Neural network based data fusion for vehicle positioning in

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1 04ANNUAL-345 Neural network based data fusion for vehicle positioning in land navigation system Mathieu St-Pierre Department of Electrical and Computer Engineering Université de Sherbrooke Sherbrooke (Québec) Canada J1K 2R1 Copyright 2003 SAE International Denis Gingras Department of Electrical and Computer Engineering Université de Sherbrooke Sherbrooke (Québec) Canada J1K 2R1 ABSTRACT Land navigation systems need a precise and continuous position in order to function properly. The sensors commonly found in those systems are differential odometer, global positioning system and 2 or 3 axis inertial measurement unit respectively. Two or more of these complementary positioning methods must be integrated together to achieve the required performance at low cost. The integration, which implies the fusion of noisy data provided by each sensor, must be performed in some optimal manner. Most positioning system designers choose the Kalman filter as the data fusion method. An interesting alternative to the Kalman filter is the artificial neural network (ANN). This paper describes the research conducted to evaluate the potential of an ANN as a centralized fusion method and as nonlinear filters for land vehicle positioning. INTRODUCTION The principal function of a land navigation system in a vehicle is to guide the driver while minimizing the trip duration and/or the traveled distance. The position must be determined with a precision of 20 meters or less and at a frequency of 1 hertz or more. alone is usually able to estimate the position with the required performance. However, possible occlusions of satellite signals by high buildings or heavy foliages may prevent computation of a solution by a receiver for several seconds. A differential odometer measures the traveled distance and the azimuth variation of a vehicle during a sampling period. A new position can be computed with these measurements and the last known position by dead reckoning. An inertial system measures the vehicle's inertia characterized by it's acceleration and it's angular velocity. By means of integration, the traveled distance and the azimuth variation can be computed and therefore a new position can be reckoned from the last known position. The recursive nature of the positioning computation cause the positioning error to grow proportionally with time. A periodic reset is needed. The Kalman filter is an optimal linear estimator which

2 minimizes the mean square error. Non-linearities can also be dealt with the extended Kalman filter. However, the performance analysis of the extended Kalman filter presents some difficulties due to the recurrence of the measure sequence into the states of the filter [6]. Also, implementation of the extended filter can be quite laborious depending on the number of states required to modelize the system. VIRTUAL ENVIRONMENT SIMULATION dˆ [] t = d [] t + s ( d [] t + d []*sin( t δ ) i i i i j ij + d []*sin( t δ )) + b[0] + w [ n] k ik i bi n= 1 + d []*sin( t δ ) + d []*sin( t δ ) + c ( d [ t] + d [ t]*sin( δ ) + d [ t]*sin( δ )) + wt [] j ij k ik i i j ij k ik i t 2 (1.2) The data used for testing the algorithms presented in this paper was generated by a virtual environment simulation. A virtual environment avoid the time, expense and difficulty encountered when conducting test track experiments. Other benefits of working in a simulated environment include the availability of ground truth measurements, the repeatability of experiments, easy modification of testing conditions and rapid turnaround of system modifications[3]. VEHICLE`S DYNAMIC Kinematics of a land vehicle in three dimensions has been simulated with the road and the vehicle's acceleration as inputs. To improve the simulation realism, the road has been taken from real digital topographic information and a real vehicle with an accelerometer has traveled this road to generate the vehicle's acceleration data. Even though the acceleration data are noisy, the simulation represent more accurately a real road situation than if the acceleration data had been generated randomly. SENSORS The simulator emulates a receiver, a 3-axis inertial measurement unit and a differential odometer. The model can be described by equation (1.1). dˆ i [] t is the measurement, di[] t is the real value, s i is the scale factor error, d j[] t is the real value relative to the axis j, dk[] t is the real value relative to the axis k, δ ij is the misalignment angle between the axis i and j, δik is the misalignment angle between the axis i and k, b [0] i is the turn-on bias, wbi[ n] is the random walk white noise characterizing the bias drift, c i is the nonlinear scale factor and wt i[] is the additive white noise component. The scale factor error, the bias parameters, the nonlinear scale factor and the additive white noise component have different value depending on the sensor type. A differential odometer is constituted of two sensors measuring the number of rotation of each wheel situated on the same axle. The total traveled distance and the azimuth of the vehicle can be computed with the equation (1.3) and (1.4) respectively. dt ˆ[] = dt ˆ[ 1] + (1 + vd * v[ t] + sd) * cd[ t] + r[] t + 2 (1 + vg * v[ t] + sg) * cg[ t] + r[ t] 2 (1.3) ˆ[ ρ t] = ρ[] t + δ [] t + δ [] t + δ [] t + δ [] t (1.1) trop iono white mult where ˆ[ ρ t] is the pseudorange measured by the emulated receiver, ρ [] t is the real pseudorange, δ trop[] t is the tropospheric delay, δ iono[] t is the ionospheric delay, δ white[] t is the white noise generated by the receiver's electronic components and δ mult[] t is the multipath problem. The 3-axis inertial measurement unit has a single model for every gyroscope and accelerometer described by equation (1.2). ˆ θ[] t = ˆ θ[ t 1] + (1 + vg * v[ t] + sg) * cg[ t] + r[] t l (1 + vd * v[ t] + sd) * cd[ t] + r[ t] l (1.4) dt ˆ[ ] is the total traveled distance at time t, dt ˆ[ 1] is the total traveled distance at time t-1, vt [] is the car's velocity, v d and v g are the gains characterizing the tire dilatation, sd and sg are the scale factors for the right and the left wheel respectively, cd[] t and cg[] t are the

3 numbers of rotation measured within the interval [t-1,t] rt is a for the right and left wheel respectively, [ ] uniform random variable describing the resolution error, ˆ[ θ t] is the azimuth estimated at time t, ˆ[ θ t 1] is the azimuth estimated at time t-1 and l is the axle length. Total traveled distance Azimuth solution availability Sampling time Differential odometer Differential odometer CENTRALIZED FUSION A centralized fusion implied the use of only one algorithm which fuses all the measurements provided by the sensors. The engineering labour requires to construct a centralized Kalman fillter can be very high. For example, most of the / systems have a decentralized filter which gains in simplicity. However, the price to pay is a loss of precision compared with a centralized one [4]. On the other hand, a centralized ANN is no more difficult to realize than a decentralized one. ANNs have already been applied to the data fusion related to positioning problem in robotic with success [1] [5]. The major advantage of an ANN compared to a Kalman filter resides in the fact that it doesn't need any a priori statistical and mathematical model to find a function which maps optimally the inputs with the outputs, in our case the absolute position. The most difficult task is to gather some sensor data which can cover adequately the different manoeuvres to be encountered by a road vehicle. The manoeuvres are a function of the road geometry and the vehicle's performance. For example, an ANN trained with measured data coming only from a straight road segment may not be able to give a good position when the vehicle meets a curve. Table 1: ANN's inputs Data Latitude Longitude Altitude PDOP Linear acceleration Sensor ANN S DESCRIPTION A feed-forward backpropagation neural network was trained with training data set for 2000 epochs. The ANN has 4 layers composed of 20, 20, 20 and 3 neurons respectively. The corresponding transfer functions are linear, log-sigmoid, tan-sigmoid and linear. This architecture was the most promising of the 12 architectures investigated with various number of layers, transfer functions and number of neurons trained initially for 100 epochs. Batch training was prefer over iterative training for it s computation efficiency. The mean square error gave the training performance at each epoch. The second order training method was the scaled conjugate gradient. This method requires only O(N) operations per epoch compared to others methods like Gauss-Newton O(N 3 ) and Levenberg-Marquardt O(N 3 ) [8]. When considering the network size, the other training methods were impractical. Table 1 contains the ANN's inputs. The receiver sampling frequency is usually lower than those of the inertial measurement unit or the differential odometer. For synchronization purpose, a boolean entry specifies to the ANN if a solution is available. A centralized Kalman filter has been realized to has a reference for the evaluation of the ANN's performances. The centralized Kalman filter has 13 states and 10 measurements. The simulation generates data samples. RESULTS The mean and variance of the positioning error during the simulation was computed. Table 2 indicates that the positions estimated by the ANN are less biased for the longitude and altitude but more biased for the latitude than the same positions estimated by the Kalman filter. The ANN s estimation biases don t exceed 10 meters. So even if the ANN is a biased estimator, it still meets the required performance. Horizontal centripetal acceleration Vertical centripetal acceleration Yaw angular velocity Pitch angular velocity As shown in table 3, the variances of the latitude errors and the altitude errors for the ANN are less than those of the Kalman filter. The variance of the longitude errors is 5 percent more for the ANN than for the Kalman filter. The performance of the ANN is generally better than the peformance of the Kalman filter in a mean square error sense. The important gain for the altitude is caused by the ANN's ability to estimate the bias of the altitude.

4 Table 2: Mean of the positioning errors by the Kalman Filter and the ANN Position Kalman (m) ANN (m) Gain (%) Latitude Longitude Altitude Table 3: Variance of the positioning errors by the Kalman filter and the ANN Position Kalman ANN Gain (%) Latitude Longitude Altitude The ANN's input signals include the measured pseudorange, the elevation angle of each tracked satellite and the sampling time. The ANN outputs a filtered pseudorange. The elevation angle gives information on the tropospheric and ionospheric noises which are greater for satellites at lower elevation. The sampling time helps the ANN to estimate tropospheric, ionospheric and multipath noises which are temporally correlated. The sampling time should be reset periodically to improve the capacity of the ANN to generalize the training data. The reset period should be set to one day or more because the tropospheric noise is larger during daytime and smaller at night. Figure 1: Pseudorange non-linear filtering by an ANN DATA PREPROCESSING In every system, some noises perturb the useful signals. It can be either white or colored, linear or nonlinear, determinist or random. Nonlinearity makes the filtering particularly difficult. An optimal filtering can be obtained with a priori knowledge of the nonlinear noise. However, this information is generally not available[7]. An ANN is an interesting solution to this problem because of it's ability to model a complex non-linear function without any a priori knowledge of the noise. PSEUDORANGE ESTIMATION A receiver measures the ranges between it's antenna and each tracked satellite. A mathematical model for the pseudorange additive noises is described by the equation (1.1). These noises depend on many factors that are difficult to measure. The position is obtained by triangulation with the measured distance and the known position of each tracked satellite. At least four satellites must be visible to get a position and the receiver's clock error. Triangulation implies solving a set of nonlinear equations. The impact of noise on the computed position is difficult to determine analytically. A feed-forward backpropagation neural network with 3 layers was trained to perform the nonlinear filtering. The layers have 3 linear neurons, 5 tan-sigmoid and 1 linear neuron respectively. The training method was the Levenberg-Marquardt that appears to be the fastest method for training moderate-sized feed-forward neural networks (up to several hundred weights) [2]. Results Figure 1 shows the important noise diminution on the pseudoranges. The filtered pseudoranges are used to reckon the position. The important error bias pertains to the receiver clock offset and has little impact on the precision of the position. So the performance gain on the computed positions shown in table 4 is much less than the gain on the pseudoranges. Table 4: Error variance of the positions Parameters No ANN filtering ANN filtering Gain (%) Latitude Longitude Altitude ANN Description DIFFERENTIAL ODOMETER The analytical relation between differential odometer measurement errors and dead reckoning computed positions is difficult to obtain. As with the, an ANN

5 seems an interesting solution to diminish the noise on data measured with a differential odometer. ANN description A feed-forward backpropagation neural network with 2 layers was trained for this purpose. The layers have 10 log-sigmoid neurons and 2 linear neurons respectively. The training method was the Levenberg-Marquardt. The ANN must estimate the traveled distance and the azimuth variation in the time interval [t-1,t]. These two values correspond to the last term of the equations (3) and (4). The time interval is usually one second or less. Table 5: Error variance on the data provided by the differential odometer Figure 3: Azimuth variation error during a sampling period Parameters Traveled distance Azimuth variation No ANN filtering ANN filtering Gain (%) 0.81 * * * * CONCLUSION In this paper, ANNs have been used as a centralized fusion method and as a nonlinear filter for the land navigation positioning problem. The results show that ANN is an attractive alternative to the Kalman filter as a centralized fusion method. The ANN's capability to learn nonlinear function was also successfully applied to and differential odometer measurement pre-filtering. The major difficulty with an ANN is to have access to ground truth data for the supervised training. Usually, the only solution is to have access to some data coming from a reference usually given by high precision, high cost sensors. Some further researches on this topic include the evaluation of the developed ANN with real sensors and the replacement of the feed-forward backpropagation ANN with a recurrent ANN. ACKNOWLEDGMENTS Figure 2: Error of the traveled distance during a sampling period Results The magnitude of the traveled distance and the azimuth variation are very small. The reasons for this are the low dynamic of a land vehicle and the short sampling period. Therefore good coverage of the input data range is easier and the trained ANN should be more robust. An important diminution of the error variance can be observed when the ANN is used. The figure 2 and 3 show the error diminution on the raw odometer data obtained with the ANN filtering for the whole path. The authors would like to thank the Canadian NCE AUTO21 for funding this research project, the University of Calgary for providing the acceleration data and Geomatic Canada for the topographical information. REFERENCES 1. T. F. Futoshi Kobayashi, Fumihito Arai. Sensor fusion system using recurrent fuzzy inference. Journal of Intelligent and Robotic Systems, 23, October M. T. Hagan and M. Menhaj. Training feedforward networks with the Marquardt algorithm. IEEE Transactions on Neural Networks, K. T. Keith A. Redmill, John I. Martin. Sensor and data fusion design and evaluation with a

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