  During retinal microsurgery, precise manipulation of the delicate retinal
tissue is required for positive surgical outcome. However, accurate
manipulation and navigation of surgical tools remain difficult due to a
constrained workspace and the top-down view during the surgery, which limits
the surgeon's ability to estimate depth. To alleviate such difficulty, we
propose to automate the tool-navigation task by learning to predict relative
goal position on the retinal surface from the current tool-tip position. Given
an estimated target on the retina, we generate an optimal trajectory leading to
the predicted goal while imposing safety-related physical constraints aimed to
minimize tissue damage. As an extended task, we generate goal predictions to
various points across the retina to localize eye geometry and further generate
safe trajectories within the estimated confines. Through experiments in both
simulation and with several eye phantoms, we demonstrate that our framework can
permit navigation to various points on the retina within 0.089mm and 0.118mm in
xy error which is less than the human's surgeon mean tremor at the tool-tip of
0.180mm. All safety constraints were fulfilled and the algorithm was robust to
previously unseen eyes as well as unseen objects in the scene. Live video
demonstration is available here: https://youtu.be/n5j5jCCelXk
