Wheels' performance of Mars exploration rovers: Experimental study from the perspective of terramechanics and structural mechanics

The soft ground and stones on the surface of Mars may cause sinkage of and damage to the wheels of a Mars rover. Therefore, we analyzed the performance of a wheel-step Mars rover from the perspective of terramechanics and structural mechanics. Using China’s Mars rover wheel prototype and wheel-soil interaction testbed, we obtained the driving performance of the wheels of a Mars rover under various conditions, including full skidding conditions. The vertical load set in the tests was determined using the gravity of Mars and the mass of the wheel-step Mars rover. The results indicate that the driving torque, sinkage, and resistance coefficient have a linear relationship with the wheel vertical load. An analysis of the structural mechanical characteristics of the wheel of the Mars rover was conducted by testing the radial, axial, torsional, and deflection stiffness. We found that the wheel ribs can improve the stiffness of the wheel but may reduce its driving performance. The analysis methods and evaluation indices can be used to analyze the performance of the wheels of other Mars rovers. Furthermore, the findings of this study can be used to optimize wheel design and motion control of wheel-step Mars exploration rovers.     

Using the Dirac approach to model the motion of a railway vehicle when its wheel flange climbs the rail

A mathematical model is proposed to analytically estimate the derailment conditions for a railway vehicle when its wheel flange climbs the rail. The structural parameters of the railway vehicle and the conditions of its motion are taken into account. This model is based on the primary Dirac constraints between the generalized coordinates and momenta and is quite different from the classical no-slip model in the general case. The primary Dirac constraints appear because of the fact that the Lagrangian becomes degenerate when the stiffness of the creep forces tends to infinity at the wheel-rail contact points and the ratio of the wheelset mass to the vehicle mass tends to zero.     

Performance evaluation of accelerometers used for penetration experiments

We present a Hopkinson bar technique to evaluate the performance of accelerometers that measure large amplitude pulses, such as those experienced during projectile penetration tests. An aluminum striker bar impacts a thin Plexiglas or copper disk placed on the impact surface of an aluminum incident bar. The Plexiglas or copper disk pulse shaper produces a nondispersive stress wave that propagates in the aluminum incident bar and eventually interacts with a tungsten disk at the end of the bar. A quartz stress gage is placed between the aluminum bar and tungsten disk, and an accelerometer is mounted to the free end of the tungsten disk. An analytical model shows that the rise time of the incident stress pulse in the aluminum bar is long enough and the tungsten disk length is short enough that the response of the tungsten disk can be accurately approximated as rigid-body motion. We measure stress at the aluminum bar-tungsten disk interface with the quartz gage and we calculate rigid-body acceleration of the tungsten disk from Newton's Second Law and the stress gage data. In addition, we measure strain-time at two locations on the aluminum incident bar to show that the incident strain pulse is nondispersive and we calculate rigid-body acceleration of the tungsten disk from a model that uses this strain-time data. Thus, we can compare accelerations measured with the accelerometer and accelerations calculated with models that use stress gage and strain gage measurements. We show that all three acceleration-time pulses are in very close agreement for acceleration amplitudes to about 20,000 G.     

Motion of a wheel on snow

A model of a snow layer represented by a continuous set of columns whose deformations are described by the nonlinear model of an ideal elastoplastic continuous medium with viscous properties is proposed. Under the action of a rigid wheel on snow, the field of shear stresses is specified by the law of dry friction. Prom the equations of motion describing the plane-parallel motion of the wheel, there are determined a zone of contact of the wheel with snow, the steady motions of the wheel, and a mode of slipping the wheel. The numerical results are given in tables and figures. These results are obtained by solving the nonlinear equations of motion containing definite integrals with variable integration limits.     

Variational principles for constrained systems: Theory and experiment

In this paper we present two methods, the nonholonomic method and the vakonomic method, for deriving equations of motion for a mechanical system with constraints. The resulting equations are compared. Results are also presented from an experiment for a model system: a ball rolling without sliding on a rotating table. Both sets of equations of motion for the model system are compared with the experimental results. The effects of various forms of friction are considered in the nonholonomic equations. With appropriate friction terms, the nonholonomic equations of motion for the model system give reasonable agreement with the experimental observations.     

Minimal models for squealing of railway block brakes

Block brakes have been used to brake railway vehicles for approximately 200 years. During the last 40 years, disk brakes have replaced the block brakes at passenger cars but block brakes are still used for all freight wagons. One problem of block brakes is that they show an enormous tendency to squeal. Although the block brake is a very common and old technical component, there exists almost no scientific work on its noise behavior regarding squeal. On the other hand, a lot of work has been done on the problem of disk brake squeal especially concerning the modeling of the excitation mechanism. The goal of this paper is to investigate whether and how models from disk brake squeal can be modified to model block brake squeal. The starting point of these investigations is a minimal model for an automotive disk brake introduced by von Wagner et al. (J Sound Vibration, 51(1–2):223–237, 2007) that is adapted to the block brake problem. It can be shown that such a simple converted model does not show any instability at all. A deeper analysis suggests that the reasons for squeal in block brakes could originate from in-plane vibrations of the brake disk or specific geometrical properties of the railway wheel. The self-excited vibrations explaining the squeal occur at relatively low rotational speeds far below the first critical rotor speed which has rarely been observed in rotor dynamics.     

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.     

Modeling motion resistance of rigid wheels

A mathematical model was developed to describe the motion resistance of rigid wheels under self-propelled conditions where no drawbar pull is developed. The modeling was based on the concept that the work required to overcome the motion resistance is equal to the energy dissipated in compacting the soil beneath the wheel. Slippage of the wheel was also considered in the modeling. Finally the validity of the proposed model was discussed.     

The effect of velocity on rigid wheel performance

A simulation model to predict the effect of velocity on rigid-wheel performance for off-road terrain was examined. The soil–wheel simulation model is based on determining the forces acting on a wheel in steady state conditions. The stress distribution at the interface was analyzed from the instantaneous equilibrium between wheel and soil elements. The soil was presented by its reaction to penetration and shear. The simulation model describes the effect of wheel velocity on the soil–wheel interaction performances such as: wheel sinkage, wheel slip, net tractive ratio, gross traction ratio, tractive efficiency and motion resistance ratio. Simulation results from several soil-wheel configurations corroborate that the effect of velocity should be considered. It was found that wheel performance such as net tractive ratio and tractive efficiency, increases with increasing velocity. Both, relative wheel sinkage and relative free rolling wheel force ratio, decrease as velocity increases. The suggested model improves the performance prediction of off-road operating vehicles and can be used for applications such as controlling and improving off-road vehicle performance.     

Constraint implementation in wheeled apparatus rolling problems

The models obtained from equations of rolling of an apparatus with small slips of wheels with respect to the supporting plane are considered by passing to the limit of infinite values of rigidity of contact forces (zero values of slip speeds). The conditions under which these equations become the classical nonholonomic wheel no-slip model are discussed. It is shown that, for small angles of turn of the apparatus front wheels about the vertical axis, neglect of slips in the transverse direction is not correct; namely, the limit model is determined by the wheel no-slip conditions in the longitudinal direction and by the primary Dirac constraints arising owing to the degeneration of the Lagrangian of the system. The methods used to decrease the order of equations, where small slips of wheels are taken into account, can be useful for qualitative analysis of motion of wheeled systems and for solving the problems of estimation and control in real time.     

Coupled shear-torsional motion of a rubber support system

The finite amplitude, coupled shear-torsional motion of a circular disk supported between identical rubber spring cylinders is studied. The material of the springs is assumed to be an incompressible elastic material. The oscillatory motion oscillatory of the disk is studied for two different cases. In the first case, the material of the spring is assumed to be an incompressible elastic material whose response functions are constants. Typical examples include the Mooney-Rivlin model. The motion of the disk in this case is governed by two independent equations whose closed form solutions are noted. For the second case, the material of the spring is assumed to be an incompressible quadratic material. The motion oscillatory of the disk in this case is governed by two coupled nonlinear differential equations. The stability properties of small shearing oscillation superimposed on finite torsion and small torsional oscillation superimposed on finite shearing are studied.     

Integration of the equations of motion of a disk on a rough plane

In the present paper, we study the problem on the motion of a thin homogeneous disk on a horizontal plane under the action of dry (Coulomb) friction. We consider the radially symmetric law of normal pressure distribution over the disk plane. The equations of motion are integrated in elementary functions, which gives exact formulas for the length of the disk path and the time until complete stop.     

带有姿态的绳系航天器混沌运动实验研究

利用地面物理仿真平台研究了绳系航天器的混沌动力学行为. 首先, 根据天地动力学相似原理, 通过对卫星仿真器施加喷气力和动量轮力矩来模拟空间动力学环境, 提出了两种等效方案, 给出了它们各自适用的实验工况. 数值结果表明, 在轨绳系航天器在一定的参数条件下系绳摆动为周期或概周期运动、航天器姿态发生混沌运动. 物理仿真验证了等效方案的有效性, 揭示了绳系航天器的混沌运动特征, 表明在阻尼力矩的作用下可以避免绳系航天器混沌运动的发生.      

Multi-objective traction optimization of vehicles in loose dry sand using the generalized reduced gradient method

A work optimization strategy is combined with algorithms within the vehicle-terrain interface (VTI) model to maximize the traction of a four-wheel vehicle operating on loose dry sand. The optimization model distributes traction among the steered and non-steered wheels with the work optimum coefficient (WOC) of each wheel treated as an independent design objective. Drawbar pull (DBP), motion resistance (MR), longitudinal traction coefficient (LTC), lateral force coefficient (LFC), tire deflection, and wheel slip are key parameters that appear in the VTI model for traction performance analysis. The analysis includes wheels of different diameters, widths, heights, and inflation pressures, under variable wheel slips. A multi-objective optimization problem is formulated over a thirteen-dimensional search space bounded by eight design constraints. The generalized reduced gradient method is used to predict optimal values of the design variables as well as ground and traction parameters such as DBP, MR, LTC, and LFC for maximum slope climbing efficiency. The WOCs are maximized for lateral slip angles between 0° and 24° to find a set of Pareto optimal solutions over a wide range of weight factors. A method to apply the optimization results for predicting vehicle performance and traction control on dry sand is presented and discussed.     

Measurement and modeling for two-dimensional normal stress distribution of wheel on loose soil

On the Moon or Mars, typical target environments for exploration rovers are covered with fine sand, so their wheels easily slip on such weak ground. When wheel slippage occurs, it is hard for the rover to follow its desired route. In the worst case, the rover gets stuck in loose soil and cannot move anymore. To reduce the risk of the rover getting stuck, analysis of the contact mechanics between the soil and wheel is important. Various normal stress distribution models for under the wheel surface have been proposed so far. However, classical models assume a uniform stress distribution in the wheel’s width direction. In this study, we measured the two-dimensional normal stress distribution of a wheel in experiments. The results clarified that the stress distribution in the wheel’s width direction is a mountain-shape curve with a peak located at the center of the wheel. Based on the results, we constructed a stress distribution model for the wheel’s width direction. In this paper, we report our measurements for the two-dimensional stress distribution of a wheel on loose soil and introduce our stress distribution model for the wheel’s width direction based on our experimental results.     

Integrability of the motion of a rolling disk of finite thickness on a rough plane

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometric functions. The integrability results are used to construct various bifurcation diagrams of the steady motion of the disk. The bifurcations of the steady motion of a disk on a rough plane complements the author's bifurcation analysis of the steady motion of the disk on a smooth plane ( [M. Batista, Steady motion of a rigid disk of finite thickness on a horizontal plane, Int. J. Non-Linear Mech. 41 (4) (2006) 605–621]).     

A mathematical model for spatial motion of tracked vehicles on soft ground

A mathematical model which predicts spatial motion of tracked vehicles on non-level terrain has been developed. The motion of the vehicle is represented by three translational and three rotational degrees of freedom. In order to incorporate the inelastic deformation of soil, a soil-track interaction model is introduced; this constitutive model relates the traction exerted on the track by soil to the slip velocity and sinkage of the track. The model is based upon available soil plasticity theories and furnishes mechanics-based interpretation of Bekker's empirical relations. For planar motion the proposed model reduces to the existing equations of motion by introducing kinematic constraints on the vertical translation, pitching and rolling degrees-of-freedom.     

Optimal control of nonholonomic motion planning for a flee-falling cat

The nonholonomic motion phnning of a free-falling cat is investigated.Nonholonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws.When the total angular momentum is zero,the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum.The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat.Based on Ritz approximation theory,the Gauss-Newton method for motion planning by a falling cat is proposed.The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Optimal control of nonholonomic motion planning for a free-falling cat

The nonholonomic motion planning of a free-falling cat is investigated. Non-holonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum. The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat. Based on Ritz approximation theory, the Gauss-Newton method for motion planning by a falling cat is proposed. The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.