diff --git a/test-bench-flexible-joints.html b/test-bench-flexible-joints.html index e896a71..73a9fc5 100644 --- a/test-bench-flexible-joints.html +++ b/test-bench-flexible-joints.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Flexible Joints - Test Bench @@ -39,50 +39,50 @@

Table of Contents

@@ -100,27 +100,27 @@ In this document, we present a test-bench that has been developed in order to me It is structured as follow:

-
-

1 Flexible Joints

+
+

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). +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

@@ -139,10 +139,10 @@ Ideally, these flexible joints would behave as perfect ball joints, that is to s

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. +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.

- +
@@ -199,12 +199,12 @@ Using a multi-body dynamical model of the nano-hexapod, the specifications in te
Table 1: Specifications for the flexible joints and estimated characteristics from the Finite Element Model

-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. +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

@@ -216,11 +216,11 @@ The material is a special kind of stainless steel called “F16PH”.

-The flexible joints can be seen on Figure 3. +The flexible joints can be seen on Figure 3.

-
+

IMG_20210302_173619.jpg

Figure 3: 15 of the 16 flexible joints

@@ -228,22 +228,22 @@ The flexible joints can be seen on Figure 3.
-
-

2 Dimensional Measurements

+
+

2 Dimensional Measurements

- +

-
-

2.1 Measurement Bench

+
+

2.1 Measurement Bench

-The axis corresponding to the flexible joints are defined in Figure 4. +The axis corresponding to the flexible joints are defined in Figure 4.

-
+

flexible_joint_axis.png

Figure 4: Define axis for the flexible joints

@@ -255,23 +255,23 @@ Similarly, the dimensions of the flexible part in the X-Z plane will contribute

-The setup to measure the dimension of the “Y” flexible beam is shown in Figure 5. +The setup to measure the dimension of the “X” flexible beam is shown in Figure 5.

-
+

flexible_joint_y_flex_meas_setup.png

Figure 5: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness

-What we typically observe is shown in Figure 6. +What we typically observe is shown in Figure 6. It is then possible to estimate to dimension of the flexible beam with an accuracy of \(\approx 5\,\mu m\),

-
+

soft_measure_flex_size.jpg

Figure 6: Image used to measure the flexible joint’s dimensions

@@ -280,8 +280,8 @@ It is then possible to estimate to dimension of the flexible beam with an accura
-
-

2.2 Measurement Results

+
+

2.2 Measurement Results

The expected flexible beam thickness is \(250\,\mu m\). @@ -293,10 +293,10 @@ The dimension of the beams are been measured at each end to be able to estimate

-All the measured dimensions are summarized in Table 2. +All the measured dimensions are summarized in Table 2.

- +
@@ -451,11 +451,11 @@ All the measured dimensions are summarized in Table 2.
Table 2: Measured Dimensions of the flexible beams in \(\mu m\)

-An histogram of these measured dimensions is shown in Figure 7. +An histogram of these measured dimensions is shown in Figure 7.

-
+

beam_dim_histogram.png

Figure 7: Histogram for the (16x2) measured beams’ thickness

@@ -464,11 +464,11 @@ An histogram of these measured dimensions is shown in Figure -

3 Measurement Test Bench - Bending Stiffness

+
+

3 Measurement Test Bench - Bending Stiffness

- +

The most important characteristic of the flexible joint that we want to measure is its bending stiffness \(k_{R_x} \approx k_{R_y}\). @@ -488,13 +488,13 @@ The application point of the force should far enough from the flexible part such

-The working principle of the bench is schematically shown in Figure 8. +The working principle of the bench is schematically shown in Figure 8. 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_principle.png

Figure 8: Test Bench - working principle

@@ -506,16 +506,16 @@ This test-bench will be used to have a first approximation of the bending stiffn Another test-bench, better engineered will be used to measure the flexible joint’s characteristics with better accuracy.

-
-

3.1 Flexible joint Geometry

+
+

3.1 Flexible joint Geometry

-The flexible joint used for the Nano-Hexapod is shown in Figure 9. +The flexible joint used for the Nano-Hexapod is shown in Figure 9. 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 9: Geometry of the flexible joint

@@ -537,8 +537,8 @@ h = 20e-3; % Height [m]<
-
-

3.2 Required external applied force

+
+

3.2 Required external applied force

The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is: @@ -570,8 +570,8 @@ The measurement range of the force sensor should then be higher than \(6.2\,N\).

-
-

3.3 Required actuator stroke and sensors range

+
+

3.3 Required actuator stroke and sensors range

The flexible joint is designed to allow a bending motion of \(\pm 25\,mrad\). @@ -593,14 +593,14 @@ Similarly, the measurement range of the displacement sensor should also be highe

-
-

3.4 Test Bench

+
+

3.4 Test Bench

-A CAD view of the measurement bench is shown in Figure 10. +A CAD view of the measurement bench is shown in Figure 10.

-
+

Here are the different elements used in this bench:

@@ -617,18 +617,18 @@ Both the measured force and displacement are acquired at the same time using a S

-
+

test_bench_flex_overview.png

Figure 10: 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 11. +A side view of the bench with the important quantities are shown in Figure 11.

-
+

test_bench_flex_side.png

Figure 11: Schematic of the test bench to measure the bending stiffness of the flexible joints

@@ -637,11 +637,11 @@ A side view of the bench with the important quantities are shown in Figure
-
-

4 Error budget

+
+

4 Error budget

- +

Many things can impact the accuracy of the measured bending stiffness such as: @@ -657,14 +657,14 @@ Many things can impact the accuracy of the measured bending stiffness such as: In this section, we wish to estimate the attainable accuracy with the current bench, and identified the limiting factors.

-
-

4.1 Finite Element Model

+
+

4.1 Finite Element Model

-From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables 3 and 4. +From the Finite Element Model, the stiffness and stroke of the flexible joint have been computed and summarized in Tables 3 and 4.

- +
@@ -701,7 +701,7 @@ From the Finite Element Model, the stiffness and stroke of the flexible joint ha
Table 3: Axial/Shear characteristics
- +
@@ -740,11 +740,11 @@ From the Finite Element Model, the stiffness and stroke of the flexible joint ha -
-

4.2 Setup

+
+

4.2 Setup

-The setup is schematically represented in Figure 12. +The setup is schematically represented in Figure 12.

@@ -761,7 +761,7 @@ The height between the joint’s center and the force application point is:

-
+

test_bench_flex_side.png

Figure 12: Schematic of the test bench to measure the bending stiffness of the flexible joints

@@ -769,8 +769,8 @@ The height between the joint’s center and the force application point is:
-
-

4.3 Effect of Bending

+
+

4.3 Effect of Bending

The torque applied is: @@ -796,8 +796,8 @@ The measured displacement is:

-
-

4.4 Computation of the bending stiffness

+
+

4.4 Computation of the bending stiffness

From equation \eqref{eq:bending_stiffness_formula}, we can compute the bending stiffness: @@ -824,8 +824,8 @@ And therefore, to precisely measure \(k_{R_y}\), we need to:

-
-

4.5 Estimation error due to force and displacement sensors accuracy

+
+

4.5 Estimation error due to force and displacement sensors accuracy

The maximum error on the measured displacement with the encoder is 40 nm. @@ -838,8 +838,8 @@ The accuracy of the force sensor is around 1% and therefore, we should expect to

-
-

4.6 Estimation error due to Shear

+
+

4.6 Estimation error due to Shear

The effect of Shear on the measured displacement is simply: @@ -868,8 +868,8 @@ The measurement error due to Shear is 0.1 %

-
-

4.7 Estimation error due to force sensor compression

+
+

4.7 Estimation error due to force sensor compression

The measured displacement is not done directly at the joint’s location. @@ -909,8 +909,8 @@ The measurement error due to height estimation errors is 0.8 %

-
-

4.8 Estimation error due to height estimation error

+
+

4.8 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: @@ -944,8 +944,8 @@ The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %

-
-

4.9 Conclusion

+
+

4.9 Conclusion

Based on the above analysis, we should expect no better than few percent of accuracy using the current test-bench. @@ -959,24 +959,24 @@ Another measurement bench allowing better accuracy will be developed.

-
-

5 First Measurements

+
+

5 First Measurements

- +

-
-

5.1 Agreement between the probe and the encoder

+
+

5.1 Agreement between the probe and the encoder

- +

-
+

-The time domain measured displacement by the probe and by the encoder is shown in Figure 13. +The time domain measured displacement by the probe and by the encoder is shown in Figure 13.

-
+

comp_encoder_probe_time.png

Figure 13: Time domain measurement

-If we zoom, we see that there is some delay between the encoder and the probe (Figure 14). +If we zoom, we see that there is some delay between the encoder and the probe (Figure 14).

-
+

comp_encoder_probe_time_zoom.png

Figure 14: Time domain measurement (Zoom)

@@ -1028,28 +1028,28 @@ 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 15. +The measured mismatch between the encoder and the probe with and without compensating for the time delay are shown in Figure 15.

-
+

comp_encoder_probe_mismatch.png

Figure 15: 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 16). +Finally, the displacement of the probe is shown as a function of the displacement of the encoder and a linear fit is made (Figure 16).

-
+

comp_encoder_probe_linear_fit.png

Figure 16: 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. @@ -1059,13 +1059,13 @@ However, there is some time delay of tens of milliseconds that could induce some

-
-

5.2 Measurement of the Millimar 1318 probe stiffness

+
+

5.2 Measurement of the Millimar 1318 probe stiffness

- +

-
+
  • Translation Stage: V-408
  • Load Cell: FC2231-0000-0010-L
    • @@ -1076,14 +1076,14 @@ However, there is some time delay of tens of milliseconds that could induce some
-
+

setup_mahr_stiff_meas_side.jpg

Figure 17: Setup - Side View

-
+

setup_mahr_stiff_meas_top.jpg

Figure 18: Setup - Top View

@@ -1097,11 +1097,11 @@ Let’s load the measurement results.

-The time domain measured force and displacement are shown in Figure 19. +The time domain measured force and displacement are shown in Figure 19.

-
+

mahr_time_domain.png

Figure 19: Time domain measurements

@@ -1117,17 +1117,17 @@ This is very close to the 0.04 [N/mm] written in the 20). +And compare the linear fit with the raw measurement data (Figure 20).

-
+

mahr_stiffness_f_d_plot.png

Figure 20: Measured displacement as a function of the measured force. Raw data and linear fit

-
+

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

@@ -1136,10 +1136,10 @@ The Millimar 1318 probe has a stiffness of \(\approx 0.04\,[N/mm]\).
-
-

5.3 Force Sensor Calibration

+
+

5.3 Force Sensor Calibration

-
+

Load Cells:

@@ -1156,11 +1156,11 @@ There are both specified to have \(\pm 1 \%\) of non-linearity over the full ran

The XFL212R has a spherical interface while the FC2231 has a flat surface. -Therefore, we should have a nice point contact when using the two force sensors as shown in Figure 21. +Therefore, we should have a nice point contact when using the two force sensors as shown in Figure 21.

-
+

IMG_20210309_145333.jpg

Figure 21: Zoom on the two force sensors in contact

@@ -1189,7 +1189,7 @@ Fc = Fc - mean(Fc(t > -
+

force_calibration_time.png

Figure 22: Measured force using both sensors as a function of time

@@ -1217,35 +1217,35 @@ fit_F = polyfit(Fc, F, 1);

-The two forces are plotted against each other as well as the linear fit in Figure 23. +The two forces are plotted against each other as well as the linear fit in Figure 23.

-
+

calibrated_force_dit.png

Figure 23: Measured two forces and linear fit

-The measurement error between the two sensors is shown in Figure 24. +The measurement error between the two sensors is shown in Figure 24. It is below 0.1N for the full measurement range.

-
+

force_meas_error.png

Figure 24: Error in Newtons

-The same error is shown in percentage in Figure 25. +The same error is shown in percentage in Figure 25. The error is less than 1% when the measured force is above 5N.

-
+

force_meas_error_percentage.png

Figure 25: Error in percentage

@@ -1253,8 +1253,8 @@ The error is less than 1% when the measured force is above 5N.
-
-

5.4 Force Sensor Noise

+
+

5.4 Force Sensor Noise

The objective of this measurement is to estimate the noise of the force sensor FC2231-0000-0010-L. @@ -1272,11 +1272,11 @@ Ts = t(2) - t(1);

-The measured force is shown in Figure 26. +The measured force is shown in Figure 26.

-
+

force_noise_time.png

Figure 26: Measured force

@@ -1297,11 +1297,11 @@ win = hanning(ceil(1/Ts));

-The results is shown in Figure 27. +The results is shown in Figure 27.

-
+

force_noise_asd.png

Figure 27: Amplitude Spectral Density of the meaured force

@@ -1310,15 +1310,15 @@ The results is shown in Figure 27.
-
-

5.5 Force Sensor Stiffness

+
+

5.5 Force Sensor Stiffness

The objective of this measurement is to estimate the stiffness of the force sensor FC2231-0000-0010-L.

-To do so, a very stiff element is fixed in front of the force sensor as shown in Figure 28. +To do so, a very stiff element is fixed in front of the force sensor as shown in Figure 28.

@@ -1331,7 +1331,7 @@ Then, having the force and the deflection, we should be able to estimate the sti

-
+

IMG_20210309_145242.jpg

Figure 28: Bench used to measured the stiffness of the force sensor

@@ -1373,10 +1373,10 @@ fit_k = polyfit(F, d, 1);

-The displacement as a function of the force as well as the linear fit are shown in Figure 29. +The displacement as a function of the force as well as the linear fit are shown in Figure 29.

-
+

force_sensor_stiffness_fit.png

Figure 29: Displacement as a function of the measured force

@@ -1392,30 +1392,30 @@ k = 0.76 [N/um]
-
-

6 Bending Stiffness Measurement

+
+

6 Bending Stiffness Measurement

- +

-
-

6.1 Introduction

+
+

6.1 Introduction

-A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure 30. -A closer view on flexible joint is shown in Figure 31 and a zoom on the force sensor tip is shown in Figure 32. +A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure 30. +A closer view on flexible joint is shown in Figure 31 and a zoom on the force sensor tip is shown in Figure 32.

-
+

picture_bending_x_meas_side_overview.jpg

Figure 30: Side view of the flexible joint stiffness bench. X-Bending stiffness is measured.

-
+

picture_bending_x_meas_side_close.jpg

Figure 31: Zoom on the flexible joint - Side view

@@ -1423,18 +1423,18 @@ A closer view on flexible joint is shown in Figure 31 -
+

picture_bending_x_meas_side_zoom.jpg

Figure 32: Zoom on the tip of the force sensor

-The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure 33. +The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure 33.

-
+

picture_bending_y_meas_side_close.jpg

Figure 33: Stiffness measurement bench - Y-d bending measurement

@@ -1442,8 +1442,8 @@ The same bench used to measure the Y-bending stiffness of the flexible joint is
-
-

6.2 Analysis of one measurement

+
+

6.2 Analysis of one measurement

In this section is shown how the data are analysis in order to measured: @@ -1478,29 +1478,29 @@ t = t(t > 0.2); t = t -<

-The obtained time domain measurements are shown in Figure 34. +The obtained time domain measurements are shown in Figure 34.

-
+

flex_joint_meas_example_time_domain.png

Figure 34: Typical time domain measurements

-The displacement as a function of the force is then shown in Figure 35. +The displacement as a function of the force is then shown in Figure 35.

-
+

flex_joint_meas_example_F_d.png

Figure 35: Typical measurement of the diplacement as a function of the applied force

-The bending stiffness can be estimated by computing the slope of the curve in Figure 35. +The bending stiffness can be estimated by computing the slope of the curve in Figure 35. The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.

@@ -1539,11 +1539,11 @@ fit_l(2) = 0;

-The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure 36. +The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure 36.

-
+

flex_joint_meas_example_F_d_lin_fit.png

Figure 36: Typical measurement of the diplacement as a function of the applied force with estimated linear fits

@@ -1558,10 +1558,10 @@ Then, the bending stroke is estimated as crossing point between the two fitted l

-The obtained characteristics are summarized in Table 5. +The obtained characteristics are summarized in Table 5.

-
Table 4: Bending/Torsion characteristics
+
@@ -1589,18 +1589,18 @@ The obtained characteristics are summarized in Table 5 -
-

6.3 Bending stiffness and bending stroke of all the flexible joints

+
+

6.3 Bending stiffness and bending stroke of all the flexible joints

Now, let’s estimate the bending stiffness and stroke for all the flexible joints.

-The results are summarized in Table 6 for the X direction and in Table 7 for the Y direction. +The results are summarized in Table 6 for the X direction and in Table 7 for the Y direction.

-
Table 5: Estimated characteristics of the flexible joint number 1 for the X-direction
+
@@ -1735,7 +1735,7 @@ The results are summarized in Table 6 for the X direct
Table 6: Measured characteristics of the flexible joints in the X direction
- +
@@ -1872,33 +1872,33 @@ The results are summarized in Table 6 for the X direct -
-

6.4 Analysis

+
+

6.4 Analysis

-The dispersion of the measured bending stiffness is shown in Figure 37 and of the bending stroke in Figure 38. +The dispersion of the measured bending stiffness is shown in Figure 37 and of the bending stroke in Figure 38.

-
+

bending_stiffness_histogram.png

Figure 37: Histogram of the measured bending stiffness

-
+

bending_stroke_histogram.png

Figure 38: Histogram of the measured bending stroke

-The relation between the measured beam thickness and the measured bending stiffness is shown in Figure 39. +The relation between the measured beam thickness and the measured bending stiffness is shown in Figure 39.

-
+

flex_thickness_vs_bending_stiff.png

Figure 39: Measured bending stiffness as a function of the estimated flexible beam thickness

@@ -1906,10 +1906,10 @@ The relation between the measured beam thickness and the measured bending stiffn
-
-

6.5 Conclusion

+
+

6.5 Conclusion

-
+

The measured bending stiffness and bending stroke of the flexible joints are very close to the estimated one using a Finite Element Model.

@@ -1926,7 +1926,7 @@ This should allow us to model them with unique parameters.

Author: Dehaeze Thomas

-

Created: 2021-04-30 ven. 14:33

+

Created: 2021-04-30 ven. 14:36

diff --git a/test-bench-flexible-joints.org b/test-bench-flexible-joints.org index adf3886..6f38d4f 100644 --- a/test-bench-flexible-joints.org +++ b/test-bench-flexible-joints.org @@ -125,7 +125,7 @@ The axis corresponding to the flexible joints are defined in Figure [[fig:flexib The dimensions of the flexible part in the Y-Z plane will contribute to the X-bending stiffness. Similarly, the dimensions of the flexible part in the X-Z plane will contribute to the Y-bending stiffness. -The setup to measure the dimension of the "Y" flexible beam is shown in Figure [[fig:flexible_joint_y_flex_meas_setup]]. +The setup to measure the dimension of the "X" flexible beam is shown in Figure [[fig:flexible_joint_y_flex_meas_setup]]. #+name: fig:flexible_joint_y_flex_meas_setup #+caption: Setup to measure the dimension of the flexible beam corresponding to the X-bending stiffness
Table 7: Measured characteristics of the flexible joints in the Y direction