Can robot run on mini Stirling engine?
There was a major revision to the question as I wrote this but here goes ...



Figure 1. The Stirling engine in question.

That device doesn't look like it will do any useful work but let's do some calculations.

The flame looks about the same size as you might get from a tea light. These are about 10 g and last for about 5 hours. Burn rate is \$ \frac {10}{5 \times 60 \times 60} = 0.55 \mu g/s \$.

The energy density of wax is 42 kJ/g so we can calculate the power in J/s or watts: \$ P = mE = 0.55 \mu \times 42k = 23~W \$.

On the face of it this looks promising but as can be seen from the photo most of the heat is bypassing the bulb and you'd be lucky to get 10% of it heating the hot bulb. That takes us down to 2.3 W.

The maximum efficiency of the Stirling engine is 50% Wikipedia and this unit is not optimised at all so I would guess 10% would be generous. We're down to 0.23 W.

Now throw in some mechanical losses and you can see that we'd be down about 0.1 W or so. 

Next hook up your generator and say you could get 50% efficiency in the mechanical to electrical conversion we're down to 50 mW. At 5 V you would have enough current to light an LED.

For comparison the power output of my Shimano bicycle alternator (usually incorrectly referred to as a dynamo) is 3 W and probably costs me 5 or 6 W of extra cycling effort - probably 3 to 4%.

Links: Measuring the energy output of a tea light.