Fourth, fifth, and sixth derivatives of position

Time-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; thus their names are not as standardized, though the concept of a minimum snap trajectory has been used in robotics.

The fourth derivative is referred to as snap, leading the fifth and sixth derivatives to be "sometimes somewhat facetiously" called crackle and pop, inspired by the Rice Krispies mascots Snap, Crackle, and Pop. The fourth derivative is also called jounce.

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Snap, 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: $$\overrightarrow{s}=\frac{d\overrightarrow{ȷ}}{dt}=\frac{d^2\overrightarrow{a}}{d{t}^2}=\frac{d^3\overrightarrow{v}}{d{t}^3}=\frac{d^4\overrightarrow{r}}{d{t}^4}.$$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: $$\begin{array}{cc}\overrightarrow{ȷ}& ={\overrightarrow{ȷ}}_0+\overrightarrow{s}t,\\ \overrightarrow{a}& ={\overrightarrow{a}}_0+{\overrightarrow{ȷ}}_{0}t+\frac{1}{2}\overrightarrow{s}{t}^2,\\ \overrightarrow{v}& ={\overrightarrow{v}}_0+{\overrightarrow{a}}_{0}t+\frac{1}{2}{\overrightarrow{ȷ}}_0{t}^2+\frac{1}{6}\overrightarrow{s}{t}^3,\\ \overrightarrow{r}& ={\overrightarrow{r}}_0+{\overrightarrow{v}}_{0}t+\frac{1}{2}{\overrightarrow{a}}_0{t}^2+\frac{1}{6}{\overrightarrow{ȷ}}_0{t}^3+\frac{1}{24}\overrightarrow{s}{t}^4,\end{array}$$

where Template:Div col

• $\overrightarrow{s}$ is constant snap,
• ${\overrightarrow{ȷ}}_0$ is initial jerk,
• $\overrightarrow{ȷ}$ is final jerk,
• ${\overrightarrow{a}}_0$ is initial acceleration,
• $\overrightarrow{a}$ is final acceleration,
• ${\overrightarrow{v}}_0$ is initial velocity,
• $\overrightarrow{v}$ is final velocity,
• ${\overrightarrow{r}}_0$ is initial position,
• $\overrightarrow{r}$ is final position,
• $t$ is time between initial and final states.

The notation $\overrightarrow{s}$ (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.

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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: $$\overrightarrow{c}=\frac{d\overrightarrow{s}}{dt}=\frac{d^2\overrightarrow{ȷ}}{d{t}^2}=\frac{d^3\overrightarrow{a}}{d{t}^3}=\frac{d^4\overrightarrow{v}}{d{t}^4}=\frac{d^5\overrightarrow{r}}{d{t}^5}$$

The following equations are used for constant crackle: $$\begin{array}{cc}\overrightarrow{s}& ={\overrightarrow{s}}_0+\overrightarrow{c}t\\ \overrightarrow{ȷ}& ={\overrightarrow{ȷ}}_0+{\overrightarrow{s}}_{0}t+\frac{1}{2}\overrightarrow{c}{t}^2\\ \overrightarrow{a}& ={\overrightarrow{a}}_0+{\overrightarrow{ȷ}}_{0}t+\frac{1}{2}{\overrightarrow{s}}_0{t}^2+\frac{1}{6}\overrightarrow{c}{t}^3\\ \overrightarrow{v}& ={\overrightarrow{v}}_0+{\overrightarrow{a}}_{0}t+\frac{1}{2}{\overrightarrow{ȷ}}_0{t}^2+\frac{1}{6}{\overrightarrow{s}}_0{t}^3+\frac{1}{24}\overrightarrow{c}{t}^4\\ \overrightarrow{r}& ={\overrightarrow{r}}_0+{\overrightarrow{v}}_{0}t+\frac{1}{2}{\overrightarrow{a}}_0{t}^2+\frac{1}{6}{\overrightarrow{ȷ}}_0{t}^3+\frac{1}{24}{\overrightarrow{s}}_0{t}^4+\frac{1}{120}\overrightarrow{c}{t}^5\end{array}$$

where Template:Div col

• $\overrightarrow{c}$ : constant crackle,
• ${\overrightarrow{s}}_0$ : initial snap,
• $\overrightarrow{s}$ : final snap,
• ${\overrightarrow{ȷ}}_0$ : initial jerk,
• $\overrightarrow{ȷ}$ : final jerk,
• ${\overrightarrow{a}}_0$ : initial acceleration,
• $\overrightarrow{a}$ : final acceleration,
• ${\overrightarrow{v}}_0$ : initial velocity,
• $\overrightarrow{v}$ : final velocity,
• ${\overrightarrow{r}}_0$ : initial position,
• $\overrightarrow{r}$ : final position,
• $t$ : time between initial and final states.

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

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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:

$$\overrightarrow{p}=\frac{d\overrightarrow{c}}{dt}=\frac{d^2\overrightarrow{s}}{d{t}^2}=\frac{d^3\overrightarrow{ȷ}}{d{t}^3}=\frac{d^4\overrightarrow{a}}{d{t}^4}=\frac{d^5\overrightarrow{v}}{d{t}^5}=\frac{d^6\overrightarrow{r}}{d{t}^6}$$

The following equations are used for constant pop: $$\begin{array}{cc}\overrightarrow{c}& ={\overrightarrow{c}}_0+\overrightarrow{p}t\\ \overrightarrow{s}& ={\overrightarrow{s}}_0+{\overrightarrow{c}}_{0}t+\frac{1}{2}\overrightarrow{p}{t}^2\\ \overrightarrow{ȷ}& ={\overrightarrow{ȷ}}_0+{\overrightarrow{s}}_{0}t+\frac{1}{2}{\overrightarrow{c}}_0{t}^2+\frac{1}{6}\overrightarrow{p}{t}^3\\ \overrightarrow{a}& ={\overrightarrow{a}}_0+{\overrightarrow{ȷ}}_{0}t+\frac{1}{2}{\overrightarrow{s}}_0{t}^2+\frac{1}{6}{\overrightarrow{c}}_0{t}^3+\frac{1}{24}\overrightarrow{p}{t}^4\\ \overrightarrow{v}& ={\overrightarrow{v}}_0+{\overrightarrow{a}}_{0}t+\frac{1}{2}{\overrightarrow{ȷ}}_0{t}^2+\frac{1}{6}{\overrightarrow{s}}_0{t}^3+\frac{1}{24}{\overrightarrow{c}}_0{t}^4+\frac{1}{120}\overrightarrow{p}{t}^5\\ \overrightarrow{r}& ={\overrightarrow{r}}_0+{\overrightarrow{v}}_{0}t+\frac{1}{2}{\overrightarrow{a}}_0{t}^2+\frac{1}{6}{\overrightarrow{ȷ}}_0{t}^3+\frac{1}{24}{\overrightarrow{s}}_0{t}^4+\frac{1}{120}{\overrightarrow{c}}_0{t}^5+\frac{1}{720}\overrightarrow{p}{t}^6\end{array}$$

where Template:Div col

• $\overrightarrow{p}$ : constant pop,
• ${\overrightarrow{c}}_0$ : initial crackle,
• $\overrightarrow{c}$ : final crackle,
• ${\overrightarrow{s}}_0$ : initial snap,
• $\overrightarrow{s}$ : final snap,
• ${\overrightarrow{ȷ}}_0$ : initial jerk,
• $\overrightarrow{ȷ}$ : final jerk,
• ${\overrightarrow{a}}_0$ : initial acceleration,
• $\overrightarrow{a}$ : final acceleration,
• ${\overrightarrow{v}}_0$ : initial velocity,
• $\overrightarrow{v}$ : final velocity,
• ${\overrightarrow{r}}_0$ : initial position,
• $\overrightarrow{r}$ : final position,
• $t$ : time between initial and final states.

Template:Div col end The dimensions of pop are LT⁻⁶. The corresponding SI unit is m/s⁶.

External links

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