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

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

On the motion of a heavy dynamically symmetric rigid body with vibrating suspension point

The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.     

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.     

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.     

Similarity and decay laws of wakes with zero momentum and angular momentum

In [1–4] the laws of decay of the average and fluctuating velocities in momentumless turbulent wakes were experimentally investigated with and without swirl. In [5, 6] unswirled momentumless wakes and in [7] wakes with a nonzero angular momentum were theoretically investigated. However, turbulent wakes with zero momentum and angular momentum were not covered by these investigations. This class of flows is the subject of the present study.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 35–41, September–October, 1993.     

Rotation of a dynamically symmetric body under constant and dissipative torques

Conclusion With the help of the variables k₁ and k₂ we have found quasistationary regimes of axial rotation and have determined their characteristics. For the general case (a>0, b>0, c>0, c<0) the quasistationary axial rotation k₂=k*₂ is asymptotically stable with respect to k₂. Similar regimes but with different values of k*₂ occur when one of the components of the dissipative torque vanishes (a=0, b>0, c>0, c<0 or a>0, b=0 c>0, c<0). Under the quadratic component of the torque (b=c=0, a>0) the axial rotation of the body is determined by the sign of the initial value of k₂. The behavior of the rotation of the body for only the dissipative torque (a>0, b>0, c=0) or only its linear component (a=c=0, b>0) depends on the ratio of the coefficients and b of the linear dissipation and can lead to axial rotation or to rotation in the equatorial plane. It follows from the above diagrams that axial rotation of a body in a medium with weak drag can be stabilized by applying a small constant torque about the axis of dynamical symmetry.Moscow. Translated from Prikladnaya Mekhanika, Vol. 29, No. 3, pp. 82–85, March, 1993.     

关于刚体角速度定义的一个注解

给出了一种角速度定义方法. 这种方法基于固连坐标系基矢量和单位矢量的导数给出,但避免使用角速度矩阵概念. 另外从刚体上速度分布的角度,用速度场旋度的方法解释角速度. 可以帮助学生从另外一个角度来理解角速度概念.     

Multimode laws of control of rigid body motions

The problem of control of motion of a rigid body (aircraft, satellite, or landing module) is studied. A multimode control law that can stabilize not only one prescribed motion of the body but any motion in a sufficiently broad class is constructed. The problem is solved in nonlinear setting under indeterminacy conditions on the basis of Lyapunov functions. The control law is constructed in the class of bounded discontinuous laws, special zero-overshoot response modes are constructed, and the specific character of mechanical systems is used.     

Solution of the optimal turn problem for a spherically symmetric rigid body with arbitrary boundary conditions in the class of generalized conical motions

We consider the problem of time- and energy consumption-optimal turn of a rigid body with spherical mass distribution under arbitrary boundary conditions on the angular position and angular velocity of the rigid body. The optimal turn problem is modified in the class of generalized conical motions, which allows one to obtain closed-form solutions for equations of motion with arbitrary constants. Thus, solving the optimal control boundary value problem is reduced to solving a system of nonlinear algebraic equations for the constants. Numerical examples are considered to illustrate the proximity between the solutions of the traditional and modified problems of optimal turn of a rigid body.     

空间运动刚体角速度的一种形式

一般空间运动刚体角速度是理论力学教学的难点和重点,传统基于欧拉转动定理和坐标转换 矩阵的角速度引出方法存在各自的不足,于是可通过有限转动矩阵直接引出角速度,保存传 统两种方法的优点,利于学生更深刻理解角速度的概念.     

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.     

On steady motions of a rigid body bearing a two-degree-of-freedom control momentum gyroscope with precession axis arbitrarily oriented in the carrying body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has 8, 12, or 16 steady motions and the number of stable motions is 2 or 4.     

一般运动刚体角速度向量引入的一种新方法

通过向量运算给出了角速度向量引入的一种较简单的方法. 利用刚体上的固连向量的导数与其自身垂直的特点,得出反映刚体整体运动特性的角速度向量,并对这样的导出方法与刚体角速度向量物理意义之间的联系进行了讨论.     

一种引出一般运动刚体角速度概念的新方法

在刚体一般运动的教学中,角速度是学生接受起来比较困难的一个概念,通过矢量运算引入这一概念,为角速度的教学提供了一种新的简单方法.     

刚体的匀角速定轴转动状态是平衡态吗?

由刚体平衡的充要条件出发,证明了一般刚体的匀角速定轴转动状态不是平衡态;刚体只有绕其对称轴或通过质心的惯量主轴的匀角速转动状态才是平衡态.     

On steady motions of a rigid body bearing three-degree-of-freedom control momentum gyroscopes and their stability

Equations of motion are obtained for a rigid body bearing N three-degree-of-freedom control momentum gyroscopes in gimbals and the entire set of steady motions in a homogeneous external field is determined. The steady motion dependence on the magnitude of the system angular momentum is studied and a detailed analysis of the secular stability is performed.     

A new class of analytic solutions in the optimal turn problem for a spherically symmetric body

The optimal turn problem for a rigid body with a spherical distribution of mass is considered in the quaternion setting. A functional combining the time and the integral magnitude of the control vector modulus used to turn the rigid body is used as the optimality criterion. This problem is solved analytically in the class of conical motions. An example of computations is given.