The convergence of Artificial Intelligence, the Internet of Things, and Robotics is no longer a futuristic vision; it is rapidly becoming the foundation of real-time, intelligent, and context-aware systems. AI enables perception and reasoning, IoT provides scalable sensing and communication, and robotics delivers embodied actuation. Despite significant progress in pairwise combinations such as AIoT and the Internet of Robotic Things (IoRT), there remains a lack of unified design frameworks that fully integrate all three. This survey synthesizes the state-of-the-art across these domains, emphasizing the emerging role of Small Language Models (SLMs) at the edge and Large Language Models (LLMs) in the cloud for distributed cognition and autonomous decision-making. We propose a modular system architecture that aligns with these trends, analyze persistent gaps in interoperability and feedback control, and classify existing work by integration depth. Our review highlights how hybrid SLM-LLM systems, when coupled with IoT infrastructure and robotic agents, can address challenges in real-time adaptation, scalability, and reliability. This work offers a conceptual and technical roadmap for designing next-generation AI-IoT-Robotic ecosystems that are modular, interpretable, and capable of learning within dynamic environments, paving the way for the emerging paradigm of Connected Robotics and Physical AI.