Managing uncertainty is a major challenge in radiation therapy treatment planning, including uncertainty induced by intrafraction motion, which is particularly important for tumours in the thorax and abdomen. Common methods to account for motion are to introduce a margin or to convolve the static dose distribution with a motion probability density function. Unlike previous work in this area, our development does not assume that the patient breathes according to a fixed distribution, nor is the patient required to breathe the same way throughout the treatment. Despite this generality, we create a robust optimization framework starting from the convolution method that is robust to fluctuations in breathing motion, yet spares healthy tissue better than a margin solution. We describe how to generate the data for our model using breathing motion data and we test our model on a computer phantom using data from real patients. In our numerical results, the robust solution delivers approximately 38% less dose to the healthy tissue than the margin solution, while providing the same level of protection against breathing uncertainty.