Among many fast exercises such as scales, arpeggios or shred exercises, certainly double picking deserves a special attention. It will give more assurance with your right hand.

Here are some advices that are worth considering.

1. Never rest your fingers on the pickguard .

2. Do not keep your palm on the bridge when playing a passage that involves strummed chords.

3. Do not dig your pick between the strings. .

4. Keep the pick as close to the string as possible at all times.

5. Use the very tip of the pick to make contact with the strings.

6. The right hand fingers not holding the pick should remain slightly curled into the palm.

7. As you hold the pick, bend slightly your thumb and index into the palm to have a better control and grip with your pick.

Double Picking exercises
Exercise 1 - Set your metronome to a slow tempo. Play the chromatic scale in continuous eighth notes using single picking. This takes the left hand out of difficult fingering and thus allows you to focus completely on the movement of your right hand.
Exercise 2 - Set your metronome to an even slower tempo. You might find easier to start playing open E, A, D, G, B, E strings and then play the scale.

My recommandation here is.

As you speed up the tempo, you might feel more comfortable if you downstroke all the way down AND up. (see symbol on the music sheet)

Hint: Start off the double picking exercises by warming up on a chromatic scale.