Our startup is working with a big company to develop a joint product. We want to request a minimum guaranteed payment from the company. Should that be a certain percent of our costs to bring the product to market (mid seven figures) or a certain percent of our revenue ceiling (low eight figures)? Note, we need the company more than they need us.
It depends on what value you saw in the partnership to begin with and whether or not things have since changed.
Generally speaking, timing to start (re)negotiation is an important component of the negotiation process. Do it too early and you might lose a valuable partner. Do it too late and you might leave a lot of money on the table.
One possible "soft" approach is to have an informal "review the project/progress" meeting - say over lunch, and see if you can get a read on their perception. If you think you are underpaid and they think they are paying way too much, then you have other issues to address first.
If you have the option, use someone who is not in the trenches - may be someone like one of your board members going to lunch with their most senior person in charge. That gives both parties some "wiggle room" to change position without losing face.
Also, a perhaps key piece of advice... Most people WANT to be fair to their partners. However, perceptions, particularly in early stages, will be key until partners get to know each other and have had a chance to develop trust.
They will not see this as valuable as you do. They will pay you significantly more in the future if it works. Do not ask for a minimum payment unless they have asked you to produce something not in your product road-map. The pain of B2B is you spend a lot of capital to solve for X and they see it not as valuable as you do. It is the elephant hunting scenario. It takes a long time to find one, but you can eat forever.
What does Joint Product mean? You'll each have the rights to sell it? The phase is confusing. Do you have a term sheet in place - if not, I propose you structure one ASAP? If you have engineering resources as part of the development effort you need to be paid. Big company doesn't matter - you are answering a "gap" in their portfolio and they clearly need you, so I would find someone that has actually structured these kinds of transactions and plug them into the partnership discussion.