This paper presents the first active object mapping framework for complex robotic manipulation and autonomous perception tasks. The framework is built on an object SLAM system integrated with a simultaneous multi-object pose estimation process that is optimized for robotic grasping. Aiming to reduce the observation uncertainty on target objects and increase their pose estimation accuracy, we also design an object-driven exploration strategy to guide the object mapping process, enabling autonomous mapping and high-level perception. Combining the mapping module and the exploration strategy, an accurate object map that is compatible with robotic grasping can be generated. Additionally, quantitative evaluations also indicate that the proposed framework has a very high mapping accuracy. Experiments with manipulation (including object grasping and placement) and augmented reality significantly demonstrate the effectiveness and advantages of our proposed framework.