Tratamientos

The Stepwise Regression algorithm has been launched and a solution has been found in 5.43 seconds.

The offline results obtained in this case are similar to the ones shown in subsection 6.5 on page 84. As this time, the force is not considered at all, the results are better: regarding the prediction

of the validation set (Figure 7.27), the error in predicting q1is ±41 nm, q2is ±42 nm and q3is

±49 nm (for the 90% of the points).

The model found in this way has been implemented in the system and a trajectory file has been imposed. The measured error in such positions is plotted in Figure 7.28. Here we obtained an error of ±58 nm for x axis, ±75 nm for y axis and ±62 nm for z axis (with a confidence interval of 90%).

Figure 7.27: Error in predicting the Agietron

Micro-Nano validation set.

Figure 7.28: Agietron Micro-Nano measured final

positioning error.

q1 q2 q3 x y z

Error ±41 nm ±42 nm ±49 nm ±58 nm ±75 nm ±62 nm

Table 7.4: Comparison of the Agietron Micro-Nano calibration results in motor coordinates and end-

7.5.2.2 Parasitic Rotations Model

In 1.5 seconds, Stepwise Regression algorithm found the coefficients to fit the Agietron Micro- Nano parasitic rotations model (Figure 7.29 and Table 7.5).

The error in predicting the parasitic rotations aroundθx is ±0.5”,θy is ±1.2”,θz is ±0.4”.

The model has in total 34 parameters, respectively composed by 11, 13 and 10 parameters for each axis. Notice that the results on axisθyare worst than the other axes. This is due to

a lower quality of measure on this axis caused by the presence of a 45° mirror on the laser beam path.

Figure 7.29: Prediction of the Agietron Micro-Nano parasitic rotation on the validation set.

θx θy θz

Error ±0.5” ±1.2” ±0.4” Parameters 11 13 10

Table 7.5: Error in predicting the parasitic rotations of the Agietron Micro-Nano.

7.6 Conclusions

In this chapter we covered the calibration of the Agietron Micro-Nano robot and the Min- Angle robot. We also suggested a strategy to mutually compensate for their parasitic DOF. The issue in calibrating the MinAngle comes from the lack of appropriate measuring devices. To allow the calibration of this robot in all its workspace we had to develop a complicate cal- ibration procedure.

We summarize here the most important results and conclusions of this experience: • The modeling and identification procedure already used in subsection 6.3.1 on page 79

has also been used to identify the parameters of the Large Stroke MinAngle model and Ultra-high-precision MinAngle model. As those three models are different, this demonstrates the universality of this approach.

7.6. CONCLUSIONS 109

• The robot Agietron Micro-Nano has been calibrated with a final accuracy better than ±75 nm for the three axes at the level of the end-effector, while thermal effects are acting on it.

• A model of the Agietron Micro-Nano parasitic rotations has been established. Those are predicted with a maximum error of ±1.2”.

• The MinAngle has been successfully calibrated. In a restricted workspace a maximum error of ±11” in rotations and ±1.1μm in translation has been obtained.

• A model to predict the parasitic translations of the MinAngle along the vertical axis and the robot temperature has been established. The model is able to predict the parasitic translations with a maximum error of ±400 nm.

• For each model described in this chapter, the data processing took less then 6 second to converge.

• The overall calibration time is 17 hours.

This experience once again underlines the limitation imposed by the measuring devices, in terms of:

• Lack of ultra-high-precision measuring devices having a large course (e.g. the autocol- limator).

• Sensitivity to movements along others DOF (e.g. the interferometer). • Time consumption of the measuring phase.

In this sense, we can say that the final accuracy of a robot not only is limited by its intrinsic characteristics, but also by the measuring devices used for its calibration.

In the next chapter we will complete the calibration of the 2-robot system introducing the nano-indentation process.

Chapter 8

Case C.2: Indentation

The Vickers hardness test or indentation is a process developed to evaluate the hardness of materials. It consists in engraving a mark by pushing a diamond tip in a test material with a known force. It is then possible to calculate the hardness by measuring the mark’s diagonal.

In [27] nano-indentation has been used for the first time as a calibration verification tool for ultra-high-precision robots. The idea behind this adaptation is simple: here we are not anymore interested in measuring the diagonal of the indent, but in the relative distance be- tween the marks. The calibrated robot is used to engrave the marks on a substrate in well- known positions. Then the distance between the indents is evaluated using an electronic microscope. As the work of Dr. Fazenda represents the state of the art on this matter, for each issue we will cite the solution adopted in [27] to compare it with the current case.

The nano-indentation concept could seem very simple on paper, but we will see that it is complex to accomplish. In fact, it implies a deep knowledge of many aspects, namely: the robot and the indentation setup geometry, the measuring principles and devices used for calibration and for measuring the indent positions. Every element that is not kept in account while developing the indentation system and procedure will irreparably add more incertitude to the final error budget.

With this preliminary remark in mind, we are going to introduce each step of the calibra- tion procedure applied to this case. Finally, we will close this chapter with the indentation results and with the conclusion to retain from this experience.

8.1 Step 1 - The Sources of Inaccuracy

The sources of inaccuracy arising from the concurrent use of the two robots and by the nano- indentation process are the following:

• Reference issues, • External forces,

• Parasitic degrees of freedom,

• Incertitude in the measure of the indents. 111

The effect of the Agietron Micro-Nano parasitic rotations is amplified by the setup for the nano-indentation. A strategy to compensate for this effect is proposed.

For reference issues we consider the loss of accuracy due to the misalignment of the two robots frames. We will solve this issue by evaluating the misalignment and compensating for it. Afterwards, a common zero for the two robots will be established.

The external forces in this case are generated by the contact between the indenter and the substrate, at the moment when the indent is engraved.

The incertitude in the measure comes from the resolution of the SEM microscope em- ployed for the measures, the incertitude in locating the center of the indent and the mis- alignment between the microscope and the indent substrate.

In the next section we will discuss all those topics in detail.