In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the fourth, fifth and sixth derivatives of position are defined as
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s of the
position vector with respect to
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
– with the first, second, and third derivatives being
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
,
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
, and
jerk, respectively. Unlike the first three derivatives, the higher-order derivatives are less common,
thus their names are not as standardized, though the concept of a
minimum snap trajectory has been used in
robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
and is implemented in
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
.
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
Rice Krispies (known as Rice Bubbles in Australia and New Zealand) is a breakfast cereal, marketed by Kellogg's in 1927 and released to the public in 1928. Rice Krispies are made of crisped rice (rice and sugar paste that is formed into rice ...
mascots
Snap, Crackle, and Pop
Snap, Crackle and Pop are the cartoon mascots of Rice Krispies, a brand of breakfast cereal marketed by Kellogg's.
History
The gnome characters were originally designed by illustrator Vernon Grant in the early 1930s. The names are onomatopoeia ...
.
These terms are occasionally used, though "sometimes somewhat facetiously".
(snap/jounce)
Snap, or jounce,
is the fourth
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
position vector with respect to
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, or the
rate of change of the
jerk with respect to time.
Equivalently, it is the second derivative of
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
or the third derivative of
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
,
and is defined by any of the following equivalent expressions:
In
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
, the design of
railway tracks
A railway track (British English and International Union of Railways, UIC terminology) or railroad track (American English), also known as permanent way or simply track, is the structure on a Rail transport, railway or railroad consisting of ...
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
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
, 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
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.
E ...
function.
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
) is not to be confused with the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
commonly denoted similarly.
The dimensions of snap are distance per fourth power of time. In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, this is "metres per second to the fourth", m/s
4, m⋅s
−4, or 100
gal per second squared in
CGS units.
The fifth
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
position vector with respect to
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
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:
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. In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, this is m/s
5, and in
CGS units, 100
gal per cubed second.
The sixth
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
position vector with respect to
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
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:
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. In
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, this is m/s
6, and in
CGS units, 100
gal per quartic second.
References
External links
*
{{Kinematics
Acceleration
Kinematic properties
Time in physics
Vector physical quantities