Bipedal Model Research Articles

A mathematical framework to study fast walking of human

The body of a walking human is an elaborated dynamic system that operates adaptively in various conditions such as fast walking. Due to dynamic redundancies, the individual motor control strategies for speeding up the walking can be different among normal subjects. However, in reality, we see that the pattern of motion is quite similar among people and it is only the profile of hip joint motion along its path which determines the speed. The objective of the current paper is to develop a mathematical framework to investigate time optimal motion of a human during walking. To this end, a nine-link planar biped model is used. The motion is considered to take place in sagittal plane and to follow a normal pattern of motion. The solution is obtained using a phase plane method to solve minimum time problem which is subjected to inequality constraints of variable maximum joint torques and stability conditions. The solution method can be used to find the maximum possible speed of a human with specific body characteristics and to obtain a hip joint trajectory which could produce that speed. The proposed method can be utilized to study quantitative effect of different parameters such as joint strength in fast walking.

Orthotic Body-Weight Support Through Underactuated Potential Energy Shaping with Contact Constraints.

Body-weight support is an effective clinical tool for gait rehabilitation after neurological impairment. Body-weight supported training systems have been developed to help patients regain mobility and confidence during walking, but conventional systems constrain the patient's treatment in clinical environments. We propose that this challenge could be addressed by virtually providing patients with body-weight support through the actuators of a powered orthosis (or exoskeleton) utilizing potential energy shaping control. However, the changing contact conditions and degrees of underactuation encountered during human walking present significant challenges to consistently matching a desired potential energy for the human in closed loop. We therefore introduce a generalized matching condition for shaping Lagrangian systems with holonomic contact constraints. By satisfying this matching condition for four phases of gait, we derive control laws to achieve virtual body-weight support through a powered knee-ankle orthosis. We demonstrate beneficial effects of virtual body-weight support in simulations of a human-like biped model, indicating the potential clinical value of this proposed control approach.

Displayed phenomena in the semi-passive torso-driven biped model under OGY-based control method: Birth of a torus bifurcation

We consider the displayed phenomena in the semi-passive dynamic walking of a torso-driven biped robot under the OGY control method as it descends inclined surfaces. The desired torso angle is the accessible control parameter. Analysis of the controlled steady semi-passive gaits are carried out mainly by means of bifurcations diagrams and Poincaré sections as the slope parameter varies. Based on Floquet multipliers, we show that a torus bifurcation was born leading hence to the generation of a quasi-periodic bipedal walking. Analysis of the controlled semi-passive dynamic walking is investigated also through a constrained controlled Poincaré map. We demonstrate that a such controlled map and the controlled impulsive hybrid nonlinear model of the biped display almost the same phenomena. We show also that the controlled constrained Poincaré map exhibits a discontinuous period-doubling bifurcation.

Energy-optimal relative timing of stance-leg push-off and swing-leg retraction in walking

SUMMARYSwing-leg retraction in walking is the slowing or reversal of the forward rotation of the swing leg at the end of the swing phase prior to ground contact. For retraction, a hip torque is often applied to the swing leg at about the same time as stance-leg push-off. Due to mechanical coupling, the push-off force affects leg swing, and hip torque affects the stance-leg extension. This coupling makes the energetic costs of retraction and push-off depend on their relative timing. Here, we find the energy-optimal relative timing of these actions. We first use a simplified walking model with non-regenerative actuators, a work-based energetic-cost, and impulsive actuations. Depending on whether the late-swing hip torque is retracting or extending (pushing the leg forward), we find that the optimum is obtained by applying the impulsive hip torque either following or prior to the impulsive push-off force, respectively. These trends extend to other bipedal models and to aperiodic gaits, and are independent of step lengths and walking speeds. In one simulation, the cost of a walking step is increased by 17.6% if retraction torque comes before push-off. To consider non-impulsive actuation and the cost of force production, we add a force-squared (F2) term to the work cost. We show that this cost promotes simultaneous push-off force and retracting torque, but does not change the result that any extending torque should come prior to push-off. A high-fidelity optimization of the Cornell Ranger robot is consistent with the swing-retraction trends from the models above.

Common formation mechanism of basin of attraction for bipedal walking models by saddle hyperbolicity and hybrid dynamics

In this paper, we investigate the mathematical structures and mechanisms of bipedal walking from a dynamical viewpoint. Especially, we focus on the basin of attraction since it determines the stability of bipedal walking. We treat two similar but different bipedal walking models (passive and active dynamic walking models) and examine common mathematical structure between these models. We find that the saddle hyperbolicity and hybrid system play important roles for the shape of the basin of attraction in both models, which are quite common for more general bipedal models and important for understanding the stability mechanism of bipedal walking.

Dynamic optimization of a biped model: Energetic walking gaits with different mechanical and gait parameters

Energy consumption is one of the problems for bipedal robots walking. For the purpose of studying the parameter effects on the design of energetic walking bipeds with strong adaptability, we use a dynamic optimization method on our new walking model to first investigate the effects of the mechanical parameters, including mass and length distribution, on the walking efficiency. Then, we study the energetic walking gait features with the combinations of walking speed and step length. Our walking model is designed upon Srinivasan’s model. Dynamic optimization is used for a free search with minimal constraints. The results show that the cost of transport of a certain gait increases with the increase in the mass and length distribution parameters, except for that the cost of transport decreases with big length distribution parameter and long step length. We can also find a corresponding range of walking speed and step length, in which the variation in one of the two parameters has no obvious effect on the cost of transport. With fixed mechanical parameters, the cost of transport increases with the increase in the walking speed. There is a speed–step length relationship for walking with minimal cost of transport. The hip torque output strategy is adjusted in two situations to meet the walking requirements.

Animal models of scoliosis.

Multiple techniques designed to induce scoliotic deformity have been applied across many animal species. We have undertaken a review of the literature regarding experimental models of scoliosis in animals to discuss their utility in comprehending disease aetiology and treatment. Models of scoliosis in animals can be broadly divided into quadrupedal and bipedal experiments. Quadrupedal models, in the absence of axial gravitation force, depend upon development of a mechanical asymmetry along the spine to initiate a scoliotic deformity. Bipedal models more accurately mimic human posture and consequently are subject to similar forces due to gravity, which have been long appreciated to be a contributing factor to the development of scoliosis. Many effective models of scoliosis in smaller animals have not been successfully translated to primates and humans. Though these models may not clarify the aetiology of human scoliosis, by providing a reliable and reproducible deformity in the spine they are a useful means with which to test interventions designed to correct and prevent deformity.

Analysis of period-1 passive limit cycles for flexible walking of a biped with knees and point feet

SUMMARYIn this paper, we investigate dynamic walking as a convergence to the system's own limit cycles, not to artificially generated trajectories, which is one of the lessons in the concept of passive dynamic walking. For flexible walking, gait transitions can be performed by moving from one limit cycle to another one, and thus, the flexibility depends on the range in which limit cycles exist. To design a bipedal walker based on this approach, we explore period-1 passive limit cycles formed by natural dynamics and analyze them. We use a biped model with knees and point feet to perform numerical simulations by changing the center of mass locations of the legs. As a result, we obtain mass distributions for the maximum flexibility, which can be attained from very limited location sets. We discuss the effect of parameter variations on passive dynamic walking and how to improve robot design by analyzing walking performance. Finally, we present a practical application to a real bipedal walker, designed to exhibit more flexible walking based on this study.

Passive turning motion of 3D rimless wheel: novel periodic gaits for bipedal curved walking

Passive dynamic walking originally was introduced to describe some natural stable periodic gaits in straight walking. However, in this research, we have extended this notion to curved walking, wherein some novel three-dimensional (3D) periodic gaits with natural stability have been realized. For this aim, the simplest bipedal walking model, i.e. a rimless spoked-wheel, in its general 3D form, has been considered on a straight slope surface. Then, two kinds of stable passive turning, i.e. limited and circular continuous have been discussed. The first kind is actually a transfer among two-dimensional (2D) periodic motions and can be implemented on such straight slope surface. It is shown that the second kind is related to novel 3D periodic motions and can be just recognized on a special surface profile namely ‘helical slope’. The latter are interpreted as 3D fixed points of a Poincare return map. Results not only show asymptotical stability of such periodic motions, but also surprisingly indicate that the stability of a 3D periodic gait (turning) is higher than 2D one (straight walking). Furthermore, the characteristics of passive turning are shown to be firmly connected with the value of the state variables such as the lean angle in each case.

Walking on a moving surface: energy-optimal walking motions on a shaky bridge and a shaking treadmill can reduce energy costs below normal.

Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a point-mass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energy-optimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a mass- spring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.

Towards active walking from compass gait: stability and hip torque requirement for level ground walking

The aim of this paper is to develop a stable biped model for level ground walking with anthropomorphic parameters. Two models for level ground walking has been simulated and studied, one with active hip joint and the other with ankle impulse. Proposed model is derived from compass gait model with additional torque only at hip joint. Level ground walking has been simulated to find torque requirement and relation between applied torque and angular variation of stance and swing leg. Stability analysis to find disturbance rejection range has also been found. Proposed model serves as a transition from passive walking to energy efficient anthropomorphic walking. Initial stability analysis and model behaviour shows satisfactory results. This concept can be modified towards complete anthropomorphic biped walking by applying minimum possible torque at knee and ankle joint along with hip and also for developing energy efficient active exoskeleton.

Towards active walking from compass gait: stability and hip torque requirement for level ground walking

The aim of this paper is to develop a stable biped model for level ground walking with anthropomorphic parameters. Two models for level ground walking has been simulated and studied, one with active hip joint and the other with ankle impulse. Proposed model is derived from compass gait model with additional torque only at hip joint. Level ground walking has been simulated to find torque requirement and relation between applied torque and angular variation of stance and swing leg. Stability analysis to find disturbance rejection range has also been found. Proposed model serves as a transition from passive walking to energy efficient anthropomorphic walking. Initial stability analysis and model behaviour shows satisfactory results. This concept can be modified towards complete anthropomorphic biped walking by applying minimum possible torque at knee and ankle joint along with hip and also for developing energy efficient active exoskeleton.

An intermittent control model of flexible human gait using a stable manifold of saddle-type unstable limit cycle dynamics.

Stability of human gait is the ability to maintain upright posture during walking against external perturbations. It is a complex process determined by a number of cross-related factors, including gait trajectory, joint impedance and neural control strategies. Here, we consider a control strategy that can achieve stable steady-state periodic gait while maintaining joint flexibility with the lowest possible joint impedance. To this end, we carried out a simulation study of a heel-toe footed biped model with hip, knee and ankle joints and a heavy head-arms-trunk element, working in the sagittal plane. For simplicity, the model assumes a periodic desired joint angle trajectory and joint torques generated by a set of feed-forward and proportional-derivative feedback controllers, whereby the joint impedance is parametrized by the feedback gains. We could show that a desired steady-state gait accompanied by the desired joint angle trajectory can be established as a stable limit cycle (LC) for the feedback controller with an appropriate set of large feedback gains. Moreover, as the feedback gains are decreased for lowering the joint stiffness, stability of the LC is lost only in a few dimensions, while leaving the remaining large number of dimensions quite stable: this means that the LC becomes saddle-type, with a low-dimensional unstable manifold and a high-dimensional stable manifold. Remarkably, the unstable manifold remains of low dimensionality even when the feedback gains are decreased far below the instability point. We then developed an intermittent neural feedback controller that is activated only for short periods of time at an optimal phase of each gait stride. We characterized the robustness of this design by showing that it can better stabilize the unstable LC with small feedback gains, leading to a flexible gait, and in particular we demonstrated that such an intermittent controller performs better if it drives the state point to the stable manifold, rather than directly to the LC. The proposed intermittent control strategy might have a high affinity for the inverted pendulum analogy of biped gait, providing a dynamic view of how the step-to-step transition from one pendular stance to the next can be achieved stably in a robust manner by a well-timed neural intervention that exploits the stable modes embedded in the unstable dynamics.

Periodic Cubic Spline on Motion of Five-Link Human Bipedal Model Using Nonlinear Predictive Control

This paper studies about the performance of periodic cubic spline applied on the nonlinear predictive control (NPC) based on receding-horizon cost function optimization. The five-link human bipedal model includes a torso and two legs which has a thigh and a shank in every leg. The periodic cubic spline is used to achieve smooth walking trajectories of every joint in the bipedal model in the sagittal plane. The results show that the tracking error after using periodic cubic spline is smaller than the tracking error before using periodic cubic spline.

Single-support heel-off: a crucial gait event helps realizing agile and energy-efficient bipedal walking

SUMMARYSingle-support heel-off occurs when the heel of the trailing leg has been lifted from the ground around its toe, while the leading leg is still swinging forward. A similar gait event occurs during human walking, and is crucial to achieve a longer step length and a higher walking speed. In this paper, this crucial gait event is studied, specifically in how it influences the agility and the energy efficiency of bipedal walking. Toward this goal, the concept of limit-cycle bipedal walking which possesses natural and energy-efficient gaits is employed. The aforementioned concept is applied to a flat-foot bipedal model which is developed and actuated by a constant hip torque only during the single-support phase to walk on the ground. The impedance of each ankle is adjusted by using two springs, one at the back-side and the other at the front-side, as well as one damper. In comparison with point/round foot bipedal models, the flat-foot bipedal model produces more versatile limit-cycle gaits comprised of a number of gait series, each of which is a sequence detected among twelve gait postures dictated by the kinetics of the unilateral constraints at the heel, toe, or both. As a result of comprehensive simulations, it is concluded that single-support heel-off significantly improves the agility of bipedal walking because of the increase in the step length and the walking speed. Furthermore, even though limit-cycle gaits including single-support heel-off require higher energy input as compared with gaits excluding such an event, single-support heel-off significantly improves the energy efficiency of bipedal walking since the increase in the step length dominates the increase in the energy input.

Torque-stiffness-controlled dynamic walking with central pattern generators.

Walking behavior is modulated by controlling joint torques in most existing passivity-based bipeds. Controlled Passive Walking with adaptable stiffness exhibits controllable natural motions and energy efficient gaits. In this paper, we propose torque-stiffness-controlled dynamic bipedal walking, which extends the concept of Controlled Passive Walking by introducing structured control parameters and a bio-inspired control method with central pattern generators. The proposed walking paradigm is beneficial in clarifying the respective effects of the external actuation and the internal natural dynamics. We present a seven-link biped model to validate the presented walking. Effects of joint torque and joint stiffness on gait selection, walking performance and walking pattern transitions are studied in simulations. The work in this paper develops a new solution of motion control of bipedal robots with adaptable stiffness and provides insights of efficient and sophisticated walking gaits of humans.

Realization of foot rotation by breaking the kinematic contact constraint

SUMMARYPrevious research has revealed that foot rotation of the supporting foot in a single support phase could increase walking speed. This paper presents a method for force-controlled bipeds to realize foot rotation by breaking the kinematic contact constraint between the supporting foot and the ground. An inverse dynamics controller is proposed to make the biped model controllable even when the constraint is broken. In addition, a linear inverted pendulum model is extended to make its ZMP adjustable so that the ZMP can be predefined as required. When the planned ZMP is in the toe, the kinematic contact constraint will be broken and foot rotation can be achieved. A walking simulation demonstrates the effectiveness of the proposed method.

Discrete‐time non‐linear state observer based on a super twisting‐like algorithm

The properties of robustness and finite-time convergence provided by sliding mode (SM) theory have motivated several researches to deal with the problems of control and state estimation. In the SM theory, the super-twisting algorithm (STA), a second-order SM scheme, has demonstrated remarkable characteristics when it is implemented as a controller, observer or robust signal differentiator although the presence of noise and parametric uncertainties. However, the design of this algorithm was originally developed for continuous-time systems. The growth of microcomputers technology has attracted the attention of researchers inside the SM discrete-time domain. Recently, discretisations schemes for the STA were studied using majorant curves. In this study, the stability analysis in terms of Lyapunov theory is proposed to study a discrete-time super twisting-like algorithm (DSTA) for non-linear discrete-time systems. The objective is to preserve the STA characteristics of robustness in a quasi-sliding mode regime that was proved in terms of practical Lyapunov stability. An adequate combination of gains obtained by the same Lyapunov analysis forces the convergence for the DSTA. The problem of state estimation is also analysed for second-order mechanical systems of n degrees of freedom. Simulation results regarding the design of a second-order observer using the DSTA for a simple pendulum and a biped model of seven degrees of freedom are presented.

The evolution of bipedal running in lizards suggests a consequential origin may be exploited in later lineages.

The origin of bipedal locomotion in lizards is unclear. Modeling studies have suggested that bipedalism may be an exaptation, a byproduct of features originally designed to increase maneuverability, which were only later exploited. Measurement of the body center of mass (BCOM) in 124 species of lizards confirms a significant rearward shift among bipedal lineages. Further racetrack trials showed a significant acceleration threshold between bipedal and quadrupedal runs. These suggest good general support for a passive bipedal model, in which the combination of these features lead to passive lifting of the front of the body. However, variation in morphology could only account for 56% of the variation in acceleration thresholds, suggesting that dynamics have a significant influence on bipedalism. Deviation from the passive bipedal model was compared with node age, supporting an increase in the influence of dynamics over time. Together, these results show that bipedalism may have first arisen as a consequence of acceleration and a rearward shift in the BCOM, but subsequent linages have exploited this consequence to become bipedal more often, suggesting that bipedalism in lizards may convey some advantage. Exploitation of bipedalism was also associated with increased rates of phenotypic diversity, suggesting exploiting bipedalism may promote adaptive radiation.

Bigtoe Sizing Design of Small Biped Robot by Using Gait Generation Method

The study of biped robot has long history and continuation. One of important moving processes is walking procedure. The walking posture is an important research field that always adapts and implements in the biped robot. The walking research field is very interesting because the walking posture of humans is flexible and stable. Additionally, the force that affect on the humans foot is also investigated. This research addresses the walking simulation of small biped robots that have tiptoe and bigtoe. The study based on the assumption that the bigtoe size affects on the walking posture and walking distance. The gait generation method, for finding the proper size of bigtoe, is utilized by varying the bigtoe size. There are two requirements of robot design: go straight and stay within setting conditions. The simulation results of all small biped robot models which have the different bigtoe sizes can walk within setting conditions. There is only one model its bigtoe width per foot width ratio equals 0.28 (or 28% of foot width) has the longest walking distance. Moreover, this ratio is equal to the ratio of humans foot.