**6. Conclusions**

454 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

Fig. 32. C-space map for the third slice pertaining to the case – study

"4"[3] {1=500, 2 =-100, 3=300, 4=970 5=200}.

*& output of the Angular Deviation Algorithm*.

layout & experiments.

It is to be noted that some of the c-space slice maps in the above figures bear similarity; in fact, those maps are bounded by rectangular regions, occupying the full rotational ranges8 of the participating joint-angles. That means, the full region is formidable, so far as the selection of safe nodes are concerned. This property is unique in the developed method, and it is helpful for obtaining the safe path in the final *go*. Now, assimilating all the three critical c-space slice maps as per figs. 30,31 & 32, we get the final safe path for the environment as the statistical *union* of the slice maps and it is represented as, S "2"[1] "3"[1] "4"[2] "4"[3]G, where the legend "N"[k] symbolizes node 'N' of the k th. slice (refer section 4.5.2 for the formulation). Thus, the final path has got four *Intermediate Points* (IP), besides 'S' & 'G'. The joint-angle combinations for these 'IPx', x=1,..,4 (as labeled from 'S' onwards) are evaluated as, IP1 "2"[1] {1=1200, 2 =-100, 3=580, 4=980 5=200}; IP2 "3"[1] {1=1200, <sup>2</sup> =1200, 3=580, 4=980 5=200}; IP3 "4"[2] {1=500, 2 =-100, 3=200, 4=970 5=200} and IP4

Table 6 presents a summary of the various important outputs pertaining to the case study, with details of the computational time (for PC-based evaluation). Here, *Elapsed Time* has been divided into elemental time-periods (computational) against 4 sub-heads, viz. "**A**": *Generation of slices in task-space with co-ordinates (x,y) & node numbering*; "**B**": *Generation of cspace maps, including the critical-most*; "**C**": *Development of the v-graph* and "**D**": *Graph searching* 

8 The *effective* rotational ranges of the five joint-angles of the RHINO robot are, 1: (-400 to 1200), 2: (-100 to 1200), 3: (100 to 900), 4: (100 to 1000) and 5: (50 to 650), as selected on the basis of our task-space The details of the visibility graph-based heuristic algorithm for *safe* path planning in 2D plane as well as 3D space have been discussed in the paper, backed up by the theoretical paradigms of the generation of c-space obstacles from their respective task-spaces. The outcome of the c-space and v-graph algorithms have been found effective in programming the robot in order to perform certain pre-specified tasks or a series of tasks, such as in somewhat off-the-track industrial applications. The *best* path needs to be selected out of the possible alternatives by considering the most feasible criteria, which is essentially application specific. The novelty of the developed method lies with the ease of computational burden as 2D c-space slices are being joined statistically (*union*). Also by not incorporating all obstacles in one c-space slice we are improving upon computational efficiency and thereby reducing undue technical details regarding the obstacles. However, the safe path obtained by the developed method may overrule some nearer nodes, because the corresponding c-space slices are based on maximum *safety margins*, as per the

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propositions of the model. In fact, the concept of *formidable zones* is introduced in our model to avert potentially dangerous joint-angle configurations and thus, at times, the entire jointangle range-space gets selected for the c-space map. The reason for taking this lemma is to safeguard the robot's motion between 'S' & 'G' to the best extent. Thus we may end up in some joint-angle (nodal) combinations, which might have been omitted, but it is always better to select a safe & secured path, rather than risking the robot motion for potential grazing and/or full collision with the obstacle(s). As per the proposed method, c-space slices often look trivial (e.g. regular geometrical shaped obstacles), although those are quite computationally intensive. Nonetheless, the geometrical simplification in appearance makes the v-graph map easy and subsequently the graph-search process too.
