- Email: [email protected]
Analogue computer system for the evaluation of hip joint moments during normal walking
Analogue computer system for the evaluation of hip joint moments during normal walking M. Grigoriadou-Koukis University of Athens, Athens, Greece
and M.T. Samarakou
Electronics
Laboratory,
Panepistimioupoli-Ktiria
TYPA,
157 71
Received July 1987, accepted November 1987
ABSTRACT Hip joint moments were determined by analogue computer techniques, u&g as inputs the an$es formed by body segments during normal walking. The control algorithm was based on the division of the walking cycle into six stages; each activated by contact of heel, midfoot and fore-joot. The problem of the accuracy of double dzyerentiation was overcome by means oj two fillers which amp@ the sain at the basicfrequency of the walking cycle, while reducing it above 10 Hz. The analogue model save reasonable results, suggesting that it could be used as an input to a real time microcomputer-based adaptive control system for the maintenance of hip joint balance in an amputee during walking. Keywords:
Hip joint,
analogue
computer,
link equations,
moments
INTRODUCTION We studied human locomotion in order to calculate the hip joint moments during normal walking; we used an analogue computer because of its known advantage for the processing of continuous signals. This is especially important in the case of a biological system, because of the absence of errors due to A/D conversion. The analogue computer can also provide fast results with adequate accuracy. The model is based on Dempster’s hypothesis, that the body can be partitioned in to a number of segments (F&re 1). The mathematical analysis of normal walking is effected by taking into account those segments which execute the most important movements. Any free body has six degrees of freedom. Hence, if we consider each of the twelve segments as a free body, we can describe its motion in space by means of six second-order differential equations. The fact that these segments are not independent of each other is further expressed by equations which form the set of constraints. However, to avoid undue complexity, an equivalent solution exists which consists of adding to the differential equations terms expressing the link between the body segments. we use an appropriate differentiation Further, method, with the aid of an analogue computer, in order to calculate the moments of the segments. The reduction of those moments on the hip joint gives the desired answer to our problem, while the complexity of the problem demands special simplifications and assumptions with respect to those made by other workers’-5.
Correspondence
and reprint
requests
8 1988 Butterworth & Co (Publishers) 0141-5425/88/030253-08 $03.00
to Dr M. Grigoriadou-Koukis.
Figure
1 Dempster’s
human
body partitioning.
Ltd J, Biomed.
Eng.
1988. Vol. 10, May
253
Evaluation of hip joint moments: M. Gri&riadou-Koukis and M. T. Samarakou
MATHEMATICAL
ANALYSIS
Consider the external forces applied to the SC segment, the muscular and frictional forces of the joint (and, in the case of the lower segments, ground reaction), as depicted in Figure 2. The three translation equations for the S, segment, according to are given by the vector d’alembert’s principles, equation:
m,i.i + 7 f, (u,, u,, 8,, 0,) = fl (Fk, t) where m, is the mass of the segment
S,
(1) and
with x,, y,, z,, the linear coordinates and a,, PI, y, the angular coordinates respectively of the S, centre of gravity. The summation C is extended over all the segments adjacent to S, and the result, i.e. the term cf, ( ), expresses the interaction between them through the corresponding joint. The term Z(Fk, t) represents the forces excerted by the appropriate muscles. Also, the three rotation equations of the same segment are:
a+.L,
k, 6,) = 4.1 (Fk, t)
Uj,
I
(2)
where Ze, =
I Zi: Z7,
11
Pa)
I,,, Ioz, IT, being the inertia moments of the S, segments, while the term c& (e,, 0,) is due to the friction between adjaceni segments. The term C&(u,,u,,&,t9,) represents the moments, due to the elasticity of the joint, as well as the Cf, ( ) term of equation (1); the term & (Fk, t) is due to muscular moments. We need 24 vector equations to describe the motion of the 12 segments, and are obliged to apply a number of simplifying assumptions. The first simplification concerns the term Cf, ( ), which may be replaced by a new term f3 (Fk,’ t) if we accept the inelasticity of the joint. Equation (1) is then replaced by the equation: + f3 (FI, t)
m, ii,
= A (Fk, t)
P)
or its equivalent: .. m u,,
+ f+ (Fk, t)
P)
We can also omit the term c&(&, 0,) because the coefficient of friction is very’small, and the term c &(Ul, uI, 8,, e,), if we accept the inelasticity of the
254
A
(14
I
+
Y
J. Biomed. Eng.
1988,
Vol.
10. May
0
Figure 2 Forces applied onto the Si segment a free body.
joint, which becomes: &(Ie,,
is not very
t) 8, = &(Fk,
important.
considered
Equation
t)
The following equation, which represents between segments, must also be included: qe,,
4,
Ut,
Uj)
=
0
as
(2)
(4) the link
(5)
We thus conclude that the three equations (3b), (4) and (5) are sufficient to describe human locomotion during walking.
LINK AND DIFFERENTIATION EQUATIONS OF BODY SEGMENTS We now calculate the link equations between adjacent segments, i.e. equations of the form of equation (5). The required data are the values of the angles formed between segments of the body during walking. The complexity of the problem demands yet further simplifications. We can assume that the arms, head and trunk consist of a compact set, which is true for slow walking, and that the trunk is displaced in translation only without rotation. The last assumption has the result that displacement of the trunk will have the same coordinates, x,y,z, as the pelvis. We shall, nevertheless, take into account the rotation of the pelvis, which plays an important role in determining the exact position of the two femur heads. Let m, be the mass of the upper set. This mass is situated at a distance H from the sacrum and approximately in a vertical line which passes from the midpoint of the distance w between the two femur heads (Figures 3 and 4). This consideration is in agreement with the fact that the total moment P, = Pb + P, must be equal to zero. The link equations are:
Eualuation of hip joinf moments: M. Gri’oriadou-Koukir and M. T. Samarakou
V
sin & = & = tan & and cos 8, = 1 The equations which relate femur head are: xb-xh+,.f
yb
-
zb
-
+
yh
zh
hyb
+
YPb
-
the pelvis to the right
=
+
f-fb
-
h
fj
-
@b
-
hat, + w
0
Ub
2
(7)
=
(8)
0
= 0
where xh,yh,zh are the coordinates of the right femur head, while the relation between right and left femur head is expressed by the equations: .+& - x,, + w&, = 0 yh’ - yh + w@,, = 0 zh
-
zh
+
w
=
(9)
0
where xh’, yh’, zh’ are the coordinates of the left femur head. The translation equations of the upper set are:
(10)
Figure
3
Coordinates
of the upper
while the rotation
body.
equations
are:
Ibe tib = h&,x
T---4!
Ib,
h A
xh
position
Ye ZP -
.?b
-
=
where g, h considered tion, from three axes. 10’; hence
0
-
h, + 1, sin yc = 0
-
zg
-
-
xp
+
yh - ye -
(6)
and w/2 are the distances of the sacrum, as the centre of the trunk-basin articulathe two femur heads with respect to the The rotation of the pelvis is less than + we can assume:
1, cos yc = 0 bh -JR) sin a,
th
-
23
-
(12)
= 0
where L, is the thigh length. . . tion sin a, - a,, xh
xb -g=o Yb -H=O
&
(11)
Mb:
yh - ys -
g”
zh
Figure 4 Dimensions of the pelvis and the relative of the upper body mass.
=
Furthermore, the equations which concern the lower segments (thigh, legs and feet) demand more detailed assumptions due to the fact that the rotations of the segments are large, while their masses and inertia moments are relatively small. We shall consider that the lower limb, during walking, is situated in a plane forming an angle a, with respect to the vertical. By the mean of this angle we can find the z coordinates and neglect the very small angles, which the lower limb forms with x axis (Fz@re 5). The equations which connect the coordinates of the knee, indicated by the index g, to those of the head of femur are:
v .j,
31
‘j/b
Or,
with the assump-
1, sin-y, = 0 1, cos -yc = 0 1,
a,
cos
yc
(13)
=
0
Now, the equations which relate the thigh’s centre of gravity to the coordinates of the hip joint are: XC -
xh
-
C,, 1, sin yc = 0
yc -
yh
+
c2c
1, cos
‘=fc
=
0
c
zh
+
cyc
1,
-fc
=
0
-
cos
J, Biomed.
(14)
Eng.
1988, Vol.
10. May
255
Eualuation of hip joint moments: M. Grigonadou-Koukis and M. T. Samarakou
We consider that the foot’s centre of gravity coincides with the ankle joint. This assumption simplifies the model without greatly affecting the results and is justified by the relatively small dimensions of the foot segment; thus we have: &h
=
x,,
=
$h
yp,
&h
=
zzp
(21)
The link equations for the foot are evaluated in terms of three stages represented by the heel contact, the mid-sole and the fore-foot, activated by logic circuits in the analogue computer, while the translation and the rotation equations of the foot are:
(22)
(xP’ YP) ZP) The
Figure 5
coordinates
(23)
of the lower limb and the angle
a, formed from the lower limb and the vertical.
where C,, is the Fisher and Braune coefficient6 for the thigh. Also, the equations which relate the knee to the ankle, indicated by the subscript ch, are:
In relation to the above, the equations which express the coordinates of the upper set, thigh, leg and foot in terms of those of the right femur head are, respectively, for the upper set: & -
xh =
-
h-n, -
y
Pb (24)
., _h - _%h -
zg -
&h
-
I, cosy, 1, cos
= 0 “/J
=
(15)
where the subscript j is used for the leg, and 1, is the leg length. Finally, the equations which relate the knee to the leg’s centre of gravity. Cg, lj sin ^yj = 0 JJ - Jg + C2j 1, COS ‘ye = 0 Zj - Zg + C2j 1, COS yj = 0 Xj
-
Xg
-
(16)
where C, is the Fisher and Braune6 coefficient for the leg and 1, the leg length. The translation equations of the thigh and the knee are respectively:
m, & = F,, m, jk = F, m, i, = F,
(17)
-
zh
=
,@,,
+
h&,
m, .Cj = Fj, m, jj = FIy m, 2, = F,,
(18)
while the rotation equations respectively:
for the thigh and leg are
L, ‘;vc = M,
w
2
x, _J’J -
=
1,
Jh
=
-
zj
r?h =
-
-
xh
equations
sin yc + CzJ lj sin yJ Ic COS yc C’zJ4 COS “/j C*j lj aj COS “/j - Ic 0, COS
(14)
(25) yc
For the foot: XP
-
xh
=
Yp - yh = ZP
-
t,, =
IJ sin yj - Lj COS -4
+ “lj
lc -
a, cos ?;
sin yc Ic COS yc -
(26)
I, a, cos yc
The value of the Fisher and Braune6 coefficients and Czj are, respectively: C,
= 0.42
C,,
(27)
The position of the upper set is situated, according to Figure 4, at a distance H = 0.335 m from the sacrum, while its projection onto the horizontal plane is situated at the middle of the distance between the two femur heads. Finally, we can work similarly for the left lower limb; its equations are the same as equations (13), (25) and (26), where the angles correspond to those of the left side.
(19)
and
(20)
J. Biomed.
-
For the thigh the above-mentioned apply. For the leg:
Cc = 0.44,
and
256
zc
0
Eng.
1988, Vol.
10, May
EVALUATION
OF MOMENTS
The
of
evaluation
hip
joint
moments
will
be
Evaluation of hip joint moments:M. Gn@riadou-Koukis
Table 1
Data used in the calculations,
Percentage of step
Angles in degrees
CYCk
ab
6b
and M. T. Samarakou
after Lamoreux’ Angles in radians
Yb
Yc
71
YP
0,
4 -
0
0
5
- 0.6
10
- 1.20 - 0.85 -0.75
+ 22.7 + 22.2 +21.5
+ 26.0
+ 32.3
+ 0.0473
- 0.0087
- 2.2
+ 4.8 +5.1 + 4.0
+ 19.5 + 11.0
+ 13.2 + 6.1
+ 0.0490 + 0.0694
15 20 25
- 0.0070 + 0.0122
-3.0 -2.6 - 1.2
+ 3.6 + 2.9 + 1.3
+ 1.35 + 0.90 + 0.75
+ 18.9 + 12.2 + 4.8
+ -
4.4 0.3 3.7
+ + +
5.4 5.0 4.1
+ 0.0775 + 0.1018 + 0.1086
30 35 40
+ 0.0209 + 0.0471 + 0.0558
- 0.2 0 0
-0.5 -1.7 - 2.6
+ 0.30 - 0.25 - 0.80
- 2.8 - 10.0 _ 18.0
- 6.8 - 10.0 - 15.3
+ + -
3.2 1.5 2.3
+0.1118 + 0.1063 + 0.0972
45 50 55 60 65 70 75 80 85 90
+ 0.0593 + 0.0524 + 0.0401
0 0 0.6 2.2 3.0 2.6 1.2 0.2 0 0
- 4.1 -4.8 - 5.1 - 4.0 - 3.6 - 3.9 - 1.3 +0.5 + 1.7 + 2.6
- 1.50 - 1.20 - 0.85 -0.75 + 1.35 + 0.90 + 0.75 + 0.30 - 0.25 - 0.80
- 22.6 - 24.6 - 22.9 _ 16.0 - 4.7 + 6.6 + 51.2 +21.4 + 24.5 + 24.4
- 20.6 -29.1 - 39.9 - 54.0 - 59.2 - 57.4 - 47.8 - 32.6 - 14.5 + 5.9
- 7.1 _ 16.1 - 34.4 _ 16.0 - 75.7 - 64.9 -49.1 - 29.6 - 9.2 + 10.9
+ + + + + + + + + +
+ -
0 0
+ 4.1 + 4.8
_ 1.50 - 1.20
+23.1 + 22.7
+ 24.6 + 26.0
+ 29.9 + 32.3
+ 0.0341 + 0.0473
+ + + + +
95 100
Table
2
Accepted
anthropometric
0.0813 0.0518 0.0208 0.0018 0.0035 0.0018 0.0163 0.0300 0.0422 0.0383
0.0227 0.0052 0.0332 0.0506 0.0558 0.0506 0.0366 0.0244 0.0192 0.0174
- 0.0209 - 0.0087
parameters
M, = -I& Parameters
+ (P, + m,r) y
Accepted values
Thigh length Leg length Pelvic dimensions (see
0.43 m 0.38 m
+ m,&( H + J&$-)
+
w C m#
Figure 4) 0.1698 0.0792
r
m m
+
C m,i,
Cyh -
y,)
(29)
0.024 m
Jr
body Mass of foot Mass of thigh Mass of leg Inertia moment Inertia moment Inertia moment Inertia moment
70 kg 1.18 kg 7.504 kg 3.346 kg 0.00806 kgm 0.04353 kgm 0.1248 kgm 0.0513 kgm
Mass of
of of of of
foot leg thigh pelvis
where w is the distance between the two femur heads. The first three terms correspond to the moment due to the upper set (trunk, pelvis, arms, head), while the two last terms correspond to the right lower limb. For the non-supporting time, the moment M, is given by:
M, = C m;i; (_JJI, -yi) expressed in two planes: the plane xy or flexionextension plane; and the plane y.z or abductionadduction plane. The resulting hip joint moments respectively. The analogue are A4, and M,, computer model is able to calculate the moments in the other joints (pelvis, knee). This calculation will take place every 5% of a full step cycle. During supporting time, i.e. from 10% till 55 % of step cycle, the moment M, is given by the equation:
M, = - C Z, ‘i/i + C mi.?,(jh -yl)
(30)
where the subscript i corresponds to the left lower limb. There are two special instants: the 5% and 55% instants of step cycle, which correspond to double support. For those instants the evaluation of moments will be given by the average of the two corresponding equations. The data that we have used for the above calculations are obtained by Lamoreux’ after several corrections, as we can see in Table I, while in Table 2 there is the accepted set of anthropometric parameters.
I
+
(C
??Zij
I
+
pi)
(Xh
-Xi)
(28)
where i corresponds to the following segments: the upper set (trunk, pelvis, arms, head) and the right lower limb. For the non-supporting time, i.e. from 60% till 100% of step cycle, the equation which describes M, is the same, except that the subscript i corresponds now to the left lower limb. Also, the Mx moment is described for the support time by the equation:
ANALOGUE COMPUTER IMPLEMENTATION AND RESULTS In Figure 15, we can see the analogue computer model, designed to calculate the linear and angular displacement of the body segments during walking. There are six stages: right heel contact, right midsole, right fore-foot, left heel contact, left mid-sole, left fore-foot, activated by the logic circuits PC, P,,,, P,’ and P,‘. The two blocks C1 and C2 represent the
J. Biomed.
Eng. 1988. Vol. 10, May
257
Evaluation of hip joint moments: M. Gri’oriadou-Koukis and M. T. Samarakou
link equations for right and left lower limbs, respectively. Now comes the most difficult part of the model, which consists of the double differentiation of the functions given by the link equations. We have chosen for the double differentiation operator the circuit of Figure 7. This analogue circuit uses two filters with transfer functions:
Hi =
0.12s l+s/32
H2 =
0.17s 1 +s/36
(31)
h
The filters ensure a satisfactory response to the unity step function and a non-oscillatory output with a gain of 40 dB at 1 Hz, which is near the principal frequency of the walking cycle. The role of the filters is also to attenuate the frequencies above 10 Hz. Using an analogue computer system we have simultaneously obtained the moments of various body segments during walking, as presented in Figures 8 and 9; we have also calculated the hip joint moments, depicted in Figure 10. Finally, in Figure II we can see the comparison of the results with a set of curves obtained by Paul’.
Q Q
Figure
6
Figure 7
258
Analogue
computer.
The differentiation
operator
J. Biomed. Eng. 1988, Vol. 10, May
of the analogue
computer
model.
Evaluation
M 4
ofhip joint
momts:
M. Grigoriadou-Koukis and M. T. Samarakou
F-
n.N
3 2 1 0 -f -2 -3
-4
4
PI
JX
Cl 5.01 t-a.: (3 --’ -. 2 \
‘, 4 _I :I
Figure
8
Pelvis moments
Mb,
AM bx
0 s --
VIN
0.6 --
Figure
9
Leg moments
MJz, MJx.
J. Biomed. Eng. 1988, Vol. 10, May
259
Evaluation
ofhip joint moments: M. Grigoriadou-Koukis and M. T. Samardcou
Figure 11 Hip joint moment M, comparison ones obtained (--) and those of Paul (-).
Figure
10
Hip joint moments
M,
and M,.
CONCLUSIONS The aim of this project was to evaluate the hip joint by means of an analogue computer moments, model, during walking. Emphasis was given to the choice of control signals and practical aspects concerned with the realization of an accurate double differentiation operator. The results have an error of 15-20%) calculated as follows. The analogue circuit includes a specific part dedicated to measuring the error due to the circuit itself. This specific circuit reconstructs the angles, already used as input to the analogue system, from the values obtained for the joint moments. The comparison of the calculated angles to the measured ones indicates the range of the error produced by the analogue circuit. Also, if we use the calculated hip joint moments M,, M, then we can calculate the hip joint forces with satisfactory results’. Paul’ has studied models for the calculation of hip joint forces and moments using a method of arithmetic differentiation. The comparison of the results of the two studies, using different methods, show that:
260
J. Biomed. Eng. 1988, Vol. IO, May
results of the
were found to have 1. The hip joint moments maximum and minimum values at instants which are near to those found by Paul’. 2. The values of the moments are well situated in the region of the set of curves. 3. The time delay (about 5%) is constant along the whole curve and we therefore conclude that this is due to the detection of the 0% instant. From all the results we can conclude that the method seems to be reliable and effective and that the algorithm of the model could be used for the design of a real time microcomputer-based system, able to act as an adaptive control system to maintain the hip joint balance during walking.
REFERENCES 1. Paul JP. Forces at the human hip joint. PhLI Thesis, Universisity of Strathcbde: 1967. LW. Kinematic measurements in the study of 2. Lamoreux human walking. Bull Prosth Res 1971; 4: 112-4. 3. Bonnemay A, Koukis M, Gouny M, Souza PK. Modsle analogique pour le calcul de forces appliqutes B la t&e de femur. SES/PlJB/SAZ/73-192. Commissariat B l’energie atomique, France, 1972: l-36. 4. Bar A, Ishai G, Meretsky P, Koreu Y. Adaptive microcomputer control of an artificial knee in level walking. J Biomed Eng 1983; 5: 145-50. 5. Grigoriadou-Koukis M. State variable approach to the function of the muscular system supporting the hip joint during normal walking. J Biotned Eng 1987; 9: 88-93. 6. Fisher 0, Braune W. Veber der Schwerpunkt der Menschen Koerpers. Abhd Koenig1 Saechs Gesellsch Wissensch Bd 1890; 26:
561-75.
and M.T. Samarakou
Electronics
Laboratory,
Panepistimioupoli-Ktiria
TYPA,
157 71
Received July 1987, accepted November 1987
ABSTRACT Hip joint moments were determined by analogue computer techniques, u&g as inputs the an$es formed by body segments during normal walking. The control algorithm was based on the division of the walking cycle into six stages; each activated by contact of heel, midfoot and fore-joot. The problem of the accuracy of double dzyerentiation was overcome by means oj two fillers which amp@ the sain at the basicfrequency of the walking cycle, while reducing it above 10 Hz. The analogue model save reasonable results, suggesting that it could be used as an input to a real time microcomputer-based adaptive control system for the maintenance of hip joint balance in an amputee during walking. Keywords:
Hip joint,
analogue
computer,
link equations,
moments
INTRODUCTION We studied human locomotion in order to calculate the hip joint moments during normal walking; we used an analogue computer because of its known advantage for the processing of continuous signals. This is especially important in the case of a biological system, because of the absence of errors due to A/D conversion. The analogue computer can also provide fast results with adequate accuracy. The model is based on Dempster’s hypothesis, that the body can be partitioned in to a number of segments (F&re 1). The mathematical analysis of normal walking is effected by taking into account those segments which execute the most important movements. Any free body has six degrees of freedom. Hence, if we consider each of the twelve segments as a free body, we can describe its motion in space by means of six second-order differential equations. The fact that these segments are not independent of each other is further expressed by equations which form the set of constraints. However, to avoid undue complexity, an equivalent solution exists which consists of adding to the differential equations terms expressing the link between the body segments. we use an appropriate differentiation Further, method, with the aid of an analogue computer, in order to calculate the moments of the segments. The reduction of those moments on the hip joint gives the desired answer to our problem, while the complexity of the problem demands special simplifications and assumptions with respect to those made by other workers’-5.
Correspondence
and reprint
requests
8 1988 Butterworth & Co (Publishers) 0141-5425/88/030253-08 $03.00
to Dr M. Grigoriadou-Koukis.
Figure
1 Dempster’s
human
body partitioning.
Ltd J, Biomed.
Eng.
1988. Vol. 10, May
253
Evaluation of hip joint moments: M. Gri&riadou-Koukis and M. T. Samarakou
MATHEMATICAL
ANALYSIS
Consider the external forces applied to the SC segment, the muscular and frictional forces of the joint (and, in the case of the lower segments, ground reaction), as depicted in Figure 2. The three translation equations for the S, segment, according to are given by the vector d’alembert’s principles, equation:
m,i.i + 7 f, (u,, u,, 8,, 0,) = fl (Fk, t) where m, is the mass of the segment
S,
(1) and
with x,, y,, z,, the linear coordinates and a,, PI, y, the angular coordinates respectively of the S, centre of gravity. The summation C is extended over all the segments adjacent to S, and the result, i.e. the term cf, ( ), expresses the interaction between them through the corresponding joint. The term Z(Fk, t) represents the forces excerted by the appropriate muscles. Also, the three rotation equations of the same segment are:
a+.L,
k, 6,) = 4.1 (Fk, t)
Uj,
I
(2)
where Ze, =
I Zi: Z7,
11
Pa)
I,,, Ioz, IT, being the inertia moments of the S, segments, while the term c& (e,, 0,) is due to the friction between adjaceni segments. The term C&(u,,u,,&,t9,) represents the moments, due to the elasticity of the joint, as well as the Cf, ( ) term of equation (1); the term & (Fk, t) is due to muscular moments. We need 24 vector equations to describe the motion of the 12 segments, and are obliged to apply a number of simplifying assumptions. The first simplification concerns the term Cf, ( ), which may be replaced by a new term f3 (Fk,’ t) if we accept the inelasticity of the joint. Equation (1) is then replaced by the equation: + f3 (FI, t)
m, ii,
= A (Fk, t)
P)
or its equivalent: .. m u,,
+ f+ (Fk, t)
P)
We can also omit the term c&(&, 0,) because the coefficient of friction is very’small, and the term c &(Ul, uI, 8,, e,), if we accept the inelasticity of the
254
A
(14
I
+
Y
J. Biomed. Eng.
1988,
Vol.
10. May
0
Figure 2 Forces applied onto the Si segment a free body.
joint, which becomes: &(Ie,,
is not very
t) 8, = &(Fk,
important.
considered
Equation
t)
The following equation, which represents between segments, must also be included: qe,,
4,
Ut,
Uj)
=
0
as
(2)
(4) the link
(5)
We thus conclude that the three equations (3b), (4) and (5) are sufficient to describe human locomotion during walking.
LINK AND DIFFERENTIATION EQUATIONS OF BODY SEGMENTS We now calculate the link equations between adjacent segments, i.e. equations of the form of equation (5). The required data are the values of the angles formed between segments of the body during walking. The complexity of the problem demands yet further simplifications. We can assume that the arms, head and trunk consist of a compact set, which is true for slow walking, and that the trunk is displaced in translation only without rotation. The last assumption has the result that displacement of the trunk will have the same coordinates, x,y,z, as the pelvis. We shall, nevertheless, take into account the rotation of the pelvis, which plays an important role in determining the exact position of the two femur heads. Let m, be the mass of the upper set. This mass is situated at a distance H from the sacrum and approximately in a vertical line which passes from the midpoint of the distance w between the two femur heads (Figures 3 and 4). This consideration is in agreement with the fact that the total moment P, = Pb + P, must be equal to zero. The link equations are:
Eualuation of hip joinf moments: M. Gri’oriadou-Koukir and M. T. Samarakou
V
sin & = & = tan & and cos 8, = 1 The equations which relate femur head are: xb-xh+,.f
yb
-
zb
-
+
yh
zh
hyb
+
YPb
-
the pelvis to the right
=
+
f-fb
-
h
fj
-
@b
-
hat, + w
0
Ub
2
(7)
=
(8)
0
= 0
where xh,yh,zh are the coordinates of the right femur head, while the relation between right and left femur head is expressed by the equations: .+& - x,, + w&, = 0 yh’ - yh + w@,, = 0 zh
-
zh
+
w
=
(9)
0
where xh’, yh’, zh’ are the coordinates of the left femur head. The translation equations of the upper set are:
(10)
Figure
3
Coordinates
of the upper
while the rotation
body.
equations
are:
Ibe tib = h&,x
T---4!
Ib,
h A
xh
position
Ye ZP -
.?b
-
=
where g, h considered tion, from three axes. 10’; hence
0
-
h, + 1, sin yc = 0
-
zg
-
-
xp
+
yh - ye -
(6)
and w/2 are the distances of the sacrum, as the centre of the trunk-basin articulathe two femur heads with respect to the The rotation of the pelvis is less than + we can assume:
1, cos yc = 0 bh -JR) sin a,
th
-
23
-
(12)
= 0
where L, is the thigh length. . . tion sin a, - a,, xh
xb -g=o Yb -H=O
&
(11)
Mb:
yh - ys -
g”
zh
Figure 4 Dimensions of the pelvis and the relative of the upper body mass.
=
Furthermore, the equations which concern the lower segments (thigh, legs and feet) demand more detailed assumptions due to the fact that the rotations of the segments are large, while their masses and inertia moments are relatively small. We shall consider that the lower limb, during walking, is situated in a plane forming an angle a, with respect to the vertical. By the mean of this angle we can find the z coordinates and neglect the very small angles, which the lower limb forms with x axis (Fz@re 5). The equations which connect the coordinates of the knee, indicated by the index g, to those of the head of femur are:
v .j,
31
‘j/b
Or,
with the assump-
1, sin-y, = 0 1, cos -yc = 0 1,
a,
cos
yc
(13)
=
0
Now, the equations which relate the thigh’s centre of gravity to the coordinates of the hip joint are: XC -
xh
-
C,, 1, sin yc = 0
yc -
yh
+
c2c
1, cos
‘=fc
=
0
c
zh
+
cyc
1,
-fc
=
0
-
cos
J, Biomed.
(14)
Eng.
1988, Vol.
10. May
255
Eualuation of hip joint moments: M. Grigonadou-Koukis and M. T. Samarakou
We consider that the foot’s centre of gravity coincides with the ankle joint. This assumption simplifies the model without greatly affecting the results and is justified by the relatively small dimensions of the foot segment; thus we have: &h
=
x,,
=
$h
yp,
&h
=
zzp
(21)
The link equations for the foot are evaluated in terms of three stages represented by the heel contact, the mid-sole and the fore-foot, activated by logic circuits in the analogue computer, while the translation and the rotation equations of the foot are:
(22)
(xP’ YP) ZP) The
Figure 5
coordinates
(23)
of the lower limb and the angle
a, formed from the lower limb and the vertical.
where C,, is the Fisher and Braune coefficient6 for the thigh. Also, the equations which relate the knee to the ankle, indicated by the subscript ch, are:
In relation to the above, the equations which express the coordinates of the upper set, thigh, leg and foot in terms of those of the right femur head are, respectively, for the upper set: & -
xh =
-
h-n, -
y
Pb (24)
., _h - _%h -
zg -
&h
-
I, cosy, 1, cos
= 0 “/J
=
(15)
where the subscript j is used for the leg, and 1, is the leg length. Finally, the equations which relate the knee to the leg’s centre of gravity. Cg, lj sin ^yj = 0 JJ - Jg + C2j 1, COS ‘ye = 0 Zj - Zg + C2j 1, COS yj = 0 Xj
-
Xg
-
(16)
where C, is the Fisher and Braune6 coefficient for the leg and 1, the leg length. The translation equations of the thigh and the knee are respectively:
m, & = F,, m, jk = F, m, i, = F,
(17)
-
zh
=
,@,,
+
h&,
m, .Cj = Fj, m, jj = FIy m, 2, = F,,
(18)
while the rotation equations respectively:
for the thigh and leg are
L, ‘;vc = M,
w
2
x, _J’J -
=
1,
Jh
=
-
zj
r?h =
-
-
xh
equations
sin yc + CzJ lj sin yJ Ic COS yc C’zJ4 COS “/j C*j lj aj COS “/j - Ic 0, COS
(14)
(25) yc
For the foot: XP
-
xh
=
Yp - yh = ZP
-
t,, =
IJ sin yj - Lj COS -4
+ “lj
lc -
a, cos ?;
sin yc Ic COS yc -
(26)
I, a, cos yc
The value of the Fisher and Braune6 coefficients and Czj are, respectively: C,
= 0.42
C,,
(27)
The position of the upper set is situated, according to Figure 4, at a distance H = 0.335 m from the sacrum, while its projection onto the horizontal plane is situated at the middle of the distance between the two femur heads. Finally, we can work similarly for the left lower limb; its equations are the same as equations (13), (25) and (26), where the angles correspond to those of the left side.
(19)
and
(20)
J. Biomed.
-
For the thigh the above-mentioned apply. For the leg:
Cc = 0.44,
and
256
zc
0
Eng.
1988, Vol.
10, May
EVALUATION
OF MOMENTS
The
of
evaluation
hip
joint
moments
will
be
Evaluation of hip joint moments:M. Gn@riadou-Koukis
Table 1
Data used in the calculations,
Percentage of step
Angles in degrees
CYCk
ab
6b
and M. T. Samarakou
after Lamoreux’ Angles in radians
Yb
Yc
71
YP
0,
4 -
0
0
5
- 0.6
10
- 1.20 - 0.85 -0.75
+ 22.7 + 22.2 +21.5
+ 26.0
+ 32.3
+ 0.0473
- 0.0087
- 2.2
+ 4.8 +5.1 + 4.0
+ 19.5 + 11.0
+ 13.2 + 6.1
+ 0.0490 + 0.0694
15 20 25
- 0.0070 + 0.0122
-3.0 -2.6 - 1.2
+ 3.6 + 2.9 + 1.3
+ 1.35 + 0.90 + 0.75
+ 18.9 + 12.2 + 4.8
+ -
4.4 0.3 3.7
+ + +
5.4 5.0 4.1
+ 0.0775 + 0.1018 + 0.1086
30 35 40
+ 0.0209 + 0.0471 + 0.0558
- 0.2 0 0
-0.5 -1.7 - 2.6
+ 0.30 - 0.25 - 0.80
- 2.8 - 10.0 _ 18.0
- 6.8 - 10.0 - 15.3
+ + -
3.2 1.5 2.3
+0.1118 + 0.1063 + 0.0972
45 50 55 60 65 70 75 80 85 90
+ 0.0593 + 0.0524 + 0.0401
0 0 0.6 2.2 3.0 2.6 1.2 0.2 0 0
- 4.1 -4.8 - 5.1 - 4.0 - 3.6 - 3.9 - 1.3 +0.5 + 1.7 + 2.6
- 1.50 - 1.20 - 0.85 -0.75 + 1.35 + 0.90 + 0.75 + 0.30 - 0.25 - 0.80
- 22.6 - 24.6 - 22.9 _ 16.0 - 4.7 + 6.6 + 51.2 +21.4 + 24.5 + 24.4
- 20.6 -29.1 - 39.9 - 54.0 - 59.2 - 57.4 - 47.8 - 32.6 - 14.5 + 5.9
- 7.1 _ 16.1 - 34.4 _ 16.0 - 75.7 - 64.9 -49.1 - 29.6 - 9.2 + 10.9
+ + + + + + + + + +
+ -
0 0
+ 4.1 + 4.8
_ 1.50 - 1.20
+23.1 + 22.7
+ 24.6 + 26.0
+ 29.9 + 32.3
+ 0.0341 + 0.0473
+ + + + +
95 100
Table
2
Accepted
anthropometric
0.0813 0.0518 0.0208 0.0018 0.0035 0.0018 0.0163 0.0300 0.0422 0.0383
0.0227 0.0052 0.0332 0.0506 0.0558 0.0506 0.0366 0.0244 0.0192 0.0174
- 0.0209 - 0.0087
parameters
M, = -I& Parameters
+ (P, + m,r) y
Accepted values
Thigh length Leg length Pelvic dimensions (see
0.43 m 0.38 m
+ m,&( H + J&$-)
+
w C m#
Figure 4) 0.1698 0.0792
r
m m
+
C m,i,
Cyh -
y,)
(29)
0.024 m
Jr
body Mass of foot Mass of thigh Mass of leg Inertia moment Inertia moment Inertia moment Inertia moment
70 kg 1.18 kg 7.504 kg 3.346 kg 0.00806 kgm 0.04353 kgm 0.1248 kgm 0.0513 kgm
Mass of
of of of of
foot leg thigh pelvis
where w is the distance between the two femur heads. The first three terms correspond to the moment due to the upper set (trunk, pelvis, arms, head), while the two last terms correspond to the right lower limb. For the non-supporting time, the moment M, is given by:
M, = C m;i; (_JJI, -yi) expressed in two planes: the plane xy or flexionextension plane; and the plane y.z or abductionadduction plane. The resulting hip joint moments respectively. The analogue are A4, and M,, computer model is able to calculate the moments in the other joints (pelvis, knee). This calculation will take place every 5% of a full step cycle. During supporting time, i.e. from 10% till 55 % of step cycle, the moment M, is given by the equation:
M, = - C Z, ‘i/i + C mi.?,(jh -yl)
(30)
where the subscript i corresponds to the left lower limb. There are two special instants: the 5% and 55% instants of step cycle, which correspond to double support. For those instants the evaluation of moments will be given by the average of the two corresponding equations. The data that we have used for the above calculations are obtained by Lamoreux’ after several corrections, as we can see in Table I, while in Table 2 there is the accepted set of anthropometric parameters.
I
+
(C
??Zij
I
+
pi)
(Xh
-Xi)
(28)
where i corresponds to the following segments: the upper set (trunk, pelvis, arms, head) and the right lower limb. For the non-supporting time, i.e. from 60% till 100% of step cycle, the equation which describes M, is the same, except that the subscript i corresponds now to the left lower limb. Also, the Mx moment is described for the support time by the equation:
ANALOGUE COMPUTER IMPLEMENTATION AND RESULTS In Figure 15, we can see the analogue computer model, designed to calculate the linear and angular displacement of the body segments during walking. There are six stages: right heel contact, right midsole, right fore-foot, left heel contact, left mid-sole, left fore-foot, activated by the logic circuits PC, P,,,, P,’ and P,‘. The two blocks C1 and C2 represent the
J. Biomed.
Eng. 1988. Vol. 10, May
257
Evaluation of hip joint moments: M. Gri’oriadou-Koukis and M. T. Samarakou
link equations for right and left lower limbs, respectively. Now comes the most difficult part of the model, which consists of the double differentiation of the functions given by the link equations. We have chosen for the double differentiation operator the circuit of Figure 7. This analogue circuit uses two filters with transfer functions:
Hi =
0.12s l+s/32
H2 =
0.17s 1 +s/36
(31)
h
The filters ensure a satisfactory response to the unity step function and a non-oscillatory output with a gain of 40 dB at 1 Hz, which is near the principal frequency of the walking cycle. The role of the filters is also to attenuate the frequencies above 10 Hz. Using an analogue computer system we have simultaneously obtained the moments of various body segments during walking, as presented in Figures 8 and 9; we have also calculated the hip joint moments, depicted in Figure 10. Finally, in Figure II we can see the comparison of the results with a set of curves obtained by Paul’.
Q Q
Figure
6
Figure 7
258
Analogue
computer.
The differentiation
operator
J. Biomed. Eng. 1988, Vol. 10, May
of the analogue
computer
model.
Evaluation
M 4
ofhip joint
momts:
M. Grigoriadou-Koukis and M. T. Samarakou
F-
n.N
3 2 1 0 -f -2 -3
-4
4
PI
JX
Cl 5.01 t-a.: (3 --’ -. 2 \
‘, 4 _I :I
Figure
8
Pelvis moments
Mb,
AM bx
0 s --
VIN
0.6 --
Figure
9
Leg moments
MJz, MJx.
J. Biomed. Eng. 1988, Vol. 10, May
259
Evaluation
ofhip joint moments: M. Grigoriadou-Koukis and M. T. Samardcou
Figure 11 Hip joint moment M, comparison ones obtained (--) and those of Paul (-).
Figure
10
Hip joint moments
M,
and M,.
CONCLUSIONS The aim of this project was to evaluate the hip joint by means of an analogue computer moments, model, during walking. Emphasis was given to the choice of control signals and practical aspects concerned with the realization of an accurate double differentiation operator. The results have an error of 15-20%) calculated as follows. The analogue circuit includes a specific part dedicated to measuring the error due to the circuit itself. This specific circuit reconstructs the angles, already used as input to the analogue system, from the values obtained for the joint moments. The comparison of the calculated angles to the measured ones indicates the range of the error produced by the analogue circuit. Also, if we use the calculated hip joint moments M,, M, then we can calculate the hip joint forces with satisfactory results’. Paul’ has studied models for the calculation of hip joint forces and moments using a method of arithmetic differentiation. The comparison of the results of the two studies, using different methods, show that:
260
J. Biomed. Eng. 1988, Vol. IO, May
results of the
were found to have 1. The hip joint moments maximum and minimum values at instants which are near to those found by Paul’. 2. The values of the moments are well situated in the region of the set of curves. 3. The time delay (about 5%) is constant along the whole curve and we therefore conclude that this is due to the detection of the 0% instant. From all the results we can conclude that the method seems to be reliable and effective and that the algorithm of the model could be used for the design of a real time microcomputer-based system, able to act as an adaptive control system to maintain the hip joint balance during walking.
REFERENCES 1. Paul JP. Forces at the human hip joint. PhLI Thesis, Universisity of Strathcbde: 1967. LW. Kinematic measurements in the study of 2. Lamoreux human walking. Bull Prosth Res 1971; 4: 112-4. 3. Bonnemay A, Koukis M, Gouny M, Souza PK. Modsle analogique pour le calcul de forces appliqutes B la t&e de femur. SES/PlJB/SAZ/73-192. Commissariat B l’energie atomique, France, 1972: l-36. 4. Bar A, Ishai G, Meretsky P, Koreu Y. Adaptive microcomputer control of an artificial knee in level walking. J Biomed Eng 1983; 5: 145-50. 5. Grigoriadou-Koukis M. State variable approach to the function of the muscular system supporting the hip joint during normal walking. J Biotned Eng 1987; 9: 88-93. 6. Fisher 0, Braune W. Veber der Schwerpunkt der Menschen Koerpers. Abhd Koenig1 Saechs Gesellsch Wissensch Bd 1890; 26:
561-75.