Synthesis of the Inverse Kinematic Model of Non-Redundant Open-Chain Robotic Systems Using Groebner Basis Theory

Abstract

One of the most important elements of a robot’s control system is its Inverse Kinematic Model (IKM), which calculates the position and velocity references required by the robot’s actuators to follow a trajectory. The methods that are commonly used to synthesize the IKM of open-chain robotic systems strongly depend on the geometry of the analyzed robot. Those methods are not systematic procedures that could be applied equally in all possible cases. This project presents the development of a systematic procedure to synthesize the IKM of non-redundant open-chain robotic systems using Groebner Basis theory, which does not depend on the geometry of the robot’s structure. The inputs to the developed procedure are the robot’s Denavit–Hartenberg parameters, while the output is the IKM, ready to be used in the robot’s control system or in a simulation of its behavior. The Groebner Basis calculation is done in a two-step process, first computing a basis with Faugère’s F4 algorithm and a grevlex monomial order, and later changing the basis with the FGLM algorithm to the desired lexicographic order. This procedure’s performance was proved calculating the IKM of a PUMA manipulator and a walking hexapod robot. The errors in the computed references of both IKMs were absolutely negligible in their corresponding workspaces, and their computation times were comparable to those required by the kinematic models calculated by traditional methods. The developed procedure can be applied to all Cartesian robotic systems, SCARA robots, all the non-redundant robotic manipulators that satisfy the in-line wrist condition, and any non-redundant open-chain robot whose IKM should only solve the positioning problem, such as multi-legged walking robots.