SYSTEMATIC DRIFTS OF A FLOATED INTEGRATING GYROSCOPE ON A ROCKING BASE

A. M. MALYSHENKO

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

Introduction

Floated integrating gyroscopes (FIG), first proposed by Associate Professor L. I. Tkachev of the Moscow Power Engineering Institute in 1945, are widely used in schemes for spatial stabilization of the platforms of such devices as inertial navigation and control systems for moving objects.

It is known that angular oscillations of the base of a FIG cause additional gyroscope drifts (in comparison with drifts on a fixed base). In this case drifts in one direction are possible, which are the most undesirable in gyroscopic systems. Below we investigate the dependence of the rate of such drifts on the parameters of the gyroscope and on the motion of the base.

Equation of Motion of a FIG on a Rocking Base and Its General Solution

To formulate the equation of motion of a FIG, let us rigidly attach to the base and to the gyroscope float (Fig. 1) right rectangular trihedra of axes \(Oxyz\) and \(OHPS\), issuing from the point \(O\) of symmetry of the gyroscope in such a way that the axes \(y\) and \(P\) coincide with the axis of rotation of the float (the precession axis), while the axis \(H\) is directed along the axis of proper rotation of the gyroscope rotor. The orientation of the other axes is seen from Fig. 1.

If in what follows we assume the center of symmetry \(O\) to coincide with the center of mass of the gyroscope, and the principal axes of the inertia ellipsoid of the float to coincide with the Rezal axes \(HPS\), then the motions of the FIG on a moving base can, with sufficient accuracy, be described by the equation

\[ A(\ddot{\varepsilon}+\dot{\Omega}_y)+H(\Omega_x\sin\varepsilon+\Omega_z\cos\varepsilon)=M_{\varepsilon}, \tag{2.1} \]

where \(\varepsilon\) is the gyroscope precession angle;

\(H\) is the gyroscope’s intrinsic angular momentum;

\(\Omega_x,\Omega_y,\Omega_z\) are the projections of the absolute angular velocity \(\boldsymbol{\Omega}\) of the base on the axes \(x, y, z\);

\(M_{\varepsilon}\) is the moment of the external forces acting on the gyroscope float about the precession axis \(P\);

\(A\) is the total moment of inertia of the float and the gyroscope rotor about the precession axis.

The moment of the external forces \(M_{\varepsilon}\) can be written in the form

\[ M_{\varepsilon}=M_y-\mu\dot{\varepsilon}+M_{\mathrm{v}}, \tag{2.2} \]

where \(M_y\) is the control moment applied to the gyroscope about the precession axis;

\(\mu\) is the coefficient of viscous friction characterizing the viscous-friction moment between the float and the case of the FIG;

\(M_{\mathrm{в}}\) is the moment of all other (parasitic) forces acting on the gyroscope about the precession axis (moments of friction, unbalance, gravity in the precession-angle sensor, etc.).

When the FIG is used as the sensing element for the angular oscillations of the platform of an inertial system, the precession angle

Fig. 1.

\(\varepsilon\) during operation of the system does not exceed a few angular minutes; therefore one may take

\[ \sin \varepsilon \cong \varepsilon; \qquad \cos \varepsilon \cong 1 . \]

Taking these relations and (2.2) into account, the equation of motion of the FIG assumes the form:

\[ A\ddot{\varepsilon}+\mu \dot{\varepsilon}+H\Omega_x \varepsilon = -H\Omega_z+M_y+M_{\mathrm{в}}-A\dot{\Omega}_y, \tag{2.3} \]

or

\[ T\ddot{\varepsilon}+\dot{\varepsilon}+\kappa\Omega_x \varepsilon = -\kappa\Omega_z+\chi\left(M_y+M_{\mathrm{в}}\right)-T\dot{\Omega}_y, \tag{2.4} \]

where

\[ T=\frac{A}{\mu}; \tag{2.5} \]

\[ \kappa=\frac{H}{\mu}, \tag{2.6} \]

\(T\) and \(\kappa\) are, respectively, the time constant and the static transfer coefficient of the gyroscope.

The time constant \(T\) of floated integrating gyroscopes does not, as a rule, exceed thousandths of a second [1, 2]; therefore, in investigating the effect on a FIG of base rocking in a frequency range bounded above by units, equation (2.4) can with sufficient accuracy be replaced by the truncated equation

\[ \dot{\varepsilon}+\kappa\Omega_x\varepsilon = -\kappa\Omega_z+\chi(M_y+M_{\mathrm{B}})-T\dot{\Omega}_y . \tag{2.7} \]

The solution of this equation with respect to the precession angle is

\[ \varepsilon = \exp\left(-\int_0^t \kappa\Omega_x\,dt\right) \left\{ \varepsilon_0+ \int_0^t \left[ -\kappa\Omega_z+ \right. \right. \tag{2.8} \]

\[ \left. \left. +\chi(M_y+M_{\mathrm{B}})-T\dot{\Omega}_y \right] \cdot \exp\left(\int_0^t \kappa\Omega_x\,dt\right) \cdot dt \right\}; \]

where \(\varepsilon_0\) is the value of the precession angle at the instant \(t=0\).

In accordance with the requirements imposed on the FIG, its angular precession velocity must be equal to

\[ \dot{\varepsilon}_{\mathrm{i}}=-\kappa\Omega_z+\chi M_y . \tag{2.9} \]

The difference between \(\dot{\varepsilon}_{\mathrm{i}}\) and the actual precession velocity of the FIG, representing the drift rate of the gyroscope with respect to the precession axis \(P\), according to (2.8) and (2.9), can be written in the form

\[ \Delta\dot{\varepsilon}(t) = -\chi M_{\mathrm{B}}+T\dot{\Omega}_y + \kappa\Omega_x \exp\left(-\int_0^t \kappa\Omega_x\,dt\right) \left\{ \varepsilon_0+ \right. \tag{2.10} \]

\[ \left. + \int_0^t \left[ -\kappa\Omega_z+\chi(M_y+M_{\mathrm{B}})-T\dot{\Omega}_y \right] \cdot \exp\left(\int_0^t \kappa\Omega_x\,dt\right) \,dt \right\}. \]

Determination of the FIG drift rate in quadratures by formula (2.10) for harmonic and random modes of platform oscillation does not appear possible. For this reason, in analyzing the behavior of a floated integrating gyroscope on a rocking base, approximate methods are used, for example, the method of successive substitutions [3], expansion in the integrand of the solution of the FIG equation of motion of the function \(\exp\left(\int_0^t \kappa\Omega_x\,dt\right)\) in a series of Bessel functions with subsequent determination of the solution in quadratures [1]. These methods are used only for a harmonic law of variation of the angular velocity \(\Omega_x(t)\). The solution given below differs from those mentioned above and makes it possible to use the final results not only for harmonic, but also for random oscillations of the base.

To determine the FIG drift rate for an arbitrary character of platform oscillations, we transform the truncated gyroscope equation of motion (2.7) into the integral equation

\[ \dot{\varepsilon}(t) = -\kappa\Omega_z(t)+\chi M_y(t)-T\dot{\Omega}_y(t)- \tag{2.11} \]

\[ -\varepsilon_0\kappa\Omega_x(t) - \kappa\Omega_x(t) \int_0^t \dot{\varepsilon}(t_1)\,dt_1, \]

which belongs to the class of linear integral equations of Volterra type of the second kind [4].

To find the solution of equation (2.11), we shall use the method of successive substitutions, which is the most convenient for solving problems of this type [5].

In order to shorten the notation, introduce the designation

\[ F(t)=-\kappa \Omega_z(t)+\kappa[M_y(t)+M_{\mathrm{v}}(t)]-T\dot{\Omega}_y(t)-\varepsilon_0\kappa\Omega_x(t) \tag{2.12} \]

and reduce equation (2.11) to the form:

\[ \dot{\varepsilon}(t)=F(t)-\kappa\Omega_x(t)\int_0^t \dot{\varepsilon}(t_1)\,dt_1 . \tag{2.13} \]

According to (2.13), the precession rate of the FIG at the instant \(t_1\) is equal to:

\[ \dot{\varepsilon}(t_1)=F(t_1)-\kappa\Omega_x(t_1)\int_0^t \dot{\varepsilon}(t_2)\,dt_2 . \tag{2.14} \]

Substituting in (2.13), instead of \(\dot{\varepsilon}(t_1)\), its value (2.14), we obtain

\[ \dot{\varepsilon}(t)=F(t)-\kappa\Omega_x(t)\int_0^t \left[F(t_1)-\kappa\Omega_x(t_1)\int_0^t \dot{\varepsilon}(t_2)\,dt_2\right]dt_1 . \tag{2.15} \]

If in (2.15), instead of \(\dot{\varepsilon}(t_2)\), we substitute an expression analogous to (2.14), and then repeat the process described above, we obtain the solution of equation (2.13) in the form of an absolutely and uniformly convergent series:

\[ \dot{\varepsilon}(t)=F(t)+\sum_{n=1}^{\infty}Q_n(t), \tag{2.16} \]

where

\[ Q_n(t)=(-\kappa)^n \int_0^t\int_0^{t_1}\cdots\int_0^{t_{n-1}} \Omega_x(t)\Omega_x(t_1)\cdots\Omega_x(t_n)F(t_n)\,dt_n\cdots dt_1 . \tag{2.17} \]

The drift rate of the floated integrating gyroscope in this case can be expressed by the formula

\[ \Delta\dot{\varepsilon}(t)= -\kappa M_{\mathrm{v}}(t)+T\dot{\Omega}_y(t)+\varepsilon_0\kappa\Omega_x(t) -\sum_{n=1}^{\infty}Q_n(t). \tag{2.18} \]

Below, the systematic drifts of the FIG are determined for certain types of base oscillations.

Drifts of the FIG under harmonic oscillations of the base

Let us determine the drifts of the floated integrating gyroscope when the projections of the angular velocities are

\[ \Omega_i=\Omega_{im}\sin(\omega_i t+\varphi_i), \tag{3.1} \]

\[ (i=x,y,z), \]

and the moment of the disturbing forces about the precession axis is

\[ M_{\mathrm{v}}=M_{\mathrm{v}0}+M_{\mathrm{v}m}\sin(\omega_M t+\varphi_M). \tag{3.2} \]

Here \(\Omega_i\) is the amplitude value of the projection of the angular velocity of motion of the base about the \(i\)-axis \((i=x,y,z)\);

\(\omega_i,\varphi_i\) are the circular frequency and the initial phase of the oscillations of the base about the \(i\)-th axis;

$M_{\mathrm{в}0}, M_{\mathrm{в}m}$ — the constant component and the amplitude of the variable component of the torque of the disturbing forces about the precession axis;

$\omega_M, \varphi_M$ — the circular frequency and the initial phase of the variations of the torque about the precession axis.

In what follows we shall take the control torque $M_y$ to be equal to zero, since the drifts of the FIG caused by its deviation from the prescribed value $M_y^*$ can be allowed for by adding the torque

\[ \Delta M_y = M_y - M_y^* \]

to the torques of the disturbing forces about the precession axis.

Under these conditions the drift rate of the FIG in the first approximation, according to (2.18), is equal to

\[ \Delta \dot{\varepsilon}_1(t) = -\chi M_{\mathrm{в}0}'-\chi M_{\mathrm{в}m}\sin(\omega_M t+\varphi_M)+ \]
\[ +T\Omega_{ym}^{0}\cos(\omega_y t+\varphi_y)+ \varepsilon_0 k\Omega_{xm}^{0}\sin(\omega_x t+\varphi_x). \tag{3.3} \]

Substituting $\Delta \dot{\varepsilon}_1(t)$ into (2.18) and carrying out elementary transformations, we obtain the value of the FIG drift rate in the second approximation, equal to

\[ \Delta \dot{\varepsilon}(t)= -\chi M_{\mathrm{в}0}-\chi M_{\mathrm{в}m}\sin(\omega_M t+\varphi_M) +\kappa\chi M_{\mathrm{в}0}\Omega_{xm}t\cdot\sin(\omega_x t+\varphi_x)- \]
\[ -N\kappa\Omega_{xm}\sin(\omega_x t+\varphi_x) +T\Omega_{ym}\cos(\omega_y t+\varphi_y)+ \]
\[ +\varepsilon_0\kappa\Omega_{xm}\sin(\omega_x t+\varphi_x) +\varepsilon_0\frac{\kappa^2}{2\omega_x}\Omega_{xm}\sin 2(\omega_x t+\varphi_x)+ \tag{3.4} \]
\[ +\frac{\kappa^2}{2\omega_z}\Omega_{xm}\Omega_{zm} \{\sin[(\omega_x-\omega_z)t+\varphi_x-\varphi_z] +\sin[(\omega_x+\omega_z)t+\varphi_x+\varphi_z]\}- \]
\[ -\frac{\kappa\chi}{2\omega_M}M_{\mathrm{в}m}\Omega_{xm} \{\sin[(\omega_x-\omega_M)t+\varphi_x-\varphi_M] +\sin[(\omega_x+\omega_M)t+\varphi_x+\varphi_M]\}- \]
\[ -\frac{\kappa T}{2}\Omega_{xm}\Omega_{ym} \{\cos[(\omega_x-\omega_y)t+\varphi_x-\varphi_y] -\cos[(\omega_x+\omega_y)t+\varphi_x+\varphi_y]\}, \]

where it is denoted that

\[ N=\frac{\kappa}{\omega_z}\Omega_{zm}\cos\varphi_z -\frac{\chi}{\omega_M}M_{\mathrm{в}m}\cos\varphi_M -T\Omega_{ym}\sin\varphi_y +\varepsilon_0\frac{\kappa}{\omega_x}\Omega_{xm}\cos\varphi_x. \tag{3.5} \]

In estimating the drifts of a floated integrating gyroscope one may restrict oneself to the second approximate solution (3.4), since the alternating series (2.18) converges rapidly. This is confirmed by the sufficiently exact agreement of the results of computations by formula (3.4) and of the solution of the original equation of motion of the FIG (2.4) on an analog computer.

The most unfavorable in inertial systems is the systematic drift of gyroscopes. For this reason we shall determine the modes of oscillation of the base for which such drifts of the FIG are possible.

According to the second approximate solution (3.4), the drift rate of the FIG, for different values of the circular frequencies of the base oscillations and of the torque $M_{\mathrm{в}}$, has no constant component. The angular oscillations of the base about the axes $x$, $y$, $z$ then produce an effect similar to beats.

When the circular frequency $\omega_x$ of the variations of the rate $\Omega_x$ is equal to one of the oscillation frequencies $\Omega_y$, $\Omega_z$ or $M_{\mathrm{в}}$, an additional

constant component of the precession rate. If it is assumed that

\[ \omega_x=\omega_y=\omega_z=\omega_M=\omega, \]

then, according to (3.4), the floated integrating gyroscope will, in operation, acquire a systematic drift rate equal to

\[ \Delta \dot{\varepsilon}_c(t) = -\chi M_{\mathrm{в}0} + \frac{\kappa^2}{2\omega}\Omega_{xm}\Omega_{zm}\sin(\varphi_x-\varphi_z) - \]

\[ -\frac{\kappa\chi}{2\omega}\Omega_{xm}M_{\mathrm{в}m}\sin(\varphi_x-\varphi_M) -\frac{\kappa T}{2}\Omega_{xm}\Omega_{ym}\cos(\varphi_x-\varphi_y). \tag{3.6} \]

It follows from this that the most unfavorable operating conditions for the FIG are those motions of the base for which

\[ \varphi_x=\varphi_y=\varphi_z-90^\circ=\varphi_M+90^\circ. \tag{3.7} \]

Under these conditions the systematic drift rate reaches its maximum value:

\[ \Delta \dot{\varepsilon}_{cm}(t) =- \left( \chi M_{\mathrm{в}0} +\frac{\kappa^2}{2\omega}\Omega_{xm}\Omega_{zm} +\frac{\kappa\chi}{2\omega}\Omega_{xm}M_{\mathrm{в}m} +\frac{\kappa T}{2}\Omega_{xm}\Omega_{ym} \right). \tag{3.8} \]

In the particular case when

\[ \Omega_y=0;\quad M_{\mathrm{в}}=0, \]

the solution obtained for the systematic drift rate of the FIG coincides, to within the notation, with the solution of B. I. Nazarov [1], if in the latter the amplitude of the angle of oscillation of the base about the \(x\)-axis is considered sufficiently small.

Drifts of the FIG for a random character of base oscillations

For random oscillations of the base, characteristic of gyrostabilized platforms installed on moving objects, the equation of motion of the FIG (2.4) is an inhomogeneous differential equation with a random coefficient and a random right-hand side.

To reveal the possibility of systematic drifts of the FIG under random oscillations of the base, we use, as in the case of harmonic oscillations of the base, the truncated equation of motion of the gyroscope (2.7) and its solution (2.15). For this purpose we determine the mathematical expectation of the drift rate \(\Delta\dot{\varepsilon}(t)\).

On the basis of known relations from the theory of random functions [6], the mathematical expectation \(m[\Delta\dot{\varepsilon}(t)]\) of the FIG drift rate may, according to (2.15), be written in the form

\[ m[\Delta\dot{\varepsilon}(t)] = -\chi m_{M_{\mathrm{в}}}(t) + T\frac{d}{dt}\left[m_{\Omega_y}(t)\right] + \varepsilon_0\frac{d}{dt} \times \]

\[ \times \left[ \exp\left(-\int_0^t \kappa m_{\Omega_x}(t_1)\,dt_1\right) \right] - \kappa^2\int_0^t R_{\Omega_x\Omega_z}(t,t_1)\,dt_1 - \kappa^2 m_{\Omega_x}(t) \times \]

\[ \times \int_0^t m_{\Omega_z}(t_1)\,dt_1 + \kappa\chi\int_0^t R_{\Omega_x M}(t,t_1)\,dt_1 + \kappa\chi m_{\Omega_x}(t)\int_0^t m_M(t_1)\,dt_1 - \]

\[ - \kappa T\int_0^t R_{\Omega_x\dot{\Omega}_y}(t,t_1)\,dt_1 - \kappa T m_{\Omega_x}(t)\cdot m_{\Omega_y}(t). \tag{4.1} \]

Here \(m_i(t)\) is the mathematical expectation of the \(i\)-th random process

\[ (i=\Omega_x,\ \Omega_y,\ \Omega_z,\ M_{\mathrm{в}},\ M); \]

\(R_{\Omega_x j}(t,t_1)\) is the cross-correlation function of the random processes \(\Omega_x(t)\) and \(j(t)\) \((j=\Omega_z,\ \Omega_y,\ \dot M)\).

In the particular case when the mathematical expectations of the projections of the angular velocity of the base onto the axes \(x, y, z\) and of the moment of the external forces about the gyroscope precession axis are identically equal to zero, the mathematical expectation of the rate of systematic drift of the FIG is

\[ m[\Delta \dot{\varepsilon}(t)] = -\kappa^2 \int_0^t R_{\Omega_x\Omega_z}(t,t_1)\,dt_1 + \kappa\chi \int_0^t R_{\Omega_x M}(t,t_1)\,dt_1 - \kappa T \int_0^t R_{\Omega_x\dot{\Omega}_y}(t,t_1)\,dt_1, \tag{4.2} \]

On the basis of (4.2) we arrive at the conclusion that, even when the mathematical expectations of the components of the angular velocities \(\Omega_x,\ \Omega_y,\ \Omega_z\) of the motion of the base and of the moment \(M\) about the precession axis are zero, a floated integrating gyroscope may have a systematic drift. Under these conditions, the systematic drift of the FIG arises due to the correlation between the angular velocity \(\Omega_x(t)\) of the base and any one of the random processes \(\Omega_z(t)\), \(\Omega_y(t)\), or \(\dot M(t)\).

REFERENCES

1. B. I. Nazarov. On errors of a two-stage integrating gyroscope caused by oscillations of the base, Izvestiya vuzov, section “Instrument Engineering,” No. 6, 1960.

2. S. S. Rivkin. Some questions in the dynamics of integrating gyroscopes, in the collected papers Problems of Applied Gyroscopy, issue 3, Sudpromgiz, L., 1962.

3. S. S. Arutyunov. On errors of a two-stage integrating gyroscope caused by angular oscillations of the object, Izvestiya vuzov, section “Instrument Engineering,” No. 2, 1959.

4. W. V. Lovitt. Linear Integral Equations, Gostekhizdat, Moscow, 1957.

5. L. Ya. Roytenberg. On the motion of a gyroscopic compass under random displacements of its support point, Applied Mathematics and Mechanics, issue 1, 1965.

6. V. S. Pugachev. Theory of Random Functions, Fizmatgiz, Moscow, 1962.