Accuracy Evaluation Method of Stable Platform Inertial Navigation System Based on Quantum Neural Network

Transcription

1 Accuracy Evaluation Method of Stable Platform Inertial Navigation System Based on Quantum Neural Network Chao Huang*, Guoxing Yi, Qingshuang Zen ABSTRACT This paper aims to reduce the structural complexity and navigation errors of stable platform inertial navigation system INS). For this purpose, a new navigation error model was proposed for the stable platform INS based on quantum neural network QNN). After reviewing several representative QNNs, the author established a QNN-based navigation error estimation algorithm on four coordinate systems, in which the angles of acceleration error were calibrated by twelve positions. Then, the QNN-based INS error model was constructed through training and testing. Next, the established model was applied to an experiment on two flight tracks with Kalman filter KF), fixedinterval smoothing filter smooth) and improved smoothing filter improved). The results show that the model can significantly improve the multi-position calibration accuracy and the self-calibration accuracy of acceleration errors. The research findings shed new light on the accuracy evaluation of stable INS. Key Words: Quantum Neural Network QNN), Stable Platform Inertial Navigation System INS), Navigation Error DOI Number: /nq NeuroQuantology 2018; 166): Introduction There are two main categories of INS: the stable platform INS and the strapdown INS Lee et al., 1993). In relatively terms, the stable platform INS is mature in technology and sophisticated in structure Song et al., 2002). Over the years, this INS has been extensively applied to rockets, missiles, unmanned aerial vehicles UAVs) and submarines, thanks to its high accuracy in navigation and ability to complete task interpedently Zhang et al., 2011). Apart from accuracy and independence, the stable platform INS outperforms the strapdown INS in resistance to external interferences and the computing load. For instance, the stable platform INS can obtain the real-time position of an object through direct integration rather than space conversion matrix. Nevertheless, this type of INS also has its weaknesses, including structural complexity and navigation errors. To overcome these defects, it is necessary to determine the navigation accuracy of the stable platform INS in a rapid and precise manner. Therefore, the influencing factors on the final navigation error, and their exact impacts on navigation must be clearly identified Zhang et al., 2006). Considering the above, this paper attempts to develop a comprehensive evaluation index on the overall performance of the stable platform INS through statistical analysis of parameter variation, and to define its relationship with each individual energy index of the stable platform INS. Considering the above, this paper attempts to develop a comprehensive evaluation method for the navigation error of the stable platform INS. Therefore, the author established a QNN-based navigation error estimation algorithm Corresponding author: Chao Huang Address: Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin , China huangchao198311@126.com Relevant conflicts of interest/financial disclosures: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Received: 2 March 2018; Accepted: 7 May 2018

2 on four coordinate systems, in which the angles of acceleration error were calibrated by twelve positions. Then, the QNN-based INS error model was constructed through training and testing. Next, the established model was applied to an experiment on two flight tracks with three filters. The results show that the model can significantly improve the multi-position calibration accuracy and the self-calibration accuracy of acceleration errors. Literature review US scholars have explored deep into the error mechanism, modeling, calibration and compensation of the INS. A massive amount of data has been accumulated through numerous realistic tests, laying the basis for comprehensive and accurate error models at device level and system level. The INS accuracy has been improved greatly through the research on short-term stability, long-term stability, repeatability and law of parameter variation of the system Weston et al., 2000; Guha et al., 2012). Russian researchers have developed a desirable physical model of inertial devices and systems through ground testing, and relied on the model to construct an exact polynomial model of the INS. The physical model discloses the error mechanism of the INS, making it possible to improve the quality of finished products and to identify, locate and resolve defects. As a result, the INS of Russian weapon systems, especially ground-based mobile missiles, can self-calibrate over 70 error items. Chinese universities and institutes have also investigated the parameter calibration of the stable platform INS. The system-level calibration methods are either multi-position calibration or continuous self-calibration. The multi-positioning calibration enjoys a relatively long history and great popularity. Liu et al., 1994) presented the self-calibration method of non-full attitude platform under arbitrary initial azimuth, and analysed the theory on multi-position selfcalibration. However, they neither identified the calibration accuracy of parameters nor provided a solution to the initial azimuth. Yang et al., 2000) pioneered the research on self-alignment in China. He introduced the self-calibration, self-alignment, self-detection technologies of a typical US inertial platform, and summarized the development trend of these technologies. Focusing on non-all attitude three-frame stable platform INS, Xu et al., 2003) proposed a calibration method for 9-position and 13-position accelerometer parameters, and successfully realized the self-calibration of accelerometer bias, scale factor and installation error. In general, the above calibration methods were not well integrated with engineering application. To sum up, the error modelling and parameter calibration of the INS still needs further improvement despite the numerous theoretical analysis and experiments on the system. More studies are needed to reveal the error mechanism of inertial devices, modify error models and improve error identification of the INS. INS error model based on Quantum Neural Network QNN) Introduction to the QNN The QNN is a hybrid of quantum computation and traditional neural networks. As an intelligent optimization algorithm, the QNN can improve the structure and performance of traditional neural networks based on quantum theory and brain science. In 1997, Purushothaman et al., proposed the QNN based on multi-layer excitation function and verified the inherent fuzziness of the network in pattern classification through theoretical analysis and experiment. The function is a superposition of multiple sigmoid functions. Compared with the excitation function of the traditional neural network Figure 1), the multilayer excitation function of the QNN maps different data to different magnitudes or ladders, giving more degrees of freedom to the classification Purushothaman et al., 1997). The multi-layer excitation function enables the QNN to detect the inherent data indeterminacy and uncertainty inherent. Taking cross data for instance, the QNN can divide them into two classes with a certain degree of membership. Based on the multi-layer excitation function, the QNN consists of such three layers as the input layer, the hidden layer and the output layer. The input and output layers are the same as those in the traditional feedforward neural network, while the hidden layer is the result of quantum superposition. The training of the QNN involves two steps. In the first step, the weight is adjusted to map the input data to a specific class space; In the second step, the quantum interval of hidden layer neurons is changed to reflect data uncertainty. 614

3 In 2000, Matsui et al., created a neural network whose neuron state is represented by quantum bits. The network retains the three-level topological structure of the traditional neural network Figure 2). In spite of the structural similarity, the quantum bits neural network differs from the traditional neural network in neuron information representation, weight and excitation function. The neuron information representation of the network is illustrated in Figure 3. Figure 1. Excitation functions of traditional neural network and the QNN based on multi-layer excitation function Figure 2. Topological structure of quantum bits neural network In 2000, Ajit et al., proposed a multiverse QNN model based on the optical double-slit interference experiment and the multiverse concept in quantum mechanics. The multiverse QNN is the superposition of multiple similar network components. During training, the network components are trained by different input modes, so that different inputs will be processed by different components. In other words, the QNN components are a linear superposition of traditional network components. The diversified QNN models mentioned above all promote quantum computation with the features of neural network. Suffice it to say that the QNN is an integration of multiple disciplines, ranging from quantum computation, artificial intelligence to brain science. QNN-based navigation error estimation algorithm The following coordinate systems were defined for parameter calibration before constructing the error model of the stable platform INS. Geographic coordinate system: OX gy gz g. The origin of the platform is the centre of the INS., OX g, Y g and Z g point to the east, the north and the sky, respectively. Three-axle table coordinate system: OX dy dz d. When the turntable is at zero, OX d, Y d and Z d correspond to the inner ring, outer ring and central axis of the turntable, respectively. The three axes will be used in the calibration test. Platform coordinate system: OX py pz p. When the aircraft is horizontally placed and the frame attitude angle is zero, OY p axis coincides with the longitudinal axis of the aircraft, OZ p points vertically upward, OX p axis points to the right rule side of the projectile. Accelerometer coordinate system: OX ay az a. X a, Y a and Z a are the input axes of axial accelerometers X, Y and Z, respectively. When the attitude angle of the aircraft is zero, the relationship between the coordinates of the aircraft and the platform is given in Figure Figure 3. Neuron information representation of quantum bits neural network The quantum bits neural network is characterized by the information phase of the state. The weight function is the rotation phase, while the excitation function operates on the phase-controlled gate, aiming to change the quantum state. Figure 4. Relationship between the coordinates of the aircraft and the platform

4 The expression of the relationship between the steady-state output and the specific force of accelerometers under online motion is called the static mathematical model of accelerometer. 1) where N is the output; K 0, K 1 and K 2 are bias, scale factor and second-order nonlinear coefficient, respectively; K ip, K io and K op are crossterm coefficients; a i, a p and a o are accelerations along the input axis, the pendulum, and the output axis, respectively. In engineering application, the above model can be simplified as: 4) where, and are the number of output pulses for X, Y and Z axes, respectively;, and are the accelerometer biases for X, Y and Z axes, respectively;, and are the accelerometer scale factor for X, Y and Z axes, respectively. 2) Considering the asymmetry of scale factor and the second-order nonlinear coefficient, considered, the model can be defined as: 3) The item with + should be taken when a i>0, and the item with - should be taken when a i<0. Taking the input axis of an accelerometer as the reference, the relative positions between the other two accelerometers and the reference axis were determined to define the installation error of the accelerometer, that is, the acceleration error. The relative positions are illustrated in Figure 5. Let, and be the accelerations of axes, and in the platform coordinate system, respectively. If only the bias, scale factor and acceleration error are taken into account, the output of acceleration can be expressed as: Figure 5. Definition of acceleration error Then, four positions were selected to Figure 6). Figure 6. Four positions for determining μzy Note: α2 is 5, 135, -135, and -45 in Figures 6a), 6b), 6c) and 6d), respectively The! of the four positions can be derived as: 616 %! & ' # # #! = ' $?! ' # # #! A ' " / , 4-1. ' :'4; +> / > > +> ' +> :'4; +@ / @ +@ ' +@ :'4; +B / B B +B ' +B :'4; 5 /0 1) 2, 3) 4-1) 6789 ' < 5 :'4; < +> 5 /01) 2 > 3) 4-1) > ' < +> 5 :'4; +> < +@ 5 /01) 3) 4-1) @ ' < +@ 5 :'4; +@ < +B 5 /01) 2 B 3) 4-1) B ' < +B 5 :'4; +B < 5)

5 The four positions help to offset the calibration error resulted from!,!,! and!. For each angle of acceleration error, four symmetrical positions were selected for calibration, so as to eliminate the impacts of the accelerometer bias and scale factor error on the calibration results. In total, 12 positions were selected to calibrate the angles of acceleration error, and the accelerometer bias and scale factor were obtained at the same time. Next, a three-layer QNN Figure 7) was constructed according to the classification of bias and scaling factor. Each neuron in the network involves an input vector, a weight vector, a quantum rotation door and a transfer function. The input vector contains the feature vector and the target vector of the sample. It can be seen from Figure 10 that the proposed QNN-based INS error model had significantly improved the multi-position calibration accuracy and the self-calibration accuracy of acceleration errors. Figure 8. Track one 617 Figure 7. Three-layer QNN The QNN-based INS error model was constructed in two phases: training and testing. Before construction, the dataset was randomly divided into a training dataset and a test dataset. In the training phase, the attribute data were analysed against the training dataset; in the testing phase, the classification accuracy of the model was evaluated against the test dataset. Results The established QNN-based INS error model was applied to an experiment on two flight tracks. As shown in Figures 8 and 9, track one and track two respectively simulates uniform linear motion and curve motion. For both track one and track two, the QNN-based INS error model was applied to calibrate the parameters of the INS in combination with one of the following three filters: Kalman filter KF), fixed-interval smoothing filter smooth) and improved smoothing filter improved). The estimation errors of all three combinations are compared in Figure 10. Figure 9. Track two Figure 10. Comparison of estimation errors Conclusions Stable platform INS is an accurate navigation system that can independently accomplish

6 navigation tasks. The system has been widely used in rockets, missiles and submarines. This paper aims to solve the only defects of the stable platform INS, namely structural complexity and navigation errors. Therefore, a QNN-based error model was developed for the stable platform INS. After reviewing several representative QNNs, the author established a QNN-based navigation error estimation algorithm on four coordinate systems, in which the angles of acceleration error were calibrated by twelve positions. Then, the QNNbased INS error model was constructed through training and testing. Next, the established model was applied to an experiment on two flight tracks with Kalman filter KF), fixed-interval smoothing filter smooth) and improved smoothing filter improved). The results show that the model can significantly improve the multi-position calibration accuracy and the self-calibration accuracy of acceleration errors. The research findings shed new light on the accuracy evaluation of stable INS. References Ajit N, Tammy M. Quantum artificial neural network architectures and components. Information Sciences 2000; 1281): Guha S, Plarre K, Lissner D. AutoWitness: Locating and tracking stolen property while tolerating GPS and radio outages. ACM Transactions on Sensor Networks 2012; 84): Lee JG, Park CG, Park HW. Multiposition alignment of strapdown inertial navigation system. Aerospace & Electronic Systems IEEE Transactions on 1993; 294): Liu X, Song Y. Self-calibration of Arbitrary Initial Position Platform Error Model. Journal of Chinese Inertial Technology 1994; 21): Matsui N, Takai M, Nishimura H. A network model based on qubitlike neuron corresponding to quantum circuit. Electronics & Communications in Japan 2015; 8310): Purushothaman G, Karayiannis N B. Quantum neural networks QNNs): inherently fuzzy feedforward neural networks. IEEE Transactions on Neural Networks 1997; 83): Shtessel YB. Decentralized sliding mode control in three-axis inertial platforms. Journal of Guidance Control Dynamics 1995; 184): Song JL, Han JQ. Research on a Fast Initial Alignment Method for Platform Inertial Navigation System. Journal of Chinese Inertial Technology 2002; 101): Weston JL, Titterton DH. Modern inertial navigation technology and its application. Electronics & Communication Engineering Journal 2000; 122): Xu JH, Qian PX, Wang LX. Self-calibration of the Finite Rotation Platform. Computer Measurement and Control 2003; 118): Yang L. "Three-autonomy" technique of inertial platform and its development. Missiles and Space Vehicles 2000; 71): Zhang DR, Bin YE, Dang J. Flight test performance error analysis of the platform inertial navigation system. Flight Dynamics 2011; 291): Zhang T. Azimuth strapdown inertial navigation platform system and error analysis. Journal of Chinese Inertial Technology 2006; 145):