**3.5 Evaluation of C-space plots for higher dimensional robot: Example**

Here we will study a specific case for a seven degrees-of-freedom robot, amidst a 2D cluttered environment as shown in fig. 17. The technical parameters of the robot, comprising link-lengths, {li, i = 1,2,…7}and joint limits {i, i = 1,2,…7), are highlighted in Table 2. Circular obstacles are considered for simplicity in computations. The locations of the respective centers and diameters of the obstacles (expressed in suitable units) are presented in Table 3.


Table 2. Technical Facets of the Higher Dimensional Robot Under Consideration

Spatial Path Planning of Static Robots Using Configuration Space Metrics 437

Fig. 18. Four c-space plots for the seven joint revolute robot amidst cluttered workspace of fig. 17 It may be noted that while c-space plot for a 2 d.o.f. robot (working in 2D or 3D task-space) can be composed of irregular non-geometric shapes (refer fig. 7), the same for higher d.o.f. robots are perfectly geometric (refer fig. 18). This is happening because of the incorporation of the concept of 'formidable zones' for higher dimensional robots, wherein we are deliberately allowing the collidable zone to engulf more regions in the c-space. In fact, in most of cases for higher d.o.f. robots, the c-space zones are perfect rectangular in shapes, between the minimum & maximum limits of the participating joint-angles. For example, in fig. 18, (1 vs. 2) c-space slice plot the final rectangle is constituted between 4 vertices, viz.

Based on the formation of c-space maps, collision-free paths are to be ascertained in 2D as well as in 3D. We will analyze the gamut in four functional *quadrants*, which have been

**4. Safe path in configuration space: Logistics & algorithm** 

1\_min, 1\_max, 2\_min & 2\_max.

**4.1 Perspective** 

Fig. 17. Workspace layout of the seven degrees-of-freedom articulated robot


Table 3. Obstacle Signature, as per Workspace Layout of Figure 16

Now, in this case of seven degrees-of-freedom revolute robot, we will have four different c-space plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7]. All of these plots use collision-detection algorithms, described earlier, for each of the obstacles separately, taking into account the concepts of *equivalent circles*. These c-space plots are illustrated in figure 18. A gross estimate reveal that [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7] plots occupy a planar area measuring (160x120), (80x95), (40,60) & (60x30) sq. units respectively. It is evident from fig. 18 that although complexity-wise both [1 -- 2] and [3 -- 4] plots are roughly at par, but the former is to be selected as the most significant c-space plot as it is also the largest in size.

Fig. 17. Workspace layout of the seven degrees-of-freedom articulated robot

1 15.88 (x =15, y = 30) 2 13 (x = 12.5, y = 37.5) 3 12.7 (x = 27.5, y = 42.5) 4 19 (x = 35, y = 30) 5 14 (x = 60, y = 15) 6 9.53 (x = 65, y = 50)

Table 3. Obstacle Signature, as per Workspace Layout of Figure 16

also the largest in size.

**Obstacle No. Diameter Location of Centre** 

Now, in this case of seven degrees-of-freedom revolute robot, we will have four different c-space plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7]. All of these plots use collision-detection algorithms, described earlier, for each of the obstacles separately, taking into account the concepts of *equivalent circles*. These c-space plots are illustrated in figure 18. A gross estimate reveal that [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7] plots occupy a planar area measuring (160x120), (80x95), (40,60) & (60x30) sq. units respectively. It is evident from fig. 18 that although complexity-wise both [1 -- 2] and [3 -- 4] plots are roughly at par, but the former is to be selected as the most significant c-space plot as it is

Fig. 18. Four c-space plots for the seven joint revolute robot amidst cluttered workspace of fig. 17

It may be noted that while c-space plot for a 2 d.o.f. robot (working in 2D or 3D task-space) can be composed of irregular non-geometric shapes (refer fig. 7), the same for higher d.o.f. robots are perfectly geometric (refer fig. 18). This is happening because of the incorporation of the concept of 'formidable zones' for higher dimensional robots, wherein we are deliberately allowing the collidable zone to engulf more regions in the c-space. In fact, in most of cases for higher d.o.f. robots, the c-space zones are perfect rectangular in shapes, between the minimum & maximum limits of the participating joint-angles. For example, in fig. 18, (1 vs. 2) c-space slice plot the final rectangle is constituted between 4 vertices, viz. 1\_min, 1\_max, 2\_min & 2\_max.
