Two-thirds power law in human locomotion: role of ground contact forces

Abstract

Are there general rules for the generation of curvilinear motion of the end-effector? Form and kinematics of the arm trajectory are typically inter-related. A relationship between velocity and curvature of the endpoint path has been previously described and quantified as the two-thirds power law. Here we show that the two-thirds power law holds also for the foot trajectory (during the swing phase) in human locomotion for a wide range of walking speeds and gravitational loads, but air-stepping. In air-stepping, it was violated systematically. The results suggest that the power law represents a general constrain of biological motion, may be attributed to both mechanical and neural factors and can depend on natural interactions with external environment.

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

“…Specifically, the exponent of the power law is significantly smaller in water than in air, indicating a tendency for velocity to be less dependent on the curvature of the trajectory than it is in air. Violations of the 2/3-PL have emerged previously from the analysis of air-stepping (Ivanenko et al 2002), articulatory speech (Perrier and Fuchs 2008), and some 3D movements (Schaal and Sternad 2001). It has also been reported that the law is inaccurate in the case of slow movements and movements with strongly non-symmetrical velocity profiles (Wann et al 1988).…”

“…Since the original demonstration, it has been reported that the law also applies to different types of motion, such as tongue movements during speech (Tasko and Westbury 2002;Perrier and Fuchs 2008), locomotive trajectories (Vieilledent et al 2001;Ivanenko et al 2002;Hicheur et al 2005;Pham et al 2007), and smooth eye movements (de'Sperati and Viviani 1997). It was also found that the 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 Dynamics of Legged Locomotion: Models, Analyses, and Challenges

“…Apparently associated with this dynamical collapse is a posture principle: the restriction of motion to a low dimensional subspace within a high dimensional jointspace. A kinematic posture principle has been discovered in mammalian walking [LGZ99], as demonstrated by planar covariation of limb elevation angles which persists in the face of large variations in steady state loading conditions [IGML02]. More directly relevant to the models to be developed in this paper, a preliminary study of kinematic posture in a running cockroaches using principal components analysis [KFK03] also reveals very low-dimensional linear covariation in joint space (cf.…”

“…For example, a power law inversely relating speed to path curvature, originally derived from observations of voluntary reaching movements [LTV83], has been proposed to describe diverse mammalian motor patterns, including walking [IGML02]. Moreover, primate motor cortex recordings of voluntary arm movements [SM99] reveal a neural velocity 'reference signal' that precedes and predicts observed mechanical trajectories, prescribing via variable time delay the power law of [LTV83].…”

Kinematic and EMG Determinants in Quadrupedal Locomotion of a Non-Human Primate (Rhesus)

“…Treadmill speed-related increases in the length (speed effect, P Ͻ 0.01) and height (speed effect, P Ͻ 0.01) of the path of the endpoint trajectories are seen for both the HL and FL. Ivanenko et al (2002) showed that in human locomotion, the foot trajectory obeys the socalled two-thirds power relationship between the instantaneous curvature (C) and angular velocity (⍀), i.e., the exponent ␤ of the ⍀-C relationship is very close to two-thirds (see METH-ODS). We assessed whether the same relationship characterizes HL and FL endpoint trajectories in the Rhesus.…”