1. Investigación

Over the last years, mobile robot platforms are having a key role in education worldwide. Among others, LEGO Robots and MATLAB/Simulink are being used mainly in universities to improve the teaching experience. Most LEGO systems used in the literature are based on NXT, as the EV3 version is relatively recent. In contrast to the previous versions, the EV3 allows the development of real-time applications for teaching a wide variety of subjects as well as conducting research experiments. The goal of the research presented in this paper was to develop and validate a novel real-time educational platform based on the MATLAB/Simulink package and the LEGO EV3 brick for academic use in the fields of robotics and computer science. The proposed framework is tested here in different university teaching situations and several case studies are presented in the form of interactive projects developed by students. Without loss of generality, the platform is used for testing different robot path planning algorithms. Classical algorithms like rapidly-exploring random trees or artificial potential fields, developed by robotics researchers, are tested by bachelor students, since the code is freely available on the Internet. Furthermore, recent path planning algorithms developed by the authors are also tested in the platform with the aim of detecting the limits of its applicability. The restrictions and advantages of the proposed platform are discussed in order to enlighten future educational applications.

The present paper shows, for the first time, the technique known as PGD-Vademecum as a global path planner for mobile robots. The main idea of this method is to obtain a Vademecum containing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map. The PGD is a numerical technique with three main advantages. The first one is the ability to bring together all the possible Poisson equation solutions for all start and goal combinations in a map, guaranteeing that the resulting potential field does not have deadlocks. The second one is that the PGD-Vademecum is expressed as a sum of uncoupled multiplied terms: the geometric map and the start and goal configurations. Therefore, the harmonic potential field for any start and goal positions can be reconstructed extremely fast, in a nearly negligible computational time, allowing real-time path planning. The third one is that only a few uncoupled parameters are required to reconstruct the potential field with a low discretization error. Simulation results are shown to validate the abilities of this technique.

A fundamental robotics task is to plan collision-free motions for complex bodies from a start to a goal position among a set of static and dynamic obstacles. This problem is well known in the literature as motion planning (or the piano mover's problem). The complexity of the problem has motivated many works in the field of robot path planning. One of the most popular algorithms is the Artificial Potential Field technique (APF). This method defines an artificial potential field in the configuration space (C-space) that produces a robot path from a start to a goal position. This technique is very fast for RT applications. However, the robot could be trapped in a deadlock (local minima of the potential function). The solution of this problem lies in the use of harmonic functions in the generation of the potential field, which satisfy the Laplace equation. Unfortunately, this technique requires a numerical simulation in a discrete mesh, making useless for RT applications. In our previous work, it was presented for the first time, the Proper Generalized Decomposition method to solve the motion planning problem. In that work, the PGD was designed just for static obstacles and computed as a vademecum for all Start and Goal combinations. This work demonstrates that the PGD could be a solution for the motion planning problem. However, in a realistic scenario, it is necessary to take into account more parameters like for instance, dynamic obstacles. The goal of the present paper is to introduce a diffusion term into the Laplace equation in order to take into account dynamic obstacles as an extra parameter. Both cases, isotropic and non-isotropic cases are into account in order to generalize the solution.