Accepted: July 14, 2022 | Published Online: July 16, 2022

Citation: Trung VA (2022) Autonomous Reduced Gyro Platform Drift Identifier. Int J Astronaut Aeronautical Eng 7:060

Copyright: © 2022 Trung VA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Abstract

The article considers a method of leaning the asymptomatic identifier (observation) of the drift-gyro platform with the minimum order using the internal generalized measurement information. The conditions for the asymptomatic stability of the identification process are determined and the structure of the perturbed identifier matrices is established. A simulation given shows the effectiveness in use of the identifier for estimating and compensating for the drift of the gyronamform while it is operating under the conditions of the action of uncertain (off-design) disturbances.

Keywords

Gyro, Drift identifier, Kalman filter method, Measuring system, High-dimensional filtering algorithms

Introduction

The accuracy of the angular stabilization of the gyronam form depends significantly on the nature and the level of air moments acting on the platforms and gyroscopes in the operating conditions of moving objects. High stabilization accuracy is ensured by the development of appropriate design and technological measures and the use of various methods of algorithmic compensation for the influence of disturbing moments. At the same time, significant efforts are spent on compensating for the influence of constant and slowly changing component moments relative to the gyroscopes precession axes, which ultimately lead to errors in object control.

Algorithmic compensation is based on various methods of determining the expected drift of the gyronam form under operating conditions. The implementation of these methods, as a rule, is associated with the use of a priori information about the nature and the level of uncertain disturbances. Several available statistical methods of identification (estimation) have been applied gyroscopy to deal with this problem. However the effectiveness of those applications largely depends on the ability to receive adequate information on the operating disturbances in the models of measuring systems. Furthermore, the implementation of identification algorithms is so complex. When using statistical methods for identifying system states (for example, optimal Kalman filter method and its modifications) [1], there are some difficulties arise in obtaining static characteristics of the disturbances occuring during operation, including their complex modelings, the inconveniences associated with the formation of a measuring system and high-dimensional filtering algorithms.

Autonomous measuring system based on a gyrostabilizer

We are more interested in the possibilities of developing simpler algorithms which can autonomously identify the gyroplatform drift operating under the conditions of the action of disturbances occuring with indefinite (off-design) characteristics. Indeed, under real operating conditions, the statistical characteristics of uncertain disturbances and their levels cannot always be reliably determined. Moreover, it is usually possible that some kinds of disturbances may not have obvious statistical properties, for example, when operating under difficult conditions in relationship with short time intervals. In these cases, the appearance of an additional (off-design) gyro platform drift should be considered. In fact, the possibility of using state identifiers for estimating such kind of drift by autonomous means (i.e., using only the current internal information of the output signals measured in the gyro platform stabilization system) has not yet been sufficiently studied.

There is a known method for constructing a state identifier based on the wave representation of uncertain disturbances, which does not require knowledge of their statistical characteristics. However, in this case, to model the disturbances, it is necessary to experimentally determine the form of the basis functions, which additionally leads to an increase in the dimension of the problem.

In article [2,3] an attempt is made to use the autonomous asymptotic identifier (observer) of the states to estimate the drift of the gyro platform under the action of stepwise and slowly changing moments. The possibility of estimating the drift with an acceptable accuracy for one of the channels of the stabilization system is shown. The difficulty in constructing the identifier consisted in the choice of its optimal parameters, which simultaneously provide a high estimation accuracy and an acceptable quality of the identification process with a high dimension of the problem.

As studies have shown, these difficulties can be overcome by building a modified identification system of the minimum dimensions. When developing a model of an autonomous measuring system based on a gyrostabilizer, we proceed from the possibility of using information about the nature of the motion of gyroscopes relative to the axis of precession. In order to simplify the problem, from the stabilization system of a triaxial gyro stabilizer with a classical (orthogonal) arrangement of gyroscopes on the platform, we will single out two interconnected (in terms of gyroscopic moment) channels (P and B), independent of the third channel.

The dimensions of the measuring system in this case can be reduced as much as possible if the order of its equations of state is made equal to two. For this purpose, we first transform the initial equations of moments with respect to the axes of stabilization and the procession and then represent them in the following vector-matrix form [4,5]:

$$\dot{\omega }\text{ = }{A}_{11}\omega +m_{\alpha }+I_{\alpha }{M}_{\alpha }\text{ (1)}$$

$$\dot{\omega }\text{ = }{B}_{11}\omega +m_{\beta }+I_{\beta }{M}_{\beta }\text{ (2)}$$

$$\omega \text{ = }\left[\begin{array}{c}{\omega }_P\\ {\omega }_B\end{array}\right];A_{11}\text{ = }\left[\begin{array}{cc}0& H_P{I}_X^{-1}\\ -H_B{I}_Y^{-1}& 0\end{array}\right];B_{11}\text{ = }\left[\begin{array}{cc}0& H_B{I}_B^{-1}\\ -H_P{I}_P^{-1}& 0\end{array}\right];I_{\alpha }\text{ = }\left[\begin{array}{cc}{I}_X^{-1}& 0\\ 0& I_Y^{-1}\end{array}\right];$$

$$I_{\beta }\text{ = }\left[\begin{array}{cc}0& I_B^{-1}\\ I_P^{-1}& 0\end{array}\right];M_{\alpha }\text{ = }\left[\begin{array}{c}{M}_{\alpha P}\\ M_{\alpha B}\end{array}\right];M_{\beta }\text{ = }\left[\begin{array}{c}{M}_{\beta P}\\ M_{\beta B}\end{array}\right];m_{\alpha }\text{ = }\left[\begin{array}{c}{m}_{\alpha P}\\ m_{\alpha B}\end{array}\right]m_{\beta }\text{ = }\left[\begin{array}{c}{m}_{\beta P}\\ m_{\beta B}\end{array}\right]$$

$$m_{\alpha P}\text{ = }{I}_X^{-1}\left[{\lambda }_P{K}_{P1}{\beta }_p+\left(H_p+{\lambda }_P{K}_{P2}\right){\dot{\beta }}_p\right]$$

$$m_{\alpha B}\text{ = }{I}_Y^{-1}\left[{\lambda }_B{K}_{B1}{\beta }_B+\left(-H_B+{\lambda }_B{K}_{B2}\right){\dot{\beta }}_B\right]$$

$$m_{\beta P}\text{ = }{\beta }_P\ddot{+}{d}_p{I}_p^{-1}+C_P{I}_p^{-1}{\beta }_p$$

$$m_{\beta B}\text{ = }{\beta }_B\ddot{+}{d}_B{I}_B^{-1}+C_B{I}_B^{-1}{\beta }_B$$

Here, ωР, ωВ - angular velocities of the gyro platform drift. βР, βВ - gyro precession angles. IX, IY - moments of inertia of the platform relative to the stabilization axes. IP, IB - the same gyroscopes relative to the axes of the procession. HP, HB - kinetic moments of gyroscopes. MαP, MαB-disturbing moments with respect to the stabilization axes. MβP, MβB - the same with respect to the axes of the procession. dP, dB - damping coefficients. CP, CB - cruelty coefficients. KP1, KP2, KB1, KB2 - amplification factors of signals representing, respectively, the precession angles and their derivatives in the channels Р и В; λP, λB - overall gains for channels P and B, relative to.

The representation of the motion equations in the form of (1) and (2) makes easier to obtain a model of the measuring system and the drift identifier. Terms of mα and mβ characterize a known part of the constituent moments acting on the platform and gyroscopes and depending on the parameters of the gyroscopes movement ( ). In principle, they can be determined from the results of measurements and used as measuring information when constructing the identifier. In what follows, without reducing the generality of the problem statement, we assume that the errors in determining these terms are insignificant.

From equ. (1) and (2), the equations of the measuring system are obtained in the following form

$$\left\{\begin{array}{c}\dot{\omega }\text{ = }{A}_1\omega +0,5\left(m_{\alpha }-m_{\beta }\right)+0,5\left(I_{\alpha }{M}_{\alpha }+I_{\beta }{m}_{\beta }\right)\\ z\text{ = }{C}_1\omega +\left(I_{\alpha }{M}_{\alpha }-I_{\beta }{m}_{\beta }\right)\end{array}\right.$$

Here, $z\text{ = }-\left(m_{\alpha }+m_{\beta }\right);A_1\text{ = 0,5}\left(A_{11}+B_{11}\right);C_1\text{ = 0,5}\left(A_{11}-B_{11}\right)$. These equations can be written in both the usual and standard forms

$$\dot{\omega }\text{ = }{A}_1\omega +BU+G{M}_B\text{ (3)}$$

$$z\text{ = }{C}_1\omega +D{M}_B\text{ (4)}$$

Here,

$$U\text{ = }{\left[\begin{array}{cccc}{\dot{\beta }}_P& {\dot{\beta }}_B& {\beta }_P& {\beta }_B\end{array}\right]}^T;M_B\text{ = }{\left[\begin{array}{cccc}{M}_{\alpha P}& M_{\alpha B}& M_{\beta P}& M_{\beta B}\end{array}\right]}^T;B\text{ = }\left[\begin{array}{c}{b}_{11}{b}_{12}{b}_{13}{b}_{14}\\ b_{21}{b}_{22}{b}_{23}{b}_{24}\end{array}\right];$$

$$\text{G = }\left[\begin{array}{cccc}0,5I_x^{-1}& \text{0}& \text{0}& \text{0,5}{I}_B^{-1}\\ 0& \text{0,5}{I}_Y^{-1}& \text{0,5}{I}_P^{-1}& \text{0}\end{array}\right];\text{ D = }\left[\begin{array}{cccc}{I}_x^{-1}& \text{0}& \text{0}& -I_B^{-1}\\ 0& I_Y^{-1}& -I_P^{-1}& \text{0}\end{array}\right];$$

$$b_{11}\text{ = 0,5}\left(H_P+{\lambda }_P{K}_{P2}\right)I_x^{-1};b_{12}\text{ = -0,5}{d}_B{I}_B^{-1};b_{13}\text{ = 0,5}{\lambda }_B{K}_{B1}{I}_x^{-1};b_{14}\text{ = -0,5}{C}_B{I}_B^{-1};$$

$$b_{21}\text{ = -0,5}{d}_P{I}_P^{-1};b_{22}\text{ = -0,5}\left(H_B-{\lambda }_B{K}_{B2}\right)I_Y^{-1};b_{23}\text{ = -0,5}{C}_P{I}_P^{-1};b_{24}\text{ = 0,5}{\lambda }_B{K}_{B1}{I}_Y^{-1}$$

Vector z represents generalized measurement information (observation). Uncertain moments MB and known controls U act at the inputs of the measuring system. Disturbances MB are also directly included in the observations z. In the measuring system (3), (4), information on the second derivatives of the procession angles is not used, since it does not significantly affect the identification accuracy.

Before constructing the identifier, we determine the observability conditions of the measuring system. Anunperturbed systemin the form of $\left\{\begin{array}{c}\dot{\omega }\text{ = }{A}_1\omega \\ z\text{ = }{C}_1\omega \end{array}\right.$ can be fully observable if the rank of the observability matrix $N\text{ = }\left[\begin{array}{c}{C}_1\\ C_1{A}_1\end{array}\right]$ is equal to two. Indeed, for the usual relations for a gyrostabilizer HP = HB = H; IP = IB, Ix≠IY determinant of a matrix C1 not equal to zero. Consequently, on the basis of the measuring system (3), (4), it is possible to construct an identifier (observer) that ensures the asymptotic stability of the identification process.

$$\dot{\stackrel{⌢}{\omega }}\text{ = }A\stackrel{⌢}{\omega }+BU+Kz\text{ (5)}$$

Here $\stackrel{⌢}{\omega }$ – vector of estimates of the gyro platform drift velocity; K-identifier coefficient matrix $K\text{ = }\left[\begin{array}{cc}{k}_{11}& k_{12}\\ k_{21}& k_{22}\end{array}\right]$;

A = A1-KC1

The identifier equation can be written in a more compact form

$$\dot{\stackrel{⌢}{\omega }}\text{ = }A\stackrel{⌢}{\omega }+B_{0}U\text{ (6)}$$

$$B\text{ = }\left[\begin{array}{ccc}{b}_{11}+c_{11}& b_{12}+c_{12}{b}_{13}+c_{13}& b_{14}+c_{14}\\ b_{21}+c_{21}& b_{22}+c_{22}{b}_{23}+c_{23}& b_{24}+c_{24}\end{array}\right]$$

$$c_{11}\text{ = }{k}_{11}{d}_{11}+k_{12}{d}_{21},c_{12}\text{ = }{k}_{11}{d}_{12}+k_{12}{d}_{22},c_{13}\text{ = }{k}_{11}{d}_{13}+k_{12}{d}_{23},c_{14}\text{ = }{k}_{11}{d}_{14}+k_{12}{d}_{24},$$

$$c_{21}\text{ = }{k}_{21}{d}_{11}+k_{22}{d}_{21},c_{22}\text{ = }{k}_{21}{d}_{12}+k_{22}{d}_{22},c_{23}\text{ = }{k}_{21}{d}_{13}+k_{22}{d}_{23},c_{24}\text{ = }{k}_{21}{d}_{14}+k_{22}{d}_{24},$$

$$d_{11}\text{ = }-\left(H_P+{\lambda }_P{K}_{P2}\right)I_X^{-1},d_{12}\text{ = }-d_B{I}_B^{-1},d_{13}\text{ = }-{\lambda }_P{K}_{P1}{I}_X^{-1},d_{14}\text{ = }-C_B{I}_B^{-1},d_{21}\text{ = }-d_p{I}_P^{-1},$$

$$d_{22}\text{ = }\left(H_B-{\lambda }_B{K}_{B2}\right)I_Y^{-1},d_{23}\text{ = }-C_P{I}_P^{-1},d_{24}\text{ = }-{\lambda }_B{K}_{B1}{I}_Y^{-1}$$

Under the action of disturbing moments along the axes of stabilization and procession, the estimation will be carried out with errors $\overline{\omega }\text{ = }\omega -\stackrel{⌢}{\omega }$ in so far as $\dot{\overline{\omega }}\text{ = }\dot{\omega }-\dot{\stackrel{⌢}{\omega }}$ then taking into account (3) - (5), we can obtain the equation $\dot{\overline{\omega }}\text{ = }A\overline{\omega }+D_0{M}_B$ for estimation errors, where

$$D_0\text{ = }\left[\begin{array}{cccc}\left(0.5-k_{11}\right)I_X^{-1}& k_{21}{I}_Y^{-1}& k_{12}{I}_P^{-1}& \left(0,5+k_{11}\right)I_B^{-1}\\ -k_{21}{I}_X^{-1}& \left(0,5-k_{22}\right)I_Y^{-1}& \left(0,5+k_{22}\right)I_P^{-1}& k_{21}{I}_B^{-1}\end{array}\right]$$

If the elements of the mairitsa K are chosen so that the asymptotic stability of the identification process is ensured, then ${\overrightarrow{\omega }}^{\ast }\to 0$ and in steady state

The minimum order of the obtained reduced identifier makes it possible to analyze its properties by not only modeling on a computer, but also by establishing analytical dependencies. Small dimensions of matrix K and rather simple structure of matrices D0 and A-1D0 facilitates the selection of optimal values of the coefficients kij identifier providing minimum (or practically acceptable) values of estimation errors ${\overrightarrow{\omega }}^{\ast }$, as well as acceptable quality of transients in the identifier.

In order to determine the relationship between the coefficients of the identifier and the parameters of the gyro platform, ensuring its stable operation, we find the conditions of asymptotic stability for the reduced identifier (6) with the matrix

$$A\text{ = }\left[\begin{array}{cc}{a}_1{k}_{12}& a_2+a_3{k}_{11}\\ a_4+a_5{k}_{22}& a_6{k}_{21}\end{array}\right]$$

$$a_1\text{ = }-\left(H_P{I}_P^{-1}-H_B{I}_Y^{-1}\right),a_2\text{ = 0}\text{.5}\left(H_P{I}_X^{-1}+H_B{I}_B^{-1}\right),a_3\text{ = }-\left(H_P{I}_X^{-1}-H_B{I}_B^{-1}\right),$$

$$a_4\text{ = }-\text{0}\text{.5}\left(H_P{I}_P^{-1}+H_B{I}_Y^{-1}\right),a_5\text{ = }-\left(H_P{I}_P^{-1}-H_B{I}_Y^{-1}\right)a_6\text{ = }-\left(H_P{I}_X^{-1}-H_B{I}_B^{-1}\right)$$

For asymptotic stability, it is necessary and sufficient that

(a1k12 + a6k21) < 0 and [a1a6k12k21-(a2 + a3k11)(a4 + a5k22)] > 0 (7)

With the relations usual for the gyro platform between the design parameters from (7), it is easy to establish a simpler stability condition k12 > k21

Based on the analysis of the matrix structure A-1D0 and the obtained ratios, as well as by joint computer simulation of the equations of motion of the gyro platform and the reduced identifier, the acceptable values of the identifier coefficients were determined, which ensure high accuracy in estimating the components of the gyro platform drift when simulating rather complex operating conditions. At the same time, it was found that one identifier can be most efficiently used to estimate the drift along only one of the two channels. Obviously, with the help of two identifiers with different settings of the coefficients, it is possible with high accuracy and simultaneously to estimate the drift of the gyro platform in two channels, as well as to construct an appropriate compensation system for the action of undefined disturbances.

Some of the simulation results related to the P channel are shown in Figure 1, Figure 2 and Figure 3. Figure 1 and Figure 2 show the perturbing moments set during the simulation along the stabilization axes of the gyro platform and the gyroscope procession. At the same time, simultaneously include constant, time-varying and random components. The nature of these moments was chosen quite arbitrarily and had to simulate the situation with abrupt changes in perturbations acting on the gyro stabilizer that arose in real conditions. Figure 3 shows the estimation process in the situation of the gyro platform drift with the optimally selected identifier coefficients k11= -0,5008, k12 = 0, k21 = -0,2, k22 = -0,501 HP = HB = H = 500 [g.cm.s]. Analysis of the simulation results for various variants of the action of disturbing moments along the precession and stabilization axes shows that at identification time intervals of up to 100 s, the errors in estimating the gyro platform drift do not exceed 1%.

Conclusions

The results of the study show that on the basis of the developed reduced identifier of the minimum order, it is possible to design an effective autonomous system for assessing and compensating for the drift of the gyro platform that occurs in difficult operating conditions. Proposals, the method of constructing a mathematical model of the measuring system and the identifier of the minimum order made it possible to obtain analytical expressions for the condition of asymptotic stability of the identifier, as well as to significantly simplify the analysis of the structure of matrices and the choice of optimal parameters for the identifier.

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