Others have already covered most of what I wanted to say:

• Working exclusively for royalties puts all the risk on you, it's best to collect an advance (or an up-front payment, or a split of both, or whatever).
• Your plain English agreement and even verbal agreements may well be upheld by courts, so don't agree to anything you're not happy with and make sure any agreements made include all the terms you want to include. Don't sign anything you're not happy with, and ideally have a lawyer look over your agreement before signing anything.

But there was one more point in your post that I don't think has been addressed yet:

They are bringing to the table an existing customer base that they want to push to upgrade to this product. I don't know exact numbers

Their existing customer base is what makes the royalty-based deal attractive -- if you're taking all the risk and are considering signing a deal based on the value that customer base provides you should absolutely have a very good idea (preferably exact verifiable totals) of those numbers before completing your negotiations. Without this information you can't accurately assess the potential risk and potential profits. If they're worried about disclosing this information to you they can have you sign an non-disclosure agreement (NDA).

I would try to obtain more information before continuing with negotiations, speak to a lawyer at least briefly for advice, and request an advance of an appropriate amount (most likely reducing the royalties slightly to compensate). If you think 15% is too little negotiate for a higher amount, and if you can't reach an agreement you're happy with don't sign anything or do any work.

I would be asking for an up front amount also.

Having done a project of this nature before, this is how i did it. I calculated privately how long it would take me, worked out how much i would want to do this project for per hour, and then multiplied the two together to get a figure. I then tripled it.

I presented this figure to the people wanting the software and said it would take no less than that to even get it off the ground.

If they want ongoing support and enhancement after this, suggest a monthly retainer and only do a certain amount of work per month within that retainer, made clear in advance. You don't want to burn yourself out, or get the short end of the stick.

This is especially true if you get a whiff of feature creep and you havent even officially started yet. Get a design up front, and factor the cost of creating this design into your costs. Get this signed off by all relevent parties and make it clear that you will work to that spec and complete only that spec before making further feature changes, and that further feature changes might require seperate quotes and be chargable.

This worked very well for me and might work well for you.

Definitely avoid percentage profit shares like the plague.

Good luck!