Space Curve Representation and Estimation from Overlapping Time-of-Flight Point Cloud Measurements using Geometric Integration on SE(3)

Extracting geometric properties of spatial curves from Time-of-Flight (ToF) sensor data is essential for autonomous navigation along extended structures such as powerlines, pipelines, and cables. Direct numerical differentiation of point cloud data yields poor estimates of curvature and torsion due to noise amplification. This paper presents a method that formulates path kinematics on SE(3) using the Frenet-Serret frame and Darboux vector, then employs an Extended Kalman Filter to fuse sequential, overlapping ToF measurement windows. The SE(3) formulation ensures geometric consistency of the estimated pose, while the EKF recursively integrates noisy measurements with the continuous-time path dynamics. Simulation of an aircraft-mounted ToF sensor tracking a powerline demonstrates that the proposed approach achieves path estimation errors of 10^{-3} m, curvature errors of 10^{-5} m^{-1}, and torsion errors of 10^{-7} m^{-1}. Compared to direct discrete differentiation methods used as a baseline, the EKF reduces curvature and torsion errors by four orders of magnitude and path errors by one order of magnitude. This framework provides a foundation for real-time path discovery in robotics and autonomous systems operating near curvilinear infrastructure.