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This paper introduces box approximations as a new tool for path planning of closed-loop linkages. Box approximations are finite collections of rectangloids that tightly envelop the robot's free space at a desired resolution. They play a similar role to that of approximate cell decompositions for open-chain robots - they capture the free-space connectivity in a multi-resolutive fashion and yield rectangloid channels enclosing collision-free paths - but have the additional property of enforcing the satisfaction of loop closure constraints frequently arising in articulated linkages. We present an efficient technique to compute such approximations and show how resolution-complete path planners can be devised using them. To the authors' knowledge, this is the first space-decomposition approach to closed-loop linkage path planning proposed in the literature.