**4.6.2 Sample workspace with seven degrees-of-freedom robot**

This example is in reference to the robotic environment shown in fig. 17 and subsequently the various c-space slice maps, as detailed in fig. 18. As we have declared in section 3.5 that [1 -- 2] plot is the most significant out of the four plots, we need to obtain the v-graph for this plot. Figure 27 shows the developed v-graph for this c-space slice plot. Here the generated v-graph is relatively simpler by default as it has only four nodes, which is due to the fact that both the joint-angles in consideration, viz. 1 & 2 are plotted in their full ranges. Thus 'S' & 'G' are also located on the boundary lines, because other locations will be infeasible.

However, it is to be noted that the exact shape of the v-graph (generated out of the most significant c-space slice plot) will depend upon the joint-angle ranges of the joint-pair under consideration and we will have distinct locations for 'S' & 'G', outside the c-space zone. It is evident from fig. 27 that S23G is the optimal path as '' is the smaller angle, which guides this path as per Angular deviation algorithm.

The generalized formulation for evaluating the angular position of 'S' & 'G' in the v-graph (using inverse kinematics routine for the manipulator) is as follows,

$$\sum\_{j=1}^{j=7} I\_j \text{Cost} \left( \sum\_{k=1}^{k=j} \boldsymbol{\Theta}\_k^m \right) = \mathbf{X}\_m \tag{43a}$$

Spatial Path Planning of Static Robots Using Configuration Space Metrics 451

We have studied one real-life case of robot path planning in 3D, based on c-space modeling and v-graph searching, as delineated in the paper so far. The study was made with a five degrees-of-freedom articulated robot, RHINO-XR 3, during its traverse between two predefined spatial locations through a collision-free path. The main focus was to maneuver this robot between 3D obstacles in reaching a goal location in a cluttered (laboratory) environment. Since RHINO is a low-payload robot, instead of standard pick-and-place tests, we designed our experiment such that it had to only *touch* the start ('S') and goal ('G') locations by the gripper end-point. The safe path in 3D, between the start and goal locations, was arrived using *c-space slice* mapping and *Angular Deviation Algorithm* (refer section 4.3). Figure 28 presents the photographic view of the experimental set-up, emphasizing the

Fig. 28. Photographic view of the test set-up for spatial path planning with RHINO robot

numbered depending on the shape of the obstacle-zone in that very slice.

Based on the obstacle zone map vis-à-vis waist rotation of the RHINO robot, we have discretized the workspace into three *non-identical* slices. The task-spaces, corresponding to these slices, are schematically shown in fig. 29. In all the sliced maps, the vertices of the combined obstacles are labeled alphabetically, with a numeric indication for the slicenumber. For example, the vertex "A1" signifies the vertex number "A" in slice number1. It is to be noted that the vertex numbers are not *obstacle-specific*, rather those are serially

**5. Case study** 

combined obstacle zone.

$$\sum\_{j=1}^{j=7} \mathbf{I}\_j \text{Sim}\left(\sum\_{k=1}^{k=j} \boldsymbol{\Theta}\_k^m \right) = \mathbf{Y}\_m \tag{43b}$$

where. 'm' : positional attribute of the end-point, i.e. either 'S' or 'G'; {lj}: link-lengths; {km}: joint-angles for 'S' or 'G' and (Xm, Ym): planar Cartesian co-ordinates for 'S' or 'G'.

Now considering the Cartesian co-ordinates for 'S' as (20, 72.5) and the constant values for {<sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> 7} as [100 50 150 60 100] we can solve for 1 & 2 using eqns. (43), which gives us 1 <sup>S</sup> 600 and 2<sup>S</sup> 00. Similarly considering 'G' as (-30, -40) with the constant values for {<sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> 7} as [50 80 180 40 130] we can solve for 1 & 2, which gives us <sup>1</sup> <sup>G</sup> 1080 and 2G 1200.Thus, as proposed in section 4.5.1, the nodes, {Nk, k=1,2,3,4} of the final collision-free path of the manipulator between 'S' & 'G' will be: N1 'S' = (600, 00, 50, 80, 180, 40, 130); N2 = (1400, 00, 50, 80, 180, 40, 130); N3 = (1400, 1200, 50, 80, 180, 40, 130) and N4 'G' = (1080, 1200, 50, 80, 180, 40, 130).

Fig. 27. Visibility graph obtained as per the most significant c-space slice plot of fig. 18
