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Department of Neuroscience, University of Minnesota, Minneapolis, Minnesota
Submitted 30 July 2004; accepted in final form 25 August 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Johansson and Cole 1992; Kawato 1999). Typically, in those studies, the object's contact surfaces were vertical and the object was grasped by the opposition of 2 digits. In that situation, it is useful to decompose the contact forces into 2 components: the grip force directed perpendicularly to the surface, and the load force directed vertically and opposing the object's weight. For a stable grasp, the ratio of the grip force to the load force must exceed a critical value determined by friction and it is now well established that this ratio is actively controlled, in a very precise manner (but see Werremeyer and Cole 1997). For example, as the object is lifted, the grip force and the load force increase in parallel (Johansson and Westling 1988). This is also true when the object is accelerated in the vertical plane, by intentional proximal arm movements (Flanagan and Wing 1997; Flanagan et al. 1993) as well as by the vertical oscillations of the center of mass during locomotion (Gysin et al. 2003). Similarly, a covariation between grip force and the tangential torque has been observed when an object grasped by 2 fingers is tilted (Goodwin et al. 1998). The control of the grip force is presumed to involve feedforward mechanisms (Kawato 1999), feedback from cutaneous afferents (Birznieks et al. 2001; Johansson et al. 1992), and expectations (Gordon et al. 1991).
When more than 2 digits are used to grasp an object, the situation becomes more complicated. In this case, the direction as well as the amplitude of the horizontal contact force (grip force) exerted by each digit needs to be controlled, and the load force on each digit is not predictable in advance (Baud-Bovy and Soechting 2002). As is the case for the 2-digit grasp, the ratio of the normal component of the grip force to the load force must exceed a safety margin. The direction of each of the grip forces is constrained under conditions of equilibrium. Specifically, the 3 grip forces must intersect at a common point termed the force focus (Flanagan et al. 1999; Yoshikawa and Nagai 1990, 1991). Recently, we (Baud-Bovy and Soechting 2001) showed that the location of the force focus was regulated in a manner that simplified the problem of force control. Specifically, the line from the thumb's contact point to the force focus passed midway between the 2 opposing digits. This result was in accord with the "virtual fingers" hypothesis put forth by Arbib and Iberall and colleagues (Arbib et al. 1985; Iberall and MacKenzie 1990), according to which the control of multifingered grasp is simplified by creating an opposition space of 2 virtual fingers, each of the virtual fingers representing the action of one or more actual digits.
In this communication, we extend this work by considering the control of the tripod grasp when an object is transported. Specifically, we consider the case of translational motion in the horizontal plane. In this situation, the load force does not change. However, the horizontal contact forces (grip forces) now include a component related to the object's acceleration (the "manipulating force"). Following Yoshikawa and Nagai (1991), we defined the grip force to be the sum of 2 components: the manipulating force and the "grasping force," the latter satisfying equilibrium constraints. If regulation of the safety margin is a paramount factor in grip force control, one would expect that the grasping force would remain constant during translational motion. If there is muscular coactivation between the digits, one might alternatively expect the grasping force to be modulated in phase with acceleration. However, as we will show, the experimental data followed neither of these predictions and the fluctuations in the amplitude of the grasping force followed more closely the velocity of the motion. In the DISCUSSION, we will relate this observation to the need to stabilize the orientation of an object as it is transported.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Instrumentation
Subjects grasped a manipulandum (described in detail in Baud-Bovy and Soechting 2001), in a tripod grasp. The manipulandum (weight 420 g) had 3 vertically oriented circular (17-mm diameter) contact surfaces for grasping. Each was instrumented with a 6-axis force–torque transducer (ATI Nano 17 US-6-2) and covered with No. 60 sandpaper. The transducers (T1–T3, Fig. 1A) were arranged at the vertices of an isosceles triangle inscribed in a circle of 44-mm radius. Consequently, the normals to the contact surfaces were all directed radially. Forces and torques from the transducers were sampled at 1,000 Hz. Three-dimensional orientation and position of the manipulandum were sampled at 120 Hz by a Polhemus Fastrak system. Before any further analysis all data were filtered with a double-exponential filter (cutoff frequency 40 Hz).
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Subjects were asked to grasp the manipulandum with their thumb, index, and ring fingers, lift it, and then to transport it in one of 8 directions. The target array consisted of 8 equally spaced circular targets (3-cm diameter) located on the perimeter of a 40-cm-diameter circle. Each target was connected to the center point of the circle (also marked with a small circle) with a straight line. The target array was affixed to a tabletop 96 cm from the floor, approximately at waist level for most subjects. Subjects stood facing the table during the experiment.
Subjects were instructed to move the manipulandum quickly when requested to do so (about 500-ms movement time). On each trial, the subject first lifted the manipulandum (typically by 15 to 20 cm), held it for several seconds, and then moved it out from the center of the target array to one of the 8 targets. The manipulandum was held at the target for about 1 s and then moved back to the center. The start of each of the 2 movements was signaled by an audible tone and a third tone instructed the subject to replace the manipulandum on the tabletop. There were 5 repetitions for each direction. Subjects were allowed to rest between trials and they had a chance to practice before the experiment began.
Data from a trial would be rejected if the manipulandum contacted the tabletop during the movement or if the subject moved in the wrong direction. Only one trial was rejected.
Grasping and manipulating forces
Our data analysis was restricted to the contact forces in the horizontal plane generated by each of the 3 digits. Following standard convention, we term them grip forces (Fgrip) and decompose them into 2 components, perpendicular (Fn) and tangential (Ft) to the contact surface. We adopted the sign convention that a positive normal component was directed toward the center of the manipulandum. The sign convention for the tangential components is illustrated in Fig. 1A. For example, at the thumb (transducer 1), positive Ft is directed in the +X-direction.
The grip forces must satisfy the equations of motion
 | (1) |
 | (2) |
 | (3) |
It is useful to partition the grip forces into 2 components: the force required to move the manipulandum (manipulating force, Fman) and the force required to hold the manipulandum steady (grasping force, Fgrasp). (Fgrasp is the solution to Eqs. 1–3 with the right side set to zero.) The decomposition into these 2 components is not unique, but we have used the scheme proposed by Yoshikawa and Nagai (1990, 1991) to compute manipulating forces that are physically plausible and as small as possible.
To derive the manipulating forces, it is useful to introduce another coordinate system, also illustrated in Fig. 1A. We define a set of vectors eij that connects the contact points of the 3 transducers, and we define the grasping forces as a set of internal forces in the coordinate system defined by these vectors
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 | (4) |
Equilibrium is satisfied if the weighting coefficients aij of oppositely directed forces are all equal (e.g., if a21 = a12). The moment equilibrium constraint (Eq. 3) is satisfied only if all 3 grasp forces intersect at a common point, the force focus. We previously showed that, in steady-state conditions, the force focus lies along a line from the thumb (T1) to the point midway between the 2 fingers (T2 and T3) for a large range of orientations and locations of the 3 contact surfaces (Baud-Bovy and Soechting 2001).
The manipulating forces were computed using the same coordinate system
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, 1991) we assume that the digits can only push and not pull to generate manipulating forces. This constraint requires that the coefficients bij > 0. A unique solution is then obtained by imposing the additional constraint that if bij 
0, then bji = 0. (If both bij and bji were nonzero, the smaller of both coefficients can be subtracted from both and these components can be added to the grasping force.) Thus at most 3 of the 6 coefficients in Eq. 5 are nonzero and their magnitude is given by solving Eqs. 1–3.
The manipulating forces for accelerations in 5 of the 8 movement directions are illustrated schematically in Fig. 1B, assuming the manipulandum is grasped with the right hand. For example, acceleration in the +Y-direction is provided by the thumb (i.e., only b12 and b13 are nonzero)
 | (6) |
is the angle between r2 and the X-axis (45°). Similarly, an acceleration to the right is provided by the thumb and index fingers (i.e., b12, b31, and b32 are nonzero). The configuration of manipulating forces for the directions not shown in Fig. 1B is mirror symmetric about the Y-axis. For example, an acceleration to the left and forward (–X, +Y) involves the thumb and ring fingers (b13 and b23). Data analysis
For each trial, we first computed acceleration by numerical differentiation of the position data. The manipulating forces for the accelerative and decelerative portions of the movement were then computed and subtracted from the measured grip forces. In so doing, we neglected off-axis accelerations; that is, we assumed ax = 0 for forwardly directed movements, and ax = ay for movements forward and to the right. (The off-axis accelerations were typically <10% of the peak acceleration in the instructed direction; see Fig. 4A.) Averaged grasping and manipulating forces for each movement direction were then obtained after normalizing for variations in movement time. For this purpose, movement onset and end were defined by the times at which movement speed was 5% of its maximum.
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 | (7) |
We also computed variations in the force focus of the grasping force for each trial. To do so, we first calculated the center of pressure, that is, the point of contact of the finger on each transducer, from the measured forces and moments (cf. Baud-Bovy and Soechting 2002). We then projected the direction of the grasp force vectors emanating from the center of pressure on each transducer and computed the locus at which pairs of these forces intersected. We defined the force focus as the average of the 3 values so computed and its uncertainty as the SD of the 3 values.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Figure 2 depicts representative results for one trial in which the manipulandum was moved forward and to the right, and then returned to the center. The initial steady holding position was somewhat below and to the right of center (Fig. 2C). About 1.5 s after recording began (in response to a tone), the subject moved rapidly forward and to the right toward the eccentric target. After remaining over the eccentric target for about 1.5 s (the middle plateau of the traces), the subject returned the manipulandum back to the central starting point. In both instances, the movements were fast (mean movement time: 556 ms), with bell-shaped velocity profiles. Although we did not impose accuracy constraints, the position traces show that the subject was quite accurate in acquiring the eccentric target. During this particular trial, the subject stopped short of the eccentric target by <2 cm in the Y-direction and was on target in the X-direction. Similarly, on the return path the subject stopped near the initial position at the beginning of the trial. This type of accuracy typified subjects' performance across trials and directions.
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During the static phase, the average normal grip force for this subject across all trials and directions was 6.4 ± 1.0 N for the thumb, whereas the ring and index fingers produced 3.5 ± 0.8 and 3.7 ± 0.7 N, respectively. These values were characteristic of those for all subjects and conditions, with mean normal forces of T1, T2, and T3 as 5.5 ± 2.1, 3.4 ± 1.3, and 3.3 ± 1.1 N, respectively.
Tangential forces can be either positive or negative, reflecting a directional bias. In this, as in all subjects, the tangential force exerted on T2 (ring for RH subjects, index for LH subjects) was consistently negative (see Fig. 1 for the sign convention), whereas those for T3 were consistently positive. Consequently, the force focus (the point of intersection of the grip forces) was closer to the thumb than it was to the other digits. For the trial illustrated in Fig. 2, the tangential forces during the hold period were 0.93 ± 0.04 N for the thumb, –1.07 ± 0.04 N for the ring finger and 0.52 ± 0.04 N for the index finger. Across all subjects and trials this pattern of forces was maintained with tangential forces of T1 0.12 ± 0.33 N, T2: –0.49 ± 0.47 N, T3: 0.30 ± 0.42 N.
The translational movements had a bell-shaped velocity profile and, accordingly, movements consisted of an accelerating phase and a decelerating phase, as illustrated in Fig. 3. Shown for comparison with acceleration (dashed lines) is the vector sum of the horizontal grip forces in the X- and Y-directions (solid lines). Because the vector sum should equal mass x acceleration, this analysis provided an internal check on the instrumentation. For this trial and direction (toward the subject and to the left), the correlation coefficients between the X- and Y-components of force and the respective accelerations were 0.975 and 0.961. For all subjects, these values averaged between 0.95 and 0.97, neglecting instances in which the nominal acceleration was zero (i.e., movements along the X- or Y-axis). The sum of the grip forces was close to zero during the static periods. In the example in Fig. 3, the Y-component of the total force is slightly positive, presumably because the manipulandum was not perfectly level. On average, the Y-component of the total steady-state grip force was somewhat more variable than the X-component (X = 0.10 ± 0.06 N, Y = –0.2 ± 0.18 N).
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To a large extent, the temporal fluctuations in the grip forces were attributable to the manipulating force components responsible for accelerating and decelerating the manipulandum. This can be ascertained in Fig. 4, which shows 2 representative trials from a left-handed subject, for a movement in the forward direction in Fig. 4A and a movement forward and to the right in Fig. 4B. The grip forces in the normal (Fn) and tangential (Ft) directions at each of the transducers are denoted by the solid lines. The grasping forces that result after the manipulating forces have been subtracted are indicated by the dashed lines. For the forwardly directed movement (Fig. 4A), the acceleratory component was provided by the thumb, and the index and ring fingers contributed to the deceleration. Before movement onset (indicated by the vertical line; see also the acceleration traces at the bottom of the figure), there was little variation in the magnitudes of the grip forces. At movement onset, the normal component of thumb grip force increased rapidly. In this example, this increase in force was slightly smaller than the manipulating force at the thumb and, consequently, the thumb grasping force decreased slightly. There was also an initial slight decrease in the grasping forces exerted by the ring finger. In the deceleratory phase of the motion, the manipulating force accounted for most of the modulation in the amplitudes of the normal and tangential components of the grip force exerted by the 2 fingers.
The example in Fig. 4B is more typical of the pattern of modulation in grasping forces exhibited by all of the subjects in that virtually all of the initial fluctuations in grip force are accounted for by the manipulating force components and the grasping force increased slowly if at all. At the thumb, the normal grasp force component was virtually flat for the first 150 ms after movement onset, whereas the tangential component increased slightly. Similarly, the normal and tangential components of the ring finger (T3) grasp force changed little over the first 150 ms of the movement.
Figure 5 illustrates that the pattern of force fluctuations was remarkably consistent from trial to trial. Shown are the results from a right-handed subject for the 5 center-out trials directed forward and to the right (i.e., in the same direction as in Fig. 4B). Note that this subject had considerable variability in the steady-state grip forces before movement onset. However, the magnitude of the fluctuations in grip force after movement onset did not reflect this variability. In accord with the results presented in Fig. 4B, the initial increase in the normal grip force at the thumb was accounted for primarily by the manipulating force and grasping force rose much more gradually, if at all, during the first 150 ms. Grasping force exerted by the index finger (T3) showed little modulation over this interval.
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Statistical analysis supported these observations. Averaged over direction, in this subject, thumb normal grasping force differed from 0 at 35% of movement time (and only thereafter, t-test, P < 0.01). Ring and index finger grasping force levels achieved statistical significance slightly later (at 40 and 45%, respectively). Averaged over direction, tangential grasping force levels deviated from baseline levels at 75% of movement time (thumb) and at movement onset for the ring and index fingers. Peak normal forces were attained in the middle of the movement (0.94 N thumb, 0.42 N ring finger, and 0.33 N index finger). Peak tangential forces were attained slightly later (at 65 to 85% of movement time), and were much smaller (–0.16 N thumb, –0.28 N ring, and 0.19 N index).
The results for this subject were representative (Fig. 7A). Averaged over all directions, peak normal grasp force was attained between 40 and 50% of movement time (top panel, Fig. 7A). Normal forces increased gradually to this peak and on average began to differ significantly (P < 0.01) from zero only after 20 to 25% of movement time. Tangential force at the thumb was uniformly small and generally did not differ significantly from zero. There was variability in the times at which peak amplitudes of the finger tangential forces were attained, with times ranging from 30 to 70% of movement time.
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Variations in the locus of the force focus
According to the requirements of static equilibrium, the 3 grasping forces must intersect at a common point termed the force focus (see Eq. 3 in METHODS). We previously (Baud-Bovy and Soechting 2001) showed that this force focus lies on a line from the thumb to a point midway between the 2 fingers. This was true for a wide range of locations and orientations of the 3 contact points and suggested a simplifying strategy for the control of grasp forces. This was also the case for the grasping forces during translational motion (Fig. 8). We computed the location of the force focus throughout the movement by first computing the intersection between pairs of grasp force vectors, defining the force focus as the average of these 3 values. The uncertainty in this measure was small, with an average SD of 1.25 mm in X and 2.63 mm in Y.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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We begin this discussion by considering the requirements of the task and how they might be expected to affect the modulation of grip forces. As an object is transported, the vector sum of the grip forces must equal the mass x acceleration. As stated above, we have found it useful to split the total grip force into 2 components: a grasping force that is adequate to hold the object statically and a manipulating force that represents the additional increment responsible for accelerating and decelerating the object. This decomposition does not lead to a unique solution but we believe the criteria we have used for computing the manipulation force (Yoshikawa and Nagai 1990, 1991) are reasonable and physiologically plausible. They follow from a simple constraint: each of the digits can push against, but not pull on, the contact surface. This constrains the normal component of the manipulating force at each digit to be positive. Using this criterion, we then found the solution that is the most efficient (i.e., it has the minimum norm) while satisfying the equations of motion (Eqs. 1–3).
One would expect the grasping forces, obtained by subtracting the manipulating force from the grip force, to remain constant. This is because the requirement for a stable grasp is that the ratio of the grasp force to the load force exceed a critical value determined by the frictional characteristics of the contact surfaces. In our experiments, the load force did not change because the motion was restricted to the horizontal plane. Most previous studies have shown that this ratio is indeed regulated precisely, both under static conditions and when the load force changes because of acceleration or tilt of the object (Goodwin et al. 1998; Gysin et al. 2003; Johansson and Westling 1988), although exceptions to this rule have been reported (cf. Werremeyer and Cole 1997). Our results are clearly at variance with this expectation.
It is possible that the algorithm implemented by the CNS to generate the manipulating forces is suboptimal in that the forces are larger than they need to be; that is, that there is the equivalent of "cocontraction" at the 3 digits. It is also possible that the manipulating forces are not scaled precisely to the expected acceleration of the hand. This acceleration derives mostly from activation of proximal arm muscles and it is believed that the grip force modulations are based on a feedforward estimate of expected acceleration based on proximal motor commands (Kawato 1999). However, in either case, one would expect the grasping forces to covary with the magnitude of the acceleration. In other words, one would expect them to be maximal at the peak of acceleration and at the peak of deceleration. This is also at variance with the observed results; the grasping force was maximal around the peak velocity, when acceleration is zero.
There is one other possible explanation that may account for the experimental data. It relates to the requirement to stabilize the orientation of the object as it is transported. If an object's center of mass is above or below the points at which it is grasped, it will act like a pendulum as it is translated in space
 | (8) |
is the tilt of the object; m and I are its mass and moment of inertia, respectively; M is the moment of force about the horizontal axis applied to the object; x is its translation in the horizontal direction; and 
is the length from the point of contact to the center of mass. It is clear that the solution to Eq. 8 depends on several variables, and that it is not easy to predict variations in the object's tilt as it is transported. However, the rotational motion must be counteracted by the moment M. This moment would be created if the grip forces are offset from each other in the vertical direction. Increasing the magnitude of the grasping forces thus may help to stabilize against object tilt because it would decrease the amount by which the vertical location of the contact points would need to be changed. In the present experiments, the manipulandum's center of mass was located in the plane of the contact surfaces. Thus the considerations described above do not apply. Nevertheless, it is possible that the observed increase in the grasping forces during object transport reflects a general strategy to stabilize the object against tilt.
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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Address for reprint requests and other correspondence: J. F. Soechting, Department of Neuroscience, 6-145 Jackson Hall, 321 Church St. SE, University of Minnesota, Minneapolis, MN 55455 (E-mail: soech001{at}umn.edu)
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Baud-Bovy G and Soechting JF. Two virtual fingers in the control of the tripod grasp. J Neurophysiol 86: 604–615, 2001.
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Flanagan JR, Burstedt MKO, and Johansson RS. Control of fingertip forces in multidigit manipulation. J Neurophysiol 81: 1706–1717, 1999.
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Flanagan JR and Wing AM. The role of internal models in motion planning and control: Evidence from grip force adjustments during movements of hand-held loads. J Neurosci 17: 1519–1528, 1997.
Goodwin AW, Jenmalm P, and Johansson RS. Control of grip force when tilting objects: effect of curvature of grasped surfaces and applied tangential torque. J Neurosci 18: 10724–10734, 1998.
Gordon AM, Forssberg H, Johansson RJ, and Westling G. Visual size cues in the programming of manipulative forces during precision grip. Exp Brain Res 83: 477–482, 1991.[ISI][Medline]
Gysin P, Kaminski TR, and Gordon AM. Coordination of fingertip forces in object transport during locomotion. Exp Brain Res 149: 371–379, 2003.[Medline]
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Johansson R, Riso R, Häger C, and Bäckström L. Somatosensory control of precision grip during unpredictable pulling loads. I. Changes in load force amplitude. Exp Brain Res 89: 181–191, 1992.[CrossRef][ISI][Medline]
Johansson RS. Sensory input and control of grip. Novartis Found Symp 218: 45–59, 1998.[ISI][Medline]
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Johansson RS and Westling G. Coordinated isometric muscle commands adequately and erroneously programmed for the weight during lifting task with precision grip. Exp Brain Res 71: 72–86, 1988.[ISI][Medline]
Kawato M. Internal models for motor control and trajectory planning. Curr Opin Neurobiol 9: 718–727, 1999.[CrossRef][ISI][Medline]
Werremeyer MM and Cole KJ. Wrist action affects precision grip force. J Neurophysiol 78: 271–280, 1997.
Yoshikawa T and Nagai K. Analysis of multi-fingered grasping and manipulation. In: Dextrous Robot Hands, edited by Venkataraman ST and Iberall T. New York: Springer-Verlag, 1990, p. 187–208.
Yoshikawa T and Nagai K. Manipulating and grasping forces in manipulation by multifingered robot hands. IEEE Trans Robot Autom 7: 67–77, 1991.
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