Dynamic equations for an $$\varvec{n}$$-link planar pendulum in absolute and relative coordinate systems and computational efficiency

Abstract For a planar pendulum with an arbitrary number of links, we derive dynamic equations in a matrix–vector form in absolute and relative coordinate systems. This is done for four pendulum systems of increasing levels of complexity known for their rich dynamics: a simple mathematical pendulum, a mathematical pendulum with arbitrary mass location along the links, a physical pendulum, and a damped physical pendulum. We explore the influence of the choice of coordinate system on the computational performance during the forward integration of planar pendulums presented in two coordinate systems, absolute and relative, by comparing computation time and error. Simulation results for chaotic and smooth, damped and undamped, transient and steady-state-behavior pendulum systems are presented. They demonstrate that the use of dynamic equations in the absolute coordinate system requires less computational time and leads to a smaller error compared to the computations with the dynamic equations in the relative coordinate system. Our results emphasize the importance of the choice of the correct coordinate system for the description of a given dynamic system. The Denavit-Hartenberg convention is commonly used to compute forward kinematics. It systematically assigns coordinate frames to the links of a robot manipulator, utilizing the relative coordinate system, and reducing the number of parameters associated with reference frames to four. However, we demonstrate that this choice is not optimal in terms of computational efficiency.