where cs and cf are the starting and final center points, and i corresponds to the trail, where i = 0 corresponds to the desired pull action. An expected divergence of less than 1 means that the action reduces uncertainty in expectation. A lower expected divergence is better.

To show the robustness of the pulling action primitive, we evaluate the reduction in uncertainty of the position of the dish after pulling. As shown in Figure 3, we move the gripper to the same initial position and place the dish at various points around the gripper 10 times, with the constraint that the gripper remains inside the dish when placed. We record the average distance of the initial states from the nominal initial position, then execute predetermined pull actions of 20cm, 30cm, and 40cm, and compute the average distance of the final states from the nominal final position. We report results in Table II, and see that the internal pulling action has an expected divergence of at most 0.28 for cups and at most 0.24 for bowls, which are both less than 1. Additionally we find that divergence decreases the longer the trajectory is. This is likely because longer trajectories give the item more time to correct itself.

Robustness Physical Experiments: We present experiments to evaluate uncertainty before and after taking an action. Avg. Initial Distance is the average distance between the trajectory’s initial location and nominal start location. Avg. Final Distance is the the average distance between the trajectory’s end location and nominal goal location. The expected divergence being less than 1 for all instances shows that the push action reduces the uncertainty in the final location of the object.

Theoretical Delays: We present the theoretical improvement in the time ratio as delays are added to simulate the bin being further away from the workspace. We find that the time ratio improves as more delay is added, strengthening the performance of the stack and pull policies compared to the baseline policy.

The improvement in time to completion is less pronounced because of two main factors. Firstly, the collection bin is placed right next to the workspace, making it so that trips to the collection bin are not as expensive in time, allowing the random policy to be competitive with our policies. In a more realistic scenario, however, the collection bin may be in an entirely different room from where the tableware is, making trips more expensive and favoring policies that make fewer trips to the bin. 

Secondly, the motion libraries of the UR5 require us to add  pauses between moving the arm and opening or closing the gripper, so as to avoid inadvertently pushing objects aside while actuating the gripper. This means our actions are punctuated by pauses while waiting for gripper actuation. This increases the execution time of especially our stack policy, which involves many gripper motions.