I will have a carriage sliding over a round metal shaft and the linear bearings will not necessarily be conductive. I want to calculate the position of the carriage by measuring the resistance from of it to a fixed point on the shaft (with a -0.3 to 0.3V 24Bit ADC and a very high resistance sense resistor). My problem is that I will need an electrical connection between the carriage and the shaft. A brush is an obvious choice but I would like to avoid the friction and wear that would be caused by that and I would therefore need a contactless method of gaining an electrical connection between the two. Induction immediately springs to mind but from what I can see from research is that it won't really help me, is this correct? Any other ideas?
Contactless electrical connection
An electrical connection is, by definition, not really contactless. For measuring resistance you need (continuous) current to flow, and for that you need a connection. You can work with capacitive or inductive coupling, but that needs AC.
For your problem you can try to use TI inductance-to-digital converter, e.g. the LDC1000 (there is also a not-so-expensive EVM available). With that you can mount a small coil on your carriage, and just by the changes in its relation to the shaft the measured inductance changes. And since it has 24 bit resolution this might be sensitive enough (depending on your resolution requirements).