The brachistochrone motion of a mechanical system with non-holonomic, non-linear and non-stationary constraints

Further to previous studies /1, 2/ of the brachistochrone motion of non-holonomic mechanical systems with linear homogeneous constraints, consideration is given here to non-holonomic, non-linear and non-stationary mechanical systems. The problem is to formulate the differential equations of the brachistochrone motion of non-holonomic, non-linear and non-stationary mechanical systems and to determine the additional forces which must be introduced in order to implement motion of this type.     

关于高阶非完整系统的一类新型运动微分方程

本文首先得到高阶非完整系统的一类新型运动微分方程,其次证明它们与已知方程的等价性,最后举例说明新方程的应用.     

On the theory of motion of nonholonomic systems. The reducing-multiplier theorem

This classical paper by S.A. Chaplygin presents a part of his research in non-holonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the “reducing-multiplier method”. The method is illustrated on two concrete problems from non-holonomic mechanics. This paper produced a considerable effect on the further development of the Russian non-holonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world. This publication contributes to the series of RCD translations of Chaplygin’s scientific heritage. In 2002 we published two of his works (both cited in this one) in the special issue dedicated to non-holonomic mechanics (RCD, Vol. 7, no. 2). These translations along with translations of his other two papers on hydrodynamics (RCD, Vol. 12, nos. 1,2) aroused considerable interest and are broadly cited by modern researches. Originally published in: Matematicheskiĭ sbornik (Mathematical Collection), 1911, vol. 28, issue 1. The content of §§ 2 and 3 of this study was presented at the session of the Moscow Mathematical Society on December 11, 1906.     

Analysis of the motion of a non-holonomic mechanical system

The motion of a plane non-holonomic mechanical system, consisting of two point masses, which move in such a way that their velocities are mutually perpendicular, is analysed [Zekovi? D. Examples of non-linear non-holonomic constraints in classical mechanics. Vestnik MGU. Ser. 1. Matematika Mekhanika, 1991; 1:100–3]. The equations of the constraints of such a system are derived, the reactions of the constraints are calculated and the cyclical first integrals are written.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

Asymptotic stability and associated problems of dynamics of falling rigid body

We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.      

The non-holonomic double pendulum: An example of non-linear non-holonomic system

An example of physically realizable non-linear non-holonomic mechanical system is proposed. The dynamical equations are written following a general method proposed in an earlier paper. In order to make this paper self-contained, an improved and shortened approach to the dynamics of non-holonomic systems is illustrated in preliminary sections.     

The motion of wheeled robots

The equations of motion of three-wheeled robots with two drive wheels and one passive caster wheel are derived and investigated. The control of longitudinal motion and turns of such a robot is implemented by appropriate control of the independent motors of the drive wheels. The research is carried out under the assumption that the robot is moving on a horizontal plane surface and that the wheels do not slip. A system of two non-linear equations with two controls is obtained for the non-holonomic system considered. The dependence of the phase portrait on the values of the constant controls and parameters of the system, taking into account the asymmetry of the robot, is investigated. The results obtained are not only of theoretical but also of practical interest.     

Controllability and stabilization of the programmed motions of a transport robot

A non-linear mathematical model for the motion of a transport robot (TR) with a caterpillar chassis and with drives based on DC motors, which is a non-holonomic electromechanical system, is considered. Non-linear canonical transformations of the coordinates of the state and control space are constructed, which reduce the initial equations of motion of the TR to a simpler canonical form, which is convenient for analysing and synthesizing control systems for the TR. The conditions for the TR to be controllable as a controlled object are found. Algorithms are given for constructing programmed motions (PMs) of the TR. Stabilizing control laws are synthesized under which the PMs of the TR are asymptotically stable and transients of a specified nature are ensured.     

Motion with a constant velocity modulus in a central gravitational field

The problem of the motion of a particle (point mass) with a constant velocity modulus in a Newtonian central gravitational field is investigated by two methods: using Lagrange's equations with a multiplier, and using the equations of dynamics proposed earlier [1] for systems with non-holonomic constraints that are non-linear with respect to velocities. A phase diagram of the motion is constructed. The structure of the trajectories as a function of the initial conditions is investigated. Formulae in the form of quadratures are obtained for calculating the time of motion along the trajectory and the angular distance of flight. A qualitative analysis of the properties of improper integrals expressing the angular distance is presented. These properties are illustrated by the results of a numerical investigation. The possibility of carrying out elementary manoeuvres in the vicinity of an attracting centre are analysed.     

Whittaker方程对非完整力学系统的推广

1904年Whittaker利用能量积分将一个完整保守力学系统问题降阶为一个带有较少自由度系统问题.并得到了Whittaker方程[1].本文推导对于非完整力学系统的这类方程.并称之为广义Whittaker方程;然后把这些方程变换为Nielsen形式;最后举例说明新方程的应用.     

Possibilities and Challenges in Kinematic Modeling of Highly Redundant,Non-Holonomic and Omnidirectional Mobile Robots

Industry as well as the private domain show an increasing interest to the field of mobile robotics. With a higher number of applications, the complexity of the required tasks rises, and therefore the community tends to use robots with many degrees of freedom (DOF). The present paper takes this topic up and focuses on challenges of the kinematical modeling of redundant, non-holonomic mobile robots by considering a 12 DOF platform. To reach an omnidirectional behaviour, the actuated wheels are diagonally mounted on the chassis. Well known problems resulting from this set-up are parametric singularities which unnecessarily restrict the motion of the mathematical model. As it turns out, this problem can be avoided by the use of a non-minimal parametrized model. An additional challenge results from the inverse kinematic (IK) problem on velocity level. The corresponding equations are highly underdetermined and, due to the non-holonomic wheels, not directly affected by the steering angle velocities. This problem is solved by performing a local optimization on acceleration level. Finally, simulation results are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

变质量高阶非完整力学系统的运动微分方程

本文建立变质量力学系统的万有D'Alembert原理,由此导出变质量高阶非完整力学系统各种形式的运动微分方程,最后举例说明这些新型方程的应用.     

Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.     

Stabilization of the motions of mechanical systems with non-holonomic constraints

Mechanical systems possibly containing non-holonomic constraints are considered. The problem of stabilizing the motion of the system along a given manifold of its phase space is solved. A control law which does not involve the dynamcal parameters of the system is constructed. The law is universal, that is, it stabilizes motion along any given manifold. It is only necessary that the manifold be feasible, that is, conform to the dynamics of the system.     

Reduction of the equations of dynamics of systems with constraints to a given structure

The problem of constructing systems of second-order ordinary differential equations, the solutions of which, with the appropriate initial conditions, satisfy given equations of the constraints, is considered. The conditions for representing the differential equations in the form of Lagrange equations of the second kind are determined. It is shown that, when the equations of the non-holonomic constraints are specified by polynomials of order no higher than two with respect to the generalized velocities, the generalized forces of a system with energy dissipation comprise the sum of the gyroscopic, potential and dissipative forces.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

非线性非完整空间变质量体的一种运动方程*

引入非线性非完整空间的约束超曲面的基矢量和密歇尔斯基方程点乘,作为非线性非完整系统变质量体的基本动力学方程。它简明、运算简便,而且由它可导出,Nielsou,Appell,Mac-Millan等已有的方程,不必附加关于虚位移的Appell-定义或牛青萍定义。本方程与D'Alembert-Lagrange微分变分原理相容。