Abrasive machining apparatuses and methods of finishing glass articles with abrasive machining apparatuses are disclosed herein. In one embodiment, an abrasive machining apparatus includes a support base, an edge finishing unit, and an edge finishing unit position sensor. The edge finishing unit includes an abrasive machining spindle having an abrasive wheel that is coupled to a motor and a pivot mechanism that is coupled to the support base. The pivot mechanism has an axis about which the abrasive machining spindle pivots. The abrasive machining spindle is pivotable between an extended position and a retracted position. The actuator is coupled to the edge finishing unit and to the support base and selectively positions the abrasive machining spindle about the axis. The edge finishing unit position sensor is coupled to the support base and is oriented to detect a position of the abrasive machining spindle.