## 5 Methods, Results and Discussion
### 2D Trajectory Validation  
To evaluate the bi-dimensional trajectory control of the DAA, a 0.416 g object, with size 20 mm \(\times\) 20 mm (the size of a 2 \(\times\) 2 DAA), was conveyed. Five target points were given to the A* trajectory control algorithm using a holding current of 1 A: (0,0); (0,-259); (296,-259); (296,0) and (0,0), all in \(\upmu\)m. These points formed a rectangular objective trajectory. The trajectory control algorithm generated the control currents and injected them into the DAA to convey the object. The initial and final position measures of each conveyance step were taken using the Grasshopper camera. The initial position of each conveyance step was defined as the position before the displacement step and the final position as the position after the return phase of the DAA. Once the rectangular trajectory was performed, the system was reset and the same trajectory executed. This process was done 8 times to obtain a mean position for each conveyance step with its standard deviation in the \(x\) and \(y\) axes. Figure 14 presents the objective rectangular trajectory with its target positions and predicted intermediate positions (noted \"model\") and the average of the experimental measurements. The standard deviation of the experimental results were small, indicating a repeatable outcome of the system. The DAA had a tendency to deviate towards the \(+x\)-axis while executing displacement steps on the \(y\)-axis. This could be the influence of the tilt error previously explained. There was a deviation towards the \(-y\)-axis while executing displacement steps along the \(-x\)-axis, but when executing displacement steps along the \(+x\)-axis there was no deviation tendency (apart from the correction step to attain the (296,-259) point). This indicated that the flatness condition only affected one axis. Studying the displacement values, the system performed: larger-than-expected steps along the \(y\)-axis (both negative and positive); shorter-than expected steps along the \(+x\)-axis; and well-expected steps along the \(-x\)-axis. The final mean experimental position was (\(-12.56\,\upmu\)m, \(-15.52\,\upmu\)m) \(\pm\) 12.75 \(\upmu\)m for the \(x\)-axis and 2.2 \(\upmu\)m for the \(y\)-axis. As the wished final position was (0,0), these values are also the final position error. Figure 13: 1D displacements along the \(x\) axis. The rectangular trajectory is divided on its component displacement steps: eight normal displacement step (four along the \(x\)-axis plus four along the \(y\)-axis) and four tuning or adjusting steps (two along the \(x\)-axis plus two along the \(y\)-axis). The tuning steps come from the trajectory control algorithm with A*: it executes the largest possible step in the wished direction, the maximum possible number of times and, once the object arrives near the wished position, the controller adjusts the last step to reach the position. Figure 14b compares the model-predicted displacement for each type of step with the corresponding experimental results. From Figure 14b, the \(x\)-axis displacement step was almost perfectly predicted (122.1 um exp. against 121.1 um model). For the \(y\)-axis step, the experimental displacement was larger than expected (128.7 um exp. against 100.1 um model). The root-mean-square error for the four type of steps was 19.8 um. Finally, to evaluate longer and more complex trajectories, the trajectory controller was given a predefined trajectory forming the Universite de Technologie de Compiegne acronym (UTC) (Figure 15). The trajectory spanned a 1700 um \(\times\) 800 um area and was 1109 um long. The A* trajectory algorithm and 1 A holding current controlled the DAA. From Figure 15, the trajectory was well executed, with a maximum position error of 50.4 um for the \(x\)-axis and 67.5 um for the \(y\)-axis. The final position error was 11.8 um for the \(x\)-axis and 31.8 um for the \(y\)-axis.