This is the other way of explaining modals.
Modal verbs are used to express ideas like ability, possibility, certainty, etc.:
can could have to may might must need ought to should will
The same ideas can also be expressed in other words such as:
be able to be obliged to it´s possible that it´s unnecessary to
ABILITY
Present or future Past
Ability
can/be able to was able to/managed to
Inability
can´t/ be unable to/not be able to couldn´t/was unable to/wasn´t able to
Questions about ability
Can he? / Is he able to? Was he able to?
Tony can speak Spanish very well.
I can´t find my pen anywhere.
Can you drive a car?
Alex can´t swim.
I know you´re a good swimmer, but how well can you swim on your back?
Ann was able to do the research, but she couldn´t write the report.
I was unable to finish all my work yesterday, but I hope to be able to do it tomorrow.
I couldn´t finish all my work yesterday, but I hope I can tomorrow.
POSSIBILITY AND CERTAINTY
Present or future Past
Possibility
may / might / could may have / might have / could have
Certainty or near certainty
must must have
Impossibility
can´t / couldn´t can´t have / couldn´t have
Questions about possibility
Could / Might it? Could / Might it have?
It could/may/might rain later.
We may not be able to get on the bus. It may be full up.
If his name´s Spiros, he can´t be Italian. He must be Greek.
It might be difficult to get a seat at such short notice.
Jane is always so punctual. She must have missed her train.
Her train might have been delayed. Or the train might have been cancelled.
PERMISSION
Present of future Past
Giving permission
can / may was allowed to
Refusing permission
can´t / mustn´t couldn´t / weren´t allowed to
Questions about permission
Can / Could / May we? Was he allowed to?
You can/may make notes during the talk if you like.
Can/Could/May I finish off this work tomorrow?
We were allowed to look round the church after the service was over.
You can´t/mustn´t smoke when you´re in a non-smoking seat.
We couldn´t/weren´t allowed to use our calculators in the maths test because it was against the rules.
OBLIGATION AND RESPONSIBILITY
Present or future Past
Obligation or responsibility
must / have (got) to / should / ought to had to / should have
Lack of obligation or responsibility
don´t have to / needn´t needn´t have
Obligation or responsibility NOT to do something
can´t / mustn´t / shouldn´t / oughtn´t to wasn´t allowed to / shouldn´t have
Questions about obligations or responsibility
Must we / Have we got to / Do we have to? / Did we have to?
Should we / Ought we to / Do we need to?
Do I have to hand the work in tomorrow?
You don´t have to/needn´t book in advance if you´re going by rail.
You must/should arrive at 4.30 if you want to get a good seat on the train.
You don´t have to/needn´t stand up on the bus unless all the seats are occupied.
In the exam do we have to write in pen or are we allowed to use pencils?
You shouldn´t spend so much time watchnig TV when you should be studying.