Higher derivatives of the position vector with respect to time
In
physics, the fourth, fifth and sixth derivatives of position 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. The higher-order derivatives are less common than the first three;[1][2] thus their names are not as standardized, though the concept of a
minimum snap trajectory has been used in
robotics and is implemented in
MATLAB.[3]
The fourth derivative is referred to as snap, leading the fifth and sixth derivatives to be "sometimes somewhat facetiously"[4] called crackle and pop, inspired by the
Rice Krispies mascots
Snap, Crackle, and Pop.[5] The fourth derivative is also called jounce.[4]
Fourth derivative (snap/jounce)
Snap,[6] or jounce,[2] is the fourth
derivative of the
position vector with respect to
time, or the
rate of change of the
jerk with respect to time.[4] Equivalently, it is the second derivative of
acceleration or the third derivative of
velocity,
and is defined by any of the following equivalent expressions:
In
civil engineering, the design of
railway tracks and roads involves the minimization of snap, particularly around bends with different
radii of curvature. When snap is constant, the jerk changes linearly, allowing for a smooth increase in
radial acceleration, and when, as is preferred, the snap is zero, the change in radial acceleration is linear. The minimization or elimination of snap is commonly done using a mathematical
clothoid function. Minimizing snap improves the performance of machine tools and roller coasters.[1]
The following equations are used for constant snap:
where
is constant snap,
is initial jerk,
is final jerk,
is initial acceleration,
is final acceleration,
is initial velocity,
is final velocity,
is initial position,
is final position,
is time between initial and final states.
The notation (used by Visser[4]) is not to be confused with the
displacement vector commonly denoted similarly.
The dimensions of snap are distance per fourth power of time (LT−4). The corresponding
SI unit is metre per second to the fourth power, m/s4, m⋅s−4.
Fifth derivative
The fifth
derivative of the
position vector with respect to
time is sometimes referred to as crackle.[5] It is the rate of change of snap with respect to time.[5][4] Crackle is defined by any of the following equivalent expressions:
The following equations are used for constant crackle:
where
: constant crackle,
: initial snap,
: final snap,
: initial jerk,
: final jerk,
: initial acceleration,
: final acceleration,
: initial velocity,
: final velocity,
: initial position,
: final position,
: time between initial and final states.
The dimensions of crackle are LT−5. The corresponding
SI unit is m/s5.
Sixth derivative
The sixth
derivative of the
position vector with respect to
time is sometimes referred to as pop.[5] It is the rate of change of crackle with respect to time.[5][4] Pop is defined by any of the following equivalent expressions:
The following equations are used for constant pop:
where
: constant pop,
: initial crackle,
: final crackle,
: initial snap,
: final snap,
: initial jerk,
: final jerk,
: initial acceleration,
: final acceleration,
: initial velocity,
: final velocity,
: initial position,
: final position,
: time between initial and final states.
The dimensions of pop are LT−6. The corresponding
SI unit is m/s6.
^
abcdefThompson, Peter M. (5 May 2011).
"Snap, Crackle, and Pop"(PDF). AIAA Info. Hawthorne, California: Systems Technology. p. 1. Archived from the original on 26 June 2018. Retrieved 3 March 2017. The common names for the first three derivatives are velocity, acceleration, and jerk. The not so common names for the next three derivatives are snap, crackle, and pop.{{
cite web}}: CS1 maint: unfit URL (
link)
^Mellinger, Daniel; Kumar, Vijay (2011). "Minimum snap trajectory generation and control for quadrotors". 2011 IEEE International Conference on Robotics and Automation. pp. 2520–2525.
doi:
10.1109/ICRA.2011.5980409.
ISBN978-1-61284-386-5.
S2CID18169351.