A positioning device for positioning a machining tool, which is arranged on a frame fixed to a rail vehicle and has an active section for machining a rail head, relative to a rail laid in the track and traversed by the rail vehicle, in particular according to claim 1, wherein the positioning device has a first positioning mechanism for changing an angular position, viewed transversely to a longitudinal direction of the rail laid in the track, of the active section of the machining tool, whereby the first positioning mechanism has a first coupling gear and a first actuator for moving the first coupling gear, wherein the first coupling gear exclusively comprises gear members in the form of swing arms and couplings connected to one another via pivot joints which can be rotated about axes of rotation parallel to one another, wherein the first coupling gear has a first swing arm pivotably mounted on a first pivot point fixed in position relative to the frame and a second swing arm pivotably mounted on a second pivot point fixed in position relative to the frame, wherein the first coupling gear further comprises a first coupling, which is pivotably mounted relative to the associated swing arm (16, 18) at a first bearing point arranged on the first swing arm and at a second bearing point arranged on the second swing arm, wherein a section formed from the respective straight connecting lines starting from the first pivot point via the first bearing point, the second bearing point, the second pivot point and back to the first pivot point results in a first parallelogram, wherein furthermore a third bearing point is formed on the first coupling, which is arranged at a distance from a connecting line between the first bearing point and the second bearing point and on which a third swing arm is pivotably mounted, wherein the first coupling gear further comprises a second coupling which is pivotably mounted on the third swing arm at a fourth bearing point arranged on the third swing arm and is pivotably mounted on the first swing arm at a fifth bearing point arranged on the first swing arm, wherein a straight line formed from the respective straight connecting lines starting from the first bearing point via the third bearing point, the fourth bearing point, the fifth bearing point and back to the first bearing point results in a second parallelogram. the first positioning mechanism has a first coupling gear and a first actuator for moving the first coupling gear, wherein the first coupling gear exclusively comprises gear members in the form of swing arms and couplings connected to one another via pivot joints which can be rotated about axes of rotation parallel to one another, wherein the first coupling gear has a first swing arm pivotably mounted on a first pivot point fixed in position relative to the frame and a second swing arm pivotably mounted on a second pivot point fixed in position relative to the frame, wherein the first coupling gear further comprises a first coupling, which is pivotably mounted relative to the associated swing arm (16, 18) at a first bearing point arranged on the first swing arm and at a second bearing point arranged on the second swing arm, wherein a section formed from the respective straight connecting lines starting from the first pivot point via the first bearing point, the second bearing point, the second pivot point and back to the first pivot point results in a first parallelogram, wherein furthermore a third bearing point is formed on the first coupling, which is arranged at a distance from a connecting line between the first bearing point and the second bearing point and on which a third swing arm is pivotably mounted, wherein the first coupling gear further comprises a second coupling which is pivotably mounted on the third swing arm at a fourth bearing point arranged on the third swing arm and is pivotably mounted on the first swing arm at a fifth bearing point arranged on the first swing arm, wherein a straight line formed from the respective straight connecting lines starting from the first bearing point via the third bearing point, the fourth bearing point, the fifth bearing point and back to the first bearing point results in a second parallelogram.