A positioning apparatus for positioning a tactile or optical roughness sensor, a probe or some other measuring instrument with respect to a workpiece can be secured to a movement device of a coordinate measuring machine. The positioning apparatus has a drive that produces a relative movement between two parts of the positioning apparatus, and an inhibiting device, which inhibits the relative movement between the two parts. For this purpose, the inhibiting device has a first friction element and a second friction element each having unlubricated friction surfaces. The friction surfaces are pressed against one another with a normal force that is not variable during the operation of the positioning apparatus. A coefficient of sliding friction that is less than 0.15 acts between the friction surfaces in the case of dryness and without lubrication. Typically, the inhibiting device is arranged in a flexspline of a strain wave gearing.