Movement Timing and Invariance Arise from Several Geometries

Abstract

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations.

Spectrum of power laws for curved hand movements

“…For example, the equi-affine-speed hypothesis, which assumes the affine-speed of human movement should be held constant, can only generate movements that follow the one-third power law for all planar movements (4,7), which is inconsistent with our experimental results. However, a recent generalization of the model, which defines speed in a combination of Euclidean, affine, and equiaffine geometries, can generate variable power law exponents (13). The equilibrium point hypothesis paired with low-passfiltering properties of muscles exhibits one-third power law behavior for ellipse drawing movements, if the equilibrium point moves along the elliptic trajectory with constant speed (5,6).…”

“…2B) (9,10,13). This frame naturally separates magnitude and directional information of a movement: (i) An instantaneous velocity vector is represented by its speed and tangent direction, and (ii) a velocity profile along the path is represented by two scalar profiles, the speed profile and the curvature profile; the latter describes how the tangent vector rotates along the path.…”

“…This is because the time parameterization, and thus the path shape defined by fκðtÞg, depends on the speed profile. In previous approaches, this problem was resolved by replacing the time coordinate with the arc-length coordinate (9,10,13). Here, we instead introduce the orientation of the tangent vector as the new coordinate system, the angle coordinate, θ (Fig.…”

Drawing ellipses in water: evidence for dynamic constraints in the relation between velocity and path curvature

“…Another kinematic constraint that can predict the power law is provided by the principle of constant equi-affine speed (Pollick and Sapiro 1997;Flash and Handzel 2007;Pollick et al 2009) or a combination of Euclidean, affine, and equi-affine geometries (Bennequin et al 2009). Interestingly, the principle of constant equi-affine speed leads to a generalization of the power law to three-dimensional (3D) movements Pollick et al 2009).…”

Affine trajectory deformation for redundant manipulators

“…Indeed, inspired by the recent finding that human movements are -to some extent -invariant under affine transformations [1], we deform a given trajectory by applying affine transformations on parts of it. Thus, the only "basis functions" are the original joint angle time series and, as a consequence, any deformed trajectory under this scheme automatically preserves some properties of the original one.…”