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Conceived and designed the experiments: VNC PRS. Performed the experiments: VNC. Analyzed the data: VNC PRS. Contributed reagents/materials/analysis tools: VNC PRS. Wrote the paper: VNC PRS.

The authors have declared that no competing interests exist.

Due to noisy motor commands and imprecise and ambiguous sensory information, there is often substantial uncertainty about the relative location between our body and objects in the environment. Little is known about how well people manage and compensate for this uncertainty in purposive movement tasks like grasping. Grasping objects requires reach trajectories to generate object-fingers contacts that permit stable lifting. For objects with position uncertainty, some trajectories are more efficient than others in terms of the probability of producing stable grasps. We hypothesize that people attempt to generate efficient grasp trajectories that produce stable grasps at first contact without requiring post-contact adjustments. We tested this hypothesis by comparing human uncertainty compensation in grasping objects against optimal predictions. Participants grasped and lifted a cylindrical object with position uncertainty, introduced by moving the cylinder with a robotic arm over a sequence of 5 positions sampled from a strongly oriented 2D Gaussian distribution. Preceding each reach, vision of the object was removed for the remainder of the trial and the cylinder was moved one additional time. In accord with optimal predictions, we found that people compensate by aligning the approach direction with covariance angle to maintain grasp efficiency. This compensation results in higher probability to achieve stable grasps at first contact than non-compensation strategies in grasping objects with directional position uncertainty, and the results provide the first demonstration that humans compensate for uncertainty in a complex purposive task.

Optimal sensorimotor control models actions as decisions that maximize the desirableness of outcomes, where the desirableness is captured by an expected cost or utility to each action sequence. These models provide explanations for many aspects of our ability to compensate for uncertainty, but they have not been applied to understanding purposive movements—movements involving the application of forces to change the relative position of objects and the actor in the environment. Using time efficiency as a natural cost function, we present a statistical optimal control analysis of uncertainty compensation strategies in a purposive movement task, grasping an object with directional position uncertainty. In accord with the predictions of the analysis, the experimental results showed that people compensate for uncertainty by adopting grasp strategies that increase the chance to produce a stable grasp at first contact. Our findings suggest that visuomotor system plans for uncertainty even in complex purposive movements.

Optimal sensorimotor control models actions as decisions that maximize the desirableness of outcomes, where the desirableness is captured by an expected cost or utility to each action sequence. These models provide explanations for many aspects of our ability to compensate for uncertainty

In grasp planning, fingers must be targeted toward points on the object's surface that will allow the application of forces sufficient for lifting and dexterously manipulating the object. In particular, the finger-object contacts should permit forces that are capable of stably lifting the objects and counterbalance external forces and torques exerted on the object – termed

Once people place their fingers on contact points supporting force-closure, they can begin to lift the object. Hence, the duration of a grasping task depends on the time to produce force-closure grasping, and this time is minimized by movement trajectories that produce force-closure at first contact. Failure to satisfy force-closure conditions at first contact requires subsequent adjustments to rearrange the contact points – a process that requires extra time and effort. Little is known about how people recognize contact points supporting force-closure or how this process is affected by uncertainty.

The purpose of our work is to study uncertainty compensation in grasping and compare human performance against normative predictions. An illustration of precision grasping objects with position uncertainty is presented in

(A) The critical aspect of grasping an object with position uncertainty is the control of the contact surfaces of the index finger and thumb (rectangular patches). These surfaces must be moved along paths that will make appropriate contact with the object at any of its possible locations (gray transparent cylinders). Appropriate contact involves the concept of force-closure (see _{i}_{i}^{2}) and fingers orientation (variance = 2.5 deg^{2}).

We test what, if any, changes in grasping strategy occur as compensations for object location uncertainty. Using a robotic arm to generate oriented distributions of cylindrical object locations, we investigate whether people adopt grasping strategies that minimize the impact of uncertainty on grasp success in terms of force-closure grasping at first contact.

A schematic representation of the apparatus is shown in

Trajectory characteristics are illustrated for a single participant on the no-motion (left), fixed-end location (middle) and random-end location (right) conditions in

We investigated whether participants modified their index finger/thumb approach direction with the orientation of the object position distribution. We predicted a change in approach direction from an analysis of the conditions for force-closure at first contact. Force-closure occurs when the fingers make contact on the object at locations that permit the application of forces in the direction of both the surface normals and surface tangents that can potentially cage the object. A necessary condition for force-closure is that the required tangential forces are less than the force applied to the surface normals, scaled by the coefficient of friction. Geometrically, this relation produces friction cones at the contact points of the thumb and the index finger with cylinder, whose boundaries are determined by the surface tangent and normal vectors of both fingers (_{m}_{i}, force-closure is only possible if the cylinder surface is in the approach path for both digits (w_{f} is less than the finger surface width) and the angle between the index finger and thumb is sufficiently large.

On the basis of this analysis, the choice of a grasp strategy can be turned into a statistical decision problem, where the objective is to select an approach that maximizes the probability of force-closure at first contact (see ^{2}) and the finger surface orientations (variance = 2.5 deg^{2}). The results are shown for two orientations of object location uncertainty, 0 (gray) and 45 deg (black). The principal effect of additional variability is to narrow the range of approaches that produce high force-closure probabilities.

Because the data suggest that locus of control is the index finger, we focused our analysis of approach direction on the index finger, shown in _{[0.05;1,3]}>12.2905 and P<0.0393), and scales almost linearly for all participants. However, all participants' slopes were less than predicted by ideal compensation (shown by the black discontinuous line). In contrast, for the fixed-end location condition, where the final position was fixed across trials, the approach direction was near constant (

To quantify how much the change in approach direction observed in the random-end location condition affected grasp efficiency, we computed the probability of force-closure for each participant and covariance angle. We estimated the force-closure probability for each trajectory by determining the proportion of sampled object locations that would satisfy the force-closure conditions at first contact, and averaged across trajectories (see

The probability of force-closure provides a measure of the benefits of modifying approach direction with uncertainty.

We can also gain insight into the benefits of compensation by focusing on the set of trajectories with low performance. We identified the set of trajectories that produced force-closure for less than 20% of sample locations (values between 20% and 50% produced similar results). Because these trajectories will require post-contact adjustments in fingers positions before lift occurs for the majority of cases, we call this measure the _{[0.05; 1,3]} = 40.4193 and P = 0.0079). In contrast, the inefficiency curves in the random-end location condition shows that by modifying approach direction, participants were able to maintain a low inefficiency rate for all covariance angles (R-square = 0.0015, Slope = 3.4900e-005, F_{[0.05; 1,3]} = 0.0044 and P = 0.9513). Moreover, a test of the two regression results shows these trends are different (ANCOVA, F_{[0.05; 1,6]} = 11.95 and P = 0.0135).

We also tested whether people modify their grip aperture profile as a function of condition and covariance orientation, by computing the mean value of the maximum grip aperture (MGA) across trials and regressed the results against covariance angles. For all conditions, we found no significant variation of MGA magnitude or time with covariance angle. However, there were significant differences across conditions for all participants (one-way ANOVA, F_{[0.05; 1, 598]}>106.4595 and P-value<0.001), excluding Participant 5, shown in

MGA averaged across trajectories and covariance angle for each condition and participants.

We have adopted principles from Statistical Decision Theory

Note that modifying the approach direction requires extra energy to rotate the hand, which means that the advantages of compensation in terms of efficiency must outweigh these costs. Non-compensation strategies increase the chance of not producing force-closure at first contact, which must be corrected by time-consuming and metabolically costly finger repositioning after contact. Finger repositioning occurs in a sensory feedback loop that takes time. Contact locations and forces must be sensed and appropriate adjustments computed and executed, with a minimum time lag around 200 ms

The probability of achieving force-closure grasping is not the only criterion used to plan grasping movements, and our results showed that participants did not fully optimize this measure. Grasp planning may attempt to optimize other cost criteria in addition to force-closure grasping at first contact. Possible biomechanical cost functions, such as energy expenditure

It is important to note that our analysis of optimal grasping behavior with position uncertainty will not hold in all contexts. In the optimal analysis we permitted differences between the thumb and index finger contact times. This is appropriate for our experiment because the cylindrical object was held in a cradle. In general, time differences will not affect sufficiently heavy objects. For objects light enough to be toppled by contacting with one of the fingers, there will be an advantage to contacting the object with both fingers simultaneously. For this class of objects, there is a trade-off between minimizing the chance of knocking over the object and maximizing the chance of contacting the object. For instance, catching a frisbee by opening the fingers wider increases the chance to contact the object, but decreases the chance to catch it by contacting with both fingers simultaneously. We can extend the current analysis using similar Statistical Decision Theory principles and adding a new cost to penalize non-simultaneous contact of both fingers with the object.

An interesting question is what cues drive the compensation strategies we observed. Like previous studies, that showed integration of visual and proprioceptive information in motor tasks

We also examined whether there was evidence of learning by dividing trajectories into “early” and “late” groups and compared their characteristics. We did not find any significant differences in trajectory characteristics between the two groups, for any attempted split. The absence of significant learning effects is likely due to trajectory variability and the number of trials (100). However, it may also indicate that uncertainty compensation strategies are relatively constant.

In conclusion, the results show that people plan for the effects of uncertainty in selecting object contacts in purposive movement tasks.

Five right-handed (25–30 years old, 4 men and 1 woman) participants with normal or corrected-to-normal vision participated in the study for monetary compensation. The appropriate institutional review board approved the study protocol and informed consent was obtained based on the Declaration of Helsinki.

Participants were instructed to reach rapidly with and then use a precision grasp to lift a cylindrical object (2.2 cm diameter and 11.5 cm height) held in a cradle on a platform mounted on a robotic arm that precisely moved the object (<1 mm error),

Participants selected a comfortable reference position by placing both fingers along the top edges of the reference block at the beginning of the experiment. At the start of each trial, participants viewed the cylinder for a period of time that depended on condition, and then vision was removed by shutting down the crystal liquid glasses. Thereafter, they were cued to rapidly reach, precision grasp and lift the object within 1200 ms while it was out of view. The trial was considered successful if the participant lifted the cylinder to trigger the switch within the timeout, however, none of the participants failed to grasp the cylinder within 1200 ms. The fingers had to be returned within 3 mm of their starting positions before the next trial was begun.

Participants were familiarized with the task by running a number of training trials on the non-motion condition. Once they were ready and felt comfortable with grasping the cylinder, the real trials began. On the no-motion condition the cylinder was stationary and the view time was 1 sec before glasses occluded the vision of the object. In the fixed-end location condition, the robotic arm moved the cylinder over a sequence of 5 random positions (1 sec each) sampled from a strongly oriented 2D Gaussian distribution. The cylinder was returned to the based (initial) position and the view was occluded after 1 sec. Thereafter, participants reached and grasped the cylinder, while it was out of view. Note that participants were told that the cylinder always returned to the base position. In the random-end location condition, the cylinder moved over a sequence of 5 positions (1 sec each) randomly generated by a strongly oriented 2D Gaussian distribution. Thereafter, the view of the object was occluded and the object moved to a new random position selected from the same Gaussian distribution. Finally, participants reached and grasped the cylinder, while it was out of view. In this condition, participants were told that the cylinder moved to a new position from within the same distribution of the 5 visible positions. For both fixed-end location and random-end location conditions, we used an 80 deg range of distribution orientations (−20°, 0°, 20°, 40°, 60°), designed to fit within, but strongly challenge participants' natural biomechanical reaching posture. The standard deviations of the major and minor axes of the covariance were 12 mm and 0.25 mm, respectively. Note that the covariance angle was defined with respect to the x-dimension of the coordinate frame. Trials from each covariance angle were blocked, and 100 trials of reaching, grasping and lifting the cylindrical object were performed for each block.

Kernel density estimation was used to analyze the spatial distribution, velocities and orientations of the thumb and index finger trajectories as illustrated

We computed approach direction for the average trajectory for each participant on the two conditions (fixed and random-end location). Because averages are strongly affected by outliers, we excluded trajectories that had substantially different temporal characteristics. Note that trajectory data were spatially variable, but timing was consistent for trajectories with similar velocity profile. From the histogram of the time of maximal x-velocities, we selected the trajectories that fell within 80% the histogram mean (i.e., 10%–90% percentiles of the distribution) and averaged them. From the average trajectory, we computed the approach direction of the main axis of the ellipse that describes the spatial variation of the fingers centroids,

In addition, we measured MGA because it serves as a measure of position uncertainty

To evaluate object-finger contact, we computed an estimate of the index finger and thumb contact surfaces relative to the sensor locations via a calibration procedure. The index finger and thumb were placed in grooves on a calibration block and the sensor locations were recorded. The orientation and position of the calibration block were also recorded by placing sensors in known locations on the calibration block. The groove angle, length and depth were precisely known due to the geometry of the block and the block required precision grip contacts similar to those made on the cylindrical object. We converted this information to approximate the index finger and thumb contact surfaces and computed a homogeneous transformation that converted between sensor and finger coordinates. In particular, finger coordinates had an origin at the center of the estimated region of potential contact, and had directions aligned so that one axis (the surface normal) was perpendicular to the calibration block surface (and hence index finger and thumb normals were parallel but opposite in direction), one axis pointed in the direction of the groove (along the length of the index finger and thumb), and the other axis roughly corresponded to finger width. Once projected into the x-y plane the surface approximation became a line segment that has a 1–2 mm error from the actual index finger/thumb contact surface.

According to an optimal statistical modeling approach, the goal is to select a visuomotor grasping strategy that chooses a desired movement trajectory by optimizing the expected gain. The gain function takes into account the costs and benefits of possible outcomes of the movement

The optimal movement policy

To keep the analysis tractable, we made the following simplifications. Due to the shapes of both cylinder and fingers, and the post-contact lift direction of the cylinder we can safely neglect the spatial dimension along the cylinder's z-axis and focus on the perpendicular plane. Within the plane, the contact surfaces of the index finger and thumb can be approximated as line segments (see lower inset of _{f}, directions _{f}, _{th}, _{f}, _{th}}, where time dependence is suppressed inside the brackets to make the notation less complex.

Possible locations for the cylinder (with radius _{i} sampled from a 2D Gaussian density _{i} into a frame of reference defined by the index finger's (or thumb's) contact surface, forming

Contact coordinates for cylinder with respect to the thumb are similarly defined. The analysis of optimal grasping is presented in lower inset of

The force-closure indicator function

The index finger and thumb are touching the cylinder on the appropriate sides:

The contact point is within the width of the index finger or thumb:

Nguyen

Due to the cylinder geometry, the 4 conditions above are equivalent to the simplified condition, Eq. (3) that the angle between the surface normals is greater than 90 deg, for a coefficient of fiction

Hence, the indicator function

To compare human performance to optimal, we estimated the probability of force-closure from trajectory data. For each participant and condition, we treated the set of trajectories as samples from

We estimated the expected probability of force-closure

We tested the optimality of participants' grasp strategies and to provide a comparison, we also estimated the deterioration in grasping performance if participants do not compensate for uncertainty. To provide a baseline comparison for this grasp performance measure, we estimated the probability of force-closure if participants had adopted a non-compensation strategy for the random-end location condition (and hence would not have compensated for the cylinder location uncertainty). We simulated the non-compensation strategy by estimating the expected probability of force-closure for no-motion condition trajectories on the random-end location condition cylinder locations.

Supplementary Materials

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