Ph.D. in the space between two fields? I'm applying to PhD programs in two fields and I am interested in the space between two fields, where one is applied to the other.

In my particular case I'm interested in researching Software Engineering as applied to Robotics. I will use a similar but slightly different case as an illustrative example between the fields of Software Engineering and Machine Learning, which I am also interested in. Software Engineering applied to Machine Learning could incorporate better ways, such as new language syntax, to design and implement Machine Learning Algorithms. The opposite would be using machine learning to solve Software Engineering problems, such as automatically discovering database regularities in a data mining application. 

Essentially, I see understanding, researching, and being able to apply software engineering to robotics as an area that will expand enormously in the next 10-15 years, much like how software engineering research has expanded as applied to mobile devices and data centers.

I am very interested in both fields, but ideally I would be studying how to apply software engineering to robotics.


How do I explain my interests to both Robotics and Software Engineering professors, respectively?
How do I communicate why this matters at all, and more importantly why it matters to them?
Should I focus primarily on Software Engineering programs, Robotics programs, or both to reach my goal? 


Answers based on analogous situations from other fields are welcome.
I know little about either of these fields, but will try to give a generally applicable answer.

I think your best bet is to find an advisor who already has some appreciation for the intersection of these two topics.  Convincing someone who only works on one of them that they are important together -- and that he/she should supervise a thesis involving both -- may be difficult. However, since the fields you mention can often be found within the same university department (computer science), the latter approach is also possible.

To decide which type of programs you should look at, ask yourself in which field will I innovate?  If you will apply standard software engineering techniques to do something new in robotics, focus on robotics.  If you will devise novel software engineering ideas that are useful with respect to existing robotics applications, then focus on software engineering.  Of course, the answer is rarely clear cut.

Important: if you're not substantially innovating within either field, it may be difficult to convince a committee that you deserve a Ph.D. (even though the combination may be innovative).  You can't usually get away with work that is worth half of a Ph.D. in each field.