J. Biomechonics

Vol. 25, No. 1, pp. 41-53,

1992.

0021~9290/92

Printed in Great Britain

Pergamon

THREE-DIMENSIONAL KNEE JOINT LOADING SEATED CYCLING

5WU+.oO Press pk

DURING

PATRICIA RUBY, M. L. HULL* and DAVID HAWKINS Department of Mechanical Engineering, University of California, Davis, CA 95616, U.S.A. Abstract-The hypothesis which motivated the work reported in this article was that neglecting pure moments developed between the foot and pedal during cycling leads to a substantial error in computing axial and varus/valgus moments at the knee. To test this hypothesis, a mathematical procedure was developed for computing the three-dimensional knee loads using three-dimensional pedal forces and moments. In addition to data from a six-load-component pedal dynamometer, the model used pedal position and orientation and knee position in the frontaf plane to determine the knee joint loads. Experimental data were collected from the right leg of 11 male subjects during steady-state cycling at 90 rpm and 225 W. The mean peak varus knee moment calculated was 15.3 N m and the mean peak valgus knee moment was 11.2 Nm. Neglecting the pedal moment about the anterior/posterior axis resulted in an average absolute error of 2.6 N m and a maximum absolute error of 4.0 Nm in the varus/valgus knee moment. The mean peak internal and external axial knee moments were 2.8 N m and 2.3 N m, respectively. The average and maximum absolute errors in the axial knee moment for not including the moment about an axis normal to the pedal were found to be 2.6 N m and 5.0 N m, respectively. The results strongly support the use of three-dimensional pedal loads in the computation of knee joint moments out of the sagittal plane.

. ..

NOMENCLATURE X, r, Z

x, Y, z

x’, y ‘, z’

A, g, C

8, 8, e

a0

. .. 4 a, a

AAB

axes of inertial coordinate system with X-Z forming a plane parallel to the plane of the bicycle (sagittal plane) and Y mutually perpendicular to X and Z axes of local pedal coordinate system with positive x directed towards the front of the pedal along the pedal platform, positive z normal to the pedal platform and y mutually perpendicular to x and z axes of local knee joint coordinate system with origin located at the point midway between the femoral condyles. The z’ axis is coincident with a line connecting the origin and the midpoint between the medial and lateral malleoli with the leg fully extended. Positive z’ is directed coronally. The x’ axis is perpendicular to z’ in the sagittal plane. Positive x’ is directed anteriorly. Positive y’ is mutually perpendicular to x’ and z’ and medially directed transformation angles from the pedal coordinate system to the ankle coordinate system, from the ankle coordinate system to the knee coordinate system and from the pedal coordinate system to the knee coordinate system, respectively angular position, angular velocity and angular acceleration, respectively, of the crank arm; angle is positive clockwise from vertical absolute angular position of the pedal; positive clockwise from horizontal angular position, angular velocity and angular acceleration, respectively, of the foot relative to the crank arm; angle is positive clockwise

. ., 44474 .

Y7% Y

9

c, p. t s, t, b 0,

P. a, k, h

L,, Lr, L,, L, 4, CG

%,

ms,

m,

I Ya

in final form 30 May 1991.

Mx.9

*Author to whom correspondence should be addressed. BM 25:1-o

CG,

CGs,

Fx.7

Received

> CG,,

41

Fyw

MY.,

Fz,

Ma

angular position, angular velocity and angular acceleration, respectively, of the foot; angle is positive clockwise from horizontal angular position, angular velocity and angular acceleration, respectively, of the shank; angle is positive counterclockwise from horizontal angular position, angular velocity and angular acceleration, respectively, of the thigh; angle is positive counterclockwise from vertical linear acceleration of the shank center of gravity along the inertial X and Z axes, respectively; positive anteriorly and vertically upward acceleration due to gravity links corresponding to the crank, pedal, foot, shank, thigh and bicycle, respectively axes of rotation of the crank spindle, pedal spindle, ankle joint, knee joint and hip joint, respectively lengths of the crank, foot, shank and thigh, respectively horizontal position of the center of the knee relative to the center of the pedal in the frontal plane positions of the center ofgravity of the foot, shank and thigh in the sagittal plane, respectively, measured from the proximal end of the corresponding link horizontal position of the shank center of gravity in the frontal plane relative to the center of the knee joint mass of the foot, shank and thigh, respectively moment of inertia of the shank about the center of gravity in the sagittal plane ankle joint forces in the inertial coordinate system ankle joint moments in the inertial coordinate system

42

P.

RUBY et al.

INTRODUCTION

The past decade has witnessed a large increase in the popularity of cycling, both as a recreational activity and as a competitive sport. Along with this increased participation has come an increase in cycling-related knee injuries (Holmes et al., 1989; Zahradnik, 1990) and, hence, an interest in the loading of the knee joint during cycling. The loads at the knee joint during cycling can be determined from a biomechanical model. Biomechanical models which have been developed previously for cycling may be divided into two categories, those that consider the leg motion to be confined to the sagittal plane and those that include motion out of the sagittal plane. Sagittal plane models include those by Bratt and Ericson (1985), Gregor et al. (1985) and Hull and Jorge (1985). Although the techniques used to measure input data differ, the model formulations are similar in that the kinematics of the bicycle/rider system are combined with both the normal and tangential pedal loads to determine intersegmental moments and for-

k

ces at the joints. The planar nature of these models, however, inherently precludes providing any information on out-of-plane knee loads (i.e. medial/lateral force, and axial and varus/valgus moments), which presumably contribute to overuse knee injuries. Although the three-dimensional motion of the leg in cycling has been studied experimentally (Knutzen and Schot, 1987; Boutin et al., 1989) as well as analyzed using the kinematic theory for robotics (Shen and Radharamanan, 1988), only the model by Ericson et al. (1984) determined any out-of-plane loads at the knee joint. Ericson et al. (1984) determined the varus/ valgus knee joint moment using only the vertical and transverse pedal forces and the knee position relative to the applied load in the frontal plane. Although not stated, such an analysis assumes that moments at the pedal in the transverse and frontal planes are negiigible. However, in a study conducted by Davis and Hull (1981), where the three-dimensional loads at the pedal were measured, these moments were on the order of 5 Nm, which is approximately 20% of the mean peak varus knee moment reported 6y Ericson et al. (1984). Therefore, it is hypothesized that an accurate evaluation of three-dimensional knee joint loads must be based on a complete set of three-dimensional pedal forces and moments. Two objectives were established to test this hypothesis. The first objective of this research was to develop a model which combined sagittal and frontal plane mechanics with three-dimensional pedal loads to determine three-dimensional knee loads. The second objective was to then use this model to evaluate the error incurred by not considering either out-ofplane motion or all six components of the load at the pedal.

METHODS The

a0

Fig. 1. Sagittal view of five-bar linkage model. Links are the crank, foot, shank, thigh and bicycle (fixed). Pivot points at o (crank axis), p (pedal axis), a, k and h. The foot link is an imaginary line connecting points p and a and not the actual foot segment. The model assumes that the motion is constrained to one plane, that the hip (h) is 6xed and that the knee joint cannot extend past the straight leg position.

analytic model

A five-bar linkage with motion in the sagittal and frontal planes was used to model the bicycle/rider system. The model can be viewed as two separate models which were superimposed upon each other to oljtain the complete description of the system. One portion of the model was a planar representation of the bicycle/rider system used to analyze the kinetics of cycling within the sagittal plane. The other portion of the model was a quasistatic model of the bicycle/leg system in the frontal plane. To develop the model, three orthogonal coordinate systems were used. One was the inertial coordinate system where the X-Z axes define the sagittal plane (Fig. 1). The other two coordinate systems were local, with the origin of one located at the center of the pedal and the origin of the other located at the midpoint of a line connecting the lateral and medial femoral epicondyles (Fig. 3). Note that, for both local systems, the y (pedal) and y’ (knee) axes were perpendicular to the sagittal plane.

43

Three-dimensional knee joint loading during seated cycling Derived from the work of Hull and Jorge (1985), the planar portion of the model consisted of a five-bar linkage constrained to the sagittal plane with fixed points of rotation at both the hip joint and crank axis (Fig. 1). Hip joint position is specified relative to the crank axis. To completely constrain the model, six kinematic inputs and one geometric constraint are needed. The kinematic inputs consisted of the angular position, velocity and acceleration of both the crank and the pedal relative to the crank represented by 0 and CL in Fig. 1 and their derivatives, respectively. The model was geometrically constrained by not allowing the knee joint to extend past the straight leg position. Equations for the sagittal plane joint loads were obtained by writing the equations of motion for each segment based on the free-body diagrams of Fig. 2. In the notations of Figs 1,2 and 3 the sagittal plane force and moment equations at the knee joint in the local

knee joint coordinate system become F,. = (F,, + M&

sin 4 -[F,, + m,(s + ~&os 9, (1)

F,. =(Fxa +msaxs) cos 4 + CF,, +m,(g +

a&in 4, (2)

M,! = ly,$ + My,--m&G,

-a&G,

cos 4 + m&CG,

cos 4) + FX,L, sin 4 - F,,L,

sin 4 cos 4. (3)

Definitions of the symbols in the above equations are given in the nomenclature. Anthropometric parameters specified for each segment were the length, mass, moment of inertia and center of gravity location. The length of the foot was defined as the distance from the head of the fifth metatarsal to the lateral malleolus. Shank length was measured from the lateral malleolus to the lateral

Myh

Fxk

Fza z4

FXS Fza

I mrg

Fig. 2. Sagittal plane free-body diagram of leg. Shown are the forces and moments acting on each leg segment in inertial coordinates. F,, and FZP are the measured pedal forces.

P. RUBYet al.

(a)

Fig. 3. Local coordinate systems at the pedal and knee joint. (a) The local pedal coordinate system is located at the center of the pedal platform with the x axis tangent to the pedal, the y axis parallel to the pedal spindle and the z axis normal to the pedal. (b) The local knee joint coordinate system is fixed with the tibia and the origin is located at the point midway between the center of the femoral condyles with the knee fully extended. The z’ axis is coincident with a line from the origin to the midpoint between the medial and lateral malleoli (ankle). The x’ axis is perpendicular to the z’ axis in the sagittal plane and the y’ axis is mutually perpendicular to both the x’ and z’ axes.

femoral epicondyle while the thigh length was measured from the lateral femoral epicondyle to the greater trochanter. The remaining anthropometric parameters were determined based on the work of Plagenhoef (1983). The experimental variables needed to solve for the sagittal plane knee joint loads of equations (l)-(3) include the crank angle, relative pedal angle with respect to the crank, normal pedal force and tangential pedal force. Both the crank angle and the relative pedal angle were measured using continuous rotation potentiometers (Hull and Davis, 1981). How these measured angles were used to determine the kinematics of the model will be described shortly. Both the normal and tangential pedal forces were measured using a six-component pedal dynamometer described by Hull and Davis (1981). A quasistatic model of the bicycle/rider system in the frontal plane was used to determine the remaining knee joint load components (Fig. 4). This model assumes that the frontal plane inertial contributions to the overall joint loads are second-order quantities. Thus, using the notation of Figs 1 and 4, the frontal plane knee joint loads become F,. = F,,

(4)

M,. = - M,(cos A cos B + sin A sin B)

+ F&

- mr&,

cos C - m,gCG,

sin 4,

A cos B)

- M,(cos A cos B + sin A sin B) - F,,g L, cos B-F,&,

cos C-F&

sin C

- m,gCG,, cos 4 - m,g&, cos 4,

(6)

where A=/l-aO,

B=/Y+4-n/2, C=a,+4-n/2.

Refer to the nomenclature for a definition of all the symbols in the above equations. For the purposes of discussion, the knee joint moment equations (5) and (6) may be written in a simplified format where each term is referred to as a component of the total moment due to a specific pedal load as follows: M,. = M,.(M,)+ + MJF,)

MJM,)

+ M,.(F,)+

M,.(F,)

+ M,,(m,g + mfg),

(7)

M,.=M,.(M,)+M,.(M,)+M,.(F,) +M,4F,)+M,4F,)+Mz4m,g+m~g).

(8)

Thus, for example, M,.(M,) M,.(F,)=

+ M,(sin A cos B - cos A sin B) - F,(Lf sin B + L,) - F,&,

M,, = M,(cos A sin B-sin

= - M,(cos A cos B + sin A sin B), -F,&,

sin C,

M,.(M,) = - M,(cos A cos B + sin A sin B),

sin C

M,.( F,) = - F,L,

sin 4 (5)

sin C.

The variables needed to solve the above frontal plane knee joint load equations are the position of the

Three-dimensional

knee joint loading during seated cycling

reflective marker

reflective marker

’ Fz Fig. 4. Frontal plane free-body diagram of lower leg. Shown are the applied pedal loads F,, F,, M, and M,, both the foot and shank weight (IV, and W,, respectively) and the loads of the femur on the tibia Fye, M,+ and M,, . Loads at the pedal and knee are shown in the local coordinate systems. In the

view above, the pedal is horizontal and the shank is vertical. The model assumes that the leg is quasistatic in the frontal plane and the center of gravity of the foot is aligned with the center of the pedal.

knee relative to the pedal, the position of the shank and foot centers of gravity relative to the knee, the normal, tangential and transverse pedal forces and the pedal moments about the local x and z pedal axes. The frontal plane position of the knee relative to the applied load at the pedal along the inertial Y axis was measured by tracking two reflective markers, one mounted over the tibia1 tuberosity and the other at the center of the pedal. The markers were tracked using a video-based motion analysis system from Motion Analysis Corporation. In the frontal plane the shank center of gravity was located along a line connecting the knee marker and a reflective marker placed on the lower shank proximal to the ankle joint, as illustrated in the frontal plane model of Fig. 4. The pedal loads were measured as previously described using a sixload-component pedal dynamometer. Test procedures Eleven male cyclists whose experience ranged from commuter to Category 3 racer participated in the study. The age of the subjects ranged from 23 to 45

45

years (mean 29 years), the heights ranged from 1.71 m to 1.84 m (mean 1.78 m), and the masses ranged from 66.4 kg to 82.7 kg (mean 74.3 kg). Data were collected from the right lower extremity while the subjects rode Trek 1500 bicycles on a Velodyne road simulation trainer. The protocol for the test was established to simulate typical riding conditions for a recreational rider. The test bicycle was set up with the same approximate frame size, seat height, seat position and stem length as the subject’s personal bicycle. All subjects were required to ride with cleated shoes and toe clips which, when necessary, were provided and aligned during a warm-up period. The tests were performed at a workload of 225 W and a cadence of 90 rpm in a 52 x 19 gear. Prior to testing, the equipment was both aligned and calibrated. Alignment of the bicycle and video camera was accomplished by matching the pedal path and the vertical axis of the video system to the line of a plumb bob. It was also necessary to calibrate both the Velodyne and the motion analysis system, and to measure offsets for the pedal dynamometer and crank and pedal potentiometers. Following alignment, calibration and a 15min warm-up period, data were collected at a sampling rate of 180 Hz for 8 s while each subject pedalled at a steady cadence. Transducer data were collected on a Macintosh II, while frontal plane position data were simultaneously collected by a motion analysis VP320/Sun 3-110 system. Synchronization of the Macintosh and motion analysis data acquisition was accomplished by using a digital trigger connected to both systems to initiate data collection. Approximately 11 cycles of data werecollected during each trial. Two trials were conducted for each subject; both trials were in the same test session. One trial of data per subject was selected for processing and calculating knee joint loads using specially written software. Processing of the pedal data was accomplished with software that calibrated voltage outputs to loads (Hull and Davis, 1981). Another program calculated both the angular position of the crank and the absolute angular position of the pedal using the crank and pedal potentiometer data (Newmiller et al., 1988). The program to calculate the knee joint loads first smoothed the crank and pedal angle data using a fourth-order, zero phase shift low-pass filter with a cutoff frequency of 5 Hz. The program then calculated the configuration (i.e. angles) of the model. Next the angular velocity and acceleration, and the center of gravity velocity and acceleration for each link were calculated using a finitedifference technique. Then, using the pedal loads, the computed kinematics and the motion analysis data, the software solved equations (l)-(6) yielding both the forces and the moments at the knee joint in the local knee joint coordinate system. Finally, the multiple cycles of joint loads were averaged together to obtain one complete cycle of results per subject.

P. RUBY et al.

46 RESULTS

Illustrative example results for one subject

To illustrate the relationships between loads applied to the pedal, the position of the knee joint and the knee joint loads, the results for one subject (subject 11) are presented as examples in Figs 5-9. Inasmuch as the six pedal load components have been described in detail elsewhere (Hull and Davis, 1981; Bolourchi and Hull, 1985), the description of these plots will focus on the knee loads. In this description, it is important to note that the load components presented are the loads exerted by the femur on the tibia. The polarity or sign of the loads may also be interpreted as tendencies for relative motion of the tibia with respect to the femur. Such movement would occur only if motion of the tibia with respect to the femur were not constrained. Thus, a positive F,, possibly causes a posterior translation of the tibia on the femur whereas a negative F,. corresponds to an anterior translation of the tibia relative to the femur. A positive F,, results in lateral translation of the tibia on the femur while a negative F,, indicates a medial translation. A force directed cranially (positive F,.) has a tendency to distend the joint while a negative F,. tends to compress the joint. For the moments, a positive M,. could possibly cause a varus rotation of the tibia with respect to the femur and a negative M,, is indicative of a valgus rotation, Accompanying a positive M,,, is a net flexive muscle moment to equilibrate the knee joint whereas a net extensive moment is indicated by a negative M,,. Finally, a positive M,, tends to cause external rotation of the tibia relative to the femur while a negative M,. tends to cause internal rotation. A comparison of the knee joint forces [Fig. 7(a)] to the pedal forces [Fig. S(a)] reveals similarities between F,. and F,, F,, and F,, and F,. and F,. In fact, F,. is

-0

l&l

90

3&l

270

Crank angle (de@

8)

-4’ 0

.

’

’

90

180 tbnk

.

’ 270

. 360

an& tde#

Fig. 5. Pedal loads for subject 11. Loads shown are the loads exerted by the foot on the pedal. (a) The tangential pedal force (F,) is a positive (forward) shear force over the entire crank cycle. The transverse pedal force (FJ pushes outward during the power phase and towards the bicycle during the recovery phase. The normal pedal force (F,) is the principal force which propels the bicycle and is greatest between 90 and 180”. (b) Pedal moments are M, due to inversion/eversion of the foot on the pedal, MYwhich balances F,, and M, due to abduction/adduction of the foot on the pedal.

0.04 ,

crank angle (deg.1

Fig. 6. Horizontal position of the knee relative to the pedal for subject 11. The horizontal position of the knee relative to the center of the pedal in the frontal plane. This offset of the knee creates a moment arm for the F, and F, pedal forces about the x’ and z’ knee joint axes. This moment arm is greatest near 90”. In this case the knee is inside the plane of the pedal (adducted) throughout the crank cycle, as indicated by the positive value. A negative value would denote a knee position outside the pedal plane (abducted).

41

Three-dimensional knee joint loading during seated cycling

identical to F, as a result of the assumption that the inertial contribution of the shank in the frontal plane Fx' Fy' is small and the y and y’ axes are parallel. One noted exception to the similarities is an increasing spike occurring in F, at approximately 160”. This spike is the result of acceleration of the shank as the shank angle begins to decrease. Differences in the magnitude and phase of F, and F,, as well as of F, and F, are partly due to the different orientations of the local coordinate systems at the knee joint and pedal and partly to the shank acceleration [see equation (I)]. -. 270 360 0 90 180 The knee moments M, and M,. in Fig. 7(b) are Crank angle (deg.1 similar to each other in a number of ways. Even though M,. is considerably smaller in magnitude than M, , both moments are negative during the first half of the crank cycle, equal to zero between 150” and 180” and reach their maximum values at approximately 190”. In fact, examining the pedal loads in Fig. 5 shows that both M,. and M,. are in phase with the transverse pedal force F,. The increase in the Mzr profile around 90” suggests the influence of pedal loads other than F,. The large normal (F,) and tangential (F,) pedal forces [Fig. 5(a)] in combination with out-of-plane motion (Fig. 6) in this region could be the source of the increased moment as could the pedal moment M, [see equation (6)]. The sagittal plane knee moment, My., is partly Crank MIJIC(deg.) responsible for providing the work which propels the bike forward. My, does not appear related to any Fig. 7. Knee joint loads for subject 11. Loads shown are single pedal load because, as shown in equation (3), the loads exerted by the femur on the tibia. this load is a superposition of ankle joint loads and (a) Anterior/posterior force (F,.), medial/lateral force (F,!) and axial force (F,,). The knee force profiles are similar in inertial terms. The magnitude of the inertial terms is both shape and magnitude to the corresponding pedal forces such that the influence of the pedal loads on F,, and in Fig. 5. (b) Varus/valgus moment (M,.), flexive/extensive F,, cannot dominate the equation. moment (MY,)and internal/external axial moment (M,.). To appreciate the contributions of the individual Note that M,,is plotted with an expanded axis to the right of terms in equations (7) and (8) to the net varus/valgus (b) for visual clarity. M,, and, to some extent, M,. are similar and axial knee moments, respectively, Figs 8 and 9 in shape to the pedal force F, in Fig. 5. (a)

I 90

180

I 270

-

Mx’

.““..“...““”

Mx’(Fx)

-

Mx’(Fy)

-

Mx’(Fz)

-

Mx’(Mx)

-

Mx’(Mz)

360

Crank angle (deg.) Fig. 8. Contributions of Individual pedal loads to the varus/valgus knee moment (M,,) of subject 11. Shown is the M,, knee moment and the contribution of each of the pedal loads to this moment as detined in equation (7). Superposition of the components results in the overall M,. moment. The three most dominant terms are MJF,), MJ F,) and M,.(MJ.

48

P. RULIY et al.

external

Mz' ___....._, Mz’(Fx)

-0

90

180

270

-

Mz'(Fy)

-

Mz’(Fz)

-

Mz’(Mx)

-

Mz’(Mz)

360

Crank angle (deg.)

Fig. 9. Contributions of individual pedal loads to the axial knee moment (M,.) of subject 11.Shown is the knee moment M,.and the contribution of each of the pedal load components to this moment as defined in equation (8). Superposition of the components results in the net M,. moment. All pedal load components contribute to the net moment M,. . plot both the net moments and the individual terms. Note that M,.(m,g +mrg) from equation (7) and M,(m,g + qg) from equation (8), the contribution to the knee moments due to the weight of the shank and foot, are not included in Figs 8 and 9, respectively. It was found that these terms were near zero throughout the crank cycle due to the small moment arm of the shank’s center of gravity about the knee and the small mass of the foot. Figure 8 illustrates that out of the five remaining pedal load terms in equation (7) only three contribute significantly to M,.. The most dominant term in Fig. 8 is M, ( F,), which is the moment created by the pedal force F, acting on the leverage offered by the shank. The second most dominant term is M,.( Fz) due to the normal pedal force acting on the leverage created by the adducted position of the knee relative to the center of the pedal. As expected, M,(F,) is approximately zero near the top of the crank cycle, where the knee is nearly vertically aligned with the pedal (see Fig. 6). The remaining term which contributes to the overall value of M,. is the component associated with the moment at the pedal, M,.(M,). The breakdown of M,, into each of the pedal load component contributions shown in Fig. 9 is much more complicated than for M,., with no one term dominating throughout the crank cycle. Again, the transverse pedal force. produces the largest component of M,. as shown by the term M,(F,,). However, both M,.( F,) and M,.( F,) of equation (8) are shown in Fig. 9 to counteract the negative moment of M,.( Fy) in the first half of the crank cycle. Both M,.(F,) and M,.(F,) reach peak positive values at 81”, which corresponds to the most adducted position of the knee relative to the pedal (Fig. 6). The only term other than M,.( F,,) which is nonzero during the recovery phase is M,.( M,) which acts in opposition to M,,(F,) to reduce the absolute value of M,, .

Components of varus/valgus and axial moments for all subjects The example results in Figs 8 and 9 for a single subject are not representative of all subjects because the importance of each of the terms in equations (7) and (8) is subject-dependent. Table 1 lists, for M,., the peak absolute value for each of the pedal load component terms in equation (7) for the eleven subjects. Examination of Table 1 shows that MJF,) has the greatest maximum absolute value for all subjects and is, therefore, the most important term in equation (7). It appears from the individual subject data in Table 1 that neglecting either M,.(F,) or M-J M,) would produce the second-largest error in calculating M,.. Neither term, however, exhibits a trend which extends over the entire sample of subjects. The results in Table 1 also suggest that the smallest error in calculating M,. would result from neglecting M,.(F,) and M,(M,), as these terms have the smallest maximum absolute values. The difference in importance of MJF,) and M,.(M,) mentioned above can be explained by the frontal plane motion of the knee relative to the pedal for subjects 1 and 10 (Table 2). Table 2 lists the position of the knee in its most adducted and abducted position and the crank angles at which these extremes occur. Note in Table 2 that the knee is adducted (i.e. inside the vertical plane of the pedal) 3.3 cm for subject 1 at the beginning of the power phase when F, is near its maximum value whereas for subject 10 the knee is only adducted 0.2 cm in this region. Because the moment arm for the normal force, F, is greater for subject 1 than for subject 10, M,( F,) from equation (7) is greatest for subject 1 as listed in Table 1. Table 3 contains the peak absolute values of the pedal load component terms in equation (8) for the axial knee moment. The values in Table 3 may also be

Three-dimensional

knee joint loading during seated cycling

49

Table 1. Peak values of individual pedal load contributions to M,,. Peak values of the components which make up the M,,

knee moment due to the pedal loads F,, F,,, F,, M,, M, as defined in equation (7). The two most dominant terms are M,.(F,) and M,.( F,) Peak values of terms in equation (7) M,(max) (N m)

Subject 1 2 3 4 5 6 7 8 9 10 11 Average of absolute values of peak terms SD.

M,.(min) Wm) -

MJF,) (N m)

M,,(F,) (N m)

M,.(Fz) (Nm)

M,,(M,) (N m)

MJM,) (N m)

17.33 5.59 9.22 13.37 15.49 16.37 11.69 16.39 -0.29* 5.98 12.45

10.93 13.19 14.72 12.05 10.08 - 1.22 -24.72 -9.63 -27.04 -28.45 - 16.22

-0.59 -0.78 -0.51 -0.44 -0.64 -0.58 0.10 0.28 0.09 0.06 -0.77

16.24 -25.30 - 22.82 - 19.59 - 12.14 13.56 -23.62 16.22 - 29.36 -24.55 -23.54

7.12 13.49 6.54 7.69 8.02 7.28 -3.95 - 1.78 1.73 -1.06 10.80

- 1.68 2.67 3.99 3.38 -2.67 2.06 3.14 2.23 1.17 -3.01 - 2.98

0.36 0.42 0.67 1.35 -0.62 1.00 0.17 -0.70 - 0.40 -0.39 -0.34

11.23 5.57

15.29 8.32

0.44 0.27

20.63 5.45

6.31 3.91

2.63 2.77

0.58 0.34

*The M,, knee moment for subject 9 remained negative over the entire crank cycle. This value is the maximum value of the M,, moment measured. S.D.: standard deviation. Table 2. Horizontal motion of the knee in the frontal plane relative to the center of the pedal. Peak horizontal deviations of the knee joint center from a vertical line through the center of the pedal. A positive value means the knee is adducted (inside the pedal plane) and a negative value indicates abduction (outside the pedal plane). The crank angle at which the medial and lateral extremes occur is included, as well as the distance between the two extremes

Subject 1 2 3 4 5 6 7 8 9 10 11

Extreme medial deviation (cm)

Crank angle at extreme (degrees)

Extreme lateral deviation (cm)

Crank angle at extreme (degrees)

Range of motion (cm)

3.3 4.8 2.3 2.8 3.1 3.1 - 1.0 0.2 0.5 0.2 3.7

62 88 85 92 144 128 82 144 95 177 85

1.5 2.1 -0.6 0.3 1.2 0.8 -2.5 -2.5 -1.8 -1.1 0.1

300 229 323 293 345 300 337 353 346 336 341

1.8 2.7 2.9 2.5 1.9 2.3 1.5 2.7 2.3 1.3 3.6

Average S.D.

considered as maximum absolute errors in the calculation of M,.. The only consistent trend in Table 3 is the relatively low maximum absolute value of M,.(M,) compared to the other values in the table. Compared to the results for M,. in Table 1, the values in Table 3 for M,, suggest that there is much more influence of pedal load components other than F,,. It is interesting to note the effect of the pedal moments on the net knee moments of equations (7) and (8). Comparing the average values of M,,(M,) (Table 1) and M,.(M,) (Table 3) it is evident that M, contributes five times more to M,, than to M,.. Similarly, the ratio between the average values of

2.3 0.7

M,,( M,) and M,.( M,) is also approximately equal to five, showing that M, primarily contributes to M,.. Thus, there appears to be a relation between the knee moments and pedal moments about similar axes in the respective coordinate systems. DISCUSSION

The inherent accuracy of the methodology presented for calculating the three-dimensional knee loads hinges on several assumptions surrounding kinematic issues. Because the methodology includes two separate models, one in the sagittal plane and the

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P. RUBYet af.

Table 3. Peak values of the individual pedal load contributions to M,.. Peak values of the components which make up the M,, knee moment due to pedal loads F,, F,, F,,M,, M, as defined in equation (8). For most subjects M,.(F,) is the largest contribution to the net load. The contribution of M,.( M,) is considerably smaller than the other components Peak values of terms in equation (8) Subject

M,.(max) (N m)

1 2 3 4 5 6 7 8 9 10 11

5.16 1.19 1.80 3.04 1.40 5.53 1.91 2.72 -0.59t 0.89 2.03

Average of absolute values of peak terms S.D.

2.28 1.79

M,.(min) (N m)

MzW,) (N m)

1.24: 2.66 1.79 1.20 2.07 0.75* - 5.84 -1.88 -7.69 -7.16 -2.17

3.37 3.88 1.92 1.53 1.54 1.33 -0.89 -0.51 0.37 -0.27 4.72 1.85 1.50

-

2.83 2.93

M,JF,) (Nm)

M,,(Fz) (N m)

M,,( M,) (Nm)

Mz,(M,) (Nm)

4.33 -7.55 - 6.42 - 5.86 -2.87 2.60 -6.12 3.71 - 7.93 -7.18 -6.18

1.28 2.70 1.60 1.97 3.19 2.16 -0.30 -0.90 0.34 -0.13 -1.61

--0.31 -0.04 0.81 0.59 -1.16 -0.55 0.20 -0.36 0.22 -0.68 -0.53

2.42 2.08 2.61 4.97 -2.25 3.47 -2.50 - 1.71 -2.61 - 1.67 -1.87

5.52 1.86

1.47 1.00

0.50 0.32

2.56 0.95

*The M,,knee moment of subjects 1 and 6 remained positive over the entire crank cycle. The values shown are minimum M,. moments calculated for subjects 1 and 6. iThe M..knee moment of subiect 9 remained negative over the entire crank cycle. The value listed is the maximum M,.knee moment calculated for subject 9. S.D.: standard deviation.

other in the frontal plane, assumptions may be categorized as pertaining to one model or the other. A detailed discussion of the assumptions pertaining to the sagittal plane model may be found in Hull and Gonzalez (1990) and will not be repeated here. Rather, the discussion here will be devoted to the assumptions pertaining to the frontal plane model. Perhaps the most basic assumption of the frontal plane model used to calculate nonsagittal plane knee loads is that the inertial contributions of the shank in the frontal plane are negligible. This assumption may not be valid if the loads at the knee to support this motion are of the same magnitude as the static loads. The results presented in Table 2 show the average range of motion during a complete crank cycle to be 2.3 cm. This motion may be caused by either a medial/lateral force or a varus/valgus moment. To estimate worst-case errors in these loads it was assumed that the motion is caused solely by either one load or the other. The medial/lateral force to support this motion was computed to be 4.37 N. Peak values of F,. are approximately 40 N, thus, the error is approximately 10% of the peak value. The varus/valgus moment to support this motion was estimated to be 1.35 Nm. The average absolute peak value of M, is greater than 10 N m; thus, the error in this moment is less than 14%. In addition, changing the direction of the angular momentum vector along the y’ axis would require additional moments not included in the quasistatic model. Next it should be noted that the frontal plane model shown in Fig. 4 assumes that the center of gravity of the foot is positioned directly in line with the center of the pedal. Thus, the model does not account for either

toeing-in or toeing-out of the foot which would shift the center of gravity of the foot to either the outside or the inside of the pedal, respectively. It was stated before that the mass of the foot contributes very little to the M,. and M,, knee moments because of its small value. It is estimated that a 10” toe-out would produce a lateral shift of the foot’s center of gravity of less than 2 cm and it would require a shift of more than 4 cm to produce an error greater than 1 N m in the knee joint moments. Another assumption which potentially affects loads computed using the frontal plane model is that the origin of the tibia-fixed knee coordinate system is located at the midpoint of the line connecting the medial and lateral epicondyles with the knee fully extended. If this origin does not lie along the line(s) defining the functional axes for either varus/valgus or axial rotation, then errors will be introduced. Examination of equations (5) and (6) reveals that the contributions of the pedal forces F, and F, to both axial and varus/valgus moments are influenced by the value of L,,. Further examination of equations (5) and (6) also reveals that both M,.( FY) and M,( Fy) depend on the length of the respective moment arms from the pedal to the knee. Hence, the potential for errors exists if these distances do not accurately locate the functional axes. While any errors introduced by assuming a fixed origin cannot be firmly assessed without kinematic measurements to determine functional axes which were not made here, some assessment can be made of the potential realization of these errors. Consider first the errors in M, due to the value of &,,. Because there is little in the literature related to the position of the

Three-dimensional knee joint loading during seated cycling functional varus/valgus axis, it will be assumed that the worst-case error would occur if either femoral condyle were to lift off the tibia1 plateau, in which case the functional axis would shift in the frontal plane approximately to the contact point (Crowninshield et al., 1976). Assuming a 6 cm separation between condylar lift-off (CLO) points in varus/valgus rotation (Markolf et al., 1981) the error in the value of Z+,,could be as much as 3 cm. If realized, then such an error could affect M,, substantially because M,.( F,) is the secondlargest contributor to the net moment (see Table 1). But if the axial compressive force due either to external force or to muscle force is sufficient to prevent CLO, then the functional axis in varus/valgus rotation will be between the CL0 points on the tibia1 plateau. In fact, assuming equal stiffness at cartilage contact regions, negligible ligament forces, and symmetric muscle stiffness about these contact regions leads to a functional axis which is centered between these regions. Since in cycling muscle forces are developed over the full crank cycle (Jorge and Hull, 1986) as is a compressive knee force (Fig. 7), it seems reasonable to assume that CL0 does not occur, so that the functional axis is between the contact regions near the fixed origin chosen herein. Consider next the error in M,. due to any change in the moment arm acted on by F,. Note that any change in this moment arm affects the elevation of the origin. Again it will be assumed that the worst case is realized at CLO. With the origin at the midpoint between the epicondyles and the functional varus/valgus axis located approximately at the tibia1 plateau at CLO, the difference in elevation would be about 3 cm (Yoshioka et al., 1987). Although this difference is the same as that potentially occurring in by at CLO, the error introduced here would be relatively minor, Because the moment arm length is about 15 times this difference, the relative error in M,.( F,,) would be restricted to 67%. Note that because the contribution of M,.(F,) to M,, dominates, a similar relative error would occur in M,. . The potential for errors in the axial knee moment M,. can be more firmly assessed than that for the varus/valgus moment M,. because of the availability of previous literature which has identified the location of the functional axial rotation axis. This axis passes through the medial intercondylar tubercle of the tibia1 plateau throughout the range of flexion and under the application of applied axial moments (Shaw and Murray, 1974). This point is within 0.5 cm in the transverse plane of the midpoint between the femoral epicondyles with the knee at full extension (Yoshioka et al., 1987, 1989). Similar to the case for M,., the moment contribution from the F, term to M,. is three to four times larger than any other term. Thus, the relative error in M,. due to errors in M,.(F,) will be comparable. Consequently, with the average value of the moment arm due to L, being approximately 12 cm, this difference would manifest as a 4-5% error. Some idea of the relative error in M,. introduced by

51

the uncertainty in L.,.ycan be gained by examining Tables 2 and 3. Recall that this uncertainty will be manifest as errors in both M,.( F,) and M,(F,). Peak values of these pedal forces occur in the crank region near 90” when the value of Lk, is the largest (extreme medial deviation in Table 2). For an average extreme medial deviation of 2.1 cm, the corresponding relative error in both M,.(F,) and M,.(F,) would be about 25%. While this is a potentially large error in these terms, note that the contributions of each of these terms to M,. is only about 25% of that by M,,( F,,). The relative error realized in M,, would be diminished accordingly to about 6-7% for each term. In summary, even though there may be differences between the location of the axes in the defined coordinate system and those of the functional axes, these differences will not introduce major errors into the computation of net varus/valgus and axial knee moments. This is because of the moments due to the three pedal forces, the moment due to the pedal force F, is both the largest contributor and the least affected by these differences. Consequently, while it is certainly desirable to improve the accuracy of the moment calculations by defining a coordinate system which better matches the functional axes, this refinement does not appear to be necessary to obtain meaningful results. A final assumption of the frontal plane model is that the plane defined by the x’-z’ axes of the local knee coordinate system remains parallel to the X-Z plane of the inertial coordinate system. The results of this assumption create potential confusion surrounding the interpretation of knee loads. This potential confusion arises because of the existence of other coordinate systems such as the one by Grood and Suntay (1983). In their coordinate system, one axis (axial) is fixed to the tibia, one axis (flexion/extension) is fixed to the femur, and the third axis (varus/valgus) is not fixed to either of the bones but is mutually perpendicular to the other two axes. Thus, as the thigh adducts and abducts in cycling, producing knee motion in the frontal plane, the coordinate system of Grood and Suntay (1983) both rotates and translates whereas the coordinate system defined herein translates only. Consequently, a transformation could be necessary to determine loads in their coordinate system. Note, however, that such a transformation is probably not warranted because the degree of rotation is small. Using a 42 cm thigh length, which is appropriate for the average-height subjects used herein, and 1.3 cm of frontal plane motion, which is half the average range in Table 2, the rotation is less than 2”. Thus, the loads computed according to the coordinate system defined herein correspond closely to those in the coordinate system of Grood and Suntay (1983). Beyond the assumptions and their impact on the inherent accuracy of the model, potential sources of error in the knee load computation are traced to measurement of input variables (i.e. loads and motions). The six-load-component dynamometer was

52

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subjected to an accuracy check by Stone (1990), whereby combinations of loads were applied and apparent loads were computed through the numerical calibration procedure (Hull and Davis, 1981). Differences give an indication of the absolute measured errors. Absolute measured errors for the forces depend on the load component. For the six load combinations studied, the average absolute error in the normal pedal force F, is largest at 11 N while the average absolute errors for the other two forces, F, and F,, are comparable at 4 N. The average absolute error of 0.2 N m is comparable for all pedal moments. Because the frontal plane movement was measured using video-based motion analysis techniques implemented with equipment from Motion Analysis Corporation, errors may be quantified by referring to their data. According to Walton (1986), the overall system performance (i.e. measurement error) ranges from about 1 part in 200 to 1 part in 1500, depending upon the camera equipment and target to field width ratio. For the approximate 1 m2 field of view used here, this corresponds to an error ranging from 5 mm at the worst to 0.7 mm at the best. Inasmuch as Walton (1986) does not give data for the camera equipment used here, a firmer assessment of error cannot be made. In addition to errors in the input variables, errors in the input parameters affect the accuracy of computed knee loads, particularly the loads computed from the sagittal plane knee model. Due to difficulties in locating the hip axis of rotation, one parameter which was investigated in some detail was the hip position relative to the crank axis of rotation. This investigation consisted of a sensitivity analysis in which both horizontal and vertical dimensions were varied and the resulting sagittal plane knee loads computed. The results indicated that an increase in either the vertical or the horizontal hip position relative to the bottom bracket from the reference position causes changes in F,., M,., M,, and M,. . The increase in both horizontal and vertical hip position is equivalent to increasing the seat height. Decreasing the seat height (lowering the hip and moving the hip forward) from the reference position, on the other hand, has little effect on the knee loads. Thus, the model is sensitive to increases in seat height above a threshold height where the model nears its geometric constraint. Since the model is sensitive to the hip position, the test procedures must attempt to reduce the possibility for error in these measurements. Repeated measurements of the horizontal and vertical hip position were taken to provide a check on consistency of the measurements. Average values of hip position were used in the knee load calculations when less than three consistent measurements were obtained. The results presented herein identify both the outof-plane motion and pure moments at the pedal as important factors in determining M, and M,.. The variability between subjects in the breakdown of the knee moments in Tables 1 and 3 shows that the

relative importance of each of the terms in equations (7) and (8) is dependent upon the pedalling mechanics of individual subjects. Some subjects have a large amount of out-of-plane motion (Table 2), which makes it possible for moments to develop at the knee due to the normal and tangential pedal forces. Other subjects apply large moments to the pedal, which result in increases in the peak values of M, and M,. . Still others develop a combination of these two cases, which cancel each other out, while some have pedalling mechanics which create only minimal knee joint moments. Thus, due to differences in pedalling mechanics between individuals, it is important that, in general, the pedal load components be included in the evaluation of the varus/valgus and axial knee moments. Specifically, the pedal loads F;, F,, M, and M, must be considered when computing the varus/valgus knee moment as per equation (7). Computation of the axial knee moment using equation (8) must include the contributions due to the pedal loads F,, F,, F,, M, and M,. CONCLUSION

In conclusion, motion of the knee in the frontal plane and moments at the pedal are important factors in the determination of the varus/valgus and axial knee joint moments. Hence, a complete description of the loading at the knee joint requires the knowledge of the three-dimensional loading at the pedal, the kinematics of the system in the sagittal plane and the motion of the knee in the frontal plane. Acknowledgement-We are grateful to the Shimano Corporation of Osaka, Japan, for continued financial support and, in particular, to Shinpei Okajima (Shimano Industrial Corporation) and Wayne Stetina (Shimano American Corporation). We also appreciate the technical assistance with the pedal dynamometer that Cal Stone provided. REFERENCES

Bolourchi, F. and Hull, M. L. (1985) Measurement of rider induced loads during simulated cycling. Int. J. Sport Biomechanics l(4), 308-329. Boutin, R. D., Rab, G. T. and Aboughaida, H. I. (1989) Threedimensional kinematics and muscle length changes in bicyclists. Proc. Am. Sot. Biomechanics 13th Annual Meeting; pp. 94-95, Burlington, Vermont. Bratt. A. and Ericson. M. 0. (1985) Biomechanical model for’ calculation of jbint loads duhng ergometer cycling. TRITA-Mek, 85-05, Royal Institute of Technology, Stockholm, Sweden. Crowninshield, R., Pope, M. H. and Johnson, R. J. (1976) An analytical model of the knee. J. Biomechanics 9, 397405. Davis, R. R. and Hull, M. L. (1981) Measurement of pedal loading in bicycling: II. Analysis and results. J. Biomechanics 14, 857-872. Ericson, M. O., Bratt, A., Nisell, R., Nemeth, G. and Ekholm, J. (1986) Load moments about the hip and knee joints during ergometer cycling. Stand. J. Rehab. Med. 18(4), 165-172. Ericson, M. O., Nisell, R. and Ekholm, J. (1984) Varus and valgus loads of the knee joint during ergometer cycling. Stand. J. Sports Sci. 6(2), 39-45.

Three-dimensional knee joint loading during seated cycling Gregor, R. J., Cavanagh, P. R. and Lafortune, M. (1985) Knee flexor moments during propulsion in cycling-a creative solution to Lombard’s paradox. J. Biomechanics 18, 307-316.

Grood, E. S. and Suntay, W. J. (1983) A joint coordinate system for the clinical description of three dimensional motions: application to the knee. J. biomech. Engng 105(S), 136-144. Holmes, J. C., Pruitt, A. L. and Whalen, N. A. (1989) Knee pain in the cyclist. 1st IOC World Congress on Sport Sciences, pp. 223-224, United States Olympic Committee, Colorado Springs, Colorado. Hull, M. L. and Davis, R. R. (1981) Measurement of pedal loading in bicycling: I. Instrumentation. J. Biomechanics 14, 843-856. Hull, M. L. and Gonzalez, H. (1990) The effect of pedal platform height on cycling biomechanics. ht. J. Sports Biomechanics 6(l), l-17. Hull, M. L. and Jorge, M. (1985) A method for biomechanical analysis of bicycle pedalling. J. Biomechanics 18,631~644. Jorge, M. and Hull, M. L. (1986) Analysis of EMG measurements during bicycle pedalling. J. Biomechanics 19, 683-694. Knutzen, K. M. and Schot, P. K. (1987) The influence of foot position on knee joint kinematics during cycling. In Biomechanics X-A (Edited by Jonsson, B.), pp. 599-603. Human Kinetics Publishers, Champaign, Illinois. Markolf, K. L., Bargar, W. L., Shoemaker, S. C. and Amstutz, M. D. (1981) The role ofjoint load in knee stability. J. Bone Jr Surg. 63A, 570-585.

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Newmiller, J., Hull, M. L. and Zajac, F. E. (1988) A mechanically decoupled two force component bicycle pedal dynamometer. J. Biomechanics 21, 375-386. Plagenhoef, S. (1983) Anatomical data for analyzing human motion. Res. Quart. Exercise Sports, 54(2), 169-178. Shaw, J. A. and Murray, D. G. (1974) The longitudinal axis of the knee and the role of the cruciate ligaments in controlling transverse rotation. J. Bone Jt Surg. 56A, 1603-1609. Shen, H. and Radharamanan, R. (1988) A 3-D kinematic model for the biomechanical analysis of lower limb cycling. Proc. ISMM Int Symp. Computer Applications in Design Simularion and Analysis, pp. 212-216, Honolulu.

Stone, C. (1990) Rider/bicycle interaction loads during seated and standing treadmill cycling. M.S. thesis, Department of Mechanical Engineering, University of California, Davis. Walton, J. S. (1986) The accuracy and precision of a videobased motion analysis system. In High Speed Photography, Videography, and Photonics IV: Vol. 693 (Edited by Ponseggi, B. G.), pp. 251-263. SPIE-The International Society for Optical Engineering, Bellingham, Washington. Yoshioka, Y., Siu, D. and Cooke, T. D. V. (1987) The anatomy and functional axes of the femur. J. Bone Jt Surg. 69A, 873-880. Yoshioka, Y., Siu, D. W., Scudamore, R. A. and Cooke, T. D. V. (1989) Tibia1 anatomy and functional axes. 1. orthop. Res. I, 132-137.

Zahradnik, F. (1990) Pivotal issue: should your feet be fixed or floating? Bicycling XxX1(6), 134-137.