**7. Conclusion**

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

Fig. 13. Singularity analysis in planes 4, 5 and 6 for both 3-legged (non-redundant) and 4-

legged (redundant) mechanisms.

The effects of redundant actuation are studied. The redundant 4-legged and non-redundant 3-legged parallel mechanisms are compared with 4 well-known architectures of Gough-Stewart platforms. It is shown that the inverse kinematics of the proposed 3-legged and 4 legged mechanisms have a closed-form solution. Also the Jacobian matrix has been determined using the concept of reciprocal screws.

From the design point of view, by replacing the passive universal joints in the Stewart platforms with active joints in the above mentioned mechanisms, the number of legs could be reduced from 6 to 3 or 4. It makes the mechanism to be lighter, since the rotary actuators are resting on the fixed platform, which allows higher accelerations to be available due to smaller inertial effects.

It is illustrated that redundancy improves the ability and performance of the non-redundant parallel manipulator. The redundancy brings some advantages for parallel manipulators such as avoiding kinematic singularities, increasing workspace, improving performance indices, such as dexterity, manipulability, and sensitivity. Finally, we conclude that the redundancy is a key choice to remove singular points, which are common in nearly all parallel mechanisms.

It is worthy to state that the applications of these robots can be found in flight simulators, high precision surgical tools, positioning devices, motion generators, ultra-fast pick-andplace robots, haptic devices, entertainment, multi-axis machine tools, micro manipulators, rehabilitation devices, etc.
