1. A system, for planning a grasp for implementation by a grasping device having multiple grasping members, comprising: a processor; and a computer-readable storage device having stored thereon computer-executable instructions that, when executed by the processor, cause the processor to perform operations comprising: generating, for each of the multiple grasping members, multiple planar polygon representations of a three-dimensional object model; transforming a planar polygon, of the multiple polygons generated, to a frame of a link of multiple links of a subject member of the multiple grasping members, forming a transformed polygon, wherein the transformed polygon is a cross-section of the object model taken along a member-curling plane of the subject member; and sweeping the link from a fully-open position for the link to a point at which the link contacts the transformed planar polygon.