INTRODUCTION

The forces exerted by the fingers when grasping and lifting an object and their control have been the focus of numerous studies (for reviews see Johansson 1998; 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.

METHODS

Six subjects, 2 left-handed and 4 right-handed, ages 22–59 yr, participated in the experiment. All subjects gave informed consent and the experimental protocol was approved by the University of Minnesota's Institutional Review Board.

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).

Experimental procedures

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 (F_grip) and decompose them into 2 components, perpendicular (F_n) and tangential (F_t) 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 F_t is directed in the +X-direction.

The grip forces must satisfy the equations of motion

where m is the mass of the manipulandum; a_x and a_y are the accelerations in the X- and Y-directions, respectively; and r_i is the vector from the center of mass to the ith contact point (Fig. 1A).

It is useful to partition the grip forces into 2 components: the force required to move the manipulandum (manipulating force, F_man) and the force required to hold the manipulandum steady (grasping force, F_grasp). (F_grasp 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 e_ij 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

Equilibrium is satisfied if the weighting coefficients a_ij 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

Following Yoshikawa and Nagai (1990, 1991) we assume that the digits can only push and not pull to generate manipulating forces. This constraint requires that the coefficients b_ij > 0. A unique solution is then obtained by imposing the additional constraint that if b_ij ≠ 0, then b_ji = 0. (If both b_ij and b_ji 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)

where θ 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 a_x = 0 for forwardly directed movements, and a_x = a_y 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.

We tested whether the grasping forces varied with time and whether they were directionally tuned using an ANOVA. Because this analysis indicated that the grasp forces were directionally tuned during the movement, we subsequently characterized the phase and amplitude of the directional tuning

using linear regression. In all of our analyses, we used 0.05 as the level of significance.

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.

RESULTS

In our analysis, we assumed planar horizontal motion of the manipulandum. Indeed, an analysis of horizontal position across subjects revealed that subjects maintained horizontal planar motion within 1.3 ± 0.3 cm (mean ± SD) as they moved to the target. Consequently, the vertical load force varied little throughout the movement and we restrict our consideration to horizontal forces.

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.

Figure 2A shows the measured normal (solid lines) and tangential (dashed lines) grip forces generated by the thumb (T₁), ring (T₂), and index fingers (T₃). Epochs spanning the movement and bracketed by dashed vertical lines are shown on an expanded timescale in Fig. 2B. There are 2 main features in these traces that typify the results from all subjects. The normal forces at each of the digits were much larger than the tangential forces; that is, the grip force at each digit was directed close to perpendicularly to the surface. Second, these normal forces increased during the translational movement, returning slowly to premovement values after the movement had stopped. During the stable holding phase, the subject exerted 6.5 ± 0.1 N of force with the thumb and returned to approximately the same force level at the end of the trial. The normal forces at the ring (4.3 ± 0.07 N) and index fingers (3.0 ± 0.06 N) during the static hold phases were less. When the subject began to move, there was a concomitant increase in normal force of the thumb to over 10 N. The normal force then gradually declined and increased again (after an initial decrease) when the subject made the return movement. This behavior is better illustrated in Fig. 2B in which the movement regions defined by the vertical dashed lines in Fig. 2, A and C have been isolated, the leftmost plot in each pair corresponding to the center-out phase and the rightmost plot of the pair corresponding to the out-center phase. Subsequent analysis showed that there was no difference in the force profiles for center-out and out-center movements in the same direction and those trials were combined for averaging.

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 × 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).

Grasping and manipulating forces

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 (F_n) and tangential (F_t) 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.

For subsequent analysis, we averaged the grasp forces from individual trials after normalizing for variations in movement time. We also subtracted baseline force levels, computed over the interval between 500 and 400 ms before movement onset from each of the forces. Data from individual trials were aligned on movement onset and averaged. The results of this analysis for one subject are illustrated in Fig. 6. The data are depicted in polar plots, the inner circle in each plot representing the baseline. Normal grasp forces are denoted by solid lines, and the tangential component of the grasp force is indicated by dashed lines. There are 2 columns for each transducer: the first column depicts forces at −25, 0, and 25%, of normalized movement time, and the second column depicts the results at a 10% increment later, at −15, 10, 35%, and so forth.

Several trends are apparent by inspection of these results. First, early on in the movement (before 25%), the grasp forces do not deviate from their baseline levels. Second, grasp forces attain their peak in the middle of the movement (i.e., around peak velocity). Finally, the grasp forces exhibit directional tuning. This directional tuning was generally unimodal; that is, it exhibited cosine tuning (Eq. 7). (In this example, the one exception to this general trend was the directional tuning of ring finger normal force, which exhibited 2 peaks, about 180° apart.) For the thumb, the normal grasp force was maximal for movements directed forward (i.e., in the direction of this force). However, the results from this right-handed subject for the 2 fingers were counterintuitive. For the ring finger, normal grasp force was maximal for movements forward and to the left, and it was maximal for movements forward and to the right for the index finger. (Recall, that by definition, the grasp forces produced by the 3 digits must sum to zero.)

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.

An ANOVA showed that the amplitudes of the grasp forces at the 3 digits varied significantly with direction [F_(7,72) >2.88, P < 0.01], generally throughout the entire movement. Accordingly, we then tested whether this directional tuning varied as a cosine function of movement direction (Eq. 7). The results of this analysis are shown in Fig. 7B for all subjects. Filled symbols indicate instances in which this fit reached statistical significance (linear regression, P < 0.05). The amplitudes of the directionally modulated normal grasp force components (F_norm) followed the same time course as did the average value (Fig. 7A), increasing gradually to reach a peak in the middle of the movement. The tangential grasp force components (F_tan) at the 2 fingers (T₂ and T₃) also followed this pattern, whereas in most subjects, F_tan at the thumb reached a peak at an earlier time. The phases (indicating the “best direction”) did not exhibit any consistent trends during the movement. Moreover, there was a considerable amount of variability in the results from the 6 subjects. For example, the best direction for the thumb normal grasp force was at 0°, corresponding to a forwardly directed movement (see Fig. 6), for 3 of the 6 subjects, but it was closer to 180 ° for the other 3 subjects. The same was true also for T₃ normal force, where the phases clustered around 0 and 180 ° in the latter half of the movement.

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.

As was the case for static conditions, the force focus during translational movements remained close to the Y-axis, the bisector of the 2 finger-contact points (Fig. 1A). This is illustrated in Fig. 8, which shows the average excursions in the force focus for all 8 directions of movements for 3 of the subjects. Its location at the start of the movement is indicated by the inverted filled triangle. The force focus ranged to a much larger extent along the Y-axis (maximum average excursion 12.0 mm) than it did along the X-axis (3.5 mm maximum). An ANOVA showed that movement direction had no significant effects (P > 0.05) on the amplitude or the direction of the variations in the locus of the force focus. Moreover, the variations found here are much smaller than those we reported previously when we changed the orientation of the contact surfaces (differences of 70 mm and more along the Y-axis).

DISCUSSION

In the present study, we have characterized the temporal and directional modulations of grip force as an object held in a tripod grasp is transported. To do so, we first decomposed the grip force into 2 components, one (manipulating force) responsible for imparting an acceleration to the object and the other (grasping force) satisfying equilibrium conditions. We found that the grasping forces at the 3 digits varied with time, reaching a maximum midway during the movement (i.e., at peak velocity). The modulation in the grasping force also exhibited directional tuning, but this tuning was idiosyncratic for each subject.

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 × 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

where θ 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 ℓ 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.

GRANTS

This work was supported by National Institute of Neurological Disorders and Stroke Grant NS-15018.

FOOTNOTES

• The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.