| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1Harrington Department of Bioengineering, 2Department of Kinesiology, and 3Biodesign Institute, Arizona State University, Tempe, Arizona
Submitted 23 May 2007; accepted in final form 10 July 2007
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
|---|
); however, it is difficult to ascertain what is actually being optimized, as well as how this optimization process is organized. We hypothesize that the CNS exploits biomechanical properties of the limbs to increase efficiency of movement control. The study specifically focuses on biomechanical factors that influence performance of multijoint arm movements. Three such factors have been recognized: interaction torque (IT), inertial resistance, and kinematic manipulability. IT results from mechanical influence of arm segments on each other during motion (Hollerbach and Flash 1982). Inertial resistance characterizes muscle effort necessary to produce a given acceleration of the arm endpoint (Hogan 1985). Kinematic manipulability characterizes angular velocity at the joints required to produce a given endpoint velocity (Yoshikawa 1985, 1990).
To produce goal-directed movements, muscular control must be adjusted to all these factors. Each factor depends on movement direction, thus imposing differential demands for muscular control across movement directions. For instance, IT assists horizontal shoulder-elbow movements in the left-diagonal direction, i.e., when one joint flexes and the other extends. However, it resists motion in the lateral direction in which both joints simultaneously flex or extend (Dounskaia et al. 2002a; Gribble and Ostry 1999). In contrast, inertial resistance is the highest in the left-diagonal direction and the lowest in the right-diagonal direction (Gordon et al. 1994). Finally, kinematic manipulability suggests that joint rotations of the same angular amplitude would be the most efficient (in terms of the production of the highest tangential velocity) when the two joints simultaneously flex or extend, i.e., during movements approximately in the lateral direction (Dounskaia 2007).
The anisotropic properties of the biomechanical factors result in variations of the effort for movement production across directions, depending on each factor. A specific hypothesis tested here is that the CNS has knowledge of "movement cost" associated with the effect of the biomechanical factors in each movement direction and that this knowledge is used at the stage of movement planning to increase efficiency of performance by preferring movements in directions characterized by low cost and by avoiding movements in directions of high cost.
To quantify "movement cost," we formulated cost functions associated with each factor (see METHODS). The cost functions characterizing the effect of inertial resistance and kinematic manipulability across movement directions are developed with use of established mathematical descriptions of the two biomechanical factors (Hogan 1985; Yoshikawa 1985, 1990). To develop a cost function addressing the effect of IT, we exploited a recently formulated leading joint hypothesis that provides interpretation of multijoint movement control (Dounskaia 2005; Dounskaia et al. 1998, 2000a). This hypothesis suggests that during movements of the shoulder-elbow linkage, one of these joints is controlled independently of IT similar to a whip handle. Motion at this (leading) joint causes powerful IT at the other (subordinate) joint, and therefore the locus of complexity of IT regulation is at the subordinate joint. These considerations predict advantage of two movement types that allow minimal interference with the effect of IT. They emerge when the shoulder (elbow) is actively accelerated/decelerated and the elbow (shoulder) moves passively, caused by IT. Movements of these types are performed approximately in the left- and right-diagonal directions, respectively. Two cost functions optimized on these two movement types are formally developed in METHODS.
The prediction that there is a tendency to move in directions of low cost can be tested if subjects have a choice of movement direction. Traditional tasks of arm movements are not appropriate for this purpose because they restrict movement direction. For instance, reaching, pointing, and drawing tasks prescribe the initial and final point or even a specific path, and thus obligate movement in a specific direction. To provide a choice of movement direction, a novel, free-stroke drawing task modified from a task used in handwriting studies (Van Sommers 1984) is used here. Subjects were instructed to make strokes on a table surface from a circle center to the perimeter in different directions, choosing the direction in a random order. This task is advantageous because it provides a choice of movement direction for each stroke. The freedom to choose movement direction allowed subjects to display directional biases despite instructions that encouraged a uniform distribution of strokes across directions. These directional biases were compared with predictions of the formulated cost functions to distinguish factors influencing the choice of movement direction.
To provide a comprehensive analysis of factors contributing to directional biases, cost functions adopted from previous research were included in addition to the cost functions associated with the three biomechanical factors. The additional cost functions represented optimization of trajectory smoothness (Flash and Hogan 1985), muscle torque change (Uno et al. 1989), and sum of squared muscle torque (Yen and Nagurka 1988). The inclusion of these optimization criteria provided a possibility to compare the traditional and biomechanical cost functions in terms of their contribution to the emergence of directional biases. The approach of this study opens new perspectives for investigation of optimization processes underlying movement control and of how control is adapted to biomechanical properties of human limbs.
|
METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
|---|
Fifteen students (denoted as S1–S15) from Arizona State University between 20 and 30 yr of age participated in this study after providing informed consent. Each participant was compensated with course credit for their involvement in the study. The institutional review board at Arizona State University approved the experimental protocol. All subjects were right-handed and were without any known neurological or musculoskeletal impairments.
Procedure and design
Subjects were seated at a height-adjustable table such that their right arm was parallel to the floor (shoulder abducted 90°). All movements were performed along the table's surface with the right arm in the horizontal plane using flexions/extensions of the elbow and shoulder. The upper arm segment was supported in a sling suspended from the ceiling to support the weight of the arm. Motion at the trunk was prevented by restraining the torso between the edge of the table and the back of the chair. The index finger was splinted to prevent motion relative to the hand, thus ensuring that arm movements consisted of 2 df. Finally, the index finger and splint were wrapped with Micropore paper tape (3M, St. Paul, MN) to reduce friction along the table surface, which was covered by a Plexiglas material. The task occurred within a circular workspace having a radius of 18 cm. The workspace was centered according to the position of the fingertip when the shoulder and elbow possessed joint angles of 
= 30° and 
= 100°, respectively, thus enabling a consistent starting arm posture across subjects (Fig. 1A). Joint angles were verified with use of the OPTOTRAK software and manual measurements with a simple goniometer. Movements from the center to the perimeter of the workspace could be performed in all directions without approaching the limits permitted by the range of motion at the joints. This measure minimized the influence of passive mechanical constraints associated with these limits for joint rotation, thus ensuring active control of joint motions throughout each stroke.
|
The task was repeated at a comfortable pace for eight 15-s trials. Several practice trials were provided to allow each subject an opportunity to familiarize themselves with the procedure. During these practice sessions, subjects were provided verbal feedback regarding whether the manner they performed strokes was consistent with the instructions. No visible trace of fingertip motion was provided to the subjects, and no comments were given regarding the order in which movement orientations should be realized. Rather, we repeatedly emphasized the instructions to randomly draw strokes in as many different directions as possible during each trial.
Apparatus and recording
Four infrared emitting diodes (IREDs) were positioned on the trunk, right shoulder, elbow, and index fingernail of each subject. Time varying position data were recorded at a sampling rate of 200 Hz using an OPTOTRAK (Northern Digital, Waterloo, Ontario, Canada) three-dimensional optoelectronic system that had been calibrated according to the manufacturer's specifications and that had accuracy of 
0.2 mm.
Data analysis
Discrete stroke segments were identified from the continuous 15 s of data collected during each trial using a series of rules. First, strokes were required to have been produced in the horizontal plane along the table surface as shown by the z-axis component of the fingertip marker. Second, no portion of the stroke could have a curvature value exceeding 50 m–1. This threshold allows for relatively high curvature represented by 2 cm of the radius of a circle instantaneously approximating the curve of the trajectory. The curvature limitation was used to exclude instances where subjects neglected to lift their finger after a center-out movement and while bringing the finger back to the center. Curvature, C, was computed as
 | (1) |
,
and 
represent the first and second derivatives of the (x,y) position data, respectively. Third, strokes were required to have been drawn in a center out fashion. This criterion was manually verified by considering the median movement initiation point as the workspace center during data processing. Strokes that were not initiated within a 4-cm radius from this point were excluded, thus ensuring that return movements were not considered in the analysis. Fourth, initial and final stroke portions during which velocity was <3% of its peak value achieved during production of this stroke were removed from the analyzed trajectories. Fifth, each stroke was required to have contained a minimum of 100 ms of data. Finally, strokes were required to have a minimum length of 12.5 cm. Figure 1A provides an illustrative example from S12 of a typical set of strokes accepted for analysis based on the aforementioned criteria. Endpoint kinematics
Endpoint motion was evaluated using the position of the fingertip marker. The angular orientation of each stroke was determined with use of the line connecting the location of the endpoint at stroke initiation and termination. The orientation 
s corresponded to 0° assigned to movements to the right in the mediolateral direction, whereas 90° was associated with movements drawn away from the subject in the anterioposterior direction. This angular coordinate was used to denote stroke orientations in the extrinsic space.
The orientation data generated during each of the subjects' eight trials were combined for simultaneous analysis to depict the distribution of stroke orientations. Polar histograms of stroke orientations were produced for each subject by placing the orientation data into 72, 5° bins. These histograms were smoothed using a standard normal kernel smoothing function having a window width of 5°, yielding a probability density estimate for each orientation (Bowman and Azzalini 1997). The obtained individual histograms of stroke orientations are shown in Fig. 2 as a curve outlining the gray area in each polar plot. This representation was developed in two steps. First, the amplitude of normal kernel probability density estimate of the binned orientation data were normalized to the amplitude of a uniform distribution of stroke orientations (designated by the small black circle shown for each subject). Thus ranges in which the amplitude of the smoothed histogram exceeded 1.0 indicated a denser distribution of strokes than that of a uniform distribution. Second, for visualization purposes, this estimate was individually scaled to the peak amplitude of the signal. The value of the estimate in each direction is proportional to the probability that a subsequent stroke will be produced along that direction. It should be noted that if a linear representation of the histograms was used, the area included within the bounds of the probability density estimate would be numerically equal to the area included within the bounds of the uniform distribution circle. However, the polar representation of this data distorted the image such that these two areas do not appear equal.
|
Identification of significant peaks and stroke clusters
Whereas the identification of an individual peak within a unimodal distribution (e.g., Gaussian distribution) is relatively straightforward, statistical methods for the identification of multiple significant peaks within a multimodal distribution is still under development. Thus multimodal procedures often entail some subjective aspects. In an attempt to minimize this subjectivity, the mode existence test was recruited to both ascertain whether stroke orientations were uniformly distributed and to study the significance of any suspected directional biases (Minnotte 1997; Minnotte and Scott 1993). This test revealed whether a peak (suspected mode) identified in the polar histogram of orientation data at direction x was either an artifact of the sample or a true feature of the population. In short, raw orientation data were first binned and smoothed using an average shifted histogram estimator to produce the probability density estimate f(x) (Minnotte 1997; Scott 1985). The shape of this estimate is dependent on the bin width, also known as the bandwidth of the kernel smoothing window h. If h is sufficiently increased, the data may be grouped under a single mode. Alternatively, as h is decreased, additional local maxima of the density estimate become apparent through a process known as splitting. Recognizing this feature of the density estimate, the value of h was decreased to the smallest value where the suspected mode at location x remained a single peak. At this bandwidth, a test statistic was calculated from the density estimate of the sample f(x) according to
 | (2) |
; Minnotte 1997). This procedure was performed for each suspected peak to ascertain whether they were indeed statistically significant. Because of the conservative nature of tests for multimodality, peaks were considered significant at P 
0.15 (Izenman and Sommer 1988; Minnotte 1997). The output of the mode existence test provided the direction of each peak, the bin size h at which they were deemed significant, and the range of orientations defining the corresponding cluster of strokes. Because this test identified different peaks at various levels of h, the direction of each peak and its adjacent minima differed slightly from that displayed in a graphical format. This variation was a consequence of using a constant 5° bin size for the graphical representation of the stroke distribution data. Therefore to maintain consistency in the representation of the data, the location of each peak was aligned to the position of the maximum defining the corresponding cluster of strokes in the 5° histogram. Accordingly, the boundaries of the stroke cluster were also adjusted to the locations of the two adjacent minima found within 10° limits from the originally detected cluster boundaries. Given this identification, directional biases could be evaluated to determine whether their emergence reflected an optimization of one or more cost functions. Seven cost functions described further were examined. Prior to presenting them, characteristics of joint kinematics and kinetics used for calculation of the cost functions are described.
Joint kinematic and kinetic data
Shoulder and elbow joint angles were trigonometrically computed from the IRED position data. All kinematic data were low-pass filtered at 7 Hz (4th-order Butterworth) and differentiated to yield angular velocity and angular acceleration values. Shoulder () and elbow () joint angles are defined in Fig. 1A, with positive angular velocities associated with joint flexions. Although data from the trunk IRED were not necessary for the computation of angular displacements, they were used to verify that trunk motion was minimal during arm motion. Because movements were achieved using flexions and extensions of the shoulder and elbow joints, analyses were performed on characteristics of motion at these two joints.
Torques acting at each joint were partitioned as IT, muscle torque (MT), and net torque (NT). IT represents the passive rotational effect attributed to reaction forces at the joints caused by motion of the adjacent limb segments. MT primarily represents the effect of muscle forces on joint rotation. NT reflects the net effect of IT and MT, i.e., NT = IT + MT. The influence of gravitation was not considered because arm movements were performed in the horizontal plane. Whereas NT and IT can be directly computed from kinematic data, MT is computed as the difference of NT and IT. For this reason, MT additionally includes any rotational effect induced by residual passive factors not considered in the calculation of the IT (e.g., viscoelastic properties of muscles, tendons, ligaments, or other periarticular structures at the joint). Torques were computed according to two-joint equations of motion presented by Dounskaia et al. (2002a) as torque definition II. For these calculations, anthropometric measurements including limb segment inertia, mass, and center of mass were estimated from regression equations using the height and weight of each subject (Chaffin and Andersson 1984).
Definition of cost functions
To study factors influencing the choice of movement direction, seven cost functions were analyzed to reveal their potential contribution to directional biases. Four of the cost functions quantified the difficulty associated with the regulation of biomechanical factors, including the IT separately at the elbow and shoulder, inertial resistance, and kinematic manipulability. Additionally, three cost functions proposed by previous research were tested, including minimum jerk, minimum change of MT, and the sum of squared MT. Each cost function is described below as an optimization index ranging from 0.0 to 1.0. Values near 1.0 corresponded to the optimal value of each index.
Index of interaction torque (IIT) at the shoulder and elbow
This cost function characterizes the necessity to regulate IT with muscular control to produce a goal-directed movement. It was developed based on the leading joint hypothesis that provides an interpretation of multijoint movement control (Dounskaia 2005; Dounskaia et al. 1998, 2000a). The hypothesis suggests that during movements involving motion at both the shoulder and elbow, the locus of control complexity associated with IT regulation is at the elbow and not at the shoulder. According to this hypothesis, shoulder motion is usually controlled as a single-joint motion, without consideration of IT generated by elbow movement. This is because shoulder IT is small with respect to the inertia of the upper arm. In contrast, elbow IT is high compared with inertia of the lower arm. For this reason, the goal of elbow MT is to regulate IT and to adjust elbow motion generated by IT to task requirements. IT is mutable and can be high, and therefore IT regulation may be demanding in terms of both muscle energy and neural resources. These demands vary across movement directions, being high in some directions and low in other directions (Dounskaia et al. 2002a,b; Galloway and Koshland 2002; Levin et al. 2001). This dependence on movement direction predicts that, having a choice in movement direction, subjects may exhibit a tendency to select directions that require minimal intervention of MT for elbow IT regulation, i.e., predominantly passive elbow rotation. Given that during horizontal arm movements, only MT and IT contribute in production of joint motion (NT = MT + IT, where NT is net torque), this tendency can be quantified with the following index
 | (3) |
The dynamic dominance of the shoulder that predicts the locus of IT regulation at the elbow is not observed when shoulder amplitude is substantially lower than elbow amplitude (Galloway and Koshland 2002). The LJH suggests that the two joints switch the roles during movements of this type. The elbow is rotated predominantly actively (through MT), whereas shoulder IT is influential and needs to be regulated. This predicts another possible preference in the choice of joint control (and hence, movement direction), i.e., active elbow rotation accompanied by passive shoulder motion. To test this prediction, IITS was included in the analysis. This index was computed in a similar fashion as IITE by replacing values of ITE and MTE with values of shoulder IT and MT, respectively.
Index of inertial resistance (IIR)
Inertial resistance characterizes the relationship between force applied at the endpoint and endpoint acceleration caused by the inertia of the arm segments. The dependence of this relationship on movement direction has been experimentally shown (Gordon et al. 1994; Mussa-Ivaldi et al. 1985) and has additionally been quantified by Hogan (1985). Specifically, the matrix TIR expressing the inertial resistance experienced at the endpoint has the form
 | (4) |
; Sabes and Jordan 1997), but were individually customized to the anthropometric properties of each subject and were computed for each stroke with use of initial joint angles. The major (minor) eigenvector of TIR denotes the direction having the greatest (least) amount of inertial resistance. For the purposes of this study, the chosen index of inertial resistance (IIR) was defined as
 | (5) |
s is the angular orientation of the stroke, and Emin,IR denotes the orientation of the minor eigenvector of TIR. Movements oriented in the direction of least inertial resistance were considered optimal. Index of kinematic manipulability (IKM)
Kinematic manipulability represents a property of the kinematic structure of the arm according to which similar angular velocities at the shoulder and elbow can realize different resultant velocities at the endpoint depending on the combination of flexion and extension at the two joints. For example, consider the case where shoulder flexion is combined with either elbow flexion or elbow extension. Let Vsf, Vef, and Vee represent the components of endpoint velocity induced by shoulder flexion, elbow flexion, and elbow extension, respectively, and |Vef| = |Vee|. The magnitude of the resultant velocity experienced at the endpoint is greater when the two joints rotate in flexion compared with the case when the shoulder flexes and the elbow extends. The dependence of endpoint velocity on the joint coordination pattern, referred to as kinematic manipulability, has been quantified for multijoint movements of robotic systems by Yoshikawa (1985, 1990) and discussed with respect to arm movements (Dounskaia 2007; Graham et al. 2003; Sabes and Jordan 1997). The matrix of kinematic manipulability (TKM) quantifying this relationship has the form
 | (6) |
 | (7) |
s is the angular orientation of the stroke, and Emaj,KM denotes the orientation of the major eigenvector of TKM. Movements performed in directions allowing for the maximization of kinematic manipulability (i.e., endpoint velocity) were considered optimal. Index of minimum jerk (IMJ)
The minimum jerk model has offered a resilient explanation for associated characteristics of endpoint trajectory formation (Flash and Hogan 1985; Hogan 1984; Todorov and Jordan 1998). The tendency to minimize jerk is tested with the use of an index (modified from Wolpert et al. 1995)
 | (8) |
 | (9) |
Index of minimum torque change (IMTC)
The minimum torque change model was proposed by Uno et al. (1989). The applicability of this model for the description of directional biases was evaluated using the following index (modified from Wolpert et al. 1995)
 | (10) |
 | (11) |
Index of minimum sum of squared torque (IMSST)
Optimal control modeling has often considered movement trajectories from the perspectives of a minimization of muscle energy (see Todorov 2004 for review). A simplified approach for approximating the energy expenditure is to compute the integrated sum of squared MT. Accordingly, we define the corresponding index as
 | (12) |
 | (13) |
It was investigated in this study to what extent optimization of any of the listed cost functions may account for directional biases revealed with the free-stroke drawing task.
Statistical analysis
In addition to the mode existence test, statistical analyses included a repeated-measures one-way ANOVA assessing differences in the number of movements described by distinct cost functions. Pairwise comparisons were performed using Tukey's honestly significant difference (HSD) test. This method was applied where appropriate using a significance (
) level of 0.05.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
|---|
Subjects produced 17.2 ± 4.6 (SD) strokes having a length of 17.4 ± 2.1 cm during each 15-s trial. The distribution of stroke orientations was first examined for each subject using a smoothed, normalized polar histogram of the fingertip marker orientation data (Fig. 2). These histograms provide the probability density for all orientations and offer a visual depiction of the subjects' tendencies to deviate from the assigned task, i.e., a uniform distribution of stroke orientations as indicated by the black circle on each plot. Apparent from a qualitative examination of Fig. 2, subjects displayed clear affinities for movements in various preferred directions. This was confirmed by the mode existence test that revealed 6.7 ± 1.7 significant histogram peaks exceeding the uniform distribution across the subjects. Table 1 summarizes the results of the mode existence test, indicating the number of significant histogram peaks that exceeded the level of uniform distribution and their respective orientations. For instance, six significant stroke clusters were distinguished in the S1 histogram. These clusters correspond to pronounced histogram peaks, revealing a preference of this subject to produce strokes of respective orientations. In contrast, movements in the directions between the distinguished peaks were consistently avoided by this subject, as suggested by low values of the histogram in these directions. Similarly, histograms of all other subjects displayed pronounced peaks exceeding the uniform distribution, confirming that all subjects had significant directional biases.
|
Although Fig. 2 and Table 1 showed that subjects did not produce a uniform distribution of stroke orientations, questions still remained regarding the consistency of the stroke orientation distributions across subjects. Visual inspection of the histograms suggests that multiple subjects achieved directional biases along similar orientations. In particular, almost every histogram includes pronounced peaks in the right- and left-diagonal directions. For instance, subjects S4 and S10–S14 each displayed respective biases of greatest amplitude near the right-diagonal direction, i.e., 240.3 ± 6.9°. The remaining subjects also produced a histogram peak in these directions, although these peaks were not of the greatest amplitude. Similar observations can be made with respect to a common preference to produce strokes in the left-diagonal direction.
To further analyze the consistency of the directional biases across subjects, group data were used. Stroke orientations computed for all subjects were assembled to yield the polar histogram shown in Fig. 3A. To obtain this histogram, the method that provided the individual histograms shown in Fig. 2 was applied to the strokes produced collectively by all subjects. The resulting distribution of stroke orientations again indicated that subjects preferred to perform movements along distinct directions. Specifically, the group data represented in Fig. 3A indicated that subjects primarily preferred movements approximately along the 60/240 and 135/315° axis, i.e., in the two diagonal directions. This conclusion was additionally supported by the mode existence test, indicating significant histogram peaks exceeding the level of uniform distribution at 32.3, 51.8, 131.0, 241.5, and 317.9°. Movements in other directions were generally sparse, having frequency lower than that predicted by the uniform distribution. Group data thus supported the observation from Fig. 2 that there was a common tendency to deviate from a uniform distribution of stroke orientations and that the directional biases were consistent across subjects.
|
Directions of minimal cost
The seven cost functions presented in the Introduction were next evaluated to ascertain whether any of them varied across directions and whether these variations were consistent with the biases observed in the group data. Figure 4 overlays the polar histogram of group data shown in Fig. 3A with the values of indices denoting the amount of jerk (A), integrated torque change (B), integrated sum of squared torque (C), IT regulation at the elbow joint (D), IT regulation at the shoulder joint (E), inertial resistance (F), and kinematic manipulability (G) associated with each of the subjects' movements. Each optimization index could range from 0.0 to 1.0, with 1.0 being indicative of the optimal value. Notably, Fig. 4, A–C, suggests that the traditionally used optimal control criteria (i.e., minimum jerk, minimum integrated torque change, and minimum sum of squared torque, respectively) realized similar values regardless of movement direction. There was no evidence that the behavior of these cost functions were directionally dependent. Moreover, values of these cost functions were not necessarily close to 1.0, suggesting that none of these functions was optimized during stroke production. Thus the three criteria do not offer any explanation why subjects preferred movements in certain directions more than others.
|
180° apart from one another. Specifically, optimal values were noted approximately along the 120/300° axis for IITE, 45/135° for IITS, 30/210° for IIR, and 15/195° for IKM. This feature of directionality was absent from the traditional cost functions. For this reason, these four biomechanically relevant cost functions were studied as potential candidates for the emergence of directional biases across individual subjects. The possible explanation of individual histogram peaks by these factors was examined next. Assessment of individual data
Figure 5 displays the histograms of individual subject stroke orientations (as displayed in Fig. 2) overlaid with traces describing the distribution of strokes optimized according to IITE (blue), IITS (red), IIR (green), and IKM (yellow). The uniform distribution level is noted by the black circle. In developing this figure, a stroke was considered quasi-optimal according to inertial resistance or kinematic manipulability if the respective index, IIR or IKM, was >0.875. Choice of this threshold was rather conservative because it included only one eighth of the possible values for each index. Additionally, strokes were considered quasi-optimal according to the criterion of elbow or shoulder IT regulation if IITE or IITS was >0.5. This threshold allowed the identification of strokes whereby |IT| on average exceeded |MT|, suggesting that IT served as the primary source of motion at the joint.
|
The next step of the analysis included a comparison across the four cost functions in terms of the contribution of each of them to the directional biases. With this purpose, contribution of each biomechanical factor to the peaks of the individual histograms was quantitatively assessed with a measure of factor contribution (FC). FC was defined as the percentage of the strokes belonging to a particular individual histogram peak that could be accounted for by a given cost function. Specifically, FC was computed for each peak as
 | (14) |
Maximum FC values resulting from the analysis of each subject's top three significant histogram peaks are provided in Table 2 for each biomechanical factor. In the majority of subjects, at least one of the three major peaks was accurately predicted by IITE. Indeed, the blue traces of Fig. 5 (i.e., the distribution of strokes optimized according to IITE) were often aligned with the exterior trace denoting the histogram peaks. FC values reflected this alignment by showing that the optimization of IITE accounted for 93.8 ± 19.6% of at least one of the top three histogram peaks. The high accuracy of the bias prediction suggests that the cost of regulating IT at the elbow joint was an influential contributor to these biases. Similarly, movements characterized by quasi-optimal values of IITS accounted for large portions of other significant histogram peaks. FC values confirmed this assessment by indicating that movements consisting of at least one of the top three significant histogram peaks were, on average, 74.5 ± 23.6% accounted for by the optimization of IITS. It should be noted that the optimization of IITE and IITS occurred in distinct directions, as indicated by the minor of overlap of the red and blue traces within the significant histogram peaks shown in Fig. 5. Accordingly, the analysis of FC revealed that at least two of the three significant histogram peaks were accounted by biomechanical factors associated with IT regulation across subjects
|
Assessment of unpreferred movement directions
The foregoing analyses focused on the identification of preferred movement directions and whether various biomechanical factors may have influenced their emergence. Optimization of IT contribution to NT at either the elbow or shoulder joint was identified as the most prominent biomechanical characteristic associated with the observed directional biases. To further study the role of this factor in the choice of movement direction, we study whether directions in which the IT contribution was optimized in neither of the two joints were consistently avoided. With this purpose, an analysis of unpreferred movement directions was performed. Specifically, a probability distribution h(x) of unpreferred directions was assessed from the function f(x) representing the group histogram of stroke orientations shown in Fig. 3A. The distribution h(x) was defined as
 | (15) |
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
|---|
Seven cost functions were examined as potential candidate factors. The optimization of jerk, torque change, and sum of squared torque offered a generalized description of all strokes. Regardless of the movement direction, each of these cost functions acquired similar values, revealing isotropic behavior. Moreover, values of each factor were spread between 0 and 1, indicating that the three cost functions were not optimized during stroke production. For these reasons, none of these traditionally considered cost functions could be used for explaining the emergence of preferred movement directions. In contrast, the four cost functions related to the biomechanical factors showed anisotropic behavior. Each of these functions acquired high values in some specific directions and low values in other directions. Based on these observations, the subsequent analyses were focused on the four biomechanical cost functions and their possible contribution to the directional biases.
Superimposing the values of each cost function on the directional group histogram (Fig. 4) suggested that the best prediction of the directional biases was provided by the two IT-related cost functions. Analysis of peaks in the individual histograms confirmed that these two cost functions accounted for the dominant portion of the observed biases. The dominant role of the two IT-related cost functions in production of directional biases was further supported by the analysis of unpreferred movement directions. Neither IITE nor IITS acquired quasi-optimal (close to 1.0) values during movements in these directions. These results suggest that the major directional biases emerged because of a tendency to perform two types of shoulder and elbow movements. Strokes that optimized IITE were produced with active rotation at the shoulder and predominantly passive motion at the elbow driven by IT generated by the shoulder motion. Conversely, strokes optimal in terms of IITS were executed with active rotation at the elbow and largely passive motion at the shoulder.
In addition to IITE and IITS, indices describing the associated inertial resistance and kinematic manipulability were studied. Figure 5 indicates the possible influence of these factors on the selection of movement direction. For instance, subjects S2 and S3 each displayed histogram peaks that were largely accounted by strokes possessing quasi-optimal values of IIR. However, optimization of IIR and IKM failed to consistently account for a large proportion of the strokes contained within the top three significant histogram peaks across subjects. These results suggest that, although the influence of inertial resistance and kinematic manipulability on directional biases cannot be immediately ruled out, these effects were minor.
The possibility that biomechanical factors influence movement direction has been previously discussed in studies that reported consistent deviations of movement trajectory from a prescribed path (Dounskaia et al. 2000b, 2002b; Ketcham et al. 2004; Pfann et al. 2002). In particular, contribution of IT and inertial resistance to the observed deviations has been hypothesized. Our results support that hypothesis. Furthermore, testing cost functions associated with potential factors undertaken in this study provided assessments of contribution of each factor to the observed biases, suggesting that the dominant factor influencing directional biases is IT.
The major bias toward the minimization of muscular interference with the IT effect was predicted by the leading joint hypothesis (LJH) (Dounskaia 2005; Dounskaia et al. 2000a). The LJH suggests that the biomechanical properties of the multijoint structure of the arm, and specifically IT emerging at the joints caused by motion of adjacent joints, are exploited to organize arm movements. In this way, leading joint motion imposes large IT at the subordinate joint, and thus creates a dynamic foundation for motion of the entire limb. Accordingly, the role of the subordinate joint's MT is to regulate the effect of IT and to adjust joint motion to the task requirements. Because of the bigger musculature and inertia of the proximal limb segment compared with the distal segment, movements in the majority of directions are shoulder-led. However, a sector of the right-diagonal directions is elbow-led because shoulder amplitude is small, and therefore the shoulder cannot be used as a motion generator in these directions. Whether shoulder- or elbow-led movements are considered, this interpretation of multijoint movement control predicts the preference to minimize effort for the regulation of the IT effect on the subordinate joint motion. Indeed, this regulation may be complex because it may require high levels of MT, fast changes in this torque, and extensive processing of sensory feedback. These considerations predict preferences to perform two types of movements. One type belongs to the category of shoulder-led movements and consists in active rotation at the shoulder and passive motion at the elbow. The other type is an elbow-led movement characterized by minimal muscle effort for regulation of passive shoulder motion. The results of this study are consistent with this prediction, thus providing support to the interpretation of the control strategy offered by the LJH.
The revealed tendency to move one joint largely passively may be interpreted as a minimization of metabolic energy expenditure. However, one of the two preferred movement directions, namely the left-diagonal direction, is one of the most demanding in terms of the required muscle force because this direction is characterized by maximal inertial resistance (Gordon et al. 1994). A possible alternative interpretation of our results is a tendency to simplify parameters of neural control not directly related to metabolic cost or mechanical load (Prilutsky and Zatsiorsky 2002). For instance, IT regulation at the subordinate joint requires intense neural feedforward computations and processing of sensory information (Dounskaia 2005). Thus the strategy to allow IT to predominantly drive subordinate joint motion may reflect a tendency to minimize energy associated with neural processing required for IT regulation by MT.
The biomechanical factors reliably accounted for major portions of the significant histogram peaks, suggesting that no other factors may have provided an important contribution to the directional biases. Nevertheless, it is possible that there were other influential factors, the effect of which may have been aligned with the effect of the tested factors. Also, there were peaks in some of the individual histograms that were not explained with any of the tested cost functions. For instance, Fig. 5 indicates that several subjects exhibited various histogram peaks along orientations that could not be fully explained (e.g., 270° peak of S1, 250° peak of S2, 240° peak of S3, etc.), suggesting a possibility of additional factors influencing the choice of movement direction. One possibility is that biases were influenced by the perceptual distinctiveness of movements in different directions. Our task limited this factor's potential influence, however, by not supplying subjects with a visual trace of their movements (Meulenbroek and Thomassen 1992). Moreover, movements in perceptually relevant orientations (i.e., vertical, horizontal, diagonal) did not seem to be a pronounced feature of individual stroke distributions (Fig. 2). Nevertheless, the role of sensory information in the emergence of the directional biases is an important question that needs to be addressed in future studies.
Another noteworthy factor is that humans are notoriously known to be bad random generators (Lopes 1982), and this characteristic may have contributed to inability of each particular subject to uniformly distribute movements. However, it is unlikely that this factor would lead to the emergence of consistent directional biases across subjects. Additionally, certain movement directions may be more favorable in terms of the associated muscle contractile dynamics, i.e., preferred movements may be in directions where the force-generating properties of the muscles are the most advantageous. The potential influence of this factor on the revealed directional biases cannot be excluded and needs to be studied with use of arm models including characteristics of muscle biomechanics.
The obtained results cast light on the organization of the optimization processes. Frequent movements in the directions of low biomechanical cost suggest that information about the limb's biomechanical structure and the way it influences movement cost is available before motion initiation. This feed-forward information may be used to select movement directions of low cost. The idea that this information is available at the stage of movement planning is further supported by the finding that the directional biases were similar regardless of whether the whole movement or only the initial portion (i.e., 1st 60 ms) of this movement was considered. Indeed, the two histograms in Fig. 3 would be different if movements deviated from the chosen directions during the course of motion caused by the biomechanical factors. Movement cost might be determined from an internal model of limb biomechanics. An alternative possibility is that the knowledge of movement cost is acquired during motor learning and stored in the form of an internal model of optimal cost (Shimansky et al. 2004). As suggested in that study, the internal model of optimal cost might be a critical functional component of a neural control system capable of self-optimization. In particular, a combination of the internal model of the limb's biomechanical properties and the internal model of optimal cost would allow the CNS to optimize movement planning. Findings of this study indicate that one of the crucial determinants of movement cost is associated with the demands for regulation of the IT effect at the subordinate joint.
The dominant role of the biomechanical cost functions in the optimization of motor performance is consistent with findings of directional preferences during handwriting movements. For instance, in a study by Van Sommers (1984), subjects were given a blank sheet of paper and asked to cover it with short lines with an emphasized instruction to draw in every possible direction. Consistent biases in the orientation of the produced lines were revealed. Other studies reported deviations of movement path toward production of right-tilted strokes and ovals (Dounskaia et al. 2000b; Meulenbroek and Thomassen 1991). The observed movement preferences were interpreted as resulting from biomechanical interaction between wrist and finger movements. This interaction can either facilitate or hinder hand movements, depending on the coordination pattern between the wrist and fingers. The similarity of the preferred strokes and the ovals with the shapes of cursive letters suggested that the motor act of handwriting has been developed with observations of these biomechanical constraints of the hand to allow fast and accurate performance for a long time without fatigue.
Although this approach has suggested an important role of biomechanical factors in the organization of handwriting movements, thorough study of this phenomenon was not performed because of the complexity of the biomechanical structure of the hand. Here, we addressed the role of biomechanical factors in directional preferences by taking advantage of horizontal shoulder-elbow movements for which dynamic models have been developed and widely used (Dounskaia et al. 2002a; Hoy and Zernicke 1985; Putnam 1993; Schneider et al. 1989) and influential biomechanical factors have been extensively explored (Graham et al. 2003; Hogan 1985; Hollerbach and Flash 1982). The obtained results support the notion that biomechanical factors, and specifically IT, influence organization of multijoint movements. Furthermore, they suggest that the CNS may exploit biomechanical properties of the limbs to increase efficiency of motor performance.
The revealing of the directional biases and the cost functions underlying them may have important practical applications. Many types of daily motor activities require a sequence of arm movements in various directions. These tasks usually allow for substantial freedom in the choice of movement direction because the position of the trunk in the working environment can be changed to provide a desired movement direction for the arm. Arrangement of objects in the working space can also be adjusted to facilitate motor performance. The findings presented here open perspectives for optimization of these types of activities in terms of improvements in the efficiency of performance and of fatigue minimization.
Application of these findings to practical tasks should be derived with caution because tested movements were limited to the horizontal plane and to rotation of only two joints, the shoulder and elbow. Also, movements were equidistant and they were initiated from a single arm posture. Removing any of these restrictions may result in modifications of the preferred movement directions and of the strength of biases associated with each particular cost function. Nevertheless, it can be expected that the existence of the directional biases and the association of them with the tendency to move the subordinate joints largely passively is a robust characteristic of multijoint movements. This follows from a consideration that these directional biases were predicted by the LJH and that the leading-subordinate organization of control has been observed in various multijoint movements and not only in those restricted to shoulder and elbow rotations in the horizontal plain (Dounskaia et al. 1998; Galloway and Koshland 2002; Galloway et al. 2004; Hirashima et al. 2003, 2007). This predicts that directional biases caused by the tendency to minimize active regulation of IT would be observed in a wide range of unconstrained multijoint movements.
In conclusion, when given a choice, subjects displayed a tendency to move in specific directions. Two biomechanical cost functions associated with demands for regulation of the IT effect with MT at the shoulder and elbow were identified as primary contributors to the revealed directional biases. Significant biases predominantly corresponded to the tendency to propel one joint (either the shoulder or the elbow) passively with IT generated by motion from the adjacent joint. These biases were predicted by the LJH, and therefore they provide support for this hypothesis. Tendencies to move in directions of low inertial resistance and high kinematic manipulability were also observed, but these movements were of minor preference. The crucial contribution of the cost functions associated with regulation of the IT effect on joint motions supports the hypothesis that biomechanical properties of the limbs are exploited to increase movement efficiency. The results allow for the association of movement efficiency with metabolic energy minimization but also offer a novel optimization criterion related to minimization of neural processing required for regulation of IT during multijoint movements. The results advance research of optimal control strategies underlying human movements, revealing the important role of biomechanical factors in movement optimization. This knowledge opens new perspectives in the area of ergonomic design and research concerning the organization of everyday tasks often requiring the performance of serial reaching movements in different directions.
|
GRANTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSRE |
|---|
