Developing chatbots as personal companions has long been a goal of artificial intelligence researchers. Recent advances in Large Language Models (LLMs) have delivered a practical solution for endowing chatbots with anthropomorphic language capabilities. However, it takes more than LLMs to enable chatbots that can act as companions. Humans use their understanding of individual personalities to drive conversations. Chatbots also require this capability to enable human-like companionship. They should act based on personalized, real-time, and time-evolving knowledge of their owner. We define such essential knowledge as the \textit{common ground} between chatbots and their owners, and we propose to build a common-ground-aware dialogue system from an LLM-based module, named \textit{OS-1}, to enable chatbot companionship. Hosted by eyewear, OS-1 can sense the visual and audio signals the user receives and extract real-time contextual semantics. Those semantics are categorized and recorded to formulate historical contexts from which the user's profile is distilled and evolves over time, i.e., OS-1 gradually learns about its user. OS-1 combines knowledge from real-time semantics, historical contexts, and user-specific profiles to produce a common-ground-aware prompt input into the LLM module. The LLM's output is converted to audio, spoken to the wearer when appropriate.We conduct laboratory and in-field studies to assess OS-1's ability to build common ground between the chatbot and its user. The technical feasibility and capabilities of the system are also evaluated. OS-1, with its common-ground awareness, can significantly improve user satisfaction and potentially lead to downstream tasks such as personal emotional support and assistance.