By integrating theories and methodologies from a diverse range of scientific disciplines (e.g., physics, neuroscience, cognitive science, psychology and robotics engineering) the present work is aimed at harnessing self-organized anticipatory synchronization in order to advance human-robotic interaction (HRI). This phenomenon is characterized by the emergence of anticipatory behavior by one system coupled to the chaotic behavior of another, following the introduction of short self-referential delays in the coordinating system. The current set of studies involved the creation of an artificial agent based on a time-delayed, low-dimensional dynamical model capable of behaving prospectively during an interaction with a human actor performing complex, unpredictable behaviors. By achieving characteristics similar to those observed during natural human interaction and coordination, the time-delayed modeling approached advocated here provides the potential for considerable future advancements in HRI.