Several design strategies for rehabilitation robotics have aimed to improve patients' experiences using motivating and engaging virtual environments. This paper presents a new design strategy: enhancing patient freedom with a complex virtual environment that intelligently detects patients' intentions and supports the intended actions. A `virtual kitchen' scenario has been developed in which many possible actions can be performed at any time, allowing patients to experiment and giving them more freedom. Remote eye tracking is used to detect the intended action and trigger appropriate support by a rehabilitation robot. This approach requires no additional equipment attached to the patient and has a calibration time of less than a minute. The system was tested on healthy subjects using the ARMin III arm rehabilitation robot. It was found to be technically feasible and usable by healthy subjects. However, the intention detection algorithm should be improved using better sensor fusion, and clinical tests with patients are needed to evaluate the system's usability and potential therapeutic benefits.