Reproducibility and Variability of the Cost Functions Reconstructed from Experimental Recordings in Multi-Finger Prehension

Abstract

The main goal of the study is to examine whether the cost (objective) functions reconstructed from experimental recordings in multi-finger prehension tasks are reproducible over time, i.e., whether the functions reflect stable preferences of the subjects and can be considered personal characteristics of motor coordination. Young, healthy participants grasped an instrumented handle with varied values of external torque, load and target grasping force and repeated the trials on three days: Day 1, Day 2, and Day 7. By following Analytical Inverse Optimization (ANIO) computation procedures, the cost functions for individual subjects were reconstructed from the experimental recordings (individual finger forces) for each day. The cost functions represented second-order polynomials of finger forces with non-zero linear terms. To check whether the obtained cost functions were reproducible over time a cross-validation was performed: a cost function obtained on Day i was applied to experimental data observed on Day j (i≠j). In spite of the observed day-to-day variability of the performance and the cost functions, the ANIO reconstructed cost functions were found to be reproducible over time: application of a cost function C_i to the data of Day j (i≠j) resulted in smaller deviations from the experimental observations than using other commonly used cost functions. Other findings are: (a) The 2^nd order coefficients K_i of the cost function showed negative linear relations with finger force magnitudes. This fact may be interpreted as encouraging involvement of stronger fingers in tasks requiring higher total force magnitude production. (b) The finger forces were distributed on a 2-dimensional plane in the 4-dimensional finger force space, which has been confirmed for all subjects and all testing sessions. (c) The discovered principal components in the principal component analysis of the finger forces agreed well with the principle of superposition, i.e. the complex action of object prehension can be decoupled into two independently controlled sub-actions: the control of rotational equilibrium and control of slipping prevention.

Majority of motor tasks can be performed in different ways; this fact is well known as the problem of motor redundancy (Bernstein 1967). In reality, the tasks are performed more or less similarly both across individuals and across repetitive trials. A widely held assumption—originally introduced in a seminal study by Nubar and Contini (1961)—is that the central controller follows the principle of optimality: It implements a solution that optimizes a certain unknown cost function (reviewed in Seif-Naraghi and Winters 1990; Collins 1995; Tsirakos et al. 1997; Prilutsky 2000; Rosenbaum et al. 2001; Raikova and Prilutsky 2001; Prilutsky and Zatsiorsky 2002b; Ait-Haddou et al. 2004; Todorov 2004; Erdemir et al. 2007). So far, cost functions have been chosen by researchers rather arbitrarily based on intuition and theoretical views. These guesses are then tested using experimental data. For the multi-finger prehension, several candidate cost functions have been suggested and tested (Hershkovitz et al. 1995, 1997; Zatsiorsky et al. 2002b; Pataky et al. 2004b; Aoki et al. 2006; Niu et al. 2009).

Recently, a novel Analytical Inverse Optimization (ANIO) method has been proposed to compute (‘reconstruct’) an unknown cost function from the experimental data. The method puts relatively soft restrictions on the constraints and specific type of the cost functions: (a) the task constraints are linear, and (b) the cost functions are additive and differentiable. The mathematical proofs are provided in Terekhov et al. (2010) and Terekhov and Zatsiorsky (2011). The method has been successfully applied to the multi-finger pressing (Park et al. 2010; 2011a,b) and prehension tasks (Niu et al. 2011).

The optimization approach predicts a single, optimal solution for each task. However, it is common knowledge that human motor performance is variable. It is not clear how the optimization and variability agree with one another. Recently, Park et al. (2010, 2011a) investigated relations between optimality and variability in redundant motor tasks. Using as a model the four-finger pressing the authors employed the ANIO method for reconstructing the cost functions and the uncontrolled manifold (UCM) theory (Scholz and Schöner 1999; Latash et al. 2007) for studying the motor variability. They applied the ANIO to the data recorded over a broad range of tasks with different magnitudes of the total target force and moment of force and the UCM method to the multiple trials at the same combinations of total force and moment. In the both cases the data were distributed on 2-D planes in the 4-dimensional space of finger forces. The UCM plane was however nearly orthogonal to the ‘optimal plane’. The authors concluded that the ANIO and UCM approaches complement each other: ANIO describes optimal locations of the centers of data distributions across repetitive trials, while the UCM approach analyzes the shapes of such distributions.

One of the research options that could potentially combine the optimization and variability approaches is analysis of the variability and reproducibility of the cost functions themselves. Developing the objective ANIO method of reconstructing cost functions from experimental observations, rather than simply assuming their existence, allows addressing this issue. Up to now, the stability/ reproducibility of the cost functions has never been an object of experimental research. In the present study we are interested in whether the reconstructed cost functions represent stable preferences by the subjects and can be considered (over a certain time period) personal characteristics of motor coordination.

We distinguish the reproducibility of the functional form of the cost functions from the reproducibility (variability) of its coefficients. To test whether the cost functions computed for individual subjects are reproducible (i.e. provide similar optimization results over repetitive tests performed in different days), we tested the subjects three times, — on Day 1, Day 2 and Day 7 — and compared the performance of the cost functions obtained in different days.

The reproducibility of a function cannot be equaled to the reproducibility (variability) of its coefficients, however. The cost functions determined in this study include eight coefficients: four second-order coefficients K_i and four first-order coefficients W_i (see equation 3 below). It is possible to imagine a situation where the individual coefficients change substantially but the cost function value does not change much, e.g. K_i and W_i for a given finger vary in opposite directions such that effects of these variations on the function value nearly negate each other.

Therefore, in addition to the reproducibility of a function (in the meaning of this term described above) we also examined whether its coefficients: (a) changed with time, and/or (b) were systematically related to finger force magnitudes. Larger magnitudes of the coefficients for a given finger indicate that using the finger is discouraged: they increase the value of the cost function, while optimal solutions correspond to its minimum. Therefore, we investigated the relations between the coefficients and finger force magnitudes, as well as the finger force sharing percentages. We expected that such an analysis would be useful for understanding the mechanical and/or physiological meaning of the cost functions computed by ANIO.

Finally, the last part of the research addresses the distribution pattern of the finger forces. The ANIO analysis includes as its step the testing of whether the distribution is planar or not. Besides its importance for the ANIO procedures, the planarity of the distribution, if observed, allows for some non-trivial considerations on the prehension control, since a low-dimensional distribution of finger forces in the 4-D space cannot be explained by mechanical constraints, which vary across different tasks (explained in the Discussion section). Hence while being a part of the ANIO computations, the planarity testing could also be regarded as a research goal on its own.

1. Experiment

In the experiment, participants held a vertically-oriented handle at rest in the air without any translation or rotation. The subjects used a prismatic grasp in which the fingers and the thumb contacted the object in the same plane (see grasp plane, Figure 1).

Figure 1.

An instrumented handle and recorded forces in the experiment. (A) Schematic drawing of the apparatus with five sensors and an air bubble level mounted on a handle with a T-shaped attachment. (B) Local coordinates on each transducer. (C) Total normal force generated in the tasks (examples: external torque 0.2 Nm with load of 0.25 Kg).

1.1. Subjects

Eight young healthy male subjects participated in the experiment (age 30.2 ± 4.1 years, weight 73.7 ± 11.0 kg, height 179.3 ± 10.3 cm, hand length from the middle finger tip to the distal crease of the wrist with hand extended 19.4 ± 1.4 cm, hand width at the MCP level with hand extended 9.3 ± 0.8 cm). They were all right-hand dominant according to their daily use of hand, such as eating, writing, and computer mouse usage. None of the subjects had a history of neuropathy or trauma to their upper limbs. None of the subjects was involved in special training, like playing a musical instrument, or professional typing. All subjects gave informed consent according to the policies of the Office for Research Protections of The Pennsylvania State University.

1.2. Apparatus

Five six-component force/moment transducers (Nano-17, ATI Industrial Automation, Garner, NC, USA) were fixed at two sides of the aluminum handle. Four transducers were used to measure forces and moments of force applied by the fingers, and the fifth transducer measured the force and moment of force produced by the thumb. The surface of each sensor was covered with sandpaper. The coefficient of friction between the fingertip skin and the sandpaper, measured in a previous study (Aoki et al. 2006), was 1.34 ± 0.05. Sensor signals were set to zero prior to each trial.

A 70-cm long load bar was attached at the bottom of the handle (Figure 1). Four loads, 0.25 kg, 0.5 kg, 0.75 kg and 1.0 kg could be attached at different points along the load bar. Suspending the loads at different locations generated five external torques: 0.2 Nm and 0.4 Nm clockwise and counterclockwise, as well as a zero torque. There were a total of 20 different load/torque conditions in the experiment. Such a design with broad distribution of the force and moment values was necessary for the optimization analysis (explained in Terekhov et al. 2010 and Terekhov, Zatsiorsky 2011).

At the top of the handle, an air bubble level was fixed to help the subjects keep the vertical orientation of the handle. The level (diameter: 32 mm) included a central circle (diameter: 15 mm) and an air bubble (diameter: 5 mm) in the enclosed liquid. When the bubble was within the central circle, the divergence of the moment of force exerted by the subjects from the external torque was less than 0.04 Nm across all conditions. An experimental trial was accepted by the researcher if the bubble did not move outside the central circle during the whole trial; otherwise the trial was rejected and repeated. The percentage of failed trails was 4.9%±2.4% (mean± standard deviation) over all subjects.

The distance between the centers of each of the two adjacent sensors was 3.0 cm, and the sensor center for the thumb was positioned at the midpoint between the centers of the middle and ring finger sensors. The combined mass of the handle, sensors, and the load bar was 1.01 kg.

1.3. Experimental Procedure

The experiment was designed to last for seven days for every participant, who performed the same experimental procedures on the first, second and seventh days. During the seven days, they were instructed to keep their upper limbs in healthy condition. For example, they could not lift heavy weights, type or play video games more than they would do on a typical day.

On each day, the subjects were instructed to perform five trials with each torque/load combination. Before each experimental session, the subjects washed their hands to normalize skin conditions. In the first trial (baseline session), the subjects applied natural grasping force while holding the handle in the air. In the next four trials, a target line was shown on the monitor with a different grasping force magnitude, which was determined by increasing the recorded natural grip force by 0% (i.e., essentially repeating the first trial but with the grasping force prescribed), 25%, 50% and 75% separately in each trial. The subjects were asked to match the target line while keeping the handle in a vertical position without any angular or linear movement.

The above mentioned force prescription was necessary for the ANIO method: if in different trials performed under the same load and torque the grasping force were different, the individual finger forces would be subjected to two sources of variability: (a) variability of the force sharing pattern and (b) variability of the total grasping force. The problem then would become splittable (i.e. there will be not one but two optimization problems combined together; the concept of splittability is explained in Terekhov et al. 2010). The same reason led us to limit the analysis to the normal forces only as the tangential forces are splittable from the normal forces for the optimization problem.

Each subject sat in a chair with the right upper arm positioned at approximately 45° abduction in the frontal plane and 30° flexion in the sagittal plane. The elbow joint was flexed approximately 90°. The forearm was pronated 90° such that the hand was in a natural grasping position. A computer monitor located in front of the subject showed the sum of the normal force of the four fingers exerted on the handle. The subjects were coached to keep the handle in a vertical position by looking quickly at the bubble level located at the top of the handle while mainly watching the monitor.

The subjects were told by the investigator to keep the handle vertical and as stationary as possible. In the first trial (baseline session), in which the natural grasping force —determined by the subject himself—was applied when the bubble stayed steadily at the center of the level, the digit force and moment of force were recorded for 10 s. There was no target value for the gripping force in this trial. If the edge of the air bubble came out from the central circle of the level, the trial was stopped and restarted after a 30-s break. The natural grip force for each torque/load combination was determined by averaging 1.5-s of the data with the smallest standard deviations, which were determined by moving a time window of 1.5-s for every 0.01 s.

In the other four trials, the subjects were asked to match the target line by increasing or decreasing the gripping force as necessary while keeping the air bubble at the level center. The targets were set with respect to the average gripping force observed in the first trial. To find the best matching performance with the target line, the 1.5-s time window with the smallest root mean square deviations from the target gripping force was selected for the analysis.

After the data collection in a trial stopped, the subjects placed the handle back on the rack and took a 30-second break. After the subject completed all five trials with the same torque/load combination, the investigator would change the load and/or the location of the load along the bar, and inform the subject that he could start the next trial. Sensor signals were set to zero prior to each trial.

The order of external torques and loads, and the grasp force percentages within each torque/load condition were randomized. The sequence of the four trials, with specified grasping force increases after each baseline session, was randomized. Each trial took 10 seconds. During the experiment, each subject performed 100 trials (4 loads × 5 torques × 5 grasping force magnitudes, including 20 trials with natural grasping force and 80 trials with the prescribed force magnitude). The total duration of each experiment was approximately two hours.

2. Data analysis and ANIO modeling

2.1 Data analysis

Even though subjects were instructed to place their digits at the sensor centers, it was impossible for them to keep their thumbs and fingers at the exact center of the transducers throughout all the trials. Therefore, in producing the moment of normal force (Mⁿ), the actual moment arms of the index, middle, ring and little fingers could differ from the nominal moment arms with respect to the thumb center (i.e., the distances between the centers of the digit force sensors). The actual moment arms were computed using the method described in Niu et al. (2009) where the center of the thumb is treated as the pivot point to calculate Mⁿ. In the subsequent analysis, the actual values of the finger force moment arms were used. Besides the forces and moments of individual fingers, the virtual finger (VF) forces and moments were also computed. The VF is an imagined finger that produces the same force and moment of force as all the fingers together (Arbib et al. 1985; Iberall, 1987; Santello, Soechting 1997; Baud-Bovy, Soechting 2001). The VF moment arm was defined as $Y_{\mathit{\text{VF}}}={\displaystyle \sum F_i^n{Y}_i}/F_{\mathit{\text{VF}}}^n,\text{where }{F}_i^n$ is the individual finger’s normal force (i=in, mi, ri, and li representing the index, middle, ring and little fingers respectively); the superscript n indicates normal force for the specific finger or VF finger, Y_i is the moment arm of the individual finger.

Software written in LabVIEW (National Instruments, NC, USA) was used to convert digital signals into the force and moment of force values. Data processing was performed using the Matlab software package (Mathworks, Inc., Natick, MA, USA). The raw force/moment data were filtered with a third-order, zero-lag Butterworth low-pass filter at 10 Hz.

2.2. Inverse optimization

The analytical inverse optimization (ANIO) method was used to compute the unknown cost functions based on the data distributions (Terekhov et al. 2010; Terekhov, Zatsiorsky 2011). The application of the ANIO method in multi-finger prehension and pressing tasks was discussed earlier in Terekhov et al. (2010), Park et al. (2010, 2011a, b) and Niu et al. (2011).

The core of the ANIO method is the Uniqueness Theorem while the computational procedures used in the present study are based on testing whether the experimental data are spread over a planar surface in the 4-dimensional finger force space. To reduce the size of the current paper, only the main summary on the Uniqueness Theorem (Terekhov et al. 2010) and the requirements for the planarity of multi-finger prehension data (Niu et al. 2011) is provided here, followed by the main procedures of the ANIO method for the current study. The readers who are interested in the proofs of the provided statements should consult the above mentioned papers.

1. Uniqueness Theorem (Terekhov et al. 2010) provides sufficient conditions for uniqueness, up to linear terms, of solutions of the inverse optimization problem: if the experimentally observed data points form a k-dimensional hypersurface, where k is the number of constraints in the problem, the solution of the inverse problem is unique up to linear terms.

2. Planarity of the distribution of the experimental data —if observed—drastically simplifies the ANIO analysis. According to the Lagrange principle for the inverse optimization problem (also proven in Terekhov et al. 2010), at the points of optimality the cost functions and the constraint equations are tangent to each other and hence have similar derivatives. In short, if for a set of tasks with linear constraints and varied constrain parameters a planar data distribution is observed the cost function derivatives have to be linear and hence the cost function itself is quadratic (with possible linear terms).

2.2.1 Computation procedure

The inverse optimization problem for multi-finger prehension was defined as:

$$\begin{array}{c}\text{Min }J={\displaystyle \sum _{i=1}^4}{g}_i(F_i^n)\hfill \\ \text{Subject to }\{\begin{array}{c}\hfill F_{\mathit{\text{in}}}^n+F_{\mathit{\text{mi}}}^n+F_{\mathit{\text{ri}}}^n+F_{\mathit{\text{li}}}^n=F_{\mathit{\text{total}}}^n\hfill \\ F_{\mathit{\text{in}}}^n{Y}_{\mathit{\text{in}}}+F_{\mathit{\text{mi}}}^n{Y}_{\mathit{\text{mi}}}+F_{\mathit{\text{ri}}}^n{Y}_{\mathit{\text{ri}}}+F_{\mathit{\text{li}}}^n{Y}_{\mathit{\text{li}}}=M^n\hfill \end{array}\hfill \end{array}$$ (1)

where g_i is an unknown scalar differentiable function with g'(·)>0 in the feasible region, $F={\left[F_{\mathit{\text{in}}}^n,F_{\mathit{\text{mi}}}^n,F_{\mathit{\text{ri}}}^n,F_{\mathit{\text{li}}}^n\right]}^T$ is a 4×1 non-negative vector of finger forces (the subscripts in, mi, ri, and li refer to the index, middle, ring, and little fingers, respectively), superscript n indicates that normal forces are considered, $F_{\mathit{\text{total}}}^n$ is the sum of four finger normal forces, Y designates the moment arms of the normal finger forces, M is a moment of force. The constraints can be written as CF=B.

$$\begin{array}{c}C=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill Y_{\mathit{\text{in}}}\hfill & \hfill Y_{\mathit{\text{mi}}}\hfill & \hfill Y_{\mathit{\text{ri}}}\hfill & \hfill Y_{\mathit{\text{li}}}\hfill \end{array}\right)\hfill \\ B=\left(\begin{array}{c}\hfill F_{\mathit{\text{total}}}^n\hfill \\ \hfill M^n\hfill \end{array}\right)\hfill \end{array}$$ (2)

The linear constraints provided two equality requirements on normal force $F_{\mathit{\text{total}}}^n$ and the moment of normal force Mⁿ.

The computational procedures followed several steps:

1. Identify whether the optimization problem is non-splittable (it was found to be non-splittable). The concept of splittability is explained in Terekhov et al.(2010).

2. Determine the mathematical formulation of the hypersurface composed by the experimental data in a 4-dimensional finger force space. To decide on whether the distribution can be considered planar the principal component analysis (PCA) was used. Based on the previous studies (Terekhov et al. 2010; Terekhov and Zatsiorsky 2011) a 90%-criterion was employed: if the first two principal components account for more than 90% of the total variance the data distribution was considered to be on a 2-D plane. In the present experiment the distribution of the finger normal forces was found to compose a 2-dimensional plane (see the Results section for the details).

   As it was explained above, according to the Lagrange Principle for the inverse optimization (proved in Terekhov et al. 2010) at the points of optimal solution the level sets of the cost function and the constraint functions are tangent to each other, hence at these points the cost function and the constraint functions have common derivatives. As the experimental data—they correspond to optimal solutions—are located in a 2-dimensional plane the cost function derivatives have to be linear and hence the cost functions themselves are quadratic.

3. Employ Uniqueness Theorem to compute a group of cost functions which are essentially similar up to linear terms, whose optimal solutions compose a hypersurface matching the experimentally determined (observed) hypersurface.

   The desired objective function is simplified to have the formulation of:

   $$Ĵ=\frac{1}{2}{\displaystyle \sum _{i=1}^4}{K}_i{(F_i^n)}^2+{\displaystyle \sum _{i=1}^4}{W}_i(F_i^n)$$ (3)

   where i =in, mi, ri and li. Coefficients K_i have a dimensionality of 1/N² and W_i coefficients have a dimensionality of 1/N in this study. This makes the present cost function dimensionless.

4. Evaluate the performance of the obtained cost functions by comparing the optimal solutions with the experimental observations.

   The objective functions were reconstructed from the experimental data following the sequence of steps described above. The procedures were applied to the data points from individual subjects obtained on each day. For the objective function reconstruction computer codes were written using the Matlab software. In order to evaluate the performance of the obtained cost functions, the optimal solutions were computed by applying the constrained nonlinear multivariate function “fmincon” from Matlab’s optimization tool box.

2.2.2. Validating and comparing the cost functions

The performance of the C_i functions was compared with the performance of two other cost functions, Energy-like function (F₁) and Cubic norm function (F₂). These functions have been used previously in the studies on optimization of the finger forces in multi-finger prehension tasks (Zatsiorsky et al. 2002b, Pataky et al. 2004b, Niu et al. 2009). Analogous cost functions —such as the sum of muscle stresses, muscle forces, excitation levels, joint torques, etc. squared or cubed across the involved muscles or joints—have been used in many optimization studies in biomechanics and motor control (Nubar and Contini 1961; Crowninshield and Brand 1981; An et al. 1984; Dul et al.1984; Seireg and Arvikar 1989; Challis and Kerwin 1993; Buchanan and Shreeve 1996; Anderson and Pandy 2001; Raikova and Prilutsky 2001; Xiang et al. 2011). In the present research, these functions were used as the benchmarks for the decision on the reproducibility of the C_i functions (explained below).

Overall, the following five cost functions were used to compute optimal solutions:

1. C₁ is the cost function computed from an individual subject’s experimental recordings on Day 1.

2. C₂ is the cost function computed for Day 2.

3. C₇ is the cost function computed for Day 7.

4. F₁ is an energy-like function over Fⁿ, $F_1={\displaystyle \sum _{i=1}^4{(F_i^n)}^2}$

5. F₂ is the cubic norm function over Fⁿ, $F_2=\sqrt[3]{{\displaystyle \sum _{i=1}^4{(F_i^n)}^3}}$

For validation, the optimal solutions for the five cost functions above were compared with the experimental recordings. In particular, the distribution of optimal solutions (e.g., orientation of the optimal plane computed from a given cost function) was compared with the plane composed by the experimental data. The optimal plane could be determined mathematically by knowing the center point of the data and the orientation of the plane. The center point of the data was treated as the average of the finger forces across trials; the orientation could be determined by the first two principal components (PCs) of the finger force distribution.

Since the VF moment arm, grasping forces, and moments of normal force changed from day to day for the same subject, the constraints used in the inverse optimization to compute cost functions, and constraints used in the direct optimizations to compute the optimal solutions were different over time.

2.3 Estimating the reproducibility of the reconstructed cost functions

The reproducibility of the cost functions over time was evaluated by comparing the optimal solutions computed from cost functions obtained on different days. In what follows, C_i (i=1, 2, 7) is the cost function computed using the ANIO method for an individual subject on Day i. For the cross-validation of the cost functions obtained via the ANIO method on Day 1, Day 2 and Day 7, the optimal solutions were computed by applying a cost function C_i to the data obtained on Day j, where i ≠ j. For example, when C₁ and C₂ were used as the cost functions but the experiment was performed on Day 7, for the cross validation the cost functions C₁ and C₂ were compared with the performance for Day 7. The reconstructed cost functions were considered reproducible if the performance of a cost function C_i at Day j (i ≠ j) was better than the performance of cost functions F₁ and F₂.

The following measures were employed to estimate the performance of different cost functions computed based on the data obtained on different days:

1. The differences between the averaged experimental observations of finger forces over 80 trials and the solutions predicted by different optimization approaches;

2. The values of the dihedral angles between the experimental plane and the optimal plane (D-angles), computed from the different cost functions for individual days;

3. The root mean square (RMS) difference between optimal solutions and the experimental recordings; and

4. The RMS errors of the finger force sharing percentages predicted by the optimal procedures with respect to the experimental data.

2.4 Statistical analysis

The Linear Mixed Model (LMM) in SPSS 16.0 (SPSS Inc., Chicago, IL, USA) was used to perform Repeated Measures ANOVA (RM-ANOVA) (Littel et al. 2006). The LMM is more appropriate for repeated measures analysis than the Generalized Linear Model (GLM) since the LMM formulates the appropriate covariance structure for the repeated measures data and applies it to the followed statistical tests (Littel et al. 2006; West et al. 2007). Akaike’s Information Criteria (AIC) as a criterion of goodness-of-fit was used to determine the best covariance structure for the followed RM-ANOVA model (Akaike 1974). The degrees of freedom and p-values were corrected based on the covariance structure. Pairwise comparisons using the Bonferroni correction were calculated to determine significant effects in the RM-ANOVA post-hoc tests. The significance level was set at 0.05.

In the analysis of natural grip force and VF moment arm, factor TORQUE has 5 levels of L2, L1, Mi, R1 and R2, which corresponded to the external torques of 0.4Nm, 0.2 Nm, 0 Nm, −0.2 Nm and −0.4 Nm, respectively (counterclockwise direction of the external torque was labeled as positive); the other factors include LOAD (Load 1, Load 2, Load 3 and Load 5), and DAY (3 levels: Day 1, Day 2, Day 7).

For the analysis of cost function reproducibility and cross-validation over time, FINGER factor includes 4 levels: Index, Middle, Ring and Little finger. Factor METHOD has 5 levels: M1, M2, M3, M4 and M5. M1 corresponds to C_i applied at day i. M₂ and M₃ correspond to C_j and C_k respectively, where j is closer to i than k (i, j, k= 1,2,7; |i-j|<|i-k|; i≠j≠k). M₄ and M₅ are for F₁ and F₂ respectively. For example, in Day 2 (i=2), M₁ corresponds to C₂; M₂ is for C₁; while M₃ represents C₇. With such a definition, DAY could be also inspected for its statistical influence on the optimization results during the cross validation process.

The linear regression model in SPSS was used to estimate the regression coefficients for each finger group, where the response variables were the 1^st and 2^nd order coefficients W and K of the cost functions computed from ANIO method. The slopes of the regression lines were compared between fingers (groups) by adding dummy variables to reference fingers and estimating interaction terms in the regression model (Fox 1997).

3. Results

3.1. Task performance

3.1.1. Performance accuracy

To inspect whether the subjects satisfied task requirements of matching the target grip force, the percentages of normal force increase were calculated across all trials. In the tasks where 0%, 25%, 50% and 75% force increases were required, the recorded values equaled 2±7%, 27±7%, 52±8%, and 76±10%, respectively (mean ± standard deviation across all trials). We concluded that the matching error was reasonably small.

3.1.2. The effects of TORQUE, LOAD and DAY on the natural grip force recorded during the baseline sessions

RM-ANOVA confirmed the significant effect of TORQUE×LOAD interaction (F_{[12, 311]} =2.037; p=0.021) and LOAD×DAY interaction (F_{[6, 396]}=2.761; p=0.012) on the natural grip force. The post-hoc analysis—the Bonferroni pair-wise comparisons—showed that for each LOAD level, the natural grip force was affected by TORQUE significantly (p<0.001 for all). For each TORQUE level, the grip force increased with LOAD significantly (p<0.001 for all) (Figure 2). The force-torque relations were approximately U-like; also the force magnitude increased with the load (Figure 2). Similar findings have been reported previously (Zatsiorsky et al. 2002a; Shim et al. 2005; Zatsiorsky and Latash. 2004). Due to non-originality of these results, we will not be discussing them further here.

Figure 2.

Relations between the natural grip force and external torque for individual loads. Grip forces for Day 1are averaged across subjects; error bars are not shown for clarity purposes.

For the same LOAD, the natural grip force averaged over all TORQUE levels on Day 1 was larger than for Days 2 and 7 (p<0.021 for all pairwise comparisons, see Figure 3), while the grip force on Day 2 did not differ statistically from the force on Day 7 (p>0.457 for all pairwise comparisons). The observed decrease in the natural grip force introduces a perturbation to equation 1 on which the ANIO model is based: when compared across days the constraints $F_{\mathit{\text{total}}}^n$ are not equal anymore. This difference may induce alterations in the reconstructed cost functions and creates an additional challenge to the estimates of their reproducibility.

Figure 3.

Natural grip force averaged over all TORQUES as function of DAY and LOAD. Averages across all subjects are presented with standard deviations.

The effects of TORQUE, LOAD, FINGER and DAY on the force magnitude of individual fingers were tested with a 4-way RM-ANOVA (covariance of AR1). It was found that the two-way interactions of TORQUE×LOAD (F_{[12, 731]}=2.519, p=0.003), TORQUE×FINGER (F_{[12, 1387]}=365.122, p<0.001), LOAD×FINGER (F_[9,1413]=2.287, p=0.015), and FINGER×DAY(F_{[6, 1453]}=3.741, p=0.001) had significant effects on finger forces.

Post-hoc Bonferroni pairwise comparisons showed that only F_mi was not affected by DAY. Other finger forces decreased: F_in was higher in Day 1 than in Day 2 and Day 7 (p<0.001 for both); F_ri also had higher value in Day 1 than in Day 2 and 7 (p<0.006 for both); F_li dropped monotonically from Day 1 to Day 7 (p=0.026 between day 1 and 2; p=0.014 between day 2 and 7).

3.1.3. Testing the planarity of the data distribution – principal component analysis

Principal Component Analysis (PCA) was performed as a precursor of the ANIO procedure. The goal was to determine whether the experimental data are located mainly on a two-dimensional hyperplane. The PCA was done on the finger normal forces for each subject/day combination.

Overall, the PCA results (the percent of explained variance and the loading coefficients) did not change substantially with DAY. It was found that the amount of variance explained by the first two principal components (PCs) was 92.91 ± 3.39%, 93.44 ± 2.72% and 93.99 ± 2.88% (average ± standard deviation over all subjects) for Day 1, Day 2 and Day 7, respectively. Because two PCs define a plane, it was concluded that the experimental data were mainly located on a two-dimensional hyperplane (for similar results see Terekhov et al. 2010, Park et al. 2010 and Niu et al. 2011). The variance explained by the first two PCs was not affected by DAY. PC1 explained 62.13 ± 3.84% of the total variance, while PC2 explained 31.31 ± 3.30% of the total variance; these values were not affected by DAY.

Two-way RM-ANOVA showed that the loading coefficients of PC1 were affected by FINGER (F_[2,83]=94.17, p<0.001) significantly but not affected by DAY; the index and little fingers had opposite loading coefficients for PC1; the coefficients decreased monotonically from the index finger to little finger (p <0.001 for all pairwise comparisons, Figure 4).

Figure 4.

Loading coefficients for the first two Principal Components (PCs). Averages across all subjects are presented with standard deviations.

The loading coefficients of PC2 were also affected by FINGER (F_[2,83]=68.28, p<0.001) but were not affected by DAY. The loading coefficients of PC2 had the smallest magnitude for the index finger among all fingers (p<0.001 for all pairwise comparison) while the other fingers all had positive loading coefficients.

3.2 ANIO results

The planarity of the experimental data distribution in the 4-dimensional finger force space indicates that the cost functions are either quadratic polynomials or can be closely approximated by them. The reasons for such a conclusion were discussed in Terekhov et al. (2010) and Terekhov, Zatsiorsky (2011). As shown by Niu et al (2011) the observed planarity of the distribution of the experimental data is not a trivial consequence of the task mechanics; it is an outcome of the motor control processes.

3.2.1. Basic results

The ANIO procedures were further performed on the data from individual subjects for each day. The experimentally determined coefficients of the cost functions are shown in Table 1. Note that all 1^st-order W coefficients are smaller than the 2^nd-order K coefficients. The day-to-day changes of the coefficients and their correlations with the finger forces will be described in section 3.3.

Table 1.

Coefficients K_i and W_i from the ANIO Method for three days.

Averages over all subjects	Day1	Day 2	Day 7
Kmi	1.65±0.24	1.77±0.32	1.70±0.27
Kri	1.71±0.22	2.16±0.47	2.02±0.48
Kli	2.95±0.67	2.92±0.50	2.58±0.49
Win	−0.93±0.31	−0.62±0.22	−0.25±0.17
Wmi	0.88±0.32	0.77±0.33	0.37±0.28
Wri	1.02±0.44	0.36±0.22	0.08±0.43
Wli	−0.97±0.37	−0.52±0.19	−0.20±0.23

Averages over all subjects are presented with standard errors. No data are presented for K_in, since it was assumed to be unity for all tests.

3.2.2. Cross-validation of the cost functions

The following cross-validation results are reported: (a) comparing averages; (b) the dihedral D-angles; (c) the RMS differences between the optimal solutions and experimental recordings; and (d) the RMS errors of the finger force sharing percentages.

1. Average forces. The average finger forces over all trials for individual subjects were computed for the experimental data of each day and then compared with the optimal solutions generated from all cost functions. The average of the force data that was predicted by the ANIO cost function for a specific day (i.e., predicted by cost function C_i for Day i) was close to the average of the observed data (p=1.0 for all fingers, see Table 2). The differences between the average experimentally recorded finger forces and the optimal solutions are shown in Figure 5. Since the averaged finger forces for Day i are almost equal to the predicted values from the cost function C_i, the results from C_i are not presented.

   Three-way RM-ANOVA showed that the DAY (3 levels) and the interaction of METHOD (4 levels) and FINGER (4 levels) had significant effects on the absolute error (F_{[2, 359]}=11.189, p<0.001 for DAY; and F_{[9, 359]}=3.328, p=0.001 for METHOD×FINGER). As seen in Figure 5, the optimal solutions generated by the cost functions C_j and applied to Day-j deviated from the experimental data observed on Day i (i≠j) to a much smaller degree than the solutions from cost functions F₁ for the index and middle fingers (p<0.046 for all pairwise comparisons in the post-hoc analysis) and the solutions from F₂ for all fingers (p<0.001 for all pairwise comparisons). Therefore, for the average forces the performance of the reconstructed cost functions was considered reproducible.

2. Dihedral angles. Figure 6 shows the values of the dihedral angles between the experimental plane and the optimal plane (D-angles) computed for the different cost functions on individual days. Two-way RM-ANOVA with factors METHOD and DAY showed that the dihedral angle was affected significantly by METHOD (F_{[4, 106]}=69.52, p<0.001) but not by DAY (F_{[2, 106]}=0.10, p=0.908). The post-hoc Bonferroni pairwise comparison confirmed that the angles from any C_i (i=1, 2, 7) were smaller than from F₁ and F₂ (p<0.001 for all comparisons); the angle between the experimental plane and the plane computed from C_i on Day i was the smallest (p=0.002 for C_j and p=0.004 for C_k where |i-j|<|i-k|) among all cost functions, the angles from C_j and C_k (i≠j≠k) were the same (p=1.0). Hence, for the dihedral angles the reconstructed with ANIO cost functions also yielded reproducible performance.

3. RMS differences for finger forces. The RMS differences between optimal solutions and the experimental recordings are presented for F₁, F₂ and all C_i on Day j, where i=1, 2, and 7 (Figure 7). Three-way RM-ANOVA with factors METHOD (5 levels), FINGER (4 levels) and DAY (3 levels) was performed, and it was found that the effects of DAY and two-way interaction METHOD×FINGER on the RMS difference were significant (F_{[2, 451]}=18.58, p<0.001 for DAY; F_{[12, 451]}=3.28, p<0.001 for METHOD×FINGER).

   The reconstructed cost functions, C_i, performed better than the F_i functions. Multiple comparison with Bonferroni corrections found that the index finger had smaller error for the forces computed from C_i (i=1, 2, 7) than the forces computed from F₁ and F₂ (p<0.001) while the errors were not significantly different among C_i (p>0.052 for all). The middle finger had the smallest error for force computed from C_i in day i (p<0.006 for all), and the largest error from F₁ and F₂ (p<0.001). The ring finger had the smallest error for C_i in day i, the largest for F₂, while not different among C_j (i≠j) and F₁. The little finger had larger error from F₁ and F₂ than from any C_i, while the RMS errors from C_i (i=1,2,7) were the same (p>0.128).

4. RMS differences in sharing percentages. Since the grasping force was different across subjects, torques, loads, and specified target magnitudes, even small differences between the optimal solution and the experimental observation might cause a considerable violation of the mechanical requirements, i.e. resultant force and total moment of force. The individual finger forces were normalized by computing their contribution to the VF normal force (% of the VF force), and compared with the sharing percentage of optimal solutions. The RMS errors of the actual finger force sharing percentages with respect to the optimal solutions are demonstrated in Figure 8.

   Three-way RM-ANOVA with factors METHOD, FINGER and DAY confirmed the significant effects of METHOD (F_{[4, 463]}=410.21, p<0.001), FINGER (F_{[3, 463]}=37.96, p<0.001) and DAY (F_{[2, 463]}=4.49, p=0.012) on the RMS errors of finger force sharing percentages, while the effects of interactions did not reach the level of significance (p >0.311 for all two-way and three-way interactions). The smallest RMS of the sharing percentages was observed for functions C_i at Day i (i=1, 2, 7) (p <0.001 for all pairwise comparisons); the errors for the two C_j were the same (p=1.0) on Day i, where i≠j, and both were smaller than those generated by F₁ and F₂ (p <0.001 for all), (Figure 8). Therefore, following the accepted definition of the cost functions reproducibility, the functions were reproducible with respect to the finger force sharing.

Table 2.

The group averages of individual finger forces (N)

Day	Finger	Experiment	Predicted from optimization using different cost functions
			C1	C2	C7	F1	F2
Day 1	Index	4.99±1.05	4.99±1.05	4.72±1.13	4.71±0.98	4.50±0.85	3.74±0.82
	Middle	3.20±0.51	3.20±0.51	3.65±0.82	3.65±0.81	4.05±0.47	4.70±0.44
	Ring	3.82±0.93	3.82±0.93	3.79±1.05	3.81±1.02	3.74±0.40	4.70±0.44
	Little	3.61±0.52	3.61±0.52	3.46±0.70	3.44±0.68	3.34±0.63	2.47±0.78
Day 2	Index	4.45±1.12	4.24±1.10	4.45±1.12	4.22±1.07	4.00±0.90	3.25±0.86
	Middle	2.92±0.61	3.10±0.66	2.92±0.61	3.24±0.75	3.62±0.39	4.29±0.34
	Ring	3.34±0.83	3.61±0.78	3.34±0.83	3.43±0.85	3.36±0.32	4.29±0.34
	Little	3.33±0.63	3.10±0.58	3.33±0.63	3.15±0.67	3.05±0.73	2.2±0.94
Day 7	Index	4.38±0.95	4.30±1.00	4.28±1.11	4.38±0.95	4.03±0.81	3.27±0.78
	Middle	3.06±0.60	3.04±0.69	3.19±0.89	3.06±0.60	3.59±0.37	4.23±0.33
	Ring	3.23±0.74	3.47±0.74	3.27±0.83	3.23±0.74	3.28±0.34	4.23±0.33
	Little	3.12±0.65	2.99±0.59	3.05±0.68	3.12±0.65	2.9±0.72	2.06±0.91

Average data over all trials in each subject/day; averages over all subjects with standard deviation presented. The experimental data and the values predicted from the application of the cost function C_i at day i are boldfaced.

Figure 5.

The absolute differences between averaged finger forces over 80 trials for experimental observations and optimal solutions (|ΔF̅|) in Day 2. Averaged data among all subjects are presented with standard deviations. The data for other days are similar and because of that are not presented here. The horizontal square brackets with the stars indicate significant differences.

Figure 6.

The dihedral angle between the actual plane of the experimental data and the optimal plane (D-angle), computed from various cost functions and averaged over all subjects. The horizontal square brackets with the stars indicate statistically significant differences.

Figure 7.

The root mean square (RMS) differences between the optimal solutions and the experimental data. Averages across subjects are presented with standard deviations.

Figure 8.

RMS errors of the finger force sharing percentages predicted by the optimal procedures with respect to the experimental data. (Averages across subjects with standard deviations are presented.)

In summary, for all four employed measures (a)–(d):

1. The optimal solutions from C_i when applied to Day i approximated the experimental data with smallest errors, and

2. Application of the cost functions C_i over the entire one-week period of testing resulted in smaller deviations from experimental observations than those obtained using F₁ and F₂. The latter finding supports the hypothesis on the reproducibility of the C_i cost functions over time and speaks in favor of consistent individual strategies used by the subjects.

3.3. The variability of the coefficients and their relations to finger forces

The day-to-day changes of the K and W cost function coefficients (see Table 1) were estimated by the RM-ANOVA tests.

Since K_in was normalized to be 1 during the inverse optimization procedures, K for the index finger was not included in the statistical two-way RM-ANOVA test with FINGER (3 levels) and DAY (3 levels). It was found that DAY did not influence K coefficients significantly (F_[2,60]=0.36, p=0.700). K coefficients were different among fingers (F_[2,60]=14.70, p<0.001) and the following relationship was found: 1<K_mi=K_ri <K_li (p<0.001 for all except p=1.000 for the test of K_mi=K_ri, Bonferroni pairwise comparison test).

Average W values for the index and little fingers were negative, while they were positive for the middle and ring fingers. The effect of DAY (2 levels) and FINGER (4 levels) on the W magnitude was tested. The effect of DAY was found to be significant (F_{[2, 83]}=7.20, p<0.001), while the effect of FINGER was not significant (F_{[3, 83]}=1.30, p=0.280). The magnitudes of |W| in Day 1 were significantly larger than in Day 2 and Day 7 (p=0.044 for Day 2, and p<0.001 for Day 7) while |W| in Day 2 was not significantly different from |W| in Day 7 (p=0.663). In other words, as can be seen from Table 1, on Day 1 there was relatively large contrast between the negative coefficients for the linear terms with index and little finger forces and positive coefficients for the terms with middle and ring finger forces. By Day 7, this contrast was reduced, all the coefficients dropped in magnitude.

The relations of K_i to (a) finger force magnitudes (N), and (b) finger force sharing ratios are shown in Figure 9. The parameter estimates of the linear regression models were significantly different from zero: (a) p<0.013 for the regressions of K_i on ${\mathrm{F̅}}_i^n$ (i=mi, ri, li) and (b) p<0.003 for the regressions of K_i on $\overline{\raisebox{1ex}{$F_i^n$}\!\left/ \!\raisebox{-1ex}{$F_{\mathit{\text{VF}}}^n$}\right.}$ (i=mi, ri, li). The slopes of the regression lines for individual fingers were not significantly different from each other (p>0.137 for ${\mathrm{F̅}}_i^n$ and p>0.263 for $\overline{\raisebox{1ex}{$F_i^n$}\!\left/ \!\raisebox{-1ex}{$F_{\mathit{\text{VF}}}^n$}\right.}$).

Figure 9.

The linear regression of K_i on:

1. ${\mathrm{F̅}}_i^n$ – average finger forces across all trials for each subject/day combination and
2. $\overline{\raisebox{1ex}{$F_i^n$}\!\left/ \!\raisebox{-1ex}{$F_{\mathit{\text{VF}}}^n$}\right.}$ – average sharing ratio contribution of individual finger force in the VF normal force.

s is the slope of the linear regression; r is the correlation coefficient between K_i and the corresponding independent variable. Note that the K_i values for the middle, ring and little fingers are normalized by the K_in.

The linear regressions of W_i and its absolute value |W_i| (i=in, mi, ri, li) were not significant on all predictors: finger force magnitude (p>0.079 for all) and finger force sharing ratio (p>0.234 for all). Negative linear relationships coefficients for K_i were observed with the corresponding finger forces and sharing ratios.

4. Discussion

In the current research, we reconstructed the (variable) cost functions from the (variable) performance results. However, if the principle of optimality is valid—and the present study is based on this postulate—the course of actual events was opposite: the cost functions varied across the days and these changes resulted in the varied performance.

4.1. Reproducibility vs. variability of the cost functions and performance over repeated tests

Multi-digit grasping in the 3-D space is a highly redundant task with 30 elemental mechanical variables (digit forces and moments), 6 equality constraints (related to components of the resultant force and moment exerted on the object), and 5 inequality constraints necessitated by the slipping prevention. Among many studies of prehension (reviewed in Zatsiorsky and Latash 2008, 2009), reproducibility of multi-finger prehension over time has not been addressed. The present study fills this gap. It proves that, while the pattern of performance varies over time, the cost function obtained with the ANIO method on one day agrees well with the performance values recorded on another day.

Measures of performance reproducibility are commonly used to track changes in test results, and evaluate the reliability of measurement in a single trial (reviewed by Hopkins et al. 2001). Previous studies of performance reliability are too numerous to make a list of them. In particular, they included static force measurements (Breger-Lee eta. 1993, Carlton and Newell 1993, Ng et al 2003, Roebroeck et al. 1993), isokinetic and dynamic force recordings (Cronin and Henderson 2004; Munich et al. 1997; Hatze 1998; Woo and Zatsiorsky 2006), standing vertical jumps (Liu et al. 1992), and short sprints (Hunter et al. 2004; Risberg et al 1995), and many other tests. All these studies estimated the reproducibility by comparing the variability of the test results over repeated trials, essentially scalar values, with the performance variability in the experimental sample group, e.g. by using intra-class correlation coefficients (ICCs).

This approach cannot be used in the present study where the goal is to estimate the reproducibility of the cost function of several variables. Neither studying reproducibility of the cost function values nor estimating the reproducibility of the function coefficients solves the problem. The precise magnitude of the cost function value is not important. Only the combination of the finger forces at which the value is minimal is informative (a cost function remains essentially the same if it is multiplied by a constant and if a constant is added to it).

Therefore, to estimate the reproducibility of the cost functions we employed the cross-validation approach and decided using the F₁ and F₂ values as benchmarks. We admit that using these benchmark values involves elements of subjectivity. The benchmarks may be considered too strict by some researchers or too lax by others. However, this is a feature of many scientific conventions. For instance, the traditional statistical convention for the Type 1 error a=0.05 is not less arbitrary. Another important factor in making comparisons among different cost functions is the number of free parameters. The cost function C_i from the ANIO method has eight parameters, while F₁ and F₂ have four parameters each. Since those two cost functions have been widely used (An et al. 1984; Dul et al.1984; Seireg and Arvikar 1989; Challis and Kerwin 1993; Raikova and Prilutsky 2001; Prilutsky and Zatsiorsky 2002; Zatsiorsky et al. 2002b), we chose them for comparison with the ANIO method. We admit, however, that the comparison might have been skewed in favor of the ANIO method because of the larger number of free parameters.

Based on the accepted measure of the cost function reproducibility, it was concluded that their day-to-day reproducibility was rather good: application of a cost function C_i obtained with the ANIO method to the data of Day j (i≠j) always resulted in smaller deviations from the experimental observations than using other popular cost functions F₁ and F₂. This claim is based on the results of application of all four measures (a)–(d) employed to estimate the performance of different cost functions (described above in section 3.2.2).

One of the goals of this study was to test whether the cost functions reconstructed with the ANIO method represented stable preferences by the subjects and could be considered personal characteristics of motor coordination. Within the accuracy and conventions accepted in this study, the results suggest that the answer should be clearly positive. On the whole, this research suggests that the cost functions in multi-finger prehension are reproducible over time and represent consistent individual strategies used by the subjects.

When thinking over the presented reproducibility data, one has to pay attention to the limitations and the goals of the ANIO analysis in the form used in the present paper. In particular, the ANIO model used here cannot predict the total grasping force. It uses this force as a constraint that has to be satisfied (see equation 1). The constraints can arise from the explicit instructions to the subjects supported by the visual feedback (as in the 80 trials with the prescribed force magnitude used for the ANIO analysis at each day) or they can be selected by the subjects themselves (as in the 20 basic trials at each experimental day). For some reasons (learning, adjustment to the known task conditions, such as friction) the subjects exerted smaller grasping forces at Days 2 and 7 that at Day 1. Formally, this introduces an additional obstacle to the application of the ANIO reproducibility analysis: the cost functions potentially can change with the basic grasping force alterations. Even if these changes were present, they did not undermine the outcome of performed cross-validation. As it was mentioned already, within the limitations of this study (its accuracy and the accepted definition of the reproducibility) the cross-validation results confirmed the reproducibility of the cost functions in the multi-finger prehension over time.

It is possible to look on the obtained cross-validation results from another point of view. If a change in the basic grasping force is a perturbation, the observed stability of the cost function performance can be described as its robustness, i.e. the ability of the cost function to yield similar results in spite of the perturbations. What term to use to describe the observed phenomenon —the reproducibility or robustness—is a matter of choice.

The day-to-day variability of the performance results, and the reconstructed from these data variable cost functions, represent only one aspect of the variability: the variability between the sets of 80 trials with the prescribed force magnitude performed at different days. While the data for a given force-torque combinations were not explicitly averaged, the method still does not account for possible between-trials variability (that would be observed if the trials were repeated, similar to what was done in Park et al. 2010). With respect to the interaction of reproducibility and variability this study complements the Park et al. research (Park et al. 2010; 2011a,b).

4.2. Tuning of the cost function coefficients

The questions formulated above were whether the coefficients: (a) changed with time, and/or (b) are systematically related with finger forces.

4.2.1 The day-to-day changes

It was found that the second order coefficients K_i did not change significantly (described in section 3.3) while the magnitude of the coefficients at the linear terms, W_i, decreased over time (Table 1 and section 3.3). The aforementioned stability/reproducibility of the cost function performance over the one-week period is in a clear disagreement with the observed decreasing trend in the W_i magnitudes. The simplest—but not necessary valid—explanation of this discrepancy is the relatively small contribution of the linear term into the cost function value over the whole range of task variables. For instance, on Day 7 the average force of the little finger was 3.12 N (see Table 2), its squared value equaled 9.73. When multiplied by K_li =2.58 (see Table 1) the value of the quadratic term in the cost function is 25.11. The magnitude of the associated linear term is only F_li×W_li=3.12×(−0.20)= −0.62. The difference is 40-fold.

Since W_i for the middle and ring fingers were positive, their decrease over time suggested a smaller contribution of the forces of these fingers to the cost function magnitude if the same finger forces were exerted. Because the optimal inter-finger force distribution corresponds to a minimal value of the cost function, the smaller values of the W_i for the middle and ring fingers allows for extra ‘tolerance’ to their force increase.

In contrast, the W_i for the index and little fingers were negative. Hence, the larger the W_i the smaller the cost function values, the closer these values to the minimum of the function. We may surmise that the central controller encourages generating relatively large index and little finger forces. This result complies with the principle of mechanical advantage according to which the effectors (muscles or fingers) with larger lever arms contribute more to the total moment of force than those with shorter lever arms (Buchanan et al. 1989; Prilutsky 2000; Gao et al. 2006; West et al. 2007; Zatsiorsky 2002). The drop in the magnitude of these coefficients suggests that, with practice, the contrast between the ’encouraged‘ forces of the index and little fingers and “punished” forces by the middle and ring fingers decreases.

Note that linear terms in the cost function dominate at low force values. To produce a broad range of moments while using low total forces requires using fingers with the larger lever arms, the index and little fingers. The attenuation of this contrast with practice suggests that the subjects learn to use a more even force distribution among the fingers that is still however compatible with the range of total force and moment magnitudes (Park et al. 2011b).

In a recent paper (Park et al. 2010), the relations between optimality and variability have been discussed. In particular, the authors of that paper have suggested that optimization defines preferred centers of data distributions (across repetitive trials at the same tasks) while shapes of the distributions reflect co-variation among finger forces that helps (or does not help) keep the performance of the hand close to the task values. We assume in this study that the performance is optimal in a sense that a single cost function describes the centers of such hypothetical data point distributions. For practical reasons, we did not repeat the same tasks many times. So, single observations were taken as representing the centers of distributions. Within this framework, variations of behavior are viewed not as variations of the cost function but as deviations from the criterion of optimality. Practice can lead to no changes in the cost function but smaller deviations of actual observations from values dictated by the function.

4.2.2 Correlation with the finger forces

The strong negative linear relations between the coefficients K_i and finger forces observed in our study agree well with the previously reported fact (Niu et al. (2011) that the 2^nd order coefficients K_i of the cost functions have a similar allocation across subjects (K_in<K_mi=K_ri<K_li). Such a pattern of K_i distribution can be explained by the different magnitudes of the finger force contributions to the grasping force.

The negative relations between the coefficients K_i and finger forces (Figure 9) inspire a question as to whether inter-individual differences in the cost functions are simply due to the different grasping forces exerted by the subjects. Such differences in the grasping forces could be due, for instance, to different friction coefficients at the fingertip-object interface (more slippery conditions are usually associated with higher grasping forces, a fact noted by Johansson and Westling 1984; Cole, Johansson, 1993; Aoki et al. 2006), individual preferences for smaller or larger safety margins, etc. Alternatively, perhaps different finger force patterns are employed by individual subjects C_i such that each cost function is a distinctive personal trait per se. This question can be extrapolated to other tasks, such as writing and gait. It is well known that people differ in terms of their detailed movement patterns. The question is whether inter-individual differences in movements are due to different cost functions employed by the performers, or the movement patterns differ due to other factors (e.g., body morphology). Unfortunately, this question cannot be answered without a special investigation.

In the literature, the interpretation of cost functions has been commonly associated with various physiological variables, such as minimum fatigue, minimum metabolic energy expenditure, minimum muscle stress (reviewed in Prilutsky and Zatsiorsky 2002), equal sharing among the contributors (Pataky et al. 2004b), etc. The cost functions themselves were, however, assumed by the researchers rather than deduced from experimental data.

The ANIO opens new opportunities: for instance, the fact that the reconstructed cost functions were quadratic makes an idea that the central controller minimizes an elastic energy of deformation, e.g. at the fingertips, quite plausible. The explanation is straightforward: if the deformation is a linear function of the force (i.e. the Hooke law is valid) the accumulated potential elastic energy should be proportional to mechanical work spent for the deformation, i.e. to the product of the deformation (distance) and the force, i.e. it should be proportional to the squared value of the force.

The reconstructed cost functions C_i worked much better than F₁ function that is also quadratic. The difference between the C_i and F₁ functions is two-fold: (a) the K_i coefficients in the C_i functions are reconstructed from experimental data and are different for each finger while in the F₁ function they all equal 1, and (b) the C_i functions include also linear terms. Which of these differences is more important for the improved C_i performance remains to be known.

The relations between K_i and finger force values shown in Figure 9 represent experimental results; nothing was assumed there. By using this method the researchers can—at least in some cases—determine the cost function first and then find a relation between its parameters and experimental measures. Such an approach is used in the present study. We think however that at this time more experimental work is necessary to shed light on the cost functions used by the central controller.

4.3. Planarity of the data distribution and the principle of superposition in the multi-digit prehension

According to the PC1 and PC2 loading coefficients (Figure 4), one may tentatively suggest that PC1 corresponds to rotation control and PC2 corresponds to grasping control. It has been found previously (the principle of superposition, Zatsiorsky et al. 2004) that the control of grasping force and the control of rotation are decoupled at the VF level, i.e. elemental variables co-vary within a group but not between groups. To maintain the rotational equilibrium, the index and little fingers act in opposition (i.e., as torque antagonists), while they work as agonists for grasping force control and slipping prevention. These fingers contribute more to the control of rotation than the middle and ring fingers due to their larger moment arms (Zatsiorsky et al. 2002a).

Principle of superposition was suggested for skilled actions of the robotic hand (Arimoto and Nguyen 2001; Arimoto et al. 2001). It states that some skilled actions of the grippers can be decomposed into several independently controlled elemental actions to save computation time for robotic control. It has been shown that a similar principle is also valid for the control of human hand, i.e., the forces and moments of force of individual digits at the four-finger level are defined by two independent additive commands: preventing object from slipping out of hand, and regulating the rotational equilibrium (Zatsiorsky et al. 2004; Latash and Zatsiorsky, 2006; Shim et al. 2005; Shim and Park 2007)

In the current study, it was found that the four finger normal forces were mainly located within a two-dimensional hyperplane in the 4-D space. This conclusion follows from the PCA findings: The first two components accounted for more than 90% of the total variance. The planarity of finger forces in multi-finger prehension was confirmed over time for all subjects, and hence it can be regarded a universal observation, at least for young healthy male participants. The finger force distribution along the 1^st PC represents the control of rotation while the 2^nd PC represents the control of grasping (see Figure 4). This finding agrees well with the previous reports on the principle of superposition in the control of multi-finger prehension (Shim et al. 2005; Shim and Park 2007). Note that in the present study the 2-D distribution of finger forces in the 4-D force space cannot be purely due to mechanical reasons, specifically the constraints on the grasping force and object orientation: The values of the grasping force and external torque varied among tasks. Therefore, the low dimension of data distribution reveals preference of the central controller.