Flexible Joints - Test Bench

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 \(\theta_{y,\text{max}}\approx 25\,mrad\).

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]

The bending \(\theta_y\) of the flexible joint due to the force \(F_x\) is:

Therefore, the applied force to test the full range of the flexible joint is:

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

And we obtain:

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

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);

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 \(0.5\,mm\).

A CAD view of the measurement bench is shown in Figure 5.

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

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

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

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]

The torque applied is:

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

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

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

For small displacement, we have

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 \(h\)

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%.

The effect of Shear on the measured displacement is simply:

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

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

The measurement error due to Shear is 0.1 %

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:

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

The measurement error due to height estimation errors is 0.8 %

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

The computed bending stiffness will be:

And the stiffness estimation error is:

h_err = 0.2e-3; % Height estimation error [m]

The measurement error due to height estimation errors of 0.2 [mm] is 1.6 %

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.

• 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.

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 8.

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

This delay is estimated using the finddelay command.

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 10.

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).

Let’s load the measurement results.

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

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

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 15).