###### 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 [PERSON]'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. kinematic problem; Inverse Kinematic Model (IKM); Groebner Basis; non-redundant open-chain robotic systems +
Footnote †: journal: Applied SoftThe robot's Forward Kinematics is the equation system that calculates the position and orientation of a specific point of the robot's structure according to the actual state of all its actuators. This point is normally an important point of its structure, such as the end effector of a robotic arm or the center of mass of a mobile robot. The Forward Kinematics is necessary for modeling the robot's movements, and it is also important for the synthesis of the Inverse Kinematic Model (IKM). The IKM function is to calculate the control references required by the robot's actuators, so the robotic system can reach a desired position or follow a predetermined path. This makes the IKM a fundamental part of the robot's control system, as presented in Figure 1.  
There are different procedures to compute the Forward Kinematics of an open-chain robotic system, which include [PERSON]'s algorithm [1; 2; 3], dual quaternions [4; 5], and the modeling by Displacement Matrices [6]. All these procedures calculate the Forward Kinematics through the execution of systematic algorithms, which are completely independent of the mechanical complexity of the robot's structure or its geometry. On the other hand, the Inverse Kinematic Problem of open-chain robotic systems presents a greater challenge because the most commonly used methods to calculate the IKM, the geometric method, and the analytical method strongly depend on the analyzed robot's geometry and do not offer a systematic procedure for the synthesis of the desired model. The geometric method divides the analyzed robot Inverse Kinematic Problem into several plane geometry problems, and solves those problems using trigonometric relations [3; 7]. It is obvious, by its own definition, that this method depends heavily on the geometry of the robot's structure, and any sequence of steps used to calculate the IKM of a robotic system may not be valid for any other different structure. The analytic method for the calculation of the IKM solves the state vector of the robot's actuators in the equation system that defines the Forward Kinematics [8; 9; 10].