
The tortoise formation was one of the prime examples of Roman
ingenuity at warfare. When deployed in such a way, the legionaries
became virtually invulnerable to arrows or objects dropped from
defensive walls.

The wedge was an aggressive formation used to 'crack open' enemy
lines. Relatively small groups of legionaries could form such a
triangle and then drive their way into the enemy ranks. As more
Roman soldiers reinforced the wedge from behind, the enemy line
could be forced apart. As breaking the enemy's formation was very
often the key to winning a battle, the wedge formation was vitally
important battlefield tactic of the Roman army

skirmish

The order to repel cavalry by Roman army officers brought about a
defensive formation, in which the front rank formed a tight wall of
shields with their pila protruding to form a line of spearheads
ahead of the wall. Undoubtedly it would be very hard to bring a
horse to break into that formation. The most likely occurrence
would be that it would come to a halt of its own will ahead of the
spearheads. It was at that moment that horse and rider would be at
their most vulnerable against the ranks behind the first line of
infantry which would then hurl their spears at them. Given the
short distance and the training legionaries received, it is likely
such halted cavalry, frantically trying to turn their horses around
to retreat, whilst colliding with horses following in the charge,
would prove very easy targets.
If one further considers the likely possibility of archers being
present, as is the case on the photo above, the effect of this
formation could indeed be devastating.

The orb was a defensive formation in the shape of a complete circle
which could be taken by a unit which had either become detached
from the army's main body and had become encircled by the enemy, or
a formation which might be taken by any number of units if the
greater army had fallen into disorder during a battle.
It can hence be seen as a formation representing a desperate 'last
stand' by units of a collapsing army. But also it can be seen as a
disciplined holding position by a unit which has been divided from
the army's main body in battle and which is waiting for the main
force to rejoin them.
In either case, it is not a formation one would like to find
oneself in, as it obviously indicates that they are surrounded by
the enemy.
Naturally any officers or archers would be positioned in the centre
of the orb.