**11. Workspace analysis**

The existence or nonexistence of the kinematics solution determines the robotic workspace. The lack of solution means that the robot is not able to obtain optimal orientation because it is located out of the workspace. These conditions are called robotic singularity states. Almost all the robots have singularity points in either the border of their workspace or in their workspace. The singularity point in the border of workspace denotes a state that occurs when the arm is fully stretched or folded on itself when the final manipulator is almost or precisely located in the border of the workspace.

On the other hand, the singularity states in the workspace signify the conditions that occur in the mechanism workspace or in general when two or some joint axes are located in one direction. When the robot is located in the singularity position, it loses all or some of its degrees of freedom in the deicardean space. It is obvious that this process is done in the border of the robotic workspace. The examination of the T-S(3-PRP) robotic workspace has shown that it has no singularity in its workspace; the robotic workspace is presented in **Figure 12** (**Figure 13**).

**Figure 12.** *X*-*Y* plots of the Achievable workspace of the T-S robot.

**Figure 13.** T-S Robot as milling machine' table.

(e)

**Figure 11.** (a, e) *X*-*Y* plots of the path of the motion point c; (b, d) variation *S*11, *S*21, *S*<sup>31</sup> (outputs) versus *S*13, *S*23, *S*<sup>33</sup> for

*θ* = 0; (c) variation *S*11, *S*21, *S*31 (outputs) versus *S*13, *S*23, *S*33 (inputs) for *θ* = *pi*/6.

82 Recent Advances in Robotic Systems

**Figure 14.** Introduction a new geometric model as (a) Reuleaux Triangle –Star Robot with telescopic and (b) arms Cir‐ cle-Star robot with telescopic arms.
