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Department of Kinesiology, The Pennsylvania State University, University Park, Pennsylvania
Submitted 10 August 2005; accepted in final form 23 November 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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In humans, the studies mainly addressed slipping prevention (Eliasson et al. 1995
; Flanagan and Wing 1995; Johansson and Westling 1984, 1988; Johansson et al. 1992; Salimi et al. 1999; Westling and Johansson 1984). It has been shown that, when performers grasp a vertically oriented handle, they increase the grip force if the load force increases (Johansson and Cole 1994; Kinoshita et al. 1995; Monzee et al. 2003; Winstein et al. 1991); this diminishes the risk of slipping. Only recently, rotational equilibrium became an object of research (Goodwin et al. 1998; Latash et al. 2004; Pataky et al. 2004a, b; Shim et al. 2003, 2005a, b; Zatsiorsky and Latash 2004; Zatsiorsky et al. 2002a, b, 2003a, b). It has been reported that during prehension, all the individual digit force variations —whether they occur in response to changes in the motor task, e.g., load and torque magnitude, or because of trial-to-trial or in-trial variability—are always interrelated, i.e., they manifest the prehension synergies (Baud-Bovy and Soechting 2002; Pataky et al. 2004a, b; Rearick and Santello 2002; Santello and Soechting 2000; Shim et al. 2003, 2005a, b; Zatsiorsky and Latash 2004; Zatsiorsky et al. 2003a). The cause–effect relations among the observed changes of the forces and moments are described by the chain effects (Shim et al. 2005a, b; Zatsiorsky et al. 2003b).
The common goal of the above research has been to understand how people control digit forces and moments to achieve stable grasps. In contrast, the goal of robotic research is to achieve the stable grasps with the technical means/grippers (for review, see Cutkosky and Wright 1986; Shimoga 1996; Svinin et al. 1999, 2000). In principle, the strategies used by humans and robots may be similar or they can be different.
The main tenets of the grasp stability theory developed in robotics literature (Al-Gallaf et al. 1993; Choi et al. 1993; Cutkosky 1985; Hanafusa and Asada 1977; Nguyen 1986a, b, 1987a, b, 1989) can be briefly described as follows.
Behavior of an individual finger i at the contact with the object is characterized by a stiffness matrix (Si), which represents an instantaneous static relation between contact forces and compliant deflections
 | (1) |
fi = (fx, fy, fz, mx, my, mz)T includes the linear force and moment with respect to the local coordinates, and xi is a small change in the position and orientation of the contact. The elements of Si are the coefficients (stiffness) of linear relation between the small linear and rotational displacements of the grasped object and the accompanying changes in force and torque. The diagonal elements of Si represent the resistance in the direction of the perturbation, whereas the off-diagonal elements represent the increase in force/torque in one direction in response to the perturbation in another direction, e.g., increase in the normal force as a result of a small displacement in a tangential direction. In the planar case, the Si's are 3 x 3 matrices (if only linear forces and displacements are considered the Si is a 2 x 2 matrix); in three dimensions, a full Si is a 6 x 6 matrix. Instead of stiffness, its inverse—compliance—can be used. Individual Si's form one concatenated matrix Sc of a grasp. For five-digit grasps, the dimensionality of the grasp concatenated Sc is 30 (5 digits x 6 force/torques). For the planar grasps, the Sc dimensionality is 15. The Sc is composed of diagonal Si's and the interfinger coupling off-diagonal matrices. The coupling matrices represent the effect of the perturbation of one finger on the force/torque exerted by another finger, e.g., the effect of the lateral displacement of the little finger on the normal force of the middle finger. The Sc is computed at the contact reference frames.
The grasp stiffness, i.e., its resistance to small perturbations, is represented by the stiffness matrix So computed with respect to the object reference frame. The grasp stiffness matrix So in the object frame can be obtained by applying a congruent transformation to the concatenated stiffness matrix Sc. Matrix So is 6 x 6 in three dimensions, and it is 3 x 3 in planar case (2 x 2 if only linear displacements and forces are considered).
For the grasp to be stable, the stiffness matrices Si, Sc, and So should be positive definite. This requirement assures that all the eigenvalues of the above matrices are positive numbers. Note that the requirement does not specify precisely the digit forces; they can be of any value provided that the requirement is satisfied.
An essential feature of the above theory is that on-line control is not necessary for the prehension stability: if the stiffness matrices satisfy the required conditions, the grasp will be stable both in terms of the contact stability (slip prevention) and the orientation stability (tilt prevention). In other words, the grasp stability is an inherent property of a given grasp. Such a theory essentially assumes that the digits and their interconnections can be replaced by "virtual springs" (Kao 1994), this is the virtual springs theory. Maintaining the grasp stability without the on-line control is convenient for robotic applications: it allows freeing the computational resources for manipulation control.
An imperative part of the above theory is that, for the grasp to be stable, the stiffness matrices should be symmetric, i.e., sij = sji, where s is an element of the stiffness matrix. For the concatenated matrix Sc, this requirement, as an example, means that an increase of the normal force of one finger in response to a lateral perturbation of another finger should be equal to an increase of the tangential force of the second finger in response to a perturbation of the first finger in the normal direction. It has been shown experimentally that the symmetry requirement is satisfied for the human arm: the matrix of the endpoint stiffness of human arm is approximately symmetric (Dolan et al. 1993; Flash and Mussa 1990; Gomi and Kawato 1997; Lacquaniti et al. 1993; Mussa-Ivaldi et al. 1985; Tsuji 1997). The symmetry of the stiffness matrices Sc and So has not been studied.
Testing the virtual springs theory of the grasp stability on human subjects in its entirety is technically difficult: for five-digit grasps in three dimensions, each of the 30 kinematic/dynamic variables should be independently perturbed, and the resulting 30 reactions to the perturbation should be measured. Thus far, simpler paradigms have been used in the literature. Milner and Franklin (1998) measured the stiffness of the index finger in the flexion–extension plane under the instruction to the subjects "not to respond to the displacement." The computed 2 x 2 matrix was approximately symmetric. The finger stiffness was anisotropic with the direction of greatest stiffness being approximately parallel to the proximal phalang of the finger. In contrast to the previous studies that examined finger joints in isolation and found that the joint stiffness increases with the joint torque (Akazawa et al. 1983; Becker and Mote 1990; Capaday et al. 1994; Carter et al. 1990, 1993; Hajian and Howe 1997), Milner and Franklin (1998) did not find a monotonic relation between joint stiffness and joint torque.
Several authors have attempted to explore the grasp stiffness, limiting the research to 1) planar tasks, 2) pinch (mainly 2-digit) grasp, 3) mechanically constrained handles (i.e., the handle cannot move in all the directions freely), and 4) zero external torque conditions. Either the stiffness matrices were 2 x 2 (only linear displacements and forces were considered; Kao 1994) or the reaction to the perturbation was recorded in only one direction (Hermsdörfer et al. 1992, 1994; Van Doren 1998). Kao (1994) perturbed the pinch grasp in two directions, proximal–distal and radial–ulnar, and computed the 2 x 2 stiffness matrix with and without the assumption on its symmetry. It was found that the matrix was approximately symmetric. Similar results were reported by Kao et al. (1997). By studying the effect of grasp force and finger span on the grasp stiffness, Van Doren (1998) found that stiffness increased significantly in proportion to initial force but was changed only slightly by initial span. In the latter study, the stiffness was understood as a scalar rather than a matrix and was measured in the direction of the linear perturbation, the change of the handle width. A three-digit grasp was used, but only the thumb normal force was measured.
In this study, we are interested in how the grasp stability—both slip prevention and tilt prevention—are preserved during slow changes in the task constraints. We used a customized handle to change the grasp width during the manipulation—increase or decrease it—and studied 1) five-digit grasps; 2) the normal and tangential digit forces as well as the points of digit points application, in total 15 outcome variables; and 3) both the zero-torque and non–zero-torque tasks. Because of the novelty of the experimental paradigm, we did not formulate testable hypotheses; this is an exploratory study. However, we expected to find that 1) rotational equilibrium in prehension is maintained by the elastic-like properties of the digits (i.e., the properties that depend solely on the deformation magnitude but not on other factors such as speed) and 2) the CNS uses an economical reaction to the perturbation, making minimal adjustments to cope with the changing task constraints. For instance, a change in the handle width leads to a change in the moment arm of the tangential digit forces. A straightforward, minimal adjustment would be to covary the difference between the tangential forces of the thumb and the fingers to keep the moment of tangential forces constant. We have to admit that we were wrong in both expectations: The real picture is much more complex.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Eight right-handed male university students served as subjects (age, 30.5 ± 3.74 yr; weight, 74.5 ± 9.5 kg; height, 1.786 ± 0.095 m; hand length from the middle fingertip to the distal crease of the wrist with the hand extended, 19.1 ± 1.3 cm; hand width, 9.2 ± 0.53 cm). The subjects had no previous history of neuropathies or trauma to the upper limbs. All subjects gave informed consent according to the procedures approved by the Office for Regulatory Compliance of The Pennsylvania State University.
Equipment
A customized motorized handle was attached to the top edge of an aluminum beam (5.0 x 85.0 x 0.6 cm) at the midpoint of the beam in the medio-lateral direction (Fig. 1). Five six-component force/moment transducers (mass 9.1 g; Nano-17, ATI Industrial Automation, Garner, NC) were mounted on the handle. An eyehook hanger was located along the bottom edge of the beam and was used to suspend a weight. The position of the hook in the medio-lateral direction could be varied by sliding the hook in the slot that ran the length of the beam. A level was attached to the top of the handle to monitor its orientation and avoid rotation of the handle/beam unit about the x and z axes.
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). The output cables from the five sensors were connected to a custom-made breakout box that separated individual signals. There were 30 force/moment signals in total (6 signals x 5 sensors). The analog signals were sent to a customized signal conditioning box and were used as input to two 64-channel 12-bit analog-digital converters (PCI-6031, National Instrument, Austin, TX). The sampling frequency was 100 Hz.
The moving assembly was fixed within a rigid frame. A stepper motor (IMS M-1718-1.5S, Intelligent Motion Systems, Marlborough, CT) controlled by a control box was connected through worm gears to the branches of the handle that can move outward (expand) or inward (contract). A laser displacement sensor (resolution, 0.015 mm; AR200-50M, Schmitt Measurement Systems, Portland, OR) was used to record the displacement of the handle branches under expansion or contraction. The uneven weight of the right and left halves of the handle—caused by the right side location of the laser sensor and the three extra sensors—was counterbalanced by a piece of metal fixed to the left of the midline of the handle (data not shown), such that the center of gravity of the handle without a suspended load was at the midline.
Testing procedure
Before testing, the subjects were given an orientation session to become familiar with the experimental apparatus and to ensure that they were able to accomplish the tasks. Their height, weight, and hand dimensions were measured. Before the experiment, the subjects cleaned their fingertips with the alcohol prep swabs.
Subjects sat in a chair alongside a table with the right upper arm positioned at 
45° abduction in the frontal plane and 45° flexion in the sagittal plane. The elbow joint was flexed 45°. The forearm, but not the wrist and hand, was constrained by a brace and rested on the table. The forearm was supinated 90° so that the hand was placed in a natural grasping position. Special attention was given to digit placement on the sensors such that the center of the digit surface coincided with the center of the sensor.
During the experiment, a 0.285-kg load was suspended from the beam at different positions with respect to the middle of the beam. Suspending the load at different positions produced "external" torques of 0.25 and 0.5 Nm in both clockwise and counterclockwise directions with respect to the middle of the handle. Suspending the load in the middle corresponded to a zero torque. The total weight of the apparatus including the load was 14.1 N.
Subjects were requested to hold the handle statically in the air while maintaining the horizontal orientation of the level located on the top of the handle. The subjects were instructed to hold the handle "naturally with minimal force exertion." When the subjects reported that they were holding the handle comfortably, the handle branches either expanded or contracted at one of the three prescribed speeds: 1.0, 1.5, or 2.0 mm/s. The different speeds were used to establish whether the stiffness-like behavior of the digits [their apparent stiffness (AS)] was affected by the speed. Note that in classical mechanics, stiffness—by definition—does not depend on the speed. The starting grip widths were 80 and 100 mm for handle expansion and handle contraction, respectively. A "do-not-intervene" paradigm was used; i.e., the subjects were told "do not adjust your commands to the hand" during changes in the handle width. To further reduce the voluntary intervention and distract the subject's attention from the motor task, the subjects were asked to count down from a large number in their mind.
The signals were recorded for 10 s. The signals were set to zero before each trial. During a single experimental session, each subject performed 90 trials in total (5 torques x 3 speeds x 2 directions x 3 repetitions). The order of the trials was pseudorandomized. Breaks of 
1 min were provided between trials to avoid fatigue. The total duration of each experiment was 2 h.
Data analysis
Customized data acquisition software written in LabVIEW (National Instruments) was used to convert the digital signals into force and moment values. The working motor induced slight oscillations of the handle. Because of that, the data were digitally low-pass filtered at 1.0 Hz with a fourth-order Butterworth filter. The cut-off frequency was determined based on the power spectrum analysis of the kinematics data of the handle (the peak spectral density was at 2.0 Hz). The kinematic measurements were collected only for one subject. A ProReflex Motion Capture Unit 240 (Qualisys Medical AB) was used and located 1 m away from the subject. Three reflective 20-mm markers were placed evenly on the horizontal bar. With such a setup, the accuracy of the measurement was 0.02 mm. The recorded oscillation was mainly in the vertical direction (1–1.5 mm in amplitude), and it was much smaller in the horizontal direction (<0.5 mm). Data reduction was performed by using Matlab (Mathworks, Natric, MA).
In the transducer-fixed reference system, the forces normal to the transducer surface corresponded to the z-direction, Fz. In this experiment, Fz was oriented horizontally with respect to the environment. The transducers were mounted on the handle such that the y-axis was aligned with vertical axes. Thus the force Fy exerted in y-directions was computed as the tangential force. Because the task was static, the tangential force always acted in the vertical direction. Upward tangential forces and counterclockwise moments (as seen from the subject) were defined as positive. Hence, the pronation torques exerted by the subject were positive and supination torques negative. Suspending the load to the right of the middle of the handle (these tasks are labeled with letter R) produced an external torque in the clockwise (negative) direction; this torque was negated by a positive torque (pronation, counterclockwise) exerted by the subjects. In L tasks, when the load was suspended to the left of the middle of the handle, the subjects exerted negative torque (in the clockwise direction, the direction of supination). In the text, if not specified otherwise, when angular direction of moment of forces is mentioned, it always refers to efforts exerted by the subject (not to the external torque generated by the suspended load). The digit contacts with the sensors were modeled as the soft finger contacts (Mason and Salisbury 1985), in particular the rolling of the fingers on the sensor surfaces was allowed. The y coordinates of the digit force application with respect to the sensor center were computed as 
where 
represents the moment of the ith digit with respect to x-axis and 
represents the normal force of the ith digit. The obtained values were then used to compute the moment arms of the digit forces in the handle reference frame. The moments of the normal finger forces were computed with respect to the y coordinate of the point of application of the thumb normal forces.
All analyses were performed at two levels: at the level of individual fingers (IF) and the virtual finger (VF) level. The VF is an imaginary finger that produces a wrench equal to the sum of wrenches produced by all the fingers (Arbib et al. 1985; Iberall 1987). To determine the VF normal and tangential forces—
and 
, respectively—the IF forces were summed up. The vertical coordinate of the point of VF normal force application Yvf was determined based on the theorem of moment (the Varignon theorem) as 
, where Yi is the vertical coordinate of the force application point of finger i (Yi = di + yi, where di is a projected vertical distance between the sensor centers of finger i and the thumb). The capital Y is used to designate a coordinate in the handle fixed reference system, whereas the lowercase yi designates the coordinate with respect to a sensor center i.
At the VF level, the moment of the normal forces Mn and the moment of the tangential forces Mt were computed. Mn is caused by normal forces of the thumb and the VF. These forces are equal, opposite, and not collinear. Hence, they form a force couple. Because the moment of a couple is a free moment, i.e., it is the same for all moment centers and remained unchanged under parallel displacements (Zatsiorsky 2002), the Mn can be added to the Mt to obtain the total moment exerted by the subject.
At each instant of time during the trial, the moment of the VF normal force was computed as
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Determination of apparent stiffness
The term stiffness is greatly misused in human science literature (for a discussion, see Latash and Zatsiorsky 1993; Zatsiorsky 2002). It is often applied with a meaning that has little in common with the connotation accepted in mechanics. To avoid an ambiguity, we will use the term AS, understanding under it the change in force or moment per unit of an externally imposed handle width change. We associate with this term neither mechanical factors affecting the force (e.g., the perturbation speed; in classical mechanics, stiffness by definition does not depend on speed) nor specific physiological mechanisms (e.g., passive, intrinsic, or reflex stiffness components as it done in some studies; Carter et al. 1990).
A small displacement (width perturbation) can be represented by a vector 
w = (w1 w2 w3 w4 w5)T, where the superscript T designates the vector transpose. In planar case, the effect of the above perturbation on the digit forces and moments can be described by a (15 x 5) stiffness matrix Sc such that
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fn, ft, and m are the (5 x 1) vectors of the changes of the normal digit forces, tangential forces, and moments, respectively. The m are changes in the local moments, i.e., the moments exerted on the sensor surfaces by the individual digit tips. To retain the object equilibrium, the concatenated vector [fn, ft m]T should be in the null space of the (6 x 15) grasp matrix [G], i.e., the matrix product
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Equation 5 represents an equilibrium constraint on the variations of the digit forces and moments.
Changes in the local moments m are not influenced by the tangential forces, which act in the plane of the sensors and hence do not generate moments about axes in this plane. Also, the digits do not stick to the sensors and can only push but not pull on them; hence they do not generate force couples (free moments) about the plane of contact. Therefore m is only caused by changes in the magnitude of the normal forces fn and the points of their application on the sensors p. Because p is mathematically independent on the normal digit forces and fn effects are already accounted for in equilibrium Eq. 5, we analyzed the effects of the handle expansion/contraction on p rather than on m. The p data were used to compute the moment arms of the VF normal forces.
We will use the symbol ASn to describe the change of the normal force per unit of the handle width change, 
, where 
is the change in the normal force of a single digit. The symbol ASt will be used to characterize the "tangential apparent stiffness," i.e., the observed changes in the tangential force per unit of displacement. For uniformity, we will take the liberty of using the symbol ASp with the meaning "displacement of the point of digit force application per unit of handle width change." Note that the ASp is dimensionless. Besides computing the AS for individual digits, the ASn, ASt, and ASp for the VF were computed. The ASp for the VF was computed as the displacement of the vertical coordinate of the point of application of the resultant normal forces of the four fingers. The difference between the ASp for the VF and thumb—i.e., the change in the moment arm of the normal forces per unit of handle width change—was calculated and further designated as ASp. The Mn and Mt (Eqs. 2 and 3) variations per unit of handle width change were also computed and designated as the ASMn and ASMt, respectively (the apparent stiffness of the moments).
Based on the preliminary data, it was shown that a linear pattern (Fig. 2) between grip force and displacement (grip width) was sufficient to capture the force–displacement relation, regardless of the handle movement direction. Hence the AS of individual digits was quantified by the slopes of linear regression (the intercepts of the relations clustered around 0; they are not analyzed in detail here). Because the motor did not reach a prescribed speed immediately, the starting point was set at 1.5 mm from the initial grip width. The average time from the instant when the motor started working to the 1.5-mm threshold was 2.5 ± 0.22 s. Six different windows—4, 5, 5.5, 6, and 6.5 mm and the whole course—were chosen and compared. The linearity, i.e., the measure of the departure from a straight line response, served as a criterion (Sirohi and Krishna 1991). No significant differences among windows were revealed by ANOVA and multiple comparison [F5,24177 = 1.78, P > 0.1; Tukey's HSD test (post hoc)]. Therefore a 6.5-mm window was chosen for the analysis. This window magnitude was close to those used in other studies—
7 mm in Van Doren (1998) and 4 mm in the study of Milner and Franklin (1998).
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Three-way repeated-measures ANOVA with the factors DIRECTION (2 levels), SPEED (3 levels), and TORQUE (5 levels) was performed. Because the handle expansion occurred at the starting handle width of 80 mm while the initial width for the handle contraction was 100 mm, the factor DIRECTION should be more properly called DIRECTION and RANGE. For brevity, we will, however, call it simply DIRECTION. Traditional statistical measures were also used. When computing coefficients of correlation, we sometimes pooled all the trials of all the subjects together. We understand that such a procedure—which implicitly assumes that intersubject and intrasubject variabilities are similar—is not recommended; we did it to increase the number of observation points to 24 (8 subjects x 3 trials). The conclusions about the statistical significance of such correlation coefficients should be viewed with caution. The average values of the coefficients of correlation were computed using the z-transform. Statistical analysis was performed in Minitab (Minitab, State College, PA) and Matlab.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Initial digit forces
Digit forces before the perturbation are shown in Fig. 3, A and B. The results are in good agreement with the previously published data (Zatsiorsky et al. 2002a) and are presented here only for easy reference and comparison with the modulationinduced by the grip width change. The grip force (the thumb and VF) changed with the torque in a V-like fashion, whereas the tangential forces of the thumb and VF increased/decreased in opposite directions. The moment arm of the normal forces changed in a systematic manner from negative in the L tasks to positive in the R tasks (Fig. 3C).
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During the trials, the rotational equilibrium of the handle was preserved (i.e., the total moment exerted on the handle was approximately constant); however, the moments of the normal and tangential forces changed in opposite directions (Fig. 4). As seen in the graphs, with the handle expansion, the Mn values changed in the direction of pronation, i.e., positive moments increased, negative moments decreased or changed to positive, whereas the Mt values changed in the direction of supination, i.e., the negative moments increased, the positive moments decreased or changed to negative. For instance, at a zero-torque task Mi, the magnitudes of Mn (it was positive) and Mt (it was negative) both increased.
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We discuss first the ASMn and ASMt data and then the results obtained on other variables.
In all tasks, the ASMns were positive, whereas the ASMts were negative (Fig. 6). The ASMt values almost ideally mirrored the ASMn values (with a minus sign). In the handle expansion trials (with the exception of L2), the systematic effects of speed factor were seen. In the handle contraction trials, the L1 and Mi tasks did not follow this rule. The ANOVA results showed that the effects of all three studied factors, TORQUE, SPEED, and DIRECTION, on the ASMn and ASMt were highly significant (P < 0.01).
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ASp systematically decreased (with the only exception for L1 at the speed 1.5 mm/s), and for the largest pronation torque R2 during the expansion tasks, it became negative (at the speed 2.0 mm/s; Fig. 8).
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Among the tasks, ASn did not show any correlation with the initial grip force, whereas ASt correlated with the initial level of the tangential force. The correlation between ASt and the initial force was negative both for the thumb (r = –0.71, Fig. 9) and VF (r = –0.72, data not shown). To check whether the correlations were statistically significant, the slopes of regression lines between initial forces and ASt were also calculated for individual subjects. All the slopes had the same sign across all the subjects, and hence we concluded that the correlations were statistically significant (sign rank test, P < .005). Note that for the thumb the increase of the initial tangential force was accompanied by the decrease of the ASt magnitude, whereas the ASt magnitude for the VF increased (because the ASt itself was negative). The independence of the ASn on the initial value of the normal force should be emphasized. The lack of correlation between ASn and the initial grip force does not agree with the study by Van Doren (1998). In the latter study, however, maintaining rotational equilibrium of the object was not an issue. The different relations of ASn and ASt with the initial force levels suggest that their changes may be mediated by different mechanisms.
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The group average data on the ASn and ASt of individual fingers are presented in Fig. 11.
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A general conclusion from the ASn and ASt analysis of the individual fingers (cf. Figs. 7 and 11) is that some of them can behave differently from the VF, i.e., the resultant force and moment exerted on the object may arise from dissimilar adjustments of individual fingers to the same perturbation. However, for the ASp of the individual fingers, the effects were similar to those observed for the VF: significant effects of the DIRECTION and TORQUE and insignificant effect of SPEED (with the exception of the little finger where the SPEED effect was significant). Such a similarity between the VF and individual fingers ASp tunings is interesting because the VF ASp alterations occur mainly because of the modification of the finger force sharing pattern, whereas the ASp for individual fingers is caused by the displacements of the points of finger force applications on the sensor surfaces.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Tilt prevention: maintaining rotational equilibrium
The handle width represents the moment arm of the tangential forces. Hence, if in the nonzero torque tasks, the width changes and the tangential forces were to stay put, the Mt would change and unless adjustments of other moment components are made, the rotational equilibrium of the object would be broken. It seems that the simplest possible solution for the central controller is 1) to keep Mt constant by adjusting the tangential forces to the handle width changes and 2) not to change anything else, in particular the normal digit forces. However, the central controller does not use this "simplest" option preferring instead multiple adjustments of all the involved variables.
We hypothesize that this happens because of the perturbing effects of the stiffness-like reactions of the individual digits to their forcible displacement. In this task, such reactions may not be mechanically necessary (because the load force does not change) but are still unavoidable because of both biomechanical (e.g., Jacobian change) and neurophysiological (e.g., stretch reflex action) factors. As a result, the central controller should ensure the object equilibrium while facing both the handle expansion (a pure geometric factor) and the increase of the gripping forces caused by the mentioned built-in mechanisms. In other words, the stiffness-like reactions of the individual fingers assist mainly in perturbing the rotational equilibrium rather than in restoring it. Consider as examples, the L2 and R2 tasks.
In both tasks, when the handle expands 1) the moment arm of the tangential forces increases (this should increase the magnitude of Mt, which is negative in the L2 task and is positive in the R2 task) and 2) the normal forces increase, by assumption because of the built-in spring-like reactions of the fingers to their forced displacement (the ASn's are positive; Fig. 7, A and B). If these two factors are not compensated, the magnitudes of Mt and Mn are expected to increase, thus disrupting the rotational equilibrium. To maintain the equilibrium, either Mn or Mt should decrease in magnitude.
The tangential force of the thumb in both tasks goes up while the VF force goes down: the ASt is positive for the thumb and is negative for the VF in all the tasks (cf. Fig. 7, C and D; see also Fig. 5 for an example of a single trial). The similar tangential force changes lead, however, to opposite mechanical effects in the L2 and R2 tasks. In the L2 task, the tangential force of the thumb is larger than the tangential VF force (Fig. 3B) and, hence, the Mt is negative (the supination moment, in clockwise direction). Because of the tangential force changes the difference between the forces increases and the Mt magnitude rises, i.e., the negative moment increases (ASMt is negative, Fig. 6C). Therefore in the L2 task, the tangential force changes do not restore the rotational equilibrium; instead they assist the perturbation. In contrast, in the R2 tasks the similar changes of the thumb and VF tangential forces lead to a decrease in the difference between the forces as well as to a decrease in Mt. Thus while the thumb and VF tangential forces change in the same directions in the L and R tasks (ASt are positive for the thumb and negative for the VF in all the tasks; Fig. 7. B and C)—and ASMt's are always negative (Fig. 6, C and D)—the mechanical effects of these responses in the L and R tasks are opposite: they increase the Mt magnitude in the L tasks and decrease it in the R tasks.
Because in the L and R tasks the thumb and VF normal and tangential forces change similarly—producing, however, opposite mechanical effects—the only mechanism that can be used to maintain the rotational equilibrium is a change of the moment arm of the normal digit forces(Yvf – Yth), mainly by a redistribution of the normal forces among the fingers. Because, in the L2 task, where the subjects produce the negative (supination) moments, the Mt magnitude increases (the ASMt is negative; Fig. 6C), to compensate for this increase, the changes of the moment arm of the normal digit forces should be positive and large (Fig. 8). On the contrary, in the R tasks where the Mt magnitude decreases during the handle expansion—and hence compensates for the equilibrium perturbations—changes of (Yvf –Yth) can be small, as seen in the R1 task, or even negative, as in the R2 task. ASMn is always positive, i.e., the magnitude of the moment of the normal forces decreases in the L tasks (where the moment itself is negative) and it increases in the R tasks. Figure 12 shows the described sequence of events.
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Some dependencies and correlations
We admit that at this time we cannot explain all the relations reported in Tables 1, 3, and 4 and Figs. 9 and 10. The following discussion addresses only some of them.
For the ASn, the lack of dependence from the initial normal force levels (Fig. 9A) is mechanically advantageous: scaling the ASn with the initial forces would change the Mn. Because the normal forces of the VF and thumb systematically increase with the torque magnitude (Fig. 3A), the same is valid for the absence of scaling of the ASn with the torque (Table 1). The positive correlations of the ASn with the ASp in the L2, L1, and Mi tasks and the negative correlations in the R1 and R2 tasks (Table 4; Fig. 10) are related with the fact that in the first group of tasks, the ASp is positive, whereas in the R2 task, it is negative (at speeds of 1.5 and 2.0 mm/s; see Fig. 7F). In the R1 task, the ASp is positive but very small. The dependence of the ASn on SPEED is not dictated by the task mechanics and will be discussed later in the text.
For the ASt, its correlations with the initial force levels are negative (Fig. 9B), but they have different meanings for the thumb where ASt are positive and for the VF where the ASt are negative. For the thumb, with the handle width expansion, the larger the initial tangential force, the smaller the tangential force increase; for the VF, the larger the tangential force, the larger the tangential force decrease. Because for the VF, the ASt values are negative, their negative correlation with the ASp means that, when the ASt magnitudes are larger, the ASp is also larger, i.e., in the tasks where the VF tangential force decreases to a larger extent, the point of force application displaces upward by a larger amount or displaces downward by a smaller amount.
In general, the differences in the ASn and ASt dependencies/correlations on the initial force levels and TORQUE (Table 1; Fig. 9) support an idea that the normal and tangential forces are influenced by different control mechanisms with the substantial contribution of local elastic-like (but SPEED dependent) mechanisms in the normal force control (stretch reflexes, changes in the digit Jacobians, passive stiffness of the constitutive tissues, e.g., the fingertips) and the lack of it in the control of tangential forces.
Elemental (local) versus synergy (global) reactions
We would like to suggest that the equilibrium maintenance in prehension is defined by two types of reactions, "local" (elemental) and "global" (synergy). In the present context, the term local (elemental) is used to designate the responses that start and end "at the same place," e.g., at the same finger: if the finger equilibrium is perturbed a restoring force arises that tends to return the finger to its previous position.
When behavior of individual fingers at the contact with the object is characterized by stiffness matrices (Si), the matrices depend on 1) the structural properties of the finger, e.g., on the deformability of the fingertip; 2) finger configuration, i.e., the finger joint angles represented by the finger Jacobian; and 3) individual joint stiffnesses that in turn depend on mechanical properties of the joint tissues, e.g., the tendon elasticity, and —what is most important—on the stiffness control (Cutkosky and Kao 1989). In robots, the control is realized through the joint servocontrol; in humans it may involve several mechanisms including but not limited to 1) changes in muscle activation levels produced through autogenic spinal reflexes triggered by changes in the activity of peripheral receptors located in the same muscles, 2) changes in supraspinal commands to muscles, and 3) changes in muscle activity through nonlocal neural circuits related to task-specific organization of multidigit synergies.
In this study, the ASn, both at the VF level and IF levels (with the exception of the little finger at L 2 task), were positive (Figs. 7, A and B, and 11, A and B), i.e., the digit force increased with the handle expansion and decreased with the handle contraction. Such a behavior can be explained by a combination of passive mechanical resistance and the action of the stretch-reflex (Akazawa et al. 1983; Capaday et al. 1994; Cook and McDonagh 1996; Kanosue et al. 1983; Wallace and Miles 1998, 2001; but see Wessberg and Vallbo 1996). A more general explanation is provided by the equilibrium-point hypothesis. According to the equilibrium point hypothesis (Feldman 1966, 1986), voluntary muscle control is done by changing the threshold (lambda) of the tonic stretch reflex for the involved muscles. For a fixed set of lambdas to participating muscles, reactions of effectors to external displacements are expected to lead to muscle behavior defined by the tonic stretch reflex characteristics of the participating muscles. Because characteristics of the tonic stretch reflex within typical physiological ranges of muscle length have positive slopes (Gottlieb and Agarwal 1988; Latash and Gottlieb 1990; Nichols and Houk 1976), all ASn values defined by this mechanism are expected to be positive.
The ASn's (as well as ASts) depended statistically significantly on the SPEED; this means that, strictly speaking, they cannot be called "stiffness." The speed effect was also reported by Van Doren (1998). To distinguish the cases when the resistive force depends solely on the deflection magnitude from the cases when it depends also on the deflection speed, such terms as static stiffness and dynamic stiffness were used (Hajian and Howe 1997; Milner and Franklin 1998). If these terms are adopted, the ASn can be classified as a dynamic stiffness measure. The well-known velocity sensitivity of the primary endings of muscle spindle is expected to lead to a velocity-dependent component in the tonic stretch reflex. These effects have been incorporated into the equilibrium point hypothesis (Feldman 1986; Feldman and Levin 1995). They are expected to lead to velocity-dependent changes in AS. Such changes were actually observed in the reported experiments, both for the ASn and ASt.
The ASn values did not correlate with the initial grip force level. In the literature, it is usually either reported or assumed that stiffness increases with muscle force (e.g., Goubel 1978; Goubel et al. 1971; Hunter and Kearney 1982; Perreault et al. 2001; Zhang et al. 1998) However, it seems that the relations between the stiffness and the initial force levels are different at the different levels of musculoskeletal organization. In particular, for the fingers and grasping 1) at individual joints, the relations are positive (Akazawa et al. 1983; Becker and Mote 1990; Capaday et al. 1994; Carter et al. 1990, 1993; Hajian and Howe 1997); 2) for individual finger(s), the relations are not monotonic; the stiffness is lower for higher values of joint torques than for the smaller values (Milner and Franklin 1998); the decrease of the stiffness at the high levels of forces was attributed to the activation of the multi-joint muscles that perform antagonist functions in neighboring joints (e.g., interossei are the metacarpo-phalangeal joint (MCP) flexors and proximal interphalangeal joint extensors); and 3) for the grasps, stiffness increased in proportion to initial force (Van Doren 1998). In the latter study, the handle was not completely free to move, the rotational equilibrium was not addressed, and only the normal force of the thumb was measured. In this study, the maintenance of the rotational equilibrium was required and, hence, the task was quite different from those used in other studies.
The handle width change was accompanied by the changes in the tangential forces (Fig. 5). With the handle expansion, the thumb tangential force increased (ASt was positive; Fig. 7C), whereas the VF tangential force decreased (ASt was negative; Fig. 7D). Evidently the VF force decrease with the handle expansion cannot be explained by the stretch reflexes, e.g., by the reflexes initiated by the finger forcible abduction or adduction. With respect to possible MCP abduction/adduction angle changes in the trials, we mention that these changes 1) are very small; 2) for the radial fingers (e.g., the index finger) and ulnar fingers
designates the vertical coordinate of the point of force application with respect to the sensor center i.
