  We investigate the online exploration problem (aka covering) of a
short-sighted mobile robot moving in an unknown cellular environment with
hexagons and triangles as types of cells. To explore a cell, the robot must
enter it. Once inside, the robot knows which of the 3 or 6 adjacent cells exist
and which are boundary edges. The robot's task is to visit every cell in the
given environment and to return to the start. Our interest is in a short
exploration tour; that is, in keeping the number of multiple cell visits small.
For arbitrary environments containing no obstacles we provide a strategy
producing tours of length S <= C + 1/4 E - 2.5 for hexagonal grids, and S <= C
+ E - 4 for triangular grids. C denotes the number of cells-the area-, E
denotes the number of boundary edges-the perimeter-of the given environment.
Further, we show that our strategy is 4/3-competitive in both types of grids,
and we provide lower bounds of 14/13 for hexagonal grids and 7/6 for triangular
grids.
