Motor equivalence and self-motion induced by different movement speeds

Abstract

This study investigated pointing movements in 3D asking two questions: (1) Is goal-directed reaching accompanied by self-motion, a component of the joint velocity vector that leaves the hand’s movement unaffected? (2) Are differences in the terminal joint configurations among different speeds of reaching motor equivalent (i.e., terminal joint configurations differ more in directions of joint space that do not produce different pointer-tip positions than in directions that do) or non-motor equivalent (i.e., terminal joint configurations differ equally or more in directions of joint space that lead to different pointer-tip positions than in directions that do not affect the pointer-tip position). Subjects reached from an identical starting joint configuration and pointer-tip location to targets at slow, moderate, and fast speeds. Ten degrees of freedom of joint motion of the arm were recorded. The relationship between changes in the joint configuration and the three-dimensional pointer-tip position was expressed by a standard kinematic model, and the range- and null subspaces were computed from the associated Jacobian matrix. (1) The joint velocity vector and (2) the difference vector between terminal joint configurations from pairs of speed conditions were projected into the two subspaces. The relative length of the two components was used to quantify the amount of self-motion and the presence of motor equivalence, respectively. Results revealed that reaches were accompanied by a significant amount of self-motion at all reaching speeds. Self-motion scaled with movement speed. In addition, the difference in the terminal joint configuration between pairs of different reaching speeds revealed motor equivalence. The results are consistent with a control system that takes advantage of motor redundancy, allowing for flexibility in the face of perturbations, here induced by different movement speeds.

Appendix

Self-motion related to the three-dimensional movement of the hand refers to a component of the joint velocity vector that does not affect hand motion (Murray et al. 1994). Self-motion is only possible if joint motions are redundant. Although self-motion could be considered wasted motion with respect to movement efficiency, the ability to produce self-motion is crucial for the performance of multiple tasks simultaneously. For example, the redundancy of angular motion of arm joints allows one to flip a light switch with the elbow while carrying and stabilizing a tray of drinks, i.e., without negatively impacting the hand’s movement. Thus, the component of the joint velocities linked to flipping the switch is self-motion when viewed with respect to hand movement. In principle, without a specific secondary task as was the case in the current experiment, one could expect self-motion to be minimal, most of the joint velocity vector acting to move the hand in space.

The conceptual framework of the method used here to differentiate between the component of the joint velocity vector that leads to hand movement, referred to by its technical term range-space motion (Murray et al. 1994), and the self-motion component is based on the uncontrolled manifold (UCM) variance analysis (Scholz and Schoner 1999). However, this analysis does not examine variances. The method can be illustrated by a simple example. Consider pointing with the hand in the horizontal plane where only three joint angles are available to transport the hand, shoulder horizontal flexion–extension, elbow flexion–extension, and wrist flexion–extension. To reach a target, at least two DOFs of joint motion are required, equivalent to the required two-dimensional hand position. Thus, there is one free joint DOF in this redundant system. That is, there exists a one-dimensional subspace (Fig. 5) within the space of the three arm joint motions within which a set of different combinations of shoulder, elbow, and wrist angles lead to an identical two-dimensional hand position. This subspace has been referred to as the uncontrolled manifold (UCM; Scholz and Schoner 1999) and a different UCM exists at each time point due to changing arm geometry. Thus, the first step in the self-motion analysis, like UCM analysis, was to estimate the UCM subspace at each data sample. First, the geometric model relating joint motion to hand motion is obtained as follows for the illustration:

$$ \left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {l_{\text{arm}} *\cos \left( {\theta_{\text{shoulder}} } \right) + l_{\text{forearm}} *\cos \left( {\theta_{\text{shoulder}} + \theta_{\text{elbow}} } \right) + l_{\text{hand}} *\cos \left( {\theta_{\text{shoulder}} + \theta_{\text{elbow}} + \theta_{\text{wrist}} } \right)} \\ {l_{\text{arm}} *\sin \left( {\theta_{\text{shoulder}} } \right) + l_{\text{forearm}} *\sin \left( {\theta_{\text{shoulder}} + \theta_{\text{elbow}} } \right) + l_{\text{hand}} *\sin \left( {\theta_{\text{shoulder}} + \theta_{\text{elbow}} + \theta_{\text{wrist}} } \right)} \\ \end{array} } \right]. $$

The Jacobian (J) of the geometric model is then computed using each instantaneous joint configuration ‘i’, [θ ⁱ_shoulder , θ ⁱ_elbow , θ ⁱ_wrist ]^T, of each trial. The Jacobian is defined as the matrix of partial derivatives of the hand coordinates relative to the joint angles,

$$ J(\theta ) = \left[{\begin{array}{*{20}c} {\partial x/\partial \theta_{\text{shoulder}} \;\partial x/\partial \theta_{\text{elbow}} \;\partial x/\partial \theta_{\text{wrist}} } \\ {\partial y/\partial \theta_{\text{shoulder}} \;\partial y/\partial \theta_{\text{elbow}} \;\partial x/\partial \theta_{\text{wrist}} } \\ \end{array}}\right ]$$

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Fig. 5

Cartoon examples of self-motion and range-space motion estimates from linear estimates of uncontrolled manifolds (UCMs). JC = joint configurations; Upper panel shows an example where there would be a relatively large amount of self-motion. The lower panel is an example of low self-motion. See “Appendix” for details

The Jacobian is used to transform changes in joint angles or angular velocities into hand or more generally end-effector velocities, i.e., dr/dt = J(θ) * dθ/dt.

The null space of the Jacobian is then computed by singular value decomposition of the Jacobian in Matlab™, [u, s, v] = svd(J^T), where ‘T’ is the transpose of the Jacobian, ‘u’ is a matrix of eigenvectors, and ‘s’ is the diagonal matrix of eigenvalues. The first two columns of u are the range-space vectors, and the last column is the null-space vector in this 3D example.

The final step in the self-motion analysis is to project the instantaneous joint velocity vector [dθ ⁱ_shoulder /dt, dθ ⁱ_elbow /dt, dθ ⁱ_wrist /dt]^T into the range space and null space and then compute the lengths of projection.

Figure 5 provides a cartoon illustration of the results of this procedure. In the figure, three hypothetical one-dimensional UCMs representing three consecutive time points are illustrated. The solid filled circles lying on each UCM represent joint angle combinations, or joint configurations, for a given trial at each time point. Note that given the definition of the UCM, combinations of joints lying anywhere along each line would produce the same 2D hand position (but a different hand position for each UCM). Thus, movement of the hand can be conceived of as a transition among sequences of UCMs. The heavy dashed line in each panel represents the change in joint configuration, or the joint velocity vector. If most of the joint velocity vectors act to move the hand in space, then this vector should be relatively perpendicular to the UCMs. If, instead, there is a significant amount of self-motion, then the vector will form a more acute angle with the UCMs. The upper panel is an illustration of the later situation. The lower panel represents a hand movement accompanied by little self-motion. The different situations are estimated by the lengths of projection of the joint velocity vector into each subspace, i.e., the null space (self-motion) and its complement (range-space motion). These projection lengths are also illustrated in the figure.