Once upon a time round yards were only to be found on the properties of professional horse people. They were very commonly used by racehorse breakers and a few old bushies. But it was rare to see a round yard being used by non-professional trainers and riders. But thanks to people like John Lyons, Pat Parelli, and Monty Roberts, round yards are almost mandatory on any horse property.
But just because many people have a round yard at home does not necessarily mean that they know why and how to use them. Some folks just use the round yard as a type of lunging ring. Others use them to ride in because their lack of confidence does not allow them to ride their horses in bigger spaces. Neither of these reasons is good enough to justify a round yard, but they don't concern me nearly as much as the people who use the round yard to drive their horses around to get them to come into them. This activity or approach to training is probably to blame for many of the problems we see in horses. Any horse person (professional or amateur) who thinks this is why they want or need a round pen should not have one.
It is a great thing to have a horse who wants to be with you. But note that I used the word, wants. Many people (including professionals) drive their horses around and around the yard until they will do almost anything to be allowed to stop. One well-known trainer even said in his books that a horse should be run until his "lungs ache". This attitude to using a round yard has become the norm. It seems that most people feel that if a horse does not respond in the appropriate way (such as facing up to you) that it should be made to run even harder.
The problem here is that such an approach does nothing to help a horse feel good about being with a human. If the horse felt that the best place in the round yard was right next to the person, that's where they would stand the moment you entered. But when a horse decides that being on the track or hanging by the gate or the side of the yard closest to his paddock is a better option than being next to the person, we make them run. We try to make being away from us such a miserable place to be that they eventually give up and resign themselves to the idea that standing near us is better than having to run more miles. Yet, we have done nothing to make them feel good about us, we have just made them feel more miserable about not being with us. We have given them the choice of a bad idea and a worse idea. We didn't make the right thing easy at all. We just make the wrong thing miserable. We are teaching obedience, not willingness.
I bring this topic up because we have seen several horses lately who have had considerable round yard work in the hands of professional trainers who have driven them around and around for a long time to make them submit. The result creates very worried horses who are always on alert to flee. If asked to just walk a circle in the round yard, these horses take off trotting or cantering as if they had been hit on the rump with a cattle prod. If you ask them to slow down they get very worried because they have learned that their job is to run. It's very sad. But it is very understandable when you consider how they have been trained.
There is no excuse for driving a horse in a round yard or any yard. It does nothing towards the education of the horse or your relationship with the horse. Directing a horse is one thing, but driving is another. Driving just brings the flight response close to the surface. So when asking a horse to walk, trot, or canter in the round yard or any circle, think of sending his thought forward. Think of directing him with a feel.
As for teaching him to come to you at liberty in a round yard, think about making yourself as least threatening as possible, while directing that his thought not drift to the gate or outside the yard to his friends. If his thought leaves the yard in a hard way, apply just enough energy to give him a reason to bring his mind back into the yard. Let him search where in the yard he should be. While he is searching do nothing. But when he stops searching and his thought begins to leave outside, do just enough to motivate him to keep searching inside the yard. How much is enough will depend. You want to do as little as possible, but as much it necessary. Sometimes, enough will just be a shift of your foot or cluck of your tongue, and other times you will erupt in a fit of energy (never at the horse). It will be just enough to get a change from the horse thinking outside the yard to inside the yard. Then you do nothing but wait.
In this way, when the horse looks at you or turns to you or walks to you, it will be because he figured out for himself that it was a good idea. It won't be because you made being on the track such a horrible place to be that he gave up trying to avoid you. In time he will begin to consider hanging out with you as the best thing that could happen to him. You will become a source of comfort in his life.