Primary v/s Modal/Universal Auxiliary verbs

My sisters studied in a convent called the "Auxilium Convent" a Catholic School. I never knew what the meaning of the name of the School was until recently. It is a Latin word that means "to aid" or "a tribute to". In essence, it means to help. 
The same applies to Verbs. Auxiliary verbs whether Primary or Modal ( also referred to by some as Universal) are helping verbs. Okay, so what do they help to do? 

Let's start with Primary Auxiliary verbs:
This is what they do:-
-they help to form tenses e.g. Kate is solving some equations.
-they help to form negatives e.g. Sally has not done her homework.
-they help form questions e.g. Did you get the job done?
-they help add emphasis e.g. Tom did enjoy the camp!
They can be any action verb.

Now let's go to the Modal/Linking/Universal  Auxiliary verbs.
Some of the common Modal Auxiliary verbs are 'can', 'could', 'may', 'might','must', 'ought', 'shall', 'should','will','would' and as you can see they are used with ordinary verbs to convey such things as 
-the future
-possibility
-probability
-permission
-right and wrong 
-necessity
Some examples are:
Peter and I shall visit you tomorrow.
Anne will hate that play.
Would he stoop so low?
You should not break the queue.
He could not reach on time.
You may arrive in time.
Carl might be anxious.
She must do the right thing.

Participial phrases

Participial phrases are groups of related words that include a participle and as a group function as an adjective. 

Some examples are:

Allowing enough time, Jane started practising her badminton several weeks before the University tournament.

Beaten by the brilliant attack, the enemy fled in disarray!

The words in colour are participial or participle phrases.