In logistics warehouse sorting, rubbish classification, and household services, scenarios exist in which rigid and soft objects are randomly piled together. In such situations, two major challenges arise in robotic picking tasks: the first is to distinguish rigid objects from soft objects, and the second is to grasp one object of each type at a time. In this study, we propose a novel robotic picking methodology for the grasping of objects mixed with towels. The proposed approach is based on a novel object detection method that can identify a rigid object placed in different directions using a rotational bounding box. Rigid objects can be separated from the mixed scene without object segmentation. Moreover, the grasping pose of a rigid object can be generated directly along its principal axis, without using a CAD model or specific pose detection method. The gripper opening width is determined according to the object size. Therefore, our method can detect whether other objects, particularly soft ones, exist around a rigid object. If no suitable grasping pose is available for the rigid objects, the grasping pose on a wrinkle of the towel is selected. The experiments demonstrate that our method can accomplish the picking task in scenes with mixed rigid and soft objects, thereby indicating its significance in robotic object detection and sorting. INDEX TERMS object detection, rotational bounding box, grasping pose, mixed objects, towels