The Mathematical Complexity of Robotics!

Anyone who has experience working with the robot's motion equations has realized the enormous complexity involved in solving them. Apparently, from the theoretical point of view of classical mechanics, obtaining the equations of motion of a set of coupled rigid solids may seem relatively straightforward. The only need is to have a reference coordinate system and apply the equations Newton-Euler's or Lagrange's to obtain the corresponding differential equations. The Lagrange approach provides an exciting high-level formulation to represent important parameters and give closed-form implementations, while the Newton-Euler formulation is used more for iterative algorithms. For instance, when the system is a robot-like humanoid, the mechanical analysis's complexity grows even more. The first reason is the enormous number of degrees of freedom. The second cause is because the stable biped locomotion problems appear. Besides, we need to solve crucial collision-free trajectory planning problems.

The need for simple formulations for the equations of motion becomes a vital issue as the complexity of robots and their control methods increases. It is desirable to have an explicit representation of these equations, which we can manipulate at a high level. The parameters of the mechanical system's kinematics and dynamics must be capable of being represented transparently. Furthermore, the algorithms used must be independent of the chosen coordinate systems (i.e., algorithms not linked to any local reference system) to be flexible in analyzing kinematics and dynamics.

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