Numerous researchers have applied differential geometry methods to approach the study of sets of rigid solids, the primary mathematical tool being the theory of screws and Lie algebras.
Murray, Li, and Sastry presented an excellent introduction to Lie groups' mathematical theory, especially to the special Euclidean group SE(3) and its algebra se(3). Furthermore, they show the geometric meaning of these theories by relating them to Ball's Theory of Screws. A crucial step in the contemporary revival of these theories came when Brockett connected Lie group theory to robot kinematics by introducing the Product of Exponentials (i.e., POE). We are incredibly grateful for Murray's works, with whom we initiated our training to screw theory techniques. The developments presented in this book follow those foundations to a great extent. Therefore, the nomenclature used for the formulations of this text is the same.
Paden and Sastry use Lie's theory to investigate the properties of manipulators. Park, Bobrow, and Ploen use Lie's mathematics for a formulation of robot dynamics. Brockett, Stokes, and Park derive the motion equations for open chains of rigid solids, using screw theory and the Lagrange equations. Selig derives iterative versions of the Newton-Euler and Lagrange formulations to solve robot dynamics based on Lie theory. The formulation in Khatib's operational space is beneficial for applications focused on robot tools' force/position control. Lilly and Orin, using Featherstone notation, develop algorithms for the dynamic simulation of robotic mechanisms. Ploen creates iterative versions for the equations of motion of sets of rigid solids, resulting in algorithms independent of the coordinate system, a geometric interpretation of Featherstone's presented. By the way, we must highlight Featherstone's great work on algorithms for robot dynamics, which we can consider pioneering for many of the later best developments, by introducing Spatial Vector notation.
In the investigation of open-chain mechanics of rigid solids, most of the work uses the Newton-Euler equations' iterative developments. However, as it could not be otherwise, closed-form formulations based on the Lagrange equations are equivalents, even though with more unsatisfactory performance in most cases. Almost all state-of-the-art techniques employ essential screw theory representation of six-dimensional vectors or some extension of this concept.
Regarding the control of chains of rigid solids, robots need a system that allows solving the inverse dynamic problem, even in the presence of disturbances and modeling errors. There are two basic paradigms in robotics: control in the joint space and control in the workspace. Murray et al. apply Lie group theory to develop various types of dynamic control for robots, using the Lagrange equations. To give a hint of these robust methodologies, we will see some examples of inverse dynamics for some industrial manipulators along with this Blog.