A rod support system has an upper gripper assembly and a lower gripper assembly. Each gripper assembly has a top plate having a first track. The first track has a first substantially arcuate portion having a first radius of curvature and a second substantially arcuate portion having a second and smaller radius of curvature. A jaw is constrained to move in a path defined by the first track. A bottom plate of the assembly has a second track that is congruent with the first track also engaging the jaw, so that the jaw is constrained to move in a path defined by the first track and the second track in cooperation. The jaws can support a rod when the jaws move from a position along the first arcuate portion of the first and second tracks to a position along the second arcuate portion of the first and second tracks.