## 3 Results  
The optimization procedure was run ten times for all combinations of the 3 trajectories, 4 link types, and 2 fitness functions. In total, 240 runs of the genetic algorithm were performed. Each run took approximately 2 h on a mid-range PC (Intel(r) i7-7700 HQ, 16 GM ram, Nvidia GTX 1060), so we had to divide the work across several computers and multiple days. The stochastic nature of the genetic algorithms can be seen in Figure 11. The algorithm almost never finds the same configuration. It is, however, rather consistent in getting to the same fitness values. In this case, the manipulators with linear links perform with the lowest cost values. All 40 runs in Figure 11 are for task 1 and cost\({}_{1}\). Putting the two together in one plot for comparison would result in an overcrowded graph; therefore, rather than showing them side-by-side as in Figure 11, we took the median values and presented them as single plots, as in Figures 12-14. The dispersion is shown as an area between the 25 th and 75 th percentile. With all tasks and cost functions, we saw better performance with simpler links (basic, linear, and rounded) and worse performance with the Hermite spline links. The differences between basic, linear and rounded links were task-dependent. Linear links performed well during task 1, however basic links were able to achieve lower cost values in the more geometrically complex task 3. Slower convergence of Hermite spline links can be expected, since they are described with more parameters, which dramatically increases the search space. They are more adaptable, therefore they should be able to achieve better cost values across a longer run. To test this hypothesis, we selected task 2 and increased the number of generations from 100 to 1000. Figure 15 shows the results of this test. Figure 11: All runs for trajectory task 1 and cost function cost\({}_{1}\): with (a) basic links; (b) linear links; (c) rounded links; (d) Hermite spline links. Figure 10: Task 3: (a) position; (b) orientation. Figure 14: Results for task 3: (**a**) \(\text{cost}_{1}\); (**b**) \(\text{cost}_{2}\). Figure 12: Results for task 1: (**a**) \(\text{cost}_{1}\); (**b**) \(\text{cost}_{2}\). Figure 13: Results for task 2: (**a**) \(\text{cost}_{1}\); (**b**) \(\text{cost}_{2}\). As is visible from Figure 15, even a 10x increase in generations did not produce manipulators with cost values low enough to be comparable to our previous test. One run of the algorithm with 1000 generations takes about 22 h, which makes any practical use difficult. To inspect how significantly different each link type performs during the different tasks, we performed a one-way ANOVA analysis on the results of each GA run. Figure 16 shows mean estimates and comparison intervals at a significance level of 0.05. In the graphs, if the lines overlap vertically, the link types did not perform with significant difference. The best manipulators for each link type optimized with function \(\text{cost}_{2}\) are shown in Figures 17-19. It is apparent from the resulting structures that Hermite spline links occupy a larger volume, which might bring additional disadvantages. Additional tests with obstacles in the workspace might provide evidence that more adaptable links, such as Hermite spline links, can adapt to more complex environments. We considered optimizing manipulators with obstacles in the workplace to be outside of the scope of this article, however it is something we are planning to work on in the future. Figure 16: ANOVA results. The links with lowest cost value are shown in blue, links that are not significantly different from the best one are shown in gray, and links that are significantly different are shown in red: (**a**) task 1, \(\text{cost}_{1}\); (**b**) task 2, \(\text{cost}_{1}\); (**c**) task 3, \(\text{cost}_{1}\); (**d**) task 1, \(\text{cost}_{2}\); (**e**) task 2, \(\text{cost}_{2}\); (**f**) task 3, \(\text{cost}_{2}\). Figure 15: Results for a longer run of Hermite spline links for task 2 and function \(\text{cost}_{1}\).