Fourth, fifth, and sixth derivatives of position

{{Short description|Higher derivatives of the position vector with respect to time}}  thumb|Time-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;<ref name="epr">{{Cite journal |last1=Eager |first1=David |last2=Pendrill |first2=Ann-Marie |last3=Reistad |first3=Nina |date=2016-10-13 |title=Beyond velocity and acceleration: jerk, snap and higher derivatives |journal=European Journal of Physics |language=en |volume=37 |issue=6 |article-number=065008 |doi=10.1088/0143-0807/37/6/065008 |bibcode=2016EJPh...37f5008E |s2cid=19486813 |issn=0143-0807|doi-access=free |hdl=10453/56556 |hdl-access=free }}</ref><ref name="PhysicsFAQ"/> thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics.<ref>{{cite web |title=MATLAB Documentation: minsnappolytraj |url=https://www.mathworks.com/help/robotics/ref/minsnappolytraj.html}}</ref>

The fourth derivative is referred to as '''snap''', leading the fifth and sixth derivatives to be "sometimes somewhat facetiously"<ref name="Visser2004" /> called '''crackle''' and '''pop''', named after the Rice Krispies mascots of the same name.<ref name="Thompson"/><ref name="jazar">{{Cite book |last=Jazar |first=Reza N. |title=Theory of Applied Robotics : Kinematics, Dynamics, and Control |date=2022 |publisher=Springer |isbn=978-3-030-93220-6 |location=Berlin}}</ref> The fourth derivative is also called '''jounce'''.<ref name="Visser2004" />

== 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. The same principle is applied by roller coaster designers, who use smooth transitions in loops and helices to enhance ride comfort.<ref name="epr" />

In mechanical engineering, controlling snap and jerk is important in fields such as automotive design, to prevent camfollowers from jumping off of camshafts, and manufacturing, where rapid acceleration changes in cutting tools can cause premature tool wear and uneven surface finishes.<ref name="epr" /> Minimum-snap and minimum-jerk trajectories are also used in trajectory optimization in robotics. Minimum-snap trajectories for quadrotors can reduce control effort,<ref name="Mellinger2011">{{cite conference |last1=Mellinger |first1=Daniel |last2=Kumar |first2=Vijay |date=2011 |title=Minimum snap trajectory generation and control for quadrotors |conference=2011 IEEE International Conference on Robotics and Automation |location=Shanghai, China |publisher=IEEE |pages=2520–2525 |doi=10.1109/ICRA.2011.5980409 |isbn=978-1-61284-386-5 |s2cid=18169351}}</ref> while minimum-jerk trajectories for robotic manipulators produce predictable motions that improve control performance and facilitate human-robot interaction.

=={{vanchor|Fourth derivative}} (snap/jounce)== Snap,<ref name="Mellinger2011" /> or jounce,<ref name="PhysicsFAQ"/> is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time.<ref name="Visser2004"/> Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions:

<math display="block">\mathbf{s} = \frac{\mathrm{d}\mathbf{j}}{\mathrm{d}t} = \frac{\mathrm{d}^2 \mathbf{a}}{\mathrm{d}t^2} = \frac{\mathrm{d}^3 \mathbf{v}}{\mathrm{d}t^3} = \frac{\mathrm{d}^4 \mathbf{r}}{\mathrm{d}t^4}.</math>The following equations are used for constant snap: <math display="block">\begin{align} \mathbf{j} &= \mathbf{j}_0 + \mathbf{s} t, \\ \mathbf{a} &= \mathbf{a}_0 + \mathbf{j}_0 t + \tfrac{1}{2} \mathbf{s} t^2, \\ \mathbf{v} &= \mathbf{v}_0 + \mathbf{a}_0 t + \tfrac{1}{2} \mathbf{j}_0 t^2 + \tfrac{1}{6} \mathbf{s} t^3, \\ \mathbf{r} &= \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a}_0 t^2 + \tfrac{1}{6} \mathbf{j}_0 t^3 + \tfrac{1}{24} \mathbf{s} t^4, \end{align}</math>

where {{div col}} *<math>\mathbf{s}</math> is constant snap, *<math>\mathbf{j}_0</math> is initial jerk, *<math>\mathbf{j}</math> is final jerk, *<math>\mathbf{a}_0</math> is initial acceleration, *<math>\mathbf{a}</math> is final acceleration, *<math>\mathbf{v}_0</math> is initial velocity, *<math>\mathbf{v}</math> is final velocity, *<math>\mathbf{r}_0</math> is initial position, *<math>\mathbf{r}</math> is final position, *<math>t</math> is time between initial and final states. {{div col end}}

The notation <math>\mathbf{s}</math> (used by Visser<ref name="Visser2004"/>) 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.

=={{vanchor|Fifth derivative}}== The fifth derivative of the position vector with respect to time is sometimes referred to as crackle.<ref name="Thompson"/> It is the rate of change of snap with respect to time.<ref name="Thompson"/><ref name="Visser2004"/> Crackle is defined by any of the following equivalent expressions: <math display="block">\mathbf{c} =\frac {\mathrm{d} \mathbf{s}} {\mathrm{d}t} = \frac {\mathrm{d}^2 \mathbf{j}} {\mathrm{d}t^2} = \frac {\mathrm{d}^3 \mathbf{a}} {\mathrm{d}t^3} = \frac {\mathrm{d}^4 \mathbf{v}} {\mathrm{d}t^4}= \frac {\mathrm{d}^5 \mathbf{r}} {\mathrm{d}t^5}</math>

The following equations are used for constant crackle: <math display="block">\begin{align} \mathbf{s} &= \mathbf{s}_0 + \mathbf{c} t \\[1ex] \mathbf{j} &= \mathbf{j}_0 + \mathbf{s}_0 t + \tfrac{1}{2} \mathbf{c} t^2 \\[1ex] \mathbf{a} &= \mathbf{a}_0 + \mathbf{j}_0 t + \tfrac{1}{2} \mathbf{s}_0 t^2 + \tfrac{1}{6} \mathbf{c} t^3 \\[1ex] \mathbf{v} &= \mathbf{v}_0 + \mathbf{a}_0 t + \tfrac{1}{2} \mathbf{j}_0 t^2 + \tfrac{1}{6} \mathbf{s}_0 t^3 + \tfrac{1}{24} \mathbf{c} t^4 \\[1ex] \mathbf{r} &= \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a}_0 t^2 + \tfrac{1}{6} \mathbf{j}_0 t^3 + \tfrac{1}{24} \mathbf{s}_0 t^4 + \tfrac{1}{120} \mathbf{c} t^5 \end{align}</math>

where {{div col}} *<math>\mathbf{c}</math> : constant crackle, *<math>\mathbf{s}_0</math> : initial snap, *<math>\mathbf{s}</math> : final snap, *<math>\mathbf{j}_0</math> : initial jerk, *<math>\mathbf{j}</math> : final jerk, *<math>\mathbf{a}_0</math> : initial acceleration, *<math>\mathbf{a}</math> : final acceleration, *<math>\mathbf{v}_0</math> : initial velocity, *<math>\mathbf{v}</math> : final velocity, *<math>\mathbf{r}_0</math> : initial position, *<math>\mathbf{r}</math> : final position, *<math>t</math> : time between initial and final states. {{div col end}}

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

=={{vanchor|Sixth derivative}}== The sixth derivative of the position vector with respect to time is sometimes referred to as pop.<ref name="Thompson"/> It is the rate of change of crackle with respect to time.<ref name="Thompson"/><ref name="Visser2004"/> Pop is defined by any of the following equivalent expressions:

<math display="block">\mathbf{p} =\frac {\mathrm{d} \mathbf{c}} {\mathrm{d}t} = \frac {\mathrm{d}^2 \mathbf{s}} {\mathrm{d}t^2} = \frac {\mathrm{d}^3 \mathbf{j}} {\mathrm{d}t^3} = \frac {\mathrm{d}^4 \mathbf{a}} {\mathrm{d}t^4} = \frac {\mathrm{d}^5 \mathbf{v}} {\mathrm{d}t^5} = \frac {\mathrm{d}^6 \mathbf{r}} {\mathrm{d}t^6}</math>

The following equations are used for constant pop: <math display="block">\begin{align} \mathbf{c} &= \mathbf{c}_0 + \mathbf{p} t \\ \mathbf{s} &= \mathbf{s}_0 + \mathbf{c}_0 t + \tfrac{1}{2} \mathbf{p} t^2 \\ \mathbf{j} &= \mathbf{j}_0 + \mathbf{s}_0 t + \tfrac{1}{2} \mathbf{c}_0 t^2 + \tfrac{1}{6} \mathbf{p} t^3 \\ \mathbf{a} &= \mathbf{a}_0 + \mathbf{j}_0 t + \tfrac{1}{2} \mathbf{s}_0 t^2 + \tfrac{1}{6} \mathbf{c}_0 t^3 + \tfrac{1}{24} \mathbf{p} t^4 \\ \mathbf{v} &= \mathbf{v}_0 + \mathbf{a}_0 t + \tfrac{1}{2} \mathbf{j}_0 t^2 + \tfrac{1}{6} \mathbf{s}_0 t^3 + \tfrac{1}{24} \mathbf{c}_0 t^4 + \tfrac{1}{120} \mathbf{p} t^5 \\ \mathbf{r} &= \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a}_0 t^2 + \tfrac{1}{6} \mathbf{j}_0 t^3 + \tfrac{1}{24} \mathbf{s}_0 t^4 + \tfrac{1}{120} \mathbf{c}_0 t^5 + \tfrac{1}{720} \mathbf{p} t^6 \end{align}</math>

where {{div col}} *<math>\mathbf{p}</math> : constant pop, *<math>\mathbf{c}_0</math> : initial crackle, *<math>\mathbf{c}</math> : final crackle, *<math>\mathbf{s}_0</math> : initial snap, *<math>\mathbf{s}</math> : final snap, *<math>\mathbf{j}_0</math> : initial jerk, *<math>\mathbf{j}</math> : final jerk, *<math>\mathbf{a}_0</math> : initial acceleration, *<math>\mathbf{a}</math> : final acceleration, *<math>\mathbf{v}_0</math> : initial velocity, *<math>\mathbf{v}</math> : final velocity, *<math>\mathbf{r}_0</math> : initial position, *<math>\mathbf{r}</math> : final position, *<math>t</math> : time between initial and final states. {{div col end}} The dimensions of pop are [LT−6]. The corresponding SI unit is m/s6.

== References == {{Reflist| <ref name="Visser2004">{{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 }}</ref>

<ref name="PhysicsFAQ">{{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 }}</ref>

<ref name="Thompson">{{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.}}</ref> }}

==External links== *{{Wiktionaryinline|snap|jounce|crackle|flounce|pop|pounce}}

Category:Acceleration Category:Kinematic properties Category:Time in physics Category:Vector physical quantities