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A dynamic model for finger interphalangeal coordination
J. Biomecknucs VOI 21. NO. 6. pp. 459468. Printed in Great Bntain
uO21-9290188 13.00 + .OO Perganm Press plc
1988
A DYNAMIC MODEL FOR FINGER INTERPHALANGEAL COORDINATION HELMUT J. BUCHNER* MARGARETJ. HINEst and HOOSHANGHEMAMI: *Siemens A.G. Forschungs Laboratorien, Munchen, Germany; t Department of Anatomy, The Ohio State University, Columbus, OH 43210, $ Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. Abstract-In this paper a dynamic model to investigate interphalangeal coordination in the human finger is proposed. Suitable models which describe the relationship between the tendon displacement and the joint angles have been chosen and incorporated into the skeletal dynamic model. A kinematic and kinetic model for interphalangeal coordination is suggested. Digital computer simulations are carried out to study interphalangeal (IP) flexion. Moreover, the effect of two different optimization methods is contrasted. The two optimization algorithms are employed to obtain a set of feasible values for the forces in the tendons or muscles of the finger.
1.
INTRODUCTION
Flexion and extension of human interphalangeal joints follow welldefined patterns of movement. These patterns result from biomechanical extensor and flexor constraints in the fingers. Despite various functional and anatomical investigations of the finger a satisfactory and unequivocal explanation of the observed phenomena is not yet available. Harris and Rutledge (1972) state that there is a rather welldefined coordinated interdependence between the proximal and distal interphalangeal joints which have approximately the same angle of flexion. Landsmeer (1949) states that extension of the proximal interphalangeal joint tenses the oblique retinacular ligament which in turn exerts a force onto the dorsal surface of the base of the distal phalanx; thus the ligament is tensed to straighten the distal interphalangeal joint. Consequently, as Landsmeer (1955) states, the flexor digitorum profundus which flexes the distal phalanx applies a force via the retinacular ligament to bend the middle phalanx. In a later treatise Landsmeer (1958) reports that the coordination of the interphalangeal joints can be attributed to the fact that muscles in contracting attempt to attain maximum decrease in length. He argues that any muscle in an articulating chain which is capable of shortening its length will do so, since no immediate force is present to prevent the muscle from contracting. In order to support his study he establishes a criterion that must be satisfied for proper coordination of the interphalangeal joints. This criterion is based upon the moment arm parameters of the biarticulate chain. Haines (1951) states in his study, ‘. . the middle and distal I.P. [interphalangeal] joints usually extend together.’ Corresponding results were obtained by Stack (1962) using a constructed model which demonstrated that both joints are always at the same angle. Receioed 22 October
1985: in
revised form 16 July 1987.
The results of investigations conducted by Harris and Rutledge (1972)differed from those of other investigations. Their conclusions were, ‘. . in all positions which the normal hand can actively assume, extension of the distal phalanx is performed entirely by the two lateral bands.’ Earlier investigations by Stack (1962) indicated that the middle band alone is unable to extend the distal phalanx; however, if the lateral bands are put under tension by the interossei the force of the long extensor is then transmitted to the base of the distal phalanx. Contrarily, Harris and Rutledge’s (1972) dissections on fresh specimens show that extension of the distal and the middle phalanx can be achieved simultaneously without support of the interosseus muscles. They explain that the ‘ingenious bypass system’ is perfectly balanced in length and incorporates both lateral tendons, the central slip (middle band) and the terminal tendon. Several investigations of the biomechanics of the hand and analytical studies of tendon forces and joint torques in static hand positions have also been conducted. Penrod et al. (1974) analyzed tendons in the hand by assuming that the total muscle effort is minimal and it can be explained theoretically by notions of reciprocal inhibition. In their treatise Chao et al. (1976) illustrated a force analysis of finger joints. They defined a set of equilibrium equations for particular static hand positions and eliminated the inadmissible solutions by applying constraint conditions. Berme et al. (1977) studied the forces at the metacarpophalangeal joint under external load conditions, while Storace and Wolf (1979) identified criteria for the static equilibrium of a finger based on the principle of virtual work. Wells et al. (1984) studied the interaction of passive elastic and muscular forces during unloaded finger movements. The purpose of this paper is to present a kinematic and dynamic model of the human finger in the sagittal plane in order to study finger movements dynamically and to clarify some of the aforementioned disagree459
460
H. J.
BUCHNER, M.
J. HINESand H. HEMAMI
ments on the mechanism of the finger. In particular the investigation considers whether interphalangeal coordination can be brought about by the extensor assembly only. This means that some anatomical structures are intentionally ignored, i.e. the retinacular ligament. Moreover, necessary kinetic and kinematical constraints are presented that arecapable ofdescribing the phenomenon of interphalangeal coordination mathematically. The paper also points out the effect of two different optimization algorithms on the muscle forces during finger flexion. In Section II the kinematic and dynamic equations of the finger-hand model are introduced. In Section III a digital computer simulation of finger flexion is presented and the results are interpreted. Summary and conclusions are given in Section IV. II.
DESCRIPTION OF THE MODEL AND
MATHEMATICAL
REPRESENTATION
A complete model of the behavior of the complex anatomical structure of the hand is at this point prohibitive. For ease of study therefore, the model is simplified such that it becomes analytically tractable, while at the same time preserving the phenomena under investigation. The motion of the dynamic model is restricted to the sagittal plane, since only interphalangeal movements of the finger are performed. The four fingers of the hand are modeled as one assuming that they all perform the same movement simultaneously. This assumption was thought to be necessary for estimating the total forces especially in the extrinsic finger muscles. Five spring and dashpotlike muscle force actuators are considered in the model and their moment arms at the joints are derived. In studying planar motion the palmar and dorsal interossei may be viewed as one muscle. The intricate extensor mechanism with its multiple muscle insertions is simplified. In particular the insertion of the interossei into the lateral band of the extensor apparatus is ignored, which means that for the purposes of this investigation the interossei are considered to insert solely into the base of the middle phalanx. Moreover, it is assumed that the lumbrical tendon joins together with the lateral band proceeding distally to form the terminal tendon. Further both lateral bands are considered as one. The joints are assumed to be ideally pivoted together in such a way that some ligaments may be neglected. A schematic diagram of the finger including its tendons is shown in Fig. 1. Although a number of anatomical structures are ignored in the present model, it is more refined than those used in the studies of Thomas et al. (1968) and Spoor and Landsmeer (1976). However, it is recognized further refinements will be necessary. In studying the kinematics of the multi-articular chain of the finger it is essential to determine the relationship between the location of the skeletal elements of the phalanges and the corresponding tendon displacements. Experimental studies of An et al. (1983),
Fig. 1. Schematic diagram of the finger model.
Armstrong and Chaffin (1978), Brand et nl. (1975), and Fischer (1969) suggest that Landsmeer’s Model I (see Landsmeer, 1960) is well suited for describing extensor tendon excursions in the finger. Moreover studies of Fischer (1969) and Brand et al. (1975) indicate that Landsmeer’s Model I may also appropriately describe the displacement of some flexor tendons. According to this Model the tendon excursion-joint angle relationship can be expressed mathematically as e=R,
(1)
where e = tendon excursion; R = radius of trochlea (assumed to be constant); (#J= joint angle in radians; (4 = 0 if the joint is in extended position). The displacement of the flexor tendons may be determined according to Landsmeer’s Model III. Although this model includes a trigonometric function that relates the flexor tendon excursion to the joint angles, measurements of this relationship which have been conducted by Fischer (1969), allow a good approximation by a second order polynomial. Hence, e = (R + R’@)rP
(2)
where R’ is a constant and the remaining variables are defined as in (1). Having defined the basic tendon displacement equations at a joint, the total tendon shift is simply the summation of the tendon excursion which takes place at each joint. On the basis of the above the following equations for the tendon displacement in the finger are determined, bearing in mind that a muscle shortens or lengthens accordingly. Let 41,..., (b3 be the, joint angles of the model and let the angles be related to the joints as follows: +1 = metacarpophalangeal joint (MP); 4b2= proximal interphalangeal joint (PIP): (p3 = distal interphalangeal joint (DIP). Hence, the excursion equations are extensor digitorum (communis) el = -RI,cP, flexor
+e12,
digitorum profundus
(3a)
461
A model for finger coordination flexor digitorum
superficialis
e, = (R,, + R;,v, interosseus
)$I + R32&.
(3c)
(palmar and dorsal)
~4 = R,,$,
- R,,4?.
(3d)
lumbricalis
-(Rx+R;z~Pz
ei = (R5, - R2, -R;,(b,bb,
',c,r, +R,z+Rz_P_)P_-(R,3+R23 + R;393)cPz.
(W
The equations above reflect the convention that the excursion assumes positive values whenever a muscle contracts and negative ones when a muscle relaxes and thereby elongates. As stated previously, the extended finger position was chosen as the reference position, i.e., (pi = 0, i = 1.2,3. In equation (3e) the moving origin of the lumbrical muscle has been incorporated. The term e, 2 in equation (3a) represents the contribution of the interphalangeal joints to the overall displacement of the extensor tendon. The value of ei2 is the larger magnitude of the tendon shift of either the medial band or the lateral band together with the terminal tendon. In the following the focus is on the kinematic model of the extensor mechanism of the middle and distal phalanx which define the pattern of the interphalangeal coordination. Figure 2 illustrates the proximal interphalangeal (PIP) joint with its extensor mechanism. In the extended position of the PIP joint, the middle and the lateral bands are aligned. As the finger flexes the lateral bands move palmwards while simultaneously the radius of flexion decreases on either side of the finger. Therefore in the planar model we can denote the tendon shift of the lateral band at the PIP joint by er= Similarly
-iR,2-R;&2)&.
In Fig. 3 the angular relationship between the distal and the proximal interphalangeal joint as given in equation (6) is plotted. The related anatomical parameters are given in Table 1. The numerical values of Table 1 were obtained from various sources as cited previously in this section and from Thomas ef al. (1968). The former values are corrected for the four finger model by a factor of z 0.833, because the original values pertained to the long finger of the hand. The coefficient R’,, was computed after rearranging equation (6) for a given value of (p3 and r/~~.The angles which limit the range of motion of the finger as reported by Tubiana (1981) were substituted into C/I: and o3 to compute R;,. Fischer (1969) describes the interphalangeal coordination by a second order poly-
80
0
Vb)
eT= -R,3~~3.
80
60
loo
+2(degrees)
Fig. 3. lnterphalangeal coordination from the lateral model: DIP angle vs PIP angle.
(5) Middle band I I
I
Fig. 2. PIP joint
40
with lateral -R13p2
/’
band
Laterat
model.
band
Table 1. Numerical values of the parameters of the tendon excursion models. (equations 3-5)
-(R,2-R’,2(p2)2-R,3(p3}.
tendon
20
(4c)
Adding the tendon shift at the DIP joint and considering the tendon shift of the middle band, the excursion of the extensor assemblage e, r becomes
Exterwr
I
IO
bands
-K,~(Pz
e ,L =min(-R,,c~~:
(6)
(4a)
for the middle and terminal e M=
In the present model the retinacular ligament is neglected. This makes the observed coordinated action at the interphalangeal joints solely dependent upon the extensor mechanism as described by Landsmeer (1958) or Harris and Rutledge (1972). Equating the two terms in the curly brackets of equation (5) and solving for (bJ yields the angular relationship of the interphalangeal joints for an ideally balanced extensor system. The angle (p3 can be expressed as
band
R, = R,,
j=
RJR;, i
joint tendon
1 2 3 4 5
ED FDP FDS Int Lum
ED FDP FDS Int Lum
= = = = =
1 MP (mm) lo/11.2/2.08 10/2.05 5/-
1IS/-
2 PIP (mm)
3 DIP (mm)
5/1.21 9.6li2.23 I/5l434’ - 0:4Sb
3,5/45510.261 ~-i--i-~
extensor digitorum. flexor digitorum profundus. flexor digitorum superficialis. interosseus. lumbricalis.
-!--
H. J. BUCHNER,M. J. HINESand H. HEMAMI
462
nomial including a linear term. Unfortunately, Fischer’s model has not been verified by performing experimental investigations gauging the interphalangeal joint angles of human fingers. Other IP joint models, such as Buchner’s (1983), may just as well describe interphalangeal coordination. The dynamics ofa finger or any other skeletal system can be modeled as a chain or tree structure of interconnected rigid bodies. The dynamical equations of such rigid bodies can be derived from the Lagrangian method with generalized coordinates or from free-body analysis using Newton Euler equations. The equations of motion of the three-link finger are established here from Newtonian dynamics as described by Hemami (1982) and Wittenburg (1977). Let 0 = (0, &BJ)‘be the generalized coordinates of the three links, where the angles are measured with respect to the vertical in clockwise direction. The angles of the phalanges are t?i, 8, and 8, from the proximal to the distal side, respectively. Let the vectors 6 and 6 be the first and second time derivatives of 0, and let F be the tendon force vector. The equation of motion of the model becomes (see Table 2) .!(@)6+.I,(@)62+G(0)
= M(O)F.
(7)
The inertia matrix J(O), the centrifugal matrix J,(O) and the gravity vector G (0) are analytic functions of 0. The latter matrices and the gravity vector are given in the appendix. The absolute link angles 0; and the joint angles (pi can be related by for
i= 1,2,3
(8)
terminal band. The vector F of the tendon or muscle forces may now be given as F=(FFFFFFFF)7 12345678 where the forces F1,. . . ,F, are associated with the tendons whose displacements are described in equations (3a-e), respectively. The forces F6, F, and F, are related to the middle, lateral, and the terminal bands. It should be noticed, however, that the forces F6, F, and Fs are not all independent variables. In fact they are related by the following two equations F,-F6-Fs
=0
(lla)
F,--F,+F,
=O.
(1lW
The first equation means that the sum of the forces in the middle and the lateral band are equal to the force in the extensor digitorum while the second one states that the sum of the forces in the lateral band and in the lumbrical is equal to the force in the terminal tendon. The matrix M(9) is given in the Appendix. Its elements describe the moment arms of the tendons about the center of gravity of link j, that is, with the partial derivatives being taken with respect to the absolute angle 19,and not with respect to $jt j = 1, 2, 3. Finding a feasible solution for the muscle forces in a static equilibrium has been frequently discussed in the literature. Often, however, the torques which are due to gravitational forces are neglected. Setting 6 = 6 = 0 in equation (7) will yield the correct equilibrium equation of the unloaded finger model G(O) = M(O) F.
where t& = x/2. Storace and Wolf (1979) have pointed out that the moment arm of a tendon or muscle crossing a joint can be derived from the principle of virtual work if the tendon or muscle displacement is known. The moment arm of a muscle or tendon i about a joint j can be evaluated by taking the partial derivative given below df?i --. Qj
(9)
Hence, the total torque at a joint j which is crossed by a number of n muscles or tendons is (10)
where Fi is the force in muscle or tendon i. If, as in the case of the extensor apparatus, some tendons branch out or join others, additional tendon forces are found. Therefore the partial differential of that branch must be related to the actual force being present in it. In this model the extensor tendon fans out proximally to the PIP joint to form the middle and the lateral band. Further distally the lateral band joins with the tendon of the lumbrical to proceed as the terminal tendon to the base of the distal phalanx. Thus three additional tendon forces must be considered. These forces are found in the middle, lateral and the
(12)
In order to solve equations (7) or (12) it is obviously possible to resort to principally equal mathematical algorithms in order to obtain a set of feasible muscle forces. It is also clear that the nature of equation (7) [and equation (ll)] is an indeterminate one in which so far five equations have been given to obtain eight forces. Suitable methods to compute a possible set of muscle forces are devised by establishing and minimizing a physiologically meaningful performance function. A conceivable performance function is n = J= (13) i = i PCSAi
xi
where PCSAi is the physiological cross-sectional area of the ith muscle as described by An et al. (1983). A second possible performance function can be expressed by (14) where the ratio Fi/PSCAi in the aforementioned two equations can be thought of as being the ‘stress’ Si in the muscle i. In the remainder of this section a conjecture is presented which encompasses the phenomenon of interphalangeal coordination. Also two optimization methods are proposed which yield minimal values of
463
A model for finger coordination the performance functtons given by equations (13) and (14). Consider Fig. 2 which shows the lateral band model of the finger. Let the finger be initially in the fully extended position and in equilibrium. From this position both the middle and the distal joint will be flexed if the flexor digitorum profundus force is increased. The increased force disturbs the equilibrium and causes acceleration of both interphalangeal joints to result in interphalangeal flexion. Suppose now that the distal phalanx would veer out faster than the middle phalanx. This in fact implies that the lateral band will be displaced by a larger magnitude than the middle one. The difference of the displacement between the lateral band combined with the terminal band and the middle band will be denoted E. Its value can be obtained from the equation below
The displacement difference between the lateral band and the middle band reduces the tension in the middle band and simultaneously increases the tension in the lateral band. The tension in the middle band may be reduced until possibly slacking of this band occurs. Any reduction of the tensile force in the middle band, however, contributes to a larger total flexing torque at the proximal interphalangeal joint and allows the middle phalanx to follow closely the distal phalanx. Conversely, suppose the middle phalanx would veer out faster than the distal phalanx. As a consequence the middle band tendon shift would be greater than the shift of the lateral band which in turn reduces the force in the terminal tendon and enhances the flexion torque of the distal phalanx. It will also reduce the total torque at the proximal interphalangeal joint. The situation explained above may be written in mathematical terms as F, = (I-
r)F, + KF
F, = rF, - KE
(16a) (16b)
where a is an unknown constant and K is the elasticity coefficient of the tendinous bands. The value of K may also depend on the surrounding structure of the extensor assembly if it contributes to the load distribution between the lateral and the middle bands. Also equation (16) must satisfy the constraint equation, equation (1 la). In order to show that this is correct, one simply adds equation (16b) to equation (16a) and obtains equation (1 la). The unknown coefficient x determines the load distribution between the middle band and lateral band solely if E = 0. If e # 0, the forces Fh and Fs also depend on the displacement difference as given in equation (15) and the value of the coefficient K. At this point two rather important facts about proper interphalangeal coordination can be inferred. First. the kinematics of finger interphalangeal coordination, i.e. the angular relationship of the IP joints, depend predominantly on the tendon displacements of
the middle band and the lateral band together with the terminal tendon. Similar observations had been reported by Landsmeer (1958) and Harris and Rutledge (1972). Second, the amount of loading and unloading of the tendinous bands of the extensor assembly determines how close the ideal kinematical relationship as given in equation (6) is maintained. The accountable factor for distributing the load between the middle and the lateral band is the internal elasticity of not only the tendinous bands but also other surrounding structures of the extensor assembly. Studying and modeling the elastic properties of the extensor apparatus may be quite difficult and requires a separate detailed experimental examination of the anatomical structures involved. Equations (16a) and (16b) are linear equations in the forces F,, but are not independent as they depend on equation (1 la). An independent linear equation can be obtained after subtracting equation (16b) from equation (16a). This yields (2% - l)F, + F6 - F, = 2Ke.
(17)
Equations (16a) and (16b) may always be derived from equations (1 la) and (17), and therefore, will be considered to be dependent equations. The employment of linear programming algorithms requires that all constraint equations are linear ones and, that all coefficients are known. However, the value of the coefficient a is not known and, in fact, z may be a nonconstant coefficient. Therefore in order to use a linear programming algorithm z must be guessed or. if possible, estimated. Based on equation (16) the bounds on tl can be stated as KeIF,
< z<
1 + Kc/F,,
F, > 0
118)
otherwise negative forces for Fh or F8 may result. When it is assumed that the coefficient r is an independent variable it is impossible to employ a linear programming algorithm, because equation (17) becomes nonlinear. Therefore, a second simulation must be conducted using a nonlinear optimization algorithm. In the following section actual finger flexion is simulated in order to assess the validity of the approach taken in this section. 111.SIMULATION
AND RESULTS
The proposed model of interphalangeal coordination is tested by conducting digital computer simulations. Figure 4 shows the simulation diagram of the overall system. The input to the system is the desired motion of the finger. This desired motion is given in terms of O’l:,, and O$,,. Only these two angles are specified because Olj:,, is related to O&. With the given desired trajectories, this dynamic model computes the necessary forces that must act on the finger in order to bring about a specified motion. In the absence of external disturbances, loads, or fatigue,
H. J. BUCHNER, M. J. HINESand H.
464
Ut
HEMAMI
F
U Fcfce optimization andIP - kinetics
Desired trajectar
Fig. 4. Simulation block diagram of the overall system. the actual angular trajectories, are the outputs of the simulation diagram, and are almost identical to the inputs, i.e. the desired trajectories. Under load or disturbance, the actual trajectories may not be identical to the desired ones and larger forces may be necessary. These forces can be computed by the diagram. The diagram therefore is capable of producing the needed forces F(t), and the actual trajectories of motion for all specified desired trajectories of motion under loading, disturbance, and fatigue conditions. The latter could be implemented by addition of appropriate saturation levels on the produced forces. The force vector F is computed using a linear and a non-linear optimization algorithm for comparison. The computation of torque u and feedback mechanisms are not discussed in this paper. A finger movement from full extension to full IPflexion was simulated by a digital computer. Initially, the fully extended finger is in the horizontal palm downwards directed position. From this position only the middle and distal phalanges are flexed to reach their final position at about 200” and 273”, respectively. In order to study the merit of the interphalangeal model, the desired reference trajectory did not coincide with the one in Fig. 3. Therefore, a reference trajectory was chosen such that the relationship between the PIP joint angles and the DIP joint angle is a linear one. A good interphalangeal model will cause the interphalangeal joints to flex according to the parabolic function in Fig. 3 despite a different desired trajectory. The force vector F was computed using two different optimization methods. A simplex linear programming algorithm was employed in the first optimization of F. The set of constraint equations consists of equations (1l), (17), and the one given below CJ = M(O)F
(18)
where M and Fare as given in equation (7) and U is the total feedforward torque as indicated in Fig. 4. The corresponding linear performance function is given in equation (13). The coefficients of the performance function are taken from Amis et al. (1979). In order to employ the linear programming algorithm the value of the coefficient u was chosen to be a = l/2 = constant.
It was found that with a = l/2 it is possible to obtain a feasible solution throughout the entire simulation. This suggests that cx= l/2 is a reasonable choice. Contrarily, if a = 1 the linear programming algorithm fails to produce a feasible solution during the whole simulation. However, it is obvious that if a is chosen to be TV= l/2 an overly severe restriction is imposed upon the force distribution between the middle and the lateral band. This problem may be avoided if a nonlinear programming algorithm is employed and if c( is considered to be an independent variable. The linear programming algorithm solves for the unknown forces Fi 3 0, i = 1, . . , I; I = 8, in such a way that the performance function equation (13) is minimized given the j = 6 linear equations (1 l), (17) and (18). Due to the nature of linear programming algorithms, 1-j = 2 muscle forces will always be set to zero. This is independent of the muscle model being implemented. In other words, a linear programming algorithm always ‘removes’ 1-j muscles from the model. Moreover, the computed forces represent the total force produced by passive and active structures of a particular muscle. A nonlinear optimization method was used to optimize the forces in F using the quadratic performance function in equation (14) subject to equations (1l), (17) and (18). The optimization procedure also computed the coefficient a in equation (17) to obtain the smallest possible muscle or tendon forces. The weighing coefficients in equations (13) and (14) are identical.
Table 2. Parameters of the dynamical equation di
ki
mi
4
Link i
mm
mm
g
Kgm’
Proximal phalanx
1
Mid phalanx Distal phalanx
2 3
46 28 20
23 14 10
125 60 40
Segment name
22E-6 3.92E-6 1.33E-6
d, = length of link i. k, = distance of joint i to center of gravity of link i. mi = mass of link i. Ii = inertia of link i.
465
A model for finger coordination The finger movement from full extension to full flexion was simulated within a time frame of T = 1 s. The results of the simulations are shown in Figs S-13. The absolute positions of the three phalanges are shown in Fig. 5. The angles of the phalanges are measured from the vertical in a clockwise direction. In Fig. 6 the velocity profiles of the three phalanges are shown. Each phalanx starts and ends with zero angular velocity, i.e. the phalanges start and end in an equilibrium. The distal interphalangeal joint angle vs the proximal interphalangeal phalangeal joint angle of the simulation is shown in Fig. 7. The elasticity coefficient K of the extensor assembly which determines the load distribution between the middle band and the lateral band was selected to be K = 500 N m- ‘, since a better value was not available from literature. However, the apparently small value of K already yields a good
20
,
-5
s
I
I
1
I
I
0
02
04
06
00
I
F5
F,:
Fig. 8. Muscle forces in the finger (linear optimization). FDP, FQ: Int, F,: Lum.
“i 25 -
20
5,
l s)
Time
-
51 0
I
I
I
I
I
0.2
0.4
06
0.6
I
(s )
Time
F, :
Fig. 9. Muscle forces in the finger (linear optimization). EC, Fh: middle band.
00 Tlmets)
Fig. 5. Position of the phalanges phalanx, 0, : middle phalanx,
vs time. 8,: proximal B3: distal phalanx.
14
.
13 7
I2 -
6
II
-
I
7 0
-I
1
0
I
I
02
04 Time
Fig. 6. Angular
velocity
(
I
I
I
06
08
I
s
I
)
of the phalanges
02
I 04 Time
I
I
I
06
06
I
(s)
Fig. 10. Muscle forces in the finger (linear optimization). middle band, F,: terminal band-lateral band.
vs time.
08
06
04 Trne
(s
joint angles. DIP angle vs PIP angle.
I
)
Fig. 11. Muscle forces in the finger (quadratic Fig. 7. lnterphalangeal
F,:
F,: EC, F,: FDP.
optimization).
466
H. J. BUCHNER, M.
3 ‘t 2’ 0
I 02
Y-2-I
I 04 Time
I
I 08
I
J. HINES and
1
I
I P,”
finger (quadratic optimization). F6: middle band, F,: terminal band-lateral band.
Fig. 12. Muscle forces in the
40 )
I
3020-
-20
-
-30 -40 -n 1_
-
0
I cl2
I 04 Time
I 06
I 08
I I
Is)
Fig. 13. Excursion of the tendons or muscles in the finger vs time. correlation between the simulation results and the theoretically established interphalangeal joint behavior as is evident from Fig. 3 and Fig. 7. Choosing a larger value for K only will improve the result-as K approaches infinity Figs 3 and 7 become indistinguishable. The digital computer simulation may confirm Harris and Rutledge’s theory that interphalangeal coordination can be brought about by the extensor mechanism only. This means that proper coordination does not necessarily depend on the intrinsic muscles or the retinacular ligament. The profiles of the tendon forces in the finger are shown in Figs 8-10 for the linear optimization function and in Figs 1l-12 for the quadratic function. Although both optimization methods yield identical IP-behavior the forces in the finger are entirely different. The forces of the flexor digitorum profundus (F2), the flexor digitorum superficialis (F3), the interosseus (F4) and the lumbricalis (F,) are plotted in Fig. 8. Figure 8 shows that during the entire period of simulation the lumbricalis muscle is ‘silent’ i.e. F, = 0, and that the FDS is active only during the first 0.22 s. Beyond 0.22 s of the simulation the interosseus muscle becomes active until its force reaches a relatively large value of about 18 N. The force profile of the FDP shows a slight decrease during the first 0.22 s and then increases to a value of about 9.5 N. Figure 9 shows the forces of the extensor digitorum (F,) and the middle band (Fb). The forces also show an initial decrease during the first 0.22 s and a subsequent increase to relatively large values of F, = 26.4 N and F6 = 13.2 N. The forces of the middle band (Fb)and the lateral band
H. HEMAMI
(F,) are given in Fig. 10. Since the force of the lumbricalis is zero during the entire simulation it follows that the force in the terminal band ( F8) is equal to the force in the lateral band. A more favourable result can be obtained if a quadratic performance function is optimized, including the a-coefficient as additional independent variable. This can be shown in Figs I l-12. Figure II shows the forces of the extensor digitorum and the flexor digitorum profundus. Contrary to the first optimization method in this simulation the forces F, and F2 decrease by about 7.5 N and 3.5 N respectively. Since forces of the flexor digitorum superficialis, interrosseus and lumbricalis remained constant at zero, their force profiles are not plotted here. The forces of the middle band and the lateral band are shown in Fig. 12. Initially the lateral band bears the larger share of the extensor digitorum force, however after approximately 0.22 s the middle band begins to carry the larger portion of the extensor digitorum force while the lateral band force decreases significantly. It is interesting to observe, that the time at crossover of the forces Fb and F7 in Fig. 12 coincides with the time where the forces F, = F4 = 0 in Fig. 8. Moreover, it seems that the corresponding forces in Figs 8-10 and Figs 11-12 are equal at that time. The results of the second optimization may be favored on the ground of some electromyographical studies as well. Long and Brown (1964) have shown in their electromyographic study that flexion of the middle and distal phalanges is primarily brought about by the flexor digitorum profundus and the extensor digitorum. Other muscles are silent during this movement. Landsmeer and Long (1965) conclude that the extensor digitorum acts as a ‘brake’ during finger flexion and that the intrinsic muscles are passive during this movement. This breaking phenomenon may also explain why the force of the extensor digitorum approaches a constant value as does the flexor digitorum profundus. It appears that flexing of the unloaded interphalangeal joints is possible when the ‘extensor brake’ is released by reducing the antagonistic force which opposes flexing and by lengthening of the extensor digitorum. Simultaneously, the flexor digitorum profundus contracts to its possible minimum length. This, however, does not imply that the flexor digitorum profundus force will increase, especially if an equilibrium cannot be found which minimizes the total muscle effort. The excursion of the extrinsic and intrinsic muscles are shown in Fig. 13. Referring to equations (3) and (5), the excursion of a contracting muscle assumes positive numerical values and an elongating one takes on negative values. The excursion of the muscles are given in mm and each trace shows a distinct excursion characteristic except the extensor digitorum and the interosseus. The excursions of the latter two muscles are identical due to the model and the particular movement being studied. The load distribution coefficient cf is plotted in Fig. 14 which shows that the lateral band bears a
467
A model for finger coordination
,::
i-h
0
02
04
(s9”
08
Time Fig. 14. The value of the coefficient
a vs time.
larger force than the middle band if the finger is fully extended while in the finger that is flexing the middle band bears the larger load. In the flexed finger position the lateral band bears about 28 ‘;/oof the total extensor digitorum force, while the middle band carries the remaining 72 ‘x, of the force.
slightly
IV. SUMMARY AND CONCLUSIONS
A dynamic model for finger intecphalangeal functional coordination was proposed in this paper. The dynamics of the skeletal structures were modeled as an open chain of coupled rigid bodies. Two distinct types of tendon elongation models were selected which suitably describe the relationship between the tendon displacements and the joint angles in the finger. One type of model is a linear one and is used for all extensor tendons and some flexor tendons as well. The second type of model describes the displacement of the lateral band and most flexor tendons by parabolic functions. It may appear that for some joints and muscles the tendon excursion models ace too rudimentary, however. experimental studies conducted in recent years. justify the approximations of the proposed models. Further extensions and uses of these models are possible. If more precise models of the tendon excursion in the finger ace required for various studies on finger and hand function. refined models should be developed in the future. There may also be a vital need for a more precise description of the kinematics and kinetics of interphalangeal coordination. This description. however, must be based on some very accurate anatomical and rheological studies on the extensor assemblage. In particular the structures and functions of small tendon-like tissues which constrain tendon shift of the lateral and middle bands in the palmar and lateral direction during finger Aexion should be precisely studied in order to develop more accurate mathematical models. The classical overdeterminancy problem that arises in specifying the tendon forces in the finger has been solved by employing two optimization methods. The outcome of the two optimization approaches were contrasted, and it was found that the results of the second approach ace more encouraging. In particular the large force of the interosseus and the increasing forces of the extensor digitocum may be opposed on
several grounds. First, electcomyogcaphical studies indicate that the intrinsic muscles are silent during IF’ flexion, which means, however, that the computed Interosseus force is unlikely to occur. Second, the large increase of the extensor digitocum force during finger flexion is physiologically inexplicable, although mathematically possible. Third, a linear programming algorithm does not allow any variation of the Ecoefficient when the value of LIis fixed. Contrarily. the nonlinear optimization process views a not as a fixed coefficient but as a variable that is computed much like the muscle forces. In fact, this difference seems to be the main reason for the so significantly distinct muscle forces. Finally, the kinematical model of the finger interphalangeal functional coordination itself was derived from a rather abstract point of view and was not based on any experimental measurements. Better kinematical models therefore may be established from experimental studies of IP coordination in order to allow for more accurate correlation with physiological models. The present results and future studies should provide for better treatment of abnormal finger function, and for more advanced prosthetic devices. Acknowledgement-The authors would like to thank the reviewers for their careful reading of this manuscript and many helpful suggestions. This work was supported by the National Science Foundation under Grant No. ECS-820-1240.
REFERENCES
Amis, A. A.. Dowson.
D. and Wright, V. (1979) Muscle strengths and musculo-skeletal geometry of the upper limb. Engng Med. 8,41-48. Ann, K. N., Kwak. B. M.,Chao, E. Y. and Morrey, B. F. (1984) Determination of muscle and joint forces: a new technique to solve the indeterminate problem. J. ho&em. Engng 106, 366367. An, K. N., Ueba, Y., Chao. E. Y.. Cooney. W. P. and Linscheid, R. L. (1983) Tendon excursion and moment arm of index finger muscles. J. Biomechanics 16, 419425. Armstrong, T. J. and Chaffin. D. 6. (1978) An investigation of the relationship between displacement of the finger and wrist joints and the extrinsic finger flexor tendons. J. Biomechanics I 1, 109- 118.
Berme. N., Paul, J. P. and Purves, W. K. (1977) A biomechanical analysis of the metacarpophalangeal joint. J. Biomechanics IO, 409412. Brand, P. W., Cranor, K. C. and Ellis, J. C. (1975) Tendon and pulleys at the metacarpo-phalangeal joint of a finger. J. Bone Jt Surg. 57A, 779-784. Buchner, H. (1983) Finger movement in the sagirtal plane and the mechanisms of touch control. M.S. Thesis. The Ohio State University, Columbus, OH. Buchner, H.. Hines, M. J.. Hemami, H. (1985) A mechanism for touch control of a sagittal Five-Link Hand-Finger. IEEE Trans. Sysrems. Man. Cvberner. 15, 69-77. Chao. E. Y., Opgiande, J. D. and Armer, F. E. (1976) Threedimensional force analysis of finger joints in selected isometric hand functions. J. Biomechanics 9. 387-396. Fischer, G. W. (1969) A treatise on the topographic anatomy of the long finger and a biomechanical investigation on its inter-joint movement. Ph.D. Dissertation. The University of Iowa, Iowa City, IA. Haines. R. W. (1951) The extensor apparatus of the finger. J. Anar. 85, 151-259.
468
H. J. BUCHNER,M. J. HINESand H. HEMAMI
Harris, C. and Rutledge, G. L. (I 972) The functional anatomy of the extensor mechanism of the finger. J. Bone Jr Surg. 54-A, 713-726. Hemami, H. (1982) Some aspects of Euler-Newton equations of motion. Ingenieur-Archio. 52, 167-176. Landsmeer, J. M. F. (1949) The anatomy of the dorsal aponeurosis of the human finger, and its functional significance. Anat. Rec. 104, 31-44. Landsmeer, 1. M. F. (1955) Anatomical and functional investigations on the articulation of the human hand. Acta anat. Suppl. 24, l-69. Landsmeer, J. M. F. (1958) A report on the coordination of the interphalangeal joints of the human finger and its disturbances. Acta morph. neer-stand. 2, 59-84. Landsmeer, J. M. F. (1960) Studies in the anatomy of articulation. Acta morph. neer-stand. 3, 287-321. Landsmeer, J. M. F. and Long, C. (1965) The mechanism of finger control based on electromyograms and location analysis. Acta anat. 60, 330-347. Long, C. and Brown, M. E. (1964) Electromyographic kinesiology ofthe Hand: muscles moving the long finger. J. Bane Jt Surg. 46-A, 1683-1706. Penrod, D. D., Davy, D. T. and Sigh, D. P. (1974) An optimization approach to tendon force analysis. J. Biomechanics
APPENDIX J(e)
a,,cos(g,
aI1
=
a2 i cos(& - 0, )
a3, c0s(e3 - 8, )
1
J,(e) =
a22
a,,cos(e,
-2R2161 + RSZ+ Rx + (2R52+ 2JG,M,
R,, -R,,
a,,
= d,(m,k,
a,,
= 12+m2k:+d:m,
+d,ms)
a23 = dzm3k3 as, = d,m3k, as1 = d,msk, ass = 13+m,k:
G=
gsinO,Cm,k, +d,(m,+m~)l gsin&[m,k2 +d,m,]
gsinB,m,k, 1 I The sensitivity 3 x 8 matrix M(0) whose element m,j is given by j= 1.2 dei de, mi’=~-drpj+, i = 1,2,3,4, 5, M, L, T and dei mi, = &
j=3 i = 1, 2, 3, 4, 5, M, L, T
is given in Table 3. For ease of reading, the transpose of the matrix is listed where the successive columns of the list are the rows of M.
0 &+2&41 - (R23 + %3,3)
0
0
&,+2R;,4,
0 I
I
-
WSZ
+
0
I
- Ru
R22)
I
-R,3-&3 -=;3+3
-W;2+2R;,k#9
+R,3+&,+%393
I
R I*
RI2 4, -2R;242
I I I
I
I
I I
1
a 12 = d, (m,k, +d,ms)
R 32
I
1
a 13 = dtm3k3
I &+%I& -R32 &I + RN
a33
a,,sin(f?, -e,) a,3sin(8, -0,) 0 a2,sin(Bz -e,) a,,sin(O, - ~9,) 0
0
a,,sin(e,-0,) a3,sm(6d -8,)
Table 3. Transpose of the sensitivity matrix. M(g)
&I +X141 - (Jh +2&w)
-f33)
a,,cos(g, - &)
a,, =I,+m,k:+d:(m,+m,)
Spoor, C. W. and Landsmeer, J. M. F. (1976) Analysis of the zigzag movement of the human finger under influence of the extensor digitorum tendon and the deep flexor tendon. J. Biomechanics 9, 561-566. Stack, A. G. (1962) Muscle function in the fingers J. Bone Jt Surg. 44-B, 899-909. Storace, A. and Wolf, B. (1979) Functional analysis of the role of the finger tendons. J. Biomechanics 12, Y&578. Thomas, D. H., Long, C. and Landsmeer, J. M. F. (1968) Biomechanical considerations of lumbricalis behavior in the human finger. J. Biomechanics. I, 107-l 15. Tubiana, R. (1981) Architecture and functions of the hand. The Hand, (Edited by Tubiana, R.) pp. 19-93. W.B. Saunders, Philadelphia. Wells, R. P., Ranney, D. A. and Keller, A. (1984) The interaction of muscular and passive elastic forces during unloaded finger movements: a computer graphics model. Proc. 4th European Society of Biomechanics, Switzerland. Wittenburg, J. (1977) Dynamics of Systems of Rigid Bodies. B. G. Teubner, Stuttgart.
I I
- e2)
a,,cos(e,
The elements aij of the previous matrices are:
7, 123-l 29.
-R,1
-Al)
-R,z+2R;zd, R 13
I I
0 -R,3
uO21-9290188 13.00 + .OO Perganm Press plc
1988
A DYNAMIC MODEL FOR FINGER INTERPHALANGEAL COORDINATION HELMUT J. BUCHNER* MARGARETJ. HINEst and HOOSHANGHEMAMI: *Siemens A.G. Forschungs Laboratorien, Munchen, Germany; t Department of Anatomy, The Ohio State University, Columbus, OH 43210, $ Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. Abstract-In this paper a dynamic model to investigate interphalangeal coordination in the human finger is proposed. Suitable models which describe the relationship between the tendon displacement and the joint angles have been chosen and incorporated into the skeletal dynamic model. A kinematic and kinetic model for interphalangeal coordination is suggested. Digital computer simulations are carried out to study interphalangeal (IP) flexion. Moreover, the effect of two different optimization methods is contrasted. The two optimization algorithms are employed to obtain a set of feasible values for the forces in the tendons or muscles of the finger.
1.
INTRODUCTION
Flexion and extension of human interphalangeal joints follow welldefined patterns of movement. These patterns result from biomechanical extensor and flexor constraints in the fingers. Despite various functional and anatomical investigations of the finger a satisfactory and unequivocal explanation of the observed phenomena is not yet available. Harris and Rutledge (1972) state that there is a rather welldefined coordinated interdependence between the proximal and distal interphalangeal joints which have approximately the same angle of flexion. Landsmeer (1949) states that extension of the proximal interphalangeal joint tenses the oblique retinacular ligament which in turn exerts a force onto the dorsal surface of the base of the distal phalanx; thus the ligament is tensed to straighten the distal interphalangeal joint. Consequently, as Landsmeer (1955) states, the flexor digitorum profundus which flexes the distal phalanx applies a force via the retinacular ligament to bend the middle phalanx. In a later treatise Landsmeer (1958) reports that the coordination of the interphalangeal joints can be attributed to the fact that muscles in contracting attempt to attain maximum decrease in length. He argues that any muscle in an articulating chain which is capable of shortening its length will do so, since no immediate force is present to prevent the muscle from contracting. In order to support his study he establishes a criterion that must be satisfied for proper coordination of the interphalangeal joints. This criterion is based upon the moment arm parameters of the biarticulate chain. Haines (1951) states in his study, ‘. . the middle and distal I.P. [interphalangeal] joints usually extend together.’ Corresponding results were obtained by Stack (1962) using a constructed model which demonstrated that both joints are always at the same angle. Receioed 22 October
1985: in
revised form 16 July 1987.
The results of investigations conducted by Harris and Rutledge (1972)differed from those of other investigations. Their conclusions were, ‘. . in all positions which the normal hand can actively assume, extension of the distal phalanx is performed entirely by the two lateral bands.’ Earlier investigations by Stack (1962) indicated that the middle band alone is unable to extend the distal phalanx; however, if the lateral bands are put under tension by the interossei the force of the long extensor is then transmitted to the base of the distal phalanx. Contrarily, Harris and Rutledge’s (1972) dissections on fresh specimens show that extension of the distal and the middle phalanx can be achieved simultaneously without support of the interosseus muscles. They explain that the ‘ingenious bypass system’ is perfectly balanced in length and incorporates both lateral tendons, the central slip (middle band) and the terminal tendon. Several investigations of the biomechanics of the hand and analytical studies of tendon forces and joint torques in static hand positions have also been conducted. Penrod et al. (1974) analyzed tendons in the hand by assuming that the total muscle effort is minimal and it can be explained theoretically by notions of reciprocal inhibition. In their treatise Chao et al. (1976) illustrated a force analysis of finger joints. They defined a set of equilibrium equations for particular static hand positions and eliminated the inadmissible solutions by applying constraint conditions. Berme et al. (1977) studied the forces at the metacarpophalangeal joint under external load conditions, while Storace and Wolf (1979) identified criteria for the static equilibrium of a finger based on the principle of virtual work. Wells et al. (1984) studied the interaction of passive elastic and muscular forces during unloaded finger movements. The purpose of this paper is to present a kinematic and dynamic model of the human finger in the sagittal plane in order to study finger movements dynamically and to clarify some of the aforementioned disagree459
460
H. J.
BUCHNER, M.
J. HINESand H. HEMAMI
ments on the mechanism of the finger. In particular the investigation considers whether interphalangeal coordination can be brought about by the extensor assembly only. This means that some anatomical structures are intentionally ignored, i.e. the retinacular ligament. Moreover, necessary kinetic and kinematical constraints are presented that arecapable ofdescribing the phenomenon of interphalangeal coordination mathematically. The paper also points out the effect of two different optimization algorithms on the muscle forces during finger flexion. In Section II the kinematic and dynamic equations of the finger-hand model are introduced. In Section III a digital computer simulation of finger flexion is presented and the results are interpreted. Summary and conclusions are given in Section IV. II.
DESCRIPTION OF THE MODEL AND
MATHEMATICAL
REPRESENTATION
A complete model of the behavior of the complex anatomical structure of the hand is at this point prohibitive. For ease of study therefore, the model is simplified such that it becomes analytically tractable, while at the same time preserving the phenomena under investigation. The motion of the dynamic model is restricted to the sagittal plane, since only interphalangeal movements of the finger are performed. The four fingers of the hand are modeled as one assuming that they all perform the same movement simultaneously. This assumption was thought to be necessary for estimating the total forces especially in the extrinsic finger muscles. Five spring and dashpotlike muscle force actuators are considered in the model and their moment arms at the joints are derived. In studying planar motion the palmar and dorsal interossei may be viewed as one muscle. The intricate extensor mechanism with its multiple muscle insertions is simplified. In particular the insertion of the interossei into the lateral band of the extensor apparatus is ignored, which means that for the purposes of this investigation the interossei are considered to insert solely into the base of the middle phalanx. Moreover, it is assumed that the lumbrical tendon joins together with the lateral band proceeding distally to form the terminal tendon. Further both lateral bands are considered as one. The joints are assumed to be ideally pivoted together in such a way that some ligaments may be neglected. A schematic diagram of the finger including its tendons is shown in Fig. 1. Although a number of anatomical structures are ignored in the present model, it is more refined than those used in the studies of Thomas et al. (1968) and Spoor and Landsmeer (1976). However, it is recognized further refinements will be necessary. In studying the kinematics of the multi-articular chain of the finger it is essential to determine the relationship between the location of the skeletal elements of the phalanges and the corresponding tendon displacements. Experimental studies of An et al. (1983),
Fig. 1. Schematic diagram of the finger model.
Armstrong and Chaffin (1978), Brand et nl. (1975), and Fischer (1969) suggest that Landsmeer’s Model I (see Landsmeer, 1960) is well suited for describing extensor tendon excursions in the finger. Moreover studies of Fischer (1969) and Brand et al. (1975) indicate that Landsmeer’s Model I may also appropriately describe the displacement of some flexor tendons. According to this Model the tendon excursion-joint angle relationship can be expressed mathematically as e=R,
(1)
where e = tendon excursion; R = radius of trochlea (assumed to be constant); (#J= joint angle in radians; (4 = 0 if the joint is in extended position). The displacement of the flexor tendons may be determined according to Landsmeer’s Model III. Although this model includes a trigonometric function that relates the flexor tendon excursion to the joint angles, measurements of this relationship which have been conducted by Fischer (1969), allow a good approximation by a second order polynomial. Hence, e = (R + R’@)rP
(2)
where R’ is a constant and the remaining variables are defined as in (1). Having defined the basic tendon displacement equations at a joint, the total tendon shift is simply the summation of the tendon excursion which takes place at each joint. On the basis of the above the following equations for the tendon displacement in the finger are determined, bearing in mind that a muscle shortens or lengthens accordingly. Let 41,..., (b3 be the, joint angles of the model and let the angles be related to the joints as follows: +1 = metacarpophalangeal joint (MP); 4b2= proximal interphalangeal joint (PIP): (p3 = distal interphalangeal joint (DIP). Hence, the excursion equations are extensor digitorum (communis) el = -RI,cP, flexor
+e12,
digitorum profundus
(3a)
461
A model for finger coordination flexor digitorum
superficialis
e, = (R,, + R;,v, interosseus
)$I + R32&.
(3c)
(palmar and dorsal)
~4 = R,,$,
- R,,4?.
(3d)
lumbricalis
-(Rx+R;z~Pz
ei = (R5, - R2, -R;,(b,bb,
',c,r, +R,z+Rz_P_)P_-(R,3+R23 + R;393)cPz.
(W
The equations above reflect the convention that the excursion assumes positive values whenever a muscle contracts and negative ones when a muscle relaxes and thereby elongates. As stated previously, the extended finger position was chosen as the reference position, i.e., (pi = 0, i = 1.2,3. In equation (3e) the moving origin of the lumbrical muscle has been incorporated. The term e, 2 in equation (3a) represents the contribution of the interphalangeal joints to the overall displacement of the extensor tendon. The value of ei2 is the larger magnitude of the tendon shift of either the medial band or the lateral band together with the terminal tendon. In the following the focus is on the kinematic model of the extensor mechanism of the middle and distal phalanx which define the pattern of the interphalangeal coordination. Figure 2 illustrates the proximal interphalangeal (PIP) joint with its extensor mechanism. In the extended position of the PIP joint, the middle and the lateral bands are aligned. As the finger flexes the lateral bands move palmwards while simultaneously the radius of flexion decreases on either side of the finger. Therefore in the planar model we can denote the tendon shift of the lateral band at the PIP joint by er= Similarly
-iR,2-R;&2)&.
In Fig. 3 the angular relationship between the distal and the proximal interphalangeal joint as given in equation (6) is plotted. The related anatomical parameters are given in Table 1. The numerical values of Table 1 were obtained from various sources as cited previously in this section and from Thomas ef al. (1968). The former values are corrected for the four finger model by a factor of z 0.833, because the original values pertained to the long finger of the hand. The coefficient R’,, was computed after rearranging equation (6) for a given value of (p3 and r/~~.The angles which limit the range of motion of the finger as reported by Tubiana (1981) were substituted into C/I: and o3 to compute R;,. Fischer (1969) describes the interphalangeal coordination by a second order poly-
80
0
Vb)
eT= -R,3~~3.
80
60
loo
+2(degrees)
Fig. 3. lnterphalangeal coordination from the lateral model: DIP angle vs PIP angle.
(5) Middle band I I
I
Fig. 2. PIP joint
40
with lateral -R13p2
/’
band
Laterat
model.
band
Table 1. Numerical values of the parameters of the tendon excursion models. (equations 3-5)
-(R,2-R’,2(p2)2-R,3(p3}.
tendon
20
(4c)
Adding the tendon shift at the DIP joint and considering the tendon shift of the middle band, the excursion of the extensor assemblage e, r becomes
Exterwr
I
IO
bands
-K,~(Pz
e ,L =min(-R,,c~~:
(6)
(4a)
for the middle and terminal e M=
In the present model the retinacular ligament is neglected. This makes the observed coordinated action at the interphalangeal joints solely dependent upon the extensor mechanism as described by Landsmeer (1958) or Harris and Rutledge (1972). Equating the two terms in the curly brackets of equation (5) and solving for (bJ yields the angular relationship of the interphalangeal joints for an ideally balanced extensor system. The angle (p3 can be expressed as
band
R, = R,,
j=
RJR;, i
joint tendon
1 2 3 4 5
ED FDP FDS Int Lum
ED FDP FDS Int Lum
= = = = =
1 MP (mm) lo/11.2/2.08 10/2.05 5/-
1IS/-
2 PIP (mm)
3 DIP (mm)
5/1.21 9.6li2.23 I/5l434’ - 0:4Sb
3,5/45510.261 ~-i--i-~
extensor digitorum. flexor digitorum profundus. flexor digitorum superficialis. interosseus. lumbricalis.
-!--
H. J. BUCHNER,M. J. HINESand H. HEMAMI
462
nomial including a linear term. Unfortunately, Fischer’s model has not been verified by performing experimental investigations gauging the interphalangeal joint angles of human fingers. Other IP joint models, such as Buchner’s (1983), may just as well describe interphalangeal coordination. The dynamics ofa finger or any other skeletal system can be modeled as a chain or tree structure of interconnected rigid bodies. The dynamical equations of such rigid bodies can be derived from the Lagrangian method with generalized coordinates or from free-body analysis using Newton Euler equations. The equations of motion of the three-link finger are established here from Newtonian dynamics as described by Hemami (1982) and Wittenburg (1977). Let 0 = (0, &BJ)‘be the generalized coordinates of the three links, where the angles are measured with respect to the vertical in clockwise direction. The angles of the phalanges are t?i, 8, and 8, from the proximal to the distal side, respectively. Let the vectors 6 and 6 be the first and second time derivatives of 0, and let F be the tendon force vector. The equation of motion of the model becomes (see Table 2) .!(@)6+.I,(@)62+G(0)
= M(O)F.
(7)
The inertia matrix J(O), the centrifugal matrix J,(O) and the gravity vector G (0) are analytic functions of 0. The latter matrices and the gravity vector are given in the appendix. The absolute link angles 0; and the joint angles (pi can be related by for
i= 1,2,3
(8)
terminal band. The vector F of the tendon or muscle forces may now be given as F=(FFFFFFFF)7 12345678 where the forces F1,. . . ,F, are associated with the tendons whose displacements are described in equations (3a-e), respectively. The forces F6, F, and F, are related to the middle, lateral, and the terminal bands. It should be noticed, however, that the forces F6, F, and Fs are not all independent variables. In fact they are related by the following two equations F,-F6-Fs
=0
(lla)
F,--F,+F,
=O.
(1lW
The first equation means that the sum of the forces in the middle and the lateral band are equal to the force in the extensor digitorum while the second one states that the sum of the forces in the lateral band and in the lumbrical is equal to the force in the terminal tendon. The matrix M(9) is given in the Appendix. Its elements describe the moment arms of the tendons about the center of gravity of link j, that is, with the partial derivatives being taken with respect to the absolute angle 19,and not with respect to $jt j = 1, 2, 3. Finding a feasible solution for the muscle forces in a static equilibrium has been frequently discussed in the literature. Often, however, the torques which are due to gravitational forces are neglected. Setting 6 = 6 = 0 in equation (7) will yield the correct equilibrium equation of the unloaded finger model G(O) = M(O) F.
where t& = x/2. Storace and Wolf (1979) have pointed out that the moment arm of a tendon or muscle crossing a joint can be derived from the principle of virtual work if the tendon or muscle displacement is known. The moment arm of a muscle or tendon i about a joint j can be evaluated by taking the partial derivative given below df?i --. Qj
(9)
Hence, the total torque at a joint j which is crossed by a number of n muscles or tendons is (10)
where Fi is the force in muscle or tendon i. If, as in the case of the extensor apparatus, some tendons branch out or join others, additional tendon forces are found. Therefore the partial differential of that branch must be related to the actual force being present in it. In this model the extensor tendon fans out proximally to the PIP joint to form the middle and the lateral band. Further distally the lateral band joins with the tendon of the lumbrical to proceed as the terminal tendon to the base of the distal phalanx. Thus three additional tendon forces must be considered. These forces are found in the middle, lateral and the
(12)
In order to solve equations (7) or (12) it is obviously possible to resort to principally equal mathematical algorithms in order to obtain a set of feasible muscle forces. It is also clear that the nature of equation (7) [and equation (ll)] is an indeterminate one in which so far five equations have been given to obtain eight forces. Suitable methods to compute a possible set of muscle forces are devised by establishing and minimizing a physiologically meaningful performance function. A conceivable performance function is n = J= (13) i = i PCSAi
xi
where PCSAi is the physiological cross-sectional area of the ith muscle as described by An et al. (1983). A second possible performance function can be expressed by (14) where the ratio Fi/PSCAi in the aforementioned two equations can be thought of as being the ‘stress’ Si in the muscle i. In the remainder of this section a conjecture is presented which encompasses the phenomenon of interphalangeal coordination. Also two optimization methods are proposed which yield minimal values of
463
A model for finger coordination the performance functtons given by equations (13) and (14). Consider Fig. 2 which shows the lateral band model of the finger. Let the finger be initially in the fully extended position and in equilibrium. From this position both the middle and the distal joint will be flexed if the flexor digitorum profundus force is increased. The increased force disturbs the equilibrium and causes acceleration of both interphalangeal joints to result in interphalangeal flexion. Suppose now that the distal phalanx would veer out faster than the middle phalanx. This in fact implies that the lateral band will be displaced by a larger magnitude than the middle one. The difference of the displacement between the lateral band combined with the terminal band and the middle band will be denoted E. Its value can be obtained from the equation below
The displacement difference between the lateral band and the middle band reduces the tension in the middle band and simultaneously increases the tension in the lateral band. The tension in the middle band may be reduced until possibly slacking of this band occurs. Any reduction of the tensile force in the middle band, however, contributes to a larger total flexing torque at the proximal interphalangeal joint and allows the middle phalanx to follow closely the distal phalanx. Conversely, suppose the middle phalanx would veer out faster than the distal phalanx. As a consequence the middle band tendon shift would be greater than the shift of the lateral band which in turn reduces the force in the terminal tendon and enhances the flexion torque of the distal phalanx. It will also reduce the total torque at the proximal interphalangeal joint. The situation explained above may be written in mathematical terms as F, = (I-
r)F, + KF
F, = rF, - KE
(16a) (16b)
where a is an unknown constant and K is the elasticity coefficient of the tendinous bands. The value of K may also depend on the surrounding structure of the extensor assembly if it contributes to the load distribution between the lateral and the middle bands. Also equation (16) must satisfy the constraint equation, equation (1 la). In order to show that this is correct, one simply adds equation (16b) to equation (16a) and obtains equation (1 la). The unknown coefficient x determines the load distribution between the middle band and lateral band solely if E = 0. If e # 0, the forces Fh and Fs also depend on the displacement difference as given in equation (15) and the value of the coefficient K. At this point two rather important facts about proper interphalangeal coordination can be inferred. First. the kinematics of finger interphalangeal coordination, i.e. the angular relationship of the IP joints, depend predominantly on the tendon displacements of
the middle band and the lateral band together with the terminal tendon. Similar observations had been reported by Landsmeer (1958) and Harris and Rutledge (1972). Second, the amount of loading and unloading of the tendinous bands of the extensor assembly determines how close the ideal kinematical relationship as given in equation (6) is maintained. The accountable factor for distributing the load between the middle and the lateral band is the internal elasticity of not only the tendinous bands but also other surrounding structures of the extensor assembly. Studying and modeling the elastic properties of the extensor apparatus may be quite difficult and requires a separate detailed experimental examination of the anatomical structures involved. Equations (16a) and (16b) are linear equations in the forces F,, but are not independent as they depend on equation (1 la). An independent linear equation can be obtained after subtracting equation (16b) from equation (16a). This yields (2% - l)F, + F6 - F, = 2Ke.
(17)
Equations (16a) and (16b) may always be derived from equations (1 la) and (17), and therefore, will be considered to be dependent equations. The employment of linear programming algorithms requires that all constraint equations are linear ones and, that all coefficients are known. However, the value of the coefficient a is not known and, in fact, z may be a nonconstant coefficient. Therefore in order to use a linear programming algorithm z must be guessed or. if possible, estimated. Based on equation (16) the bounds on tl can be stated as KeIF,
< z<
1 + Kc/F,,
F, > 0
118)
otherwise negative forces for Fh or F8 may result. When it is assumed that the coefficient r is an independent variable it is impossible to employ a linear programming algorithm, because equation (17) becomes nonlinear. Therefore, a second simulation must be conducted using a nonlinear optimization algorithm. In the following section actual finger flexion is simulated in order to assess the validity of the approach taken in this section. 111.SIMULATION
AND RESULTS
The proposed model of interphalangeal coordination is tested by conducting digital computer simulations. Figure 4 shows the simulation diagram of the overall system. The input to the system is the desired motion of the finger. This desired motion is given in terms of O’l:,, and O$,,. Only these two angles are specified because Olj:,, is related to O&. With the given desired trajectories, this dynamic model computes the necessary forces that must act on the finger in order to bring about a specified motion. In the absence of external disturbances, loads, or fatigue,
H. J. BUCHNER, M. J. HINESand H.
464
Ut
HEMAMI
F
U Fcfce optimization andIP - kinetics
Desired trajectar
Fig. 4. Simulation block diagram of the overall system. the actual angular trajectories, are the outputs of the simulation diagram, and are almost identical to the inputs, i.e. the desired trajectories. Under load or disturbance, the actual trajectories may not be identical to the desired ones and larger forces may be necessary. These forces can be computed by the diagram. The diagram therefore is capable of producing the needed forces F(t), and the actual trajectories of motion for all specified desired trajectories of motion under loading, disturbance, and fatigue conditions. The latter could be implemented by addition of appropriate saturation levels on the produced forces. The force vector F is computed using a linear and a non-linear optimization algorithm for comparison. The computation of torque u and feedback mechanisms are not discussed in this paper. A finger movement from full extension to full IPflexion was simulated by a digital computer. Initially, the fully extended finger is in the horizontal palm downwards directed position. From this position only the middle and distal phalanges are flexed to reach their final position at about 200” and 273”, respectively. In order to study the merit of the interphalangeal model, the desired reference trajectory did not coincide with the one in Fig. 3. Therefore, a reference trajectory was chosen such that the relationship between the PIP joint angles and the DIP joint angle is a linear one. A good interphalangeal model will cause the interphalangeal joints to flex according to the parabolic function in Fig. 3 despite a different desired trajectory. The force vector F was computed using two different optimization methods. A simplex linear programming algorithm was employed in the first optimization of F. The set of constraint equations consists of equations (1l), (17), and the one given below CJ = M(O)F
(18)
where M and Fare as given in equation (7) and U is the total feedforward torque as indicated in Fig. 4. The corresponding linear performance function is given in equation (13). The coefficients of the performance function are taken from Amis et al. (1979). In order to employ the linear programming algorithm the value of the coefficient u was chosen to be a = l/2 = constant.
It was found that with a = l/2 it is possible to obtain a feasible solution throughout the entire simulation. This suggests that cx= l/2 is a reasonable choice. Contrarily, if a = 1 the linear programming algorithm fails to produce a feasible solution during the whole simulation. However, it is obvious that if a is chosen to be TV= l/2 an overly severe restriction is imposed upon the force distribution between the middle and the lateral band. This problem may be avoided if a nonlinear programming algorithm is employed and if c( is considered to be an independent variable. The linear programming algorithm solves for the unknown forces Fi 3 0, i = 1, . . , I; I = 8, in such a way that the performance function equation (13) is minimized given the j = 6 linear equations (1 l), (17) and (18). Due to the nature of linear programming algorithms, 1-j = 2 muscle forces will always be set to zero. This is independent of the muscle model being implemented. In other words, a linear programming algorithm always ‘removes’ 1-j muscles from the model. Moreover, the computed forces represent the total force produced by passive and active structures of a particular muscle. A nonlinear optimization method was used to optimize the forces in F using the quadratic performance function in equation (14) subject to equations (1l), (17) and (18). The optimization procedure also computed the coefficient a in equation (17) to obtain the smallest possible muscle or tendon forces. The weighing coefficients in equations (13) and (14) are identical.
Table 2. Parameters of the dynamical equation di
ki
mi
4
Link i
mm
mm
g
Kgm’
Proximal phalanx
1
Mid phalanx Distal phalanx
2 3
46 28 20
23 14 10
125 60 40
Segment name
22E-6 3.92E-6 1.33E-6
d, = length of link i. k, = distance of joint i to center of gravity of link i. mi = mass of link i. Ii = inertia of link i.
465
A model for finger coordination The finger movement from full extension to full flexion was simulated within a time frame of T = 1 s. The results of the simulations are shown in Figs S-13. The absolute positions of the three phalanges are shown in Fig. 5. The angles of the phalanges are measured from the vertical in a clockwise direction. In Fig. 6 the velocity profiles of the three phalanges are shown. Each phalanx starts and ends with zero angular velocity, i.e. the phalanges start and end in an equilibrium. The distal interphalangeal joint angle vs the proximal interphalangeal phalangeal joint angle of the simulation is shown in Fig. 7. The elasticity coefficient K of the extensor assembly which determines the load distribution between the middle band and the lateral band was selected to be K = 500 N m- ‘, since a better value was not available from literature. However, the apparently small value of K already yields a good
20
,
-5
s
I
I
1
I
I
0
02
04
06
00
I
F5
F,:
Fig. 8. Muscle forces in the finger (linear optimization). FDP, FQ: Int, F,: Lum.
“i 25 -
20
5,
l s)
Time
-
51 0
I
I
I
I
I
0.2
0.4
06
0.6
I
(s )
Time
F, :
Fig. 9. Muscle forces in the finger (linear optimization). EC, Fh: middle band.
00 Tlmets)
Fig. 5. Position of the phalanges phalanx, 0, : middle phalanx,
vs time. 8,: proximal B3: distal phalanx.
14
.
13 7
I2 -
6
II
-
I
7 0
-I
1
0
I
I
02
04 Time
Fig. 6. Angular
velocity
(
I
I
I
06
08
I
s
I
)
of the phalanges
02
I 04 Time
I
I
I
06
06
I
(s)
Fig. 10. Muscle forces in the finger (linear optimization). middle band, F,: terminal band-lateral band.
vs time.
08
06
04 Trne
(s
joint angles. DIP angle vs PIP angle.
I
)
Fig. 11. Muscle forces in the finger (quadratic Fig. 7. lnterphalangeal
F,:
F,: EC, F,: FDP.
optimization).
466
H. J. BUCHNER, M.
3 ‘t 2’ 0
I 02
Y-2-I
I 04 Time
I
I 08
I
J. HINES and
1
I
I P,”
finger (quadratic optimization). F6: middle band, F,: terminal band-lateral band.
Fig. 12. Muscle forces in the
40 )
I
3020-
-20
-
-30 -40 -n 1_
-
0
I cl2
I 04 Time
I 06
I 08
I I
Is)
Fig. 13. Excursion of the tendons or muscles in the finger vs time. correlation between the simulation results and the theoretically established interphalangeal joint behavior as is evident from Fig. 3 and Fig. 7. Choosing a larger value for K only will improve the result-as K approaches infinity Figs 3 and 7 become indistinguishable. The digital computer simulation may confirm Harris and Rutledge’s theory that interphalangeal coordination can be brought about by the extensor mechanism only. This means that proper coordination does not necessarily depend on the intrinsic muscles or the retinacular ligament. The profiles of the tendon forces in the finger are shown in Figs 8-10 for the linear optimization function and in Figs 1l-12 for the quadratic function. Although both optimization methods yield identical IP-behavior the forces in the finger are entirely different. The forces of the flexor digitorum profundus (F2), the flexor digitorum superficialis (F3), the interosseus (F4) and the lumbricalis (F,) are plotted in Fig. 8. Figure 8 shows that during the entire period of simulation the lumbricalis muscle is ‘silent’ i.e. F, = 0, and that the FDS is active only during the first 0.22 s. Beyond 0.22 s of the simulation the interosseus muscle becomes active until its force reaches a relatively large value of about 18 N. The force profile of the FDP shows a slight decrease during the first 0.22 s and then increases to a value of about 9.5 N. Figure 9 shows the forces of the extensor digitorum (F,) and the middle band (Fb). The forces also show an initial decrease during the first 0.22 s and a subsequent increase to relatively large values of F, = 26.4 N and F6 = 13.2 N. The forces of the middle band (Fb)and the lateral band
H. HEMAMI
(F,) are given in Fig. 10. Since the force of the lumbricalis is zero during the entire simulation it follows that the force in the terminal band ( F8) is equal to the force in the lateral band. A more favourable result can be obtained if a quadratic performance function is optimized, including the a-coefficient as additional independent variable. This can be shown in Figs I l-12. Figure II shows the forces of the extensor digitorum and the flexor digitorum profundus. Contrary to the first optimization method in this simulation the forces F, and F2 decrease by about 7.5 N and 3.5 N respectively. Since forces of the flexor digitorum superficialis, interrosseus and lumbricalis remained constant at zero, their force profiles are not plotted here. The forces of the middle band and the lateral band are shown in Fig. 12. Initially the lateral band bears the larger share of the extensor digitorum force, however after approximately 0.22 s the middle band begins to carry the larger portion of the extensor digitorum force while the lateral band force decreases significantly. It is interesting to observe, that the time at crossover of the forces Fb and F7 in Fig. 12 coincides with the time where the forces F, = F4 = 0 in Fig. 8. Moreover, it seems that the corresponding forces in Figs 8-10 and Figs 11-12 are equal at that time. The results of the second optimization may be favored on the ground of some electromyographical studies as well. Long and Brown (1964) have shown in their electromyographic study that flexion of the middle and distal phalanges is primarily brought about by the flexor digitorum profundus and the extensor digitorum. Other muscles are silent during this movement. Landsmeer and Long (1965) conclude that the extensor digitorum acts as a ‘brake’ during finger flexion and that the intrinsic muscles are passive during this movement. This breaking phenomenon may also explain why the force of the extensor digitorum approaches a constant value as does the flexor digitorum profundus. It appears that flexing of the unloaded interphalangeal joints is possible when the ‘extensor brake’ is released by reducing the antagonistic force which opposes flexing and by lengthening of the extensor digitorum. Simultaneously, the flexor digitorum profundus contracts to its possible minimum length. This, however, does not imply that the flexor digitorum profundus force will increase, especially if an equilibrium cannot be found which minimizes the total muscle effort. The excursion of the extrinsic and intrinsic muscles are shown in Fig. 13. Referring to equations (3) and (5), the excursion of a contracting muscle assumes positive numerical values and an elongating one takes on negative values. The excursion of the muscles are given in mm and each trace shows a distinct excursion characteristic except the extensor digitorum and the interosseus. The excursions of the latter two muscles are identical due to the model and the particular movement being studied. The load distribution coefficient cf is plotted in Fig. 14 which shows that the lateral band bears a
467
A model for finger coordination
,::
i-h
0
02
04
(s9”
08
Time Fig. 14. The value of the coefficient
a vs time.
larger force than the middle band if the finger is fully extended while in the finger that is flexing the middle band bears the larger load. In the flexed finger position the lateral band bears about 28 ‘;/oof the total extensor digitorum force, while the middle band carries the remaining 72 ‘x, of the force.
slightly
IV. SUMMARY AND CONCLUSIONS
A dynamic model for finger intecphalangeal functional coordination was proposed in this paper. The dynamics of the skeletal structures were modeled as an open chain of coupled rigid bodies. Two distinct types of tendon elongation models were selected which suitably describe the relationship between the tendon displacements and the joint angles in the finger. One type of model is a linear one and is used for all extensor tendons and some flexor tendons as well. The second type of model describes the displacement of the lateral band and most flexor tendons by parabolic functions. It may appear that for some joints and muscles the tendon excursion models ace too rudimentary, however. experimental studies conducted in recent years. justify the approximations of the proposed models. Further extensions and uses of these models are possible. If more precise models of the tendon excursion in the finger ace required for various studies on finger and hand function. refined models should be developed in the future. There may also be a vital need for a more precise description of the kinematics and kinetics of interphalangeal coordination. This description. however, must be based on some very accurate anatomical and rheological studies on the extensor assemblage. In particular the structures and functions of small tendon-like tissues which constrain tendon shift of the lateral and middle bands in the palmar and lateral direction during finger Aexion should be precisely studied in order to develop more accurate mathematical models. The classical overdeterminancy problem that arises in specifying the tendon forces in the finger has been solved by employing two optimization methods. The outcome of the two optimization approaches were contrasted, and it was found that the results of the second approach ace more encouraging. In particular the large force of the interosseus and the increasing forces of the extensor digitocum may be opposed on
several grounds. First, electcomyogcaphical studies indicate that the intrinsic muscles are silent during IF’ flexion, which means, however, that the computed Interosseus force is unlikely to occur. Second, the large increase of the extensor digitocum force during finger flexion is physiologically inexplicable, although mathematically possible. Third, a linear programming algorithm does not allow any variation of the Ecoefficient when the value of LIis fixed. Contrarily. the nonlinear optimization process views a not as a fixed coefficient but as a variable that is computed much like the muscle forces. In fact, this difference seems to be the main reason for the so significantly distinct muscle forces. Finally, the kinematical model of the finger interphalangeal functional coordination itself was derived from a rather abstract point of view and was not based on any experimental measurements. Better kinematical models therefore may be established from experimental studies of IP coordination in order to allow for more accurate correlation with physiological models. The present results and future studies should provide for better treatment of abnormal finger function, and for more advanced prosthetic devices. Acknowledgement-The authors would like to thank the reviewers for their careful reading of this manuscript and many helpful suggestions. This work was supported by the National Science Foundation under Grant No. ECS-820-1240.
REFERENCES
Amis, A. A.. Dowson.
D. and Wright, V. (1979) Muscle strengths and musculo-skeletal geometry of the upper limb. Engng Med. 8,41-48. Ann, K. N., Kwak. B. M.,Chao, E. Y. and Morrey, B. F. (1984) Determination of muscle and joint forces: a new technique to solve the indeterminate problem. J. ho&em. Engng 106, 366367. An, K. N., Ueba, Y., Chao. E. Y.. Cooney. W. P. and Linscheid, R. L. (1983) Tendon excursion and moment arm of index finger muscles. J. Biomechanics 16, 419425. Armstrong, T. J. and Chaffin. D. 6. (1978) An investigation of the relationship between displacement of the finger and wrist joints and the extrinsic finger flexor tendons. J. Biomechanics I 1, 109- 118.
Berme. N., Paul, J. P. and Purves, W. K. (1977) A biomechanical analysis of the metacarpophalangeal joint. J. Biomechanics IO, 409412. Brand, P. W., Cranor, K. C. and Ellis, J. C. (1975) Tendon and pulleys at the metacarpo-phalangeal joint of a finger. J. Bone Jt Surg. 57A, 779-784. Buchner, H. (1983) Finger movement in the sagirtal plane and the mechanisms of touch control. M.S. Thesis. The Ohio State University, Columbus, OH. Buchner, H.. Hines, M. J.. Hemami, H. (1985) A mechanism for touch control of a sagittal Five-Link Hand-Finger. IEEE Trans. Sysrems. Man. Cvberner. 15, 69-77. Chao. E. Y., Opgiande, J. D. and Armer, F. E. (1976) Threedimensional force analysis of finger joints in selected isometric hand functions. J. Biomechanics 9. 387-396. Fischer, G. W. (1969) A treatise on the topographic anatomy of the long finger and a biomechanical investigation on its inter-joint movement. Ph.D. Dissertation. The University of Iowa, Iowa City, IA. Haines. R. W. (1951) The extensor apparatus of the finger. J. Anar. 85, 151-259.
468
H. J. BUCHNER,M. J. HINESand H. HEMAMI
Harris, C. and Rutledge, G. L. (I 972) The functional anatomy of the extensor mechanism of the finger. J. Bone Jr Surg. 54-A, 713-726. Hemami, H. (1982) Some aspects of Euler-Newton equations of motion. Ingenieur-Archio. 52, 167-176. Landsmeer, J. M. F. (1949) The anatomy of the dorsal aponeurosis of the human finger, and its functional significance. Anat. Rec. 104, 31-44. Landsmeer, 1. M. F. (1955) Anatomical and functional investigations on the articulation of the human hand. Acta anat. Suppl. 24, l-69. Landsmeer, J. M. F. (1958) A report on the coordination of the interphalangeal joints of the human finger and its disturbances. Acta morph. neer-stand. 2, 59-84. Landsmeer, J. M. F. (1960) Studies in the anatomy of articulation. Acta morph. neer-stand. 3, 287-321. Landsmeer, J. M. F. and Long, C. (1965) The mechanism of finger control based on electromyograms and location analysis. Acta anat. 60, 330-347. Long, C. and Brown, M. E. (1964) Electromyographic kinesiology ofthe Hand: muscles moving the long finger. J. Bane Jt Surg. 46-A, 1683-1706. Penrod, D. D., Davy, D. T. and Sigh, D. P. (1974) An optimization approach to tendon force analysis. J. Biomechanics
APPENDIX J(e)
a,,cos(g,
aI1
=
a2 i cos(& - 0, )
a3, c0s(e3 - 8, )
1
J,(e) =
a22
a,,cos(e,
-2R2161 + RSZ+ Rx + (2R52+ 2JG,M,
R,, -R,,
a,,
= d,(m,k,
a,,
= 12+m2k:+d:m,
+d,ms)
a23 = dzm3k3 as, = d,m3k, as1 = d,msk, ass = 13+m,k:
G=
gsinO,Cm,k, +d,(m,+m~)l gsin&[m,k2 +d,m,]
gsinB,m,k, 1 I The sensitivity 3 x 8 matrix M(0) whose element m,j is given by j= 1.2 dei de, mi’=~-drpj+, i = 1,2,3,4, 5, M, L, T and dei mi, = &
j=3 i = 1, 2, 3, 4, 5, M, L, T
is given in Table 3. For ease of reading, the transpose of the matrix is listed where the successive columns of the list are the rows of M.
0 &+2&41 - (R23 + %3,3)
0
0
&,+2R;,4,
0 I
I
-
WSZ
+
0
I
- Ru
R22)
I
-R,3-&3 -=;3+3
-W;2+2R;,k#9
+R,3+&,+%393
I
R I*
RI2 4, -2R;242
I I I
I
I
I I
1
a 12 = d, (m,k, +d,ms)
R 32
I
1
a 13 = dtm3k3
I &+%I& -R32 &I + RN
a33
a,,sin(f?, -e,) a,3sin(8, -0,) 0 a2,sin(Bz -e,) a,,sin(O, - ~9,) 0
0
a,,sin(e,-0,) a3,sm(6d -8,)
Table 3. Transpose of the sensitivity matrix. M(g)
&I +X141 - (Jh +2&w)
-f33)
a,,cos(g, - &)
a,, =I,+m,k:+d:(m,+m,)
Spoor, C. W. and Landsmeer, J. M. F. (1976) Analysis of the zigzag movement of the human finger under influence of the extensor digitorum tendon and the deep flexor tendon. J. Biomechanics 9, 561-566. Stack, A. G. (1962) Muscle function in the fingers J. Bone Jt Surg. 44-B, 899-909. Storace, A. and Wolf, B. (1979) Functional analysis of the role of the finger tendons. J. Biomechanics 12, Y&578. Thomas, D. H., Long, C. and Landsmeer, J. M. F. (1968) Biomechanical considerations of lumbricalis behavior in the human finger. J. Biomechanics. I, 107-l 15. Tubiana, R. (1981) Architecture and functions of the hand. The Hand, (Edited by Tubiana, R.) pp. 19-93. W.B. Saunders, Philadelphia. Wells, R. P., Ranney, D. A. and Keller, A. (1984) The interaction of muscular and passive elastic forces during unloaded finger movements: a computer graphics model. Proc. 4th European Society of Biomechanics, Switzerland. Wittenburg, J. (1977) Dynamics of Systems of Rigid Bodies. B. G. Teubner, Stuttgart.
I I
- e2)
a,,cos(e,
The elements aij of the previous matrices are:
7, 123-l 29.
-R,1
-Al)
-R,z+2R;zd, R 13
I I
0 -R,3