A Fortran package for the planar analysis of limb intersegmental dynamics from spatial coordinate-time data

K L A U S S C H N E I D E R A N D R O N A L D F. Z E R N I C K E

Department o f Kinesiology, University o f California, Los Angeles, CA 90024-1568, USA

Limbs in humans and animals are systems of linked bodies, in which motion of any one part of the linkage exerts forces on the remaining parts. Thus, during natural movements, forces can act on a limb segment even if a segment is not exposed to active muscle forces. Passive reactions arise from forces in tendons, ligaments, and peri- articular tissues, as well as from dynamical limb reactions related to inertial, Coriolis, and centri- petal forces. Consequently, limb trajectories are influenced not only by active muscle forces but also by passive, intersegmental forces. For a more complete understanding of movement control, it is important to explain the mechanisms by which active and passive forces are coordinated. To achieve this, an analysis of limb intersegmental dynamics is essential, and here we provide a detailed description of the equations used for the planar analysis of limb intersegmental dynamics from spatial coordinate-time data, as well as the FORTRAN software to perform those calculations.

Key Words: Limb dynamics, human movement

INTRODUCTION

Natural movements of humans or animals require the coordination of interconnected and interacting limb segments moving in three-dimensional space. Deter- mining how such coordinated actions are controlled, however, is a profoundly complicated problem. Of the many studies directed at understanding movement control, most, unfortunately, involve simple, restricted movements, and kinematics are generally only ana- lyzed. Very little is known about the dynamical aspects of movements (e.g., Hollerbach and Flash1), but dynamical analyses are essential for quantifying the mechanical causes of a movement, because kinematical analyses only quantify the effect, the overt behavior.

Nevertheless, building on the work of Bernstein 2 and

Accepted April 1990. Discussion closes January 1991.

Manter 3, researchers have recently made significant advances in quantifying limb dynamics by focusing on how limb motions are influenced by joint torques resulting from active and passive forces (e.g., Smith and Zernicke4). Torques at a limb's joint arise not only from muscles that are actively contracting but also from gravity and passive limb reactions to muscle actions. Those passive reactions include forces in ten- dons, ligaments, and periarticular tissues, as well as dynamical limb reactions related to inertial, Coriolis, and centripetal forces. The dynamical limb reactions originate because limbs are systems of linked bodies, in which the motion of any one segment exerts forces on the remaining parts of the linkage. Thus, torques can act on a limb segment, even if that segment is not exposed to active muscle forces. Recent studies have examined these intersegmental or motion-dependent torques, and thus, have begun to explain mechanisms by which intersegmental torques significantly influence limb trajectories (e.g., Hoy and ZernickeS; Schneider et al.6).

The methods used in these recent studies generally consisted of." (1) the recording of the movement in three dimensions (e.g., via high-speed cin6 film or video), and (2) the calculation of the torques at each of the limb joints using inverse dynamics, where the limb (e.g., lower extremity) is modeled as a set of interconnected- planar rigid links (e.g., thigh, leg, foot) with frictionless joints (e.g., hip, knee, ankle). To reduce the complexity of the dynamical analyses and still maintain the fidelity of interpretation, a planar rather than a spatial analysis of limb intersegmental dynamics has been used, with the dynamical analysis restricted to the moving-local plane containing the limb segments. Thus, at each limb joint torques were calculated about axes that were normal to the moving-local plane. Here we describe: (I) the equations used for such a planar analysis of limb intersegmental dynamics from spatial coordinate-time data, and (2) the FORTRAN subroutines to perform those calculations.

DATA COLLECTION

The equations of motion assume that at least four points are marked on the moving limb (contrasting markers, infrared diodes, or reflective markers) for which spatial coordinates with respect to time are

© 1990 Computational Mechanics Publications Adv. Eng. Software, 1990, Vol. 12, No. 3 123

recorded (via high-speed cin6 film, video, or opto- electronic devices). Markers are assumed to be at the rotational centers of the limb joints, and one marker is placed distal to the most distal joint. With the nomen- clature: M3, M2, M1, and Mo, M3 is the most proximal and Mo the most distal marker. For example, going proximal to distal for an upper limb, M3 is at the rota- tional center of the glenohumeral joint, M2 of the elbow joint, M1 of the wrist joint, and Mo of the third metacarpophalangeal joint. For a lower limb, M3 is placed at the rotational center of the hip joint, M2 of the knee joint, M1 of the ankle joint, and Mo of the fifth metatarsophalangeal joint.

muscular forces are embedded within this compo- nent, the generalized muscle torque comprises the actively-controlled elements of limb-trajectory motor programs, as well as passive effects of muscle and connective tissues.

While the net, gravitational, and interactive torques are calculated directly from the limb kinematics, the generalized muscle torque is calculated as a "residual" term, because the sum of the generalized muscle torque and the other torques equals the net torque. There is no ubiquitious term for this torque component, and the term "generalized muscle torque ''4'~'7 or "residual torque" has also been called "joint torque" 9.

L I M B M O D E L

The limb is modeled as three interconnected-planar rigid links or segments (S3, Sz, S1) with frictionless joints (J3, J2, J1), where $3 and J3 are most proximal, and S1 and J1 are most distal. J3 is allowed to translate freely, and thus, the resulting torques due to the linear accelerations of J3 are included in the equations of motion. For example, for an upper limb from proximal to distal: $3 = upper arm, $2 = forearm, and $1 = hand, J3 = glenohumeral joint, J2 = elbow joint, and J1 = wrist joint. For a lower limb, $3 - $1 correspond to the thigh, leg, and foot, while J3 - J1 correspond to hip, knee, and ankle joints. Figure 1 illustrates a human arm moving in an inertial (x-y-z) coordinate system. The orientation of the (x-y-z) coordinate system is arbi- trary and is common to all the elements.

I N T E R S E G M E N T A L D Y N A M I C S

Rigid-body equations of motion can be formulated in several ways, but the form of the equations used here allows not only gravity and muscle influences on limb motion to be quantified, but also how the motion of one segment affects other segments. To reduce the com- plexity of the analysis, limb dynamics can be calculated in a moving-local plane that contains the limb joints. Thus, at each successive instant in time and at each seg- ment, joint torques are calculated about axes that are normal to the moving-local plane. At each of the three joints, torques are partitioned into four categories that can be generally defined as4'5'7'8:

1. Net Joint Torque: The sum of all the positive and negative torque components (gravitational, interactive, and muscle) that act at a joint.

2. Gravitational Torque: A passive torque resulting from gravity acting at the center of mass of each segment.

3. Interactive Torques: Passive torques arising from dynamical interactions among segments, such as inertial forces proportional to segmental accelera- tions or centripetal forces proportional to the square of.segmental velocities.

4. Generalized Muscle Torque: A "generalized" torque that includes forces arising from active muscle contractions and from passive defor- mations of muscles, tendons, ligaments, and other periarticular tissues. Because the effects of active

Equations o f motion The equations of motion for a three-segment model

are (with segments $3, $2, and $1, and joints J3, J2, and J1 from proximal to distal):

. Joint J3: The net joint torque [NET] at J3 is equal to the sum of the torque components: generalized muscle torque M3 [MUS], and the torque compo- nents due to segment $3 angular acceleration [3AA], $3 angular velocity [3AV], $2 angular acceleration [2AA], $2 angular velocity [2AV], S~ angular acceleration [1AA], S1 angular velocity [1AV], J3 linear acceleration [3LA], and gravity [GRA]. The torque equation for J3 is:

( / 3 + f~3),~;3

= [NET]

M3 [MUS]

- [/3s +/39 +/31 cos(01 - 03) q- (/34 + /35)COS(02 -- 03)] ~3 [ 3 A A ]

- [/31sin(01 - 03) + (/34 +/3~)sin(02 - 03)]~] [3AV]

- [ h + ~22 +/36 +/32 cos(01 - 0~)] + (/34 +/35)c0s(02 - 0~)] ~2 [2AA]

- [/32 sin(0~ - 02) - (/34 +/3~)sin(02 - 0 ~ ) ] ~ [2AV]

+ [11 + fll -/32 CO$(01 -- 02) --/3~ COS(01 -- 03)]~I [1AA]

+ [/31 sin(0~ - 03) +/32 sin(01 - 0z)]~Zl [1AV]

+ [/31o sin 03 +/37 sin 02 + /33 sin 01] 3?3 [3LA]

- [/31o cos 0 3 - / 3 7 c o s 0 ~ + / 3 ~ cos 01] ~ 3 [3LA]

- [/311 cos 03+/37 cos 02+/33 cos 0 1 ] g ' [GRA]

(1)

2. Joint J2: The net joint torque [NET] at J2 is equal to the sum of the torque components: generalized muscle torque M2 [MUS], and the torque compo- nents due to $3 angular acceleration [3AA], $3 angular velocity [3AV], $2 angular acceleration [2AA], Sz angular velocity [2AV], $1 angular acceleration [1AA], $1 angular velocity [1AV], J3 linear acceleration [3LA], and gravity [GRA].

124 Adv. Eng. Software, 1990, Vol. 12, No. 3

The torque equation for J2 is:

(12 + ~']2)¢~2 = [NET]

M2 [MUS]

- [34 c0s(02 - 03) + 35 cos(O~ - 4,3). "+" 31 COS(01 -- 03)]03 [ 3 A A ]

- [~4 sin(02 - 03) + 3~ sin(02 - 03) +31 sin(01 - t.b3)]$32 [3AV]

-- [~6 d- 32 COS(01 -- 02)] ~2 [2AA]

- [32 s in (01- 02)] ~22 [2AV]

- - [I1 + f]l +/$2 COS(01 -- 02)] t~l [1AA]

+ [32 s in(01- 02)] ~ [1AV]

+ [33 sin 01 + 37 sin 02] )?3 [3LA]

-- [33 COS 01 + 37 COS 02].~3 [3LA]

- [/33 cos 01 + 37 cos ~b2] g' [GRA]

(2)

3. Joint Jl: The net joint torque [NET] at J~ is equal to the sum of the torque components: generalized muscle torque M1 [MUS], and the torque compo- nents due to $3 angular acceleration [3AA], $3

angular velocity [3AV], $2 angular acceleration [2AA], $2 angular velocity [2AV], J3 linear acceleration [3LA], and gravity [GRA]. The torque equation for J1 is:

([1 -~- ~1)~1 = [NET]

MI [MUS]

-- [31 COS(~I -- 63)] (~3 [3AA] - [/31 sin(01 - 03)] q~a 2 [3AV]

- [32 c os (01 - 4)2)] ~2 [2AA]

- [ 3 2 s in (01- 02)]~22 [2AV]

+ [33 sin 0 1 ] ~ 3 [3LA]

-- [33 COS 0 1 ] ) 3 [ 3 L A ]

- [33 cos 01] g ' [GRA]

(3) where

m3, m2, ml r3, r2, rl

13, lz, 11 13, 12, 11

~3, 62, 61

~3, t~2, ~1 ¢3, ~z, ~1 X3, Y3 g'

31 ~2

B3 B4 3~

= masses of S3, $2, and Sl = distances f rom the proximal joint to

center of mass of $3, $2, and $1 = lengths of $3, $2, and $1 = moments of inertia at center of mass of

$3, $2, and $1 = orientation angles (at proximal end of

segment from the right horizontal) for $3, $2, and $1

= angular velocities for $3, $2, and S1 = angular accelerations for $3, $2, and $1 = linear accelerations for J3 = component of the gravitational constant

( g=9 .81 ms -z) acting at a segment's center of mass in the moving-local plane that contains $3, $2, and $1

= m l r l l 3

= m l r l l 2

= m l r l

= m11213 = m2r213

3e = m112~ 3 7 "~" m112 + m2r2 38 = m21] 39 = md] [31o = m3r3 + m21~ + mli~ ~11 = m2h + mlh + mara fll = mlr~ f12 = m2r~ fl~ = m~r]

Anthropomorphic data

Segmental lengths (li), center-of-mass locations (ri), masses (mi), and moments of inertia (Ii) can be obtained by direct measurement using standard pro- cedures as described in the literature 1°'n'12, and consequently, variables 3i and fli in (1)-(3) can be calculated.

Orientation angles At each instant in time, the moving-local plane (P1)

(see Figure 1 for the human arm) containing J3, J2, and J1 can be mathematically described with

A l x + B i y + Clz + D1 = 0 (4)

where A1, B1, C1, and D1 can be determined from

X -- Xl Y -- Yl Z -- Zl

X 2 - - X 1 Y 2 - - Y l Z 2 - - Z l = 0

x 3 - x l Y 3 -Y l Z3-Zl

to

(5)

A1 = (y2 - yl)(Z3 - zl) - (y3 - yl)(Z2 - Zl)

B1 = (x3 - xl)(z2 - Zl) - (x2 - xl)(z3 - Zl)

C1 = (X2 -- x~)(y3 - Yl) - (x3 - x1)(y2 - Yl) D1 = - x I A 1 - y I B 1 - Z l C 1 (6)

with (x3, Y3, Z3), (x2, Y2, Z2), and (xl, yl , zl) being the three-dimensional coordinates of joint markers M3, M2, and MI.

Z ~

Z

Y

Fig. 1. Human arm moving in an inertial (x-y-z) coor- dinate system. The limb is modeled as three interconnected-planar rigid segments ($3, $2, SD with frictionless joints (J3, J2, Jr). A moving-local plane (P1) contains Js, J2, and Jr. The orientation angle ( 0 ) f o r each segment (S) is calculated with respect to the right horizontal (x '), and the torque at jo int (J) is calculated with respect to axis (z ') which passes through the joint and is perpendicular to the moving-local plane (Pr)

Adv. Eng. Software, 1990, Vol. 12, No. 3 125

To calculate the orientation angles (~ ) in the moving-local plane (P1) at the proximal end of segment (S~) a right horizontal has to be determined within the moving-local plane. It is assumed that the three- dimensional coordinates of the joint markers M~, M2, and M1 at J3, J2, and J1 are recorded in a Cartesian coordinate system ( x -y - z ) that is oriented with respect to anatomical axes, with: (1) the x-axis being anterior- posterior, (2) the y-axis being medial-lateral, and (3) the z-axis being inferior-superior. Furthermore, it is assumed that the analyzed motion occurs in the first octant (x ~> 0, y ~> 0, z ~> 0) of a Cartesian coordinate system. The right horizontal is then defined as the inter- section between the moving-local plane (P1) and a plane which is parallel to the x -y plane containing the joint at which the orientation angle is calculated. With the cosines (cos a, cos ~, cos 30 describing the spatial orientation of the intersection, the intersecting line becomes a "right" horizontal for cos o~ ~> 0.

Any plane (P2) which is mathematically described by

A z x + B 2y + Czz + D2 = 0 (7)

is parallel to the x -y plane if A2 = B2 = 0. The intersec- ting lines between plane (Px) and any plane ( P z ) which is parallel to the x -y plane are oriented parallel to vector rx, (I~,, mx,, nx,) where

Ix' = B1C1 ] = B1C2 BzCz I

[ C 1 A l l = - A I C 2 (8 ) m x ' = C2A2

lAmB1[ = 0 n x ' = A z B z

The directional cosines for vector rx, are

cos a x ' = lx'(lZ~ ' + rn~x ' + nZx,) -°'5 =BI(A21 + B~) -°'5

cos Bx' = mx'(l~x ' + m~x ' + nZx') -° '~ = - A I ( A ~ + B ~ ) - ° ' ~ (9)

cos 7x' = 0

To' calculate the orientation angles (~ ) in the moving-local plane (P1) at the proximal end of the segment (S~) from the right horizontal, a Cartesian coordinate system ( x ' - y ' - z ' ) has to be established with its origin at joint (J i) where the orientation angle (q~) is calculated, and where the x ' -axis is the right hori- zontal, thus, being parallel to vector rx,. The orien- tation of the y ' -axis and the z '-axis of the coordinate system ( x ' - y ' - z ' ) can be found by determining vector ry, (ly,, my , , ny,) which is normal to the moving-local plane (P1), and vector rz, ( l z , , m z , , n z , ) which is per- pendicular to vectors rx, and i'y, fulfilling the right hand rule. Any vector ry, ( l y , , m y , , n y , ) which is normal to the moving-local plane (Px), and thus, normal to vector rx, is described by directional cosines

COS Oty' = ly,(lZy , + m2y , + n~,,) -°'5

= A 1 ( A ~ + B~ + C~) -°'~

cos 3y' = my,( l~, + mZy , + nZy,) -°'5 (10) = B1(A21 + B21 + C21) -0"5

cos ")'y' = ny,(l~y , + m~, + n~,) -°'5

: C l (A ~z + B~ + C~) -°'5

Any vector r~, (lz', mz , , nz') which is .perpendicular to

vectors rx, and ry, fulfilling the right hand rule is defined by

i j k i j k

rz' = rx, xry, = Ix, rex' nx, = 91 - A ~ 0

ly' my, ny, A~ B~ C1 (ll)

where i, j , and k are the unit vectors. Thus,

lz, = - A 1 C 1

mz, = - B I C 1 (12)

nz, = A ~ + B~

and the directional cosines for vector rz, are

cos c~z, = lz,(l~, + mZz , + n~z,) - ° 5 = - A I C I [ ( A ~ + B ~ ) ( A ~ + B ~ + C12)]-°"5

cos 3z' = mz'(lZz ' + mY, + nZz,) -°'5 (13) = _ B 1 C I [ ( A 2 1 + B a ~ ) ( A z z 1+61 + C21)] -0.5

cos •z' = nz'(12z ' + mZz ' + n~z') -°'5 = ( A Z ~ + B ~ ) [ ( A Z ~ + B ~ ) ( A ~ + B 2 ~ + C l ) ] 2 - 0 . 5

In the Cartesian coordinate system ( x ' - y ' - z ' ) : the x '-axis is chosen parallel to vector rx,, the y ' -axis is chosen parallel to vector ry,, and the z ' - a x i s is chosen parallel to vector rz,. The x '-axis becomes the "right" horizontal if A1, 61, and C1 from (6) are

A1 for B1 /> 0 A I =

-AI for B1 < 0

B1 f o r B i d > 0 B1 = (14)

-B1 f o r B l < 0

C1 for BI ~> 0 C1 =

-C1 for B 1 < 0

A parallel transformation brings the origin of the, co- ordinate system ( x ' - y ' - z ' ) to joint (J i) where the orien- tation angle (4~i) is calculated, and for each joint (Ji) respectively for each joint marker ( M i ) with coor- dinates (x~, yi, z~), the coordinates ( x l - 1, y [ - ~, z[- 1 ) of the distal end of segment (SD for which the orientation angle (q~) is calculated in coordinate system ( x ' - y ' - z ' ) are

X[- 1 = (X i - 1 -- Xi)COS Otx' + ( Y i - 1 -- y i ) c o s /~x'

+ (Zi- 1 -- Zi)COS "Yx'

Y[-1 = (X i -1 -- Xi)COS Oly' "t- (Y i -1 - yi )COS 3 y ' (15) q- (Z i - 1 -- Zi)COS "/y'

Z[- 1 = (X i - 1 -- Xi)COS Ol z' + ( Y i - 1 -- y i )COS 3z '

"}- (Zi- I -- Zi)COS ~/z'

where (x~- 1, y~- 1, z i - 1) are the coodinates of the distal marker in coordinate system (x -y - z ) . Because the calcu- lation of the orientation angle (4~i) is performed in a plane, the x ' -coordinate and z ' -coordinates only have to be considered. Thus, the orientation angle (4~) for segment (SD calculated at joint (J~) from the right hori- zontal is given as

¢i = arc cos [x/- a (x[-21 + z/ i l l ) -°"5] (16)

where the sign of the orientation angle (positive = counterclockwise, and negative= clockwise) depends on the quadrant of the ( x ' - z ' ) coordinate system in which segment (S~) is positioned, and how the orien- tation angle (¢D changes with time.

The magnitude of the component of the gravitational constant (g) acting at a segment's center of mass in the

126 A d v . Eng. So f tware , 1990, Vol. 12, N o . 3

moving-local plane (P~) can be determined from the orientation of vector rz'.The inclination of vector rz' with respect to the z-axis of coordinate system (x-y-z) describes the inclination of the moving-local plane (Pl ) with respect to gravity, and thus with (13), the compo- nent ( g ' ) of the gravitational constant acting at a seg- ment 's center of mass in the moving-local plane (P~) is defined as

g ' =9.81 ms -2 cos 3~z, (17)

Time derivatives To determine the segmental an.~ular velocities (~i),

segmental angular accelerations (~1), and linear accel- erations (23, .P3) of joint J3 time derivatives may be obtained using standard procedures (e.g., spline smoothing and differentiation as described by Woltr- ing13).

A P P L I C A T I O N

We demonstrate the application of the method described above with data from a study 6 in which we examined Bernstein's 2 hypothesis that practice alters the motor coordination among the muscular and passive joint torques. In our study, male human volun- teers performed maximal-speed, unrestrained vertical arm movements between two target endpoints resulting

E Z

o

60

30

0

-30

Flexion ~

UAA ~

GRI

MUS ~

/ I

/

• Extension

\, • ~

~ .

\

", , ~ •

" ~ s S~ ~

/ /

F~u~ ~,

~'~,~. . ~ ~

-60 I t

0.4 0.5 0.6

Time (sec)

Fig. 2. Time series o f the shoulder torque components during the flexion-extension phase o f an unrestrained vertical arm movement at the beginning (solid lines) and end (dashed lines) o f practice. The torque compo- nents are: muscle torque (MUS), torque components due to gravity (GRA), upper-arm-angular acceleration (UAA), and forearm-angular acceleration (FAA). As subjects practiced the motion, shoulder-extensor muscle torques (MUS) counterbalanced an interactive torque (UAA) that caused the shoulder to flex

in complex shoulder, elbow, and wrist joint move- ments. These movements were recorded by high-speed cin6 film. The arm was modeled as described above, so that dynamical interactions among the upper arm, forearm and hand could be calculated. Our results showed that with practice subjects achieved sig- nificantly shorter movement times. The analysis of intersegmental dynamics indicated that the underlying dynamical cause was that subjects were able to use muscle torques as "complementary forces ' '2 to the passive-interactive torques of the moving limb. For example, in their first trials (see Fig. 2 solid lines) the torque profile for the shoulder joint showed a near static equilibrium when the subjects reversed their motion at the upper target displaying torques close to zero; small muscular activities (see muscle torque MUS) primarily counterbalanced gravity (GRA). With prac- tice (see Fig. 2 dashed lines), however, the torque profile changed to a state of dynamic equilibrium where increased muscle torques (MUS) "complemented" similarily-increased interactive torque components created by segment movements (see torque components UAA and FAA due to the angular acceleration of the upper arm and forearm).

FORTRAN SUBROUTINES

The FORTRAN subroutines to perform the calcu- lations described above are given in the Appendix.

A C K N O W L E D G E M E N T

This research was supported in part by grants from the U.$. National Institute of Health (N$ 19864 and HD 22803).

REFERENCES

1 Hollerbach, J. M. and Flash, T. Dynamic interactions between limb segments during planar arm movements, BioL Cybern., 1982, 44, 67-77.

2 Bernstein, N. The co-ordination and regulation of movements, Pergamon Press, Oxford, 1967. (A commented re-publication is available as: Bernstein, N. Biodynamics of Locomotion, Human Motor Actions: Bernstein Reassessed, H. T. A. Whiting (Ed.), North-Holland, Amsterdam, 1984)

3 Manter, J. T. The dynamics of quadrupedal walking, J. Exp. BioL 1938, 15, 522-540

4 Smith, J. L. and Zernicke, R. F., Predictions for neural control based on limb dynamics, Trends Neurosci., 1987, 10, 123-128

5 Hoy, M. G. and Zernicke, R. F. The role of intersegmental dynamics during rapid limb oscillations, J. Biomech., 1986, 19, 867-877

6 Schneider, K., Zernicke, R. F., Schmidt, R. A. and Hart, T. J. Changes in limb dynamics during the practice of rapid arm movements, J. Biomech., 1989, 22, 805-817

7 Hoy, M. G. and Zernicke, R. F. Modulation of limb dynamics in the swing phase of locomotion, J. Biomech. 1985, 15, 49-60

8 Hoy, M. G., Zernicke, R. F. and Smith, J. L. Contrasting roles of inertial and muscular moments at ankle and knee during paw- shake response, J. Neurophysiol., 1985, 54, 1282-1294

9 Atkeson, C. G. and Hollerbach, J. M. Kinematic features of unrestrained vertical arm movements, J. Neurosci., 1985, 5, 2318-2330

10 Clauser, C. E., McConville, J. T. and Young, J. W. Weight, volume, and center of mass of segments of the human body, Technical report AMRL-TR-69-70, Wright-Patterson Air Force Base, Ohio, 1969

11 Dempster, W. T. Space requirements of the seated operator, WADC Technical report, Wright-Patterson Air Force Base, Ohio, U.S.A., 1955, 55-159

Adv. Eng. Software, 1990, Vol. 12, No. 3 127

12 Hatze, H. A mathematical model for the computational deter- mination of parameter values of anthropomorphic segments, J. Biomech., 1980, 13, 833-843

13 Woltring, H. J. A Fortran package for generalized, cross valida- tory spline smoothing and differentiation, Adv. Eng. Software,

A P P E N D I X

SUBROUTINE HLP IX1, Y1, Z t . X2, Y2, Z2, X3o Y3. Z3, • A, B C)

C~.~ .e •e • • •e •ooeeeo .o . .QeQ~ ,eoeeee •oo •o looeoeoome•oe ,e lee leeeeeeo~eB• .~e C THIS SUBROUTINE OET£NMIN£S A MOVINO-LOCAL PLANE THAOUGH JOINT C HA.qK(RS M | . 142. M3 ( N t MOlT DISTAL - M3 HOST PNOXII~kL). A NIGHT C HOriZONTAL IS DEFINE• AS THE INT[RSECTION SET,TEN THE HOVING- C -LOCAL PLANE ANO A PLAN( ~ I C H IS PARALLEL TO THE X-Y PLANE OF C THE IN(NTiAL COORDINATE SY|T(H. C e v o e e e e e e ~ o t o e s e o s e s e e e e o ~ e t s e t e e e o ~ e e o e e a s e s e e s e e o e e ~ e e e e o e e e e s e ~ ; e t e ~ C I N P U T - X1~ Y l , ZI~ N2, Y2, Z2 , X3o Y3, Z | C X I . . . Z 3 : X- , Yo, Z-COOrDINATES fOR JOINT HARK[RS H1, M2, C H3 C OUTPUT " A, 80 C G A . . . C : PARAMETERS DES(RIDING THE PLAN( GIVEN BY EQUATION C AeX + SeY ~ CeZ ~ 0 • 0 0 ~ ~ • ~ ~ ~ ~ . ~ • ~

IMPLICIT REAL*8 ( A-Z ) A = ( Y 2 - Y I ) • ( Z $ - Z 1 ) - ( V 3 - ¥ 1 | e ( Z 2 o Z T ) B = ( ~ 3 - X t p ( Z 2 - Z I | - ( N 2 - N 1 p ( Z 3 - Z I ) C = ( X 2 - X 1 ) e ( Y 3 " Y t ) ' ( X 3 o X 1 ) e ( Y 2 " Y 1 )

C ~ e ~ O ~ e o e ~ o e O l O O O O O e O ~ e ~ e e e o ~ o e e l e ~ e e e ~ e e o e e e e e e ~ e e ~ e ~ t e e ~ e e ~ e v ~ e ~ e e ~ C TH( iNTENS(OTION BECOMES THE URIGHTn HORIZONTAL IF: C ~ * ~ * * e e ~ l * ~ * ~ e e e ~ o e l * e e e l e • o o * ~ e l ~ e e l l e l e l ~ l ~ e ~ l ~ e ~ * e ~ e * e ~ = * * * ~ * *

IF ( S . L T . 0 . D 0 ) A=-A IF (S.LT.O.DO) C=oC IF ( S . L T . 0 . O 0 ) 8=-8 RETURN END * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * SUBROUTINE ANO IX, Y, Z, U, V, W, A, S, C,

• IO, AOo AG) C ~ e ~ e ~ e ~ e e ~ e ~ e ~ e e ~ e e e ~ e e ~ e ~ e e e ~ ~ C THIS SUBROUTINE DETERMINES THE ORIENTATION ANGL( BETWEEN SEGMEKT S C AND THE RIGHT HORIZONTAL~ ANO THE COSINE OF THE INCLINATION OF THE C MOVING-LOCAL PLANE MITH RESPECT TO GRAVITY. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C INPUT " X. Y, Z, U, V. W. A. B, C C X. V, Z: X ' , Y ' , Z'COOStOINATES OF OISTAL SEGM[NTMARKEI C U° V, ~: X ' , Y 'o Z'CO~DINATES OF PROXIMAL JOINT C A. S, C: PANAMETENS DES(RISING THE MOVING-LOCAL PLAN( C OUTPUT " I 0 . AO. AG C IQ: INOIP~kTOR IN ~HIGH OUAONANT OF THE COORD~HAT[ SYSTEM C SEGMENT S IS LOCAT(O C (~ITH THE OfllOIN IN TNE JOINT, THE MOVING-LOCAL PLANE C AS X ' - Y * PLANE ANO THE RIGNT HORIZONTAL AS NtoAXIS) C AO: ONIENTATION ANGL( OF S(GM(NT S ~ T RIGHT HORIZONTAL c ( IN R~) C AG: INCLINATION OF TH( MOVING'LOCAL PLAN( ~RT GRAVITY C (COSINE OF INCLINATION ANGLE) C ~ * ~ • e ~ e ~ e * ~ e ~ e e e ~ e e e ~ e e ~ e e ~ e e e ~ e ~ e ~ * * e ~ e e ~

IHPLICIT REALES ( A-H, J 'Z ) INTEGER IG TI=D$~T(A~A+B~B) T2=OSGRT(A~A÷B~B+C'C) T3=Tt~T2 L I=B/T1 N I - o A / t I L3~oA•C/ t3 N3-*SeC/T~ N3=(AOA+NeB)/T~ CI•XoU C2=yoV C3 :Z -~ XN=LleCI+MI~C2 ZN=L3eCltMJ~C2~N]~C~ AO:OACOS(XN/DS(~T(XNeXNtZNeZN)) IF (XN.GT.OG0.ANO.ZN.G£.O.~O) | Q : I IF (XN.LE.000.ANO.ZN.G[ .O.D0) IQ:2 IF (XN.LT.OO0.ANO.ZN.LT,O.DO| IQ:$ IF |XN.G(.0000ANO.ZN.~T.O.DO| IG:~

CeeB~Olooolee~ooeeeeel•Ot~eeeeeOeeeeoeoeoe~eol leoeeeeeeee~eeeeeeeeee~eee G INCLINATION VERSUS ORAVITV Ceeeee•eoeeeeeeooeooee•eele~oeo~oeeeeeeeoeeeeeoeeeeeeeeoeeoeeeBoeeees~ee

AGz(AeA÷B*S)/((DS~T(AOA+S°S))o(OSQOT(A~A~BeS~CeC)|) RETURN END

SUBROUTIN( kOJ (10 : IO. ~A) C ~ e e ~ e • ~ e • e e ~ * ~ o e ~ e e ~ l ~ l ~ e ~ l ~ e o ~ e o e o e e e ~ e e e e ~ e e e ~ e ~ e ~ o * e e ~ e ~ o C THiS SUSNOUTINE N)JUSTS TNE OflI[NTATION ANGLES FOR V~RYING C SEGMENT POSITIONS (POSiTiVE ANGLES COUNTERCLOCNWISC " NEGATW[ C ANGLES CLOCKWISE).

C INPUT o IO, I q , AA C ID: NUI'I~ER OF OATA POINTS C IQ: V[CTO~ O~ LENGTH ( tO) CONTAINING TNE POSITION C INOICAT~S FOR MA~KER C AN: VECTOR OF LENGTH ( ID) CONTAINING THE ORIENTATION 0 ANGLES FOR SEGMENT S~ C OUTPUT - ~A C AA: VECTOR OF LENGTH (10) CONTAINING THE ADJUSTED C OR I(NTATION AISLES FOR SE~I[NT S I C l e e e o e o e e * e e e o e e • e ~ e ~ o o • e • e l l e e e e o l e o l ~ e l e l e s l e e l e e o * * s o * e l l e o e s e ~ e * ~ e ~

IMPLICIT NEALeS ( A-H, J*Z ) INTEGER tO( tO) OINENSION AA(ID) P I=$ .1~15926~DO HP=PI /2.O0 OP=2.DO*PI TP=$.OOePI/2.O0 IF ( I Q ( 1 } . L T . 3 | GOTO tO IF (tO(1).•[.3) ~(1)=OP-~(1)

I0 GONTINU( DO ! I~2.10 •ohio(i-l) INelG()) AO~AA(J-I) IF IO.EQ. 1 .A~. IN.E•. I .A~.AO.L(,MP ~TO I IF IO.(I.I.ANO.iN.[I.I.*~.~.~.~ ~(1)~AA(1)÷OP IF IO.EQ.I.~.IN.(O.2.~.~.LE.~ ~iO i IF IO.(G.I.A~.tN.EQ.~.~.AO.~.~ ~|I)-~(I)*OP mF IO.EQ.I,~.m.EO.~.~.~.~(.~ ~(m)--~(1) mP tO.EG.I.A~omN.EO.N~.~.~.~ I~(1)-OP-~¢I) I~ IO,EG.~.Am.mN.E¢.I.AIe~.~.L(.PI "~TO i IF IO.(G.2.A~.IN.EQ.I,ANO.~.O/:.OP ~(I|-~(1)+OP IP IO.(~.2.AND.IN.E~.~.~.AO.L(.PI ~iO ! IT IO.(G.~.~.IN.[~.~,~.G4:.OP ~(I|~A~MI)÷OP IP IO.(Q.~.A~.IN.E~.|~.~.LE.~| AA(1)-OP-~(t)

~TO I tF io. EG. 3.~. I N.(O.~.AND,AO.~.P! I~ tO.(G.).~.m.(O.),~.~.~.~t ~(m)-DPo~Uql) I~ I O. El. $.ANO. IN,(O.).AN0.~.LT.O. ~| J)--~(I) I~ IO.EGJ.A~.IN.(O.~,~.~.~.PI ~(1)-OPo~(I| )P tO.(Q.S.A~.IN.(O.N.~.AO.LT.O. ~(*)--~(1) I~ mO.l~.~.~.)N.(~.i.~.~.C(.i@ ~())-~(m)+~ Ir )O.(~.~.~.mN.(~.i.~.~.~i.O. ~iO I IF ~O.(Q.~.A~, ~.(Q.).A~.~.~(.TP ~I t )~DP-~( ( ) IF *O.(Q.~.A~. #~.(Q. 3.A~.AO.LT.O. ~(~ I--~(m)

1986, 8, 104-107 (A generalization is available from NETLIB as package VSPLINE. The authors are accessible as: FESSLER ISE.STANFORD.EDU; Jeffrey A. Fessler and Albert Macovski, Information Systems Lab., Dept. of Electr. Eng., Stanford Univ., Stanford, CA 94305, USA)

IF ( IO . [Q .k .AND. IN.EQ.q.ANO.AO.G(.TP) AA( I )=DP-AA( I ) IF ( IO .EQ. I I .ANO. IH .EQ. I I .AN0.AO.LT.0 . ) AA( I )= '~J~( I )

1 CONT ! NUE RETURN [NO

C ~ , ~ . ~ . • ~ . ~ ~ ~ ~ ~ ~ ~ ~ . e ~e~e~ ~ ~ • ~ m ~e ~ t . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ SUBROUTIN( T ~ (P1, PIAV~ P I ~ Pg, PgAV, P ~ P]~ P]AV, P3AA,

~ PGRA, X~LA. Z~LA, m ~1~ H2, H3, I 1 . 12~ 13, Ol~ D~. D3. LI~ L2. L3. ; T 1NIT ~ T1HUS ~T 1 3 ~ , T13AV, T 1 2 ~ , T 12AV. T 1 ;LA, T I ~ A ,

T~N~ T, T ~ , T~ 3AA ~ T ~3AV, T ~ A A , T ~ A V , T~ 1 ~ , T ~ ~ AV, ~ T2~LA, T 2 ~ A . ~ T3N~T, T3MU~,T33~, T33AV.T32~ , T32AV, T3 I ~ , T 3 1 A V .

T33LA, T 3 ~ A ) ~ ~ ~ ~ ~ ~ ~

C THIS SUBROUTINE C~CULATES THE T~QU£ C~PONgNTS AT JOINTS J1, J2, C J3 (J1 ~ S T D I S T ~ - J~ ~ T P R O X I ~ ) . Ceeeeeeeeeeeeeeeeee~eeeeeeeeeeeeeeeeseeeeeee~ee~eee~eeeee~eeeeee~eeet Bee C INPUT - P1. PTAV, P1AA, P2, P2AV, P 2 ~ P~, P3AV, P 3 ~ , PGRA, C X3LA, Z$LA, C H1. H2. H$~ I I ~ 12, 13. Dle D2. D3, L t , L2, L3 C P# : ~IENTATION A N ~ AT PROXIM~ EN0 OF SEGMENT ~ T C THE RIGHT H ~ I Z O N T ~ F ~ SEGMENT S I c ( IN RAP) C PIAV : FIRST TIME O[RIVATIV( OF ORIENTATION ANGLE F ~ C S[GM(NT S I C ( A N ~ V ~ L ~ I T Y ; IN RAO/S) C P # ~ : S ~ T IN[ ~ R I V A T I V [ OF ~ I [NTATION A N ~ F ~ C S ~ N T S I C ( ~ ACCELERATION; IN RAO/S~S) ; C P ~ A : COSINE ~ INTIMATION ANGL[ OF ~VING-LOCAL PLANE C ~ T ~ A V I TY C X3LA, ZSLA : L I N [ ~ ACCELERATION OF ~ K E R H3 C ( IN ~ / S ~ S } . C N# : ~ S S ~ S E ~ N I S I C ( IN 0) C i l : ~ N T ~ I ~ R T i A AT CENTER OF ~ S OF S E ~ N T S I C ( IN O ~ ) G Ol : O I S T ~ F R ~ TH[ P R O X I ~ JOINT TO CENTER OF ~ S S C OF S £ ~ N T S l O ( IN ~ ) C L I : L E I T H OF S ( ~ N T S I

C ~IN ~ ) 1 ~ T1 AV T1 T A C OUTPUT - T 1 N ( T . T ~ . T I 3 ~ . T 1 3 A V . T 2 . 2 . 3LA~ 1 ~ . G T2N( T. T ~ . T ~ 3 ~ . T23AV. T 2 2 ~ . T22AV ~ T 2 1 ~ . T21AV. C T~3LA T ~ A . C T3N(T T ~ . T ) 3 ~ T 3 3 A V , T 3 2 ~ , T 3 2 A V , T 3 1 ~ , T3 tAV, C T33LA T 3 ~ C T t ~ T . . T 3 ~ A : T ~ C ~ O ~ N T S ( IN ~ ) WiTH THE C ~ ( N T $ A~ ~ I N T ~ 8 ( 1 ~ ) C TINET NET T ~ ( C T ~ S ~ ( T ~ ( C T I I ~ S ( ~ N T St A ~ A ~ ( L E R A T I ~ T ~ ( C ~I1AV $ ( ~ N T $1 ~ ~ L ~ I T Y T ~ C T I 2 ~ S E ~ N t $2 ~ ~ L ( R A ~ I ~ T ~ ( C TI2AV S ( ~ T S2 ~ ~ L ~ I T Y T ~ C T I 3 ~ ~ N T SJ ~ ~ E R A T I ~ T ~ ( C TI3AV ~ N T S3 ~ ~ L ~ I ~ T ~ C T I ~ A ~ I N T $3 L I ~ ~ L ( R A T I ~ T ~ C T ~ A ~ A V I ~ T ~ ( C ~ e o e o o e o e o ~ o ~ e s e ~ o e e e ~ e o e e e o e e e o e e e o e e o e e e o e De ~e e~e ~ee eee~ e ee

I~PLICIT REALES ( A-Z ) P ~ t . D ? G=9. 8066S02

B I=~ Ie01*L3 O2~H t*O 1"L2 83=M 1~01 8~BMleL2eL3 OSm~*O2eL3 BB-~I~L2eL2 87~MlOL2~H2oD2 BOtM2~L 3 "L 3 BD=MteL3oL3 8 lO~M3*O3~M2eL3 ~M 1"L3 B1 IsN2°k3~Ml~L3*M3~3 OimMleOleO1 O 2 - ~ * 0 2 " D 2 O3=N3*O3~O3

Coooeeoe~ooeo~oee~e=oeoe eoeooooeeoe*eoee*oeo~eeee~=e=oeeeoeee~e*eee ~ =o~o C JOINT 13 C ~ e e e ~ e ~ e ~ e ~ e ~ e ~ e ~ e ~ e ~ e ~ e ~ e ~ v ~ , , , ~ e ~ e ~ v ~

T3NtT- ( [ I ] * O 3 ) ' P ) ~ ) / e T 3 3 ~ - ( * ( 8 0 ~ 9 ~ l O ~ S (P 1 -P3)+ ( ~ + B S ) * ~ S ( P 2 - P 3 ) ) e P 3 ~ ) / P T33AV: ( - [ B 1 0 ~ 1 N(P I - P ] ) ~(Ok*BS)eO$1N(P~*P3) ) ~P$AV*P3AV)/P T 3 2 ~ = ( * { 1 2 - ~ 6 ~ ( P leP2)*(Ok~B5 ) ~ ( p ~ - P 3 } ) e p 2 ~ ) / P T32AV: ( - ( 8 2 o ~ iN(P 1 -P~) - ( O k ~ 5 ) ~ I N ( P ~ - P 3 ) ) SP~AYeP~AV)/P T 3 1 ~ = ( ( I I ~ l - a B ~ ( P l - P ~ t . 8 1 ~ ( ~ t - P ~ ) ) ~ P l ~ ) / P X3 tAV= { ( 8 1 ~ 8 I N ( P Z , P ] J ~ I~ (p l - ~ ] J J ~ 1 ~ IAV) /~ T33LA= ( ( O l O ~ ! N ( P S J * B ~ t ~ ] ~ ) ~ N f F t ) ) ~ X ~ A

" - ( O l O ~ ( P l ) 4 ~ t ~ ] ~ ( P t ) ) ~ Z ~ ) / e T 3 ~ A = ( - ( 0 1 1 ~ P) ~ ~ 3 ~ ( P 1 ~ A ) / P . . , - , , , . - . , ,~: , t~-d.~, , ,=. , ; ; . .~. . , ,~, )

C ~ * o ~ w o ~ e ~ o ~ e o o o e ~ e e e o e e e e e e e e e e ~ e o o o e e o ~ o e ~ e e e ~ e e e e e o e e ~ e e o e e e e e ~ e * ~ C ~ I N T ~2 Ceeeoeoeooee eeoeeees&eoete~oeeeoe~o~eseooeeese ooeeeeeeseeeoeoe~eeee e see

T ~ T : ( ( 1 2 ~ ) e F ~ ) / F T ~ 3 ~ - ( - ( S t ~ l ~ ' P 9 I ~ ( P2.P 3 ) ~ 1 ~ ( P t . P 3 ) ) *e 3 ~ ) /P T23AV: ( " ( 8q~D$ I N (P~P3 ) ~ 5 ~ I N( P2"P 3 ) ~ I eOS I N( P 1 "P3 ) ) eP~AVeP 3~.V )

• / p T ~ 2 ~ ( " ( B 6 ~ ( P I - P 2 ) ) o P ~ ) / p T 2 ~ V : ( - ( 8 ~ I ~ P I " ~ J S ~ J A ~ V l / P T ~ I ~ ( - ( I I ~ I ~ ( p ~ o p 2 ) ) o p I ~ ) / P T~ tAV= ( ( B ~ iN(P ~ J ) ~ ~ P t A V ) / P

, T ~ I L A : ( 1 8 3 ~ ( ~ I H I ~ I ) ~ ) L A ~ - ( e 3 ~ l P l ) ~ ( e ~ j j Z ~ ) l e 8 ~ = ( - ( B l e E P t J q ~ t J ) ~ J / P

Y m - T2NET- ( / ~ t t ~ ~ t ~ 1 ~ * T21AV*T=3L A * T 2 ~ A ) C o o e e ~ e e e e e ~ e e ~ o B e e e ~ e e e o e w o ~ s o w O e e e e o e e o e e o e e e e e e e ~ e e e o o ~ e ~ e e e e e e C ~ I N T ~1 Ceee~t e~o~eee~eeeeeoeoeteoeoeeeo~e~eeoooe eeeeeeooeooe eeeeeeeesee~e~ese

T t ~ T - ( ( I I ~ 1 ) ~ 1 ~ ) ~ T I 3 ~ = ( " ( B l e E P l ' P 3 ) ) ~ ] ~ ) / P T I ~ V = ( - ( 8 t ~ ( P t -P~ ) J ~ 3 ~ F 3 A V ) / e T I ~ = ( - ( 8 ~ ( @ I - ~ ) ) ~ I ~ ~ T t ~ v = ( - ( O = ~ ( ~ ~ - ~ ) ) ~ = ~ P = ~ V ) / P ' t i = L ~ = ( ( O = ~ ( ~ i ) p ~

e - ( O 3 ~ ( P 1) ] e Z ~ A J / P T t ~ = ( - ( 1 3 ~ - I P 1) ) ~ ) / e T I ~ - T t N£T- ( T I $ ~ # T 13AV+T 1 2 ~ t T 12AV~T 13LA~ I ~ A ) R ( T ~ N ( ~

125 Adv. Eng. Software, 1990, Vol. 12, No. 3