Oscullation/"anticipation" in transitions is so gimmicky and cartoonish. Living things don't move like that, and mechanical things don't unless they are broken, chintzy, or poorly designed (underdamped).
Oscillations typically happen when moving quickly and then stopping at a certain position.
A prime example is the ISSF 25m rapid fire pistol event, where you start with the pistol pointing down at 45 degrees, and within 8, 6 and 4 seconds have to raise the pistol and take 5 aimed shots.
Beginners will often find they oscillate vertically above and below the center of the target as they raise the pistol, and it usually takes a fair bit of practice not to oscillate. This is more pronounced at the rapid pistol events due to the tight time constraint forcing them to move quickly, especially at the final 4 second round.
Another example is gymnastic rings, especially when they go from a swinging motion to stopping vertically or horizontally.
Anticipation-like movement often occurs when we want to get a bit of extra momentum. For example, casually tossing a base ball or tennis ball often results in a bit of backwards movement first, before the main swing. Typically it's one smooth movement.
We see this, not only in mammals and muscles, but in organic systems which evolved to trade speed for power by storing mechanical, electrical, or chemical potential energy over time.
Overshoot and damping is the same in reverse: dissipating kinetic energy when you, for example, bend your knees when you land from a jump (hopefully).
Most living beings aren't trying maximize performance, they're trying to maximize survival. That crouch gives away that you're about to jump, and it allows prey to avoid you more easily or predators to adjust their attack to counter for it. A rabbit trying to get away from a predator simply bounds away in its given direction without giving away what its doing.
Even the human jumping example again... Suggesting people only bend their knees for performance would mean the "normal" way of jumping involves keeping your legs straight at all times and using your ankles/toes for all your thrust?
"Anticipation" in character animations is a source of persistent friction between game programmers and animators because if incorporated as part of user/player action it simply kills responsiveness. OTOH being able to see NPCs prepare to do something (like pulling the bow) is incredibly useful.