Fourth, fifth, and sixth derivatives of position

In the physics field of kinematics, the fourth, fifth and sixth derivatives of position are generalizations of velocity and acceleration. They are defined as derivatives of the position vector with respect to time – with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively. These higher-order derivatives are less common than the first three; thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics.

The fourth derivative is referred to as snap, leading the fifth and sixth derivatives to be "sometimes somewhat facetiously" called crackle and pop, named after the Rice Krispies mascots of the same name. The fourth derivative is also called jounce. According to Don Eyles, a former NASA computer programmer, the use of snap, crackle, and pop for derivatives of jerk was suggested by NASA engineer Allan Klumpp during the Apollo Program.

Applications

Minimizing snap and jerk is useful in mechanical and civil engineering because it reduces vibrations and ensures smoother motion transitions. In civil engineering, railway tracks and roads are designed to limit snap, particularly around bends with varying radii of curvature. When snap is constant, the jerk changes linearly, producing a gradual increase in radial acceleration; when snap is zero, acceleration changes linearly. These profiles are often achieved using mathematical clothoid functions.