.I. liiornecha~~rcs.lY75. Vol. R. pp. 8~102.

Pergamon Press.

Prmted I” Great Britam

THE PREDICTION OF MUSCULAR LOAD SHARING AND JOINT FORCES IN THE LOWER EXTREMITIES DURING WALKING* A. SEIREGtand R. J. ARVIKAR+, Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin, U.S.A. Abstract-Previous work by the authors on the mathematical modelling of the lower extremities is extended in this paper to the prediction of muscle load sharing and the corresponding hip, knee and ankle joint reactions during walking when the motion pattern is known beforehand. Using the same criterion for muscle load sharing as in the previous work on leaning and squatting, the results show excellent agreement with typical electromyographic patterns for all the major muscles. The extent of alleviating the load on certain joints without changing the motion pattern is also investigated.

INTRODUCTION The skeletal system, connected together through ligaments and muscles, provides the vital structural support for the human body. With the aid of muscular actions the human body can perform a plethora of coordinated limb movements through the numerous articulating joints. In performing such movements the joints are inescapably subjected to forces, the severity of which varies with the type of action performed. In recent years the mechanical aspects of the human body have been attracting increasing attention of investigators who have attempted to study the function and architecture of the musculoskeletal system, the mechanical properties of the joint constituents, the forces transmitted by the joints during normal human activities, and the mechanical stresses produced in the various bones, ligaments and muscles necessary to maintain the body structural equilibrium. Walking is one of the most common of all human activities. Simple though as it may appear, it is controlled by complicated and meticulous coordination between various elements. Interest in the study of muscular control of locomotion was sparked off by the early works of the Weber brothers (1836) who claimed that during the swing phase of walking, muscular control was not necessary and the motion of the leg occurred much like a simple pendulum. Further contributions to study of human gait were made by Marey rt al. (I 885. 1887, 189.5)in France, Braune and Fischer (1889) in Germany, and Bernstein (1935) in Russia. Elftman (1934, 1939) studied the distribution of pressure in the human foot, the function of arms in walking, the rotation of the body and the functions of muscles in locomotion. Murray ef al. (1964) investigated the displacement

associated

with locomotion

for

normal men spanning a wide range of age and height. Clinical fractures in the lower extremities placed emphasis on the analysis of joint reactions. Bresler and Frankel (1950) using the force plate developed by * Receiwd 8 Auglrst 1974. t Professor. t Post Doctoral Fellow.

Cunningham and Brown (1952) to measure the ground-to-foot forces and simultaneously recording the positions of the leg in space obtained curves showing the variation with time of the three force components and the three moment components transmitted at the ankle, knee and hip joints during walking on a level surface but did not estimate the values of muscle forces and joint reactions. Direct determination of joint force has been performed by Rydell (1965, 1966) who fitted two of his patients, one male and one female, with a hip-joint prosthesis carrying strain gages. The greatest value of joint force recorded was 4.33 times the body weight which occurred in the female subject when running. For level walking, the greatest force was 3.3 times the body weight. Pauwels (1935) estimated the total hip joint force to be three times body weight for a person standing on one leg. He also states that under the dynamic conditions ofwalking, the force may increase to 4.5 times the body weight during the stance phase. Other analyses of hip forces include those of Inman (1947); Blount (1956); Strange (1963); McLeish and Charnley (1970); Williams and Svenson (1968). There have also been extensive studies of the electromyographic signals from all important muscles during the walking cycle. Comprehensive data can be found, for example. in reports from the University of California (1947. 1953). Such studies provide information about the pattern of muscle participation but do not quantify the muscle forces. Recently. the reaction forces at the hip joint during walking have been analyzed by Paul (1966, 1967, 1971) and at the knee joint by Morrison (1969. 1970). Both Paul and Morrison measured photographically the three dimensional configuration of the leg segments during a walking cycle in which the ground-to-foot actions were measured by a force plate dynamometer. Using the technique of Bresler and Frankel (lot. cit.) the resultant forces and moments transmitted between segments were calculated. The moments are transmitted by tensions in muscles or ligaments. Allotting these forces to the respective muscles at the joints (shown to be active by electromyographic

signals which were also

90

A. SEIREG and R. J. ARVIKAR

recorded) and making various simplifying assumptions to facilitate solution of the statically indeterminate system of equilibrium equations, the joint reactions were calculated. Maximum loads in the range of 2.358 times the body weight at the hip joint are reported by Paul for a series of male and female subjects of varying structure. Morrison reports a value between 2 and 4 times the body weight at the knee joint. Numerous mathematical models can be found in the literature for analysis of human body dynamics using rigid body idealization with active controls at the different joints. Examples of such studies are those of Murray, Seireg and Shultz (1967), Smith and Kane (1968), Kane and Scher (1970), Huston and Passerello (1971), Frank and Vukobratovic (1969), etc. Townsend and Seireg (1972a, 1972b, 1973) developed a computer based procedure for the trajectory synthesis and control of bipedal locomotion for optimum stability and energy expenditure. The procedure calculates the motion patterns and the necessary resultant controls at the different joints corresponding to any particular motion criterion without consideration of individual muscle actions and load sharing. The problem of determining the load sharing between the different muscles in order to maintain a particular posture has been investigated by Seireg and Arvikar (1973). They developed a mathematical model which simulates the configuration and actions of the different bones, joints and all the significant muscles of the lower extremities. The lower extremities are modelled as a system of seven rigid bodies or skeletal segments (pelvis, left and right thighs, legs and feet), kept in equilibrium by the pull on the muscles and ligaments. The muscle forces are assumed to be directed along lines joining the corresponding points of origin and insertion on the skeletal segments. A system of equilibrium equations can be obtained for each individual segment in terms of muscle forces acting on the segment, reaction forces and moments at the articulating intersegmental joints, and any external forces or moments such as segment weights, inertia forces etc. Using a linear programming procedure, the model can be let to predict, without prior selection, the participating muscles and the forces they exert based on a selected criterion for the control of muscle load sharing. A brief illustration of the applicability of the model to the study of muscular control of both posture and locomotion can be found in a paper by Arvikar and Seireg (1973) A criterion which appears to control the muscle load sharing during static posture and quasi-static locomotion is the minimization of a weighted sum of all the muscle forces and the forces in the ligaments (which are assumed to carry any unbalanced moment at the joints). This paper describes the general procedure for extending the previous work to the analysis of muscular load sharing as well as joint reactions corresponding to a particular walking pattern. Quasi-static walking is considered because of the lack of complete dyna-

mic data on all the body segments and corresponding

ground reactions during fast walking. Such information when available can be easily incorporated in the reported procedure by application of d’Alemberts principle to satisfy equivalent static equilibrium for the total body at each instant of the walking cycle. THE MUSCUL@SKELETAL MODEL

The upper torso is treated as one segment in this study. The lower extremities are modelled as a system of seven segments connected by 31 significant muscles on each side of the sagittal plane as reported previously by the authors (see Table 1). The main considerations for developing the model can be summarized as: 1. The muscles are assumed to produce tensile forces only. 2. The action of each muscle is represented by one or more lines to simulate the capabilities of the muscle in three dimensional space. As an example, a model for the quadriceps group with its four components is taken as shown in Fig. 1. 3. Whenever a straight line representing a muscle is interrupted by some interposing structure, the direction of the line is changed to wrap around the structure and a resultant reaction is assumed on both the muscle and the structure to simulate the expected pressure between them. For example, the quadriceps muscle is connected to the tibia through the patellar ligament.

Table 1. Important muscles of the lower extremities Muscle No. Muscle name

i

1

2 3 4 5 6 I 8 9 IO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2.5 26 27 28 29 30 31

Gracilis Adductor longus Adductor magnus (adductor part) Adductor magnus (extensor part) Adductor brevis Semitendinosus Semimembranosus Biceps femoris (long head) Rectus femoris Sartorius Tensor fasciae latae Gluteus maximus Iliacus Gluteus medius Gluteus minimus Biceus femoris (short head) Vastbs medial& Vastus intermedius Vastus lateralis Gastrocnemius (medial head) Gastrocnemius (lateral head) Soleus Tibialis anterior Tibialis posterior Extensor digitorum longus Extensor hallucis longus Flexor digitorum longus Flexor hallucis longus Peroneus longus Peroneus brevis Peroneus tertius

Muscular

load sharing

during

91

walking

c&J = K+1l$=41 CT,,]= CTclld,=a,,,.di=“,d,=a,;

so that

Similarly, a point of interest on body 5 is linked to the ground axes through the following transformation matrices: vostus intermed. vastus

lateralis

Fig.

’

I. Muscle model for quadriceps

The patella therefore has contact with the femur and consequently introduces a reaction on it. The coordinates of the points of origin and insertion of the muscles are obtained with respect to a selected reference-axes system utilizing anatomical data. These as well as the weights and the centers of mass of the various segments are identical to those used in the earlier paper by Seireg and Arvikar (1973).

CO-ORDINATE

TRANSFORMATION

CT,,]

MATRICES

For quasi-static walking, the cycle is treated as a progression of static postures at successive intervals. The data for the angles between segments in the plane of progression sagittal plane) and in the plane perpendicular to the direction of walking (coronal plane) are obtained from the University of California report (1947). In order to calculate the coordinates of the points of origin and insertion of the muscles (on various segments) with respect to fixed ground axes (Fig. 2), reference axes are chosen on each segment. The coordinates with respect to the ground axes are then obtained using a set of coordinate transformation matrices. For example, the changed coordinates of a point on body 3 resulting from rotation about the Y axis is given by:

=

[~lldl=O.h=O,dx=AB

C~,l = CT&=@, CT,,+] = [~lldl=O.d>=O.d,=BC C&l = CTJ=+, CT,,1 = [~lld,=O.d,=CD.d,=O cTf31 = C~ll4=@, &l = [Tclld,=0.d2”0.d3=-DE,etc. EQUILIBRIUM

EQUATIONS

The muscular force system keeps the lower extremities in equilibrium at all times. Each of the seven segments of the lower extremities is assumed as a rigid body in space and is therefore described by six equations of equilibrium: three for force balance along the three coordinate axes and three for moment balance about the three coordinate axes. The force equations include the muscle force components acting on each segment, inertia forces, gravity forces, joint reactions and ground reactions where applicable. The moment equations include the moments of all forces about the three reference axes as well as the inertia moments and any unbalanced moments at the joints which would be carried by the ligaments.

= cTm,l.cT,,1.c~,1.cT,,1.c~,1. (

where I

0

0

0

0

sin 4

0

cos 4

and 1000

CT1=

11 1

0

0

dz 0

1

0

d,

L4

0 0 1-l

’ D!rection

Side-view of

Front

-view

walking-

Fig. 2. Co-ordinate transformation axes. Q1, G2, OS, Qj,, Q5, G6 and @, are angles between axes pairs (X,,X,), (X*,X,), (X3.X4), (X5,X6X (X7,X8), (X,J,,) and (X, ,,X,,) mpectlvely measured counterclockwise.

92

A. SEIREG and R. J. ARVIKAR

D

based on which to select one of these solutions that best satisfies the stated objective or criterion. Since all the equations describing the system are linear one can resort to linear programming techniques to obtain a solution, provided the objective function selected be linear. Based on the authors’ earlier investigation on static posture (e.g. leaning forwards and backwards, stooping, etc.) the criterion selected for this study is minimization of the sum of all the muscle forces plus four times the sum of the moments at all the joints, i.e. i=62 U

=

C

Fi

+

4(Mj,

i= I

+

Mj, + MjZ)R

+ 4(Mjx + Mj, + MjZ)L,

Fig. 3. Free-body diagrams of the various segments for a typical posture during quasi-static walking. The inertia forces and moments are neglected in the case considered in this paper as the walking is assumed to be performed quasistatically. For faster rates of walking, terms representing the inertia forces and moments can be readily included in the equations when all linear and angular accelerations of all the body segments are known. The free body diagrams of the various segments at a particular instant of the cycle are shown in Fig. 3. TECHNIQUEOF SOLUTION

where U = objective function; Fi = force in muscle i; (Mj,,Mj,,M,), = right-side moments at the joint j (hip, knee or ankle) about the X, Y or Z axis, respectively; (Mjx,hlju,Mjz)L = left-side moments at the joint j (hip, knee or ankle) about the X, Y or Z axis, respectively. This criterion which represents a combined objective of minimizing the total sum of all muscle forces as well as the forces in the joint ligaments (assumed to carry the unbalanced joint moments) is linear. A solution can be conveniently obtained by the Simplex algorithm. Detailed discussion of this procedure can be found in many references (e.g. Dantzig, 1968). The simplex technique requires that all variables be constrained to be 20. In our case, certain variables may take a positive or negative value, namely, the reactions and the moments, which have been arbitrarily assigned directions. The solution will automatically indicate whether the assumed directions are correct or should be reversed. To allow these variables to take up either a positive or a negative sign in the program, they are divided into two parts, both constrained to be 20; but the net value of the variable being the difference of the two parts can be either positive or negative depending on whichever part is greater. The total number of variables is thereby increased to 146.

RESULTSAND

DISCUSSIONS

The seven segments of the musculo-skeletal model (a) Muscle forces would yield 42 equilibrium equations. With 31 muscles Since the model is used to evaluate muscular forces on either side of the sagittal plane, 3 joint reaction components along the three reference axes at each of during quasistatic walking (i.e. the inertia forces and the six joints, 3 moment components at each joint and moments are assumed to be relatively insignificant with 3 patellar reactions on each side, the total compared to the gravity forces), the net ground-to-foot reactions in X and Y directions are zero. This also imnumber of unknown variables is 104. Consequently there are more unknowns than equations to solve. If plies that net ankle moment about the vertical axis (Z62 of the unknown variables (the number obtained by axis) is also zero. The variation in the ground-to-foot subtracting the number of equations from the total vertical supportive force is approximately assumed as number of unknown variables) are arbitrarily assigned given in Fig. 4. The force on each foot is taken to vary some value then the solution for the remaining 42 linearly from a zero value at heel strike to a value equal becomes feasible. However, since there are infinite to weight of the body at the beginning of one-legged ways in which the 62 variables can be assigned values stance phase. At this time the other leg is off the arbitrarily, infinite number of solutions are implied. ground and swinging free while the entire weight of the Therefore, one must establish a ‘criterion’ or ‘objective’ body is borne by the leg on the ground. As soon as the

Muscular load sharing during walking

RHS

LTO

LHS -%

Fig.

4.

of WALKING

93

RTO

RHS

CYCLE

Simplified ground reaction R, during quasi-static walking. (RHS: right heel strike; LTO: left toe off; LHS: left heel strike; RTO: right toe off.)

Fig. 5. Path of center of support

during

quasi-static walking. (Each interval the Walking cycle.)

free-swinging extremity reaches forward for its heel strike the weight starts shifting from the other leg. Consequently, the supportive force starts decreasing linearly from a value equal to weight of the body at the instant of heel strike of the other leg to a zero value at the instant of take-off, and stays zero till the next heel strike occurs. The corresponding path of center of support is shown in Fig. 5 as calculated from conditions of total body equilibrium at different intervals of the quasistatic walking cycle. Figures 6-10 show a representative sample of the forces in various muscles during the walking cycle obtained from the model. The figures also show the corresponding EMG results as reported by Institute of Engineering Research, University of California (1953) obtained from tests on 5 subjects.

Y

IOOr

Stance

Phase

,

Swing Phase

three per cent of

The calculated force patterns of the tibialis posterior, the flexor digitorum longus, the flexor hallucis longus. and the calf group (soleus and gastrocnemius) are shown in Figs. 6(b), 7(b), 8(b), 9(b) and 9(c), respectThe experimental EMG responses from these muscles are shown in Figs. 6(b), 7(b), 8(b), 9(b) and 9(c), respectively. All these muscles are situated in the back of the leg and with the exception of gastrocnemius which is a two-joint muscle, originate on tibia and insert on the foot. All seem to act exclusively in the stance phase. The theoretical results for the abductor group are shown in Fig. la(a). The result plotted is the sum of the individual muscle forces-the gluteus medius, the gluteus minimus and the tensor fasciae latae. The experimental EMG patterns (Figs. 1Obd) appear to closely match the results obtained from the model. Stance

d

represents

Phase

,

Swing Phase

,

1

(a) 20

30

40

50

60

of WALKING

-%

70

60

90

100

CYCLE

-%

t

of WALKING

CYCLE

(b)

Fig. 6. (a) Theoretical results for the tibialis posterior. (b) EMG response from the tibialis posterior during normal

.

level

.

.. .

Walking.

(b)

Fig. 7. (a) Theoretical result for the flexor digitorum longus. (b) EMG response from the flexor digitorum longus during . .

normal level walking.

A. SEWECand R. J. ARVIKAR

94

::

I

Stance

Phase

$200

Swing Phase

r

Stance

Swing

Phose

Phase

.E (a)

t

---+%

of

WALKING

z

s t

CYCLE

(a)

100

0 0

IO 20 -%

30

40

of

50

60

WALKING

70 80 CYCLE

90

100

(b) (b)

Fig. 8. (a) Theoretical result for the flexor hallucis longus. (b) EMG response from the flexor hallucis longus during normal level walking.

In general the theoretical results for every muscle of the lower extremities showed good correlation with the reported averages of the EMG pattern obtained experimentally from different subjects. Any deviations between the theoretical results and the reported averages is of a smaller magnitude than the deviations between the data from the different subjects. It should also be emphasized that magnitude variations and phase shifts in the EMG patterns can be expected due to dynamic effects which have been neglected in this quasi-static investigation. (b) Jo& reactions A plot of the resultant joint reaction and its respective components during a complete walking cycle, along X (the direction of walking), Y (medio-lateral direction) and 2 (the vertical direction) for the hip, knee and ankle joints is shown in Figs. 11, 12 and 13, respectively. The maximum values for the joint reaction are 5.4, 7.1 and 5.2 times the body weight re,spectively for the hip, knee and ankle joint.

(a)

20 -%

30

40

of

50

60

WALKING

70

80

90

100

CYCLE

(b)

(c)

Fig. 9. (a) Theoretical result for the calf group (gastrocnemius and soleus). (b) EMG response from the gastrocnemius during normal level walking. (c) EMG response from the soleus during normal level walking.

(cl

(d)

Fig. 10. (a) Theoretical result for the abductor group (gluteus minimus, gluteus medius and tensor fascial latae). (b) EMG response from the g. minimus during normal level walking. (c) EMG response from the g. medius during normal level walking. (d) EMG response from the tensor fasciae latae during normal level walking.

The variation in magnitude and direction of the resultant hip joint reaction is demonstrated by a threedimensional view of the femoral head (Fig. 14). The joint forces obtained from the model are somewhat different from those obtained by Paul (1965, 1967) for the hip joint, as shown in Fig. 15 and by Paul (1967) and Morrison (1969) for the knee joint, as shown in Fig. 16. Since at the hip joint explicit solutions were not possible, Paul reports the results in form of limit curves between which the actual value of the joint force could lie at any interval in the cycle. In his earlier work, Paul (1965) calculated the maximum value for the joint force to be 5.8 times the body weight for a 1801b subject occurring after heel strike in the transition from partial to complete weight-bearing. In his later work (1967) he reports the two limit curves corresponding to the maximum and minimum possible values of joint reaction. In his analysis Paul combines the 22 hip muscles into 6 major groups and assumes that the forces in the muscle groups are directed along lines joining the centroid of the areas of origin to the centroid of the areas of insertion. In our model, however, muscles have been individually modelled by lines in space joining their points of origin and insertion. This allows two (or more) portions of the same muscle (or muscle group) which may be serving different functions in the body to be represented individually. Instead of the usual six equations of equilibrium for a

Muscular load sharing during walking -D--O.0,-Qx--x-

95

hip-joint reaction Joint force along X-direction Joint force Joint force

Resultont

2 Y X

t

------+

% of WALKING

CYCLE

itrike

Toe Off

Heel Strike

Fig. 11. Hip joint reaction and components. 7-

6-

-

Resultont

-O--D-

Joint force along X-direction

knee join1 reaction

.-m-m-

Joint force along Y-direction

--x---x-

Joint force

along Z-direction

54-

3-

% of WALKING

CYCLE \ ’ b’

Heel strike

Heel &ike

Toe’ off

Fig. 12. Knee joint reaction and components. -

Resultant

-o--o-

Joint force along X-direction

.-m-.o-

Joint force along Y-direction

--X---X-

Heel ktrike

I’

0

ankle-joint

reaction

Joint force along Z-direction

Toe off

Fig. 13. Ankle joint reaction and components.

Heel strike

A. SEIREG and R. J. ARVIKAR

96

‘;Zoff4h

HIP

JOINT

REACTION

maximum value of 3.03. These values for the knee joint force are lower than the 7.1 times the body weight as obtained from the model. This value occurs in the double-leg support phase when antagonistic activity is generally agreed to be maximum. However, both Paul and Morrison have assumed no antagonistic activity in their calculations. (c) Results obtained when the objectiuefinction

includes terms requiring the minimization ofa joint reaction

Right heel strike.

To illustrate the versatility of the optimizatiotrtechnique employed in the development of the model the objective function was modified by adding a term requiring the minimization of the reaction on one of the joints. Since the vertical component at each of the three joints is predominantly higher than the lateral components it was decided to include this component only with a suitable weighting factor to describe the desire for alleviating the load on the joint. The objective function, for the hip joint, takes the form: i=62

minimize

U =

[ Fig. 14. Variation of total hip joint force vector in a walking cycle.

+

tions and omitted the equation for moment balance about the vertical axis (long axis of the segment). For the knee joint, the results of Paul (1967) and Morrison (1968) for the joint force are in close agreement. From an analysis of 15 tests, Paul found the average of the greatest peak values of the ratio ofjoint force to body weight to be 3.39 with a maximum value of 4.46. Morrison (1968), from experiments on nine males and three females in the age group 18-36 reports a value for the maximum joint force (the vertical component alone) to be 4 times body weight with a mean

HIP

JOINT

4%

-2 Heel strike

of WALKING

+

Mjr

REACTION

-

-I

4(MjX

q”jX

+

+

MjY

MjZ)L

+

MjZ)R

+

k sR ZhlD’

1

where R+ = hip joint force along 2 axis and k = a weighting factor. The solutions corresponding to values of k = 2,3,4,5 and 10 for the hip joint at 15 per cent of the cycle (where the maximum force occurs) are shown in Table 2. Similarly, solutions were obtained with the same values of k for the knee joint (both at 39 and 53 per cent of the cycle where the two large peaks occur) and the ankle joint (at 48 per cent of the cycle). The results are shown in Tables 3(a, b) and 4, respectively. When the objective function is modified to include minimization of the vertical component of the hip joint force, the same is reduced from its peak value of 384.5 kg with k = 0 to 299.7 kg with k = 2 (see

rigid body in space, Paul formulated only five equa-

6

i;I E +

Paul

(19651

CYCLE Toe off

Heel strike

Fig. 15. Reported hip joint reaction: Paul (1965, 1967).

Muscular Table 2. Muscle

load sharing

during

forces (kg) and joint reaction (kg) at 15 per cent of the cycle with modified U = Wi + 4Cjoint moments) + kRZhlp k=O

Muscle

Right leg

Abductor group* Adductor group Calf group Hamstrings Quadriceps Gracilis Sartorius lliopsoas Gluteus maximus Tibialis anterior Tibialis posterior Extensor digitorum longus Extensor hallucis longus Flexor digitorum longus Flexor hallucis longus Peroneus longus Peroneus brevis Peroneus tertius Hip joint reaction: s Y Z Knee joint reaction: .Y Y Z Ankle joint reaction: s Y Z Hip moment Xi.YZ

154.9 0 0 1562 78.2 0 0 0 0

Knee moment X.y.Z Ankle moment (kg-em) Xand k Z

97

walking

k=2

Left leg 13.1 0 0.4 36.1 9.9 0 0 0 1.0

Right leg 83.7 0 70.3 58.1 148.6 0 0 0 0

objective

function:

minimize

k = 3,4,5 and IO Left leg 13.1 0 0.4 36.1 9.9 0 0 0 1.0

Right leg 97.0 0 92.4 41.7 146.9 0 47.8 0 0

49.3

172.6

21 1.2

0

0

0

40.8

866

100.9

Left leg 13.1 0 0.4 36.1 9.9 0 0 0 I ,o

0

0.3

0

0.3

0

0.3

0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0.2

02

0.2

50.8 35.7 384.5

-7.2 9.4 35.5

29.9 1.5 299.7

- 7.5 9.4 35.5

5.7 -4.2 241.6

31.9 52.6 431.2

- 12.9 2.7 37.2

-27.3 47.5 3354

- 12.9 2.7 37.2

- 32.7 55.9 386.4

- 13.9 2.7 37.2

-3.4 13.6 160.2

- 0.72 0.02 0.6

- 223 46.8 396.1

- 28.2 57.2 469.9

-0.72 0.02 06

0.72 0.02 0.6

-7.5 9.43 35.5

0

0

0

0

0

0

0 -31.8

0 0

0 - 28.2

Table 2). With subsequent increase in li. the force is further reduced to 241% kg with k = 3. However, any further increase in the value of k does not alter the solution suggesting that no further reduction in the hip force is possible without changing the walking pattern. The corresponding changes in muscle forces can also be seen in the table. For the knee joint, inclusion of the vertical force on the knee in the objective function does not produce

any changes in the muscle forces or the joint reaction at 53 per cent of the cycle for values of li between 2 and 10. The joint force and the muscle force patterns are altered however, at 39 per cent of the cycle with the modified criterion as shown in Table 3(a) where a reduction in the vertical force on the knee from 445.3 to 326.8 kg was obtained for a weighting factor k of 3, 4,5 and 10. Similar results were obtained for the ankle joint as shown in Table 4.

A. SEIREGand R. J. ARVKAR

98

Table 3(a). Muscle forces (kg) and joint reactions (kg) at 39 per cent of the cycle with modified objective function: minimize U = ZF, + 4 (joint moments) + kR,,“_ k=O

k=O

Muscle Abductor group* Adductor group Calf group Hamstrings Quadriceps Gracilis Sartorius Iliopsoas Gluteus maximus Tibialis anterior Tibialis posterior Extensor digitorum longus Extensor hallucis longus Flexor digitorum longus Flexor hallucis longus Peroneus longus Peroneus brevis Peroneus tertius Hip joint reaction: X Y Knee joint reaction: X Z

Ankle joint reaction: X Y Hip ioment X,W Knee moment X,W Ankle moment

(k-cm) X,ZZ

k = 3, 4, 5 and 10

Right leg

Left leg

Right leg

Left leg

Right leg

Left leg

118.5 0 172.5 116.9 0 0 49.1 0 0

11.1 4.0 3.7 0 2.6 0 16.4 0 0

169.7 33.4 172.5 85 1 1.8 0 0 0 0

11.1

4.0 3.7 0 2.6 0 16.4 0 0

179.7 57.4 129.8 1104 23.8 0 0 0 0

11.1 4.0 3.1 0 2.6 0 16.4 0 0

0

5.9

59

0

5.9

0

0

0

0

0

0

2.9

2.9

0

2.9

90.5

1.5

90.5

1.5

38.9

1.5

3.54

0

35.4

0

85.1

0

0

0

0

0

0

0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 63.4

0 0 0

- 22.2 -2.2 304.6

12.4 3.8 18.4

- 26.5 -30.1 341.7

124 3.8 18.4

-37.9 - 50.4 419.1

12.4 3.8 18.4

-643 20.7 445.3

9.0 0.8 23.8

- 67.9 12.8 366.4

9.0 0.8 23.8

-43.6 12.5 326.8

9.0 0.8 23.8

-38.0 41.8 359.7

0.7 0.6 12.8

-38.0 41.8 359,7

0.7 0.6 12.8

-35.2 43.7 377.9

0.7 0.6 12.8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Muscular load sharing during walking

99

Table 3(b). Muscle forces (kg) and joint reactions (kg) at 53 per cent of the cycle with modified objective function: minimize U = Wi + 4 (joint moments) + kRz,nss k=O

Muscle Abductor group* Adductor group Calf group Hamstrings Quadriceps Gracilis Sartorius Iliopsoas Gluteus maximus Tibialis anterior Tibialis posterior Extensor digitorum longus Extensor hallucis longus Flexor digitorum longus Flexor hallucis longus Peroneus longus Peroneus brevis Peroneus tertius Hip joint reaction: X Y Z Knee joint reaction: X Y Z Ankle joint reaction: X Y Hip cement X, Y, 2 Knee moment X, Y. Z Ankle moment (kg-cm) Xand Y Z

k = 2. 3. 4. 5 and 10

Right leg

Left leg

Right leg

Left leg

0.8 4.2 2.7 106.8 0 5.8 19.6 0 0

44, I 653 17.9 4.9 272.1 41.7 84.0 0 0

0.8 4.2 2.7 106.8 0 5.8 19.6 0 0

43.5 80.0 74.2 0 271.4 44.4 88.5 0 0

0

0

0

0

0

78.9

0

71.2

2.7

0

2.7

0

12.0

0

12.0

0

0

0

0

0

0

0

0

9.2

0

0 0

0 0 0

0 0 0

0 0 0

65.1 - 13.5 121.6

12.9 - 22.4 276.2

65.1 - 13.5 121.6

16.2 -261 291.9

64.7 -11.9 127.2

- 176.9 -71.9 416.2

64.7 -11.9 121.2

- 175.3 -71.8 4752

9.4 0.2 28.4

- 93.2 - 32.6 174.1

9.4 0.2 28.4

- 94.9 - 34.4 177.3

0

0

0

0

0

0

0

0

0

0 0

0

0

16.9

0 12.1

A. SEIREG and R. J. ARVIKAR

100

Table 4. Muscle forces (kg) and joint reactions (kg) at 48 per cent of the cycle with modified objective function: minimize U = CF, + 4 (joint moments) + kRI,n,e k=O

Muscle Abductor group* Adductor group Calf groug _ Hamstrings Quadriceps Gracilis Sartorius Iliopsoas Gluteus maximus Tibialis anterior Tibialis posterior Extensor digitorum longus Extensor hallucis

longus Flexor digitorum longus Flexor hallucis longus Peroneus longus Peroneus brevis Peroneus tertius Hip joint reaction: X Y Knee joint reaction: X Y Ankle joint reaction: X

.>

; Hip moment X,XZ Knee moment x,r,z Ankle moment (kg-cm) X and Y Z

k = 2, 3, 4, 5 and 10

Right leg

Left leg

Right leg

Left leg

122.3 0 155.7 60.3 0 0 0 0 0

15.6 45 0 13.7 0 0 21.3 6.6 0

154.2 32.5 101.7 70.4 0 0 7.2 0 0

15.6 4.5 0 13.7 0 0 21.3 6.6 0

0

0.4

0

0.4

0

0

0

0

0

1,3

0

1.3

86.5

1.0

0

1.0

68.3

0

88.1

0

0

0

0

0

0

0 0

0 0 0

0

0 0

0

0 0

- 5-o -13.5 201.9

21.0 6.4 38.1

-21.4 -33.2 316.1

21 6.4 38.1

- 56.4 13.7 318.8

21.4 3.4 40.8

-35.8 11.7 260.9

21.4 3.4 408

- 57.4 53.6 3668

1-6 0 1.14

- 50.0 29.9 2506

1.6 0 1.14

0

0

0

0

0

0

0

0

0 0

0

0

0

48.6

0 0

It should be noted here that in Tables 2, 3 and 4 some of the muscles have been grouped in the tabulation of the results in order to quantify their combined effort. These are: abductor group (gluteus medius, gluteus minimus and tensor fasciae latae); adductor group adductor brevis, longus and magnus); calf group (gastrocnemius and soleus); hamstrings (semimembranosus, semitendinosus and biceps femoris); quadriceps (rectus femoris, vastus medialis, intermedius and lateralis).

CONCLUSIONS

It can be concluded from the reported study that: 1. The objective function that minimizes a weighted sum of forces in all the significant muscles of the lower extremities and the moments at all the joints (which has been previously used by the authors for static posture) yields solutions for all the muscle forces which are consistent with typical electromyographic patterns for level walking. The results from the model also con-

Muscular load sharing during walking

Knee

Joint

101

Reaction

-

Paul

----

Morrison

1

(1967

I

(1970

b4

0

m

t

IO

20

30

40

50

60

70

80

90

100

-I -% -2

of

WALKING

CYCLE

c

I

Heel strike

Toe off

I

Heel strike

Fig. 16. Reported knee joint reaction: Paul (1967). Morrison (1970).

firm the phasic activity of the plantar flexors (the calf group, tibialis posterior, flexor digitorum longus, flexor hallucis longus, peroneus longus and peroneus brevis) which are active during the middle and latter parts of the stance phase and, therefore indirectly assist in stabilizing the knee as discussed by Sutherland (1966). Although quasi-static walking is considered,for illustration, forces during dynamic activities can be readily calculated by including the corresponding inertia forces and moments at the different body segments. 2. Maximum joint reaction forces during the considered quasi-static walking cycle are 5.4. 7.1 and 5.2 times the body weight for the hip, knee and ankle joint, respectively. 3. Modifying the objective function to include the minimization of joint reactions, can only produce a certain reduction in the joint forces. A corresponding change also occurs in the muscle-force-patterns. Increasing the emphasis on minimizing the joint reaction without changing the walking pattern does not appear to produce further reduction suggesting that an alteration in the kinesiological pattern would be necessary in order to attenuate the intensity of pain at a particular joint beyond a certain level. REFERENCES

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Elftman, H.(1939)The force exerted by the ground in walking. Arbeitsphysiologie 10, 485. Elftman. H. (1939) The function of the muscles in locomotion. km. j. Phjsiol. 125, 357. Frank, A. A. and Vukobratovic. M. (1969) On the synthesis of biped locomotion machines. 8th Int. Conf Medical and Biological Engng. Evanston, Illinois. Huston, R. L. and Passerello, C. E. (1971) On the dynamics of a human body model. J. Biomechanics 4, 369. Inman, V. T. (1947) Functional aspects of the abductor muscles of the hip. J. Bone Jnt Surg. 39 (3), 607. Kane, T. R. and &her, M. P. (1970) Human self-rotation by means of limb movements. J. Biomechanics 3, 39-40. Marey, E. J. (1883) La Methode Graphique dans les Sciencc~s Experimentales. Marson, Paris, France. Marey, E. J. and Demeny, G. (1887) Etudes experimentales de la locomotion humaine. C. Y.hebd. Shnnc. Acad. Sci. 105, 544. Marey, E. J. (1895) Movement. Appleton, New York. McLeish, R D. and Charnlev. J. (1970) Abduction forces in the one-legged stance. J. ~~omk&&s3(2), 191l209. Morrison, B. B. (1969) Bioengineering analysis of force actions transmitted by the knee joint. Bio-med. Enyng 4, 164.

Morrison, J. B. (1970) The mechanics of the knee joint in relation to normal walking. J. Biomechanics 3, 51. Murray, M. P., Drought, A. B. and Kory, R. C. (1964) Walking patterns of normal men. J. Bone Jnt Surg. 46A, 335. Murray, M. P., Seireg, A. and Scholz, R. C. (1967) Centre of gravity, centre of pressure and supportive forces during human activities. J. appl. Physiof. 23(6). Paul, J. P. (1965) Bio-engineering studies of the forces transmitted by joints: (II). Engineering analysis. In Biomechanics and Related Bio-Engineering Topics (Edited by R. M. Kenedi). Pergamon Press, Oxford. (Proceedings of a Symposium held in Glasgow, Sept. 1964.)

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Paul, J. P. (1967) Forces transmitted by joints in the human body, Proc. Inst. Me&. Engrs 181 (3J), 8. Presented at the Inst. of Mech. Eng. Symposium in Lubrication and Wear in Living and Artificial Human Joints. Paper 8, London, April 1967. Paul, J. P. (1971) Load actions on the human femur in walking and some resultant stresses. Exp. Mech. 3, 121. Pauwels, F. (1935) Der Schenkelhalsbruch Ein mechanisches Problem. Ferdinand Enke, Stuttgart, Germany. Rydell, N. (1965) Forces in the hip joint (ii); intravital Studies. In Biomechanics and Related Bio-Engineering Topics (Edited by Kenedi R. M.). Pergamon Press, Oxford. Rydell, N. (1966) Forces acting on the femoral head pros~thesis. Acta. Orthop. Stand. SuppI. 88. Seirea. A. and Arvikar. R. J. (1973). A mathematical model fo;evaluation of forces in ‘lower extremities of the muscult-skeletal system. J. Biomechanics 6, 313. Smith, P. G. and Kane, R. R. (1968) On the dynamics of the human body in free fall. J. appl. Mech. 35, 167-168. Strange, F. G. St. C. (1963) The Hip. Heinemann, London. Sutherland, D. H. (1966) An electromyographic study of the

plantar flexors of the ankle in normal walking on the level. J. Bone Jnt Surg. 48-A (1) 66. Townsend, M. A. and Seireg, A. (1972a) The Synthesis of bipedal locomotion. J. Biomechanics 5, 71. Townsend, M. A. and Seireg, A. (1972b) Optimal trajectories and controls for systems of coupled rigid bodies. J. Engng Industry 94 (B-2), 472.

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