The gemara says that "Rebbi Chanina ben Dosa placed his head between his knees" and davened - why this strange position to daven ? Also isnt this the position one uses not to fall asleep ?
Rav Moshe Shternbuch, shlit'a, in Sefer "P'shat v'Iyun l'Maseches Berachos," explains that this was a form of body expression used when Davening, but only by great Tzadikim, just like the expression of "Nefilas Apayim" which could be done (in its full form) only by great Tzadikim (those who are "Ra'uy l'Kach"). (Presumably, besides the Kabbalistic intentions, this bodily expression probably enhances focus and concentration.)
Rav Avraham Ehrman, shlit'a, in Sefer "Razei Berachah," explains that this form of expression was used in order to arouse Hash-m's mercy. It is similar to the way that a fetus rests in the womb of its mother, and it is similar to the way an animal acts, and thus it is an expression that is saying, "Just like You have mercy on these (the fetus and the animal) even though they can give You nothing, please, too, have mercy on the sick person."
We find also that Eliyahu placed his head between his knees ("va'Yasem Panav Bein Birkav") when Davening to Hash-m to send rain (Melachim I 18:42). The Ralbag there explains that he did this in order to humble himself, by putting his face low, towards the ground.
This position is also that of someone who is in great anguish, such as a mourner, and thus it is appropriate for a Tzadik to Daven in this position.