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.     

COMPARISON OF GRANULAR MATERIAL BEHAVIOUR UNDER DRAINED TRIAXIAL AND PLANE STRAIN CONDITIONS USING 3D DEM SIMULATIONS

Three dimensional (3D) DEM (discrete element method) simulations of drained triaxial compression and plane strain tests are presented for both dense and loose assemblies of polydisperse spheres using a...     

DEM SIMULATION OF LIQUEFACTION FOR GRANULAR MEDIA UNDER UNDRAINED AXISYMMETRIC COMPRESSION AND PLANE STRAIN CONDITIONS

Based on three dimensional (3D) Discrete Element Method (DEM), the paper presents simulation results of undrained tests on loose assemblies of polydisperse spheres under axisymmetric compression and plane strain conditions using a periodic cell. In the present work, undrained tests were modelled by deforming the samples under constant volume conditions. The undrained (effective) stress paths are shown to be qualitatively similar to experimental results in literature. A microscopic parameter in terms of redundancy factor (RF) is used to identify the onset of liquefaction (or temporary liquefaction), with the condition of RF equal to unity defining the transition from ’solid-like’ to ’liquid-like’ behaviour. It is found that the undrained behaviour is governed by the evolution of redundancy factor under both undrained axisymmetric compression and plane strain conditions, and a reversal of deviatoric stress in stress path for medium loose systems occurs due to the fact that the system becomes a structural mechanism (RF<1) transiently at the microscopic level during the evolution.     

Multiple three-dimensional equilibrium solutions for cantilever beams loaded by dead tip and uniform distributed loads

The development of multiple solutions for orthotropic cantilever beams in a fully three-dimensional setting is investigated. The governing equations are solved using an iterative shooting procedure that converts the original boundary value problem into a sequence of initial value problems that converge to the desired solution. This method is well suited to finding multiple equilibrium solutions. Several classes of equilibrium configurations are described and illustrated including planar shapes, buckled planar shapes and fully three-dimensional configurations which appear far removed from the initial plane of loading. The solutions for the planar shapes and the buckled configurations compare favourably to previously published results. The development of the far-removed shapes is shown to be qualitatively similar to that of the planar shapes. The behaviour is shown to be highly dependant upon the aspect ratio of the cross-section. For certain aspect ratios it is shown, somewhat surprisingly, that out-of-plane equilibrium solutions can exist at loads below those required for multiple planar solutions.     

New analytical results of energy-based swing-up control for the Pendubot

In this paper, we revisit the energy-based swing-up control solutions for the Pendubot, a two-link underactuated planar robot with a single actuator at the base joint. The control objective is to swing the Pendubot up to its unstable equilibrium point (at which two links are in the upright position). We improve the previous energy-based control solutions by analyzing the motion of the Pendubot further. Our main contributions are threefold. First, we provide a bigger control parameter region for achieving the control objective. Specifically, we present a necessary and sufficient condition for avoiding the singular points in the control law. We obtain a necessary and sufficient condition on the control parameter such that the up–down equilibrium point (at which links 1 and 2 are in the upright and downward positions, respectively) is the only undesired closed-loop equilibrium point. Second, we prove that the up–down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively. We show that the Pendubot will eventually enter the basin of attraction of any stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero provided that these improved conditions on the control parameters are satisfied. Third, we clarify the relationship between the swing-up controller designed via the partial feedback linearization and that designed by the energy-based approach. We present the simulation results for validation of these results.     

Free-floating closed-chain planar robots: Kinematics and path planning

The study of free-floating manipulators is important for the success of robotics program in space and in the design of innovative robot systems which can operate over a large workspace. In order to study the fundamental theoretical and experimental issues encountered in space robotics, a closed-chain planar manipulator was built at Ohio University (OU) which floats on a flat table using air bearings. Due to the absence of external forces in the plane of the table and couples normal to this plane, the linear momentum in the plane and the angular momentum normal to this plane are conserved. It is well known that the linear momentum equations are holonomic while the angular momentum equation is nonholonomic. Due to this nonholonomic behavior, the path-planning schemes commonly used for fixed-base manipulators do not directly apply to free-floating manipulators. In this paper, we present an algorithm for motion planning of planar free-floating manipulators based on the inverse position kinematics of the mechanism. It is demonstrated that the inverse position kinematics algorithms, commonly used for fixed-base manipulators, can be successfully applied to free-floating manipulators using an iterative search procedure to satisfy the nonholonomic angular momentum constraints. This procedure results in paths identical to those predicted by inverse rate kinematics. The inverse position kinematics algorithm is then used to avoid singularities during motion to result in successful paths. The results of the simulation of this algorithm using parameter estimates of the OU free-floating robot are presented.     

Order and chaos in a new 3D dynamical model describing motion in non-axially symmetric galaxies

We present a new dynamical model describing 3D motion in non-axially symmetric galaxies. The model covers a wide range of galaxies from a disk system to an elliptical galaxy by suitably choosing the dynamical parameters. We study the regular and chaotic character of orbits in the model and try to connect the degree of chaos with the parameter describing the deviation of the system from axial symmetry. In order to obtain this, we use the Smaller ALingment Index (SALI) method to extensive samples of orbits obtained by integrating numerically the equations of motion, as well as the variational equations. Our results suggest that the influence of the deviation parameter on the portion of chaotic orbits strongly depends on the vertical distance z from the galactic plane of the orbits. Using different sets of initial conditions, we show that the chaotic motion is dominant in galaxy models with low values of z, while in the case of stars with large values of z the regular motion is more abundant, both in elliptical and disk galaxy models.     

Mechanical properties of lattice grid composites

An equivalent continuum method only considering the stretching deformation of struts was used to study the in-plane stiffness and strength of planar lattice grid com- posite materials. The initial yield equations of lattices were deduced. Initial yield surfaces were depicted separately in different 3D and 2D stress spaces. The failure envelope is a polyhedron in 3D spaces and a polygon in 2D spaces. Each plane or line of the failure envelope is corresponding to the yield or buckling of a typical bar row. For lattices with more than three bar rows, subsequent yield of the other bar row after initial yield made the lattice achieve greater limit strength. The importance of the buckling strength of the grids was strengthened while the grids were relative sparse. The integration model of the method was used to study the nonlinear mechanical properties of strain hardening grids. It was shown that the integration equation could accurately model the complete stress-strain curves of the grids within small deformations.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: I. Mechanical Equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.     

Global motion analysis of energy-based control for 3-link planar robot with a single actuator at the first joint

This paper studies nonlinear control of a 3-link planar robot moving in the vertical plane with only the first joint being actuated while the two other revolute joints are passive (called the APP robot below). A nonlinear energy-based controller is proposed, whose objective is to drive the APP robot into an invariant set where the first link is in the upright position and the total mechanical energy converges to its value at the upright equilibrium point (all three links are in the upright position). By presenting and using a new property of the motion of the APP robot, without any condition on its mechanical parameters, this paper proves that if the control gains are larger than specific lower bounds, then only a measure-zero set of initial conditions converges to three strictly unstable equilibrium points instead of converging to the invariant set. This paper presents numerical results for a physical 3-link planar robot to validate the obtained theoretical results and to demonstrate a switch–and–stabilize maneuver in which the energy-based controller is switched to a linear state feedback controller that stabilizes the APP robot at its upright equilibrium point.     

Fracture in plates of finite thickness

Application of the plane theory of elasticity to planar crack or angular corner geometries leads to the concept of stress singularity and stress intensity factor, which are the cornerstone of contemporary fracture mechanics. However, the stress state near an actual crack tip or corner vertex is always three-dimensional, and the meaning of the results obtained within the plane theory of elasticity and their relation to the actual 3D problems is still not fully understood. In particular, it is not clear whether the same stress field as found from the well-known 2D solutions of the theory of elasticity do describe the corresponding stress components in a plate made of a sufficiently brittle material and subjected to in-plane loading, and what effect the plate thickness has. In the present study we adopt, so called, first order plate theory to attempt to answer these questions. New features of the elastic solutions obtained within this theory are discussed and compared with 2D analytical results and experimental studies as well as with 3D numerical simulations.     

Orbit classification in the Hill problem: I. The classical case

The case of the classical Hill problem is numerically investigated by performing a thorough and systematic classification of the initial conditions of the orbits. More precisely, the initial conditions of the orbits are classified into four categories: (i) non-escaping regular orbits; (ii) trapped chaotic orbits; (iii) escaping orbits; and (iv) collision orbits. In order to obtain a more general and complete view of the orbital structure of the dynamical system, our exploration takes place in both planar (2D) and the spatial (3D) version of the Hill problem. For the 2D system, we numerically integrate large sets of initial conditions in several types of planes, while for the system with three degrees of freedom, three-dimensional distributions of initial conditions of orbits are examined. For distinguishing between ordered and chaotic bounded motion, the Smaller Alignment Index method is used. We managed to locate the several bounded basins, as well as the basins of escape and collision and also to relate them to the corresponding escape and collision time of the orbits. Our numerical calculations indicate that the overall orbital dynamics of the Hamiltonian system is a complicated but highly interested problem. We hope our contribution to be useful for a further understanding of the orbital properties of the classical Hill problem.     

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.     

Epsilon-continuation approach for truss topology optimization

In the present paper, a so-called epsilon-continuation approach is proposed for the solution of singular optimum in truss topology optimization problems. This approach is an improved version of the epsilon-relaxed approach developed by the authors previously. In the proposed approach, we start the optimization process from a relaxation parameter with a relatively large value and obtain a solution by applying the epsilon-relaxed approach. Then we decrease the value of the relaxation parameter by a small amount and choose the optimal solution found from the previous optimization process as the initial design for the next optimization. This continuation process is continued until a small termination value of the relaxation parameter is reached. Convergence analysis of the proposed approach is also presented. Numerical examples show that this approach can alleviate the dependence of the final solution on the initial choice of the design variable and enhance the probability of finding the singular optimum from rather arbitrary initial designs. The project supported by the National Natural Science Foundation of China (10102003, 10032010 and 10032030)     

Asymmetric three-link passive walker

Passive walkers are dynamically stable robots with a gait that resembles the human locomotion. These walkers can be studied to better understand the dynamic behavior of the human gait and design efficient active walkers and assistive devices. In this paper, we study the walking dynamics of a three-link passive walker with an asymmetrical structure where one leg has a knee while the other is knee-less. After finding a 2-periodic steady gait for the three-link walker with humanlike inertial parameters for both legs, the possibility of a gait with symmetrical step lengths is discussed where the half inter-leg angles at the beginning of every step are made equal by altering the physical parameters of the knee-less leg. We further study the gaits with symmetrical step lengths and show that by replacing one leg of a four-link symmetric walker with the knee-less leg of the three-link walker with the symmetrical half inter-leg angles, the dynamic behavior of the kneed leg remains unchanged. This approach can be adapted in the field of gait rehabilitation and prosthesis design to obtain a more symmetrical gait and preserve the motion of the healthy leg.

     

On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)

Strain and damage interactions during tearing of a ductile Al-alloy with high work hardening are assessed in situ and in 3D combining two recently developed experimental techniques, namely, synchrotron laminography and digital volume correlation. Digital volume correlation consists of registering 3D laminography images. Via simultaneous assessments of 3D strain and damage at a distance of 1-mm ahead of a notch root of a thin Compact Tension-like specimen, it is found that parallel crossing slant strained bands are active from the beginning of loading in a region where the crack will be slanted. These bands have an intermittent activity but are stable in space. Even at late stages of deformation strained bands can stop their activity highlighting the importance of plasticity on the failure process rather than damage softening. One void is followed over the loading history and seen to grow and orient along the slant strained band at very late stages of deformation. Void growth and strain are quantified. Gurson–Tvergaard–Needleman-type simulations using damage nucleation for shear, which is based on the Lode parameter, are performed and capture slant fracture but not the initial strain fields and in particular the experimentally found slant bands. The band formation and strain distribution inside and outside the bands are discussed further using plane strain simulations accounting for plastic material heterogeneity in soft zones.     

Initial planar deformation of dilatant granular materials

A theory for the initial planar deformation of dilatant granular materials based on a kinematic proposal of R. Butterfield and R.M. Harkness (1972) is presented. The theory introduces an additional parameter called the angle of dilatancy into the traditional structure of plasticity theories for granular materials and soils. When the angle of dilatancy is zero, the present theory reduces to the theory introduced by A.J.M. Spencer in 1964. When the angle of dilatancy is equal to the angle of internal friction, the present theory reduces to the planar form of the theory introduced by D. C. Drucker and W. Prager in 1952. The properties of the theory presented here include coincidence of the stress and velocity characteristics, realistic energy dissipation predictions, and, in general, non-coincidence of the principal axes of stress and strain-rate. However, the angle of dilatancy is assumed to be a constant in this analysis and it does not decrease to zero with increased monotonic shearing deformation as experiment requires that it should, the theory therefore being limited to the initial deformation of dilatant granular materials.     

三维扰动波的非平行边界层稳定性研究

导出了三维扰动波的原始变量形式的抛物化稳定性方程（PSE）,研究了三维空间模态TS波的非平行边界层稳定性问题.采用了法向四阶紧致格式,以提高计算精度.通过给出不会导致奇性的坐标变换、修改外边界条件以及克服平行流初始值的瞬态影响和推进步长的限制,保证了计算的数值稳定.用补全元素带状矩阵法求解块三对角矩阵,大大提高了速度.计算结果清楚地显示了三维扰动波的演化过程和非平行性对边界层稳定性的影响,特别是,观察到非平行性对三维扰动波的影响,有时会使其稳定性出现逆转的现象.还研究了逆压梯度的作用.算例的结果与其他结果符合良好.     

Measurement of Orthogonal Stress Gradients Due to Impact Load on a Transparent Sheet using Digital Gradient Sensing Method

A full-field optical method called Digital Gradient Sensing (DGS) for measuring stress gradients due to an impact load on a planar transparent sheet is presented. The technique is based on the elasto-optic effect exhibited by transparent solids due to an imposed stress field causing angular deflections of light rays quantified using 2D digital image correlation method. The measured angular deflections are proportional to the in-plane gradients of stresses under plane stress conditions. The method is relatively simple to implement and is capable of measuring stress gradients in two orthogonal directions simultaneously. The feasibility of this method to study material failure/damage is demonstrated on transparent planar sheets of PMMA subjected to both quasi-static and dynamic line load acting on an edge. In the latter case, ultra high-speed digital photography is used to perform time-resolved measurements. The quasi-static measurements are successfully compared with those based on the Flamant solution for a line-load acting on a half-space in regions where plane stress conditions prevail. The dynamic measurements, prior to material failure, are also successfully compared with finite element computations. The measured stress gradients near the impact point after damage initiation are also presented and failure behavior is discussed.