SLAM on the Hexagonal Grid

Hexagonal grids have many advantages over square grids and could be successfully used in mobile robotics as a map representation. However, there is a lack of an essential algorithm, namely, SLAM (simultaneous localization and mapping), that would generate a map directly on the hexagonal grid. In this paper, this issue is addressed. The solution is based on scan matching and solving the least-square problem with the Gauss-Newton formula, but it is modified with the Lagrange multiplier theorem. This is necessary to fulfill the constraints given by the manifold. The algorithm was tested in the synthetic environment and on a real robot and is entirely fully suitable for the presented task. It generates a very accurate map and generally has even better precision than the similar approach implemented on the square lattice.