###### Abstract
The drawback of this method is that the aforementioned equation system is composed of nonlinear equations, which forbids the use of traditional matrix algebra procedures. Instead, individual relations inside the analyzed equation system must be found, relations that are specific for each case. This implies that the analytic method is also non-systematic because the relations found to calculate the IKM of a robot with a certain structure, may not be valid or suitable for a different robot. All the problems presented before have prompted the development of several projects that use different artificial intelligence techniques to solve the Inverse Kinematic Problem, such as Neural Networks [11; 12; 13], Particle Swarm Optimization (PSO) [14], PSO-optimized Neural Networks [15], Neuro-evolutionary Algorithms (combination of Neural Networks and Genetic Algorithms) [16], and PSO variants [17]. These techniques normally calculate satisfactory solutions for the Inverse Kinematic Problem (IKP), without having any problems with the geometry of the robot's structure. However, it is important to clarify that, while these methods solve the IKP, they do not synthesize an Inverse Kinematic Model (IKM). This implies that the position references that are offered as output do not come from a differentiable function; therefore, these procedures are not capable of computing the speed or acceleration references for the robot's actuators. In addition, these methods could suffer from  
Figure 1: Inverse Kinematic Model (IKM) in the robot’s control system. The IKM’s function is to calculate the required references for the robot’s multi-axis control. the known training problems of artificial intelligence techniques, such as the over-fitting of Neural Networks and Genetic Algorithms. Nowadays, several projects are being developed in the field of Robotics that use Groebner Basis theory [18] to implement systematic methods that solve the Kinematic Problem, in a way that is independent of the geometry of the robot's structure. Among these projects, the works of [PERSON] [19] and [PERSON] et al. [20] stand out, which solve the Inverse Kinematic Problem of robotic manipulators using Groebner Bases. [PERSON] et al. [21] use this theory for the kinematic analysis of cable-driven robots. [PERSON] et al. [22] employ it to calculate the mobility conditions of several mechanisms that can be used in robotic arms or parallel robots. The works of [PERSON] et al. [23] and [PERSON] et al. [24] use Groebner Basis theory for the resolution of the Kinematic Problem of parallel robots, while [PERSON] et al. [25] employ it to triangularize the kinematic constraint equations of closed-chain robotic systems. Most of the aforementioned works employ Groebner Bases to solve their corresponding Kinematic Problems, proving that the full set of solutions can be obtained using this theory. However, they do not provide a procedure to synthesize a Kinematic Model that can be used in the control system of the robot (see Figure 1). This work objective is to use Groebner Basis theory to develop a systematic procedure for the synthesis of the IKM of non-redundant open-chain robotic systems. This procedure is explained in Section 3 and was used to synthesize the IKMs of two robots: a PUMA manipulator and a walking hexapod. Section 2 presents the calculation by traditional methods of the kinematic models of the two mentioned robots, which are later used as reference models in the performance analysis of the developed procedure. Section 4 shows the procedure's performance analysis, comparing the outputs of the IKMs synthesized in this project with those of the reference models. The possible future works that arise from this project are summarized in Section 5. In addition, finally Section 6 presents the conclusions of the obtained results and the final remarks of this work.