Yang Wange-mail: yang-wang@uiowa.edu

Salam Rahmatalla1

e-mail: salam-rahmatalla@uiowa.edu

Department of Civil and

Environmental Engineering,

College of Engineering,

The University of Iowa,

Iowa City, IA 52242;

Center for Computer-Aided Design,

College of Engineering,

The University of Iowa,

Iowa City, IA 52242

Three-Dimensional Modelingof Supine Human and TransportSystem Under Whole-BodyVibrationThe development of predictive computer human models in whole-body vibration hasshown some success in predicting simple types of motion, mostly for seated positions andin the uniaxial vertical direction. The literature revealed only a handful of papers thattackled supine human modeling in response to vertical vibration. The objective of thiswork is to develop a predictive, multibody, three-dimensional human model to simulatethe supine human and underlying transport system in response to multidirectional whole-body vibration. A three-dimensional dynamic model of a supine human and its underlyingtransport system is presented in this work to predict supine-human biodynamic responseunder three-dimensional input random whole-body vibration. The proposed supine-human model consists of three interconnected segments representing the head, torso-arms, and pelvis-legs. The segments are connected via rotational and translational jointsthat have spring-damper components simulating the three-dimensional muscles and tis-suelike connecting elements in the three x, y, and z directions. Two types of transport sys-tems are considered in this work, a rigid support and a long spinal board attached to astandard military litter. The contact surfaces between the supine human and the underly-ing transport system are modeled using spring-damper components. Eight healthy supinehuman subjects were tested under combined-axis vibration files with a magnitude of0.5 m/s2 (rms) and a frequency content of 0.5–16 Hz. The data from seven subjects wereused in parameter identification for the dynamic model using optimization schemes in thefrequency domain that minimize the differences between the magnitude and phase of thepredicted and experimental transmissibility. The predicted accelerations in the time andfrequency domains were comparable to those gathered from experiments under differentanthropometric, input vibration, and transport conditions under investigation. Based onthe results, the proposed dynamic model has the potential to be used to provide motiondata to drive a detailed finite element model of a supine human for further investigationof muscle forces and joint dynamics. The predicted kinematics of the supine human andtransport system would also benefit patient safety planners and vibration suppressiondesigners in their endeavors. [DOI: 10.1115/1.4024164]

Keywords: spinal board, litter, coupled model, biodynamics, vibration

1 Introduction

Whole-body vibration (WBV) has been recognized as a stressorto supine humans during ground and aerial transportation [1–4].In a series of experimental studies, Huang and Griffin [5,6] inves-tigated the biodynamic response of humans in supine positionsunder WBV. The information gained from these and similar ex-perimental studies are normally used in conjunction with existingbiomechanical measures such as apparent mass, transmissibility, andimpedance to predict discomfort and injury risk during transporta-tion. While experimental studies can provide considerable insightinto human biodynamics under WBV, they are expensive and limitedto the availability of the testing and measuring equipment.

The development of predictive computer human models inWBV has shown some success in predicting simple types ofmotion, mostly for seated positions and in the uniaxial verticaldirection under specific boundary conditions [7–9]. In these mod-els, discrete mass-spring-dampers [10–13], rigid bodies connected

by viscoelastic elements and muscle groups [8,14], and finiteelement models [15–17] were used to model the human bodyresponse to vibration. Some of the models [8,15,18] showed goodpotential in predicting muscle forces and stresses, in some instan-ces [19] using very detailed finite-element-based schemes. How-ever, the validity of these models has yet to be shown.

The literature revealed only a handful of papers that tackledsupine-human modeling in response to vibration [20,21]. Vogtet al. [20] proposed a multidegree-of-freedom mass-spring-damper supine-human model to reproduce the measured imped-ance and transmissibility of nine human subjects under sinusoidalvertical vibration and a constant acceleration magnitude. Theirmodel has four segments: the head is represented by a pure mass,and the chest, abdomen, and legs are each represented by a multi-degree-of-freedom lumped parameters model. The study revealedthat the chest, because of its anatomical configuration, reacted dif-ferently than the rest of the body segments. The transmissibilitycurve for the chest showed lower value than the abdomen, and theimpedance curve for the chest showed one dominant resonancefrequency compared to the two pronounced resonances of the ab-domen. Peng et al. [21] proposed a 14-degree-of-freedom mass-spring-damper impedance-based human-berth coupled dynamicsystem adopted by ISO 5892-1981, which simulates the human

1Corresponding author.Contributed by the Bioengineering Division of ASME for publication in the

JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received November 30, 2012;final manuscript received March 28, 2013; accepted manuscript posted April 8,2013; published online May 9, 2013. Editor: Beth Winkelstein.

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body with three segments representing the head, buttock, and leg.The model was used to predict the dynamic response of the humanbody on a railway sleeper carriage as a result of the random trackvertical vibration. The latter models [20,21] represent simplifiedsupine-human models that can capture, to a certain degree, the rel-ative vertical motion between uncoupled adjacent segments. How-ever, both models lack the ability to capture the relative angularmotion between the adjacent segments, work only under verticalinput vibration, and do not consider the effect of the underlyingtransport system.

A new predictive, multibody, three-dimensional (3D) humanmodel is introduced to simulate the supine human and underlyingtransport system in response to multidirectional WBV. The humanbody is modeled by three segments, the head-neck, torso-arms,and pelvis-legs, connected via joints that have six degrees of free-dom (DOFs) with translational and rotational components. Theunderlying transport system and contact surfaces between thehuman body and the transport system are considered and modeledusing spring and damper elements. Optimization schemes basedon minimizing the differences between the experimental and pre-dicted transmissibility magnitudes and phases were used in thefrequency domain to identify the parameters of the model and theunderlying transport system.

2 Methods

2.1 Human Volunteer Tests

2.1.1 Participants. Eight healthy male subjects participatedin this study. The participants had a mean age of 23 6 3.16 years,a mean weight of 81 6 14.98 kg, and a mean height of182 6 8.36 cm. The participants had no history of musculoskeletaldisorders or injury. The study was approved by the InstitutionalReview Board for human subject studies, and an informed consentwas obtained for each participant prior to the study. The data fromseven participants were used in parameter identification for themodel, while anthropometric and response data from the eighthsubject were used for testing the model’s capability to predictsupine-human response under different anthropometries, inputvibrations, and transport conditions.

2.1.2 Experiments. A six-DOF Moog-FCS 628-1800 electri-cal motion platform system was used to generate combined-axisvibration files. Vibration files of white noise random signals with

a frequency content of 0.5–16 Hz and a vibration magnitude of0.5 m/s2 rms were applied at the platform level. All files repre-sented a different random vibration time history of 60 s in length.Two experimental conditions were conducted on the participants.In the first condition, called the rigid case, the participants werelying down and strapped to the rigid motion platform (Fig. 1(a))with straps over the shoulders, chest, pelvis, mid-thigh, and mid-shank [22]. In the second condition, called the litter-board case,the participants first lay down and were strapped to a long spinalboard (Spine Board 50-013, North American Rescue, Greer, SC)in a manner similar to that used in the rigid case, and then thehuman and board were strapped together to a standard military lit-ter (Talon II Model 90 C Litter, North American Rescue, Greer,SC), rigidly attached to the motion platform, using straps on thechest and mid-thigh (Fig. 1(b)) [22]. The output displacement andacceleration at the surfaces of the head (at the forehead), torso (atthe sternum), and pelvis were measured using a Vicon motioncapture system [23] and inertial sensors [24].

2.1.3 Biodynamic Measure. The experimental multiple-inputmultiple-output transmissibility He fð Þ is defined as the complexratio between the output acceleration €xe ¼ €xe

1; €ye1; €z

e1; €x

e2; €y

e2;

�€ze

2; €xe3; €y

e3; €z

e3g

T, measured at the segment levels and the input tri-

axis acceleration xTo ¼ €xo; €yo; €zof g

� �, measured at the shaker’s

rigid-platform level,

Heðf Þ ¼ S�1x0x0ðf ÞSx0xe

ðf Þ (1)

where Sx0x0represents a matrix of the auto- and cross-spectral

density of the input acceleration, and Sx0xerepresents a matrix of

the auto- and cross-spectral density of the input and outputaccelerations.

€xe1; €y

e1; €z

e1

� �are the coordinates of the head measured at E1

(Fig. 2), €xE2 ; €y

E2 ; €z

E2

� �, represent the coordinates of the torso meas-

ured at E2, and €xE3 ; €y

E3 ; €z

E3

� �, are the coordinates of the pelvis

measured at E3.The geometric mean ðHg

ijðf ÞÞ is used in this work to calculatethe mean transmissibility of seven subjects (Eq. (2)). Previouswork [25] has shown that Hg

ij may help reduce the scatter or inter-subject variability and may give an unbiased estimation of thetransfer function better than the arithmetic mean. i in Eq. (2) rep-resents the directions of the measured output-acceleration on the

Fig. 1 Supine-human setups and experimentation (a) rigid case, (b) litter-board case

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human body segments (i ¼ 1; 2, and 3 for the head xh; yh; zhð Þ;i ¼ 4; 5, and 6 for the torso xt; yt; ztð Þ; i ¼ 7; 8, and 9 for the pelvisðxp; yp; zpÞÞ, j represents the direction of input signal (j ¼ 1 forx-axis, j ¼ 2 for y-axis, and j ¼ 3 for z-axis), k represents a coun-ter (k ¼ 1! n), and n is the number of subjects,

Hgij fð Þ ¼

Yn

k¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHe

ij fð Þh i

k

n

r(2)

2.2 Computational Modeling

2.2.1 Model Description. Figure 2 is a schematic representa-tion of the proposed supine-human model and the underlying sup-

port for the rigid case. The human is modeled with three segments,the head (spherical segment), representing the head and neck with acenter at position F1, the torso (ellipsoid segment), representing thethorax, abdomen, and arms with a center at position F2, and the pel-vis (ellipsoid segment), representing the hips and legs, with a centerat position F3. The position of the segment’s centers is expressed as

Fm ¼xm

ym

zm

8<:

9=;

with m¼ 1 for the head, m¼ 2 for the torso, and m¼ 3 for the pel-vis. Due to the difficulty of experimentally measuring the motion

Fig. 2 Components of the supine-human model. S1, S2, and S3 comprise damping and stiffness components of the contactsurfaces between the head, torso, and pelvis, respectively.

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of the center of each segment directly, the position and accelera-tion of selected points on the surfaces of the head E1ð Þ (forehead),torso E2ð Þ (sternum), and pelvis E3ð Þ were measured and thentransformed to the segment centers. The position of the experi-mental accelerometers can be expressed as

Em ¼xe

m

yem

zem

8<:

9=;

The relationship between the position of the center of the segmentFmð Þ and the experimental (testing) position Emð Þ on each seg-

ment can be expressed using the following transformation:

Fm ¼ Em � ATmPe

m

1

� �(3)

where

A ¼1 0 0 0

0 1 0 0

0 0 1 0

24

35; Pe

m ¼Pe

mx

Pemy

Pemz

8><>:

9>=>;

is the Cartesian distance between the center of the segment andthe location of the sensors on the mth segment. The transformationmatrix Tm, based on the Denavit-Hartenberg (DH) method [26],can be obtained using Eq. (4),

Tm ¼Y3

j¼1

Tjm (4)

Tjm (Eq. (A1) in the Appendix) is the transformation matrix

between the adjacent rotational DOF for the mth segment accord-ing to Denavit and Hartenberg [26] with a rotation order of z�x�yas shown in Table 1. The head, torso, and pelvis are connected viathe rotational and translational joints J1 and J2. J1 represents thejoint between the head and torso, and J2 represents the jointbetween the torso and pelvis. J1 includes two positions: J1

1 repre-sents the location of J1 on the head side, and J2

1 represents thelocation of J1 on the torso side. Similarly, J2 has two components:J2

2 represents the location of J2 on the torso side, and J32 represents

the location of J2 on the pelvis side. The Cartesian components ofthe position of each joint can be represented as

Jmo ¼

Jmox

Jmoy

Jmoz

8><>:

9>=>;

where o is the joint’s number, o¼ 1, 2. The vectors from the cen-ter of each segment Fm to the connection joint Jm

o between twobody segments are

Lbs ¼Lbsx

Lbsy

Lbsz

8><>:

9>=>;

and

Lat ¼Latx

Laty

Latz

8><>:

9>=>;

(s¼ 1, 2, and t¼ 2, 3). The position of Jmo can be found using

Eq. (A2).For the rigid case, S1, S2, and S3 in Fig. 2 represent the contact

components (soft tissue and rigid platform) between the platformand the human body segments. The contacts are represented bylinear translating spring and damper elements with

km ¼ krm ¼ diag kr

xm; krym; k

rzm

n ocm ¼ cr

m ¼ diag crxm; c

rym; c

rzm

n okr

m and crm are the spring and damper systems.

The difference between the configuration of the rigid case andthe litter-board case in Fig. 2 is in the contact elements. For thelitter-board case, the contact elements represent the human soft-tissue interference with the board. It also comprises the flexibilityand damping of the litter-board system [27,28]. The experimentalobservation indicated that the litter fabric demonstrated a majorcontribution to the system flexibility and damping, while theboard showed a relatively smaller contribution. For the litter-board system, the contacts are represented by linear translating

spring and damper elements with km ¼ klm ¼ diagfkl

xm; klym; k

lzmg,

cm ¼ clm ¼ diagfcl

xm; clym; c

lzmg. kl

m and clm represent the spring and

damper system representing the litter-board case.

2.2.2 Dynamic Equation. In Fig. 2, €x0; €y0; €z0f g represents thex, y, and z directions of the input platform random accelerations.€x1; €y1; €z1f g, €x2; €y2; €z2f g, and €x3; €y3; €z3f g represent the linear accel-

eration of the geometric center of the head (at F1), torso (at F2),and pelvis (at F3), respectively. The proposed supine-humanmodel has 18 DOFs with xm; ym; zmf g, m ¼ 1; 2; 3ð Þ representingthe translational motion of the head, torso, and pelvis, respec-tively, and hm ¼ hxm; hym; hzm

� �, representing the rotation motion

about the x, y, and z axes of the mth segments (head, torso, andpelvis). The segments have a mass mm and a moment of inertia Im.The basic mass and inertia information of the human body seg-ments were obtained from the literature [29–31] and were basedon the ratio of the segment mass relative to that of the total bodymass. Table 2 shows the mass and length of each segment.

The stiffness and damping characteristics of the vertebras, liga-ments, tendons, and muscles at each joint are modeled as transla-tional and rotational spring and damper elements and werelumped between the adjacent segments at joint Jm

o . The transla-tional stiffness components at joints J1 and J2 are represented bykt1 ¼ diag ktx1; kty1; ktz1

� �and kt2 ¼ diag ktx2; kty2; ktz2

� �, respec-

tively, and the rotational stiffness components are represented bykr1 ¼ diag krx1; kry1; krz1

� �and kr2 ¼ diag krx2; kry2;

�krz2g, respec-

tively. The translational damping components are represented byct1 ¼ diag ctx1; cty1; ctz1

� �and ct2 ¼ diag ctx2;f cty2; ctz2g, and the

rotational damping components by cr1 ¼ diag crx1; cry1; crz1

� �and

cr2 ¼ diag crx2; cry2; crz2

� �. The contact points between the body

Table 1 DH table for one-link 3D model

Joint number (j) h d a a

1 90 degþ hmz 0 90 deg 02 90 degþ hmx 0 90 deg 03 90 degþ hmy 0 90 deg 0

Table 2 Body segments’ masses and lengths based on workby Dempster [26]

Head and neck Torso Hip, legs, and feet(Link 1) (Link 2) (Link 3)

Ratio mass/total mass 0.081 0.455 0.464Segment length/height 0.182 0.288 0.530Proximal length (La/L) — 0.630 0.10Proximal length (Lb/L) 0.118 0.370 0.895Gyration/segment 0.495 0.496 0.326

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segments and the underlying rigid support (shown as circles S1,S2, and S3 in Fig. 2) are represented by springs with coefficientkm ¼ diag kxm; kym; kzm

� �and dampers with coefficients

cm ¼ diag cxm; cym; czm

� �. km and cm represent the stiffness and

damping coefficient of the human muscle and tissue at the areaconnecting to the rigid platform.

The forces between the adjacent segments at joints J1 and J2

are illustrated in Eqs. (A3)–(A8).The contact forces

fcm ¼fcmx

fcmy

fcmz

8<:

9=;

between each segment and the rigid platform are shown inEq. (5),

fcm ¼ km Fm � x0ð Þ þ cm_Fm � _x0

� �(5)

The dynamic equation of motion for the model is derived usingthe Lagrangian equation. A linear form of the equation is shownin Eq. (6) and is derived by taking the Taylor’s series first expan-sion at the equilibrium state at

x̂ ¼ x1o; y1o; z1o; h1xo; h1yo; h1zo; x2o; y2o; z2o; h2xo; h2yo

�; h2zo; x3o;

y3o; z3o; h3xo; h3yo; h3zogT

where x̂ is the initial position of

x ¼ x1; y1; z1; h1x; h1y; h1z; x2; y2; z2; h2x; h2y

�; h2z; x3; y3; z3; h3x;

h3y; h3zgT ¼ xif g i ¼ 1;…; 18

and x0 ¼ €x0; €y0; €z0f gTis the input acceleration.

M€xþC _xþKx ¼ akx0 þ ac _x0 (6)

where M is a diagonal matrix of (m1; m1; m1; I1x, I1y, I1z, m2, m2,m2, I2x, I2y, I2z, m3; m3; m3; I3x, I3y; I3z). The details of ak and ac

are shown in the Appendix (Eqs. (A9)–(A10)). The theoreticaltransmissibility Hth fð Þ

� �between the output acceleration at the

testing points and the input 3D acceleration at the rigid-platform

level is demonstrated in Eq. (7) [32] with zero initial condition. Inthis equation, matrix R shown in Eq. (A9) represents the transfor-mation between the accelerations at the segment surface and thecenter of each segment,

Hth ¼ R Ms2 þ CsþK� ��1

ak þ acsð Þ (7)

2.2.3 System Identification

2.2.3.1 Rigid case. The unknown biomechanical model’s pa-rameters for the rigid case are the translational and rotationalspring and damping coefficients at each joint and the contactspring and damping coefficients between the body segments andthe rigid platform. These unknown parameters are identified byminimizing the error between the magnitude and phase of the the-oretical transmissibility (Hth fð Þ) and the magnitude and phase ofthe geometrical transmissibility (Hg fð Þ).

The nonlinear least square method [33] with the trust region re-flective algorithm is used to solve the nonlinear minimizationproblem to identify the system parameters. The optimization cal-culations are conducted in MATLAB. The variable tolerance andobjective tolerance are both 1.0� 10�6.

The objective function of the optimization problem f obj shownin Eq. (8) represents the normalized differences between the pre-dicted and experimental magnitude and the phase of thetransmissibility,

f obj ¼X

f

Mag Hth fð Þ� �

�Mag Hg fð Þð ÞMag Hth fð Þ

� ��Mag Hg fð Þð Þ

( )2

þX

f

Pha Hth fð Þ� �

� Pha Hg fð Þð ÞPha Hth fð Þ� �

� Pha Hg fð Þð Þ

( )2

(8)

where Mag :ð Þ represents the transmissibility’s magnitude andPha :ð Þ represents the transmissibility’s phase. It should be notedhere that the optimization problem considered only the diagonalcomponents of the transmissibility of each segment and three non-diagonal transmissibility components (H13, H43, and H73) basedon their contribution to the motion.

The independent variables of the optimization problem are theunknown parameters xr

xr ¼ x1; x2;…; x42f gT¼ krx1; k

rx2; k

rx3; k

ry1; k

ry2; k

ry3; k

rz1; k

rz2; k

rz3; ktx1; ktx2; kty1; kty2;

hktz1; ktz2; krx1; krx2; kry1; kry2; krz1; krz2; c

rx1; c

rx2; c

rx3; c

ry1; c

ry2; c

ry3; c

rz1;

crz2; c

rz3; ctx1; ctx2; cty1; cty2; ctz1; ctz2; crx1; crx2; cry1; cry2; crz1; crz2

T(9)

The upper boundary limits on the design variables are considered as

UBr , UBr ¼ 1;000;000;…; 1;000;000f gT|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}42 components

. The lower boundary lim-

its are considered as LBr , LBr ¼(

1;…; 1|fflfflffl{zfflfflffl}21 components

; 0:001;…; 0:001|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}21 components

)T

.

The upper and lower bounds were selected based on the results ofstudies in the literature [18,34,35]. The starting points for the designvariables are denoted as xr

0 ¼ 1;…; 1f g|fflfflfflfflfflffl{zfflfflfflfflfflffl}42 components

.

2.2.3.2 Litter-board case. For the litter-board case, thesupine-human parameters at the joints (kt1, kt2, kr1, kr2, ct1, ct2,

cr1, and cr2) are considered to be similar to those of the rigid case.Therefore, the optimization process will focus on identifying thecontact parameters kl

m and clm based on the difference between the

model prediction and the experimental data. The objective functionform is similar to the rigid-case model (Eq. (8)). The independentvariables of the optimization are xl

� �as shown in Eq. (10),

xl ¼ xl1; x

l2;…; xl

18

� �T

¼ kl1x; k

l1y; k

l1z; k

l2x; k

l2y; k

l2z; k

l3x; k

l3y; k

l3z; k

l1x; k

l1y;

nkl

1z; kl2x; k

l2y; k

l2z; k

l3x; k

l3y; k

l3z

oT(10)

The upper boundaries on the design variables are considered asUBl,

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UBl ¼ kr1x; k

r1y; k

r1z; k

r2x; k

r2y; k

r2z; k

r3x; k

r3y; k

r3z; k

r1x; k

r1y;

nkr

1z;

kr2x; k

r2y; k

r2z; k

r3x; k

r3y; k

r3z

oT

The lower boundary limits are considered as LBl,

LBl ¼ 0;…; 0f gT|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}18 components

, and the starting point for the optimization is

denoted as xl0,

xl0 ¼ kr

1x; kr1y; k

r1z; k

r2x; k

r2y; k

r2z; k

r3x; k

r3y; k

r3z; k

r1x; k

r1y;

nkr

1z; kr2x; k

r2y;

kr2z; k

r3x; k

r3y; k

r3z

oT

As in the rigid case, the optimization calculations are conductedby MATLAB. The variable tolerance and objective tolerance areboth 1.0� 10�6.

Fig. 3 Transmissibility between the input acceleration at the rigid-platform level and the output accelerations at points on thehead, torso, and pelvis under the rigid condition. The light gray lines represent the transmissibility of the individual subjects,the solid dark line represents the geometrical transmissibility of the subjects, and the dotted line represents the optimizedtransmissibility of the model.

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3 Results

3.1 Experimental Results

3.1.1 Rigid Case. The magnitude of the experimental trans-missibility of the seven tested subjects in the rigid case (gray linesin Fig. 3) showed observable subject intervariability, especially atthe first peak and after that. The solid black lines in each graphrepresent the geometrical transmissibility of the seven subjects.The transmissibility graphs show 27 components, with each seg-

ment having nine components representing the relationshipbetween three input acceleration components and three outputacceleration components at the segment surface. In general, thediagonal components of each segment showed greater contribu-tion than the out-of-diagonal components. Among the nondiago-nal transmissibility, the head transmissibility in the x direction inresponse to input z direction Hg

13

� �showed a large magnitude after

9 Hz compared to the other nondiagonal transmissibilitycomponents.

Fig. 4 Transmissibility between the input acceleration at the rigid-platform level and the output accelerations at points on thehead, torso, and pelvis under the litter-board condition. The light gray lines represent the transmissibility of the individual sub-jects, the solid dark line represents the geometrical transmissibility of the subjects, and the dotted line represents the opti-mized transmissibility of the model.

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3.1.2 Litter-Board Case. The transmissibility components forthe litter-board case showed similar characteristics to those of therigid case in terms of the variability between the subjects after thefirst peak (Fig. 4). Similar to the rigid case, the diagonal compo-nents of the transmissibility for each body segment have shownlarger magnitudes than the nondiagonal components. One noticea-ble observation in this case is that all segments showed peaksaround 5 Hz due to the input vertical vibration (H33, H63, and H93).

3.2 Model Results. Table 3 demonstrates the optimizationresults of the supine-human model stiffness and damping coeffi-cients identified under the rigid condition. Table 4 shows the stiff-ness and damping coefficients of the contact elements under therigid and litter-board conditions.

Table 3 Joint stiffness and damper coefficients of the three-dimensional human model under the rigid-case condition

X Y Z

Springa Dampingb Spring Damping Spring Damping

RJ-1c 50005.3 1.0 772.9 11.0 48964.0 2140.5RJ-2 50005.6 1.0 18832.4 278.9 49596.6 4992.1TJ-1d 31967.2 14335.5 14553.6 591.9 122987.8 8.1TJ-2 38485.4 273.8 22719.8 66.5 68341.0 46061.6

aSpring coefficient unit: N/m (translate) and N m/rad (rotational).bDamper coefficient unit: N s/m (translate) and N m s/rad (rotational).cRJ: rotational joint.dTJ: translational joint.

Table 4 Rigid and litter-board spring and damper softening coefficients

X Y Z

Case Joints Springa Damping Spring Damping Spring Damping

Rigid Head (m¼ 1) 21042.5 6.2 24968.7 1.9 109881.9 118.4Torso (m¼ 2) 13626.3 230.8 8756.4 425.7 83841.3 3574.1Pelvis (m¼ 3) 9147.7 487.4 6042.0 319.3 107865.1 0.156

Litter-board Head (m¼ 1) 21042.5 0.004 21382.4 0.9 43148.0 117.1Torso (m¼ 2) 13626.3 160.1 7337.7 337.3 0.8 488.8Pelvis (m¼ 3) 4859.3 458.3 6041.9 166.4 66421.1 0.023

aSpring coefficient unit: N/m and damper coefficient unit: N s/m.

Fig. 5 Comparison between the predicted and experimental 3D acceleration of thehead, torso, and pelvis in the frequency domain under the rigid condition. Figuresin the right column represent a snapshot of the circles shown in the left columns.

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The power spectral density (PSD) [36] of the predicted and ex-perimental acceleration under the rigid condition is depicted inFig. 5. As shown in Fig. 5, the predicted PSD of the three seg-ments in the fore-aft x-direction showed weak performance incomparison to the lateral and vertical directions. For the litter-board case (Fig. 6), the model showed weak performance in pre-dicting the vertical component of the torso.

In the time domain, the small time window for the predictedacceleration of the pelvis in the lateral y-direction during the rigidcondition (Fig. 7) showed weaker performance than the rest of thesegments and directions. The small time window for the litterboard case (Fig. 8) showed generally weaker performance thanthe rigid case. Table 5 shows the differences between the rmsmagnitude of the predicted and experimental acceleration of therigid and litter-board conditions for all segments in the x, y, and zdirections during the 60 s of random motion. The coefficient ofdetermination (R2) magnitude of the rms value of all segments inTable 5 is found to be 0.608 and 0.677 for the rigid and litter-board cases, respectively.

4 Discussion

A 3D dynamic supine-human model coupled with its underly-ing transport system under multiple-axis random WBV is pre-sented in this work. The model simulates the human body with

three segments (head, torso-arms, and pelvis-legs) connected viarotational and translational joints and therefore is expected to cap-ture the relative angular and translational motion between adja-cent segments. The proposed model considers the coupling effectbetween the human body and the underlying transport systems,which presents a practical approach to investigating humanresponse under different transport conditions. A potential applica-tion of the proposed model is to provide kinematics informationfor existing kinematics-driven muscle models, such as those withdetailed finite element models [16,17,19] where comfort andinjury risk may be investigated.

While this study presented a supine-human response under mul-tiple-input-multiple-output vibration signals, the results stillshowed some similarity between the vertical components of thetransmissibility for the sternum and pelvis (H63 and H93) and thosepresented in the literature [6,20]. In the current study, the rigidcase for the sternum showed two peaks around 6 Hz and 11 Hz,which are comparable to the impedance results by Vogt et al. [20]and the transmissibility results by Huang and Griffin [6]. Thetransmissibility of the sternum-pelvis for the litter-board case ofthe current study showed a dominant frequency around 5 Hz,which is also comparable with the transmissibility peaks in thework by Vogt et al. [20].

The results (Fig. 5) showed that the predicted acceleration PSDfor the rigid case support captured most of the frequency

Fig. 6 Comparison between the predicted and experimental 3D acceleration of the head, torso,and pelvis in the frequency domain under the litter-board condition. Figures in the right columnrepresent a snapshot of the circles shown in the left columns.

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components and their energies, except for that of the pelvis in thex and y directions. This could be related to the pitching and rollingmotions of the pelvis that present some challenges to the predictedmodel in these directions. However, the predicted motion for thetorso in the x-direction showed better agreement with the experi-ments as the torso is expected to have lesser pitching and rollingmotions than the pelvis. It is expected that better results could beachieved by modeling the torso with two masses to capture thecharacteristics of the chest and the abdomen. However, the ab-dominal area has a lot of skin and tissues, and it is expected to bevery hard to measure its motion due to the extensive relativemotion of the skin and tissues. Also, the motion of all segments inthe x-direction will normally contain global rigid-body and slidingmotions that may be hard to capture.

For the litter-board case, the predicted acceleration PSD of thetorso and pelvis in the z-direction (Fig. 6) have demonstratedweak energies for frequencies above 5 Hz. This could be attrib-uted to the effect of the resonance and damping caused by thelitter-board system around and after 5 Hz. In the vertical direction,the litter-board system has shown a resonance around 5 Hz, butgenerates heavy damping after that.

The predicted acceleration for the rigid and litter-board condi-tions of all segments (Figs. 7 and 8) for the time segment between26 and 31 s has shown characteristics comparable to that in theexperiments with some discrepancies in the phase. The time seg-ment 26-31 s was randomly selected to show observable differen-ces between the predicted motion and the experiments, however,

these discrepancies may vary depending on the selected time seg-ment on the acceleration-time graph.

Several assumptions have been used in the present model’s pa-rameter estimation that may impose some limitations on theapplicability of the model. First, each body segment was consid-ered as a rigid body. This assumption may be reasonable for thehead segment, but may be inadequate for the rest of the body seg-ments, such as the torso and pelvis where soft tissues and wob-bling masses [13,37,38] play considerable roles. However,wobbling masses seem to be more affected by impact [37] than bythe vibration encountered in this work. Second, the linearizationprocess of the equation of motion adds another limitation to theproposed model, and therefore a model with nonlinear parametersestimation would be more effective.

There is a tendency for the supine human to slide along the sup-porting surfaces during vibration, however, the straps imposed onthe subjects during the experiments may play a positive role inreducing the sliding motion and increasing the contact forcesbetween the human and the supporting surfaces. The straps alsoplay a role in minimizing the global motion of the subject and thetransport system in the transverse directions. While in this workthe strap forces are considered to contribute to the contact forces,this assumption may not be representative, and a detailed modelof the straps would add more accuracy to the model. There wereconsiderable attempts during the experiments to make the tensionin the straps comparable between subjects, however, that assump-tion may involve some subjective errors. In the future, it would be

Fig. 7 Comparison between the predicted and experimental 3D acceleration of the head, torso,and pelvis in the time domain under the rigid condition. Figures in the right column represent asnapshot of the circles shown in the left columns.

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better to use tensioning devices to control the strap tension andmake it more comparable between subjects.

The experiments for the vertical input motion showed that thetransmissibility of the supine human calculated between the inputacceleration at the rigid platform and the output acceleration atpoints on the human body (in the rigid case) were comparable tothose calculated between the input acceleration at the human/board interface (in the litter-board case) and the output accelera-tion on the human body. This indicates that the board has littleeffect on the human response and that the board behaved in a way

comparable to that of the rigid platform. Nevertheless, a betterdynamic model may be introduced if the flexibility of the board isconsidered in the model parameter identification process. The pro-posed model and the experiments have also indicated that thelitter-board system has minimal effect on the supine humanresponse in the fore-aft x-direction and the lateral y-direction, butthat it plays a major role in the vertical z-direction. As shown inFig. 3, the litter-board system was able to filter out the peaks ofthe head, torso, and pelvis, however, it created a prominent peakaround 5 Hz. While 5 Hz may generate large transmissibility for

Fig. 8 Comparison between the predicted and experimental 3D acceleration of the head, torso, and pelvis in the time domainunder the litter-board condition. Figures in the right column represent a snapshot of the circles shown in the left columns.

Table 5 Experimental and predicted acceleration (rms(m/s2)) of the head, torso, and pelvis in the x, y, and z directions under therigid and litter-board conditions

X Y Z

Case rms acceleration (m=s2) Head Torso Pelvis Head Torso Pelvis Head Toro Pelvis

Rigid Experiment 0.197 0.178 0.163 0.171 0.174 0.178 0.126 0.130 0.140Predicted 0.140 0.141 0.146 0.146 0.150 0.163 0.121 0.123 0.124

Litter-board Experiment 0.228 0.178 0.228 0.186 0.201 0.210 0.133 0.151 0.157Predicted 0.144 0.142 0.152 0.143 0.158 0.165 0.123 0.119 0.120

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the human segments due to the input motion, different human seg-ments moved together at that frequency, indicating small relativemotion between the body segments. This finding would supportthe linear assumption used in this study. Nevertheless, the headunder the litter-board condition showed another peak around10–11 Hz as shown in Fig. 4. This peak could be an indication ofthe effect of the coupling motion between the head and the torso;the latter has a resonance around 10 Hz, as indicated in Fig. 3. Tothe authors’ knowledge, there is no reference to any specifichealth issues related to 5 Hz and 10–11 Hz. However, 5 Hz and10–11 Hz will cause large motion, and that could be problematicto injured patients, such as those with traumatic head and spinalcord injuries.

The combined inertial and marker-based systems [24] used inthis work to acquire the subject’s motion accurately captured theposition and acceleration of selected points on the human body.Still, the inertial sensors showed sensitivity to ferrous surfaces inthe environment, which may affect the reading of the magnetome-ter and cause a drift in the acceleration readings. The latter prob-lem was minimized by keeping the sensors away from the ferroussurfaces. Also, both the markers and the inertial sensors are proneto introducing relative skin motion that may affect the resultingmotion [39,40], however, care was taken during experiment setupto minimize such motion (for example, wide adhesive straps wereused to mount the sensors). Additionally, initial investigationshowed that the inertia and marker systems have resonance fre-quencies above 25 Hz, which is higher than the frequency underconsideration.

5 Conclusion

There is a critical need for 3D dynamic models that can predictthe response of a supine human and the underlying transport sys-tems during real-world transportation. The proposed model hasthe potential to be augmented with existing detailed finite elementmodels to investigate joint dynamics and muscle forces of supinehumans under different types of transport systems in WBV. Themodel would also benefit patient safety planners and vibrationsuppression designers in their effort to develop better and safertransportation systems.

Acknowledgment

This study was conducted at the 3D Biomotion Research Lab atthe Center for Computer-Aided Design (CCAD) at The Universityof Iowa. Mr. Jonathan DeShaw and Mr. John Meusch assisted inthe human experimentation.

Appendix

Tjm ¼

cos hmj � cos aj sin hmj sin aj sin hmj aj cos hmj

sin hmj cos aj cos hmj � sin aj cos hmj aj sin hmj

0 sin aj cos aj dj

0 0 0 1

26664

37775

(A1)

J11 ¼ F1 þ AT1Lb1

J21 ¼ F2 þ AT2La2

J22 ¼ F2 þ AT2Lb2

J32 ¼ F3 þ AT3La3

(A2)

fJ1 ¼ kt1l12 þ ct1_l12 (A3)

fJ2 ¼ kt2l23 þ ct2_l23 (A4)

mJ1 ¼ kr1 h2 � h1ð Þ þ cr1_h2 � _h1

� �(A5)

mJ2 ¼ kr2 h3 � h2ð Þ þ cr2_h3 � _h2

� �(A6)

where

l12 ¼l12x

l12y

l12z

8<:

9=;

represents the displacement between joints J21 and J1

1, and

l32 ¼l32x

l32y

l32z

8<:

9=;

represents the displacement between joints J32 and J2

2. The equa-tions for l12 and l32 are shown in Eqs. (A7) and (A8),

l12 ¼ J21 � J1

1 ¼ F2 þ AT2La2 � F1 � AT1Lb1 (A7)

l32 ¼ J32 � J2

2 ¼ F3 þ AT3La3 � F2 � AT2Lb2 (A8)

R ¼R1 O3�6 O3�6

O3�6 R2 O3�6

O3�6 O3�6 R3

24

35 (A9)

where

Rm ¼1 0 0 0 pe

mz pemy

0 1 0 �pemz 0 pe

mx

0 0 1 pemy �pe

mx 0

264

375 m ¼ 1; 2; 3ð Þ

and O3�6 is a 3� 6 matrix of zeros.

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