**Abstract**:

**Abstract**
**When** **we** **grasp an object, our visuomotor system has to solve an intricate problem: how to find the best out of an infinity of possible contact points of the fingers with the object?** **The contact point selection model (CoPS) we present here solves this problem and predicts human grasp point selection in precision grip grasping by combining a few basic rules that have been identified in human and robotic grasping.** **Usually, not all of the rules can be perfectly satisfied. Therefore, we assessed their relative importance by creating simple stimuli that put them into conflict with each other in pairs. Based on these conflict experiments we made model-based grasp point predictions for another experiment with a novel set of complexly shaped objects. The results show that our model predicts the human choice of grasp points very well, and that observers' preferences for their natural grasp angles is as important as physical stability constraints. Incorporating a human grasp point selection model like the one presented here could markedly improve current approaches to cortically guided arm and hand prostheses by making movements more natural while also allowing for a more efficient use of the available information.**

*γ*(the sum of the angular deviances of the grasp axis from both friction cone center axes). The smaller

*γ*, the better force closure is fulfilled. Force closure is optimal if points on the object's surface are chosen that align the grasp axis with the central axes of both friction cones. In this case,

*γ*is zero.

*τ*(the product of object mass and torsion arm length, which is proportional to torque). When the torque rule is perfectly satisfied,

*τ*is zero and the grasp axis passes through the barycenter.

*α*between the NGA and the actually realized grasp angle is zero.

*g*be a grasp (i.e., an ordered pair of contact points of thumb and index finger). Every

*g*is then associated with a certain violation of the force closure rule

*γ*(

*g*), the torque rule

*τ*(

*g*), and of the NGA rule

*α*(

*g*). The preference for certain values of

*α*(

*g*),

*τ*(

*g*), and

*γ*(

*g*) can be modeled with a set of penalty functions f

*(*

_{α}*α*), f

*(*

_{τ}*τ*), and f

*(*

_{γ}*γ*). A simple penalty function, which can cover a wide variety of different shapes while having only very few free parameters, would be of the general form f(x) =

*a*x

*. Here*

^{b}*a*is a parameter responsible for the weighting of the rule and

*b*defines how quickly penalty values increase when rule deviation increases. Generally the function is symmetrical around 0, such that rule deviations are punished equally in both directions. The most simple way to combine the individual rule penalties so that each rule makes an independent contribution to the final penalty value for a particular grasp p(

*g*) is to sum them up: The grasp associated with the lowest penalty value is then chosen for grasping.

*a*and

*b*associated with each grasp rule. They were designed as rule-conflict experiments. Participants were forced to reveal to what extent they prefer to satisfy one rule at the cost of the other. Experiment 3 served as a validation of the model.

*SD*: 3, 4, and 3 years). Informed consent was obtained according to the Declaration of Helsinki. Methods and procedures followed the guidelines of the APA (American Psychological Association). Participants were paid eight euro (approx. $10.42) per hour.

^{3}. In Experiment 1 we used a disc of 2.5 cm radius and nine square blocks of 5 cm edge length. All objects had a height of 1.5 cm. In Experiment 2 we used a disc of 2.5 cm radius and nine ellipsoids with an extent of 10 cm along the major and 5 cm along the minor axis. Embedded in each ellipsoid was a clearly visible lead cylinder of 1.5 cm radius and 0.8 cm height, which was varied between objects along the major axis of the ellipse such that the barycenter moved from −2 to +2 cm in steps of 0.5 cm relative to the ellipse center. The weight of the ellipses was 89 g. All objects of this experiment and Experiment 3 had a height of 1 cm. Stimuli of Experiment 3 were one disc of 2.5 cm radius and nine objects of complex shape. The contours of these objects are pictured in Figure 5b. Their weight ranged between 38 and 56 g.

*γ*= 0,

*τ*= 0,

*α*= 0). Additionally, we used eight blocks rotated away from the participant's NGA, such that participants had to decide whether to follow the rotation with their digits. Following the rotation would ensure good force closure but would increase the deviation from the NGA. Distance to the block's barycenter (corresponding to the value of

*τ*) could always freely be chosen and thus did not influence the values of the other two rules in this experiment.

*γ*= 0,

*τ*= 0,

*α*= 0). For the remaining eight ellipsoids, the barycenter was shifted along the major axis. Participants had to choose whether to follow this shift with their grasp. Doing so would ensure a small distance to the barycenter and thus a small value of

*τ*. Due to the curved ellipse contour, however, it would result in a larger deviance from perfect force closure and thus enlarge

*γ*. Because of the objects' alignment to the NGA the value of

*α*did not influence the values related to the other two rules in this experiment.

_{α}relative to f

_{γ}. We used f

_{γ}as reference function and thus set its weight to 1 and its power to 2, the lowest power, which would be used in a Taylor expansion to approximate a function symmetrically increasing around x = 0. As the value of

*τ*could be chosen independently from

*γ*and

*α*in this experiment, the penalty function for Experiment 1 reads As the value of

*γ*was completely dependent on the chosen value of

*α*and the object's angle of rotation

*r*(Equation 2) can also be expressed as We estimated the coefficients

*a*,

*b*of the penalty function by numerically minimizing the criterion value

*c*of the objective function The value of

*α*depends on the configuration of the digits relative to each other but also on the rotation of the wrist. As the ease of a rotation in the wrist likely depends on rotation direction, we estimated separate penalty functions for the objects rotated clockwise and counterclockwise away from the NGA respectively.

*γ*depended on the chosen distance

*d*of the intersection point of the grasp axis with the major ellipse axis to the ellipse center. The value of

*τ*was dependent on the distance between this intersection point and the barycenter of the ellipse. Thus,

*τ*could also be written as a function of

*d*and the position

*k*of the barycenter on the major ellipse axis. As the influence of

*α*was negligible in this experiment, the penalty function thus could be expressed as Inspecting the data of Experiment 2, we saw that participants' choice of contact points was biased towards shorter movement distances (see Results). Therefore we also included a penalty term for distance (

*λ*). For the average rotation of the ellipse,

*λ*could also be expressed as a function of

*d*thus Equation 6 was extended to From the data of Experiment 2 we estimated for the torque rule the values of coefficients

*a*and

*b*of the penalty function For the distance rule, however, as distance had not been subject to a stepwise conflict with another rule, the observable average undershoot just allowed for the estimation of one coefficient In order to estimate the coefficients we minimized the objective function

*α*= .05 were used to test if differences are normally distributed. As this was not always the case, we used Wilcoxon's signed-rank test for the evaluation of the CoPS model using a quality index (see below).

_{α}) relative to the penalty function for deviance from perfect force closure (f

_{γ}) with f

*(*

_{α}*α*) = 1.77

*α*

^{1.76}for the clockwise and f

*(*

_{α}*α*) = .78

*α*

^{1.9}for the counterclockwise direction of grasp axis rotation away from NGA. We also tested if participants adapted their grasp to the objects over the course of repeated presentation of the same object. We found that the chosen grasp angle did not depend on the number of object presentations,

*F*(2.356, 37.7) = 0.356,

*p*= 0.737, on Greenhouse-Geisser corrected

*df*s.

*) relative to the function for deviance from perfect force closure (f*

_{τ}*).*

_{γ}*λ*) as an additional rule f

*(*

_{λ}*λ*) into the CoPS model. Only one parameter could be estimated for this distance rule, because it had not been subject to systematic variation (see Data analysis). It should also be noted that, depending on the orientation of the ellipsoid, barycenter distance and movement distance could covary. From our data we arrived at an estimate for the penalty functions f

*(*

_{τ}*τ*) = 5.52

^{3}

*τ*

^{1.82}and f

*(*

_{λ}*λ*) = 4.87

*λ*. The complete penalty function for a given grasp according to the CoPS model thus reads for the clockwise direction of

*α*and for the counterclockwise direction of

*α*. Values for

*γ*and

*α*are specified in rad,

*λ*in m, and

*τ*in kg × m. Figure 2 shows a set of example grasps on a rectangular object in order to demonstrate equal penalties arising from different rule deviations.

*F*(4.508, 81.139) = 1.117,

*p*= 0.356, on Greenhouse-Geisser corrected

*df*s.

*q*), which indicates for each individual grasp how close it was to the prediction of the model. For each grasp, it is calculated which percentage of possible grasps would have received higher penalty values by the model. A value of

*q*= 100% corresponds to a perfect prediction (participants always choose the grasp with the lowest penalty value; i.e., no other possible grasp has a lower penalty value). The mean value of the quality index across all objects and participants amounted to 98.02%, the lowest mean value for a single object being 96.97% and the highest being 99.11%.

*q*for every possible combination of object and map. The correct combination of object and map had a significantly higher quality index (

*q*= 98.02%) than the average of the control combinations (

*q*= 92.7%,

*V*= 171,

*p*< 0.001). Furthermore, there was no single control combination performing better than the correct combination of object and map.

*V*= 171,

*p*< 0.001) and 92.63% (

*V*= 171,

*p*< 0.001), respectively. Excluding the torque rule had a relatively small effect on performance (

*q*= 98.00% as compared to

*q =*98.02% in the complete CoPS model). Nevertheless, all participants improved on the majority of objects such that this difference was also significant in the nonparametric rank-based test we used (

*V*= 158,

*p <*0.001). A slightly larger decrease in performance was found when excluding the distance rule (

*q*= 97.93%,

*V*= 167,

*p*< 0.001). Here, only one participant did not perform better on the majority of objects when including the rule. With both rules, there was a tendency that including the rule had a higher impact on performance with the more elongated objects (like e.g., objects eight and nine in Figure 5b), as compared to the rounded objects (e.g., objects one or two). Using a symmetrical NGA rule instead of two separate rules for clockwise and counterclockwise deviations from NGA lead to a midsize drop in prediction quality (

*q*= 95.09,

*V*= 162,

*p*< 0.001).

Measure | Experiment 1 | Experiment 2 | Experiment 3 |

Reaction time, in ms | 352 (76) | 351 (80) | 332 (47) |

Movement time, in ms | 861 (112) | 898 (131) | 895 (164) |

Maximum grip aperture, in mm | 67 (5) | 65 (5) | 78 (9) |

Time at maximum grip aperture, in ms | 632 (88) | 559 (88) | 556 (96) |

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