A Lie symmetry approach for the solution of the inverse kinematics problem

In this paper, it is considered the inverse kinematics problem, which is faced from a differential point of view. In particular, it is shown that an asymptotic inverse kinematics can be interpreted as a Lie symmetry of the direct kinematics. A parameterization of all Lie symmetries of the direct kinematics is proposed, and the classical Newton and gradient method are obtained as particular cases.     

A general solution for Stokes flow and its application to the problem of a rigid plate translating in a fluid

A general solution for 3D Stokes flow is given which is different from, and more compact than the existing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape translating along the direction normal to it in an unbounded fluid.The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the velocity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and elliptic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.     

Quasi-Euler-Poinsot motion of a rigid body

The phase-plane method of nonlinear oscillation is used to discuss the influence of the small dissipation upon the Euler-Poinsot motion of a rigid body about a fixed point. The equations of phase coordinates are applied instead of Eulerian equations, and the global characteristics of the motion of rigid body are analysed according to the distribution and the type of the singular points. A Chaplygin's sphere on a rough plane, a rigid body in viscous medium and one with a cavity filled with viscous fluid are discussed as examples. It is shown that the motions of rigid bodies dissipated by various physical factors have a common qualitative character. The rigid body tends to make a permanent rotation about the principal axis of the largest moment of inertia. The transitive process can change from oscillatory to aperiodic with the decrease in dissipation.     

New control laws for stabilization of a rigid body motion using rotors system

This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced.     

Kinematic problem of optimal nonlinear stabilization of angular motion of a rigid body

The problem of optimal transfer of a rigid body to a prescribed trajectory of preset angular motion is considered in the nonlinear statement. (The control is the vector of absolute angular velocity of the rigid body.) The functional to be minimized is a mixed integral quadratic performance criterion characterizing the general energy expenditure on the control and deviations in the state coordinates.Pontryagin’s maximum principle is used to construct the general analytic solution of the problem in question which satisfies the necessary optimality condition and ensures the asymptotically stable transfer of the rigid body to any chosen trajectory of preset angular motion. It is shown that the obtained solution also satisfies Krasovskii’s optimal stabilization theorem.     

Averaging method in the motion stability problem for a rotating body suspended on a long rigid string

We use the averaging method to study the stability of the vertical rotation of a rigid body suspended on a long rigid string.     

Chaotic attitude motion of a magnetic rigid spacecraft and its control

This paper deals with chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. The dynamical model of the problem is established. The Melnikov analysis is carried out to prove the existence of a complicated non-wandering Cantor set. The dynamical behaviors are numerically investigated by means of time history. Poincare map, power spectrum and Lyapunov exponents. Numerical simulations indicate that the onset of chaos is characterized by the intermittency as the increase of the torque of the magnetic forces and decrease of the damping. The input-output feedback linearization method is applied to control chaotic attitude motions to the given fixed point and periodic motion.     

The inverse identification problem and its technical application

This paper presents an overview of a loading force identification technique. Load identification methods are based on the solution of the inverse identification problem. Many different approaches for linear systems have been developed in this area. For both linear and nonlinear systems, methods based on the minimization of assumed objective functions are formulated. The least square error between the simulated and measured system responses is mainly used as the objective function. The dynamic programming optimization method formulated by Bellman is commonly used for the minimization of the objective function to estimate the excitation forces. The inverse identification problem in most practical cases is ill-posed because not all the state variables or initial conditions are known. Ill-posed inverse identification problems can be solved using several techniques, the most useful of which are: the generalized cross-validation method, the dynamic programming technique and Tikhonov’s method. This article presents the theoretical background and main limits to the application of inverse identification methods. Numerical and experimental tests on a laboratory rig were made to verify the formulated procedures. The method is applied to the identification of wheel–rail contact forces during rail vehicle operation. The method can be applied for indirect measurements of contact forces in railway equipment testing.     

一类柱铰链动约束力问题的解答与探讨

利用达朗贝尔原理解决动力学问题在工程实践中得到重要应用,也是理论力学教学中的一个重要内容,但具有机械式调速原理机构中的一类柱铰链处的动约束力问题需要更深入的探讨,通过对此类动力学问题的求解,得到了系统之间的约束力、速度与结构参数之间的关系及应用达朗贝尔原理解题的条件,并给出了在教学中的几点建议.     

一类柱铰链动约束力问题的解答与探讨1）

利用达朗贝尔原理解决动力学问题在工程实践中得到重要应用,也是理论力学教学中的一个重要内容,但具有机械式调速原理机构中的一类柱铰链处的动约束力问题需要更深入的探讨,通过对此类动力学问题的求解,得到了系统之间的约束力、速度与结构参数之间的关系及应用达朗贝尔原理解题的条件,并给出了在教学中的几点建议.     

Extremals in the kinematic control of motion

A possible classification of kinematic controls of motion is proposed for extremal problems in the mechanics of control systems.     

New integrable problems of motion of a rigid body with a particle oscillating or bouncing in it

Some qualitative aspects of the problem of motion about a fixed point of a rigid body with a particle moving in it in a prescibed (sinusoidal) way was treated in [1–3]. The mechanical system comprised of a rigid body containing an internal mass that moves along a fixed line in the body was considered in several works [4–5]. Recently, an integrable case of this system was found, in which the body is dynamically axisymmetric and moves under no external forces while the particle moves on the axis of dynamical symmetry under the action of Hooke's force to the fixed point [5].In the present note we introduce a more general integrable case in which the particle moves on the axis of dynamical symmetry and is subject to an arbitary conservative force that depends only on the distance from the fixed point. Separation of variables is accomplished and the solution is reduced to quadratures. As a special version of this problem, the case when the particle bounces elastically between two points is briefly discussed.     

From the Kovalevskaya to the Lagrange case in rigid body motion

We study the motion of a family of symmetric tops in which the center of mass is located between the symmety plane and the symmetry axis of the inertia matrix. We analyze the transition from the Kovalevskaya to the Lagrange integrable cases using Poincaré sections and symmetry lines. The fate of periodic orbits as a function of the location of the top's center of mass is analyzed. The critical points of the Kovalevskaya constant are calculated in terms of the energy, of the angular momentum about the vertical, and of the Kovalevskaya constant itself.     

Differential equations for librational motion of gravity-oriented rigid body

We consider a gravity-oriented rigid body on a circular Keplerian orbit in a central gravitational field. The motion of the body is affected by a perturbation torque given by a cubic approximation. With the inclusion of the third infinitesimal terms, we introduce a new notation for the differential equations of disturbed motion. This form generalizes the familiar equations in canonical variations extending them to the case where both the potential and the non-potential disturbing forces are operative. This form is convenient for the analysis of non-linear oscillations of a body about its center of mass with the use of the asymptotic methods of non-linear mechanics.     

Averaging method for discrete systems and its application to control problems

We propose and justify algorithms for partial and complete averaging of systems of discrete equations and inclusions. On the basis of the averaging schemes obtained, we construct algorithms for the numerical asymptotic solution of problems of optimal control for discrete systems.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 241–254, April–June, 2004.