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Technical Brief

Modeling Wheelchair-Users Undergoing Vibrations OPEN ACCESS

[+] Author and Article Information
Korkut Brown

Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: korkutbr@alumni.usc.edu

Henryk Flashner

Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: hflashne@usc.edu

Jill McNitt-Gray

Biological Sciences and Biomedical Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: mcnitt@usc.edu

Philip Requejo

Rehabilitation Engineering,
Rancho Los Amigos National Rehabilitation Center,
Downey, CA 90242
e-mail: requejo@usc.edu

Manuscript received November 21, 2016; final manuscript received June 20, 2017; published online July 20, 2017. Assoc. Editor: David Corr.

J Biomech Eng 139(9), 094501 (Jul 20, 2017) (7 pages) Paper No: BIO-16-1474; doi: 10.1115/1.4037220 History: Received November 21, 2016; Revised June 20, 2017

A procedure for modeling wheelchair-users undergoing vibrations was developed. Experimental data acquired with a wheelchair simulator were used to develop a model of a seated wheelchair user. Maximum likelihood estimation procedure was used to determine the model complexity required to characterize wheelchair-user's response. It was determined that a two segment rotational link model is adequate for characterization of vibratory response. The parameters of the proposed model were identified using the experimental data and verified using additional experimental results. The proposed approach can be used to develop subject-specific design criteria for wheelchair seating and suspension.

FIGURES IN THIS ARTICLE
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Early studies on wheelchair-vibrations [13] have shown that on certain road conditions the accelerations experienced by the wheelchair-user exceed the standardized levels set for the general population [4]. The primary health concern that is associated with whole-body vibrations is back pain [57]. Wheelchair-users with spinal cord injuries are particularly vulnerable to the negative effects of vibrations due to their limited capacity to stabilize their trunk [8]. Wheelchair manufacturers have responded to the growing concerns of wheelchair-vibrations with suspension elements installed in the rear or caster wheels. Vibration studies conducted on a range of these suspension elements [913] have found that they are capable of reducing the vibrations transmitted to the wheelchair-user. The performance of the suspension elements was noted to be dependent on the relative orientation of the suspension elements with respect to the direction of the vibratory excitations. It was reported, under certain conditions [14], that the suspension elements even amplified the vibratory excitations. These findings highlight the lack of available guidelines for evaluating the vibrations transmitted to the wheelchair-user. The principal organizations in charge of wheelchair design standards, the American National Institute/Rehabilitation Engineering and Assistive Technology Society of North America, do not administer vibration tests. Vibration exposure levels prescribed in the ISO 2631, intended for the general population, have been used as a reference in the previous wheelchair vibration studies. Since the ISO 2631 standards were developed from the data collected on able-bodied individuals, they may not be applicable to wheelchair-user populations.

The majority of contributions to the area of wheelchair vibrations to date have been based on empirical studies. Participants in the studies are often fully abled persons and not actual wheelchair-users. Using fully abled individuals may not sufficiently capture the unique characteristics of the wheelchair-user population. In addition, to adequately study whole-body wheelchair vibrations, development of an analytical model is important. Models can be used to predict system responses for operational conditions that cannot be recreated in the laboratory or without endangering the subjects. Numerous models of a seated person have been created to better understand the resonant behavior of the human body in nonwheelchair related fields [1520]. A number of models [21] were used in the development of the ISO 2631 standards. Similar models are needed to study wheelchair vibrations [22] and to eventually develop standards for wheelchair-users. In this paper, we propose an approach for developing experimentally based models of wheelchair-users, with different levels of injury, subjected to periodic excitations. The modeling procedure consists of (i) postulating the order and configuration of a mechanical model, (ii) identification of the model transfer function (TF), and (iii) computation of model parameters representing physical characteristics of the wheelchair-user. Maximum likelihood estimation [23] was used to identify the model transfer functions. Various degrees-of-freedom analytical models were evaluated, and a two segment-trunk (ST) model, corresponding to a fourth-order transfer function representation, was found to adequately represent the wheelchair-users' dynamic response. The experimental data from a previous vibration study that evaluated different rear wheel suspension elements [11] were used to validate the method.

The paper is organized as follows: Section 2 details the modeling methodology, including experimental setup and the proposed mechanical model. Section 3 presents the results of the identified transfer function representation of two subjects with different levels of spinal-cord injury (SCI). Section 4 presents the results of model parameters of the two subjects, computed from their respective identified transfer function representation. Discussion of the results is presented in Sec. 5.

Experimental Setup.

The experimental data used in this paper were collected in a study described in Ref. [11]. Subjects were seated on wheelchairs and exposed to vibratory excitations from a novel vibration simulator. Subjects with C6, T4, T7, T12, and L4L5 levels of SCIs participated in the study. Participants were asked to face forward and keep their arms and hands inside the wheelchair during the experiments. The vibration simulator, shown in Fig. 1, consisted of a rotating drum with an affixed bump used to impact the rear wheel of the wheelchairs, while the wheelchairs were constrained from translating in the fore/aft direction. Single-axis accelerometers (Model 2210, Silicon Design), with 300 Hz bandwidth, were placed along the wheelchair and on a helmet worn by the subjects. Transverse accelerations were measured at the subject's head (Fig. 1). Load cells (MLP Series, Transducer Techniques, Temecula, CA), oriented tri-axially, were located at the seat to measure the combined effect of forces at the backrest and seat-user interface. All measurements were sampled at 2000 Hz on PCMCIA DAQ cards (DAQP-12H, DAQP-16, Quantech, Inc., Rockville, MD). Each vibration trial lasted approximately 30 s. The experimental data collected at the 30 rpm (0.85 m/s) drum rotation speed setting on the rigid frame Quickie GPV wheelchair, absent of any suspension elements, were used for model identification. Experimental data collected at the 35 rpm (1.03 m/s) and 40 rpm (1.21 m/s) rotation speeds were used for validation purposes in the development of the model. Informed consent was obtained from the subjects, including consent about the publication of results in subsequent or related studies.

Dynamic Model of Wheelchair-User.

To investigate the model complexity needed to capture the dynamics of the wheelchair-user's motion, analytical models of varying degrees-of-freedom were constructed. The corresponding transfer functions of the analytical models were identified from experimental data, using the procedure described in Sec. 2.3 and evaluated as described in Sec. 2.4. A two degrees-of-freedom analytical model was found sufficient to describe the vibratory response of the seated wheelchair-users and is derived in Sec. 2.2.1. The derivation of the model is an extension of a one ST model given in Ref. [24]. The model consists of translational and rotational parts. The translational part represents the lower anatomy of the wheelchair-user. It is modeled as a rigid element with only translational degrees-of-freedom. The rotational part accounts for the rest of the upper anatomy of the wheelchair-user that is not included in the translational part, and consists of two rotational degrees-of-freedom. The dynamics of the wheelchair couples into the modeled wheelchair-user system via vertical and horizontal input forces. The tip of the most distal rotational link of the model is treated as the top of the wheelchair-user's head.

Two Segment-Trunk Model.

The rigid elements of the two-ST model are parameterized as seen in Fig. 1. The center of mass of the model's translational part is described by the horizontal and vertical positions, xb,yb, the angle of the lower rotational link, θ1, is relative to the global Y axis, and the angle of the upper rotational link, θ2, is expressed relative to the radial axis of the lower rotational link. Gravity, g, acts along the negative Y-axis. The nonlinear equations of motion of the model are derived using Euler–Lagrange equations and can be found in Ref. [24]. After linearization about an arbitrary equilibrium point θ1*,θ2*, the relation between the head transverse accelerations and seat forces is given by the following equations: Display Formula

(1)translationalpart{mtotx¨b+[l1m2cos(θ1*)+m1r1cos(θ1*)+m2r2cos(θtot*)]θ¨1+[m2r2cos(θtot*)]θ¨2=Fhmtoty¨b+[l1m2sin(θ1*)+m1r1sin(θ1*)+m2r2sin(θtot*)]θ¨1+[m2r2sin(θtot*)]θ¨2=Fv
Display Formula
(2)rotationalpart{M[θ¨1θ¨2]+C[θ˙1θ˙2]+K[θ1θ2]=B[FvFh]
Display Formula
(3)headaccel.{x¨bcos(θtot*)y¨bsin(θtot*)+[gcos(θtot*)·(θtot)]+l1θ¨1cos(θ2*)+l2(θ¨1+θ¨2)=atran

where M={mij},C={cij}, and K={kij} are 2 × 2 symmetric matrices, and B is a 2 × 2 matrix. All matrices are a collection of the model parameters and equilibrium point, θ1*,θ2*. atran is the linearized expression for the transverse accelerations at the head derived from Fig. 1. By taking the Laplace transform of Eq. (3) and substituting terms from Eqs. (1) and (2), a fourth-order, two-input, single-output, transfer-function of the two-ST model is obtained Display Formula

(4)Atran(s)=[TFv(s)TFh(s)][Fv(s)Fh(s)]
Display Formula
TFm(s)=bm4s4++bm1s+bm0s4+a3s3+a2s2+a1s+a0m=v,hanan(mij,cij,kij),n=0,,3bmnbmn(mij,cij,kij),n=0,,4

The M, C, K, B matrices, denominator coefficients, an, and numerator coefficients, bmn, are detailed in Ref. [24].

Identified Model Transfer Functions.

System identification was used to estimate the transfer functions of Eq. (3). The identification was based on maximum likelihood estimation [25] and Gauss–Newton gradient optimization [26] methods. The measurements from the vibration experiment corresponding to the inputs and output of the model TF in Eq. (3) were used in the system identification procedure.

Using the values of the identified transfer function, stiffness and damping coefficients, ki,ci, were computed from the analytical expressions of the denominator coefficients in Eq. (3). Inertial properties of the various elements of the model, Icgi,mi,ri,li, were computed using anatomical descriptions from de Leva [27], for a given total mass and height of the wheelchair-user.

Data Analysis.

The experimental data were filtered to cover the range of primary resonant frequencies reported in previous wheelchair vibration studies [1]. A fourth-order Butterworth filter with a 20 Hz cutoff frequency was used. Mean TF estimates were obtained for input–output data sets spanning from 5 to 10 bump excitations. Model estimates were further processed by excluding estimates with denominator coefficients that were outside one standard-deviation of the computed mean. The experimental data collected at 30 rpm drum speed and 0.5 in diametered bumps were used in the system identification procedure. The resulting mean TF estimates were validated by simulating the model using data at 35 rpm and 40 rpm rotation speeds. Performance of the model was quantified by calculating the weighted-root-mean-squared-error (WRMSE) of the model simulations to the corresponding experimental transverse accelerations, Display Formula

(5)WRMSE=[1N(εTε)]12

where ε=W·[atranexpâtran], atranexp are the experimental transverse accelerations, âtran are the simulated transverse accelerations, and N is the length of the data set. The weights, W, were weighted 1 for regions defined from 200 data points prior to the first peak, up to when the accelerations cross 1m/s2 when the signal has died down and 0 at all other regions, as seen in Fig. 2(a).

The proposed modeling methodology is demonstrated using two subjects from the experimental study presented in Ref. [11]: a subject with cervical C6 and a thoracic T7 SCI level subject. Second- and fourth-order transfer functions were identified, corresponding to one- and two-ST mechanical models. In addition, sixth-order TFs were identified for each subject to investigate the improvement that can be attained by further increasing the number of degrees-of-freedom. The WRMSE values of each subject's transfer function response, as defined in Eq. (4), are shown in Table 1. It can be observed that the identified fourth-order TF has a smaller WRMSE value than the second-order TF by 25% for the C6 subject, and by 45% for the T7 subject. The sixth-order TF was found to yield a higher WRMSE value than the fourth-order TF for the C6 subject, and was within the standard deviations of the fourth-order transfer function WRMSE value for the T7 subject. It was therefore concluded that increasing the number of degrees-of-freedom beyond a fourth-order TF, or equivalently a two-ST mechanical model, does not substantially improve the system model accuracy. The simulated response of the second- and fourth-order transfer functions is shown in Figs. 2(b) and 2(c). It can be observed that for both subjects, the fourth-order TF captures higher harmonic dynamics present in the experimentally measured response.

The poles of the identified transfer functions in Eq. (3) for subjects C6 and T7 are shown in Table 2. For both subjects, the second-order TF representation consists of a complex pole pair. One pair of poles of the fourth-order TF appears in a close location to the pole of the second-order TF. The re-occurring nature of this pole implies that it represents a distinct characteristic of each subject's dynamic response. The additional pole pair of the fourth-order TF, along with the zeros, helps to characterize features of the response not captured by the second-order TF representation. Based on these results, it was concluded that a fourth-order TF is an adequate model for representing the response of wheelchair-users.

In order to check the feasibility of the identified fourth-order TFs for other external perturbations, the TFs were validated by simulating them using experimental trials collected at the medium and fast drum rotation settings, 35 rpm and 40 rpm. Comparison of the TF simulations and transverse head accelerations for subjects C6 and T7 are shown in Fig. 3. As seen in the simulations, the characteristics represented in the transfer functions identified using the 30 rpm trials stay valid for input excitations experienced under the medium and fast drum rotation speeds for both subjects. The TF simulations of the subjects reflect most of the features found in their medium and faster drum rotation trial responses and support the validity of the identified TFs.

A representation of the wheelchair-user dynamics can be made by the stiffness and damping coefficients, ki,ci, of the model in Fig. 1. Solutions for the coefficients can be computed from the identified transfer functions of the model. Details of the computation of the dynamic parameters are demonstrated on the two-ST mechanical model as follows. The denominator coefficient values of the identified transfer function were equated to the analytical denominators in Eq. (3), and by assuming values for the anatomical properties of the model, the unknown dynamic parameters of the two-ST mechanical model were computed from the denominator equations. The anatomical properties, such as segment masses and lengths, were calculated from de Leva [27]. Based on the experimental observations, the equilibrium angles (θ1*,θ2*) for the rotational links of the model (see Fig. 1) were taken to be θ1*=20deg,θ2*=20deg. Since the height of the subjects was not provided in Ref. [11], it was assumed to be 1.8 m. The total mass of the subjects was computed using the experimental vertical force measurements at static conditions. The rotational stiffness and damping coefficients in Eq. (3) were obtained by solving fourth-order polynomial equations. Values were computed for three anatomical configurations to check the suitability of different mass distributions and corresponding ki,ci solutions to each wheelchair-user. The top rotational link in the configurations was chosen to represent the head, and the lower link represented the different parts of the wheelchair-user's trunk. Each configuration differed in the location at which the translational and rotational parts connect anatomically: at the xyphoid, omphalion, or hip, respectively, as seen in Fig. 4.

The mean and standard-deviation values of the rotational stiffness and damping coefficient solutions for the three configurations are given in Fig. 5. The spring and damper coefficient values for the lower rotational link, k1,c1, increase with each model configuration. This is as expected, since the mass and moment of inertia of the lower rotational link increases with each model configuration and would require a stiffer rotational spring to maintain the balance at the equilibrium point. The rotational spring and damper solutions of the upper rotational link, k2,c2, maintain their relative values at each model configuration. This is also as expected, since the upper rotational link is always modeled as the head of each subject, and its inertia is constant throughout the three model configurations.

The rotational stiffness and damping coefficients in Fig. 5 and the corresponding anatomical parameters were used to calculate the coefficients of the mechanical model's TF in Eq. (3). Simulation of the response obtained using the mechanical model transfer functions of both subjects are shown in Fig. 6. It can be seen that the mechanical model TF responses are qualitatively similar to the experimental measurements. The peak amplitudes of the simulations occur at instances close to the peaks of the experimental data. However, simulation amplitudes are lower than the experimental responses, and simulations between the different configurations are indistinguishable.

A methodology for constructing models of passively seated wheelchair-users with varying levels of SCI experiencing rear wheel vibratory excitations was developed. The procedure consisted of a maximum likelihood estimation method for estimating model parameters. The model complexity required to characterize the wheelchair-user response to whole-body vibration excitation was determined. It was found that a two degrees-of-freedom model is the best representation of the dynamics of a wheelchair-user. The WRMSE values in Table 1 indicate that the complexity of the underlying wheelchair-user system is better represented by the two-ST mechanical model. Furthermore, the re-occurring poles of the second- and fourth-order identified TFs imply that the characteristics of the second-order TF are contained in the fourth-order TF.

Each pair of stiffness and damping coefficients of the two-ST model characterize the dynamic behavior of a rotational link in the model. Based on the assumed mass properties with each anatomical configuration, in Fig. 4, the coefficient solutions can be interpreted as an indication of the motor control of the anatomical segments represented by each link. Comparing the relative magnitudes of the stiffness coefficient solutions between the two subjects in Fig. 5, the solutions can be associated with each subject's level of SCI. Then, the lower rotational link stiffness, k1, values of the C6 subject imply little trunk control. The upper rotational link spring coefficient, k2, values show that the C6 subject exhibits more neck control than the T7 subject. This may indicate that the T7 subject is more capable of distributing the required control demand across their trunk. The above results correlate with the findings reported in the original vibration study [11]. Using the stiffness and damping coefficient solutions, parameterized TFs of the two-ST mechanical model were obtained. The response of the TFs in Fig. 6 reflected the main characteristics in each subject's experimental measurements, but resulted in smaller magnitudes and found to be insensitive to change in model configurations.

To better understand the modeling results, more accurate anthropometric source measurements are required [28]. Due to the physical changes wheelchair-users develop [29,30], the anatomical segment inertial parameters from Ref. [27] may not represent the wheelchair-users demographic. The availability of more accurate anthropometrics would improve the model parameter solutions, but would not affect the validity of the proposed methodology. Assuming linear springs and dampers also may contribute to estimation error. Other researchers have had success modeling muscles and joints with nonlinear characteristics [3133]. Experimental data collected from the vibration simulator in Ref. [11], for purposes of modeling wheelchair-users, provide advantages over previous wheelchair vibration experiments. The excitations produced by the simulator are repeatable, as compared to experiments conducted on an overground road-course or side-walks, where the excitations induced by the act of propulsion may vary the experimental results. The consistent nature of the vibration simulator lends itself to systematic studies of different wheelchair configurations and seat orientations, albeit ones assuming no upper body interaction with the wheelchair. Unlike some previous wheelchair vibration studies, the participants in the vibration simulation study were actual wheelchair-users. Using experimental data collected on wheelchair-users are essential for the purposes of creating wheelchair-user models. The methodology will require testing on a wider range of subject population with more accurate anthropometric data. Larger data sets can be used to test for common traits between subjects with similar levels of SCI. Thoroughly validated models can be used as references in clinical applications or in the development of standards for wheelchair-users. The presented models are valid for the duration of the experimental data used in this paper. However, once the model parameters are validated, the models will not change for longer durations of vibratory disturbances. As future developments, the scope of the models can be expanded to include explicit vibration excitations experienced at the footrest and backrest. Modifications can also be made to the models to accommodate for active propulsion states by adding upper limbs and nonlinear model parameters.

  • ki,ci =

    stiffness and damping coefficients of the torsional spring of link i

  • mi =

    mass of element i

  • mtot =

    the sum total of the mass of the portions of the model

  • ri,li,Icgi =

    position of the center of gravity, length, and mass moment of inertia of link i, measured with respect to the proximal end of the link

  • SCI =

    spinal cord injury

  • ST =

    segment trunk

  • TF =

    transfer function

  • WRMSE =

    weighted root mean squared error

  • θtot =

    the sum total of the angular displacements of the rotational links

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References

DiGiovine, C. , Cooper, R. , and Boninger, M. , 1999, “ Comparison of Absorbed Power to Vertical Acceleration When Measuring Whole-Body Vibration During Wheelchair Propulsion,” Joint BMES/EMBS Conference, Atlanta, GA, Oct. 13–16, Vol. 1, p. 610.
DiGiovine, C. P. , Cooper, R. A. , Wolf, E. J. , Hosfield, J. , and Corfman, T. , 2000, “ Analysis of Vibration and Comparison of Four Wheelchair Cushions During Manual Wheelchair Propulsion,” RESNA Annual Conference, Orlando, FL, June 28–July 2, Vol. 20, pp. 429–431. http://www.varilite.com/pdfs/k_knowledge_reports_upitt_vib.pdf
VanSickle, D. , Cooper, R. , and Boninger, M. , 2000, “ Road Loads Acting on Manual Wheelchairs,” IEEE Trans. Rehabil. Eng., 8(3), pp. 371–384. [CrossRef] [PubMed]
ISO, 1997, “ Mechanical Vibration and Shock. Evaluation of Human Exposure to Whole-Body Vibration—Part 1: General Requirements,” International Organization for Standardization, Geneva, Switzerland, Standard No. ISO 2631-1:1997. https://www.iso.org/standard/7612.html
Bongers, P. M. , Hulshof, C. T. J. , Dijkstra, L. , Boshuizen, H. C. , Groenhout, H. J. M. , and Valken, E. , 1990, “ Back Pain and Exposure to Whole Body Vibration in Helicopter Pilots,” Ergonomics, 33(8), pp. 1007–1026. [CrossRef] [PubMed]
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Figures

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Fig. 1

The two ST model and the corresponding translational and rotational portions of the modeled wheelchair-user

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Fig. 2

An example of the span of the data set that was emphasized by weights 1 (shaded in gray) in the system identification procedure (a) C6 (b) and T7 (c) subject's experimental data, and identified second and fourth-order TF simulations

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Fig. 3

Experimental data at medium (35 rpm) and fast (40 rpm) drum rotation speeds, and identified fourth-order TF simulations for subjects C6 (a) and (b) and T7 (c) and (d). The TFs were identified using the 30 rpm drum rotation trials and simulated using the medium and fast drum rotation trial input forces.

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Fig. 4

The three anatomical configurations of the two-ST model

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Fig. 5

Rotational stiffness and damping coefficient solutions of the two-ST model for the C6 and T7 subjects

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Fig. 6

The response of the two ST model using the computed rotational stiffness and damping coefficients for the C6 (a) and T7 (b) subjects

Tables

Table Grahic Jump Location
Table 1 The WRMSE of the second-, fourth-, and sixth-order TF responses to the experimental data, for each subject
Table Grahic Jump Location
Table 2 The poles, natural frequencies, and damping ratios of the averaged one-ST and two-ST continuous TFs identified from Sec. 2.3 for C6 and T7 subjects

Errata

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