## 3 Conventional Offline Motion Preparation
### Performance Evaluation  
The state of the art offers few hints on how motion preparation is performed in similar applications.  
In the obvious case of offline motion preparation, both coordinates (x and y in the TCP-coordinate system; \(\varphi_{1}\) and \(\varphi_{2}\) in axis coordinate system, cf. Figure 1a) are considered together. In this case, the length of the spline segments in both coordinate directions is always identical compared to the normalized cycle time \(\tau\). Thus, the specifications are \"coupled\" over the segment lengths, similar to a 2D spline. The advantage of this is that each segment can be transformed completely from the axis level to TCP level and vise-versa.  
Figure 7 shows the performance of the evaluated algorithms in two 2D performance diagrams similar to [9]. As described in Section 2, the aim is to obtain a small amount of data with low calculation effort. This means, in the diagram, if a point is close to the origin, the assigned algorithm is more performant than one far away. It is then more suitable for the use in example. If two points are close to each other, it is proposed to enhance this qualitative performance evaluation method by looking at the concrete measurement results. In our example, the decision about the performance of an algorithm will be made in favor of the greater savings achieved.  
Figure 6: Monomial spline-based recursive algorithm msRECUR.  
Figure 7a shows the performance curve of the algorithms over several accuracies between 1 \(\upmu\)m and 500 \(\upmu\)m. As expected, the performance decreases with increasing accuracy (small \(\varepsilon\)), Figure 7a. A closer look reveals that the performance curve of the monomial spline algorithms is approximately linear with respect to the accuracy. The gradient of the msRECUR-curve is flatter, resulting in a positive effect on the computational effort if the accuracy is raised. For the B-spline algorithms, the performance decreases following a power function with an exponent of about 2.2 (bsLSQmax, bsITERmax) and 1.3 (bsLSQall, bsITERall). This trend speaks against using B-spline algorithms in our example. From the location of the curves it can be deduced that the performance of the monomial spline algorithms is always better than the performance of the B-spline algorithms.  
For an accuracy of 1 \(\upmu\)m, Figure 7b, the B-spline based algorithms produce up to 90% more data while using up to about 46,000 times more calculation effort compared to msRECUR. Furthermore, bsITER requires about 10 times of the effort compared to bsLSQ to get sufficiently close to the exact solution. Therefore, bsITER shows the worst performance, followed by bsLSQ. This performance of the B-spline algorithms is primarily due to the very complex least square fitting in the background of those algorithms.  
In the diagram Figure 7b, msITER appears closer to the origin than msRECUR, but their difference is small. In concrete numbers, for an accuracy of 1 \(\upmu\)m msITER is 10% more efficient in terms of coefficients, while msRECUR saves 19% effort. Due to the greater savings, msRECUR is the most suitable algorithm for our example, closely followed by msITER.