Autonomous Strapdown Stellar-Inertial Navigation Systems: Design Principles, Operating Modes and Operational Experience

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1 ISSN , Gyroscopy and Navigation, 213, Vol. 4, No. 4, pp Pleiades Publishing, Ltd., 213. Published in Russian in Giroskopiya i Navigatsiya, 213, No. 3, pp Autonomous Strapdown Stellar-Inertial Navigation Systems: Design Principles, Operating Modes and Operational Experience G. A. Avanesov a, R. V. Bessonov a, A. N. Kurkina a, M. B. Lyudomirskii b, I. S. Kayutin b, and N. E. Yamshchikov b a Space Research Institute of Russian Academy of Sciences, Moscow, Russia b Electrooptics, Moscow, Russia Received March 4, 213 Abstract The paper considers stellar-inertial navigation systems (Stellar INS for airborne and ground applications based on strapdown star tracker, strapdown inertial navigation system (SINS and GNSS receiver. Specifications for optical and electronic components of star tracker are defined. Results from ground and onboard tests of Stellar INS model are provided. Various options of SINS stellar corrections are discussed. DOI: /S INTRODUCTION Some applications require high precision navigational measurements to be done by fully autonomous onboard instruments. Traditionally inertial navigation systems are applied, which comprise the sensors determining all attitude and navigation parameters by measuring angle and linear velocity increments. However, even the most accurate INS accumulates trajectory measurement errors with time. Therefore, alternative aids measuring the directions to natural astronomical bodies like stars are needed for INS autonomous correction. The main challenge in development of these instruments is stellar fixing, especially in daylight. Earlier, round-the-clock star observation for navigation purposes was performed by aiming the Stellar INS optoelectronic tools at the brightest stars and long-term holding them using gyrostabilized platforms [1, 2, 3]. Airborne Stellar-Inertial Navigation System LN-12G by Northrop Grumman Corp. (USA is the most advanced system of this type, declared to be the most accurate heading source in the world. LN-12G contains high precision laser gyros, quartz accelerometers, GPS receiver, and star tracker with mechanical driving gear providing aiming at 57 brightest stars (up to second stellar magnitude. To reduce dimensions and cost of Stellar INS systems and enhance their reliability and accuracy, a new system without precision mechanical guidance device and gyrostabilized platform should be developed [2, 4]. Strapdown Stellar INS without a guidance device needs to observe groups of 2 3 stars in system field of view (FOV at any orientation, therefore it requires a star tracker able to distinguish much dimmer stars. The required star tracker sensitivity vs. its FOV is shown in Fig. 1. It has been calculated using the star catalogue. From the Figure, the star tracker detects the stars up to 5th stellar magnitude even with the widest FOV (3 deg. Thus, with no mechanical guidance device, stars 15 3 times less bright than the second stellar magnitude stars should be detected. Besides, replacing narrow-field star tracker with a wide-angle strapdown one brings technological problems associated both with optics and sensitive elements. This paper discusses the design philosophy of a strapdown airborne Stellar INS requiring no guidance. Optoelectronic star tracker in strapdown Stellar INS observes the group of stars or the Sun, identifies them in onboard star catalogue, and determines angular orientation at min 1 km altitude at any time. Stellar correction is possible at lower altitudes as well if astronomical bodies are directly visible in cloudless sky. Strapdown Stellar INS comprises a star tracker and a SINS based on high precision laser gyros, quartz pendulum accelerometers, and embedded GLO- NASS/GPS receiver. The star tracker features two Stellar magnitude FOV 2α, deg Fig. 1. Star tracker desired sensitivity vs. FOV. 24

2 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 25 Fig. 2. 3D model of strapdown Stellar INS. optoelectronic channels for star and Sun observations. Stellar INS automatically calculates the vehicle attitude and position. SINS provides high-frequency jamproof attitude and navigation solution and generates a priori data for the star tracker. The star tracker performs periodic SINS stellar corrections, calculating the three-axis attitude during constellation observation or 2 angles of direction to the Sun if the star channel is flared. 3D model of Stellar INS is shown in Fig. 2. The solar channel of the star tracker has ±5 deg FOV, and the star channel is flared at 4 deg angle between the Sun and the star tracker optical axis, so both channels can function simultaneously at certain intervals. At night and in daytime when Sun angles are more than 4 deg the star tracker observes the group of stars and determines the attitude. Otherwise, only solar channel is functioning. Development of strapdown Stellar INS presents a number of challenges. The most difficult task is to create a celestial navigation instrument able to observe and automatically detect group of stars in its FOV both in night and daytime with small Sun angles. Attitude determination using the known fixed stars is a wellstudied task: experience in development and operation of celestial navigation instruments for space applications is abundant [5, 6]. The next step is the integration of the star tracker with navigation-grade SINS, and realization of SINS stellar correction algorithms and SINS aiding for star tracker functioning. Additionally, the solar sensor should be designed, able to determine the direction to the Sun in wide FOV, and solar corrections with account for the Sun ephemerises should be developed. SELECTION OF THE KEY PARAMETERS OF STAR TRACKER Possibility to fix the stars on the sky image and accuracy of attitude determination using these images depend on the signal to noise ratio (SNR of the star signal. Background noise is the sum of electronic and heat noises of video channel, photon noise from side solar flare (after dampening by the lens hood and photon noise caused by diffused light in atmosphere column in star tracker FOV. Electronic noise in modern optoelectronics is units and tens of electrons, heat noise is removed by cooling of CCD matrix, and side flare is eliminated by the lens hood (protection factor is defined by its size, so diffused light in atmosphere column in the device FOV turns to be the major source of noise. From the fundamental characteristics of light, the RMS photon noise can be considered equal to square root of the background level Q P expressed in photoelectrons. To fix a star on the image, signal level Q S in the brightest pixel of star image should be 3 5 times as much as noise RMS, i.e. SNR should be more than 1: QS SNR =, (1 3 QP where Q S is the quantity of photoelectrons in the star brightest pixel; Q P is the quantity of photoelectrons accumulated by CCD matrix because of background radiation. From (1, SNR grows proportionally to square root of exposure time, i.e. the more photoelectrons are taken by the star tracker camera, the larger is SNR for the stars on the frame. Exposure in its turn is limited by potential electron well of CCD, signal accumulation time, objective lens aperture, CCD quantum sensitivity, and optical system bandwidth. It should be noted that the signal providing proper SNR can be accumulated in analog form in CCD potential electron well, or using digital combination of frame series in processing module. Angular resolution of photocamera is one of the main parameters defining the star tracker ability to observe the stars against bright background. Increasing the angular resolution reduces background influence GYROSCOPY AND NAVIGATION Vol. 4 No

3 26 AVANESOV et al. Signal level, relative units Star tracker FOV 2α, deg Fig. 3. Useful signal from fixed star vs. star tracker FOV. Spectral luminosity, photon/cm 2 s Å Spectral class Κ Spectral class Α λ, Å Fig. 4. Spectral luminosity of zero magnitude stars. Spectral brightness, photon/cm 2 s Å s 4e e e e λ, Å Fig. 5. Spectral brightness of atmosphere at 1 km altitude and 4 deg Sun angle. proportionally to square root of bin angular size. Then, saving the ability of optical system to collect the light from star to one pixel, SNR can be significantly increased. It should be noted that angular resolution depends both on CCD pixel size and focal length, and on objective lens ability to focus the energy from point light source. With limited angular resolution of objective lens, decreasing the pixel size becomes ineffective and barely improves angular resolution of the whole optoelectronic system. The other equally important parameter of the star tracker is its field of view. Obviously, the wider FOV is, the brighter stars can be fixed (see Fig. 3. Widening FOV from 8 to 3 deg increases the useful signal by an order of magnitude. Star tracker operation in infrared and red spectrum can also increase SNR. Spectral luminosities of zero magnitude stars and spectral brightness of atmosphere are presented in Figs. 4 and 5. From Fig. 5, the spectral brightness of atmosphere decreases dramatically as wave length increases. To reduce the background signal, atmosphere maximum spectrum should be removed from optical channel sensitivity spectrum by using the red optical filter. For example, KS14 filter cutting the spectrum region below 6 Å increases SNR by 3.2 times for red K stars and by 2 times for blue A stars. Then exposure time should be increased by 4.2 times to receive the same background level on the frames. Using a 9 Å filter increases SNR ninefold for red stars and fivefold for blue stars, but requires a 66 times longer accumulation time. It should be noted that signal accumulation time strongly depends on the stationarity of star light on CCD matrix during the exposure in conditions of vehicle angular motions. Figure 6 shows the measured aircraft attitude angles during its motion in autopilot mode. Angular rate can be as high as 1 deg/s at some flight segments. With such a high angular rate, the star image can spread for tens of pixels, which is unallowable in daytime survey. Therefore, aircraft angular disturbances critically restrict the image accumulation time and possibility of reducing background radiation with an optical filter. The exposure time is also associated with lens aperture: doubled lens aperture reduces exposure time by 4 times. Focal length affects nearly all optimization conditions, and its choice brings controversial results. Increased focal length, on the one side, enhances angular resolution, which reduces background noise in the pixels, and on the other side, narrows the star tracker FOV, which necessitates registration of signals from much dimmer stars. Figure 7 shows how SNR changes depending on the focal length. From the figure, with the focal length increased from 5 to 5 mm, SNR increased 2.2 times only. It caused higher angular resolution and, as a consequence, higher vibration sensitivity (star image blurring. Besides, it is technologically difficult to provide sharp one pixel focusing for long-focus objectives with a wide FOV. Thus, increased focal length can degrade SNR even worse in some cases. To sum up, the star tracker photocamera should feature big CCD matrix providing wide FOV and small pixel providing high angular resolution. CCD matrix should have high quantum sensitivity in IR- and red spectrum and big potential electron well to accumulate sufficient quantity of photoelectrons. Objective lens of photocamera should be of diffraction quality and offer utmost performance as regards angular reso- GYROSCOPY AND NAVIGATION Vol. 4 No

4 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 27 Roll, deg Yawing, deg Pitch, deg Fig. 6. Aircraft angular disturbances. 964 Time, s lution and aberrations throughout the FOV. Big objective lens aperture guarantees accumulation of sufficient quantity of light during short exposures. Red filter which excludes blue, green and yellow regions of radiation spectrum is used in the optical system. Period of video data accumulation in star tracker should be chosen with consideration of aircraft angular motion and shutter operation. To optimize the star tracker design, its key components: CCD matrix, objective lens, optical filter, and shutter were selected according to the above criteria. We considered the most advanced candidates and performed detailed photometric calculations taking into account the spectral brightness of atmosphere in daytime, star spectral luminosity and star distribution in the sky, and spectral characteristics of CCD matrix and objectives. Besides, we considered the available function of objective pixel blurring limited by diffraction limit. Calculations reveal that none of modern CCD or CMOS matrices can provide observation of group of stars in any part of firmament in daylight conditions on one frame: in least star populated areas, the star tracker is unable to extract the useful star signal against photon noise produced by diffused light in atmosphere column in star tracker FOV with small Sun angles. Moreover, such high sensitive elements and objectives are not likely to appear in the foreseeable future, or these will very large scale systems. As an example we can take the Kepler space telescope with focal length over 1 m, relative aperture 1/1.2, and 42 CCD matrices with total size of 25 mm mounted in the lens focal plane. Such a large sensitive plane jointly with a Schmidt camera provides a 12 deg FOV. If the Kepler telescope was used not for its designated purpose but to observe a group of stars in daytime, SNR would not exceed 1 for the least star populated sky areas. This SNR, relative units Fig. 7. SNR vs. focal length. 4 5 Focal length, mm SNR can be reached only using star tracking. It means that the Kepler telescope would realize the objective from photometry viewpoint, but would fail from the viewpoint of vibration stability. The system with an objective of acceptable dimensions can fail to observe a group of stars on one frame in IR spectrum. Appearing big sensitive CCD matrices with spectral range over 1.5 microns will provide wide FOV and functioning with dramatically decreased atmosphere glow. However increased Airy disk on these wave lengths makes the pixel size grow, and, consequently, background enhances. It is important to note that using available IR matrices makes it impossible to solve the problem without tracking. High SNR is achieved by using a star tracker built according to a classical optical scheme with a 125 mm objective with lens aperture of 1/4, and 4mp CCD matrix (pixel size is 6.8 microns. The star tracker operates in 7 to 11 nm wavelength. Figure 8 shows the results of calculation for the chosen elements. The signal from the dimmest star to be observed by a star tracker with 22 deg circular FOV is shown in Fig. 8, top left. It would be about 25 3 electrons. Then the background level accumulated during exposure will be about 3 electrons with photon noise RMS about 17 electrons. The signal from the star against noise background is shown in Fig. 8, top right. With a.4 SNR the star is inobservable. As it was mentioned above, useful signal can be accumulated by averaging the series of frames. Averaging reduces the random photon noise by a factor equal to the square root of the quantity of averaged frames. SNR can be improved by a factor equal to the square root of the quantity of averaged frames by averaging of frames of the same sky regions around the stars and converging the signal from the star to one pixel of CCD matrix on different frames. Figure 8 (bottom right shows the result from averaging of 16 frames with a 27 electron star with initial background level of 173 electrons. Averaging reduced the noise level by four times, increased SNR up to 2, and the star became visible against the daytime sky. The result barely depends on internal noise of the camera: even with a high level noise of 3 electrons the GYROSCOPY AND NAVIGATION Vol. 4 No

5 28 AVANESOV et al The frame of 6m star of A spectral class 283 electrons Background: 3 electrons Noise RMS: 173 electrons One frame. SNR = averaged frames SNR = 1.8 (four times improvement Fig. 8. Results of star tracker calculations. resulting noise on the frame is 175 electrons. The main target is to reduce external noise independent from instrument sensitivity. STUDYING THE STELLAR INS MODEL After the selection of star tracker components, its model and software were developed, and integration with SINS was realized. Figure 9 shows the star tracker model mounted with the SINS on a common mounting frame. Star tracker-sins algorithmic integration consists in mutual information support. SINS generates a priori data for the star tracker required to converge the star signal in frame series with precision up to 1 pixel so that the star tracker can observe the stars and correct SINS drifts using its calculations. Converging the star signal is performed at certain intervals during which SINS does not accumulate significant relative errors. That is why SINS accuracy requirements mostly concern short-term errors. Position of the stars in CCD matrix plane during the exposure is calculated using the measured attitude of SINS frame with respect to local level and Earth Centered frames, the known time and transformation matrix between SINS and star tracker frames. Figure 1 presents SINS relative errors accumulated during 5 minutes integrating intervals expressed in terms of fractions of pixel of star tracker CCD matrix. The following experiment was conducted during Stellar INS ground tests. The system mounted on a fixed foundation with a star tracker aimed at nadir was rotating with the Earth in inertial space at 15 deg/h rate. Both experimentally and theoretically it was revealed that during observations at 1 km altitude and 4 deg angle between the Sun and the star tracker optical axis the luminosity of atmosphere column in star tracker FOV is equivalent to the background brightness during observations from the Earth with Sun angle above horizon of 3 to 5 deg, i.e., right after the sunrise. The last is conditioned by serious difference between air layers and sprays in the star tracker FOV where sunlight dispersion occurs. It is sufficient to say that 9% of atmosphere lies lower than 1 km. Figure 11 shows the background and noise in the frame growing proportionally with time and Sun angle. With 3 5 deg Sun angles, the background is 25 3 electrons and noise is close to the square root of background level. This confirms that the noise is photonic and stochastic. RMS background is electrons. Processing of star images obtained at night shows that the dimmest stars to be observed by the star tracker have integral brightness of 2 3 electrons, which fully agrees with the calculations. In Fig. 12, RMS noise decreases proportionally to quantity of frames (dotted line denotes experimental data, and solid line, calculated data. GYROSCOPY AND NAVIGATION Vol. 4 No

6 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 29 Star tracker electronics Star tracker astrocamera SINS GLONASS/GPS antenna Plate Fig. 9. Models of star tracker and SINS on a common foundation. Fractions of pixel Fractions of pixel SINS error scaled to columns of CCD matrix SINS error scaled to columns of CCD matrix Time, s Time, s Fig. 1. SINS errors over 5 minute intervals scaled to CCD matrix plane. GYROSCOPY AND NAVIGATION Vol. 4 No

7 21 AVANESOV et al. Background, electrons :: 5:3: 6:: 6:3: 7:: 7:3: Noise, electrons Time, h RMS noise 5 Square root of background 5:: 5:3: 6:: 6:3: 7:: 7:3: Sun angle above horizon, deg Time, h :: 5:3: 6:: 6:3: 7:: 7:3: 1 Time, h Fig. 11. Background and noise on the frame vs. Sun angle above horizon. Figure 13 shows the calculated attitude of star tracker body frame with respect to equatorial frame obtained during the experiment. As the star tracker within the system was rotating with the Earth, its right ascension and azimuth angles should remain unchanged if effects of Earth axis precession and nutation are compensated. Figure 13 presents the measurements obtained over 5 hours the period covering all possible levels of background brightness at 1 km altitude. The measurements on single frames without averaging are shown with lighter color, and the angles received by averaging of 3 frames, by darker color. It is seen that Ratio between RMS noise on averaged frame to RMS noise on initial frame Quantity averaged frames Fig. 12. Averaged noise vs. the quantity of averaged frames. with a Sun angle above horizon of about deg the algorithm using only single frames fails to observe the number of stars required for recognition. Under further flare the algorithm does not make any false recognition which could arise because of high noise level. The algorithm processing the frame series performs well even under higher background level. From Fig. 13, attitude angles include a low-frequency component caused by temperature deformations of the structure. These deformations can be eliminated in a thermally stable structure. The achieved accuracy measured in half an hour periods without averaging at night is about 1 arc sec for two angles that define CCD matrix plane orientation in inertial space, and 4 arc sec for the angle of rotation about optical axis. Accuracy at night with averaging is 1.5 arc sec and 1 arc sec, respectively. During daytime the accuracy barely changes (2 arc sec and 12 arc sec. These accuracies in a high resolution system would be impossible without geometrical calibration of the star tracker and consideration of distortion and aberration effects. Analysis of experimental useful signals and background flares revealed that to observe the stars in all sky areas at 1 km altitude and 4 deg angle between the Sun and the optical axis, about 5 frames should be processed. Software and hardware solutions applied in the star tracker provide accelerated reading of CCD GYROSCOPY AND NAVIGATION Vol. 4 No

8 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 211 Ascension, deg :3 4: 4:3 5: 5:3 6: 6:3 7: 7:29 Averaged over 3 frames Without averaging Azimuth, deg :3 4: 4:3 5: 5:3 6: 6:3 7: 7:29 Sun angle above horizon, deg Equivalent background during the fliqht 2 3:3 4: 4:3 5: 5:3 6: 6:3 7: 7:29 Fig. 13. Accuracy of star tracker measurement vs. Sun angle above horizon. matrix, which increases video data update rate and averaging frequency to about 2 Hz. Then the required frame series is obtained in about 3 seconds. Stellar INS can also be used in ground conditions. It has been experimentally proved that about 3 frames received in 2.5 minutes are needed to sight the stars in the least star populated sky area at sea level with 4 deg angle between the Sun and the optical axis. Obviously, attitude parameters can be determined by processing the frame series only under fair weather. In real flight conditions, the time interval required for accumulation of frames can be prolonged. Vehicle vibrations make the star energy spread over CCD matrix plane, which is unacceptable for the solution of a problem with lowest possible SNR. The frames fixed at high angular rate of vehicle are inappropriate for averaging since they practically do not have the useful signal in one pixel and enhance the noise. Therefore, a vehicle dynamics analysis mode is provided in the system software: under high angular rate, exposure is not performed, and on the completion of exposure the angular motion during signal accumulation is checked. The frame is considered unsuitable for further processing if angular motions exceed the angular equivalent of 3 pixels of CCD matrix during exposure (in terms of angular rate, more than.3 deg/s. Figure 14 shows the measured angular motions occurring under uniform linear motions during 2 ms exposure, recalculated in CCD matrix plane. Stationarity condition is met in most of time points, and the star tracker can fix the frames suitable for attitude determination. So the star tracker observes a group of stars, also in daytime, by averaging the frame series using SINS measurements (if required, and provides a highaccuracy attitude solution. Star tracker measurements are used for SINS stellar corrections. SINS STELLAR CORRECTIONS The principle of SINS correction uses the difference between the spectrums of autonomous SINS errors and external sensor errors, and stellar correction is not an exception. In autonomous SINS, the errors have low-frequency range (Schuler and daily fluctuations and tend to accumulate with time, while the errors in the star tracker and embedded GNSS receiver have higher frequency range with nearly zero RMS, which generally makes it possible to detect SINS errors [7 1]. SINS stellar correction is based on the relationship between various coordinate systems used in Stellar INS operations. The main coordinate systems are: ECI Earth Centered Inertial frame of Epoch J2.; ECEF Earth Centered Earth Fixed frame (Greenwich Geocentric coordinate system; Topocentric (Local-Level frame (also known as East-North-Up; B SINS body frame (right-handed orthogonal coordinate system aligned with SINS axes; B ST star tracker body frame (right-handed orthogonal coordinate system aligned with the optical axis and CCD matrix plane. Relationships between these frames can be mathematically represented as a simple matrix equation describing ECI to B ST transformation: GYROSCOPY AND NAVIGATION Vol. 4 No

9 212 AVANESOV et al. Angular oscillations expressed in terms of CCD matrix pixels (rows Angular oscillations expressed in terms of CCD matrix pixels (columns Time, s Fig. 14. Vehicle angular oscillations during the exposure. where (2 is the matrix describing B ST angular position with respect to ECI; C is B to B ST transformation matrix, determined during the system technological adjustment; C, C ECEF, C ECI are to ECEF B, ECEF to, and ECI to ECEF transformation matrices. Matrices C and C ECI from Eq. (2 can be defined as follows: where ST ST ECEF CECI = CC CECEFCECI ST C ECI (3 (4 cosγ sinγ 1 Cγ = 1, Cϑ = cosϑ sinϑ, sinγ cosγ sinϑ cosϑ cosψ sinψ Cψ = sinψ cosψ 1 are the matrices of rotations through roll γ, pitch ϑ and heading ψ angles; R pol is the polar motion matrix at epoch t (current time; R S is the sidereal time matrix; N, P are the nutation and precession matrices at epoch t. ST ECEF C = CγCϑCψ ECEF ECI = pol S C R R NP, ;, With account of (3 and (4, Eq. (3 can be presented as a relationship: ST ST ECEF CECI = CCγCϑCψCECEFCECI (5 or ST ST CECI = CC CECEF RpolRSNP. (6 ST The elements of matrix C ECI are the main data coming from the star tracker to SINS, and parameters ST of C, R pol are known a priori. Various modes of SINS stellar correction are realized based on relationships (2, (5, and (6. SINS ATTITUDE ERRORS COMPENSATION MODE The mode is realized if GNSS receiver and star tracker data are jointly processed in SINS. Geodetic latitude B and latitude L periodically fed from GNSS receiver make it possible to generate C ECEF sinl cosl CECEF = sin BcosL sinbsinl cos B. cosbcosl cosbsinl sin B ECEF Matrix C ECI is calculated according to [11] using reference data and Universal Time Coordinated (UTC received from GNSS receiver. Then all elements of ( C within C can be determined i, j using the modified equation (3: GYROSCOPY AND NAVIGATION Vol. 4 No

10 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 213 deg (a (b (c Time, s Corrected channel Autonomus channel Fig. 15. Roll (a, pitch (b and true heading (c angles by data of autonomous and corrected SINS channels during attitude errors compensation mode (within ground tests of Stellar INS model. ( ( ( ST 1 ST ECEF 1 1 = ECI ECI ECEF. C C C C C (7 On the other side, elements of C are known to depend on the sought estimates of pitch ϑ, roll γ, and yawing ψ [7], which are easily defined as ( C ( C ( C ( C ( C 13 ϑ= arcsin ; γ= arctan ; 23 ψ= arctan Positioning accuracy corresponds to GNSS receiver accuracy (units of meters, and accuracy of attitude determination with respect to corresponds to the star tracker accuracy (units of angular seconds. Figure 15 shows the roll, pitch and true heading angles by data of autonomous and corrected channels of a low-accuracy SINS during one of the first ground tests of Stellar INS model. Outliers in corrected channel measurements occur during attitude corrections (total 3 corrections over 1.5 h. SINS POSITIONING AND YAWING ERRORS COMPENSATION MODE This mode is realized in case of GNSS data failure due to signal loss, receiver failure, etc. UTC required ECEF to calculate C ECI is determined by SINS computer by counting the cycles of its microprocessor since GNSS data loss. Errors in roll and pitch generated by an autonomous SINS do not exhibit linear trends and intensive growth (fluctuation amplitudes are max 2 arc sec unlike yawing and positioning errors, so this mode provides correction of position data and yawing angles using (5 and the available data Relation (5 can be easily reduced to ST 1 ST ECEF ( ( ( ( CψCECEF = Cϑ Cγ C CECI CECI. (8 The left-hand part of (8 can be expressed in terms of sought estimates of latitude B, longitude L and heading ψ, which makes it possible to calculate these estimates using the elements of the product C ψ C ECEF : ( ψ ECEF ( ( CψCECEF ( CψCECEF 32 B = arcsin C C, L = arctan, 33 CψCECEF 13 ψ= arctan. ( CψCECEF 23 The obtained estimates are further used to calculate SINS corrections to the current position and yawing angles. SINS errors in determination of local vertical and the star tracker errors arouse maximum positioning errors of about 45 m and heading errors of about 2 arc sec. Figure 16 shows the true heading angles, latitudes and longitudes by data of autonomous and stellar inertial channels of a low-accuracy SINS during one of the first ground tests of Stellar INS model. The curves in Figs. 15 and 16 confirm that a rather simple stellar correction from the algorithmic viewpoint reduces the rate of output errors accumulation in the system. SINS TIME DETERMINATION MODE Both modes described above directly or indirectly use GNSS receiver data, which limits the system autonomy and applications. In the most unfavourable case, if GNSS data are lost since the system start-up it would have to function autonomously. Then the loss of UTC data essential for integration with the star tracker becomes a critical factor. 31 GYROSCOPY AND NAVIGATION Vol. 4 No

11 214 AVANESOV et al. (a (b (c deg Time, s Corrected channel Autonomus channel Fig. 16. True heading angles (a, latitude (b and longitude (c by data of autonomous and stellar inertial SINS channels in positioning and yawing errors compensation mode (within ground tests of Stellar INS model. In this situation a method of Greenwich stellar time and UTC determination by the star tracker data can be applied to initialize time determinations aids in Stellar INS. Like the standard modes considered above, this mode is algorithmically based on simple algebraic manipulations with equations of type (7 with further commonly known numerical procedures. It is known [11, 12] that components of Eq. (6 such as R S, N, and P depend on the current time, and this dependency is very complicated, generally nonlinear and insignificant as regards the nutation and precession effects on the system accuracy in critical mode. In further calculations of matrix R S by the simplified and modified Eq. (7 these effects are neglected: 1 1 ST 1 ST S = pol ECEF ECI 1 ( ( ( ( R R C C C C (9 To calculate the elements of C ECEF, C, only SINS measurements should be applied. On the other side, the sidereal time matrix in common representation is a directional cosine matrix of the form cos( SGMT + Nα sin( SGMT + Nα RS = sin( SGMT + Nα cos( SGMT + N α, (1 1 where S GMT is the Greenwich mean sidereal time (GMT for epoch t; N α is nutation in right ascension at epoch t. Greenwich mean sidereal time can be calculated by (9 and (1: SGMT = arcsin( RS N (11 12 α, and depends on the universal time UT1 as follows [11]: SGMT = d M τ ( τ, where d is the time interval from epoch J2 to epoch t differing from time UT1 expressed in fractions of day. by a known (a priori calculated [13] time interval from the epoch J2 to the beginning of the current day; M is the time UT1 of epoch t expressed in fractions of a day; τ is the time interval from the epoch J2 to epoch t expressed in julian centuries mean solar days long: τ = d Despite the constant natural growth of variable τ, the highest term in (12 will remain negligibly small as regards time (fractions of microseconds for a number of years. Therefore, determination of UT1 is reduced to solving a common quadratic algebraic equation for variable d. UTC estimate is finally calculated as follows: UTC = UT1 ΔUT1, where ΔUT1 is the daily correction to universal time published in the Russian bulletin Universal Time an Pole Coordinates, in the bulletins of International Earth Rotation and Reference System Service (IERS and transmitted in radio time signals [11, 14]. With unknown N α error in time calculations can exceed 1 s, so it is proposed to use a mean value of N α for the current day calculated by SINS computer in advance in (11 instead of unknown current value. It reduces the error to permissible level (hundredth fractions of second. It is expedient to use UTC determination procedure during the first stellar session of the system, before SINS errors become large. Determination of UTC by AINS computer aids since the start initialization is supposed to enable positioning and yawing errors compensation in SINS during further stellar sessions. CONCLUSIONS Use of high accuracy star trackers and SINS, and rational combination of SINS stellar correction methods are expected to provide an autonomous and effective fully strapdown airborne and near-earth Stellar INS described in this paper. GYROSCOPY AND NAVIGATION Vol. 4 No

12 AUTONOMOUS STRAPDOWN STELLAR-INERTIAL NAVIGATION SYSTEMS 215 Similar small-sized systems featuring high accuracy, reliability, and noise immunity will surely be further developed and become popular in the nearest future. REFERENCES 1. Vorob ev, L.M., Astronomicheskaya navigatsiya letatel nykh apparatov (Aircraft Stellar Navigation, Moscow: Mashinostroenie, Gregerson, C., Bangert, J., and Pappalardi, F., Celestial Augmentation of Inertial Navigation Systems: A Robust Navigation Alternative. U.S. Naval Observatory/Space and Naval Warfare Systems Command, USNO/SPAWAR white paper, n. d. 3. Branets, V.N. and Shmyglevskii, I.P., Vvedenie v teoriyu besplatformennykh inertsial nykh navigatsionnykh sistem (Introduction to the Theory of Strapdown Inertial Navigation Systems, Moscow: Nauka, Ishlinskii, A.Yu., Orientatsiya, giroskopy i inertsial naya navigatsiya (Orientation, Gyroscopes, and Inertial Navigation, Moscow: Nauka, Avanesov, G.I. et al., BOKZ-M Star Tracker and Its Evolution, 14th St. Petersburg International Conference on Integrated Navigation Systems, St. Petersburg: Elektropribor, 27, pp Avanesov, G.A. et al., Integrated Instruments for Spacecraft Autonomous Navigation, 7th International Symposium on Reducing the Costs of Spacecraft Ground Systems and Operations, Moscow, Grewal, M., Weil, L., and Andrews, A., Global Positioning Systems, Inertial Navigation and Integration, Wiley, Titterton, D. and Weston, J., Strapdown Inertial Navigation Technology, The Institution of Electrical Engineers, 24, 2nd edition. 9. Salychev, O., Inertial Systems in Navigation and Geophysics, Moscow: Bauman MSTU Press, Stepanov, O.A., INSS/GNSS Navigations Systems: Design Philosophy and Development Prospects, in Integrirovannye inertsial no-sputnikovye sistemy navigatsii (Integrated INS/GNSS Navigation Systems, St. Petersburg: Elektropribor, 21, p RD Normative document. Recommended Practices. Artificial Earth Satellites. Main Reference Frames for Ballistic Provision of Flights and Procedure of Star Time Calculation, Zharov, E.V., Sfericheskaya astronomiya (Spherical Astronomy, Fryazino: Vek-2, Meeus, J., Astronomical Formulae for Calculators, Audoin, C. and Guinot, B., The Measurement of Time, 21.l GYROSCOPY AND NAVIGATION Vol. 4 No

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