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D* is a planning method that always routes a robot in initially unknown terrain from its current location to a given goal location along a shortest presumed unblocked path. The robot moves along the path until it discovers new obstacles and then repeats the procedure. D* has been used on a large number of robots. It is therefore important to analyze the resulting travel distance. Previously, there has been only one analysis of D*, and it has two shortcomings. First, to prove the lower bound, it uses a physically unrealistic example graph which has distances that do not correspond to distances on a real map. We show that the lower bound is not smaller for grids, the kind of map-based graph on which D* is usually used. Second, there is a large gap between the upper and lower bounds on the travel distance. We considerably reduce this gap by decreasing the upper bound on arbitrary graphs, including grids. To summarize, we provide new, substantially tighter bounds on the travel distance of D* on grids, thus providing a realistic analysis for the way D* is actually used.