Physics:Fourth, fifth, and sixth derivatives of position

Short description: Higher derivatives of the position vector with respect to time

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 and is implemented in MATLAB.

The fourth derivative is often referred to as snap or jounce. The name "snap" for the fourth derivative led to crackle and pop for the fifth and sixth derivatives respectively, inspired by the Rice Krispies mascots Snap, Crackle, and Pop. These terms are occasionally used, though "sometimes somewhat facetiously".

Fourth derivative (snap/jounce)

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: [math]\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}. }[/math]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: [math]\displaystyle{ \begin{align} \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{align} }[/math]

where

• [math]\displaystyle{ \vec s }[/math] is constant snap,
• [math]\displaystyle{ \vec \jmath_0 }[/math] is initial jerk,
• [math]\displaystyle{ \vec \jmath }[/math] is final jerk,
• [math]\displaystyle{ \vec a_0 }[/math] is initial acceleration,
• [math]\displaystyle{ \vec a }[/math] is final acceleration,
• [math]\displaystyle{ \vec v_0 }[/math] is initial velocity,
• [math]\displaystyle{ \vec v }[/math] is final velocity,
• [math]\displaystyle{ \vec r_0 }[/math] is initial position,
• [math]\displaystyle{ \vec r }[/math] is final position,
• [math]\displaystyle{ t }[/math] is time between initial and final states.

The notation [math]\displaystyle{ \vec s }[/math] (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.

Fifth derivative

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: [math]\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} }[/math]

The following equations are used for constant crackle: [math]\displaystyle{ \begin{align} \vec s &= \vec s_0 + \vec c \,t \\ \vec \jmath &= \vec \jmath_0 + \vec s_0 \,t + \tfrac{1}{2} \vec c \,t^2 \\ \vec a &= \vec a_0 + \vec \jmath_0 \,t + \tfrac{1}{2} \vec s_0 \,t^2 + \tfrac{1}{6} \vec c \,t^3 \\ \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 \\ \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{align} }[/math]

where

• [math]\displaystyle{ \vec c }[/math] : constant crackle,
• [math]\displaystyle{ \vec s_0 }[/math] : initial snap,
• [math]\displaystyle{ \vec s }[/math] : final snap,
• [math]\displaystyle{ \vec \jmath_0 }[/math] : initial jerk,
• [math]\displaystyle{ \vec \jmath }[/math] : final jerk,
• [math]\displaystyle{ \vec a_0 }[/math] : initial acceleration,
• [math]\displaystyle{ \vec a }[/math] : final acceleration,
• [math]\displaystyle{ \vec v_0 }[/math] : initial velocity,
• [math]\displaystyle{ \vec v }[/math] : final velocity,
• [math]\displaystyle{ \vec r_0 }[/math] : initial position,
• [math]\displaystyle{ \vec r }[/math] : final position,
• [math]\displaystyle{ t }[/math] : 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. It is the rate of change of crackle with respect to time. Pop is defined by any of the following equivalent expressions:

[math]\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} }[/math]

The following equations are used for constant pop: [math]\displaystyle{ \begin{align} \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{align} }[/math]

where

• [math]\displaystyle{ \vec p }[/math] : constant pop,
• [math]\displaystyle{ \vec c_0 }[/math] : initial crackle,
• [math]\displaystyle{ \vec c }[/math] : final crackle,
• [math]\displaystyle{ \vec s_0 }[/math] : initial snap,
• [math]\displaystyle{ \vec s }[/math] : final snap,
• [math]\displaystyle{ \vec \jmath_0 }[/math] : initial jerk,
• [math]\displaystyle{ \vec \jmath }[/math] : final jerk,
• [math]\displaystyle{ \vec a_0 }[/math] : initial acceleration,
• [math]\displaystyle{ \vec a }[/math] : final acceleration,
• [math]\displaystyle{ \vec v_0 }[/math] : initial velocity,
• [math]\displaystyle{ \vec v }[/math] : final velocity,
• [math]\displaystyle{ \vec r_0 }[/math] : initial position,
• [math]\displaystyle{ \vec r }[/math] : final position,
• [math]\displaystyle{ t }[/math] : time between initial and final states.

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