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This paper presents a sampling-based approach to computing and executing feedback-motion strategies by defining a global navigation function over a collection of neighborhoods in configuration space. The collection of neighborhoods and their underlying connectivity structure are captured by a sampling-based neighborhood graph (SNG), on which navigation functions are built. The SNG construction algorithm incrementally places new neighborhoods in the configuration space, using distance information provided by existing collision-detection algorithms. A termination condition indicates the probability that a specified fraction of the space is covered. Our implementation illustrates the approach for rigid and articulated bodies with up to six-dimensional configuration spaces. Even over such spaces, rapid online responses to unpredictable configuration changes can be made in a few microseconds on standard PC hardware. Furthermore, if the goal is changed, an updated navigation function can be quickly computed without performing additional collision checking.