Example One:
Round One:
A Frigate is moving through empty space. Frigates have a speed of six
squares, so from a stationary start, these are the Frigate‚s options
for moving (if it only moves once a round)
The Orange circle is the Frigate, and the shaded squares are the
squares that the Frigate can move to this turn (ie, everywhere within
six squares). The Frigate decides to move three squares south and once
square east, which is well within the area it can move to. It leaves
behind a ghost marker in the original square.
Round Two:
The Frigate places another ghost marker three squares south and one square east of its position at the start of round two, the same distance and direction it moved last turn. The Frigate can move to any square within six squares of the new ghost marker this turn.
The Frigate decides to turn, and goes three squares east and three squares north of the new ghost marker (or a total distance of four squares east of its current position).
The Frigate moves to the dark orange circle, which is within its ring of possible moves. It leaves behind a ghost marker at the light orange circle for next turn‚s movement.
Round Three:
Last turn, the Frigate moved a total of four squares east. So this turn, it places a ghost marker another four squares east of itself, and can move to within six squares of that marker.
Example Two:
Round One:
In this example, the Frigate has to narrowly dodge an obstacle. As
before, the orange circle is the frigate, the shaded area shows the
Frigate‚s possible moves, and the dark rectangle represents a wall the
Frigate will try to avoid. 
Foolishly, the Frigate flies towards the wall at full speed this round. It leaves behind a ghost token for next turn.
Round Two:
Since the distance at the start of the turn between the Frigate and
the ghost token is 6 squares, place another token six squares in front
of the Frigate‚s new position. The frigate can move anywhere within six
squares of that token.
As you can see, this leaves the Frigate with limited options to avoid a collision. Note that even though the ghost is now on the far side of a wall, this does not necessarily mean a collision. The frigate is still travelling at a speed where it can avoid a crash. If it were going much faster, it would have to use a Jackrabbit stunt or possibly fire afterburners to avoid a crash. It chooses to fly south one square and east two squares.
By doing this, it avoids a collision with the wall. As always, leave behind a ghost token for next turn‚s movement.
Round Three:
The distance at the start of round three between the frigate and its ghost is one square south and two squares east, so place another token one square sound and two squares east of the Frigate.
Many of the available moves will still result in a collision, but the
helmsman decides to fly away at full speed.