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, ΔF ^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.

Appendix

Optimization procedure

We used the methods of optimization and cost functions employed previously in our laboratory for studying finger forces in multi-finger tasks (Pataky et al. 2004b; Zatsiorsky et al. 2002a). The optimization problem was formulated in terms of design variables F as follows:

$$ \begin{aligned}{} & {\text{minimize \{ }}f{\text{(}}{\mathbf{F}}{\text{): }}{\mathbf{F}} \in \Omega {\text{\} }} \\ & {\text{where }}\Omega {\text{ = \{ }}{\mathbf{F}}{\text{: }}g_{j} {\text{(F)}} \le 0{\text{, }}h_{k} {\text{(}}{\mathbf{F}}{\text{) = 0\} ; }}j{\text{ = 1,}} \ldots {\text{,10; }}k{\text{ = 3}}{\text{.}} \\ \end{aligned} $$

The function f is a cost function, the g _j ’s and h _k ’s are the inequality and equality constraints, respectively, and F is the vector of normal and tangential digit forces. The inequality constraints g _j are the no-slip constraint (five) and the constraints on non-negativity of normal digit forces (five); and the equality constraints h _k are the equilibrium constraints (totally three: for the horizontal forces, vertical forces, and the moment in the frontal plane). The set Ω is the feasible solution region defined by the constraints. Four convex cost functions f(F) were selected. The friction values for each subject were determined experimentally and the optimization was performed for each subject, friction condition, and task separately. The obtained data were averaged. Only the individual finger forces (n=4) were subjected to optimization. The thumb and VF forces were not included explicitly in the cost functions.

(CF1) Energy-like function over F^r

The measure is proportional to the potential energy of deformation for a linear (i.e. Hookian) system

$$ f_{1} = {\sum\limits_{i = 1}^4 {{\left( {F_{i}^{r} } \right)}^{2} } }, $$

where (F ^r _i ) is the sum of the vector magnitudes: |F ⁿ _i | + |F ^t _i |.

(CF2) Energy-like function over Fⁿ

The energy-like functional can be expressed in terms of only normal digit forces F ⁿ as:

$$ {\mathop f\nolimits_2 } = {\sum\limits_{i = 1}^4 {{\left( {{\mathop F\nolimits_i^{\text{n}} }} \right)}} ^{2}} . $$

(CF3) Entropy-like function

Since it is conceivable that the CNS prefers to spread efforts as evenly as possible among redundant effectors, an entropy-motivated measure:

$$ {\mathop f\nolimits_3 } = {\sum\limits_{i = 1}^4 {{\left[ {{\mathop F\nolimits_i^{\text{n}} } + 1} \right]}} }\log {\left[ {{\mathop F\nolimits_i^{\text{n}} } + 1} \right]} - {\sum\limits_{i = 1}^4 {{\mathop F\nolimits_i^{\text{n}} }} } $$

was employed (Hershkovitz et al. 1995, 1997).

(CF4) Motor command function

The procedure for computing motor commands (c) from observed F ⁿ has been described elsewhere (Zatsiorsky et al. 2002a). The N×1 vector c represents the intended activation level of each finger (each c _i ∈[0, 1]) and the manifested F ⁿ is the result of anatomical and physiological interconnections among fingers. Thus, based on the neural network model of Li et al. (2002) and Zatsiorsky et al. (1998), an interconnection matrix [M] was constructed that linearly mapped c to F ⁿ:

$$ F^{{\text{n}}} {\text{ = [}}{\mathbf{M}}{\text{]}}c. $$

The c can be computed based on the observed F ⁿ by inverting the square [M] matrix. Thus, the cost function:

$$ f_{4} = {\sum\limits_{i = 1}^4 {c^{2}_{i} } } $$

represents a minimization of finger activation.

All optimization problems were solved using the procedure: fmincon from MATLAB’s optimization toolbox (The MathWorks Inc., Natick, MA, USA). The optimum (F*) results were unaffected by random variations in initial (F ⁰) values submitted to the routines.