +%% Intialize Laplace variable +s = zpk('s'); + +addpath('./mat/'); + +% Results :ignore: +% Let's load the measurements. + +load('meas_probe_against_encoder.mat', 't', 'd', 'dp', 'F') + +%% Sampling time [s] +Ts = (t(end) - t(1))/(length(t)-1); + +%% Remove first second +t = t(ceil(1/Ts):end); +d = -d(ceil(1/Ts):end); +dp = -dp(ceil(1/Ts):end); +F = F(ceil(1/Ts):end); + + + +% The time domain measured displacement by the probe and by the encoder is shown in Figure [[fig:comp_encoder_probe_time]]. + + +%% Time Domain plots +figure; +hold on; +plot(t, d, 'DisplayName', 'Encoder'); +plot(t, dp, 'DisplayName', 'Probe'); +hold off; +xlabel('Time [s]'); ylabel('Displacement [m]'); + + + +% #+name: fig:comp_encoder_probe_time +% #+caption: Time domain measurement +% #+RESULTS: +% [[file:figs/comp_encoder_probe_time.png]] + +% If we zoom, we see that there is some delay between the encoder and the probe (Figure [[fig:comp_encoder_probe_time_zoom]]). + + +%% Zoom +figure; +hold on; +plot(t, d, 'DisplayName', 'Encoder'); +plot(t, dp, 'DisplayName', 'Probe'); +hold off; +xlabel('Time [s]'); ylabel('Displacement [m]'); +xlim([7.7, 7.9]) + + + +% #+RESULTS: +% : The time delay is approximately 15.8 [ms] + +% 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]]. + + +figure; +hold on; +plot(t, d-dp, 'DisplayName', 'Raw Mismatch'); +plot(t(1:end-finddelay(d, dp)), d(1:end-finddelay(d, dp))-dp(finddelay(d, dp)+1:end), 'DisplayName', 'Removed Delay'); +hold off; +xlabel('Time [s]'); ylabel('Measurement Missmatch [m]'); + + + +% #+name: fig:comp_encoder_probe_mismatch +% #+caption: Measurement mismatch, with and without delay compensation +% #+RESULTS: +% [[file:figs/comp_encoder_probe_mismatch.png]] + +% 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]]). + + +figure; +hold on; +plot(1e3*d, 1e3*dp, 'DisplayName', 'Raw data'); +plot(1e3*d, 1e3*d*(d\dp), 'DisplayName', sprintf('Linear fit: $\\alpha = %.5f$', (d\dp))); +hold on; +xlabel('Encoder [mm]'); ylabel('Probe [mm]'); +legend('location', 'southeast') diff --git a/test-bench-flexible-joints.html b/test-bench-flexible-joints.html index 02b505c..5465c41 100644 --- a/test-bench-flexible-joints.html +++ b/test-bench-flexible-joints.html @@ -3,9 +3,9 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + -Flexible Joint - Test Bench +Flexible Joints - Test Bench @@ -34,33 +34,40 @@ | HOME

-

Flexible Joint - Test Bench

+

Flexible Joints - Test Bench

@@ -68,95 +75,186 @@

This report is also available as a pdf.


-
-

1 Flexible Joints - Requirements

-
- +

+In this document, we present a test-bench that has been developed in order to measure the bending stiffness of flexible joints. +

+

+It is structured as follow: +

+
    +
  • Section 1: the geometry of the flexible joints and the expected stiffness and stroke are presented
  • +
  • Section 2: the measurement bench is presented
  • +
  • Section 3: an error budget is performed in order to estimate the accuracy of the measured stiffness
  • +
  • Section 4: first measurements are performed
  • +
  • Section 5: the bending stiffness of the flexible joints are measured
  • +
+ +
+

1 Flexible Joints

+
+

+ +

+ +

+The flexible joints that are going to be measured in this document have been design to be used with a Nano-Hexapod (Figure 1). +

+ + +
+

nano_hexapod.png +

+

Figure 1: CAD view of the Nano-Hexapod containing the flexible joints

+
+ +

+Ideally, these flexible joints would behave as perfect ball joints, that is to say: +

+
    +
  • no bending and torsional stiffnesses
  • +
  • infinite shear and axial stiffnesses
  • +
  • un-limited bending and torsional stroke
  • +
  • no friction, no backlash
  • +
+ + +

+The real characteristics of the flexible joints will influence the dynamics of the Nano-Hexapod. +Using a multi-body dynamical model of the nano-hexapod, the specifications in term of stiffness and stroke of the flexible joints have been determined and summarized in Table 1. +

+ +
++ + + - + + + - + + - + + + +
Table 1: Specifications for the flexible joints and estimated characteristics from the Finite Element Model
  SpecificationFEM
Axial Stiffness> 200 [N/um]> 100 [N/um]94
Shear Stiffness > 1 [N/um]13
Bending Stiffness< 100 [Nm/rad]< 100 [Nm/rad]5
Torsion Stiffness< 500 [Nm/rad]< 500 [Nm/rad]260
Bending Stroke > 1 [mrad]24.5
Torsion Stroke > 5 [urad] 
+ +

+Then, the classical geometry of a flexible ball joint shown in Figure 2 has been optimized in order to meet the requirements. +This has been done using a Finite Element Software and the obtained joint’s characteristics are summarized in Table 1. +

+ + +
+

flexible_joint_fem_geometry.png +

+

Figure 2: Flexible part of the Joint used for FEM - CAD view

+
+ +

+The obtained geometry are defined in the drawings of the flexible joints. +The material is a special kind of stainless steel called “F16PH”.. +

-
-

2 Test Bench Description

+
+

2 Measurement Test Bench - Bending Stiffness

-The main characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\). + +

+

+The most important characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\).

-To do so, a test bench is used. -Specifications of the test bench to precisely measure the bending stiffness are described in this section. +To do so, we have to apply a torque \(T_x\) on the flexible joint and measure its angular deflection \(\theta_x\). +The stiffness is then +

+\begin{equation} +k_{R_x} = \frac{T_x}{\theta_x} +\end{equation} + +

+As it is quite difficult to apply a pure torque, a force will be applied instead. +The application point of the force should far enough from the flexible part such that the obtained bending is much larger than the displacement in shear.

-The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque. +The working principle of the bench is schematically shown in Figure 3. +One part of the flexible joint is fixed. On the mobile part, a force \(F_x\) is applied which is equivalent to a torque applied on the flexible joint center. +The induced rotation is measured with a displacement sensor \(d_x\).

-
-

test-bench-schematic.png +

+

test_bench_principle.png

-

Figure 1: Schematic of the test bench to measure the bending stiffness of the flexible joints

-
+

Figure 3: Test Bench - working principle

-
-

2.1 Flexible joint Geometry

+ +

+This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints. +Another test-bench, better engineered will be used to measure the flexible joint’s characteristics with better accuracy. +

+
+ +
+

2.1 Flexible joint Geometry

-The flexible joint used for the Nano-Hexapod is shown in Figure 2. -Its bending stiffness is foreseen to be \(k_{R_y}\approx 20\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 20\,mrad\). +The flexible joint used for the Nano-Hexapod is shown in Figure 4. +Its bending stiffness is foreseen to be \(k_{R_y}\approx 5\,\frac{Nm}{rad}\) and its stroke \(\theta_{y,\text{max}}\approx 25\,mrad\).

-
+

flexible_joint_geometry.png

-

Figure 2: Geometry of the flexible joint

+

Figure 4: Geometry of the flexible joint

@@ -167,16 +265,16 @@ The height between the flexible point (center of the joint) and the point where Let’s define the parameters on Matlab.

-
  kRx = 20; % Bending Stiffness [Nm/rad]
-  Rxmax = 20e-3; % Bending Stroke [rad]
-  h = 20e-3; % Height [m]
+
kRx = 5; % Bending Stiffness [Nm/rad]
+Rxmax = 25e-3; % Bending Stroke [rad]
+h = 20e-3; % Height [m]
 
-
-

2.2 Required external applied force

+
+

2.2 Required external applied force

The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is: @@ -193,154 +291,465 @@ Therefore, the applied force to test the full range of the flexible joint is: \end{equation}

-
  Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
+
Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N]
 

And we obtain:

-\begin{equation} F_{max} = 20.0\, [N] \end{equation} +\begin{equation} F_{x,max} = 6.2\, [N] \end{equation}

-The measurement range of the force sensor should then be higher than \(20\,N\). +The measurement range of the force sensor should then be higher than \(6.2\,N\).

-
-

2.3 Required actuator stroke and sensors range

+
+

2.3 Required actuator stroke and sensors range

-The flexible joint is designed to allow a bending motion of \(\pm 20\,mrad\). -The corresponding actuator stroke to impose such motion is: -

- -

+The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\). +The corresponding stroke at the location of the force sensor is: \[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]

-
  dxmax = h*tan(Rxmax);
+
dxmax = h*tan(Rxmax);
 
-\begin{equation} d_{max} = 0.4\, [mm] \end{equation} +\begin{equation} d_{max} = 0.5\, [mm] \end{equation}

-In order to test the full range of the flexible joint, the stroke of the actuator should be higher than \(0.4\,mm\). -The measurement range of the displacement sensor should also be higher than \(0.4\,mm\). +In order to test the full range of the flexible joint, the stroke of the translation stage used to move the force sensor should be higher than \(0.5\,mm\). +Similarly, the measurement range of the displacement sensor should also be higher than \(0.5\,mm\).

-
-

2.4 First try with the APA95ML

+
+

2.4 Test Bench

-The APA95ML as a stroke of \(100\,\mu m\) and the encoder in parallel can easily measure the required stroke. +A CAD view of the measurement bench is shown in Figure 5.

+

-Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be: +Here are the different elements used in this bench:

-
-
  Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N]
-
-
- -\begin{equation} F_{max} = 5.0\, [N] \end{equation} - -

-And the tested angular range is: -

-
-
  Rmax = tan(100e-6/h);
-
-
- -\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation} -
-
- -
-

2.5 Test Bench

-
- -
-

test_bench_flex_overview.png -

-

Figure 3: Schematic of the test bench to measure the bending stiffness of the flexible joints

-
- - - -
-

test_bench_flex_side.png -

-

Figure 4: Schematic of the test bench to measure the bending stiffness of the flexible joints

-
- -
+
+ +

+Both the measured force and displacement are acquired at the same time using a Speedgoat machine. +

+ + +
+

test_bench_flex_overview.png +

+

Figure 5: Schematic of the test bench to measure the bending stiffness of the flexible joints

+
+ +

+A side view of the bench with the important quantities are shown in Figure 6. +

+ + +
+

test_bench_flex_side.png +

+

Figure 6: Schematic of the test bench to measure the bending stiffness of the flexible joints

-
-

3 Agreement between the probe and the encoder

+
+

3 Error budget

+

+ +

+

+Many things can impact the accuracy of the measured bending stiffness such as: +

+
    +
  • Errors in the force and displacement measurement
  • +
  • Shear effects
  • +
  • Deflection of the Force sensor
  • +
  • Errors in the geometry of the bench
  • +
+ +

+In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors. +

-
-

3.1 Setup

+
+

3.1 Finite Element Model

+From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables 2 and 3. +

+ + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + +
Table 2: Axial/Shear characteristics
 Stiffness [N/um]Max Force [N]Stroke [um]
Axial944695
Shear1324219
+ + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + +
Table 3: Bending/Torsion characteristics
 Stiffness [Nm/rad]Max Torque [Nmm]Stroke [mrad]
Bending511824
Torsional26015086
+
+
+ +
+

3.2 Setup

+
+

+The setup is schematically represented in Figure 7. +

+ +

+The force is applied on top of the flexible joint with a distance \(h\) with the joint’s center. +The displacement of the flexible joint is also measured at the same height. +

+ +

+The height between the joint’s center and the force application point is: +

+
+
h = 25e-3; % Height [m]
+
+
+ + +
+

test_bench_flex_side.png +

+

Figure 7: Schematic of the test bench to measure the bending stiffness of the flexible joints

+
+
+
+ +
+

3.3 Effect of Bending

+
+

+The torque applied is: +

+\begin{equation} + M_y = F_x \cdot h +\end{equation} + +

+The flexible joint is experiencing a rotation \(\theta_y\) due to the torque \(M_y\): +

+\begin{equation} + \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x \cdot h}{k_{R_y}} +\end{equation} + +

+This rotation is then measured by the displacement sensor. +The measured displacement is: +

+\begin{equation} + D_b = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \label{eq:bending_stiffness_formula} +\end{equation} +
+
+ +
+

3.4 Computation of the bending stiffness

+
+

+From equation \eqref{eq:bending_stiffness_formula}, we can compute the bending stiffness: +

+\begin{equation} + k_{R_y} = \frac{F_x \cdot h}{\tan^{-1}\left( \frac{D_b}{h} \right)} +\end{equation} + +

+For small displacement, we have +

+\begin{equation} + \boxed{k_{R_y} \approx h^2 \frac{F_x}{d_x}} +\end{equation} + +

+And therefore, to precisely measure \(k_{R_y}\), we need to: +

+
    +
  • precisely measure the motion \(d_x\)
  • +
  • precisely measure the applied force \(F_x\)
  • +
  • precisely now the height of the force application point \(h\)
  • +
+
+
+ +
+

3.5 Estimation error due to force and displacement sensors accuracy

+
+

+The maximum error on the measured displacement with the encoder is 40 nm. +This quite negligible compared to the measurement range of 0.5 mm. +

+ +

+The accuracy of the force sensor is around 1% and therefore, we should expect to have an accuracy on the measured stiffness of at most 1%. +

+
+
+ +
+

3.6 Estimation error due to Shear

+
+

+The effect of Shear on the measured displacement is simply: +

+\begin{equation} + D_s = \frac{F_x}{k_s} +\end{equation} + +

+The measured displacement will be the effect of shear + effect of bending +

+\begin{equation} + d_x = D_b + D_s = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right) +\end{equation} + +

+The estimated bending stiffness \(k_{\text{est}}\) will then be: +

+\begin{equation} + k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} +\end{equation} + +
+The measurement error due to Shear is 0.1 %
+
+
+
+ +
+

3.7 Estimation error due to force sensor compression

+
+

+The measured displacement is not done directly at the joint’s location. +The force sensor compression will then induce an error on the joint’s stiffness. +

+ +

+The force sensor stiffness \(k_F\) is estimated to be around: +

+
+
kF = 50/0.05e-3; % [N/m]
+
+
+ +
+k_F = 1.0e+06 [N/m]
+
+ + +

+The measured displacement will be the sum of the displacement induced by the bending and by the compression of the force sensor: +

+\begin{equation} + d_x = D_b + \frac{F_x}{k_F} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_F} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_F} \right) +\end{equation} + +

+The estimated bending stiffness \(k_{\text{est}}\) will then be: +

+\begin{equation} + k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} +\end{equation} + +
+The measurement error due to height estimation errors is 0.8 %
+
+
+
+ +
+

3.8 Estimation error due to height estimation error

+
+

+Let’s consider an error in the estimation of the height from the application of the force to the joint’s center: +

+\begin{equation} +h_{\text{est}} = h (1 + \epsilon) +\end{equation} + +

+The computed bending stiffness will be: +

+\begin{equation} + k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x} +\end{equation} + +

+And the stiffness estimation error is: +

+\begin{equation} +\frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2 +\end{equation} + +
+
h_err = 0.2e-3; % Height estimation error [m]
+
+
+ +
+The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %
+
+
+
+ +
+

3.9 Conclusion

+
+

+Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench. +This is well enough for a first estimation of the bending stiffness of the flexible joints. +

+ +

+Another measurement bench allowing better accuracy will be developed. +

+
+
+
+ +
+

4 First Measurements

+
+

+ +

+
    +
  • Section 4.1:
  • +
  • Section 4.2:
  • +
+
+
+

4.1 Agreement between the probe and the encoder

+
+

+ +

+ +

The measurement setup is made such that the probe measured the translation table displacement. It should then measure the same displacement as the encoder. Using this setup, we should be able to compare the probe and the encoder.

-
-
- -
-

3.2 Results

-

Let’s load the measurements.

-
load('meas_probe_against_encoder.mat', 't', 'd', 'dp', 'F')
 

-The time domain measured displacement by the probe and by the encoder is shown in Figure 5. +The time domain measured displacement by the probe and by the encoder is shown in Figure 8.

-
+

comp_encoder_probe_time.png

-

Figure 5: Time domain measurement

+

Figure 8: Time domain measurement

-If we zoom, we see that there is some delay between the encoder and the probe (Figure 6). +If we zoom, we see that there is some delay between the encoder and the probe (Figure 9).

-
+

comp_encoder_probe_time_zoom.png

-

Figure 6: Time domain measurement (Zoom)

+

Figure 9: Time domain measurement (Zoom)

@@ -353,28 +762,28 @@ The time delay is approximately 15.8 [ms]

-The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure 7. +The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure 10.

-
+

comp_encoder_probe_mismatch.png

-

Figure 7: Measurement mismatch, with and without delay compensation

+

Figure 10: Measurement mismatch, with and without delay compensation

-Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure 8). +Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure 11).

-
+

comp_encoder_probe_linear_fit.png

-

Figure 8: Measured displacement by the probe as a function of the measured displacement by the encoder

+

Figure 11: Measured displacement by the probe as a function of the measured displacement by the encoder

-
+

From the measurement, it is shown that the probe is well calibrated. However, there is some time delay of tens of milliseconds that could induce some measurement errors. @@ -383,12 +792,14 @@ However, there is some time delay of tens of milliseconds that could induce some

-
-
-

4 Measurement of the Millimar 1318 probe stiffness

-
-
+
+

4.2 Measurement of the Millimar 1318 probe stiffness

+
+

+ +

+
  • Translation Stage: V-408
  • Load Cell: FC2231-0000-0010-L
  • @@ -399,23 +810,18 @@ However, there is some time delay of tens of milliseconds that could induce some
-
+

setup_mahr_stiff_meas_side.jpg

-

Figure 9: Setup - Side View

+

Figure 12: Setup - Side View

-
+

setup_mahr_stiff_meas_top.jpg

-

Figure 10: Setup - Top View

+

Figure 13: Setup - Top View

-
- -
-

4.1 Results

-

Let’s load the measurement results.

@@ -425,14 +831,14 @@ Let’s load the measurement results.

-The time domain measured force and displacement are shown in Figure 11. +The time domain measured force and displacement are shown in Figure 14.

-
+

mahr_time_domain.png

-

Figure 11: Time domain measurements

+

Figure 14: Time domain measurements

@@ -450,17 +856,17 @@ This is very close to the 0.04 [N/mm] written in the 12). +And compare the linear fit with the raw measurement data (Figure 15).

-
+

mahr_stiffness_f_d_plot.png

-

Figure 12: Measured displacement as a function of the measured force. Raw data and linear fit

+

Figure 15: Measured displacement as a function of the measured force. Raw data and linear fit

-
+

The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).

@@ -470,13 +876,18 @@ The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
-
-

5 Experimental measurement

+
+

5 Bending Stiffness Measurement

+
+

+ +

+

Author: Dehaeze Thomas

-

Created: 2021-02-16 mar. 21:23

+

Created: 2021-02-18 jeu. 11:33

diff --git a/test-bench-flexible-joints.org b/test-bench-flexible-joints.org index 79a09b4..a7599c3 100644 --- a/test-bench-flexible-joints.org +++ b/test-bench-flexible-joints.org @@ -1,4 +1,4 @@ -#+TITLE: Flexible Joint - Test Bench +#+TITLE: Flexible Joints - Test Bench :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com @@ -46,43 +46,101 @@
#+end_export -* Flexible Joints - Requirements +* Introduction :ignore: +In this document, we present a test-bench that has been developed in order to measure the bending stiffness of flexible joints. -| | *Specification* | -|-------------------+-----------------| -| Axial Stiffness | > 200 [N/um] | -| Shear Stiffness | > 1 [N/um] | -| Bending Stiffness | < 100 [Nm/rad] | -| Torsion Stiffness | < 500 [Nm/rad] | -| Bending Stroke | > 1 [mrad] | -| Torsion Stroke | > 5 [urad] | +It is structured as follow: +- Section [[sec:flexible_joints]]: the geometry of the flexible joints and the expected stiffness and stroke are presented +- Section [[sec:test_bench_desc]]: the measurement bench is presented +- Section [[sec:error_budget]]: an error budget is performed in order to estimate the accuracy of the measured stiffness +- Section [[sec:first_measurements]]: first measurements are performed +- Section [[sec:bending_stiffness_meas]]: the bending stiffness of the flexible joints are measured -* Test Bench Description +* Flexible Joints +<> + +The flexible joints that are going to be measured in this document have been design to be used with a Nano-Hexapod (Figure [[fig:nano_hexapod]]). + +#+name: fig:nano_hexapod +#+caption: CAD view of the Nano-Hexapod containing the flexible joints +#+attr_latex: :width 0.7\linewidth +[[file:figs/nano_hexapod.png]] + +Ideally, these flexible joints would behave as perfect ball joints, that is to say: +- no bending and torsional stiffnesses +- infinite shear and axial stiffnesses +- un-limited bending and torsional stroke +- no friction, no backlash + + +The real characteristics of the flexible joints will influence the dynamics of the Nano-Hexapod. +Using a multi-body dynamical model of the nano-hexapod, the specifications in term of stiffness and stroke of the flexible joints have been determined and summarized in Table [[tab:flexible_joints_specs]]. + +#+name: tab:flexible_joints_specs +#+caption: Specifications for the flexible joints and estimated characteristics from the Finite Element Model +#+attr_latex: :environment tabularx :width 0.5\linewidth :align Xcc +#+attr_latex: :center t :booktabs t :float t +| | *Specification* | *FEM* | +|-------------------+-----------------+-------| +| Axial Stiffness | > 100 [N/um] | 94 | +| Shear Stiffness | > 1 [N/um] | 13 | +| Bending Stiffness | < 100 [Nm/rad] | 5 | +| Torsion Stiffness | < 500 [Nm/rad] | 260 | +| Bending Stroke | > 1 [mrad] | 24.5 | +| Torsion Stroke | > 5 [urad] | | + +Then, the classical geometry of a flexible ball joint shown in Figure [[fig:flexible_joint_fem_geometry]] has been optimized in order to meet the requirements. +This has been done using a Finite Element Software and the obtained joint's characteristics are summarized in Table [[tab:flexible_joints_specs]]. + +#+name: fig:flexible_joint_fem_geometry +#+caption: Flexible part of the Joint used for FEM - CAD view +#+attr_latex: :width 0.5\linewidth +[[file:figs/flexible_joint_fem_geometry.png]] + +The obtained geometry are defined in the [[file:doc/flex_joints.pdf][drawings of the flexible joints]]. +The material is a special kind of stainless steel called "F16PH".. + +* Measurement Test Bench - Bending Stiffness +:PROPERTIES: +:header-args:matlab+: :tangle ./matlab/bench_dimensioning.m +:END: +<> ** Introduction :ignore: +The most important characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$. -The main characteristic of the flexible joint that we want to measure is its bending stiffness $k_{R_x} \approx k_{R_y}$. +To do so, we have to apply a torque $T_x$ on the flexible joint and measure its angular deflection $\theta_x$. +The stiffness is then +\begin{equation} +k_{R_x} = \frac{T_x}{\theta_x} +\end{equation} -To do so, a test bench is used. -Specifications of the test bench to precisely measure the bending stiffness are described in this section. +As it is quite difficult to apply a pure torque, a force will be applied instead. +The application point of the force should far enough from the flexible part such that the obtained bending is much larger than the displacement in shear. -The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque. +The working principle of the bench is schematically shown in Figure [[fig:test_bench_principle]]. +One part of the flexible joint is fixed. On the mobile part, a force $F_x$ is applied which is equivalent to a torque applied on the flexible joint center. +The induced rotation is measured with a displacement sensor $d_x$. -#+name: fig:test-bench-schematic -#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints -[[file:figs/test-bench-schematic.png]] +#+name: fig:test_bench_principle +#+caption: Test Bench - working principle +[[file:figs/test_bench_principle.png]] + + +This test-bench will be used to have a first approximation of the bending stiffnesss and stroke of the flexible joints. +Another test-bench, better engineered will be used to measure the flexible joint's characteristics with better accuracy. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) - <> +<> #+end_src #+begin_src matlab :exports none :results silent :noweb yes - <> +<> #+end_src ** Flexible joint Geometry The flexible joint used for the Nano-Hexapod is shown in Figure [[fig:flexible_joint_geometry]]. -Its bending stiffness is foreseen to be $k_{R_y}\approx 20\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 20\,mrad$. +Its bending stiffness is foreseen to be $k_{R_y}\approx 5\,\frac{Nm}{rad}$ and its stroke $\theta_{y,\text{max}}\approx 25\,mrad$. #+name: fig:flexible_joint_geometry #+caption: Geometry of the flexible joint @@ -92,9 +150,9 @@ The height between the flexible point (center of the joint) and the point where Let's define the parameters on Matlab. #+begin_src matlab - kRx = 20; % Bending Stiffness [Nm/rad] - Rxmax = 20e-3; % Bending Stroke [rad] - h = 20e-3; % Height [m] +kRx = 5; % Bending Stiffness [Nm/rad] +Rxmax = 25e-3; % Bending Stroke [rad] +h = 20e-3; % Height [m] #+end_src ** Required external applied force @@ -110,96 +168,312 @@ Therefore, the applied force to test the full range of the flexible joint is: \end{equation} #+begin_src matlab - Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N] +Fxmax = kRx*Rxmax/h; % Force to induce maximum stroke [N] #+end_src And we obtain: #+begin_src matlab :results value raw replace :exports results - sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax) +sprintf('\\begin{equation} F_{x,max} = %.1f\\, [N] \\end{equation}', Fxmax) #+end_src #+RESULTS: -\begin{equation} F_{max} = 20.0\, [N] \end{equation} +\begin{equation} F_{x,max} = 6.2\, [N] \end{equation} -The measurement range of the force sensor should then be higher than $20\,N$. +The measurement range of the force sensor should then be higher than $6.2\,N$. ** Required actuator stroke and sensors range -The flexible joint is designed to allow a bending motion of $\pm 20\,mrad$. -The corresponding actuator stroke to impose such motion is: - +The flexible joint is designed to allow a bending motion of $\pm 25\,mrad$. +The corresponding stroke at the location of the force sensor is: \[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \] #+begin_src matlab - dxmax = h*tan(Rxmax); +dxmax = h*tan(Rxmax); #+end_src #+begin_src matlab :results value raw replace :exports results - sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax) +sprintf('\\begin{equation} d_{max} = %.1f\\, [mm] \\end{equation}', 1e3*dxmax) #+end_src #+RESULTS: -\begin{equation} d_{max} = 0.4\, [mm] \end{equation} +\begin{equation} d_{max} = 0.5\, [mm] \end{equation} -In order to test the full range of the flexible joint, the stroke of the actuator should be higher than $0.4\,mm$. -The measurement range of the displacement sensor should also be higher than $0.4\,mm$. - -** First try with the APA95ML - -The APA95ML as a stroke of $100\,\mu m$ and the encoder in parallel can easily measure the required stroke. - -Suppose the full stroke of the APA can be used to bend the flexible joint (ideal case), the measured force will be: -#+begin_src matlab - Fxmax = kRx*100e-6/h^2; % Force at maximum stroke [N] -#+end_src - -#+begin_src matlab :results value raw replace :exports results - sprintf('\\begin{equation} F_{max} = %.1f\\, [N] \\end{equation}', Fxmax) -#+end_src - -#+RESULTS: -\begin{equation} F_{max} = 5.0\, [N] \end{equation} - -And the tested angular range is: -#+begin_src matlab - Rmax = tan(100e-6/h); -#+end_src - -#+begin_src matlab :results value raw replace :exports results - sprintf('\\begin{equation} \\theta_{max} = %.1f\\, [mrad] \\end{equation}', 1e3*Rmax) -#+end_src - -#+RESULTS: -\begin{equation} \theta_{max} = 5.0\, [mrad] \end{equation} +In order to test the full range of the flexible joint, the stroke of the translation stage used to move the force sensor should be higher than $0.5\,mm$. +Similarly, the measurement range of the displacement sensor should also be higher than $0.5\,mm$. ** Test Bench +A CAD view of the measurement bench is shown in Figure [[fig:test_bench_flex_overview]]. + +#+begin_note +Here are the different elements used in this bench: +- *Translation Stage*: [[file:doc/V-408-Datasheet.pdf][V-408]] +- *Load Cells*: [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]] +- *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]] +#+end_note + +Both the measured force and displacement are acquired at the same time using a Speedgoat machine. + #+name: fig:test_bench_flex_overview #+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints #+attr_latex: :width 0.8\linewidth [[file:figs/test_bench_flex_overview.png]] +A side view of the bench with the important quantities are shown in Figure [[fig:test_bench_flex_side]]. #+name: fig:test_bench_flex_side #+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints #+attr_latex: :width 0.25\linewidth +#+attr_html: :width 300px [[file:figs/test_bench_flex_side.png]] -#+begin_note -- *Translation Stage*: [[file:doc/V-408-Datasheet.pdf][V-408]] -- *Load Cells*: [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]] and [[file:doc/FRE_DS_XFL212R_FR_A3.pdf][XFL212R]] +* Error budget +:PROPERTIES: +:header-args:matlab+: :tangle ./matlab/error_budget.m +:END: +<> +** Introduction :ignore: + +Many things can impact the accuracy of the measured bending stiffness such as: +- Errors in the force and displacement measurement +- Shear effects +- Deflection of the Force sensor +- Errors in the geometry of the bench + +In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors. + + +** Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +** Finite Element Model +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]]. + +#+begin_src matlab :exports none +%% Stiffness +ka = 94e6; % Axial Stiffness [N/m] +ks = 13e6; % Shear Stiffness [N/m] +kb = 5; % Bending Stiffness [Nm/rad] +kt = 260; % Torsional Stiffness [Nm/rad] + +%% Maximum force +Fa = 469; % Axial Force before yield [N] +Fs = 242; % Shear Force before yield [N] +Fb = 0.118; % Bending Force before yield [Nm] +Ft = 1.508; % Torsional Force before yield [Nm] + +%% Compute the corresponding stroke +Xa = Fa/ka; % Axial Stroke before yield [m] +Xs = Fs/ks; % Shear Stroke before yield [m] +Xb = Fb/kb; % Bending Stroke before yield [rad] +Xt = Ft/kt; % Torsional Stroke before yield [rad] +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable([1e-6*ka, Fa, 1e6*Xa; 1e-6*ks, Fs, 1e6*Xs], {'Axial', 'Shear'}, {'Stiffness [N/um]', 'Max Force [N]', 'Stroke [um]'}, ' %.0f '); +#+end_src + +#+name: tab:axial_shear_characteristics +#+caption: Axial/Shear characteristics +#+attr_latex: :environment tabularx :width 0.6\linewidth :align Xccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | Stiffness [N/um] | Max Force [N] | Stroke [um] | +|-------+------------------+---------------+-------------| +| Axial | 94 | 469 | 5 | +| Shear | 13 | 242 | 19 | + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable([kb, 1e3*Fb, 1e3*Xb; kt, 1e3*Ft, 1e3*Xt], {'Bending', 'Torsional'}, {'Stiffness [Nm/rad]', 'Max Torque [Nmm]', 'Stroke [mrad]'}, ' %.0f '); +#+end_src + +#+name: tab:bending_torsion_characteristics +#+caption: Bending/Torsion characteristics +#+attr_latex: :environment tabularx :width 0.7\linewidth :align Xccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | Stiffness [Nm/rad] | Max Torque [Nmm] | Stroke [mrad] | +|-----------+--------------------+------------------+---------------| +| Bending | 5 | 118 | 24 | +| Torsional | 260 | 1508 | 6 | + +** Setup + +The setup is schematically represented in Figure [[fig:test_bench_flex_side_bis]]. + +The force is applied on top of the flexible joint with a distance $h$ with the joint's center. +The displacement of the flexible joint is also measured at the same height. + +The height between the joint's center and the force application point is: +#+begin_src matlab +h = 25e-3; % Height [m] +#+end_src + +#+name: fig:test_bench_flex_side_bis +#+caption: Schematic of the test bench to measure the bending stiffness of the flexible joints +#+attr_latex: :width 0.25\linewidth +#+attr_html: :width 300px +[[file:figs/test_bench_flex_side.png]] + +** Effect of Bending +The torque applied is: +\begin{equation} + M_y = F_x \cdot h +\end{equation} + +The flexible joint is experiencing a rotation $\theta_y$ due to the torque $M_y$: +\begin{equation} + \theta_y = \frac{M_y}{k_{R_y}} = \frac{F_x \cdot h}{k_{R_y}} +\end{equation} + +This rotation is then measured by the displacement sensor. +The measured displacement is: +\begin{equation} + D_b = h \tan(\theta_y) = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) \label{eq:bending_stiffness_formula} +\end{equation} + +** Computation of the bending stiffness +From equation eqref:eq:bending_stiffness_formula, we can compute the bending stiffness: +\begin{equation} + k_{R_y} = \frac{F_x \cdot h}{\tan^{-1}\left( \frac{D_b}{h} \right)} +\end{equation} + +For small displacement, we have +\begin{equation} + \boxed{k_{R_y} \approx h^2 \frac{F_x}{d_x}} +\end{equation} + +And therefore, to precisely measure $k_{R_y}$, we need to: +- precisely measure the motion $d_x$ +- precisely measure the applied force $F_x$ +- precisely now the height of the force application point $h$ + +** Estimation error due to force and displacement sensors accuracy +The maximum error on the measured displacement with the encoder is 40 nm. +This quite negligible compared to the measurement range of 0.5 mm. + +The accuracy of the force sensor is around 1% and therefore, we should expect to have an accuracy on the measured stiffness of at most 1%. + +** Estimation error due to Shear +The effect of Shear on the measured displacement is simply: +\begin{equation} + D_s = \frac{F_x}{k_s} +\end{equation} + +The measured displacement will be the effect of shear + effect of bending +\begin{equation} + d_x = D_b + D_s = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_s} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_s} \right) +\end{equation} + +The estimated bending stiffness $k_{\text{est}}$ will then be: +\begin{equation} + k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_s h^2}} +\end{equation} + +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('The measurement error due to Shear is %.1f %%', 100*abs(1-1/(1 + kb/(ks*h^2)))) +#+end_src + +#+RESULTS: +: The measurement error due to Shear is 0.1 % + +** Estimation error due to force sensor compression +The measured displacement is not done directly at the joint's location. +The force sensor compression will then induce an error on the joint's stiffness. + +The force sensor stiffness $k_F$ is estimated to be around: +#+begin_src matlab +kF = 50/0.05e-3; % [N/m] +#+end_src + +#+begin_src matlab :results value replace :exports results +sprintf('k_F = %.1e [N/m]', kF) +#+end_src + +#+RESULTS: +: k_F = 1.0e+06 [N/m] + +The measured displacement will be the sum of the displacement induced by the bending and by the compression of the force sensor: +\begin{equation} + d_x = D_b + \frac{F_x}{k_F} = h \tan\left( \frac{F_x \cdot h}{k_{R_y}} \right) + \frac{F_x}{k_F} \approx F_x \left( \frac{h^2}{k_{R_y}} + \frac{1}{k_F} \right) +\end{equation} + +The estimated bending stiffness $k_{\text{est}}$ will then be: +\begin{equation} + k_{\text{est}} = h^2 \frac{F_x}{d_x} \approx k_{R_y} \frac{1}{1 + \frac{k_{R_y}}{k_F h^2}} +\end{equation} + +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('The measurement error due to height estimation errors is %.1f %%', 100*abs(1-1/(1 + kb/(kF*h^2)))) +#+end_src + +#+RESULTS: +: The measurement error due to height estimation errors is 0.8 % + +** Estimation error due to height estimation error +Let's consider an error in the estimation of the height from the application of the force to the joint's center: +\begin{equation} +h_{\text{est}} = h (1 + \epsilon) +\end{equation} + +The computed bending stiffness will be: +\begin{equation} + k_\text{est} \approx h_{\text{est}}^2 \frac{F_x}{d_x} +\end{equation} + +And the stiffness estimation error is: +\begin{equation} +\frac{k_{\text{est}}}{k_{R_y}} = (1 + \epsilon)^2 +\end{equation} + +#+begin_src matlab +h_err = 0.2e-3; % Height estimation error [m] +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('The measurement error due to height estimation errors of %.1f [mm] is %.1f %%', 1e3*h_err, 100*abs(1-(1 + h_err/h)^2)) +#+end_src + +#+RESULTS: +: The measurement error due to height estimation errors of 0.2 [mm] is 1.6 % + +** Conclusion +Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench. +This is well enough for a first estimation of the bending stiffness of the flexible joints. + +Another measurement bench allowing better accuracy will be developed. + +* First Measurements +<> +** Introduction :ignore: + + +- Section [[sec:test_meas_probe]]: +- Section [[sec:meas_probe_stiffness]]: + +** Agreement between the probe and the encoder +:PROPERTIES: +:header-args:matlab+: :tangle ./matlab/probe_vs_encoder.m +:END: +<> +*** Introduction :ignore: + +- *Load Cells*: [[file:doc/A700000007147087.pdf][FC2231-0000-0010-L]] (and [[file:doc/FRE_DS_XFL212R_FR_A3.pdf][XFL212R]]) - *Encoder*: [[file:doc/L-9517-9448-05-B_Data_sheet_RESOLUTE_BiSS_en.pdf][Renishaw Resolute 1nm]] - *Displacement Probe*: [[file:doc/Millimar--3723046--BA--C1208-C1216-C1240--FR--2016-11-08.pdf][Millimar C1216 electronics]] and [[file:doc/tmp3m0cvmue_7888038c-cdc8-48d8-a837-35de02760685.pdf][Millimar 1318 probe]] -#+end_note -* Agreement between the probe and the encoder -** Introduction :ignore: -** Setup +*** Setup :ignore: The measurement setup is made such that the probe measured the translation table displacement. It should then measure the same displacement as the encoder. Using this setup, we should be able to compare the probe and the encoder. -** Matlab Init :noexport:ignore: +*** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src @@ -217,9 +491,8 @@ addpath('./matlab/'); addpath('./mat/'); #+end_src -** Results +*** Results :ignore: Let's load the measurements. - #+begin_src matlab load('meas_probe_against_encoder.mat', 't', 'd', 'dp', 'F') #+end_src @@ -333,8 +606,12 @@ From the measurement, it is shown that the probe is well calibrated. However, there is some time delay of tens of milliseconds that could induce some measurement errors. #+end_important -* Measurement of the Millimar 1318 probe stiffness -** Introduction :ignore: +** Measurement of the Millimar 1318 probe stiffness +:PROPERTIES: +:header-args:matlab+: :tangle ./matlab/probe_stiffness.m +:END: +<> +*** Introduction :ignore: #+begin_note - *Translation Stage*: [[file:doc/V-408-Datasheet.pdf][V-408]] @@ -353,7 +630,7 @@ However, there is some time delay of tens of milliseconds that could induce some #+attr_latex: :width \linewidth [[file:figs/setup_mahr_stiff_meas_top.jpg]] -** Matlab Init :noexport:ignore: +*** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src @@ -371,7 +648,7 @@ addpath('./matlab/'); addpath('./mat/'); #+end_src -** Results +*** Results :ignore: Let's load the measurement results. #+begin_src matlab load('meas_stiff_probe.mat', 't', 'd', 'dp', 'F') @@ -456,4 +733,5 @@ exportFig('figs/mahr_stiffness_f_d_plot.pdf', 'width', 'wide', 'height', 'normal The Millimar 1318 probe has a stiffness of $\approx 0.04\,[N/mm]$. #+end_summary -* Experimental measurement +* Bending Stiffness Measurement +<> diff --git a/test-bench-flexible-joints.pdf b/test-bench-flexible-joints.pdf index 902f3089038d6f287c2af6560aeba9cc224dd028..87a19089383f0b927826268e4009b718663ef2f5 100644 GIT binary patch delta 397771 zcmXuKWmFtZ*EJlR;O-iN4DJ?OgL`lY?(W(IcO7hScXx+i!5xAIfmrQlpkdFZyMsc@#S(5)+o zw6Blg{gT_>wlr%JDw?zL0v43a%59Jd^(9~U!&t938{#o*A0(Pg#Te*PN2D9L7;y8) z>otGK1=;%>J{_7OOB!q%Pq;k7;AJYYs75jKTx<&QrajoUrw-DPMY`jo>@fuXQE@c9 zfmMPR-$t=;Gq`@5ss27KM*51qKrrj#Gdm#RLTe=qY$*^WtqiYuAUmq0GfZ zG}^h^=N*ta{azI?{3?@MVDXhP-{mc!F7ay+Ojkq5LPZY;8qWOw^SQpv@`%LpJ~4@* zX!Z1#;lnrSL_+wXwl7{^eh7uPKy8F z+ocZ+0Cj;wEYtMVAN4&8Y6Yc&*y|~5L+k^3N8FtcM_eIq<5B&1kS{t z8dgrq(x@(F4-g$m)1FMyPwM?xEp}Ax70A_`PH$y2|1nB$PlTzZC-qBmlOU@5A{X1_ zei@mc;Y#IVSl}b^)#7JmIMOqhO{0)}ag*l1lKH*kjA&Zsfl1 zJz1NfZnL1sf$*Uz=rymSKgM~d%@V%<+*cDBxsNWza3wsh@`#Y|)je(Xw0XnIrBKpn z5eV-VPbQ_8W7*wXpOEg`clpB^wEdpn$Vr+xp_z!lr)0~L7#%aOY)3Eb))^K3t~l9x2bqp*D^?DB@m z0(L%RWb*njDlB-Mw-g)-?l0c^hpuxHH38D6U-!iSlERBGrw;Tv789An3Z|eslyC1` zXx|z{`h)+R6>N_>j9aOPx9VHcedGF((i)pK8U-&PJY|ad0Kcr&fmCzs!Xe*O5s+$0 z+-|U&&oQhv8o8tNFOcDp?j}?W)97b*fWvgtsDx5X6)?-~rHqG|y$-JupJ0#Y9t?;% zvm*ucf$T!OsvIg?cfX3Q#V2>3slWbiM|&O-ytPj~Bhy5p7<$@7F>PKauqr*XO-q=X zbg;kUSO~@b%JSyO`y;xCLg)0*rnZbwUANV!dUhoTam2odrxD#&vfnH_B^}N`buGlJ zW1)?GTZO_W)u>C8@{`6?v4tICqcTwWH2c7|Ud=Iq(lV*NiF_zTf4-FhlgWAtT+n^FnZk(4D_wdxG~fyeAXYgcNKdtVB&JOiCHn946cf279-9+R)(=}yNW5jdxg4}#_ z87jKzO^{9*SBf~9+pidgZ#p<(9}+&Sz158^?p-gfinhpCCZ}`oBI3euq|Y59;=*%q zvGb*vz%u|X+WJoGym*24b=x%PnrL;&I%o)MJjZ(Z2(8{KHWYX<4J~@=bYmX##(je^ z<_c=TwVPN}IMTTn{z6_F&Ej}H`mIjodTqvx-T4JVjv(M}dva|{Wg zfm4g2zrEZ>Iik%|Y4z70KJchNN>Lh1S$LPs_TOXd-8s}fsJgHBZ;*`W1#uLnUJ*Qi zwLE~Etd5!U?C`bM8NVV1Up4z}oUP)o0rfP&R>|_uidqnqMf+I|eV|)EpDAJ(ne-7s z6HbVxDB-XQA%rlR$TS%{Y3Q@90zu8h@GsQD;)8i|ejgA09-Rm0yI%7nhn@%Ww^Ep8 z$~a&AIkgDX-q2ilG~DFBMBX2RhZOya5Q+e)4<3YmHf$X4xl^%}Y8dJNo}RXC6&s_v z)jQ#=v;~+WqzfNLd7KT>FjZo?hva7u)$+$}YV0cK@nYa4m*Ytr1ougp_RJ`?t_w5o z*s>u?pV>ox$lAEpyXaqdJu*<*^Ow=@Fgq{%K7OKXnyPWJnML5BV(%G8Wt%;u{-_D$ zWm3QBr*mPl(XUdTYFNn^((-4c77ludy2<1*5#2dBoAFFvGg4+JY57p;@Hwby>$Z!n zd1F~Q1CVP}HXuB|q9Eqjie7<984YP&%s zOR<%`$ZOZ=VFiJ$?sa?(R{*@9m`f{ESnR_5*pbXL0?_qEP;v4s>P7In-d%Lp-j(WQ zVP}s#N~Wf&*k!TtniI=m?~3YQkYW+t$Nv1))SSL{(fOfMAT{tEHm&c}G;BSj~!{7YmP2uOW;Jyml$ivmF6hJ_$ziEiY=aI&yb?q zpVnWx_u(jf+}pC~>y1p6BBIowG#e!d8U06*6gBkD?%ulR`W-zq+jqCEpL}$sDU819u=9 zz+EoN^CpHO(Q5x89bX&}2J)(X z2f5)w|{RQl}D6CPdxR(1{!Do$=5RvvDCLlic33pXbZS2GJYDj^{hHXRjH zTMIMy{~gOYuv2kx@c!>7L`2wBU7gG{EZnK|sn}GdWT@CQExg^S*yJ2ctSls{3{gZx zP}meL9If1~skj9GJ9A_CKNjqQ^8df|zX1N$n`Z3}pXED{NZnruPxf)Kuy z@1wV;KwGU#@m6Q$uI~ZT7v;gZbXG+42 z(Gs+Kmn|2AN&Y<7!3%ybjzPhB$Neomph(_rmPXYaIP?Xd`y zUs@W1jz&AqE7fQ1ws=(j`8Y@6yX|R^8UgBkVCVCcI^#XIddJNtLG=ZvvXWo&yF6#g zMqGagSq1jo2ugt1s3T;b8wSKe%15dv@#RyRPbHM*YEK2XBUv-|=Q4i<+&4Vb)`kzg zu>(G3&FDt<;zu1#ewLCgbMM3;Eos8ep@_1UJk~VxMVxD^_0|wvFu8q;oK{4e0}WuE z0n*2$4Ga#{g8NabPr)Jc$A=ml`O^sU&)=S0kry*r<*i2Gd9*0u7yV7kn#FQR5EgB` zK8TYGUXhsf>5O%t<}efj(LSLKFAULy0l)4@rou4Pp5}UxMMS$#?+W3l)|70J3Y~U8 z%SyW+)Li6%dmjTn7CL~Zf5>pl+J<8V0Q4+>W{OKKs0An5)t#(XbEbqss0A00<}wAu z-Noed5+deTBQXT#RifuXNnng`YjVfeBH%m3+f@FI;|Xo9UH3Al;X}1D!Oi2Nd<=p0 zg_5IFl#SQ{G_Z4bIuTeokTmJzMP|4tv1V9u45=>ijo`rpdGZce5*{<@Y)4q#ET34S>m+)+06>Rh-C7OeBRQ^1X> z6Og`15T)%u%tk1r6nymmHDX0c2&@rDdP)mWgkn30o{zd02%FBS!IAUzV3F@g$c7g! 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