Nowadays dangerous, repetitive or precision requiring jobs are done by robots like flying drones, industrial assembly arms or medical assistants. In all these cases, human beings can interact with the machines, therefore it is essential to guarantee that every part of the robot's software and hardware will produce a safe behavior: both the overall behavior and the local behaviors of such embedded systems have to be carefully analyzed. The complexity of embedded systems software architectures increase with more and more tasks involved; state-machines are applied to implement more functional capabilities of the tasks; and the task models used in the analyses gain in complexity. The analysis techniques have to be adapted in order to face such new complexities. This paper focuses on the real-time analysis of state-machine-based software architectures. We propose a method to analyze the temporal behavior of a component-based architecture in which the components are described by state-machines. The method computes an accurate worst-case response time by taking into account the state-machines of the components. Finally, we validate our approach with a real real-time robotic case study.