The cerebellum is known to be critical for accurate adaptive control and motor learning. We propose here a mechanism by which the cerebellum may replace reflex control with predictive control. This mechanism is embedded in a learning rule (the delayed eligibility trace rule) in which synapses onto a Purkinje cell or onto a cell in the deep cerebellar nuclei become eligible for plasticity only after a fixed delay from the onset of suprathreshold presynaptic activity. To investigate the proposal that the cerebellum is a general-purpose predictive controller guided by a delayed eligibility trace rule, a computer model based on the anatomy and dynamics of the cerebellum was constructed. It contained components simulating cerebellar cortex and deep cerebellar nuclei, and it received input from a middle temporal visual area and the inferior olive. The model was incorporated in a real-world brain-based device (BBD) built on a Segway robotic platform that learned to traverse curved paths. The BBD learned which visual motion cues predicted impending collisions and used this experience to avoid path boundaries. During learning, the BBD adapted its velocity and turning rate to successfully traverse various curved paths. By examining neuronal activity and synaptic changes during this behavior, we found that the cerebellar circuit selectively responded to motion cues in specific receptive fields of simulated middle temporal visual areas. The system described here prompts several hypotheses about the relationship between perception and motor control and may be useful in the development of general-purpose motor learning systems for machines.