Discrete trajectory checks can miss collisions between sampled robot states. Reachability-based certification bounds motion between states, but uniform time partitions waste computation where clearance is large. We present lazy adaptive reachability certification (LARC), which checks a planned trajectory by bisecting only intervals with an inconclusive clearance test. For piecewise-cubic Hermite joint trajectories, the method bounds link occupancy using midpoint capsules inflated by exact componentwise speed maxima. Certified intervals covering the trajectory provide continuous-time external-obstacle clearance, subject to geometric containment, static obstacles, and a prescribed margin. On 160 AgileX PIPER trajectories from 80 start-goal pairs, LARC matched all decisions of the fixed-fine baseline at depth nine. It used 20328 interval evaluations (24.8% of baseline work), with a median paired speedup of 10.28x. A separate MoveIt/FCL audit checked 158051 states and detected collisions in 21 direct-interpolation controls, none of which LARC certified. The method reduced computation under a shared certificate model, but 27 of 139 sampled-clear trajectories remained uncertified. The sampled audit cannot independently prove continuous-time clearance.
Robotic grippers face substantial challenges in grasping and manipulating thin objects. Most existing grippers rely on highly precise approach and grasp motions, which limits robustness and reduces applicability. This paper explores thin-object grasping using books as a representative example. Here, we propose a novel solution that integrates an active surface with underactuated compliance to achieve stable grasping of thin objects without complex control. First, an underactuated gripper with an active surface is designed. The active-surface thumb performs in-hand repositioning of the target book without requiring adjustments of the robot arm or the other fingers, while the underactuated fingers establish compliant contact conditions with the environment, and the reconfigurable structure enables reliable grasping of books under different configurations. Second, we establish a kinematic model of the gripper, and determine the initial grasp postures for two representative scenarios (books lying flat on a desktop and books vertically packed in a shelf). Third, by analyzing the physical model of a book lying on a table and its interaction with the gripper and the environment, we systematically optimize the structural parameters and grasping strategy. Finally, extensive experiments validate the effectiveness of the proposed gripper and strategy. The results demonstrate strong robustness and adaptability when grasping thin objects placed flat (including books, paper, fabric, plastic film, and mouse pad), as well as a high success rate when grasping vertically packed books. Moreover, the proposed gripper can reliably complete long sequential"grasp-place"tasks.
