Rigid Body Motion is a basis for Robotics!

A rigid body is a solid object in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant regardless of external forces exerted. In that sense, the typical robotics manipulator is a rigid multibody, and we must have some mathematical language to deal with such a mechanism. For the chain of linked rigid bodies that constitute a robot, we are looking for effective kinematics solutions to our robotics challenges to permit the implementation of advanced projects & applications. The problem is that the mathematical difficulty for solving the Kinematics increases with the number of Degrees of Freedom (DoF) complex mechanisms.

For a mechanism with few DoF, like the historical case of the study of a bird's wing of Leonardo Da Vinci, the Kinematics can be solved quite well with geometric or algebraic solutions. It is more difficult to solve the inverse Kinematics for a typical industrial arm with six DoF. The trouble increases exponentially for a humanoid robot such as the newest "NAO"robot with 25 DoF.

Given the mathematical difficulties, the most common approach is to use numerical procedures. They are iterative algorithms, and their convergence is not guaranteed, or the resolution speed is not sure, which is not the right approach for real-time robot applications.

In many projects, we need to solve the kinematics problem in real-time (e.g., tracking a trajectory), and therefore we need to get closed-form solutions. It would be a lot easier to have a mathematical approach to get this kind of geometric solution for different robotics structures and mechanisms with many DoF. Therefore, we propose and defend this elegant approach of the screw theory for robotics. Nonetheless, we get started with some mathematical basics for the representation of any rigid body motion.

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