On One Method for Reducing Gyrostabilizer Drift
Abstract Generated abstract
This paper analyzes drift in a three-degree-of-freedom gyrostabilizer with inter-gimbal correction when mounted on an oscillating base. Starting from linearized equations of motion for small precession angle, it derives expressions for drift rate and shows that correlated or phase-shifted base oscillations can produce systematic drift, with the kinematic component often dominating other terms. The authors propose compensating this kinematic drift by measuring the angular velocity of the outer gimbal about its normal axis, multiplying it by the precession-angle signal, and applying a corresponding corrective moment about the inner suspension axis. The method is presented as suitable for gyrostabilizers installed directly on moving objects subject to substantial periodic angular oscillations.
Full Text
ON ONE METHOD FOR REDUCING DRIFT IN GYROSTABILIZERS
A. M. MALYSHENKO, L. I. TKACHEV
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
A gyrostabilizer, which is a three-degree-of-freedom gyroscope with inter-gimbal correction, has found wide application in gyroscopic instruments and systems. Its principle forms the basis of directional gyroscopes, gyro-semicompasses, gyroframes, etc. [1–3].
Studies of the operation of a gyrostabilizer have shown [3, 4] that oscillations of the base on which it is mounted cause its drift relative to the stabilization axis. These drifts may be so large that the error of the instrument exceeds the permissible value. For this reason, for example, directional gyroscopes and gyro-semicompasses are mounted on stabilized bases [1].
In order to determine the dependence of the gyrostabilizer drift on the parameters of the base oscillations, let us write the equations of its motion.
Let us attach (Fig. 1) to the base a right rectangular trihedron of axes \(xyz\) so that the \(z\)-axis is perpendicular, and the \(x\)- and \(y\)-axes lie in the plane of the base. The gyroscope is installed in such a way that the axis of its outer gimbal coincides with the \(z\)-axis of the base. A trihedron of axes \(x_n\), \(y_n\), \(z_n\) is associated with the outer gimbal, and a trihedron \(HPS\) (Rezal axes) with the inner gimbal. The orientation of the above-mentioned axes is indicated in Fig. 1. The \(H\)-axis is the axis of proper rotation of the gyroscope; the \(P\)-axis will hereafter be called the precession axis, and the \(S\)-axis the stabilization axis.
Introduce the following notation:
\(\vartheta\) — the angle of rotation of the gyroscope outer gimbal relative to the base, measured between the axes \(x\) and \(z_n\) (or between the axes \(y\) and \(y_n\));
\(\varepsilon\) — the deviation of the angle between the outer and inner gimbals of the gyroscope from \(90^\circ\) (the precession angle).
During operation of the gyrostabilizer, the angle \(\varepsilon\) remains small in absolute value; therefore one may take
\[ \sin \varepsilon \simeq \varepsilon; \qquad \cos \varepsilon \simeq 1. \]
The angle \(\vartheta\) of rotation of the outer gimbal relative to the base may take any value in the range \(0\)–\(360^\circ\); however, during operation of the gyrostabilizer the angle \(\vartheta\) cannot change abruptly, and therefore it may be represented as the sum
\[ \vartheta = \theta_0 + \theta, \]
where \(\theta_0\) is the mean value of the angle \(\vartheta\);
Fig. 1.
\(\theta\) is the small deviation of the angle \(\vartheta\) from its mean value.
Taking the above remarks into account, the equations of motion of the gyrostabilizer can be represented in the form:
\[
C\frac{d^2\theta}{dt^2}
-H(\Omega_x\sin\theta_0-\Omega_y\cos\theta_0)\cdot\theta
+H\frac{d\varepsilon}{dt}
=M_\theta-C\frac{d\Omega_z}{dt}
\]
\[
{}-H(\Omega_x\cos\theta_0+\Omega_y\sin\theta_0)
+(A-Д)\Omega_x\Omega_y\cos 2\theta_0
\tag{1}
\]
\[
{}-\frac{1}{2}(A-Д)(\Omega_x^2-\Omega_y^2)\sin 2\theta_0;
\]
\[
A\frac{d^2\varepsilon}{dt^2}
-H(\Omega_x\sin\theta_0-\Omega_y\cos\theta_0)\varepsilon
-H\frac{d\theta}{dt}
=M_\varepsilon+H\Omega_z
\]
\[
{}-A\left(\frac{d\Omega_x}{dt}\cos\theta_0+\frac{d\Omega_y}{dt}\sin\theta_0\right)
-B\Omega_z(\Omega_x\sin\theta_0-\Omega_y\cos\theta_0).
\tag{2}
\]
Here the following notation is adopted:
\(M_\theta, M_\varepsilon\) are the moments of the external forces acting, respectively, on the outer and inner gimbal rings of the gyroscope about the axes \(x\) and \(P\) of their own rotation;
\(H\) is the intrinsic angular momentum of the gyroscope;
\[ A=I_\э+I_p; \tag{3} \]
\[ B=I+I_\н-I_s-I_\э; \tag{4} \]
\[ C=I_\mathrm{e}+I_S+I_{x_\mathrm{n}}; \tag{5} \]
\[ D=I+I_\mathrm{n}+I_{z_\mathrm{n}}-I_{y_\mathrm{n}}; \tag{6} \]
\(I\) is the axial and \(I_\mathrm{e}\) the equatorial moment of inertia of the gyroscope rotor;
\(I_H, I_P, I_S\) are the moments of inertia of the inner ring about the axes \(H, P, S\), respectively;
\(I_{x_\mathrm{n}}, I_{y_\mathrm{n}}, I_{z_\mathrm{n}}\) are the moments of inertia of the outer ring of the gyroscope, respectively about the axes \(x_\mathrm{n}, y_\mathrm{n}, z_\mathrm{n}\).
The moments acting on the gyroscope about the axes of its suspension may be written in the form
\[ M_\theta=M_{\mathrm{mc}}+M_{\mathrm{v}\theta}; \tag{7} \]
\[ M_\varepsilon=M_y+M_{\mathrm{v}\varepsilon}, \tag{8} \]
where \(M_{\mathrm{mc}}\) is the moment produced by the torque generator of the interframe correction circuit and applied to the gyroscope in order to keep the rings of its suspension in a mutually perpendicular position;
\(M_y\) is the control moment used to control rotations of the gyroscope about the stabilization axis;
\(M_{\mathrm{v}\theta}, M_{\mathrm{v}\varepsilon}\) are the moments of harmful forces with respect to the precession and stabilization axes of the gyroscope.
From equation (2) we obtain the following formula for determining the drift rate of the gyroscope in inertial space relative to its stabilization axis:
\[ \frac{d\theta}{dt}+\Omega_z = -\frac{1}{H}\left(M_y+M_{\mathrm{v}\varepsilon}\right) -\left(\Omega_x\sin\theta_0-\Omega_y\cos\theta\right)\varepsilon + \]
\[ +\frac{B}{H}\Omega_z\left(\Omega_x\sin\theta_0-\Omega_y\cos\theta_0\right) +\frac{A}{H}\left( \frac{d^2\varepsilon}{dt^2} +\frac{d\Omega_x}{dt}\cos\theta_0 +\frac{d\Omega_y}{dt}\sin\theta_0 \right). \tag{9} \]
It is seen from this that the drift rate of the gyrostabilizer contains components that depend on the oscillations of the base. If the precession angle and the projections of the angular velocity of the base oscillations on the axes \(x, y, z\) have phase-shifted harmonic components of the same frequency, then systematic drift of the gyrostabilizer will occur with angular velocity
\[ \Omega_c = -\frac{1}{2}\varepsilon_m \left(\Omega_{xm}\sin\theta_0-\Omega_{ym}\cos\theta_0\right) \left(\cos\psi_{\varepsilon x}-\cos\psi_{\varepsilon y}\right) + \]
\[ +\frac{B}{H}\Omega_{zm} \left(\Omega_{xm}\sin\theta_0-\Omega_{ym}\cos\theta_0\right) \left(\cos\psi_{zx}-\cos\psi_{zy}\right), \tag{10} \]
where the index \(m\) denotes the amplitude values of the harmonics, and \(\psi_{ij}\) is the phase shift between the harmonics of the quantities \(i\) and \(j\) \((i=\varepsilon,z;\ j=x,y)\).
The rate of systematic drift of a gyrostabilizer due to oscillations of the object (base) can reach impermissible values. As an example, let us consider the particular case when
\[ H=50000\ \mathrm{g\,cm/sec}; \]
\[ B=1\ \mathrm{g\,cm/sec^2}; \]
\[ \Theta_0=90^\circ; \]
\[ \Omega_{xm}=\Omega_{ym}=\Omega_{zm}=0.1\ \mathrm{1/sec}; \]
\[ \varepsilon_m=0.001\ \mathrm{rad}; \]
\[ \cos \psi_{\varepsilon x}=\cos \psi_{\varepsilon y}=1. \]
Under these conditions the rate of systematic drift of the gyrostabilizer will be equal to
\[ \Omega_c=10.28\ \text{deg/hour}, \]
which in many cases is completely inadmissible.
It should be noted that the last two components of the drift rate of the gyrostabilizer in (9) are much smaller than the component
\[ \Omega_k=(\Omega_x\sin\theta_0-\Omega_y\cos\theta_0)\cdot\varepsilon, \tag{11} \]
which has been called the kinematic drift [3, 4].
For random oscillations of the base, the magnitude of the kinematic component of the gyrostabilizer drift rate will also be a random quantity; however, as shown in [5], the mathematical expectation of the rate \(\Omega_k\) will differ from zero if the angular velocities of the base oscillations \(x, y, z\) are mutually correlated.
It follows from this that in those cases when the gyrostabilizer base undergoes substantial angular oscillations, reaching amplitudes of several angular degrees, inadmissibly large kinematic drifts are possible. In this connection, measures must be taken to reduce the level of kinematic drifts or to compensate them.
It is easy to see that the quantity
\[ \Omega_x\sin\theta_0-\Omega_y\cos\theta_0\simeq\Omega_x\sin\theta-\Omega_y\cos\theta=\Omega_{z_n} \tag{12} \]
is nothing other than the projection of the absolute angular velocity of motion of the outer gimbal ring of the gyroscope onto the axis \(z_n\) (see Fig. 1), perpendicular to its plane.
It is therefore evident that, in order to compensate the kinematic component of the gyrostabilizer drift, it is necessary to measure the angular velocity of motion of the outer ring about the axis \(z_n\), multiply the obtained signal by the signal of the precession-angle sensor \(\varepsilon\), and apply to the gyroscope, along the inner axis of its suspension, a moment equal to
\[ M=k(\Omega_x\sin\theta-\Omega_y\cos\theta)\cdot\varepsilon, \tag{13} \]
where
\[ k=k_1\,k_2\,k_3\,k_4 \tag{14} \]
is the product of the gain (transfer) coefficients, respectively, of the precession-angle sensor, the angular-velocity sensor, the torque generator about the precession axis, and the multiplier-amplifier device. The gain coefficient of the multiplier-amplifier device must be equal to
\[ k_4=\frac{H}{k_1\,k_2\,k_3}. \tag{15} \]
Under this condition, the precession rate of the gyrostabilizer under the action of the moment (13) will be equal to the component (11) of its drift rate.
To implement this method of compensation, a gyroscopic angular-velocity sensor \(\Omega_{z_n}\) must be installed on the outer ring of the gyroscope’s gimbal suspension. A functional diagram of a gyrostabilizer with a device for compensating kinematic drifts according to the method described above is shown in Fig. 2.
The indicated method of compensating kinematic drifts is advisable to use in those cases when the gyrostabilizer is installed directly on a moving object undergoing substantial periodic oscillations.
Fig. 2. Scheme for compensating kinematic drifts of a gyrostabilizer.
Legend: DU — precession-angle sensor; U — amplifier; ZM — torque-setting device about the stabilization axis; ZM — torque-setting device about the precession axis; DUS — sensor of the absolute angular velocity of the outer gimbal; UMU — amplifying-multiplying device.
References
-
V. A. Pavlov. Aviation Gyroscopic Instruments. Oborongiz, Moscow, 1954.
-
S. S. Rivkin. Theory of Gyroscopic Devices, Part I. Sudpromgiz, Leningrad, 1962.
-
A. Yu. Ishlinskii. Mechanics of Gyroscopic Systems. Publishing House of the Academy of Sciences of the USSR, Moscow, 1963.
-
R. H. Cannon. “Kinematic Drift of a Two-Degree-of-Freedom Gyroscope.” Periodical collection of translations of foreign articles Mechanics, No. 1, Moscow, 1960.
-
A. M. Malyshenko. “Systematic Drifts of a Float Gyroscope on an Oscillating Base.” Proceedings of the Tomsk Polytechnic Institute, vol. No. 171, 1969.