- There are of course robotic lawn mowers available, probably best suited for the smaller suburban yard.
I have considered the same problem, as I have quite a large area (1/3 hectare / 0.8 acres), half kudzu, half native.
For now, I have a ride-on lawn mower that does the area in about 3 hours every 2 weeks to one month. I cut at the highest setting. I also leave the clippings just where they fall. Collecting them in the catcher and emptying it on the compost pile easily makes the chore a whole-day exercise.
If you do go for biological lawn robots (animals), in my opinion the way to go would be to set up an intensive grazing system. You divide your area into a couple of enclosures and have your animals graze everything in it to the ground. Then rotate them to the next enclosure while that area recovers and grows lush from all the nice manure :-) In this way all plants are grazed equally (not only the tasty kudzu - while the bitter weeds are left alone to thrive). Do read the Wikipedia article. This is actually used in some areas to improve pasture by suppressing weeds (while lessening the use of weed killer).
Animals I have considered:
- Cows. Not really. Maybe a very small breed. Rejected the idea.
- Sheep. Probably preferably a mutton breed as you do not want to do the shearing... Else a milking breed. Look for what is available and thriving in your area. (If you are in the USA, I know mutton is all but unknown, but quite tasty, I assure you, if you need to cull some stock.) An ewe-only flock would be easier to handle.
- Goats. Much the same as for sheep, but since they also browse (eat tree leaves - and branches, and bark), you need to keep them away from what you don't want eaten. Quite intelligent animals with a mind of their own. I would prefer a milk breed like the Swiss Saanens or the Nigerian/Cameroon dwarf goats. The latter are excellent milk producers, cute and make good pets, and may be more suited for a smaller space. Again, beginners especially should rather avoid rams.
- Chickens. I have actually kept a couple of Dutch bantams, and am very fond of those little birds (but am now looking at a larger breed that is quite hardy in local conditions). Nice for the eggs, much better than bought. If handled regularly from an early age, can be quite tame and make nice pets. In addition, can help to control insects and slugs in your veggie patch (but should be kept separated from the plants especially cabbages and lettuce - it is quite tasty to them too). I found them quite low-maintenance, I could go away for a week or two and just leave them with a waterer and a feeder. Occasional trimming of claws and wings, maybe once or twice a year. Of course, you would need a lot more individuals for a given area than the larger animals. Also, they should not be combined with sheep (as sheep manure contains a lot of copper, which chickens are sensitive to). Depending on neighbors and city ordinances, roosters may not be a good idea due to noise - although I had a Pekinese that did not make himself all that noticeable.
Droppings are not that much of a problem. The grazers and browsers have fairly fibrous pellets, due to the free nature of their confinement they dry out quickly and break down to provide plant nourishment. The poultry may be a slightly bigger problem, but simply avoid the grazed area for a few days until dried out :-)
Of course, an enclosure would disrupt your lawn's flow of movement. But you do not absolutely need permanent enclosures, you may make movable ones (examples: chicken tractor, electrified tape or mesh). You would also have patches in various stages of grazing/recovery so your lawn would not always look suburb-pristine. You would need to find out (or determine through experimentation) your property's carrying capacity to determine how many animals to get, and then you need to keep in mind that grazing in spring and summer will feed more animals than in fall and winter - so you would need to cull your flock in winter and/or buy feed, then get additional animals in spring (new offspring is nature's way, but all the advice against the males may make that difficult to achieve).
Milking breeds may need daily milking (if they were pregnant), which can be a lot of trouble (milking machines do exist), but can at the same time be rewarding in the self-sufficiency department.
Obviously, animals should always have access to drinking water (which luckily can be made movable and/or automated) and depending on your climate, shelter. So some thinking should go into that. Also, chickens should have a nesting box, which makes it easier to find and collect eggs.
- You can always do away with the lawn and put it to use for (most of) your food needs, perhaps similar to the One Acre Farm concept. Obviously, this depends a lot on how many people in your family, how much land, climate and soil fertility, and not least, your expertise. I guess something like that is more like an ideal to work towards...
Additional note regarding chickens combined with grazers: I know of this Zimbabwean farmer, having bred an extremely hardy indigenous breed of chicken (Boschvelder), that would accompany the cattle and peck ticks off them, thus helping to keep them more healthy without the need for dips and antibiotics.