 |
||||
|
||||
|
||||
|
||||
Previous Article | Next Article
Volume 17, Number 6, Issue of March 15, 1997 pp. 2128-2142
Copyright ©1997 Society for NeurosciencePostural Dependence of Muscle Actions: Implications for Neural Control
Christopher A. Buneo, John F. Soechting, and Martha Flanders Department of Physiology, University of Minnesota, Minneapolis, Minnesota 55455
ABSTRACT
The neural control of reaching entails the specification of a precise pattern of muscle activation distributed across the many muscles of the arm. Musculoskeletal geometry limits the possible solutions to this problem. Insight into the nature of this constraint was obtained by quantifying the postural variation in the mechanical actions of six human shoulder muscles. Estimates of muscle mechanical actions were obtained by electrically stimulating muscles to the point of contraction and recording the resulting forces and torques with a six-degree-of-freedom force-torque transducer. In a given experiment, data were obtained for up to 29 different arm postures. The mechanical actions of each muscle varied systematically with arm posture, regardless of the frame of reference used to define these actions. The nature of this dependence suggests that a relatively simple strategy can be used by the nervous system to account for the changing mechanical actions of arm muscles.
Key words: muscle; torque; internal model; arm movement; reaching; posture
INTRODUCTION
At first glance, the processes underlying the production of reaching movements appear dauntingly complex. The problems are typified by the concept of kinetic redundancy: because the number of force-generating elements of the arm exceeds the number of mechanical degrees of freedom, it is not clear how the motor system apportions activity among the various muscles during the performance of a particular motor task. Nevertheless, reaching movements tend to be performed in a relatively stereotypic manner, both with regard to their kinematics and, to a lesser extent, their muscle activation patterns (Morasso, 1981
; Soechting and Lacquaniti, 1981
Implicit in the above scenario is the idea that the motor system possesses an internal representation, or model, of the mechanical action of each muscle that it draws on in producing a particular movement. Information about muscle mechanical actions would be critical not only for the specification of the spatial and temporal aspects of the muscle activation pattern but also at earlier stages of movement planning. To plan a reaching movement to a particular target, the motor system must have some sense of the starting location of the hand or, alternatively, the initial arm configuration (Bizzi et al., 1984
Most existing data regarding muscle mechanical actions were obtained via gross anatomical dissection. In these methods, the origin and insertion of a muscle are measured, and assumptions are made about the center of joint rotation and the line of action to obtain an estimate of the mechanical action of that muscle (An et al., 1984
In the present paper we report estimates of the mechanical actions of six single-joint human shoulder muscles, obtained using an approach similar to the one used successfully by Lan and Crago (1992)
MATERIALS AND METHODS
General. Data were obtained from three adult human subjects. Subject A was a 5 ft, 9 in (175 cm), 130 lb (59 kg) female; Subject B was a 6 ft, 0 in (183 cm), 190 lb (86 kg) male; and Subject C was a 5 ft, 10 in (178 cm), 143 lb (65 kg) female. Informed consent was obtained from all subjects before their participation in this study. In addition, all experimental procedures were approved by the Institutional Review Board of the University of Minnesota.
Subjects sat in a modified dental chair that was coupled to a rigid stereotaxic frame through a 1-cm-thick steel plate with stabilizing runners (Fig. 1A). The main pieces of the frame consisted of a 7-cm-diameter steel drill press column with a rack and several segments of 4- to 6-cm-diameter steel pipe. The frame allowed the six-degree-of-freedom force-torque sensor (JR3, Woodland, CA) to be positioned anywhere within a large portion of the three-dimensional workspace of the right arm. The sensor had a maximum load rating of 15 lbs [66.72 N (Newtons)] for the forces and 45 in-lbs [5.08 N-m (Newton-meters)] for the torques with 12-bit resolution. The sensor was coupled to the frame at one end and to a prefabricated, hinged, upper extremity orthosis at the other. The location of the sensor with respect to the orthosis is indicated in Figure 1A. In this figure, the upper arm portion of the orthosis is oriented vertically and the forearm portion horizontally. When the arm was in the orthosis, the sensor was located just distal to the elbow joint. (The precise alignment of the sensor with respect to the upper arm and forearm is described below under Coordinate Systems). The orthosis served to secure the sensor to the arm; this was facilitated by inflating air bladders lining its molded plastic segments. On average, the upper arm portion of the orthosis extended from just distal to the deltoid tuberosity of the humerus to just proximal to the elbow joint. The forearm segment began just distal to the elbow joint and ended just distal to the wrist. These two segments were connected by a steel hinge joint that was locked during all experiments at 90° of elbow flexion; elbow angle was not varied in these experiments. 
Fig. 1. Experimental apparatus (A), angles defining arm posture (B), and a free-body diagram of the forces and torques exerted on the upper arm (C). A, A six-degree-of-freedom force-torque sensor was rigidly coupled to an upper extremity orthosis at one end and a stereotaxic frame at the other. Two coordinate frames are illustrated: an x, y, z, frame that was fixed to the arm/sensor, and an x, y
, a rotation about the vertical z
, a rotation in a vertical plane passing through the upper arm (measured relative to the vertical z
, a rotation about the long axis of the humerus. C, A cylinder representing the upper arm is depicted along with the forces (Fx, Fy, Fz) and torques (Tx, Ty, Tz) recorded at the transducer and the corresponding torques at the shoulder (Mx, My, Mz). The shoulder, elbow, and length of the upper arm are labeled S, E, and La, respectively.
[View Larger Version of this Image (41K GIF file)]
The following shoulder muscles of the right arm were examined in this study: anterior deltoid (AD), middle deltoid (MD), posterior deltoid (PD), latissimus dorsi (LaD), clavicular head of pectoralis major (CPec), and the upper portion of the sternocostal head of pectoralis major (SPec). Muscle contractions were elicited with a commercially available dual-channel neuromuscular electrical stimulation unit (Empi, St. Paul, MN). Bipolar, circular (3 cm diameter) surface electrodes were used to deliver the stimulation. The best position for the active electrode (cathode) was determined by moving this electrode to various points on the muscle belly and looking for the largest contraction for a given stimulus. The inactive electrode (anode) was placed ~4 cm away from the cathode on the muscle belly. The unit was programmed to produce an asymmetrical biphasic waveform with a fixed pulse width of 300 µsec at a stimulation frequency of 50 Hz. The peak current was chosen to achieve a tetanic contraction (determined by visual inspection and palpation) while maintaining subject comfort and typically ranged from 15-20 mA. The unit was preprogrammed to follow a trapezoidal modulation of current: a 1 sec ramp up to peak current, followed by a 1 sec hold, followed by a 0.5 sec ramp down to 0 current. Stimulation was controlled by means of a hand switch that accompanied the stimulator. The stimulus was not painful, and subjects were instructed to relax and to not intervene intentionally in response to the stimulus. Five trials of stimulation were delivered at each arm posture, with an intertrial delay period of ~30 sec to minimize fatigue. In each experiment, up to 29 postures of the upper arm were examined. (Throughout all experiments, motion of the scapula was not restricted.)
Data acquisition and processing. Muscle contractions resulted in the production of forces and torques that were transduced by foil strain gauges within the sensor into millivolt analog signals. These signals were sent to a signal conditioning board that accompanied the sensor and then digitized at 100 Hz. A complete trial produced a 5-sec-long record consisting of ~1 sec of baseline before stimulation, 2.5 sec of data during stimulation, and 1.5 sec of poststimulation baseline. In addition to the six channels of force-torque data, a single channel carrying signals indicating the depression and release of the hand switch controlling the stimulator also was sent to the microcomputer. This signal was used to align the force and torque traces obtained from the repeated trials of stimulation at each arm posture (see Data Analysis below). Arm posture was derived from the recorded location of spherical reflective markers (placed on the orthosis at the approximate positions of the shoulder, elbow, and wrist), using a video-based, three-dimensional motion analysis system (VP110, Motion Analysis).
Coordinate systems. Because the transducer moved with the arm/orthosis during changes in arm posture, forces and torques applied to the transducer initially were defined in an arm-fixed frame of reference. In this coordinate system, the z-axis of the transducer was always aligned with the long axis of the upper arm. The x-axis was oriented parallel to the long axis of the forearm. The y-axis was then defined as being perpendicular to the plane containing the upper arm and forearm. These axes are indicated on the orthosis in Figure 1A. Forces applied in the anterior (x) direction were defined as positive, as were forces applied in the medial (y) and superior (z) directions. By definition, torques about these positively directed axes were also positive.
When the upper arm was oriented vertically and the forearm horizontally within a parasagittal plane passing through the shoulder (as in Fig. 1A), the axes of the transducer were parallel to the axes used to define upper arm posture (Fig. 1B). These latter axes were fixed in space and are labeled in Figure 1A,B as x
Upper arm azimuth and elevation were computed from the measured location of the elbow relative to the shoulder by using the following relations:
(1) in which e is a vector from the shoulder to the elbow, the subscripts x
(2) If
is defined to be the angle that p makes with the horizontal plane, then humeral rotation can be calculated from the following relation:
Upper arm azimuth was 0° when the shoulder and elbow were in the parasagittal plane passing through the shoulder and was positive when the elbow moved left relative to the shoulder. Upper arm elevation was 0° when the upper arm was oriented vertically with the elbow down and was positive when the elbow moved forward relative to the shoulder. Humeral rotation was 0° when the wrist was in a vertical plane passing through the shoulder and was positive when the arm was rotated internally. The 29 postures examined in an experiment typically covered a range of ~80° of azimuth and elevation and 120° of humeral rotation.
(3)
Data analysis. The following procedure was used to calculate the muscle torques acting at the shoulder. First, cross-coupling between channels of the force-torque sensor was removed numerically by means of a calibration matrix. Then the average value during the baseline period was subtracted from each force and torque trace. Repeated trials of stimulation at each arm posture were aligned at stimulation onset and averaged over the 2 sec corresponding to the stimulation ramp and hold periods. Torques at the shoulder were calculated from these force and torque traces with the following relations, in accord with the demands of static equilibrium:
(4) in which La is the length of the upper arm, Tx, Ty, and Tz are the torques recorded at the transducer, Fx and Fy are forces recorded at the transducer, and Mx, My, and Mz are the corresponding muscle torques at the shoulder. These quantities are indicated on a free-body diagram in Figure 1C. Torques about the positive Mx, My, and Mz-axes are referred to throughout this manuscript as shoulder adduction (ADD), extension (EXT), and internal rotation (IR) torques, respectively. Torques about the corresponding negative axes are referred to as shoulder abduction (ABD), flexion (FLEX), and external rotation (ER) torques, respectively.
Figure 2A depicts the calculation of shoulder muscle torques graphically. The left and center plots show 1.5 sec of force and torque traces generated during stimulation of MD at a single arm posture. Data from five consecutive trials are depicted in each plot. The data are aligned at stimulation onset. In this example, only the forces changed substantially during the stimulation period. The Fx and Fy traces began to rise above baseline ~0.5 sec after the onset of stimulation and then continued to rise monotonically and plateau after ~1.1 sec (see below). Thus, the time course of force modulation followed the time course of current modulation; after an initial delay, forces ramped from a baseline value to a steady-state value over an ~1 sec interval. It also can be seen in this example that both the sign and magnitude of the force and torque traces were consistent over consecutive trials obtained at the same arm posture. With regard to these findings, the data presented in this figure are representative of the data presented throughout this manuscript. The force and torque traces shown in the left and center plots were averaged (data not shown) and combined by using Equation 4 to yield the traces on the right, representing the components of torque at the shoulder with respect to an arm-fixed frame of reference. A complete set of these components for a particular arm posture will be referred to as a data set. 
Fig. 2. Graphical representation of the calculation of shoulder torques and the method used to evaluate the constancy of torque direction. A, The left and center plots show 1.5 sec of force (Fx, Fy) and torque (Tx, Ty, Tz) traces recorded at the transducer during the stimulation of MD at a single arm posture. Data from five consecutive trials aligned at stimulation onset are shown. These traces were averaged and combined using Equation 4 (see Materials and Methods) to obtain the traces on the right, representing the components of torque at the shoulder (Mx, My, Mz) in an arm-fixed frame of reference. Force units are Newtons (N), and torque units are Newton-meters (N-m). B, A three-dimensional torque trace obtained from AD during the ramp period of stimulation. The SD of the torque direction during this period had to be within the bounds of the 10° cone depicted here to be included in subsequent analyses.
[View Larger Version of this Image (35K GIF file)]
Each data set was evaluated to assess the extent to which torque direction changed during the ramp period of stimulation. The period of evaluation extended from 33% of the torque magnitude at steady state until the acquisition of steady state (steady state was chosen to be at 100 msec past the end of the stimulation ramp; 33% of this value was chosen to exceed baseline fluctuations in torque direction). This period is indicated in the shoulder torque traces of Figure 2A by the dashed rectangles. If the SD of the three angles representing torque direction were <10° during this period, then the torque direction was judged to be constant. Data sets with torque directions that were not constant during the ramp period were excluded from further analyses (~5% of the data sets).
This elimination process is depicted graphically in Figure 2B. In this three-dimensional plot, a single torque trace resulting from the stimulation of AD is shown. This trace was obtained by plotting the three components of shoulder torque against each other for the evaluation period described above. Each point in the trace represents the value of torque at a particular instant during this period. The relative amount of shoulder flexion and adduction at each instant can be ascertained from the reflection of the trace onto the Mx/My plane. The relative amount of internal rotation is indicated by the height of each point above this plane. In this arm posture AD acted primarily to adduct and internally rotate the upper arm. Also depicted is a 10° cone centered on the mean torque direction of this trace. The SD of the torque direction during the ramp period of stimulation had to be within the bounds of this cone for the direction to be judged constant.
The SD of torque direction was generally much less than 10° for those data sets that were retained. For example, for the trace indicated in Figure 2B, the SDs of the Mx, My, and Mz components of torque direction were 1.03, 1.57, and 0.75°, respectively. Figure 3 shows histograms of the SDs for all the retained data sets. A separate histogram for each muscle is shown. Each histogram was constructed by grouping the SDs of the three components of torque direction from each data set. For all muscles, the vast majority of these SDs were <5°. In fact, the median SD ranged from 1.43° (for MD) to 2.59° (for LaD). Thus, our stimulation protocol produced minimal changes in torque direction as current intensity increased during the ramp. 
Fig. 3. SD (in degrees) of the angles defining torque direction for each of the six muscles. Data from all retained data sets are shown.
[View Larger Version of this Image (33K GIF file)]
Because torque direction was nearly constant during the ramp period of stimulation, each data set was reduced to a single point representing the instantaneous torque direction at steady state. This was accomplished by modeling each component of torque as a linear function of time during the ramp and evaluating this model for the time corresponding to steady state. In Results, the three components of torque at steady state are plotted against each other in separate two-dimensional plots; data are represented as vectors emanating from the center of each plot (see Fig. 4). These two-dimensional vectors can be thought of as projections of the three-dimensional torque vector. The torque direction can be inferred on these plots by applying the right-hand rule: if the thumb points along a particular vector (originating at the center), the fingers curl in the direction of the torque, which is also the direction the arm would rotate in response to this applied torque. 
Fig. 4. Shoulder torque vectors resulting from the stimulation of AD. Data were obtained from Subject A in one experimental session. Three separate views are shown, obtained by plotting the individual components of shoulder torque (Mx, My, My) against each other. Each vector is an average of five stimulations at a single arm posture. Data from 28 different arm postures are shown. Torques are defined in an arm-fixed frame of reference; units are Newton-meters (N-m).
[View Larger Version of this Image (11K GIF file)]
During subsequent analysis we transformed these torque data from an arm-fixed frame of reference to a body-fixed (or trunk-fixed) frame of reference. This was accomplished by multiplying the muscle torques at the shoulder with the rotation matrix for each posture. Thus, in which C and S are shorthand for cosine and sine, respectively, and Mx,b, My,b, and Mz,b represent the components of muscle torque at the shoulder with respect to a body-fixed frame of reference. For this analysis, we retained our previous convention for naming the torques: (e.g., adduction torque for torques about the Mx,b-axis, etc.).
![<FENCE><AR><R><C>M<SUB><UP>x,b</UP></SUB></C></R><R><C>M<SUB><UP>y,b</UP></SUB></C></R><R><C>M<SUB><UP>z,b</UP></SUB></C></R></AR></FENCE> =[<AR><R><C>C&zgr;C&thgr;C&eegr;<UP>−</UP>S&zgr;S&eegr;</C><C><UP>−</UP>S&zgr;C&thgr;C&eegr;<UP>−</UP>C&zgr;S&eegr;</C><C><UP>−</UP>S&thgr;C&eegr;</C></R><R><C>C&zgr;C&thgr;S&eegr;<UP>+</UP>S&zgr;C&eegr;</C><C><UP>−</UP>S&zgr;C&thgr;S&eegr;<UP>+</UP>C&zgr;C&eegr;</C><C><UP>−</UP>S&thgr;S&eegr;</C></R><R><C>C&zgr;S&thgr;</C><C><UP>−</UP>S&zgr;S&thgr;</C><C> C&thgr;</C></R></AR>] <FENCE><AR><R><C>M<SUB><UP>x</UP></SUB></C></R><R><C>M<SUB><UP>y</UP></SUB></C></R><R><C>M<SUB><UP>z</UP></SUB></C></R></AR></FENCE>,(5)](/content/vol17/issue6/fulltext/2128/img010.gif)
Multiple regression analysis. The dependence of muscle torque direction on arm posture was examined quantitatively by means of a multiple regression analysis. For this analysis, data from all subjects and experiments were grouped together (the rationale for doing so appears in Results). Rather than fit three normalized components of torque, we chose to represent torque direction with two angles. is the angle that the projection of the three-dimensional torque vector onto the Mx/My plane makes with the
Mx-axis.
is the angle that the three-dimensional torque vector makes with the Mx/My plane. Stated mathematically (for the arm-fixed frame of reference),
(E6) The angle
![<UP>tan</UP>(<UP>&bgr;</UP>)<UP> = </UP>(<IT>M</IT><SUB><UP>z</UP></SUB><UP>/</UP><IT>M</IT><SUB><UP>xy</UP></SUB>) [<IT>M</IT><SUB><UP>xy</UP></SUB><UP> = </UP>(<IT>M</IT><SUB><UP>x</UP></SUB><SUP><UP>2</UP></SUP><UP> + </UP><IT>M</IT><SUB><UP>y</UP></SUB><SUP><UP>2</UP></SUP>)<SUP><UP>1/2</UP></SUP>]<UP>.</UP>](/content/vol17/issue6/fulltext/2128/img012.gif)
Each angle originally was fit to a four-term model containing only linear functions of in which a0 represents a constant, a1-a3 represent the coefficients of the various terms in the model, and
(E7) represents the error (i.e., the difference between the actual value of
in which SSm is the sum of squares of the model (i.e., the variation in the data that is explained by the model) and SSt is the total sum of squares (the total variation in the data). The r2 value represents the proportion of variance in the data that can be accounted for by the model.
(E8)
RESULTS
The results demonstrate the systematic manner in which the action of each of the six muscles changed with arm posture. Within each experimental session, both the magnitude and direction of shoulder muscle torques resulting from electrical stimulation varied with changes in arm posture. Figure 4 depicts data from AD of Subject A, obtained in one experimental session. Data are plotted in a vector format; the components of torque at the end of a stimulation ramp are plotted against each other and represented as vectors emanating from the center of each plot (see Materials and Methods for details). Each vector represents a single data set, i.e., an average of five stimulations at a single arm posture. Data from 28 different arm postures are shown, covering a range of 73° of azimuth, 78° of elevation, and 110° of humeral rotation. Magnitude variations across postures are apparent from the different lengths of the individual vectors. Torque direction variations (i.e., the relative amounts of flexion, adduction, and internal rotation) are evident from the wide range of vector orientations in each of the three plots of this figure. For most of the arm postures examined in this experiment, AD produced a combination of adduction, flexion, and internal rotation torques. These actions are generally consistent with previous investigations of this muscle (Basmajian and Deluca, 1985
For some of the arm postures depicted in Figure 4, AD produced substantial internal rotation torques, whereas for other postures these torques were negligible. The variation in the Mz component of torque perhaps is best appreciated in the middle plot of Figure 4. Some vectors are oriented parallel or nearly parallel to the Mx-axis, indicating little or no component of humeral rotation for those arm postures. However, AD produced substantial internal rotation torques in other postures, as indicated by those vectors oriented at larger angles relative to the Mx-axis. The range of
As mentioned above, torque magnitude varied with arm posture within an experimental session. For example, for the data presented in Figure 4, torque magnitude varied from 0.97 N-m to 2.54 N-m, with a mean of 1.64 N-m and a SD of 0.39 N-m. Variations in torque magnitude were observed for all the muscles examined in this study, although the extent of this variation differed for different muscles. Because the focus of this manuscript is on the postural variation in the directions of exerted torques, there is no further description of torque magnitude variations. So that variations in torque direction may be visualized independently of torque magnitude variations, subsequent plots depict vector components that have been normalized by torque magnitude.
Torque directions varied with arm posture for all of the muscles examined in this study. Figures 5 and 6 depict muscle torque vectors for all six muscles. In these and subsequent vector plots, only the top two views pictured in Figure 4 are shown, and all vectors are of unit magnitude. The data illustrated were obtained from Subject C in three separate experiments. In Figure 5, 29 postures are shown for both AD and MD, covering a range of 67° of azimuth, 70° of elevation, and 130° of humeral rotation. For PD, 28 postures are illustrated, covering a comparable range of 66, 72, and 117°. Torque directions varied for all three of these muscles, but these variations were limited to discrete regions of the vector space. This can be seen in the top plots of this figure, where vectors span a range of 90° for AD, 74° for MD, and 99° for PD. Similar degrees of variation were observed for the other muscles. In Figure 6, data for LaD (21 postures), CPec (29 postures), and SPec (28 postures) are shown. These postures varied to an extent similar to that described above. In the top plots of Figure 6, vectors vary over 101° for LaD, 107° for CPec, and 125° for SPec. For all muscles, internal and external rotation torques generally showed less variation. This can be appreciated from the bottom plots of Figures 5 and 6. For the experiments corresponding to these two figures, the range of 
Fig. 5. Normalized shoulder torque vectors for AD, MD, and PD. The data in each plot were obtained from Subject C in one experimental session. Data are plotted in the same format as those in the top two panels of Figure 4.
[View Larger Version of this Image (23K GIF file)]
Fig. 6. Normalized shoulder torque vectors for LaD, CPec, and SPec. The data in each plot were obtained from Subject C in one experimental session. Date are plotted in the same format as in Figure 5.
[View Larger Version of this Image (22K GIF file)]
Although torque direction varied, the "average" torque directions resulting from electrical stimulation generally agreed with previous descriptions of the actions of these muscles (Basmajian and Deluca, 1985
Muscle torque directions depended systematically on the angles defining arm posture. For example, Figure 7 depicts plots of ,
,
) denoting the three subjects. Figure 7 reveals a strong dependence of
Fig. 7. Dependence of torque direction (ALPHA) on humeral rotation [ROT(
[View Larger Version of this Image (33K GIF file)]
Fig. 8. Dependence of torque direction (BETA) on humeral rotation [ROT (
[View Larger Version of this Image (26K GIF file)]
Whereas the torque angle
Torque direction also depends on the other two arm angles (elevation and especially azimuth), and this dependence accounts for much of the scatter of the data points in Figures 7 and 8. In Figure 9, we show plots of 
Fig. 9. Dependence of torque direction (ALPHA) on humeral rotation [ROT (
[View Larger Version of this Image (23K GIF file)]
These trends and others are evident in the results of the multiple regression analysis, presented in Table 1. As described in Materials and Methods, both
Table 1. Multiple regression analysis (arm-fixed frame of reference)
Muscle Constant r2 (linear) r2 (cubic) AD 143.45 (2.59) (0.07) * (0.05) 0.76 0.76 MD 19.85 (3.17) (0.07) 0.24 (0.07) (0.05) 0.55 0.60 PD (3.40) (0.09) * (0.06) 0.57 0.59 LaD 269.62 (4.79) (0.08) (0.08) (0.06) 0.76 0.79 CPec 171.50 (2.84) (0.06) 0.01 (0.06) (0.04) 0.85 0.88 SPec 167.80 (6.11) (0.11) 0.27 (0.12) (0.08) 0.70 0.77 AD 13.94 (1.53) (0.03) (0.03) (0.02) 0.45 0.55 MD 1.79 (1.36) (0.03) 0.16 (0.03) (0.02) 0.62 0.64 PD (1.78) (0.04) 0.10 (0.04) (0.03) 0.65 0.65 LaD 12.26 * * * 0.00 0.00 CPec 21.49 * (0.03) (0.02) 0.31 0.38 SPec 19.32 * (0.02) (0.01) 0.42 0.43 Constants and parameter coefficients (±1 SD) from the multiple linear regression of the angles defining torque direction (
A dependence of muscle torque direction on arm posture also was evident when torque vectors were analyzed in a body-fixed frame of reference. In previous plots all data were represented in an arm-fixed frame of reference. The observed variation in muscle torque direction could have been a consequence of this chosen frame of reference; in another frame of reference torque directions could appear relatively constant across arm postures. To explore this possibility, we transformed the data from all experiments into a body-fixed frame of reference by multiplying each data set with the rotation matrix for that posture (Eq. 5). The results of this analysis are presented in Figure 10 for the AD of Subject C and can be compared with the data for the arm-fixed frame of reference shown in Figure 5. On the left are the two standard views of the torque vectors, now in a body-fixed frame of reference. It can be seen that there is still a substantial degree of variability in the orientations of the torque vectors. Although the variability in the abduction-adduction/flexion-extension plane is less than in Figure 5, the variability in internal-external rotation is greater (internal-external rotation now corresponds to a torque about a fixed vertical z-axis rather than about a moving humeral axis). To the right of these vector plots are scatterplots of 
Fig. 10. Dependence of torque direction (ALPHA and BETA) on the angles defining arm posture (ROT (/My
[View Larger Version of this Image (20K GIF file)]
The results of a multiple regression analysis of the dependence of muscle torque direction on arm posture for the body-fixed frame of reference are depicted in Table 2. Here again, quadratic and linear terms did not improve the fit of the data substantially; therefore, only linear regression coefficients are shown. For
Table 2. Multiple regression analysis (body-fixed frame of reference)
Muscle Constant r2 (linear) r2 (cubic) AD 136.00 (1.88) * * * 0.00 0.09 MD 21.00 (2.92) 0.82 (0.07) 0.30 (0.06) 0.51 (0.05) 0.61 0.67 PD (2.90) * * * 0.00 0.27 LaD 269.48 (2.94) 0.65 (0.06) * * 0.48 0.51 CPec 160.09 (4.18) 0.16 (0.07) 0.56 (0.08) * 0.35 0.43 SPec 155.14 (5.27) * 0.76 (0.13) * 0.35 0.43 AD 13.37 (1.78) (0.04) 0.74 (0.04) (0.03) 0.78 0.82 MD (1.88) (0.04) (0.04) (0.03) 0.40 0.48 PD (2.34) (0.05) (0.05) (0.03) 0.74 0.76 LaD 23.40 (3.66) 0.27 (0.06) (0.06) * 0.20 0.26 CPec 26.51 (1.69) 0.11 (0.03) 0.70 (0.03) (0.02) 0.83 0.88 SPec 29.69 (2.15) 0.24 (0.03) 0.51 (0.04) * 0.77 0.85 Constants and parameter coefficients (±1 SD) from the multiple linear regression of the angles defining torque direction (
In summary, the mechanical actions of all six muscles varied substantially with arm posture when these actions were defined either in the frame of reference of the insertions of these muscles (the upper arm) or the frame of reference of their origins (the body or trunk). These variations were not random but depended strongly on arm posture. For both frames of reference, the dependence on arm posture could be described mathematically by a four-term model containing only linear functions of the angles defining arm posture. In the arm-fixed frame of reference, torque direction varied strongly with humeral rotation and, to a lesser extent, upper arm azimuth. In the body-fixed frame of reference, torque direction instead varied most strongly with upper arm elevation.
Although the dependence of muscle torque direction on arm posture was mathematically simple in both frames of reference, the three-dimensional nature of the torque vectors and the large number of postures used may make it difficult to get an intuitive feel for the main trends in the data. To assist the reader in this respect, in Figure 11 we present three-dimensional plots of the predicted torque direction vectors for AD and SPec in the body-fixed frame of reference. These vectors originate at the right shoulders of the depicted "mannequins" and are colored according to upper arm elevation (yellow vectors, 15°; orange, 45°; red, 75°). The direction in which the arm would rotate in response to an applied torque can be ascertained by applying the right-hand rule to these vectors (see Data Analysis). The top panel demonstrates that when the upper arm is nearly vertical (yellow vectors), contraction of AD will cause the humerus to move forward, upward, and inward (toward the midline of the body). When the arm is oriented more horizontally (red vectors), AD will rotate the humerus inward in a plane close to the horizontal plane. This postural change in the action of AD reflects the dependence of 
Fig. 11. Predicted torque direction vectors for anterior deltoid (AD, top) and the sternocostal head of pectoralis major (SPec, bottom), viewed in the body-fixed frame of reference. Vectors from 28 different arm postures are shown, representing three values each of upper arm azimuth, elevation, and humeral rotation and one additional posture. For the 27 postures,
[View Larger Version of this Image (102K GIF file)]
The effects of upper arm elevation, azimuth, and humeral rotation on the action of SPec are shown in the bottom panel of Figure 11. With regard to the effects of elevation, this muscle acts to move the humerus primarily upward and inward (toward the midline of the body) when the upper arm is oriented more vertically (yellow vectors) and downward and inward when the arm is oriented more horizontally (red vectors). The illustrated change in action corresponds to a strong dependence of both
DISCUSSION
In this paper we describe the mechanical actions of six superficial shoulder muscles across a range of arm postures. Estimates of muscle mechanical actions were obtained by electrically stimulating muscles to the point of contraction and simultaneously measuring the resulting forces and moments with a six-degree-of-freedom force-torque transducer. Muscle mechanical actions, as indicated by the direction of exerted torques, varied systematically with arm posture.
ABSTRACT
