In this paper a new method is proposed to quantify and reduce the radiation beam position uncertainty due to the radiotherapy treatment machine gantry deflection. A new tool has been designed and manufactured to provide the means to measure the alignment of the collimator axis and of the beam central axis in space, using the NDI Polaris optical tracking system and Gafchromic films. The tool can be mounted onto the accessory tray of the linacs from different manufacturers. The approach has been demonstrated with measurements of the mechanical isocentre being performed on ten linacs from three major manufacturers at four clinical sites. Measurements of the radiation isocentre were performed on a single linac. The collimator axis trajectory is modelled using a vector-end effector in order to provide more information than standard mechanical assessment methods. The method uses a mathematical optimization technique to calculate the position of the mechanical isocentre and the 'size' of the collimator axes intersection locus. Deviations of the collimator axes from the isocentre are expressed in terms of systematic and random error. The effects of measurement uncertainties are evaluated both via simulations and experimentally. The use of optical tracking and optimization techniques combined with an operator-induced measurement error compensation algorithm leads to a faster measurement of the mechanical isocentre (20 min for 24 angles) and eliminates operator-induced uncertainties. The uncertainty of the measurement of the mechanical isocentre was between 40 microm and 70 microm in terms of standard deviation. For some of the linacs assessed, the mechanical isocentre obtained using a standard approach with an adjustable pointer was displaced by over 1 mm from that found with the proposed method. The radii of the collimator axes intersection locus found with the proposed method were between 0.4 mm and 0.72 mm for the linacs assessed. Film measurement revealed a misalignment of the mechanical isocentre and the radiation isocentre by 0.4 mm. The proposed method potentially enables a reduction of the beam position uncertainty. This may be achieved at the planning stage by compensating for the identified systematic collimator axes deviations which were found to be reproducible. The method also creates a potential for using different setup margins independently for each axis and for each gantry angle, calculated specifically for a given linac.