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Control mechanism performance criteria for an above-knee leg prosthesis
1. Biomechonics. Vol. 3. pp. 87-97.
Pergamon Press. 1970.
Printed in Great Britain
CONTROL MECHANISM PERFORMANCE CRITERIA FOR AN ABOVE-KNEE LEG PROSTHESIS* JOHN WALLACH Mechanical Engineering Laboratory, General Electric Research and Development Center, Schenectady,N.Y. 12301,U.S.A. and EDWARD
SAIBEL
Department of Applied Mechanics, Carnegie-Mellon
University, Pittsburgh, Pa 15213, U.S.A.
Abstract-A primary requirement of an above-knee leg prosthesis is that it enables the amputee to walk with a normal appearing walking pattern. The leg motion during the swing and stance phase of level walking should approximate that of a non-amputee. To accomplish this, a control mechanism is added to the conventional leg prosthesis. This study is concerned with the performance of the mechanism during the swing phase. The normal walking pattern for a normal man was obtained from the literature for average, tall, medium, and short men. The data, consisting of tabulated experimental data for the linear and angular displacements of the normal leg, were smoothed and differentiated twice to obtain velocities and accelerations. A mathematical mode1 was chosen that typifies the current mechanical-hydraulic mechanisms. The physical constants were calculated from data collected fi-om many sources. The force, velocity, and power were calculated for the control mechanism. INTRODUCTION
The most promising mechanisms are linearlyactuated mechanical-hydraulic devices which are attached behind the knee joint and about half-way down the shank. A device typical of those with which this investigation is concerned is the DUPACO ‘Hermes’ hydraulic control unit (Fig. 1). This type of device uses a combination of spring force and fluid damping to control the motion of the shank relative to the thigh. Springs, at one or both ends of the piston stroke, aid in reversing the shank motion without the expenditure of energy. Fluid damping is generated by forcing an incompressible fluid through a series of staggered holes in the cylinder. The holes are staggered to provide a damping force which is a function of the relative position of the piston, as well as of the relative velocity of the piston. That is, the device is programmed to provide an axial
widely accepted criterion for devices to rehabilitate lower-extremity amputees is that the prosthetic replacement must have a normal external shape and appearance, and must permit the amputee to walk comfortably and safely at normal rates on level ground, without undue mental or physical effort and with a normal-appearing gait, Wagner (1954). A recent comprehensive study of the walking patterns of normal men defines this normal-appearing gait and makes possible an improvement in the design data for prostheses, Murray (1964). The motion of the thigh section of the prosthesis is precisely controlled by the amputee. However, the amputee has no voluntary control over the motion of the shank relative to the thigh. Therefore, various types of control mechanisms have been developed. AN
ESTABLISHED,
*Received5 March
1969. 87
88
J. WALLACH
force which is a function of its length and the time rate of change of its length. Considering the triangle formed by the knee joint and the upper and lower mounting points of the control device (Fig. 2), the device provides a knee moment that is a function of the angle between the shank and thigh, and the time rate of change of this angle.
and E. SAIBEL
sary to determine these variables as a function of percent of the walking cycle and also to determine the first and second derivatives of these variables. Dr. Murray made an extensive study of the walking patterns of normal men, Murray (1964). Sixty men were divided into three equal groups by height: tall, medium, and short. Each man was observed walking at a normal cadence on a level surface. This data had to be adjusted in order to determine xH, yH, AT and AK as defined in this study. Dr. Murray very kindly furnished the constants necessary to adjust the data as well as a complete tabulation of all the data. ’ xH is the horizontal displacement of the hip. The forward displacement data, as measured at the neck, were used for xH. This assumes that the motion of the hip is satisfactorily approximated by that of the neck. A comparison of the neck displacement data of Murray (1964) with the hip displacement data of Eberhart showed that this assumption is sufficiently accurate for this study. A study of the data showed that it can be approximated by a linear term plus a very small sinusoidal variation. This was done for each height group. The results are given in Table 1. Table 1. Forward displacement of the hip
Fig. 2. Mathematical model of above-knee prosthesis.
This investigation is concerned with the determination of the performance criteria for such a control mechanism. It is restricted to the performance during the swing phase of normal level walking. Although the same mechanism can perform a control function during both the swing and stance phase, these phases can be separated for study. NORMAL
WALKING
PAlTERN
This study is based upon the known walking pattern of a normal man. In terms of the variables of this study, this pattern is defined by xH, yH, AT and A,. Therefore. it was neces-
x,, = b sin (&p-n)
@-ad.) b(in.) c(in.)
+ cp, where a. b and c are:
Tall
Medium
Short
Average
0@099 0.3592 65.34
0.0825 0.3 198 62.38
0.1421 0.3694 60.9
0.0787 0.3491 62.87
The vertical dis@lacement of the hip is yH. The vertical displacement data as measured at the neck, were used for yW Again, it is assumed that the motion of the hip is satisfactorily approximated by that of the neck. A comparison of the neck displacement data of Murray (1964) with the hip displacement data of Ryker (1952) showed that this assumption is valid. A study of the data showed that it can
Fig.
(furingp.
88)
1. DUPACO
‘Hermes’ hydraulic control unit.
CONTROL
MECHANISM
be approximated by a sinusoid. This was done for each height group. The results are given in Table 2. Table 2. Vertical disalacement of the hio y, = b cos (4rp - a) + c, where a, b and c are:
&ad.) Min.) c(in.l
Tall
Medium
Short
Average
0.5138 -0.8913 0.8862
0.5774 -0.9272 0.8581
0.5409 -0.9017 0903
0.5445 -0%&l 0.8824
AT is the angle between the long axis of the thigh and the vertical, as shown in Fig. 2. This data consisted of twenty equally spaced points for the walking cycle. As the tirst and second derivatives of AT were required at every point, the data was first smoothed. Using the method of least squares ,a cubic was sequentially fitted to each set of five adjacent experimental points. A new value for AT was calculated for the center point and for 49 equally spaced points between the center point and the adjacent experimental point, both to the left and to the right. As points were computed to each side of an experimental point, there were two calculated values of AT for each point. The smoothing was considered complete when the difference between the two values of AT at each point was less than one degree. The double values from the last smoothing were then averaged using a weighted average. The result was a smoothed value of AT for every OXMH fraction of the walking cycle. Using the method of least squares a cubic was sequentially fitted to each set of 101 adjacent points. This cubic was differentiated to obtain the first and second derivatives for the central point. The smoothed values for AT and the two derivatives ofA, are presented for each height group in Figs. 3-6. The experimental values for AT are also included. In this study AS is the angle between the line joining the center of gravity of the shank and foot assembly to the knee joint, and the vertical, Fig. 2. A more commonly used angle
PERFORMANCE
CRITERIA
89
is that between the long axis qf the shank and the vertical &. This second definition is used in Figs. 7-10. The angles defined by these definitions differ by a constant of 6”. That is, the angle to the center of gravity is 6” larger than the angle to the long axis of the shank. The smoothing and differentiation of AS was done identically to that for AT. The smoothed values for & and the two derivatives are presented for each height group in Figs. 7-10. The experimental values for & are also included. It is seen that the smoothed values differ very little from the experimental values. CONTROL MECHANISM PERFORMANCE CRITERIA
The design of a control mechanism depends on an accurate knowledge of the performance characteristics required. For the linear type of mechanical-hydraulic mechanism considered in this study, these characteristics are presented in the form of a force-displacement curve (fhlmbg vs. r), a velocity-displacement curve i vs. r), and a power curve (Pt vs. r). These results are presented for the average, tall, medium, and short man in Figs. 1 l-14, respectively. The displacement, r, is the length of the mechanism, LD, divided by the distance from the knee joint to the lower attachment point in the shank, h. The velocity, k, is likewise i,/h. Normalizing the force,f. by dividing by the weight of the shank brought the forcedisplacement curves for the average, medium, and short man into fairly close agreement. The curve for the tall man shows a general increase in the absolute value of (fh/m,g). This is probably due to the fact that the group of tall men used for the physical constants were proportionately heavier than the other height groups. This difference also appears in the power curves. The velocity curves for all the groups are very close. The force on the control mechanism, f, is positive when the mechanism is in tension. The velocity. F, is positive when r is increasing. The power, Pf, is positive when a tensile
,
30
I
60
I
40
I
60
,
70
I
00
, -30 SO 100
Fig. 3. Thigh angle and its derivatives for the average man.
PERCENT OF WALKtNG CYCLE
I
20
I
20’
-2a
10
.I0
-la
>
0
20
10
a
0’
G
I
I
1
I
I
IO PO 30 40 10 PLRCENT Of WALIINO
I
70 CYCLE
I
60
1
00
I
so
ab
I0
20
10
1
0
to
10
Fig. 4. Thigh angle and its derivatives for the tall man. 0, Experimental data.
-30’
-20
-10
0
IO
20
3c
:
J
CONTROL
MECHANISM
PERFORMANCE
CRITERIA
91
1
4
2
a
-2
-4
a,--
>-
-4a ,- .
40 c-
,- -
da
20
IO
PERCENT
1
I
40
I
10
,
I
60
OF WALKING
30
I
80
1
CYCLE
70
,
90
,
r”
10
40
20
0
CO
IO
d
3
3
Fig. 7. Shank angle and its derivatives for the average man.
-2C
a
2Cc-
40
60
2
e
3
P
r^
PtRCLWT
I 20
I D
I
30
I
I
60 OF WALltIYO
40
I
80
,
c7cl.c
70
1
SO
,
m
,I)0 0
‘60
40
*PO
0
LO
Fig. 8. Shank angle and its derivatives for the tall man. 0, Experimental data.
40
40
-40
0
20
40
60
:2
CONTROL
MECHANISM
PERFORMANCE
CRITERIA
93
Fig.
-61
3
7
I I. ‘Ideal’
-20
I I
.91
I
93
r
I
95
I
4?
I
I.01
I
.ss
:”
-67
.69
-93
r
DHEEL-STRIKE
I.03
Fig. 12. ‘Ideal’ mechanism performance characteristics for the tall man.
-6C
-3c
~6
mechanism performance characteristics for the average man.
.69
C
3c
60
80
120
I60
.6
I
Sf
-101
c
I
9
12(
I!5C
16C
b.
-0
P
CONTROL
B.M. Vd.
3 No. I-G
MECHANISM
PERFORMANCE
CRITERIA
95
96
J. WALLACH
force is elongating the mechanism. Positive energy is energy input to the mechanism. Thus, in the first part of the swing phase the compressive force acting on the mechanism causes it to shorten and results in positive energy. This part of the swing phase is referred to as toe-off to maximum heel rise. In the second part of the swing phase r increases from a minimum to a maximum and the force changes from compression to tension. The power is first negative and then positive. This part of the swing phase is referred to as maximum heel rise to maximum extension. In the last part of the swing phase r decreases while the force remains tensile, and the power is negative. This part is referred to as maximum extension to heel-strike. The energy curves show that the energy input to the mechanism during toe-off to maximum heel rise would more than offset the energy required during the subsequent part of the swing phase. The calculations were done on a digital computer using the equations in the Appendix. Because of the tabular form of the data for AT and As, the expressions forf, i, P,, and r were derived as a function of these variables for ease of computation. The physical constants for the leg prosthesis are based on unpublished data taken during the studies reported in VA (1964) and VA (1965). To compute these constants it was necessary to assume a physical model for the prosthesis which is compatible with the available data. The methods and complete results are in Wallach (1967). A summary of the results is given in Table 3. CONCLUSIONS
This investigation takes the new data on walking patterns for normal men and the new data for the physical properties of a prosthesis, and determines specific performance criteria for a control mechanism for an above-knee prosthesis. Specifically, the investigation determined the following for the swing phase of level walking: (1) The linear force-displacement relation-
and E. SAIBEL Table 3. Physical constants for the tall, medium, short and average man Height group Constant
Tall
Medium
Short
mD f% t
0.145 8.25 5.52 0.075 4.26 0.0433 0.0033 o-0135 4.4 7.56
0.145 8.25 4.43 0.075 3.55 0.0387 0.0033 0.0122 4.4 7.39
0.145 8.25 3.69 0.075 294 0.0344 0.0033 O-011 4.4 7.23
b: LT Tc To
16.3 17.1 1.01 0.6161
15.7 16.4 1.07 0.6527
E h
In ID
IK mT
15-l 15.6 1.02 0.612
Average 0.145 8.25 4.43 0.075 3.55 0.0387 O-0033 O-0122 4.4 7.39 15.7 16.4 1.03 0.6283
ship required of a control mechanism so that the walking pattern of a normal man can be duplicated. (2) The power requirements for a control mechanism that makes possible the duplication of the walking pattern of a normal man. These results indicate that a powered meclianism is not required. authors are grateful to Dr. E. F. Murphy of the VA Prosthetics and Sensory Aids Service for the data he furnished the authors, and to Dr. M. P. Murray of the VA Center, Wood, Wisconsin for the data she furnished. This investigation was supported by a Public Health Service Fellowship 5-Fi-GM-30,439-02 from the National Institute of General Medical Sciences.
Acknowledgements-The
REFERENCES Bresler, B., Radcliffe, C. W. and Berry, F. R., Jr. (1957) Energy and power in the legs of above-knee amputees during normal level walking. Lower Extremity Amputee Research-Project. Inst. Engng Res., University of California, Serkeley, Series I 1, Issue 3 I. Eberhart, H. D. The forces and moments in the leg during level walking. Prosthetic Devices Research Project. Inst. Engng Res., University of California, Berkeley. Murphy, E. F. (1960) Lower-extremity components. In Orthopaedic Appliances Atlas, Artificial Limbs, Vol. 2, Chap. 5. Edwards, Ann Arbor, Michigan. Murray, M. P.. Drought, A. B. and Kory, R. C. (1964) Walking patterns of normal men. J. Bone Jr Surg. 46A. 335-360. Rvker, N. J., Jr. (1952) Glass walkway studies of normal subjects during normal level walking. Prosthetic Devices Research Project. Inst. Engng Res., University of California, Berkeley, Series 1 I, Issue 20.
CONTROL
MECHANISM
VA (1964) Clinical application study of the HenschkeMauch ‘Hydraulik’ swing control system. Research & Development Division, Prosthetic and Sensory Aids Service, Department of Medicine and Surgery, Tech. Rep. No. 3. VA (1965) Clinical application study of the DUPACO ‘Hermes’ hydraulic control unit. Research & Development Division, Prosthetic and Sensory Aids Service, Department of Medicine and Surger), Tech. Rep. No. 4. Wagner, E. M. and Catranis, J. Cl. (1954) New developments in lower-extremity prosthesis. In Human Limbs and Their Substitutes. (Edited by P. E. Klopstey and P. D. Wilson), Chap. 17. McGraw-Hill, New York. Wallach, J. (1967) Control mechanism performance criteria for an above-knee leg prosthesis. Dissertation Abstracts V28,4, p. 1468B. A Ax As A, b bn bs br d
r” g’ h I ID IH Ix L LO LT m mD ms mT Pr P r
NOMENCLATURE angle, rad. of knee of shank of thigh distance to center of gravity, in. of mechanism from upper attachment point of shank assembly from knee joint of thigh from hip joint distance of upper attachment point of force mechanism behind the knee joint, in. d/h force on mechanism, positive in tension, lb gravitational acceleration,ln./secZ distance of lower attachment point of mechanism below the knee joint, in. moment of inertia, in.-lb-set* of mechanism about upper attachment point of thigh about hip joint of shank assembly about knee joint overall length, in. of mechanism of thigh mass, lb-sec*/in. of mechanism of shank assembly of thigh power for mechanism, in.-lblsec. fraction of walking cycle LB/h I (1 --EsinA,)l,+
(1 +p-22EsinA,)lx rEcosAK
f=i;[
97
CRlTERlA
APPENDIX Derivation of equations The derivation of the equations is based on the mathematical model shown in Fig. 2. A two-dimensional model was chosen because most of the prostheses currently used have a single-axis knee joint (and sometimes a single-axis hip joint). (Murphy, 1960). Also, the rotations and displacements neglected are small (Ryker, 1952). The shank and foot were considered as one rigid body. Actually, most prostheses have a single-axis ankle joint with rubber bumpers to restrain the motion of the foot relative to the shank. However, during the swing phase the angle between the foot and shank has been shown to be nearly constant and the moment has been shown to be zero (Bresler, 1957). The location of the centroid of the combined mass of the shank and foot lies ahead of the line joining the knee and ankle joints. Also, the lower attachment point of the control mechanism is on a line joining the location of the combined center of mass to the knee joint. This is slightly ahead of the actual location, but the effect is negligible. From free-body diagrams of the various parts of the prosthesis the dynamical equations and geometrical relations can be written directly. Letting:
E=$ r=-.
LD h
The law of cosines gives: r= V/(l+F--2EsinA.). Differentiation gives: i=-;A’,cosA,. The control mechanism force is found by manipulation of the geometric and dynamical equations. This is a laborious, but direct procedure. The exception is that the trigonometric substitutions make use of the fact that the angular motions of the prosthesis are limited, in order to eliminate any ambiguities in signs.
sinAx)lr,~~_(EP-l)(l-EsinA.)I&$ A”,- (E -sinAx)(l--E r5 r3cos Ax
+mr,gbD(l-EsinA.)[sinA,-E(sinAssinA.+cosA,cosAX)] rPE cos AK time for one walking cycle, sec. TO time to toe-off. sec. xi, yH horizontal and vertical coordinates respectively of hip joint, in. Derivatives with respect to time are denoted by dots. TC
PERFORMANCE
+rmsgbssinA, EcosA,
I
The knee angle, AK, in the above expressions
is:
AK= w+As-Ar. The power can be calculated from the above results.
dx -&
=
.d=x x. -@ =
”
x.
P, = fr.
Pergamon Press. 1970.
Printed in Great Britain
CONTROL MECHANISM PERFORMANCE CRITERIA FOR AN ABOVE-KNEE LEG PROSTHESIS* JOHN WALLACH Mechanical Engineering Laboratory, General Electric Research and Development Center, Schenectady,N.Y. 12301,U.S.A. and EDWARD
SAIBEL
Department of Applied Mechanics, Carnegie-Mellon
University, Pittsburgh, Pa 15213, U.S.A.
Abstract-A primary requirement of an above-knee leg prosthesis is that it enables the amputee to walk with a normal appearing walking pattern. The leg motion during the swing and stance phase of level walking should approximate that of a non-amputee. To accomplish this, a control mechanism is added to the conventional leg prosthesis. This study is concerned with the performance of the mechanism during the swing phase. The normal walking pattern for a normal man was obtained from the literature for average, tall, medium, and short men. The data, consisting of tabulated experimental data for the linear and angular displacements of the normal leg, were smoothed and differentiated twice to obtain velocities and accelerations. A mathematical mode1 was chosen that typifies the current mechanical-hydraulic mechanisms. The physical constants were calculated from data collected fi-om many sources. The force, velocity, and power were calculated for the control mechanism. INTRODUCTION
The most promising mechanisms are linearlyactuated mechanical-hydraulic devices which are attached behind the knee joint and about half-way down the shank. A device typical of those with which this investigation is concerned is the DUPACO ‘Hermes’ hydraulic control unit (Fig. 1). This type of device uses a combination of spring force and fluid damping to control the motion of the shank relative to the thigh. Springs, at one or both ends of the piston stroke, aid in reversing the shank motion without the expenditure of energy. Fluid damping is generated by forcing an incompressible fluid through a series of staggered holes in the cylinder. The holes are staggered to provide a damping force which is a function of the relative position of the piston, as well as of the relative velocity of the piston. That is, the device is programmed to provide an axial
widely accepted criterion for devices to rehabilitate lower-extremity amputees is that the prosthetic replacement must have a normal external shape and appearance, and must permit the amputee to walk comfortably and safely at normal rates on level ground, without undue mental or physical effort and with a normal-appearing gait, Wagner (1954). A recent comprehensive study of the walking patterns of normal men defines this normal-appearing gait and makes possible an improvement in the design data for prostheses, Murray (1964). The motion of the thigh section of the prosthesis is precisely controlled by the amputee. However, the amputee has no voluntary control over the motion of the shank relative to the thigh. Therefore, various types of control mechanisms have been developed. AN
ESTABLISHED,
*Received5 March
1969. 87
88
J. WALLACH
force which is a function of its length and the time rate of change of its length. Considering the triangle formed by the knee joint and the upper and lower mounting points of the control device (Fig. 2), the device provides a knee moment that is a function of the angle between the shank and thigh, and the time rate of change of this angle.
and E. SAIBEL
sary to determine these variables as a function of percent of the walking cycle and also to determine the first and second derivatives of these variables. Dr. Murray made an extensive study of the walking patterns of normal men, Murray (1964). Sixty men were divided into three equal groups by height: tall, medium, and short. Each man was observed walking at a normal cadence on a level surface. This data had to be adjusted in order to determine xH, yH, AT and AK as defined in this study. Dr. Murray very kindly furnished the constants necessary to adjust the data as well as a complete tabulation of all the data. ’ xH is the horizontal displacement of the hip. The forward displacement data, as measured at the neck, were used for xH. This assumes that the motion of the hip is satisfactorily approximated by that of the neck. A comparison of the neck displacement data of Murray (1964) with the hip displacement data of Eberhart showed that this assumption is sufficiently accurate for this study. A study of the data showed that it can be approximated by a linear term plus a very small sinusoidal variation. This was done for each height group. The results are given in Table 1. Table 1. Forward displacement of the hip
Fig. 2. Mathematical model of above-knee prosthesis.
This investigation is concerned with the determination of the performance criteria for such a control mechanism. It is restricted to the performance during the swing phase of normal level walking. Although the same mechanism can perform a control function during both the swing and stance phase, these phases can be separated for study. NORMAL
WALKING
PAlTERN
This study is based upon the known walking pattern of a normal man. In terms of the variables of this study, this pattern is defined by xH, yH, AT and A,. Therefore. it was neces-
x,, = b sin (&p-n)
@-ad.) b(in.) c(in.)
+ cp, where a. b and c are:
Tall
Medium
Short
Average
0@099 0.3592 65.34
0.0825 0.3 198 62.38
0.1421 0.3694 60.9
0.0787 0.3491 62.87
The vertical dis@lacement of the hip is yH. The vertical displacement data as measured at the neck, were used for yW Again, it is assumed that the motion of the hip is satisfactorily approximated by that of the neck. A comparison of the neck displacement data of Murray (1964) with the hip displacement data of Ryker (1952) showed that this assumption is valid. A study of the data showed that it can
Fig.
(furingp.
88)
1. DUPACO
‘Hermes’ hydraulic control unit.
CONTROL
MECHANISM
be approximated by a sinusoid. This was done for each height group. The results are given in Table 2. Table 2. Vertical disalacement of the hio y, = b cos (4rp - a) + c, where a, b and c are:
&ad.) Min.) c(in.l
Tall
Medium
Short
Average
0.5138 -0.8913 0.8862
0.5774 -0.9272 0.8581
0.5409 -0.9017 0903
0.5445 -0%&l 0.8824
AT is the angle between the long axis of the thigh and the vertical, as shown in Fig. 2. This data consisted of twenty equally spaced points for the walking cycle. As the tirst and second derivatives of AT were required at every point, the data was first smoothed. Using the method of least squares ,a cubic was sequentially fitted to each set of five adjacent experimental points. A new value for AT was calculated for the center point and for 49 equally spaced points between the center point and the adjacent experimental point, both to the left and to the right. As points were computed to each side of an experimental point, there were two calculated values of AT for each point. The smoothing was considered complete when the difference between the two values of AT at each point was less than one degree. The double values from the last smoothing were then averaged using a weighted average. The result was a smoothed value of AT for every OXMH fraction of the walking cycle. Using the method of least squares a cubic was sequentially fitted to each set of 101 adjacent points. This cubic was differentiated to obtain the first and second derivatives for the central point. The smoothed values for AT and the two derivatives ofA, are presented for each height group in Figs. 3-6. The experimental values for AT are also included. In this study AS is the angle between the line joining the center of gravity of the shank and foot assembly to the knee joint, and the vertical, Fig. 2. A more commonly used angle
PERFORMANCE
CRITERIA
89
is that between the long axis qf the shank and the vertical &. This second definition is used in Figs. 7-10. The angles defined by these definitions differ by a constant of 6”. That is, the angle to the center of gravity is 6” larger than the angle to the long axis of the shank. The smoothing and differentiation of AS was done identically to that for AT. The smoothed values for & and the two derivatives are presented for each height group in Figs. 7-10. The experimental values for & are also included. It is seen that the smoothed values differ very little from the experimental values. CONTROL MECHANISM PERFORMANCE CRITERIA
The design of a control mechanism depends on an accurate knowledge of the performance characteristics required. For the linear type of mechanical-hydraulic mechanism considered in this study, these characteristics are presented in the form of a force-displacement curve (fhlmbg vs. r), a velocity-displacement curve i vs. r), and a power curve (Pt vs. r). These results are presented for the average, tall, medium, and short man in Figs. 1 l-14, respectively. The displacement, r, is the length of the mechanism, LD, divided by the distance from the knee joint to the lower attachment point in the shank, h. The velocity, k, is likewise i,/h. Normalizing the force,f. by dividing by the weight of the shank brought the forcedisplacement curves for the average, medium, and short man into fairly close agreement. The curve for the tall man shows a general increase in the absolute value of (fh/m,g). This is probably due to the fact that the group of tall men used for the physical constants were proportionately heavier than the other height groups. This difference also appears in the power curves. The velocity curves for all the groups are very close. The force on the control mechanism, f, is positive when the mechanism is in tension. The velocity. F, is positive when r is increasing. The power, Pf, is positive when a tensile
,
30
I
60
I
40
I
60
,
70
I
00
, -30 SO 100
Fig. 3. Thigh angle and its derivatives for the average man.
PERCENT OF WALKtNG CYCLE
I
20
I
20’
-2a
10
.I0
-la
>
0
20
10
a
0’
G
I
I
1
I
I
IO PO 30 40 10 PLRCENT Of WALIINO
I
70 CYCLE
I
60
1
00
I
so
ab
I0
20
10
1
0
to
10
Fig. 4. Thigh angle and its derivatives for the tall man. 0, Experimental data.
-30’
-20
-10
0
IO
20
3c
:
J
CONTROL
MECHANISM
PERFORMANCE
CRITERIA
91
1
4
2
a
-2
-4
a,--
>-
-4a ,- .
40 c-
,- -
da
20
IO
PERCENT
1
I
40
I
10
,
I
60
OF WALKING
30
I
80
1
CYCLE
70
,
90
,
r”
10
40
20
0
CO
IO
d
3
3
Fig. 7. Shank angle and its derivatives for the average man.
-2C
a
2Cc-
40
60
2
e
3
P
r^
PtRCLWT
I 20
I D
I
30
I
I
60 OF WALltIYO
40
I
80
,
c7cl.c
70
1
SO
,
m
,I)0 0
‘60
40
*PO
0
LO
Fig. 8. Shank angle and its derivatives for the tall man. 0, Experimental data.
40
40
-40
0
20
40
60
:2
CONTROL
MECHANISM
PERFORMANCE
CRITERIA
93
Fig.
-61
3
7
I I. ‘Ideal’
-20
I I
.91
I
93
r
I
95
I
4?
I
I.01
I
.ss
:”
-67
.69
-93
r
DHEEL-STRIKE
I.03
Fig. 12. ‘Ideal’ mechanism performance characteristics for the tall man.
-6C
-3c
~6
mechanism performance characteristics for the average man.
.69
C
3c
60
80
120
I60
.6
I
Sf
-101
c
I
9
12(
I!5C
16C
b.
-0
P
CONTROL
B.M. Vd.
3 No. I-G
MECHANISM
PERFORMANCE
CRITERIA
95
96
J. WALLACH
force is elongating the mechanism. Positive energy is energy input to the mechanism. Thus, in the first part of the swing phase the compressive force acting on the mechanism causes it to shorten and results in positive energy. This part of the swing phase is referred to as toe-off to maximum heel rise. In the second part of the swing phase r increases from a minimum to a maximum and the force changes from compression to tension. The power is first negative and then positive. This part of the swing phase is referred to as maximum heel rise to maximum extension. In the last part of the swing phase r decreases while the force remains tensile, and the power is negative. This part is referred to as maximum extension to heel-strike. The energy curves show that the energy input to the mechanism during toe-off to maximum heel rise would more than offset the energy required during the subsequent part of the swing phase. The calculations were done on a digital computer using the equations in the Appendix. Because of the tabular form of the data for AT and As, the expressions forf, i, P,, and r were derived as a function of these variables for ease of computation. The physical constants for the leg prosthesis are based on unpublished data taken during the studies reported in VA (1964) and VA (1965). To compute these constants it was necessary to assume a physical model for the prosthesis which is compatible with the available data. The methods and complete results are in Wallach (1967). A summary of the results is given in Table 3. CONCLUSIONS
This investigation takes the new data on walking patterns for normal men and the new data for the physical properties of a prosthesis, and determines specific performance criteria for a control mechanism for an above-knee prosthesis. Specifically, the investigation determined the following for the swing phase of level walking: (1) The linear force-displacement relation-
and E. SAIBEL Table 3. Physical constants for the tall, medium, short and average man Height group Constant
Tall
Medium
Short
mD f% t
0.145 8.25 5.52 0.075 4.26 0.0433 0.0033 o-0135 4.4 7.56
0.145 8.25 4.43 0.075 3.55 0.0387 0.0033 0.0122 4.4 7.39
0.145 8.25 3.69 0.075 294 0.0344 0.0033 O-011 4.4 7.23
b: LT Tc To
16.3 17.1 1.01 0.6161
15.7 16.4 1.07 0.6527
E h
In ID
IK mT
15-l 15.6 1.02 0.612
Average 0.145 8.25 4.43 0.075 3.55 0.0387 O-0033 O-0122 4.4 7.39 15.7 16.4 1.03 0.6283
ship required of a control mechanism so that the walking pattern of a normal man can be duplicated. (2) The power requirements for a control mechanism that makes possible the duplication of the walking pattern of a normal man. These results indicate that a powered meclianism is not required. authors are grateful to Dr. E. F. Murphy of the VA Prosthetics and Sensory Aids Service for the data he furnished the authors, and to Dr. M. P. Murray of the VA Center, Wood, Wisconsin for the data she furnished. This investigation was supported by a Public Health Service Fellowship 5-Fi-GM-30,439-02 from the National Institute of General Medical Sciences.
Acknowledgements-The
REFERENCES Bresler, B., Radcliffe, C. W. and Berry, F. R., Jr. (1957) Energy and power in the legs of above-knee amputees during normal level walking. Lower Extremity Amputee Research-Project. Inst. Engng Res., University of California, Serkeley, Series I 1, Issue 3 I. Eberhart, H. D. The forces and moments in the leg during level walking. Prosthetic Devices Research Project. Inst. Engng Res., University of California, Berkeley. Murphy, E. F. (1960) Lower-extremity components. In Orthopaedic Appliances Atlas, Artificial Limbs, Vol. 2, Chap. 5. Edwards, Ann Arbor, Michigan. Murray, M. P.. Drought, A. B. and Kory, R. C. (1964) Walking patterns of normal men. J. Bone Jr Surg. 46A. 335-360. Rvker, N. J., Jr. (1952) Glass walkway studies of normal subjects during normal level walking. Prosthetic Devices Research Project. Inst. Engng Res., University of California, Berkeley, Series 1 I, Issue 20.
CONTROL
MECHANISM
VA (1964) Clinical application study of the HenschkeMauch ‘Hydraulik’ swing control system. Research & Development Division, Prosthetic and Sensory Aids Service, Department of Medicine and Surgery, Tech. Rep. No. 3. VA (1965) Clinical application study of the DUPACO ‘Hermes’ hydraulic control unit. Research & Development Division, Prosthetic and Sensory Aids Service, Department of Medicine and Surger), Tech. Rep. No. 4. Wagner, E. M. and Catranis, J. Cl. (1954) New developments in lower-extremity prosthesis. In Human Limbs and Their Substitutes. (Edited by P. E. Klopstey and P. D. Wilson), Chap. 17. McGraw-Hill, New York. Wallach, J. (1967) Control mechanism performance criteria for an above-knee leg prosthesis. Dissertation Abstracts V28,4, p. 1468B. A Ax As A, b bn bs br d
r” g’ h I ID IH Ix L LO LT m mD ms mT Pr P r
NOMENCLATURE angle, rad. of knee of shank of thigh distance to center of gravity, in. of mechanism from upper attachment point of shank assembly from knee joint of thigh from hip joint distance of upper attachment point of force mechanism behind the knee joint, in. d/h force on mechanism, positive in tension, lb gravitational acceleration,ln./secZ distance of lower attachment point of mechanism below the knee joint, in. moment of inertia, in.-lb-set* of mechanism about upper attachment point of thigh about hip joint of shank assembly about knee joint overall length, in. of mechanism of thigh mass, lb-sec*/in. of mechanism of shank assembly of thigh power for mechanism, in.-lblsec. fraction of walking cycle LB/h I (1 --EsinA,)l,+
(1 +p-22EsinA,)lx rEcosAK
f=i;[
97
CRlTERlA
APPENDIX Derivation of equations The derivation of the equations is based on the mathematical model shown in Fig. 2. A two-dimensional model was chosen because most of the prostheses currently used have a single-axis knee joint (and sometimes a single-axis hip joint). (Murphy, 1960). Also, the rotations and displacements neglected are small (Ryker, 1952). The shank and foot were considered as one rigid body. Actually, most prostheses have a single-axis ankle joint with rubber bumpers to restrain the motion of the foot relative to the shank. However, during the swing phase the angle between the foot and shank has been shown to be nearly constant and the moment has been shown to be zero (Bresler, 1957). The location of the centroid of the combined mass of the shank and foot lies ahead of the line joining the knee and ankle joints. Also, the lower attachment point of the control mechanism is on a line joining the location of the combined center of mass to the knee joint. This is slightly ahead of the actual location, but the effect is negligible. From free-body diagrams of the various parts of the prosthesis the dynamical equations and geometrical relations can be written directly. Letting:
E=$ r=-.
LD h
The law of cosines gives: r= V/(l+F--2EsinA.). Differentiation gives: i=-;A’,cosA,. The control mechanism force is found by manipulation of the geometric and dynamical equations. This is a laborious, but direct procedure. The exception is that the trigonometric substitutions make use of the fact that the angular motions of the prosthesis are limited, in order to eliminate any ambiguities in signs.
sinAx)lr,~~_(EP-l)(l-EsinA.)I&$ A”,- (E -sinAx)(l--E r5 r3cos Ax
+mr,gbD(l-EsinA.)[sinA,-E(sinAssinA.+cosA,cosAX)] rPE cos AK time for one walking cycle, sec. TO time to toe-off. sec. xi, yH horizontal and vertical coordinates respectively of hip joint, in. Derivatives with respect to time are denoted by dots. TC
PERFORMANCE
+rmsgbssinA, EcosA,
I
The knee angle, AK, in the above expressions
is:
AK= w+As-Ar. The power can be calculated from the above results.
dx -&
=
.d=x x. -@ =
”
x.
P, = fr.