Effects of friction at the digit-object interface on the digit forces in multi-finger prehension

Abstract

The effects of surface friction at the digit-object interface on digit forces were studied when subjects (n=8) statically held an object in a five-digit grasp. The friction conditions were SS (all surfaces are sandpaper), RR (all are rayon), SR (S for the thumb and R for the four fingers), and RS (the reverse of SR). The interaction effects of surface friction and external torque were also examined using five torques (-0.5, -0.25, 0, +0.25, +0.5 Nm). Forces and moments exerted by the digits on a handle were recorded. At zero torque conditions, in the SS and RR (symmetric) tasks the normal forces of the thumb and virtual finger (VF, an imagined finger with the mechanical effect equal to that of the four fingers) were larger for the RR than the SS conditions. In the SR and RS (asymmetric) tasks, the normal forces were between the RR and SS conditions. Tangential forces were smaller at the more slippery side than at the less slippery side. According to the mathematical optimization analysis decreasing the tangential forces at the more slippery sides decreases the cost function values. The difference between the thumb and VF tangential forces, DeltaF (t), generated a moment of the tangential forces (friction-induced moment). At non-zero torque conditions the friction-induced moment and the moment counterbalancing the external torque (equilibrium-necessitated moment) could be in same or in opposite directions. When the two moments were in the same direction, the contribution of the moment of tangential forces to the total moment was large, and the normal forces were relatively low. In contrast, when the two moments were in opposite directions, the contribution of the moment of tangential forces to the total moment markedly decreased, which was compensated by an increase in the moment of normal forces. The apparently complicated results were explained as the result of summation of the friction-related (elemental) and torque-related (synergy) components of the central commands to the individual digits.

Fig. 1

Schematic of the point about which the moments are computed (left panel) and schematic of experimental handle/beam apparatus (right panel). In the left panel, w is the handle width (w = 82 mm); y_i is vertical coordinate of the index finger force application with respect to the sensor center; d_i is a projected vertical distance between the sensor centers of the index finger and the thumb; Y_i is vertical coordinate of the point of the index finger normal force application. Note that the lower case y_i designates the coordinate with respect to a sensor center i while the uppercase Y is used to designate a coordinate in the handle fixed reference system. In the right panel, for L1 and L2 torque conditions, the external torque was positive but the torque produced by the subjects was negative. In contrast, for R1 and R2 torque conditions, the external torque was negative but the torque produced by the subject was positive

Fig. 2

Normal force (a), tangential force (b), and safety margin (c) for the thumb and VF at different friction conditions in the zero torque tasks. Group averages and standard errors

Fig. 3

M^t (moment of tangential forces) (a) and Mⁿ (moment of normal forces) (b) at different friction conditions in the zero torque tasks. Group averages and standard errors

Fig. 4

Normal forces by the individual fingers (% of the VF) at different friction conditions in the zero torque tasks. Group averages

Fig. 5

Normal force for the thumb (a), thumb and VF tangential forces for symmetric (b) and asymmetric (c) tasks, and safety margin for the thumb (d) and VF (e) at different torque tasks. Group averages and standard errors

Fig. 6

M^t (moment of tangential forces) (a) and Mⁿ (moment of normal forces) (b) at different friction conditions in the different torque tasks. Group averages and standard errors

Fig. 7

The difference between the position of the point of application of the VF normal force Y_vf and the position of the point of application of the thumb normal force Y_th in different tasks. The (Y_vf − Y_th) difference equals the moment arm of the force couple formed by the normal VF and thumb forces

Fig. 8

Actual and predicted from optimization differences between the tangential forces of the VF and thumb, ΔF^t, zero torque tasks. Group average data and standard errors. The optimization results are for the following cost functions: CF1—energy-like function over F^r; CF2—energy-like function over Fⁿ; CF3—entropy-like function; CF4—motor command function. The positive values of the ΔF^t signify the positive moment of the tangential forces (i.e. the friction-induced moment in the counterclockwise direction) while the negative values of the ΔF^t represent the moment of tangential forces in the negative (clockwise) direction. The figure illustrates generation of the friction-induced moments. See text for the further discussion

Fig. 9

Actual and predicted from optimization differences between the tangential forces of the VF and thumb, ΔF^t, non-zero torque tasks. a L2 task, b R2 task. Group average data and standard errors. The figure illustrates the joint effects of the friction-induced moment and the equilibrium-necessitated moment on the VF and thumb tangential forces. All the four cost functions yielded the ΔF^t values that corresponded to the moments of the tangential forces in the same direction as the actual data while the moment magnitudes were different. See text for the further discussion

Fig. 10

The friction-induced and equilibrium-necessitated moments and their interaction, a schematic. a The zero torque tasks. In the SS and RR tasks (the central picture), the tangential forces of the thumb and VF are equal and the moment of the tangential forces M^t is zero. In the RS and SR tasks, due to the opposite changes of the tangential forces F^t, the friction-induced moments arise. To prevent the object tilting, this moment must be compensated by an equal and opposite moment of the normal forces. b Non-zero (L2) torque tasks. In the SS and RR tasks, the moment resisting the external torque is generated (the equilibrium-necessitated moment). The moments seen in the RS and SR tasks are due to the blending of the friction-induced and equilibrium-necessitated moments