Kinematics of Serial Robotics - Algorithms for simplified calculation of Direct & Inverse Kinematics in a Consistent Coordinate Reference system
Authors: L.  Brodtmann Norbert , Schilberg Daniel 
Published: October 2023

A novel algorithm structure of direct and inverse kinematics for the motion calculation of articulated robots is presented. These algorithms are based on a 3D rotation matrix, which is known in itself but not established in robotics, as well as the principle of a normalized vector orientation, introduced here. The algorithms can handle any number of rotation and telescop axes, can be fully parameterized according to individual hardware and are also much clearer than the established & classic Denavit-Hartenberg conventions.State of the art According to current research (2022/23), the (published) robot mathematics are still based on the classical Denavit-Hartenberg conventions from 1955, which formalize the calculation of the "Direct" kinematics. For this purpose, they require a separate Local Coordinate System (LCS) for each robot arm in addition to the stationary (world) reference coordinate system. On the one hand, the spatial position of individual links to each other is therefore not directly comparable. On the other hand, the movement of a single robot arm changes the spatial point position of each following link in the kinematic chain accordingly - which is not represented by the LCS.For decades, the DH conventions have been considered state of the art for forward and direct kinematics in science and technology. Standard algorithms for backward and inverse kinematics, however, have not been disclosed to the general public. Companies that manufacture robots consider the computational core of their own algorithms to be a "trade secret". Only the operator level is published.Solution approachIn this paper, a general inverse kinematics trajectory control algorithm for serial robot systems (articulated arm and SCARA) is presented. These algorithms systematically avoid the problem of uncontrolled singularity of inverse kinematics. The combination of similar rotation matrices working consistently in the (world) reference coordinate system as well as the normalized vector orientation – introduced in this work – offer a number of advantages. They not only allow a doubtless parameter assignment without danger of confusion of the two length parameters required for each DH matrices. It becomes possible to specify a generally valid algorithm of inverse kinematics for trajectory control.The solution to be published here does not require any additional location coordinates for each moving robot element, it consistently references one and the same (world) reference coordinate system. Any vector position and its orientation in 3D-Space can be compared directly - an essential prerequisite for the snapping algorithms of the inverse kinematics also disclosed here and the additional option of an integrated traversability of the robot on the gantry system. The consistency of the coordinate system enables perspective representation in imaging processes.The vector input values required for this are directly available – in the (world) reference coordinate system (CRS) – for each point of the kinematic chain.For hardware control, kinematic angular values of the robot arm movement are output via a motion protocol. A specially developed simulation visualizes the moving robot silhouette and the path to be traced in a freely selectable perspective.