Fourth, fifth, and sixth derivatives of position

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]

Time-derivatives of position

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:

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

 

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.^[1]

The following equations are used for constant snap:

$${\displaystyle {\begin{aligned}{\vec {\jmath }}&={\vec {\jmath }}_{0}+{\vec {s}}t,\\{\vec {a}}&={\vec {a}}_{0}+{\vec {\jmath }}_{0}t+{\tfrac {1}{2}}{\vec {s}}t^{2},\\{\vec {v}}&={\vec {v}}_{0}+{\vec {a}}_{0}t+{\tfrac {1}{2}}{\vec {\jmath }}_{0}t^{2}+{\tfrac {1}{6}}{\vec {s}}t^{3},\\{\vec {r}}&={\vec {r}}_{0}+{\vec {v}}_{0}t+{\tfrac {1}{2}}{\vec {a}}_{0}t^{2}+{\tfrac {1}{6}}{\vec {\jmath }}_{0}t^{3}+{\tfrac {1}{24}}{\vec {s}}t^{4},\end{aligned}}}$$

 

where

• ${\displaystyle {\vec {s}}}$  is constant snap,
• ${\displaystyle {\vec {\jmath }}_{0}}$  is initial jerk,
• ${\displaystyle {\vec {\jmath }}}$  is final jerk,
• ${\displaystyle {\vec {a}}_{0}}$  is initial acceleration,
• ${\displaystyle {\vec {a}}}$  is final acceleration,
• ${\displaystyle {\vec {v}}_{0}}$  is initial velocity,
• ${\displaystyle {\vec {v}}}$  is final velocity,
• ${\displaystyle {\vec {r}}_{0}}$  is initial position,
• ${\displaystyle {\vec {r}}}$  is final position,
• ${\displaystyle t}$  is time between initial and final states.

The notation ${\displaystyle {\vec {s}}}$  (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:

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

 

The following equations are used for constant crackle:

$${\displaystyle {\begin{aligned}{\vec {s}}&={\vec {s}}_{0}+{\vec {c}}\,t\\[1ex]{\vec {\jmath }}&={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\tfrac {1}{2}}{\vec {c}}\,t^{2}\\[1ex]{\vec {a}}&={\vec {a}}_{0}+{\vec {\jmath }}_{0}\,t+{\tfrac {1}{2}}{\vec {s}}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {c}}\,t^{3}\\[1ex]{\vec {v}}&={\vec {v}}_{0}+{\vec {a}}_{0}\,t+{\tfrac {1}{2}}{\vec {\jmath }}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {s}}_{0}\,t^{3}+{\tfrac {1}{24}}{\vec {c}}\,t^{4}\\[1ex]{\vec {r}}&={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\tfrac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\tfrac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\tfrac {1}{120}}{\vec {c}}\,t^{5}\end{aligned}}}$$

 

where

• ${\displaystyle {\vec {c}}}$  : constant crackle,
• ${\displaystyle {\vec {s}}_{0}}$  : initial snap,
• ${\displaystyle {\vec {s}}}$  : final snap,
• ${\displaystyle {\vec {\jmath }}_{0}}$  : initial jerk,
• ${\displaystyle {\vec {\jmath }}}$  : final jerk,
• ${\displaystyle {\vec {a}}_{0}}$  : initial acceleration,
• ${\displaystyle {\vec {a}}}$  : final acceleration,
• ${\displaystyle {\vec {v}}_{0}}$  : initial velocity,
• ${\displaystyle {\vec {v}}}$  : final velocity,
• ${\displaystyle {\vec {r}}_{0}}$  : initial position,
• ${\displaystyle {\vec {r}}}$  : final position,
• ${\displaystyle t}$  : 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:

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

 

The following equations are used for constant pop:

$${\displaystyle {\begin{aligned}{\vec {c}}&={\vec {c}}_{0}+{\vec {p}}\,t\\{\vec {s}}&={\vec {s}}_{0}+{\vec {c}}_{0}\,t+{\tfrac {1}{2}}{\vec {p}}\,t^{2}\\{\vec {\jmath }}&={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\tfrac {1}{2}}{\vec {c}}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {p}}\,t^{3}\\{\vec {a}}&={\vec {a}}_{0}+{\vec {\jmath }}_{0}\,t+{\tfrac {1}{2}}{\vec {s}}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {c}}_{0}\,t^{3}+{\tfrac {1}{24}}{\vec {p}}\,t^{4}\\{\vec {v}}&={\vec {v}}_{0}+{\vec {a}}_{0}\,t+{\tfrac {1}{2}}{\vec {\jmath }}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {s}}_{0}\,t^{3}+{\tfrac {1}{24}}{\vec {c}}_{0}\,t^{4}+{\tfrac {1}{120}}{\vec {p}}\,t^{5}\\{\vec {r}}&={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\tfrac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\tfrac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\tfrac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\tfrac {1}{120}}{\vec {c}}_{0}\,t^{5}+{\tfrac {1}{720}}{\vec {p}}\,t^{6}\end{aligned}}}$$

 

where

• ${\displaystyle {\vec {p}}}$  : constant pop,
• ${\displaystyle {\vec {c}}_{0}}$  : initial crackle,
• ${\displaystyle {\vec {c}}}$  : final crackle,
• ${\displaystyle {\vec {s}}_{0}}$  : initial snap,
• ${\displaystyle {\vec {s}}}$  : final snap,
• ${\displaystyle {\vec {\jmath }}_{0}}$  : initial jerk,
• ${\displaystyle {\vec {\jmath }}}$  : final jerk,
• ${\displaystyle {\vec {a}}_{0}}$  : initial acceleration,
• ${\displaystyle {\vec {a}}}$  : final acceleration,
• ${\displaystyle {\vec {v}}_{0}}$  : initial velocity,
• ${\displaystyle {\vec {v}}}$  : final velocity,
• ${\displaystyle {\vec {r}}_{0}}$  : initial position,
• ${\displaystyle {\vec {r}}}$  : final position,
• ${\displaystyle t}$  : time between initial and final states.

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

References

1. 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.
2. 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.
3. "MATLAB Documentation: minsnappolytraj".
4. 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.
5. Thompson, Peter M. (5 May 2011). (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)
6. 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