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In this paper, we propose a novel and coherent framework for fast footstep planning for legged robots on a flat ground with 3-D obstacle avoidance. We use swept volume approximations that are computed offline in order to considerably reduce the time spent in collision checking during the online planning phase, in which a rapidly exploring random tree variant is used to find collision-free sequences of half-steps (which are produced by a specific walking pattern generator). Then, an original homotopy is used to smooth the sequences into natural motions, gently avoiding the obstacles. The results are experimentally validated on the robot HRP-2.