# Task-Based Motion Planning Using Optimal Redundancy for a Minimally Actuated Robotic Arm
## 5 Simulation Results
### Last Link vs. Any Link  
One of the main advantages of the MASR over other manipulators is its ability to move the end-effector along the arm, i.e., any link (AL) and not just by the last link (LL) can grab an object. In the previous simulation, we showed that in the LL case, which is applicable to any other traditional manipulator, our algorithm exhibited a significant improvement in terms of computational efforts and the feasibility of the solution. The next simulation was conducted to test the AL advantage in terms of numerical performance. The arm's model, initial position, and environment were the same as in the previous simulation. Twenty targets to pick were randomly generated in the arm's vicinity, and the algorithm  
Figure 7: Simulation results for the pick and place task. (**a**) For the given targets, minimal redundancy only involves 4 out of 5 joints (in magenta). In (**b**,**d**), an optimal path was found for the target to pick (**b**) and place (**d**), using full areal coverage as the search graph for the A* search. The steps shown in black were downsized by half. (**c**) Shows the path from pick to place after modification for a minimally actuated arm.  
was executed twice for each target, once as LL, and once as AL. Naturally, for the same set of joints, the AL workspace is larger than the LL, so we expected that for a given target, the AL approach would require fewer active joints than the LL. Fewer active joints means fewer configurations for full areal coverage, which translates into a smaller search graph, and hence a more effective path planning. Out of the 20 targets, the algorithm found that 10 were out of reach for both AL and LL (marked in red, Figure 8). This emphasizes another advantage of the preliminary search for minimal redundancy; namely, that it can predetermine whether a path does not exist, and thus economize on a significant amount of unnecessary computing efforts. As for the target within the workspace boundary (marked in green, (Figure 8)), once the optimal assembly was found for LL and AL separately, the A* search was executed for each.  
The results presented in Table 3 show that in 8 out of 10 searches, the AL approach was more efficient (up to 12.2 times improvement) than LL. The two targets for which the LL outperformed the AL indicate that in cases where the target is near the tip of the arm (target No. 16, Figure 8), the LL did better, although the difference was negligible in terms of the number of expansions (4 as compared to 9 expansions, Table 3). The case where the target lies near the outer boundary of the maximal joint assembled arm (5 active joints in our case, target No. 7, Figure 8) is also instructive. If both AL and LL use the same joints to create a fully covered workspace, the number of nodes in the search graph will be the same, which directly leads to a similar number of expansions. Thus, these results make it clear that while the minimal redundancy through the LL approach can make optimal path planning possible, the AL approach can make it practical.  
Figure 8: Last link vs. any link simulation. The algorithm indicated that the targets in red were out of reach. The targets in green were within the workspace boundary and were carried on to the path planning stage.