Research On Attitude Algorithm Of Strapdown Inertial Navigation System

The Strapdown Inertial Navigation System (SINS) uses the rigidly connected inertia sensors (including accelerometer and gyroscope) and provide the vessel navigation information: speed, position, course, attitude information. The source of SINS's error includes: instrument errors, installation error, initial condition error, computation error, the dynamic motion error and disturbance which come from the vessel angular motion (mainly include impact, vibration and random disturbance) and so on.Since SINS performance base on the mathematical platform, the vessel motion based dynamic errors and the computation errors are mostly important. Because of this, SINS have to compensate the error signals of gyroscopes and the accelerometers, and then implements the calculation of attitude matrix representation. The error compensation model and the attitude matrix computation method are different according to the basic technology of gyroscopes and accelerometers that used in the system.This dissertation emphasize in the study of the error mechanism of SINS. Based on the self potential of the system, study the new theory, new method of suppression inertial navigation system error.The original application of inertial navigation technology is based on the stable platform techniques in which the inertial sensors are mounted on a platform and it is isolated from the rotational motion of the vehicle using three gimbals arranged to provide three degrees of rotational freedom. SINS simplify the mechanical complexity of PINS and provide the more accurate navigation information of position, velocity and attitude of vehicle which carries it. The inertial sensors of accelerometers and gyroscopes are attached rigidly to the body of host vehicles and not mechanically moved. Therefore the signals produced by these sensors are resolved mathematically. By using computer software, SINS keep track the orientation of inertial measurement unit to rotate the measurement from body frame to the navigation frame. Thus attitude and heading can be computed rather than being provided directly by electrical pick-off device of PINS. There are two main tasks of SINS computer: attitude computing and navigation data computing. Attitude computer not only computes attitude angle but also updates these attitude information at each navigation step. This thesis describes the development of quaternion based attitude updating algorithm for SINS. And also develop the quaternion based non-linear error model of SINS.To achieve the goal of the development of algorithm and error model, following issue are addressed in this thesis:Develop the navigation equation operating in the local geographic reference frame for terrestrial navigation systemDevelop the attitude quaternion updating algorithm by using the fourth order Adams-Moulton Method.Develop the estimated the algorithm error from the predictor value and corrector value. And analyze the performance of updating algorithm utilized the coning motion as the numerical test.Develop the quaternion based non-linear error model according to the misalignment between true navigation coordinate and computed mathematical coordinate.The attitude computation is the main task of SINS computation because it is the basic algorithm of error computation and the navigation information process. It involves the real time resolution of vessel attitude and also relates with "mathematics platform" i.e. the updating of attitude matrix information. Therefore, this Strapdown attitude algorithm performance directly affects the navigation accuracy of strapdown system.The attitude computer takes the measurements of body rate about three orthogonal axes provided by trio gyroscopes. These angular rates are executed into attitude integration function, so-called attitude algorithm. And these attitude data are used to resolve measured acceleration into a suitable navigation coordinate frame where it is integrated into velocity and position with respect to navigation frame. The attitude is usually represented as a set of direction cosine matrix or quaternion parameters, either of which is appropriate for on-line attitude computation.Because the quaternion algorithm requires less computing time and gives less truncation error than the corresponding direction cosine matrix (DCM), the 4-elements quaternion is used as the attitude information. To transfer the measured acceleration of body frame into n-frame for solving the navigation equation, the attitude quaternion has to be update according to the orientation of the vehicle at each navigation step. In order to update this quaternion, it is necessary to numerically solve the closed form solution of the quaternion propagation differential equation.In this thesis, the attitude quaternion q represents the coordinate transformation from the accelerometers measurement of b-frame into n-frame to solve the terrestrial navigation equation in the navigation computer and this attitude quaternion differential equation is solved by using Adams-Moulton 4th order Method. This method is one of the multi-step methods to solve the numerical differential equations and it can't be self started. Thus the first initial 3 steps are applied by single-step Modified Euler's Method. Moreover, both of this Adams-Moulton Method and Modified Euler's Method are also the predictor-corrector method, i.e. they have the back calculation process to make the confirmation of first step by using the next step input. Adam-Bashforth 4^th order Method is utilized as predictor step for Adams-Moulton Method and Euler's 1^st order Method is for Modified Euler's Method respectively.All applied methods are directly used the angular rate measurements of gyroscopes. Therefore, there is small error rather than the rotation angle calculation based attitude algorithm. However it has to be notice that the gyroscope output must be higher speed process then the attitude algorithm process speed because this predictor-corrector method is utilized the next step input value in corrector stage.Another advantage of predictor-corrector method is the algorithm updated quaternion error can be estimated from the difference between the predicted value and corrected value. Although the order of predictor stage and corrector stage of Adams-Moulton Method is same, generally the corrector stage has the lower truncation error than predictor stage because of their original polynomial interpolation points have differ in one step size. Under this awkward situation of adapting one step side between prediction and correction, we assume that there is nearly one step side of error order between them. By using this advantage, the algorithm drift error is derived and analyzed the algorithm performance.Development of SINS error model is one of the vital processes of system performance. The accuracy of SINS is limited as the result of errors in the data which are passed from the initial navigation information as well as the imperfections in the various components which combine to make up the system. It is developed by perturbing the differential equation whose situation yields the system output of velocity, position and attitude. The error differential equations are divided into translational error propagation and attitude error propagation. The translational error equations have two components of velocity error component and position error component.There are two common approaches for the derivation of error models:Φangle approach andΨangle approach. When the attitude errors are large enough, these conventional error model cannot be describe the non-linear characteristic of the system. In this thesis, all error models are based on the misalignment between true navigation coordinate and computed mathematical coordinate.Modeling the attitude quaternion error is required to correct and update the navigation equations in real time at the sampling time. There are two main type of modeling quaternion errors: additive quaternion error and multiplicative quaternion error. Since the additive quaternion error model is applicable to the system with large attitude errors, both of the translational error and attitude error of non-linear models are developed based on the additive error method of quaternion differences between navigation coordinate propagation and mathematical coordinate propagation.The attitude quaternion error propagation are composed the function of the attitude updating computer output, the gyros angular rate measurements and the navigation computer output of navigation frame rate. The translational error of the velocity errors and position errors are predominantly functions of specific force measurement. Among them, the velocity error propagation is vital of translational error and can be applied from the navigation equation. There is no difference between small error model and large error model in position error model and its error propagation equation depends on the velocity error and the local geographic frame rate (or transport rate).The above theoretical work is verified by the experiments using coning motion as the numerical data. The algorithm is analyzed in the following facts:quaternion normalizationthe algorithm drift in a specific frequency &sampling timethe variation of algorithm drift according with difference frequencies and sampling timeFor the numerical test, let coning motion of angular rate areω_x=O.5sin(2πft) rads^-1,ω_y=0.5cos(2πft) rads^-1 andω_<sub>z=0.01rads^-1 and the initial quaternion is q(0) = [1 0 0 0]^T . Andthe motion with frequency 2.5Hz, 5Hz, 7.5Hz, 10Hz,-----, 25Hz and the sampling time (At) of0.0025s, 0.005s, 0.0075s and 0.01s are conducted for numerical simulation.According to the experiment, updated quaternion is entirely normalized at frequency 2.5Hz and 5Hz. However, the normalization is deviate at 7.5Hz and 10Hz and its deviation is increase as the propagation time increase. From the sampling time point of view, the algorithm drift is rising nearly seven times with 1.5 times increased in time steps, thirty times with double increase in time step and two hundred times with three times increased in time steps for all frequencies of motion. Therefore we can say the drift is rise by increasing the frequencies of motion. However their rising rate of specific sampling time is gradually declined. For example, considerΔt=0.005s, although the drift rise 15 times with the frequency from 2.5Hz to 5Hz, it is only rise 1.5 times from 17.5 Hz to 20Hz.According to the result, we can be concluded that both of the algorithm stable normalization and drift is reliable for low frequencies motion and can be obtained the good accuracy. And also summarize as the sampling time is reduced, the accuracy of algorithm improved, i.e. drift was decreased, especially for low frequencies.