Bidirectional measuring head (1) comprising: a stationary frame (12), a feeler (2), a kinematic motion assembly (13) which is supported by the stationary frame and carries the feeler to enable the feeler to move along two measuring directions (D1, D2) perpendicular to each other, and at least one position sensor (3a, 3b) that is mounted on the kinematic motion assembly and detects the position of the feeler along at least one measuring direction. The kinematic motion assembly has four columns, each of which runs perpendicular to the two measuring directions: a first column (14) is rigidly linked to the support frame, a second column (15) is adapted to translate along both measuring directions and supports the feeler, a third column (16) and a fourth column (7) each performs a displacement almost exclusively along one of the two measuring directions. The kinematic motion assembly may be a deformable mechanism. The kinematic motion assembly further comprises two balancing elements (60) each of which is hinged to the stationary frame so as to rotate around a rotation axis (33) perpendicular to the measuring directions and comprises one end which is mechanically constrained to the third column or the fourth column.
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