Flexible Joints - Test Bench

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

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 5.

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 \(\approx 5\,\mu m\),

The expected flexible beam thickness is \(250\,\mu m\). However, it is more important that the thickness of all beams are close to each other.

The dimension of the beams are been measured at each end to be able to estimate the mean of the beam thickness.

All the measured dimensions are summarized in Table 2.

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

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

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 10.

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 11.

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

The setup is schematically represented in Figure 12.

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 13.

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

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 15.

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).

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 19.

Now we can estimate the stiffness with a linear fit.

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 20).

A picture of the bench used to measure the X-bending stiffness of the flexible joints is shown in Figure 21. A closer view on flexible joint is shown in Figure 22 and a zoom on the force sensor tip is shown in Figure 23.

The same bench used to measure the Y-bending stiffness of the flexible joint is shown in Figure 24.

In this section is shown how the data are analysis in order to measured:

• the bending stiffness
• the bending stroke
• the stiffness once the mechanical stops are in contact

The height from the flexible joint’s center and the point of application force \(h\) is defined below:

h = 25e-3; % [m]

%% Load Data
load('meas_stiff_flex_1_x.mat', 't', 'F', 'd');

%% Zero the force
F = F - mean(F(t > 0.1 & t < 0.3));

%% Start measurement at t = 0.2 s
d = d(t > 0.2);
F = F(t > 0.2);
t = t(t > 0.2); t = t - t(1);

The obtained time domain measurements are shown in Figure 25.

The displacement as a function of the force is then shown in Figure 26.

The bending stiffness can be estimated by computing the slope of the curve in Figure 26. The bending stroke and the stiffness when touching the mechanical stop can also be estimated from the same figure.

%% Determine the linear region and region when touching the mechanical stop
% Find when the force sensor touches the flexible joint
i_l_start = find(F > 0.3, 1, 'first');
% Reset the measured diplacement at that point
d = d - d(i_l_start);
% Find then the maximum force is applied
[~, i_s_stop] = max(F);
% Linear region stops ~ when 90% of the stroke is reached
i_l_stop = find(d > 0.9*d(i_s_stop), 1, 'first');
% "Stop" region start ~1N before maximum force is applied
i_s_start = find(F > max(F)-1, 1, 'first');

%% Define variables for the two regions
F_l = F(i_l_start:i_l_stop);
d_l = d(i_l_start:i_l_stop);

F_s = F(i_s_start:i_s_stop);
d_s = d(i_s_start:i_s_stop);

%% Fit the best straight line for the two regions
fit_l = polyfit(F_l, d_l, 1);
fit_s = polyfit(F_s, d_s, 1);

%% Reset displacement based on fit
d = d - fit_l(2);
fit_s(2) = fit_s(2) - fit_l(2);
fit_l(2) = 0;

The raw data as well as the fit corresponding to the two stiffnesses are shown in Figure 27.

Then, the bending stroke is estimated as crossing point between the two fitted lines:

d_max = fit_l(1)*fit_s(2)/(fit_l(1) - fit_s(1));

The obtained characteristics are summarized in Table 5.

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 dispersion of the measured bending stiffness is shown in Figure 28 and of the bending stroke in Figure 29.

The relation between the measured beam thickness and the measured bending stiffness is shown in Figure 30.