## 5 Methods, Results and Discussion
### Driving and Holding Current Influence on a Displacement Step  
Using the model described in Section 3, we simulated the object displacement for different driving currents along \(x\) and \(y\)-axes without and with a 1 A holding current. We injected a driving current in one of the two pair of wires of each actuator and, simultaneously, a holding current into the other pair of wires, to obtain a linear motion of the mobile permanent magnet (MPM) and the object along that axis. Both current pulses were longer than the displacement step time to ensure that the MPM accelerated throughout its displacement. Figure 9 presents the displacement of the conveyed object as a function of the driving current intensity for both the \(x\) and \(y\) axes with and without a 1 A holding current. Each experimental point in Figure 9 was measured 20 times under the same conditions to obtain a mean value and a standard deviation for the point's conditions. The figure shows the point as the mean value, and its standard deviation as error bars. From Figure 9, the object displacement decreased as the driving current increased on both axes and for both conditions (with and without the 1 A holding current). When the injected current increases, the MPM displacement time and the energy transfer through friction to the object are reduced, which leads to a decrease of the object displacement. For the same driving current value, the obtained displacement is lower along the \(x\)-axis than along the \(y\)-axis. This is due to the lower distance between the wires actuating the MPM along the \(x\)-axis than for the \(y\)-axis (\(d_{2}\) and \(d_{3}\) in Table 1). Therefore, the electromagnetic force is higher for the \(x\)-axis, inducing a shorter MPM displacement time and consequently a lower energy transfer to the object, as explained before. For example, with a 2 A driving current (without holding current), the object displacement along \(x\) and \(y\)-axis are 151 um and 192.3 um, respectively. The holding and no holding results followed a similar tendency. The holding current induced a shift towards higher driving currents (shift of the initial  
Figure 9: Driving and holding current influence on a displacement step: Object displacement vs. a varying driving current with and without a 1 A holding current. (**a**) \(x\)-axis step. (**b**) \(y\)-axis step. curve to the right). This is due to the higher driving current intensity needed to overcome the increased friction force generated by the holding current. The error bars decreased as the driving current increased for both conditions. This is attributed to the important influence of the friction conditions at lower currents and its reduced influence with faster kinematics as the current increases. The model predicted the experimental tendency correctly. The root-mean-square error (RMSE) between the experimental and simulated results without holding current was \(29.9\,\mathrm{\SIUnitSymbolMicro m}\) for the \(x\)-axis and \(19\,\mathrm{\SIUnitSymbolMicro m}\) for the \(y\)-axis. The RMSE between the model prediction and experimental results with 1 A holding current were \(38.6\,\mathrm{\SIUnitSymbolMicro m}\) for the \(x\)-axis and \(49.3\,\mathrm{\SIUnitSymbolMicro m}\) for the \(y\)-axis. In order to implement our optimum control strategy, the minimum current values ensuring a motion were experimentally sought and found to be \(1.5\,\mathrm{A}\) (corresponding displacement: \(179\,\mathrm{\SIUnitSymbolMicro m}\)) and \(1.75\,\mathrm{A}\) (\(205\,\mathrm{\SIUnitSymbolMicro m}\)) for \(x\) and \(y\)-axes without holding current, respectively. When using a 1 A holding current, the minimum driving currents to ensure a motion were higher and found to be \(2.5\,\mathrm{A}\) (\(189\,\mathrm{\SIUnitSymbolMicro m}\)) and \(2.75\,\mathrm{A}\) (\(177\,\mathrm{\SIUnitSymbolMicro m}\)) for \(x\) and \(y\)-axes, respectively. The straightness error of all the experimental points of Figure 9 was measured.