Robot path planning using family of SOR iterative methods with laplacian behaviour-based control - Azali Saudi (2015)
A truly autonomous robot must have the capability to find path from its start point to a specified goal point. This study proposed a robot path planning technique that relies on the use of Laplace's equation to constrain the generation of potential values. It is based on the theory of heat transfer, when there exist a temperature gradient within a surface, heat energy will flow from the region of high temperature at heat source to the region of low temperature at heat sink. In this model, high Laplacian potentials are assigned to outer boundary, inner walls and obstacles. Whilst, the goal point is assigned the lowest and no Laplacian potentials are assigned to all other free spaces. The Laplacian potentials for nodes on free spaces are then computed iteratively using numerical techniques. In the literature, computing these Laplacian potentials using numerical techniques produced encouraging results. The numerical implementations of these previous works, however, were only based on family of point iterative methods i.e. Jacobi, Gauss-Seidel and Successive Overrelaxation (SOR). These standard methods are too slow when handling large environment. Therefore, this study introduces the concepts of half-sweep and quarter-sweep iterations, and initiates the first application of using family of Point SOR and family of Four Point-Block SOR iterative methods for computing the Laplacian potentials to solve the path planning problem. The implementations employ two finite difference discretization schemes that are based on 5-Point and 9-Point Laplacian. Within the family of Point SOR iterative methods, the simulation results shows that the application of half-sweep and quarter-sweep concepts reduced the computational complexities of the algorithms by approximately 50% and 75%, respectively. Significantly, simulations with family of Four Point-Block SOR iterative methods provide even faster computation. In terms of iterations count, the iterative methods based on the 9-Point Laplacian give the less number of iterations than the 5-Point Laplacian. Whilst, in terms of execution time, the speed difference between iterative methods based on 5-Point and 9-Point Laplacian is very minimal. Once the Laplacian potentials are obtained, the standard Gradient Descent Search (GDS) technique is performed for path tracing to the goal point. The existing GDS, however, suffers from the occurrence of flat region in a more difficult environment which causing the path generation to fail. Thus, this study proposes a new control known as Laplacian Behaviour-Based Control (LBBC) to overcome such problem. Due to its robustness, the LBBC successfully generated smooth path even in a more complex configuration space. Therefore in conclusion, the significant contribution of this study is in introducing for the first time the fast half-sweep and quarter-sweep iterative methods using families of Point SOR and Four Point-Block SOR methods via 5-Point and 9-Point Laplacian. These faster iterative methods overcome the slow performances of the existing standard methods, particularly when handling large environment. In addition, the newly proposed LBBC overcomes the drawback of the existing GDS that face difficulty when handling complex environment. Finally, the path planning problem is solved by combining the fast iterative method with the robust path searching LBBC technique, so that the path planning algorithm is capable of handling large and complex environment.