optimization of pedaling rate

8/11/2019 Optimization of Pedaling Rate

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ORIGIN L NVESTIG TIONS

INTERNATIONAL JOURNAL OF SPORT BIOMECHANICS 1988 4 1-20

Optimization of pedaling Rate

in Cycling Usihg a Muscle

Stress-Based Objective Function

Maury

L

Hull Hiroko K Gonzalez and Rob Redfield

Relying on a five-bar linkage model of the lower limb hicy cle system, inter-

segmental forces and moments

re

computed over a

full

crank cycle. Experi-

mental

data

enabling the solution of intersegm ental oads consist of measured

crank arm

and pedal angles together with the driving pedal force components.

Intersegmental loads are computed as a function of pedaling ra te while hold-

ing

the average power over a c rank cycle constant. Using an algorithm that

avoids redundant equations, stresses are computed in 12 lower limb muscles.

Stress computations serve to evaluate a muscle stress-based objective func-

tion. T he pedaling

rate

that minim izes the objective function is found to be

in the range of

95 100

rpm. In solving for optimal pedaling rate, th e muscle

stresses are examined over a complete crank cycle. This examination pro-

vides insight into the functional roles of individual muscles in cycling.

In

the field of

sport

biomechanics, one subject of primary interest is

maxi-

mizing the performance of competitors. With regard to cycling, one factor known

to affect performance is the pedaling rate or cadence. Consequently, in the in-

terest of maximizing performance, it is useful to conduct analytical andlor experi-

mental studies thats k o understand the relation

between

pedaling

rate

and cyclist

performance.

Other

researchers have recognized the utility of such studies, which

have resulted in the development of a body of research. To our knowledge,

all

previous studies have been experiments of human performance e.g., see Coast

Welch, 1985). The typical protocol is to have subjects pedal at different rates

while an ergometer is adjusted to provide constant average power. Oxygen up-

take is measured, and the optimal pedaling rate is that

rate

for which uptake is

a

minimum.

Although thereare some contradictory

data,

the majority of studies show that

at a power level of 200 W, which is typical of steady-state cycling over level

M.L.

Hull

and

H K

onzalez rewithithe Department of Mechanical Engineering

at the University of California, Davis, CA 95616. Rob Redfield is with the Department

of Mechanical Engineeting at Texas A M University.


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HULL GON ZALEZ AN D REDFlEtD


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compotrents In OYB&YOg~ re

W

liilik

W

if.elati&s atld~~~bleraTiofft;

these angle* erms timei.plots mu& differentiaEd twice. 'At steady-state cyk

cling, the first deriirativk of tfie crank mgle is a constant. Inspection of the pe iil

angle revealed that it is approximately sinusoidal. Accordingly, derivatives were

obtained analytically.

Additional input*@i%nclnd'eif values of:the St li,tleP~thropomdtrictriala&

mefeb Mass, moment of iaertia, and center+ofDavity location values for the

leg segments were estimated usingiZhe Work df M s nd Continir(1966).Actu-

al subject body mass and lower limb segment lengths were used from the study

of Hull and Jorge (1985) as input to Drillis' empirical formulas to obtain model

parameters. Hull and Jorgeused six subjer3s in their study, and

data

frdm one

of those subjects who provided average anthropometryas well as fjlpical and coni

sistent cycling dynamics were used herein. Table 1 lists the data for this subject.

To determine the iatersegmental forces and moments illustrated in Figure2

the equations of motion derived by Redfield and Hull (1986a) were used. With

all input data available, the equations of motion were solved to yield the inter-

segmental forces and moments of all three joints over a complete crank cycle

at 5 intervals.

In

order to compute muscle forces, the procedure for computing

contributions of muscles to the interlsegmental moments devised by Redfield and

Hull (1986b) was~followed.Muscles of the leg were lumped into functional groups

(e.g.,

knee

extensor). The muscles used in this study, their groupings, and their

abbreviations are given in Table 2 Then the contribution of muscle groups to

the total joint moment was-determined by assuming a lack of cocontraction in

agonisttantagonist muscles as necessary to avoid the redundant and hence indeter-

minate problem (Crowninshield Brand, 1981).

To illustrate, the ankle was considered first. Depending on moment direc-

tion, the ankle moment was attributed to either the tibialis anterior or gastroc-

nemius. Next thekneemoment was considered. Because it is a two-joint muscle,

egment


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HULL GON ZAL EZ AN D REDFIELD

the gastrocnemius, if active, produces a moment about the knee as well as the

ankle. Therefore the gastrocnemius contribution to the knee moment, if any,

was subtracted from the net knee moment to create a remainder moment about

the knee Positive (extensive) remainder kneemoment was attributed to the quad-

riceps muscle group, while negative (flexive) remainder moment was attributed

to the hamstrings group. Similarly, because muscles of both the hamstrings group

and the rectus femoris are two-joint muscles, the remainder torque at the hip was

found by summing either the hamstrings or the rectus femoris moment contribu-

tion with the net hip moment. Finally, the remainder hip torque was ascribed

to either the gluteus maximus or the illio-psoas. Table 3 summarizes the

agonistlantagonist muscle groups for which cocontraction is both permitted and

not permitted by this procedure.

Implementing the procedure for muscle force computations required that

muscle moment rm engths be specified. For the purposes of determining mo-

Table

Model Leg Muscles

Ankle dorsi-flexor

Ankle plantar-flexor

Knee flexors

Hip flexors

Hip extensors

- ibialis anterior TA)

- gastrocnemius G) medial and lateral head)

- gastrocnemius G) medial and lateral head)

- semimembranosus SM)

-

semitendinosus (ST)

-

biceps femoris BF) long head)

- ectus femoris RF)

- psoas P)

- lliacus I)

- gluteus maximus GM)

-

semirnembranosus SB)

-

semitendinosus ST)

-

biceps femoris BF) long head)

Note. After Redfield Hull 1986b).

Table 3

Cocontraction of Lower Limb gonistI ntagonist Muscles

Permitted

Not permitted

Gastrocnemiuslquadricepsat the knee

Rectus femorislgluteus maximus at the hip

Illio-psoaslharnstrings at the hip

Gastrocnemiusltibialis anterior at the ankle

Quadricepslhamstrings at the knee

Rectus femorislhamstrings at the hip

Gluteus maximus/illio-psoas at the hip


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P n

EEl

igure3 Normal ndtangentialpedal

forces

versuscr nkanm angle 0 vertical .

Results nd iscussion

The

data

that result from analysis of muscle stresses have a number of uses. As-

suming that the muscle stress computing algorithm gives an accurate picture of

muscle stress, one use is to provide an understanding of muscle function in cy-

cling. The function of muscles as indicated by the algorithm will be discussed

first, followed by an assessment of the efficacy of the algorithm. Figures 4a, 4b,

and 4c illustrate the muscle stresses for the tibialis anteriorlgastrocnemius,quad-

ricepslhamstrings, and illio-psoas/gluteus maximus, respectively, computed at

a pedaling rate of

80

rpm.

n

Figure 4a, which plots the stress in two major muscles

crossing the ankle joint, notice the long range of activity for the gastrocnemius.

Because this range extends essentially over the full crank cycle, and the muscle

force computing algorithm does not allow for cocontraction of the tibialis anteri-

or, the stress in the tibialis is bound to zero. Also notice that the gastroc stress

closely tracks the normal pedal force

see

Figure 3 . Accordingly, the activity

of the gastroc muscle produces a moment at the ankle which acts to equilibrate

that due to the normal

pedal

force.

Unlike the picture at the ankle oint, Figure

4b

shows that both the quadri-

ceps

knee

xtensors and hamstrings

knee

lexors experience stress over different

regions of the crank cycle. Muscles included in the quadriceps group are rectus


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HULL GONZALEZ AND REDFIELD

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,

8 . a ~

68.08 1b 00 83.0e1 241 /.~0 ' d . 00 ' ~ 1 / . 0 a

CRAWK WGLE DEG)

b)

- 2 . 5 m m

9.08 69 -09 129.88 18s.BD 24O.BO Js8-DO 368.Oa

CRMK WGLE DEG

Figure Muscle stms s at

8 rpm ped ling

rate. a)

ibialis

anterior TASTR)

and gastrocnemius GASSTR); b) Hamstrings group HAMSTR) and quadriceps

group QUDSTR). cont.)


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PED LING

RATE

IN CYCLING

Figure 4c Gluteus m ximus GLTSTR)

nd illio psoas

PSOSTR).

fernoris and the three vastii, while those in the hamstrings grouparebiceps femoris,

semimembranosus, and semitendinosus. Stresses of the two groups are neces-

sarily confined to different regions because the muscle force algorithm does not

allow cocontraction of these groups at the knee. Note that the algorithm does

provide for cocontraction of the gastroc at the knee, however. Because the gastroc

is a two-joint muscle, it acts not only as an extensor of the ankle but also as a

flexor of the knee. Accordingly, this muscle is antagonistic to the knee extensors.

Superposition of Figures4a and 4b indicates stress

in

these muscles simultaneously.

n interesting observation in Figure 4b surrounds the timing of the stresses

in the two groups. The m ximum stress in the quadriceps group occurs at a crank

angle of about 15 while that in the hamstrings group occurs at

200 .

Note that

neither group experiences significant stress at about 110 . According to Figure

3, this is the instant of greatest absolute normal pedal force and hence peak instan-

taneous power. The interest in this observation is that neither muscle group ap-

pears to play a significant role in developing this power.

The muscle primarily responsible for developing peak power is evident

from Figure 4c. The stress data in Figure 4c indicate that the gluteus maximus

stress ranges over virtually the entire downstroke while the stress of the illio-

psoas group ranges over the opposite region, the upstroke. Because the peak

gluteus maximus stress coincides with the peak instantaneous power, it can be

concluded that this muscle plays a dominant role

in

developing peak power. Note

that because of the massive size of this muscle, the stress is relatively low. Ac-

cordingly, the gluteus maximus muscle appears well suited for the demand placed

upon it in cycling. The fact that the illio-psoas is active only during the upstroke,


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HULL G ONZA LEZ AN D REDFIELD

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64.0980-

51 SO08

29.OWB

b

24.0000 1

I I

60.00 80.00 100.00 i20.00 140.00

RPW

EM2

igure5

-

Average musclestr ss versus

pedaling

rate. a) Tibialis anterior TASTR)

and

gastrocnemius GASSTR); b) Hamstrings group HAMSTR)

and

quadriceps

group QUDSTR) cont.)

13.5989-

11.625L5e

1.750,

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W.00 lW.011 120.00 148.98

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PED LING R TE IN CYCLING

Figure

c

Gluteus

maximus

GLTSTR) and illia-psow VSOSTR).

RPM

F gure

Kinematic

and

static

momenttrends spedal@ rate isvaried afterRed

field Hull,

1986a .


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PEDALING

RATE

IN CYCLING

17

in the downstroke region remains about equal. Inasmuch as the pedal forces are

small in the upstroke region, and the illio-psoas stress exists,only in that region

(see Figure

4c ,

the apparent quadratic increase in the illio-psoas average stress

with pedaling rate indicates that the stress in this muscle group i~~dominatedy

the kinematic contribution. This is with the earlier interpretation of

the function of those muscles.

To gain a clear picture of the e muscle stresses, refer to Fi$ure

7.

Note that the joint stresses for the ankle,

knee

and hip include str6sses from ll

muscles crossing a particular joint. According to this procedure, the stress from

ll two-joint muscles will be considered twice. From Figure

7

it is apparent that

the sum totals of stresses in muscles crossing the hip and

knee

oints are similar,

while the sum totals of stresses in muscles crossing the ankle joint are anywhere

from 2 to 10 times lower depending on the pedaling rate. This result emphasizes

the importance of including the stress from muscles crossing the hip and knee

joints in the objective function, but suggests that the stresses in muscles crossing

the ankle joint are of lesser importance.

The fin l result of this study, namely the detembtiion of the,aptim_alpedal

ing rate using the muscle stress-based objective function given in Equation 1,

is apparent from Figure

8.

This optimal rate fa1ls.h the range of

95

to

100

rpm


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HULL GONZALEZ AND REDFIELD

RP

Figure Muscle stress cost function versus pedaling rate

Among the cycling variables that might influence the value of the optimal

pedaling rate, one that readily comes to mind is the power output level. Although

the effect on the optimal rate of changing power level has not been studied here,

some insight into what effect might result can be gained by referring to Redfield

and Hull 1986a). These authors studied the dependence of the optimal pedaling

rate on power level and found that the rate increased with increasing power. This

result was explained by noting that, at higher power levels, the static contribu-

tion illustrated in Figure 6 would shift upward relative to the kinematic contribu-

tion, thus moving the trough of the superimposed contributions to higher rpm

values. Since the trends shown in Figure 6 for the moment contributions hold

for the stress contributions as well, the optima rpm for the stress-based objec-

tive function would be expected to shift similarly to the moment-based objective

function.

oncluding

Remarks

In considering extending the optimization analysis of cycling biomechanics to in-

clude additional variables e.g., seat height), it is desirable to rely on an objec-

tive function that offers ease of computation without compromising the accuracy

of results. Although Redfield and Hull 1986b) showed that the muscle stress-

based objective function better predicted measured ped l forces and intersegmental

moments than the joint moment-based objective function, the close comparison

of the results herein to those of Redfield and Hull 1986a) suggests that the

moment-based function may be used in lieu of the stress-based function. The joint

moment-based objective function is attractive because of its computational sim-

plicity,


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Coast, J:R:, &+WeIch, N.G. (1985). Lineair kcteak in optha1pedalling mte4ith in-

heabed power on'tlYlif in$icycld rg~metij:~Europ&anournal of Applied Physibl-

O J, 53, 339-342. I A s e

h

~ r o ~ s l i f e l d ~.Dr,

BWd,

RIA. (19814 : A pflysiolagically Wdrcriterion

of muscle^

force prklictiietl

n

locotklotion. Jounidl of Bidtnechani~s, 4, 793-801.

'

th,

Education, and Wel-

r * t i . *

Gregm

R.J., Green,

D.

.'of selected muscle

Sctivity

in-el

K. Kedzior, &A. Wit @d .),%B

' sify Park press.

Gfbsd'eiLordernann, Hgi

- ~ U I I ~ P

.&. ($937). D

Arbeitsgeschftrindrgkeit auf

d % & r b e i & d

l'jfahren [The influence of

power

and velocity

in bicycling]. Arbeitsphysiologie, 9, 454-475.

in, J.P., Giese, M.D Spitznage:l@. (1981). Effecthf w i n g

exercise respbndes of competitivecfrclis&. Journal s Applied

- > y

i i ~ ~ 1 7

f

optimization of pedaling rate

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