Get the character jumps in his local position

I have other problem with my game inspired by gravity rush, and it is the next:

When the character is in a surface different of the ground, you have to jump perpendicularly to the surface, and to do this normally you have to taking in account the axis up of the player, which I currently know, the problem is that I use the velocity of the rigidbody in the character to make him jump, so I want to know how to calculate the up velocity of the rigidbody which is the axis where it has to impulse, I never explain to you clearly, sorry for that.

The code to jump:

  velocity.y=0;    
      if (moveInput.magnitude == 0) {
         velocity.x = 0;
         velocity.z = 0;
     }
     
     if (jumpInput) {
         // jump!
         onGround = false;
         velocity = moveInput * airSpeed;
         velocity += jumpPower *normal;
         jump=true;
         gravity=false;
     }

The velocity that actually it has to be put to 0, at the beginning of the code, is the velocity of the character in its transform.up, and the other is his right and forward velocity, but I don't know how calculate it correctly.

velocity += jumpPower *normal; I think that doesn't work properly

My idea to know the velocity in the up direction of the character is the next:

velocity+=jumpPower* (new Vector3(1,1,1)-transform.up);

After this code, I check when the velocity of the player in up direction is almost 0, to apply again the gravity, and like this end the jump:

if (velocity.y < jumpPower * .5f) , but as I have said, this is wrong.

This is not the right form to add the jumpPower to the velocity.

Any idea how to get the velocity in the up direction of the character?

Thanks in advance and sorry for so much text.