Abstract
Gottlieb, Gerald L., Qilai Song, Gil L. Almeida, Dian Hong, and Daniel Corcos. Directional control of planar human arm movement. J. Neurophysiol. 78: 2985–2998, 1997. We examined the patterns of joint kinematics and torques in two kinds of sagittal plane reaching movements. One consisted of movements from a fixed initial position with the arm partially outstretched, to different targets, equidistant from the initial position and located according to the hours of a clock. The other series added movements from different initial positions and directions and >40–80 cm distances. Dynamic muscle torque was calculated by inverse dynamic equations with the gravitational components removed. In making movements in almost every direction, the dynamic components of the muscle torques at both the elbow and shoulder were related almost linearly to each other. Both were similarly shaped, biphasic, almost synchronous and symmetrical pulses. These findings are consistent with our previously reported observations, which we termed a linear synergy. The relative scaling of the two joint torques changes continuously and regularly with movement direction. This was confirmed by calculating a vector defined by the dynamic components of the shoulder and elbow torques. The vector rotates smoothly about an ellipse in intrinsic, joint torque space as the direction of hand motion rotates about a circle in extrinsic Cartesian space. This confirms a second implication of linear synergy that the scaling constant between the linearly related joint torques is directionally dependent. Multiple linear regression showed that the torque at each joint scales as a simple linear function of the angular displacement at both joints, in spite of the complex nonlinear dynamics of multijoint movement. The coefficients of this function are independent of the initial arm position and movement distance and are the same for all subjects. This is an unanticipated finding. We discuss these observations in terms of the hypothesis that voluntary, multiple degrees of freedom, rapid reaching movements may use rulebased, feedforward control of dynamic joint torque. Rulebased control of joint torque with separate dynamic and static controllers is an alternative to models such as those based on the equilibrium point hypotheses that rely on a positionally based controller to produce both dynamic and static torque components. It is also an alternative to feedforward models that directly solve the problems of inverse dynamics. Our experimental findings are not necessarily incompatible with any of the alternative models, but they describe new, additional findings for which we need to account. The rules are chosen by the nervous system according to features of the kinematic task to couple muscle contraction at the shoulder and elbow in a linear synergy. Speed and load control preserves the relative magnitudes of the dynamic torques while directional control is accomplished by modulating them in a differential manner. This control system operates in parallel with a positional control system that solves the problems of postural stability.
INTRODUCTION
Common voluntary tasks involve moving the hand from one stationary position to another. These may be labeled as reaching movements and are studied frequently and easily in the laboratory. Although studies usually avoid explicitly controlling other kinematic features of the movement, it has long been recognized that reaching movements share several distinctive and relatively invariant kinematic properties (Lacquaniti and Soechting 1982; Morasso 1981). For example, the relatively straight paths and bellshaped tangential velocity profiles between the movements' end points in Cartesian space are little affected by either the intended speed of movement or the addition of loads. Under some conditions, linear relations also have been found between joint angles (Atkeson and Hollerbach 1985; Hong et al. 1994; Lacquaniti et al. 1986; Soechting and Lacquaniti 1981).
If the CNS uses an explicit, extrinsic, kinematic representation of the intended movement, then it must transform this representation into a set of intrinsic muscle commands to produce appropriate muscle torques. Alternatively, the CNS might avoid the transformation by working directly in dynamic terms. A premise of the torquebased approach is that the CNS has an internal dynamic model of the movement task that allows it to predict a satisfactory kinematic outcome (Gomi and Kawato 1996; Gottlieb 1991; Gottlieb et al. 1989; Shadmehr and MussaIvaldi 1994). The kinematic features of the movements emerge from trialanderror adjustments to the model by higher centers acting in a supervisory manner.
Most natural movements involve the contraction of muscles about several joints. This entails the coordination of torques between joints that may have entirely different kinematic trajectories, a problem that potentially can create a great deal of complexity for any control system. This complexity emerges both from the nonlinearity of the equations of motion and from the excess in degrees of freedom of the biomechanical system (Bernstein 1967). In an earlier work (Gottlieb et al. 1996b), we proposed an interjoint coordination rule we termed “linear synergy.” This rule postulates that to make common, loosely constrained movements of a limb, the CNS uses a single command that is distributed to all of the joints in proper proportion. The temporal pattern of this command is intended to produce muscle activation patterns that lead to torques of similar shape at each joint, scaled in amplitude to the dynamical demands of the task. This system is assumed to operate in parallel with a postural system that maintains the static stability of the limb configuration in the face of external forces. This implies that the joint torques for reaching movements involving the shoulder and elbow can be described by Eq. 1.
The biphasic torque patterns of our previous studies all started into flexion at both shoulder and elbow. Our specific aims in this work are to test two necessary consequences of Eq. 1. The first is that the torque patterns at the two joints are similar to each other, regardless of movement direction or the initial signs of the joint torques. The second is that the constant of proportionality varies systematically with movement direction. We performed experiments that varied the direction of hand movement and, consequently, the relative signs and magnitudes of the muscle torques. We considered first how the muscle torques are modulated when only the direction of a fixed distance movement, outward from a common central point was systematically varied over 360° around the sagittal plane. We then examined whether movements in different directions, performed over different and longer distances and from different starting positions, used the same rules. We found that the linear synergy between the dynamic components of the elbow and shoulder muscle torque (that is after the removal of gravity dependent terms) is maintained for movements in most directions. We confirmed that at both joints, the dynamic muscle torques during pointtopoint reaching movements are usually synchronous, biphasic pulses but find that when linear synergy is violated, this pattern has changed at the joint with the smallest torque.
A third and unexpected finding is that in spite of the complex dynamic relations between joint torque and elbow motion, the rules that appear to be used by the nervous system give rise to a linear relationship between movement distance, measured in joint angle space, and the magnitudes of the muscle torques.
METHODS
Subjects stood at ease and faced a series of small targets (cotton balls, 2 cm diam) arranged on the perimeter of either a circle or an ellipse that lay in a parasagittal plane aligned with the right shoulder. In some movement sets, subjects started their movements from the center of the circle and moved outward to one of 12 targets, located at the hours of an imaginary clock face. On this clock face, 12 o'clock was upward and 9 o'clock was toward the subject's shoulder. The forearm was approximately aligned with a 2–8 o'clock axis. These are termed centerout (CO) movements, and Fig. 1 A shows typical hand paths. In other movement sets, the targets were located around a larger, elliptical region ∼50 × 80 cm. The movements started from the perimeter and moved across to a target located on the opposite side as illustrated in Fig. 1 B. These are termed centercrossing (CX) movements.
In the first series of experiments, six subjects performed CO movements to targets on a 20cmradius circle. The movements all started from the same central location near shoulder height. The subjects were allowed three to four practice movements for each target, after which data were collected for 10 repetitions.
The second series of experiments, performed by three subjects, consisted of two sets of CO movements (on 10 and 20cm circles) and one set of CX movements, each to 12 targets centered about a shoulderhigh origin as used in the first experiment. They then repeated the three sets with the target centered about an origin closer to waist height. Subjects moved to a total of 72 different targets, with two to three practice trials followed by three to five recorded movements for each target. Subjects rested for 3–4 min between each of the target sets.
Subjects in both series were told to which target they should move rapidly. No other instructions were given about the hand path. On a verbal “get ready” signal, the subject positioned the right arm at the starting point and waited until the experimenter said “go”. Subjects moved rapidly to and touched the soft target and stayed there until they heard a computer generated tone. A total of eight adult male subjects and one adult female were tested after they gave informed consent according to protocols approved either at Rush Medical Center or Boston University.
Kinematic/dynamic analysis
A threedimensional, electrooptical motion measurement system (OPTOTRAK3010) recorded at 200 samples/s the locations of four markers attached to the shoulder, elbow, wrist, and index finger tip.
A fivedf model of the kinematic linkage of the human arm was used to analyze the data. This model included shoulder, elbow, and wrist joint rotations and horizontal and vertical shoulder translation, all in a sagittal plane through the shoulder. Joint angles and their derivatives were calculated from the measured coordinate data of the distal and proximal segment endpoints. Muscle torques were computed by Newtonian equations of motion shown in Eq. 2, A and B, in simplified form, similar to the approach used by Putnam (1993). The data presented in this manuscript are two of those degrees of freedom, shoulder and elbow rotation. The absolute angles of the joint segments θ_{s} and θ_{e} are defined in Fig. 1
A. The relative angle of the elbow joint is given by φ = θ_{e} − θ_{s}.
The focus of this paper is on the transient pulses of torque that propel the limb toward and arrest it at its intended target. These are superimposed on position dependent torque requirements for resisting gravity. We assumed the separability of the two components, a static one proportional to gravity and a dynamic one independent of it (Atkeson and Hollerbach 1985; Hollerbach and Flash 1982). The gravitational component is a function of angle and load and is computed directly from Eq. 2 with all derivatives set to zero. All analyses were performed after removing the gravitational components of the torque. This residual we refer to as the dynamic torque. In presenting the data from the first experiment, we have averaged the records after aligning them on the point at which the tangential velocity of the hand reached 5% of its peak. In the second experiment, we present singlemovement data records to show that none of our observations are an artifact of the averaging procedure.
Hypothesis testing
The first necessary consequence of linear synergy is that the patterns of torque at the elbow and shoulder are very similar in all movement directions. We tested this prediction in three ways: one qualitative and two quantitative. On a qualitative level, we visually compared the torque patterns of the shoulder and elbow, computed by Eq. 2. Because the torques are of changing magnitudes and signs as the hand moves in different directions, this is very difficult. To address this, we removed the gravitational terms and normalized the torque at each joint with respect to its first peak. The comparison was simplified further for the averaged data of the first experiment by scaling the time base for both joints to the time at which the shoulder torque crossed zero.^{1} This brought all normalized waveforms to uniform amplitude and temporal scales without altering their shapes or the relative timing between joints. For perfect linear synergy, the two normalized waveforms should be identical. This also could be examined qualitatively by plotting shoulder torque versus elbow torque. Perfect linear synergy predicts that this should result in a straight line, but, as we showed in Gottlieb et al. (1996b), small deviations in timing between the two torques will result in narrow elliptical or figureeight shapes.
Our primary quantitative analysis was to define and compute a figure of linear merit (Φ_{LM}). This measure is equivalent to plotting dynamic elbow torque versus shoulder torque and then rotating the resulting curve about the origin until its projection on the x axis is maximized. The x and y variance (σ^{2}
_{x}, σ^{2}
_{y}) then are calculated. After rotation, the standard deviation σ_{x} is termed σ_{max} and σ_{y} is termed σ_{min}. The ratio of these two values is used to compute the figure of linear merit according to Eq. 3. The figure has a value of unity for data lying on any straight line in the torque plane and zero for data uniformly distributed about the origin. The computation is described in greater detail in the appendix where the figure of merit is shown to be a more conservative estimate of “linearity” than is linear regression.
The second prediction of linear synergy is that the constant of proportionality in Eq. 1 (K _{d}) will vary uniformly with movement direction. The computation of K _{d} accompanies the computation of Φ_{LM}. It is the tangent of the angle through which the torquetorque plot must be rotated to maximize σ_{x}. We expect that angle to continuously vary from 0 to 360° in the joint torque plane as the direction of movement makes a similar rotation about the sagittal plane.
A simple measure of the mechanical output of the muscles that we have used previously is the impulse at each joint. This is the time integral of the dynamic muscle torque from movement onset to its first sign reversal. Because there is extensive overlap of force production between the opposing muscle groups, this is only a fraction of the total mechanical output by the muscle groups, but it is the only one we can estimate from kinematic measurements. According to linear synergy, a plot of shoulder impulse versus elbow impulse for the 12 directions should lie on an ellipse that is similar in shape to the ellipse that would contain the torquetorque plots.
After examining the data, we performed a multiple linear regression analysis, which found that most of the impulse at each joint could be accounted for by the net change in joint angles between initial and final positions. We used this relationship to reexamine previous findings for movements of various loads, speeds, and distances.
RESULTS
The two types of movement tasks are illustrated in Fig. 1. Heavy lines show the configuration of the arm at movement onset. In Fig. 1 A, the thin curves radiating out show the average path that was followed by the finger tip of one subject (T, see Table 1) to each of the 12 targets of CO movements. The paths are typical of our six subjects. Figure 1 B shows the paths of six individual CX movements that started near the perimeter of the work space, using targets that were approximately centered at shoulder height. All CX figures are drawn from this subject.
In terms of fingertip kinematics, all movements were kinematically simple, being fairly straight^{2} with bellshaped tangential velocity profiles. All CO movements were also simple in terms of joint kinematics with monotonic angle changes, as shown in Fig. 2 A, and bellshaped velocity profiles, which are not illustrated. Most of the CX movements were made with similarly simple joint and tangential kinematic patterns but this depended on initial and final positions. Figure 2 B is an example of a movement with more complex joint kinematics. It illustrates a single movement from the 6 o'clock to the 12 o'clock target. Although the tangential velocity profile remains typically bell shaped, elbow kinematics are quite different from those at the shoulder.
Figure 3 illustrates the joint torques, computed from Eq. 2 for the subject illustrated in Figs. 1 A and 2 A. They include the gravitational component. The dynamical and gravitational components both vary with movement direction. In some directions of movement, the two torques have different shapes or do not appear to be biphasic. The 10 o'clock movement is a good example of where the biphasic nature of the shoulder torque is not selfevident. It is not easy to infer any consistency in the torque patterns across movement direction from this kind of torque representation.
Linear synergy: consequence 1—similarity of torque waveforms
Removing the gravitational component and normalizing the dynamic torque makes the consistency clearer. Figure 4 A shows the normalized average elbow and shoulder torques at each direction of movement for another subject (S). In most directions, there is a strong similarity between the patterns but this is not true for every direction. The differences are greatest for movements in which the torque at one joint or the other is very small. Elbow torque is smallest at 2 and 8 o'clock and is least biphasic or like the shoulder torque. There are noticeable timing differences between the joints near 4 and 10 o'clock, where the shoulder torque is minimum.
Figure 4 B shows the normalized elbow and shoulder torques for individual CX movements made in each of the 12 directions. These records also show similar torque patterns in most directions. The largest differences in shape are where elbow torque is smallest.
The two principal implications of Eq. 1 are that plots of shoulder torque versus elbow torque will lie along a straight line and that the slope of that line, i.e., the relative magnitudes of the muscle torques at the shoulder and elbow, will change in a systematic manner as the direction of movement is altered. Figure 5 A illustrates the covariation of elbow and shoulder torque over all 12 directions of CO movements for subject S. We have drawn only the first halves of the average torque trajectories because their second halves would overlap with the first halves of movements made in the opposite direction. Except for their orientation, the torque ellipses are comparable with those found for movements of different distances (Gottlieb et al. 1996a) and different speeds and loads (Gottlieb et al. 1996b). In those earlier studies, the shoulder/elbow ratio of the acceleration impulse was independent of load, speed, or angular distance. We do not expect that invariance to be preserved for different directions of movement. Impulse (scaled by 10) is shown by the open circles in Fig. 5 A that lie along the long axis of the torquetorque plots.
Figure 5 B shows the torques for individual CX movements in six directions. These curves are plotted over the full time course of acceleration and deceleration. As would be expected from Fig. 4, these curves are similar to those of the CO movements and rotate about the origin as movement direction is altered.
Although Figs. 4 and 5 provide a qualitative sense of what linear synergy implies for the relationship between joint torques, they offer no quantitative measure. Linear regression or crosscorrelation coefficients can be used but these measures are sensitive to the relative magnitudes of the torques (see appendix). Figure 6 plots Φ_{LM} for subject S (dashed line) as well as the mean (heavy line ± SD) of all six subjects in the first experiment. Note that even in the directions at which the poorest visual correspondence exists between the two torque time series (2 and 8 o'clock), this measure is high. Its minima occur where the torque ellipses are widest. The figure also shows that Φ_{LM} for 12 individual, CX movements of one subject (dotted line) are similarly high.
When two functions of time are similar, we expect that their recognizable landmarks (e.g., their peaks, valleys, and zero crossings) should occur simultaneously. We have shown previously that for movements at different speeds or with different loads (Gottlieb et al. 1996b) or over different angular distances (Gottlieb et al. 1996a), there is a strong synchrony in the timing of the peaks and zero crossings of the two joint torques. To quantify the relative timing of these patterns for movements in different directions, we have plotted the temporal coincidence of the peaks and the zero crossings of the elbow and shoulder torque of subject S in Fig. 7 A for 10 of 12 CO movement directions (Fig. 4 A, omitting 2 and 8 o'clock). The linear regression curve for the pooled data, t _{s} = 0.002 + 0.987t _{e}, r = 0.944 is drawn with a heavy line. The thin dotted lines show the regression curves for each of the 10 directions. For all individual directions, r values are 0.99. Figure 7 B shows times for the same 10 movement directions of the individual CX movements in Fig. 4 B. The linear regression curve for the pooled data,t _{s} = −0.008 + 1.04t _{e}, r = 0.994 is drawn with a heavy line. The thin dotted lines show the regression curves for each of the 10 directions (9 of 10 r values are 0.99).
Relationship between kinetics and kinematics
The foregoing data specifically relate to linear synergy, an interjoint relationship between torques. We also have observed an unexpected relationship among kinematic variables and between kinematics and torque. The open circles in Fig. 8 show the angular change at the joints as direction varied in the CO movements of subject T. The solid line is a cosine function that has been fit to the data (r > 0.95). This function is a geometric constraint of two joint planar movement because there was little motion of the wrist. The box and × symbols show peak velocity and acceleration respectively. Both are also well fit by cosine functions (r > 0.95). The relative phases of the cosine functions fit to the three kinematic variables were within 5° of each other at both joints. There is no biomechanical requirement that these three variables covary in this way. That is, although it is not surprising that peak angular acceleration and peak angular velocity at each joint should be greatest when the angular excursion of that joint is greatest, this need not happen.^{3}
The filled circles in Fig. 8 show that impulse also has a cosine dependence on direction (shown by the dashed line), but the largest shoulder impulse occurs in a movement direction that is 20° out of phase with the largest shoulder angular displacement, and the largest elbow impulse occurs in a movement direction that is 70° out of phase with the largest elbow angular displacement. This is a consequence of the fact that the motion about each joint is not exclusively due to the muscles acting about it.
The separate relations in the two parts of Fig. 8 can be combined into a single relation between impulse and limb segment rotation. Figure 9 demonstrates that the impulse of subject T, computed by integration of Eq. 2, lies close to a planar ellipse in this threedimensional space. Multiple linear regression (MLR) gives Eq. 4, A and B, where Î, which represents the regression model impulse, is expressed in terms of the angular change (units in degrees) of each limb segment relative to vertical. The coefficients of Eq. 4 are for this subject and the MLR equation coefficients for all six subjects are shown in Table 1.
To determine whether the MLR equation depends on either movement distance or initial arm position, this approach can also be applied to CX movements. We computed the MLR equations for the second series of experiments. Equation 5 is based on six sets of 12 movement directions including the four sets of CO movements (4 × 12) and the two sets of CX movements (2 × 12) for one subject. The MLR equations for each of the six subsets of 12 movements were similar to Eq. 5. All angle coefficients are significant (P < 0.001) but the intercepts are not (P > 0.1). The MLR equations from CO and CX movements all describe similar planar relationships between impulse and joint angles.
Linear synergy: consequence 2—constant of proportionality varies with direction
The computation of the figure of linear merit also leads to a value for K _{d} for Eq. 1. Figure 10 plots the results of this computation for subject S. The curve is tangentlike but highly asymmetric with only three negative values. The results for all CO movements in both experiments were very similar with the exception that at the discontinuities ∼2 and 8 o'clock, the slopes were large but of variable sign. The slopes for the CX movements followed a similar pattern.
An inverse dynamic computation of joint torque such as Eq. 2 is a nonlinear transformation of joint angles, velocities, and accelerations. We had not expected that the time integral of this equation would be predictable from the net change in limb segment angles. Because impulse computed by integrating Eq. 2 is proportional to the inertial load and the intended speed (Gottlieb et al. 1996b) but Eq. 4 is not, we might expect different MLR coefficients if we experimentally manipulate kinematic features other than direction. However, according to linear synergy, the ratio of the impulse at the two joints (K
_{d}) should not be sensitive to either load or speed. Therefore, we can use the ratio of the two parts of Eq. 4 to estimate K
_{d}. This ratio is shown by Eq. 6.
We can compute a reliable MLR equation only from a series of movements that cover a sufficiently wide range of joint angles. However, Eq. 6 can be applied to an individual movement. Therefore, it enables us to use Eq. 6 to determine by inference whether Eq. 4 applies to earlier findings for which we cannot determine an MLR equation. This can be done by comparing the value of K _{d} as calculated by integrating the time series (Eq. 2) with that calculated statically for an individual movement using Eq. 6. Figure 11 shows K _{d} computed from the ratio of the integrals of Eq. 2 on the abscissa and from Eq. 6 on the ordinate. These two computed values of K _{d} are plotted against each other for subject T making CO movements in 12 directions, shown by the filled circles. The numbers along side the circles indicate the direction of movement. For ratios with magnitudes less than about ±5, the symbols lie along a line of unity slope showing that the correspondence is very good. For some higher ratios (at 2 and 8 o'clock in this case), there is sometimes a difference in the magnitudes of the two calculations, but this only occurs when the denominator of a ratio approaches zero.
We also used these two methods to compute K _{d} from two other experiments that had different starting conditions and longer movement distances (Almeida et al. 1995; Gottlieb et al. 1996a). This was done for nine of the subjects from those two experiments and are denoted by the other symbols in Fig. 11. Because K _{d} by either computational method can range from plus to minus infinity, those experiments explored a very narrow portion of the torque work space. Nevertheless, for the data they provide, the ratios all lie close to the line of unity slope. Thus the figure shows that Eq. 6, originally derived from a single subject performing a series of 20 cm CO movements in 12 different directions from one initial arm configuration, can estimate K _{d} for nine other subjects, making movements from other initial arm configurations, over different distances, at different speeds and with different inertial loads. This suggests that the relationship between impulse and angular displacement is robust.
DISCUSSION
The present findings confirm a number of documented and distinctive features of joint torque patterns. The dynamic torques are biphasic pulses^{4} of relatively invariant shape, regardless of load or speed (Hollerbach 1982; Soechting and Lacquaniti 1981). The physical justification for load and speed invariance was explained by Atkeson and Hollerbach (1985), who pointed out that the gravitational torque component would have different directional dependencies. The present data show in greater detail evidence that these pulses are not only similar in shape at the shoulder and the elbow for movements over different distances and in different directions but are also in close temporal synchrony. These features can be found in the sagittal plane movements reported in (Gottlieb et al. 1996a,b) and in the observations of Bock (1994) and of Buneo et al. (1995), who showed torque patterns of horizontal plane movements in different directions. The relative amplitudes of the torques at the two joints vary in a systematic manner with the direction of movement (Buneo et al. 1995). Movements in which the torque does not have the biphasic shape (e.g., at 2 and 8 o'clock in Fig. 2) at one joint are also those in which the synchrony between joints is lost. It is noteworthy however that these movements are the ones in which the torque in the “deviant” joint is very small. When the torques are substantial at both joints, linear synergy also may be observed with other than biphasic torques. We have shown preliminary data that this is true for at least some reversal movements (Gottlieb 1997).
Linear synergy
We have termed this widely found proportionality between the joint torques linear synergy. We proposed that linear synergy reflects similar central commands that activate the motoneuron pools of elbow and shoulder muscles. The observed relationship between joint torques is a consequence of these central commands. It is clear, however, that exact linearity is not observed nor is exact synchrony seen between the torque patterns, even when they both are biphasic pulses. One can ask therefore whether near linearity supports the proposition or inexact linearity contradicts it. We have used a figure of linear merit to show in a quantitative manner similar to linear regression, that linearity is preserved in all directions. We also can address this question by considering, not whether the torque patterns at the elbow and shoulder are similar, but by considering instead the question of if a common joint torque command of central origin does indeed exist, what differences between joint torques should we expect?
The known properties of the neuromuscular system will produce deviations between observed joint torques even with a common central command. There are at least four reasons for this. The first is the fact that different muscles contract at different rates. This will cause a shift in the timing of the torques and when one joint torque is plotted versus the other. The expected result would be a narrow ellipse or figure eight not a straight line. This is consistent with Fig. 5. We have demonstrated (Gottlieb et al. 1996b) that noticeable ellipses are seen with timing discrepancies of only 15 ms in a 300ms movement. The range of discrepancy between the landmark times can be seen in Fig. 7. Timing differences depend on which landmark and which direction is compared. They are generally greatest during the deceleration peaks that occur 150–425 ms after movement onset.
The second reason is that active muscle force production depends on its length and its velocity of shortening or lengthening. This compliant behavior will cause muscles with identical patterns of activation to produce significantly different amounts of force if they undergo different changes in length as they interact with external forces. These compliant torque components may produce measurable torque patterns that are not biphasic, especially when the dynamic central command to a muscle is small because K _{d} = 0 or 1/K _{d} = 0. We expect therefore that in some directions of movement, a small compliant component may be dominant and different from the biphasic central command. That is consistent with what we observe in the 2 and 8 o'clock directions where the above conditions on K _{d} are satisfied most closely.
The third reason is that muscles are activated by length and velocity sensitive reflexes as well as by central command signals. Thus, in principle, a common central command cannot assure a common pattern of muscle activation. Reflexdriven joint torque components will differ if the joint kinematics differ, but the relative contributions of reflex and central components are a subject of controversy. One would expect that if muscle activation were dominated by length and velocity sensitive reflexes, the electromyogram and force patterns would differ when the kinematic patterns substantially differ. The data here imply that reflex contributions are small. Figure 8 shows that the kinematic variables that drive these reflexes depend very differently on movement direction than does the torque. Figures 2 B and 4 B illustrate an example of a movement (6 o'clock) where the kinematic patterns differ between the joints while the torque patterns are similar. This is also shown in Almeida et al. (1995).
Finally, large passive torque components will develop as a joint approaches the limits of its range of motion. Our movements were designed to try to avoid this region of the work space. One of our subjects in the second experiment made movements to the 9 o'clock target with extreme elbow flexion at the end of the movement. The elbow extension torques during deceleration were much larger than predicted by Eq. 1, using the acceleration phase of the movement as a reference. Deviations from linearity are usually greatest during the last 25% of the movement when the dynamic torque is returning to its static level. This is usually within a few degrees of final position. This allows ample time for corrections to be made by the CNS. Movement corrections may be in directions that differ from the original movement direction and will therefore produce joint torque patterns with different relative amplitudes and appear as nonlinearities in the torquetorque plot.
We conclude this section by suggesting that a linear relationship between centrally planned dynamic torque components of different joints will produce an approximate, not an exact, linear relationship between the actual joint torques. The deviations from linearity that we observe are consistent with the muscular, biomechanical, and neural mechanisms we know can produce deviations from linearity and are not inconsistent with the linear synergy proposition.
Is this evidence for how movements are planned?
One interpretation of the principle of linear synergy is that movement planning specifies the overall timing and magnitude of a preselected, common pattern for the torque pulses. The scaling of the pattern is based on the distance, load, and speed of the intended movement, whereas direction is determined by the relative apportionment of the torques across joints. We cannot directly verify this interpretation, but we can observe that many of the experimental observations that have been presented in the preceding section are logical and necessary consequences of such an interpretation. If Eq. 1 reflects a plan for interjoint torque relations and if K _{d} varies smoothly with movement direction, then the data in Figs. 47 and 10 are predictable.
One objection that might be raised to inferring torquebased movement rules from these data is that a linear relationship between joint torques might be the consequence of linear relations between kinematic variables rather than the cause of them. The CO movements demonstrate fairly linear relations between joint angles and of course the hand path itself is fairly straight. Atkeson and Hollerbach (1985) proposed what they called a joint interpolation strategy for kinematic movement planning in which both the joint angular trajectories were proportional to a common function, temporally shifted between joints. If the joint kinematics share a common pattern, it is possible that over a small enough range of motion (perhaps 10 to 20cm CO movements), the torques would be similar as well. However, for the large CX movements, the nonlinear nature of Eq. 2 indicates that elbow and shoulder kinematic and torque timeseries cannot both be linearly related at the same time. Furthermore, for the 6 o'clock movement for which kinematics are illustrated in Fig. 2 and torques in Fig. 4, as well as for some of the movements in Gottlieb et al. (1996a,b), the shoulder and elbow kinematics are quite different, whereas the torque profiles are very similar and linear synergy is preserved.
Nevertheless, planning in kinematic terms might produce the same torque patterns. Kinematic plans might result from optimization strategies based on criteria such as energy or smoothness (Flash and Hogan 1985; Nelson 1983) or the minimization of torque change (Uno et al. 1989) and produce torque patterns that, pari passu, demonstrate linear synergy.^{5} The existence of a hypothetical torque controller in association with a stable, compliant neuromuscular system implies the existence of a “virtual trajectory,” partitioned across joints, and this could be hypothesized as the “positional” central command (Gottlieb 1996). What those virtual trajectories might look like depends on whether they are assumed to be monotonic (Feldman and Levin 1995) or N shaped (Latash 1992). Won (Won and Hogan 1995) and Gomi (Gomi and Kawato 1996) have shown for movements slower than those performed here that a hypothetical virtual trajectory will be more complex than the trajectory of the hand itself.^{6} Moving equilibrium point models posit a control variable that is expressed in positional terms. It remains to be shown that their predicted joint torques are consistent with the data discussed above. We have discussed elsewhere how torque planning might be done for single joint movements (Gottlieb 1993) and speculated how it might be extended to multijoint movements (Almeida et al. 1995; Hong et al. 1994). The problem that the CNS confronts is how to plan a central command, be it in terms of force or positional variables, given the features of the task. If the plan is kinematic, then it must be transformed into patterns for muscle activation. The data here are not incompatible with either approach.
How might torques be planned?
We have noted above that the joint torque patterns might be a consequence of planned trajectory and a scheme for converting an extrinsic plan into an intrinsic command. The other side of that proposition is that the CNS plans torque patterns and kinematics are “emergent” properties. What are the components of a torque plan? The first is the pattern. The use of a rule such as linear synergy dramatically simplifies the search for a suitable pattern both by reducing the number of patterns needed (i.e., 1 per movement instead of 1 per joint) and by reducing the number of patterns that can accomplish the task. Thus much of the complexity associated with surplus degrees of freedom and redundant mechanical and kinematic solutions is removed.
Even with a pattern (such as a biphasic pulse), the mover must scale the pattern appropriately to each joint. The ratio determines the direction of motion, whereas the magnitude determines the speed. The observation that the dynamic torque can be apportioned according to a linear function of the distance each joint moves is an unexpected and remarkable simplification of the problem.
Muscle selection and torque partitioning
Our findings also can give us some insight into some other recent observations concerning the control of horizontal arm movements in two dimensions. Karst and Hasan examined the onset of muscle activation and determined that “the choice of muscles to be activated for initiating multijoint arm movements could be accomplished through the use of relatively simple rules, which are based on positional variables and do not specifically take into account the dynamic effects” (Karst and Hasan 1991, p. 1592). Their rule was specified in terms of an angle Ψ, the angle between the initial orientation of the forearm and a line drawn from the finger tip to the target. The angle at which elbow flexor and extensor muscles switched roles was Ψ = 0° and 180 ± 20° (mean ± SD) and the switch at the shoulder took place about Ψ = 110° and 260°. These switching angles are similar to the angles at which we have found the dynamic torques at the joints to go through zero and reverse the sign of the impulse produced. Thus our findings, based on dynamic muscle torque patterns, suggest that in fact the “positional” rules that determine the switching between agonist and antagonist muscles have dynamical causes.
To move directly to a target as we normally do, Eq. 4 implies that we must know or be able to estimate the net angular displacement of two joints. This could be from knowledge of our initial and our final joint angles or equivalently the present location of our hand and the location of the target in Cartesian space. Soechting (1982) found that subjects could more accurately reproduce the orientation of the forearm in Cartesian space than they could reproduce elbow angles. The accurate knowledge of forearm orientation is of great importance in our hypothesized control scheme. In the absence of such information, such as in patients who lack proprioceptive input from their limbs, we would be expect movements to be launched with incorrect relative joint torques and consequently to often move in wrong directions. Having made these initial errors, such patients also would be at a disadvantage in correcting them. These kinds of movement errors are prominent in some recent work (Bastian et al. 1996; Sainburg et al. 1995) the authors of which concluded that the differences in the movement trajectories between their patient populations and neurologically normal individuals was due to the inability of their patients to adapt to the “interaction torques”.
Our results are consistent with the experimental findings of those studies. However, interaction torques are only one of the kinematically dependent torque components of the freely moving arm, the sum of which is equal and opposite to the muscle torque. Hence, the assertion that subjects cannot adapt to interaction torques is equivalent to saying that they cannot generate the correctly shaped and scaled dynamic muscle torques. The available data do not require that we attach a special importance to any one component.
We suggest that the partitioning of kinematically dependent torques into different components (such as self, interaction, or net) is an exercise that is useful in the analysis of a movement but is not necessarily performed by the nervous system in its planning or execution of movement. The dynamic torques at each joint are at every instant proportional to a nonlinear transformation (Eq. 2) of the velocities and accelerations of all limb segments, weighted by moment coefficients that depend on the instantaneous angles of the joints. The individual components themselves vary with the coordinate system in which the transformation is written. If the CNS specifically plans for an individual component of the dynamic torque such as the interaction component, then the CNS must be capable of estimating all the components at both joints and then computing the residual terms such that they add up to biphasic, linearly related pulses. We suggest as a reasonable alternative that the CNS plans in terms of two components, the dynamic torque and the static or gravitation torque, functionally separating movement and posture.
Conclusions
It is interesting to consider why movements tend to have certain invariant kinematic patterns. Independently of that, we can ask how it finds the torque patterns. Singlejoint movements can be produced by modulating muscle activation pattern generators using rules base on taskspecific features such as distance, load, or planned speed. In the same manner, multijoint movements can be planned using the same kinds of rules and features of the intended task. The dynamic problem of multijoint movement is more complex than that faced for moving a single joint, for which the problems of stability are quite different and segment interactions nonexistent. As such, we cannot expect multijoint solutions to be as simple as those for a single joint. Nevertheless, the analysis of the torques over a variety of planar arm movements demonstrates the existence of an often simple relationship between muscle torques across joints. This relationship we have called linear synergy was not anticipated and is not obvious. Linear synergy could be the consequence of some optimization strategy. It could be an emergent property of equilibrium point control. Whether either is true remains to be shown. We speculate that these torque patterns are a solution to the problem of controlling movements that emerges from trial and error in the early stages of life because it is discovered easily. It is retained because it is adequate to satisfy the loosely defined criteria of everyday movement. Movements in which linear synergy is not an adequate rule are learned when necessary, assuming we have sufficient skill and endure sufficient practice. It is also possible that the covariation of torques across joints is an inborn pattern, and it is the sculpting, timing, and scaling of the dynamic torque pulses that we learn first (Daigle et al. 1996). The existence of linear synergy as an inborn feature might explain why complex tasks that cannot be accomplished under such a linear constraint are difficult to learn. An important next question is to ask how the nervous system recruits the muscles to produce these patterns of torques. That is a question we will consider in future work.
Acknowledgments
This work was supported in part by National Institutes of Health Grants RO1 AR33189, RO1 NS28176, KO4 NS01508, and RO1 NS28127.
Footnotes

Address for reprint requests: G. L. Gottlieb, NeuroMuscular Research Center, Boston University, 44 Cummington St., Boston, MA 02215.

1 Normalized dynamic torque is τ̂_{x}(t) = K _{x}τ_{x}(t/t _{sz}) where x = s for the shoulder or e for the elbow. K _{x} is the reciprocal of the first extremum of τ_{x}, and t _{sz} is the time at which τ_{s} reverses sign.

2 The description of these paths as “straight” is traditional terminology. They could as well be called “moderately curved.” Movements in a sagittal plane are often less straight than those in a horizontal plane, depending on direction (Atkeson and Hollerbach 1985).

3 The variation in speed was not sufficient to keep movement time (MT) constant. In experiment one, the mean movement time was 335 ± 49 ms and varied smoothly with direction, being greatest in the 1 o'clock direction (406 ms) and smallest in the 10 o'clock direction (246 ms) with similar extrema 180° from each of those directions. Thus the directions of peak MT were near the line of the initial orientation of the forearm and the minimum MT directions were 90° away, consistent with the findings of Flanders et al. (1996).

4 This biphasic feature is specific to the type of movement. When subjects make deliberately more complex movements, different patterns of torque will be required. One example is the class of reversal movements used by Sainburg (Sainburg et al. 1995) where subjects moved to a target and returned to their starting position. For these, the torques were triphasic pulses. A less symmetrical biphasic pattern will be produced for loading conditions that have greater viscous or elastic properties.

5 But investigators also have shown that subjects who observe their movements through a medium that causes kinematically straight paths to appear curved will spontaneously curve the actual paths to straighten their appearance (Flanagan and Rao 1995; Wolpert et al. 1994).

6 As movements get faster, the difference between the actual and virtual trajectories are likely to grow in proportion with the inertially dominated dynamic torque components. This is a more than sixfold difference in the magnitude of the dynamic torque components between movements taking 750 ms (e.g., Won and Hogan 1995) and those here, which took ∼300 ms.

7 The actual algorithm used to determine Φ_{LM} uses the eigenvalues of the covariance matrix of the two joint torques. The criterion Φ_{LM} can be expressed as a function of ζ, the fraction of the total variation explained by the first prncipal component of the covariance matrix. The ratio ζ is σ^{2} _{max}/(σ^{2} _{max} + σ^{2} _{min}). We thank Hiroaki Gomi for pointing out this efficient method of computing ΦLM and Sue Leurgans for pointing out that this is the same as the the fraction of the total variance explained by first principal component in principal components analysis. The “best” line lies along the direction of the first principal component.
 Copyright © 1997 the American Physiological Society
APPENDIX
Quantifying joint torque linearity
If two variables bear a linear relationship to one another, we can describe that by Eq.
EA1.
If we regard g as measurement noise, then r ^{2} will be degraded by the relative magnitude of g compared with Kx. Suppose we have a system that is described by Eq. EA1 and that g is fixed. Also suppose that K is an independently controlled variable of the system. Then the reality of the linear relationship is not in question but the goodness of fit of Eq. EA1 will be a function of K. In these circumstances, we suggest that a superior alternative to regression is to use principal components analysis to find the best straight line through the data.
We now explain how principal components analysis applies to our data sets. For example, if K = 0 in Eq. EA1, all data points will lie along the x axis running through the origin. The x axis might be regarded as the “best” line, whereas a correlation coefficient that was computed for such data would be close to zero. The same zero correlations would be true for data distributed along the y axis, yet both data sets would lie “close” to an easily defined straight line. In neither case, however, would the variation of x be predictive of the variation in y. Below we describe how to find that straight line, how to measure its goodness of fit, and then compare that measure to a correlation coefficient.
In this manuscript we have used a figure of linear merit (Φ_{LM}) that is described by Eq.
EA2.
Below we have compared Φ_{LM} with the correlation coefficient (r) for two model data sets. The first set consists of data described by Eq. EA1 where x is evenly distributed between 0 and 1, K = 1 and g is a normally distributed random variable. Figure FA1 A shows a plot of x versus y when the standard deviation of g (σ_{g}) is 0.1. Figure FA1 B shows Φ_{LM} versus r for a series of data sets when σ_{g} varied from 0 to 0.25. As σ_{g} increases from zero, both r and Φ_{LM} decrease from 1.0 to about 0.6. The value of Φ_{LM} is less then r for these data with high “linearity” and is thus a more “conservative” measure. For Φ_{LM} > 0.7, it is also a more conservative measure than r ^{2}. The filled symbol corresponds to the case where σ_{g} = 0.1.
If we believe that the data in Fig. FA1 A indicates the existence of a linear relationship between x and y, that linearity should survive both translation and rotation of the data about the origin. The correlation coefficient is independent of translation (i.e., changes in the mean) but is extremely sensitive to rotation. An advantage of Φ_{LM} is that it is a measure of linearity that is independent of the orientation of the data with respect to the axes of the coordinate system. The data generated by Eq. EA1 has a unity slope. In Fig. FA1 C, we show the effect of rotating the data. Rotating the data ±45° makes r = 0 and rotating it ±90° makes r = −1. The straight line at the top of the graph shows that Φ_{LM} is constant under rotation.
In Fig. FA2, we examine nonlinearities similar to those found in our torque data. The values of x and y are given by Eq.
A3 and are plotted for one period of the sinusoid.
In considering whether our data demonstrates or contradicts the existence of a linear relation between joint torques, Fig. FA2 A presents the viewer with a pattern which can be named. It is an ellipse. Figure FA1 A presents no such opportunity. Yet the degrees of linearity of both can be measured and are high. We conclude that Φ_{LM} is a robust measure of linearity in the sense that if the variance about a straight line is small, Φ_{LM} is close to unity, regardless of the orientation of the line or the shape of the distribution about that straight line. This does not address the question of whether x and y are in fact similar functions of an unmeasured control variable. It does, however, provide a quantitative measure of the similarity of two functions that is less sensitive to the relative magnitudes of the variables than is the correlation coefficient.