The Strange Attractor Model of Bipedal Locomotion and its Consequences on Motor Control

TL;DR

This work advances a strange attractor model of human gait where the center of mass (CoM) dynamics are described as a harmonic oscillator on a saddle-based manifold under gravity. It derives closed-form equations for the CoM trajectory, foot swing (vCoRs), and ankle strategies, calibrated from motion capture data of 12 healthy subjects walking at three speeds, and validates the model against experimental trajectories. The results show that ankle strategies regulate the vertical CoM motion and double-support duration, producing human-like 3D CoM and vCoR trajectories and speed-dependent gait adjustments, consistent with empirical findings and prior theory. The authors propose an integrated motor-control framework linking dynamic primitives, task-space planning on Saddle Space, decentralised control with IBoS/BBoS concepts, and CPG-based execution, with implications for rehabilitation and assistive technologies that leverage gravity-driven attractor dynamics for energy-efficient, stable gait.

Abstract

Despite decades of study, many unknowns exist about the mechanisms governing human locomotion. Current models and motor control theories can only partially capture the phenomenon. This may be a major cause of the reduced efficacy of lower limb rehabilitation therapies. Recently, it has been proposed that human locomotion can be planned in the task-space by taking advantage of the gravitational pull acting on the Centre of Mass (CoM) by modelling the attractor dynamics. The model proposed represents the CoM transversal trajectory as a harmonic oscillator propagating on the attractor manifold. However, the vertical trajectory of the CoM, controlled through ankle strategies, has not been accurately captured yet. Research Questions: Is it possible to improve the model accuracy by introducing a mathematical model of the ankle strategies by coordinating the heel-strike and toe-off strategies with the CoM movement? Our solution consists of closed-form equations that plan human-like trajectories for the CoM, the foot swing, and the ankle strategies. We have tested our model by extracting the biomechanics data and postural during locomotion from the motion capture trajectories of 12 healthy subjects at 3 self-selected speeds to generate a virtual subject using our model. Our virtual subject has been based on the average of the collected data. The model output shows our virtual subject has walking trajectories that have their features consistent with our motion capture data. Additionally, it emerged from the data analysis that our model regulates the stance phase of the foot as humans do. The model proves that locomotion can be modelled as an attractor dynamics, proving the existence of a nonlinear map that our nervous system learns. It can support a deeper investigation of locomotion motor control, potentially improving locomotion rehabilitation and assistive technologies.

Figures (6)

• Figure 1: (a) The mechanical bipedal structure used to formulate the model is composed of two identical legs connected to the CoM via spherical joints. Each leg comprises two rigid bodies ($L_{Leg}$ and the foot) connected via the revolute joint at the ankle. The feet are modelled as two triangles, with one vertex in the ankle joint, the second congruent to the middle point of the segment connecting the metatarsal joints and the last vertex in the heel. $CoR_R$ and $CoR_L$ are the projections on the foot sole of the triangle formed by the heel, first and fifth metatarsal markers. $vCoR_R$ and $vCoR_L$ are the projections on the ground of the CoRs used for the planning. They coincide with the CoRs when the feet are in contact with the ground. The lengths of the two pendula ($LP_R$ and $LP_L$) are defined as the Cartesian distances between the CoM and the vCoRs. (b) The Heel-Strike is modelled as a rotation about the $CIR_{HS}$ of the foot. (c) The Toe-Off is modelled as a rotation about the $CIR_{TO}$. (d) The planner uses two reference systems. The Task-Space frame (TS) is aligned with the anatomical planes, and it is used to describe walking trajectories. The Saddle-Space frame (Saddle) is used to describe posture-dependent quantities, which are later projected in the TS.
• Figure 2: The CoM trajectories generated by the proposed planner in the transverse, sagittal and frontal planes for speed between 0.6 and 2.2 m/s. It is important to notice how the frontal view shows the typical butterfly shape of the strange attractors. Furthermore, Orendurff et al observed in humans a similar behaviour to the generated by the proposed mode, where the vertical amplitude of the CoM trajectory increases with speed, the mediolateral amplitude decreases, and the CoM approaches its minimum height closer to zero.
• Figure 3: The anteroposterior trajectories of the vCoRs for different walking speed shows the intrinsic regulation of the support and swing phases. A lower speeds, the support phase dominates the gait cycle, but this dominance gets mitigated at higher speeds.
• Figure 4: The model can generate foot swing, mediolateral and vertical trajectories that are within the 95% confidence level of the human data. Furthermore, the curves are very close to the average human behaviour.
• Figure 5: The data at the slow, normal and fast speed for one of the subjects' trajectories on the 3 planes. The trajectories show confirms that the model describes idealised locomotion strategies, which are then affected by local conditions (e.g., body state and environmental interaction). The One Step Frontal Plane highlights this, showing how, when a single step is isolated, we observed a similar trajectory adaptation to the velocity change as in the trajectories generated in the model shown in \ref{['fig2']}.