On Two Methods of Stabilizing the Gyroscopic Moment of a Control Moment Gyroscope
Abstract Generated abstract
The paper examines the problem of making the gyroscopic moment of a control moment gyroscope independent of the precession angle, since the usual moment expression contains a cosine dependence on the angular position of the angular momentum vector. It compares two approaches: feedback stabilization based on measured moment-related signals, including analysis of tachogenerator dead zone and dry friction effects, and a compensation method that varies precession velocity according to an inverse-cosine law using a noncircular gear mechanism. The analysis indicates that increasing feedback improves stabilization but leaves accuracy dependent on loop gain and component errors, while both methods exhibit dead zones caused by dry friction. The authors conclude that the mechanical compensation scheme can provide higher reliability and accuracy, limited mainly by reducer manufacturing and assembly errors.
Full Text
On Two Methods for Stabilizing the Gyroscopic Moment of a Control Moment Gyroscope
E. I. Gromakov, V. A. Gerasimenko, A. M. Malyshenko
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
It is known that the gyroscopic moment of a gyrostabilizer depends on the angular position of the angular-momentum vector of the gyroscope and is determined by the expression
\[ M_{\mathrm{g}} = H \dot{\theta}\cos\theta, \tag{1} \]
where \(H\) is the intrinsic angular momentum of the gyroscope;
\(\theta\) is the precession angle of the gyroscope.
In gyrostabilization systems there sometimes arises the problem of using the gyroscopic moment to control objects. In this case the nonlinear dependence of this moment on the angle of rotation of the angular-momentum vector, i.e., on the precession angle, is undesirable.
In order to eliminate the influence of the angle of rotation of the gyroscope gimbal on the level of the output moment, let us consider and evaluate two possible methods for solving the problem posed: a method of stabilizing the level of the gyroscopic moment according to the angular position of the angular-momentum vector, and a compensation method. A system for stabilizing the level of the gyroscopic moment can be constructed on the basis of using information on the magnitude of the gyroscopic moment. One version of such a scheme is shown in Fig. 1.
We shall assume that the torque motor is a direct-current machine, while the tachogenerator (TG) and the sine-cosine rotating transformer (SCRT) are alternating-current devices. Therefore the signals from the TG and the SCRT, after demodulation, pass through filters.
The quantity \(h\) determines the feedback coefficient. The differential equations describing the dynamics of the system for stabilizing the level of the gyroscopic moment have the form:
\[ I_0 \frac{d^2 \theta}{dt^2} + I_{\mathrm{d}} i^2 \frac{d^2 \alpha_{\mathrm{d}}}{dt^2} = i M_{\mathrm{d}}; \]
\[ M_{\mathrm{d}} = k_{\mathrm{d}}\Delta U + S i \frac{d\alpha_{\mathrm{d}}}{dt}; \]
\[ \Delta U = U - U_{\mathrm{z}}; \]
\[ U_{\mathrm{z}} = h U_1 U_2; \]
\[ U_1+T_1\frac{dU_1}{dt}=K_{\mathrm{tg}}K_{\phi 1}\dot{\alpha}_{\mathrm{d}}; \]
\[ U_2+T_2\frac{dU_2}{dt}=K_{\phi 2}\cos\Theta; \]
\[ \dot{\Theta}=\frac{1}{i}\dot{\alpha}_{\mathrm{d}}. \tag{2} \]
Here the following notation has been used:
\(J_0\)—moment of inertia of the gyromotor;
\(J_{\mathrm{d}}\)—moment of inertia of the torque-motor rotor;
\(K_{\mathrm{d}}\)—transmission coefficient of the torque motor;
Fig. 1. Functional diagram of the system for stabilizing the level of the gyroscopic moment of a control moment gyroscope
\(i\)—reduction ratio of the reducer;
\(S\)—coefficient of viscous friction;
\(K_{\mathrm{tg}}\)—transmission coefficient of the tachogenerator;
\(\Theta\)—angle of rotation of the gyroscope frame;
\(T_1, T_2, K_{\phi 1}, K_{\phi 3}\)—filter parameters.
On the basis of equations (2), dependences of the steady-state value \(M_{\mathrm{g}}\) as a function of the angle of rotation of the gyroscope frame can be constructed for various values of the feedback coefficient \(h\) (Fig. 2). From these graphs it may be concluded that, in order to reduce the influence of the angle of rotation of the kinetic moment on the value of \(M_{\mathrm{g}}\), the feedback coefficient \(h\) should be increased.
To assess the quality of stabilization systems of this type for the level of the gyroscopic moment, the influence of nonlinearities of the type of the tachogenerator dead zone and the dry-friction moment on the value of \(M_{\mathrm{g}}\) was considered. Graphs showing this influence are presented in Figs. 3 and 4. From these graphs it is evident that the indicated nonlinearities affect the dependence \(M_{\mathrm{g}}(\Theta)\) in different ways. Increasing the dead zone of the tachogenerator leads to an increase in the output moment, but at the same time increases the nonlinearity of the dependence \(M_{\mathrm{g}}(\Theta)\). This can be explained as follows. For small values of the input voltage \(U\) (Fig. 1), the angular velocity of the torque motor is close to the dead zone of the tachogenerator.
In this connection, the feedback signal becomes weak, and, consequently, the control moment proves to be higher than with a smaller dead zone of the tachogenerator.
On the other hand, reducing the feedback signal leads to a lower effectiveness of the feedback circuit of the stabilization system and, consequently, to a greater influence of the angle of rotation of the angular-momentum vector on the level \(M_{\mathrm g}\).
Fig. 2
Fig. 3. Dependence of the gyroscopic moment \(M_{\mathrm g}\) of the gyroscope on the angle \(\Theta\) of rotation of its angular-momentum vector for various levels of the tachogenerator dead zone \(\Delta_{\mathrm{tg}}\)
Dry friction determines the dead zone of the characteristic \(M_{\mathrm g}(U)\) in the steady-state regime. In addition, an increase in the dry-friction moment leads to a decrease in the rotational speed of the motor, which in turn leads to a decrease in the magnitude of the gyroscopic moment (Fig. 2b). The graphs shown were obtained for the following values of the system parameters:
\[ \kappa_{\mathrm d}=10\ \mathrm{g\,cm/V};\qquad H=10^6\ \mathrm{g\,cm/sec^2};\qquad \kappa_{\phi 1}=\kappa_{\phi 2}=0.8; \]
\[ \kappa_{\mathrm{tg}}=0.1\ \mathrm{V\,sec};\qquad i=2400. \]
In view of the fact that the main purpose of the analysis was to determine the qualitative characteristics of the stabilization system, the values of the dead zone, the magnitude of the dry-friction moment, and the magnitude of the voltage applied to the torque motor were taken as variable.
Another method of maintaining the gyroscopic moment at a level independent of the angle of rotation of the angular-momentum vector is to compensate for the cosine dependence of \(M_{\mathrm g}\) by varying the angular velocity \(\dot{\Theta}\) according to an inverse-cosine law as a function of the angle of rotation of the gyro chamber.
Let us consider one variant of such compensation—a mechanical scheme constructed on noncircular gear wheels [2]. According to equation (1), the mechanical device must create a velocity applied to the gyro chamber and depending on its angular position, i.e.,
\[ \dot{\Theta}=\frac{\dot{y}}{\cos\Theta}, \tag{3} \]
where
\[ \dot{y}=\frac{\alpha_{\mathrm{d}}}{i}. \tag{4} \]
Let us set ourselves the goal of creating a functional device that makes it possible to obtain the following dependence between the output and input angles:
\[ y=\sin\theta . \tag{5} \]
Then the angular velocities of the output will be related by
\[ \dot{y}=-\dot{\theta}\cos\theta . \tag{6} \]
Consequently, if the input and output are interchanged, expression (3) is obtained. The device that makes it possible to obtain function (5) is considered in detail in [3] and has the form shown in Fig. 5. In this figure the following designations are adopted:
1, 2 — noncircular wheels, whose shape consists of closed convex centroids;
3, 4 — a pair of circular wheels (gear wheels);
5 — differential.
Taking \(\dot{y}\) as the input and \(\dot{\theta}\) as the output, we obtain the required dependence (3). This scheme can be represented as the closed loop shown in Fig. 6, where
\[ W_1=\frac{1}{2} \]
is the transfer function of transmission on the circular wheels;
\[ W_2=1+\frac{1}{2}\cos\Theta \]
is the transfer function of transmission on the noncircular wheels;
Fig. 4. Dependence of the gyroscopic moment \(M_{\mathrm{g}}\) of the gyroscope on the angle \(\Theta\) of rotation of the vector of its angular momentum for various dead zones \(\Delta_{\mathrm{dv}}\) of the motor, caused by the moment of dry friction.
Fig. 5
Fig. 6
\[ W_3=2 \]
is the transfer function of the connection between the driver and the differential.
An important advantage of such a compensation scheme is the reliability and accuracy of reproducing the functional dependence (5), and, consequently, the high accuracy of compensation. With such a scheme
compensation, its accuracy depends on the error in introducing the independent variable \(y\) and on the error in the manufacture and assembly of the gear meshes. The latter error can be reduced to the level of 0.02–0.04 percent.
Conclusion
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The static characteristics of both the system for stabilizing the level of the gyroscopic moment and the compensation system have a dead zone determined by the dry-friction moment. From this standpoint the two schemes are equivalent.
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The accuracy of maintaining the independence of the gyroscopic moment from the angular position of the angular-momentum vector in the first scheme is determined by the gain of the open-loop system. In the case of a small gain in the forward path, the accuracy in this scheme will be determined by the feedback coefficient and will be the higher, the larger its value. In the scheme constructed on the basis of compensation, the accuracy is determined primarily by the manufacturing error of the reducer.
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For reasons of reliability, preference should be given to the compensation scheme.
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The elements included in the loop of the system for stabilizing the level of the gyroscopic moment introduce their own errors, which reduce the accuracy of maintaining the level of the gyroscopic moment.
References
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V. A. Bessekerskii, E. A. Fabrikant. Dynamic Synthesis of Gyroscopic Stabilization Systems. Leningrad, Sudostroenie Publishing House, 1968.
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F. L. Litvin. Theory of Gear Meshing. Moscow, Nauka Publishing House, 1968.
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F. L. Litvin. Noncircular Gears. Leningrad, Mashgiz, 1956.