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

Abstract

Several types of continuous human movements comply with the so-called Two-Thirds Power Law (2/3-PL) stating that velocity (V) is a power function of the radius of curvature (R) of the endpoint trajectory. The origin of the 2/3-PL has been the object of much debate. An experiment investigated further this issue by comparing two-dimensional drawing movements performed in air and water. In both conditions, participants traced continuously quasi-elliptic trajectories (period T = 1.5 s). Other experimental factors were the movement plane (horizontal/vertical), and whether the movement was performed free-hand, or by following the edge of a template. In all cases a power function provided a good approximation to the V-R relation. The main result was that the exponent of the power function in water was significantly smaller than in air. This appears incompatible with the idea that the power relationship depends only on kinematic constraints and suggests a significant contribution of dynamic factors. We argue that a satisfactory explanation of the observed behavior must take into account the interplay between the properties of the central motor commands and the visco-elastic nature of the mechanical plant that implements the commands

The speed–curvature power law in Drosophila larval locomotion

“…Therefore, not only do we find in the larvae the geometric–kinematic constraint dictated by the power law, but also hints of dynamic constraints in the power exponent as recently found in humans [2,5,8,17], where its specific value depends on the viscosity of the medium (air or water for hand drawing, agar support and thin liquid coat for larval locomotion) and the trajectory shape.…”

“…The power exponent for human hand drawing is generally close to 0.66 (so-called two-thirds power law [1]), but it becomes 0.73 when drawing in water [17], the latter value being close to the present values in larvae. Therefore, not only do we find in the larvae the geometric–kinematic constraint dictated by the power law, but also hints of dynamic constraints in the power exponent as recently found in humans [2,5,8,17], where its specific value depends on the viscosity of the medium (air or water for hand drawing, agar support and thin liquid coat for larval locomotion) and the trajectory shape.…”

Do bumblebees have signatures? Demonstrating the existence of a speed-curvature power law in Bombus terrestris locomotion patterns

“…In support of this notion, the power exponent recorded for human drawing shifts closer to this value of three-quarters (0.73) when drawing underwater [28], suggesting that power laws can indeed be governed by kinematic constraints. Our analyses suggest that, in walking bumblebees, a power law exponent between 0.55 and 0.66 (two-thirds) better defines movements than the near 0.75 exponents previously reported for Drosophila [5] and constrained human movements [28]. Our evidence further supports the idea that exponents are forced closer to the three-quarters value when kinematic constraints are present, as our constraint-free bees have a generally much lower exponent at closer to two thirds.…”

“…The researchers suggest that these findings prove a role for dynamic effects adding on purely kinematic constraints [5]. In support of this notion, the power exponent recorded for human drawing shifts closer to this value of three-quarters (0.73) when drawing underwater [28], suggesting that power laws can indeed be governed by kinematic constraints. Our analyses suggest that, in walking bumblebees, a power law exponent between 0.55 and 0.66 (two-thirds) better defines movements than the near 0.75 exponents previously reported for Drosophila [5] and constrained human movements [28].…”

The velocity-curvature power law inDrosophilalarval locomotion

“…A few individuals did not comply with the law (especially in the dispersal condition, see outliers in Figures 1H,I), corroborating that it is not an obligatory outcome of our analyses. For the sake of comparison, the exponent for human hand-drawing is generally close to 0.66 (so called 2/3-powerlaw [1]), but it becomes 0.73 when drawing in water [12], the latter value being close to the present values in larvae. Therefore, not only do we find in the larvae the geometric-kinematic constraint dictated by the power law, but also hints of dynamic constraints in the exponent [2,5,12], as recently found in humans, where the specific value of the exponent appears to depend on the viscosity of the medium (air or water for hand-drawing, agar support and thin liquid coat for larvae).…”