Fig 1 - uploaded by Baojun Chen
Content may be subject to copyright.
Passivity-based dynamic bipedal walking model with flat feet and compliant ankle joints. The biped is powered by hip torque and ankle actuation. The thigh and the shank are connected at the knee joint, while the foot is mounted on the ankle with a torsional spring. Similar to Wisse et al. , 12 a kinematic coupling has been used in the model to keep the body midway between the two legs. The knee joints and ankle joints are modeled as passive joints that are constrained by torsional springs.
Source publication
This paper presents a seven-link dynamic walking model that is more close to human beings than other passivity-based dynamic walking models. We add hip actuation, upper body, flat feet, and ankle joints with torsional springs to the model. Walking sequence of flat-feet walkers has several substreams, which forms bipedal walking with dynamic series...
Contexts in source publication
Context 1
... bipedal walking is restricted to stop in three cases, including falling down, running, and shank releasing. We define that the walker falls down if the angle of either leg exceeds the normal range, which is from − 1 rad to 1 rad in this study. And the model is considered to be running when the stance leg lifts up, which means that the ground force acted on the stance leg orthogonal to the floor decreases to zero, while the swing foot has no contact with ground. Shank releasing is the case that the shank of the swing leg is not locked before heel-strike. Foot scuffing, which means that the foot of the swing leg travels below the floor, often appears when the knee joint locks the shank too early. It is another case that the walker maybe have to stop. However, this case is likely avoided for a real three-dimensional walker with lateral motion. Thus, similar to other related studies, 12, 15, 16 we allow it if the foot travels below the floor not very seriously. We suppose that the x -axis is along the ground while the y -axis is vertical to the ground upward. The configuration of the walker is defined by the coordinates of the point mass on hip joint and several angles, which include the swing angles between vertical axis and each thigh and shank, the angle between vertical axis and the upper body, and the foot angles between horizontal axis and each foot (see Fig. 1 for details), which can be arranged in a generalized vector q = ( x h , y h , θ 1 , θ 2 , θ 3 , θ 2 s , θ 1 f , θ 2 f ) T . The positive directions of all the angles are counterclockwise. Note that the dimension of the generalized vector in different phases may be different. When the knee joint of the swing leg is locked, the freedom of the shank is reduced and the angle θ 2 s is not included in the generalized coordinates. Consequently, the dimensions of mass matrix and generalized active force are also reduced in some phases. In the following paragraphs, we will focus on the Equation of Motion (EoM) of the proposed bipedal walking model. The model can be defined by the Euclidean coordinates x , which can be described by the x - and y -coordinates of the mass points and the corresponding angles (suppose leg 1 is the stance ...
Context 2
... m h , m l , m t , m b , m s , and m f are the masses of hip, each leg, each thigh, upper body, each shank, and each foot, respectively. I components are moments of inertia of corresponding parts. Since the mass of the model is distributed as point masses, the angles in x and the moments of inertia in M could be taken off for simplification. Denote F as the active external force vector in rectangular coordinates. The constraint function is marked as ξ ( q ), which is used to maintain foot contact with ground and detect impacts. Note that ξ ( q ) in different walking phases may be different since the contact conditions change. For example, the constraint function ξ ( q ) in the single-support phase (the full foot of the stance leg keeps contact with ground, as shown in Fig. 1) can be written as ...
Context 3
... x ankle is the x -coordinate of the ankle of leg 1, l is leg length as shown in Fig. 1, and l f h and l f t are the distances from heel to ankle and from ankle to toe, respectively. Each component of ξ ( q ) should keep zero to satisfy the contact condition. The contact of stance foot is modeled by one ground reaction force (GRF) along the floor and two GRFs perpendicular to the ground act on the two endpoints of the foot, respectively. If one of the forces perpendicular to the ground decreases below zero, the corresponding endpoint of the stance foot will lose contact with ground and the stance foot will rotate around the other endpoint. Each element of the constraint function corresponds to the generalized constrain force F f . We can obtain the EoM by Lagrange’s equation of the first ...
Context 4
... Section 5, we discuss the effects of ankle stiffness on gait selection and compliant leg behavior. We conclude in Section 6. To obtain further understanding of real human walking, we propose a passivity-based bipedal walking model that is more close to human beings than other passivity-based dynamic walking models. We add flat feet and compliant ankle joints to the model. As shown in Fig. 1, the two-dimensional model consists of two rigid legs interconnected individually through a hinge with a rigid upper body (mass added stick) connected at the hip. Each leg includes thigh, shank, and foot. The thigh and the shank are connected at the knee joint, while the foot is mounted on the ankle with a torsional spring. A point mass at hip represents the pelvis. The mass of each leg and foot is simplified as point mass added on the Center of Mass (CoM) of the shank, the thigh, and the foot, respectively. Similar to Wisse et al. , 12 a kinematic coupling has been used in the model to keep the body midway between the two legs. In addition, our model adds compliance to the knee joints and ankle joints. Specifically, knee joints and ankle joints are modeled as passive joints that are constrained by torsional springs. The springs are mounted at the joints in the torsional way, which means that the torque generated by the spring is proportional to the deviation of the angle between links from the equilibrium position (see Fig. 2). To simplify the motion, we have several assumptions, including (1) shanks and thighs suffering no flexible deformation; (2) hip joint and knee joints with no damping or friction; (3) the friction between the walker and the ground is enough. Thus, the flat feet do not deform or slip; (4) strike is modeled as an instantaneous, fully inelastic impact where no slip and no bounce occurs. The bipedal walker travels forward on level ground with hip torque and ankle actuation. The stance leg keeps contact with ground while the swing leg pivots about the constraint hip. When the flat foot strikes the ground, there are two impulses, “heel-strike” and “foot- strike,” representative of the initial impact of the heel and the following impact as the whole foot contacts the ground. 16, 17 The shank of the stance foot is always locked and the whole leg can be modeled as one rigid stick, while the knee joint of the swing leg will release the shank immediately after foot- strike. The shank will be locked when it swings forward to a relatively small angle to the ...
Similar publications
Walking in adults seems to rely on a small number of modules allowing to reduce the number of degrees of freedom effectively regulated by the central nervous system (CNS). However, the extent to which modularity evolves during development remains unknown, particularly regarding the ability to generate several strides in an optimized manner. Here we...
[Purpose] This study aimed to elucidate the effects of upper extremity loading on pelvic movements during wheeled upright walker use. [Participants and Methods] Thirteen healthy male adults participated in this intervention study. Participants walked under five conditions with targeted loads on their upper extremities of 0%, 10%, 20%, 30%, and 40%...
Background:
The study of human walking patterns mainly focuses on how control affects walking because control schemes are considered to be dominant in human walking.
Objective:
This study proposes that not only fine control schemes but also optimized body segment parameters are responsible for humans' low-energy walking.
Methods:
A passive dyn...
Precisely manipulating the center of mass (CoM) of the underactuated locomotion robot can’t be easily achieved by common control mechanisms which apply only joint torques. A novel and indirect method has been recently introduced using an active wobbling mass attached to limit cycle walkers. The next important issue is to design an optimal control i...
Citations
... The human-prosthesis heterogeneous coupled system walks in the sagittal plane (the orange plane in Fig. 1(a)) identified by a vertical XOY coordinate frame [67,68]. ...
Because of distinct differences in structure and drive, the lower limb amputee that walks with prosthesis forms a heterogeneous coupled dynamic system. The strongly coupled nonlinearity makes it difficult for the lower limb prosthesis (LLP) to adapt to complex tasks, such as variable-speed walking and obstacle crossing. As a result, the typical behavior can be seen as gait incoordination or even gait instability. This paper proposes a new gait-coordination-oriented adaptive neural sliding mode control (GC-ANSMC) for the heterogeneous coupled dynamic system. At the high level, the controller adopts the homotopy algorithm, which inherits the intelligence of the healthy lower limb (HLL), to create the GC-oriented desired trajectory for the LLP. The embedding parameter of the homotopy algorithm is updated online based on the mean difference between the lab-based target trajectory and the HLL's delayed motion, resulting in better GC performance. In addition, the new GC-planning strategy has sufficient environmental adaptability with a limited lab-based target trajectory for complex tasks. At the low level, radial basis function neural network (RBFNN) is employed to model the human-prosthesis heterogeneous coupled system uncertainties online and generate the controlled torques for simultaneous uncertainty compensation and gait driving. According to Lyapunov's theory, the sliding mode gains and the cubic order evolution rules of the network's weight are carried out. As a result, the global convergence of the proposed control approach can be ensured, and the dynamic motion could be quickly tracked. Applications for the variable-speed walking and the obstacle crossing show that the present GC-ANSMC could achieve better control accuracy, faster convergence speed, lower controlled torques, and higher GC performance than traditional methods. These advantages, as a result, indicate a convincing potential for the adaptive control for the nonlinear human-prosthesis heterogeneous coupled dynamics in complex tasks.
... Walking motions have several gaits depending on the followed scenarios. Adopted gaits have been studied in detail in [15]. Our study focuses on two gaits which are closer to human walking with a moderate speed [15]. ...
... Adopted gaits have been studied in detail in [15]. Our study focuses on two gaits which are closer to human walking with a moderate speed [15]. e phases of these two gaits are as follows: (1) push-off phase: the robot rotates around two contact points (the heel of the leading leg and the toe of the trailing leg). is phase ends when the trailing leg leaves the ground. ...
... Lagrange's equations of the first kind could be used to get the EoM [12,15]: ...
In this paper, we propose a central pattern generator-based model to control the walking motion of a biped robot. The model independently controls the joint torque and joint stiffness in real time. Instead of the phase-dependent neural model used by Huang in 2014, we adopt the same structure for all the walking phases, reducing the number of connections between neurons. This reduction enables the employment of the particle swarm algorithm to find the optimal values of these parameters which lead to different solutions with different performance criteria. The simulation of the proposed method on a seven-link bipedal walking model gave a good performance in the range of walking speeds, which is referred to as versatility, and in walking pattern transition. The achieved walking gaits are 1-period cyclic motions for all the input control signals except for few gaits. Besides, these 1-period cyclic motions have a good local and global stability. Finally, we expanded our neural model by adding connections that work only when the robot walks on uneven terrains, which improved the robot’s performance against this kind of perturbation.
... The human gait cycles involve multiple phases; a single support phase initiated by a push-off where the body's center of mass (CoM) moves as an inverted pendulum, and a double support phase initiated by a heelstrike which decelerates the body's CoM displacement in the propulsion direction. Even very simple biomechanical models have to include transformations of the equation of motion between the single and double support phase, thereby changing dynamic stability properties (Norris et al., 2008;Srinivasan et al., 2008;Huang et al., 2012). At the transformation, reaction size d(t) is a non-exponential function of time and cannot be estimated by an exponential function (Andronov and Pontrjagin, 1937; see intersection in Figure 3). ...
... With the knowledge of the ε, the difference, ε-d(t) > 0, can be considered as a margin of dynamic stability and the time for d(t) to reach ε can be computed by the rate of change in d(t) quantified by λ f , α, and β. The ε separating a fall and a non-fall can be evaluated for walking models where ε can be estimated as the threshold of d (0), where values above this threshold diverge to a fall and values below this threshold converge to walking (Karssen and Wisse, 2009;Huang et al., 2012). The value of ε may depend on anthropometric factors, like segment mass, length and inertia, and environmental factors, like walking surface. ...
Over the last decades, various measures have been introduced to assess stability during walking. All of these measures assume that gait stability may be equated with exponential stability, where dynamic stability is quantified by a Floquet multiplier or Lyapunov exponent. These specific constructs of dynamic stability assume that the gait dynamics are time independent and without phase transitions. In this case the temporal change in distance, d(t), between neighboring trajectories in state space is assumed to be an exponential function of time. However, results from walking models and empirical studies show that the assumptions of exponential stability break down in the vicinity of phase transitions that are present in each step cycle. Here we apply a general non-exponential construct of gait stability, called fractional stability, which can define dynamic stability in the presence of phase transitions. Fractional stability employs the fractional indices, α and β, of differential operator which allow modeling of singularities in d(t) that cannot be captured by exponential stability. The fractional stability provided an improved fit of d(t) compared to exponential stability when applied to trunk accelerations during daily-life walking in community-dwelling older adults. Moreover, using multivariate empirical mode decomposition surrogates, we found that the singularities in d(t), which were well modeled by fractional stability, are created by phase-dependent modulation of gait. The new construct of fractional stability may represent a physiologically more valid concept of stability in vicinity of phase transitions and may thus pave the way for a more unified concept of gait stability.
... Nevertheless, these two studies used relatively simple models with point feet and did not analyze the influence of ankle joint compliance in dynamic walking. Our previous studies realized multiple dynamic walking gaits of a passivity-based biped and analyzed the motion characteristics of each gait [17], [18]. ...
Stable bipedal walking is one of the most important components of humanoid robot design, which can help us better understand natural human walking. In this paper, to study gait selection and gait transition of efficient bipedal walking, we proposed a dynamic bipedal walking model with an upper body, flat feet and compliant joints. The model can achieve stable cyclic motion with different walking gaits. The hip actuation and ankle stiffness behavior of the model are quite similar to those of human normal walking. In simulation, we studied the influence of hip actuation and ankle stiffness on walking performance of each gait. The effects of ankle stiffness on gait selection are also analyzed. Gait transition is realized by adjusting ankle stiffness during walking.
... Compared with bipedal walking based on trajectory-control methods [1], which are commonly applied in industrial robots, passive dynamic walking pioneered by McGeer [2] shows more natural gaits and more efficient motions [3]. In order to understand motion characteristics of passive walkers with more natural anthropomorphic features, researchers have added the upper body [4], knee joints [5], feet with different shapes [6,7] and compliant ankle joints [8][9][10]. Although passivity-based walkers can achieve higher energetic efficiency, they have limitations of practical uses [11]. ...
... Owaki et al. added hip torsional spring and leg springs to a two-link passive bipedal model and obtain multiple gaits with different stiffness [14]. Our previous studies indicated the important effects of ankle compliance on gait selection [10] and realized speed and step length control of bipedal walkers with adaptable joint stiffness [15]. However, few efforts have been made in systematic control methods in variable-joint-stiffness bipeds and the comparison between the control performance and human natural gaits. ...
... Different from a lot of existing passivity-based bipedal walking models with round feet or point feet, flat-foot walkers have the ability of standing stably and more complex walking phase sequences [10]. When the flat foot strikes the ground, there are two impulses, "heel-strike" and "foot-strike", representing the initial impact of the heel and the following impact as the whole foot contacts the ground, respectively. ...
In this paper, we propose a seven-link passivity-based dynamic walking model, in order to further understand the principles of real human walking and provide guidance in building bipedal robots. The model includes an upper body, two thighs, two shanks, flat feet and compliant joints. A bio-inspired central pattern generator (CPG)-based control method is applied to the proposed model. In addition, we add adaptable joint stiffness to the motion control. To validate the effectiveness of the proposed bipedal walking model, we carried out simulations and human walking experiments. Experimental results indicate that human-like walking gaits with different speeds and walking pattern transitions can be realized in the proposed locomotor system.
... Second, the state space dynamics for nonlinear models of human gait are not similar to homogeneous chaotic systems like the Lorenz or Rössler attractor (e.g., Gates and Dingwell, 2009). In contrast to homogeneous chaotic systems, dynamic walking models are ''hybrid'' dynamic systems with a different set of equations of motion for the single and double support phases (e.g., Huang et al., 2012;Srinivasan et al., 2008). The sequential foot impact from the single to double support phase leads to a sequential dissipation of kinetic energy in contrast to the continuous dissipation of energy described by a single equation of motion for prototypical chaotic systems. ...
... However, Kang and Dingwell (2009) reported proximal-distal segmental differences in l. Second, dynamic models of humanoid robotic gait indicate that the degrees of freedom of the gait kinematics couple and decouple according to the transitions between single support and double support phases (Huang et al., 2012). Such coupling and decoupling of joint kinematics will lead to intra-cyclical changes in the dimensionality of the attractor manifold and, consequently, to the breakdown of the generic assumption in Takens's embedding theorem. ...
This study investigates local stability of a four-link limit cycle walking biped with flat feet and compliant ankle joints. Local stability represents the behavior along the solution trajectory between Poincare sections, which can provide detailed information about the evolution of disturbances. The effects of ankle stiffness and foot structure on local stability are studied. In addition, we apply a control strategy based on local stability analysis to the limit cycle walker. Control is applied only in the phases with poor local stability. Simulation results show that the energy consumption is reduced without sacrificing disturbance rejection ability. This study may be helpful in motion control of limit cycle bipedal walking robots with flat feet and ankle stiffness and understanding of human walking principles.
This article proposes a design method of legged walking robot hardware capable of performing passive dynamic walking with its desirable characteristics. Passive dynamic walking has a relatively good energy efficiency, and is said to be similar to the walking style of animals. However, most legged robot hardware capable of passive dynamic walking is designed through trial and error on the basis of experience. One of the major problems of designing through trial and error is the difficulty of verifying walking for the legged robot hardware that has many degree of freedom. It is relatively easy to determine the initial condition for compass-type robot hardware. However, it often takes long time to determine the appropriate initial conditions and slope angles for complicated robots such as legged robots with knees. We proposed and verified a method to design a legged robot with knees that has a desired leg length and leg mass from a compass-type legged robot.
In this article, we propose a method to design a passive dynamic walker that has a desired leg angle, step length, leg mass, etc., and verify the resulting design. More specifically, the physical parameters, such as the leg length, leg mass, and joint friction, are defined as “physical parameters” and the parameters acquired as the result of walking, such as the leg angle, step length, and walking cycle, are defined as “variable parameters.” By observing variable parameters while the robot is walking and by changing the physical parameters according to the observed variable parameters, the variable parameters are indirectly changed to desired values.
