Retransmisión de TYLCV a Solanum lycopersicum

For each model introduced in section 6.3, we used the Stepwise Regression algorithm to find the coefficients to fit the calibration set. Afterwards, we use the first validation set (in which temperatures, forces and poses are inside the calibration range) to fill Table 6.1 with the cali- bration results. Notice that the values used for the confirmation are not used to find the value of the coefficients. The results are presented for each axis in motor coordinates considering an interval of confidence of 90 % (1.645σ); we also give the number of parameters employed to allow such prediction. Finally, we show the computational time needed to calculate each complete model. In comparison with Case A, we improved the data processing performed before launching the Stepwise Regression algorithm. For this reason, even if we have a more complex model to find, the total computational time is lower than in the first case study.

For the pure geometric model we obtained a final accuracy in motor coordinates around ±0.5μm on the three axes. This means that without keeping into account external forces and

6.5. STEP 5 - RESULTS 85 Model→ G G + F G + T G + T + F q1 ±449 nm ±360 nm ±449 nm ±41 nm par amq1 13 18 13 22 q2 ±572 nm ±390 nm ±173 nm ±42 nm par amq2 12 21 24 24 q3 ±533 nm ±192 nm ±274 nm ±49 nm par amq3 12 19 25 27 Computational time 1.60 s 3.51 s 3.84 s 4.82 s

Table 6.1: Error in predicting the validation set inside the calibration range using a pure geometric

model (G), a geometric + force model (G + F), a geometric + thermal model (G + T) and a complete model (G + T + F). The number of parameters used to obtain this result for each axis and the computational time for each complete model is also indicated.

F F W 10’’ q1 q2, q3 y x z

Figure 6.11: Effect of the force on the robot geometry, when F=3.84 N and W=0.67 N.

thermal effects, we can guarantee an end-effector positioning precision up to 1μm.

For the models G + F and G + T, the explanation of the results need further investiga- tion. As expected, the effect of the force is different on each axis. This happens because the force acting on the end-effector deforms the robot in a non-symmetric way. In fact, the cube mounted asymmetrically in respect to the center of mass and the force creates a momen- tum on the end-effector, which deforms the robot structure between the end-effector and the motor and causes a rotation of the end-effector of 10” around y axis (Figure 6.11). For example, notice that the prediction of q1for the model G + T is at the same level of model

G. As the model G + T has no access to the force reading, it is not capable of handling such interaction. Also on q3 we can find a way similar effect, but with reduced consequences.

An effect on those results is also imputable to the Stepwise Regression algorithm falling in a local minimum.

The model G + T + F expresses the full predictional power of the calibration algorithm proposed in this thesis. By allowing the use of all the readings of the sources of inaccuracy, the Stepwise Regression is capable of predicting the set of data with an error under ±50 nm. In Figure 6.12 we plot the error in predicting the first validation set for the three axes. Finally, we tested this model on the second validation data set (having temperatures and forces out-

Figure 6.12: Plot of the error in predicting the

first validation set (points inside the calibration range).

Figure 6.13: Plot of the error in predicting the sec-

ond validation set (points outside the calibration range).

Axis Force inside the calibration range Force outside the calibration range

q1 ±41 nm ±164 nm

q2 ±42 nm ±97 nm

q3 ±49 nm ±93 nm

Table 6.2: Comparison of the error in motor coordinates obtained while applying a force inside the

calibration range with an error outside the calibration range.

side the calibration range). Logically, we expected a worse result than the previous one. In fact, we obtained an error in predicting the motor coordinates of ±164 nm on q1, ±97 nm

on q2and ±93 nm on q3, considering a confidence interval of 90 % (Figure 6.13 and Table

6.2). With such results we demonstrate that the model is also effective outside its range, as the parametric approaches as the Stepwise Regression algorithm are known to be good in finding general solutions.

Also in this case, all the models considered here are linear in the parameters and non- linear in the variables. As the thermal behavior is responsible for changing the robot ge- ometry during the time flow, the models G and G + F are time variant. On the other side, the thermal compensation cancel this effect, making the models G + T and G + T + F time invariant.

As done for Case A, cross parameters relevant to each case have been tested as well: cross thermal and geometric parameters have been tested for the model G + T and G + T + F, while cross force and geometric parameters have been employed for the model G + F and G + T + F. Also in this case all the cross parameters have been excluded by all the models suggesting once again that the thermal compensation act as a “moving offset”, shifting the geomet- ric model prediction dependently by the thermal changes. For this specific case, the force compensation behaves exactly like the thermal compensation, shifting the geometric model prediction dependently by the force applied on the end-effector.

6.5. STEP 5 - RESULTS 87

Figure 6.14: End-effector final positioning error.

For a practical use of the robot, we can draw the following conclusions:

• Forces acting on the robot can dramatically affect the final accuracy of a robot. They have to be compensated for depending on their value, the final accuracy required and the stiffness of the robot.

• Thermal effects must be imperatively compensated for applications that produce heat or for applications that last in time. In the case where a robot performs a fast indus- trial process without generating heat, it is possible to avoid thermal compensation by initializing the zero of the robot before starting each process, initializing de facto the offset due to thermal drift.