The specific date of the invention and it's inventor is unknown, but it was believed to be around 1900. It is widely used because the initial learning of the grip is very easy and the operation of the grip is very simple that it takes relatively short amount of time to master the technique. The mallets crossed inside the palm with the outside mallet underneath (with palm facing down). The thumb and 1st finger go between the mallets and function as the mechanism to increase interval by spreading the mallets apart. The 3rd and 4th finger apply counterpressure to hold the interval in place and with the aid of the thumb they function as the mechanism to decrease interval by pushing the mallets together. It is widely used by orchestral percussionist to play four mallet orchestra excerpts for the strength and security it offered.

It was invented around 1960s by the famous Vibraphonist, Gary Burton.  The Burton grip is also a cross stick grip as the traditional except that the outside mallet is on top of the inside mallet instead.   The interval is open by pulling the end of the inside mallet with the 3rd and the 4th finger away from the palm and the closing of the interval is achieved by  performing  the contrary. The middle finger touches the shaft of the outside mallet slightly lower then where the two mallets cross.  The middle finger  holds the outside mallet in place and provides a great amount of strength and security for the outside mallet.  A Burton grip player normally make use of this advantage and use the outside mallet to play  melodic passages, passages that require a lot of speed or loud dynamic range.  The Burton Grip is normally use by Vibraphonist.  Very rarely would one find a marimbist playing the marimba with the Burton Grip, partly because it is just not the tradition and  most importantly the bars on a marimba is so much wider  that it adds a great level of technical difficulty to the operation of the grip.  

It was invented around 1920s by the probably greatest keyboard percussion player of the 20th century, Clair Omar Musser.  The grip operates with the palm facing down position  as the Traditional  and Burton Grip, except for rolls when the hand position switch to vertical( thumb facing up).  The mallets do not cross in the hand.  The 3rd and 4th finger held the outside mallet and the thumb, 1st and 2nd finger held the inside mallet.  The interval is opened by pushing the thumb and the 1st finger out and to the side, pivoting the inside mallet on  one point  in the palm, closing the interval is achieved by performing  the contrary.  As a result of the mallet independence of the grip, it allows the application of finger control( using fingers to aid the acceleration of  the mallets) over a single mallet.  This is a great advantage  for playing transcription of classical and early Romantic literature, or composition in these style.    However, this grip did not maintain its great popularity because it is not very easy to learn and the grip is too weak  and insecure for loud  dynamic passages.  

It was introduce in 1971 by the renown marimbist Leigh Howard Stevens.   Mr. Stevens prefer to  call his grip the "modified Musser Grip".   As it is implied in Mr. Stevens' term, the grip is very similar to the Musser Grip except that it operates with a thumb-facing-up  position at all times with some exception,  and  that the inside mallet does not pivot at one point in the palm but rather moving in a curved line in the palm when the interval is spreading.  The 3rd and 4th finger held the outside mallet and the thumb 1st and 2nd finger held the inside mallet with the tip of thumb on the inside mallet at all times.  The interval spread by  rotating the inside mallet up and out with the thumb, 1st and 2nd finger, "drawing"  a smooth curve line in the air. The weight of the inside mallet transfer from the 1st finger to the 2nd finger as the interval spreads.  It is one of the most  if not the most employed grip nowadays, due to its great  ability to execute large intervals such as tenth, rapid interval changes, fine independent control over each mallet individually, ability to play one handed roll,  just  to name a few.