Minimum-Length Coverage Path Planning for Grid Environments With Approximation Guarantees

We focus on planning minimum-length robot paths to cover environments using the robot's sensor or coverage (e.g., cleaning) tool. Many algorithms use the following framework: (i) compute a grid decomposition of the environment, (ii) partition the grid to be covered by non-overlapping <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">coverage lines</i> (straight-line paths), and (iii) compute a cost-minimizing tour of the coverage lines to get a coverage path. While this framework aims to minimize turns in the path, it does not yield guarantees on the resulting path length. In this paper, we show that this framework guarantees a coverage path of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1 + 1.5\gamma)$</tex-math></inline-formula> times the optimal, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma \gt 1$</tex-math></inline-formula> is the approximation factor to solve the metric traveling salesman problem (metric-TSP). Following this, we propose the Minimum Length Coverage Approx (MLC-Approx) approach that modifies this framework to achieve an approximation factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1.5 + \epsilon)$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon \ll 1$</tex-math></inline-formula> depends on the number of coverage lines. Instead of computing a tour of the coverage lines, MLC-Approx merges minimum-length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sub-tours</i> of coverage lines while minimizing the turns added by the merges. We also propose a lazy variation of MLC-Approx that achieves the same result with faster empirical runtime. We validate MLC-Approx in simulations using maps of real-world environments and compare against state-of-the-art CPP approaches.