Shaping of arm configuration space by prescription of non-Euclidean metrics with applications to human motor control

Armin Biess

Phys. Rev. E 87, 012729 – Published 31 January 2013

Abstract

The study of the kinematic and dynamic features of human arm movements provides insights into the computational strategies underlying human motor control. In this paper a differential geometric approach to movement control is taken by endowing arm configuration space with different non-Euclidean metric structures to study the predictions of the generalized minimum-jerk (MJ) model in the resulting Riemannian manifold for different types of human arm movements. For each metric space the solution of the generalized MJ model is given by reparametrized geodesic paths. This geodesic model is applied to a variety of motor tasks ranging from three-dimensional unconstrained movements of a four degree of freedom arm between pointlike targets to constrained movements where the hand location is confined to a surface (e.g., a sphere) or a curve (e.g., an ellipse). For the latter speed-curvature relations are derived depending on the boundary conditions imposed (periodic or nonperiodic) and the compatibility with the empirical one-third power law is shown. Based on these theoretical studies and recent experimental findings, I argue that geodesics may be an emergent property of the motor system and that the sensorimotor system may shape arm configuration space by learning metric structures through sensorimotor feedback.

Received 21 July 2012

DOI:https://doi.org/10.1103/PhysRevE.87.012729

Authors & Affiliations

Armin Biess^*

Bernstein Center for Computational Neuroscience and Max-Planck-Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

^*armin@nld.ds.mpg.de

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Vol. 87, Iss. 1 — January 2013

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Images

Figure 1
An arm configuration can be defined by different sets of coordinates, for example, by four joint angular coordinates $q = {(θ, η, ζ, ϕ)}^{T}$ . Another possible set of generalized coordinates consists of the center of mass of the hand location $x$ and the swivel angle $α$ around the shoulder-hand axis, i.e., $q^{'} = {(x, y, z, α)}^{T}$ . The shoulder joint is fixated and located at the origin of the $X Y Z$ coordinate system. The vector $p$ denotes the elbow location, $d = | x |$ is the shoulder-hand distance, $r$ is the radial distance to the elbow location from the shoulder-hand axis, $l_{1}$ denotes the upper arm length, and $l_{2}$ is the distance from the elbow joint to the center of mass of the hand (forearm and hand are modeled as one rigid body). The insert shows the definition of the swivel angle $α$ .Reuse & Permissions
Figure 2
Predictions of the geodesic model with task space metric (32) for movements between pointlike targets in a fronto-parallel plane. (a) Predictions of hand paths to pointlike targets arranged in a fronto-parallel plane. The hand paths (shown in gray) vary and lead to different final arm postures at the target locations. The straight hand paths (shown in blue) correspond to the shortest geodesics connecting the initial with the final targets. (b) Normalized speed profiles, $\tilde{v} = v / (S / T)$ , for movements to the upper-middle target. $S$ is the Euclidean arc-length of the hand path and $T$ denotes the total movement time. (c) Swivel angle trajectories for movements to the upper-middle target. (d) Radial distance to the elbow location from the shoulder-hand axis for movements to the upper-middle target. The profiles shown in black correspond to the movement along the shortest geodesic to the upper-middle target. Simulation parameters are provided in Table .Reuse & Permissions
Figure 3
Predictions of the geodesic model with kinetic energy metric for movements between pointlike targets in a fronto-parallel plane. Same settings as in Fig. 2. The components of the metric tensor are provided in [23]. Predictions of hand paths to the same point-like targets as in Fig. 1. (a) The shortest geodesics connecting the initial with the final targets are shown in blue. Normalized speed profiles (b), swivel angles (c), and elbow radial distances (d) to the upper-middle target. The profiles shown in black correspond to the movement along the shortest geodesic to the upper-middle target. Simulation parameters are provided in Table .Reuse & Permissions
Figure 4
Predictions of the geodesic model with task space metric (56) for movements between pointlike targets on a sphere. (a) Predictions of hand paths. Note that the great circle path is obtained for constant swivel angle. (b) Normalized speed profiles (gray), $\tilde{v} = v / (S / T)$ , for movements along the hand paths shown in (a). The solid line shows the speed profile along the great circle path. The dashed line is the speed profile predicted by the classical MJ model when constrained to the sphere. (c) Swivel angle trajectories. (d) Radial distance to the elbow location from the shoulder-hand axis. Simulation parameters are provided in Table .Reuse & Permissions
Figure 5
Predictions of speed profiles resulting from the geodesic model for periodic (solid) and nonperiodic (dashed) tracing movements of an ellipse (a), cloverleaf (b), limaçon (c), and asymmetric lemniscate (d). In the case of nonperiodic boundary conditions the contours were traced once with zero initial and final velocity. All contours have the same perimeter. Note that for the lemniscate contour the generalized speed-curvature relations, (102) and (103), are used. The dot denotes the zero angle position and the arrow indicates the direction of increasing angle. Simulation parameters are provided in Table .Reuse & Permissions
Figure 6
Comparison of the original one-third power law (dashed), $v / g = k^{- 1 / 3}$ , with the modified power law (solid), $v / g = {(k^{2} + μ^{2})}^{- 1 / 6}$ , for $μ = 0.1$ . The modified power law is a symmetric function in $k$ and has no singularity at $k = 0$ .Reuse & Permissions

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