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

### **4.1 Perspective**

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

Spatial Path Planning of Static Robots Using Configuration Space Metrics 439

The workspace of the robot has been modeled by formulating the *Visibility Matrix* of the *visibility graph*6 of the cluttered environment. This matrix has been conceived as a kind of 'adjacency' relationship and framed by knowing the necessary visibility information about the nodes of the graph. Figure 21 illustrates the visibility matrix, [Vij], as developed from the environment shown in fig. 20, which depicts a typical sub-visibility graph, generated out of

**4.2 Logistics of visibility graph formulation: Our model** 

several polygonal obstacles known a-priori (i.e. with pre-fixed locations).

Fig. 20. A representative sub-visibility graph in two dimensions

node) can 'see' nodes 5, 6, 8, 16 & 21 only.

M-line.

The matrix is of the order (N + 2) x (N + 2), 'N' being the total number of intermediate nodes of the graph. The full matrix is formed by adding the 'start' (S) and 'goal' (G) nodes, making [V ij] a square matrix. Each row of [Vij] gives the visibility information of that very node. For example, the fifth row of the matrix signifies that the node no. 4 (considering 'S' as the first

6 In order to reduce computational burden, 'Sub-visibility Graph' technique has been adopted in the present study. It considers only those obstacles in the robot workspace, which collide directly with the

conceptualized from the point of view of a] disposition of the obstacles (i.e. planar or spatial) and b] kinematics of the manipulator (i.e. its degrees-of-freedom). Following checker-box illustrates the situation pictorially (refer fig. 19).

Fig. 19. Schematic of the robotic path planning scenario using Configuration Space approach

In fact, the methodologies to be adopted for different situations of robot path planning vary significantly and those depend on the task-space nature (i.e. planar or spatial) and the robot kinematics (i.e degrees-of-freedom). Besides, the 3D path planning is also dependent upon the nature of the slices produced from the task-space. Those can be identical in nature or non-identical. Table 4 presents the scenario, highlighting the methods used for the collisionfree path planning,


Table 4. Illustrative Summary of the Methods Used for Collision-free Path Planning of Robots
