Microsoft Word - anton_sinelnikov,+ASSA+Word+Template_Sin (1) Adv Syst Sci Appl 2025; 1; 120-131 Published online at https://ijassa.ipu.ru. Analysis of Ring Laser Gyroscope Cavity Thermal Deformations Anton Sinelnikov* Patrice Lumumba RUDN University, Moscow, Russia Abstract: Ring laser gyroscopes are crucial for high-precision orientation, stabilization, and autonomous inertial navigation systems. However, their operational accuracy is significantly affected by temperature-induced cavity deformations, leading to potential performance degradation and data loss. This study addresses this challenge by developing a robust algorithm in the MATLAB environment designed to simulate the temperature deformations of a ring laser cavity. Leveraging experimental data, the methodology enables the precise modeling of cavity deformation behavior at various stages of the ring laser gyroscope assembly. The developed mathematical model effectively demonstrates how the addition of structural elements influences the cavity's thermal response and overall deformation. This research provides a novel approach to quantitatively estimate the contribution of individual structural components to the total thermal deformation. The findings offer critical insights for optimizing ring laser gyroscope design, thereby enhancing their accuracy, reliability, and resilience in diverse operational conditions. Keywords: inertial navigation systems, inertial measurement unit, ring laser gyroscope, cavity, control system, thermal deformations, MATLAB, mathematical model, simulation 1. INTRODUCTION Due to their high accuracy, reliability and resistance to external influences, ring laser gyroscopes (RLG) are actively used in high-precision systems for orientation, stabilization and navigation [1]. Today, they are employed in aircraft and spacecraft for calculating the angular rotation of a broad range [2]. In the case of inertial navigation guidance, the aerospace vehicles use onboard sensors to determine their motion and acceleration with the help of RLG or other gyroscopes and accelerometers [3]. A gyroscope is used to measure angular rotation, whereas an accelerometer is used to measure linear motion. The two are combined into a single unit along with a control mechanism, and the unit is called inertial measurement unit (IMU), or inertial navigation systems (INS). RLG, is an important optronic rate sensor and, in essence, the heart of any INS [4]. Based on application, RLG market can be split into platform stabilizations as well as aeronautic, missile, submarine and satellite types of navigation [5]. Based on the key end users, RLG commercial market can be divided into business jets, civil aircraft, helicopters and UAVs, general aviation, satellites, spacecraft, and rockets. According to the forecasts, the RLG market volume should exceed $1 billion by 2030 [4,5]. Today optical gyroscopes are one of the dominant technologies in the IMUs and INSs. They are divided into active RLGs and passive Fiber Optical Gyroscopes (FOG) [6]. They uses the Sagnac effect to calculate the rotation rate by measuring the phase shift of two counter propagating light beams in a passive interferometer and frequency difference in an active laser interferometer [7]. According to the Sagnac equation, frequency difference is obtained by fundamental formula (1.1): * Corresponding author: mr.sinelnikov.a@mail.ru ANALYSIS OF RING LASER GYROSCOPE CAVITY THERMAL DEFORMATIONS 121 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) Δf=4SΩ/λL (1.1) where Δf is the frequency difference between two counter-propagating light waves in the ring laser; S and L are the square and length of the ring laser cavity; Ω is the angular rate; λ is the wavelength of the laser. So, the ideal input-output characteristic of a RLG is described by a simple expression (1.2): Δf=KΩ (1.2) where K=4S/λL is called a scale factor. For a more compact form of this expression a scale factor K is often included into Ω, thereby making the direct relation between the RLG output signal and the measured angular rate more obvious. Let us consider the analytical review of the Yole Development company to assess the market of inertial navigation systems over the past 10 years [8,9]. The main competitive optical gyroscopes are Hemispherical Resonator Gyroscopes (HRGs), Micro-machined gyros (MEMS) and Dynamically Tuned Gyros (DTGs). Figure 1.1 presents ring diagrams of the evolution of high-end inertial technology market from 2011 to 2019 in monetary value. As can be seen from the left diagram in Figure 1.1, RLG occupied 40 percent of the inertial systems market in 2011. (a) (b) Fig. 1.1. Global inertial technology market evolution from 2011 (a) to 2019 (b) year It was about $510 million. The entire market was estimated at $1,3 billion. In 2019, the share of RLG increased to 52% as can be seen from the right diagram. This corresponds to $1,7 billion and a threefold increase in a cash volume. The share of FOG decreased by 3 percent, the share of MEMS increased by 5%, the share of HRGs increased by 5%, and the share of other gyros types amounted to 1 percent. So, considering the presented data, RLG technology is dominant in the inertial technology market. Figure 1.2 shows distribution diagrams of gyroscope types by accuracy class in 2011 and 2019. It can be seen that MEMS gyros consistently dominate in commercial and industrial navigation. Optical gyros are dominating in tactical navigation. The share of RLG in Midl- term navigation researches 61%. The use of RLG in high-end and strategic navigation is 78%. Give the conservatism of inertial navigation market, the dominance of RLG will continue in the next 10-20 years. 122 A. SINELNIKOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) (a) (b) Fig 1.2. Gyro technology breakdown per bias stability categories from 2011 (a) to 2019 (b) year There are top-3 types of the most common RLG high-end navigation accuracy class suitable for use in spacecraft among a variety of optical gyroscopes. They are shown in Figure 1.3. (a) (b) (c) Fig. 1.3. Most common types of RLG Commonly used ring laser cavity structure is triangular geometry, as shown in figure 1.3.a. This type of geometry is popular with RLGs from Honeywell [10]. Figure 1.3.b shows square ring laser geometry, which is one form of geometry in use [11]. These two types of RLG using forced dithering to overcome lock-in effect [12]. Figure 1.3.c shows non-polar square-type ring laser geometry that allows the using magneto-optical frequency shift based on Zeeman effect to overcome lock-in effect [13]. Due to the absence of dither, the Zeeman RLG has increased resistance to external mechanical influence [14,15]. This allows using the term in many areas of aerospace industry. However, this may be due to sudden changes in external temperature, which may have a negative impact on the Zeeman RLG operation [16]. The most important condition for achieving all types of RLG stable operation and high precision characteristics, including super-high stability of the scale factor K, is to ensure the constancy of the laser cavity so that the ring laser generates on a single mode [17,18]. The accuracy of timing and stabilizing the laser cavity to keep the generation frequency in the center of the gain curve should be less than a hundredth of die generation wavelength, i.e, units of nanometers (λ~ 1-10nm). Figure 1.4 illustrates active and passive compensation methods been used to ensure stable operation of the RLG under temperature influences. ANALYSIS OF RING LASER GYROSCOPE CAVITY THERMAL DEFORMATIONS 123 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) (a) (b) Fig. 1.4. Active and passive methods for the RLG cavity thermal stabilization The adjustment of the RLG to the working mode and the time of continuous operation in it is provided by the cavity control system (CCS), which is an active thermal compensation system [18, 19]. At the same time, a wide range of operating temperatures and heating of the Zeeman RLG during its operation impose significant restrictions on the dynamic range of the CCS, therefore, in addition to the active method, a passive method of thermal compensation is used [20]. To provide passive thermal compensation, the gyroscope cavity materials should have ultra-low thermal expansion coefficient (TEC) [21]. This materials include glass ceramics of the brands Zerodur, Clearceram, Cervit and SO-115M Sitall [22-24]. In particular, the housings of the ring laser cavity and mirror substrates in Russia are made of grade SO-115M Sitall. The use of this material in the manufacture of RLG structural elements is determined by a number of indisputable advantages, which include an ultra-low TEC, optical transparency and increased retention of the active medium inside the cavity [24]. The main disadvantage of this material is the non-linear nature of the TEC in the range of operating temperatures for measuring systems based on ring laser sensors, which limits the possibility of cavity perimeter active adjustment [25]. In addition to the cavity itself, there are other structural elements in the RLG. These include fasteners, piezoelectric actuators (executive elements of an active CCS), magnetic screens, etc. These structural elements are made of various materials, which have different TEC than glass- ceramic material, which directly affects the temperature stability of the ring laser cavity [26]. To varying degrees, this problem is inherent in all types of modern optical gyroscopes, and mathematical modeling methods are actively used to solve it and temperature correction [27- 29]. The purpose of this work is to develop an algorithm in the MATLAB environment, with which it is possible to simulate the Zeeman RLG cavity temperature deformations. To achieve this goal, the following tasks are solved in the work: • Obtaining initial data for modeling; • Development and implementation of the algorithm in the MATLAB environment; • Development of a temperature drift model for path length of a ring laser cavity. 2. THEORETICAL AND EXPERIMENTAL PART A sensitive RLG is a He-Ne laser with a non-planar optical cavity (Fig. 1.3.c) that generates laser beams with circular polarization at a wavelength of λ= 632,8 nm. This makes it possible to use a magneto-optical frequency bias based on the Zeeman effect to remove the RLG from the capture region and form a reference signal for the active CCS. The spectrum of natural 124 A. SINELNIKOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) frequencies of longitudinal modes of a non-planar four-mirror cavity Zeeman RLG is half the beams wavelength λ/2, which corresponds to 316,4 nm [30]. Two movable mirrors of the RLG resonator are equipped with piezoelectric actuators, which ensure the operation of the active CCS [31]. A part of the beam is put out through a semitransparent mirror to the mixing unit to form output signals and obtain information about rotation. The cavity is installed on the base, fixed with special fasteners and placed in magnetic screens. The CCS inputs receive a cavity detuning signal Ap from a special photodetector installed in the RLG, and a reference signal synchronous with the current of the alternating frequency bias. Cavity stabilization is carried out by integrating the signal rectified by a synchronous detector from the photodetector, followed by converting the output voltage of the integrator into a voltage proportional to it on the actuating piezoelectric actuators that move the movable mirrors in the direction of decreasing the error signal. From this it follows that the signal amplitude Ap is proportional to the cavity detuning in units of the generation frequency, which in turn is proportional to the temperature deformation of the ring laser cavity. Setting the path length to the center of the RLG amplification circuit in the open CCS mode corresponds to the minimum value of the cavity signal amplitude, and the detuning corresponds to the maximum. Thus, in the studied RLG, from the amplitude of the signal Ap, it is possible to estimate, with an accuracy of one fourth of the generation wavelength λ/4 the cavity temperature deformation and its direction from the change in the phase of the signal Ap. To build a model of the RLG cavity thermal deformations, studies of the ZLK-16 type device were carried out under long-term temperature effects according to the methodology presented in and obtained the necessary initial data [32]. Figure 2.1 demonstrates a general view of the experimental setup for obtaining initial data for computer modeling. Fig. 2.1. Experimental study and initial data It shows the dependence of the amplitude of the cavity detuning signal with an non-working CCS. At a linearly changing temperature T in the range from –55°С to 75°С at a rate of 1°/min inside the Heat and Cold Chamber, this characteristic reflects the sequential rearrangement of the RLG frequency spectrum relative to the center of the amplification circuit, caused by the cavity temperature deformation ΔL, nm. It is known from paper that the typical value of the path length drift ΔL for Zeeman RLG type ZLK-16 in the operating temperature range is 3λ-4λ (from 1900 to 2500 nm) and may depend on many factors: TEC of glass-ceramic SO-115M and other structural elements, the length of the cavity, the method of pumping the active medium, the magnitude of the working ANALYSIS OF RING LASER GYROSCOPE CAVITY THERMAL DEFORMATIONS 125 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) current of the discharge, and other factors [33,34]. Thus, having the time dependences of the signal Ap on temperature T as initial data, it becomes possible to create a model of the cavity temperature drift using a software method. 3. MATHLAB SIMULATION To build the model, a special algorithm was developed, which was implemented in the MATLAB environment. Figure 3.1 shows a block diagram of the modeling of ring laser cavity temperature deformations algorithm. Fig. 3.1. Block diagram of the algorithm Data from the text file WORK.txt obtained during the experimental study is written to the array "A" (3.1): A=dlmread('WORK.txt') (3.1) In the array "t" the 3rd column of the array "A" is written, containing the readings of the temperature sensor. The 2nd column of the array "A" is written to the "e" array, containing the amplitude values of the detuning signal (3.2): t=A(:,3) e=A(:,2) (3.2) The first empty intermediate arrays "e1" and "t1" are filled with values from the inflection point search algorithm. Next, the search for points of the array "e", the values of which are 126 A. SINELNIKOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) greater or less than the values of their previous and subsequent points, which are entered into the intermediate array, also giving these points the values of the array "t" (3.3): e1=[] t1=[] indx=1 for i = 2:size(e)-1 if ( ( e(i-1)< e(i))&& ( e(i+1)< e(i)))||( ( e(i-1) > e(i) ) && ( e(i+1)> e(i) ) ); e1(indx) = e(i); t1(indx) =t(i); indx = indx + 1; end (3.3) After that, the values of the "e1" array are replaced by their corresponding serial numbers and converted to nanometers (3.4): e1=1:size(e1’); e1=633*0.25*e1 (3.4) Figure 3.2 shows an intermediate graph of the dependence of the signal Ap on temperature T, containing the key points of "peaks" and "dips" necessary to build the final model of the temperature deformation of the RLG cavity. On the intermediate graph (Fig. 3.1) the number of the inflection point "ind" is selected, namely the number 9. This is the point at which the phase of the signal Ap changes. Fig. 3.2. Intermediate chart corresponding to the step of determining the Extremes of the cavity signal amplitude From the values of the points of the "e1" array, the value of the magnitude of the amplitude change at the point of change in the direction of deformation is subtracted, which allows you to move the graph to zero coordinates at the inflection point (3.5): ind=9 (3.5) ANALYSIS OF RING LASER GYROSCOPE CAVITY THERMAL DEFORMATIONS 127 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) ind=9 e1=e1-e1(ind) Next, the values "e1" and "t1" from the cycle are entered into the empty intermediate arrays "ex" and "te", which reflects the values of the change up to the inflection point relative to the abscissa axis (3.6): ex=[] te=[] i1=1 for k=1:size(e1') if k=ind ex(i1)=e1(k)' te(i1)=t1(k)' i1=i1+1 end (3.6) Based on the values from the "ex" and "te" arrays, a final graph of the dependence of the absolute change in the perimeter ΔL on temperature t is constructed. Figure 3.3 shows the temperature deformation model of the RLG resonator obtained in the MATLAB environment, which reflects the dependence of the absolute temperature change in the path length. Fig. 3.3. Model of the ring laser cavity temperature drift Having an idea about the nature of the path length temperature drift, it becomes possible to estimate the contribution of all structural elements of the RLG, which has TEC different from the glass-ceramic SO-115M, to the resulting temperature deformation of its cavity ΔL. 128 A. SINELNIKOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) 4. DISCUSSION OF THE RESULTS Based on the initial data and thanks to the algorithm implemented in the MATLAB environment, it becomes possible to simulate and obtain the form of the dependence of the cavity temperature deformations at different stages of the RLG assembly, which will be described by expression (4.1): ΔL(T)=a·T3+ b·T2+c·T+ d (4.1) where, ΔL is the value of thermal deformation, nm; T is the temperature value, °C; a, b, c, d are the polynomial coefficients determined by the design parameters of the RLG. In the present model (4.1), the coefficient a characterizes the nonlinear TEC of the glass-ceramic SO-115M in temperature range from –55°С to 75°С. The coefficient b takes into account the contribution of piezo actuators and c determines the influence to other RLG structural elements. Figure 4.1 (Approximation of the ring laser cavity temperature drift) presents a graph of an approximating function of the type and its formula obtained using Microsoft Excel. Fig 4.1. Approximation of the ring laser cavity temperature drift It has a confidence level R=0,9932, which indicates the reliability of the obtained functional dependence. This opens up the possibility of using formula (4.1) to calculate the RLG cavity drift in the operating temperature range of the spacecraft, as well as to use it as a governing law for the CCS active stabilization (4.2). Uccs(t) =Up·ΔL(T(t)) (4.2) where, Uccs is the CCS control voltage, V; t is the operating time, sec, °C; Up·is the piezo actuators transmission coefficient, V/nm. 5. CONCLUSION The presented mathematical model demonstrates the behavior of the temperature deformation of the RLG cavity at different stages of its generation. When adding new structural elements, a shift of the inflection point and a change in the resulting cavity deformation are observed. ANALYSIS OF RING LASER GYROSCOPE CAVITY THERMAL DEFORMATIONS 129 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) This makes it possible to estimate the contribution of each structural element to the resulting thermal deformation of the ring laser cavity. The next step in this work will be to check the operability of the approximating function as the governing law of the active CCS. 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