Do changes in the orbit of the Earth contribute to global warming "Researchers have largely put to rest a long debate on the underlying mechanism that has caused periodic ice ages on Earth for the past 2.5 million years – they are ultimately linked to slight shifts in solar radiation caused by predictable changes in Earth's rotation and axis." 

How do we know that the amount of solar irradiation has increased since the ice age. Do current changes in the orbit path still cause the average temperature to still be increasing until another such event causes another orbit change? 

This appears to be the article.
They found in 1976 that the axial tilt and precession changes matched the geologically known ice age time periods. (see this pic for precise details)

The graphs say the world will get a tiny bit warmer for a few hundred years and then colder, if it weren't for CO2 being at the highest level for many million years:


Figure 1:  Orbital Parameters:  Eccentricity, Precession and Obliquity- click for larger image


How is it measured? It's mostly straight mathematics, say if the earth is 10% nearer to the sun it casts a larger shadow into outer space and collects more solar radiation. 

This figure is developed from A.L.Berger, 1978, Long Term Variations of Daily Insolation and Quaternary Climatic Changes, Journal of the Atmospheric Sciences, volume 35 (12), 2362-2367.


M.F.Loutre and A.Berger, 2000, Future Climate Changes: Are we entering an exceptionally long interglacial?, Climatic Change 46, 61-90

The effects of galactic cosmic rays on the atmosphere (via cloud nucleation) and those due to shifts in the solar spectrum towards the ultraviolet (UV) range, at times of high solar activity, are largely unknown. The latter may produce changes in tropospheric circulation via changes in static stability resulting from the interaction of the increased UV radiation with stratospheric ozone. More research to investigate the effects of solar behaviour on climate is needed before the magnitude of solar effects on climate can be stated with certainty.