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An evaluation of the kinematics of gait by minimum energy
1. Bwrwchurricr.
Vol. I. pp. 147-159.
Pcrgamon Press. 1968.
Printed in Great Britain
AN EVALUATION OF THE KINEMATICS BY MINIMUM ENERGY*
OF GAIT
ROYCE BECKETT Department of Mechanics & Hydraulics, University of Iowa. Iowa City. Iowa. 52240. U.S.A. and KURNG CHANG Department of Mechanics & Hydraulics. University of Iowa, lowa City, Iowa. 52240. U.S.A. Abstract-This paper is an anlaysis of the motion of the leg and foot in the swing phase of a’ step. The analysis is based on the hypothesis that the behavior will be such as to minimize the. amount of mechanical work done. Results of the analysis are the geometric position of the leg and the equivalent forces in the hip and knee joints as functions of time. Also included is the amount of work done as a function of cadence. Comparison is made with experimental results that are available in the literature. INTRODUCTION
well-learned body tasks such as walking and running, it seems reasonable to expect that the movement of the body components would be made in such a way as to minimize the amount of mechanical work that is done. As early as 1836, the brothers Wilhelm and Eduard WebeJ studied the mechanism of walking and running and concluded that the motion during the swing phase of a step was pure pendulum motion. In the 1870’s Marey, in his extensive work on the problem of human locomotion did some computations of energy expended while walking. In the period 18941904 Fischer, a German mathematician working with Braune, an anatomist, published a classical work on human locomotion. This work is an exhaustive investigation of the problems of locomotion. Fenn, in the period around 1930, studied the changes in kinetic and potential energies of the body during walking and running. He concluded that the work done in producing the periodic fluctuations of legs, arms, etc. and in overcoming wind resistance accounts for most of the energy used in running. Correlation of these IN
CERTAIN
*Received
13 July
results of analysis with practice was studied independently by Hill ( 1913, 1922). Elftman ( 1939. 1940) continued in this vein and made careful evaluations of the energy expenditure for various. body parts by using cinephotography. He further used the information to determine necessary moments in the joints to produce the motion and then to approximate muscle behavior. Bernstein and CO-Workers of Russia studied the dynamics of human locomotion and in 1935 published a comprehensive report of their work. Bernstein took issue with some of the results of Fischer and Braune on the actual path of the body components in locomqtion. lnman andassociates (1954, 1953,1947), at the University of California, have made extensive studies of body motion during level walking for both normal and prosthetic gait. Principal results have come from careful analysis of stick diagrams and force plates used to measure the reaction of the foot with the ground. Details of their work are available in reports to the Veterans Administration division in prosthetic devices which sponsored much of the program. Bresler and Frankel in 1950 reported on force plate analysis of the
1967. 147
148
R. BECKETT
forces and moments in the leg db_ing level walking. Cunningham and Brown ( 1952) later perfected the force plate so that more reliable results could be obtained. Considerable effort in this direction has also been made at the Veterans Administration prosthetic devices laboratory under Murphy. Ralston (1958, 1961) has made studies of the energy expended in walking by measuring the oxygen consumed. His work is relevant here in that he shows that there is an ideal cadence for an individual that will give a minimum energy expended per unit of distance traveled. Nubar and Contini (1961) contend that the individual will determine his motion to reduce the muscular effort to a minimum consistent with imposed conditions of constraint. This paper is an analytical investigation of the behavior of the leg in the swing phase of a normal walking step. Forces and moments are imposed at the joints of the leg to produce motion that is consistent with the geometrical constraint and in such a way as to give a minimum expenditure of energy. The analysis gives the motion of the leg and foot, the equivalent moments in the hip and knee to produce the motion and the energy expended in the swing phase of the step. The results obtained are compared with some of the experimental results that have been reported in the research noted above.
and K. CHANG
Fig. 1. Positions of the leg in level walking.
swing phase of the step is considered in this study the action of the foot in lifting the body is ignored, except that the motion of the leg and foot is restrained to follow a prescribed path. The ankle joint moves freely prior to toe-off so that the toe remains in contact with the floor. After toe-off it is assumed to be locked until the end of the cycle at heel-strike. The analysis begins during the push-off phase of the step and proceeds until heelstrike at the end of the swing phase as illustrated in Fig. 2. The position of the hip, point 0 at time t, is prescribed by giving the forward velocity, assumed constant, and h, the
MECHANICAL MODEL
A simplified model of the leg is assumed for the analysis and is shown in Fig. 1. Point 0 represents the hip joint and is assumed to move on a sinusoidal curve that approximates actual body motion. Link 1 is the thigh and is connected by joint 0’ to link 2 which represents the shank. The foot is connected to the leg by the ankle joint labeled P. The two couples, H and M, are produced by the muscle complex associated with each of the joints. Figure 2 shows how an idealized version of H will act to give the required motion. Also shown is the behavior of the two principal muscle groups that produce H. Since only the
25
35
50
go
82
ea
par Cent of Cycle
(0)
I I
(cl
c-m Pmceare
Fig. 2. One complete cycle in level walking.
100
AN
EVALUATION
OF THE
KINEMATICS
vertical motion of 0. The angular position of the thigh is given by C$ which is the angle between the position of the thigh in maximum extension (beginning of heel rise) and its position at time t. The angle between the shank and the thigh is y and the rise of the foot during toe-off is measured by (Y. The initial positions of each of these angles is &, y*, and cr,, and the height of the hip joint is h,. Each of these variables and its initial value is shown in Fig. 1. In order to have the motion simulate the swing phase of a typical walking step, certain geometric restraints must be imposed. It is required that the foot pivot about the toe until toe-off much like the actual case and then to force the ankle to lock. The foot is then constrained to follow through in the swing phase so that the toe moves on a prescribed curve shown by ATB in Fig. 1. The precise form of this curve is determined in the analysis. When the toe swings under the knee, point B in Fig. 1, the thigh will have reached maximum flexion and will remain in this angular position while the shank comes smoothly into position for heel-strike. The acceleration and deceleration moments at the hip joint are each assumed to be of a constant magnitude, and to act for a given interval of time, so that the moment profile is rectangular when plotted against time (see Fig. 2). On the basis of known data on the action of the muscle groups that supply the power for the motion of the leg during swing phase. this is a reasonable assumption. Figure 2a shows a complete cycle of the leg from heel-strike to heel-strike. The portion considered irerein is the latter half. i.e. from 50 to 100 per cent of the cycle. For purpose of analysis this latter half is broken down into three phases that depend upon the restraints that are imposed on the motion. In the initial phase designated phase I in Fig. 2(c) the toe is fixed on the ground and the foot rotates around it (Fig. 1). In Phase II the toe is restrained to move on the curve ATB in Fig. I. The general form of this curve is obtained from experiment
OF GAIT
BY MlNlMUM
ENERGY
149
but the analysis provides some help in the choice of the particular coefficients in the equation of the curve. When the toe glides under the knee joint the thigh is in maximum flexion and will remain in this position until heel-strike. When the thigh reaches maximum flexion. or the equivalent. when the toe moves directly under the knee. the constraint conditions change and this marks the start of Phase I II. The condition of constraint in Phase I I1 is that the thigh should remain fixed in the positioh of maximum flexion. In Fig. 2(b) a plot of the hip moment as a function of time is shown. The positive moment over the initial part of the cycle is supplied primarily by the quadriceps group of muscles and the deceleration moment at the end of the cycle comes from the hamstring group. Superposed in this portion of the figure are dotted curves which approximate the actual behavior of the two groups of muscles (College of Engineering, University of California). EQUATIONS
OF MOTION
The equations of motion for the leg are obtained by the Lagrangian Method and the conditions of constraint imposed by means of the Lagrangrian Multiplier. This requires the kinetic and potential energies of the system and the constraints in equation form. Consider first the kinetic and potential energies. It is assumed that motion is confined to the x-g or sagittal plane shown in Fig. 1. The forward (X direction) component of velocity z+,is constant and the vertical motion of the hip joint 0 is given by h where h = sin
27r(r,+t) 7
(1)
r is the time for one step, i.e. 50 per cent of the full scale in Fig. 2(a). t,, is a phase shift which corresponds to the starting value for time. When h = h,, t,, = O-067 and t is time measured from to. For purpose of finding the equations of motion the limb is approximated by the two
R. BECKE-IT and K.
150
links in Fig. 1 representing the thigh, link 1, and the shank and foot, link 2. The foot is combined with the shank in evaluating the energy of the system since its relative motion with the shank occurs during toe-off when the motion of the foot is small. The coordinates of the center of gravity of each of the two links 1 and 2 are found from Lissner (1962) and are given by (x1, yJ and (x2, y2) respectively. x1 = uot + a, sin ( $I - $vJ
(2a)
Y1= h+a,
(2b)
xi=
cos (+-cb”)
vof+IIsin
(d,-&)
-ua,sin
(y+&-4) (2c)
y,=h+&cos
(4-&)+%cos
(Y+&-+) (2d)
where li is the length and af the distance from the upper joint to the center of gravity of each link, i = 1,2, +, is the angle between a vertical through 0 and the thigh when in the position of maximum extension. The velocity of the center of gravity of each link is obtained by differentiating equations (2) with respect to time. The angular velocity of link 1 is d and of link 2, (Q-i). Assuming the foot is fixed rigidly to the shank the kinetic energy is written as follows T = 311~+312(~-~)2+~m,u,2+~m,v~2.
CHANG
the potential energy of the system is V = --m&r& cos (4 - 40) + hl - mzg[l, x cos (b,-40)
Where Ii is the mass moment of inertia of link i about its c.g. mf is the mass of link i. The conditions of constraint that are imposed on the motion are such that 4 and y cannot be independent but are related by an algebraic equation. The particular condition of constraint depends upon the position of the toe, T, in Fig. 1. When the toe is fixed during toe-off the point P moves on an arc with the center at T. Two equations involving vertical and horizontal distances between T and 0 can be written. dsina=Icos&,-&cos
dcosa!=
h,,- h + d sin CQ
(7)
sin (+-&) do + &)
uoc + d cos a,,.
(8)
f(& y) = 12+ 1*2+ 1*2- d2 + v02t2+ d2 cos2 a 0 -
211,cos 4 - 211,cos (4 - y)
+ 21,12cos y - 2vor d cos QO +2(dcosa,
(9-bo)+u,‘~-2~u,~sin(~-9,)
(c$-&)
- uot)[--I sin +.
-l,sin(+-f#~~)+l~sin(y-++4~0)1 +2K[Icos~,-l,cos(~-~o)
cosy+?&“]
(4--4,)-u,t+Q~
+
(y-~+4d
(4)
+h2] +3m2{[1,2r$+u+-r&
xcos
12 cos
-lsin&)-f,
-
T = $11@+312 (Q--)2+dnrl[v,2+2v,u,d;
Xsin (y-cp+&)]
-
+ 1, sin (y -
the kinetic energy is
-2jI[l$sin
(4-&J
Eliminating o! from these two equations gives the constraint condition for Phase I as a function of $ and y.
D, = d(_&‘+i,2)
-2f,u,&+$)
+h]. (6)
(3)
If 11~and u2 are replaced by
xcos
+a, cos (y-f$b&)
(5)
+a,(;-&)
where
+jz2}+2vo[11G cos
-f2cos(y-~+~o)1+-*=0
(r-++&)l
K = ho-hhdsina,. 1 = v (I,2 + 1.,2-t-21112cos ys).
(9)
AN EVALUATION
OF THE KINEMATICS
The end of Phase I is toe-off and occurs when the angle between the foot and the shank reaches a given limiting value. If the angle is 8 then a must satisfy the equation. cr= y++,,-mts-;.
y = Y(X) = 5 AiX4-i i=O
(12)
where x is measured from A. The value for x in equation 12 may be replaced from equation 2(c) to obtain a function in 4 and y. The five conditions for evaluating A,, i = 0, 1, . .4 are
y’(C) = 0
?J’(0) = G/i at toe-off y(D) = y’(D) = 0.
i13)
Both analysis and experiment have been used as a guide in choosing C and D. The velocities + and X-of the toe come from differentiating equations (2) when x and y are the coordinates of T and then substituting the value of each of the variables at toe-off. *=-l,t$sin
(4-&,)-l,
Xsin (y-++f&,-8)+i i=V,~+f,&os x cos (y-&-t
(14)
40-6)
where
(15)
I+= ~(1,2+d2-212dcosi3 @= arcsin [dsin (F--6)/f4].
ii=
At the end of Phase II the hip flexion angle is
V-Af.
( 18)
The two equations of motion in the variables 4 and y are then given by substituting T and v into the equations. d aT aT ---_--+-_=H dt a4 a4 d aT ---_ dt a+
av a4
(19)
aT av_M ar+z-
(20)
.
Where H and M are respectively the hip and knee flexion moments. Substituting T and I/ from equations (5) and (6) gives +[I1 + I, + m,a,2 + m21z2-t m2az2 + 2m211a2cos y] - i;[i2 + m2az2
+ m21,a2 cos 7-J + m,ga, sin ($J- +o) + mzlla2+ ($- 26) sin Y+ m,gEl, -a,sin
+i[-_(m,a,+m21,~
(r-++h)l sin (
(y-d+&,,3-*$=
H. (21)
+[I2 + m2a2”) - ;f; [I, + m2az2+ m,l,a, cos y] + m211a2~2sin y+
(I61
(17)
The equations of motion may now be obtained from Lagranges equation where the potential energy function V is modified to include the condition of constraint. If the condition of constraint is defined by f= 0 where f is some function relating the coordinates of the system, i.e. 4 and y. then a new function p is defined. Lanczos ( 1949).
+m,a2sin
(+#J,,-I,(+&,
151
&j&O.
X sin (+-&)
(G-6)
ENERGY
a maximum and the toe is directly below the knee in the swing through. This marks the beginning of Phase 3. The condition of constraint for Phase 3 is that the angular position of the thigh remain fixed, i.e.
(11)
After toe-off, the foot leaves the ground and point T moves on the curve AT6. This curve must have a slope at A which will permit the foot to rise smoothly and it must pass near to the floor at point B with a horizontal slope. An additional condition that is imposed is that the curve have a maximum value at a distance C from A. The value of D = AB and C will depend upon the length of step. In order to satisfy the five conditions summarized below a fourth degree polynomial is chosen to represent the curve.
y(O)= 0
OF GAIT BY MINIMUM
-i[m,a,sin
m,ga., sin (y - I#I-t &)
(y-$+&,)]-hzy=M. (22)
152
R. BECKE’TT and K. CHANG
Equations (21) and (22) with the appropriate conditions of constraint for the functionfand the proper initial conditions are sufficient to solve the problem in each phase of the motion. NUMERICAL
COMPUTATIONS
Equations (21) and (22) are solved simultaneously for 4 and $ to give equations of the form (23) 4 = F,(+, i, Y, +, A, H) .. (24) Y=F&P,+,Y,;JM. These equations in combination with the appropriate constraint condition are solved for $, y and A. The numerical method used to solve the system is the second order RungeKutta method. For the first phase of the motion the constraint condition is equation (9). The starting conditions for Phase I are the initial values for the angle C#Iof the thigh, the angle y at the knee joint and the corresponding angular velocities (time derivatives). In this study the starting values for Phase I are chosen so that the time from the start of the analysis to heel-strike will make up just 50 per cent of a complete cycle. This starting time corresponds roughly to the initiation of action by the quadriceps group of muscles, and is therefore assumed to be the time at which H is applied. Starting values are obtained from available experimental studies. Bresler et al. (1950). 4(O) = (bb= 6.0” Y (0) = Ys =f(&)
d(0) =$(O) These correspond
=
13”
(25)
= 0
to values of LX.,, and & of CYO = 35.5” d” = 23.0”
(26)
The power to drive the leg is supplied by the couple H at the hip joint. The magnitude and *From
stick diagrams furnished
duration of H are such as to bring the leg through the cycle in accordance with the limitations of geometry and strength of the muscle group and with the least amount of work. For this study both the accelerating and decelerating couples in the hip joint are assumed to have a square profile when plotted against time. This is the form to give the least work for a given maximum value of H and agrees qualitatively with experimental results Sumen et al. ( 1954). The magnitude of the couple is limited by the forces that can be developed by the muscle complex that generates it although this does not appear to be a limiting factor in walking. The important limitation on H is imposed by the reactions that are created by a large accelerating moment in the hip joint. High heel rise will occur after toe-off necessitating large control moments in the knee and this will cause an excessive expenditure of energy. A smooth motion and one which gives good agreement with a natural gait is obtained when the duration of H is roughly 10 per cent of a cycle and is independent of the cadence. With the form and duration of H known it is still necessary to find its magnitude. This is done by assuming a starting value for H and then computing the motion of the system up to the limiting angle of hip flexion which corresponds to #I = 45”. If H is too large the leg comes through too fast for the assumed cadence and if it is too small then the leg is slow in reaching the limiting angle. The time to reach the limiting angle is about 64 per cent of the time for a step+ and is therefore fixed when the cadence, one of the given conditions, is fixed. In order to find the correct starting value for H an iteration method is empolyed whereby a correction is made to H that is based on a linear interpolation (secant) method, to give the desired end conditions on 4. In this way a suitable value for H is obtained with few iterations. Toe-Off is the beginning of Phase 2 and is
by Dr. E. Murphy. Veterans
Administration
Prosthetics
Center, New York.
AN EVAULATION
OF THE KINEMATICS
identified by a limiting angle d between the foot and the shank (see equation (1 I )). An arbitrary value of 1 IO”, which is based upon the analysis of several stick diagrams from level walking experiments, is selected. The point at which the angle reaches 1IO” is identified by equation ! 1 I ) with S = 1IO”. and where cxcomes from the equation i, cos i@-@,,I +/?COS (y+&-+)
+h
+ d sin (Y= I cos &,+ ho + 6 sin (Y,,. (27) When the angle reaches the limiting value the ankle is locked until heel-strike. During Phase 2 the toe moves on the curve AB which is defined by equations (12) and ( 13). The distance D is determined by finding the forward travel of the toe at the time the thigh reaches maximum flexion. This is when the toe swings through directly under the knee and is about 64 per cent of the time for one step.
- d cos ffO+ 064rz, + I, sin (22”)
(28)
where T is the time for one step and depends upon the cadence. The angle of 22” is the approximation used for the maximum flexion angle at the hip. The value of C is about 40 per cent of D and comes in part from experimental studies but may be adjusted on the basis of the numerical results. The angular velocity of the thigh should come to zero at the same time that the toe swings under the knee. This will use less energy and as expected it agrees with experimental observations. The time at which the toe swingsunder the knee is the end of Phase 2 and the start of Phase 3 of the motion. In Phase 3 the shank is brought into the position for heel-strike. At the time of heelstrike the angular velocity ; of the shank is roughly zero. The deceleration of the shank must be brought about smoothly and in a way to use the least amount of energy. This. like the acceleration phase. is accomplished by a step input of moment at the hip joint. The
OF GAIT
BY MINIMUM
ENERGY
IS3
duration of the hip moment in the deceleration phase is found to be about 12 per cent of a cycle and again is independent of the cadence. Like the acceleration moment, the magnitude of the deceleration moment is found by the Secant method. The condition that determines the magnitude of the decelerating moment is that + must be brought to zero at heel-strike. lt might appear that the knee moment could be adjusted to bring i, to zero at the required time but this is not the case. Rather the knee moment takes on values that will keep $I fixed during Phase I I1 of the motion. The integration of equations (23) and (24) is made by the second-order Runge-Kutta method. The procedure in finding the solution is as follows: For Phase I:
Find from equation (20) the starting value for A at time t = 0. This is the Lagrangian multiplier that represents the force (less body weight and acceleration) that is applied at T between the foot and the floor. Equation (19) is then solved by the Runge-Kutta method for do. 4 and 4 at a time At. These are then used in the constraint condition to find y, -j and i;. Equation (20) again gives h at time At. When the solution is found at At the process is repeated for the solution at 2A.r. The technique is continued until the angle between the foot and shank is 110” at which point the ankle is locked and the toe leaves the floor. The condition of motion at this instant in time provides starting values for the second phase. For Phase 2 :
The equations of behavior in Phase 2 are the same as those for Phase I but the constraint condition changes. During Phase 2 the toe is constrained to move on the curve AB. The moment in the knee joint provides the constraining couple to force the toe to move on the curve. The end of Phase 2 is marked by the condition that the toe is directly under the knee jojnt.
154
R. BECKEIT
For Phase 3: Phase 3 of the motion begins with the toe in the lowest position of swing-through. This last phase of the motion is carried out with the thigh fixed against rotation, i.e. d; = ;p’= 0 and d, with a fixed value of 45”. This leaves only the equation of motion in y. The solution for y is carried on from the terminal condition of Phase 2. The decelerating hip moment is applied shortly after the beginning of Phase 3 (approximately 12 per cent of the time for one step) and serves to slow the shank. The knee moment is computed to maintain Q = 0. For the computations made herein the numerical data for the body parameters have been obtained by using information from Lissner (1962). Two sets of parameters have been used, one for a body weight of 150 lb. and the other for 120 lb. W = body weight = 150 and 120 lb ml = (9.7%. W) 11= 17.4 in. a1 = 7-53 in. d= 9~1 in. 1, = 864 16 lb-in-sec2 11 = 6.9133 lb-in-sec2 Computations for each weight are made for several different walking speeds. The speed depends upon the cadence which is varied by 10 steps per min from a slow walk of 60 steps per min to a fast walk of 120 steps per min. Initially the length of step was held constant at 33 ins. This is not natural and was noted in the results -particularly for the slow walk where it was necessary to apply a flexion moment in the hip in order to extend the leg to heel-strike. Changing the length of the step to correspond with the cadence gave a more natural gait. The length of step for each cadence is given in Table 1. Variations in the step length with weight were not made. For each cadence the work done by the two moments in the hip and knee was computed according to the equation
and K. CHANG
Absolute values are used for the integrands since they represent effort that is not recoverable. The integrals are evaluated by summing over the finite intervals of time At that are used for the integration of the equations of motion. The important results of the computations are the configuration path of the leg, given by 4 and y; the moments in the hip and knee joints, given by H and M; and the work done for each cadence (walking speed). RESULTS
The acceleration and deceleration moments H in the hip are shown in Fig. 2(b). The time of application and the duration of these moments are determined somewhat from the interpretation of experimental results reported in the bibliography and by the analysis. m, = (6.4%. W) 1z = 16.9 in. a2 = 9.81 in. Independent of W 1 = 7.0727 lb-in-sec2 for W = 150# 12 = 5.6582 lb-in-sec2 for W = 120#. 12 An objective in the analysis is a smooth motion of the leg that will require the least input of energy for control. This requirement places some restriction upon the duration of both the accelerating and decelerating moments in the hip. Once the two parameters of time of application and the duration of the moment are set, then the magnitude of the accelerating moment is determined by the analysis to give the proper value of 4 and 4 at the end of Phase II; the decelerating moment is found to give y = + = 0 at the end of Phase III. Figure 2(b) shows graphically the quantitative results of the analysis and they are given for the two cases considered in Table 1. In each case the duration of the accelerating moment is roughly 10 per cent of a full cycle. Termination of the moment comes at toe-off. The decelerating moment acts over 12 per cent of the cycle just before heel-strike.
AN EVALUATION
OF THE KINEMATICS
OF GAIT
BY MINIMUM
155
ENERGY
Table 1
Cadence 60
70 80 90 100 110 120
Length of step (in.) 27.3 28.5 29.9 31.5 33.3 35.3 37.5
Act. couple (in.-lb) (15Olb) (1201b) 20.56 41.16 65.15 %.OO 133*00 175.29 22967
24.93 SO.45 81.18 119.98 166.24 218.00 299.00
Values for 4 and y are plotted against time measured as a per cent of one step for a cadence of 90 steps per minute in Fig. 3. Variations in $ and y for changes in body weight and cadence will give onfy small changes from the values shown. The experimental results shown in Fig. 3 were obtained from stick diagrams of normal subjects walking at 90 steps per min. furnished by the Veterans Administration Prosthetics Centef in New York. Table 2 gives results for 4, y, 9 and q for each of the two body weights used and for each cadence. Figure 4 shows the time derivative of $J and y plotted against time as a per cent of one step for the case of body weight equal to 150 lbs. and a cadence of 90 steps per min. Variations of the velocity in the separate
60 50 !4O .{ ‘0 30 a 20
Fig. 3.
Energy exp. I33 in. (120lb) (1501b)
Dec. couple (in.-lb) (150lb) (120lb) -40@0 -22.36 28.22 54.42 90.82 143.71 195*00
-38.04 -899 20.28 47.50 75+0 132.00 157.50
7690 76-40 76.00 85.90 105~00 131Xlo 158.20
94.30 94.00 97. IO 106.90 131.10 163.40 219.00
‘7
-15
0
20
40 Time,
I 60
I 80
I 100
per cent of step
Fig. 4. Angular velocity vs. time for thigh and shank.computation; -. - from stick diagrams;---Moffatt’s results.
runs are much greater than for the displacements, which is expected. Experimental results for the angular velocity of the shank as deduced from correlated stick diagrams are given in Fig. 4. Plotted also is the angular velocity of the shank as obtained by Moffatt ez al, by using accelerometers on normal subjects. These results are included because of the excellent experimental techniques employed. From the results in Fig. 4 it appears that the calculated values for the angular velocity of the shank are low. The form of the curve agrees qualitatively, however, with the experimental curve and except for the results of Moffatt agree reasonably 20 40 60 80 100 well with the magnitude. In the computation routine it was tedious to obtain a smooth lime ,per cent of step transition from Phase II to Phase III. FurtherRotation vs. time for the thigh and shank.computation:----experiment. more. the jump in the hip moment caused
156
R. BECKETT
and K. CHANG
Table 2 Time (% of swing phase) Cadence
Weight
0
10
120 150 120 150 120 1.50 120 150 120 I.50
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.5 8.5 0.8 O-8 26.1 26.0 2.5
120 150 120 150 120 150 120 150 120 150
20
30
40
50
60
70
80
20-o
44.2
90
100
-2:.: -32.0
13.8 14.7 I.0 I.0 41.3 41.2 2.0 2.0 -31.9 -40.2
19.8 I.3 I.3 51.4 51.2 I.5 1.6 47.9 59.7
29.0 28.8 1.7 I.7 60.0 59.9 I.3 1.3 46.0 57,6
38.4 38.2 1.5 1.5 63.9 63.8 -0.1 -0.1 33.3 41.4
44.1 0.5 0.5 56.2 56.2 -2.8 -2.8 19.9 23.4
43.9 44.5 0.4 0.0 35. I 37.4 4.0 -2.8 44.5 66.0
43.4 44.5 0.0 O-0 21.3 24.7 -1.7 -1.8 16.5 23.2
43.4 44.5 0.0 0.0 13.5 15.9 -1.0 -1.0 11.5 19.2
43.4 44.5 0.0 o-o IO.4 14.0 -0.1 0.0 IO.4 19.0
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.4 8.4 0.9 0.9 25.6 25.5 3.1 3.1 -20.5 -25.7
15.3 13.7 1.4 I.3 44.5 40.9 2.4 2.7 -17.2 -25.5
22.3 22.1 1.7 1.7 54.4 54.1 I.9 1.9 48.6 60.6
32.8 30.5 I.8 1.8 63.3 61.9 1.0 I.3 40.5 54.4
40.4 38.5 1.4 1.6 64.3 64.8 -0.7 -0.2 22.9 35.0
44.6 44.5 0.3 0.3 55.0 55.1 -3.5 -3-5 13.8 16.1
44.1 44.7 -0.4 0.0 35.5 37.9 -4.6 -3-3 43.3 68.1
43.6 44.7 0.0
43.6 44.7 0.0 0.0
2y.i 25.3 -2.2 -2.3 26.2 39.0
43.6 44.7 0.0 0.0 13.5 17.0 -1.4 -1.6 20.3 32.6
11.2 13.2 -0.0 -0-2 18.7 29.6
120 150 120 150 120 I50 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
9.2 9. I 1.2 l-2 28.4 28.4 3,7 3.7 -14.4 -16.2
14.2 14.2 l-7 I.7 46.2 46.2 2.8 2.8 5.1 4.6
22.5 22.5 2.0 2.0 55.0 55.0 2.3 2.3 48.2 60.2
32.0 31.9 2.0 2.0 63.5 63.5 1.3 1.3 38.8 48.5
40.3 40.3 1.5 l-5 65.1 65.1 -0.8 -0.8 14.3 17.9
44.5 44.5 0.2 0.2 53.8 53.9 -4.3 -4.3 5.4 6.5
43.4 44.5 -0.8 0.0 35.5 39.0 -6.3 -3.9 25.1 71.9
42.3 44.5 0.0 0.0 17.4 24.0 -2.5 -1.9 38.0 54.6
42.3 44.5 0.0 0.0 9.4 15.3 -0.9 -1.1 31.2 52.0
42.3 44.5 0.0 o-o 8.4 13.5 0.1 0.0 32.1 52.8
120 150 120 I50 120 150 120 I50 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.5 8.5 1.3 1.3 26.1 26.1 4.3 4.3 -6.3 -7.9
16.6 16.6 2.1 2.1 47.1 47.1 3.3 3.3 35.0 43.7
24.2 24.2 2.3 2.3 57,o 57.0 2.5 2.5 47.2 59.0
32.1 32.1 2.2 2.2 63.9 63.9 1.4 1.4 36.0 45.1
41-I 41.1 I.6 1.6 65.1 65.1 -1.1 -1.1 4.3 5.4
44.7 44.8 0.5 0.5 56.3 56.3 -4.2 -4.2 -6.9 -8.6
44.9 45.0 -0.1 0.0 39.7 40.0 4.6 4.4 65.3 76.9
44.9 45.0 0.0 0.0 22.3 22.6 -3.0 -3.0 58.8 67.7
44.9 45.0 0.0 0.0 14.5 14.7 -1.6 -1.6 52.4 63.6
44.9 45.0 0.0 0.0 11.7 Il.7 -0~0 -0. I 52.8 63.7
120 150 120 I50 I20 I50 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
9.1 9. I I.7 1.7 28-6 28.6 5.1 5.1 7.0 8.7
16.3 16.3 2.4 2.4 46.9 46.9 3.9 3.9 67.4 84.3
25.0 25.0 2.6 2.6 58.3 58.3 2.8 2.8 45.6 56.9
33.7 33.7 2.4 2.4 65.3 65.3 I.2 I.2 .27.8 34.7
41.0 41.0 1.8 I.8 65.9 65.9 -1.1 -1.1 -4.5
45.0 45.0 0.4 0.5 56.2 56.2 -4.7 4.7 -18.0 -22.5
45. I 45.2 -0.1 0.0 37.8 38.2 -5.1 4.9 67. I 76.0
45.0 45.2 0.0 0.0 22.9 23.3 -3.6 -3.6 70.3 92.1
45.0 45.2 0.0 0.0 13.9 13.7 -1.9 -1.9 62.0 80.7
45.0 45.2 0.0 0.0 9-5 IO.4 -0-o -0.0 62.3 81.5
-5.6
PIN EVALUATION
OF THE
KINEMATICS
OF GAIT
BY MINIMUM
157
ENERGY
Table 2. (cont.) Time (%,) Cadence
@J i I10 Y ;
M
CtJ i I20
-Y ; M
0
10
20
30
40
120 I50 I20 150 120 I50 120 150 120 150
6.4 6.4 0.0 0-O 13-O 13.0 0.0 0.0 0.0 0.0
9.8 9.8 2.0 2.0 31.3 31.2 5.8 5.8 28.1 34.8
15.3 15.3 2.7 2.7 45.2 45.1 4.8 4.8 94.8 117.5
24.9 24.9 2.8 2.8 58,7 58.4 3.? 3.; 43.6 55.1
34.2 34.2 2.5 2.5 66.6 66.0 I.2 1.2 17.9 25.1
41.6 41.8 1.7 1.7 66.1 65.6 -1.8 -1.7 -28.1 -26.5
120 150 120 150 120 150 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
10.7 11-o 2.5 2.6 34.2 34.6 6-6 6.8 63.6 84.0
17.3 17.7 3.1 3.2 48.9 49,4 4.7 4,9 42.4 53.1
28.0 28.7 3.0 3.1 63.1 63.1 2.9 3.0 36.3 43.4
34.7 35.5 2.8 2.8 66.9 68.0 1.3 1.2 11.7 8.7
42.6 43.3 I.7 1.6 65.6 65.9 -2.3 -2.8 -42.5 -72.0
Weight
6 atidy are in degrees.
4 and i are in rad./sec.
50
60
70
80
90
100
44.3 44.6 0.6 0.7 58.6 58.6 -4.9 -4.5 -48.3 -42.9
43.1 45.4 -1.3 o-0 34-2 39.9 -9.2 -5.4 -0.8 82.9
41.9 45.4 0.0 o-o 14-9 23.3 -3.7 -4-o 77.6 115-s
41.9 45.4 0.0 0.0 7.5 13.3 -2.4 -1.8 79.1 104.1
41.9 45.4 0.0 0.0 3.0 11.1 -0.0 -0.1 77.5 105.1
45.1 45.3 0.4 0.0 56.4 54.9 -5.8 -7.1 -42.0 -86.0
45.1 45.3 -0.1 0.0 34.2 33.3 -6.0 -5.6 68.6 70.2
45.1 45.3 0.0 0.0 22.2 21-4 -4.5 -4.5 109.1 137.2
45.1 45.3 0.0 0.0 11-O IO.5 -1.9 -1.9 97.5 124.0
45.1 45.3 0.0 0.0 7.5 8.4 -+I 0.0 100-O 126.9
M is in in.-lb.
irregular behavior. particularly in the knee moment. The knee moment is the couple that imposes the proper geometric restraint in each step of the computation. Its value is quite sensitive to small changes in the parameters. For example, when the form of the curve on which the toe moves in Phase II is altered only slightly. the knee moment may be changed by a significant amount. See Fig. 5. This was a disturbing situation until it was realized that the knee moment is not really a very important factor in level walking. In fact. the leg acts very much like a penduhrm as it moves through the swing phase of the step. The small knee moment obtained is for guidance, and in walking it will be small. Thus any change in the conditions on the problem may be expected to produce large percentage changes in knee moment. An ideal gait would require a minimum of guidance by the knee moment. Figure 5 shows a plot of the knee moment for each body weight used and a cadence of 90 steps
100
I-
Y \
100 -
\ \
’ ‘-/ 150
l
0
I
I 20
40 Time,
2
I 60
80
100
per cent of step
Fig. 5. Knee moment vs. time.-..M’ = 120 lb. C = C = 0.40;-.--w = 150 lb, C = 0.4&--w = 150 lb. 0.331):~---Moffatt’s results.
per min. Also included is an experimental result from Moffatt that has been obtained from accelerometer records. The analytical
R. BECKElT
15%
results only roughly approximate the experiment. However, considering the impulsive form of the hip moment, the transition between phases in the motion and the fact that the knee moment is not an overriding factor in gait, the results may be regarded as encouraging. Figure 6 shows the energy expended in the swing phase of the step. The energy expended per unit of distance traveled (33 in.) is plotted for each body weight as a function of the
Speed. Seps/min
Fig. 6. Energy expended in level walking.----constant W = 150 lb, natural step size: -‘-IV = step size;120 lb, natural step size.
cadence. The curves show a sharp decline in energy expended as the speed is reduced until an ideal cadence is reached and then for slower speeds of walking there is a slow increase in the energy required. The ideal cadence is different when the body weight is changed. For a body weight of 150 lbs the ideal cadence is about 73 steps per min, while for a weight of 120 Ibs it goes to about 84 steps per min. Changes in other body parameters such as the length of the leg, would be expected to have an effect on the ideal cadence. These results agree qualitatively with the experimental work of Ralston (1958) who measured the energy expended in level walking by measuring the oxygen consumption. The form of the curve obtained by
and K. CHANG
Ralston is plotted in the it illustrate the similarity. CONCLUSION
The analysis in this paper is based on plane motion and it appears that results which check reasonably well with natural gait can be obtained. The most important factor in gait seems to be the hip moment. In the acceleration phase this moment is furnished primarily by the muscle complex called the quadriceps group while in the decelerating phase the hamstring group is predominant. This would suggest that these two muscle groups along with the muslces used in toe-off are the really significant power units in walking. Other muscle groups such as those which give the knee moment are largely for control. The energy consumed is obtained by evaluating the work done in traveling a given distance. This showed a sharp decrease as the cadence was reduced from 120 steps per min., reached a minimum around 80 steps per min. and then increased as the cadence was decreased. This would seem to indicate that for a given individual there is a natural gait at which he can travel a given distance with a minimum effort. Given the parameters of the body one can determine this gait by analysis. Useful results from effort along the line of this paper might be the effects upon locomotion caused by certain deficiencies such as immobilizing certain muscles. It may be possible by an extension of the general procedures pointed up here, to study the effect of certain corrective measures. This could be of great help in clinical work.
REFERENCES Bernstein, N. A. ef al. (1935) Biodynamics of locomotion. VlEM Moscow, 1.
Bresler, B. and Frankel J. P., (1950) The forces and moments in the leg during level walking. Trans. Am. Sot. me&. engrs. 72. 27-36. University of California College of Engineering, University of California. Fundamental studies of human locomotion and other information relating to design of artificial limbs. Rep. Natn. Res. Corm. ArtificialLimbs.
AN EVALUATION
OF THE
KINEMATICS
Cunningham. D. M. and Brown G. W.. (1952) Two devices for measuring the forces acting on the human body during walking. Proc. SESA 9,75-90. Elftman, H. I 1939) Forces and energy changes in the leg during level walking. Am.J. Physiol. 125.339-356. Elftman. H. (1939) The function of muscles in locomotion. Am. J. Physiol.
125.357-366.
Elftman. H. ( 1940) The work done by muscles in running. Am. J. Pkysiol. 129.672-684. Fenn. W. 0. (1930) Frictional and kinetic factors in the work of sprint running. Am. J. Physiol. 92,583. Fenn. W. d. (1930) W&h against gravity and work due to velocitv changes in running. Am. J. Phwiol. 93.433-61. Fenn. W: 0. (1538) The mechanics of &uscular contraction in man. J. appl. Phys. 9. 165. Fischer. 0. (1898-1904) Der Gung des Menschen (“Human Gait”). Abhandhmgen der Saechs. Gese//schufi der Wissenschaji Vols. 11-28. Hill. A. V. I 1913) The energy degraded in the recovery processes of stimulated muslces. J. Physiol. Lond. 46. ‘8-80.
Hill. A. V. ( I9 13) The absolute mechanical effi,ciency of the contraction of an isolated muscle. J. Physiol. Lond. 46.435.
Hill. A. V. (1929) The maximum work and mechanical efficiency of human muscles, and their most economical speed. J..Physiol. Lond. 56. 19-4 1. Inman. V. T.. Eberhart. H. D. and Bresler. B. (1954)The principal elements in human locomotion. In Human Limbs and Their Substirutes (Edited by P. E. Klopsteg. P. D. Wilson et al.) McGraw-Hill. New York. Inman. V. T.. Dec. J. B. Saunders, M. and Eberhart H. (19531 The major determinants in normal and pathological gait. J. Bone Jt. Srrr,?.. 35A. 543-58.
OF GAIT
BY MINIMUM
ENERG\
159
Inman. V. T. ( 1947) Functional Aspects of the abductor muscles of the hip. J. Bone Jt. Suq. 29.607-619. Lanczos. C. ( 1949) The variational principles of mechanics. University of Toronto Press, Toronto. Lissner. H. R. and Williams W. ( 1962) Biomechanics of Human Marion. W. B. Sanders. Philadelphia. Penn. Marey. E. J. ( 1873) De la locomotion terestre chez les bipedes et les quadrupeds. (Terrestrial locomotion of bipeds and quadrupeds). J. Anat. Physiol. Paris 9. 4180. Marey. E. J. ( 1874) Animal mechanism. A Trearise on Trrreslrial and Aereal Locomotion. AppletonCentury-Crofts. New York. Marey. E. J. and Demeny G. (1887) Etudes experimentales de la locomotion humaine (Experimental studies on human locomotion). C. R. Acud. des Sciences 105. 544-522.
Moffatt. C. A. ( 1966) An experimental determination of prosthetic knee moment for normal gait. Am. SW. mech. en.grs Paper No. 66-WAIBHF-8. Nubar. Y. and Contini, R. (I 961) A minimal principle in biomechanics. Bull. math. Biophys. 23.377-390. Ralston. H. J. (1958) Energy-speed relations and optimal speed during level walking. Inr. A. AuRew. PIgxioI. einschl Arbeirsphysiol. Bd 17.277-283. Ralston. H. J.. Ewald. B. A. and Lucas. E. B. ( 1961) Effect of immobilization of the hip on energy expenditure during level walking. Biomechanirs Lab.. Univ. Co/$
Rep.-No.
44.
-
Veterans Admmistration Prosthetic Center. N. Y. Reports of Cissephotographic Studies. Unpublished. Weber. Wilhelm and Eduard ( 1836) Mechonil der Menschlichen Gehwerkzeuge (Mechanics of human locomotion). Gottingen.
Vol. I. pp. 147-159.
Pcrgamon Press. 1968.
Printed in Great Britain
AN EVALUATION OF THE KINEMATICS BY MINIMUM ENERGY*
OF GAIT
ROYCE BECKETT Department of Mechanics & Hydraulics, University of Iowa. Iowa City. Iowa. 52240. U.S.A. and KURNG CHANG Department of Mechanics & Hydraulics. University of Iowa, lowa City, Iowa. 52240. U.S.A. Abstract-This paper is an anlaysis of the motion of the leg and foot in the swing phase of a’ step. The analysis is based on the hypothesis that the behavior will be such as to minimize the. amount of mechanical work done. Results of the analysis are the geometric position of the leg and the equivalent forces in the hip and knee joints as functions of time. Also included is the amount of work done as a function of cadence. Comparison is made with experimental results that are available in the literature. INTRODUCTION
well-learned body tasks such as walking and running, it seems reasonable to expect that the movement of the body components would be made in such a way as to minimize the amount of mechanical work that is done. As early as 1836, the brothers Wilhelm and Eduard WebeJ studied the mechanism of walking and running and concluded that the motion during the swing phase of a step was pure pendulum motion. In the 1870’s Marey, in his extensive work on the problem of human locomotion did some computations of energy expended while walking. In the period 18941904 Fischer, a German mathematician working with Braune, an anatomist, published a classical work on human locomotion. This work is an exhaustive investigation of the problems of locomotion. Fenn, in the period around 1930, studied the changes in kinetic and potential energies of the body during walking and running. He concluded that the work done in producing the periodic fluctuations of legs, arms, etc. and in overcoming wind resistance accounts for most of the energy used in running. Correlation of these IN
CERTAIN
*Received
13 July
results of analysis with practice was studied independently by Hill ( 1913, 1922). Elftman ( 1939. 1940) continued in this vein and made careful evaluations of the energy expenditure for various. body parts by using cinephotography. He further used the information to determine necessary moments in the joints to produce the motion and then to approximate muscle behavior. Bernstein and CO-Workers of Russia studied the dynamics of human locomotion and in 1935 published a comprehensive report of their work. Bernstein took issue with some of the results of Fischer and Braune on the actual path of the body components in locomqtion. lnman andassociates (1954, 1953,1947), at the University of California, have made extensive studies of body motion during level walking for both normal and prosthetic gait. Principal results have come from careful analysis of stick diagrams and force plates used to measure the reaction of the foot with the ground. Details of their work are available in reports to the Veterans Administration division in prosthetic devices which sponsored much of the program. Bresler and Frankel in 1950 reported on force plate analysis of the
1967. 147
148
R. BECKETT
forces and moments in the leg db_ing level walking. Cunningham and Brown ( 1952) later perfected the force plate so that more reliable results could be obtained. Considerable effort in this direction has also been made at the Veterans Administration prosthetic devices laboratory under Murphy. Ralston (1958, 1961) has made studies of the energy expended in walking by measuring the oxygen consumed. His work is relevant here in that he shows that there is an ideal cadence for an individual that will give a minimum energy expended per unit of distance traveled. Nubar and Contini (1961) contend that the individual will determine his motion to reduce the muscular effort to a minimum consistent with imposed conditions of constraint. This paper is an analytical investigation of the behavior of the leg in the swing phase of a normal walking step. Forces and moments are imposed at the joints of the leg to produce motion that is consistent with the geometrical constraint and in such a way as to give a minimum expenditure of energy. The analysis gives the motion of the leg and foot, the equivalent moments in the hip and knee to produce the motion and the energy expended in the swing phase of the step. The results obtained are compared with some of the experimental results that have been reported in the research noted above.
and K. CHANG
Fig. 1. Positions of the leg in level walking.
swing phase of the step is considered in this study the action of the foot in lifting the body is ignored, except that the motion of the leg and foot is restrained to follow a prescribed path. The ankle joint moves freely prior to toe-off so that the toe remains in contact with the floor. After toe-off it is assumed to be locked until the end of the cycle at heel-strike. The analysis begins during the push-off phase of the step and proceeds until heelstrike at the end of the swing phase as illustrated in Fig. 2. The position of the hip, point 0 at time t, is prescribed by giving the forward velocity, assumed constant, and h, the
MECHANICAL MODEL
A simplified model of the leg is assumed for the analysis and is shown in Fig. 1. Point 0 represents the hip joint and is assumed to move on a sinusoidal curve that approximates actual body motion. Link 1 is the thigh and is connected by joint 0’ to link 2 which represents the shank. The foot is connected to the leg by the ankle joint labeled P. The two couples, H and M, are produced by the muscle complex associated with each of the joints. Figure 2 shows how an idealized version of H will act to give the required motion. Also shown is the behavior of the two principal muscle groups that produce H. Since only the
25
35
50
go
82
ea
par Cent of Cycle
(0)
I I
(cl
c-m Pmceare
Fig. 2. One complete cycle in level walking.
100
AN
EVALUATION
OF THE
KINEMATICS
vertical motion of 0. The angular position of the thigh is given by C$ which is the angle between the position of the thigh in maximum extension (beginning of heel rise) and its position at time t. The angle between the shank and the thigh is y and the rise of the foot during toe-off is measured by (Y. The initial positions of each of these angles is &, y*, and cr,, and the height of the hip joint is h,. Each of these variables and its initial value is shown in Fig. 1. In order to have the motion simulate the swing phase of a typical walking step, certain geometric restraints must be imposed. It is required that the foot pivot about the toe until toe-off much like the actual case and then to force the ankle to lock. The foot is then constrained to follow through in the swing phase so that the toe moves on a prescribed curve shown by ATB in Fig. 1. The precise form of this curve is determined in the analysis. When the toe swings under the knee, point B in Fig. 1, the thigh will have reached maximum flexion and will remain in this angular position while the shank comes smoothly into position for heel-strike. The acceleration and deceleration moments at the hip joint are each assumed to be of a constant magnitude, and to act for a given interval of time, so that the moment profile is rectangular when plotted against time (see Fig. 2). On the basis of known data on the action of the muscle groups that supply the power for the motion of the leg during swing phase. this is a reasonable assumption. Figure 2a shows a complete cycle of the leg from heel-strike to heel-strike. The portion considered irerein is the latter half. i.e. from 50 to 100 per cent of the cycle. For purpose of analysis this latter half is broken down into three phases that depend upon the restraints that are imposed on the motion. In the initial phase designated phase I in Fig. 2(c) the toe is fixed on the ground and the foot rotates around it (Fig. 1). In Phase II the toe is restrained to move on the curve ATB in Fig. I. The general form of this curve is obtained from experiment
OF GAIT
BY MlNlMUM
ENERGY
149
but the analysis provides some help in the choice of the particular coefficients in the equation of the curve. When the toe glides under the knee joint the thigh is in maximum flexion and will remain in this position until heel-strike. When the thigh reaches maximum flexion. or the equivalent. when the toe moves directly under the knee. the constraint conditions change and this marks the start of Phase I II. The condition of constraint in Phase I I1 is that the thigh should remain fixed in the positioh of maximum flexion. In Fig. 2(b) a plot of the hip moment as a function of time is shown. The positive moment over the initial part of the cycle is supplied primarily by the quadriceps group of muscles and the deceleration moment at the end of the cycle comes from the hamstring group. Superposed in this portion of the figure are dotted curves which approximate the actual behavior of the two groups of muscles (College of Engineering, University of California). EQUATIONS
OF MOTION
The equations of motion for the leg are obtained by the Lagrangian Method and the conditions of constraint imposed by means of the Lagrangrian Multiplier. This requires the kinetic and potential energies of the system and the constraints in equation form. Consider first the kinetic and potential energies. It is assumed that motion is confined to the x-g or sagittal plane shown in Fig. 1. The forward (X direction) component of velocity z+,is constant and the vertical motion of the hip joint 0 is given by h where h = sin
27r(r,+t) 7
(1)
r is the time for one step, i.e. 50 per cent of the full scale in Fig. 2(a). t,, is a phase shift which corresponds to the starting value for time. When h = h,, t,, = O-067 and t is time measured from to. For purpose of finding the equations of motion the limb is approximated by the two
R. BECKE-IT and K.
150
links in Fig. 1 representing the thigh, link 1, and the shank and foot, link 2. The foot is combined with the shank in evaluating the energy of the system since its relative motion with the shank occurs during toe-off when the motion of the foot is small. The coordinates of the center of gravity of each of the two links 1 and 2 are found from Lissner (1962) and are given by (x1, yJ and (x2, y2) respectively. x1 = uot + a, sin ( $I - $vJ
(2a)
Y1= h+a,
(2b)
xi=
cos (+-cb”)
vof+IIsin
(d,-&)
-ua,sin
(y+&-4) (2c)
y,=h+&cos
(4-&)+%cos
(Y+&-+) (2d)
where li is the length and af the distance from the upper joint to the center of gravity of each link, i = 1,2, +, is the angle between a vertical through 0 and the thigh when in the position of maximum extension. The velocity of the center of gravity of each link is obtained by differentiating equations (2) with respect to time. The angular velocity of link 1 is d and of link 2, (Q-i). Assuming the foot is fixed rigidly to the shank the kinetic energy is written as follows T = 311~+312(~-~)2+~m,u,2+~m,v~2.
CHANG
the potential energy of the system is V = --m&r& cos (4 - 40) + hl - mzg[l, x cos (b,-40)
Where Ii is the mass moment of inertia of link i about its c.g. mf is the mass of link i. The conditions of constraint that are imposed on the motion are such that 4 and y cannot be independent but are related by an algebraic equation. The particular condition of constraint depends upon the position of the toe, T, in Fig. 1. When the toe is fixed during toe-off the point P moves on an arc with the center at T. Two equations involving vertical and horizontal distances between T and 0 can be written. dsina=Icos&,-&cos
dcosa!=
h,,- h + d sin CQ
(7)
sin (+-&) do + &)
uoc + d cos a,,.
(8)
f(& y) = 12+ 1*2+ 1*2- d2 + v02t2+ d2 cos2 a 0 -
211,cos 4 - 211,cos (4 - y)
+ 21,12cos y - 2vor d cos QO +2(dcosa,
(9-bo)+u,‘~-2~u,~sin(~-9,)
(c$-&)
- uot)[--I sin +.
-l,sin(+-f#~~)+l~sin(y-++4~0)1 +2K[Icos~,-l,cos(~-~o)
cosy+?&“]
(4--4,)-u,t+Q~
+
(y-~+4d
(4)
+h2] +3m2{[1,2r$+u+-r&
xcos
12 cos
-lsin&)-f,
-
T = $11@+312 (Q--)2+dnrl[v,2+2v,u,d;
Xsin (y-cp+&)]
-
+ 1, sin (y -
the kinetic energy is
-2jI[l$sin
(4-&J
Eliminating o! from these two equations gives the constraint condition for Phase I as a function of $ and y.
D, = d(_&‘+i,2)
-2f,u,&+$)
+h]. (6)
(3)
If 11~and u2 are replaced by
xcos
+a, cos (y-f$b&)
(5)
+a,(;-&)
where
+jz2}+2vo[11G cos
-f2cos(y-~+~o)1+-*=0
(r-++&)l
K = ho-hhdsina,. 1 = v (I,2 + 1.,2-t-21112cos ys).
(9)
AN EVALUATION
OF THE KINEMATICS
The end of Phase I is toe-off and occurs when the angle between the foot and the shank reaches a given limiting value. If the angle is 8 then a must satisfy the equation. cr= y++,,-mts-;.
y = Y(X) = 5 AiX4-i i=O
(12)
where x is measured from A. The value for x in equation 12 may be replaced from equation 2(c) to obtain a function in 4 and y. The five conditions for evaluating A,, i = 0, 1, . .4 are
y’(C) = 0
?J’(0) = G/i at toe-off y(D) = y’(D) = 0.
i13)
Both analysis and experiment have been used as a guide in choosing C and D. The velocities + and X-of the toe come from differentiating equations (2) when x and y are the coordinates of T and then substituting the value of each of the variables at toe-off. *=-l,t$sin
(4-&,)-l,
Xsin (y-++f&,-8)+i i=V,~+f,&os x cos (y-&-t
(14)
40-6)
where
(15)
I+= ~(1,2+d2-212dcosi3 @= arcsin [dsin (F--6)/f4].
ii=
At the end of Phase II the hip flexion angle is
V-Af.
( 18)
The two equations of motion in the variables 4 and y are then given by substituting T and v into the equations. d aT aT ---_--+-_=H dt a4 a4 d aT ---_ dt a+
av a4
(19)
aT av_M ar+z-
(20)
.
Where H and M are respectively the hip and knee flexion moments. Substituting T and I/ from equations (5) and (6) gives +[I1 + I, + m,a,2 + m21z2-t m2az2 + 2m211a2cos y] - i;[i2 + m2az2
+ m21,a2 cos 7-J + m,ga, sin ($J- +o) + mzlla2+ ($- 26) sin Y+ m,gEl, -a,sin
+i[-_(m,a,+m21,~
(r-++h)l sin (
(y-d+&,,3-*$=
H. (21)
+[I2 + m2a2”) - ;f; [I, + m2az2+ m,l,a, cos y] + m211a2~2sin y+
(I61
(17)
The equations of motion may now be obtained from Lagranges equation where the potential energy function V is modified to include the condition of constraint. If the condition of constraint is defined by f= 0 where f is some function relating the coordinates of the system, i.e. 4 and y. then a new function p is defined. Lanczos ( 1949).
+m,a2sin
(+#J,,-I,(+&,
151
&j&O.
X sin (+-&)
(G-6)
ENERGY
a maximum and the toe is directly below the knee in the swing through. This marks the beginning of Phase 3. The condition of constraint for Phase 3 is that the angular position of the thigh remain fixed, i.e.
(11)
After toe-off, the foot leaves the ground and point T moves on the curve AT6. This curve must have a slope at A which will permit the foot to rise smoothly and it must pass near to the floor at point B with a horizontal slope. An additional condition that is imposed is that the curve have a maximum value at a distance C from A. The value of D = AB and C will depend upon the length of step. In order to satisfy the five conditions summarized below a fourth degree polynomial is chosen to represent the curve.
y(O)= 0
OF GAIT BY MINIMUM
-i[m,a,sin
m,ga., sin (y - I#I-t &)
(y-$+&,)]-hzy=M. (22)
152
R. BECKE’TT and K. CHANG
Equations (21) and (22) with the appropriate conditions of constraint for the functionfand the proper initial conditions are sufficient to solve the problem in each phase of the motion. NUMERICAL
COMPUTATIONS
Equations (21) and (22) are solved simultaneously for 4 and $ to give equations of the form (23) 4 = F,(+, i, Y, +, A, H) .. (24) Y=F&P,+,Y,;JM. These equations in combination with the appropriate constraint condition are solved for $, y and A. The numerical method used to solve the system is the second order RungeKutta method. For the first phase of the motion the constraint condition is equation (9). The starting conditions for Phase I are the initial values for the angle C#Iof the thigh, the angle y at the knee joint and the corresponding angular velocities (time derivatives). In this study the starting values for Phase I are chosen so that the time from the start of the analysis to heel-strike will make up just 50 per cent of a complete cycle. This starting time corresponds roughly to the initiation of action by the quadriceps group of muscles, and is therefore assumed to be the time at which H is applied. Starting values are obtained from available experimental studies. Bresler et al. (1950). 4(O) = (bb= 6.0” Y (0) = Ys =f(&)
d(0) =$(O) These correspond
=
13”
(25)
= 0
to values of LX.,, and & of CYO = 35.5” d” = 23.0”
(26)
The power to drive the leg is supplied by the couple H at the hip joint. The magnitude and *From
stick diagrams furnished
duration of H are such as to bring the leg through the cycle in accordance with the limitations of geometry and strength of the muscle group and with the least amount of work. For this study both the accelerating and decelerating couples in the hip joint are assumed to have a square profile when plotted against time. This is the form to give the least work for a given maximum value of H and agrees qualitatively with experimental results Sumen et al. ( 1954). The magnitude of the couple is limited by the forces that can be developed by the muscle complex that generates it although this does not appear to be a limiting factor in walking. The important limitation on H is imposed by the reactions that are created by a large accelerating moment in the hip joint. High heel rise will occur after toe-off necessitating large control moments in the knee and this will cause an excessive expenditure of energy. A smooth motion and one which gives good agreement with a natural gait is obtained when the duration of H is roughly 10 per cent of a cycle and is independent of the cadence. With the form and duration of H known it is still necessary to find its magnitude. This is done by assuming a starting value for H and then computing the motion of the system up to the limiting angle of hip flexion which corresponds to #I = 45”. If H is too large the leg comes through too fast for the assumed cadence and if it is too small then the leg is slow in reaching the limiting angle. The time to reach the limiting angle is about 64 per cent of the time for a step+ and is therefore fixed when the cadence, one of the given conditions, is fixed. In order to find the correct starting value for H an iteration method is empolyed whereby a correction is made to H that is based on a linear interpolation (secant) method, to give the desired end conditions on 4. In this way a suitable value for H is obtained with few iterations. Toe-Off is the beginning of Phase 2 and is
by Dr. E. Murphy. Veterans
Administration
Prosthetics
Center, New York.
AN EVAULATION
OF THE KINEMATICS
identified by a limiting angle d between the foot and the shank (see equation (1 I )). An arbitrary value of 1 IO”, which is based upon the analysis of several stick diagrams from level walking experiments, is selected. The point at which the angle reaches 1IO” is identified by equation ! 1 I ) with S = 1IO”. and where cxcomes from the equation i, cos i@-@,,I +/?COS (y+&-+)
+h
+ d sin (Y= I cos &,+ ho + 6 sin (Y,,. (27) When the angle reaches the limiting value the ankle is locked until heel-strike. During Phase 2 the toe moves on the curve AB which is defined by equations (12) and ( 13). The distance D is determined by finding the forward travel of the toe at the time the thigh reaches maximum flexion. This is when the toe swings through directly under the knee and is about 64 per cent of the time for one step.
- d cos ffO+ 064rz, + I, sin (22”)
(28)
where T is the time for one step and depends upon the cadence. The angle of 22” is the approximation used for the maximum flexion angle at the hip. The value of C is about 40 per cent of D and comes in part from experimental studies but may be adjusted on the basis of the numerical results. The angular velocity of the thigh should come to zero at the same time that the toe swings under the knee. This will use less energy and as expected it agrees with experimental observations. The time at which the toe swingsunder the knee is the end of Phase 2 and the start of Phase 3 of the motion. In Phase 3 the shank is brought into the position for heel-strike. At the time of heelstrike the angular velocity ; of the shank is roughly zero. The deceleration of the shank must be brought about smoothly and in a way to use the least amount of energy. This. like the acceleration phase. is accomplished by a step input of moment at the hip joint. The
OF GAIT
BY MINIMUM
ENERGY
IS3
duration of the hip moment in the deceleration phase is found to be about 12 per cent of a cycle and again is independent of the cadence. Like the acceleration moment, the magnitude of the deceleration moment is found by the Secant method. The condition that determines the magnitude of the decelerating moment is that + must be brought to zero at heel-strike. lt might appear that the knee moment could be adjusted to bring i, to zero at the required time but this is not the case. Rather the knee moment takes on values that will keep $I fixed during Phase I I1 of the motion. The integration of equations (23) and (24) is made by the second-order Runge-Kutta method. The procedure in finding the solution is as follows: For Phase I:
Find from equation (20) the starting value for A at time t = 0. This is the Lagrangian multiplier that represents the force (less body weight and acceleration) that is applied at T between the foot and the floor. Equation (19) is then solved by the Runge-Kutta method for do. 4 and 4 at a time At. These are then used in the constraint condition to find y, -j and i;. Equation (20) again gives h at time At. When the solution is found at At the process is repeated for the solution at 2A.r. The technique is continued until the angle between the foot and shank is 110” at which point the ankle is locked and the toe leaves the floor. The condition of motion at this instant in time provides starting values for the second phase. For Phase 2 :
The equations of behavior in Phase 2 are the same as those for Phase I but the constraint condition changes. During Phase 2 the toe is constrained to move on the curve AB. The moment in the knee joint provides the constraining couple to force the toe to move on the curve. The end of Phase 2 is marked by the condition that the toe is directly under the knee jojnt.
154
R. BECKEIT
For Phase 3: Phase 3 of the motion begins with the toe in the lowest position of swing-through. This last phase of the motion is carried out with the thigh fixed against rotation, i.e. d; = ;p’= 0 and d, with a fixed value of 45”. This leaves only the equation of motion in y. The solution for y is carried on from the terminal condition of Phase 2. The decelerating hip moment is applied shortly after the beginning of Phase 3 (approximately 12 per cent of the time for one step) and serves to slow the shank. The knee moment is computed to maintain Q = 0. For the computations made herein the numerical data for the body parameters have been obtained by using information from Lissner (1962). Two sets of parameters have been used, one for a body weight of 150 lb. and the other for 120 lb. W = body weight = 150 and 120 lb ml = (9.7%. W) 11= 17.4 in. a1 = 7-53 in. d= 9~1 in. 1, = 864 16 lb-in-sec2 11 = 6.9133 lb-in-sec2 Computations for each weight are made for several different walking speeds. The speed depends upon the cadence which is varied by 10 steps per min from a slow walk of 60 steps per min to a fast walk of 120 steps per min. Initially the length of step was held constant at 33 ins. This is not natural and was noted in the results -particularly for the slow walk where it was necessary to apply a flexion moment in the hip in order to extend the leg to heel-strike. Changing the length of the step to correspond with the cadence gave a more natural gait. The length of step for each cadence is given in Table 1. Variations in the step length with weight were not made. For each cadence the work done by the two moments in the hip and knee was computed according to the equation
and K. CHANG
Absolute values are used for the integrands since they represent effort that is not recoverable. The integrals are evaluated by summing over the finite intervals of time At that are used for the integration of the equations of motion. The important results of the computations are the configuration path of the leg, given by 4 and y; the moments in the hip and knee joints, given by H and M; and the work done for each cadence (walking speed). RESULTS
The acceleration and deceleration moments H in the hip are shown in Fig. 2(b). The time of application and the duration of these moments are determined somewhat from the interpretation of experimental results reported in the bibliography and by the analysis. m, = (6.4%. W) 1z = 16.9 in. a2 = 9.81 in. Independent of W 1 = 7.0727 lb-in-sec2 for W = 150# 12 = 5.6582 lb-in-sec2 for W = 120#. 12 An objective in the analysis is a smooth motion of the leg that will require the least input of energy for control. This requirement places some restriction upon the duration of both the accelerating and decelerating moments in the hip. Once the two parameters of time of application and the duration of the moment are set, then the magnitude of the accelerating moment is determined by the analysis to give the proper value of 4 and 4 at the end of Phase II; the decelerating moment is found to give y = + = 0 at the end of Phase III. Figure 2(b) shows graphically the quantitative results of the analysis and they are given for the two cases considered in Table 1. In each case the duration of the accelerating moment is roughly 10 per cent of a full cycle. Termination of the moment comes at toe-off. The decelerating moment acts over 12 per cent of the cycle just before heel-strike.
AN EVALUATION
OF THE KINEMATICS
OF GAIT
BY MINIMUM
155
ENERGY
Table 1
Cadence 60
70 80 90 100 110 120
Length of step (in.) 27.3 28.5 29.9 31.5 33.3 35.3 37.5
Act. couple (in.-lb) (15Olb) (1201b) 20.56 41.16 65.15 %.OO 133*00 175.29 22967
24.93 SO.45 81.18 119.98 166.24 218.00 299.00
Values for 4 and y are plotted against time measured as a per cent of one step for a cadence of 90 steps per minute in Fig. 3. Variations in $ and y for changes in body weight and cadence will give onfy small changes from the values shown. The experimental results shown in Fig. 3 were obtained from stick diagrams of normal subjects walking at 90 steps per min. furnished by the Veterans Administration Prosthetics Centef in New York. Table 2 gives results for 4, y, 9 and q for each of the two body weights used and for each cadence. Figure 4 shows the time derivative of $J and y plotted against time as a per cent of one step for the case of body weight equal to 150 lbs. and a cadence of 90 steps per min. Variations of the velocity in the separate
60 50 !4O .{ ‘0 30 a 20
Fig. 3.
Energy exp. I33 in. (120lb) (1501b)
Dec. couple (in.-lb) (150lb) (120lb) -40@0 -22.36 28.22 54.42 90.82 143.71 195*00
-38.04 -899 20.28 47.50 75+0 132.00 157.50
7690 76-40 76.00 85.90 105~00 131Xlo 158.20
94.30 94.00 97. IO 106.90 131.10 163.40 219.00
‘7
-15
0
20
40 Time,
I 60
I 80
I 100
per cent of step
Fig. 4. Angular velocity vs. time for thigh and shank.computation; -. - from stick diagrams;---Moffatt’s results.
runs are much greater than for the displacements, which is expected. Experimental results for the angular velocity of the shank as deduced from correlated stick diagrams are given in Fig. 4. Plotted also is the angular velocity of the shank as obtained by Moffatt ez al, by using accelerometers on normal subjects. These results are included because of the excellent experimental techniques employed. From the results in Fig. 4 it appears that the calculated values for the angular velocity of the shank are low. The form of the curve agrees qualitatively, however, with the experimental curve and except for the results of Moffatt agree reasonably 20 40 60 80 100 well with the magnitude. In the computation routine it was tedious to obtain a smooth lime ,per cent of step transition from Phase II to Phase III. FurtherRotation vs. time for the thigh and shank.computation:----experiment. more. the jump in the hip moment caused
156
R. BECKETT
and K. CHANG
Table 2 Time (% of swing phase) Cadence
Weight
0
10
120 150 120 150 120 1.50 120 150 120 I.50
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.5 8.5 0.8 O-8 26.1 26.0 2.5
120 150 120 150 120 150 120 150 120 150
20
30
40
50
60
70
80
20-o
44.2
90
100
-2:.: -32.0
13.8 14.7 I.0 I.0 41.3 41.2 2.0 2.0 -31.9 -40.2
19.8 I.3 I.3 51.4 51.2 I.5 1.6 47.9 59.7
29.0 28.8 1.7 I.7 60.0 59.9 I.3 1.3 46.0 57,6
38.4 38.2 1.5 1.5 63.9 63.8 -0.1 -0.1 33.3 41.4
44.1 0.5 0.5 56.2 56.2 -2.8 -2.8 19.9 23.4
43.9 44.5 0.4 0.0 35. I 37.4 4.0 -2.8 44.5 66.0
43.4 44.5 0.0 O-0 21.3 24.7 -1.7 -1.8 16.5 23.2
43.4 44.5 0.0 0.0 13.5 15.9 -1.0 -1.0 11.5 19.2
43.4 44.5 0.0 o-o IO.4 14.0 -0.1 0.0 IO.4 19.0
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.4 8.4 0.9 0.9 25.6 25.5 3.1 3.1 -20.5 -25.7
15.3 13.7 1.4 I.3 44.5 40.9 2.4 2.7 -17.2 -25.5
22.3 22.1 1.7 1.7 54.4 54.1 I.9 1.9 48.6 60.6
32.8 30.5 I.8 1.8 63.3 61.9 1.0 I.3 40.5 54.4
40.4 38.5 1.4 1.6 64.3 64.8 -0.7 -0.2 22.9 35.0
44.6 44.5 0.3 0.3 55.0 55.1 -3.5 -3-5 13.8 16.1
44.1 44.7 -0.4 0.0 35.5 37.9 -4.6 -3-3 43.3 68.1
43.6 44.7 0.0
43.6 44.7 0.0 0.0
2y.i 25.3 -2.2 -2.3 26.2 39.0
43.6 44.7 0.0 0.0 13.5 17.0 -1.4 -1.6 20.3 32.6
11.2 13.2 -0.0 -0-2 18.7 29.6
120 150 120 150 120 I50 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
9.2 9. I 1.2 l-2 28.4 28.4 3,7 3.7 -14.4 -16.2
14.2 14.2 l-7 I.7 46.2 46.2 2.8 2.8 5.1 4.6
22.5 22.5 2.0 2.0 55.0 55.0 2.3 2.3 48.2 60.2
32.0 31.9 2.0 2.0 63.5 63.5 1.3 1.3 38.8 48.5
40.3 40.3 1.5 l-5 65.1 65.1 -0.8 -0.8 14.3 17.9
44.5 44.5 0.2 0.2 53.8 53.9 -4.3 -4.3 5.4 6.5
43.4 44.5 -0.8 0.0 35.5 39.0 -6.3 -3.9 25.1 71.9
42.3 44.5 0.0 0.0 17.4 24.0 -2.5 -1.9 38.0 54.6
42.3 44.5 0.0 0.0 9.4 15.3 -0.9 -1.1 31.2 52.0
42.3 44.5 0.0 o-o 8.4 13.5 0.1 0.0 32.1 52.8
120 150 120 I50 120 150 120 I50 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
8.5 8.5 1.3 1.3 26.1 26.1 4.3 4.3 -6.3 -7.9
16.6 16.6 2.1 2.1 47.1 47.1 3.3 3.3 35.0 43.7
24.2 24.2 2.3 2.3 57,o 57.0 2.5 2.5 47.2 59.0
32.1 32.1 2.2 2.2 63.9 63.9 1.4 1.4 36.0 45.1
41-I 41.1 I.6 1.6 65.1 65.1 -1.1 -1.1 4.3 5.4
44.7 44.8 0.5 0.5 56.3 56.3 -4.2 -4.2 -6.9 -8.6
44.9 45.0 -0.1 0.0 39.7 40.0 4.6 4.4 65.3 76.9
44.9 45.0 0.0 0.0 22.3 22.6 -3.0 -3.0 58.8 67.7
44.9 45.0 0.0 0.0 14.5 14.7 -1.6 -1.6 52.4 63.6
44.9 45.0 0.0 0.0 11.7 Il.7 -0~0 -0. I 52.8 63.7
120 150 120 I50 I20 I50 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
9.1 9. I I.7 1.7 28-6 28.6 5.1 5.1 7.0 8.7
16.3 16.3 2.4 2.4 46.9 46.9 3.9 3.9 67.4 84.3
25.0 25.0 2.6 2.6 58.3 58.3 2.8 2.8 45.6 56.9
33.7 33.7 2.4 2.4 65.3 65.3 I.2 I.2 .27.8 34.7
41.0 41.0 1.8 I.8 65.9 65.9 -1.1 -1.1 -4.5
45.0 45.0 0.4 0.5 56.2 56.2 -4.7 4.7 -18.0 -22.5
45. I 45.2 -0.1 0.0 37.8 38.2 -5.1 4.9 67. I 76.0
45.0 45.2 0.0 0.0 22.9 23.3 -3.6 -3.6 70.3 92.1
45.0 45.2 0.0 0.0 13.9 13.7 -1.9 -1.9 62.0 80.7
45.0 45.2 0.0 0.0 9-5 IO.4 -0-o -0.0 62.3 81.5
-5.6
PIN EVALUATION
OF THE
KINEMATICS
OF GAIT
BY MINIMUM
157
ENERGY
Table 2. (cont.) Time (%,) Cadence
@J i I10 Y ;
M
CtJ i I20
-Y ; M
0
10
20
30
40
120 I50 I20 150 120 I50 120 150 120 150
6.4 6.4 0.0 0-O 13-O 13.0 0.0 0.0 0.0 0.0
9.8 9.8 2.0 2.0 31.3 31.2 5.8 5.8 28.1 34.8
15.3 15.3 2.7 2.7 45.2 45.1 4.8 4.8 94.8 117.5
24.9 24.9 2.8 2.8 58,7 58.4 3.? 3.; 43.6 55.1
34.2 34.2 2.5 2.5 66.6 66.0 I.2 1.2 17.9 25.1
41.6 41.8 1.7 1.7 66.1 65.6 -1.8 -1.7 -28.1 -26.5
120 150 120 150 120 150 120 150 120 150
6.4 6.4 0.0 0.0 13.0 13.0 0.0 0.0 0.0 0.0
10.7 11-o 2.5 2.6 34.2 34.6 6-6 6.8 63.6 84.0
17.3 17.7 3.1 3.2 48.9 49,4 4.7 4,9 42.4 53.1
28.0 28.7 3.0 3.1 63.1 63.1 2.9 3.0 36.3 43.4
34.7 35.5 2.8 2.8 66.9 68.0 1.3 1.2 11.7 8.7
42.6 43.3 I.7 1.6 65.6 65.9 -2.3 -2.8 -42.5 -72.0
Weight
6 atidy are in degrees.
4 and i are in rad./sec.
50
60
70
80
90
100
44.3 44.6 0.6 0.7 58.6 58.6 -4.9 -4.5 -48.3 -42.9
43.1 45.4 -1.3 o-0 34-2 39.9 -9.2 -5.4 -0.8 82.9
41.9 45.4 0.0 o-o 14-9 23.3 -3.7 -4-o 77.6 115-s
41.9 45.4 0.0 0.0 7.5 13.3 -2.4 -1.8 79.1 104.1
41.9 45.4 0.0 0.0 3.0 11.1 -0.0 -0.1 77.5 105.1
45.1 45.3 0.4 0.0 56.4 54.9 -5.8 -7.1 -42.0 -86.0
45.1 45.3 -0.1 0.0 34.2 33.3 -6.0 -5.6 68.6 70.2
45.1 45.3 0.0 0.0 22.2 21-4 -4.5 -4.5 109.1 137.2
45.1 45.3 0.0 0.0 11-O IO.5 -1.9 -1.9 97.5 124.0
45.1 45.3 0.0 0.0 7.5 8.4 -+I 0.0 100-O 126.9
M is in in.-lb.
irregular behavior. particularly in the knee moment. The knee moment is the couple that imposes the proper geometric restraint in each step of the computation. Its value is quite sensitive to small changes in the parameters. For example, when the form of the curve on which the toe moves in Phase II is altered only slightly. the knee moment may be changed by a significant amount. See Fig. 5. This was a disturbing situation until it was realized that the knee moment is not really a very important factor in level walking. In fact. the leg acts very much like a penduhrm as it moves through the swing phase of the step. The small knee moment obtained is for guidance, and in walking it will be small. Thus any change in the conditions on the problem may be expected to produce large percentage changes in knee moment. An ideal gait would require a minimum of guidance by the knee moment. Figure 5 shows a plot of the knee moment for each body weight used and a cadence of 90 steps
100
I-
Y \
100 -
\ \
’ ‘-/ 150
l
0
I
I 20
40 Time,
2
I 60
80
100
per cent of step
Fig. 5. Knee moment vs. time.-..M’ = 120 lb. C = C = 0.40;-.--w = 150 lb, C = 0.4&--w = 150 lb. 0.331):~---Moffatt’s results.
per min. Also included is an experimental result from Moffatt that has been obtained from accelerometer records. The analytical
R. BECKElT
15%
results only roughly approximate the experiment. However, considering the impulsive form of the hip moment, the transition between phases in the motion and the fact that the knee moment is not an overriding factor in gait, the results may be regarded as encouraging. Figure 6 shows the energy expended in the swing phase of the step. The energy expended per unit of distance traveled (33 in.) is plotted for each body weight as a function of the
Speed. Seps/min
Fig. 6. Energy expended in level walking.----constant W = 150 lb, natural step size: -‘-IV = step size;120 lb, natural step size.
cadence. The curves show a sharp decline in energy expended as the speed is reduced until an ideal cadence is reached and then for slower speeds of walking there is a slow increase in the energy required. The ideal cadence is different when the body weight is changed. For a body weight of 150 lbs the ideal cadence is about 73 steps per min, while for a weight of 120 Ibs it goes to about 84 steps per min. Changes in other body parameters such as the length of the leg, would be expected to have an effect on the ideal cadence. These results agree qualitatively with the experimental work of Ralston (1958) who measured the energy expended in level walking by measuring the oxygen consumption. The form of the curve obtained by
and K. CHANG
Ralston is plotted in the it illustrate the similarity. CONCLUSION
The analysis in this paper is based on plane motion and it appears that results which check reasonably well with natural gait can be obtained. The most important factor in gait seems to be the hip moment. In the acceleration phase this moment is furnished primarily by the muscle complex called the quadriceps group while in the decelerating phase the hamstring group is predominant. This would suggest that these two muscle groups along with the muslces used in toe-off are the really significant power units in walking. Other muscle groups such as those which give the knee moment are largely for control. The energy consumed is obtained by evaluating the work done in traveling a given distance. This showed a sharp decrease as the cadence was reduced from 120 steps per min., reached a minimum around 80 steps per min. and then increased as the cadence was decreased. This would seem to indicate that for a given individual there is a natural gait at which he can travel a given distance with a minimum effort. Given the parameters of the body one can determine this gait by analysis. Useful results from effort along the line of this paper might be the effects upon locomotion caused by certain deficiencies such as immobilizing certain muscles. It may be possible by an extension of the general procedures pointed up here, to study the effect of certain corrective measures. This could be of great help in clinical work.
REFERENCES Bernstein, N. A. ef al. (1935) Biodynamics of locomotion. VlEM Moscow, 1.
Bresler, B. and Frankel J. P., (1950) The forces and moments in the leg during level walking. Trans. Am. Sot. me&. engrs. 72. 27-36. University of California College of Engineering, University of California. Fundamental studies of human locomotion and other information relating to design of artificial limbs. Rep. Natn. Res. Corm. ArtificialLimbs.
AN EVALUATION
OF THE
KINEMATICS
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