By networking mobile robots, personal smart devices, and smart space networks, we can provide for a more accurate data for patient care than when the former are used individually. We call this network of personal and smart space devices and robots "Robot-Assisted Ambient Intelligence (RAmI)". Even more, with the application of distributed network optimization, not only can we improve the assistance of an individual patient, but we can also minimize conflict or congestion over multiple patients' usage of limited resources that are spatially and temporally constrained in such a system. The emphasis of RAmI is on the efficiency and effectiveness of the physical assistance of multiple users and on the influence of individual robot actions on the desired system's performance. In this paper, we propose a distributed RAmI system and put the basis for the architectural setup of such a system. This distributed system should be modular and should facilitate fast decision-making to multiple agents over limited available resources. The proposed architecture is showcased by means of a case study.