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 ...
, 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. ...
s of the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
with respect to
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
– 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 i ...
,
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 ...
, 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, implementat ...
.
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".
(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. ...
of the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
with respect to
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
, 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 ...
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 i ...
,
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, sewa ...
, 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
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.
Eu ...
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 commonly denoted similarly.
The dimensions of snap are distance per fourth power of time. In
SI units, 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. ...
of the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
with respect to
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
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, 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. ...
of the
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
with respect to
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
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, 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