Dear Amir Arshia,
I have plotted the functions given as arguments to FindRoot. It seems that the real solution does not exist as given curves do not intersect, so q1[t] == -q2[t] is not fulfilled.
Plot[{q1[t], -q2[t]} /. lagrangesolveg2, {t, 0, .5}]
If you are sure the solution exists it may mean there is an error somewhere or the interpolation is used improperly. Sometimes the interpolation may distort the funcition supposed to be interpolated. Please notice that the algorithm stops at the point where the minimal distance between curves occurs, so it works fine for sure.
