Analytical and numerical analysis of inverse optimization problems: conditions of uniqueness and computational methods

Do Tangential Finger Forces Utilize Mechanical Advantage During Moment of Force production?

This study investigated the beneficial effects of the utilization of mechanical advantage (MA) of finger tangential forces during the moment production. Subjects produced the resistive moment of force against the external torque while the moment arms of the tangential forces were systematically changed. We observed a relatively large contribution to the net moment by the tangential forces with the increased moment arms, whereas the vector sum of normal and tangential forces decreased. The indices of multi-finger coordination for the stabilization of the moment of forces and force direction increased with the moment arms. The current results provide evidence that the utilization of MA is associated with both the efficiency of force production and the stabilization of performance variables.

Optimality and stability of intentional and unintentional actions: I. Origins of drifts in performance

We address the nature of unintentional changes in performance in two papers. This first paper tested a hypothesis that unintentional changes in performance variables during continuous tasks without visual feedback are due to two processes. First, there is a drift of the referent coordinate for the salient performance variable toward the actual coordinate of the effector. Second, there is a drift toward minimum of a cost function. We tested this hypothesis in four-finger isometric pressing tasks that required the accurate production of a combination of total moment and total force with natural and modified finger involvement. Subjects performed accurate force-moment production tasks under visual feedback, and then visual feedback was removed for some or all of the salient variables. Analytical inverse optimization was used to compute a cost function. Without visual feedback, both force and moment drifted slowly toward lower absolute magnitudes. Over 15 s, the force drop could reach 20% of its initial magnitude while moment drop could reach 30% of its initial magnitude. Individual finger forces could show drifts toward both higher and lower forces. The cost function estimated using the analytical inverse optimization reduced its value as a consequence of the drift. We interpret the results within the framework of hierarchical control with referent spatial coordinates for salient variables at each level of the hierarchy combined with synergic control of salient variables. The force drift is discussed as a natural relaxation process toward states with lower potential energy in the physical (physiological) system involved in the task.

Analytical Inverse Optimization in Two-Hand Prehensile Tasks

The authors explored application of analytical inverse optimization (ANIO) method to the normal finger forces in unimanual and bimanual prehensile tasks with discrete and continuously changing constraints. The subjects held an instrumented handle vertically with one or two hands. The external torque and grip force changed across trials or within a trial continuously. Principal component analysis showed similar percentages of variance accounted for by the first two principal components across tasks and conditions. Compared to unimanual tasks, bimanual tasks showed significantly more frequent inability to find a cost function leading to a stable solution. In cases of stable solutions, similar second-order polynomials were computed as cost functions across tasks and condition. The bimanual tasks, however, showed significantly worse goodness-of-fit index values. The authors show that ANIO can be used in tasks with slowly changing constraints making it an attractive tool to study optimality of performance in special populations. They also show that ANIO can fail in multifinger tasks, likely due to irreproducible behavior across trials, more likely to happen in bimanual tasks compared to unimanual tasks.

Effects of Parkinson's disease on optimization and structure of variance in multi-finger tasks

We explored the role of the basal ganglia in two components of multi-finger synergies by testing a group of patients with early-stage Parkinson's disease and a group of healthy controls. Synergies were defined as co-varied adjustments of commands to individual fingers that reduced variance of the total force and moment of force. The framework of the uncontrolled manifold hypothesis was used to quantify such co-variation patterns, while average performance across repetitive trials (sharing patterns) was analyzed using the analytical inverse optimization (ANIO) approach. The subjects performed four-finger pressing tasks that involved the accurate production of combinations of the total force and total moment of force and also repetitive trials at two selected combinations of the total force and moment. The ANIO approach revealed significantly larger deviations of the experimental data planes from an optimal plane for the patients compared to the control subjects. The synergy indices computed for total force stabilization were significantly higher in the control subjects compared to the patients; this was not true for synergy indices computed for moment of force stabilization. The differences in the synergy indices were due to the larger amount of variance that affected total force in the patients, while the amount of variance that did not affect total force was comparable between the groups. We conclude that the basal ganglia play an important role in both components of synergies reflecting optimization of the sharing patterns and stability of performance with respect to functionally important variables.

Optimization and variability of motor behavior in multifinger tasks: what variables does the brain use?

The neural control of movement has been described using different sets of elemental variables. Two possible sets of elemental variables have been suggested for finger pressing tasks: the forces of individual fingers and the finger commands (also called finger modes or central commands). The authors analyzed which of the 2 sets of the elemental variables is more likely used in the optimization of the finger force sharing and which set is used for the stabilization of performance. They used two recently developed techniques-the analytical inverse optimization (ANIO) and the uncontrolled manifold (UCM) analysis-to evaluate each set of elemental variables with respect to both aspects of performance. The results of the UCM analysis favored the finger commands as the elemental variables used for performance stabilization, while ANIO worked equally well on both sets of elemental variables. A simple scheme is suggested as to how the CNS could optimize a cost function dependent on the finger forces, but for the sake of facilitation of the feed forward control it substitutes the original cost function by a cost function, which is convenient to optimize in the space of finger commands.

Optimal isn't good enough

The notion that biological systems come to embody optimal solutions seems consistent with the competitive drive of evolution. It has been used to interpret many examples of sensorimotor behavior. It is attractive from the viewpoint of control engineers because it solves the redundancy problem by identifying the one optimal motor strategy out of many similarly acceptable possibilities. This perspective examines whether there is sufficient basis to apply the formal engineering tools of optimal control to a reductionist understanding of biological systems. For an experimental biologist, this translates into whether the theory of optimal control generates nontrivial and testable hypotheses that accurately predict novel phenomena, ideally at deeper levels of structure than the observable behavior. The methodology of optimal control is applicable when there is (i) a single, known cost function to be optimized, (ii) an invertible model of the plant, and (iii) simple noise interfering with optimal performance. None of these is likely to be true for biological organisms. Furthermore, their motivation is usually good-enough rather than globally optimal behavior. Even then, the performance of a biological organism is often much farther from optimal than the physical limits of its hardware because the brain is continuously testing the acceptable limits of performance as well as just performing the task. This perspective considers an alternative strategy called "good-enough" control, in which the organism uses trial-and-error learning to acquire a repertoire of sensorimotor behaviors that are known to be useful, but not necessarily optimal. This leads to a diversity of solutions that tends to confer robustness on the individual organism and its evolution. It is also more consistent with the capabilities of higher sensorimotor structures, such as cerebral cortex, which seems to be designed to classify and recall complex sets of information, thereby allowing the organism to learn from experience, rather than to compute new strategies online. Optimal control has been a useful metaphor for understanding some superficial aspects of motor psychophysics. Reductionists who want to understand the underlying neural mechanisms need to move on.

Movements that are both variable and optimal

This brief review addresses two major aspects of the neural control of multi-element systems. First, the principle of abundance suggests that the central nervous system unites elements into synergies (co-variation of elemental variables across trials quantified within the framework of the uncontrolled manifold hypothesis) that stabilize important performance variables. Second, a novel method, analytical inverse optimization, has been introduced to compute cost functions that define averaged across trials involvement of individual elements over a range of values of task-specific performance variables. The two aspects reflect two features of motor coordination: (1) using variable solutions that allow performing secondary tasks and stabilizing performance variables; and (2) selecting combinations of elemental variables that follow an optimization principle. We suggest that the conflict between the two approaches (a single solution vs. families of solutions) is apparent, not real. Natural motor variability may be due to using the same cost function across slightly different initial states; on the other hand, there may be variability in the cost function itself leading to variable solutions that are all optimal with respect to slightly different cost functions. The analysis of motor synergies has revealed specific changes associated with atypical development, healthy aging, neurological disorders, and practice. These have allowed formulating hypotheses on the neurophysiological mechanisms involved in the synergic control of actions.

Forces and moments generated by the human arm: variability and control

This is an exploratory study of the accurate endpoint force vector production by the human arm in isometric conditions. We formulated three common-sense hypotheses and falsified them in the experiment. The subjects (n = 10) exerted static forces on the handle in eight directions in a horizontal plane for 25 s. The forces were of 4 magnitude levels (10, 20, 30 and 40 % of individual maximal voluntary contractions). The torsion moment on the handle (grasp moment) was not specified in the instruction. The two force components and the grasp moment were recorded, and the shoulder, elbow, and wrist joint torques were computed. The following main facts were observed: (a) While the grasp moment was not prescribed by the instruction, it was always produced. The moment magnitude and direction depended on the instructed force magnitude and direction. (b) The within-trial angular variability of the exerted force vector (angular precision) did not depend on the target force magnitude (a small negative correlation was observed). (c) Across the target force directions, the variability of the exerted force magnitude and directional variability exhibited opposite trends: In the directions where the variability of force magnitude was maximal, the directional variability was minimal and vice versa. (d) The time profiles of joint torques in the trials were always positively correlated, even for the force directions where flexion torque was produced at one joint and extension torque was produced at the other joint. (e) The correlations between the grasp moment and the wrist torque were negative across the tasks and positive within the individual trials. (f) In static serial kinematic chains, the pattern of the joint torques distribution could not be explained by an optimization cost function additive with respect to the torques. Plans for several future experiments have been suggested.

Muscle coordination is habitual rather than optimal

When sharing load among multiple muscles, humans appear to select an optimal pattern of activation that minimizes costs such as the effort or variability of movement. How the nervous system achieves this behavior, however, is unknown. Here we show that contrary to predictions from optimal control theory, habitual muscle activation patterns are surprisingly robust to changes in limb biomechanics. We first developed a method to simulate joint forces in real time from electromyographic recordings of the wrist muscles. When the model was altered to simulate the effects of paralyzing a muscle, the subjects simply increased the recruitment of all muscles to accomplish the task, rather than recruiting only the useful muscles. When the model was altered to make the force output of one muscle unusually noisy, the subjects again persisted in recruiting all muscles rather than eliminating the noisy one. Such habitual coordination patterns were also unaffected by real modifications of biomechanics produced by selectively damaging a muscle without affecting sensory feedback. Subjects naturally use different patterns of muscle contraction to produce the same forces in different pronation-supination postures, but when the simulation was based on a posture different from the actual posture, the recruitment patterns tended to agree with the actual rather than the simulated posture. The results appear inconsistent with computation of motor programs by an optimal controller in the brain. Rather, the brain may learn and recall command programs that result in muscle coordination patterns generated by lower sensorimotor circuitry that are functionally "good-enough."

The development of lower limb musculoskeletal models with clinical relevance is dependent upon the fidelity of the mathematical description of the lower limb. Part I: Equations of motion

Contemporary musculoskeletal modelling research is based upon the assumption that such models will evolve into clinical tools that can be used to guide therapeutic interventions. However, there are a number of questions that must be addressed before this becomes a reality. At its heart, musculoskeletal modelling is a process of formulating and then solving the equations of motion that describe the movement of body segments. Both of these steps are challenging. This article argues that traditional approaches to musculoskeletal modelling have been heavily influenced by the need to simplify this process (and in particular the solution process), and that this has to some degree resulted in approaches that are contrary to the principles of classical mechanics. It is suggested that future work is required to understand how these simplifications affect the outputs of musculoskeletal modelling studies. Equally, to increase their clinical relevance, the models of the future should adhere more closely to the classical mechanics on which they are based.

Reproducibility and variability of the cost functions reconstructed from experimental recordings in multifinger prehension

The study examines whether the cost functions reconstructed from experimental recordings are reproducible over time. Participants repeated the trials on three days. By following Analytical Inverse Optimization procedures, the cost functions of finger forces were reconstructed for each day. The 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 2nd order coefficients of the cost function showed negative linear relations with finger force magnitudes; (b) the finger forces were distributed on a 2-dimensional plane in the 4-dimensional finger force space for all subjects and all testing sessions; (c) the data agreed well with the principle of superposition, i.e. the action of object prehension can be decoupled into the control of rotational equilibrium and slipping prevention.

Optimality versus variability: effect of fatigue in multi-finger redundant tasks

We used two methods to address two aspects of multi-finger synergies and their changes after fatigue of the index finger. Analytical inverse optimization (ANIO) was used to identify cost functions and corresponding spaces of optimal solutions over a broad range of task parameters. Analysis within the uncontrolled manifold (UCM) hypothesis was used to quantify co-variation of finger forces across repetitive trials that helped reduce variability of (stabilized) performance variables produced by all the fingers together. Subjects produced steady-state levels of total force and moment of force simultaneously as accurately as possible by pressing with the four fingers of the right hand. Both before and during fatigue, the subjects performed single trials for many force-moment combinations covering a broad range; the data were used for the ANIO analysis. Multiple trials were performed at two force-moment combinations; these data were used for analysis within the UCM hypothesis. Fatigue was induced by 1-min maximal voluntary contraction exercise by the index finger. Principal component (PC) analysis showed that the first two PCs explained over 90% of the total variance both before and during fatigue. Hence, experimental observations formed a plane in the four-dimensional finger force space both before and during fatigue conditions. Based on this finding, quadratic cost functions with linear terms were estimated from the experimental data. The dihedral angle between the plane of optimal solutions and the plane of experimental observations (D (ANGLE)) was very small (a few degrees); it increased during fatigue. There was an increase in fatigue of the coefficient at the quadratic term for the index finger force balanced by a drop in the coefficients for the ring and middle fingers. Within each finger pair (index-middle and ring-little), the contribution of the "central" fingers to moment production increased during fatigue. An index of antagonist moment production dropped with fatigue. Fatigue led to higher co-variation indices during pronation tasks (index finger is an agonist) but opposite effects during supination tasks. The results suggest that adaptive changes in co-variation indices that help stabilize performance may depend on the role of the fatigued element, agonist or antagonist.

Reconstruction of the unknown optimization cost functions from experimental recordings during static multi-finger prehension

The goal of the research is to reconstruct the unknown cost (objective) function(s) presumably used by the neural controller for sharing the total force among individual fingers in multifinger prehension. The cost function was determined from experimental data by applying the recently developed Analytical Inverse Optimization (ANIO) method (Terekhov et al. 2010). The core of the ANIO method is the Theorem of Uniqueness that specifies conditions for unique (with some restrictions) estimation of the objective functions. In the experiment, subjects (n = 8) grasped an instrumented handle and maintained it at rest in the air with various external torques, loads, and target grasping forces applied to the object. The experimental data recorded from 80 trials showed a tendency to lie on a 2-dimensional hyperplane in the 4-dimensional finger-force space. Because the constraints in each trial were different, such a propensity is a manifestation of a neural mechanism (not the task mechanics). In agreement with the Lagrange principle for the inverse optimization, the plane of experimental observations was close to the plane resulting from the direct optimization. The latter plane was determined using the ANIO method. The unknown cost function was reconstructed successfully for each performer, as well as for the group data. The cost functions were found to be quadratic with nonzero linear terms. The cost functions obtained with the ANIO method yielded more accurate results than other optimization methods. The ANIO method has an evident potential for addressing the problem of optimization in motor control.

Evidence for composite cost functions in arm movement planning: an inverse optimal control approach

An important issue in motor control is understanding the basic principles underlying the accomplishment of natural movements. According to optimal control theory, the problem can be stated in these terms: what cost function do we optimize to coordinate the many more degrees of freedom than necessary to fulfill a specific motor goal? This question has not received a final answer yet, since what is optimized partly depends on the requirements of the task. Many cost functions were proposed in the past, and most of them were found to be in agreement with experimental data. Therefore, the actual principles on which the brain relies to achieve a certain motor behavior are still unclear. Existing results might suggest that movements are not the results of the minimization of single but rather of composite cost functions. In order to better clarify this last point, we consider an innovative experimental paradigm characterized by arm reaching with target redundancy. Within this framework, we make use of an inverse optimal control technique to automatically infer the (combination of) optimality criteria that best fit the experimental data. Results show that the subjects exhibited a consistent behavior during each experimental condition, even though the target point was not prescribed in advance. Inverse and direct optimal control together reveal that the average arm trajectories were best replicated when optimizing the combination of two cost functions, nominally a mix between the absolute work of torques and the integrated squared joint acceleration. Our results thus support the cost combination hypothesis and demonstrate that the recorded movements were closely linked to the combination of two complementary functions related to mechanical energy expenditure and joint-level smoothness.

To reach an object, the brain has to select among a set of possible arm trajectories that displace the hand from an initial to a final desired position. Because of the intrinsic redundancy characterizing the human arm, the number of admissible joint trajectories toward the goal is generally infinite. However, many studies have demonstrated that the range of actual trajectories can be limited to those that result from the fulfillment of some optimal rules. Various cost functions were shown to be relevant in the literature. A peculiar aspect of most of these costs is that each one of them aims at optimizing one specific feature of the movement. The necessary motor flexibility of everyday life, however, might rely on the combination of such cost functions rather than on a single one. Testing this cost combination hypothesis has never been attempted. To this aim we propose a reaching task involving target redundancy to facilitate the comparisons of different candidate costs and to identify the best-fitting one (possibly composite). Using a numerical inverse optimal control method, we show that most participants produced movements corresponding to a strict combination of two subjective costs linked to the mechanical energy consumption and the joint-level smoothness.

Finger coordination under artificial changes in finger strength feedback: a study using analytical inverse optimization

A recently developed method of analytical inverse optimization (ANIO) was used to compute cost functions based on sets of experimental observations in 4-finger pressing tasks with accurate total force and moment production. In different series, feedback on total force and moment was provided using the index finger force at its value, doubled, or halved. Finger force data across different force-moment combinations formed a plane. This allowed reconstructing cost functions as 2nd-order polynomials with linear terms. Changes in the coefficients of the cost function across the 3 series allowed the authors to offer a biomechanical interpretation related to constraints on finger forces with different lever arms. ANIO allows the authors to describe preferred regions within the space of solutions for redundant tasks in terms of cost functions.