On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis.We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable.     

Motion of a three-degree-of-freedom gyro with an electric spring and with zero angular momentum

We consider the motion of a three-degree-of-freedom gyro with an electric spring and with zero angular momentum mounted on a fixed base. The gimbal rotation angles are constrained by an elastic ring that is attached to the base and confines a rod attached to the internal gimbal. The equations of motion of the gyro are supplemented with equations allowing for the delay in the response of the gyro feedback circuit. By numerical integration we find a steady-state mode of the gyro motion, in which the rod alternately slides along the ring and moves in the interior of the ring. This motion can be used to check the gyro performance after launching a spacecraft to a near-Earth orbit by comparing the gyro motions in ground tests and on the orbit.     

Stability analysis of steady rotations of a rigid body bearing two-degree-of-freedom control moment gyros with dissipation in gimbal suspension axes

To study the stability of steady rotations of a control moment gyro system with internal dissipation, we use the Barbashin-Krasovskii theorem and the relation, established in [1], between the Lyapunov function and steady motions. Taking into account the special properties of the original problem, we reduce it to a lower-dimensional problem.We give a detailed presentation of an algorithm for analyzing the stability of steady motions of a gyrostat and use this algorithm to perform a complete study for two systems consisting, respectively, of one and two gyros whose gimbal axes are parallel to the principal axis of inertia of the system. Each steady motion is identified as either asymptotically stable or unstable. We find periodic motions that exist only in the presence of dynamic symmetry and which are regular precessions. For the system with two gyros, we prove the asymptotic stability of quiescent states and prove that in the angular momentum range where these states are defined the system does not have any other stable motions.     

On limit motions of a system of control moment gyros with intrinsic dissipation in a homogeneous gravitational field

We study the limit motions of a free rigid body bearing n two-degree-of-freedom control moment gyros with dissipation in the gyro gimbal suspension axes. We show that, in the absence of dynamic symmetry, the limit motions of the system are only steady rotations at a constant angular velocity. In the case of dynamic symmetry, the gyros can be arranged so that, in addition to steady rotations, the system exhibits limit motions that are regular precessions.     

外环干扰力矩对陀螺加速度计测试精度的影响

摆式积分陀螺加速度计的外环干扰力矩包括仪表外环轴的摩擦力矩和交叉轴加速度引起的交变力矩。作者分析了引起摆式积分陀螺加速度计外环干扰力矩的主要原因,提出一种在高精度三轴测试转台上分离摆式积分陀螺加速度计外环干扰力,测试摆式积分陀螺加速度计精度的试验方法。该试验采用三轴转台中环转动速度随动摆式积分陀螺加速度计外环进动角速度,同时摆式积分陀螺加速度计陀螺摆的输出轴在整个试验中保持水平,从而分离仪表外环干扰力的方案。通过对试验数据进行分析,得出外环干扰力的存在影响了摆式积分陀螺加速度计测试精度,为改善摆式陀螺加速度计工艺以提高摆式陀螺加速度计的测试精度提供了依据。     

Attraction property of a gimbal gyro with an electric motor

We consider an unbalanced gimbal gyro with vertical outer suspension axis mounted on an immovable base in the gravity field and supplied with an electric motor. The equations of motion of such a system admit a family of solutions describing its steady-state motions (regular precessions and rotor uniform rotations). We show that, in the case of an isolated minimum of the reduced potential energy, the perturbed motions tend in the course of time to steady-state motions in the same family (even in the critical cases of stability).     

Stability of stationary motions of a gimbal gyro with an electric motor

A gimbal gyro on an immovable base in the gravity field and with vertical outer suspension axis is considered. The rotor is driven by an asynchronous or synchronous electric motor, and there is no friction or any controlling torques on the suspension axes. Such a systemhas a family of steady-state motions (regular precessions and the rotor uniform rotations). It is shown that a necessary and sufficient condition for stability of the steady-statemotions with respect to generalized velocities and the angle of rotation of the inner gimbal frame is the existence of an isolated minimum of the reduced potential energy with respect to the angle of rotation of the inner frame.     

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.     

控制力矩陀螺框架伺服系统的超低速测速方法

在超低转速下,有限的转速测量分辨率是影响控制力矩陀螺框架伺服系统控制性能的一个重要因素.在分析转速分辨率对超低速控制性能影响的基础上,提出了一种基于多周期后向差分的转速计算方法,可提高测速分辨率并保证测速带宽.针对后向差分带来的相位延迟会限制测速分辨率进一步提高的问题,在选取合适的差分周期基础上,采用Kalman预估器对框架转速进行预估,进一步提高了框架转速的测速分辨率及测速带宽.通过仿真表明,所提出的测速方法将转速分辨率提高了两个数量级,解决了框架伺服系统在超低转速下由于有限的转速测量分辨率造成的爬行现象.     

Steering laws analysis of SGCMGs based on singular value decomposition theory

The steering laws of single gimbal control moment gyros （SGCMGs） are analyzed and compared in this paper for a spacecraft attitude control system based on singular value decomposition （SVD） theory. The mechanism of steering laws escaping singularity, especially how the steering laws affect singularity of gimbal configuration and the output torque error, is studied using SVD theory. Performance of various steering laws are analyzed and compared quantitatively by simulation. The obtained results can be used as a reference for designers.     

微机械框架陀螺仪的动力学分析

本文根据欧拉动力学方程导出微机械框架陀螺仪的动力学方程，给出陀螺仪的运动规律，并探讨惯性质量对测量灵敏度的影响。     

The output torque estimation of MCMG for agile satellites

Studied in this paper are the attitude control law design and the output torque estimation problem of micro control moment gyros （MCMGs） for the agile satellites executing rapid attitude maneuver mission. An algorithm is proposed for estimating the output torques and the gimbal angular rates of MCMGs, which can help engineers to choose reasonable size for actuators so that the cost of satellite can be decreased. According to some special maneuver missions, a numerical example of attitude control system for a small satellite with MCMGs in pyramid configuration is studied, and the simulation results validate the proposed estimation algorithm.     

基于扫频振动环境的燃气陀螺仪动态漂移控制技术

燃气陀螺仪是一种依靠火药燃气驱动的陀螺本体为机械框架式结构的惯性陀螺仪,它能够准确地测出弹体在飞行过程中随时间变化的姿态角,是弹体控制回路中的关键测量部件,其在不同振动环境中的漂移控制技术对制导武器弹体姿态的准确测量非常重要。本文利用欧拉方法,推导了陀螺仪的测量原理和漂移产生机理,分析了摩擦力矩、不平衡力矩,以及弹体环境施加的外力矩对陀螺漂移度的影响,并设计燃气陀螺仪的扫频振动试验找到陀螺对外加力矩的敏感频率。试验结果表明,在敏感频率400～800Hz范围内,通过改善弹体安装环境,可将陀螺仪漂移从大于3°减少至0.5°范围内。该方案具有较高的工程实用价值。     

Attitude control for part actuator failure of agile small satellite

The stability and singularity problem of agile small satellite （ASS） with actuator failure is discussed in this paper. Firstly, the three-axis stabilized controller of an ASS is designed, where micro control moment gyros （MCMG＇s） in pyramid configuration （PC） is used as the actuator. By using the same controller and steering law, the control results before and after one gyro fails are compared by simulation. The variation of singular momentum envelope before and after one gyro fails is also compared. The simulation results show that the failure intensively decreases the capacity of output torque, which leads to the emergence of more singular points and the rapid saturation of MCMG＇s. Finally, the parameters of system controller are changed to compare the control effect.     

Primary resonance of a beam-rigid body microgyroscope

We present the non-linear dynamical features of a beam-rigid-body microgyroscope operating under the electrostatic force. A comprehensive system of discretized governing equations is solved using the method of multiple scales and the shooting method. For an amplitude-modulation gyro, the effects of the quality factor, AC voltage, and the rotation rate on the response are investigated and the fold-bifurcation points are identified. A frequency-modulation gyro is examined and in addition to the bifurcation points, the entrainment region is identified. The effects of a positive and a negative internal detuning parameter on the frequency-response of a frequency-modulation gyro are investigated.     

Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.     

小型静电陀螺仪转子变形及干扰力矩分析

推导了小型静电陀螺仪转子非球形干扰力矩公式，仿真分析了两种不同结构的实心转子的温度变形、离心变形、压力变形及综合变形，计算了其变形产生的干扰力矩及漂移。     

A generalized mathematical model to analyze the nonlinear behavior of a controlled gyroscope in gimbals

In this paper, the effect of the parameter variation on the stability and dynamic behavior of a gyroscope in gimbals with a feedback control system, formed by a Proportional + Integral $+$ Derivative (PID) controller and a DC motor with an ideal train gear is researched. The generalized mathematical model of the gyro is obtained from the Euler-Lagrange equations by using the nutation theory of the gyroscope. The use of approximated models of the control system are deduced from the mathematical model of the gyro, taking into account that the integral action of the PID controller is constrained and that the inductance of the DC motor may be negligible. The analysis and choice of appropriate state variables to simulate the dynamic behavior of different models of the gyro are also considered. The paper shows that from the analysis of the equilibrium points, a Bogdanov Takens and a Poincaré-Andronov-Hopf bifurcation can appear. These bifurcations are analyzed from the calculation of a parameter which is known as the first Lyapunov value, showing that it is possible to deduce a procedure to find out when a complicated model can be substituted by a simpler one. In particular, the possibility of self-oscillating and chaotic behavior for different models of the system by using the first Lyapunov value as a function of the parameters of the PID controller is researched. Numerical simulations have been performed to evaluate the analytical results and the mathematical discussions.     

Attitude control of a rigid spacecraft with one variable-speed control moment gyro

Nonlinear controllability and attitude stabilization are studied for the underactuated nonholonomic dynamics of a rigid spacecraft with one variable-speed control moment gyro(VSCMG), which supplies only two internal torques.Nonlinear controllability theory is used to show that the dynamics are locally controllable from the equilibrium point and thus can be asymptotically stabilized to the equilibrium point via time-invariant piecewise continuous feedback laws or time-periodic continuous feedback laws. Specifically,when the total angular momentum of the spacecraft-VSCMG system is zero, any orientation can be a controllable equilibrium attitude. In this case, the attitude stabilization problem is addressed by designing a kinematic stabilizing law, which is implemented through a nonlinear proportional and derivative controller, using the generalized dynamic inverse(GDI)method. The steady-state instability inherent in the GDI controller is elegantly avoided by appropriately choosing control gains. In order to obtain the command gimbal rate and wheel acceleration from control torques, a simple steering logic is constructed to accommodate the requirements of attitude stabilization and singularity avoidance of the VSCMG. Illustrative numerical examples verify the efcacy of the proposed control strategy.     

Optimum energy extraction from rotational motion in a parametrically excited pendulum

A pendulum rotating under vertical base excitation is considered from the viewpoint of energy extraction. Since the uncontrolled system can exhibit complex dynamics, we consider an added control torque and seek the optimal period-1 rotational motion for maximum energy extraction. We find, and confirm through complementary methods, that the limiting optimal motion for harmonic base excitation is piecewise-constant: there are extended dwells at the top and bottom positions with rapid transitions in between. The limiting optimal solution gives about a quarter more energy extraction than uniform rotation, in the limit of no damping. Approximating motions with finite-speed transitions can be almost as good. Base excitations other than pure sinusoids are also considered and the corresponding optima determined.