Trapdown Inertial Navigation Systems SDINS Computer Science Essay

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Abstract- The last two decades have shown an increasing trend in the use of navigation technologies such as strapdown inertial navigation systems (SDINS) in several applications including land vehicles and automated car navigation. On the other hand it can cause large position errors over short time, due to the low quality of the inertial measurement unit (IMU). These errors determine the performance and the navigation accuracy of the INSs. Although the huge efforts to improve SDINS in terms of its mechanization equations, it could not cover the remaining drawbacks of SDINS; such as the impact of INS short term errors, model dependency, prior knowledge dependency, sensor dependency, and computational errors.

This paper proposed an intelligent navigator to overcome the limitations of existing INS algorithms. The intelligent navigator is based on Adaptive Neuro-Fuzzy Inference System (ANFIS).

The proposed conceptual intelligent navigator consisted of SDINS architecture that was developed using adaptive fuzzy system networks to acquire the navigation knowledge. In addition, a navigation information DataBase, and a temporal windowing-based learned parameters updating method were implemented to store and accumulate navigation knowledge. This paper optimizes the operation of the SDINS by implementing a temporal windowing updating method. The results discuss the merits, suggestions, and limitations of the proposed intelligent navigator.

Keywords: vehicular navigation, inertial navigation system (INS), DataBase, adaptive neuro fuzzy inference system (ANFIS).

1. Introduction

Most of the present vehicle navigation instruments rely mainly on the global positioning system (GPS) as the primary source of information to provide the vehicles position. GPS is capable of providing precise positioning information to an unlimited number of users any where on the planet. However, GPS can provide this type of information only when there is a direct line of sight to four or more satellite [1]. In other words, the system does not work properly in urban areas due to signal blockage and attenuation that may deteriorate the overall positioning accuracy.

An INS is a self-contained positioning and attitude device that continuously measures three orthogonal linear accelerations and three angular rates. By measuring vehicle acceleration and angular velocity in an inertial frame of reference, integrating it with respect to time and transforming it to the navigational frame, velocity, attitude and position components can be obtained. Sensors used to implement such a system are accelerometers for the measurement of a vehicles linear acceleration (specific force) and gyroscopes for monitoring vehicle rotation (angular velocity) with respect to an inertial frame [2]. Since specific force measurements contain the effect of the earth's gravity field, a gravity model is used to extract vehicle acceleration from the measurements. Because they employ three translational (accelerometers) and three rotational (gyroscopes) sensors, inertial measuring units (IMU) can be used as positioning and attitude monitoring devices [3].

In fact, INS can not operate as a stand-alone navigation system like GPS. Residual bias errors in both the accelerometers and the gyroscopes may deteriorate the long-term positioning accuracy. A comparison between the two navigation systems is illustrated in table (1). In addition, these bias errors are random in nature and need to be modeled using empirical and adaptive model processes. And since there is a lack of researches towards the conceptual intelligent navigator, this paper is devoted to develop an intelligent navigator that consists of adaptive neuro fuzzy inference system (ANFIS) based on Terrestrial SDINS algorithm described in [2, 4]. As each of these limitations contributes to certain amount of positional error accumulation during computational errors, therefore, the proposed new algorithm are expected to reduce the impact of these limitations by reducing the positional error accumulation during navigation phase. In land vehicle and submarine navigation. Regular updates are needed to limit the rapidly growing positional error. It is often possible to considerably improve the accuracy of the SDINS by building a conceptual intelligent navigator to accumulate the navigation knowledge and retrieve it to trim down the SDINS error.

1.1 Objectives and Motivation

This paper aims at introducing a novel method based on adaptive neuro-fuzzy inference system (ANFIS) to fuse the outputs of IMU and provide accurate positioning and velocity information for the moving vehicle.

In addition, this paper suggests a navigation DataBase to retrieve the navigation knowledge to provide an accurate real-time computing system for the strapdown INS algorithm for vehicular navigation.

In other words, this paper introduces a conceptual intelligent navigator for next generation navigation systems to accumulate the navigation knowledge and retrieving the stored navigation knowledge to be able to provide the real time prediction. Ultimately, the conceptual intelligent navigator is expected to overcome or, at least, reduce the limitations of the conventional based SDINS algorithms.

2. Adaptive Neuro Fuzzy Inference Systems

Ever since the artificial intelligence, considered as a powerfull and applicable tool in engineering modeling, computation, nonlinear function approximation, system identification and estimation theory. The neuro fuzzy models have the connectionist structure of neural networks combined with flexibility and intuitive learning capabilities of fuzzy system. It has hybrid learning method based on gradient descent and least square estimation [5].

In order for an FIS to be mature and well established so that it can work appropriately in prediction mode, its initial structure and parameters (linear and non-linear) need to be tuned or adapted through a learning process using a sufficient input-output pattern of data. One of the most commonly used learning systems for adapting the linear and non-linear parameters of an FIS, particularly Takagi and Sugeno (TS) type, is the ANFIS. ANFIS is a class of adaptive networks that are functionally equivalent to fuzzy inference systems [6]. Different interpretations for the fuzzy IF-THEN rules result in different mappings of the fuzzy inference engine, also there are different types of fuzzifier and defuzzifier. Several combinations of the fuzzy inference engine, fuzzifier, and defuzzifier may constitute useful fuzzy logic system. If the fuzzy logic system can be represented as a feed forward network, then the idea of back propagation training algorithm can be used to train it. The structure of ANFIS and the main concepts and algorithm adopted during its learning process will be introduced later.

3. Proposed Conceptual Intelligent Navigator based on ANFIS

The proposed conceptual intelligent navigator integrates the data from IMU and mimics the dynamical model of the vehicle to generate navigation knowledge. Thus the latest acquired navigation knowledge can be applied to predict the vehicles velocity and position during IMU errors in real time.

The resulting ANFIS has the structure depicted in figure (1) . It consists of 5 layers. Layer 1 and layer 5 define the input and output spaces respectively. Layer 2 and layer 3 are used to perform the IF part of fuzzy rules. Layer 4 performs the normalization of each node in layer 3. The THEN part of the fuzzy rules is completed in the fifth layer. Detailed descriptions and equations for each layer are given here [7].

Layer 1: Input-variable layer. This is the layer where the IMU data first enter the neural network, and each node in layer 1 represents an input linguistic variable.

Layer 2: Each node in layer 2 represents a membership function (MF), which is a Gaussian function of the following form:

…(1)

where r is the number of input variables and u is the number of membership functions.

Layer 3: The rule layer associated with the input variables is given by eq.(2). Each node in this layer is a radial basis function (RBF) unit which represents a possible IF- part of the fuzzy rule. The outputs are given by:

…(2)

where

Layer 4: This layer consists of normalized nodes. The number of nodes is equal to that of RBF units. The output is given by:

…(3)

Layer 5: This is the output layer, which comprises of output nodes, each of which is weighted according to eq. (4). This layer performs defuzzification (weighted average) of the output as follows:

…(4)

The weight is of linear structure and can be expressed as follows:

…(5)

where are real-valued parameters.

4. ANFIS learning using hybrid technique

As mentioned earlier, both the premise (non-linear) and consequent (linear) parameters of the FIS should be tuned, utilizing the so-called learning process, to optimally represent the factual mathematical relationship between the input space and output space. Normally, as a first step, an approximate fuzzy model is initiated by the system and then improved through an iterative adaptive learning process.

The training algorithm, namely ANFIS, was developed by [6]. Basically, ANFIS takes the initial fuzzy model and tunes it by means of a hybrid technique combining gradient descent back-propagation and mean least-squares optimization algorithms figure (2). At each epoch, an error measure, usually defined as the sum of the squared difference between actual and desired output, is reduced. Training stops when either the predefined epoch number or error rate is obtained. The gradient descent algorithm is mainly implemented to tune the non-linear premise parameters while the basic function of the mean least-squares is to optimize or adjust the linear consequent parameters. Table (2) shows the two stages of hybrid learning process of ANFIS.

5. Intelligent Navigator Architecture

The general architecture of ANFIS is shown in figure (1), the input vectors for the ANFIS are the raw accelerations at each current epoch, and angular velocity at current epoch while the output of ANFIS was the instant position and velocity at each current epoch for both position and velocity in the three directions for the moving vehicle.

ANFIS and SDINS algorithms receive raw outputs of accelerometers and gyros, respectively as inputs and generate navigation state as outputs as shown in figure (3). Thus, SDINS mechanization illustrated in [2, 4] is replaced by the proposed ANFIS.

So, the navigation knowledge can be learnt, stored and accumulated during the availability of the INS signal (i.e., no sensors error). On the other hand, during IMU errors, the latest acquired navigation knowledge can be retrieved from the navigation information DataBase.

6. Navigation Information DataBase

The second step towards building the intelligent navigator is to store the learnt navigation knowledge provided by the intelligent navigator. As a result, a navigation information DataBase that contains the acquired and learnt navigation knowledge can serve as the "brain" of the intelligent navigator. Therefore, several issues regarding the DataBase are addressed as follows:

DataBase Contents: it consists of three main parts, the input accelerometers and gyroscopes (input vectors), position and velocity outputs (desired outputs), and the optimized learned parameters during training phase. Thus, these contents can be considered as navigation knowledge. And these contents varies depends on the type of navigation algorithm structure that will be used.

Distributing acquired learned parameters: the navigation knowledge stored in the proposed DataBase becomes more complicated, therefore, considering the efficiency of DataBase maintenance and retrieval; the acquired learned parameters should be stored individually in a distributed DataBase, as shown in figure (4).

DataBase maintenance: Due to expensive storage element and to reduce the storage requirement, during the offline training the redundant learned parameters (including the input vectors and its desired outputs) should be removed to reduce the stored navigation information. It is worth to mention that the implementation of the navigation information DataBase was done with matlab application. Up to now, the intelligent navigator has been given the ability to generate, and learn navigation knowledge and it also has been given the "space" to store the navigation learned parameters.

7. Training the ANFIS based Temporal windowing method

As the learned parameters (x, c, and σ ) are the core components of the navigation knowledge, the final step towards building the intelligent navigator is to develop a technique to accumulate the acquired navigation knowledge by updating the learned parameters whenever the IMU signal is available (i.e., no sensors).

In most of their applications, ANFIS are trained using some known training data set (input/desired output) to obtain the optimal values of the learned parameters via off-line training. For any other set of inputs, different from those used in training, the learned parameters can then be applied to provide prediction of the network outputs. It is worth mentioning that ANFIS parameters are frozen after completing the training procedure and no further modification will be made during the prediction process.

In fact, off-line training can work well in case of slowly changing time sequences [8]. In the case of INS navigation applications, it is required to track direction changes and mimic the motion dynamics utilizing the latest available INS data. In other words, the learned parameters should be updated during the navigation process to adapt the network to the latest INS sensor readings whenever the INS signal is available.

A temporal windowing method was used to built a reliable navigation system that can retrive the navigation information obtained during the conventional off-line training phase (or probably from previous navigation missions). This technique utilizes the latest available navigation information provided by the INS to adapt the stored learned parameters so that they can apply to mimic the latest motion dynamic. The learned parameters are stored after each training procedure. They are then used as initial values for the parameters to be estimated during next training session or for prediction during INS sensor errors. The learning parameters acquired for the temporal windowing based parameters updating strategy can be arranged through using the following two procedures [9].

One step training procedure: the training parameters acquired for each INS window during navigation are the combination of stored training parameters and available training parameters obtained at the end of each window. In other words, as the size of training parameters increases during navigation mission, the size of DataBase grows during navigation as well which is not preferred.

The advantage of the one step training procedure is that it can provide better generalization of the navigation knowledge by incorporating stored and previous training parameters during navigation. The one step training procedure is recommended at the early stage for building the intelligent navigator as the navigation knowledge acquired by the navigator at this moment might not be enough to provide acceptable accuracy during long term INS error. As the size of DataBase is quite small and its high cost, the incorporation of stored training parameters doesn't slow down the learning process during each window; actually, it can provide better generalization of the navigation knowledge, but when the size of DataBase increases this will slow down the learning process and increase the cost.

Two steps training procedures: the learning parameters acquired for each INS temporal window during navigation are obtained at the end of each INS window. After navigation, all training parameters acquired during the navigation are recalled and combined with the stored training parameters then fed into the navigator to improve the generalization of navigation knowledge using a conventional off-line batch training method [11]. This procedure is recommended for the regular operational stage for building the intelligent navigator. After several field tests, the navigator might accumulate enough navigation knowledge to provide navigation solutions during navigation without incorporating stored learning parameters.

In other words, the size of training samples is the same as the INS temporal window. Therefore, the training speed during each window is expected to be faster than the previous procedure. After navigation, all the training parameters acquired during the current navigation are recalled and combined with the stored training parameters first to remove redundant navigation knowledge, and are then re-trained to improve the generalization of the navigation knowledge for future navigation missions. This will ensure to keep DataBase at specified size to avoid slowing down in learning process.

The key factor that can accelerate the learning is the generalization of navigation knowledge. The perfect solution is to obtain the most generalization navigation knowledge that can then be fed into the navigator in one field test. However, that is not the case for real life applications. Therefore, the navigator must have the ability to evolve during each navigation mission to provide generalized navigation knowledge for future missions. Thus, using the proposed intelligent navigator, DataBase, and temporal windowing technique for updating learned parameters, the intelligent navigator has the ability to generate and accumulate the navigation knowledge.

As shown in figure (1) the temporal windowing technique for training learned parameters consists of three main parts as shown below:

1- Learned parameters initialization: the initial learned parameters P(0) can be obtained using previously stored parameters in the DataBase or can be generated through random initialization. In this paper the initial parameters were obtained using random initialization. And after each training mission the learned parameters can be stored and then used in next navigation mission or can be used in long term navigation mission also to reduce the unbounded INS error. It is worth to say that accurate initial parameters may significantly reduce the training time required and the MSE.

2- Short-term navigation: with in the first INS window (t=1), INS (t), the learned parameters are not updated, thus the stored learned parameters P(0) are still the initial learned parameters. And in the next INS window, INS (t+1), the stored parameters P(t-1) (i.e., P(0)) are updated utilizing the previous stored learned parameters in P(t). and these steps are repeated until long term navigation is detected.

3- Long-term navigation: in the case of long-term navigation, then the previous learned parameters (P(t-1)) is first applied for real time prediction and then p(t) is utilized to replace P(t-1) and carry on real time prediction during long-term navigation.

Since the ANFIS training procedure needs time for training, updating the learned parameters immediately at the latest available sample of IMU signal before long-term navigation is difficult. However, the utilization of the proposed technique can still provide reasonable prediction accuracy during IMU long term navigation since it provides the latest updated parameters instead of real time updated parameters for real time prediction. Therefore, failure in providing real time updated learned parameters doesn't mean the intelligent navigator is not able to provide real time prediction.

On the contrary, it can utilize the latest acquired and learned navigation knowledge to provide real time navigation solutions. Combining the latest INS window learned parameters; the stored parameters can be adaptively updated to follow the latest motion dynamics thus improving the prediction accuracy during long-term navigation.

As mentioned previously the 5th layer (output layer) generates velocity and position in the local level frame at the current epoch. Thus, the navigation knowledge can be learnt, stored and accumulated during the availability of the INS signal. On the other hand, during IMU errors, the latest acquired navigation knowledge can be retrieved from the "brain" (navigation information DataBase) of the intelligent navigator to predict the velocity and position in real time. The proposed navigator consist of two main procedures, the initialization and training procedures so, if the network well initialized and trained a good prediction will be obtained.

During the initializition of ANFIS network, where the parameters x, c, and σ are the ANFIS learning parameters computed during the training phase and they are determine the input/output functionality of the network. The initial values for the ANFIS learning prarmeters are obtained by trail and error and they are stated as in table (3). in contrast, the selection of the number of rules and the learning rate are very important to get fast and accurate training as illustrate below:

i Selection of the learning rate.

The selection of different values of the learning rate which affect the speed of convergence during the training process will be presented below.

To illustrate the relationship between the value of the learning rate and the MSE for all networks of position and velocity, the learning rate was varied from 0.1 to 0.9 in a step of 0.1. For each of these values of the learning rate, the networks were initialized randomly over specified ranges of x, c, and σ. The same initial ranges will be used for all values of the learning rate.

For each network, the same initial values listed in table (3) were used to obtain the results shown in figure (7). For each value of the learning rate, the number of epochs was 10 and the number of rules was 15 in all networks.

It was noticed that when the learning rate was small, the network will adjust its parameters gradually but in this case convergence might be slow, on the other hand, a high learning rate might make drastic changes that are not desirable; therefore, medial values of the learning rate are preferable to choose.

Also, it can be seen from figure (7) that the value of learning rate equal to 0.4 or 0.5 gives small error value in most of the networks; therefore, the value of 0.4 for the learning rate was used to obtain the results from all networks in the training process.

The ANFIS used in this paper to predict the SDINS position and velocity in real time. So after completing the initialzintion procedure the ANFIS network can be trained and according to figure (3) which illustrates the training procedure of the ANFIS, the network output is compared to the SDINS algorithm output. figure (8) shows the error between the desired and actual output of the ANFIS, this error is feed to the ANFIS network, which adjust the network learning parameters in a way to minimize the mean square value of error.

The output is obtained and compared to the target (desired performance) to determine the estimation error. This error is propagated through the network in the backward direction (opposite to the flow of the input data) starting from output layer and is utilized to update the computation of the network parameters. The forward and backward computations are repeated until the optimal value of the learning parameters are achieved, which correspond to certain objective mean square estimation errors, The network parameters are updated according to certain learning rules to minimize the mean square value of the estimation error. the training process continued for 10 epochs to reduce the MSE value and the relation between the number of epochs and MSE is shown in figure (9).

After the training is completed the network is ready to work in the prediction mode. However the parameters of the networks are modified during the availability of the IMU signals (i.e. the training procedure continues) and network is considered working in the update mode. During IMU errors, the network will use the latest estimation parameters saved in the DataBase to perform the prediction process as shown in figure (4).

ii. Selection of the number of rules.

As mentioned previously, the arbitrarily selection of the number of rules (M) is also based on the trial-and-error procedures. The purpose of this illustration to show the relationship between the number of rules and the mean square error (MSE) for the ANFIS network.

The value of M was varied from 5 to 50 in a step of 5. For each value of M, the network is initialized randomly over specified ranges of the parameters x, c, and σ. These ranges will be the same for all values of M.

The results shown in figure (6) were obtained after implementing the six networks of position and velocity components using the same initial values listed in table (3). For each value of M, the number of epochs was 10 and the value of the learning rate was 0.4 in all networks.

It was noticed that, as the number of rules increases as the training process becomes slower; therefore, if the value of the error is decreasing then because of this slow training process, the convergence to a minimum error value will also be slow and the specified number of iterations might be ended without reaching that minimum error value. Hence it can be noticed from figure (6) that, in general, as the number of rules increases as the value of the MSE increases. Even if a high value of M achieves small error value, because of the randomly initialized parameters, it is not recommended to use this value because the training process will be too slow and requires a lot of time which is not practical in real time prediction, i.e. it is better to look for the value of M that achieves minimum error and with minimum time (small M values).

It can also be seen from figure (6) that the values of M equal to 5, 10, 15, and 20 give minimum error values in all networks; therefore, M=15 was used to implement all the networks in learning process.

9. Conclusions

The conclusions drawn from the results presented in this paper are:

In this paper, an attempt to build a reliable navigation system is made by combining the merits of INS, and ANFIS.

The parameters of the intelligent navigator are included in the navigation knowledge. Thus they can be updated without a human expert during navigation whenever newly updated navigation knowledge is acquired.

The long procedure of trial-and-error in finding the optimal number of layers in the network, the number of nodes in each layer, and the activation function in each node, which exists in Artificial Neural Network (ANN), has been avoided in this paper by using the proposed ANFIS with its constant structure as described in this paper.

The appropriate selection of the initial values of the parameters x, c, and σ has a significant effect on the performance of the ANFIS network and plays an important role in decreasing the convergence time to the best solution (minimum error value).

The selection of M (number of fuzzy rules) is essential in achieving good results. It was noticed that using large number of rules results in slow training and large error values whereas the small M values lead to small error values and fast training performance.

The speed of convergence during the training process is remarkably affected by the value of the learning rate. Small values of learning rate cause slow convergence, on the other hand, undesirable results may appear when using large values; therefore, most reasonable performance is achieved with medial values of learning rate.

The results presented in this paper strongly indicate the potential of including the intelligent navigator as the core navigation algorithm for the next generation navigation system.

10. Suggestions for Future Work

From the previous work presented in this paper, some ideas may be introduced for future work:

Using neural network adaptive wavelets (wavenet) to filter out the noise that exists at the IMU outputs.

Finding the appropriate initial values of the parameters x, c, and σ by means of the Genetic Algorithm.

Using other types of fuzzy logic system structure that can be built based on different combinations of the fuzzy inference engine, fuzzifier, defuzzifier, and membership function.

The proposed conceptual intelligent navigator can be used instead of Kalman filter or others traditional techniques to support INS/GPS integrated system, and to generate more reliable navigation solutions.

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