# Task-Based Motion Planning Using Optimal Redundancy for a Minimally Actuated Robotic Arm
## 3 Workspace Analysis
### Contour Coverage  
As the number of joints comprising the arm increases, the workspace area grows as well, and more cloud points are needed for an accurate boundary description. Since more cloud points results in more extensive computational effort, the goal was to find the minimal number of points per number of joints that can form clear boundaries. The optimal superposition parameters ((\(\alpha_{1}\),\(\beta_{1}\)) and (\(\alpha_{2}\),\(\beta_{2}\)) in Equation (4)) provide full contour coverage, i.e., the area enclosed by the workspace boundary, with the minimal number of cloud points \(N_{p}\). To compute the contour coverage, we used MATLAB's _boundary_ function, which creates a polygon shape around the cloud, and then calculated the area within it using the _polyarea_ function. The determination of the optimal parameters was performed by examining different high-medium-low combinations of values within the range (\(\alpha,\beta\)) \(\in\) (0,1) to find the best (\(\alpha,\beta\)) pairing, while comparing the simulation results from the sub-regions method to those obtained when using a single optimal beta value (i.e., \(\alpha=\beta=0.43\)) and a uniform distribution (i.e., \(\alpha=\beta=1\)). The principle guiding the simulations involved computing the contour coverage for different combinations of values of the parameters, with increasing numbers of cloud points. Statistically, for a sufficiently large number of points, the contour coverage of all combinations should converge to the same constant value. Since this number could be infinitely large, we took \(N_{p}=\) 1,000,000 to be the upper bound, whereas the maximal area calculated across all combinations was taken as the true contour coverage value. To determine which values converged the fastest, we set a criterion of 99% of the full contour coverage, i.e., the minimal number of cloud points needed to reach 99% of the true coverage value.  
The simulation results presented in Figure 4 show that the low-high combination ((\(\alpha,\beta\)) = 0.1\(\cup\)0.9) achieved full contour coverage the fastest, since it was the first (210,00 cloud points) to enter and remain within the 99% zone (grey in Figure 4). Compared to previous methods, the single optimal beta value (0.43) ranked third for 99% coverage using 610,00 cloud points, compared to the low-medium combination ((\(\alpha,\beta\)) = 0.1\(\cup\)0.5) at 260,000 points, but behind the medium-high ((\(\alpha,\beta\)) = 0.5\(\cup\)0.9) combination at 1,000,000 points, whereas the uniform distribution did not meet the criteria at all.  
These simulations tested the minimal number of cloud points needed to achieve full coverage in case of a 1 m arm, composed of five links. In subcases where the arm has fewer links (or fewer joints that are active), the workspace area will naturally decrease, and a smaller number of cloud points will be needed. To set a criterion for the number of points  
Figure 4: Contour coverage simulation results. Top—different sets of (\(\alpha\),\(\beta\)) were combined with low–medium–high values of 0.1–0.5–0.9 and compared to single beta values of \(\alpha=\beta=0.43\) and to \(\alpha=\beta=1\) (uniform distribution). Bottom—pairs of (\(\alpha\),\(\beta\)) values between 0.1 and 0.9 combined with a constant pair with value of 0.9. All simulations were based on the 5R MASR model.  
per number of links, we ran similar convergence simulations for a 1 m arm composed of two, three, and four links of equal length. The results confirm that the low-high combination (0.1 and 0.9) produced the fastest convergence, and in addition (as indicated in Table 1), the superposition method dramatically decreases (up to 90.1%) the number of points needed to reach full workspace coverage--compared to the optimal single value of (\(\alpha\),\(\beta\)) = 0.43.