Precisely manipulating an unseen articulated object is a common yet challenging task, primarily due to the ambiguity in predicting the affordance and the articulation model of the object. Current data-driven methods learn spurious correlations through passive observation, which limits their adaptability to new objects and precision in manipulation tasks. To achieve robust generalization andprecisemanipulation,weproposeanovelActive KinematicModelingmethod(AKM),whererobotsactivelyinteract with objects to overcome spurious correlations and achieve precise kinematicmodeling.Ourmethodiscomposedofthreestages.First, hypothesis-drivenexploration identifies contact points where inter actions with the object trigger detectable relative motion between its parts. Next, unsupervised articulation modeling analyzes the recorded motion frames to construct an articulation model of the object. Finally, the model enables closed-loop manipulation via on-the-fly relocalization and replanning for precise execution. We validate AKM in simulation across a diverse benchmark of 116 articulated objects and demonstrate its generalization ability by deploying it to a real-world robotic system, consistently achieving sub-centimeter manipulation accuracy.