A Dynamic System Model of an Off-Road Cyclist

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Furthermore, Minetti and Belli (19) found that neglecting themotion of the visceral mass can cause serious errors in locomo-tion studies.

Thus, it appears from the reviewed literature that it is possibleto model pars of the body such as the ars, legs, and visceralmass using traditional lumped parameter models with linearelements. The objective of the work reported in this aricle wasto develop a dynamc system model of an off-road cyclist andidentify the parameter values that resulted in the best matchbetween simulated and experimentally measured frequency re-sponse functions for seven subjects.

A Dynamic System Model of anOff-Road CyclistTo optimize the performance of off-road bicycle suspension systems, a dynamic modelof the bicycle/rider system would be usefuL. This paper takes a major step towardthis goal by developing a dynamic system model of the cyclist. To develop the cyclistmodel, a series of four vibrational tests utilizing random inputs was conducted onseven experienced off-road cyclists. This allowed the transfer functions for the armsand legs to be determined. To reproduce the essential features (i.e., resonance peaks)of the experimental transfer functions, the system model included elements represent-ing the visceral mass along with the arms and legs. Through simulations, the frequencyresponses of the system model of the rider in each of the four ~sts were comp~ted.Optimal stifess and damping parameter values for each subject were determinedby minimizing the diference between the experimental and simulation results. Goodagreement between experimental and simulation results indicates that modeling therider as a lumped parameter system with linear springs and dampers is possible.

E. L. WangAssistant Professor,

University of Nevada, Reno,Department of Mechanical Engineering,Reno, NV 89557

M. L. HullProfessor,University of California, Davis,

Department of Mechanical Engineering,Davis, CA 95616

IntroductionThe recent growth in popularity of off-road cycling, com-

bined with the desire to increase comfort and control, has ledto an increasing use of suspension systems in off-road bicycles.To optimize. the performance of suspension systems when tra-versing rough terrain, a two-dimensional dynamic system modelof a bicycle and rider would be usefuL. To achieve this, however,an accurate dynamic description of the rider is necessar.Based on previous studies of the ars (1-3), legs (4-6),and whole body (7 -12), it appears that modeling the rider asa lumped parameter system with linear springs and dampers ispossible. While nonlinearities have been found to exist (13-15), the nonlinearties may be neglected if the model is re-

strcted to a single body configuration, loading condition, andsmall range of motion. Thus, most of the previous studies arenot directly relevant to this study because the flexion anglesused and joint moments are not typical of off-road cycling.Since the ars and legs provide significant vibration isolationfor an off-road cyclist, they should be included as model compo-nents. Wong and Hull (2) determined that for an on-road cy-cling position, the ars could be modeled as a spring anddamper in parallel from the shoulders to the hands. Similarly,Greene and McMahon ( 4 ) found that the legs could be modeledas a spring and damper in parallel from the hip to the foot. Ofthe above-mentioned previous studies dealing with the arms andlegs, these are the only two that had even remotely similar bodyconfigurations and load lvels to those seen in off-road cycling.Wilczynski and Hull (16) used the studies of Wong and Hull(2) and Greene and McMahon (4) to create a dynamic systemmodel of the rider. Their simulations showed that the range ofmotion of the.imbs during off-road cycling is sufficiently smallto justify the use of linear elements.Along with the ars and legs, several researchers have foundthe motion of the visceral mass to be important. Zagorski et al.(17) and Coermann (7) found that the internal organs had anatural frequency of 2-5 Hz and 3 Hz, respectively, for seatedsubjects. Garg and Ross (8) found the internal organs to havea natural frequency of about 2 Hz for a standing subject. Muk-sian and Nash (18) used a nonlinear model of a seated humanand found that the thoracic organs resonated at about 3 Hz.

Contributed by the Bioengineering Division for publication in the JOURALOF BIOMECHANICAL ENGINEERIG. Manuscript received by the BioengineeringDivision March 27,1995; revised manuscript received September 16, 1996. Asso-ciate Technical Editor: A. G. Erdman.

248 I Vol. 119, AUGUST 1997

MethodsThe model development involved three distinct stages: vibra-tory testing to determne transfer funtions for the ars andlegs, development of computer-based dynamic system models,and an optimization on stiffness and damping parameter values.Vibratory Tests.. To determne the frequency responsefunctions (FRs) of the ars and legs experimentally, a seriesoffour vibratory tests was conducted on seven experienced off-

road cyclists. Their masses ranged from 72.7 to. 79.5 kg (avg= 76.3 kg, std dev = 2.2 kg) and their heights ranged from1.5 to 1.83 m (avg = 1.80 m, std dev = 0.03 m). The firsttwo tests were used to determne the FRFs of the arms in flexionand the second two were used to determne the FRs of thelegs in flexion. Each subject assumed his normal riding position(i.e., flexion angles were not specified).Depending on whether the ars or legs were being tested,either a set of handlebars or a set of bicycle crank ars andpedals was mounted on the ram of a servohydraulic actua~or.Although the inputs to the bicycle/rider system can be as highas 20 Hz (16), preliminar tests showed no significant riderdynamics above 10 Hz. Thus, the ram was displaced with arandom signal (white noise) bandlimited to 12 Hz using a low-pass filter ( 120 db/ decade rolloff). The peak-to-pe~. amplitudewas set to approximately 12 mmto produce arealistic off-roadriding level of comfort (as judged by the subjects).Three riding positions were tested: seated, standing, anddownhilL. In the seated position approximately 30 percent ofthe rider's mass was supported by the hands (the rest wasdistributed over the seat and pedals). In the standing positiononly the hands (30 percent) and feet (70 percent) supportedthe rider's mass. Finally, in the downhill position, the rider'sfeet supported most (approximately 90 percent) of the rider'sweight with the hands supporting the rest.

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In the first two tests, the ars were evaluated in both theseated and standing positions. In each test, reflective markerswere placed on the ram (the input) and the subject's spinebetween the shoulders (the output) at approximately the fourththoracic vertebra. A motion analysis system recorded the mo-tions of the markers. For the seated case, the subject was seatedon a custom fixture, which positioned the seat and pedals (whichwere stationary) such that the relative positions between theseat, pedals, and handlebars matched that of a standard 46 cmoff-road bicycle (Fig. 1). In the second test the subjects stoodstraight-legged and pinched the saddle between their legs tominimize any hip motion. The neutral vertical position of thehandlebars was increased 1.27 cm to accommodate the fact thatthe hips were higher than normal (due to the straight legs).In both the standing and seated tests an instrumented handle-bar monitored the average vertical force on the right hand.This value was used to maintain the same weight distribution. between tests. A real-time .output of this force was also dis-played to help the subjects maintain a constant weight distribu-tion.The third and fourth vibratory tests were used to determnethe FRFs of the legs in both the standing and downhil positions.For both cases the subject used a pedal! shoe interface thatrigidly fixed the rider's feet to the pedals. The reflective markerused to monitor the motion ofthe rider was placed on the rider'slower spine near the third lumbar vertebra.For the standing position, the subjects were instructed to keeptheir ars locked so that the ars would not contribute tothe FRF. The instrumented handlebar ensured the same weightdistribution as in the second arm test. In the downhill positionthe subjects did nt have their ars locked, but the motion ofthe shoulders was observed to be minimaL.

Dynamic ModeL. Based on the literature search and thevibratory tests, the model included stiffness and damping ele-ments to represent the ars, legs, and visceral mass. Two differ-

Fig. 1 Experimental test setup for the arms while in the seated cyclingPosition. The reflective markers are located on the subject's spine be-tween the shoulders (near T4) and on the ram of the actuator (handle-bars).- JOurnal of Biomechanical Engineering

ent dynamic system models of the rider were developed tosimulate the four vibratory tests. The model shown in Fig. 2simulated the two vibratory tests involving the arms and themodel shown in Fig. 3 simulated the two vibratory tests involv-ing the legs. In all cases, the segment lengths were experimen-tally measured and the inertial properties calculated via theregression equations given by Chandler et al. (20).In the dynamic model used to simulate the ar tests, thears were modeled as two rigid bodies connected by frictionlessrevolute joints with a linear spring and damper in parallel span-ning the handlebars and shoulders. The arm segments weremodeled as being in a plane perpendicular to the plane of thebicycle and oriented such that the plane contained both thehands and the shoulders (note that, for clarity, the ars are notshown in this configuration in Fig. 2). Because the shouldergirdle was modeled as translationally rigid, but actually hadsignificant flexibilty, the actual mechanics of the shoulder gir-dle were included in the arm rnodel. The inertia properties ofthe shoulder were included with those of the torso. The handswere modeled as point masses attached to the end of the lowerars.The rider's head was modeled as rigidly connected to thetorso. The torso, in turn, was modeled as a rigid body (con-taining the visceral mass, discussed below) pivoting about theischial tuberosities, which were. modeled as a fixed point. Inthe seated case, the location of the ischial tuberosities coincidedwith the seat. In the standing position, the location of the ischialtuberosities was measured during the vibratory tests.In the dynamic model that simulated the leg tests, each legwas modeled as two rigid segments connected by frictionlessrevolute joints with a linear spring and damper connecting thehip to the crank spindle (bottom bracket). However, becausethe model contained only one degree of freedom associatedwith the legs, the anke was treated as a revolute joint whoselocation was fixed relative to the cranks. The feet were modeledas fixed links connecting the ankle to the cranks (also fixed).Like the shoulders, the hip joint also has some translationalflexibility. Thus, similar to the shoulder girdle, the hips weremodeled as translationally rigid, but the actual mechanics ofthe hips were included in the leg portion of the modeL.In both the standing and downhil positions, the rider's torsoand ars were modeled as rigid links. The ars were modeledas a single link connected to the handlebars and shoulders viafrictionless revolute joints. This configuration resulted in thetorso having pin joints at both ends (hips and shoulders), whichallowed translation in the horizontal direction. To prevent thismotion, the rider's hips were constrained to move vertically.For the downhill position, a torsional spring and damper wereintroduced between the legs and torso because the hip was alsomodeled as a frictionless pin joint. These elements reflected. that fact that the rider creates a moment about the hip to supporthis upper body (recall that the hands supported only about 10percent of the weight).As shown in both Figs. 2 and 3, the visceral mass was mod-eled as a point mass located inside the torso. It was attached tothe torso at the midpoint between the shoulders and hip by alinear spring and damper oriented to allow motion perpendicularto the long axis of the torso. The visceral mass was assumedto be 15 percent of the total body mass (19, 31). The inertialparameters for the torso given by Chandler et al. (20) werereduced accordingly to account for this discretization of thetorso. The resulting inertial properties are presented in Table 1.

Parameter Determination. Once the vibration tests werecompleted and the dynamic models developed, the stiffness anddamping parameter values were calculated for all four tests foreach of the seven test subjects. To accomplish this, the ampli-tude ratios were calculated for both the experiments and thesimulations. An error function was defined and the stiffness anddamping parameters were chosen to minimize the error function.AUGUST 1997, Vol. 119 I 249


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/Q:...Fig. 2 Dynamic system model representing the tests involving the arms.The input, x(t), represents the motion of the handlebars, and the output,y(t), represents the motion of the reflective. marker. Note that for claritythe arms are shown in a position different than that in which they weremodeled. Solid circles represent revolute joints.

Because of the random input, random data analysis tech-niques were applied (22). An estimate of the amplitude ratio,AR ,at frequency f was defined as

AR(f) = I ~Xy(f) ISxx(f)where Sxy(f) is the cross-spectral density function of the inputand output, and Sxx(f) is the autospectral density function ofthe input. SXy(f) and Sxx(f) were calculated by averaging 45subrecords of 2 seconds each (leading to 0.5 Hz resolution).For all of the results presented, a normalized random error ofless than 10 percent was maintained.To determne the amplitude ratio for the models, the equa-tions of motion for each model were developed using Kane'smethod (23). The equations were then integrated using a fourth-

(1I. x(t) 1

Fig. 3 Dynamic system model representing the tests involving the legs.The input, x(t), represents the motion of the crank spindle, and the out-put, y(t), represents the motion of the reflective marker. For clarity, thetorsional hip spring and damper used in the downhil position are notshown. Solid circles represent revolute joints.250 I Vol. 119, AUGUST 1997

Table 1 Averages and standard deviations of anthropometric parame_ters for the seven subjectsSel!ent Prooert Ave (std devl Seirent Prooert Ave (st devlupper ar lengt (m) 0.32 (0,01) lower ar lengt (m) 0.32 (0.1)

ma (kg) 4,19 (0.08) ma (kg) 2.S5 (0.07)iIert (kg-m') 0.031 (0,00) inerta (kg-m2) 0.016 (0.00)0.00 (0,00) 0.002 (0,00)

0.029 (0.000 0.015 0.001upper leg lengt (m) 0.44 (0.02) lower leg lengt (m) 0,44 (0.01)

ma (kg) 8.0:5 (0.28) ma (kg) 3.13 (0,09)inertallo-m2 0.144 (0.005) inerta r1~m2, 0.045 (0.01)

tors lengt (m) 0.53 (0.02) hand mass ekel 0,96 (0.03)ma (kg) 32.79 (0.92) foot mass ekgl 1.9 (0.04)inerta eke.m'i 2.221 (0.141) viscera mas (kel 11.5 (0.33)

Notes: Mass and inerta for an segments is for both left and right segments combined.Inerta for ar segments is about segment principa axes.Inerta for al.other segments is about axs perpndicuar to saggta plane.Ma and inerta for leg segments is for right only (left is ased to be the sa):Torso ma and inerta include the bead.

(1 )

order Runge- Kutta routine. The input displacements used inthe simulations were the actual displacement data taken duringthe vibration tests.The same data reduction routine that was used to calculatethe amplitude ratios for the experimental data was used on theoutput data from the simulations. The difference between the

experimental and simulation results formed the basis for thecost function:N

CF = i. (ARexp(;) - ARmodel(f ))2i=l (2)where ARexp and ARmodel are the estimates of the experimental.and model amplitude ratios respectively,f is the ith frequency,and N is the total number of discrete frequencies for which thenormalized random error was less than 10 percent. However,because the estimate given by Eq. (I) does not estimate the"valleys" of the FRF well (22), for the cases where a definite"valley" existed, the lowest frequency between the two majorresonance peaks and its two neighboring frequencies were ex-cluded from the cost function. Ths exclusion prevented "val-leys" from erroneously enlarging the cost function. The costfunction given in Eg. (2) was then minimized using Powell'smultivarate search technique (24).. Since it is not possible voluntarly to alter the properties ofthe visceral mass, it was assumed that the stiffness and dampingof the visceral mass did not depend on riding position. Thus,once the optimal parameter values were obtained for all fourtest cases, the parameter values associated with the visceralmass were averaged over the four tests for each subject individu-ally. The parameter optimizations were then performed againwith the visceral parameters constrained to the averaged values.Results

The optimal values for the stiffness and damping parameters .for the ars, legs, and visceral mass are presented in Table 2.The stiffness and damping values for the hip (downhill caseonly) are not shown because the parameter optimization didnot cause changes from initial values (300 N-m/rad and 50 N-

Table 2 Stiffness and damping parameter values

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(0.01)(0.07)(0.000)(0.00)

(0.01)(0.09)

(0.03)(0.04)

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r;m-slrad). Furthermore, manual adjustment of these values bya factor of ten had no effect on the results.The varation between subjects was quite large for most ofthe parameter values (Table 2). For example, for the stiffnessof the legs in the standing position, the largest value was 2.5times larger than the smallest value.

The damping for the ars increased for every subject whencomparng the seated and the standing cases. The stiffnessshowed a similar trend for six of the seven subjects. The legsdisplayed a consistent trend of decreasing stiffness and dampingfor every subject when comparing the standing and the downhilriding positions.The stiffness values for the visceral mass were fairly consis-tent both within a subject (four cases) and among subjects, withthe average being about 19 kN /m. Within a subject the stiffnessvalues vared a maximum of 13 percent (relative to maximum)between test cases. The damping values, however, were ex-tremely inconsistent (both within and between subjects) withthe largest value being nearly 20 times that of the smallest.The large varation in the damping of the visceral mass canbe seen iu the plots of the amplitude ratios (Figs. 4 and 5). Ingeneral, subjects could be divided into two categories based onthe prominence of the second resonance peak (five of sevensubjects exhibited prominent second peaks). Because of thesimilarty of the results, those for only two of the subjects (onefrom each category) are shown.The agreement between the experimental data and the com-puter simulations was quite good (Figs. 4 and 5). The largestdiscrepancy was in the prediction of the second peak (Fig. 4).

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The discrepancy was apparent in all the subjects that displayeda prominent second peak.Although the amplitude ratio plots are not shown for all thesubjects, the frequency at which the resonance peaks occurredwas fairly consistent. For the legs in the downhill position, thefirst peak ranged froID 1 to 2.5 Hz. For the legs in the standingposition, the peak always occurred between 2 and 4 Hz. Forthe arms in the seated position, the peaks occurred between 2and 3.5 Hz. For the arms in the standing position, the peak wasbetween 2.5 and 3.5 Hz. Finally, for the visceral mass, the peakvared from 6 to 8.5 Hz.DiscussionIn creating the dynamic models, several assumptions weremade. The most basic assumption was that the system could bemodeled using linear elements. While the elements were linear,nonlinearities in the model could exist due to geometrical fac-tors.Preliminary tests using a swept sine wave to determine theamplitude ratios indicated another source of nonlinearities. Fora single-component sine wave, the rider was able to adapt hisresponse by varing the muscle forces without varying flexionangles (through co-contraction). To minimize the possibility ofadaptation, random inputs were used. Qualitatively, each subjectstated that the inputs felt very similar to actual off-road cycling.This nonlinear behavior of muscles also prevents a meaningfulcomparison of the results presented in this study to the resultsof the previous studies by Wong and Hull (2) and Greene andMcMahon (4).

Arms (standing position)0.90.8

0 0.7=i:.. 0.6't=;i 0.5'ae0.4-0.30.2 a

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o 2 4 6 8 10 12Frequency (Hz)Fig.4 Experimental (points) and simulation (solid line) amplitude ratios for all four test cases for Subject7. These plots are representative of those subjects for which a prominent second peak was evident. Onlythe amplitude ratios for which the normalized random error was less than 10 percent are reported.

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Arms (seated position) Arms (standing position)1.2 1.0

1.0 0.8~ 0'l~ 0.8 ~Il Il 0.6.. ..'t 0.6 't= .~":. . 0.4e 0.4 e~ ~

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0.0 0.00 2 4 6 8 1 a 12 0 2 4 6 8 10 12Frequency (Hz) Frequency (Hz)

Legs (standing position) Legs (downhil position)2.00 2.001.75 1.75

0 1.50 01.50'l 'l~ ~Il 1.25 Il 1.25.. ..'t 1.00 'g 1.00=.": ~. 0.75 lO.75~ 0.50 ~ 0.50

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0 2 4 6 8 10 0 2 4 6 8 10Frequency (Hz) Frequency (Hz)Fig.5 Experimental (points) and simulation (solid line) amplitude ratios for all four test cases for Subject5. These plots are representative of those subjects for which a prominent second peak was not evident.Only the amplitude ratios for which the normalized random error was less than 10 percent are reported.

Another assumption was that the second degree of freedom(DOF) was due to the motion of the visceral mass. While it ispossible that the second DOF was the head/neck in the artests and the feet! ankles for the leg tests, the likelihood thateach subject had the same ankle and neck stiffness tended torule out this possibility especially in light of the varations thatexisted for the ar and leg parameter values.There is also the possibility that during the ar tests, theDOF was due to the compliance of the seat fixture for the seatedcase and of the legs (knees locked straight) for the standingcase. However, the likelihood of this is remote when consider-ing the fact that this would explain the second resonance peaksfor only the ar tests. The second resonance peak for the legtests would have to come from another source (e.g., the head),and it is highly improbable that it would have the same reso-nance frequency as in the ar tests.To prevent horizontal translation of the torso in both thestanding and downhil positions, the hips were constrained tovertical movement. In reality, moments generated at the shoul-ders and hips are used to balance the rider, but these momentsare ultimately resisted by horizontal forces at the handlebarsand pedals. Thus, creating a constraint force between the riderand bicycle frame seems reasonable.While the logic behind the use of the hip spring (to supportthe upper body) is reasonable, the insensitivity of the model tothis parameter seems paradoxicaL. For a particular case andsubject, the initial angle of rotation was such that the inclinationof the torso was the same for each simulation. Although thespring must be present to support the weight of the torso, therange of motion for the hip angle was so small (a few degrees)252 I Vol. 119, AUGUST 1997

that the results were insensitive to this parameter within therange tested.The lack of interaction between the ars and legs was anotherassumption. Correct body positioning required that the subjectdistribute his weight on both hands and feet, which allowed forthe possibility of the arms and legs interacting during the stand-ing and downhill tests. To minimize this possibility, the subjectslocked either their ars or legs as required by the pariculartest. Since the model of the rider in the downhill position isessentially the same as the model of the rider in the standingposition minus the ars and the relative accuracy of each modelwas similar (Figs. 4 and 5), it appears that little interactionbetween the arms and legs existed.Inherent in. the modeling of the visceral mass motion werethe assumptions that the torso acted as a rigid body and thatthe visceral mass moved perpendicular to the long axis of thetorso. In each of the studies cited earlier involving whole bodyvibration, the visceral mass moved vertically (along the longaxis of the torso) with a natural frequency of 2-5 Hz (17 -20).Although this motion may have been present, it was probablysmall because the only mechanisms that would cause this mo-tion were the centrifugal forces from the rotation of the torso.Thus, neglecting the longitudinal mode was probably justified.As for the flexibility of the torso, any motion would havebeen perpendicular to the long axis. This would have acted toalter the effective mass and location of the viscera. While theuse of a different visceral mass and location would change theresulting ptimal stiffness and damping parameter values, thereis no guarantee that the discrepancy between simulation andexperimental results would be decreased.

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In addition to modeling assumptions, the possibilty of mea-surement errors due to soft tissue motion existed. However, themarkers were placed on the subject's spine where the skin isvery thin. Additionally, large motion of the skin in the directionof travel (perpendicular to the spine in the plane of the bicycle)is not possible. Consequently, it was concluded that soft tissuemotion was negligible.The issue of the model's uniqueness is also important. Whilethe model developed is most likely not unique in that othermodel forms could attain similar accuracy, the model developedis physically plausible.A related concern is that the number of parameters (four foreach case) also allows for the possibility of non-unique set ofparameter values for the model developed. That is, a differentset of parameter values may result in a similar cost functionvalue. While local minima were checked for (by restaring theoptimization at different initial conditions), uniqueness of thesolution was not.While the possibility of non-unique solutions makes interpre-tation of the results (e.g., trends in stiffness values) difficult,the relatively good agreement between the experimental andsimulation results encourages some discussion regarding theutility of the models. Recall that the motivation for this studywas to develop a dynamic system model for an off-road cyclistso that off-road bicycle suspension systems can be optimized.The model of the cyclist described herein could be combinedwith a dynamic model of an off-road bicycle (25) to create adynamic system model of the bicycle and rider. The combinedmodel could then be "ridden" over simulated terrain and vari-ous quantities such as suspension motion, acceleration of spe-cific points, and rider-bicycle interaction loads could be calcu-lated. These quantities could be used in an objective functionto optimize suspension characteristics.

To determne the robustness of the optimal design, a sensitiv-ity analysis to the input varables also would be usefuL. Becausethe input varables (i.e., stiffness and damping parameters) as-sociated with the cyclist are not independent varables, in thiscase a sensitivity analysis would require changing all the var-ables simultaneously (i.e., use a different rider). The parametervalues given in this paper would serve for such an analysis.

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ConclusionsThe stiffness and damping constants for the ars, legs, andvisceral mass were determned for seven subjects in three off-road cycling positions. A series of four vibrational tests utilizingrandom inputs provided the experimental amplitude ratios.Computer-based dynamic models of the rider in each of thefour tests were developed and used to determne the optimalstiffness and damping parameter values for each subject. Basedon the relatively good agreement between the experimental andsimulation results, it is worthwhile to use the model in simulat-ing the response of a rider/bicycle system to optimize off-roadbicycle suspension performance.

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Journal of Biomechanical Engineering

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A Dynamic System Model of an Off-Road Cyclist

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