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The purpose of this paper is to present some qualitative results related to collision-free path planning in the joint space for a planar n-DOF linkage. It will be shown that if each of the joints of the linkage has full range, then the forbidden regions in the joint space due to obstacles which obstruct only the last link of the linkage are closed surfaces in the joint space. The consequences of limited joint ranges are also discussed. The connectedness of the collision-free joint space for this special case is also studied.