# Combining the Advantages of Different Processing Solutions Using a Novel Motion Processing Approach
## 4 Operating-Speed-Dependent Motion Processing Approach
### Motion Processing Approach  
For executing the characteristic map of motion profiles, different approaches were considered and investigated in [16]. The requirement in switching between the optimized motion profiles is that no additional disturbances can be induced into the process. The motion processing approaches can be categorized into discrete and continuous motion processing.  
In discrete motion processing, two motion profiles are replaced by each other when the operating speed passes a given switching frequency \(f_{\text{s}}\). 'Discrete' thus refers to the handling  
Figure 4: Principle of using a characteristic map with three optimized motion profiles [16].  
of changing operating speeds. Different approaches can be considered for switching [17]. One is switching on a point \(\tau_{s}\) within the motion profiles, as shown in Figure 5a. The prerequisite for using this approach is that a suitable point with special properties exists on two neighboring optimized motion profiles. In this case, all motion parameters (e.g., \(s\), \(v\), and \(a\)) have to be equal to achieve C\({}^{2}\) continuity when switching between the motion profiles. When using a weighting function (Figure 5b), it is possible to switch between any given motion profiles. A special point for changing the motion profiles is not required. An important requirement for the application of the weighting function is that it needs to be applied in a part of the motion profiles where it is not disturbing the process. In our example, this could be the return stroke. In several works [7; 16; 17; 18; 19] the discrete processing has been applied and validated. It was shown that the available process window can be increased considerably in certain applications.  
Another approach to switching between optimized motion profiles is continuous motion processing. 'Continuous' means here that with the operating speed continually changing, motion values are interpolated between two optimized motion profiles. In [16; 17] it was assumed that this can improve the motion quality compared to the discrete motion processing approach. In [7; 13; 16] it was experimentally proven that the interpolated value of the continuous processing might be closer to the optimal value compared to discrete processing. However, the success of this depends on the process characteristics and the used interpolation method [13]. In [7], a 1D linear interpolation was validated for the intermittent transport method, shown in Figure 6a. In [19], a novel 2D linear interpolation approach was developed and validated on another process with fundamentally different characteristics, shown in Figure 6b. Using C\({}^{2}\)\(\times\) C\({}^{2}\) spline surfaces, shown in Figure 6c, was investigated in [16]. This approach was not pursued further due to the higher online calculation effort [16]. Continuous motion processing has been used several times and validated [7; 13; 16]. Here, it was shown that compared to discrete switching, it can improve motion accuracy particularly [16]. This effect can be used to further extend the process window.  
Figure 5: Principle of the discrete processing of motion profiles. (**a**) Switching on a point. (**b**) Switching using a weighting function. Adapted from [17].  
Figure 6: Principle of the continuous processing of motion profiles. (**a**) 1D linear interpolation [7], (**b**) 2D adaptive interpolation [19], and (**c**) C\({}^{2}\)\(\times\) C\({}^{2}\) spline surface [17].