ROS (Robot Operating System) is currently the most popular robotic software development framework. Robotic software in safe-critical domain are usually subject to hard real-time constraints, so designers must formally model and analyze their timing behaviors to guarantee that real-time constraints are always honored at runtime. This paper studies real-time scheduling and analysis of processing chains in ROS2, the second-generation ROS with a major consideration of real-time capability. First, we study response time analysis of processing chains on ROS2 executors. We show that the only existing result of this problem is both optimistic and pessimistic, and develop new techniques to address these problems and significantly improve the analysis precision. Second, we reveal that the response time of a processing chain on an executor only depends on its last scheduling entity (callback), which provides useful guidance for designers to improve not only the response time bound, but also the actual worst-case/average response time of the system at little design cost. We conduct experiments with both randomly generated workload and a case study on realistic ROS2 platforms to evaluate and demonstrate our results.