Absement
When an object moves, its motion can be described by the integrals of displacement, including absement, absity, abseleration, abserk, etc., as well as the derivatives of displacement, including velocity, acceleration, jerk, jounce, etc.
Common symbols	A
SI unit	metre-second
In SI base units	m·s
Dimension	L T

Classical mechanics
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Integrals and derivatives of displacement, including absement, as well as integrals and derivatives of energy, including actergy. (Janzen et al. 2014)

In kinematics, absement (or absition) is a measure of sustained displacement of an object from its initial position, i.e. a measure of how far away and for how long. The word absement is a portmanteau of the words absence and displacement. Similarly, absition is a portmanteau of the words absence and position.^[1]^[2]

Absement changes as an object remains displaced and stays constant as the object resides at the initial position. It is the first time-integral of the displacement^[3]^[4] (i.e. absement is the area under a displacement vs. time graph), so the displacement is the rate of change (first time-derivative) of the absement. The dimension of absement is length multiplied by time. Its SI unit is meter second (m·s), which corresponds to an object having been displaced by 1 meter for 1 second. This is not to be confused with a meter per second (m/s), a unit of velocity, the time-derivative of position.

For example, opening the gate of a gate valve (of rectangular cross section) by 1 mm for 10 seconds yields the same absement of 10 mm·s as opening it by 5 mm for 2 seconds. The amount of water having flowed through it is linearly proportional to the absement of the gate, so it is also the same in both cases.^[5]

Occurrence in nature

Whenever the rate of change f′ of a quantity f is proportional to the displacement of an object, the quantity f is a linear function of the object's absement. For example, when the fuel flow rate is proportional to the position of the throttle lever, then the total amount of fuel consumed is proportional to the lever's absement.

The first published paper on the topic of absement introduced and motivated it as a way to study flow-based musical instruments, such as the hydraulophone, to model empirical observations of some hydraulophones in which obstruction of a water jet for a longer period of time resulted in a buildup in sound level, as water accumulates in a sounding mechanism (reservoir), up to a certain maximum filling point beyond which the sound level reached a maximum, or fell off (along with a slow decay when a water jet was unblocked).^[1] Absement has also been used to model artificial muscles,^[6] as well as for real muscle interaction in a physical fitness context.^[7] Absement has also been used to model human posture.^[8]

As the displacement can be seen as a mechanical analogue of electric charge, the absement can be seen as a mechanical analogue of the time-integrated charge, a quantity useful for modelling some types of memory elements.^[4]

Applications

In addition to modeling fluid flow and for Lagrangian modeling of electric circuits,^[4] absement is used in physical fitness and kinesiology to model muscle bandwidth, and as a new form of physical fitness training.^[9]^[10] In this context, it gives rise to a new quantity called actergy, which is to energy as energy is to power. Actergy has the same units as action (joule-seconds) but is the time-integral of total energy (time-integral of the Hamiltonian rather than time-integral of the Lagrangian). Just as displacement and its derivatives form kinematics, so do displacement and its integrals form "integral kinematics".^[9]

Fluid flow in a throttle:

A vehicle's distance travelled results from its throttle's absement. The further the throttle has been opened, and the longer it's been open, the more the vehicle's travelled.

Relation to PID controllers

PID controllers are controllers that work on a signal that is proportional to a physical quantity (e.g. displacement, proportional to position) and its integral(s) and derivative(s), thusly defining PID in the context of integrals and derivatives of a position of a control element in the Bratland sense^[11]

depending on the type of sensor inputs, PID controllers can contain gains proportional to position, velocity, acceleration or the time integral of position (absement)…
— Bratland et al.

Example of PID controller (Bratland 2014):

P, position;
I, absement;
D, velocity.

Strain absement

Strain absement is the time-integral of strain, and is used extensively in mechanical systems and memsprings:

a quantity called absement which allows mem-spring models to display hysteretic response in great abundance.^[3]

Anglement

Absement originally arose in situations involving valves and fluid flow, for which the opening of a valve was by a long, T-shaped handle, which actually varied in angle rather than position. The time-integral of angle is called "anglement" and it is approximately equal or proportional to absement for small angles, because the sine of an angle is approximately equal to the angle for small angles.^[12]

Phase space: Absement and momentement

In regards to a conjugate variable for absement, the time-integral of momentum, known as momentement, has been proposed.^[13]^[14]^[15]^[16]

This is consistent with Jeltsema's 2012 treatment with charge and flux as the base units rather than current and voltage.^[17]

References

^ ^a ^b Mann, Steve; Janzen, Ryan; Post, Mark (2006). Hydraulophone design considerations: absement, displacement, and velocity-sensitive music keyboard in which each key is a water jet. MM06: The 14th ACM International Conference on Multimedia. Santa Barbara, CA: Association for Computing Machinery. pp. 519–528. doi:10.1145/1180639.1180751.
^ Amarashiki (2012-11-10). "LOG#053. Derivatives of position". The Spectrum Of Riemannium. Retrieved 2016-03-08.
^ ^a ^b Pei, Jin-Song; Wright, Joseph P.; Todd, Michael D.; Masri, Sami F.; Gay-Balmaz, François (2015). "Understanding memristors and memcapacitors in engineering mechanics applications". Nonlinear Dynamics. 80 (1–2): 457–489. doi:10.1007/s11071-014-1882-3. S2CID 254891059. for example, a new concept and state variable called "absement," the time integral of deformation, emerge
^ ^a ^b ^c Jeltsema, Dimitri (2012). "Memory Elements: A Paradigm Shift in Lagrangian Modeling of Electrical Circuits". IFAC Proceedings Volumes. 45 (2): 445–450. arXiv:1201.1032. doi:10.3182/20120215-3-AT-3016.00078. S2CID 119564676. Although time-integrated charge is a somewhat unusual quantity in circuit theory, it may be considered as the electrical analogue of a mechanical quantity called absement.
^ Maya Burhanpurkar. Absement: Direct Evidence of the Time-Integral of Distance. Canada-Wide Science Fair 2014.
^ ROBUST CONTROL LAW FOR PNEUMATIC ARTIFICIAL MUSCLES, Jonathon E. Slightam and Mark L. Nagurka, Proceedings of the ASME/Bath 2017 Symposium on Fluid Power and Motion Control, FPMC 2017, October 16-19, 2017, Sarasota, USA
^ Effectiveness of Integral Kinesiology Feedback for Fitness-based Games, Steve Mann, Max Lv Hao, Ming-Chang Tsai, Maziar Hafezi, Amin Azad, and Farhad Keramatimoezabad, 2018 IEEE Games, Entertainment, Media Conference (GEM), pages 43-50
^ Postural strategy for mediolateral weight shifting in healthy adult, J Tousignant, C Cherriere, A Pouliot-Laforte, É Auvinet, Gait & Posture, 2018 - Elsevier
^ ^a ^b Janzen, Ryan; Mann, Steve (Oct 2014). "Actergy as a reflex performance metric: Integral-kinematics applications". IEEE: 1–2. doi:10.1109/GEM.2014.7048123. ISBN 978-1-4799-7545-7. ((cite journal)): Cite journal requires |journal= (help)
^ "Integral Kinematics (Time-Integrals of Distance, Energy, etc.) and Integral Kinesiology", by Mann etal, in Proceedings of the IEEE GEM 2014, pp. 270-2.
^ Bratland, Magne; Haugen, Bjørn; Rølvåg, Terje (2014). "Modal analysis of active flexible multibody systems containing PID controllers with non-collocated sensors and actuators". Finite Elements in Analysis and Design. 91: 16–29. doi:10.1016/j.finel.2014.06.011.
^ Integral Kinesiology Feedbackfor Weight and Resistance Training, 2019 15th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS), http://wearcam.org/sitis2019.pdf
^ Mann, S., Janzen, R., Ali, M. A., Scourboutakos, P., & Guleria, N. (2014, October). Integral kinematics (time-integrals of distance, energy, etc.) and integral kinesiology. In Proceedings of the 2014 IEEE Games, Entertainment, Media Conference (GEM), Toronto, ON, Canada (pp. 22-24).
^ Mann, S., Defaz, D., Pierce, C., Lam, D., Stairs, J., Hermandez, J., ... & Mann, C. (2019, June). Keynote-Eye Itself as a Camera: Sensors, Integrity, and Trust. In The 5th ACM Workshop on Wearable Systems and Applications (pp. 1-2).
^ Biolek, Z., Biolek, D., Biolková, V., & Kolka, Z. (2021). Lagrangian and Hamiltonian formalisms for coupled higher-order elements: theory, modeling, simulation. Nonlinear Dynamics, 1-14.
^ Mann, S., Pierce, C., Zheng, B. C., Hernandez, J., Scavuzzo, C., & Mann, C. (2019, November). Integral kinesiology feedback for weight and resistance training. In 2019 15th International Conference on Signal-Image Technology & Internet-Based Systems (SITIS) (pp. 319-326). IEEE.
^ Jeltsema, D. (2012). Memory elements: A paradigm shift in Lagrangian modeling of electrical circuits. IFAC Proceedings Volumes, 45(2), 445-450.

External links

Kinematics

Classical mechanics SI units

Dimensions	1	L	L²	Dimensions	1	$θ$	$θ$ ²
Linear/translational quantities				Angular/rotational quantities
T	time: $t$ s	absement: $A$ m s		T	time: $t$ s
1		distance: $d$ , position: $r$ , $s$ , $x$ , displacement m	area: $A$ m²	1		angle: $θ$ , angular displacement: $θ$ rad	solid angle: $Ω$ rad², sr
T⁻¹	frequency: $f$ s⁻¹, Hz	speed: $v$ , velocity: $v$ m s⁻¹	kinematic viscosity: $ν$ , specific angular momentum: $h$ m² s⁻¹	T⁻¹	frequency: $f$ , rotational speed: $n$ , rotational velocity: $n$ s⁻¹, Hz	angular speed: $ω$ , angular velocity: $ω$ rad s⁻¹
T⁻²		acceleration: $a$ m s⁻²		T⁻²	rotational acceleration s⁻²	angular acceleration: $α$ rad s⁻²
T⁻³		jerk: $j$ m s⁻³		T⁻³		angular jerk: $ζ$ rad s⁻³
M	mass: $m$ kg	weighted position: $M ⟨ x ⟩ = \sum m x$		ML²	moment of inertia: $I$ kg m²
MT⁻¹	Mass flow rate: ${\dot {m))$ kg s⁻¹	momentum: $p$ , impulse: $J$ kg m s⁻¹, N s	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s	ML²T⁻¹		angular momentum: $L$ , angular impulse: $Δ L$ kg m² s⁻¹	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s
MT⁻²		force: $F$ , weight: $F g$ kg m s⁻², N	energy: $E$ , work: $W$ , Lagrangian: $L$ kg m² s⁻², J	ML²T⁻²		torque: $τ$ , moment: $M$ kg m² s⁻², N m	energy: $E$ , work: $W$ , Lagrangian: $L$ kg m² s⁻², J
MT⁻³		yank: $Y$ kg m s⁻³, N s⁻¹	power: $P$ kg m² s⁻³, W	ML²T⁻³		rotatum: $P$ kg m² s⁻³, N m s⁻¹	power: $P$ kg m²s⁻³, W

Kinematics
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Absement Displacement (Distance) Velocity (Speed) Acceleration Jerk Higher derivatives