Crackle (physics)

Crackle is the facetious name of a high-order derivative, and more specifically, the fifth derivative of the displacement.^[1] There is little consensus on what to call derivatives past the 4th derivative, jounce, due to there being few well-defined practical applications. The terms are, however, utilized within the fields of robotics and human motion.^[2]

Notation

Crackle is given by the notation:

{\vec {c}}={\frac {d^{5}{\vec {r}}}{dt^{5}}},

meaning crackle is equal to the 5th-order derivative of position vector over time, equal to the vector s.

The following equations are used for constant crackle:

{\vec {s}}={\vec {s}}_{0}+{\vec {c}}\,t

{\vec {\jmath }}={\vec {\jmath }}_{0}+{\vec {s}}_{0}\,t+{\frac {1}{2}}{\vec {c}}\,t^{2}

{\vec {a}}={\vec {a}}_{0}+{\vec {\jmath }}_{0}\,t+{\frac {1}{2}}{\vec {s}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {c}}\,t^{3}

{\vec {v}}={\vec {v}}_{0}+{\vec {a}}_{0}\,t+{\frac {1}{2}}{\vec {\jmath }}_{0}\,t^{2}+{\frac {1}{6}}{\vec {s}}_{0}\,t^{3}+{\frac {1}{24}}{\vec {c}}\,t^{4}

{\vec {r}}={\vec {r}}_{0}+{\vec {v}}_{0}\,t+{\frac {1}{2}}{\vec {a}}_{0}\,t^{2}+{\frac {1}{6}}{\vec {\jmath }}_{0}\,t^{3}+{\frac {1}{24}}{\vec {s}}_{0}\,t^{4}+{\frac {1}{120}}{\vec {c}}\,t^{5}

where

{\vec {c}}

: constant crackle,

{\vec {s}}_{0}

: initial jounce,

{\vec {s}}

: final jounce,

{\vec {j}}_{0}

: initial jerk,

{\vec {j}}

: final jerk,

{\vec {a}}_{0}

: initial acceleration,

{\vec {a}}

: final acceleration,

{\vec {v}}_{0}

: initial velocity,

{\vec {v}}

: final velocity,

{\vec {r}}_{0}

: initial position,

{\vec {r}}

: final position,

t

: time between initial and final states.

References

↑ Uhlik, Christopher Richard (1990). Experiments in high-performance nonlinear and adaptive control of a two-link, flexible-drive-train manipulator. Stanford University. p. 81. Retrieved 8 November 2015. Jerk is the technical term for the third derivative of position- snap, crackle, and pop correspond to the fourth, fifth, and sixth derivatives of position.
↑ Nagengast, Arne; Braun, Daniel; Wolpert, Daniel (26 June 2009). "Optimal Control Predicts Human Performance on Objects with Internal Degrees of Freedom". PLoS Comput Biol. 5 (6). doi:10.1371/journal.pcbi.1000419. Retrieved 9 February 2016.

Kinematics

← Integrate … Differentiate →

Absement Displacement (Distance) Velocity (Speed) Acceleration Jerk Jounce Crackle Pop

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Crackle (physics)

Notation

See also

References