Artemis BJJ (MYGYM Bristol), Can Sönmez, Bristol, UK - 23/03/2016
On top of half guard, your opening goal is to get them flat on the mat: there are passes you can do while they are on their side, but generally speaking it is much easier if their back is pinned to the floor. A simple method, drawing on the Ribeiro brothers, is to drive your free knee into their hip, block their head with your same side arm, then step your trapped leg up and away from you. Having generated some space, drive the trapped knee forwards as your return it to the mat, which should also help you drive your opponent to the mat as well.
If you are able to get the cross-face and an underhook, there is now the option of generating lots of shoulder pressure. This is the simplest way I've learned to pass the half guard: both Saulo and his brother refer to this as the 'esgrima pass', but I call it the shoulder pressure pass in the interests of clarity. Cross-face their head (if you can't get the cross-face, you can also use your own head), so that they can't turn in that direction. Put your own head on the other side (or your arm, if you're already using your head to cross-face), locking their head into place: your shoulder and head work together to form a vice. Combined with your underhook, it should now become hard for them to move their upper body, because their head is stuck.
From here, come up on your feet so that all your weight is driving through your shoulder. Even if you're small, this should maximise your weight. I'm only 66kgs, but if I can get all of that weight against somebody's head, it becomes more significant. From there, bounce your trapped knee to wriggle it free (if you're having trouble and need additional leverage, rotate your free leg back to hook their leg with your instep). As soon as it is clear of their legs, twist in the direction of your cross-facing arm and put that knee on the mat. You can then kick their leg off your foot: some people prefer to kick the top leg, but I would generally go for the bottom leg. Turning your hips to the ceiling can also help if you're struggling to get that foot loose.
Teaching Notes: I modified this a bit, trying out the lower version Saulo does on BJJ Library. In that, the shoulder is across the ribs rather than high by the head. The lower position also leaves a hand free to push on their knee, a handy addition that I think may make that variation superior to the one I usually teach. I'll try it again in the mixed class next time half guard month comes around.