Ocular tracking of targets in biological systems involves switching between two strategies: slow pursuit and fast corrective saccades producing pursuit nystagmus. Here, a symmetric (bilateral) controller is used as a model for the oculomotor control system (OCS) to drive two cameras on a robotic head. It relies, as in biology, on internal switching in shared premotor circuits to alternate automatically between the two types of movements comprising nystagmus. The symmetric structural concept is gaining acceptance as evidence points to sharing of both fast phase and slow phase control in brainstem structures previously thought to be solely involved in one mode alone. This bilateral OCS model is a parsimonious design that is at once biomimetic and analytically simple. We extend prior results by incorporating more biological clues from floccular projections to establish rudimentary prediction mechanisms for both slow and fast phases; prediction is achieved by using retinal slip, which contains target velocity information. This provides a more accurate replication of the difference between fast phase and slow phase dynamics, and considers neural activity profiles in the superior colliculus to refine the controller performance. The resulting controller eliminates the need for saccades in steady state for low frequency inputs, and each saccade now has better accuracy, despite visual delays.