I have been working as a freelancer for a few months, providing R&D services for small companies. My preferred mode of collaboration is billing per project, which usually lasts a few weeks. It gives me the flexibility to solve the problems as a want and in the order that I prefer. The disadvantage is that I systematically under-estimate the time required for accomplishing a project, and consequently find myself underpaid. 

I have tried for about a month charging per week, but it seems to provoke the dynamics where I am more concerned with showing my client that I do work than focusing on getting good results. Specifically, I avoid the risks of trying new ways that may potentially take a lot of time without certain positive outcome, and I feel pressure from my client to report on the intermediate steps rather then on the final outcome.

I am looking for suggestions on improving collaboration, as well as for the pros and cons of different modes of collaboration that I might not have considered.
['https://freelancing.stackexchange.com/questions/9607', 'https://freelancing.stackexchange.com', 'https://freelancing.stackexchange.com/users/23884/']
I started my consultancy twenty years ago and struggled with this problem a lot in the past. I found that everyone is generally happier in the long run with hourly contracts billed weekly or monthly. If you are running into issues where customers want quantifiable proof that you are working on their projects as much as you say that you are, then you should utilize a time tracking solution that monitors your screen, mouse, and keyboard movements like [HubStaff](http://ssqt.co/m5fIOsy).

I also bill on a per project basis, but I only do this when I am absolutely certain of how long it will take and how much it will cost in order to complete the project.
It depends on the scope of the project. If you think that after completion of project multiple revisions will be required and client can ask for multiple modifications, you need to go for the hourly payment method. 

If all the requirements are crystal clear and you are clear in your mind regarding the requirements and can estimate the time duration for the project, go for the project wise payment. 

As project wise payments are better when it comes to bigger projects.