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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:
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.
References
- ^ a b Eager, David; Pendrill, Ann-Marie; Reistad, Nina (2016-10-13). "Beyond velocity and acceleration: jerk, snap and higher derivatives". European Journal of Physics. 37 (6): 065008. Bibcode:2016EJPh...37f5008E. doi:10.1088/0143-0807/37/6/065008. hdl:10453/56556. ISSN 0143-0807. S2CID 19486813.
- ^ a b c Gragert, Stephanie; Gibbs, Philip (November 1998). "What is the term used for the third derivative of position?". Usenet Physics and Relativity FAQ. Math Dept., University of California, Riverside. Retrieved 2015-10-24.
- ^ "MATLAB Documentation: minsnappolytraj".
- ^ a b c d e f g Visser, Matt (31 March 2004). "Jerk, snap and the cosmological equation of state". Classical and Quantum Gravity. 21 (11): 2603–2616. arXiv:gr-qc/0309109. Bibcode:2004CQGra..21.2603V. doi:10.1088/0264-9381/21/11/006. ISSN 0264-9381. S2CID 250859930.
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.
- ^ a b c d e f Thompson, 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.
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: 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. ISBN 978-1-61284-386-5. S2CID 18169351.
External links
- The dictionary definition of jounce at Wiktionary