**3.4 C-space maps for higher dimensionality**

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

Fig. 7. Final c-space mapping for the environment filled with non-circular polygonal

Fig. 8. Final c-space mapping for the environment congested with circular obstacles

obstacles

Figures 7 and 8 show the final c-space for the above two environments.

The development of c-space is relatively simpler while we have two degrees-of-freedom robotic manipulators. As described in earlier sections, irrespective of the nature of the environment, we can generate simple planar maps, corresponding to the variations in the joint-angles (in case of revolute robots). Thus the mapping between task space and c-space is truly mathematical and involves computationable solutions for inverse kinematics routines. The procedure of generating c-space can be extrapolated to three degrees-of-freedom robots at most, wherein we get 3D plot, i.e. mathematically speaking, *c-space surface*. However, this procedure can't be applied to higher dimensional robots, having degrees-of-freedom more than three. In fact, c-space surface of dimensions greater than three is unrealizable, although it is quite common to have such robots in practice amidst cluttered environment. For example, let us take the case of a workspace for a seven degrees-of-freedom articulated robot, as depicted in fig. 9. Here we need to consider variations in each of the seven jointangles, viz. 1, 2,……, 7 (the last two degrees-of-freedoms are attributed to the wrist rotations) towards avoiding collision with the obstacles.

Fig. 9. A representative cluttered environment with seven degrees-of-freedom revolute robot

So, the question arises as how can we tackle this problem of realizing higher dimensionality of c-space mapping. Hence it is a clear case of building *composite c-space map,* with proper characterization of null space. We propose a model for this mapping in higher dimensionality as detailed below.

### **3.4.1 Lemma**


Spatial Path Planning of Static Robots Using Configuration Space Metrics 431

As it may be apparent from fig. 10, either [1 -- 2] or [3 -- 4] plot can be significant, considering *larger area* or *presence of multiple loops* criterion (refer serial no. 8 & 9 of 3.4.1).

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

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.

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

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

all the dummy links, located before /after the active links.

manipulator.


#### **3.4.2 Schematic of the model**

Let us take an example of a seven degrees-of-freedom articulated robot, similar to one illustrated in fig. 9. According to the lemma proposed in 3.4.1, there will be four planar cspace plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] and [6 -- 7]. Figure 10 shows a sample view of these segmental c-space maps.

Fig. 10. Sample view of the composite C-space map for a seven degrees-of-freedom robot

5. If 'n' is odd, then the pairs for the planar c-space plots will be: [q1 – q2], [q3 – q4],……[qn-

6. In a way, we are considering several *virtual 2-link mini manipulators,* located at the

8. One way of accessing the most significant c-space map is to consider finite measurement of the planar area of the c-space. A larger area automatically indicates more complex dynamics of the joint-variables so far as the collision avoidance is

9. Alternatively, significant c-space plots will be those having multiple disjointed loops, i.e. regions of formidable area. Individually the regions may be of smaller area, but the

10. Once the most significant c-space map is selected, the locations corresponding to 'S' and 'G' are to be affixed in that plot. This will be achieved using inverse kinematics routine

11. For the most significant plot so obtained, all joint-variables, except the two used in the plot will be *constant*. For example, if [q3 – q4] plot is the most significant one, then q1,

12. In general, if [qi –qj] plot be the most significant, then the set {q1, q2,……qn-1, qn}, except [qi –qj] , will be constant. And, the value of the set {q1, q2,……, qi-1} will be ascertained by the inverse kinematic solution of 'S' while the other set, {qj+1, qj+2,…..,qn} will be

Let us take an example of a seven degrees-of-freedom articulated robot, similar to one illustrated in fig. 9. According to the lemma proposed in 3.4.1, there will be four planar cspace plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] and [6 -- 7]. Figure 10 shows a sample view

Fig. 10. Sample view of the composite C-space map for a seven degrees-of-freedom robot

2 – qn-1], [qn-1 – qn].

concerned.

from 'S'(x,y) and 'G' (x,y).

**3.4.2 Schematic of the model** 

of these segmental c-space maps.

respective joints of the original manipulator.

q2,……qn-1, qn are constant except q3 and q4.

determined by the inverse kinematic solution of 'G'.

7. Out of the plots so generated, select the *most significant* c-space map.

multiplicity of their occurrence adds complexity to the scenario.

As it may be apparent from fig. 10, either [1 -- 2] or [3 -- 4] plot can be significant, considering *larger area* or *presence of multiple loops* criterion (refer serial no. 8 & 9 of 3.4.1).
