Optimality vs. variability: an example of multi-finger redundant tasks

Abstract

Two approaches to motor redundancy, optimization and the principle of abundance, seem incompatible. The former predicts a single, optimal solution for each task, while the latter assumes that families of equivalent solutions are used. We explored the two approaches using a four-finger pressing task with the requirement to produce certain combination of total normal force and a linear combination of normal forces that approximated the total moment of force in static conditions. In the first set of trials, many force–moment combinations were used. Principal component (PC) analysis showed that over 90% of finger force variance was accounted for by the first two PCs. The analytical inverse optimization (ANIO) approach was applied to these data resulting in quadratic cost functions with linear terms. Optimal solutions formed a hyperplane (“optimal plane”) in the four-dimensional finger force space. In the second set of trials, only four force–moment combinations were used with multiple repetitions. Finger force variance within each force–moment combination in the second set was analyzed within the uncontrolled manifold (UCM) hypothesis. Most finger force variance was confined to a hyperplane (the UCM) compatible with the required force–moment values. We conclude that there is no absolute optimal behavior, and the ANIO yields the best fit to a family of optimal solutions that differ across trials. The difference in the force-producing capabilities of the fingers and in their moment arms may lead to deviations of the “optimal plane” from the subspace orthogonal to the UCM. We suggest that the ANIO and UCM approaches may be complementary in the analysis of motor variability in redundant systems.

Appendices

Appendix 1

Uniqueness theorem (for the mathematical proof see Terekhov et al. 2010)

The core of the ANIO approach is the theorem of uniqueness that specifies conditions for unique (with some restrictions) estimation of the objective functions. The main idea of the theorem of uniqueness is to find necessary conditions for the uniqueness of solutions in an inverse optimization problem. An optimization problem (i.e., direct optimization problem) with an additive objective function and linear constraints are defined as:

$$ \begin{gathered} {\text{Let }}J:R^{n} \to R^{1} \hfill \\ {\text{Min}}:J(x) = g_{1} (x_{1} ) + g_{2} (x_{2} ) + \cdots + g_{n} (x_{n} ) \hfill \\ {\text{Subject to}}:CX^{T} = B \hfill \\ \end{gathered} $$

(14)

where $X = (x_{1} ,x_{2} , \ldots ,x_{n} ) \in R_{n} ,g_{i}$ is an unknown scalar differentiable function with $g^{\prime } ( \cdot ) > 0.$ g _i came from the Lagrange minimum principle, which has a unique solution. On the contrary, the functions of g _i can be computed from the set of solutions X* (e.g., experimental data). This inverse procedure is called the inverse optimization problem. C is a k × n matrix and B is a k-dimension vector, k < n.

First, assume that the optimization problem Eq. 14 with k ≥ 2 is non-splittable. If the inverse optimization is splittable, the preliminary step is to split it until a non-splittable subproblem is acquired. If the functions g _i (x _i ) in problem Eq. 14 are twice continuously differentiable (i.e., twice continuously differentiable functions f _i ) and $f_{i}^{\prime }$ is not identically constant, complying $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (X) = 0$ for all X ∈ X ^*,

$$ f^{\prime } (X) = (f_{1}^{\prime } (x_{1} ), \ldots ,f_{n}^{\prime } (x_{n} ))^{T} $$

(15)

and

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} = I - C^{T} (CC^{T} )^{ - 1} C $$

(16)

then

$$ g_{i} (x_{i} ) = rf_{i} (x_{i} ) + q_{i} x_{i} + {\text{const}}_{i} $$

(17)

for every $x_{i} \in X_{i}^{*}$, where $X_{i}^{*}$ = {s| there is X ∈ X ^*: x _i  ∈ s} and X ^* is the set of the solutions for all B ∈ R ^k. The constants q _i satisfy the equation $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} q = 0$ where q = (q ₁,…,q _n )^T. Primes designate derivatives.

If the experimental data correspond to solutions of an inverse optimization problem with additive objective function (g _i ) and linear constraints, equation $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (X) = 0$ (X ∈ X ^*) must be satisfied (i.e., the Lagrange principle). The Uniqueness Theorem provides sufficient condition (i.e., $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} f^{\prime } (X) = 0$) for solving the inverse optimization problem in a unique way up to linear terms.

Appendix 2

Uncontrolled manifold (UCM) analysis (see Latash et al. 2002, 2007 for details)

For F _TOT, changes in the elemental variables (finger forces) sum up to produce a change in F _TOT:

$$ {\text{d}}F_{\text{TOT}} = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {{\text{d}}F_{i} } & {{\text{d}}F_{m} } & {{\text{d}}F_{r} } & {{\text{d}}F} \\ \end{array}_{l} } \right]^{T}. $$

(18)

The UCM was defined as an orthogonal set of the vectors e _i in the space of the elemental forces that did not change the net normal force, i.e.:

$$ 0 = \left[ {\begin{array}{*{20}c} 1 & 1 & 1 & 1 \\ \end{array} } \right]e_{i}. $$

(19)

These directions were found by taking the null-space of the Jacobian of this transformation ([1 1 1 1] e _i ). The mean-free forces were then projected onto these directions and summed to produce:

$$ f_{||} = \sum\limits_{i}^{n - p} {\left( {e_{i}^{T} \cdot {\text{d}}f} \right)e_{i} } , $$

(20)

where n = 4 is the number of degrees-of-freedom of the elemental variables, and P = 1 is the number of degrees-of-freedom of the performance variable (F _TOT). The component of the de-meaned forces orthogonal to the null-space is given by:

$$ f_{ \bot } = {\text{d}}f - f_{||}. $$

(21)

The amount of variance per degree-of-freedom parallel to the UCM is:

$$ V_{\text{UCM}} = {\frac{{\sum {\left| {f_{||} } \right|^{2} } }}{{(n - p)N_{\text{trials}} }}}. $$

(22)

The amount of variance per degree-of-freedom orthogonal to the UCM is:

$$ V_{\text{ORT}} = {\frac{{\sum {\left| {f_{ \bot } } \right|^{2} } }}{{pN_{\text{trials}} }}} $$

(23)

The normalized difference between these variances is quantified by a variable ΔV:

$$ \Updelta V = {\frac{{V_{\text{UCM}} - V_{\text{ORT}} }}{{V_{\text{TOT}} }}} ,$$

(24)

where V _TOT is the total variance, also quantified per degree-of-freedom. If ΔV is positive, V _UCM > V _ORT, caused by negative co-variation of the finger forces, which we interpret as evidence for a force-stabilizing synergy. In contrast, ΔV = 0 indicates independent variation of the finger forces, while ΔV < 0 indicates positive co-variation of the individual finger forces, which contributes to variance of F _TOT.

A similar procedure was used to compute the two variance components related to stabilization of M _TOT. The only difference was in using a different Jacobian corresponding to the lever arms of individual finger forces, [d _i d _m d _r d _l ].

We also analyzed the data with respect to the stabilization of both F _TOT and M _TOT simultaneously. In that case, the Jacobian was [1 1 1 1 d _i d _m d _r d _l ]. The dimensionality of V _UCM for the analysis with respect to F _TOT and M _TOT separately is three (one constraint), while the dimensionality of V _UCM with respect to F _TOT and M _TOT simultaneously is two (two constraints).