Quantitative model of transport-aperture coordination during reach-to-grasp movements

Abstract

It has been found in our previous studies that the initiation of aperture closure during reach-to-grasp movements occurs when the hand distance to target crosses a threshold that is a function of peak aperture amplitude, hand velocity, and hand acceleration. Thus, a stable relationship between those four movement parameters is observed at the moment of aperture closure initiation. Based on the concept of optimal control of movements (Naslin 1969) and its application for reach-to-grasp movement regulation (Hoff and Arbib 1993), it was hypothesized that the mathematical equation expressing that relationship can be generalized to describe coordination between hand transport and finger aperture during the entire reach-to-grasp movement by adding aperture velocity and acceleration to the above four movement parameters. The present study examines whether this hypothesis is supported by the data obtained in experiments in which young adults performed reach-to-grasp movements in eight combinations of two reach-amplitude conditions and four movement-speed conditions. It was found that linear approximation of the mathematical model described the relationship among the six movement parameters for the entire aperture-closure phase with very high precision for each condition, thus supporting the hypothesis for that phase. Testing whether one mathematical model could approximate the data across all the experimental conditions revealed that it was possible to achieve the same high level of data-fitting precision only by including in the model two additional, condition-encoding parameters and using a nonlinear, artificial neural network-based approximator with two hidden layers comprising three and two neurons, respectively. This result indicates that transport-aperture coordination, as a specific relationship between the parameters of hand transport and finger aperture, significantly depends on the condition-encoding variables. The data from the aperture-opening phase also fit a linear model, whose coefficients were substantially different from those identified for the aperture-closure phase. This result supports the above hypothesis for the aperture-opening phase, and consequently, for the entire reach-to-grasp movement. However, the fitting precision was considerably lower than that for the aperture-closure phase, indicating significant trial-to-trial variability of transport-aperture coordination during the aperture-opening phase. Implications for understanding the neural mechanisms employed by the CNS for controlling reach-to-grasp movements and utilization of the mathematical model of transport-aperture coordination for data analysis are discussed.

Appendix

Derivation of an equation describing coordination between hand transport and grasp aperture

Reach-to-grasp movements can be viewed as consisting of two independently and optimally controlled processes (hand transport and grasping), which are required to finish simultaneously. Note that their independence here is understood as the independence of the corresponding optimality criteria. Simple examples of such criteria are described, e.g., in Hoff and Arbib (1993). The final state for hand position and aperture is determined by the target’s location and size, respectively. A control action for regulating the transport and the aperture can be presented, e.g., as hand transport acceleration and as finger aperture acceleration, respectively (Hoff and Arbib 1993). Hence, an optimal control law for hand transport and grasp aperture can be described by the following set of equations

$$ A_{\text{w}} = f_{\text{w}} (D,\,V_{\text{w}} ,\,T) $$

(10)

and

$$ A_{\text{g}} = f_{\text{g}} (G,\,V_{\text{g}} ,\,T), $$

(11)

where D, V _w, and A _w are the instantaneous values of hand–target distance, wrist velocity, and wrist acceleration, respectively; G, V _g, and A _g are grip aperture and the corresponding velocity and acceleration, respectively; and T is the amount of time left to target contact. Note that the explicit inclusion of T in the above control laws does not imply that that parameter is prescribed. On the contrary, in this case, it is a variable requiring optimization (Naslin 1969). If two control processes are required to finish simultaneously, the optimization of movement time left to finish has to be performed according to a generalized optimality criterion that can be expressed, e.g., as a weighted sum of such criteria corresponding to the control of hand transport and grip aperture if optimized separately from each other. By excluding T from the above equation set (Eqs. 10, 11), one obtains an equation describing a functional relationship between all other movement parameters:

$$ f(D,\,V_{\text{w}} ,\,A_{\text{w}} ,\,G,\,A_{\text{g}} ,\,V_{\text{g}} ) = 0. $$

(12)

This equation is rather general because it does not depend on the results of optimizing T. By solving Eq. 12 with respect to D, one obtains

$$ D = D(A_{\text{w}} ,\,V_{\text{w}} ,\,G,\,A_{\text{g}} ,\,V_{\text{g}} ). $$

(13)

It should be emphasized that the existence of Eq. 13 is based on an assumption that both the hand transport and grasp aperture are regulated optimally. Therefore, the precision with which any related experimental data can be approximated based on that equation strongly depends on how close to optimality are the control actions regulating the above two processes.