�����������������

Citation: Wieczorek, B.; Kukla, M.;

Warguła, Ł. Describing a Set of Points

with Elliptical Areas: Mathematical

Description and Verification on

Operational Tests of Technical

Devices. Appl. Sci. 2022, 12, 445.

https://doi.org/10.3390/app12010445

Academic Editor: Redha Taiar

Received: 28 September 2021

Accepted: 23 December 2021

Published: 3 January 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

applied sciences

Article

Describing a Set of Points with Elliptical Areas: MathematicalDescription and Verification on Operational Tests ofTechnical DevicesBartosz Wieczorek * , Mateusz Kukla and Łukasz Warguła

Faculty of Mechanical Engineering, Institute of Machine Design, Poznan University of Technology,60-965 Poznan, Poland; mateusz.kukla@put.poznan.pl (M.K.); lukasz.wargula@put.poznan.pl (Ł.W.)* Correspondence: bartosz.wieczorek@put.poznan.pl

Abstract: The purpose of this article was to present an algorithm for creating an ellipse for anydata set represented on a two-dimensional reference frame. The study objective was to verify thedeveloped method on real results of experimental tests with different subject matter. This articlecontains a mathematical algorithm to describe a set of points with elliptical areas. In addition, fourresults of tests with different subject matter are cited, based on which the developed method wasverified. The verification of the method included checking the deviation of the geometric dimensionsof the ellipse, the number of points contained within the ellipse, and the area of the ellipse. Theimplemented research methodology allowed to demonstrate the possibility of using the methodof describing a set of points with elliptical areas, in order to determine quantitative parametersevaluating the results of the test. The presented results show the method’s applicability for theresults obtained in four different operational tests: measurement of the human body’s gravity centerposition for a person propelling a wheelchair, measurement of marker position using motion capturemethods, measurement of particulate emissions when using equipment powered by an internalcombustion engine, and measurement of the muscle activity of the upper limb when propelling ahybrid manual-electric wheelchair. The performed experiments demonstrated that the method allowsto describe about 85% of all measurement points with an ellipse.

Keywords: method of data description; exploitation research; point cloud; research analysis

1. Introduction

Currently developed technical devices are characterized by the complexity of theirfunctional structure. Such a trend can be observed in all types of equipment, ranging frommechanisms [1–3], to rehabilitation equipment [4–6], machines [7,8], vehicles [9,10], andtransport infrastructure [11]. The complex functionality and variety of configurations ofoperating parameters characterizing the innovative technical solutions result in studies oftheir effect on the surrounding environment consisting of several experiments measuringthe same physical quantity. Tests of exhaust emissions into the atmosphere [12,13] and testsof the effects of technical devices on the biomechanics of anthropotechnical systems [14–16]are excellent examples of such tests. In both cases, operating parameters and design featuresthat affect the value of one physical quantity can be controlled during operational tests.This translates into the ability of the device to perform a single function that, depending onthe configuration, generates different values of the studied physical quantities.

When analyzing the results of modern tests, one can notice this is caused by differentfunctions describing the same physical value measured for different configurations andparameters of the tested device. Examples of such tests include measurements of emissionsof various chemical substances into the atmosphere, depending on the engine design, fueltype, or operating conditions [17–19] and tests of human muscular activity during operationof technical equipment with different design features [20,21]. The performance of tests that

Appl. Sci. 2022, 12, 445. https://doi.org/10.3390/app12010445 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 445 2 of 18

depend on numerous test parameters and design features is problematic when analyzingthe results and when drawing conclusions from the tests performed. Furthermore, in sometests, it is necessary to determine the range of variation of the analyzed physical quantity fordifferent device settings. Examples of such tests include energy consumption tests [22,23]during grinding of different materials with a single device, or tests comparing the effect ofdifferent configurations on technical device operation [24]. In the majority of such tests,the experiment result can be presented on two axes: the vertical axis showing the variationof tested physical quantity, and the horizontal axis showing the change of argument ofthe function. The results placed on such a graph are represented by measurement pointsdefined on the vertical axis by the measured physical value, and on the horizontal axisby the value of the measurement sample variable. Because of the possibility of multipleoperational parameters and design features, these points can be grouped by transformingthem into a set of linear functions. In the case of such a result presentation, it is difficultto compare two independent experiments measuring the same physical value for twodifferent devices characterized by an individual set of parameters affecting the way themain function of the device is performed. Because of that, there is a noticeable need for amethod that allows converting a set of points or a set of functions into a single quantitativeparameter or a set of several parameters describing a certain area in which the measuredphysical value changes.

When having two sets of points relating to the same physical quantities, the simplestway to compare them is to determine the area of presence for the points in the samereference frame. According to this, any issue related to assessing the results of operationaltests for technical devices can be reduced to the evaluation of geometric properties of planefigures representing the area of presence for measured points [25]. There are numerousmethods known for describing a set of points with an envelope forming a plane figure.One such method is the alpha shape method [26,27], which is characterized by plotting theperimeter of a plane figure based on all extreme points from the analyzed set. This methoddemonstrates the advantage of possible application in sets of points generating a planefigure with a concave perimeter. The disadvantage of this method is its complex algorithmthat can only be implemented in a separate, complex software used for post-processingof the data measured in the study experiment. Another disadvantage of the method is itsinability to generate a function defining the perimeter of the determined plane figure. Theconvex hull method [28,29] and the Delaunay triangulation method [30,31] are similar tothe alpha shape method. These methods also have the same disadvantages as the alphashape method. Moreover, the convex hall method makes it impossible to represent planefigures with a concave perimeter. The main disadvantage of these three methods is definingthe delimited region of the plane figure by a set of vertex points. The irregularity of theperimeter plotted on the basis of these points makes it impossible to correctly approximatethe perimeter of the figure determined by a mathematical function.

In the case of operational tests aimed at comparing two devices or two configurationsof a single device, it is important that the geometric areas replacing the analyzed setsof measurement points are expressed by a mathematical function. Such an assumptionresults in the possibility of fitting such an area to various sets of points, and mathematicaloperations performed on it require less complicated mathematical methods. Havingconsidered the foregoing, it is possible to develop an algorithm to describe any two-dimensional data set with a plane figure as an ellipse. The choice of the ellipse as theboundary describing the set of points results from the possibility of its description by meansof two parametric equations, and its ease of implementation in the measurement resultscharacterized by a normal distribution. The main objective of this paper was to presentan algorithm for creating an ellipse for any data set represented on a two-dimensionalreference frame. Verifying the developed method on real results of experimental tests withdifferent subject matter was an additional objective of the study.

Appl. Sci. 2022, 12, 445 3 of 18

2. Materials and Methods

The method of describing a set of n points with elliptical areas allows us to replace anynumber of points with an ellipse, representing the area of their occurrence in a single plane(Figure 1). The described method should be applied to sets of points in which the numberof radius vectors of the points Ri (1) larger than the mean length of the radius vector R (2)is close to the number of vectors smaller than the mean length of the radius vector. Thisdependence can be verified by determining the value of the distribution uniformity ratio∆P (3) of the analyzed points Pi (xi; yi) with respect to the geometric center of gravity of theanalyzed set P(x; y) (4, 5). For the method described, the value of ∆P should be close to 0.5.

Ri =√

x2i + y2

i (1)

R =∑n

i=1 Ri

n(2)

∆P =nmin

n(3)

x =∑n

i=1 xi

n(4)

y =∑n

i=1 yi

n(5)

Figure 1. Schematic illustration of the method replacing an arbitrary set of points (a) with an ellipsedefining the area of presence for points on the analyzed plane (b).

Applying the method on a set of points satisfying the properties discussed aboveresults in replacing it with an ellipse defined by five parameters for the position of thecenter of the ellipse xCOG and yCOG; the angle of inclination of the directional line α; thelength of the semiaxis a, parallel to the directional line; and the length of the semiaxis b,perpendicular to the directional line.

2.1. Mathematical Description of the Method

The algorithm of the method (Figure 2) requires providing it with a set of Cartesiancoordinates describing the position of the analyzed points Pi (xi; yi) from the analyzed setof points (E1). The provided set is used to determine the linear trend function y(x) (6) first,with the inclination angle of the trendline α (7) (E2) being determined subsequently basedon it. The inclination angle of the trendline is the target ellipse inclination angle, whichis the angle between the ellipse’s longer semiaxis and the horizontal axis of the referenceframe.

y(x) =

(∑n

i=1(xi − x)(yi − y)

∑ni=1(xi − x)2

)x +

(y−

(∑n

i=1(xi − x)(yi − y)

∑ni=1(xi − x)2

)x

)(6)

Appl. Sci. 2022, 12, 445 4 of 18

α = tan−1

(∑n

i=1(xi − x)(yi − y)

∑ni=1(xi − x)2

)(7)

where y(x)—the value of the linear function of the trend, n—the number of points in theanalyzed set, xi—the coordinate of the position on the horizontal axis of any point from theset, x—the mean coordinate of the position on the horizontal axis from the coordinates ofthe analyzed set of points, yi—the coordinate of the position on the vertical axis of any pointfrom the set, y—the mean coordinate of the position on the vertical axis for the coordinatesof the analyzed set of points, x—any coordinate on the horizontal axis, and α—the angle ofinclination of the trendline with respect to the horizontal axis.

Figure 2. Schematic of the algorithm allowing to describe a set of points with an ellipse.

In the next step (E3), the position of the ellipse center on the horizontal xCOG (8) andvertical yCOG (9) axes is calculated. These coordinates correspond to the position of thecenter of the plotted COG ellipse.

xCOG =∑n

i=1 xi

n(8)

yCOG =∑n

i=1 yi

n(9)

where xCOG—mean position on the horizontal axis of the analyzed set of points, yCOG—mean position on the vertical axis of the analyzed set of points, xi—position coordinate onthe horizontal axis of any point from the set, yi—position coordinate on the vertical axis ofany point from the set, and n—the number of points in the analyzed set.

The next step of the algorithm (E4) requires rotating the analyzed points by thenegative value of the angle of inclination of the ellipse. This step is only a mathematicaltransformation necessary to calculate the dimensions of the ellipse and it does not affect theshape or position of the area approximating the set of points in the next steps. The rotation

Appl. Sci. 2022, 12, 445 5 of 18

should result in a set of points with their trendline parallel to the horizontal axis. Therotation of the analyzed points is performed using the rotation matrix R(−α) (10), which ismultiplied by all the analyzed points Pi, resulting in points P′i (11) with coordinates rotatedby angle α.

R(−α) =

[cos(−α) − sin(−α)sin(−α) cos(−α)

](10)[

cos(−α) − sin(−α)sin(−α) cos(−α)

][xiyi

]=

[xi cos(−α)− yi sin(−α)xi sin(−α) + yi cos(−α)

]= P′i (11)

where R (α)—rotation matrix, α—the angle of inclination of the ellipse, xi—position coordi-nate on the horizontal axis of any point from the set, yi—position coordinate on the verticalaxis of any point from the set, and P′i —points after rotation by negative value by the angleof inclination of the ellipse α.

Knowing the horizontal and vertical coordinates for the rotated points P′i is necessaryto determine the dimensions of the semiaxis a and the semiaxis b of the ellipse (E5). In orderto determine the dimensions of the semiaxis of the ellipse, the method uses the standarddeviation of the horizontal x′i and vertical y′i coordinates of the analyzed set of points Pi.Depending on the type of distribution, this approach allows to control the percentage ofpoints inside the ellipse by controlling a multiple of the standard deviation. For a set ofpoints with a normal distribution, the dimension of the semiaxis a (12) corresponds to twicethe standard deviation of the horizontal coordinate 2σx and the dimension of the semiaxisb (13) corresponds to twice the standard deviation of the vertical coordinate 2σy. Assuminga double standard deviation for normal distributions results in 95.4% of all analyzed Pipoints being inside the plotted ellipse. In order to implement the method for non-normalpoint distributions, it is advisable to express the lengths of the semiaxes a and b as theproduct of the conformity coefficient $ of the semiaxis dimensions to the point distributionand the standard deviation (14, 15). In this case, changes in the coefficient ρ translate intothe percentage of measurement points that will be included in the plotted ellipse, and thusinto the degree of similarity in the representation of the space of presence for the analyzedpoints Pi.

a = 2σx = 2

√√√√∑ni=1

(x′i − x′

)2

n(12)

b = 2σy = 2

√√√√∑ni=1

(y′i − y′

)2

n(13)

a = ρσx = ρ

√√√√∑ni=1

(x′i − x′

)2

n(14)

b = ρσy = ρ

√√√√∑ni=1

(y′i − y′

)2

n(15)

where a—the longer semiaxis of the ellipse parallel to the trendline of the analyzed setof points Pi, b—the shorter semiaxis of the ellipse perpendicular to the trendline of theanalyzed set of points Pi, σx—standard deviation of horizontal coordinates of the set ofinverted points P′i , σy—standard deviation of vertical coordinates of inverted set of pointsP′i , x′i—horizontal coordinate of any point from set of inverted points P′i , y′i—vertical coor-dinate of any point from the set of inverted points P′i , x′—mean of horizontal coordinatesof the inverted set of points P′i , y′—mean of vertical coordinates of the inverted set ofpoints P′i , and ρ—conformity coefficient for fitting the semiaxis dimensions to the analyzeddistribution.

Appl. Sci. 2022, 12, 445 6 of 18

The last step of the presented algorithm is to plot the ellipse (E6). In order to dothis, it is necessary to follow the previous steps, which will provide information about theparameters describing the ellipse. These parameters include the angle of inclination ofthe directional line of the ellipse α (E2), the coordinates of the position of the center of theellipse xCOG and yCOG (E3), the dimension of the semiaxis a (E5), and the dimension of thesemiaxis b (E5). In the adopted method, the parametric Equation (16) with parameter t∈〈0; 2π〉 was used to determine the coordinates of the points Ei located on the perimeter ofthe ellipse.

Ei =

[xE

iyE

i

]=

[a cos(t)b sin(t)

](16)

where Ei—point located on the perimeter of the ellipse, xiE—horizontal position of the

ellipse’s envelope point, yiE—horizontal position of the ellipse’s envelope point, a—length

of the ellipse’s long axis, b—length of the ellipse’s short axis, and t—parameter of theparametric equation of the ellipse.

Based on the known values of horizontal coordinates xEi and vertical coordinates

yEi of the ellipse, taking into account the dimensions of the semiaxes a and b, one should

then rotate the ellipse by the angle of inclination of the directional line α and its shiftby the coordinates of the position of the center of the ellipse xCOG and yCOG (17). Thesemathematical operations result in the calculation of coordinates for the perimeter of theellipse E′i rotated in accordance with the angle of inclination of the trendline of analyzedpoints, and shifted relative to the beginning of the coordinate system, so that it covers 95.4%of all measured points.

E′i = R(α)Ei + COG =

[cos(α) − sin(α)sin(α) cos(α)

][a cos(t)b sin(t)

]+

[xCOGyCOG

]=

[[a cos(t) cos(α)− b sin(t) sin(α)] + xCOG[a cos(t) sin(α) + b sin(t) cos(α)] + yCOG

](17)

where E′i—point located on the perimeter of the ellipse by the angle α, Ei—point locatedon the perimeter of the ellipse, R(α)—rotation matrix, α—the angle of inclination of theellipse, COG—midpoint of the ellipse and the set of analyzed points Pi, a—length of thelonger semiaxis of the ellipse, b—length of the shorter semiaxis of the ellipse, t—parameterof the parametric equation of the ellipse, xCOG—mean position on the horizontal axis of theanalyzed set of points, and yCOG—mean position on the vertical axis of the analyzed setof points.

2.2. Procedures for the Verification of the Method

The developed method of describing sets of points with elliptical areas was verifiedusing real measurement results from four different tests. Using these data, the effect of thenumber of measured points for the same measurement sample on the dimensions of theellipse was checked, as well as the effect of the value of the conformity coefficient $ forthe distribution of points on the percentage of points contained in the plotted ellipse. Inaddition, the practical application of the method in operational testing of mechanical equip-ment was evaluated based on two independent tests with different subject matter. Datafrom a measurement of the gravity center position for the human body while propellinga wheelchair (SAMPE 1), a motion capture measurement of a point moving on a circle ofconstant radius (SAMPLE 2), a ground-based measurement of PM 2.5 emissions from aninternal combustion engine designed for low-power wood chipping machines (SAMPLE3), and a measurement of muscle activity resulting from propelling a hybrid wheelchair(SAMPLE 4) were used for verification.

The methodology for verifying the algorithm provided that, for each data set analyzed,the dimensions of the semiaxes of the ellipse a and b, the angle of inclination of the ellipseα, the horizontal coordinate xCOG and vertical coordinate yCOG of the location of the centerof the ellipse, and the distribution uniformity ratio ∆P were determined. Additionally, inorder to assess the correctness of mapping a set of points with an ellipse, a set of criteriaindividually selected to the type of test providing the measurement points was checked.

Appl. Sci. 2022, 12, 445 7 of 18

The classification of the determined parameters and evaluated criteria assigned to eachdataset is provided in Table 1.

Table 1. Summary of measured parameters and evaluated criteria for analyzed data sets, where a—length of the longer semiaxis of the ellipse, b—length of the shorter semiaxis of the ellipse, α—angleof inclination of the directional line of the ellipse, xCOG—mean position on the horizontal axis ofthe analyzed set of points, yCOG—mean position on the vertical axis of the analyzed set of points,∆P—distribution uniformity ratio, k—percentage of analyzed points inside the area plotted by theellipse ∆R—difference between the standard radius of curvature and the mean radius of curvature ofthe ellipse, A—area of the plotted ellipse σa—standard deviation of the semiaxis a, and σb—standarddeviation of the semiaxis b.

Dataset Distribution TypeDetermined Parameters Assessment Criteria

a b α xCOG yCOG ∆P k ∆R A

SAMPLE 1 normal distribution + + + + + + + – +SAMPLE 2 uniform distribution + + + + + + + + –SAMPLE 3 normal distribution + + + + + + + – +SAMPLE 4 normal distribution + + + + + + + – –

(+)—element taken into account in the analysis, (–)—item not included in the analysis.

The evaluation of the effect of different sampling frequencies for a single measurementsample on the geometric dimensions of the determined ellipse was carried out on theexample of the variation measurement for the position of the human body’s center ofgravity when propelling a wheelchair [32–35] (SAMPLE 1). The data were obtained from themeasuring stand (Figure 3), which consisted of weighing scales enabling the determinationof coordinates for the center of gravity on a plane parallel to the surface on which thestand was located. The weighing scales were equipped with traction rollers, allowingmeasurement of the human body’s center of gravity in dynamic conditions correspondingto the propulsion of a wheelchair [36].

Figure 3. Test stand for the implementation of biomechanical tests with the specified elements formeasuring the position of the center of gravity (A) and the test stand during the test (B), where 1—weighing pan, 2—strain gauges supporting the test stand, 3—linear guides stabilizing the weighingpan, and 4—rollers that connect the wheel of the wheelchair with the weighing scales.

The set of analyzed points contained coordinates for the position of the human body’scenter of gravity measured in the Cartesian system located in the horizontal plane. Theanalyzed points were the result of the wheelchair user performing five full propulsion

Appl. Sci. 2022, 12, 445 8 of 18

cycles. However, because of the type of evaluation performed, this measurement samplewas recorded with eight different frequencies f, taking the following values: 25 Hz, 29 Hz,33 Hz, 40 Hz, 50 Hz, 67 Hz, 100 Hz, and 200 Hz. The variation in sampling frequencyduring the test translated into the number of points provided to the algorithm of the verifiedmethod.

The effect of the conformity coefficient value of the semiaxis dimensions for thedistribution of points $ on the percentage of points contained inside the plotted ellipse wasevaluated with the example of data obtained from the motion capture measurement ofthe point ID1 position in space (SAMPLE 2), moving with constant speed v on a circle ofconstant radius d [37].

The test station (Figure 4) used to generate the data consisting of a measuring wheelwith a fixed reference diameter d of 520 mm. The ID1 marker was placed on this wheelmoving with a constant velocity v of 0.2 m/s2. The position data collected were thecoordinates of the ID1 marker’s position relative to the ID0 marker. These data wereobtained after processing the image recorded by a camera capturing video at 240 fps, withoriginal software.

Figure 4. Schematic of the test station (A,B) for ArUco markers (C) used for motion capture tests,where 1—reference wheel; 2—motor setting the reference wheel in rotation; 3—camera recordingthe image; xc, yc, zc—coordinates of the camera position relative to the stationary marker ID0; andv—rotational speed of the movable marker ID1.

An application example of the method for describing a set of points with elliptical areasfor real operational tests of technical devices is presented using the example of measuringPM 2.5 particulate matter concentration (SAMPLE 3) and measuring the muscular activityof upper limb muscles (SAMPLE 4). The concentration of PM 2.5 particulate matterwas measured in a city park, simulating the performance of greenery maintenance usingequipment driven by an internal combustion engine (Figure 5). The test was performed ina circular working area (WA) with a 17 m diameter. In the center of this area, there was aLifan internal combustion engine, with a power of 10 KW, 390 cm3 capacity, powered bygasoline, which was a particulate matter emitter [38]. Measurement of PM 2.5 concentrationwas performed in different points of WA area using an ATMON FL sensor.

Muscle activity (SAMPLE 4) was measured on a wheelchair dynamometer (Figure 3),where a manual-hybrid wheelchair [39] was tested (Figure 6). In these tests, a wheelchairuser propelled the wheelchair on a dynamometer that generated three different momentsof resistance to motion: 8.14 Nm, 9.67 Nm, and 11.19 Nm. The procedure for propelling thewheelchair was to push off the push rings using the upper limb. During this propulsivemovement, an electric motor built into the hub of the drive wheel additionally supportedthe propulsive force generated by the upper limb. During the performed measurementtests, the electric motor assist gain factor was 0%, 25%, 30%, 40%, 50%, 60%, 70%, and75%. The value of maximal muscular activity MVCmax expressed in percentage was

Appl. Sci. 2022, 12, 445 9 of 18

determined for the parameters changed as such. Muscle activity was determined by surfaceelectromyography using a Noraxon Mini DTS device.

Figure 5. Photograph of the used engine and PM 2.5 sensor (A) and a view of the test area (B), where1—engine, 2—particle sensor, and WA—area of the analysis.

Figure 6. A hybrid cart used in muscle activity tests, where 1—drive wheel, 2—push rings, and3—electric motor built into the hub of the drive wheel.

3. Results

One of the most important factors affecting the quality of the presented tests is thesampling frequency. This is particularly noticeable when measuring physical quantitiescharacterized by large changes in a short time. The tests in which the sampling frequency af-fects the final result of the analysis include the measurement of speed [40,41], measurementof force in dynamic conditions [42,43], and measurement of the change in position [44,45].The results of the experiment checking the effect of sampling frequency on changes indimensions of the ellipses are presented using the example of a set of points illustrating thevariability of the human body’s gravity center position after performing five complete pro-pelling cycles [46] (Figure 7) on a manual wheelchair. It should be noted that the presentedresults refer to the same measurement sample recorded simultaneously with eight differentsampling frequencies.

Changes in parameters describing the shape of the ellipse, the percentage of pointsdescribed by the ellipse, the area and standard deviations in the dimensions of the semiaxesa and b were checked by evaluating the effect of sampling frequency (Table 2).

Appl. Sci. 2022, 12, 445 10 of 18

Figure 7. Plots showing variation in the human body’s gravity center during manual wheelchairpropulsion measured with different sampling frequencies f, with marked ellipses approximating thearea of presence for the set of the analyzed points.

Table 2. List of ellipse parameters and assessed criteria for one measurement test of the gravity centerposition made with different frequencies, where f —sampling frequency, n—number of points in theset of analyzed points, a—semiaxis of the ellipse parallel to the direction line, b—semiaxis of theellipse perpendicular to direction line, α—angle of inclination of the direction line, xCOG—positionof the center of the ellipse on the horizontal axis, yCOG—position of the center of the ellipse on thevertical axis, ∆P—uniformity ratio of points distribution in relation to the mean value of the radiusvector for the points of the analyzed set, k—percentage of points located inside the area defined bythe ellipse, A—ellipse area, µ—mean, and σ—standard deviation.

f n a b α xCOG yCOG ∆P k A

Hz - mm mm ◦ mm mm - % mm2

25 230 61.46 16.00 −6.80 117.35 290.60 0.58 83.9 3066.1929 263 61.41 15.74 −6.75 117.34 290.63 0.58 83.3 2874.5633 306 61.43 16.05 −6.75 117.30 290.62 0.58 84.0 3066.1940 367 61.46 16.09 −6.78 117.29 290.56 0.57 84.5 3066.1950 459 61.40 15.86 −6.81 117.28 290.58 0.57 84.3 2874.5667 612 61.39 15.86 −6.83 117.27 290.56 0.58 84.0 2874.56

100 918 61.37 15.84 −6.84 117.26 290.56 0,58 84,5 2874,56200 1835 61.37 15.83 −6.84 117.24 290.55 0.61 84.1 2874.56

µ - 61.41 15.86 −6.80 117.29 290.58 0.58 84.08 2946.42σ - 0.04 0.08 0.04 0.04 0.03 0.01 0.39 99.18

Appl. Sci. 2022, 12, 445 11 of 18

The performed operational tests do not always end with measuring a set of measure-ment points characterized by normal distribution. Motion capture tests are such a test type,in which the analyzed markers and their spatial position depend on the testing scenario,and in their case, the distribution of measured points is random. Measurement of humanbody segment kinematics can be taken as an example of such tests [47,48]. The effect of theconformity coefficient ρ value of the ellipse semiaxis dimensions for the set of points withdistribution other than normal was performed with the example of data obtained duringthe motion capture measurement of a marker moving with a constant speed v = 0.2 m/s2 ona circle with a constant radius of curvature d = 520 mm. The data provided to the algorithmof the evaluated method were horizontal and vertical coordinates of the marker measuredrelative to a single stationary point (Figure 8). The performed evaluation included criteriato check the difference between the radius of the reference circle and the radius determinedby the plotted ellipse ∆R (18), as well as the percentage of points included inside the areaplotted by the ellipse k (Table 3).

∆R =

∣∣∣∣ a + b2− d

2

∣∣∣∣ (18)

where ∆R—difference between the reference radius of curvature and the mean radius ofcurvature of the ellipse, a—semiaxis of the target ellipse parallel to the trendline of theanalyzed set of points, b—semiaxis of the target ellipse perpendicular to the trendline of theanalyzed set of points, and d—diameter of the reference circle on which the ArUco markerwas placed.

Figure 8. Graphs of the motion capture measurement of a point moving on a constant-radius circlewith ellipses plotted taking into account different values of the conformity coefficient $ for thedimensions of the ellipse semiaxes.

Appl. Sci. 2022, 12, 445 12 of 18

Table 3. Summary of the parameters of ellipses plotted with different values of the ellipse semiaxisdimension conformity coefficient ρ and the evaluated criteria for a single motion capture measurementsample, where ρ—the conformity coefficient of the ellipse’s semiaxis dimensions, a—semiaxis ofthe ellipse parallel to the directional line, b—semiaxis of the ellipse perpendicular to the directionalline, α—angle of inclination of the directional line, xCOG—location of the ellipse’s center on thehorizontal axis, yCOG—location of the ellipse’s center on the vertical axis, ∆P—uniformity ratio of thedistribution of points with respect to the mean value of the radius vector, k—percentage of pointsinside the area plotted by the ellipse, and ∆R—difference between the reference radius of curvatureand the mean radius of curvature of the ellipse.

$ a b α xCOG yCOG ∆P k ∆R

- mm mm ◦ mm mm - % mm1.35 251.37 254.89 −0.28 −0.45 −6.39 0.64 0.2 6.871.4 260.67 264.33 −0.28 −0.45 −6.39 0.64 33.5 2.501.45 269.99 273.77 −0.28 −0.45 −6.39 0.64 89.8 11.881.5 279.30 283.21 −0.28 −0.45 −6.39 0.64 96.8 21.261.55 288.61 291.65 −0.28 −0.45 −6.39 0.64 99.9 30.131.6 297.92 302.09 −0.28 −0.45 −6.39 0.64 100.0 40.01

µ 279.30 283.01 - - - - - 17.46σ 14.72 14.77 - - - - - 16.48

An example of an application of the method in real operational tests is presented usingthe example of PM 2.5 particulate matter emission measurement. The data used to presentthe method consisted of a set of coordinates for the sensor’s position in the horizontalplane, with assigned particulate matter concentration values. The applied test verified howperforming maintenance work using tools with an internal combustion engine affects thevalue of air pollutant emissions. Based on such a data set, elliptical areas were plotted, withone of the five adopted ranges of PM 2.5 concentration measured in them (Figure 9).

Figure 9. Graph of areas of different concentrations of PM 2.5 solid particles during maintenancework in the green area with the use of tools powered by an internal combustion engine.

Using the method of describing a set of points with elliptical areas, it was possibleto replace the points with areas of known centers xCOG and yCOG, semiaxis dimensions aand b of the ellipse, angle of inclination α of the directional line of the ellipse, and totalarea. The fact that sorting of points taken into account when plotting the ellipse was used

Appl. Sci. 2022, 12, 445 13 of 18

is important in the implementation of the method. The criterion according to which sortingwas performed was the concentration of PM 2.5 particulate matter (Table 4).

Table 4. List of parameters describing dimensions of areas with different values of PM 2.5 particulatematter concentration, where a—semiaxis of the ellipse parallel to the direction line, b—semiaxis of theellipse perpendicular to direction line, α—angle of inclination of the direction line, xCOG—positionof the center of the ellipse on the horizontal axis, yCOG—position of the center of the ellipse on thevertical axis, ∆P—uniformity ratio of points’ distribution in relation to the mean value of the radiusvector for the points of the analyzed set, k—percentage of points located inside the area defined bythe ellipse, and A—area of the plotted ellipse.

PM2.5 a b α xCOG yCOG ∆P k A

µg/m3 m m ◦ m m - % m<23 8.43 8.33 7.90 8.71 9.05 0.61 82.7 220.50<20 8.26 8.26 6.17 8.79 8.97 0.58 82.6 214.21<17 8.69 8.55 18.00 9.21 8.83 0.59 88.2 233.63<14 6.84 4.05 19.10 4.79 2.01 0.79 79.3 87.10<11 2.44 0.50 −2.94 3.01 0.77 0.31 87.5 3.87

µ 6.56 3.83 10.47 6.45 5.15 0.57 84.4 134.70σ 2.86 3.83 10.47 3.04 4.37 0.20 4.2 108.77

Biochemical tests are another example for the applicability of the method of describinga set of points with elliptical areas. The adopted tests measured the effect of wheelchairresistance moment values on the muscle activity resulting from propelling the wheelchair.The tests involved a hybrid wheelchair that could assist the user with electric motors. Thesemotors could have different settings for the assistance gain factor w. This complexity ofvariables meant that there were several curves on a single graph representing changes inmuscle activity values for different gain factor settings. Using the described method, allthese curves were included in a single area, which defines the possible physical activity fora given range of resistance moment values and any values of the assistance gain factor w(Figure 10).

Figure 10. Graph showing muscle activity values during propulsion of a manual wheelchair depen-dent on the value of the moment of resistance to motion for different assistance gain factor values wfrom the electric motor assistance system.

4. Discussion

The presented description of the method of describing a set of points with ellipticalareas can be implemented to describe the results of operational tests in which the test

Appl. Sci. 2022, 12, 445 14 of 18

results can be presented on two-dimensional graphs. The axes of these graphs may presentany units, which is confirmed by the provided examples of method applicability. Theapplication of the method is shown on data obtained during tests measuring the emissionsof the internal combustion engine and the muscular activity of the human–wheelchairanthropotechnical system during the propulsion of the push drive. When implementingthe method for individual measurement results, one should note that the values presentedon the horizontal and vertical axes are of the same order of magnitude because, otherwise,inaccuracies may arise in mapping the set of points with an ellipse. This was noted inthe applied example of the muscle activity tests. In this case, the original muscle activityvalues were expressed as a percentage; therefore, the measured effort ranged from 0 to 1.However, assuming such a size interval on the vertical axis resulted in excessive elongationof the semiaxis a. Because of that, the muscle activity value MVCmax was converted intoa percentage value ranging from 0 to 100%. A better ellipse-shaped representation ofthe area of the measured points was obtained by doing so, and the location of points wasdetermined by the values of the resistance moment on the horizontal axis and by the muscleactivity MVCmax on the vertical axis. The results of muscle activity of disabled wheelchairusers implemented in the verification are characterized by a similar range of findings as inarticles analyzing these results in terms of other aspects [49,50].

When applying the method, the uniformity of distribution for the set of analyzedpoints with respect to the mean value is important. This uniformity is defined using theuniformity ratio ∆P of the distribution of points with respect to the mean value of the radiusvector of the points in the analyzed set. The effect of the ∆P coefficient was particularlynoticeable when analyzing the PM 2.5 particulate matter emission areas. In this test, thebest elliptical representation of a set of points was obtained for sets in which ∆P was closeto 0.5; areas with PM 2.5 concentrations <20 µg/m3 and <17 µg/m3 demonstrate this. Intheir case, the value of ratio ∆P was 0.58 and 0.59. The least accurate mapping of the set ofpoints with ellipse was obtained for a PM 2.5 concentration of <14 µg/m3. For this set ofpoints, the value of the ratio ∆P was 0.79. The presented example of applying the methodon data describing PM 2.5 particulate matter emission has demonstrated the advantageof the described method consisting of simplicity to determine the area occupied by theanalyzed set of points. Using this method, determining the area of presence of the analyzedpoints requires only calculating the area of an ellipse with two known parameters definingthe lengths of the semiaxes a and b. The results obtained during the verification of themethod confirm that the highest emission of air pollutants is closest to the emission source,i.e., the internal combustion engine [51,52].

The experiments carried out to verify the operation of the method checked how thedistribution type and the number of measurement samples affect its applicability. Theexperiment testing the effect of the number of points on the dimensions of the ellipseshowed that, when analyzing a set characterized by a normal distribution, the number ofpoints has little effect on changes in the dimensions of the ellipse. When measuring thevariation of gravity center position for the human body propelling the wheelchair, usingthe set consisting of 230 points, the dimension of semiaxis a was 61.46 mm, while it was16.00 mm for semiaxis b. However, for the same measurement test measured with thefrequency enabling to measure 1835 points, the dimension was 61.37 mm for semiaxis a and15.83 mm for semiaxis b. Having included these results, it is noticeable that the mappingaccuracy of the point presence area increases with the increase in the number of points.Despite that, the obtained small difference in the dimensions of the semiaxes confirmsthe thesis saying that the increase in the number of points of the analyzed set has littleeffect on the final dimensions of the ellipse. Based on that, it was found that the methodis characterized by high flexibility when the analyzed set of points demonstrates variablesize. Additionally, using the example of data from the measurement of the human body’sgravity center position, it was found that, for the distribution of points characterized bythe uniformity ratio ∆P at the level of 0.57–0.61, there is an average of 84.08% of all pointsdelivered to the algorithm of the method included in the ellipse. The data on the center of

Appl. Sci. 2022, 12, 445 15 of 18

gravity position used for verification during wheelchair propulsion were consistent with theresults of other tests measuring and analyzing this parameter in a different aspect [53,54].

An experiment conducted on motion capture data was supposed to demonstrate theeffect of the distribution type on the representation of a set of points by an ellipse. Pointscharacterized by a uniform distribution were used in this data set. In the case of suchdistributions, determining the length of the ellipse semiaxis as a value of double standarddeviation is not very precise. It ensures covering over 95.4% of the points with the areaplotted by the ellipse, but the plotted area is noticeably larger than the area of presencefor the set of analyzed points. In order to also fit this method to uniform distributions, themethod of calculating the semiaxis ellipse dimensions was modified. In this case, they arethe quotient of the conformity coefficient $ value of ellipse semiaxis dimensions and thestandard deviation value. In one variant of the performed verification, the optimum valueof $ was chosen by searching for the smallest difference between the radius of the referencecircle and the mean radius of curvature of the ellipse ∆R. The performed experimentshowed that, for the coefficient ρ = 1.4, the smallest difference between the standard radiusand the mean radius of ellipse curvature was ∆R equal to 2.50 mm. For such a value of thecoefficient ρ, a dimension close to the dimension of the reference wheel diameter d wasdetermined analytically; however, only 33.5% of all measured points were included in theplotted ellipse. This small percentage of points included in the ellipse area is due to thecharacteristics of the performed test. The use of motion capture testing was characterizedby a proportional scatter of the measured marker position relative to a circle definedby a reference diameter d to which the marker was attached. In the case of the secondverification variant, the number of points contained within the plotted ellipse was adoptedas the criterion of optimality. In this case, the best results were obtained for a coefficient$ of 1.60, for which the percentage of points inside the ellipse k was 100%. Nevertheless,the difference between the reference radius and the mean radius of curvature of the ellipse∆R was 40.01 mm in this verification variant. The performed experiment demonstratedthat, by controlling the value of the coefficient ρ, the method can be adjusted to differenttypes of distribution, and it is possible to extract various parameters—important for theperformed description of the experimental test—from the analyzed set of points. Studieshave confirmed the feasibility of analyzing motion capture data over a specific range ofspeeds [55,56]. As part of the work, the method was verified for four different experimentalstudies. On this basis, it was found that the method is the most accurate for the results witha normal distribution. In the case of other distributions, the coefficient ρ should be selectedindependently, which may introduce an additional error.

5. Conclusions

The presented algorithm of the method provides a complete description, enabling itsapplication to the presentation of the experimental results of various physical phenomena.The advantage of the method is describing a set of analyzed points by means of an ellipse,with the possibility to perform various mathematical operations on it. One such operationis calculating the area that, when appropriate units are juxtaposed on the horizontal andvertical axes, can result in the determination of a quantitative coefficient. For example,when measuring the operation of a mechanical system (expressed inwatts) as a functionof time (expressed in seconds), the calculated area of the ellipse will represent the energyrequirement of the device (expressed in joules). The verification tests performed showedthat the method can be successfully applied to sets with any distribution. However, fullautomation of the described algorithm is achieved for normal distributions characterized bythe uniformity ratio of distribution ∆P close to 0.5. In the case of other point distributions,it is necessary to introduce the method of conformity coefficient $ of the ellipse semiaxisdimensions into the algorithm. This coefficient is chosen with individual optimizationcriteria for each test. The criteria adopted should result from the research objective adopted,and the target parameters for the analyzed set of points. There are plans to modify theparametric equations of the ellipse used in the further work on developing this method.

Appl. Sci. 2022, 12, 445 16 of 18

Successful realization of such an assumption will allow to increase the accuracy of themapping with the plotted area of the set of analyzed points.

Author Contributions: Conceptualization, B.W.; methodology, B.W.; software, B.W.; validation, B.W.,M.K., and Ł.W.; formal analysis, B.W.; investigation, B.W.; resources, B.W.; data curation, B.W., M.K.,and Ł.W.; writing—original draft preparation, B.W.; writing—review and editing, B.W.; visualization,B.W.; supervision, B.W.; project administration, B.W.; funding acquisition, B.W. All authors have readand agreed to the published version of the manuscript.

Funding: The test was performed as part of the project LIDER VII “Testing the manual wheelchairpropulsion biomechanics for innovative manual and hybrid propulsions” (LIDER/7/0025/L-7/15/2016)financed by the Polish National Centre for Research and Development.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

References1. Krawiec, P.; Warguła, Ł.; Małoziec, D.; Kaczmarzyk, P.; Dziechciarz, A.; Czarnecka-Komorowska, D. The Toxicological Testing

and Thermal Decomposition of Drive and Transport Belts Made of Thermoplastic Multilayer Polymer Materials. Polymers 2020,12, 2232. [CrossRef] [PubMed]

2. Krawiec, P.; Rózanski, L.; Czarnecka-Komorowska, D.; Warguła, Ł. Evaluation of the thermal stability and surface characteristicsof thermoplastic polyurethane V-belt. Materials 2020, 13, 1502. [CrossRef]

3. Sydor, M. Geometry of wood screws: A patent review. Eur. J. Wood Wood Prod. 2019, 77, 93–103. [CrossRef]4. Rui, J.; Gao, Q. Design and Analysis of A Multifunctional Wheelchair. In IOP Conference Series: Materials Science and Engineering;

IOP Publishing: Bristol, UK, 2019; Volume 538, p. 12045.5. Rebsamen, B.; Guan, C.; Zhang, H.; Wang, C.; Teo, C.; Ang, M.H.; Burdet, E. A brain controlled wheelchair to navigate in familiar

environments. IEEE Trans. Neural Syst. Rehabil. Eng. 2010, 18, 590–598. [CrossRef]6. Tanaka, K.; Matsunaga, K.; Wang, H.O. Electroencephalogram-based control of an electric wheelchair. IEEE Trans. Robot. 2005, 21,

762–766. [CrossRef]7. Minav, T.A.; Heikkinen, J.E.; Pietola, M. Direct driven hydraulic drive for new powertrain topologies for non-road mobile

machinery. Electr. Power Syst. Res. 2017, 152, 390–400. [CrossRef]8. Lajunen, A.; Sainio, P.; Laurila, L.; Pippuri-Mäkeläinen, J.; Tammi, K. Overview of powertrain electrification and future scenarios

for non-road mobile machinery. Energies 2018, 11, 1184. [CrossRef]9. Cižiuniene, K.; Matijošius, J.; Cereška, A.; Petraška, A. Algorithm for Reducing Truck Noise on Via Baltica Transport Corridors in

Lithuania. Energies 2020, 13, 6475. [CrossRef]10. Szpica, D.; Kusznier, M. Model Evaluation of the Influence of the Plunger Stroke on Functional Parameters of the Low-Pressure

Pulse Gas Solenoid Injector. Sensors 2021, 21, 234. [CrossRef]11. Kilikevicius, A.; Bacinskas, D.; Selech, J.; Matijošius, J.; Kilikeviciene, K.; Vainorius, D.; Ulbrich, D.; Romek, D. The Influence of

Different Loads on the Footbridge Dynamic Parameters. Symmetry 2020, 12, 657. [CrossRef]12. Wang, F.; Li, Z.; Zhang, K.; Di, B.; Hu, B. An overview of non-road equipment emissions in China. Atmos. Environ. 2016, 132,

283–289. [CrossRef]13. Pirjola, L.; Rönkkö, T.; Saukko, E.; Parviainen, H.; Malinen, A.; Alanen, J.; Saveljeff, H. Exhaust emissions of non-road mobile

machine: Real-world and laboratory studies with diesel and HVO fuels. Fuel 2017, 202, 154–164. [CrossRef]14. Minetti, A.E. Bioenergetics and biomechanics of cycling: The role of ‘internal work’. Eur. J. Appl. Physiol. 2011, 111, 323–329.

[CrossRef] [PubMed]15. Vanlandewijck, Y.; Theisen, D.; Daly, D. Wheelchair propulsion biomechanics. Sports Med. 2001, 31, 339–367. [CrossRef] [PubMed]16. Kotajarvi, B.R.; Sahick, M.B.; An, K.N.; Zhao, K.D.; Kaufman, K.R.; Basford, J.R. The effect of seat position on wheelchair

propulsion biomechanics. J. Rehabil. Res. Dev. 2004, 41, 403–414. [CrossRef]17. Van Mierlo, J.; Maggetto, G.; Van de Burgwal, E.; Gense, R. Driving style and traffic measures-influence on vehicle emissions and

fuel consumption. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2004, 218, 43–50. [CrossRef]18. Brouwer, R.; Falcão, M.P. Wood fuel consumption in Maputo, Mozambique. Biomass Bioenergy 2004, 27, 233–245. [CrossRef]19. Lijewski, P.; Merkisz, J.; Fuc, P.; Ziółkowski, A.; Rymaniak, Ł.; Kusiak, W. Fuel consumption and exhaust emissions in the process

of mechanized timber extraction and transport. Eur. J. For. Res. 2017, 136, 153–160. [CrossRef]20. Ivanenko, Y.P.; Poppele, R.E.; Lacquaniti, F. Five basic muscle activation patterns account for muscle activity during human

locomotion. J. Physiol. 2004, 556, 267–282. [CrossRef]

Appl. Sci. 2022, 12, 445 17 of 18

21. Mulroy, S.J.; Gronley, J.K.; Newsam, C.J.; Perry, J. Electromyographic activity of shoulder muscles during wheelchair propulsionby paraplegic persons. Arch. Phys. Med. Rehabil. 1996, 77, 187–193. [CrossRef]

22. Howey, D.A.; Martinez-Botas, R.F.; Cussons, B.; Lytton, L. Comparative measurements of the energy consumption of 51 electric,hybrid and internal combustion engine vehicles. Transp. Res. Part D Transp. Environ. 2011, 16, 459–464. [CrossRef]

23. Warguła, Ł.; Kukla, M. Determination of maximum torque during carpentry waste comminution. Wood Res. 2020, 65, 771–784.[CrossRef]

24. Shrivastava, P.; Verma, T.N.; Pugazhendhi, A. An experimental evaluation of engine performance and emisssion characteristics ofCI engine operated with Roselle and Karanja biodiesel. Fuel 2019, 254, 115652. [CrossRef]

25. Pedzik, M.; Stuper-Szablewska, K.; Sydor, M.; Rogozinski, T. Influence of Grit Size and Wood Species on the Granularity of DustParticles during Sanding. Appl. Sci. 2020, 10, 8165. [CrossRef]

26. Liang, J.; Edelsbrunner, H.; Fu, P.; Sudhakar, P.V.; Subramaniam, S. Analytical shape computation of macromolecule. InPROTEINS: Structure, Function, and Genetics; Wiley Periodicals LLC: The Hoboken, NJ, USA, 1998; Volume 33, pp. 1–17.

27. Giesen, J.; Cazals, F.; Pauly, M.; Zomorodian, A. The conformal alpha shape filtration. Vis. Comput. 2006, 22, 531–540. [CrossRef]28. Avis, D.; Bremner, D.; Seidel, R. How good are convex hull algorithms? Comput. Geom. 1997, 7, 265–301. [CrossRef]29. Eddy, W.F. A new convex hull algorithm for planar sets. ACM Trans. Math. Softw. (TOMS) 1977, 3, 398–403. [CrossRef]30. Lee, D.T.; Schachter, B.J. Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci. 1980, 9, 219–242.

[CrossRef]31. Chen, L.; Xu, J.C. Optimal delaunay triangulations. J. Comput. Math. 2004, 22, 299–308.32. Wieczorek, B.; Górecki, J.; Kukla, M.; Wojtokowiak, D. The analytical method of determining the center of gravity of a person

propelling a manual wheelchair. Procedia Eng. 2017, 177, 405–410. [CrossRef]33. Wieczorek, B.; Kukla, M. Methods for measuring the position of the centre of gravity of an anthropotechnic human-wheelchair

system in dynamic conditions. In IOP Conference Series: Materials Science and Engineering; IOP Publishing: Bristol, UK, 2020;Volume 776, p. 12062.

34. Wieczorek, B.; Kukla, M.; Warguła, Ł. The Symmetric Nature of the Position Distribution of the Human Body Center of Gravityduring Propelling Manual Wheelchairs with Innovative Propulsion Systems. Symmetry 2021, 13, 154. [CrossRef]

35. Wieczorek, B.; Kukla, M. Biomechanical Relationships between Manual Wheelchair Steering and the Position of the HumanBody’s Center of Gravity. J. Biomech. Eng. 2020, 142, 81006. [CrossRef]

36. Wieczorek, B.; Warguła, Ł. Problems of dynamometer construction for wheelchairs and simulation of push motion. In MATECWeb of Conferences; EDP Sciences: Les Ulis, France, 2019; Volume 254, p. 1006.

37. Wieczorek, B.; Warguła, Ł.; Kukla, M.; Kubacki, A.; Górecki, J. The effects of ArUco marker velocity and size on motion capturedetection and accuracy in the context of human body kinematics analysis. Tech. Trans. 2020, 117, 1–10. [CrossRef]

38. Warguła, Ł.; Kukla, M.; Lijewski, P.; Dobrzynski, M.; Markiewicz, F. Influence of Innovative Woodchipper Speed Control Systemson Exhaust Gas Emissions and Fuel Consumption in Urban Areas. Energies 2020, 13, 3330. [CrossRef]

39. Wieczorek, B.; Warguła, Ł.; Rybarczyk, D. Impact of a hybrid assisted wheelchair propulsion system on motion kinematics duringclimbing up a slope. Appl. Sci. 2020, 10, 1025. [CrossRef]

40. Veeger, H.E.; Van Der Woude, L.H.; Rozendal, R.H. Effect of handrim velocity on mechanical efficiency in wheelchair propulsion.Med. Sci. Sports Exerc. 1992, 24, 100–107. [CrossRef]

41. Jiang, R.; Wu, Q.; Zhu, Z. Full velocity difference model for a car-following theory. Phys. Rev. E 2001, 64, 17101. [CrossRef]42. Bregman DJ, J.; Van Drongelen, S.; Veeger, H.E.J. Is effective force application in handrim wheelchair propulsion also efficient?

Clin. Biomech. 2009, 24, 13–19. [CrossRef]43. Robertson, R.N.; Boninger, M.L.; Cooper, R.A.; Shimada, S.D. Pushrim forces and joint kinetics during wheelchair propulsion.

Arch. Phys. Med. Rehabil. 1996, 77, 856–864. [CrossRef]44. O’Brien, J.F.; Bodenheimer, R.E., Jr.; Brostow, G.J.; Hodgins, J.K. Automatic Joint Parameter Estimation from Magnetic Motion Capture

Data; Georgia Institute of Technology: Atlanta, GA, USA, 1999.45. Horprasert, T.; Haritaoglu, I.; Wren, C.; Harwood, D.; Davis, L.; Pentland, A. Real-time 3d motion capture. In Second Workshop on

Perceptual Interfaces; 1998; Volume 2.46. Van der Woude, L.H.V.; Veeger, H.E.J.; Rozendal, R.H.; Sargeant, A.J. Optimum cycle frequencies in hand-rim wheelchair

propulsion. Eur. J. Appl. Physiol. Occup. Physiol. 1989, 58, 625–632. [CrossRef] [PubMed]47. Richter, W.M.; Rodriguez, R.; Woods, K.R.; Axelson, P.W. Stroke pattern and handrim biomechanics for level and uphill wheelchair

propulsion at self-selected speeds. Arch. Phys. Med. Rehabil. 2007, 88, 81–87. [CrossRef]48. Collinger, J.L.; Boninger, M.L.; Koontz, A.M.; Price, R.; Sisto, S.A.; Tolerico, M.L.; Cooper, R.A. Shoulder biomechanics during the

push phase of wheelchair propulsion: A multisite study of persons with paraplegia. Arch. Phys. Med. Rehabil. 2008, 89, 667–676.[CrossRef] [PubMed]

49. Dubowsky, S.R.; Rasmussen, J.; Sisto, S.A.; Langrana, N.A. Validation of a musculoskeletal model of wheelchair propulsion andits application to minimizing shoulder joint forces. J. Biomech. 2008, 41, 2981–2988. [CrossRef]

50. Van Drongelen, S.; Van der Woude, L.H.; Janssen, T.W.; Angenot, E.L.; Chadwick, E.K.; Veeger, D.H. Mechanical load on theupper extremity during wheelchair activities. Arch. Phys. Med. Rehabil. 2005, 86, 1214–1220. [CrossRef] [PubMed]

51. Magagnotti, N.; Picchi, G.; Sciarra, G.; Spinelli, R. Exposure of mobile chipper operators to diesel exhaust. Ann. Occup. Hyg. 2014,58, 217–226.

Appl. Sci. 2022, 12, 445 18 of 18

52. Neri, F.; Foderi, C.; Laschi, A.; Fabiano, F.; Cambi, M.; Sciarra, G.; Aprea, M.C.; Cenni, A.; Marchi, E. Determining exhaust fumesexposure in chainsaw operations. Environ. Pollut. 2016, 218, 1162–1169. [CrossRef]

53. Lemaire, E.D.; Lamontagne, M.; Barclay, H.W.; John, T.; Martel, G. A technique for the determination of center of gravity androlling resistance for tilt-seat wheelchairs. J. Rehabil. Res. Dev. 1991, 28, 51–58. [CrossRef] [PubMed]

54. Asahara, S.; Yamamoto, S. A method for the determination of center of gravity during manual wheelchair propulsion in differentaxle positions. J. Phys. Ther. Sci. 2007, 19, 57–63. [CrossRef]

55. Ukida, H.; Kaji, S.; Tanimoto, Y.; Yamamoto, H. Human motion capture system using color markers and silhouette. In Proceedingsof the 2006 IEEE Instrumentation and Measurement Technology Conference Proceedings, Sorrento, Italy, 24–27 April 2006; IEEE:Piscataway, NJ, USA, 2006.

56. Kok, M.; Hol, J.D.; Schön, T.B. An optimization-based approach to human body motion capture using inertial sensors. IFAC Proc.Vol. 2014, 47, 79–85. [CrossRef]