Passive Walking Model Research Articles

Multistability of gaits, the basin of attraction and its external topology in the simplest passive walking model on stairs

For the passive walking robot, the basin of attraction is very small and thin, and thus makes it a great challenge to produce stable passive dynamic walking. This paper reports systematically the effect of initial conditions on gait dynamics of the simplest passive walking model on stairs. We find two new gaits with period-3 and period-4, and they both lead to higher periodic gaits and chaos by period-doubling bifurcations. The critical period-3 unstable orbit generated via the cyclic-fold bifurcation takes part in the annihilation of chaotic attractors twice, and this double boundary crisis becomes the main cause to the falling down on stairs of the passive walker. In particular, we compute the Lyapunov exponents to verify the effectiveness of the bifurcation process of these new gaits. More importantly, through analyzing the topology of phase space, the external domain of basin of attraction is classified in detail and marked precisely with different falling patterns. We believe such findings to add significantly to the knowledge of the selection of initial values and to be potentially helpful in the design and optimization of passive walking robots.

Passive walking biped robot model with flexible viscoelastic legs

To simulate the complex human walking motion accurately, a suitable biped model has to be proposed that can significantly translate the compliance of biological structures. In this way, the simplest passive walking model is often used as a standard benchmark for making the bipedal locomotion so natural and energy efficient. This paper is devoted to a presentation of the application of internal damping mechanism to the mathematical description of the simplest passive walking model with flexible legs. This feature can be taken into account by using the viscoelastic legs, which are constituted by the Kelvin–Voigt rheological model. Then, the update of the impulsive hybrid nonlinear dynamics of the simplest passive walker is obtained based on the Euler–Bernoulli’s beam theory and using a combination of Lagrange mechanics and the assumed mode method, along with the precise boundary conditions. The main goal of this study is to develop a numerical procedure based on the new definition of the step function for enforcing the biped start walking from stable condition and walking continuously. In our previous work, it was shown that the period-one gait cycle can be produced by adding the proportional damping model with high damping ratio to the elastic legs. The present paper overcomes the practical limitations of this damping model and similarly demonstrates the stable period-one gait cycles for the viscoelastic legs. The study of the influence of various system parameters is carried out through bifurcation diagrams, highlighting the region of stable period-one gait cycles. Numerical simulations clearly prove that the overall effect of viscoelastic leg on the passive walking is efficient enough from the viewpoint of stability and energy dissipation.

Gait cycles of passive walking biped robot model with flexible legs

Passive dynamic walking of bipeds exhibits a new concept of robot locomotion that is well- known for its success in human mimicry. Many efforts have been devoted in the literature to implement flexible and anthropomorphic passive bipeds. However, utilization of flexible characteristics of elastic structures has not been considered yet. This paper presents a modified simplest passive walking model with flexible legs which are a pair of elastic beams with small deflections instead of usual rigid links. The main contributions of this paper lie in modeling the impulsive hybrid dynamics of the flexible passive biped and applying a numerical technique based on finite difference formula to seek for appropriate initial conditions of walking gait cycles. Extended Hamilton's principle and assumed mode method along with Euler–Bernoulli's beam theory and updated transition rules at impact are used to derive the mathematical model of this biped. Numerical simulations reveal non-periodic patterns of motion for small slope angles due to vibrations of elastic legs. It is also shown that the period-one gait cycles with quicker steps can be generated on slower slopes for different values of elasticity and damping coefficients compared to the rigid model.

Analyzing Bifurcation, Stability and Chaos for a Passive Walking Biped Model with a Sole Foot

Human walking is an action with low energy consumption. Passive walking models (PWMs) can present this intrinsic characteristic. Simplicity in the biped helps to decrease the energy loss of the system. On the other hand, sufficient parts should be considered to increase the similarity of the model’s behavior to the original action. In this paper, the dynamic model for passive walking biped with unidirectional fixed flat soles of the feet is presented, which consists of two inverted pendulums with L-shaped bodies. This model can capture the effects of sole foot in walking. By adding the sole foot, the number of phases of a gait increases to two. The nonlinear dynamic models for each phase and the transition rules are determined, and the stable and unstable periodic motions are calculated. The stability situations are obtained for different conditions of walking. Finally, the bifurcation diagrams are presented for studying the effects of the sole foot. Poincaré section, Lyapunov exponents, and bifurcation diagrams are used to analyze stability and chaotic behavior. Simulation results indicate that the sole foot has such a significant impression on the dynamic behavior of the system that it should be considered in the simple PWMs.

Influence of the swing ankle angle on walking stability for a passive dynamic walking robot with flat feet

To achieve high walking stability for a passive dynamic walking robot is not easy. In this article, we aim to investigate whether the walking performance for a passive dynamic walking robot can be improved by just simply changing the swing ankle angle before impact. To validate this idea, a passive bipedal walking model with two straight legs, two flat feet, a hip joint, and two ankle joints was built in this study. The walking dynamics that contains double stance phase was derived. By numerical simulation of the walking in MATLAB, we found that the walking performance can be adjusted effectively by only simply changing the swing ankle angle before impact. A bigger swing ankle angle in a reasonable range will lead to a higher walking stability and a lower initial walking speed of the next step. A bigger swing ankle angle before impact leads to a bigger amount of energy lost during impact for the quasi-passive dynamic walking robot which will influence the walking stability of the next step.

Towards Biomimetic Virtual Constraint Control of a Powered Prosthetic Leg.

This brief presents a novel control strategy for a powered prosthetic ankle based on a biomimetic virtual constraint. We first derive a kinematic constraint for the "effective shape" of the human ankle-foot complex during locomotion. This shape characterizes ankle motion as a function of the Center of Pressure (COP)-the point on the foot sole where the resultant ground reaction force is imparted. Since the COP moves monotonically from heel to toe during steady walking, we adopt the COP as a mechanical representation of the gait cycle phase in an autonomous feedback controller. We show that our kinematic constraint can be enforced as a virtual constraint by an output linearizing controller that uses only feedback available to sensors onboard a prosthetic leg. Using simulations of a passive walking model with feet, we show that this novel controller exactly enforces the desired effective shape whereas a standard impedance (i.e., proportional-derivative) controller cannot. This work provides a single, biomimetic control law for the entire single-support period during robot-assisted locomotion.

New bifurcations in the simplest passive walking model

This paper uncovers several new stable periodic gaits in the simplest passive walking bipedal model proposed in the literature. It is demonstrated that the model has period-3 to period-7 gaits beside the period-1 gaits found by Garcia et al. By simulations, this paper shows that each of these new gaits leads to chaos via period-doubling bifurcation and loses its stability by cyclic-fold bifurcation. This interesting phenomenon suggests a series of new bifurcation scenarios that have not been observed before. To confirm the new gaits and their bifurcations, this paper presents computer assisted proofs on the existence and stability of each periodic gait and its period-doubling gaits with the interval Newton method. To verify that the routes indeed lead to chaos, computer-assisted proofs are also given by means of topological horseshoes theory.

Passive Dynamic Biped Walking—Part I: Development and Validation of an Advanced Model

Passive dynamic walking is a manner of walking developed, partially or in whole, by the energy provided by gravity. Studying passive dynamic walking provides insight into human walking and is an invaluable tool for designing energy-efficient biped robots. The objective of this research was to develop a continuous mathematical model of passive dynamic walking, in which the Hunt–Crossley contact model, and the LuGre friction model were used to represent the normal and tangential ground reactions continuously. A physical passive walker was built to validate the proposed mathematical model. A traditional impact-based passive walking model was also used as a reference to demonstrate the advancement of the proposed passive dynamic walking model. The simulated gait of the proposed model matched the gait of the physical passive walker exceptionally well, both in trend and magnitude.

The introduction of the passive walking testing system and experimental study

In this paper, we represent the testing system of the planar compass-like passive walking model. The testing parameters in the experiment are the swing angles and angular velocities of the legs as well as the contact forces between the feet and the ground. The sensors we chose are the gyros and the Flexiforce film sensors. With the testing system we established, the contact process in passive walking is described in detail.

Passive Dynamic Biped Walking—Part II: Stability Analysis of the Passive Dynamic Gait

Passive dynamic walking is an excellent tool for evaluating biped stability measures, due to its simplicity, but an understanding of the stability, in the classical definition, is required. The focus of this paper is on analyzing the stability of the passive dynamic gait. The stability of the passive walking model, validated in Part I, was analyzed with Lyapunov exponents, and the geometry of the basin of attraction was determined. A novel method was created to determine the 2D projection of the basin of attraction of the model. Using the insights gained from the stability analysis, the relation between the angular momentum and the stability of gait was examined. The angular momentum of the passive walker was not found to correlate to the stability of the gait.

A Study of a Back-and-Forth Swing Type Bipedal Passive Walking Model as a Teaching Tool : Examining the Optimum Walking Condition of the Model and Pupils' Recognition of the Process of Making It

A Study of a Back-and-Forth Swing Type Bipedal Passive Walking Model as a Teaching Tool : Examining the Optimum Walking Condition of the Model and Pupils' Recognition of the Process of Making It

The experimental study on the contact process of passive walking

The passive dynamic walking is a new concept of biped walking. Researchers have been working on this area with both theoretical analysis and experimental analysis ever since McGeer. This paper presents our compass-like passive walking model with a new set of testing system. Two gyroscopes are used for measuring the angles of two legs, and ten FlexiForce sensors are used for measuring the contact forces on the feet. We got the experimental data on the passive walking process with the validated testing system. A great emphasis was put on the contact process between the feet and the slope. The contact process of the stance leg was divided into four sections, and differences between the real testing contact process and the classic analytical contact process with no bouncing and slipping were summarized.

J163012 一定リズムを刻む胴を伴う一次元受動歩行波動アルゴリズム : 尻尾への加振を用いた3体結合衝突振動子モデル

We formulate one dimensional biped passive walking model with periodic rhythmic movement of a tail and a torso. The legs and the torso were modeled by globally coupled vibro-impact oscillators. The Grover like wave algorithm realizes indirect movement of a swing leg by transporting energy from the driven tail. Periodic leg exchange attains stable passive walking like movement. By using the designed mechanism, one central pattern generator can control multiple limbs movement.

New walking dynamics in the simplest passive bipedal walking model

This paper revisits the simplest passive walking model by Garcia et al. which displays chaos through period doubling from a stable period-1 gait. By carefully numerical studies, two new gaits with period-3 and -4 are found, whose stability is verified by estimates of eigenvalues of the corresponding Jacobian matrices. A surprising phenomenon uncovered here is that they both lead to higher periodic cycles and chaos via period doubling. To study the three different types of chaotic gaits rigorously, the existence of horseshoes is verified and estimates of the topological entropies are made by computer-assisted proofs in terms of topological horseshoe theory.

313 バネ機構を有する受動歩行ロポットによる着地時のエネルギー損失低減

Passive walking robot loses kinetic energy on landing process of walk. Therefore, energy loss of landing process decreased, walk speed and energy efficiency will improve. In this paper, at first, speed ratio of typical passive walking model before and after plastic impact the leg and the ground is discussed theoretically. Next, we proposed a walking model with the spring for saving kinetic energy as potential energy that next walk can reuse. And so that, we showed that the result of numerical simulation based rigid multibody dynamics and evaluated effects of spring for saving kinetic energy. Finally, we proposed new spring passive walking with the friction brake, and designed the brake controller for reusing energy.

1P1-Q05 受動歩行ロボットの着地時のエネルギー損失を低減・再利用する機構の提案(【機械力学・計測制御部門】ロボットシステムのダイナミクス&デザイン)

Passive walking robot loses kinetic energy on landing process of walk. Therefore, energy loss of landing process decreased, walk speed and energy efficiency will improve. At first, speed ratio of typical passive walking model before and after plastic impact is discussed theoretically. Next, we proposed a walking model with the spring for saving kinetic energy as potential energy that next walk can reuse.

The effects of parameter variation on the gaits of passive walking models: simulations and experiments

SUMMARYWe have made a systematic study of the gait of a straight leg planar passive walking model through simulations and experiments. Three normalised parameters, which represent the foot radius, the position of the mass centre and the moment of inertia, are used to characterise the walking model.In the simulation, we have obtained the fixed points and the basins of attraction of the walking models with different parameter combinations by the aid of the cell mapping method. With the results of fixed points, we investigated the effects of parameter variations on the gait descriptors, including step length, period, average speed and energy inefficiency. A model that has a large basin of attraction has been obtained, and it can start walking far from its fixed point. However, the size of the basin of attraction is not a good measurement of robustness. Thus, we proposed floors with random slope angles or stairs with random heights to test robustness. Five hundred times of simulations with 100 non-dimensional time units were implemented for each parameter combination. The times that the walker failed to arrive at the end were recorded. The simulation results showed that the model with a larger foot radius and higher position of mass centre has a lower possibility of falling on uneven floors. A large moment of inertia is helpful for walking on a random slope angle floor, while low values of moment of inertia are good for navigating random stairs.Prototype experiments have validated the simulation results, which showed that models with larger feet have a longer step length and high speed. However, period differences were difficult to obtain in the experiments since the differences were very small. We have tested the sensitivity with the initial conditions of the models with different foot radii on a flat floor, and have also tested the robustness of the models on a floor with random slope angles. The times that the model reached the end of the floor were recorded. The experimental results showed that a large foot radius is good for improving the basin of attraction and robustness on uneven floors. Finally, the exceptions of the experiment are explained.

The effects of parameter variation on the gaits of passive walking models

We have made a systematic study of the gait of a straight leg planar passive walking model through simulations and experiments. Three normalised parameters, which represent the foot radius, the pos...

Period-doubling gait and chaotic gait of biped walking model

Passive walking model can exhibit natural gait on a slope， depending only on the gravity. The gait changes with the changing of parameters. The fixed point of the periodical gait can be obtained by the combination of cell mapping method and Newton-Raphson iteration. By the aid of cell mapping method，periodical cells can be used as the initial value of iteration，and the basins of attraction can be obtained. Simulations of models with different parameters show that the increasing of moment of inertia will result in the appearance of period-doubling and chaotic gaits. The same effect can be obtained by the decreasing of foot radius or position of center of mass.

Passive walker that can walk down steps: simulations and experiments

A planar passive walking model with straight legs and round feet was discussed. This model can walk down steps, both on stairs with even steps and with random steps. Simulations showed that models with small moments of inertia can navigate large height steps. Period-doubling has been observed when the space between steps grows. This period-doubling has been validated by experiments, and the results of experiments were coincident with the simulation.