**3.4.3 Generation of C-space plots: Concept of equivalent circle**

In order to compute c-space data points for any particular combination of consecutive jointvariable pair for the higher dimensional robot, we would use a new concept, viz. the formulation of *Equivalent Circle* at the end of amidst the pair of links. Since we are considering *virtual two-link mini-manipulators* for the generation of c-space maps in pair, we would theoretically divide the links in two groups. The links, directly related to the generation of the specific c-space map, are termed as *active links*, while the others are known as *dummy links*. The philosophy of this equivalent circle is to re-represent the higher dimensional manipulator with only the active links and the joints therein, interfaced with circular zone(s) either at the bottom of the first active link or at the tip of the second active link. In general, the equivalent circles are constructed considering full rotational freedom of all the dummy links, located before /after the active links.

For example, if we wish to generate [1 -- 2] plot for the seven d.o.f. manipulator, then the *equivalent circle* alias *equivalent formidable zone* is to be constructed adjacent to the end the second link and circumscribing the remaining links. Figure 11 schematically presents the concept of *equivalent formidable zone*, with first two links as active links for a seven d.o.f. manipulator.

Fig. 11. Schematic view of equivalent formidable zone for a seven d.o.f. manipulator

However it is possible to have two *equivalent formidable zones* in cases where some intermediate links are considered for c-space plots. For example, if the third & fourth links of the manipulator become active links, then there will be two formidable zones, as

Spatial Path Planning of Static Robots Using Configuration Space Metrics 433

Fig. 13. Schematic showing the analytical layout for the evaluation of equivalent radius

Fig. 14. Locations of the base for the two link virtual robot with respect to equivalent circle

viz. when the obstacle is a] within the range of both Lk & Lk+1 and b] outside the range of Lk. Now, considering the first link of the virtual robot is just able to touch the obstacle we can get two extreme positions for the base, e.g. 'C1' & 'C2'. As explained earlier, the point 'C' is the other location for the base, situated on the least path. From geometry, we can assign 'D',

[type I]

[type I]

illustrated in fig. 12. Of course, as per this proposition, there can't be more than two formidable zones for any higher dimensional manipulator.

Fig. 12. Occurrence of two equivalent formidable zones for a seven d.o.f. manipulator

It is essential to locate the center of the *equivalent circle* vis-à-vis its radius (equivalent radius, Req.), in order to start computing for colliding combinations. However, the formulations for equivalent radii are not same in the two cases, as cited in figs. 11 & 12. In fact, the center of the equivalent circle, for cases wherein active links are followed by dummy links, will be at the base of the manipulator and its radius will be the summation of the lengths of the dummy links till we reach the first active link. For example, Req. 1 in fig. 12 will be the added sum of L1 & L2. Figure 13 shows the computational backup for the evaluation of the *radius* of the equivalent formidable zone /circle in this case, nomenclated as *equivalent radius [type I].* 

Although finding center and calculating equivalent radius [type I] is straight -forward, evaluation of the new *base* for the *virtual two-link robot* is critical. For example, as shown in fig. 12, we need to find out the possible location of the base for the virtual robot, comprising of L3 & L4. This has been explained in fig. 14, wherein we adopt a methodology, called *Least Path*. The least path(s) is/are the straight line-segment(s) joining the centers of equivalent circle [type I] and the obstacles in the vicinity of the virtual robot. The respective locations for the base of the virtual robot will be the point of intersection of the least path and the equivalent circle. Thus, as per fig. 14, there will be three base-points, viz. 'Cp', 'Cq' & 'Cr' corresponding to three obstacles, vide 'p', 'q' & 'r', which are situated within the limit-zone of the virtual robot with links Lk & Lk+1. However, this location of the base can range between two extreme points, as detailed in the inset of fig. 14. Two cases may appear here,

illustrated in fig. 12. Of course, as per this proposition, there can't be more than two

Fig. 12. Occurrence of two equivalent formidable zones for a seven d.o.f. manipulator

as *equivalent radius [type I].* 

It is essential to locate the center of the *equivalent circle* vis-à-vis its radius (equivalent radius, Req.), in order to start computing for colliding combinations. However, the formulations for equivalent radii are not same in the two cases, as cited in figs. 11 & 12. In fact, the center of the equivalent circle, for cases wherein active links are followed by dummy links, will be at the base of the manipulator and its radius will be the summation of the lengths of the dummy links till we reach the first active link. For example, Req. 1 in fig. 12 will be the added sum of L1 & L2. Figure 13 shows the computational backup for the evaluation of the *radius* of the equivalent formidable zone /circle in this case, nomenclated

Although finding center and calculating equivalent radius [type I] is straight -forward, evaluation of the new *base* for the *virtual two-link robot* is critical. For example, as shown in fig. 12, we need to find out the possible location of the base for the virtual robot, comprising of L3 & L4. This has been explained in fig. 14, wherein we adopt a methodology, called *Least Path*. The least path(s) is/are the straight line-segment(s) joining the centers of equivalent circle [type I] and the obstacles in the vicinity of the virtual robot. The respective locations for the base of the virtual robot will be the point of intersection of the least path and the equivalent circle. Thus, as per fig. 14, there will be three base-points, viz. 'Cp', 'Cq' & 'Cr' corresponding to three obstacles, vide 'p', 'q' & 'r', which are situated within the limit-zone of the virtual robot with links Lk & Lk+1. However, this location of the base can range between two extreme points, as detailed in the inset of fig. 14. Two cases may appear here,

formidable zones for any higher dimensional manipulator.

Fig. 13. Schematic showing the analytical layout for the evaluation of equivalent radius [type I]

Fig. 14. Locations of the base for the two link virtual robot with respect to equivalent circle [type I]

viz. when the obstacle is a] within the range of both Lk & Lk+1 and b] outside the range of Lk. Now, considering the first link of the virtual robot is just able to touch the obstacle we can get two extreme positions for the base, e.g. 'C1' & 'C2'. As explained earlier, the point 'C' is the other location for the base, situated on the least path. From geometry, we can assign 'D',

Spatial Path Planning of Static Robots Using Configuration Space Metrics 435

Fig. 16. Schematic showing the analytical layout for the evaluation of equivalent radius

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

l1=15.52; l2=14.87; l3=13.04; l4=12.54; l5=9.0; l6=5.22 & l7=8.0

1: -200 to 1400; 2: 00 to 1200; 3: 00 to 800; 4: -50 to 900;

5: 100 to 500; 6: -50 to 550; 7: 50 to 350

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

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

[type II]

in Table 3.

units]

(Anticlockwise)

Type of Robot Considered Revolute

Co-ordinates of the Robot Base xb = 10; yb = 10

Resolution of Joint Rotation 3 deg. (for all joints)

No. of Links 7 No. of Degrees -of- Freedom 7 Length of the Links [in suitable

Ranges of Rotation of the Joints

which is numerically equal to either Lk or Lk+1, depending upon which link is colliding with that obstacle. The collidable range of joint-angles, corresponding to 'C', 'C1' & 'C2' will be (- ), (1-1) & (2-2) respectively. Thus, in general a formidable range from 2 to 1 should be selected for c-space mapping in (k --k+1) plot.

In contrary to this situation, the other one, namely where dummy links are followed by active links, is more intricate so far the thematic is concerned. Figure 15 explains this case of evaluating *equivalent radius [type II],* the corresponding circular zone being *optimized* between two extremes, viz. maximum and minimum formidable zones. The minimum formidable zone is a circular space, tangent to the work-zone limit circle at 'A' while the maximum formidable zone is a semi-circular area, with two opposite extremities as 'Z1' & 'Z2'. Several feasible formidable zones are theoretically possible in-between, with pair of *chordal endpoints* as [B1 – B1\*] or [B2 – B2\*] etc. It may be noted that all the three formidable zones, namely the minimum, maximum & optimum, share the common vertex 'V'.

Fig. 15. Schematic showing the disposition of the equivalent formidable circle [type II]

The *equivalent radius [type II]* is evaluated using geometrical attributes, as detailed in fig. 16. Here, the points 'A', 'C' & 'Cm' represent the locations of the centers of the maximum, equivalent & minimum formidable zones respectively. As evident from the figure, the ratio between the two line-segments, viz. the semi-chordal length of the equivalent circle and the radius of the maximum formidable zone is 'k', where 0<k<1. In-line with the numerical evaluation of 'Req.', the location of the center, 'C' can be determined also.5

<sup>5</sup> The location of the center is determined by evaluating the length of the line-segment, AC, which is numerically equal to [(1-k2)/2]Lj and it is also at a distance of (k2/2)Lj from the center of the minimum formidable zone, i.e. 'Cm'.

which is numerically equal to either Lk or Lk+1, depending upon which link is colliding with that obstacle. The collidable range of joint-angles, corresponding to 'C', 'C1' & 'C2' will be (- ), (1-1) & (2-2) respectively. Thus, in general a formidable range from 2 to 1 should be

In contrary to this situation, the other one, namely where dummy links are followed by active links, is more intricate so far the thematic is concerned. Figure 15 explains this case of evaluating *equivalent radius [type II],* the corresponding circular zone being *optimized* between two extremes, viz. maximum and minimum formidable zones. The minimum formidable zone is a circular space, tangent to the work-zone limit circle at 'A' while the maximum formidable zone is a semi-circular area, with two opposite extremities as 'Z1' & 'Z2'. Several feasible formidable zones are theoretically possible in-between, with pair of *chordal endpoints* as [B1 – B1\*] or [B2 – B2\*] etc. It may be noted that all the three formidable zones,

namely the minimum, maximum & optimum, share the common vertex 'V'.

Fig. 15. Schematic showing the disposition of the equivalent formidable circle [type II]

evaluation of 'Req.', the location of the center, 'C' can be determined also.5

minimum formidable zone, i.e. 'Cm'.

The *equivalent radius [type II]* is evaluated using geometrical attributes, as detailed in fig. 16. Here, the points 'A', 'C' & 'Cm' represent the locations of the centers of the maximum, equivalent & minimum formidable zones respectively. As evident from the figure, the ratio between the two line-segments, viz. the semi-chordal length of the equivalent circle and the radius of the maximum formidable zone is 'k', where 0<k<1. In-line with the numerical

5 The location of the center is determined by evaluating the length of the line-segment, AC, which is numerically equal to [(1-k2)/2]Lj and it is also at a distance of (k2/2)Lj from the center of the

selected for c-space mapping in (k --k+1) plot.

Fig. 16. Schematic showing the analytical layout for the evaluation of equivalent radius [type II]
