**3.1.1 C-space transformation of "POINT"**

This algorithm gives the C-space data for an obstacle, considered theoretically as a *point* in Cartesian space. The robot with line representation is being considered here. (refer fig.3).

 4 Concave obstacles are modeled as a combination of several convex polygons.

Spatial Path Planning of Static Robots Using Configuration Space Metrics 423

Figure 4 shows five different cases of collision, considering only the first link. However, the

Case III: Obstacle is collidable by the first link, with its centre inside the range of the first

Modifications of the previous algorithms for *POINT*, *LINE* and *CIRCLE* obstacles have been considered with finite dimensions of the robot arms (refer fig. 3). Also, planar 3-link manipulator is considered now, where the third link is nothing but the end-effector. Since planar movements are taken into account, only pitch motion of the wrist is considered along with shoulder and elbow rotations. The exhaustive list of input parameters for these cases will be as follows, viz. (i) Robot base co-ordinates (xb, yb); (ii) Length of the upper arm or shoulder (l1); (iii) Length of the fore arm or elbow (l2); (iv) Length of the end-effector or *wrist*  (lw); (v) Width of the upper arm (d1); (vi) Width of the fore arm (d2); (vii) Width of the endeffector (dw); (viii) Radii of curvature for upper and fore arm (r1 & r2 respectively); (ix) The co-ordinates of the *Point*: (x1, y1) or *Line*: [(x1, y1) & (x2, y2) as end - points ] or *Circle*: [centre

The analytical models of various C-space mapping algorithms, described earlier, have been grouped into two categories, viz. (a) Model for *LINE* obstacles and (b) Model for *CIRCLE* 

The ideation of collision detection phenomena for *Line* obstacle is based on the intersection of the line- segments in 2D considering only the kinematic chain of the manipulator. The positional information, i.e. co -ordinates (xk, yk) of the manipulator joints can be generalized

1

1

lk Sin (

1

*j*

 j )]T

*j k*

k = 1,2,3,..............,n & j= 1,2,.........., k (1)

*k*

*k n*

*j*

The slope of the *line* (i.e. the edge of the obstacle) with (x1, y1) & (x2, y2) as end-points is

m0 = | (y2 - y1 ) | / ( x2 - x1 ) | = | y / x | (2)

 j ),

*j k*

1

lk Cos (

*k*

[ xb, yb ]T : Robot base co - ordinate vector in Newtonian frame of reference;

*k n*

Case II: Obstacle is within the first link's range, touching the *range circle* internally.

Case IV: Same as before, but centre is outside the range of the first link. Case V: Obstacle is touching the range circle of the first link externally.

**3.1.4 C-space transformation with finite dimensions of the robot** 

paradigm is valid for subsequent links also. These cases are: Case I: Obstacle is fully within the range of the first link.

link.

as,

where,

given by,

at (xc, yc) & radius : 'r'].

**3.2 Analytical model of the mapping algorithms** 

[xk, yk ]T = [ xb, yb ]T + [

lk : k th. link - length of the manipulator; j : j th. joint - angle of the manipulator.

obstacles. These are being described below.

**3.2.1 Model for LINE obstacles** 
