## Apparatus

We asked subjects to grasp and manipulate a custom-made grip device using the thumb and index fingertip. The inverted T-shape device consisted of a sensorized vertical handle connected to a horizontal structure (**Fig. 1A**). The handle has two parallel graspable surfaces (length = 80 mm; width = 24 mm; distance = 59.2 mm) covered with sandpaper (100-grit). Two light-emitting diode (LED) arrays matching the length of the graspable surfaces were placed on the frontal plane of the object to cue subjects about the location of digit placement for each experimental condition (**Fig. 1B**). The LED arrays were controlled via a microcontroller board (Arduino Uno Rev 3, Arduino, Boston, Massachusetts, USA). Each graspable surface was instrumented with one six-component force/torque (F/T) transducer (Nano 25, ATI Industrial Automation, Garner, NC, USA; nominal force resolution: 0.0625 N; nominal torque resolution: 0.076 N-cm). The transducers measure forces and moments of forces exerted by each digit on the graspable surfaces (sampling rate: 1 kHz).

The center of mass (CM) of the object was configured to be symmetrical (Center CM, *C*) or asymmetrical (Left or Right CM, *L* or *R*, respectively). For each of these conditions, we placed a mass (0.3 Kg) in one of the three compartments (center, left or right) of the horizonal base of the grip device (**Fig. 1B**, CM). The torques caused by the added mass about the *x*-axis of the handle’s center of geometry in the handle coordinates were − 0.17, 0, and + 0.17 N–m when the mass was located in the left, center or right compartment, respectively. The cables of the force transducers and the LED arrays ran away from the handle with a slight offset to the left of the handle center and created a small torque 0.01 N–m. Therefore, the actual task torques were − 0.18, − 0.01, and 0.16 N–m for CM L, C, and R, respectively. Note that the terms “left” and “right” denote the thumb and index finger side of the vertical handle, respectively. The total mass of the object was 0.7 Kg.

We used five thumb-index fingertip vertical distances (**Fig. 1B**, offsets): 0 mm (0), 8.84 mm (Small, *S*), 29.7 mm (Medium, *M*), and 55.4 mm (Large, *L*). An additional experimental condition (Unconstrained, *U*) consisted in allowing subjects to grasp the object anywhere along the vertical graspable surfaces. The rationale for testing combinations of CM and offsets is described in the *Effect of digit offset on grasp force procedures* section.

The position and orientation of the object and hand were tracked by a ten-cameras active infrared marker motion capture system (Impulse, PhaseSpace Inc., San Leandro, CA, USA). We used four infrared LED markers on the object (**Fig. 1A**, black dashed circles, sampling rate: 480 Hz; spatial accuracy: ~1 mm; spatial resolution: 0.1 mm). On each trial, the LED arrays and an audio signal cued the participants about the upcoming task event and grasp condition (see *Experimental procedures*). A customized LabVIEW program (National Instruments, Austin, TX, USA) was used to drive the microcontroller board and stream kinetic and kinematic data to the hard drive of the host computer.

## Experimental procedures

Each participant reached and manipulated the object for a total of 12 experimental conditions. These conditions consisted of 12 combinations of three CM locations and five offsets: C0, CU, L0, LS, LM, LL, LU, R0, RS, RM, RL, and RU. These combinations of external torques and offsets were selected by taking into consideration the extent to which participant could perform the task comfortably. Note that we presented participants with opposite offsets when grasping and manipulating the object with the L or R CM. Specifically, for three L CM conditions (LS, LM and LL), the LED cued participants to place the thumb higher than the index fingertip, whereas for the R CM conditions (RS, RM and RL) subjects were required to place the thumb lower than the index fingertip (see example at the bottom left of **Fig. 1B**). The LED arrays indicated the designated contact locations of the thumb and index fingertip by lighting up according to the offset condition (see examples in **Fig. 1B**). The distance spanned by each set of three LEDs (18.84 mm) was sufficiently large to enable positioning of the tip of the thumb and index finger. Participants were instructed to position their fingertips as close as possible to the active LED arrays. In the constrained offset blocks (0, S, M, and L), three LEDs lighted up on each side of the graspable surface. For the unconstrained grasping block (U), all LEDs in both arrays lighted up to inform participants they could choose digit placement.13

Before the experiment, participants washed their hands with soap and warm water to normalize skin condition. The coefficient of friction between the fingertips and the graspable surfaces was estimated from slip force measurement as the ratio between the minimal finger force normal to the graspable surface required to prevent slip to the tangential finger force measured at the object slip onset.22 This method requires fingertip tangential (vertical) forces to be equal and aligned with the gravitational force. Therefore, we performed the slip force measurements using the object in the symmetrical (center) CM configuration. Participants sat in front of the table and held the object with the mass added to the center slot (0 N–m torque) 5 cm above the table while following the instructions to “slowly move the index finger and thumb apart and let the object drop freely when it slips.” Participants performed three object release trials at the beginning and end of the experimental session.

To familiarize with the object, cues and task, subjects performed three CU trials, followed by 20 consecutive trials for each experimental condition (20⊆12 = 240 trials). The pseudo-randomization of the 12 experimental condition blocks was designed to present a different CM across consecutive blocks. The object was placed at a distance of 30 cm in front of the participant on a leveled tabletop 30 cm below the participant’s shoulder joint. This configuration ensured subjects could perform a comfortable grasp. The first auditory cue (*Ready*) prompted the subject to prepare for the reach. After 1.5 seconds, the second cue from the LED arrays informed subjects where to place their digits and cued them to start the reach, grasp and lift at a self-paced, natural speed. Participants were instructed to (a) grasp the object by placing, as accurately as possible, the tip of their thumb and index finger of their right hand such that each coincided with the active LEDs while extending the other digits, (b) lift the object while preventing it from tilting, as if the object were a cup filled with liquid, and (c) hold it straight for 2 seconds. We note that fulfilling the object tilt minimization criterion required participants to plan and generate a compensatory torque (*T**COM*) at object lift onset to compensate for the object’s asymmetrical mass distribution (R and L CM conditions13). When the object reached a height of 20 cm above the table, a third auditory cue (*Hold*) was triggered, after which subjects were required to hold the object stationary and in a vertical orientation for ~ 2 seconds. The next auditory cue (*Relax*) informed subjects to replace the object on the table and move the hand back to the initial position. Subjects were given ~ 10-s breaks between trials and ~ 30-s breaks between blocks.

As the CM of the object cannot be visually inferred (the added mass is hidden from view), subjects do not exert adequate *T**COM* at lift onset on the first left or right CM trial, thus causing the object to tilt during the lift. However, we previously found that subjects can learn to generate an anticipatory *T**COM* to minimize object tilt within the first three trials.13 This phenomenon was further confirmed in our data. Therefore, to quantify differences in anticipatory *T**COM* at lift onset across experimental conditions, we excluded the first three trials in each condition block from analysis.

## Data analysis

*Object kinematics and task epochs.* Object kinematics was computed from the data obtained through the infrared LED markers (**Fig. 1A**). Object translation was estimated as the displacement of the origin of the object’s fixed frame, whereas the object orientation was extracted as the Z-Y-X Euler angle from the rotation matrix using the MATLAB function *rotm2eul*. Object lift onset was identified using the maximum object height as the starting point of our algorithm and defined as the time at which the object’s height was higher than 1 mm above the table and the object’s vertical velocity was greater than 5 mm/s. Object hold phase was defined as a one-second window 0.5 second after the *Hold* cue. As subjects were asked to minimize object tilt, we quantified the quality of dexterous manipulation by measuring peak object tilt at object lift onset and during object hold. For the former epoch, peak object tilt was quantified as the largest angle of the vertical axis of the grip device relative to the vertical occurring within 250 ms from object lift onset. During object hold, peak object tilt was computed by averaging the object’s angle relative to the vertical throughout the one-second object hold window.

To determine the trial after which task performance became stable, the resultant torque (external torque minus compensatory torque, *T**RES*; see *Analysis of manipulation force components’ contribution to T**COM*) was tested by linear mixed-effects models (LMMs) with three-way fixed effects of *CM* x *Digit Offset* × *Trial*. The data and *Trial* factor were incrementally increased starting from trial 10 to 20, followed by 9 to 20, and so forth, until a significant *Trial* effect was found. We found task performance became stable from trial 4 to 20 for all subjects and conditions.

*Coefficient of friction estimation.* The translational and torsional coefficients of friction at each contact were estimated for constructing the friction cone at each contact to approximate the minimum grasp (normal) force required to prevent object slip.43 Slip onset was identified as the time of the peak change of the resultant vertical force rate closest to the first drop in the rate of resultant vertical force. This approach was preferred to defining slip onset as the time at which the first peak change in the resultant vertical force occurred as subjects might reflexively re-grip the object after slip onset. The accuracy of this algorithm was verified by also examining object position data. The translational coefficients of friction at the thumb and index fingertip were quantified as the largest value of the ratio between the vertical tangential and normal forces at the time of slip onset estimated at the thumb and index fingertip. We found no significant difference between the translational coefficient of friction measured at the beginning versus end of the experimental session (t15 = 0.078, p = 0.9389). Therefore, we averaged these values for each subject for the analysis of grasp and manipulation forces (below). The coefficient of translational friction averaged across subjects was 1.56 ± 0.06. The torsional coefficient at contacts was assumed to be 6.59 times the translational coefficient.44

*Analysis of grasp and manipulation forces.* A stable grasp that can resist any force vectors applied on the grasped object (i.e., a grasp is force-closure45) requires internal forces that are exerted inside the friction cone. Unlike *T**COM*, the internal forces have no direct effect on the object kinematics and are only limited by the friction cone, the object rigidity, and the amount of force each digit can exert. Therefore, manipulation and internal forces play very different roles in dexterous manipulation: the former is devoted to controlling object position and orientation (collectively defined as ‘object pose’), whereas the latter is devoted to preventing object slip. The procedure to decompose the contact forces at individual digit contacts into manipulation and internal forces are described below.

All forces measured by the two transducers were first spatially rotated and aligned from individual transducer coordinates to the object coordinate frame fixed at the center of the handle (inset, **Fig. 1A**) and defined following the recommendations for reporting kinematic data.46 Each digit force was assumed to be applied at a point denoted as the center of pressure (*CoP*, \({\left[\begin{array}{ccc}{\text{CoP}}_{\text{x}}& {\text{CoP}}_{\text{y}}& {\text{CoP}}_{\text{z}}\end{array}\right]}^{\text{T}}\)) of the force vector application on each graspable surface and computed from the measured forces (\(\stackrel{\text{⃑}}{\text{F}}\text{ }\text{=}{ \left[\begin{array}{ccc}{\text{f}}_{\text{x}}& {\text{f}}_{\text{y}}& {\text{f}}_{\text{z}}\end{array}\right]}^{\text{T}}\)) and moment of force (\(\stackrel{\text{⃑}}{\text{M}}\text{ }\text{=}{ \left[\begin{array}{ccc}{\text{m}}_{\text{x}}& {\text{m}}_{\text{y}}& {\text{m}}_{\text{z}}\end{array}\right]}^{\text{T}}\)) using the cross product equality \(\stackrel{\text{⃑}}{\text{r}}\text{ }\text{×}\text{ }\stackrel{\text{⃑}}{\text{F}\text{ }}\text{=}\text{ }\stackrel{\text{⃑}}{\text{M}}\) where \(\stackrel{\text{⃑}}{\text{r}}\text{ }\text{=}\text{ }{\left[\begin{array}{ccc}\text{Co}{\text{P}}_{\text{x}}& \text{Co}{\text{P}}_{\text{y}}& \text{Co}{\text{P}}_{\text{z}}\end{array}\right]}^{\text{T}}\), *T* denotes transpose, and |*CoP**z*| indicates the distance from the graspable surface to the geometrical center of the sensorized vertical handle, i.e., half of the grasp width normal to the grasp surface. The vertical distance between the tip of the thumb and index finger was defined as \(\text{Δ}{\text{CoP}}_{\text{y}}\text{ }\text{=}\text{ }{\text{CoP}}_{\text{y}}^{\text{IN}} \text{–}\text{ }{\text{ CoP}}_{\text{y}}^{\text{TH}}\). We note that *∆CoP**y* is equivalent to the digit placement offset that we either systematically changed across experimental conditions or was chosen by the subject (U condition; **Fig. 1B**).

The soft-finger contact model assumes that the finger can exert forces in three directions and one moment of force normal to the contact surface.47 The contact forces and moments of forces exerted by each digit at the contact surface in the digit-contact coordinate frame can be computed from \(\stackrel{\text{⃑}}{\text{F}}\) and \(\stackrel{\text{⃑}}{\text{M}}\) in the object coordinate frame with \(\stackrel{\text{⃑}}{\text{r}}\). The generalized force vector in ℝ4: \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\text{ }\text{=}\text{ }{\left[\begin{array}{cccc}{\text{f}}_{\text{x}}& {\text{f}}_{\text{y}}& {\text{f}}_{\text{z}}& {\text{m}}_{\text{z}}\end{array}\right]}^{\text{T}}\) represents the contact force vectors applied on the contact surface at \({\text{CoP}}^{\text{i}}\text{ }\text{=}\text{ }{\left[\begin{array}{cc}\text{Co}{\text{P}}_{\text{x}}^{\text{i}}& \text{Co}{\text{P}}_{\text{y}}^{\text{i}}\text{ }\end{array}\right]}^{\text{T}}\), where *i* = TH, thumb or IN, index finger, in the digit-contact coordinate frame.

The effect of \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\) on the object can be determined by transforming them to the object coordinate frame using the adjoint transformation matrix and the wrench basis.21,47,48 The adjoint transformation matrix was

\({\text{Ad}}_{{\text{g}}_{\text{i}}^{\text{-1}}}^{\text{T}}\text{=}\left[\begin{array}{cc}{\text{R}}^{\text{i}}& \text{0}\\ {\widehat{\text{p}}}^{\text{i}}{\text{R}}^{\text{i}}& {\text{R}}^{\text{i}}\end{array}\right]\) [1]

where, for each digit, *R**i* is the rotation matrix from the object frame to the contact frame and \(\widehat{\text{p}}\) is the skew-symmetric matrix of \(\stackrel{\text{⃑}}{\text{r}}\). The wrench basis was

\({\text{B}}^{\text{i}}=\left[\begin{array}{cccc}\text{1}& \text{0}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{1}& \text{0}\\ \text{0}& \text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{0}& \text{1}\end{array}\right]\) [2]

where each column represents elements of \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}\) and each row is the element of the generalized force vector acting on the object. Based on the force application locations of the thumb and index fingertip with the soft-finger contact model assumption,21 the grasp map (G) can be written as:

\(\text{G =}{\text{Ad}}_{{\text{g}}_{\text{i}}^{\text{-1}}}^{\text{T}}{\text{B}}^{\text{i}}\text{= }\left[\begin{array}{cccccccc}\text{1}& \text{0}& \text{0}& \text{0}& \text{-1}& \text{0}& \text{0}& \text{0}\\ \text{0}& \text{1}& \text{0}& \text{0}& \text{0}& \text{1}& \text{0}& \text{0}\\ \text{0}& \text{0}& \text{1}& \text{0}& \text{0}& \text{0}& \text{-1}& \text{0}\\ \text{0}& \text{-}{\text{CoP}}_{\text{z}}^{\text{TH}}& {\text{CoP}}_{\text{y}}^{\text{TH}}& \text{0}& \text{0}& \text{-}{\text{CoP}}_{\text{z}}^{\text{IN}}& \text{-}{\text{CoP}}_{\text{y}}^{\text{IN}}& \text{0}\\ {\text{CoP}}_{\text{z}}^{\text{TH}}& \text{0}& \text{-}{\text{CoP}}_{\text{x}}^{\text{TH}}& \text{0}& \text{-}{\text{CoP}}_{\text{z}}^{\text{IN}}& \text{0}& {\text{CoP}}_{\text{x}}^{\text{IN}}& \text{0}\\ \text{-}{\text{CoP}}_{\text{y}}^{\text{TH}}& {\text{CoP}}_{\text{x}}^{\text{TH}}& \text{0}& \text{1}& {\text{CoP}}_{\text{y}}^{\text{IN}}& {\text{CoP}}_{\text{x}}^{\text{IN}}& \text{0}& \text{-1}\end{array}\right]\) [3]

where \({\text{CoP}}_{\text{i}}^{\text{TH}}\) and \({\text{CoP}}_{\text{i}}^{\text{IN}}\) denote the points of thumb and index force application, respectively.

The grasp map was used to transform \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\) at individual digit contacts to the object coordinate frame and consequently mapped the digit forces to the total resultant forces (\({\stackrel{\text{⃑}}{\text{F}}}_{\text{O}}\)) acting on the object, \({\stackrel{\text{⃑}}{\text{F}}}_{\text{O}}\text{ =}\text{ }\text{G}{\left[\begin{array}{cc}{\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{TH}}& {\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{IN}}\end{array}\right]}^{\text{T}}\). The null space of G was denoted as G0. Projecting \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\) on to G0 resulted in *F**G* defined as \({\stackrel{\text{⃑}}{\text{F}}}_{\text{G}}\text{ = }{\text{G}}_{\text{0}}{\text{G}}_{\text{0}}^{\text{T}}{\left[\begin{array}{cc}{\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{TH}}& {\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{IN}}\end{array}\right]}^{\text{T}}\), i.e., the internal force that lies in G0 and *F**M* can be computed as \({\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\) minus the grasp force, \({\stackrel{\text{⃑}}{\text{F}}}_{\text{M}}\text{ = }{\left[\begin{array}{cc}{\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{TH}}& {\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{IN}}\end{array}\right]}^{\text{T}}-{\stackrel{\text{⃑}}{\text{F}}}_{\text{G}}\). As the grasp tasks were successfully performed in all trials, we assumed all grasps satisfied the force closure criterion.45 Based on the force closure assumption, in order to maintain proper *F**M* without slipping, there must exist an *F**G* to guarantee that contact forces satisfy friction cone constraints. The magnitude of all thumb and index finger force vectors was quantified as \(\left|\text{F}\right|\text{ }\text{=}\text{ }\sum _{\text{i}\text{ }\text{=}\text{ }\text{TH, IN}}\left|{\stackrel{\text{⃑}}{\text{F}}}_{\text{C}}^{\text{i}}\right|\). The magnitude of *F**G* and *F**M* was computed as \(\left|{\text{F}}_{\text{G}}\right|\text{ }\text{=}\text{ }\sum _{\text{ }\text{i=}\text{ }\text{TH, IN}}\left|{\stackrel{\text{⃑}}{\text{F}}}_{\text{G}}^{\text{i}}\right|\) and \({\text{|F}}_{\text{M}}\text{|}\text{ }\text{=}\text{ }\sum _{\text{i}\text{ }\text{=}\text{ }\text{TH, IN}}\left|{\stackrel{\text{⃑}}{\text{F}}}_{\text{M}}^{\text{i}}\right|\), respectively. The magnitude of a generalized force vector for each digit was calculated as \(\left|{\overrightarrow{F}}_{general}\right|= \sqrt{{f}_{x}^{2}+{f}_{y}^{2}+{f}_{z}^{2}}\). The data clusters of |*F**G*|, |*F**M*| and the minimum *F**G* required to prevent object slip (see below) are plotted for each experimental condition as an ellipse measured at object lift onset and hold (**Fig. 3B**). The ellipses are based on clusters of data from trials 4–20 from all subjects in each digit offset condition. The length of the principal axes of each ellipse was computed using principal component analysis. Half-length of the two principal axes denote the standard deviation along the corresponding axes.

*Effect of digit offset on grasp force*. We note that the larger the digit offset, the closer the direction of *F**G* vector (± 43º relative to *z*O axis for the largest digit offsets, L) to the boundary of object slip (± 57º relative to *z*O axis; the edge of friction cone computed from the translational friction coefficient averaged across subjects; **Fig. 5**) due to the mechanically-obligatory requirement of *F**G* being in the null space of the grasp map G (see *Analysis of grasp and manipulation forces*). *F**G* is responsible for preventing object slip by steering the digit force vectors at individual contacts to remain within the friction cone. Therefore, the closer the *F**G* vector is to the edge of the friction cone, the greater the risk of object slip. At the same time, an asymmetric object CM requires an active control of object orientation during manipulation, i.e., object lift. Therefore, digit forces had to be accurately coordinated to simultaneously address the challenges of increased risk of object slip caused by increasing digit offset and large object tilt caused by the need to counter the external torque caused by the asymmetric CM.

*Estimation of minimally-required grasp forces.* We used the *F**M**−F**G* decomposition to determine how the CNS coordinates these two functionally distinct aspects of dexterous manipulation: slip prevention and object pose control. With regard to slip prevention, it is well known that humans can efficiently attain a small safety margin by exerting small normal force above the minimum required to prevent object slip.9,10,12,26 However, how the CNS deals simultaneously with object pose control and slip prevention remains unknown. To address this gap, we first estimated excessive grasp force (*F**G**XS*) as the difference between the magnitude of *F**G* and the minimal grasp force required to prevent object slip (*F**G**min*). *F**G**min* was estimated by an optimization process based on the manipulation force, the grasp map, and the coefficients of friction.49 The optimization objective was set as minimizing the magnitude of *F**G* with translational and torsional friction cone constraints. The magnitude of *F**G**min* was computed as \(|{\text{F}}_{\text{G}}^{\text{min}}\text{| = }\sum _{\text{i = TH, IN}}\left|{\stackrel{\text{⃑}}{\text{F}}}_{\text{G}}^{\text{min, i}}\right|\).

We defined relative grasp safety margin (*SM**G*) as the ratio between *F**G**XS* and *F**G* to quantify the proportion of excessive force normalized to exerted grasp force. Note that this definition of *SM**G* is different from how relative safety margin (*SM*) has been defined in the grasping literature.27,50 Specifically, the traditional definition is based on decomposing the digit force vector into normal and tangential components. As such, *SM* denotes how far the digit force vector is from the boundary of its friction cone at the digit contact. We note that previous grasping literature focused primarily on manipulation of symmetrical objects at zero digit offsets, where there are no external torque or torques induced by digit forces acting at non-zero digit offsets. Therefore, the magnitude of normal forces above the minimum normal force required to prevent object slip can take any value up to physiological limits with no risk of changing object orientation. In contrast, our *SM**G* definition uses the *F**G* component of the digit force vector obtained from the *F**M**−F**G* decomposition. Therefore, while *SM* and *SM**G* both quantify digit force control efficiency, *SM**G* captures the excessive force above that required to satisfy both object slip prevention and tilt minimization.

*Analysis of manipulation force components’ contribution to T* *COM* . Next, we computed *T**COM* required to counteract the external torque (*T**EXT*) caused by the added mass (L or R CM) as the resultant moment of force about the *x*-axis. *T**COM* consists of the two moment of forces generated by the difference of the tangential and normal manipulation forces between the thumb and index fingertip:

$${\text{T}}_{\text{COM}}\text{ = }\text{–}{\text{CoP}}_{\text{z}}\text{∙}\left({\text{F}}_{\text{M,y}}^{\text{IN}}\text{ }\text{– }{\text{F}}_{\text{M,y}}^{\text{TH}}\right)\text{ }\text{+}\text{ 0.5 }\text{Δ}{\text{CoP}}_{\text{y}}\text{∙}\left({\text{F}}_{\text{M,z}}^{\text{IN}}\text{ }\text{– }{\text{F}}_{\text{M,z}}^{\text{TH}}\right)$$

\(\text{=}{\text{ }\text{–}\text{CoP}}_{\text{z}}\text{∙∆}{\text{F}}_{\text{M,y}}\text{ }\text{+}\text{ 0.5 }\text{Δ}{\text{CoP}}_{\text{y}}\text{∙∆}{\text{F}}_{\text{M,z}}\) [4]

The resultant torque (*T**RES*) was computed as the sum of *T**COM* and *T**EXT*. As the task requirement was to attain a vertical pose from object lift onset to static hold, *T**RES* should be zero to attain the rotational equilibrium about the x-axis.

It can be noticed from Eq. [4] that there is a constant (*CoP**z*, i.e.,half-object width) and three variables (*∆CoP**y*, *∆F**M,y*, and *∆F**M,z*). For a prescribed *∆CoP**y* (digit offset), the CNS has one degree of freedom to produce *T**COM* through arbitrary *∆F**M,y* and *∆F**M,z* sharing patterns, i.e., their magnitudes are indeterminate as long as they are within the *T**COM* manifold (**Fig. 4**). Our previous work13 on unconstrained grasping where subjects used relatively small digit offsets has shown that successful performance of our manipulation task is attained by distributing these *F**M* components near the task manifold (**Fig. 4A;** see also Fig. 8 of Fu et al.13). In the present work, changes in digit offset lead to changes in the slope of the task *T**COM* manifold. This is because the slope of the *T**COM* line increases as a function of digit offset. However, how the two *F**M* components are coordinated as a function of digit offset is indeterminate. Specifically, subjects could satisfy our task requirements through one of two alternative strategies. In the first scenario, one of the *F**M* components, e.g., *∆F**M,z*, could be invariant to changes in digit offset whereas the other component (*∆F**M,y*) would have to compensate to ensure attainment of *T**COM* (**Fig. 4B**). Alternatively, *∆F**M,y* and *∆F**M,z* could covary across digit offsets (**Fig. 4C**).

*Solution space of non-slippery manipulation forces.* Changes in the magnitude of *F**G* effectively modify the volume of the solution space of non-slippery *F**M* .10,23 Thus, as |*F**G*| increases, it enables the exertion of a larger tangential component of *F**M* without slip at individual digit contacts. To quantify these effects, we computed the volume of (a) the *F**M* solution space enabled by the grasp forces (*V**M*) in the 3D force space and (b) the *F**M* solution space enabled by *F**G**XS* (*V**M**XS*). We then computed the ratio, *R**vol*, between *V**M* and *V**M**XS* which represents the relative change of the non-slippery *F**M* solution space. *R**vol* was computed on the small, medium and large digit offset only. The rationale for excluding the U condition was that participants adopted very small and variable digit offsets (see **Fig. 2B**), and therefore were not suitable to be pooled with data obtained from systematic changes in digit offset.