SyncHeC: Markerless Spatio-Temporal Hand-Eye Calibration via Differentiable Rendering and Continuous-Time Joint Optimization

Markerless hand-eye calibration under continuous motion requires jointly resolving camera geometry and the time offset between images and joint states. We present SyncHeC (Synchronized Hand-Eye Calibration), a dense differentiable-rendering framework that estimates both from a single robot trajectory. Continuous-time forward kinematics makes latency differentiable, metric depth complements mask contours, and an offline planner combines spatial sensitivity with temporal excitation. Alternating updates support Eye-to-Hand, Eye-in-Hand, and multi-camera calibration. On a 7-DoF arm, one $6.5$\,s motion yields $0.21$\,cm translation error against a marker-board reference, $0.34$\,cm touch-test error, and $2.7$\,ms latency error relative to an $87$\,ms reference with $95%$ uncertainty of ±1.5\,ms. Against CM-Calib on matched recordings, translation and nominal latency errors improve by 3.7× and 2.3×. Experiments span seven arms and four depth sensors. Compensating the recovered offset during policy training and deployment increases success by $8.1$ percentage points across six families and six tasks (1,800 matched pairs; paired $95%$ confidence interval $[6.3,9.9]$ points).