Assume square A1 is black, and that the floor is covered in a checkerboard pattern. Then, the entire grid contains 253 squares, 127 black and 126 white. A total of 42 squares are removed, leaving 211 squares.
Since the tiles alternate colour whenever the robot moves to a new tile, any path that ends up back on itself must be of even length. But there are an odd number of tiles on the grid, so no cycle is possible.
Now, can we do slightly worse and have there at least be a path where the robot can clean each tile?
From columns A and W, four black and four white tiles each are removed, for a total of eight black and eight white tiles.
From columns G and Q, each of the three-tile pillars contains two white tiles and one black, so a total of eight white and four black tiles are removed.
Finally, from columns I to O, each of the seven-tile blocks contains four black tiles and three white tiles, so a total of eight black tiles and six white tiles are removed.
In total, 20 black tiles and 22 white tiles are removed, leaving 107 black tiles and 104 white tiles. Any path cannot have a difference of more than one tile between the number of black and white, but there are three more black than white tiles. Thus, not even a path is possible.