Geometric Reconstruction of Extrinsic Contact Trajectories using Tactile Sensing and Proprioception for Tool Manipulation

Tactile sensing enables robots to perceive rich contact information at the grasp, supporting tasks such as object recognition, in-hand pose estimation, and slip detection. However, in many tool-mediated manipulation tasks, the interaction that determines task success occurs at the tool tip, away from the tactile sensor, making direct sensing of tool-environment contact difficult, particularly when the contact moves during interaction. In this work, we reconstruct the trajectory of extrinsic tool-tip contact using tactile sensing and robot proprioception. We formulate tool-tip trajectory reconstruction as a geometric inference problem under a single-point contact assumption. Our method first estimates the global tool-tip contact location from a calibration segment designed to approximate fixed-point behavior, and then reconstructs the full trajectory by composing relative tool motion estimated from tactile marker observations under continuous contact. Across n=51 trials with multiple trajectories, tools, wrist poses, and grasp configurations, the proposed pipeline achieves a trajectory RMSE of 8.59 +/- 2.41 mm in the world frame and a shape RMSE of 5.96 +/- 1.16 mm, while operating online at 14.00 +/- 4.11 Hz. Overall, the results show that extrinsic tool-tip trajectory geometry can be recovered consistently from grasp-level tactile sensing, with trajectory shape remaining stable across variations in tools, wrist poses, and grasp configurations.