Mobile robots with two independent coaxial drive wheels: Dynamics and the schemes of control

A two-wheeled robot and a robot composed of a snakelike chain of two-wheeled objects are considered. The problem of trajectory synthesis and the problem of motion planning and control are studied for these mobile robots. The mobile robots with two independently controlled coaxial wheels and with one or several passive wheels are called robots with differential drive. These problems are formulated in such a way that the construction of dynamically correct and precise control becomes possible for such robots.     

Studies of dynamics of a lightweight wheeled mobile robot during longitudinal motion on soft ground

The paper is concerned with the study of longitudinal motion of a lightweight wheeled mobile robot on soft ground. The study is focused on the influence of the desired longitudinal velocity of a robot on both the longitudinal slip of the wheels and the ratio of wheel-terrain contact angles. Design of the four-wheeled skid-steered robot and research environment are described. Experimental investigations were conducted on a dedicated test stand with dry sand. A dynamics model of the robot-ground system taking into account properties of soft ground is presented. The classical terramechanics models of Bekker and Janosi-Hanamoto are used. Results of simulation research of robot motion and of the analogous experimental investigations are presented. Actual motion parameters of the robot and the values of longitudinal slip ratio of the wheels are determined. The results of simulation and experimental investigations are compared and discussed. A formula to describe front-to-back wheel-terrain contact angle ratio dependency on the desired velocity is proposed.     

Dynamics modelling of the four-wheeled mobile platform

The model of dynamics of the four-wheeled mobile platform has been presented. Model of construction of prototype of has also been presented. The proposed model is useful to examine different configurations of the drive wheels and to analyze the relations between causes and effects of the motion parameters. The solution presented in the work allows to study the behavior of the platform also under slippage and in the circumstances to refrain the platform from falling into the skid. The problem of the forced motion and free motion of the platform with the possibility of modification the drive modulus positions has been considered. Analysis of the active forces with the resistive forces is also included. The formulated initial problem has been solved numerically with use of the Runge-Kutta method of the fourth order. The sample simulation results for the solution and conclusions are in the final sections.     

Modeling and motion simulation of a three-wheeled mobile robot with front wheel driven and steered taking into account wheels�� slip

The problem of modeling of dynamics of a three-wheeled mobile robot with front wheel driven and steered is analyzed in this paper. Kinematical structure and kinematics of the robot are described. A universal methodology of analytical modeling of robot??s dynamics is applied. This methodology takes into account wheel-ground contact conditions and wheels?? slip. Its essence is the use of a contact model of deformable tire with rigid ground and division of the robot??s dynamics model into parts connected with wheels, including tire model, and with the mobile platform. The tire model used in this paper results from empirical dependencies determined during investigations of car tires. Ground geometry and type are specified in the environment model. Tire-ground interface is characterized by coefficients of friction and rolling resistance. The robot model takes into account the presence of friction in kinematical pairs. The model of servomotors is included as well. The important part of this work is simulation research performed using Matlab/Simulink package. Simulation research includes solving of the forward and inverse dynamics problems as well as the tracking control task. During simulations, the robot was moving on concrete and on a piece of ice. The simulation research enabled verification of the elaborated solutions.     

Stability and Bifurcation of Longitudinal Vehicle Braking

The longitudinal braking dynamics of a two-wheel vehicle model on an incline are considered using techniques from nonlinear dynamics. The model is planar and incorporates the coupled dynamics of two independently braked wheels and the vehicle body, and takes into account the slip dynamics of each wheel. By using the wheel slip values and the vehicle speed as dynamic states, it is shown that the qualitative behavior of the system can be completely captured by studying a relatively simple phase plane problem described in terms of the slip values. A systematic bifurcation analysis is carried out in which the brake torques of the two wheels are varied, and it is shown how the system transitions from stable braking to the possibility of lockup in one or both wheels, to guaranteed lockup in both wheels. In this manner a quite complete picture of the dynamic behavior is obtained as a function of the two brake torques, including regions with multiple possible steady braking outcomes, depending on the initial conditions. This analysis provides new insights into the dynamics of vehicle braking, and it provides a correction to the standard result for the critical values of the brake torques at which the wheels undergo lockup. This approach may also prove useful for evaluating brake proportioning schedules, or for investigating anti-lock braking systems and other methods of traction control.     

Adaptive robust control of Mecanum-wheeled mobile robot with uncertainties

This paper presents a novel implementation of an adaptive robust second-order sliding mode control (ARSSMC) on a mobile robot with four Mecanum wheels. Each wheel of the mobile robot is actuated by separate motors. It is the first time that higher-order sliding mode control method is implemented for the trajectory tracking control of Mecanum-wheeled mobile robot. Kinematic and dynamic modeling of the robot is done to derive an equation of motion in the presence of friction, external force disturbance, and uncertainties. In order to make the system robust, second-order sliding mode control law is derived. Further, adaptive laws are defined for adaptive estimation of switching gains. To check the tracking performance of the proposed controller, simulations are performed and comparisons of the obtained results are made with adaptive robust sliding mode control (ARSMC) and PID controller. In addition, a new and low-cost experimental approach is proposed to implement the proposed control law on a real robot. Experimental results prove that without compromising on the dynamics of the robot real-time implementation is possible in less computational time. The simulation and experimental results obtained confirms the superiority of ARSSMC over ARSMC and PID controller in terms of integral square error (ISE), integral absolute error (IAE), and integral time-weighted absolute error (ITAE), control energy and total variance (TV).     

Dynamic Stability and Random Vibrations of Rigid and Elastic Wheelsets

A stability analysis and vibration studies are presented for a passenger coach model which is equipped with rigid or elastic wheels. The elastic components between wheel rim and disc act as third suspension reducing the unsprung mass and isolating the passenger coach from the high frequency motion of the wheel rim. The vertical and lateral motion by such a design requires a thorough analysis of the system dynamics. The excitation of the vertical vibrations by stochastic track irregularities results in acceleration amplitudes of the carbody that may generate droning noise. A parameter study of spring and damper coefficients of the system with radialelastic wheels leads to considerably reduced droning noise and lower force level between wheel and rail. Furthermore, the eigenmotion of a rigid and an elastic wheelset rolling on a track, the so-called hunting, is investigated. The variation of the spring and damper coefficients shows limits to guarantee the stability of the system.     

Modeling and Simulation of a Three-Wheeled Mobile Robot on Uneven Terrains with Two-Degree-of-Freedom Suspension Mechanisms

A wheeled mobile robot (WMR) will move on an uneven terrain without slip if its torus-shaped wheels tilt in a lateral direction. An independent two degree-of-freedom (DOF) suspension is required to maintain contact with uneven terrain and for lateral tilting. This article deals with the modeling and simulation of a three-wheeled mobile robot with torus-shaped wheels and four novel two-DOF suspension mechanism concepts. Simulations are performed on an uneven terrain for three representative paths—a straight line, a circular, and an ‘S’-shaped path. Simulations show that a novel concept using double four-bar mechanism performs better than the other three concepts.     

Characteristics of normal and tangential forces acting on a single lug during translational motion in sandy soil

Lugs (i.e., grousers) are routinely attached to the surfaces of wheels/tracks of mobile robots to enhance their ability to traverse loose sandy terrain. Much previous work has focused on how lug shape, e.g., height, affects performance; however, the goal of this study is to experimentally confirm the effects of lug motion on lug–soil forces. We measured normal and tangential forces acting on a single lug as functions of inclination angle, moving direction angle, sinkage length, horizontal displacement, and traveling speed. The experimental results were mathematically fitted by using least square method to facilitate quantitative analyses on effects of changes in these motion parameters. Moreover, we compared the measured tangential forces to values calculated from a conventional tangential force model to evaluate the effects of the lug-tip surface, which is generally ignored in existing terramechanics models. The conclusions from this study would be useful for estimating the traveling performance of locomotive mechanisms equipped with lugs, modeling interaction mechanics between lugged wheels and soil, etc.     

OPTIMAL MOTION PLANNING FOR A RIGID SPACECRAFT WITH TWO MOMENTUM WHEELS USING QUASI-NEWTON METHOD

An optimal motion planning scheme based on the quasi-Newton method is proposedfor a rigid spacecraft with two momentum wheels. A cost functional is introduced to incorporatethe control energy, the final state errors and the constraints on states. The motion planning fordetermining control inputs to minimize the cost functional is formulated as a nonlinear optimalcontrol problem. Using the control parametrization, one can transform the infinite dimensionaloptimal control problem to a finite dimensional one that is solved via the quasi-Newton methodsfor a feasible trajectory which satisfies the nonholonomic constraint. The optimal motion planningscheme was applied to a rigid spacecraft with two momentum wheels. The simulation results showthe effectiveness of the proposed optimal motion planning scheme.     

Using the Dirac approach to simulate the rolling of a wheeled vehicle

The rolling of a wheeled vehicle is considered in the case when the turning angles of the front wheels about the vertical axis are small. The small relative slip is taken into account in the adopted model of contact between the wheels and the road surface. It is shown that, when the stiffness of the wheels tends to infinity, the system of equations of motion may become nonclassical and its form is specified by the no-slip conditions along the longitudinal direction of motion and by the primary Dirac constraints arising because of the degeneracy of the Lagrangian.     

REVIEW OF DYNAMIC MODELING AND STABILITY ANALYSIS OF A BICYCLE$^{\bf 1)}$

自行车发明于两个多世纪前. 这一看似古老的交通工具在为人们提供出行便利的同时,其独特的运动特性及动力学性质 也吸引了来自数学、物理及力学等多个学科相关学者的兴趣. 大体上,自行车可以描述为具有 7 个自由度和 4 个非完整约束的多刚体系统. 但由于前后车轮之间复杂的运动耦合关系,使得自行车的约束方程和动力学模型变得异常复杂, 导致对自行车的稳定性存在一些模糊认识. 本文针对经典的 Carvallo-Whipple 自行车构型,系统回顾了历史上自行车动力学研究中的相关问题,这些问题包括：(1) 自行车在复杂曲面上的几何约束和非完整约束的数学描述;(2) 自行车系统内在的对称性及守恒量; (3) 自行车动力学的各类建模方法; (4) 自行车运动的相对平衡点及稳定性分析,包括水平面上的匀速直线运动及旋转对称曲面上的匀速圆周运动;(5) 影响自行车自稳定性的结构参数等. 本文最后对自行车动力学实验和控制方面的研究工作进行了回顾,并对自行车今后的研究给出了展望.     

自行车动力学建模及稳定性分析研究综述

自行车发明于两个多世纪前. 这一看似古老的交通工具在为人们提供出行便利的同时,其独特的运动特性及动力学性质 也吸引了来自数学、物理及力学等多个学科相关学者的兴趣. 大体上,自行车可以描述为具有 7 个自由度和 4 个非完整约束的多刚体系统. 但由于前后车轮之间复杂的运动耦合关系,使得自行车的约束方程和动力学模型变得异常复杂, 导致对自行车的稳定性存在一些模糊认识. 本文针对经典的 Carvallo-Whipple 自行车构型,系统回顾了历史上自行车动力学研究中的相关问题,这些问题包括:(1) 自行车在复杂曲面上的几何约束和非完整约束的数学描述;(2) 自行车系统内在的对称性及守恒量; (3) 自行车动力学的各类建模方法; (4) 自行车运动的相对平衡点及稳定性分析,包括水平面上的匀速直线运动及旋转对称曲面上的匀速圆周运动;(5) 影响自行车自稳定性的结构参数等. 本文最后对自行车动力学实验和控制方面的研究工作进行了回顾,并对自行车今后的研究给出了展望.      

论轮式移动结构的非完整约束及其运动控制

当质点沿光滑曲线运动时,必须满足横向速度为零的条件.同样地,不同轮式移动结构在平面上做光滑曲线运动时都需要满足该非完整约束条件.本文结合轮子转速和它们运动速度的完整约束关系,理清各轮式移动结构的完整和非完整约束,然后利用 Euler-Lagrange 方程方便地推导出相应的动力学方程.另外,通过该非完整约束,将目标轨迹曲线转化为速度目标的形式,然后引入目标轨迹曲线的相对曲率设计合适的动态跟踪目标.进一步,通过采用该动态跟踪目标可以将轮式移动结构的运动规律和动力学方程有机结合起来,并将原运动任务简化为一般的 轨迹 控制问题.基于该动态跟踪目标可以为轮式移动结构设计合适的鲁棒跟踪控制器,通过跟踪目标轨迹曲线的相对曲率来实现对目标曲线的精确跟踪.最后,理论分析和仿真结果显示,采用动态目标跟踪方法能够从根本上解决初始速度误差过大和位置误差不断被累积的问题,即使前向速度误差系统不稳定的,实际运动轨迹也几乎能和目标轨迹曲线重合.      

ON THE NONHOLONOMIC CONSTRAINTS AND MOTION CONTROL OF WHEELED MOBILE STRUCTURES1)

当质点沿光滑曲线运动时,必须满足横向速度为零的条件.同样地,不同轮式移动结构在平面上做光滑曲线运动时都需要满足该非完整约束条件.本文结合轮子转速和它们运动速度的完整约束关系,理清各轮式移动结构的完整和非完整约束,然后利用 Euler-Lagrange 方程方便地推导出相应的动力学方程.另外,通过该非完整约束,将目标轨迹曲线转化为速度目标的形式,然后引入目标轨迹曲线的相对曲率设计合适的动态跟踪目标.进一步,通过采用该动态跟踪目标可以将轮式移动结构的运动规律和动力学方程有机结合起来,并将原运动任务简化为一般的 轨迹 控制问题.基于该动态跟踪目标可以为轮式移动结构设计合适的鲁棒跟踪控制器,通过跟踪目标轨迹曲线的相对曲率来实现对目标曲线的精确跟踪.最后,理论分析和仿真结果显示,采用动态目标跟踪方法能够从根本上解决初始速度误差过大和位置误差不断被累积的问题,即使前向速度误差系统不稳定的,实际运动轨迹也几乎能和目标轨迹曲线重合.     

Dynamical behavior of a mobile system with two degrees of freedom near the resonance

The design of mobile robots that can move without wheels or legs is an important engineering and technological problem.Self-propelling mechanisms consisting of a body that has contact with a rough surface and moveable internal masses are considered.Mathematical models of such systems are presented in this paper.First,a model of a vibration driven robot that moves along a rough horizontal plane with isotropic dry friction is studied.It is shown that by changing the off-resonance frequency detuning in sign,one can control the direction of motion of the system.In addition,a locomotion system which moves in an environment with anisotropic viscous friction is considered.For all models,the method of averaging to obtain an algebraic equation for the steady-state"average"velocity of the system is used. Prototypes were constructed to compare the theoretical results with experimental ones.     

A method for predicting the internal motion resistance of rubber-tracked undercarriages,Pt. 2. A research on the motion resistance of road wheels

In spite of an increasing number of rubber-tracked crawlers, the literature provides few guidelines and calculation models suitable for minimizing their internal motion resistance. This article presents a model where the internal resistance of double-flanged road wheels for rubber-tracked vehicles is calculated as a sum of the losses resulting from the indentation of the wheels into the track surface and friction of the wheels against the track guide lugs. The model allows for vertical and lateral load of the wheels, the non-uniform distribution of the wheel pressure on the track, and the relationship between the friction coefficient and normal reaction force in the interface between the wheel and track guide lugs. The model has been verified by experiments. According to the results of model computations and experiments discussed in the article, the internal losses of a given rubber-tracked undercarriage might be reduced if: the road wheels are covered with a material that exhibits low friction coefficient and mechanical hysteresis, the vehicle suspension system features oscillating bogie wheels, the undercarriage is fitted with the largest possible number of road wheels, and the vehicle weight is evenly distributed to all of the road wheels.     

The motion equations of rotors,gyrostats and gyrodesics: application to powertrain-vehicle mechanical systems

The kinematics of wheels and rotors is described using a new auxiliary frame called gyrodesic frame or simply gyrodesic. By this, the absolute motion of the wheel becomes a serial composite of two motions: (1) the gyrodesic motion and (2) the wheels eigenmotion (or spin), i.e., the motion relative to the gyrodesic. The eigenmotion is described by an equation called rotor-equation. Gyrodesic coordinates turn out to be a particular useful tool in powertrain- and vehicle-dynamics as well as for general multibody systems. They allow a proper separation of the rotor- from the vehicle-equations and provide a rigorous method of coupling the powertrain-model into full spatial multibody-systems vehicle model. Some common misconceptions regarding this subject are identified and dispelled. The method is generalized to be applicable to the study of motion of general systems of rigid bodies with gyrostats or rotors as subsystems. The usefulness of the formalism is demonstrated by means of an illustrative example of non-trivial nature: the gyrostatic chain. Gyrodesic coordinates lead to a better grasp and deeper understanding of the structure of the dynamic equations of spatial vehicles in particular and of the motion of multibody-systems with rotors in general. The investigation reveals an interesting analogy to concept of parallel transport of vector fields in the sense of Levi-Civita. Dedicated to Prof. J. Wittenburg at the occasion of his 70th birthday.     

车辆纵向非光滑多体动力学建模与数值算法研究

在建立车辆纵向多体系统的动力学模型中, 将车身与车轮视为刚体, 两者通过减振器链接; 将传动系统视为一个圆盘通过扭簧和阻尼器与驱动轮连接; 将车轮与路面间的接触力简化为法向约束力、摩擦力和滚阻力偶,其中摩擦力的模型采用库仑干摩擦模型. 采用笛卡尔坐标作为该系统的广义坐标用于描述该系统的位形, 给出系统单双边的约束方程, 应用第一类拉格朗日方法建立了系统的动力学方程. 由于摩擦与滚阻的非光滑性, 使得该系统动力学方程不连续. 为便于计算, 建立了车轮与路面接触点的相对切向加速度与摩擦力余量的互补条件、车轮角加速度与滚阻力偶余量的互补条件, 以及车轮轮心法向加速度与路面法向约束力的互补条件. 将接触—分离、黏滞—滑移的判断问题转化成线性互补问题的求解, 并给出了具有约束稳定化的基于事件驱动法的数值计算方法. 最后, 应用该方法对车辆纵向多体系统进行了仿真, 分析了输出扭矩、摩擦及滚阻系数对其动力学行为的影响.