## 4 Simple Approach for Optimizing the Offline Motion Preparation
### Performance Evaluation  
With this approach and the previous knowledge about the workspace of the mechanism, there are many possibilities for optimizing the offline motion preparation. For the sake of simplicity, a single scenario will be considered, which could be extended to other applications in the following studies. In this scenario, two basic definitions are made. On the one hand, a constant transmission \(\lambda_{2}\) over the entire workspace is estimated. In doing so, all kinematic transformations used to check for compliance with the tolerance can be outsourced to the commissioning of the mechanism. They are no longer required for the preparation of a specific motion. Only the motion itself has to be calculated once from the TCP to axis level. On the other hand, the axes are decoupled from each other by using \(\lambda_{\text{2.corr.ax}}\) at axis level. This allows an optimized segmentation and a better adaptation of the splines to the motion specification. For performance evaluation purposes, firstly the reduction of the number of calculated kinematic transformations is examined. As can be seen in Figure 14, this number has been reduced significantly for all algorithms. For the B-spline algorithms, 87.5% (bsLSQall, bsITERall) to 95.8% (bsLSQmax, bsITERmax) transformations are saved on average. This percentage is so high because the algorithms perform many transformations for searching the global maximum. For the algorithms that divide all segments (bsLSQall, bsITERall), the saving is smaller since the computational effort itself is smaller. However,  
Figure 13: Estimation of a minimum \(\lambda_{2}\) (blue) by extrapolation between the minimum (black) and the surrounding maximum (red). since the kinematic transformations in the B-spline approaches only have a minimal impact on the total computation effort (Figure 8), no significant performance increase is expected. With the monomial spline algorithms, in average 63.8% (msITER) and 57.2% (msRECUR) of the kinematic transformations can be saved on average (66.7% and 58.7% for an accuracy of 1 \(\upmu\)m). Due to the great influence of the transformations (Figure 8), this should be apparent in the overall performance. The next step is to consider how the accuracy of motion changes over the path. As described above, two factors lead to the calculation of a more accurate path than required. One is the reduction to a constant \(\lambda_{2}\), which leaves up to 14% of the accuracy unused. The other is splitting \(r_{\varphi}\) to the two axes via Equation (8). Here an error of up to 64% is expected. Figure 15 shows the deviation of the calculated spline from the optimized set points. (a) shows the conventional approach where all algorithms come very close to the allowed accuracy. In (b) the simple approach is shown. It can be seen that for the given motion a large part of the allowed accuracy is unused (>60%). Furthermore, it can be seen that for the algorithms that divide all segments (bsLSQall, bsITERall) a much higher accuracy than necessary is achieved in wide ranges of the motion (e.g., between \(\tau\) = 0.2-0.4). This reduces the performance of those approaches. The results of the performance evaluation of the algorithms for decoupled axes are shown in Figure 16. The qualitative trend in (a) shows that the basic characteristic of the performance curves does not change compared to the coupled case. Due to the simplification made, the B-spline algorithms have a worse performance than before. At 1 \(\upmu\)m accuracy, the number of generated data increases slightly by +13%, while the computational effort rises significantly by +30%. Figure 14: Reduction of the number of kinematic transformations by using the approach. Figure 15: Motion accuracy of the calculated splines. (**a**) Using conventional an (**b**) simple motion preparation approach. effort can be explained by the fact that the simplifications require a much higher accuracy than before.