There are two fundamental timpani mallet grips in use among the worlds’ timpani players. I use the French Grip which has a thumbs up and palms facing each other profile and there are many German Grip players who play with a palms down and thumbs to the side profile.
The French Grip that I play along with many other timpanists has the mallet gripped in a firm manner between the thumb and the fore finger with little or no space between the thumb and the rest of the hand. The other three fingers are in loose contact with the mallet and move in unison with the stick and the entire hand and wrist.
The stroke with the French Grip is basically a rotation of the wrist with the arm only having limited movement most of the time. The arm will come into play for very loud playing but for the most part, the wrist is the only thing that moves.
With the fulcrum between the thumb and fore finger the other three fingers act in support. They don’t pull or snap the stick unless you are trying to make a very sharp sounding note or in some circumstances for very fast single strokes when the fingers are used to generate speed.
Watch this video to see this exercise
I play 4 notes with one hand then 4 notes with the other hand followed by 8 alternating notes. This is a simple but extremely important thing to practice to develop a consistent sound and to strengthen your grip.