The simple chaotic model of passive dynamic walking

Recent findings on the dynamical analysis of human locomotion characteristics such as stride length signal have shown that this process is intrinsically a chaotic behavior. The passive walking has been defined as walking down a shallow slope without using any muscular contraction as an active controller. Based on this definition, some knee-less models have been proposed to present the simplest possible models of human gait. To maintain stability, these simple passive models are compelled to show a wide range of different dynamics from order to chaos. Unfortunately, based on simplifications, for many years the cyclic period-one behavior of these models has been considered as the only stable response. This assumption is not in line with the findings about the nature of walking. Thus, this paper proposes a novel model to demonstrate that the knee-less passive dynamic models also have the ability to model the chaotic behavior of human locomotion with some modifications. The presented novel model can show chaotic behavior as a stable and acceptable answer using a chaotic function in heel-strike condition. The represented chaotic model is also able to simulate different types of motor deficits such as Parkinson’s disease only by manipulating the value of chaotic parameter. Our model has extensively examined in complexity and chaotic behavior using different analytical methods such as fractal dimension, bifurcation and largest Lyapunov exponent, and it was compared with conventional passive models and the stride signal of healthy subjects and Parkinson patients.     

LCP method for a planar passive dynamic walker based on an event-driven scheme

The main purpose of this paper is to present a linear complementarity problem (LCP) method for a planar passive dynamic walker with round feet based on an event-driven scheme. The passive dynamic walker is treated as a planar multi-rigid-body system. The dynamic equations of the passive dynamic walker are obtained by using Lagrange’s equations of the second kind. The normal forces and frictional forces acting on the feet of the passive walker are described based on a modified Hertz contact model and Coulomb’s law of dry friction. The state transition problem of stick-slip between feet and floor is formulated as an LCP, which is solved with an event-driven scheme. Finally, to validate the methodology, four gaits of the walker are simulated: the stance leg neither slips nor bounces; the stance leg slips without bouncing; the stance leg bounces without slipping; the walker stands after walking several steps.     

摩擦与滚阻对被动行走器步态影响的研究

本文研究了圆弧足被动行走器支撑足与地面间的摩擦系数和滚阻系数对被动行走器步态的影响. 首先分别利用扩展的 赫兹接触力模型和LuGre摩擦模型描述了支撑足与地面接触点处的法向支撑力和切向摩擦力,并考虑了行走过程中支撑足 所受的滚动摩阻;其次利用第二类Lagrange方程推导出了该系统的动力学方程,并通过与已有成果的对比确定 了合适的LuGre摩擦模型参数;最后仿真分析了摩擦系数和滚阻系数对被动行走器步态的影响. 研究发现:摩擦系数的改变 虽然对被动行走器行走的平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较小,但摩擦系数的减小 会改变其行走步态类型,如发生倍周期分岔甚至混沌现象;然而,滚阻系数的改变会对行走器行走的 平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较大,尚未发现滚阻系数的改变会引起其行走步态的变化.      

弹簧刚度对被动步行的稳定性影响

由人类步行的生物力学研究得到启发,在被动双足步行机器人的髋关节处引入了扭簧,并通过仿真和试验研究了弹簧刚度对被动步行稳定性的影响.在仿真中,用胞映射方法计算被动步行机器人的吸引盆,并用吸引盆来衡量机器人的稳定性,研究了弹簧刚度对被动步行吸引盆大小的影响. 仿真结果表明, 吸引盆随着弹簧刚度的增大而增大. 在试验中,使机器人在各弹簧刚度参数下沿斜坡向下行走100次,记录下行走到头的次数作为稳定性的度量. 试验结果表明, 存在一个大小适中的弹簧刚度使机器人稳定性最大. 对弹簧提高机器人稳定性的原因进行了分析,对造成仿真与试验之间差异的原因进行了分析.      

A data recovery algorithm for a video analysis of human movements on the basis of force plate measurements

A data recovery method is proposed for a video analysis of human movements using force plate measurements. An algorithm is formulated to determine the missing angles of a three-link anthropomorphic device on the basis of a mathematical model of motion, a parameter identification model, and force plate measurements. This algorithm is used to recover the missing inclination angle between a shank and a horizontal line during the process of sitting down.     

3D Passive Walkers: Finding Periodic Gaits in the Presence of Discontinuities

This paper studies repetitive gaits found in a 3D passivewalking mechanism descending an inclined plane. By using directnumerical integration and implementing a semi-analytical scheme forstability analysis and root finding, we follow the corresponding limitcycles under parameter variations. The 3D walking model, which is fullydescribed in the paper, contains both force discontinuities andimpact-like instantaneous changes of state variables. As a result, thestandard use of the variational equations is suitably modified. Theproblem of finding initial conditions for the 3D walker is solved bystarting in an almost planar configuration, making it possible to useparameters and initial conditions found for planar walkers. The walkeris gradually transformed into a 3D walker having dynamics in all spatialdirections. We present such a parameter variation showing the stabilityand the amplitude of the hip sway motion. We also show the dependence ofgait cycle measurements, such as stride time, stride length, averagevelocity, and power consumption, on the plane inclination. The paperconcludes with a discussion of some ideas on how to extend the present3D walker using the tools derived in this paper.     

Transparency enhancement for an active knee orthosis by a constraint-free mechanical design and a gait phase detection based predictive control

This paper proposes a novel mechanical design of a lower limb exoskeleton device which prevents the residual stresses due to arthro-kinematics movements of synovial joints and by the way allows effective compensation for dynamic disturbances in osteo-kinematic movements of the wearer. Here, the exoskeleton is only actuated at the knee joints to provide assistive torques, which are required to assist the anatomical joint motion and to increase the transparency of the device. Dynamic simulations of a virtual human equipped with this exoskeleton are used to quantify the disturbances induced by the device during locomotion and to show the benefit of passive mechanisms introduced in the mechanical attaches as well. The authors also demonstrated how the device’s transparency can be improved by providing the motor torques in order to compensate the inertial and gravitational effects. This can be done rely on the knowledge of the locomotion movement phases. A robust gait phase detection method was implemented on the experimental device in order to identify specific gait phases in real time. This method exploits the K-nearest neighbors algorithm to identify the k-closest trained vectors, coupling with a discrete time Markov chain to determine the phases shift probability during the gait cycle. This gait detection algorithm was tested with a percentage of success of more than 95% when the subjects walked with constant and variable stride lengths.     

Dynamic Analysis of Human Gait Disorder and Metabolical Cost Estimation

For the design and improvement of orthotic and prosthetic devices the biomechanical effort is an important criterion to obtain a more comfortable and natural gait of humans with gait disorders. In the first part of the paper the inverse dynamic analysis based on measurements of the human gait for subjects with different kinds of disorders is presented. The second part is devoted to a method to estimate the energy expenditure for human motions. This approach allows the computation of metabolical cost for human locomotion using Hill-type muscle models.     

Impactless biped walking on a slope

Walking without impacts has been considered in dynamics as a motion/force control problem. In order to avoid impacts, an approach for both the specified motion of the biped and its ground reaction forces was presented yielding a combined motion and force control problem. As an application, a walker on a horizontal plane has been considered. In this paper, it is shown how the control of the ground reaction forces and the energy consumption depend on the gradient of a slope. The biped dynamics and the constraints within the biped system and on the ground are discussed. A motion control synthesis is developed using the inverse dynamics principle proven to be most efficient for human walking research, too. The impactless walking with controlled legs is illustrated by a seven-link biped. The “flying” biped has nine degrees of freedom, with six control inputs. During locomotion, the standing leg has three scleronomic constraints, and the trunk has three rheonomic constraints. However, there are three rheonomic constraints for the prescribed leg motion or three scleronomic constraints for reaction forces of the trailing leg, respectively. The nominal control action for impactless walking can be precomputed and stored. The model proposed allows the investigation of several problems: uphill and downhill walking, optimization of step length, stiction of the feet on the slope and many more. All these findings are also of interest in biomechanics.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

3D-Simulation of human walking by parameter optimization

The simulation of human gait is a complex dynamical problem that requires accounting for energy consumption as well as dealing with a redundantly actuated multibody system. If muscle forces and generalized coordinates are parameterized, optimization techniques allow the simulation of the muscle forces and of the walking motion. An optimization framework is presented for non-symmetrical gait cycles found in the presence of one-sided gait disorders. The motion of each leg is independently parameterized for a whole walking cycle. The non-linear constraints used to fulfill the equations of motion and the kinematical constraints of the different walking phases are implemented in an efficient way. Fifth-order splines are used for the parameterization to reduce the oscillatory behavior coming from non-periodicity conditions. To achieve the computational performance required for three-dimensional simulations, the spline interpolation problem has been split in two parts, one is performed in a preprocessing stage and the other during the optimization. Numerical differentiation via finite differences is avoided by implementing analytical derivatives of the splines functions and of the contractile element force law. The results show good numerical performance, and the computational efficiency for 3D-simulations with one-sided gait disorders is highlighted.     

Influence of the flexibility of links on the dynamics of a multilink robot manipulator

The influence of the distributed flexibility of links on the dynamic behavior of a cantilevered multilink robot manipulator is studied. Two dynamic models of the robot are considered. In one of the models, the effectors are flexible. It is described by a hybrid system of ordinary and partial differential equations. In the other model, the manipulator links are perfectly rigid. It is based on the Lagrange formalism and described by a system of ordinary differential equations. To estimate the positioning accuracy, the results of modeling the manipulator in both cases are compared. A three-link manipulator is examined as an example __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 126–137, May 2007.     

Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear

Some complex engineering structures can be modeled as multiple beams connected through coupling elements. When the coupling element is elastic, it can be simplified as a mass-spring system. The existing studies mainly concentrated on the double-beam coupled through elastic connectors, where the connector is simplified as the equivalent linear stiffness element or linear mass-spring system. Furthermore, many researches ignore rotational boundary restraints in analyzing dynamic behavior of the double-beam connected through elastic connectors, limiting their engineering generality. Considering the above limitations, this study attempts to employ the cubic nonlinear stiffness in the coupling mass-spring system and study the potential application of the mass-spring system that is nonlinear on the vibration control of the double-beam system. Using the variational method and the generalized Hamiltonian method build the corresponding system’s governing functions. Applying the Galerkin truncation method (GTM) obtains the dynamic behavior of the double-beam connected through a mass-spring system that is nonlinear. According to this study, the change of the mass-spring system that is nonlinear significantly influences the dynamic behavior of the double-beam system, where the complex dynamic behavior occurs under certain parameters of the mass-spring system that is nonlinear. Suitable parameters of the mass-spring system that is nonlinear are good at the vibration suppression at the boundary of the vibration system. Furthermore, the mass-spring system that is nonlinear can change the characteristics of the double-beam system’s kinetic energy transfer. For the vibration model established in this work, a quasi-periodic vibration state can be regarded as a sign of the occurrence of the targeted energy transfer of the double-beam connected through a mass-spring system that is nonlinear.

     

A simple and quick control strategy for a class of first-order nonholonomic manipulator

This paper presents a simple and quick control strategy for a class of first-order nonholonomic manipulator: planar n-link manipulator with a passive first joint (no gravity). The control target is driving its endpoint from any initial position to any target position. First, we reduce the planar n-link manipulator to a planar three-link one by maintaining the states (angles and angular velocities) of n-3 active links in the initial value all the time. That is, we only adjust the states of the remaining two active links in the whole control process, and these two adjusted active links are chosen to guarantee that the target position is in the reachable region of the planar n-link manipulator by using the enumeration method. Then, we divide the whole control process into two stages for the reduced planar three-link manipulator. In each stage, the manipulator becomes a planar two-link one by maintaining the states of one adjusted active link to be the constant value. The state constraint existing between the passive first link and the adjusted active link is obtained by using the integral characteristics of a planar two-link manipulator. Meantime, the geometric constraint between the position of the endpoint and angles of all joints is obtained based on the homogeneous coordinate transformation method. According to the above two kinds of constraint, the target angles of the two adjusted active links are calculated by employing the particle swarm optimization algorithm. When the two adjusted active links are controlled to their target angles in turn, the control target of the planar n-link manipulator is completed. Finally, simulation results demonstrate that the proposed control strategy is valid and rapid.     

Frequency–energy plot and targeted energy transfer analysis of coupled bistable nonlinear energy sink with linear oscillator

The nonlinear energy sink (NES), which is proven to perform rapid and passive targeted energy transfer (TET), has been employed for vibration mitigation in many primary small- and large-scale structures. Recently, the feature of bistability, in which two nontrivial stable equilibria and one trivial unstable equilibrium exist, is utilized for passive TET in what is known as bistable NES (BNES). The BNES generates a nonlinear force that incorporates negative linear and multiple positive or negative nonlinear stiffness components. In this paper, the BNES is coupled to a linear oscillator (LO) where the dynamic behavior of the resulting LO-BNES system is studied through frequency–energy plots (FEPs), which are generated by analytical approximation using the complexification-averaging method and by numerical continuation techniques. The effect of the length and stiffness of the transverse coupling springs is found to affect the stability and topology of the branches and indicates the importance of the exact physical realization of the system. The rich nonlinear dynamical behavior of the LO-BNES system is also highlighted through the appearance of multiple symmetrical and unsymmetrical in- and out-of-phase backbone branches, especially at low energy levels. The superimposed wavelet frequency spectrums of the LO-BNES response on the FEP have verified the robustness of the TET mechanism where the role of the unsymmetrical NNM backbones in TET is clearly observed.

     

A free gait algorithm for quadrupedal walking machines

A free gait is a computer generated, rule-based gait for a walking machine to walk on rough terrain. Based on a given terrain map, the gait algorithm selects footholds for leg placements and determines the movements of legs and body. In the past, a few free gaits for hexapods have been developed. For quadrupeds, the only report on free gait was briefly mentioned in a paper by Hirose [Int. J. Robotics Res., 3(2) (1984)]. In this paper, a free gait algorithm for a quadrupedal walking chair is developed. For quadrupeds, the stability margin is small due to a small number of legs and the choices of a leg to be lifted are limited. Hence, deadlock situations may occur quite often. Many special techniques are incorporated into the algorithm in order to reduce deadlocks. This free gait algorithm adopts the wave-crab gaits as the primary gait because they are periodic and can provide good stability. The algorithm also adopts a non-periodic free gait to handle terrain with higher concentration of forbidden areas. This algorithm is evaluated under different terrain conditions using computer simulations. The results show that the performance is satisfactory on randomly generated rough terrain and needs improvement on manually generated rough terrain.     

A Note on a Walking Machine Model

The paper continues studies intended to find out whether it is possible to create a prototype walking machine with relatively simple components. In this connection, the control problem is solved for a two-dimensional model of biped machine. It has a torso and two telescopic legs. Each leg includes a ponderable section of constant length and an imponderable section of variable length. The machine, regarded as a system with variable constraints, implements a single-stance gait (one stance leg at a time) with a step of constant duration. The contact of the swing leg with the ground is analyzed within the framework of Carnot's theorem (perfectly inelastic impact). It is assumed that the force developed in the stance leg is due to the deformation of the leg's spring and that this deformation can be controlled. An algorithm is proposed to synthesize a control system that takes into account collisions occurring at reverse of the roles of the legs. This algorithm is based on methods of optimizing periodic systems. The algorithm is compared with approaches used by other authors     

Kinematic analysis of a 5-R SP parallel mechanism with?centralized motion

The paper presents a kinematic analysis of a parallel mechanism, referred to here as a mechanism with centralized motion. The paper includes a proof, based on the geometry of the mechanism, that the platform exhibits centralized motion. An interesting feature of this parallel mechanism is that it is partially collapsible which may be beneficial in practical applications where storage space is limited. The platform is connected to a base, regarded as fixed in this paper, by five identical legs where each leg is a three-link chain connected by a revolute joint, a spherical joint, and a prismatic joint. The result is that the platform has a screw motion about an axis which is perpendicular to the base and passes through the centroids of the base and the platform, for all positions of the platform. The pitch of the instantaneous screw depends on the platform assembly configuration and is a function of the platform position and orientation. To complete the kinematic study, the paper includes closed-form solutions to the inverse and forward position and velocity problems. Finally, the paper includes several numerical examples to illustrate some of the key features of this novel parallel mechanism.     

A dynamic gait stabilization algorithm for quadrupedal locomotion through contact time modulation

In this paper, we propose a stabilization method for dynamic gaits of quadrupedal walking robots covering a wide range of speeds and various types of gait. Our stabilization method is based on adjusting the contact time between the four legs and ground. By modulating the contact time, the impact applied to the body can be controlled and stabilized. The stability provided by the proposed algorithm was proved in the sense of Lyapunov. The proposed algorithm also demonstrated robust performance under large external disturbances, and the performance was compared with other algorithms through simulations. Simulation results of bounding gaits under different ground conditions were compared, and the various types of stable gait implemented by the proposed algorithm are also presented.

     

Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam

Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever beam–impactor systems. For off-resonance excitation at two and a half times the fundamental frequency, the response of the oscillating cantilever experiences period doubling as the separation distance or clearance between the beam axis and the contact surface is decreased. The nonlinear phenomenon is studied by using phase portraits, Poincaré sections, and spectral analysis. Motivated by atomic force microscopy, this general dynamic behavior is studied as a means to locating the separation distance corresponding to grazing where the contact force is minimized.