Flexible Joint - Test Bench
+Flexible Joints - Test Bench
Table of Contents
-
-
- 1. Flexible Joints - Requirements -
- 2. Test Bench Description +
- 1. Flexible Joints +
- 2. Measurement Test Bench - Bending Stiffness
-
-
- 2.1. Flexible joint Geometry -
- 2.2. Required external applied force -
- 2.3. Required actuator stroke and sensors range -
- 2.4. First try with the APA95ML -
- 2.5. Test Bench +
- 2.1. Flexible joint Geometry +
- 2.2. Required external applied force +
- 2.3. Required actuator stroke and sensors range +
- 2.4. Test Bench
- - 3. Agreement between the probe and the encoder +
- 3. Error budget
-
-
- 3.1. Setup -
- 3.2. Results +
- 3.1. Finite Element Model +
- 3.2. Setup +
- 3.3. Effect of Bending +
- 3.4. Computation of the bending stiffness +
- 3.5. Estimation error due to force and displacement sensors accuracy +
- 3.6. Estimation error due to Shear +
- 3.7. Estimation error due to force sensor compression +
- 3.8. Estimation error due to height estimation error +
- 3.9. Conclusion
- - 4. Measurement of the Millimar 1318 probe stiffness +
- 4. First Measurements -
- 5. Experimental measurement +
- 5. Bending Stiffness Measurement
This report is also available as a pdf.
-
1 Flexible Joints - Requirements
-
+| Specification | +FEM | ||
|---|---|---|---|
| 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. +
+ + +
+
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 \(d_x\).
-
+
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 \(\theta_{y,\text{max}}\approx 25\,mrad\).
-
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);
-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] --
-And the tested angular range is: -
-Rmax = tan(100e-6/h); --
2.5 Test Bench
-
-
Figure 3: Schematic of the test bench to measure the bending stiffness of the flexible joints
-
-
Figure 4: Schematic of the test bench to measure the bending stiffness of the flexible joints
-- Translation Stage: V-408 -
- Load Cells: FC2231-0000-0010-L and XFL212R +
- Load Cells: FC2231-0000-0010-L
- Encoder: Renishaw Resolute 1nm -
- Displacement Probe: Millimar C1216 electronics and Millimar 1318 probe
+Both the measured force and displacement are acquired at the same time using a Speedgoat machine. +
+ + +
+
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. +
+ + +
+
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. +
+ +| + | Stiffness [N/um] | +Max Force [N] | +Stroke [um] | +
|---|---|---|---|
| Axial | +94 | +469 | +5 | +
| Shear | +13 | +242 | +19 | +
| + | Stiffness [Nm/rad] | +Max Torque [Nmm] | +Stroke [mrad] | +
|---|---|---|---|
| Bending | +5 | +118 | +24 | +
| Torsional | +260 | +1508 | +6 | +
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] ++
+
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
+ +4.1 Agreement between the probe and the encoder
+-
+
- Load Cells: FC2231-0000-0010-L (and XFL212R) +
- Encoder: Renishaw Resolute 1nm +
- Displacement Probe: Millimar C1216 electronics and Millimar 1318 probe +
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.
-
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).
-
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.
-
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).
-
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
Figure 9: Setup - Side View
+Figure 12: Setup - Side View
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.
-
Figure 11: Time domain measurements
+Figure 14: Time domain measurements
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 \(\approx 0.04\,[N/mm]\).Created: 2021-02-16 mar. 21:23
+Created: 2021-02-18 jeu. 11:33
#+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 +<