Flexible Joint - Test Bench

1. Flexible Joints - Requirements
2. Test Bench Description
3. Agreement between the probe and the encoder
- 3.1. Results
4. Measurement of the Millimar 1318 probe stiffness
- 4.1. Results
5. Experimental measurement

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1 Flexible Joints - Requirements

	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]

2 Test Bench Description

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, a test bench is used. Specifications of the test bench to precisely measure the bending stiffness are described in this section.

The basic idea is to measured the angular deflection of the flexible joint as a function of the applied torque.

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

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 \(\theta_{y,\text{max}}\approx 20\,mrad\).

Figure 2: Geometry of the flexible joint

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

2.2 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:

\begin{equation} F_{max} = 20.0\, [N] \end{equation}

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

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:

\[ d_{x,\text{max}} = h \tan(R_{x,\text{max}}) \]

  dxmax = h*tan(Rxmax);

\begin{equation} d_{max} = 0.4\, [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 \(0.4\,mm\).

2.4 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:

  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

Figure 3: 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
Encoder: Renishaw Resolute 1nm
Displacement Probe: Millimar C1216 electronics and Millimar 1318 probe

3 Agreement between the probe and the encoder

3.1 Results

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

Figure 4: Time domain measurement

Figure 5: Time domain measurement (Zoom)

finddelay(d, dp)

Ts*finddelay(d, dp)

0.0158

Figure 6: Measurement mismatch, with and without delay compensation

Figure 7: Measured displacement by the probe as a function of the measured displacement by the encoder

4 Measurement of the Millimar 1318 probe stiffness

Translation Stage: V-408
Load Cell: FC2231-0000-0010-L
Encoder: Renishaw Resolute 1nm
Displacement Probe: Millimar C1216 electronics and Millimar 1318 probe

Figure 8: Setup - Side View

Figure 9: Setup - Top View

4.1 Results

load('meas_stiff_probe.mat', 't', 'd', 'dp', 'F')

The time domain measured force and displacement are shown in Figure 10.

Figure 10: Time domain measurements

Now we can estimate the stiffness with a linear fit.

Stiffness is 0.039 [N/mm]

This is very close to the 0.04 [N/mm] written in the Millimar 1318 probe datasheet.

And compare the linear fit with the raw measurement data (Figure 11).