In recent years, edge computing has served as a paradigm that enables many future technologies like AI, Robotics, IoT, and high-speed wireless sensor networks (like 5G) by connecting cloud computing facilities and services to the end users. Especially in medical and healthcare applications, it provides remote patient monitoring and increases voluminous multimedia. From the robotics angle, robot-assisted therapy (RAT) is an active-assistive robotic technology in rehabilitation robotics, attracting many researchers to study and benefit people with disability like autism spectrum disorder (ASD) children. However, the main challenge of RAT is that the model capable of detecting the affective states of ASD people exists and can recall individual preferences. Moreover, involving expert diagnosis and recommendations to guide robots in updating the therapy approach to adapt to different statuses and scenarios is a crucial part of the ASD therapy process. This paper proposes the architecture of edge cognitive computing by combining human experts and assisted robots collaborating in the same framework to help ASD patients with long-term support. By integrating the real-time computing and analysis of a new cognitive robotic model for ASD therapy, the proposed architecture can achieve a seamless remote diagnosis, round-the-clock symptom monitoring, emergency warning, therapy alteration, and advanced assistance.