I received a really good question about the relative advantages of bipedal and quadrupedal launch at the Royal Tyrrell Museum. Essentially, they asked what the relative performance tradeoffs of each one might be, and why pterosaurs might have ended up locked into a quadrupedal launch style.
As it turns out, there are really two parts to the answer. In terms of which launch mode ends up as the dominant method of takeoff within a clade, it is likely that phylogenetic inertia plays an important role: birds inherited an obligate bipedal stance from their ancestors, and so every bird (so far as we know) has been a biped and launches bipedally, at least from the ground (more on this later, but some birds are quad launchers from the water, which is pretty neat stuff).
Bats seem to have inherited an obligate quadrupedal lifestyle. Pterosaur origins are more fuzzy, but they probably arose from one of a few different groups where bipedal to quadrupedal transitions were more common, and so their early evolution may have been more phylogenetically plastic with regards to stance. They eventually ended up as obligate quadrupeds, with most species placing most of their weight on the forelimbs (this is apparent because manus tracks from pterosaurs are typically deeper than the pes prints).
In terms of the relative performance advantages, it turns out that we can solve the question algebraically:
1) Both forms of launch will start as a leap (or a run ending in a leap). This means that the immediate post-launch cycle is ballistic. That's handy - ballistic math is easy.
2) Quad launch adds an upstroke immediately after push-off. Bipeds can (and do) raise the wings as they toe-off, so they can engage the first downstroke as soon as the wings have clearance.
3) Quad launch adds more power for initial push-off, so this goes into the ballistic equation. This will mean more height and speed, but at the cost of the added upstroke (the extra upstroke is accomplished with folded wings, so it's quite quick).
So, all we have to do is have an idea of max acceleration and unload time for takeoff, which gives launch speed, combined with the launch angle. We can vary these a bit to get a range of plausible values, and these give us the ballistic trajectory. So, for example, the maximum height gain can be calculated from the launch velocity squared x sin(launch angle) squared, divided by 2 x gravitation acceleration.
It turns out that for just about any flying animal, quadrupedal launch does better in nearly every way. They get a lot more power (because the flight muscles are so strong and can add to the launch in a quad takeoff, whereas in a biped takeoff they add very little). This means more clearance, ballistic time, and speed. It also allows for a greater range of starting wing attack angles, and is essentially "safer" because of the much greater clearance for the wings and body (larger margin of error, as it were). The extra time in the air from the greater push more than makes up for the extra upstroke time. For example, even in a giant like Quetzalcoatlus northropi, the initial upstroke would only take about a tenth of a second. It would have almost a third of second to reach the top of the ballistic leap, however, giving plenty of time to spare.
So, on the whole, quad launch is just "better" - with one exception. A bipedal launcher with short wings and a very short flapping time can switch from ballistic phase to flapping phase a bit earlier. This is not as efficient as the quad option, but it can mean a steeper and more immediate climb-out. This is only useful for a burst-launching specialist at moderate or small sizes (at giant sizes quad launch is king), but it is perhaps noticeable that this exact set of morphological features and takeoff strategy is extremely common among living birds - it is highly typical of galliform birds (pheasants, grouse, etc), pigeons and doves, and many of the passerines. It is also perhaps telling that there are no particularly short-winged pterosaurs. For a quadrupedal launching animal, very short wings don't do nearly as much good. The closest example might be anurognathids, and as noted in my GSA abstract, they are quite unique among pterosaurs. More on that later...