For a larger company that manufactures wooden windows, we developed an extensive iPad iOS application for optimizing their internal processes (measurements, installation, etc..). The development of the application went smoothly and the project was completed successfully. However, we received a new order from our client to upgrade the application with new features/functions. Some of the new functions which they require are special and unfortunately, we can't estimate exactly how much further coding and development is needed. Of course, the client's desire is to have a fixed price offer.
Since we do not have a lot of experience in the field of estimating the scope for unknown work/functions we want to be sure before we give our client a new offer. So help is needed - we're looking for an expert who will help us with the consultation about how to approach r&d for unknown tasks, how to estimate the project, how to explain to client etc. This knowledge will be also needed for future projects.
Have you considered splitting the research phase and the development phase? You can quote a fixed amount for the research phase (I'm sure you can estimate this in weeks and take the hit if you go over). You can say to the client that you will quote the development once the research phase is over.
Plus, they can take the research phase output to other providers if you do not or cannot do the work or the quote is too much.
Call me to discuss further if it helps.