A non-Cartesian coordinate positioning machine is provided that comprises an extendable leg assembly for positioning a component such as a measurement probe within a working volume of the machine, and a constraint member associated with the extendable leg assembly for providing a predetermined part of the extendable leg assembly with substantially a same orientation relative to gravity for a same position of the component within the working volume. In a preferred embodiment, the orientation relative to gravity is maintained substantially constant, so that a plane defined by the predetermined part is substantially aligned with gravity, as the component is moved around the working volume.