Fourth, fifth, and sixth derivatives of position

Snap,{{Cite book | doi=10.1109/ICRA.2011.5980409| chapter=Minimum snap trajectory generation and control for quadrotors| title=2011 IEEE International Conference on Robotics and Automation| year=2011| last1=Mellinger| first1=Daniel| last2=Kumar| first2=Vijay| pages=2520–2525| isbn=978-1-61284-386-5| s2cid=18169351}} or jounce, is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. Equivalently, it is the second derivative of acceleration or the third derivative of velocity,

and is defined by any of the following equivalent expressions:

$$\backslash vec\; s\; =\; \backslash frac\{d\; \backslash ,\backslash vec\; \backslash jmath\}\{dt\}\; =\; \backslash frac\{d^2\; \backslash vec\; a\}\{dt^2\}\; =\; \backslash frac\{d^3\; \backslash vec\; v\}\{dt^3\}\; =\; \backslash frac\{d^4\; \backslash vec\; r\}\{dt^4\}.$$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.

The following equations are used for constant snap:

$$\backslash begin\{align\}$$

\vec \jmath &= \vec \jmath_0 + \vec s t, \\

\vec a &= \vec a_0 + \vec \jmath_0 t + \tfrac{1}{2} \vec s t^2, \\

\vec v &= \vec v_0 + \vec a_0 t + \tfrac{1}{2} \vec \jmath_0 t^2 + \tfrac{1}{6} \vec s t^3, \\

\vec r &= \vec r_0 + \vec v_0 t + \tfrac{1}{2} \vec a_0 t^2 + \tfrac{1}{6} \vec \jmath_0 t^3 + \tfrac{1}{24} \vec s t^4,

\end{align}

where

{{div col}}

• $\backslash vec\; s$ is constant snap,
• $\backslash vec\; \backslash jmath\_0$ is initial jerk,
• $\backslash vec\; \backslash jmath$ is final jerk,
• $\backslash vec\; a\_0$ is initial acceleration,
• $\backslash vec\; a$ is final acceleration,
• $\backslash vec\; v\_0$ is initial velocity,
• $\backslash vec\; v$ is final velocity,
• $\backslash vec\; r\_0$ is initial position,
• $\backslash vec\; r$ is final position,
• $t$ is time between initial and final states.

{{div col end}}

The notation $\backslash vec\; s$ (used by Visser) 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.

The fifth derivative of the position vector with respect to time is sometimes referred to as crackle. It is the rate of change of snap with respect to time. Crackle is defined by any of the following equivalent expressions:

$$\backslash vec\; c\; =\backslash frac\; \{d\; \backslash vec\; s\}\; \{dt\}\; =\; \backslash frac\; \{d^2\; \backslash vec\; \backslash jmath\}\; \{dt^2\}\; =\; \backslash frac\; \{d^3\; \backslash vec\; a\}\; \{dt^3\}\; =\; \backslash frac\; \{d^4\; \backslash vec\; v\}\; \{dt^4\}=\; \backslash frac\; \{d^5\; \backslash vec\; r\}\; \{dt^5\}$$

The following equations are used for constant crackle:

$$\backslash begin\{align\}$$

\vec s &= \vec s_0 + \vec c \,t \\[1ex]

\vec \jmath &= \vec \jmath_0 + \vec s_0 \,t + \tfrac{1}{2} \vec c \,t^2 \\[1ex]

\vec a &= \vec a_0 + \vec \jmath_0 \,t + \tfrac{1}{2} \vec s_0 \,t^2 + \tfrac{1}{6} \vec c \,t^3 \\[1ex]

\vec v &= \vec v_0 + \vec a_0 \,t + \tfrac{1}{2} \vec \jmath_0 \,t^2 + \tfrac{1}{6} \vec s_0 \,t^3 + \tfrac{1}{24} \vec c \,t^4 \\[1ex]

\vec r &= \vec r_0 + \vec v_0 \,t + \tfrac{1}{2} \vec a_0 \,t^2 + \tfrac{1}{6} \vec \jmath_0 \,t^3 + \tfrac{1}{24} \vec s_0 \,t^4 + \tfrac{1}{120} \vec c \,t^5

\end{align}

where

{{div col}}

• $\backslash vec\; c$ : constant crackle,
• $\backslash vec\; s\_0$ : initial snap,
• $\backslash vec\; s$ : final snap,
• $\backslash vec\; \backslash jmath\_0$ : initial jerk,
• $\backslash vec\; \backslash jmath$ : final jerk,
• $\backslash vec\; a\_0$ : initial acceleration,
• $\backslash vec\; a$ : final acceleration,
• $\backslash vec\; v\_0$ : initial velocity,
• $\backslash vec\; v$ : final velocity,
• $\backslash vec\; r\_0$ : initial position,
• $\backslash vec\; r$ : final position,
• $t$ : time between initial and final states.

{{div col end}}

The dimensions of crackle are LT⁻⁵. The corresponding SI unit is m/s⁵.

The sixth derivative of the position vector with respect to time is sometimes referred to as pop. It is the rate of change of crackle with respect to time. Pop is defined by any of the following equivalent expressions:

$$\backslash vec\; p\; =\backslash frac\; \{d\; \backslash vec\; c\}\; \{dt\}\; =\; \backslash frac\; \{d^2\; \backslash vec\; s\}\; \{dt^2\}\; =\; \backslash frac\; \{d^3\; \backslash vec\; \backslash jmath\}\; \{dt^3\}\; =\; \backslash frac\; \{d^4\; \backslash vec\; a\}\; \{dt^4\}\; =\; \backslash frac\; \{d^5\; \backslash vec\; v\}\; \{dt^5\}\; =\; \backslash frac\; \{d^6\; \backslash vec\; r\}\; \{dt^6\}$$

The following equations are used for constant pop:

$$\backslash begin\{align\}$$

\vec c &= \vec c_0 + \vec p \,t \\

\vec s &= \vec s_0 + \vec c_0 \,t + \tfrac{1}{2} \vec p \,t^2 \\

\vec \jmath &= \vec \jmath_0 + \vec s_0 \,t + \tfrac{1}{2} \vec c_0 \,t^2 + \tfrac{1}{6} \vec p \,t^3 \\

\vec a &= \vec a_0 + \vec \jmath_0 \,t + \tfrac{1}{2} \vec s_0 \,t^2 + \tfrac{1}{6} \vec c_0 \,t^3 + \tfrac{1}{24} \vec p \,t^4 \\

\vec v &= \vec v_0 + \vec a_0 \,t + \tfrac{1}{2} \vec \jmath_0 \,t^2 + \tfrac{1}{6} \vec s_0 \,t^3 + \tfrac{1}{24} \vec c_0 \,t^4 + \tfrac{1}{120} \vec p \,t^5 \\

\vec r &= \vec r_0 + \vec v_0 \,t + \tfrac{1}{2} \vec a_0 \,t^2 + \tfrac{1}{6} \vec \jmath_0 \,t^3 + \tfrac{1}{24} \vec s_0 \,t^4 + \tfrac{1}{120} \vec c_0 \,t^5 + \tfrac{1}{720} \vec p \,t^6

\end{align}

where

{{div col}}

• $\backslash vec\; p$ : constant pop,
• $\backslash vec\; c\_0$ : initial crackle,
• $\backslash vec\; c$ : final crackle,
• $\backslash vec\; s\_0$ : initial snap,
• $\backslash vec\; s$ : final snap,
• $\backslash vec\; \backslash jmath\_0$ : initial jerk,
• $\backslash vec\; \backslash jmath$ : final jerk,
• $\backslash vec\; a\_0$ : initial acceleration,
• $\backslash vec\; a$ : final acceleration,
• $\backslash vec\; v\_0$ : initial velocity,
• $\backslash vec\; v$ : final velocity,
• $\backslash vec\; r\_0$ : initial position,
• $\backslash vec\; r$ : final position,
• $t$ : time between initial and final states.

{{div col end}}

The dimensions of pop are LT⁻⁶. The corresponding SI unit is m/s⁶.

{{Reflist|

{{cite journal |last=Visser |first=Matt |date=31 March 2004 |title=Jerk, snap and the cosmological equation of state |journal=Classical and Quantum Gravity |volume=21 |issue=11 |pages=2603–2616 |issn=0264-9381 |doi=10.1088/0264-9381/21/11/006 |quote=Snap [the fourth time derivative] is also sometimes called jounce. The fifth and sixth time derivatives are sometimes somewhat facetiously referred to as crackle and pop.|arxiv = gr-qc/0309109 |bibcode = 2004CQGra..21.2603V |s2cid=250859930 }}

{{cite web

| last1 = Gragert

| first1 = Stephanie

| last2 = Gibbs

| first2 = Philip

| title = What is the term used for the third derivative of position?

| url = http://math.ucr.edu/home/baez/physics/General/jerk.html

| publisher = Math Dept., University of California, Riverside

| work = Usenet Physics and Relativity FAQ

| date = November 1998

| access-date = 2015-10-24

}}

{{cite web | url = https://info.aiaa.org/Regions/Western/Orange_County/Newsletters/Presentations%20Posted%20by%20Enrique%20P.%20Castro/AIAAOC_SnapCracklePop_docx.pdf |archive-url=https://web.archive.org/web/20180626030437/https://info.aiaa.org/Regions/Western/Orange_County/Newsletters/Presentations%20Posted%20by%20Enrique%20P.%20Castro/AIAAOC_SnapCracklePop_docx.pdf |archive-date=26 June 2018 |url-status=unfit | title = Snap, Crackle, and Pop | last = Thompson | first = Peter M. | date = 5 May 2011 | website = AIAA Info | publisher = Systems Technology | location = Hawthorne, California | page = 1 | access-date = 3 March 2017 | quote = 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.}}

}}

• {{Wiktionaryinline|snap|jounce|crackle|flounce|pop|pounce}}

{{Kinematics}}

Category:Acceleration

Category:Kinematic properties

Category:Time in physics

Category:Vector physical quantities