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A collision-free trajectory planning method based on speed alternation strategy for multijoint manipulators in overlapped working envelopes is proposed. Since the shape of a robot's link is usually rectangular or cylindrical approximately, the proposed method models a robot's link mathematically by quadric primitives, such as ellipsoids and spheres. The occurrence of collisions between links can be predicted easily by means of relative coordinate transformations and geometric deformations between those ellipsoids. Furthermore, the collision-trend index which is defined by projecting the ellipsoids geometrically onto the Gaussian distribution plays a significant role in searching the optimal resolution in the proposed collision-avoidance method. Experiments with two Motoman robots from the YASUKAWAI Company are conducted to demonstrate the performance of the proposed methods.