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For a grasping task planner, the pre-planning of the reachable tip pose in a manipulator’s workspace is important. On this basis, for a seven-degree-of-freedom (7-DOF) redundant manipulator, it is very meaningful to study how to give full play to its redundant characteristics to achieve more dexterous grasping. In this paper, through the improved shape primitive method, the reachability spheres of the 7-DOF manipulator studied were more accurately captured, and a more accurate reachability capability map was generated. Then, based on the idea of redundancy angle discretization, the concept of the obstacle avoidance ability (OAA) index is innovatively proposed to measure the dexterity of the grasping task of the manipulator in a certain tip pose. Based on the OAA index axis, the distribution of the OAA index in each reachability sphere was analyzed. The prediction models of the OAA index of all reachability spheres in the workspace of the manipulator were obtained by the Levenberg–Marquardt algorithm and finally formed a dexterity capability map of the manipulator, which provides a new idea for the pre-planning of the dexterous grasping task of the redundant manipulator. Finally, we give the general frameworks of different grasping pre-planning by combining two kinds of capability maps and verify the effect.