It is recognizable that the longitudinal axis of the Hummingbird Hawkmoth (Macroglossum stellatarum) is sloped in an angle of about 45°.
The wings are not just simply moved up and down but in a more complicated course. Four phases of the beat of the wings can be distingushed: beat downwards, lower point of return, beat upwards, upper point of return.

Beat Downwards The moth moves its wing diagonally from above and behind to the front below. First the wing is accelerated, then it keeps a relatively constant speed for a while. At the end of the beat downwards the wing is slowed down while being rotated around its longitudinal axis. At the lower point of return the speed of rotation reaches its maximum.

Lower Point of Return It seems as if the beat stops for a moment at the lower point of return. At the same time the speed of rotation is maximum. The wing is whirled around its longitudinal axis. So the front edge of the wing which is reinforced by veins can take the lead again for the beat upwards.

Beat Upwards The wings are diagonally moved back above and behind. The fast rotation which turned the front edge of the wing to take the lead is finished now and the speed of the beat grows. Farther above the speed of the beat is reduced until the wing has reached the upper point of return.

Upper Point of Return The speed of rotation reaches a maximum again. The wings are turned the other way round compared to the lower point of return and so rotated back to the starting position. In the following beat downwards the front edge of the wing takes the lead again. So the oscillation of beat and rotation are coupled.

"Standing" in the air the moth accelerates air downwards with its wings. This air causes forces pointing upwards which hold the insect in its position. The relations of forces of a moth standing in the air are similar to those of a helicopter standing in the air. The helicopter produces lift by rotating its blades whereas the moth oscillates its wings. This is much more complicated.
Few researchers tried to "color" the currents of air under flying insects with colored dust. The pictures of currents they got were completely different from those of helicopters and very hard to evaluate.
Probably the animal causes a current around its wings with each beat. At the following point of return it makes the currents drift downwards.