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Actergy as a Reflex Performance Metric: - Applications Ryan Janzen, Steve Mann Department of Electrical and Computer Engineering, University of Toronto                  Abstract—Actergy, the -integral of , is employed in  $  a human kinetic sensing system. This paper presents hardware  and signal-processing mathematics to measure actergy in a health n "#   j "#    science, training or gaming environment, as a more detailed x  a "#   metric of performance than reflex time-delay. Fundamental quan- v "#   tities of Integral-Kinematics are compared alongside actergy, with x "#   x "#  further potential applications in aerospace. A "# !  B "# $ !   C "# $ ! 

NTRODUCTION  I. I D "# $ !  

We introduce a new type of physical fitness metric for  training, human health science, and interactive gaming.   $  We propose actergy, the time-integral of energy expendi- P "#   ture (unlike , the time- of energy), as a new K "#     metric for reflex tests and reflex-based interactive games.  "#    Rather than previous reflex-measurement methods which use time-delay between stimulus and response as the metric for Fig. 1. The time- of are fundamental kinematic reflex performance [1]–[3], actergy takes into account the quantities that complete the sequence of “kines” of displacement in both directions. Energy can also be time-integrated, leading to actergy. Actergy complete waveform of energy expenditure, not merely two we propose as a generalized analogue to Lagrangian , which itself is data points (start time and end time). Actergy, as the time- specifically an integral of minus of a system. integral of energy, is biased towards rewarding early power Subject(1) Subject(2) expenditure, making it well suited for continuously-varying 150 150 reflex measurements. 100 100 ass

lM 50 50 By processing a continuously-varying signal, actergy ac- rtua 0 0 Dynamic (kg) Dynamic (kg) Vi Virtual counts for different users who may have the same overall reflex −50 −50 0 500 1000 1500 2000 0 500 1000 1500 2000 time-delay, but different waveforms of energy reaction from Time step (ms) Time step (ms) one millisecond to the next. For example, one user may have 3000 3000 s) s) ⋅ Stimulus ⋅ Stimulus 2000 2000 zero reaction at first followed by a large reaction, while another Time Time user may have a weak, broadened curve of power expenditure. 1000 1000 (J = W Work (J = W Physical fitness training often includes: (1) Training over 0 0 0 500 1000 1500 2000 0 500 1000 1500 2000

short bursts for maximum power, as measured in ; (2) ) )

2 2000 Assessment 2 2000 Assessment s s Endurance training over longer periods for maximum energy, ⋅ Time ⋅ Time   s=W s=W as measured in . We propose a third class of physical ⋅ 1000  ⋅ 1000   fitness training, to maximize actergy, creating three types of   energetic fitness: Actergy (J 0 Actergy (J 0 0 500 1000 1500 2000 0 500 1000 1500 2000 (1) Power as measured in watts; Time step (ms) Time step (ms) (2) Energy as measured in · or joules; Fig. 2. Reflex test data from gas-cylinder scale, converted to actergy for (3) Actergy as measured in watt·seconds2 or ·seconds. two subjects, using the conversion formulas derived in this paper. Actergy is a Each of these is the time-integral of the one before it. A metric that favours early energy expenditure, while it also considers all energy physical fitness regimen could focus on all three: Power, data points in a range rather than merely a time delay measurement. Energy, and Actergy, to enhance different types of muscle and metabolic activity in short, medium and long segments of time. measurement into actergy. While actergy is the integral of Our actergy measurement system, which consists of hard- energy, the conversion to actergy on our system is more ware and signal-processing software, therefore is designed both involved than a simple integration. We derive the actergy as a metric for reflex tests and as a platform for rapid-action conversion as follows. gaming to enhance physical fitness. We took an initial sample of DC mass, m0 (static mass with no movement), whose static is F0 = m0g, where g is the II. TRANSFORMING WEIGH-SCALE SIGNALS TO ACTERGY local gravity constant. In a dynamic situation (user jumping on the scale), a different, virtual mass, MV , is measured in real- We used a gas-cylinder weigh scale as an interface device time. The total force FT sensed is the rest-mass force F0, plus (Fig. 3). The scale’s output signal was connected to a National the dynamic -causing force, F : Instruments analog-to-digital converter, for signal processing on a computer where we converted the time-varying “mass” F = FT − F0 =(MV − m0) · g (1)

978-1-4799-7546-4/14/$31.00 ©2014 IEEE Aircraft dynamics is a further potential application of integral-kinematics, through a mathematical model for computer-efficient flight-control systems, as well as for physics education and gaming. We propose that - quantities can relate response of an aircraft to the pilot’s control yoke , as follows. First, we define the following: cz control yoke displacement in the vertical elev. direction θE elevator control surface TP in the pitch axis, caused by elevator θP pitch angle, and ωP angular IP aircraft’s of in the pitch-axis. Z altitude; βKA kinetic-altitude conversion rate pitch factor To a first-order approximation, the following relationships Fig. 3. GCS300 gas cylinder load cell, providing a continuous-time signal hold, and build up to a new conclusion using absement: from standing, jumping, and a variety of . The purpose of the gas is to distribute pressure for equal sensitivity at all points on the top surface. θE ∝ cz for a displacement-sensitive yoke (6) This device which ordinarily measures static , we repurpose to measure dynamic motion, and we convert the signal into actergy (time-integrated TP ∝ θE for small θE, θP , controlled airspeed (7) energy).  1 t 2 1 θP (t)  TP (τ)dτ + θP (t0) under conditions IP t However, the acceleration from force F depends on the rest- 0  (8) mass, not the virtual mass: F = m0a, where a is acceleration t 2 of the body. We then examine the work W done along a pro- Z(t)  βKAθP (τ)dτ + Z(t0) under conditions (9) gression of time. While constant-force work is W = Fx (for t0 displacement x), dynamic work over a slice of displacement δx is: Therefore, to a first-order approximation for this aircraft dx type, the change in altitude is proportional to the third-order δW = F (t)δx = F (t)δt · (2) dt integral of the displacement of the control yoke: Integrating: ΔAltitude ∝ ΔAbseleration of the control yoke (10)     W = F (t)v(t)dt = F (t) a(τ) dτ dt (3) Abseleration is illustrated in Fig. 1. Even though pitch is an angle and not a linear measurement (i.e. absity-analogue of    or, F (τ) yoke ), the final result of altitude returns to a linear W = F (t) dτ dt (4) measurement (i.e. abseleration of yoke position). m0 Applications may include: reducing the computing load Therefore, the actergy is:       of flight-control computers; flight simulation and gaming; and 2 MV (τ) 2 education (teaching ’s laws with reference to aircraft ℵ = Wdt = g (MV (t)−m0) − 1 dτ dt m0 motion, in a simplified first-order model). (5) REFERENCES This simple analysis provided the basis for signal- processing of the electronic signal from the foot pad scale. [1] T. Ooi, R. Whitlock, P. Frengley, and H. Ibbertson, “Systolic time intervals and ankle reflex time in patients with minimal serum tsh One future extension of this analysis is to correct for the elevation: Response to triiodothyronine therapy,” Clinical Endocrinology, fact that the accelerated mass (above the legs) is slightly vol. 13, pp. 621–7, 1980. smaller than the total body mass, for a brief moment when [2] P. M. Clarkson, “The relationship of age and level of physical activity the legs are pushing the body upward but not yet themselves with the fractionated components of patellar reflex time,” Journal of Gerontology, vol. 33, no. 5, pp. 650–6, 1978. lifting off the ground. An anatomical analysis would lead to a [3] J. Viitasalo and P. Komi, “Interrelationships between electromyographic, slightly modified effective m0 for that brief moment in time. mechanical, muscle structure and reflex time measurements in man,” Acta Physiol Scand, vol. 111, pp. 97–103, 1981. III. FURTHER WORK:INTEGRAL-KINEMATICS TO MODEL [4] S. Mann, R. Janzen, and M. Post, “Hydraulophone design considerations: PHYSICAL PROCESSES Absement, displacement, and velocity-sensitive music keyboard in which each key is a jet,” in Proc. ACM International Conference on Absement, the time-integral of displacement, was intro- Multimedia, October 23-27, Santa Barbara, USA., 2006, pp. 519–528. duced in [4] as a new fundamental kinematic quantity, creating [5] D. Jeltsema, “Memory elements: A paradigm shift in lagrangian model- a new unexplored part of the sequence of: displacement, ing of electrical circuits,” in Proc. 7th Vienna Conference on Mathemat- velocity, acceleration, , jounce, ... . The of dis- ical Modelling, Nr. 448, Vienna, Austria, February 15-17, 2012. placement are well-established, but the integrals of displace- ment have not been fully studied as fundamental kinematic 1Assuming a large aircraft whose dominates over pitch restora- quantities. Absement also has applications in electromagnetics tive and turbulence. 2Assuming engines set on auto-throttle to hold airspeed constant; this and circuit theory [5], and, like actergy, may also provide a models conversion of forward-moving kinetic energy to vertical potential more detailed metric for kinetic performance than reflex time- energy, according to a rate in proportion to pitch, for small pitch ; delay. Fig. 1 shows higher-order integrals of displacement. this model requires symmetry for the case of zero rolling and yawing motion.

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