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Static collision checking amounts to testing a given configuration of objects for overlaps. In contrast, the goal of dynamic checking is to determine whether all configurations along a continuous path are collision-free. While there exist effective methods for static collision detection, dynamic checking still lacks methods that are both reliable and efficient. A common approach is to sample paths at some fixed, prespecified resolution and statically test each sampled configuration. But this approach is not guaranteed to detect collision whenever one occurs, and trying to increase its reliability by refining the sampling resolution along the entire path results in slow checking. This paper introduces a new method for testing path segments in c-space or collections of such segments, that is both reliable and efficient. This method locally adjusts the sampling resolution by comparing lower bounds on distances between objects in relative motion with upper bounds on lengths of curves traced by points of these moving objects. Several additional techniques and heuristics increase the checker's efficiency in scenarios with many moving objects (e.g., articulated arms and/or multiple robots) and high geometric complexity. The new method is general, but particularly well suited for use in probabilistic roadmap (PRM) planners, where it is critical to determine as quickly as possible whether given path segments collide, or not. Extensive tests, in particular on randomly generated path segments and on multisegment paths produced by PRM planners, show that the new method compares favorably with a fixed-resolution approach at "suitable" resolution, with the enormous advantage that it never fails to detect collision.