Control of the rotational motion of the rigid body with the help of internal rotors using Rodrigues-Caley parameters

The purpose of this paper is to study the control of the rotational motion of the rigid body with the help of three rotors attached to the principal axes of the body. In such study the asymptotic stability of this motion is proved by using the Lyapunov technique. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia of the body coincide with the inertial axes, is proved to be asymptotically stable. The control moments that impose the stabilization of the rotational motion and equilibrium position are obtained.     

Open-plus-closed-loop control for chaotic motion of 3D rigid pendulum

An open-plus-closed-loop （OPCL） control problem for the chaotic motion of a 3D rigid pendulum subjected to a constant gravitationM force is studied. The 3D rigid pendulum is assumed to be consist of a rigid body supported by a fixed and frictionless pivot with three rotational degrees. In order to avoid the singular phenomenon of Euler＇s angular velocity equation, the quaternion kinematic equation is used to describe the motion of the 3D rigid pendulum. An OPCL controller for chaotic motion of a 3D rigid pendulum at equilibrium position is designed. This OPCL controller contains two parts： the open-loop part to construct an ideal trajectory and the closed-loop part to stabilize the 3D rigid pendulum. Simulation results show that the controller is effective and efficient.     

Optimal control of a rotational motion of a gyrostat on circular orbit

A control scheme is proposed to guarantee an optimal stabilization of a given rotational motion of a symmetric gyrostat on circular orbit. The gyrostat controlled by the control action generated by rotating internal rotors. In such study the asymptotic stability of this motion is proved using Barbachen and Krasovskii theorem's and the optimal control law is deduced from the conditions that ensure the optimal asymptotic stability of the desired motion. As a particular case, the equilibrium position of the gyrostat, which occurs when the principal axes of inertia coincide with the orbital axes, is proved to be asymptotically stable. The present method is shown to more general than previous ones.     

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published companion paper on finite-element appendage vibration equations, this paper provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.     

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

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.     

Quasi-optimal deceleration of rotational motion of a dynamically symmetric rigid body in a resisting medium

We study the problem of quasi-optimal (with respect to the response time) deceleration of rotational motion of a free rigid body which experiences a small retarding torque generated by a linearly resisting medium. We assume that the undeformed body is dynamically symmetric and its mass is concentrated on the symmetry axis. A system of nonlinear differential equations describing the evolution of rotation of the rigid body is obtained and studied.     

Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body

We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations.     

On the exponential stability of the permanent rotational motion of a gyrostat

This paper is devoted to the study of the problem of exponential asymptotic stability of the rotational motion of a gyrostat using servo-control moments which are applied to the internal rotors. The servo-control moments which impose the rotational motion are obtained. The stabilizing servo-control moments are obtained from the conditions to ensure exponential asymptotic stability of the desired motion. Estimations of the phase coordinations as exponential functions are presented. The method based on a choice of the structural form of the servo-control moments such that the equations of motion reduce to a system of differential equations with exponential asymptotic stability of an special solution.     

Taking into account the influence of elastic elements of a free structure on the accuracy of its stabilization under a bang-bang control

We consider the problem of stabilization with respect to a prescribed position for the translational motion of a rigid body with interior material points connected with each other and with the exterior body by linear viscoelastic constraints. The motion occurs under the action of a constant exterior perturbation and a bang-bang control force that are directed along the line of motion. We assume that the bang-bang force control channel has a fixed delay, so that arbitrarily frequent switchings are impossible. We suggest a positional control ensuring the solution of this problem. We estimate the amplitude of the rigid body vibrations about the center of mass of the entire structure and the accuracy of stabilization of the prescribed position of the rigid body depending on the mechanical characteristics of the system and the control force magnitude. We also consider the problem of maximizing the stabilization accuracy depending on the control parameters. By way of example, we consider the controlled motion of a two-mass oscillatory system. This work is closely related to [1–3] and continues the studies of the guaranteed optimal bang-bang controllers with delay in the control channel [4–9]. The dynamics of a rigid body with elastic and dissipative elements was studied in [10] under the assumption that the period of natural vibrations and their decay time are small compared with the characteristic time of motion.     

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

The invariant manifold method and the controllability of nonlinear control system

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A.M.Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics of rigid body to nonlinear control system for the first time and obtained a series of efficient results. Based on Kovalev’s oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition was discussed. At last, the motion of a rigid body with two rotors was investigated and the necessary condition which is satisfied by this system was proved.     

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kine- matics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their incre- ments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motion- deformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.     

Dynamic Analysis of Spatial Linkages: A Recursive Approach

In this paper, recursive equations of motion of spatial linkages are presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. Closed-chain system is transformed to open-chain by cutting suitable kinematic joints and introducing cut–joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

On the stability of the rotational motion of a rigid body having a liquid filled cavity under finite initial disturbance

In this paper the problem of the stability of rotational motion of a rigid body which has a liquid filled cavity and a fixed point is investigated without any approximation. Criteria of stability and instability under finite disturbance are obtained. The region of stability is found out explicitly.     

大型偏心结构吊装欠约束问题研究

吊装施工过程中被吊模块的水平度是作业要求的重要指标,通常需要增加配重调平。传统有限元方法需要补充约束以消除单元刚体位移,且需要重复计算平衡方程来求解调平载荷,效率不高。将模块的运动分解为随动坐标系的整体运动以及相对该坐标系的弹性变形,可将欠约束问题化为多体系统的静平衡问题。基于虚功率原理推导了吊装平顺时刻的节点力平衡方程以及相应的切线刚度矩阵,并将配重表示为基础配重与载荷系数相乘的形式。通过对节点力平衡方程求导,得到一组以载荷系数为自变量的微分方程,通过求解微分方程并结合水平度判据,可快速搜寻满足水平度要求的载荷系数。数值算例表明,该方法在解决偏心模块吊装欠约束问题方面具有明显的优势,在确定配重载荷方面具有较快的速度和合理的精度。     

The control of a permanent rotational motion and equilibrium position of a spacecraft

This article has adopted an analytical method to obtain a non-linear control law to reach the exponential asymptotic stablity of the permanent rotational motion of a spacecraft. The control moments achieving this rotational motion are obtained. The control moments to establish exponential asymptotic stablity of the mentioned motion are obtained as non-linear functions of the phase coordinates of the spacecraft. The general solution of the equations of perturbed motion is derived. Furthermore, analysis and numerical simulation study of this solution are presented. For numerical examples the time needed for control is calculated. An equilibrium position of the spacecraft is proved to be exponentially asymptotically stable as a special case of the above-studied problem.     

The Method of Averaging for Euler's Equations of Rigid Body Motion

We formulate the method of averaging for perturbations of Euler's equations of rotational motion. Euler's equations are three strongly nonlinear coupled differential equations that can be viewed as a three dimensional oscillator. The method of averaging is used to determine the long-term influence of perturbation terms on the motion by averaging about the nominal rigid body motion. The treatment is applicable to a large class of motions including precession with large nutation – it is not restricted to small motions about simple spins or nearly axi-symmetric bodies. Three examples are shown that demonstrate the accuracy of the method's predictions.