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The application of 4 × 4 matrix method to the correction of the measurements of hip joint rotations
1. Biomrchunics.
Vol. 3, pp. 459-47
I.
Pcqamon
Press. 1970.
Printed inGreat
Britain
THE APPLICATION OF 4 x 4 MATRIX METHOD TO THE CORRECTION OF THE MEASUREMENTS OF HIP JOINT ROTATIONS* EDMUND
Y. S. CHAOi. KWAN RIMS. GARY L. SMIDTP RlCHARD C. JOHNSTON]/ University of Iowa, Iowa City. lowa 52140. U.S.A.
and
Abstract-The three rotations of the human hip joint can be approximated by using the electrogoniometric technique. Because the measuring device is not directly located at this point. potential errors exist between the recorded and the actual motions. The objective of this paper is to determine the magnitude and the distribution of these errors. The properly attached goniometer linkage and the femur form a 7-link spatial mechanism. The motion of this spatial mechanism is solved by using the 4 X 3 matrix merhod with three input motions defined from potentiometers attached to the goniometer linkage. From rhe solution of this mechanism, the corrected rotations of the hip-joint are obtained. The maximum error is less than 5 deg at any point in the walking cycle for a normal subject. The largest corrections occur at maximum internal and external rotations, Measurements of hip motion for several activities of daily living are also corrected. and the errors are magnified. compared to walking, since larger angles of rotation are involved. INTRODCCTION
joint is one of the most inaccessible joints in the human body. The joint is composed of the femoral head which is juxtaposed with a concave surface at the lower aspect of the pelvis known as the acetabulum. The joint is surrounded by ligaments, muscle and other soft tissue structures. The femoral head is nearly circular with the greatest deviation from true circularity in the coronal plane. By virtue of its unique sphericity, movement at the hip joint requires minimal shift of the instant centers. Therefore, the hip joint can be closely approximated as a ball-and-socket joint. The three independent rotations of this joint are measured with respect to three orthogonal reference planes as shown in Fig. 1. Because of the inaccessibility of the hip joint, measurement of motion necessitates the THE
HIP
*Receiced
use of external landmarks for photographic methods and external linkage systems for electrogoniometric methods. Since these indirect methods are not applied in the exact location of the hip joint, potential differences exist between actual hip motion and recorded motion. For gait. Levens, Inman and Blosser (1948) procured measurements for hip motion of rotation about the long axis of the femur by taking photographs of landmarks, fixed to bone and projecting lateraily, from the pelvis and the distal aspect of the femur. By computing space co-ordinates for the projecting landmarks and correcting for errors due to parallax, these investigators demonstrated concern for the accuracy of these measurements. The maximum difference of 5-6 deg between the actual angular measurements and the angular measurements obtained directly
9 July 1969.
YInstructor. Department of Mechanics and Hydraulics. SProfessor. Department of Mechanics and Hydraulics. &Assistant Professor, Department of Orthopedics, and Director, Master’s Degree Program for Physical Therapy. College of Medicine. /Assistant Professor. Department of Orthopedics. College of Medicine. 459
460
EDMUND
Y. S. CHAO
Fig. I. Reference planes for hip joint motion. A = Sagittal plane; B = Coronal plane: C = Transverse plane; D = Center of hip joint.
from motion picture frames was not considered significant. Johnston and Smidt (1969). using potentiometers incorporated in an external linkage system, obtained measurements for hip motion in three planes during gait. The recognition that potential error in the method existed because the measuring device was not located within the joint caused this study to be undertaken. The purpose of this study is to investigate the magnitude and distribution of the difference (error) between recorded and actual rotation measures in each of the three planes of hip joint motion for walking and selected activities of daily living.
et al.
way that the rotating axes of the potentiometers for the sagital and coronal pianes are parallel with two of the orthogonal axes intersecting at the center of the femoral head. With the subject standing in standard position, the main links of the resulting spatial mechanism are oriented in a plane initially. The center of the femoral head is located by reference to external body landmarks in a standard fashion which is validated radiologically. Details of this method have been previously reported (Johnston and Smidt, 1969). Since two of the seven links are connected by a spherical joint, this mechanism possesses three degrees of freedom according to Gruebler’s Criterion of Movability (Hartenberg and Denavit. 1964). With any three motions recorded simultaneously from the potentiometers, this spatial mechanism is uniquely determined kinematically. The potentiometers located in the sagittal. transverse planes record coronal, and extension-flexion, abduction-adduction, internal-external rotation. respectively. Appropriate symbols for these motions are defined in Table 1. The differences between the measured and corrected rotations represent an error of the electrogoniometric device. This error was expected to differ in magnitude at different positions throughout the walking cycle. Therefore, evaluation of the error for multiple points in the gait cycle reveals the magnitude and the distribution of the correction involved. Table I. Symbols and signs used for hip joint rotation measuremenrs
METHOD OF OBTAINING MEASLREAIENTS
The goniometric assembly is attached to the pelvis by a molded leather girdle and to the distal thigh by firm elastic straps as illustrated in Fig. 2. If the relative motion between the attachments and bone due to soft tissue motion is neglected, the goniometer linkage and femur constitute a closed 7-link spatial mechanism as seen in Fig. 3. The goniometer is applied to the subject in such a
Hip joint motion
Measured values
Corrected values
Extension Flexion Abduction Adduction internal rotation External rotation
6, positive 0, negative 0, positive & negative
6, positive Bi negative i35positive e3 negative
6, positive
0, positive
tJ1negative
O4negative
Fig. 2. Electrogoniometric apparatus and landmarks. A = Sagittal and transverse plane reference point for center of femoral head; B = Coronal plane reference point for center of femoral head: C = Potentiometer in sagittal plane: D = Potentiometer in coronal place: E = Potentiometer in transverse plane: F = Plesiglas rod: G = Arruchment of Electrogoniometer to pelvis belt: H = .Attachment of Elssrogoniometer apparatus to suprapatel!or cuff.
HIP JOINT
Fig. 3. A schematic of the 8R-IP
ROTATIONS
spatial mechanism with the pair variables illustrated (equivafent jR-IG-IP mechanism).
The kinematic analysis of spatial linkages is usually quite complicated and tedious because of the three-dimensional motions encountered. There are several existing methods available for attacking this problem. However, the method of 4 x 4 matrix is most attractive because it is simple, systematic and advantageous for digital computer application.
Since the method of 4 X4 matrix is not original with the authors, only a brief summary of it will be introduced here. For the details, readers are referred to the original publications (Denavit, Hartenberg, 1955; and Uicker, 1963). This method is based on the concepts of deriving the relative displacement relationships of any lower-pair mechanism from a system of manipulative symbolic notations.
B.M.Vol.3.No.4-F
the original
This relationship is in the form of a symbolic equation in which the terms are equivalent to From the basic principles of matrices. kinematics, the motion of any simple-closed lower-pair linkage must satisfy the following symbolic equation.
asi
fff 1 THE METHOD OF 4 X 4 MATRIX IN SPATIAL LINKAGE ANALYSIS
to
RI
f,
i
ai &
.._Ri ei
_Si_
(1) where Ri, Pj and SLn are the symbolic notations for the types of pairs involved, and ai, ai, 8i and si are four pair variables describing the shape and orientation of the ith link relative to the I’+ fth link. These pair variables are illustrated and defined in Fig. 3. Any closed linkage can be regarded as a
462
EDMUND
Y. S. CHAO
kinematic chain. The symbolic equation assumes that a right-handed Cartesian coordinate system is fixed to each link. Thus, the coordinate transformation equation from one link to the next can be written in terms of a 4 x 4 matrix of the form
et al.
values. If the increment is small in a prescribed motion of the linkage, the values for the previous point can be used as the initial estimates for the next point. The degree of accuracy and the rate of convergence of the iterative scheme depend f
I
cli =
aicos 8i atsin gi Lsi
0
0
cos 8< -sin B~ZOS Lyi sin 8i. COS 8iCOS (Yi 0 sin Cyi
where the transformation pair variables ai, 8i, LYE and Sr are defined in Fig. 4. In a closed kinematic chain, the successive transformation of coordinate systems will lead to an identity transformation because of return to the original system. Therefore, the symbolic equation (1) gives rise to the matrix loop equation G,G,G,.
. . G, = I,
(3)
where I is a unit matrix. The displacement analysis of a given closed linkage with proper specified input motion is performed by solving equation (3). Once the pair variables are defined from the motion of the input links, equation (3) will be modified to calculate the values of the remaining pair variables for that one point of the mechanism operation. The number of specified independent input motions has to match the degrees of freedom of the mechanism. Otherwise, either the linkage does not
0 -
sin Blsin ai 1 cos 8isin CQ cos (Yi !.
I?\ \‘I
on the initial estimates and the increment chosen between adjacent points. THE DERIVATION OF THE MATHEMATICAL IMODEL OF THE 7 LINK SPATIAL hll!XHANIS~M
Based on the general 4 X 4 matrix method introduced above, a specific mathematical model is derived to describe the spatial mechanism resolved from the hip joint motion measuring assembly. This 7-link mechanism is classified as a 5R-lG-I P spatial linkage. The symbols R, G and P denote revolute pair, spherical pair and prismatic pair. respectively. Since the spherical pair. which represents the hip joint, cannot be handled directly by the matrix method, it is decomposed into three equivalent revolute pairs designated as R,, R5 and Rs as shown in Fig. 3. This mechanism now can be regarded as an 8R- 1P linkage. The general symbolic loop equation for this 8R-1 P mechanism is given by
I
Rf~~~-~]R~ -;~&~f’f-j&jR~ =I
possess constrained motion or it may be forced into an unobtainable configuration. Initial estimates for the unknown variables are assumed. Then, an iterative method, similar to the Newton-Raphson method of successive approximation, is applied to continuously correct the original estimated
(4)
In general, there is only one unknown pair variable for each link. In revolute pair (R) the unknown pair variable is 8, and in prismatic pair (P). the unknown variable is s. Other variables wili either be zero or constants throughout the entire cycle of the mechanism. Equation (4) represents a multiplication of
JOINT
HIP
463
ROTATlONS
.th set of right-handed Xi' Yi' oi = The I cartesian coordinate system.
I
the unique cperp@ndiculu betvcen tvo z-axis, ry 2i ad Li+l*
* - Length of
‘i
I - Mstuw
.
I I I
aloag r-ufr b&van
interaectiora pedLhihrs .
I ’
‘L, - Ji
of c-n
pr-
%- Angle batveen the projeetiom of ‘a’ on the ry plane and x-uls.
\
a.
\
I \
o = Angle betveen z-axes around 'a'.
1 '..I
mi f Axes of reation of rwoluta pain OF the axes of rectilinear translation of prismatic pain.
Eaaic Definitfoa
b.
Illmtratioa
Fig. 4. Illustration of the pair variables tli. a,. Biand si.
nine 4 X 4 transformation matrices involving only six unknown variables since three out of the nine pair variables are defined inputs. Certain modifications are required to obtain a unique solution to the system. Recall that, for the purpose of achieving constrained motion, a total of three inputs to the system are required. If f3*,6, and 8, are the inputs, then the first three matrices in (4) are completely defined for each set of inputs. The
remaining unknown pair variables can be expressed as the sum of an initial estimate and an error term. The symbolic equation can be written as A1A2(8,+de2)A3(iJ3+de3)A~(~*+de-l). Ag(Bj+dej)As(eg+des)A,(S;+dr7) where
= I
(5)
464
EDMUND
Y. S. CHAO
et al. 4
ai ai
A, = R,
R2
Ai = Ri+2
I _
i=2,3,...,6
A, = P,
8i + dBi
4 ST+ ds,
&
8, and & are the estimated values for the unknown pair variables and de1 and ds, are the error terms. If the increments are small for the input angles, 01, 0, and &,, the higher order terms of the corresponding error terms d& (i=2,3,... ,6) and dr, may be neglected. Thus, equation (5) can be reduced to
0 0 0 1 1 &21 &2z B,ZX&, 1
= I - A,A,A,...&.
(9) 0
0
Bi21
Bi22Bi23 Bt24
Bi31
(6) In this matrix equation, Ai denote the transformation matrices based on the estimated pair variables, and Qe and Q, are two matrices of the form
I Ii00 00 00-10
QL9= 01 00
00 00
0000 0000
Qs=
0000 1 ooo_.
(‘)
Solving (6) and utilizing the translational and rotational characteristics of the transformation matrices, nine simultaneous equations for 6 unknowns, d&, dt$, . . . ,dt& and dsT, are obtained. In matric form this equation can be written as MD=V (8) where
I
- -
0
0 doi = AlA:,
+ (A,A,&...&Q,&)ds,
I a7
Bi32
Bi33
Bi34
m. e Ai_,QeA;
. . . A;
(10)
_BM, Bi.rz Bid3 Bi+t _
when i = 7, replace QOby Q,! In order to obtain an approximated solution of this overdetermined system of equations, the method of least square is used. With this method, a system of six equations can be resolved to yield the approximated solutions. These six equations are written in matrix form as D = N-‘W. (11) where A = M’M W = M’V. COMPUTATIONAL
ALGORITHM
The mathematical model of the previously described 7-link spatial mechanism is programmed on an IBM 360 model 65 computer for approximated solution. This mathematical model and the resulting computer program apply only to the linkage described herein.
M=
The components in the M matrix are defined from the following equations.
However, the general technique of the 4 X 4 matrix method is not limited to any particular
HIP JOiNT
type of spatial linkage. For the convenience of debugging and following the computational logic. eleven subroutines are used to construct the main skeleton of the program. A generalized algorithm describing the computational sequence of the program follows: Step 1. Read in the preliminary input information for the system and set all the initial parameters and indices for the transformation matrices. Step 2. Define the seven A matrices by combining and interchanging indices of the original nine transformation matrices. Step 3. Calculate the elements of the B matrices defined in equations (9) and (10) so that all the matrix components are known. Step 4. Calculate the matrices M and V and define the column matrix D of the error terms. Step 5. Obtain the square matrix N from multiplying the transpose of M by &I. Calculate the W matrix which is the product of I%+ and V. Step 6. Invert matrix N and multiply it by W. The resultant column matrix is the solution of the unknown error terms. Srep 7. Add the error terms obtained from step 6 to the corresponding pair variables to obtain the corrected new estimates. Step 8. Test whether the error terms are less than the specified tolerances. If they are, then initial estimated values for the unknown pair variables have converged to the solutions.
10 1
fo1
Pl
Pl
165
ROTATIONS
Step 10. After all the pair variables of the mechanism are determined for each point of the moving cycle, a table and three digital plots are prepared to illustrate the magnitude and the distribution of the corrections involved. The rate of convergence of this scheme is quite satisfactory. The rotational increment of the input variables reaches 10 deg at some points, yet the number of iterations is kept below six. The details of this program can be found in a previous publication (Rim and Chao, 1969). RESULTS
The goniometer is installed initially in such a way that the center lines (same as the z axis) of links 1, 3, 4, 6, 7 and 9 are in a plane. Consequently, the origins 4, OS,0; and 0, form a trapezoid as shown in Fig. 5. The pair variables sI, sg, sg and S? are constants depending on the physical dimensions of the subject. The rest of the pair variables are either constants or can be easily estimated from the input information. A complete set of results for walking for one of the 26 normal subjects tested is illustrated in this section. The initial dimensions of this subject are indicated at the bottom of Fig. 5. With these dimensions given, the symbolic equation can be simplified as
10
After printing out the solutions for this point, renew the input for the new point and go back to step 2 with the present solutions as the initial estimates. Step 9. If the error terms are larger than the specified tolerances, use the present values of the pair variables for the initial estimates and go to step 2 with matrix A, unchanged.
where &, 8? and f13 denote the input pair variables. In order to start the iterative scheme, the rest of the unknown variables have to be approximated corresponding to the input set for the beginning point. A total of 34 discrete points are selected for one complete walking cycle. The comparison of the measured and the corrected hip
446
EDMUND
Y. S. CHAO et al.
2
'1
=a
T”
-T
‘3
‘7
1,
Hip Joint 1 ‘6
O7
=L
‘6 ‘I
l
10~00 Q
- 19.00 Q
.3 - c2.00 Q ‘6 s 7
l
a9 = 32'10 a 82 - 87.57 de6 e8 - 87.57 de6
16.00 a
Fig. 5. A plane view of the initial configuration of the 8R- IP mechanism.
joint rotations for the 34 points are shown in Figs. 6.7 and 8, respectively. The abrupt fluctuations observed in the results are probably due to the soft tissue motion at the hip and thigh during gait. This motion affects the rigidity of the imaginary links of the spatial mechanism. As a result, small additional motions are introduced into the linkage. Since these motions are small, they may be disregarded in the process of obtaining hip joint motion corrections. Examining the results for walking, for all 26 subjects, the magnitude of correction for abduction and adduction (6,) as well as internal and external rotations (8,) are quite small. The errors related to extension and flexion (0,) are relatively large but the magnitude has never exceeded 5 deg. The maximum correction occurring in a walking cycle follows a general trend. The largest error of correction for extension and flexion coincides
with large values of internal and external rotations. The same corrections for several activities of daily living of a normal subject were also investigated. The results of these observations are summarized in Table 2. In each activity examined, only extreme values of rotations are illustrated. The magnitude of correction in this case is magnified since larger angles of rotation are involved, but exceeded eight degrees only for tying the shoe with the foot on the opposite thigh. The general pattern of larger errors matching with larger internal and external rotations still holds. The rate of convergence for the iterative scheme is quite satisfactory even with sizable errors in the initial estimates. This characteristic has greatly increased the merit of this program to evaluate the points of interest in any activity regardless of the magnitude of the increment between any of these points.
HIP JOINT ROTATIONS
467
468
EDMUND
Y. S. CHAO et al.
HIP
JOIXT
ROT.ATlOSS
469
J
:3 L
B.M.VoI.3.No.4-G
470
EDMUND Table 2. Comparisons Sample point No.
Activities
Y. S. CHAO
er al.
between the measured and corrected hip joint motions for activities of daily living
01
0,
Error
&
e5
Error
&
-0.16 0.58 - 1.30 l-45
-58 -100 -80 -60
66.02 - 104+X
- 93-45 -64.16
8.02 4.84 13.45 4-16
ga
Error
Tying shoe across knee
1 2 3 4
-20 - 12 -33 - 10
- 17.39 - IO.41 - 27.76 - 8.35
-260 - 1.58 - 5.23 - 164
20 20
2.5 25
20.16 19.41 26.30 23.54
Squatting
1 2 3
- 10 20 - 10
-899 16.74 -9.31
- l*oO 3.25 - 0.68
15 25 10
14.84 23.38 10.20
O-15 0.61 - 0.20
-100 - 120 -80
- 103.94 -111.72 - 83.87
3.94 - 8.27 3.87
Sitting
1 2 3
-20 -20 -20
- 17.39 - 17.29 - 17.39
- 2.60 - 260 - 2.60
20 20 20
20.16 20.16 20.16
-0.16 -0.16 -0.16
-% -76 -104
- 104.02 - 84.02 - 112.02
8.02 8.02 8.02
Tying shoe on floor
1 2 3 4
- 10 -20 -12 - 15
-9.31 - 18.03 - 10.41 - 13.99
-
10 15 20 10
10.20 1574 19.41 10.59
-0.20 -0.74 0.58 - 0.59
-60 -90 - 126 -60
-63.87 - 97.82 - 140.84 - 65.78
3.87 7.82 4.84 5.78
Stooping
1 2
-20 -20
- 17.00 - 1999
-3.00 0.00
23 0
22.71 1.32
0.28 - 1.32
-120 -30
- 128.16 - 37.52
8.16 7.52
3
10 8 3
964 7.93 299
0.35 0.07 o-01
5 1 0
5.37 1.23 o-02
- 0.37 -0.23 -0.02
10 -57 .5
13.82 - 53.94 6.14
- 3.83 - 3.05 - 1.14
1 2
5 0
4.82 0.00
O-18 0.00
5 6
5.12 6.03
-0.12 - 0.03
-30 -27
- 28.08 - 27.00
- 1.92 0.00
Going Upstairs Going downstairs
1 2
0.68 I.96 1.58 l+?O
SUMMARY AND DISCUSSION
The exact hip joint rotations can be determined from electrogoniometric measurements with additional corrections. The need for these corrections is due to the relative motions between links of the equivalent spatial mechanism described herein. The 4 X 4 matrix method is effective in obtaining the magnitude and distribution of these corrections. The calculated results of the corrections reveal that the goniometric technique without correction is adequate for approximating the hip joint motion for gait. However, when other activities of daily living are examined, the magnitude of error is greater and may be significant. Dynamic information of the thigh with respect to the pelvis is important in studying human gait. The present analysis can be
extended to include the velocity and acceleration analysis for the femur if the input velocities and accelerations are known. After the kinematic analysis of the femur is obtained, the forces and moments occurring at the hip joint can be calculated with the knowledge of the mass and the moment of inertia of the thigh. Acknowledgements-The
authors wish to thank Dr. G. Jayaraman, a former research assistant in the Department of Mechanics and Hydraulics, University of Iowa, for his initial programming work done in the matrix technique used here. This investigation was also partially sponsored by the U.S. Army Weapons Command, Rock Island. Illinois. REFERENCES
Denavit, J. and Hartenberg, R. S. (1955) A kinematic notation for lower pair mechanisms based on matrices. .I. appl. Mech. 22; Trans. ASME 77.2 13-22 1. Hartenberg. R. S. and Denavit, J. (1964) Kinematic Synthesis of Linkages. pp. 347-355. McGraw-Hill, New York.
HIP JOINT Johnston, Richard C. and Smidt. Gary L. ( 1969) Measurement of hip joint motion during walking: evaluation of an electrogoniometric method. J. Bone Jt Surg. 51-A. 1083- 1094. Levens. A. S., Inman, V. T. and Blosser. J. A. (1948) Transverse rotation of the segments of the lower extremity in locomotion. J. Bone Jr Surg. 30-A. 859-872. Rim. K. Chao. E. Y. S. (1969) A feasibility study of mathematical modeling of musculo-skeletal systems for
ROTATIONS
37 1
bio-mechanics applications. Final report submitted to U.S. Army Weapons Command. Rock Island. Illinois under Contract No. DAAFOI-6%Xl-B71 I-POOI, Department of Mechanics and Hydraulics. Univeristy of Iowa. Iowa City. Iowa. March. Uicker. John J.. Jr. (1963) Displacement analysis of spatial mechanisms by an iterative method based on 4 x 4 matrices. MS. Thesis. Department of Mechanical Engineering, Northwestern University. Evanston. Illinois. June.
Vol. 3, pp. 459-47
I.
Pcqamon
Press. 1970.
Printed inGreat
Britain
THE APPLICATION OF 4 x 4 MATRIX METHOD TO THE CORRECTION OF THE MEASUREMENTS OF HIP JOINT ROTATIONS* EDMUND
Y. S. CHAOi. KWAN RIMS. GARY L. SMIDTP RlCHARD C. JOHNSTON]/ University of Iowa, Iowa City. lowa 52140. U.S.A.
and
Abstract-The three rotations of the human hip joint can be approximated by using the electrogoniometric technique. Because the measuring device is not directly located at this point. potential errors exist between the recorded and the actual motions. The objective of this paper is to determine the magnitude and the distribution of these errors. The properly attached goniometer linkage and the femur form a 7-link spatial mechanism. The motion of this spatial mechanism is solved by using the 4 X 3 matrix merhod with three input motions defined from potentiometers attached to the goniometer linkage. From rhe solution of this mechanism, the corrected rotations of the hip-joint are obtained. The maximum error is less than 5 deg at any point in the walking cycle for a normal subject. The largest corrections occur at maximum internal and external rotations, Measurements of hip motion for several activities of daily living are also corrected. and the errors are magnified. compared to walking, since larger angles of rotation are involved. INTRODCCTION
joint is one of the most inaccessible joints in the human body. The joint is composed of the femoral head which is juxtaposed with a concave surface at the lower aspect of the pelvis known as the acetabulum. The joint is surrounded by ligaments, muscle and other soft tissue structures. The femoral head is nearly circular with the greatest deviation from true circularity in the coronal plane. By virtue of its unique sphericity, movement at the hip joint requires minimal shift of the instant centers. Therefore, the hip joint can be closely approximated as a ball-and-socket joint. The three independent rotations of this joint are measured with respect to three orthogonal reference planes as shown in Fig. 1. Because of the inaccessibility of the hip joint, measurement of motion necessitates the THE
HIP
*Receiced
use of external landmarks for photographic methods and external linkage systems for electrogoniometric methods. Since these indirect methods are not applied in the exact location of the hip joint, potential differences exist between actual hip motion and recorded motion. For gait. Levens, Inman and Blosser (1948) procured measurements for hip motion of rotation about the long axis of the femur by taking photographs of landmarks, fixed to bone and projecting lateraily, from the pelvis and the distal aspect of the femur. By computing space co-ordinates for the projecting landmarks and correcting for errors due to parallax, these investigators demonstrated concern for the accuracy of these measurements. The maximum difference of 5-6 deg between the actual angular measurements and the angular measurements obtained directly
9 July 1969.
YInstructor. Department of Mechanics and Hydraulics. SProfessor. Department of Mechanics and Hydraulics. &Assistant Professor, Department of Orthopedics, and Director, Master’s Degree Program for Physical Therapy. College of Medicine. /Assistant Professor. Department of Orthopedics. College of Medicine. 459
460
EDMUND
Y. S. CHAO
Fig. I. Reference planes for hip joint motion. A = Sagittal plane; B = Coronal plane: C = Transverse plane; D = Center of hip joint.
from motion picture frames was not considered significant. Johnston and Smidt (1969). using potentiometers incorporated in an external linkage system, obtained measurements for hip motion in three planes during gait. The recognition that potential error in the method existed because the measuring device was not located within the joint caused this study to be undertaken. The purpose of this study is to investigate the magnitude and distribution of the difference (error) between recorded and actual rotation measures in each of the three planes of hip joint motion for walking and selected activities of daily living.
et al.
way that the rotating axes of the potentiometers for the sagital and coronal pianes are parallel with two of the orthogonal axes intersecting at the center of the femoral head. With the subject standing in standard position, the main links of the resulting spatial mechanism are oriented in a plane initially. The center of the femoral head is located by reference to external body landmarks in a standard fashion which is validated radiologically. Details of this method have been previously reported (Johnston and Smidt, 1969). Since two of the seven links are connected by a spherical joint, this mechanism possesses three degrees of freedom according to Gruebler’s Criterion of Movability (Hartenberg and Denavit. 1964). With any three motions recorded simultaneously from the potentiometers, this spatial mechanism is uniquely determined kinematically. The potentiometers located in the sagittal. transverse planes record coronal, and extension-flexion, abduction-adduction, internal-external rotation. respectively. Appropriate symbols for these motions are defined in Table 1. The differences between the measured and corrected rotations represent an error of the electrogoniometric device. This error was expected to differ in magnitude at different positions throughout the walking cycle. Therefore, evaluation of the error for multiple points in the gait cycle reveals the magnitude and the distribution of the correction involved. Table I. Symbols and signs used for hip joint rotation measuremenrs
METHOD OF OBTAINING MEASLREAIENTS
The goniometric assembly is attached to the pelvis by a molded leather girdle and to the distal thigh by firm elastic straps as illustrated in Fig. 2. If the relative motion between the attachments and bone due to soft tissue motion is neglected, the goniometer linkage and femur constitute a closed 7-link spatial mechanism as seen in Fig. 3. The goniometer is applied to the subject in such a
Hip joint motion
Measured values
Corrected values
Extension Flexion Abduction Adduction internal rotation External rotation
6, positive 0, negative 0, positive & negative
6, positive Bi negative i35positive e3 negative
6, positive
0, positive
tJ1negative
O4negative
Fig. 2. Electrogoniometric apparatus and landmarks. A = Sagittal and transverse plane reference point for center of femoral head; B = Coronal plane reference point for center of femoral head: C = Potentiometer in sagittal plane: D = Potentiometer in coronal place: E = Potentiometer in transverse plane: F = Plesiglas rod: G = Arruchment of Electrogoniometer to pelvis belt: H = .Attachment of Elssrogoniometer apparatus to suprapatel!or cuff.
HIP JOINT
Fig. 3. A schematic of the 8R-IP
ROTATIONS
spatial mechanism with the pair variables illustrated (equivafent jR-IG-IP mechanism).
The kinematic analysis of spatial linkages is usually quite complicated and tedious because of the three-dimensional motions encountered. There are several existing methods available for attacking this problem. However, the method of 4 x 4 matrix is most attractive because it is simple, systematic and advantageous for digital computer application.
Since the method of 4 X4 matrix is not original with the authors, only a brief summary of it will be introduced here. For the details, readers are referred to the original publications (Denavit, Hartenberg, 1955; and Uicker, 1963). This method is based on the concepts of deriving the relative displacement relationships of any lower-pair mechanism from a system of manipulative symbolic notations.
B.M.Vol.3.No.4-F
the original
This relationship is in the form of a symbolic equation in which the terms are equivalent to From the basic principles of matrices. kinematics, the motion of any simple-closed lower-pair linkage must satisfy the following symbolic equation.
asi
fff 1 THE METHOD OF 4 X 4 MATRIX IN SPATIAL LINKAGE ANALYSIS
to
RI
f,
i
ai &
.._Ri ei
_Si_
(1) where Ri, Pj and SLn are the symbolic notations for the types of pairs involved, and ai, ai, 8i and si are four pair variables describing the shape and orientation of the ith link relative to the I’+ fth link. These pair variables are illustrated and defined in Fig. 3. Any closed linkage can be regarded as a
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kinematic chain. The symbolic equation assumes that a right-handed Cartesian coordinate system is fixed to each link. Thus, the coordinate transformation equation from one link to the next can be written in terms of a 4 x 4 matrix of the form
et al.
values. If the increment is small in a prescribed motion of the linkage, the values for the previous point can be used as the initial estimates for the next point. The degree of accuracy and the rate of convergence of the iterative scheme depend f
I
cli =
aicos 8i atsin gi Lsi
0
0
cos 8< -sin B~ZOS Lyi sin 8i. COS 8iCOS (Yi 0 sin Cyi
where the transformation pair variables ai, 8i, LYE and Sr are defined in Fig. 4. In a closed kinematic chain, the successive transformation of coordinate systems will lead to an identity transformation because of return to the original system. Therefore, the symbolic equation (1) gives rise to the matrix loop equation G,G,G,.
. . G, = I,
(3)
where I is a unit matrix. The displacement analysis of a given closed linkage with proper specified input motion is performed by solving equation (3). Once the pair variables are defined from the motion of the input links, equation (3) will be modified to calculate the values of the remaining pair variables for that one point of the mechanism operation. The number of specified independent input motions has to match the degrees of freedom of the mechanism. Otherwise, either the linkage does not
0 -
sin Blsin ai 1 cos 8isin CQ cos (Yi !.
I?\ \‘I
on the initial estimates and the increment chosen between adjacent points. THE DERIVATION OF THE MATHEMATICAL IMODEL OF THE 7 LINK SPATIAL hll!XHANIS~M
Based on the general 4 X 4 matrix method introduced above, a specific mathematical model is derived to describe the spatial mechanism resolved from the hip joint motion measuring assembly. This 7-link mechanism is classified as a 5R-lG-I P spatial linkage. The symbols R, G and P denote revolute pair, spherical pair and prismatic pair. respectively. Since the spherical pair. which represents the hip joint, cannot be handled directly by the matrix method, it is decomposed into three equivalent revolute pairs designated as R,, R5 and Rs as shown in Fig. 3. This mechanism now can be regarded as an 8R- 1P linkage. The general symbolic loop equation for this 8R-1 P mechanism is given by
I
Rf~~~-~]R~ -;~&~f’f-j&jR~ =I
possess constrained motion or it may be forced into an unobtainable configuration. Initial estimates for the unknown variables are assumed. Then, an iterative method, similar to the Newton-Raphson method of successive approximation, is applied to continuously correct the original estimated
(4)
In general, there is only one unknown pair variable for each link. In revolute pair (R) the unknown pair variable is 8, and in prismatic pair (P). the unknown variable is s. Other variables wili either be zero or constants throughout the entire cycle of the mechanism. Equation (4) represents a multiplication of
JOINT
HIP
463
ROTATlONS
.th set of right-handed Xi' Yi' oi = The I cartesian coordinate system.
I
the unique cperp@ndiculu betvcen tvo z-axis, ry 2i ad Li+l*
* - Length of
‘i
I - Mstuw
.
I I I
aloag r-ufr b&van
interaectiora pedLhihrs .
I ’
‘L, - Ji
of c-n
pr-
%- Angle batveen the projeetiom of ‘a’ on the ry plane and x-uls.
\
a.
\
I \
o = Angle betveen z-axes around 'a'.
1 '..I
mi f Axes of reation of rwoluta pain OF the axes of rectilinear translation of prismatic pain.
Eaaic Definitfoa
b.
Illmtratioa
Fig. 4. Illustration of the pair variables tli. a,. Biand si.
nine 4 X 4 transformation matrices involving only six unknown variables since three out of the nine pair variables are defined inputs. Certain modifications are required to obtain a unique solution to the system. Recall that, for the purpose of achieving constrained motion, a total of three inputs to the system are required. If f3*,6, and 8, are the inputs, then the first three matrices in (4) are completely defined for each set of inputs. The
remaining unknown pair variables can be expressed as the sum of an initial estimate and an error term. The symbolic equation can be written as A1A2(8,+de2)A3(iJ3+de3)A~(~*+de-l). Ag(Bj+dej)As(eg+des)A,(S;+dr7) where
= I
(5)
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et al. 4
ai ai
A, = R,
R2
Ai = Ri+2
I _
i=2,3,...,6
A, = P,
8i + dBi
4 ST+ ds,
&
8, and & are the estimated values for the unknown pair variables and de1 and ds, are the error terms. If the increments are small for the input angles, 01, 0, and &,, the higher order terms of the corresponding error terms d& (i=2,3,... ,6) and dr, may be neglected. Thus, equation (5) can be reduced to
0 0 0 1 1 &21 &2z B,ZX&, 1
= I - A,A,A,...&.
(9) 0
0
Bi21
Bi22Bi23 Bt24
Bi31
(6) In this matrix equation, Ai denote the transformation matrices based on the estimated pair variables, and Qe and Q, are two matrices of the form
I Ii00 00 00-10
QL9= 01 00
00 00
0000 0000
Qs=
0000 1 ooo_.
(‘)
Solving (6) and utilizing the translational and rotational characteristics of the transformation matrices, nine simultaneous equations for 6 unknowns, d&, dt$, . . . ,dt& and dsT, are obtained. In matric form this equation can be written as MD=V (8) where
I
- -
0
0 doi = AlA:,
+ (A,A,&...&Q,&)ds,
I a7
Bi32
Bi33
Bi34
m. e Ai_,QeA;
. . . A;
(10)
_BM, Bi.rz Bid3 Bi+t _
when i = 7, replace QOby Q,! In order to obtain an approximated solution of this overdetermined system of equations, the method of least square is used. With this method, a system of six equations can be resolved to yield the approximated solutions. These six equations are written in matrix form as D = N-‘W. (11) where A = M’M W = M’V. COMPUTATIONAL
ALGORITHM
The mathematical model of the previously described 7-link spatial mechanism is programmed on an IBM 360 model 65 computer for approximated solution. This mathematical model and the resulting computer program apply only to the linkage described herein.
M=
The components in the M matrix are defined from the following equations.
However, the general technique of the 4 X 4 matrix method is not limited to any particular
HIP JOiNT
type of spatial linkage. For the convenience of debugging and following the computational logic. eleven subroutines are used to construct the main skeleton of the program. A generalized algorithm describing the computational sequence of the program follows: Step 1. Read in the preliminary input information for the system and set all the initial parameters and indices for the transformation matrices. Step 2. Define the seven A matrices by combining and interchanging indices of the original nine transformation matrices. Step 3. Calculate the elements of the B matrices defined in equations (9) and (10) so that all the matrix components are known. Step 4. Calculate the matrices M and V and define the column matrix D of the error terms. Step 5. Obtain the square matrix N from multiplying the transpose of M by &I. Calculate the W matrix which is the product of I%+ and V. Step 6. Invert matrix N and multiply it by W. The resultant column matrix is the solution of the unknown error terms. Srep 7. Add the error terms obtained from step 6 to the corresponding pair variables to obtain the corrected new estimates. Step 8. Test whether the error terms are less than the specified tolerances. If they are, then initial estimated values for the unknown pair variables have converged to the solutions.
10 1
fo1
Pl
Pl
165
ROTATIONS
Step 10. After all the pair variables of the mechanism are determined for each point of the moving cycle, a table and three digital plots are prepared to illustrate the magnitude and the distribution of the corrections involved. The rate of convergence of this scheme is quite satisfactory. The rotational increment of the input variables reaches 10 deg at some points, yet the number of iterations is kept below six. The details of this program can be found in a previous publication (Rim and Chao, 1969). RESULTS
The goniometer is installed initially in such a way that the center lines (same as the z axis) of links 1, 3, 4, 6, 7 and 9 are in a plane. Consequently, the origins 4, OS,0; and 0, form a trapezoid as shown in Fig. 5. The pair variables sI, sg, sg and S? are constants depending on the physical dimensions of the subject. The rest of the pair variables are either constants or can be easily estimated from the input information. A complete set of results for walking for one of the 26 normal subjects tested is illustrated in this section. The initial dimensions of this subject are indicated at the bottom of Fig. 5. With these dimensions given, the symbolic equation can be simplified as
10
After printing out the solutions for this point, renew the input for the new point and go back to step 2 with the present solutions as the initial estimates. Step 9. If the error terms are larger than the specified tolerances, use the present values of the pair variables for the initial estimates and go to step 2 with matrix A, unchanged.
where &, 8? and f13 denote the input pair variables. In order to start the iterative scheme, the rest of the unknown variables have to be approximated corresponding to the input set for the beginning point. A total of 34 discrete points are selected for one complete walking cycle. The comparison of the measured and the corrected hip
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2
'1
=a
T”
-T
‘3
‘7
1,
Hip Joint 1 ‘6
O7
=L
‘6 ‘I
l
10~00 Q
- 19.00 Q
.3 - c2.00 Q ‘6 s 7
l
a9 = 32'10 a 82 - 87.57 de6 e8 - 87.57 de6
16.00 a
Fig. 5. A plane view of the initial configuration of the 8R- IP mechanism.
joint rotations for the 34 points are shown in Figs. 6.7 and 8, respectively. The abrupt fluctuations observed in the results are probably due to the soft tissue motion at the hip and thigh during gait. This motion affects the rigidity of the imaginary links of the spatial mechanism. As a result, small additional motions are introduced into the linkage. Since these motions are small, they may be disregarded in the process of obtaining hip joint motion corrections. Examining the results for walking, for all 26 subjects, the magnitude of correction for abduction and adduction (6,) as well as internal and external rotations (8,) are quite small. The errors related to extension and flexion (0,) are relatively large but the magnitude has never exceeded 5 deg. The maximum correction occurring in a walking cycle follows a general trend. The largest error of correction for extension and flexion coincides
with large values of internal and external rotations. The same corrections for several activities of daily living of a normal subject were also investigated. The results of these observations are summarized in Table 2. In each activity examined, only extreme values of rotations are illustrated. The magnitude of correction in this case is magnified since larger angles of rotation are involved, but exceeded eight degrees only for tying the shoe with the foot on the opposite thigh. The general pattern of larger errors matching with larger internal and external rotations still holds. The rate of convergence for the iterative scheme is quite satisfactory even with sizable errors in the initial estimates. This characteristic has greatly increased the merit of this program to evaluate the points of interest in any activity regardless of the magnitude of the increment between any of these points.
HIP JOINT ROTATIONS
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HIP
JOIXT
ROT.ATlOSS
469
J
:3 L
B.M.VoI.3.No.4-G
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Activities
Y. S. CHAO
er al.
between the measured and corrected hip joint motions for activities of daily living
01
0,
Error
&
e5
Error
&
-0.16 0.58 - 1.30 l-45
-58 -100 -80 -60
66.02 - 104+X
- 93-45 -64.16
8.02 4.84 13.45 4-16
ga
Error
Tying shoe across knee
1 2 3 4
-20 - 12 -33 - 10
- 17.39 - IO.41 - 27.76 - 8.35
-260 - 1.58 - 5.23 - 164
20 20
2.5 25
20.16 19.41 26.30 23.54
Squatting
1 2 3
- 10 20 - 10
-899 16.74 -9.31
- l*oO 3.25 - 0.68
15 25 10
14.84 23.38 10.20
O-15 0.61 - 0.20
-100 - 120 -80
- 103.94 -111.72 - 83.87
3.94 - 8.27 3.87
Sitting
1 2 3
-20 -20 -20
- 17.39 - 17.29 - 17.39
- 2.60 - 260 - 2.60
20 20 20
20.16 20.16 20.16
-0.16 -0.16 -0.16
-% -76 -104
- 104.02 - 84.02 - 112.02
8.02 8.02 8.02
Tying shoe on floor
1 2 3 4
- 10 -20 -12 - 15
-9.31 - 18.03 - 10.41 - 13.99
-
10 15 20 10
10.20 1574 19.41 10.59
-0.20 -0.74 0.58 - 0.59
-60 -90 - 126 -60
-63.87 - 97.82 - 140.84 - 65.78
3.87 7.82 4.84 5.78
Stooping
1 2
-20 -20
- 17.00 - 1999
-3.00 0.00
23 0
22.71 1.32
0.28 - 1.32
-120 -30
- 128.16 - 37.52
8.16 7.52
3
10 8 3
964 7.93 299
0.35 0.07 o-01
5 1 0
5.37 1.23 o-02
- 0.37 -0.23 -0.02
10 -57 .5
13.82 - 53.94 6.14
- 3.83 - 3.05 - 1.14
1 2
5 0
4.82 0.00
O-18 0.00
5 6
5.12 6.03
-0.12 - 0.03
-30 -27
- 28.08 - 27.00
- 1.92 0.00
Going Upstairs Going downstairs
1 2
0.68 I.96 1.58 l+?O
SUMMARY AND DISCUSSION
The exact hip joint rotations can be determined from electrogoniometric measurements with additional corrections. The need for these corrections is due to the relative motions between links of the equivalent spatial mechanism described herein. The 4 X 4 matrix method is effective in obtaining the magnitude and distribution of these corrections. The calculated results of the corrections reveal that the goniometric technique without correction is adequate for approximating the hip joint motion for gait. However, when other activities of daily living are examined, the magnitude of error is greater and may be significant. Dynamic information of the thigh with respect to the pelvis is important in studying human gait. The present analysis can be
extended to include the velocity and acceleration analysis for the femur if the input velocities and accelerations are known. After the kinematic analysis of the femur is obtained, the forces and moments occurring at the hip joint can be calculated with the knowledge of the mass and the moment of inertia of the thigh. Acknowledgements-The
authors wish to thank Dr. G. Jayaraman, a former research assistant in the Department of Mechanics and Hydraulics, University of Iowa, for his initial programming work done in the matrix technique used here. This investigation was also partially sponsored by the U.S. Army Weapons Command, Rock Island. Illinois. REFERENCES
Denavit, J. and Hartenberg, R. S. (1955) A kinematic notation for lower pair mechanisms based on matrices. .I. appl. Mech. 22; Trans. ASME 77.2 13-22 1. Hartenberg. R. S. and Denavit, J. (1964) Kinematic Synthesis of Linkages. pp. 347-355. McGraw-Hill, New York.
HIP JOINT Johnston, Richard C. and Smidt. Gary L. ( 1969) Measurement of hip joint motion during walking: evaluation of an electrogoniometric method. J. Bone Jt Surg. 51-A. 1083- 1094. Levens. A. S., Inman, V. T. and Blosser. J. A. (1948) Transverse rotation of the segments of the lower extremity in locomotion. J. Bone Jr Surg. 30-A. 859-872. Rim. K. Chao. E. Y. S. (1969) A feasibility study of mathematical modeling of musculo-skeletal systems for
ROTATIONS
37 1
bio-mechanics applications. Final report submitted to U.S. Army Weapons Command. Rock Island. Illinois under Contract No. DAAFOI-6%Xl-B71 I-POOI, Department of Mechanics and Hydraulics. Univeristy of Iowa. Iowa City. Iowa. March. Uicker. John J.. Jr. (1963) Displacement analysis of spatial mechanisms by an iterative method based on 4 x 4 matrices. MS. Thesis. Department of Mechanical Engineering, Northwestern University. Evanston. Illinois. June.