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 · School of Electronics Engineering, Kyungpook National University, Daegu, 702701, South Korea · Biomechatronics Laboratory. Escola Politecnica da Universidade de Sao Paulo, Sao Paulo, 05508-030, Brazil
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 (λ0-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, LLeg , 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 ( CoRR and CoRL ) are defined as the midpoints of the plantar segments connecting the metatarsals to the heel. The virtual CoRs ( vCoRR and vCoRL ) 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 ( LPR and LPL ) 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 ( CIRHS ). (c) The toe-off phase is modelled as a rotation about the CIRTO . (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.
vW (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.
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.
Franek Stark, Felix Wiebe, Shubham Vyas +2
Robotics Innovation Center at the German Research Center for Artificial Intelligence (DFKI), Bremen, Germany · University Bremen, Germany
A class of planar bipedal robots with unique mechanical properties has been proposed, where all links are balanced around the hip joint, preventing natural swinging motion due to gravity. A common property of their equations of motion is that the inertia matrix is a constant matrix, there are no nonlinear velocity terms, and the gravity term contains simple nonlinear terms. By performing a Taylor expansion of the gravity term and making a linear approximation, it is easy to derive a linearized model, and calculations for future states or walkability determination can be performed instantaneously without the need for numerical integration. This paper extends the method to a planar biped robot model with knees. First, we derive the equations of motion, constraint conditions, and inelastic collisions for a planar 6-DOF biped robot, design its control system, and numerically generate a stable bipedal gait on a horizontal plane. Next, we reduce the equations of motion to a 3-DOF model, and derive a linearized model by approximating the gravity term as linear around the expansion point for the thigh frame angle. Through numerical simulations, we demonstrate that calculations for future states and walkability determination can be completed in negligible time. By applying control inputs to the obtained model, performing state-space realization, and then discretizing it, instantaneous walkability determination through iterative calculation becomes possible. Through detailed gait analysis, we discuss how the knee joint flexion angle and the expansion point affect the accuracy of the linear approximation, and the issues that arise when descending a small step.
Fumihiko Asano, Ning Lei, Taiki Sedoguchi
Graduate School of Advanced Science and Technology, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan
This paper presents a three-stage offline command generation framework for reproducing human lower-limb motion on a suspended bipedal robot while matching torque trajectories computed from the robot dynamic model. First, State-Dependent Riccati Equation (SDRE) control derives the reference torque trajectory for the measured motion. Second, parameterized optimization converts this trajectory into trapezoidal joint velocity commands under motor speed and acceleration limits. Third, a proportional-integral-derivative linear quadratic regulator (PID-LQR) compensation scheme refines these commands using experimental tracking data. The platform executes the resulting profiles to reproduce human walking and squatting motions recorded by a Vicon system, allowing evaluation of tracking accuracy and repeatability. Results show that the average root mean square error (RMSE) and standard deviation (STD) of joint angles across repeated trials remain below 7° and 0.33°, respectively. Joint angle and torque trajectory comparisons show lower maximum RMSE and STD values than those for MPC and IPSO-PID in every reported case. The framework enables accurate and repeatable motion reproduction within actuator limits, providing controlled and measurable conditions that can reduce reliance on human participation and associated risks during preliminary evaluation of devices for assistive walking, gait training, and rehabilitation.
Ping-Kong Huang, Chien-Wu Lan, Chin-Tien Wu +1
Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan · Department of Electrical Engineering, National Central University, Taoyuan City 320317, Taiwan