ON THE KINEMATICS OF THE GIMBAL SUSPENSION OF A SPATIAL GYROSTABILIZER

Yu. M. KAMASHEV

(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)

The article determines the relationship between platform-stabilization errors measured with respect to the suspension axes and with respect to the reference coordinate system. Expressions are given for the projections of the absolute angular velocities of the suspension rings and of the platform, taking into account the rotation of the base.

In published works [1, 2, 3, 4, etc.] various methods are given for determining the projections of the absolute angular velocities of the mechanical elements of a spatial gyrostabilizer and various ways of specifying the platform-stabilization errors. It is advisable to obtain more general expressions for the angular velocities and to find the relationship between the different variants of the stabilization errors.

Fig. 1

Let us choose the following right-handed orthogonal coordinate systems (Fig. 1): $\xi_c \eta_c \zeta_c$, $x_n y_n z_n$, $x_v y_v z_v$, $x_p y_p z_p$, rigidly connected respectively with the movable base, the outer (OR), inner (IR) suspension rings, and the platform (P); we shall call the coordinate system $\xi_0 \eta_0 \zeta_0$ the reference system. This coordinate system is modeled by a stabilized platform on the movable base and, in the particular case, may be fixed.

We shall consider the coordinate system \(\xi_c \eta_c \zeta_c\) to be the initial one, and the system \(\xi_0 \eta_0 \zeta_0\) to be the final one. We shall define the mutual position of the coordinate systems \(\xi_c \eta_c \zeta_c\), \(x_h y_h z_h\), \(x_v y_v z_v\), \(x_p y_p z_p\) by the angles \(\gamma, \beta, \alpha\) of their relative rotation about the suspension axes \(CC, BB, AA\), respectively. The angles \(\gamma, \beta, \alpha\) may take arbitrary values. We shall define the position of the reference coordinate system \(\xi_0 \eta_0 \zeta_0\) relative to the platform by the angles \(\alpha_p^p, \beta_p^p, \gamma_p^p\). These angles are small and represent the errors of platform stabilization in the reference coordinate system. The platform axes \(x_p y_p z_p\) have been called the stabilization axes.

We denote the projections of the absolute angular velocities of rotation of the base, the NK, the VK, and the P onto their axes, respectively, by:

\[ \omega_{\xi_c}^{c}\quad \omega_{\eta_c}^{c}\quad \omega_{\zeta_c}^{c},\qquad \omega_{x_h}^{h}\quad \omega_{y_h}^{h}\quad \omega_{z_h}^{h},\qquad \omega_{x_v}^{v}\quad \omega_{y_v}^{v}\quad \omega_{z_v}^{v},\qquad \omega_{x_p}^{p}\quad \omega_{y_p}^{p}\quad \omega_{z_p}^{p}. \]

The projections of the absolute angular velocities of rotation of the gimbal rings and the platform are determined by the obvious relations:

\[ \begin{aligned} \omega_{x_h}^{h} &= \omega_{\xi_c}^{c}+\dot{\gamma},\\ \omega_{y_h}^{h} &= \omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma,\\ \omega_{z_h}^{h} &= -\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma. \end{aligned} \tag{1} \]

\[ \begin{aligned} \omega_{x_v}^{v} &=(\dot{\gamma}+\omega_{\xi_c}^{c})\cdot \cos\beta-\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right)\sin\beta,\\ \omega_{y_v}^{v} &=\dot{\beta}+\omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma,\\ \omega_{z_v}^{v} &=(\dot{\gamma}+\omega_{\xi_c}^{c})\cdot \sin\beta+\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right)\cdot \cos\beta. \end{aligned} \tag{2} \]

\[ \begin{aligned} \omega_{x_p}^{p} &= \left[(\dot{\gamma}+\omega_{\xi_c}^{c})\cdot \cos\beta-\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right)\cdot \sin\beta\right]\cdot \cos\alpha\\ &\quad+\left(\dot{\beta}+\omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma\right)\cdot \sin\alpha,\\[4pt] \omega_{y_p}^{p} &= -\left[(\dot{\gamma}+\omega_{\xi_c}^{c})\cdot \cos\beta-\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right)\cdot \sin\beta\right]\cdot \sin\alpha\\ &\quad+\left(\dot{\beta}+\omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma\right)\cdot \cos\alpha,\\[4pt] \omega_{z_p}^{p} &= \dot{\alpha}+(\dot{\gamma}+\omega_{\xi_c}^{c})\cdot \sin\beta+ \left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right)\cdot \cos\beta. \end{aligned} \tag{3} \]

Here \(\dot{\alpha}, \dot{\beta}, \dot{\gamma}\) are the relative angular velocities of rotation of the gimbal rings, caused by the rotation of the base and the platform. From (3) we find:

\[ \begin{aligned} \dot{\alpha} &=\omega_{z_p}^{p} -\operatorname{tg}\beta\left(\omega_{x_p}^{p}\cdot \cos\alpha-\omega_{y_p}^{p}\cdot \sin\alpha\right) -\frac{1}{\cos\beta}\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right),\\ \dot{\beta} &=\omega_{x_p}^{p}\cdot \sin\alpha+\omega_{y_p}^{p}\cdot \cos\alpha -\left(\omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma\right),\\ \dot{\gamma} &=\frac{1}{\cos\beta}\left(\omega_{x_p}^{p}\cdot \cos\alpha-\omega_{y_p}^{p}\cdot \sin\alpha\right) +\operatorname{tg}\beta\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right) -\omega_{\xi_c}^{c}. \end{aligned} \tag{4} \]

We eliminate the relative velocities from expressions (1), (2), (3), substituting their values from (4), and obtain:

\[ \begin{aligned} \omega_{x_h}^{h} &=\frac{1}{\cos\beta}\left(\omega_{x_p}^{p}\cdot \cos\alpha-\omega_{y_p}^{p}\cdot \sin\alpha\right) +\operatorname{tg}\beta\left(-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma\right),\\ \omega_{y_h}^{h} &=\omega_{\eta_c}^{c}\cdot \cos\gamma+\omega_{\zeta_c}^{c}\cdot \sin\gamma,\\ \omega_{z_h}^{h} &=-\omega_{\eta_c}^{c}\cdot \sin\gamma+\omega_{\zeta_c}^{c}\cdot \cos\gamma, \end{aligned} \tag{5} \]

\[ \begin{aligned} \omega^{v}_{x_v}&=\omega^{p}_{x_p}\cdot\cos\alpha-\omega^{p}_{y_p}\cdot\sin\alpha,\\ \omega^{v}_{y_v}&=\omega^{p}_{x_p}\cdot\sin\alpha+\omega^{p}_{y_p}\cdot\cos\alpha, \end{aligned} \tag{5} \]

\[ \omega^{v}_{z_v}=\operatorname{tg}\beta\,(\omega^{p}_{x_p}\cdot\cos\alpha-\omega^{p}_{y_p}\cdot\sin\alpha) +\frac{1}{\cos\beta}(-\omega^{s}_{\eta_s}\cdot\sin\gamma+\omega^{s}_{\xi_s}\cdot\cos\gamma). \tag{6} \]

\[ \begin{aligned} \omega^{p}_{x_p}&=\omega^{p}_{x_p},\\ \omega^{p}_{y_p}&=\omega^{p}_{y_p},\\ \omega^{p}_{z_p}&=\omega^{p}_{z_p}. \end{aligned} \tag{7} \]

For the platform we have obtained obvious identical equalities. On the other hand, the angular velocities of the platform in the reference coordinate system are determined by the relations:

\[ \begin{aligned} \omega^{p}_{x_p}&=\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\cdot\beta^{p}_{p},\\ \omega^{p}_{y_p}&=\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\gamma^{p}_{p},\\ \omega^{p}_{z_p}&=\dot{\alpha}^{p}_{p}+\dot{\beta}^{p}_{p}\cdot\gamma^{p}_{p}. \end{aligned} \tag{8} \]

Substituting the values of the projections (8) of the absolute angular velocity of the platform into expressions (5) and (6), we obtain:

\[ \begin{aligned} \omega^{n}_{x_n} &=\frac{1}{\cos\beta}\cdot \left[(\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\beta^{p}_{p})\cdot\cos\alpha -(\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\gamma^{p}_{p})\cdot\sin\alpha\right]\\ &\quad+\operatorname{tg}\beta(-\omega^{s}_{\eta_s}\cdot\sin\gamma+\omega^{s}_{\xi_s}\cdot\cos\gamma),\\ \omega^{n}_{y_n} &=\omega^{s}_{\eta_s}\cdot\cos\gamma+\omega^{s}_{\xi_s}\cdot\sin\gamma,\\ \omega^{n}_{z_n} &=-\omega^{s}_{\eta_s}\cdot\sin\gamma+\omega^{s}_{\xi_s}\cdot\cos\gamma. \end{aligned} \tag{9} \]

\[ \begin{aligned} \omega^{v}_{x_v} &=(\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\cdot\beta^{p}_{p})\cdot\cos\alpha -(\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\cdot\gamma^{p}_{p})\sin\alpha,\\ \omega^{v}_{y_v} &=(\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\cdot\beta^{p}_{p})\cdot\sin\alpha +(\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\cdot\gamma^{p}_{p})\cos\alpha,\\ \omega^{v}_{z_v} &=\operatorname{tg}\beta\left[(\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\cdot\beta^{p}_{p})\cdot\cos\alpha -(\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\cdot\gamma^{p}_{p})\sin\alpha\right]\\ &\quad+\frac{1}{\cos\beta}(-\omega^{s}_{\eta_s}\cdot\sin\gamma+\omega^{s}_{\xi_s}\cdot\cos\gamma). \end{aligned} \tag{10} \]

Expressions (8), (9), (10) are the projections of the absolute angular velocities of the platform, the outer ring, and the inner ring on their axes, expressed through the angles and angular velocities of the platform relative to the reference coordinate system, taking into account the motion of the base. For a fixed base these expressions take the form obtained in work [4].

Since the resistance moments in the suspension axes depend on the relative angular velocities of the suspension rings, we shall determine the values of the relative velocities (4) also through the angles and velocities relative to the reference system:

\[ \dot{\alpha}=\dot{\alpha}^{p}_{p}+\dot{\beta}^{p}_{p}\gamma^{p}_{p} -\operatorname{tg}\beta\left[(\dot{\gamma}^{p}_{p}+\dot{\alpha}^{p}_{p}\cdot\beta^{p}_{p})\cos\alpha -(\dot{\beta}^{p}_{p}-\dot{\alpha}^{p}_{p}\gamma^{p}_{p})\sin\alpha\right] -\frac{1}{\cos\beta}(-\omega^{s}_{\eta_s}\cdot\sin\gamma+\omega^{s}_{\xi_s}\cdot\cos\gamma), \]

\[ \dot{\beta}=(\dot{\gamma}_{\mathrm{p}}^{\mathrm{p}}+\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot \beta_{\mathrm{p}}^{\mathrm{p}})\cdot \sin\alpha+ (\dot{\beta}_{\mathrm{p}}^{\mathrm{p}}-\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot \gamma_{\mathrm{p}}^{\mathrm{p}})\cos\alpha- \]
\[ -(\omega_{\eta c}^{c}\cdot \cos\gamma+\omega_{\xi c}^{c}\cdot \sin\gamma), \]
\[ \dot{\gamma}=\frac{1}{\cos\beta} \left[(\dot{\gamma}_{\mathrm{p}}^{\mathrm{p}}+\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot \beta_{\mathrm{p}}^{\mathrm{p}})\cos\alpha -(\dot{\beta}_{\mathrm{p}}^{\mathrm{p}}-\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\gamma_{\mathrm{p}}^{\mathrm{p}})\sin\alpha\right]+ \]
\[ +\operatorname{tg}\beta(-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma)-\omega_{\xi c}^{c}. \tag{11} \]

Following [3], let us introduce into consideration the angles of rotation \(\alpha^{\mathrm{p}}\), \(\beta^{\mathrm{p}}\), \(\gamma^{\mathrm{p}}\) and the velocities \(\dot{\alpha}^{\mathrm{p}}\), \(\dot{\beta}^{\mathrm{p}}\), \(\dot{\gamma}^{\mathrm{p}}\) of the platform about the suspension axes \(AA\), \(BB\), \(CC\), respectively.

The angles \(\alpha^{\mathrm{p}}\), \(\beta^{\mathrm{p}}\), \(\gamma^{\mathrm{p}}\) are small and represent the platform stabilization errors, measured about the suspension axes. The angular velocities of the platform about the suspension axes have the obvious values:

\[ \dot{\alpha}^{\mathrm{p}}=\omega_{z\mathrm{p}}^{\mathrm{p}}, \]
\[ \dot{\beta}^{\mathrm{p}}=\omega_{x\mathrm{p}}^{\mathrm{p}}\cdot \sin z+\omega_{y\mathrm{p}}^{\mathrm{p}}\cdot \cos z, \]
\[ \dot{\gamma}^{\mathrm{p}}=\frac{1}{\cos\beta}\left(\omega_{x\mathrm{p}}^{\mathrm{p}}\cdot \cos z-\omega_{y\mathrm{p}}^{\mathrm{p}}\cdot \sin z\right). \tag{12} \]

Whence

\[ \omega_{x\mathrm{p}}^{\mathrm{p}}=\dot{\gamma}^{\mathrm{p}}\cdot \cos\beta\cdot \cos z+\dot{\beta}^{\mathrm{p}}\cdot \sin z, \]
\[ \omega_{y\mathrm{p}}^{\mathrm{p}}=-\dot{\gamma}^{\mathrm{p}}\cdot \cos\beta\cdot \sin z+\dot{\beta}^{\mathrm{p}}\cdot \cos z, \]
\[ \omega_{z\mathrm{p}}^{\mathrm{p}}=\dot{\alpha}^{\mathrm{p}}. \tag{13} \]

Substituting the values (13) into (5) and (6), we obtain:

\[ \omega_{x\mathrm{p}}^{\mathrm{n}}=\dot{\gamma}^{\mathrm{p}}+\operatorname{tg}\beta(-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma), \]
\[ \omega_{y\mathrm{p}}^{\mathrm{n}}=\omega_{\eta c}^{c}\cdot \cos\gamma+\omega_{\xi c}^{c}\cdot \sin\gamma, \]
\[ \omega_{z\mathrm{p}}^{\mathrm{n}}=-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma. \tag{14} \]

\[ \omega_{x\mathrm{v}}^{\mathrm{v}}=\dot{\gamma}^{\mathrm{p}}\cdot \cos\beta, \]
\[ \omega_{y\mathrm{v}}^{\mathrm{v}}=\dot{\beta}^{\mathrm{p}}, \]
\[ \omega_{z\mathrm{v}}^{\mathrm{v}}=\dot{\gamma}^{\mathrm{p}}\cdot \sin\beta+ \frac{1}{\cos\beta}(-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma). \tag{15} \]

Expressions (13), (14), (15) represent the projections of the absolute angular velocities of the platform, the outer ring, and the inner ring onto their axes, expressed through the angular velocities of the platform about the suspension axes with allowance for the motion of the base.

The relative angular velocities (4) of the suspension rings, expressed through (13), have the values:

\[ \dot{\alpha}=\dot{\alpha}^{\mathrm{p}}-\dot{\gamma}^{\mathrm{p}}\cdot \sin\beta- \frac{1}{\cos\beta}(-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma), \]
\[ \dot{\beta}=\dot{\beta}^{\mathrm{p}}-(\omega_{\eta c}^{c}\cdot \cos\gamma+\omega_{\xi c}^{c}\cdot \sin\gamma), \]
\[ \dot{\gamma}=\dot{\gamma}^{\mathrm{p}}+\operatorname{tg}\beta(-\omega_{\eta c}^{c}\cdot \sin\gamma+\omega_{\xi c}^{c}\cdot \cos\gamma)-\omega_{\xi c}^{c}. \tag{16} \]

Substituting (8) into (12), we determine the relation between the coordinates \(\alpha^{\mathrm{p}}, \beta^{\mathrm{p}}, \gamma^{\mathrm{p}}, \alpha_{\mathrm{p}}^{\mathrm{p}}, \beta_{\mathrm{p}}^{\mathrm{p}}, \gamma_{\mathrm{p}}^{\mathrm{p}}\) and the velocities:

\[ \dot{\alpha}^{\mathrm{p}}=\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}+\dot{\beta}_{\mathrm{p}}^{\mathrm{p}}\cdot\gamma_{\mathrm{p}}^{\mathrm{p}}, \]

\[ \dot{\beta}^{\mathrm{p}}=(\dot{\gamma}_{\mathrm{p}}^{\mathrm{p}}+\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot\beta_{\mathrm{p}}^{\mathrm{p}})\sin\alpha+ (\dot{\beta}_{\mathrm{p}}^{\mathrm{p}}-\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot\gamma_{\mathrm{p}}^{\mathrm{p}})\cos\alpha, \]

\[ \dot{\gamma}^{\mathrm{p}}=(\dot{\gamma}_{\mathrm{p}}^{\mathrm{p}}+\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot\beta_{\mathrm{p}}^{\mathrm{p}})\frac{\cos\alpha}{\cos\beta} -(\dot{\beta}_{\mathrm{p}}^{\mathrm{p}}-\dot{\alpha}_{\mathrm{p}}^{\mathrm{p}}\cdot\gamma_{\mathrm{p}}^{\mathrm{p}})\cdot\frac{\sin\alpha}{\cos\beta}. \tag{17} \]

The relative velocities (4) can be represented in the form:

\[ \dot{\alpha}=\dot{\alpha}_{\mathrm{k}}^{\mathrm{p}}+\dot{\alpha}_{\mathrm{k}}^{\mathrm{c}}, \]

\[ \dot{\beta}=\dot{\beta}_{\mathrm{k}}^{\mathrm{p}}+\dot{\beta}_{\mathrm{k}}^{\mathrm{c}}, \tag{18} \]

\[ \dot{\gamma}=\dot{\gamma}_{\mathrm{k}}^{\mathrm{p}}+\dot{\gamma}_{\mathrm{k}}^{\mathrm{c}}, \]

where \(\dot{\alpha}_{\mathrm{k}}^{\mathrm{p}}, \dot{\beta}_{\mathrm{k}}^{\mathrm{p}}, \dot{\gamma}_{\mathrm{k}}^{\mathrm{p}}\) are the rotation velocities of the suspension gimbals caused by the rotation of the platform, while the components \(\dot{\alpha}_{\mathrm{k}}^{\mathrm{c}}, \dot{\beta}_{\mathrm{k}}^{\mathrm{c}}, \dot{\gamma}_{\mathrm{k}}^{\mathrm{c}}\) are caused by the rotation of the base.

As is evident from (18), in the case of motion of the base, the relative angles and angular velocities of the suspension gimbals do not reflect the platform-stabilization errors.

From (4) we obtain:

\[ \dot{\alpha}_{\mathrm{k}}^{\mathrm{p}}=\omega_{z\mathrm{p}}^{\mathrm{p}}-\operatorname{tg}\beta(\omega_{x\mathrm{p}}^{\mathrm{p}}\cdot\cos\alpha-\omega_{y\mathrm{p}}^{\mathrm{p}}\cdot\sin\alpha), \]

\[ \dot{\beta}_{\mathrm{k}}^{\mathrm{p}}=\omega_{x\mathrm{p}}^{\mathrm{p}}\cdot\sin\alpha+\omega_{y\mathrm{p}}^{\mathrm{p}}\cdot\cos\alpha, \]

\[ \dot{\gamma}_{\mathrm{k}}^{\mathrm{p}}=\frac{1}{\cos\beta}(\omega_{x\mathrm{p}}^{\mathrm{p}}\cdot\cos\alpha-\omega_{y\mathrm{p}}^{\mathrm{p}}\cdot\sin\alpha). \tag{19} \]

Substituting (12) into (19), we obtain:

\[ \dot{\alpha}_{\mathrm{k}}^{\mathrm{p}}=\dot{\alpha}^{\mathrm{p}}-\dot{\gamma}^{\mathrm{p}}\cdot\sin\beta, \]

\[ \dot{\beta}_{\mathrm{k}}^{\mathrm{p}}=\dot{\beta}^{\mathrm{p}}, \]

\[ \dot{\gamma}_{\mathrm{k}}^{\mathrm{p}}=\dot{\gamma}^{\mathrm{p}}. \tag{20} \]

For a fixed base, relations (18) take the form:

\[ \dot{\alpha}=\dot{\alpha}_{\mathrm{k}}^{\mathrm{p}}, \]

\[ \dot{\beta}=\dot{\beta}_{\mathrm{k}}^{\mathrm{p}}, \tag{21} \]

\[ \dot{\gamma}=\dot{\gamma}_{\mathrm{k}}^{\mathrm{p}}. \]

Equating (20) and (21), we find:

\[ \dot{\alpha}=\dot{\alpha}^{\mathrm{p}}-\dot{\gamma}^{\mathrm{p}}\cdot\sin\beta, \]

\[ \dot{\beta}=\dot{\beta}^{\mathrm{p}}, \]

\[ \dot{\gamma}=\dot{\gamma}^{\mathrm{p}}, \tag{22} \]

i.e., for a fixed base the relative rotation angles and angular velocities of the suspension gimbals are functions of the platform-stabilization errors, resolved about the suspension axes, and are determined by relations (22).

To determine the relation of the relative velocities, with a fixed base, to the stabilization errors relative to the reference coordinate system, the value (17) should be substituted into (22).

Conclusions

1. In studying the accuracy of operation of a spatial stabilizer, the stabilization errors of the platform are considered both relative to the stabilization axes or the axes of the reference system, and relative to the axes of the platform suspension. The relation between the stabilization errors in the first and second cases is determined by relations (17).

2. When the stabilization errors in the reference coordinate system are considered as independent variables, the equations of motion of the stabilizer should be formulated using the angular velocities determined by expressions (8), (9), (10), and (11).

3. When the stabilization errors relative to the axes of the platform suspension are considered as independent variables, the equations of motion of the stabilizer should be formulated using the angular velocities determined by expressions (13), (14), (15), and (16).

4. With a fixed base, the relative angles of rotation of the suspension rings may be used as independent variables in the equations of motion of the stabilizer, bearing in mind that their relation to the stabilization errors of the platform is determined by relations (22).

References

1. A. Yu. Ishlinskii. Mechanics of Gyroscopic Systems. Academy of Sciences of the USSR, 1963.

2. P. I. Saidov. Theory of Gyroscopes, Part 1. Vysshaya Shkola, 1965.

3. D. S. Pelpor. Theory of Gyroscopic Stabilizers. Mashinostroenie, 1965.

4. A. S. Ponyrko, L. A. Severov. On engineering methods for formulating the equations of motion of a spatial gyrostabilizer. Proceedings of LIAP, issue 49, 1966.