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

Carlo Tiseo; Ming Jeat Foo; Kalyana C Veluvolu; Arturo Forner-Cordero; Wei Tech Ang

cs.ROFeb 10, 2018

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

Authors: Carlo Tiseo, Ming Jeat Foo, Kalyana C Veluvolu, Arturo Forner-Cordero, Wei Tech Ang

Organizations: School of Engineering and Informatics, University of Sussex, Sussex House, Falmer, Brighton, BN1 9RH, England, United Kingdom · Rehabilitation Research Institute of Singapore, Nanyang Technological University, 50 Nanyang Avenue N3-01a-01, 639798, Singapore · School of Mechanical & Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue N3-01a-01, 639798, Singapore · School of Informatics, University of Edinburgh, , 1.17 Bayes Centre 47 Potterrow, Edinburgh, EH8 9BT, Scotland, United Kingdom ·

Abstract

Despite decades of research, the dynamic mechanisms governing human locomotion remain incompletely understood. Current motor control models only partially capture these complex dynamics, a theoretical gap that likely limits the clinical efficacy of modern lower-limb rehabilitation therapies. Recently, it has been proposed that human locomotion is planned in task-space by explicitly exploiting the gravitational pull acting on the Center of Mass (CoM). Building upon this concept, this paper presents a novel deterministic task-space planner for straight, constant-speed locomotion. Our model formulates the CoM trajectory in the transverse plane as a harmonic oscillator moving at a constant forward velocity, while demonstrating that the vertical CoM trajectory is actively regulated through localized ankle strategies. We derive closed-form equations that successfully generate human-like trajectories for both the CoM and the swing foot. The model's predictive accuracy is empirically validated against the kinematic behavior of 12 healthy subjects walking at three self-selected speeds. Furthermore, the proposed planner is mathematically compatible with an extended formulation of the Passive Motion Paradigm ( $λ_0$ -PMP), establishing a comprehensive theoretical foundation for decentralized, hierarchical architectures of semi-autonomous controllers in both human motor control and assistive robotics.

Figures & tables

Figure 1: (a) The mechanical bipedal structure used to formulate the model consists of two identical legs connected to the CoM via spherical joints. Each leg comprises two rigid bodies (a leg segment, $L_{Leg}$ , and a foot) connected via a revolute joint at the ankle. The feet are modelled as rigid triangles, with the first vertex at the ankle joint, the second coincident with the midpoint between the metatarsal joints, and the third at the heel. The Centres of Rotation ( $CoR_{R}$ and $CoR_{L}$ ) are defined as the midpoints of the plantar segments connecting the metatarsals to the heel. The virtual CoRs ( $vCoR_{R}$ and $vCoR_{L}$ ) are the ground projections of the CoRs used for trajectory planning, and they coincide with the physical CoRs when the feet are in full 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 respective vCoRs. (b) The heel-strike phase is modelled as a rotation about the foot’s instantaneous centre of rotation ( $CIR_{HS}$ ). (c) The toe-off phase is modelled as a rotation about the $CIR_{TO}$ . (d) The task-space planner utilizes two reference systems. The Task-Space (TS) frame is aligned with the standard anatomical planes and is used to describe the walking trajectories. The Saddle-Space (Saddle) frame is used to describe posture-dependent quantities, which are subsequently projected onto the TS frame.

Figure 2: CoM trajectories generated by the proposed planner in the transverse, sagittal, and frontal planes for walking speeds ranging from 0.6 to 2.2 m/s. Notably, the frontal view exhibits the typical butterfly shape characteristic of a strange attractor. Furthermore, the proposed model successfully replicates the speed-dependent behavior observed in vivo by Orendurff et al. : as walking speed increases, the vertical amplitude of the CoM trajectory increases, the mediolateral amplitude decreases, and the CoM reaches its minimum height closer to zero.

Figure 3: Anteroposterior trajectories of the vCoRs across different walking speeds, demonstrating the intrinsic regulation of the support and swing phases. At lower speeds, the support phase dominates the gait cycle, whereas this dominance diminishes at higher speeds.

Figure 4: The proposed model generates swing foot, mediolateral, and vertical trajectories that fall within the 95% confidence intervals of the experimental data, closely matching the mean human behavior.

Figure 5: Kinematic trajectories of a single representative subject across the three anatomical planes at slow, normal, and fast walking speeds. These experimental data confirm that the proposed model captures idealized locomotion strategies, which are subsequently modulated by local conditions (e.g., somatic state and environmental interactions). The isolated single-step view in the frontal plane further highlights this phenomenon: speed-dependent trajectory adaptations in the in vivo data closely mirror the simulated model behaviors depicted in Figure 2 .

Figure 6: Anatomophysiological parallels between the proposed control architecture and the human central nervous system, synthesized from existing literature [ 17 , 42 , 8 , 47 , 16 , 11 , 13 , 5 , 10 ] . Solid lines represent active control information, whereas dotted lines denote supervisory signals. Both types of neural pathways are frequently mediated by additional brain regions not depicted in this simplified schematic. Notably, the pre-motor cortex is positioned across the boundary of the task-space and joint-space planners to reflect our hypothesis that it functions as a functional bridge between the frontal and motor cortices.

$v_{W}$ (m/s)	Double Support	Foot Ground Contact
	[% Gait Cycle]	[% Gait Cycle]
0.6	46	73
0.8	32	66
1	24	62
1.2	20	60
1.4	18	59

Table 1: The intrinsic regulation of the durations of foot-ground contact and the double support phase emerges directly from the coordinated foot swing and ankle strategies.

Bipedal Locomotion

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Bipedal Locomotion Nonlinear MPC Whole-Body Control Model Predictive Control Robot Locomotion Optimal Control

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This paper presents a multi-phase whole-body model predictive control (MPC) approach for bipedal walking, combining a detailed whole-body model in the near horizon with a simplified single-rigid-body model in the later prediction steps. This reduces computational complexity while retaining prediction capabilities. The resulting nonlinear optimal control problem is solved entirely within the general-purpose, off-the-shelf nonlinear MPC framework acados, using sequential quadratic programming (SQP). Given a contact schedule and a target walking speed, the controller optimizes joint torques without depending on preselected footstep locations. The controller is validated in MuJoCo simulation on the 18-DoF bipedal robot HyPer-2.

Abstract

Bipedal Locomotion

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