# An Efficient Stochastic Constrained Path Planner for Redundant Manipulators
## 5 Results
### Simulation Study
As the number of hypotheses K increases, we observe a higher success rate, while, at the same time, increasing the manipulability along the whole trajectory. It is worth noting that, for \(K=10\) hypotheses, the proposed algorithm achieves a success rate of 100%. Furtherincreasing \(K=20,50\) hypotheses maintains the same success rate and allows us to obtain an even higher manipulability index along the trajectories. Figure 2: A comparison of the manipulability metric evolution for case studies I, II and III. Cases IA, IIA and IIIA were carried out over a total of \(N=2000\) sample waypoints along the desired task-space trajectory. The blue line represents the trajectory that presents the maximum value in terms of mean manipulability along the line that the algorithm is able to achieve, red is the minimum and green the mean value across all the runs. The corresponding simulations IB, IIB and IIIB were obtained by using SCO with K=20 samples. (**a**) Case study I A, (**b**) Case study I B, (**c**) Case study II A, (**d**) Case study II B, (**e**) Case study III A, (**f**) Case study III B. Table 1 also presents the results of case study II. As before, the \"Simple Planner\" in the first row refers to the results of the path obtained with Algorithm 2, where the simulation samples the path at a total of \(N=2000\) and, for a successful run, collisions with itself and the environment for a given trajectory must be avoided. The success rate of this algorithm is, again, very low. Given the increased complexity of the task when compared to run IA, the success rate is even lower. The results with SCO are collected under case study IIB, and as below different cases are investigated with varying initial hypotheses along the path: \(K=\{1,5,10,20,30\}\). It can be observed how the proposed SCO is successful in producing a 56% of feasible solutions for \(K=1\) and 80% for \(K=5\), a marked improvement over the \"Simple Planner\" even with these low number of hypotheses. Moreover, as the number of hypotheses \(K\) increases, so does the success rate, while, at the same time, increasing the manipulability along the length of the trajectory. It is worth noting that, for \(K=10\), the proposed algorithm is able to achieve full success (rate of 100%). Further increasing \(K\) to 20 or 50 maintains flawless success and allows us to obtain even marginally higher manipulability along the trajectories. Table 1 also collects the results of case study III. As before, \"Simple Planner\" in the first row refers to the results of the path obtained with Algorithm 2. The success rate of this algorithm is, again, very low. The task in case study III is more demanding, which explains the low success rate, compared to case studies IA and IIA. Again, the results with SCO are collected under simulation IIIB with \(K=\{1,5,10,20,30\}\). Following the same trends observed before, the proposed SCO produces a 33% of successful solutions for \(K=1\) and 80% for \(K=5\). The proposed algorithm also succeeds in this more demanding scenario, and as the number of hypotheses \(K\) increases, the success rate and the manipulability are increased. It is worth noting that for \(K=20\) the proposed algorithm is able to achieve full success (rate of 100%). As observed earlier, further increasing \(K\) beyond this point (to the maximum of 50 hypotheses in this case) only slightly increases the optimized manipulability while maintaining the same success rate.