1. A system, for use in 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: determining a member-curling plane for each of the multiple grasping members; generating for each of the member-curling planes, a respective planar polygon representation of a three-dimensional object model, yielding multiple planar polygon representations; transforming a first planar polygon representation, of the multiple planar polygon representations, corresponding to a first grasping member of the multiple grasping members, to a frame of a link, of multiple links, of the first grasping member, yielding a transformed planar polygon representation, wherein the transformed planar polygon representation is a cross-section of the three-dimensional object model taken along the member curling plane of the first grasping 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 representation.