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This paper presents a novel approach for navigation of cleaning robots in an unknown workspace. To do so, we propose a new map representation method as well as a complete coverage navigation method. First, we discuss a triangular cell map representation which makes the cleaning robot navigate with a shorter path and increased flexibility than a rectangular cell map representation. Then, we propose the complete coverage navigation and map construction methods which enable the cleaning robot to navigate the complete workspace without complete information about the environment. Finally, we evaluate the performance of our proposed triangular cell map via the existing distance-transform-based path-planning method comparing it to that of the rectangular cell map. Also, we verify the effectiveness of the proposed methods through computer simulations.