THE UNIV ERS I T Y OF MI C HI G AN MEDICAL SCHOOL Department of Physical Medicine and Rehabilitation Orthetics Research Project Technical Report DYNAMIC ANALYSIS OF THE UPPER EXTREMITY FOR PLANAR MOTIONS J. R. Pearson D. R. McGinley L. M. Butzel ORA Project o4468 under contract with: OFFICE OF VOCATIONAL REHABILITATION DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE CONTRACT NO. 216 WASHINGTO:N, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1961

FOREWORD This report is composed of a paper submitted to a journal, Human Factors, for consideration for publication, and of supporting material, included here as an appendix, relevant here but not appropriate for submission to the journal. In the interest of economy, the paper was not retyped for the report; the stylistic peculiarities, such as the senior author's name on the upper left-hand corner of each page, indicate conformity to the journal's requirements, not egocentricity. It is hoped that this arrangement will not annoy the reader excessively. ii

Pearson 1 Dynamic Analysis of the Upper Extremity For Planar Motions1 J. RAYMOND PEARSON Dale R. McGinley, and Leo M. Butzel, Department of Mechanical Engineering, The University of Michigan, Ann Arbor. ABSTRACT The free body diagrams for the elements of the arm establish the forces involved in planar motions of the arm. The principle of D'Alembert is applied in graphical vector diagrams to represent the condition of equilibrium, The latter lead to equations for determination of the joint forces and torque reactions to weight and inertia forces. These equations indicate a need for accelerations and physical constants, A graphical vector acceleration diagram indicates the manner of determining linear accelerations from angular accelerations which are in turn derived from displacement time data of experimental methods by finite differences, Experimental methods to determine kinematic data and constants are described. The rationale of the analysis is used to establish a computational procedure to evaluate the equations. The procedure was developed as the basis of an algorithm for programmed computation by digital computer.

Pearson 2 INTRODUCTION To deal realistically with the mechanics involved in the body linkages in motion, the investigator must be concerned with such intrinsic properties of the various segments as their linear dimensions, their masses, and distribution of mass, It is also necessary to know the velocity and acceleration of each part from the beginning to end of the motion to determine the inertial effects of each part, Adequate analysis of the motion of the body system continuously throughout the motion in terms of forces and moments at critical points depends on knowledge of the measurements of these parameterso The most important work on this problem has been done by Fischer (1906) and by Taylor and Blaschke (1949, 1950, 1951, 1953, 1955). This study extends that work by developing an analysis adapted to the use of modern computers, and by presenting the approach and results graphically to make the information more readily accessible to a wider audience. Such an approach should permit the treatment of more subjects than has been feasible in the past. This type of study can be made most profitably on a system of several segments for which motion is limited to a plane. Certain motions of the upper-limb segments alone were selected for study, namely, voluntary ones in the sagittal plane, in so far as the joints of the shoulder, elbow, and wrist permitted~ Experimental records of the motions

Pearson 3 measured and analyzed were strobe-photos of an entire phase of voluntary motion~ The subject was hidden from view by a black velvet screen with only the free limb showingo Strips of Scotch-Lite reflective tape were attached like fins to the rear surface of each of the three limb segments. The motion was then illuminated and the camera was provided with a rotating slotted shutter; the photographic film was strictly parallel to the plane of motion. ASSUMPTIONS In the analysis which follows, the elements of the complex are treated as solid bodies. Obviously this is not precisely so, as the soft tissue may be subject to deformation in extreme motions, and there may be some blood displacement. But the relatively large external forces iflvolved, however, make such an assumption reasonable. The transverse axes of the joints were assumed to be pinned. The joints are, in fact, held together by collagenous tissue which acts in tension. The extensivity of this tissue permits some displacement of the axes of head and base of adjacent bones. In normal subjects, however, the degree of displacement is small compared to the magnitude of total arm movements involved, and is presumed to have little influence of the final value of dynamic forces.

Pearson 4 The joints were considered frictionless, The existence of synovial fluid with low viscosity and -the experiments of Wright and Johns (1960) give evidence that this assumption is also reasonable. Although initial experimental. work in connection with this study treated the hand as a separate element, it became evident that the relative motion between hand and forearm was small for the motions used. Hence it is assumed that this motion is negligible, and an equivalent mass representing the forearm plus hand. is used. The method of analysis can be extended to separate treatment if desired, SYMBOLS Analysis Definition >So initial position of forearm-hand combination with respect to downward vertical, degree angular change of forearm from initial postion, radians angular velocity of the forearm-hand combination, radians/sec angular acceleration of the forearm-hand combination, radians/ sec2 go initial position of upper am-hand, combination with respect to downward vertical, degree Gi angular change of upper arm from initial position, radians G angular velocity of the upper arm, radians/sec

Pearson 5 Analysis Definition G angular acceleration of the upper arm, radians/sec2 angle of elbow reaction, degrees His angle of shoulder reaction, degrees time, see x subscript denotes x direction y subscript denotes y direction Gu- gravity center of the upper arm (Gf gravity center of the forearm Gh gravity center of the hand Gc gravity center of the forearm-hand combination SE shoulder-elbow length, cm SGu shoulder to upper arm center of gravity length, cm EGC elbow to forearm-hand combination center of gravity length, cm We forearm-hand combination weight, grams Wu upper arm weight, grams Ic forearm-hand combination moment of inertia with respect to center of gravity, gram-cm-sec2 Iu upper arm moment of inertia with respect to center of gravity, gram-cm- sec2 Ac total absolute acceleration of the center of gravity of the forearm-hand combination, cm/sec2 Agu acceleration of the center of gravity of the upper arm

Pearson 6 Analysis Definition Au acceleration of the elbow, cm/sec2 Fc the total force at the center of gravity of the forearm-hand comb inat ion Fu the total force at the center of gravity of the upper arm Re reaction at the elbow, grams Rs reaction at the shoulder, grams Sfc the inertia (D'Alembert) force at the center of gravity of the forearm-hand combination, grams Sfu the inertia (D'kAlembert) force at the center of gravity of the upper arm Tc torque about the elbow due to the total force at the center of gravity of the forearm-hand combination, gramn-cm Teu Torque about the shoulder axis due to the reaction of the elbow, Re, gram-cm Tic inertial torque about the gravity center of forearm-hand combinat ion TIu inertial torque about the gravity center of upper arm Ts shoulder torque reaction, gram-cm rTul torque about the shoulder due to total force at the center of gravity of the upper arm, Fu, gram-cm asterisk denotes cross product superbar denotes vector quantity

Pearson 7 GRAPHICAL ANALYSIS The so-called free body diagrams in which the elements in question are isolated with their forces, torques, and reactions are shown qualitatively in Fig, 1 for a representative instantaneous phase of the motion being considered, Since these forces are vector quantities, they can be represented by graphical vectors in a polygon which gives a clear visual representation of their relations one to another and provides a base for writing equations for the numerical analysis which follows. A typical polygon representing the vector summation of all forces is shown in Fig. 20 TAI magnitudes of this illustration are arbitrary and the polygon will vary in size and shape for each position of the configurat iono The closing of the polygon indicates that the sum of the horizontal components and the sum of the vertical components are equal to zero, -thereby satisfying two of'the three conditions of equilibrium, The inertia force vectors, Sfu and Sfc, are in a direction opposite to the acceleration, following the principle of D'Alembert where ZFx-ma = 0 or ZFx+S = 0 and S = -mao The summation of forces in Fig~ 2 shows graphically how the output values Re and Rs, are determinedo

Pearson 8 Re = W + Sfc, (a) and Rs = WC + Sfc + Wu + Sfu, (b) or Rs = Re + Wu + SYu (c) The bar over the symbols indicates vector quantities in which both direction and magnitude are significant. The third condition of equilibrium is that the sum of the torques on each free body must equal zero. Torques are also vector quantities and are normally represented by vectors perpendicular to the plane of motion, into and out of the plane of the paper, in this case. The convention used here is a vector out of the plane of motion for a plus vector or a counter-clockwise torque and into the plane for a negative (clockwise) torque vector. Since all torque vectors are perpendicular to the plane of motion, their graphical representation will be along a straight line. Mathematically, this results in simple arithmetical addition and subtraction. Figure 3 shows the graphical representation of the torques, in which Te = TI +T c (d) and Ts = TIc + Tc + Tu + Tu (e) or T = Te + TIu + Tu' (f)

Pearson 9 thereby furnishing two more ourput values, Te and Ts. As has been noted, S = -ma. Therefore the magnitude of the inertial forces, Sfc and Sfu, are dependent upon the magnitude of the accelerations involved, The accelerations of the centers of gravity of the two elements must then be determined for each position of the motion of the complex. It can be seen from Fig. 4, a graphical representation of the typical position, that the acceleration of the elbow, Au, is the vector sum of accelerations tangential to and normal to the path of the elbow axis, E. In this instance the path is a circular arc of radius SE with center at S. Since tangential acceleration is angular acceleration times radius, and normal acceleration is angular velocity squared times radius, Au = [G SE] + [(6)2SE]. (g) Since both tangential and normal accelerations are proportional to radius, total accelerations of all points along the radius SE will be proportional to radius and will have'the same direction. Hence the acceleration of the gravity center Gu will be Agu = ku (SGu/SE), (h) and will have the same direction as Au. The proportional triangle SEe demonstrates that the vector length Gj-gu:Ee as SGu:SE. The inertial force vector SfU is shown opposed in directional sense to Agu.

Pearson 10 The acceleration of the gravity center of the forearm-hand combination is the vector sum of the acceleration of the elbow plus the relative acceleration of the gravity center to the elbow. This relative acceleration has components tangential to and normal to the path of the gravity center Gc relative to the elbow, E. Hence Ac = Au + EG [()2EG]. (i) The angular velocities, 9 and i, and the angular accelerations, 9 and I, can be determined by the method of finite differences from the displacement-time plot of the motion ~in question, Determination of the displacement-time data is an experimental problem, the procedure of which is explained below. The derivation of the finite difference equations is discussed in the next section. Again the inertial force vector, Sfc, is shown opposed in directional sense to the acceleration-vector Ac. As indicated above, the magnitude of the two inertial force vectors Sfc and Sfu are obtained by Sfu = (866) Agu () and We rfc =-980.616 (k) where 980.616 is the gravity acceleration constant in cm/sec2.

Pearson 11 ALGORITHM Although the graphic analysis serves well as a visual means of communication and has established the rationale, it is necessary to established a procedural system of equations suitably adapted to the numerical capabilities of the digital computers Such a procedure, commonly called an algorithm, will be more or less in reverse order to the analyses above, moving from known quantities to unknowns. For the problem at hand, the procedure with pertinent explanations is as follows. 9i = f(ti) (1)3 i = f(ti) (2) Equations (1) and (2) represent the positions of the forearm and upper arm, respectively, at a time ti, The curves indicating the nature of the function are drawn through experimentally determined data points as indicated in Fig. 5. As the mathematical description of these functions will be unknown, the derivatives for velocity and acceleration cannot be computed directly. Expressions for them can be derived, however, as followso Since it is obvious that the functions in question and their derivatives are continuous, the functions at t-l,t, and t+l can be related in a Taylor's series: i~ = P i ii + Xti 2 ) )i-l = 5i ~ At i A +- *- (m)

Pearson 12 Subtracting Eq. (m) from ( ) yields the first derivative or velocity, = i+l - i-1l X~~~~~i = i > ~~(n) 2At Adding Eqs. (a) and (b) yields the second derivative, or acceleration, ii - (o) (At)2 The accuracy of these finite difference equations depends upon the magnitude of At. This accuracy can be optimized by comparison of the values of the derivatives of known functions, similar in nature to those in question, with the values determined by the finite difference equations. For the case at hand this procedure leads to the conclusion that the time difference for acceleration, Eq. (o), should be twice that for velocity, Eq. (n). The foregoing derivation and optimization procedure lead, then, to finite difference equations (p) and (q). = (i+l - Xi-)/2At (p) = (i+2 + i-2 - 20i)/(2t) () The time difference, At, is the time between exposures in the photographic experimental method, and was 0.0298 seconds for the experimental procedure of this study. Use of this value continues the analytical procedure with: = (a i+d - io_)/0e0596 (

Pearson 13 / = (/i+2 + i- 2/i)/000555 (4) Equation (5) establishes a basic axis of reference for subsequent vector operations and converts initial positions readings from degrees to radians: Xi ~i + 001745335o, (5) where the numerical constant is the degree-to-radian conversion factor. This is not an equality, but a substitution command which means that the values on the right should be used for Si until further notice. The algorithm continues with: = (Giol -.i-1)/0.0596, (6) G = (Gi+2 + Gi-2 - 2Gi)/0.00355, (7) i< — Qi + 0o0174533G0, (8) which are similar to (3), (4), and (5). The position of the upper arm as shown in Fig. 6 is established by SEX = (SE)sin Gi, (9) SEy = (TE)cos Gi, (10) and the components of the acceleration of the elbow are then determined by Aux = -. SEy- g (g * SEx) (11)

Pearson 14 Auy= +. SE- ( Ey) (12) Here we digress to explain Eqs. (11) and (12), which are derived in the following manner. SE is a position vector of magnitude ISEI and direction 9, from the downward vertical which rotates about the fixed point S with angular velocity 6 and angular acceleration 9. The vector acceleration of the point E, Au, can be expressed by the vector equation Au = Aut + Aun, (r) Au = G * SE + G * G * SE, (s) where the bar over the symbol indicates a vector quantity and the multiplication asterisk represents a vector cross product, wherein the product of two perpendicular vectors produces a third vector whose direction is mutually perpendicular to the multiplier and multipicand vectors, i.e., the product vector is perpendicular to the plane of the multiplying vectors. By definition of a cross product, the magnitude of the product vector is!GI. SEI (sine angle between vectors 9 and SE). The direction of the vectors 9 and 9 are normal to the plane of rotation, and by convention are positive and out of the plane of rotation for counterclockwise rotation. Thus the 9 and 9 vectors are always parallel to the z axis of Fig. 6. By dealing with x and y components of all vectors, the angle between vectors is always 90~, The vectors may now be expressed in terms of their components by use

Pearson 15 of unit vectors i, j, k in the x, y, z directions, respectively. Hence the quantities of Eq. (s) become 9 = Gk (t) SE = (SEx)i + (SEy)] (u) 9 = ~+ Gk (v) Substitution of these equations into (s) leads to Au = SE = 93k * [(SEx)+i + (SEy)I] + k * [(SEX)i + (SEy)]) (w) Au = SE = - [I(SEy) + (6)2(SEx)]i + [6(SEX) - (4)2(SE)]] (x) The components of the acceleration Au with respect to S are as previously given in Eqs. (11) and (12): Aux - - * SEy ( (9 * SEx) (11) Auy = + * SEx - * (9 * SEy) (12) This approach will simplify the programming procedure. Continuing now with the force analysis, the values from Eqs. (11) and (12) are used to determine the components of the inertial force at the center of gravity of the upper arm, sfux = - (Wu/98O.6l6)Tux(SGu/SE) (13) sfuy = (w u/98o0 616) y(SGu/sE) (l4)

Pearson 16 A similar procedure is followed for the forearm-hand combination. The components of the distance from the elbow axis to the center of gravity of'the forearm-hand combination are expressed in Eqs. (15) and (16). EGcx = (EGc)sin ~i (15) EGcy = (Ec) Cos i (16) Values from Eqs. (15), (16), (3), and (4) are substituted in Eqs. (17) and (18) to give the components of acceleration of the forearm-hand combination's gravity center relative to the elbow axis. 7icex = * E-Gcy -. * EGcx (17) Acey = EGcx, * EGcy (18) These in turn are added vectorially to the components of acceleration of the elbow in: Acx = + Aux + Acex (19) Acy =Auy + Acey (20) which lead again to inertial forces: Sfcx = - (Wc/98.0)Acx (21) Sfcy = - (Wc/98.0o)lcy (22) From the free-body diagram (Fig. 1), it is evident that Rex = - Sfcx, (23)

Pearson 17 and Rey = - Sfcy + Wc (24) Values from the latter equations are combined with those of Eqs. (13) and (14) to give the torque at the elbow due to weight and inertia translatory effects. Tc = - (EGcx * Rey - EGcy * Rex) (25) Output of this equation combines with inertial resistance to rotation to give the torque reaction at the elbow. Te = - To + IcJ (26) The weight and inertial forces of the upper arm are added vectorially in: Fux = Sfux (27) FUY = Sfuy - Wu (28) These components are added to the elbow reaction components to give the shoulder reaction forces. Rsx = - Fux + Rex (29) Rsy = - Fuy + Rey (30) The distances from elbow axis to upper arm gravity center are: SGux = Sx(SGu/SE) (31) SGuy = S-(SCj/SE) (32)

Pearson 18 and combine with the forces at the gravity center to give a torque effectTu = SGux * Fuy - SGuy * Fux (33) The torque effect of the elbow reaction on the upper arm is: Teu = - (x * Rey SEy * Rex) (34) These two torques and the inertial resistance to rotation render the shoulder torque reaction Ts = - Tu - Teu + Te + Iug (35) MEASUREMENTS Examination of Eqs. (1)-(35) in the algorithm will show that the three kinematic quantities, t, i, G, and seven physical constants must be determined by measurement. The constants are SE, SGu, EGc, Wu, Wc, Iu, and I.c They present problems of measurement of angular displacement, time, length, weight, and weight distribution. Kinematic data were collected photographically with a sequence of exposures taken at constant time intervals. This resulted in a multiple exposure picture in which several positions of the upper and forearm for the motion in question were recorded on the film. The film recorded positions of fluorescent tape attached to the posterior aspect of the arm, minimizing movement of the skin relative to the arm axes. The pho

Pearson 19 tographs were then magnified and thrown onto a screen. The position of the axes were determnined by use of a template which established the relative position of tape and axis, resulting in "stick" diagrams of the type shown in Fig. 7. The same figure shows the relative position of tape and axes, and the direction of the downward vertical. Measurement of the angular positions of the axes by vernier-scaled protractor gave the positions in degrees. The values were fed into the program and converted to radians by the computer. Since the time interval between exposures was kept constant, the photographs furnished position-time data. A disk with 4 slots whose width could be varied for light control was substituted for the camera shutter. The disk was rotated at 503 rpm, giving a frequency of 2012 time intervals per minute or 33p533 time intervals per second or 0.0298 seconds per interval. The angular values measured from the stick diagrams with their corresponding time intervals were used to establish curves of the type shown in Fig. 8. Values were taken from these curves and arranged on the input data cards of the program. The length measurement SE was made from an X-ray photograph of the subjects arm showing both the joints and the fluorescent fin tapes used to establish positions of the elements of the arm. The joint axes were established on the photographs by Dr. W. T. Dempster and the distance was measured between the axes. A similar measurement was taken between elbow

Pearson 20 to-wrist axes. A previous study by Dr. Dempster on the location of the center of gravity of limb elements established a ratio for proximal-axis-to-gravitycenter distance to over-all length of limb segment of three to seven for normal subjects. This ratio was used to determine the length, SGU. The position of the gravity center of the forearm-hand combination was computed by the moment equation EGc = [Wf(EGf) + Wh(EGh) ]/(Wf + Wh), where EGf is the elbow-to-forearm gravity center and EGh is the elbowto-hand gravity center. EGf was determined in the same manner as SGu. The gravity centers of the hand were determined by suspending bioplastic casts of the subjects' hand in a relaxed position. This gave a wristto-hand gravity-center distance which was added to the elbow-wrist length to give EGh. The weights of the hand, forearm, and upper arm were determined by the water-displacement method. Landmarks at the wrist, elbow, and shoulder permitted determination of the separate elements. To determine the moments of inertia of the arm segments, models made of cork and linoleum disks, giving a good approximation of density and mass distribution, were swung as pendulums and the periods were then used in the equation: T 2 Io = w Nd,

Pearson 21 where lo represents moment of inertia about the axis of suspension; W is the weight of the segment and d is the distance from suspension axis to gravity center SGu or EGf or EGc, depending upon which moment of inertia was needed. The moment of inertia about the gravity center was then computed from W2 or IG = Wd [( -)2 _ d]. The moment of inertia of the forearm-hand combination was determined by first setting up an equation of dynamic equivalence: Ief + Ieh = Iec (Igf + mfE) + (Igh + mh) (Igc + ) in which the two unknowns are Igc and EGc. The length EGc is then deterfrom the relation Wf(EGf) + Wh(EGh) EG, = (Wf + Wh) The equation of dynamic equivalence is then solved for Igc. The physical constants for the five subjects treated are tabulated in Table 1.

Pearson 22 TYPICAL RESULTS The output of the computer is in two forms. The first part is in tabular form, indicating the subject and motion, the values of the constants, their units, and lists the position, number, the position angle, the angular velocity and acceleration, the (elbow) reaction force magnitude, and its angular direction for the forearm-hand. The same quantities for the upper arm follow. The second part is a graphical presentation of the first, an example of which can be seen in Fig. 9. Here the velocity is plotted with angular position in polar coordinates. The inner circular arc gives the position numbers and represents the datum circle from which the angular velocity values are plotted. The outer curve is drawn through points designated by letters, the radial distance between the curve and the datum circle representing the velocity. The scale can be deduced from the value and length given on the horizontal axis. In this instance the scale is 40 cm/sec/ll.5 cm = 3.478 cm/sec/ cm. Since the points involved lie anywhere within the field outlined by the letter or numeral, this method of plotting is not precise, but the rapidity with which it is performed by the computer warrants acceptance for initial studies. In instances where greater accuracy is required, the values can be replotted by conventional precise methods. It might be noted that the velocity has a small but significant value at the initial (zero) position of the motion. This came about be

Pearson 23 cause of the lag between motion start and exposure lighting. A synchronizing mechanism was used but it had an inherent and constant lag. This meant that the zero position was not the position of zero velocity. The motion of the forearm is further described in the acceleration curve of Fig~ 10o The scale of this curve is 400 cm/seC2/cm. The motion described by the kinematic curves results in a reaction at the elbow, the variation of which is indicated in Fig. 11. In these force diagrams the position arc with numerals is again included. The lengths of the force vectors, however, are now measured radially from the pole of the polar diagram. The direction of the force vector,-ey, is defined by the direction of the line drawn from pole to letter. Numerals and letters are in correspondence, O-A, 1-B, 2-C, etc. The torque reaction at the elbow is shown in Fig. 12 where the position arc is again used as a datum. Corresponding numerals and letters lie on the same radius. Positive values of torque are plotted radially outward from the position arco The scale of Fig. 12 is 53.158 gm-cm/cm. The description of motion for the upper arm and the resulting reactions at the shoulder are shown in Figs. 13-16. The scales for these figures are 31,478 cm/sec/cm for 15, 400 cm/se2/cm for 16, 2631.6 gm/cm for 17, and 100,439 gm-cm/cm for 18.

Pearson 24 REFERENCES Blaschke, A. C. General energy considerations and determination of muscle forces in the mechanics of human bodies. U. C. L. A. Dept. of Engineering, Memorandum Report Noo 9, Sept., 1950. Blaschke, A. C. A dynamical analysis of the shoulder, arm, and hand complexo Unpublished PhoD dissertation. U. C. L. A. Dept. of Engineering, 1953. Fischer, 0. Theoretische Grundlagen fUr eine Mechanik der Lebender Kdrper. Berlin: B. G. Teubner, 1906. Taylor, C. L., Schwartz, R. J,N and Blaschke, A. C. Anatomy and kinematics of upper extremity. Uo Co L. A, Engineering Prosthetics Research, Spec. Techo Report No. 10, 1949. Taylor, C. L, and Blaschke, A. C. A method for kinematic analysis of the shoulder, arm, and hand complex. Human Eng., Annals N. Y. Acad. Sci., 1951, 51, 1251-1265. Taylor, C. L. The biomechanics of control in upper extremity amputee. Artific. Limbs, 1955, 2, 4-25. Wright, Verna, and Johns, R. J. Physical factors concerned with the stiffness of normal and diseased joints. Bullo Johns Hopkins Hosp., 1960, 106, 215-231

Pearson 25 LIST OF ILLUSTRATIONS Fig. 1. Free-body diagram isolating elements and showing forces and torques on each. Fig. 2. Summation of forces. Fig. 3. Summation of torques. Figo 4. Acceleration diagrams. Fig. 5. Angular displacement as a function of time. Fig. 6. Orientation of vectors. Fig. 7. "Stick" diagram of arm motion. Fig. 8. Displacement-time relationship. Fig. 9. Angular velocity at the elbow for Subject No. 3, executing Motion No. 3. Fig. 10. Angular acceleration at the elbow for Subject No. 3, executing Motion No. 3. Fig. 11. Force in grams at the elbow for Subject No. 3, executing Motion Noo 3. Fig. 12. Torque in gram-centimeters at the elbow for Subject No. 3, executing Motion No. 3.

Pearson 26 LIST OF ILLUSTRATIONS (Concluded) Fig. 13. Angular velocity at the shoulder for Subject No. 3, executing Motion No. 3. Fig~ 14i Angular acceleration at the shoulder for Subject No. 3, executing Motion No, 3. Fig. 15. Force in grams at the shoulder for Subject No, 3, executing Motion No. 3. Fig, 16. Torque in gram-centimeters at the shoulder for Subject No. 3, executing Motion No, 3. Table 1. Physical constants.

Pearson 27 Rs Iu \ ( I DIRECTION,~E OF MOTION Wu C SFU -RE WC SFC Fig. 1. Free-body diagram isolating elements and showing forces and torques on each.

Pearson 28 Wu OF Rs RE Wu Sfu Fig. 2. Summation of forces.

Pearson 29 +Z OT +Ts +TE Y Tic - TIu -Tt Fig. 3. Summation of torques.

Pearson 30 g e- 9- e -$EGF A /Au "GF SFC Fig. 4. Acceleration diagrams.

Pearson C') / | I I I ti-ti-i-l ti ti+l ti+z t, sec Fig. 5. Angular displacement as a function of time.

tY~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t +y S~~~~~ +8 1~~~~~~~~~~~~~~~~~~~~~~~~~ Aun~~~A +6E +00 Auy Au~~~Au + )E Fig. 6. Orientation of vectors. -Y E~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~R

Pearson 14 13 12 II 16 i,5 l 5 C 15 5 CH 14 G1 3V 5 motion. Fig. 7.' "Stick" diagram of arm motion.

50. I I I I I I I I I I I I a, ~0 o-I13o0' F'X! -X= - Xe27067' 8.0 2.0 0 i.o I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 POSITION IN SECONDS * 0.0298 Fig. 8. Displacement-time relationship.

----— L E —-------------------— EOIT AT THE ELBOW FOR SUBJECT NO. 3 EXECUTING MOTION NO. 30 + —------------------------------------------------- I I -II I L I E I I A I I I R --------... I I —---------------- ---— I —------- -----------------—' —----------- I ----- ----------— _____________ 7;;; —-;; I —. — I I EI I - ----— I -2-I --- --- I 0.7I BHI - 5 — I - I F I I I I I I I I - I N —-------- II 7 I II --— D -------------------------------- I-I 0 I I O I 5 1 Vi —.-b -- - -- - - -- - -- - - - - - - -- - ---- ---- ---- -------- --- A I I In --------------- -- I O-1..... I I: V I I V - - - - - - - - - - - - - - - I R I~~~~ ~~ I ___ __ C - - - - I - - - - - - - - - - - -E — - - - I S 4 I I IC/ I I I I -~I —----- i I I I O I 1~ -----------— I I I I I I W --------------------- ---------------------------------- --------------'9 —-------------- s I I I E 40 O -- - - --- - - - - - - - - - - - - --- -- - - - —.00- - -- - -- - - 40.00 -- --- - -- - --------------- ----------------------------- __~s_ -------— /I I__________ _'~ ~ ~ ~- ---------------------—' —-------------------- I ---------------------------------- ---— L —-----— I ------------- 1 11 —---------- -40.00 -— ~ —"" "'' —---------------------------------------------------- -40, 00. ---------------— __ J, 40. 00 L Fig* 9. Angular velocity at the elbow for Subject No. 35, executing MoctiOn No. 5.

ANGULAR ACCELERATION AT TRE ELBOW FOR SUBJECT NO. 3 EXECUTING MOTION NO. 3. 460.00 ---------------------------------------- I I I 0 A" ---— r- — I I I L I I I E I I I I I I s I A I- I I E I I R —T —-------------------------------- — I S I I I U I -I I OFI F( I I R I 1~ T01~3 I 0F I// I 1 I I N I I H7 I I Il-l I I _________ ------- ----------------------------------------- I- -- - O I I \\ I A N ~~~~~~~~~~~~~I I ---------------------------------------------------------- I R I_ I I - I --- - I I I I F r I C I I I IR~~~~ ~~~~~~ I IZ~~~~~~~ 4 I p I ---------------------------- --------------- ----- L I I / J I E I -I —------ -------------------- I.-.. -- I I. — II I I I ------------------ I I I 3 / I W I AI - S - I —r E I____ _______________________ _ I I 7...I I I I I I -460.00 + —-------------------------------------- -460.00 u. C0 460.00 Fig. 10. Angular acceleration at the elbow for Subject No. 3, executing Motion No. 3.

FORCE IN GRAMS AT THE ELBOW FOR SUBJECT NO. 3 EXECUTING MOTION NO. 3 ~+~4I I I I 6 I I I r L I I I E I I I S I I II_ I I I -- - - - - I I I E I I I R I I I R \ I E I F 1 0 L5 I__ —_ —___ - -- -.. — - - -- - I I r Z r2 I I — F ---------- -------------- R I / I T tI C7 I D I r P/ \I ___D_ —-------— I —---------------------- ---------------------- I —----- WI I I I I | I I O. _ —----------— _ —------------------------- ---- __________ -------- -------- -------------— tI ----------------- R I I M I I I -r ——. — -/ —--------------- - -------- -----' —------------- I I' I L - - -- I E I _.lK I 7 — -— R —---- ------------- --- I —-------------- --— r —--- ----- n —------------------------------------------ r C I I I R I t\U I L I 0, I 2 I ------------- -- -------— r -------------------------------- --------------- -- ------------------ -------------------------------------- " —- -------- I — - -- ------ I —--— IL I I I S I I I I I' —------------— T —-------------------------------------------------- ---------------------------------------------------- T —- --— _-_ ------- I I I I I I I I I I -..- -N. -+ —--—. _- -- - _ ___ _ _ __+ _. ___ __ __ _ _ _ _ _- _ _ _ _ Fig. 11. Force in grams at the elbow for Subject No. 35 executing Motion No. 3.

TORQUE IN GRAM-CEitTIMETERS AT THE ELBOW FOR SUBJECT NO. 3 EXECUTING NOTION NO. 3 I I I O Q I I I N I I I' ------— r —-- — I —------------------- I I L I I I ----------------------------— I I S I I I I I I 4 I I I ---- ------------ --------------------------------------------------- I —----- ----------------------- I —------------- U I I I I I I R I I E I I 2 -—..... —---- T -------- IH......... I R I I \ I I p I I r - I 7... I W I I 0 I I 6| I w O. + — + R I I ------------— r —----------------------- --------------------------------------------- --------- ------------- r —------- R I E II I E I I/ F I R ----- - -- I -—!- — t- - T I I// I r ---— r... ------------------------------ ---------------------------------- - ------------ -— r C I Q I I /- / I L I I I C I T I "- c I E....... I E I LE ~ ~ ~ ~ ~ ~ ~ ~ ~ D I I E I \ I II B I I -r..... - II I ----------------------------------- I - - --- I I I ___ I _......~-~i~3 —-. ------------------------------------ -------- -- - ------- -- -- ------------------------— + CO Fig. 12. Torque in gram-centimeters at the elbow for Subject No. 3, executing Motion No. 3.

- ---—. —-— ~I.-# — Yi' * E —OULDfR FOR SUBJECT NOW. 3 EXECUTING NOTION N-. 3 (' 40.00 ----- ------ ------ -----— _-_ — _ j - JT I 1 ______ _____________ —---------- --------------------------- ---------------— I -------------------- _ E I I I ------------ ------------------------- -- S I I I G —--------------------- ------------------------- - - - E I I R I I ____ —-------------------------------------------------------------- --------------------------------------------------— I —--------------- 0 I D I I I ---------- --- ----------- I F t I I R I I I - ------------ I —------------------------------------------------— I —-------------------------------------------------- I I l o I I I....T....... —. —- - - ------------------ 0 I I I 1 I I I - I a I I 1 \ - ------------ r ——' —---------------------------------------------- I —--------------- ------------ r —---------------- D I I O \ I.I. IH c I I I ------------ --------- ----------------- - - - I-I I I I E r I I I I I —------------------------------- I -------------------------— 40.00 —----- --------------------------— + —--------- -------------------------— +Fig. 13. Angular velocity at the shoulder for Subject No. 3, executing Motion No. 3.

— "'-'~~ —-----— ~.'A —---—,O' —-~-AT- THE-SHOULDER FOR SUBJ ECT NO. 3 EXECUTING N MOTION NO. 3 460.00 + —-------------------------------------— + I I I h I. I ---- N I I I I I........ L I I I E I I II S I I I I I I M I I I E I I -I E I I I $ I I I u I I I R I I I E I I I 0 I I I I I I F I I I O I I I I'I I -— ~ — ---- - ----- --- ---- ---- -------------- ------- - - -------------------- - - -------- ------------ ------------— i III I O —---- I. I.... — I —-----------—. —----------------------—....I... -— ii- - ---------------------------------- ---------- 0 I II' I O I I I -- - - - - - - - Td....-.- -0 -+- - - -............................................-,- - -- - - - - - - - - -- - - - - - - - - - A I iI' I mI E I i 0 r I -— ~" —-----—' —---- ---— ~ —---------— ~ —-— ~-~ —- r —----------------- -------—` —------— r —------------------ 0 ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~ j/'(KO ~~~~~I ~~~~~~~~~~~~~~/ I E — -- V I I I I I H / I _ _ _ _ C I I' P I I _ __I...................................................... —-------------------------------------—...... L 1 -- I: 9- I L I I w~~~~~~~~~~~~~~~~~ E~~~~~~ C I i a, ------------ --------------- ----— ~~ ~ -------------- ---- ~ —------------— ~ — ---- ---— ~~ —--------------— ~ —-~ —----------— r —------- a —— 5 —---- ------ -------- ----- I I -- E I..I I I I I I I_~. / I E I I I' —- ------------- -- - - ---------------------- -- ----------- - ---------- ------------------- I —-- ------------------ -- ------- -------- ------- ------------- -I - - - - - - - - - -460.00 + —------------------------------------- -460.00 u.00'460. 00 Fig. 14. Angular acceleration at the shoulder for Subject No. 5, executing Motion No. 5.

FORCE IN GRA'iS AT THE SHOULDER FOR SUBJECT Ni. 3 EXECUTING MOTION NO. 3'~C)-. —— i —- --------------------- I I I C Pl I I I N --------------------------- ------------ -----------— I —----------- - -~~~~~~~~~~~~~~~~~~~~~~~I I M I I I -- ------— ~ —— r —I —----------------------------- ----------- I —----------------------------------- I --- ------ ------------------------— I —---------------------— T —--------— I —------------- G I'1I I R I I I E I - I 0 I D I I I ------------- — I —------ RI I R I I _ I 0 I H I M I I C I — I ——.-. —--— I —E -- I R ~ ~ ~ ~ ~ ~ ~ ~ ~ I -I T~~~~ - - -- - - --- - - - - - - - - - - - - - - -I — - - - - - - - - - - N- - - - - - - - - - - - - E I / I 0~~~~~~~.... T I I I -— ~A --------- - ----------- ----------------- ---------- --- -~~- ---------------- ----— t —----------------- - ----- I I~~ ~ —------------- ------------------------ --- ---- --------------------- T —------ C I i I...... I v C L I I I D I-....- - - -. - - I —------------ I -R ---------- ---— T~ ~ ------ -------- ---------—' —- ---------------------------------- - - - -- - - - C I I I I ---- ----------------------- ------------------ S I I I EI ----— I _........ -T- ----------------------------------------— I —------------- I I I II —--------------------------------------------------------------------------------------- L I I I.... - II y I vo W I I I I I I I I I III~~~~~~a --- -- ------------------- - ~~- ~ ------------------ -------------- -- I --- ------------- --- -- ------- ---------------------------- I ——' —------- Motion No. 1. IoI Mti n Io I

TORQUE IN GRAM-CENTIMETERS AT THE SHOULDER FOR U' —;PJFCT NO. 3 EXECUTING MOTION NO. 3 - -3 --- "' —— r~~~~ —- -- --- -- --- ------ - -- - - - - -- -- - - - -- -- - -------- -------- - - - -- - - - I I A I I I MN - - - - I -------------------------— I —---------- N I I I k --- --- - ---------------------------- I I L I I I I I I M I I I ----------- --------- -----------------------—! —------------ I - -------------- -------------------------— I —---------- ----- f~- I A I I I R —---------------- I IE I I I R I I I E I I I - - - - I I I R I I I o I I I M I I I I I I oI —- -I I I M ----------- I —------------------ ----— ~~~ —- --------------------- I —------------------- -- -— ~ ------------------------ I —- ----- N I I 5F I N I W O. + -M —- ----------------------------------- -2+ -~ —-----— r —-— ~~~~ —--- ---- -— ~ —-------------------------- ---------- --------------------- R I I I E - - - - - - - ---- - - - - --- -- - - - - - - I — - - - T HT I'...- -.......-I - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - E I I'/ I ~ —R —- -I — - --- - - 4I' I T I C~~~~~~~~~1 —-— ~~~ _ I ------..-. I....... I...... TC ~~~~~~~~~~~ I ~~~I I/ L I O12 /I - I C I I I R I S II I + z.-:- Z. —-Z --- ------ I I I. I I i T~~~~~~~~~~~~~~ I I I I I I 4FzFig. 16. Torque in gram-centimeters at the shoulder for Subject No. 3, executing Motion No. 3.

Pearson 43 TABLE 1 PHYSICAL CONSTANTS Quantity Units 1 2 3 4 5 Part CM 29.00 29.70 32.10 30.30 29.50 Upper arm 25.70 24.70 28.90 29.30 27.10 Forearm CM 18. oo 17.50 19.50 19.40 18.00 Hand EH CM 43.70 42.20 48.40 48.70 45.10 Forearm-hand comb. SG- CM 12.40 12.73 13.76 12.99 12.64 Upper arm E-f CM 11.00 10.60 12.40 12.60 11.61 Forearm W-Gh CM 4.70 5.00 5.48 5.10 5.28 Hand E-c CM 16.30 15.03 17.58 17.22 17.02 Combination WU Grams 1472.3 2510.2 3147.9 2456.7 2334.7 Upper arm Wf Grams 858.0 1331.0 1676.0 1526.0 1219.0 Forearm Wh Grams 314.6 402.9 506.0 643.6 429.6 Hand WC Grams 1179.7 1741.8 2196.1 2009.6 1661.3 Combination Isu G-CM-sec2 334.1 584.8 869.0 584.5 525.8 Ief G-CM-sec2 141.7 207.1 371.6 335.6 223.3 Iwh(F-E) G-CM-sec2 11.18 15.5 23.4 19.6 16.4 Flexion-Ext. Iwh(A-A) G-CM-sec2 12.02 16.7 24.2 20.3 17.6 Abduction-Ad. Iec G-CM-sec2 458.3 585.9 977.9 894.1 716.2 Igu 103.3 170.1 261.4 161.97 145.6 Igf 35.75 54.2 108.9 90.1 55.75 Igh(F-E) 3.93 5.2 7.5 6.77 5.88 Igh(A-A) 4.77 6.40 8.3 7.47 7.08 Igc 138.68 184.7 317.7 286.5 225.2

Pearson FOOTNOTES'The work reported here was performed as part of the Orthetics Research Project, Department of Physical Medicine and Rehabilitation, Medical School, The University of Michigan, under Contract No. 216 with the Office of Vocational Rehabilitation, Department of Health, Education, and Welfare, administered through the UniversityVs Office of Research Administration. 2The experimental work was conducted under the guidance of Dr. W. Co Dempster, Department of Anatomy, Medical School, The University of Michigan. 3Numbered equations are included in the algorithmic sequence. Lettered equations are explanatory digressions.

APPENDIX

I. COMPUTER PROGRAM The analysis in the foregoing paper sets forth the mathematical procedure for determination of the values sought and gives the data and results for one of the subjects and motions treated. As five subjects and 17 basic motions were treated, the amount of computation warranted the preparation of a program for the digital computer. Furthermore, this offered the possibility of handling the large amounts of data associated with analysis of continuous action of this type. The particular program described here enables the computer to do a number of things. It accepts the data of the particular subject and motion involved, and computes the angular velocity and acceleration for each element of the extremity, thereby describing the motion. It then computes the magnitudes and directions of torque and force reactions at the, oints in. questiono All this is then tabulated as output. The program goes on to arrange scales for optimum size plotting of each of these in cartesian coordinates. The forces and torques are then assembled as vectors. Again scales are arranged for optimum size plotting of the vectors in polar coordinates. The net result is a fund of information in graphical form, which will make possible a comparative study of the causes and effects of dynamic actions of the upper extremity~ The symbols used in the program are as follows: A-1

Program Definition MO'I~NO motion number ISUBJ subject number PHIO initial position of forearm-hand combination with respect to downward vertical,degrees THETAO initial position of upper arm-hand combination with respect to downward vertical., degree SE shoulder-elbow length, cm SGU shoulder to upper arm center of gravity length, cm EGC elbow to forearm-hand combination center of gravity length, cm WC forearm-hand combination weight, grams WU upper arm weight, grams ENERTC forearm-hand combination moment of inertia with respect to center of gravity, gram-cm-sec2 ENERTU upper arm moment of inertia with respect to center of gravity, gram-cm-sec2 PHI(I) angular change of forearm from initial position, radians THETA(I) angular change of upper arm from initial position, radians AU acceleration of the elbow, cm/sec2 AGC relative acceleration of the forearm-hand combination center of gravity relative to the elbow AC total absolute acceleration of the center of gravi~ty of't>e forearm-hand combination, cm/sec2 SFC the inertia (D'Alembert) force at the center of gravity of the forearm-hand combination) grams SFU the inertia (DeAlesembert) force at the center of gravity of the upper arm FU the total force at the center of gravity of the upper arm

Program Definition RE reaction at the elbow, grams GAMMAE angle of elbow reaction, degrees RS raction at the shoulder, grams GAMMAS angle of shoulder reaction, degrees TC torque about the elbow due to the total force at the center of gravity of the forearm-hand combination, gram-cm TEU torque about the shoulder axis due to the reaction of the elbow, Re, gram-cm TU torque about the shoulder due to total force at the center of gravity of the upper arm, Fu, gram-cm VPHI angular velocity of the forearm-hand combination, radians/ sec VTHETA angular velocity of the upper arm, radians/sec APHI angular acceleration of the forearm-hand combination, radians/ sec2 ATHETA angular acceleration of the upper arm, radians/sec2 TORQE elbow torque reaction, gram-cm TORQS shoulder torque reaction, gram-cm IIo FLOW DIAGRAM The step-by-step procedure of the program is described in the'block flow diagram which appears on page A-6 of this appendix. In this diagram blocks 1-15 are devoted to various input requirements, assemblage of data, printing instructions, arrangement of sublscriptts, etc Items 16-37 give in.structions for computing the values as set forth in the analyses~ Blocks 38-i40 and the

so-called external function convert the rectangular components of the joint reaction forces to polar components. The flow diagram continues with blocks 41 and 42 which give instructions to store the polar components of the elbow reaction vector. With item 43 and 44 the diagram calls for computation of elbow torque reaction. Block 45 starts a similar procedure for the upper arm and continues through to the end of the analysis, giving elbow and shoulder force and torque reactions, Blocks 59-68 are conversion, printing, and storage instructions. The rest of the flow diagram is concerned with arranging and printing the output values in graphical form, the details of which are discussed in the section of the paper entitled, "Typical Resultso" IIIo INPUT DATA The flow diagram (Fig. A-I) is followed by a typical data sheet~, Table A-l, for the subject and motion used as an example in the article manusecript. It gives the angular displacements of the upper and forearm for -the sixteen different positions involved, The:initial positions are indicated in the upper right-hand corner for use in the axis transformation. These data and the physical. constants of the subject were transferred to the IBM cards in the manner shown in Fig. A-2 on page A-I.L Sincre the angles were measured in degrees, but computed in radians, both appear on the data sheet. The input data cards, however, record -the angles in radians only. A-4

IV. PROGRAM The program, which is on pages A-12-A-24, is an expression of the flow diagram in the Michigan Algorithm Decoder language, commonly termed MAD. It was run on an IBM 707. The punched cards for the program are available for future use for any set of data. The manner of arranging the input data is demonstrated in Fig. A-2. Although this particular program was prepared for a two-link system, a subsequent generalized program in which any number of links can be specified has been prepared. V. OUTPUT The output information is tabulated in Table A-2 of this appendix. It records the input data as well with a list of the units for each item. The position angles, phi and theta, should have 360~ subtracted from each value to give the position of the arm from the downward vertical, i.e., o = 34736o = -13~; ~4 = 405-360 = 450, etc. Underlined values are simply maximums and minimums. The output values were then treated for scaling and plotting, resulting in curves of the nature shown in Figs. 9-16 in the paper. The output of the program calls for expression of kinematic and dynamic results in cartesian coordinate form as well as polar coordinates, By slight changes in the program, either or both forms can be produced. A-5

4 6 Read all Print title Constants and con- Print desc Read Start1 0, ISII, stants on of units No. data pts PJi, THtIVkAO, output and Lab'e3 N, TIWOH and etc. sheet for cols. SQd 10 1 1 1 13 14 15 Dimengularon Rad angular all incremental component component-2 3 supbscripte I of upper of upper variables for I -e 1in I > 16 2I 12 19 Compute AX Compute Conte rt Compute VP onentI compo incremental VT component angulartion N. o tangulation of inertia of inertia vAclocity elbowf accl. of absolute velocity C.G. lof upper arm louer arm alue in upper arm RAD/8EC RAD/S/SIzC radians RAD/SEC 28 2 22 31 Comapute EOCX = Compu rt ~Compute ACXE = Compute ACY = ATeECmA ncremental X component component angular oAf len accl. at C.G.th of length C.G. of accl. oe f aboler to tat o of upper a upper arm.alue in a lower arm RADSECLSEC radians ComUY - Compute S1 Compute SWY - X compute AC = Comput e AC = X component Y component of translation. of translationent of inertia of inertia cl. of elbow accl. of elbowl. force at C.G. force at C.G. duc to rot. due to rot. of ofupper.G.of upper of upper arm foupper arm am arm 28 29 30 31 Compute EGCX Compute EGCY = Compute ACXE | Compute ACYE X component Y component | X component of Y component of I; of length L_] of length | _| accl. at C.G. of accl. at C.G. of Fig. A-. F low diagram for original prograloer am duelot. { arm l l arm i | to rotation of to rotation of I I lower arm lower arm 32 33 3 Compute ACX = Compute ACY = I I SFC I c n X component Y component of total accl. of total accl. of fortiat o fortiat t 9 | foreana l l forea forearm forearm I Fig. A-1. Flow diagram for original program with plot. A- 6

C~ll toe pet G'IA(L) - Zmt torque rto 36 4 46 4 Com=pute Com) C eCputoe aute mt C t r compon enty p component of 2torque at of *total force neatore ralue of at of reaction of reaction at elbow. at elbow [ store the utore tar ou r tor result Presult pr e 1 cop the set set Ca fuaction See flow dief em tvetor. f orin Results wogrm for are Z details 44 45 46 4T etSetet C,tompute ompute MCW(L) -'X -syx 4L)- G UY S- lFU- (L) - x reaction x cvZptonent U, y compo- component of torque at Of to tal for nent of total reaction at elbow actin action on force action shoulder. on the forearm upper arm on upper arm store relalt Compute S set Call the IWY(L) - y X - RBX Y - BOY external compnent Prepare function uf reaction for epe o a ctrre at shoulder. asbly sly Store result as in no- 13T &tCon v Set Compute Compute RS(L) - Z GAEAS(L) - 8G UX -x = I - y ] store value ZETA, store Component of component of _ re c of reaction reac TORQ C.G. dis C Cstace elbow acting aUtA in sh our.udnbern f at shoulder at shoulder on upper arm on upper arm for post'n(L) r for pot'n(L) for post'n(L) for tn(L) Compute Compute Compute Set | TU - TC<E Z | T ME1J - TtORCUE | CRQS(L) = PHI(I) due to forese due to rea- totl reac- retain PI at C.G. w ttion at elbow t ion torque in radi upper arm on upper at shoulder | for use in Sm rult L aarm plot routines 60 61 62 63 Convert Set Cosnvert Set PHI(I) to THETA(L) to ETA(I ) K = I-3 degrees TuHETA, retai n to degrees position plot ig d t rout-neprinting printing.e

results of PI E(L) Convert THAAT(L) - in tabular Save PHI to incre- maw T M A form in degrees mental -in degrees for plot radian value for plot 70 69 Convert /e\A(I) back L pL - L+1 ato incre- c,\ used inalusofpsinmental radian value to 112 page 1 From no. 15 page 1 Declare Store in Secute values program S fQoPM. * BREAK to be a coamon instructs integer (permanent the machine I, J, IJ, K, storage) for to execute etc. use in Sec II Section II SAC. II Set I Restate Dimension Set J = J-5 L program all values Nlscale Sets no. common usel in Values. of positions values for Section II Scale factors to be Section II for graphs analyzed _ -, r 78-87 88 89-123 11 1 lo LT(M) = $M15,i MT(ML) $ A$| >J LT(I+l) 32 > J etc. M >? e rical? \e alphabetic M+0l ye plotting char. | M+3 35 plotting. L~~~ j t { character 124 125 126 17 Define Execute Set Print ldbtl for Plot 1. Set maximum and title for ordinate Grid spacing inium Graph. of graphs for polar values for Polar graph called by graphs Graph 1 of elbow L Yaxis Plot 2 force 128 129-130 131 152 Li = 1 C ompute VWLX tExecute Plot 3 Execute Plot 3 m i8/ | Pnd VLY x and y Mot the end Plot the end L > IJ N 1 components of the posi- of the? L [ of position tion vector force vector ~L = L+Il [ vector by a number by a letter LT(L) MT(L) Ie page 4 Fig. A-1 (Continued). A-8

$13334b-142 143-142 145 Execute Plot Construct Print title which writepolargraExecute plotf grah asforce siilar g storque soutput to Grapwih L torque forL graph L+1\ es po npo lar grapha f Execute dompute Execute Cotmpute is and RLY plot 3. Plot andn graph 6onents leto e x and y Wt he he positionl of the torque vect graponent of vector by vector as L L Yes I position a nsumber or - from lot 3. Plot output graph 54 fsunt for gptor by7 graph as torque of graph1 a lof tter )ofsoulder similar to called by MT(L) output graph 3 Yord 170 171 172-174 Execute plot Print L = 1 Reduce 1 and 2. Sets up title for is all angles grid spacing graph 5 I D to 560e for graph 5 Cartesian? or less elbow force vs graph L Lor ease of arm position plotting 175 176 177 178-187Execute Execute I I | i Construct Plot 3. Plot plot 4. Print graph 6 letter on Write the ble for shoulder the Cartesian |abscissa - force as a graph as output on graph 5 function of arm position 188-194 195-201 202-208 209-215 Construct Construct Construct Construct | graph 7. graph 8. graph 9. graph 10. - )of elbow I,> of shoulder Elbow force | Shoulder force force action force action as a function as a function vs elbow vs shoulder of K of elbow of p of shoulder I force force force action force action 216-222 225-229 20 Construct Construct Execute graph I | graph 12 SEQPGM. Declare elbow torque shoulder Instructs integer as a function torque vs machine to values of arm arm return to position position first sec Fig. A-1 (Concluded). A-9

TABLE A-i DISPLACEMENT VALUES Subject: Coleman THETAO = -27067 Motion: 3 PHIO = -13~0 4 in degrees radians Values from Tracing Position 0~ 0R eoR o'R 0 0.000 0.000 1 + 4o0' +.070 + 6010'.108 2 60'.105 + 17010'.300 H H 3, 12020'.215 36~20'.634 A A N N 4 22030'.393 58020' 1.018 D D 5 36010'.631 82050' 1.498 V V A A 6 51050'.905 107020' 1.873 L L U U 7 69010' 1.207 129010' 2.254 E E S S 8 83010' 1.451 150050' 2.632 N N 9 94020' 1.646 17200' 3.002 0 0 T T 10 o104010o' 1.818 191050' 3.348 U U 11 112010' 1.958 210020' 3.671 5 S E E 12 118020' 2.065 225040' 3.939 D D 13 12200' 2.129 238040' 4.166 14 124050' 2.179 246020' 4.299 15 126040' 2.211 251040' 4. 392 16 138020' 2.414 234010' 4.436 A-10

Subject: Coleman Card No. 1 Card No. 2 1 2 3 I 1 2 3 4 5 olo 31 + MOTION NUMBER 1 3 2. 1o 0 l SE 4 5 6 6 7 8 9 10 0 ~ I 1 03 1 + SUBJECT NUMBER 11 1. 71 6 1 + SGU 7 8 9 10 11 12 l 12 13 14 15 3 1417.1 0 10 1 PHIO 17 1. 1 518 EGC 13 14 15 16 17 18 16 17 18 19 20 21 3 3 1 2 i31 3 + THETAO 1211 LL61 1 WC 22 23 24 25 26 27 1 1 4 1 7. 191 + Wu 28 29 30 31 32 33 Card No. 3 3 1 71. 1 71 ENERTC 1 2 3 34 35 36 37 58 39 0 2 1 | + N O 2 | 6 |1.1 41 ENERTU Variable Cards M(I) PHI( I ) TIETA(I) M(I) PHI(I) THETA(I) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 H1 i-' 00 I i-ol1i i olo0 11 - 041.159503181131 02 21'10 2 -6130 - 0 1 0 2 81 1 1 19 01 4 1 41 36 0 0 12 1 1 2 3 1 3 10 000000oolo 0 0lolo00 0 0141. 4 11lo o121.4152 41710 4 o 10lol80 0lolo090 21 0lo14.14600 012N.l 261010 5I 3 03-loo o0oi5 15 l10210 22 I 00.609 olo0.2150 5 1 1 1 1 1 1.12 1 7 10 2 O. 13 9 14 0 214 8 O 01 4 750 6500 2 O I I Z61 1 I I 9 01.8750o o0 090280 216_1 I - 10 0121.l25160lo 0o 111.l1 l9 l2 o0 217 I i I I 1 11 l 01 2.6lL2 0 019 17 44o6 28 1 1V.! 1 — __ o12/.]lsislol o 53 1 1 1 1 2 1 0 9 0 1. 10 I m113 o 13L 3 3 9 4o o I1 1 l o 31 L 15 13. 9 1 1 o 1 1 1 6 l O 3 1164 _0 1. 1l0 1 S o3 o 12 ],,1 1I3 13 O I3 1II3 167 04.154 10 53 12. 1l _ 18 lo.! 3 4 Fig. A-2. Input data. A-il

PROGRAM FOR IBM 707 DIGITAL COMPUTER IN MICHIGAN ALGORITHM DECODER (MAD) LANGUAGE CP........ P. L...COP.LE. MA1:.,...E X E(U T D:. DUMP., PUNCH OBJECT, PUNCH LIBRARY CONER R 1 R PROGRAM FOR OVR FORCE ANALYSIS (PROJECT 03655) 2 R 3 R 4 R Core Load No. 1 5 QQ002 READ FORMAT QQOOO3,MOTNO,ISUBJPHIO,THETAO,SE,SGU,EGC,WC,WUE 0006 1NERTC ENERTU 7 VECTOR VALUES QQ00003 $ 213,2F6*2/3F5.2t4F6.1 *$ 8 QQ0004 PRINT FORMAT QQ0005,MOTNOISUBJSEWCENERTC9 SGUWU#ENERTU, 9 iEGC 10 VECTOR VALUES QQ0005 $ 16H1 MOTION NUMBER 12921H 11 1SUBJEcT NUMBER I2/8H SE, F5.2,8H WC= F6el,12H ENERTC 0012 1= F6*1/SH SGU= F5.2s8H WUZ F6*11l2H ENERTU= F6.1/8H 13 1 EGC= F5*2 *$ 14 QQGO06 PRINT FORMAT QQOO7 15 VECTOR VALUES QQO007 = $ 23H0 SESGU,EGC ARE IN CM./26H 16 I WU,WC ARE IN GRAMS FORCE/40H ENERTCENERTU ARE IN GM*-CM.-S 0017 1EC.*SFC./40H PHIgTHETAGAMMAE,GAMMAS ARE IN DEGREES/37H VPH 0018 1IVTHFTA ARE IN RADIANS PER SECe/46H APHIATHETA ARE IN RADI 0019 1ANS PFR SEC. PER SECo/2~H RE.RS ARE IN GRAMS FORCE/29H TORO 0020 1EIOROS ARE IN GRAM-CM/10O9HOPOSITION PHI VPHI APHI 21 1 RE GAMMAE TORQE THETA VTHETA ATHETA RS 22 1 GAMMAS TORQS *$ 23 QQ0008 READ FORMAT QQ0009,N 24 VECTOR VALUES 00QQ009 $ I3 *$ 25 QQ0010 DIMENSION M(50)gPHI(50),THETA(50),REX(50),REY(50)TORQE(50) 0026 1,RSX(50}),RSY(50),TORQS(50).~HE(50).THATA(50}. 27 A-12

1 RS(50), RE(50), GAMMAE(50),GAMMAS(50), P 0028 1HEE(S5) #THAAT (50) 29 THROUGH QQ0011FOR In 11 I *G. N 30 QQ0011 READ FORMAT QQ0012 9 M(I). PHI(I), THETA(I) 31 VECTOR VALUES QQOO12 $ 1292F7,4 *$ 32 QQ0013 J=N-~ 33 QQ014 Lzl 34 QQ0015 THROUGH QQO070 9FOR I = 3 I9 eG 35 1IJ 36 QQ0016 VPHI=(PHI(I+1)-PHI(I-1))/00596 37 QQOO17 APHI=(PHI(I+2)+PHI(I-2)-2,0*PHI(I))/.00355 38 QQ0018 PHI('!)=PH.I(I)+PHIO* 0 174533 39 QQOO19 VTHETA=(THETA(I+1)-THETA(I-'1))/0596 40 QQO020 ATHETAa(THETA(I+2)+THETA(I-2)-2.0*THETA(I))/900355 41 QQ0021 THETA(I)=THETA(I)+THETA0O**0174533 42 QQ0022 $EX=SF*SIN.( THETA(I) 43 QQ0023 SEY=-SE*COS ( THETA(I)) 44 QQ0024 AUX=-ATHETA*SEY-VTHETA*VTHETA*SEX 45 QQ0025 AUYA THE T A*S E X-VTHE T A*VTHETA*S E Y 46 QQ0026 SFUX=-(WU/981-O)*AUX*SGU/SE 47 QQ0027 SFUY=-(WU/981eo0)*'AUY*SGU/SE 48 0QQO028 EGCX=FGC*SIN.( PHI(I)) 49 QQ0029 EGCY'-EGC*COS.( PHI(I)) so 50 QQO030 ACXE=-APHI*EGCY-VPHI*VPHI*EGCX 51 a0031 ACYE=APHI*EGCX- VPHI*VPHI*EGCY 52 QQ0032 ACX=ACXE+AUX 53 0QQ0033 ACY=ACYE+AUY 54 QQ0034 SFCX=-(WC/981*0)*ACX 55 0QQ0O035 SFCY=.(WC/981i0)*ACY 56 A-:13

000036 REX(L)=-SFCX 57 Q00037 REY(L)t-SFCY+WC 5.8 000038 XxREX(L) 59 000039 Y=REY(L) 60 6Q0040 EXECUTE VECTOR,( XY*ZETA#Z) 61 QQ0041 RE(L)uZ 62 QQ0042 GAMMAF L)=ZETA 63 QQ0043 TC=-(EGCX*REY(L)- EGCY*REX(L)) 64'U~-~`-~c"~- ~ -— T —-~-~~ —L. ~ —- -- ----- -- - - - - -- --------- -- - - - - - - - - - - -- - - - - -- - - -- - - - -- - - - -- - ----- ~-11 — ~ -1 - --- -___ __C -__ QQ0044 TORQE (L) a-TC+ENERTC*APHI 65 000045 FUX=SFUX 66 0QQ0046 FUYsSFUY-WU 67 Q00047 RSX(L)=-FUX+REX(L) 68 QQ0048 RSY(L)u-FUY+REY(L) 69 QQ0049 X=RSX(L) 70 QQO050 YaRSY(L) 71 QQ0051 EXECUTE VECTOR-( X*Y*ZETA*Z) 72 000052 RS(L)=Z 73 Q0053 GAMMA, (L) -ZETA 74 QQ000054 SGUXsEX*SGU/SE 75 000o055 SGU~ =EY*SGU/SE 76 000056 TUSG!,X* F-UY-SGUY*FUX 77 00QQ0057 TEU=-(SEX*REY(L)-SEY*REX(L)) 78 Q000058 TORQS(L)=-TU-TEU+ENERTU*ATHETA+TORQE(L) 79 000059 PHE(L)=PHI(l) 80 000060 PHI(I =PHIIH)/,0174533 __ 81 QQ0061~ THATA(L)=THETA(!) 82 00062 THETA~I),THETA(I)/,0174533 83 000063 Ku!-3 84 000064 PRINT FORMAT 00QQ0065,K*PHI(I)VPHIAPH:IRE(L),GAMM1AE(L),TORQE( 0085 A-14

1L),TH-FTA(I),VTHETA,ATHETARS(L),GAMMAS(L) TORQS(L) 86 VECTOR VALUES QQ00065 4H I2,F1O.2,F9#3,F9g2,F9.1,F 0087 17e2,F9e1,F9.2,F9e3,F9g2,F9.1 1F7.2,F9. 1 *$ 88 QQ0066 PHEE(L)=PHI( ) 89 QQ0067 PHI(I)=(PHI(I)-PHIO)*0'174533 90 QQO068 THAAT(L)=THETA(I) 91 QQ0069 L=L+1 92 QQO070 THETA(I)=(THETACI)-THETAO)*.0174533 93 INTEGFR I, J, IJt K, Lo Mt N 94 PROGRAM COMMON ISUBJ9 MOTNO. PHE, REX. REY. THATA, RSX. RSY. 95 1TORQE * TORQS, RE, RS, PHEE, THAAT, GAMMAE, GAMMAS,N 96 EXECUTE SEQPGM. 97 END OF PROGRAM 98 * COMPILE MAD. EXECUTE, DUMP, Subroutine Vector 99 * PUNCH OBJECT. PUNCH LIBRARY 100 QQOO02 EXTERNAL FUNCTION (X.YZETA.Z) 101 ENTRY TO VECTOR. 102 QQ0003 WHENEVER X *L.O.,TRANSFER TO QQ0014 103 WHENEVER X *G.O.,TRANSFER TO QO.021 104 QQ0004 WHENEVER Y *L.O.,TRANSFER TO QQ00O11 105 WHENEVER Y *G.O.. TRANSFER TO QQOO08 106 QQO0.05 Z=O0 107 QQOO07 FUNCTION RETURN 109 QQ0008 Z=Y 110 QQOO09 ZETA=180.00 111 QQ001O FUNCTION RETURN 112 QQ0011 Z= *ABS.( Y) 113 QQOOi2 ZETA=nOO 114 A-K5

0Q0013 FUNCTION RETURN _____ ___ ____ 115 Q00014 WHENEVER Y *NE.Oo, TRANSFER TO Q00Q18 116 QQ0015 Z= *ABS.( X) 117 QQ0016 ZETA=270,00 118 QQ000017 FUNCTION RETURN 119 QQ0018 ZETA=270.O0+ATAN.( Y/X)/,0174533 120 000019 Z=SQRT.( X*X+Y*Y) 121 QQ000020 FUNCTION RETURN 122 Q000021 WHENEVER Y *NE.O0, TRANSFER TO 0Q0025 123 QQ0022 Z:X 124 QQ0023 ZETA=90:00 125 Q000024 FUNCTION RETURN 126 00QQ0025 ZETA=QO0OO+ATAN.( Y/X)/~0174533 127 00QQ0026 TRANSFER TO QQ000019 128 0300001 END OF FUNCTION 129 BREAK, COMPILE MAD, EXECUTE, DUMP Core Load No. 2 130 PUNCH OBJECT9 PUNCH LIBRARY 131 PROGRAM COMMON ISUBJ, MOTNO, PHE, REX, REY, THATA, RSX, RSY9 132 1TORQE, TORQS, RE, RS, PHEE, THAAT, GAMMAE, GAMMASN 133 DIMENsION PHE(50), REX(50), REY(50), THATA(50), RSX(50),RSY(5 0134 10, TORQE(50), TORQS(50), RE(50), RS(50), PHEE(50)*I'HAAT(50), 0135 1GAMMAF(50), GAMMAS(50)9 POLAR(883), CARTE(839), YORD(8), YAXI 0136 1S(8), LT(55)9 MT(72)9, NSCALE(5) 137 000071 NSCALF:1 138 NSCALF()=0O 139 NSCALF(2)=O 140 NSCALF(3)O0 141 NSCALF (4)=0 142 QQ000076 J=N-5 143 A-16

QQO077 THROUGH 0Q0087 *FOR M = 1 10 * 144 iM *G. J 145 QQ0078 LT(M)=$0$ 146 QQ0079 LT(M+1)=zl$ _147 OQO080 LT(M+?)-$2$ 148 QQ081 LT(M+I),$3S 149 QQO082 LT(M+4)=$4$ 150 QQQ083 LT(M+5)=$5$ 151 QQO084 LT(M+6)=$6$ 152 QQ0085 LT(M+7)=$7$ 153 QQ0086 LT(M+)S=$8$ 154 QQ0087 LT(M+9)=$9$ 155 QQ00088 THROUGH QQ0123,FOR M = 1 35 156 iM *G. J 1i57 QQ0089 MT(M)=$A$ 158 QQO090 MT(M+1)=$B$ 159 0QQ0O091 MT(M+P)=$C$ 160 QQ0092 MT(M+i)=$D$ 161 QQO093 MT(M+4)=$E$ 362 QQ0094 MT(M+S)=$F$ 163 _QQ095 MT(M+6)=$GS 16.4 Q0096 MT(M+7)=$H$ 165 QQ0097 MT(M+B)=$I$ 166 QQ0098 MT(M+9)=$J$ 167 QQO099.. MT(M+10)=$K$ 1. ___68 QQ0100 MT(M+11)=$L$ 169 __ 000101 MT(M+12)=$M$ 170 QQ0102 MTI(M+13)= $N$ S.......... 171 QQ0103 MT (M+14) =$OS 172 A-17

Q00104 MT(M+15)zSP$ 173 QQ 105 MT(M+I6)=QS0 174 QQO106 MT(M+i7)=$SRS 175 QQ0107 MT(M+18)=SSS 176'Q0108 MT(M+19)=$T$ 177 QQ0109 MT(M+20)zSUS 178 QQ0110 MT(M+21)=$VS 179 QQ0111 MT(M+22)=$W$ 180 QQ0112 MT(M+73)=$XS 181 QQ0113 MT ( M+ 24 ) = SYS 182 0O011-4 MT(M+?5)=$Z$ 183 QQO115 MT(M+?6)S1S 1$84 QQ0116 MT(M+27)=S2S 185 00QQ0117 MT(M+?8)=$3S 186 QQO118 MT(M+?9)=$4S 187 QQO119 MT(M+iO)=S5S 188 0Q0120 MT(M+1)=$6S 189 0QQ0121 MT(M+12)=S7S 190'-~` —~'~-` -— ~' — — ~~` ~`~~~~`~`~ —~~-` " ~'` " ~- ---- ~ —- -------— ~ —---- — c —- — L- ~ ^ I~ —~ —~0QQ0122 MT(M+13)=8S5 191 QQ0123 MT(M+'4)=$9S 192 0QO124 VECTOR VALUES YAXIS = $ ANGLES MEASURED FROM DOWNWARD 193 1VERTICAL CCLWSE$ 194 Q000125 EXECUTE PLOT1.( NSCALE,2o, 27-. 2os,45o) 195 GQ0126 EXECUTE PLOT2.( POLAR 22000, -22000o 22000. -22000 ~ ) 196 QQ000127 PRINT FORMAT QQ0231#ISUBJ9MOTNO 197 0Q0128 THROUGI{ 0Q0132 9FOR L a 1 lt. L *Go 198 ~~~1dJ~~~~~~~~~ P~~~199 QQ0129 VLX1O0000*SIN ~( PHE(L)) 200 000130 VLY=-10000.*COS ~. PHE(L)) 201 A-18

QQ0131 EXECUTE PLOT3.( LT(L),VLXVLY,1*O) 202 QQ0132 EXECUTE PLOT3.( MT(L),REX(L),REY(L),11,0O) 203 QQ0133 EXECUTE PLOT4. POLAR#YAXIS, 8*,91,Oe) 204 QQ0134 EXECUTE PLOT1.( NSCALE,2.,27..2.,45*) 205 0Q0135 EXECUTE PLOT2 *( POLAR 300000-300**3~30000.-30000) 206 QQGI36 PRINT FORMAT QQ02329ISUBJtMOTNO 207 QQ00137 THROUGH 00QQ0141 *FOR L 1 1 L *G. 208 209 QQ0138 VUX=15000 *SIN ~ ( THATA(L)) 210 QQ0139 VUY-15000OO*COS ( THATA(L)) 211 QQO 140 EXECUTE FLOT3 ( LT ( L ), VUXVUY. 1 O 0) 212 QQ0141 EXECUTE PLOT3.( MT(L)qRSX(L)}RSY(L).9190o) 213 QQ0142 EXECUTE PLOT4 ( POLARYAXIS, 8.,1.0p.) 214 0Q0143 EXECUTE PLOT1.( NSCALE 2.,27. *2.,4:5 ) 215 0Q00144 EXECUTE PLOT2 ( POLAR,606000, t-606000.606000, t-6060 216 10,) 217 000145 PRINT FORMAT QQ02339ISUBJMOTNO 218 Q000146 THROUGH 0001.53 #FOR L 1 I 1, L,Go 219 1J 220 QO147 RLX37.0000o*S[N.( PHE(L)) 221 000148 RLY —20000.*COS ~( PHE(L)) 222 QQ0149 EXECUTE PLOT3.( LT(L),RLXRLY,1.0*.) 223 00Q0150 TRL=3?0000.+TORQE(L) 224 OQ0151 TRLX=TRL*SIN.( PHE(L)) 22'5 QQ0152 TRLY-TRL*COS ( PHE(L)) 226 000153 EXECUTE PLOT3.( MT(L)oTRLX*TRLY,1.,O.) 227 Q000154 EXECUTE PLOT4.( POLARYAXIS, 89,1.,0.) 228 000155 EXECUTE PLOT1.( NSC'ALE9,2.,27.2.,45o) 229 QQO1S6 EXECUTE PLOT2.( POLAR,1145000.,-1145000,111 45000t-.11 0230 A-19

1450006) 231 QQ0157 PRINT FORMAT QQ0234,ISUBJMOTNO 232 QQ000158 THROUGH QQ0165,FOR L 1 919 L *G 233 _1J 234 0QQ0159 RUX=5;5000**SIN ('THATA(L)) 235 000160 RUY=-9550000*COS.( THATA(L)) 236 QQO161 EXECUTE PLOT3.( LT(L),RUXRUY,1.O90) 237 QQ0162 TRU=555000.+TORQS(L) 238 QQ0163 TRUX=TRU*SIN.( THATA(L)) 239 000164 TRUY=-TRU*COS.( THATA(L)) 240 QQO165 EXECUTE PLOT3.( MT(L),TRUXqTRUYs1 O0.) 241 0QO166 EXECUTE PLOT4.( POLAR,YAXISP 8*.,1,0.) 242 QQ000167 VECTOR, VALUES YORD = $ FORCE IN GRAMS OR TORQUE IN 0243 1 GRAM CENTIMETERS$ 244 0QQ0168 EXECUTE PLOTI.( NSCALE,4.,12.,5.,20,) 245 QQ0169 EXECUTE PLOT2.( CARTE,300*,-40.,22000.,0*) 246 QQ00170 PRINT FORMAT QQ0235,ISUBJMOTNO 247 000171 THROUGH QQ0175,FOR L = 1,1, L *G. 248 1r~~~~~~~~~~~~~~~J ~249 QQ0172 WHENEVER (PHEE(L)-360.o) *L.O TRANSFER TO QQ0175 250 QQ0173 PHEE(L)=PHEE(L)-360, 251 0QQ0174 TRANSFER TO QQO0172 252 000175 EXECUTE PLOT3.( MIL),PHEE(L)9,RE(L),l.,0) 253 QQO176 EXECUTE PLOT4.( CARTEYORD,8o.O.,O.) 254 0QQO177 PRINT FORMAT 0QQ0236 255 QQ0178 EXECUTE PLOTi.( NSCALE4~, 12i.,5,20 ) 256 00QQ0179 EXECUTE PLOT2.( CARTE,300.,-40.,30000.,O) 257 QQ0180 PRINT FORMAT QQ0237ISUBJMOTNO 258 QQ00181 THROUGH QQ0185 9FOR L = 1 L *G. 259 A-20

1J __ _ _ __ _ _ _ __ _ _260 QQ0182 WHENEVER (THAAT(L)-360.) *L.O0 TRANSFER TO QQ0185 261 QQ0183 THAAT(L)=THAAT(L)-360, 262 QQ0184 TRANSFER TO QQO182 263 QQ0185 EXECUTE PLOT3.( MT(L),THAAT(L)vRS'(L)91,0O) 264 QQ0186 EXECUTE PLOT4.[ CARTE#YORDO8.,0,Ot0) 265 QQ0187 PRINT FORMAT QQ0236 266 QQ0188 EXECUTE PLOT1.[ NSCALE,4.t12.,592o20) 267 0QQ0189 EXECUTE PLOT2.( CARTE,22000.,0o.,360.,0.) 268 QQO190 PRINT FORMAT QQ0238*ISUBJPMOTNO 269 QQ0191 THROUGH QQ0192,FOR L = 1.1, L *G. 270 I~~~~~~~~~~~~~~~J ~~271 QQ0192 EXECUTE PLOT3 _( MT(L),RE(L)9GAMMAE(L)t1,90.) 272 Q000193 EXECUTE PLOT4.( CARTE#YAXISs 89.,O.,O) 273 -'" c`~'` ~ ~~`~~`~~'~~- ~r" —'`~~~ -`- ~ ~'-~ ~`-~~`~r"-~ —-- -— r` —- -- -- -- -- - QQ0194 PRINT FORMAT QQ0239 274 Q000195 EXECUTE PLOTi.( NSCALE,4.t12.,5.,20,) 275 QQ0196 EXECUTE PLOT2.( CARTE,30000.,O#,360..O*) 276 GQ0197 PRINT FORMAT QQ02+40ISUBJMOTNO 277 Q0198 THROUGH QQ0199 *FOR L = 91. L *Ge 278 1J 279 QQ0199 EXECUTE PLOT3.( MT(L),RS(L),GAMMAS(L)sleeO.) 280 Q00200 EXECUTE PLOT4.( CARTE,YAXIS, 8.,O.O.9 ) 281 QQ0201 PRINT FORMAT QQ0239 282 QQ0202 EXECUTE PLOT1.( NSCALE,4.o12o,5,,20o) 283 QQ0203 EXECUTE PLOT2.( CARTE,360.0.o,22000.90. ) 284 QQ0204 PRINT FORMAT QQ0241#ISUBJMOTNO 285 QQ0205 THROUGH QQ0206,FOR L = 1 919 L *G._. 286 1J _____287 QQ0206 EXECUTE PLOT3.( MT(L)#GAMMAE(L)9RE(L),1,0.o) 288 A-21

QQ0207 EXECUTE PLOT4.( CARTEYORD98.O.,0.O) 289 QQ0208 PRINT FORMAT QQ0236 290 QQ0209 EXECUTE PLOT1.( NSCALE94.912.5.9520,) 291`~~ ~~ ~``` ~~" ~'~~~ ~~r`` ~ ----------—' cl —-1 —---- ------------------- ------ QQ0210 EXECUTE PLOT2.( CARTE,360.,Q.,30000.0O.) 292 0Q0211 PRINT FORMAT QQ02429ISUBJ#MOTNO 293 I-~~ —~~- --—`~~ -- ----— "~ —- — ~c~- -- -- ---- ---- - - - - - -- - - - - - -- - - - - - - QQ0212 THROUGH QQ0213 9FOR L = 1 i19 L *G. 294`- ~`c-~ — ~ r~ ~~~~-`~~~ ----— ~~ —-`` I~ —------ ------------ ----- -- --------------- ------------ --- 1J 295 QQ0213 EXECUTE PLOT3,( MT(L)9GAMMAS(L),RS(L),1le0.) 296 QQ0214 EXECUTE PLOT4.( CARTE,YORD,8.tO0,O.) 297 ---------—` —------- ----- — I —----- QQ0215 PRINT FORMAT QQ0236 298 QQ0216 EXECUTE PLOTI,( NSCALE,4.,12.,5.v20*) 299 000217 EXECUTE PLOT2.( CARTE,300.,-40.,300000.,-300000.) 300`~`~`' "~'~~~- ll~~c'~ —c~ —~~ — --— I —--— I —----------— ~ —QQ000218 PRINT FORMAT QQ0243,ISUBJ,MOTNO 301 000219 THROUGH QQ0220 #FOR L 1 ll L *G. 302 1J 303 0Q0220 EXECUTE PLOT3.( MT(L),PHEECL)sTORQE(L)s,1,90) 304 QQ0221 EXECUTE PLOT4.( CARTEYORD, 80,.00.) 305 QQ0222 PRINT FORMAT QQ0236 306 QQ0223 EXECUTE PLOTI e( NSCALE,4*,12,5*9,20*) 307 QQ0224 EXECUTE PLOT2.( CARTE,300.,-40.,550000.,-550000.) 308 QQ0225 PRINT FORMAT QQ0244,ISUBJ#MOTNO 309 QQ0226 THROUGH QQ000227,FOR L = 1.1 L *G. 310 " 1J 311 Q0227 EXECUTE PLOT3.( MT(L),THAAT(L),TORQS(L),1,O#0) 312 QQ0228 EXECUTE PLOT4.( CARTEqYORD t 8.,0.,0.) 313 QQ0229 PRINT FORMAT QQ0236 314 0Q0230 EXECUTE SEQPGM, 315 VECTOR VALUES QQ0231 = $ 1H1,24(1H ),44H FORCE IN GRAMS 316 1AT THE ELBOW FOR SUBJECT NO.13,22H EXECUTING MOTION NO.13 317 A-22

1*$ 318 VECTOR VALUES QQ0232 = $ 1H1t22(1H ),47H FORCE IN GRAMS 319 1AT THF SHOULDER FOR SUBJECT NO.I3922H EXECUTING MOTION NO.I 320 13 *' 321 VECTOR VALUES QQ0233 = $ 1H1917(1H )956H TORQUE IN GRAM 322 lCENTIMETERS AT THE ELBOW'FOR SUBJECT NO1I3t22H EXECUTING MOT 0323 1ION NOeIl *$ 324 VECTOR VALUES QQ0234 = $ 1H1915(1H )*59H TORQUE IN GRAM 325 iCENTIMETERS AT THE SHOULDER FOR SUBJECT NO..13,22H EXECUTING 326 IMOTION NO.I3*S 327 VECTOR VALUES QQ0235 = $O1H12(H1H )*67H ELBOW FORCE IN 328 1GRAMS AS A FUNCTION OF ARM POSITION FOR SUBJECT NO.I3922H EX -03-2:9 lECUTING MOTION NO I3/1HO*$ 33Q VECTOR VALUES QQ0236 = $026(1H ),68H ANGLES IN DEGREES M 0331 1EASURFD FROM DOWNWARD VERTICAL COUNTER-CLOCKWISE*$ 332 VECTOR VALUES QQ0237 $SO1Hll111(1H )970H SHOULDER'FORCE 333 1IN GRAMS AS A FUNCTION OF ARM POSITION FOR SUBJECT NOoI3,22H 334 1 EXECUTING MOTION NO.I3/iHO *$ 335 VECTOR VALUES QQ0238 = $O1Hl.12(1H ),67H ANGLE OF ELBOW 336 iFORCE ACTION AS A FUNCTION OF FORCE FOR SUBJECT NO*I3,22H EX 0337 1ECUTING MOTION NO I3,/1HO *$ 338 VECTOR' VALUES QQ0239 = $052(1H ),15H FORCE IN GRAMS 339 1*5 340 VECTOR VALUES QQ0240 = $OlHlll(1H ),70H ANGLE OF SHOULD 0341 1ER FORCE ACTION AS A FUNCTION OF FORCE FOR SUBJECT NOeI3,22H 342 1 EXECU.TING MOTION NO.I3/1HO *$ 343 VECTOR VALUES QQ0241 = $01H1,12(1H ),67H. ELBOW FORCE AS 344 1A FUNCTION OF ANGLE OF FORCE ACTION FOR SUBJECT NO*I3,22H EX 0345 A-23

VECTOR VALUES QQ0242 = $OlH1,11(1H )#70H SHOULDER FORCE 347 [AS A FUNCTION OF ANGLE OF FORCE ACTION FOR SUBJECT NOoI3t22H 348 1 EXECUTING MOTION NO*I3/IHO *5 349 VECTOR VALUES 00QQ0243 =$01H,15( 1H ),59H ELBOW TORQUE AS 0350 1 A FUNCTION OF ARM POSITION FOR SUBJECT NO,13,22H EXECUTING 351 1MOTION NOI3/IHO *$ 352 VECTOR VALUES QQ0244 = SO1H1,13(1H ),62H SHOULDER TORQUE 0353 1 AS A FUNCTION OF ARM POSITION FOR SUBJECT NO*13922H EXECUTI 0354 1NG MOTION NO.I3/1HO *$ 355 INTEGFR IJ, N, L I 356 1K 357 INTEGFR NSCALE J M, LT * 358 1MT 359 QQ0001 END OF PROGRAM 360 BREAK9 DATA A-24

TABLE A-2 OUTPUT DATA Motion Number 3 Subject Number 3 SE = 32.10 WC = 2196.1 ENERTC = 317.7 SGU = 13.76 WU = 3147.9 ENERTU = 261.4 EGC = 17.58 SE, SGU, EGC are in cm. PHI, VTHETA are in radians per sec. WU, WC are in grams force. PHI, ATHETA are in radians per see per sec. ENERTC, ENERTU are in gm-cm-sec *sec. RE, RS are in grams force. PHI, THETA, GAMMAE, GAMMAS are in degrees. TORQE, TORQS are in gram-cm. Positi-_I PHI VPHI APHI RE GAMMAE TORQE THETA VTHETA ATHETA RS GAMMAS TORQS'~> 0 347.00 2.869 62.82 3941.7 111.35 77171.1 332.33 1.126 18.03 6127.8 134.08 155708 1 353.19 5.084 92.96 6143.8 109.32 126498.6 334.56 1.711 30.11 8491.1 123.82 274103 2 364.36 8.406 119.15 9391.3 118.38 188655.3 338.17 2.919 53.52 12745.4 123.76 411993 3 381.89 12.181 102.82 11648.9 144.42 205330.3 344.53 4.899 68.45 16256.4 142.16 381852 4 405.96 14.530 33.80 12050.2 181.81 158292.5 354.90 6.997 60.85 17286.4 171.73 160819 5 431.51 14.195 - 23.94 10993.0 211.12 117636.8 368.43 8.523 40.85 16582.1 195.27 23365 6 454.43 13.104 - 26.20 10944.8 231.73 122169.4 384.01 9.430 10.14 16829.7 215.53 - 14790 7 476.26 12.634 - 11.27 11113.7 258.42 116285.3 400.63 9.128 -28.45 16023.0 242.68 - 98838 8 497.57 12.433 - 11.83 9738.5 287.91 80964.0 415.18 7.735 -48.17 12649.8 270.38 -174110 9 518.72 11.930 - 20.85 7190.7 315.09 44031.4 427.04 6.258 -43.38 8012.5 292.25 -167874 10 538.31 11.191 - 32.96 4957.7 344.90 9746.4 436.55 5.151 -35.21 4010.4 315.86 -137799 11 556.93 9.966 - 50.14 3869.9 23.35 - 23531.0 444.63 4.161 -37.75 2086.1 17.19 -122150 12 572.34 8.205 - 60.56 3234.7 62.22 - 47569.2 450.76 2.903 -38.03 2512.6 89.01 86907 13 584.95 6.359 - 69.58 3143.6 109.84 - 72146.4 454.54 1.896 -26.76 4184.8 136.28 - 25598 14 594.06 4.060 - 71.83 3636.4 142.47 - 86724.2 457.23 1.309 -16.90 5734.9 157.07 24431 15 598.81 2.081 - 52.39 3287.2 156.89 - 73863.0 459.01 0.889 -11.83 5806.1 166.68 48000 16 601.16 0.940 - 28.17 2703.1 166.45 - 54788.6 460.27 o.604 - 7.32 5496.4 172.95 60848