I've been doing independent, non-funded, self-chosen research for years. The only real issue is that most of the papers of interest have been paywalled ($40 for a 4 page paper). I only have a Master's degree.
This is something I've been wondering if possible for a while too; and it is one of the reason I resigned from my previous job, to spend my full time in reading papers, reaching out to researcher, implement/test existing papers. Fortunately, the software side of robotics is still possible to test/conduct independent research with help of simulations, open source code/platforms and papers.
But I'm very new to this, so I would like to know more about your experience (limitation, downside/pros) doing this!