Human-robot collaborative assembly can leverage the unique capabilities of humans and robots to provide a flexible and efficient way for complex tasks. In this context, robots are expected to be endowed with the perception ability to collaborate with humans. For flexible processes, robotic grasping for a desired assembly part from variable multi-parts is essential but remains challenging, especially in situations of complex backgrounds including disordered placement, occlusion, similarity, and large-scale differences in size. To improve the perception and decision-making abilities of robots in collaborative assembly, firstly, a multi-stage approach integrating 2D part recognition and 6D pose estimation with perspective-focusing is proposed for desired part grasping under complex backgrounds. Secondly, to efficiently and accurately recognize a desired part from variable multi-parts under complex backgrounds, based on RGB data, an enhanced detector is adopted, which lays the foundation for the subsequent pose estimation. Thirdly, based on point cloud data, a pose estimation method based on perspective-focusing is proposed to enhance the success rate and efficiency of pose acquisition. Additionally, a modular autonomous robotic grasping system is designed. Finally, taking decelerator assembly as a case, the proposed approaches are comprehensively validated and compared on a real collaborative robot platform. Under complex backgrounds with occlusion, our approach shows stable performance with an average grasping success rate of 90.0 % and an average cycle time of 6.4175 s. Our work is expected to improve robots' perception and decision-making capabilities while further reducing the cycle time of collaborative assembly tasks.