Original article
Multisensor integration using neuron computing for land-vehicle navigation Kai-Wei Chiang Æ Aboelmagd Noureldin Æ Naser El-Sheimy
users anywhere on the planet. Since its advent, the number of applications using GPS has increased dramatically, including tracking people, a fleets of trucks, trains, ships or planes and how fast they are moving; directing emergency vehicles to the scene of an accident; mapping where a city’s assets are located; and providing precise timing for endeavors that require large-scale coordination. However, GPS can provide this type of information only when there is direct line of sight to four or more satellites. In other words, the system does not work well in urban areas due to signal blockage and attenuation, which may deteriorate the positioning accuracy. For the moment, any sophisticated urban application, which essentially demands continuous position determination, cannot depend on GPS as a standalone system. More recently, and accepting that these techniques must inevitably cost more than GPS as a standalone system, the concept of combining complimentary navigation systems, such as a dead reckoning (DR), inertial navigation system (INS), or a navigation aid, such as digital map database, have been integrated in commercial applications. Certainly it has always been a maxim in safety-related applications that it is imprudent to depend on a single navigation technique. The complimentary navigation systems should have error mechanisms that are disjoint. The resultant system design is then driven by a trade-off between cost and performance (El-Sheimy and Introduction Naser 2000). This article focuses on the implementation of a multisensor integration based on utilizing multilayer Today, most vehicle navigation systems mainly rely on feed-forward neural networks with a back propagation global positioning system (GPS) receivers as the primary learning algorithm. source of information to provide the position of the veThe basis of multisensor integration is to fuse all available hicle (Shin and El-Sheimy 2002). GPS is a satellite-based data from various sensors in order to obtain an optimal all-weather radio navigation system, developed by the navigation solution (Ashkenazi et al. 1995). Traditionally, United States Department of Defense (DoD), which a Kalman filter is used to combine data from various became fully operational in 1994. The system can provide sensors, which may contain different sources of errors. precise positioning information to an unlimited number of Figure 1 shows a simplified scheme of the Kalman filter process. Before the estimation process starts, values for the initial error state ^xþ Received: 17 May 2002 / Accepted: 27 July 2002 0 and the corresponding error covariance Pþ are assumed. Consequently, the filter projects the Published online: 5 November 2002 0 ª Springer-Verlag 2002 state and error covariance ahead to estimate x^ k andPk , this is called the prediction mode. If new measurements at time epoch k are available, the filter starts the updating K.-W. Chiang (&) Æ A. Noureldin Æ N. El-Sheimy mode by computing the Kalman gainKk and updating the Department of Geomatics Engineering, þ error state and error covariance to estimate x^þ k and Pk . The University of Calgary, 2500 University Dr. NW, The Kalman filter incorporates all of this information Calgary, Alberta T2N 1N4, Canada together to provide an optimal estimate of the error states E-mail:
[email protected] Tel.: +1-403-2208794 or +1-403-2106263 at time k (Brown and Hwang 1992). Abstract Most of the present navigation sensor integration techniques are based on Kalman-filtering estimation procedures. Although Kalman filtering represents one of the best solutions for multisensor integration, it still has some drawbacks in terms of stability, computation load, immunity to noise effects and observability. Furthermore, Kalman filters perform adequately only under certain predefined dynamic models. Neuron computing, a technology of artificial neural network (ANN), is a powerful tool for solving nonlinear problems that involve mapping input data to output data without having any prior knowledge about the mathematical process involved. This article suggests a multisensor integration approach for fusing data from an inertial navigation system (INS) and differential global positioning system (DGPS) hardware utilizing multilayer feedforward neural networks with a back propagation learning algorithm. In addition, it addresses the impact of neural network (NN) parameters and random noise on positioning accuracy.
DOI 10.1007/s10291-002-0024-4
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Fig. 1 Outline of Kalman filter
Fig. 2 Basic model of neuron
Although the Kalman filter represents one of the best solutions for multisensor integration, it still has some drawbacks. The Kalman filter only works well under certain pre-defined models. If the filter is exposed to input data that does not fit the model, it will not result in reliable estimates (Forrest et al. 2000). Another problem related to the Kalman filter is the observability of the different states. The system is considered to be non-observable if there are one or more state variables that are hidden from the view of observer (i.e., the measurements). Consequently, if the unobserved process is not stable, the corresponding estimation errors will be similarly unstable (Brown and Hwang 1992; Ibrahim et al. 2000). For example, if the error state equation of an INS is examined, one can determine an azimuth error state that is weakly coupled with the velocity error states (Salychev 1998). Therefore, optimal estimates of the velocity errors provided by the Kalman filter due to GPS position or velocity updates will not benefit the azimuth accuracy. Therefore, the azimuth error state is a weakly observable component (Noureldin 2002). The objectives of this paper are to (1) suggest a new multisensor integration method utilizing multilayer feedforward neural networks with a back propagation learning algorithm, (2) evaluate the proposed architecture utilizing field test data, and (3) investigate the impact of the neural 210
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network parameters and the random noise on position errors.
Artificial neural network-based method for multisensor integration Artificial neural networks (ANNs) are designed to mimic the human brain and duplicate its intelligence. Based on their highly parallel architecture, ANNs are powerful tools for solving nonlinear problems that involve mapping input data to output data (Abhijit and Robert 1996). It has been shown that an ANN can approximate any continuous and differentiable function to any degree of accuracy and it can model complex problems without any prior knowledge of the mathematical processes involved (Chansarkar 1999). An ANN will internally adjust the processing structure according to the discrepancy between the network’s output value(s) and the desired target output value(s) in the training session. The adaptability, the nonlinear processing and the parallel processing are characteristics that make the ANNs
Original article
Fig. 3 Sigmoid (logsig) activation function
Fig. 4 Proposed multilayer neural network topology
enhanced by extensive interconnectivity. This provides concurrent processing as well as parallel-distributed information storage. In general, ANNs could be divided into two classes. The first class is the supervised NN, which is trained by exposing the network to a series of training samples. These training samples contain an input data set as well as the desired output data set (i.e., multilayer feed-forward network trained by a back propagation algorithm). The second class is the unsupervised NN, which groups similar input vectors together without the use of desired output data to specify what a typical member of each group looks like or to which group each vector belongs (i.e., selforganizing map; Haykin 1999). Basic model of neurons ANNs are constructed from small processing units (neurons) that are interconnected within the network using weighted links. Figure 2 shows the basic model of the neuron, which contains three major components: (1) weight links wkj; (2) an adder for summing the input signals that are weighted by respective synapses of the neuron (mk) and external bias (bk); and (3) an activation function uðÞ for limiting the amplitude of the neuron output and the final output yk. A nonlinear activation function is utilized so that the non-linearities can serve to enhance the network’s classification, approximation capabilities, and reduce the impact of noise (Reed and Marks 1999). Figure 3 shows the sigmoid (logsig) activation function, which is defined as follow: uðvÞ ¼
important in wide variety of applications (Dumville and Tsakiri 1994). Adaptability is a powerful learning algorithm utilized by ANNs in order to adapt to a continually changing environment. Nonlinear processing is the ability of ANN to perform tasks involving nonlinear input/output mapping relationships (Ham and Kostanic 2001). The parallel processing property allows the ANN to have architectures with a large number of neurons
1 1 þ expðavÞ
ð1Þ
where (a) is the slope parameter. ANN design criterion The proposed architecture uses a three layer feed-forward NN with a back propagation learning algorithm to integrate the data from INS and differential global positioning system (DGPS) and mimic the dynamical model of the
Fig. 5 Proposed multilayer neural network training
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Fig. 6 activation function, enables the ANN to solve a nonlinear a Smooth curve trajectory. b Harsh curve trajectory. c Training plot input/output relationship. example for smooth curve (512 neurons). d Training plot example for When the input layer receives the input, its neurons harsh curve (512 neurons)
vehicle. After training the NN, it can be used to predict the vehicle’s position during GPS signal blockage. As shown in Fig. 4, the network inputs are the INS velocity V(t–1), and the INS heading wðt 1Þ. The network outputs the difference in coordinates between two different epochs, i.e., N(t) and N(t–1) for the north component and E(t) and E(t–1) for the east component. The proposed architecture contains an input layer, a hidden layer, and an output layer. The input layer consists of neurons that receive input from the external environment. The output layer consists of neurons that communicate the output of the system to the user or external environment. There are usually a number of hidden layers between these two layers, yet the proposed architecture consists of only one hidden layer. A hidden layer, together with a nonlinear
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produce an output. The output is then input to the other layers of the system. The process continues until a certain condition is satisfied or until the output layer is invoked and fires their output to the external environment. In general, the optimal architecture is empirically chosen. Hence, there is no guideline for specifying how many hidden layers and neurons should be used. Yet it might be appropriate to use more hidden layers and neurons for complex problems. Insufficient neurons might result in the divergence of the network while too many neurons could result in over-constraining the model (Lippman 1987). The INS navigation parameters are continuously computed and applied to the NN. Position differences between the current and the previous DGPS solutions, DN(t) and DE(t), are set at the output side of the NN as the desired target. It was decided to use the position differences instead of the position itself during the training procedure
Original article
Fig. 7 Impact of number of neurons on position errors
to simplify the learning process. In fact, the differences Back propagation algorithm helped to reduce the complexity of the input/output function relationship as they provide a more efficient NN ANNs do not need any prior knowledge of the mathetraining and reduce the required training time. matical model of the problem. They learn by training samples, which include the input data and the desired output data. As shown in Fig. 5, the signals are propagated Real-time learning process The learning process is performed in real-time to deter- through the network. The final network outputs yk(t), mine the NN parameters (the weights and the biases). The [DN(t), DE(t)] are compared with the desired outputs INS velocity and heading information are used as inputs Dk(t) [GDN (t), GDE (t)] and the network error Ek(n) is and the NN outputs are compared with the DGPS position computed. differences. As long as the DGPS signal is available, the Ek ðtÞ ¼ Dk ðtÞ Yk ðtÞ ð2Þ learning process is continuously improving the estimation error in order to obtain optimal values of the NN Standard back propagation is a gradient descent algorithm parameters. During and beyond a GPS outage, the NN and the term ‘‘back propagation’’ refers to the manner in parameters are used in prediction mode to provide esti- which the gradient is computed for nonlinear multilayer mates for the position components along the east and the networks. The back propagation algorithm runs backwards north directions. from the output layer through all the hidden layers to the Table 1 Factors in different simulated cases
Cases
Standard deviation
Velocity errors
Heading errors
A
B
C
D
a
1 (m/s)
10 (m/s)
50 (m/s)
100 (m/s) 1
b
c
3
5
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Fig. 9 Effect of heading errors on position errors Fig. 8 Effect of velocity errors on position errors
The weights are adjusted according the generalized delta rule as follows: ð‘Þ
input layer. The network error is used to adjust the weights associated with the connection and neurons by applying a generalized Delta rule (Haykin 1999). This rule states that the learning process is proportional to the difference between the NN output and desired output. The whole procedure starts with initializing the NN parameters (the weights and the biases) and the learning-rate parameter (c¸). The learning parameter is a small positive constant that controls the step size of the iterative changes during the learning process (0
