# Manipulation Planning for Large Objects through Pivoting, Tumbling, and Regrasping
## 5 Hierarchical Manipulation Planning
### Task Level Planning  
In task level planning, we sample the object poses on a table, and save information on the object configurations and the related grasp configurations in the nodes. The object poses are not sampled randomly on the table because pivoting includes a kinematic constraint that the object must be rotated around a fixed point on the table, which is also the object's rotational vertex. In the graph, we sample new object poses with respect to the pivoting and tumbling motions (see Figure 7). In pivoting motion, we discretize the object pose when rotating about the vertical line including the object vertex. The rotation is discretized in \(\theta_{t}\) intervals. We repeat such sampling for all the vertices in the current base surface. By sequentially changing the rotational vertices, the object can be moved through pivoting gait along a certain direction (see Figure 8). Figure 6: Object, EEFs, and environment. To sample new object configurations by tumbling, the object is rotated around an object edge until the object moves from one SP to another (see Figures 2b and 7). Tumbling enables a change in the support surface, which provides more choice in the path planning of the object, in case an obstacle is present (see Figure 3). In this study, most of the object poses are designed in SP. However, within a limited table area, such as the boundary of the table, it is difficult to place the object in SP. For such an area, we sample the object pose in USP where an EEF of the robot assists in holding the object. To retain an object in USP, we check the grasp stability, discussed in Section 4.3, to select the contact points of the EEFs. Here, it is to be noted that sampling the object poses in USP corresponds to pivoting the object assuming the double support (DS) gait mode, as discussed in [3]. On the other hand, sampling the object poses in SP corresponds to pivoting the object assuming the quadruple support (QS) gait mode. Figure 9 shows the difference between the DS and QS gait modes. Refer [3] for the details on the DS and QS gait modes. Figure 8: Examples of moving an object to certain direction through pivoting gait. The red dashed arrows indicate rotating directions. \(\theta_{i}\) indicates the amount of yaw rotation in the \(i\)-th sequence. Figure 9: Changes in the object’s rotational vertices (**a**) in the QS mode when four vertices are in contact with the table and (**b**) in the DS mode when two vertices are in contact with the table. Figure 10a shows the sampled object poses on the table. Each node indicates an object pose; the red ones denote SP, whereas the blue ones denote USP. The red and blue edges indicate pivoting and tumbling motion, respectively. For better manipulation performance in the real physical world, we also evaluate the change in the angle between the EEF and the contact surface normal during pivoting motion. Combining the corresponding grasp configurations with the object configuration, we can generate the graph shown in Figure 10b. In this graph, each node includes information on the grasp configurations and the object pose related to Figure 10a. The detailed explanations of the graph are shown in Figure 11, where the blue rectangular part in Figure 10b is amplified. A node contains information on both an object and a grasp configuration. Nodes insides a light blue circle (which is drawn only for explanations in Figure 11 and is not included in the graph) indicate they share the same object pose but different grasp configurations. Three primitive motions are represented by edges where yellow, red, and blue edges imply regrasping, pivoting, and tumbling, respectively. The configurations saved in nodes can be transferred by the edges connecting them. For example, in Figure 11b, the grasp configurations of two nodes are changed by a yellow edge which implies regrasping. The design of SP and USP is also reflected in the graph where blue nodes indicate USP, see Figure 11b and red nodes indicate SP, see Figure 11c,d.