2.1. Dynamics Model

The system error dynamic model of integrated navigation used in the Kalman filter is designed based on the INS error equations. The insignificant terms are neglected in the process of linearization (Titterton, Reference Titterton2004). The psi-angle error equations of INS are as follows (Han and Wang, Reference Han and Wang2012):

(1)$$\delta {\bi \dot r} = - {\bi \omega} _{en} \times \delta {\bi r} + \delta {\bi v}$$

(2)$$\delta {\bi \dot v} = - (2{\bi \omega} _{ie} + {\bi \omega} _{en} ) \times \delta {\bi v} - \delta {\bi \psi} \times {\bi f} + {\bi \delta} $$

(3)$$\delta {\bi \dot \psi} = - ({\bi \omega} _{ie} + {\bi \omega} _{en} ) \times \delta {\bi \psi} + {\bi \varepsilon} $$

where δ r, δ v and δψ are the position, velocity and orientation error vectors, respectively. ω_en is the rate of navigation frame with respect to earth, and ω_ie is the rate of earth with respect to inertial frame. The system error dynamics of GPS/INS integration is obtained by expanding the accelerometer bias error vector η and the gyro drift error vector ε.

The accelerometer bias error vector η and the gyro drift error vector ε are regarded as the random walk process vectors, which are modelled as follows (Wang et al., Reference Wang, Lee, Hewitson and Lee2003):

(4)$${\bi \dot \eta} = {\bi u}_\eta $$

(5)$${\bi \dot \varepsilon} = {\bi u}_\varepsilon $$

where u_η and u_ε are white Gaussian noise vectors.

By combining Equations (1) to (5), the system dynamical model becomes:

(6)$$\left\{ \matrix{\delta {\bi \dot r} = - {\bi \omega} _{en} \times \delta {\bi r} + \delta {\bi v} \cr \delta {\bi \dot v} = - ({\bi \omega} _{ie} + {\bi \omega} _{in} ) \times \delta {\bi v} - \delta {\bi \psi} \times {\bi f} + {\bi \eta} \cr \delta {\bi \dot \psi} = - {\bi \omega} _{in} \times \delta {\bi \psi} + {\bi \varepsilon} \cr {\bi \eta} = {\bi u}_\eta \cr {\bi \varepsilon} = {\bi u}_\varepsilon } \right.$$

which can be generalized in matrix and vector form:

(7)$${\bi \dot X} = {\bi \Phi X} + {\bi u}$$

wherein X is the error state vector, Φ is the system transition matrix, and u is the process noise vector.

2.2. GPS/INS Observation Model

The observation model in GPS/INS integrated navigation is composed of the position and velocity difference vector between the GPS solutions and the INS computation value:

(8)$$\eqalign{Z_r (t) = & r_{GPS} (t) - r_{INS} (t) \cr = & r_{GPS} (t) - \left( {r_{INS} (t - \Delta t) + v_{INS} (t - \Delta t) \cdot \Delta t + \displaystyle{1 \over 2}\alpha (t) \cdot \Delta t^2} \right)} $$

(9)$$\eqalign{Z_v (t) = & v_{GPS} (t) - v_{INS} (t) \cr = & v_{GPS} (t) - (v_{INS} (t - \Delta t) + \alpha (k) \cdot \Delta t)} $$

where Z _r(t) is the position error measurement vector at t time, Z _v(t) is the velocity error measurement vector, r _GPS(t) is the GPS position vector, r _INS(t) is the INS position vector, v _GPS(t) is the GPS velocity vector, v _INS(t) is the INS velocity vector, α(t) is the acceleration vector determined by the INS alone, and Δt is the sample time of IMU.

The generic measurement equation system of the Kalman filter can be written as:

(10)$${\bi Z}_k = \left[ {\matrix{ {Z_r (t)} \cr {Z_v (t)} \cr}} \right] = {\bi B}_k \left[ {\matrix{ {{\bi X}_{Nav}} \cr {{\bi X}_{Acc}} \cr {{\bi X}_{Gyo}} \cr}} \right] + \left[ {\matrix{ {\tau _r} \cr {\tau _v} \cr}} \right]$$

where B_k is the observation matrix, and τ is the measurement noise vector, assumed to be white Gaussian noise.

2.3. INS/odometer Observation Model

2.3.1. Odometer Position and Velocity

An odometer can only measure the overall velocity of the vehicle. Suppose the vehicle velocity output by odometer is v _O and its direction is the same as the front of the vehicle. The velocity of the vehicle observed by odometer can be written in the navigation frame (Yan, Reference Yan2006):

(11)$${\bi v}_O^n = {\bi C}_b^n \left[ {\matrix{ 0 & {v_O} & 0 \cr}} \right]^T $$

where C_bⁿ is the direction cosine matrix which defines the attitude of the body frame with respect to the navigation frame. In INS/odometer integrated navigation, C_bⁿ can be obtained by INS computation.

The integration of velocity in the navigation frame is the position of the vehicle. The differential equation of position is expressed as:

(12)$$\dot L_O = \displaystyle{{v_{ON}^n} \over {R_M + h_O}} $$

(13)$$\dot \lambda _O = \displaystyle{{v_{OE}^n} \over {(R_N + h_O )\cos L_O}} $$

(14)$$\dot h_O = - v_{OD}^n $$

where L _O, λ_O and h _O are the longitude, latitude and height information calculated by INS/odometer integration system respectively. R _M and R _N are meridian and traverse radii of curvature in the earth ellipsoid. v ⁿ_ON, v_OEⁿ and v _ODⁿ are the velocity component of north, east and down directions in the local level navigation frame.

Because the position and velocity information of the odometer are sampled discretely, the computing method of the odometer position can be deduced based on the differential equation of position:

(15)$$L_O (k) = L_O (k - 1) + \displaystyle{{v_{ON}^n \cdot \Delta t} \over {R_M + h}}$$

(16)$$\lambda _O (k) = \lambda _O (k - 1) + \displaystyle{{v_{OE}^n \cdot \Delta t} \over {(R_N + h)\cos L_O (k - 1)}}$$

(17)$$h_O (k) = h_O (k - 1) - v_{OD}^n \cdot \Delta t$$

where (k−1) is the k−1 time of the odometer position, (k) is the k time of the odometer position, and Δt is the sample interval of the odometer.

2.3.2. INS/odometer Observation Model

The INS/odometer observation model is similar to the GPS/INS observation model. The position difference between the odometer and INS which is transformed from geodetic coordinate system to navigation frame, with the velocity difference, is the observation input of the Kalman filter. Because the ability of the odometer to sensor the velocity in the down direction is weak (Yan, Reference Yan2006), so in the INS/odometer integrated system, the observation input of the Kalman filter only includes the measurement in the north and east directions:

(18)$${\bi Z}_k^O = \left[ {\matrix{ {\Delta r_N} \cr {\Delta r_E} \cr {\Delta v_N} \cr {\Delta v_N} \cr}} \right] = {\bi B}_k^O \left[ {\matrix{ {{\bi X}_{Nav}} \cr {{\bi X}_{Acc}} \cr {{\bi X}_{Gyo}} \cr}} \right] + \left[ {\matrix{ {\nu _r} \cr {\nu _v} \cr}} \right]$$

where B_k^O is the observation matrix of INS/odometer observation model.

2.4. Kalman Filter

The optimal estimates of the state vector from the Kalman filter can be reached through a time update and a measurement update at a time instant:

(19)$${\bi \hat X}_k = {\bi \hat X}_{k,k - 1} + {\bi K}_k ({\bi Z}_k - {\bi {\bi B}}_k {\bi \hat X}_{k,k - 1} )$$

(20)$${\bi K}_k = {\bi P}_{k,k - 1} {\bi B}_k^T ({\bi B}_k {\bi P}_{k,k - 1} {\bi B}_k^T + {\bi R}_k )^{ - 1} $$

(21)$${\bi \hat X}_{k,k - 1} = {\bi \Phi} _{k,k - 1} {\bi \hat X}_{k - 1} $$

(22)$${\bi P}_{k,k - 1} = {\bi \Phi} _k {\bi P}_{k - 1} {\bi \Phi} _k^T + {\bi Q}_k $$

(23)$${\bi P}_k = ({\bi I} - {\bi K}_k {\bi B}_k ){\bi P}_{k,k - 1} $$

In a closed loop integration scheme, a feedback loop is used for correcting the systematic errors. In this way, the mechanization performs simple navigation calculation under the assumption of small errors. In this case, the error states will be reset to zero after every measurement update (Godha, Reference Godha2006). Thus, the navigation resolution is expressed by:

(24)$${\bi \hat X}_k = {\bi P}_{k,k - 1} {\bi B}_k^T ({\bi B}_k {\bi P}_{k,k - 1} {\bi B}_k^T + {\bi R}_k )^{ - 1} {\bi Z}_k $$

4.1. Fuzzy Neural Network

In the study, a FNN scheme is proposed to learn and compensate for the odometer observation error and to improve the odometer observation accuracy during GPS outages. The T-S fuzzy system is a self-adaptive fuzzy system (Hsiao et al., Reference Hsiao, Chen, Liang, Xu and Chiang2005). The T-S model not only updates by itself, but also corrects the membership function of the fuzzy subset continually. This T-S fuzzy system is described by fuzzy IF-THEN rules:

(25)$${\rm if}\;{\rm} x_1 \;{\rm is}\;A_1^i, x_2 \;{\rm is}\;A_2^i, {\rm \ldots}, x_k \;{\rm is}\;A_k^i \;{\rm then}\;y_i = p_0^i + p_1^i x_1 + \cdots + p_k^i x_k $$

where A _jⁱ(j=1,2, … k) is the fuzzy set of the fuzzy system, p _jⁱ is the fuzzy system parameters, y _i is the output by fuzzy rule. In a fuzzy system, the input part is fuzzy; otherwise, the output is definite.

Suppose the input is x=[x ₁,x ₂, … x _k]. At first, the membership of the input is computed according to the fuzzy rule:

(26)$$u_{A_j^i} = \exp ( - (x_j - c_j^i )^2 /b_j^i )\quad j = 1,2, \cdots, k;i = 1,2, \cdots, n$$

where c _jⁱ and b _jⁱ are the centre and width of membership function respectively, k is the number of input parameters, n is the number of fuzzy subsets. Gaussian membership functions provide more continuous transition from one interval to another and hence provide a smoother control surface from the fuzzy rules. At the same time, they are able to provide a system with fewer degrees of freedom and hence more robustness. Thus, Gaussian membership functions are suitable for problems which require continuously differentiable curves and therefore smooth transitions, whereas the others (such as triangular) do not posses these abilities (Hameed, Reference Hameed2011).

The fuzzy factor is computed by membership value using the continuous multiplication:

(27)$$\omega ^i = \prod\limits_{g = 1}^k {u_{A_j^g} (x_g )\quad i = 1,2, \cdots, n} $$

Then the output y of the fuzzy model can be shown:

(28)$$y_i = {{\sum\limits_{i = 1}^n {\omega ^i \left( {\,p_0^i + p_1^i x_1 + \cdots + p_k^i x_k} \right)}} \Big / {\sum\limits_{i = 1}^n {\omega ^i}}} $$

A fuzzy neural network is composed of three layers, which are the input (pre-processing), hidden, and output (post-processing) layers. The hidden layers are composed of the fuzzy layer and the fuzzy rule calculation layer. The input value is fuzzy-processed to get the membership value by Equation (26) in the fuzzy layer. The fuzzy factor will be computed in the fuzzy rule calculation layer by Equation (27).

The training algorithm of fuzzy neural network is as follows (Wang et al., Reference Wang, Shi, Yu and Li2012):

1. (1) Calculate error:

   (29)$$e = \displaystyle{1 \over 2}(y_d - y_c )^2 $$
   where y _d is the exception output of neural network, y _c is the real output of neural network, e is the error between exception output and real output.

2. (2) Modify coefficient

   (30)$$p_j^i (k) = p_j^i (k - 1) - \alpha \displaystyle{{\partial e} \over {\partial p_j^i}} $$
   (31)$$\displaystyle{{\partial e} \over {\partial p_j^i}} = {{(y_d - y_c )\omega ^i} \Big/ {\sum\limits_{i = 1}^n {\omega ^i x_j}}} $$
   where α is the neural network training rate.

3. (3) Modify parameters

   (32)$$c_j^i (k) = c_j^i (k - 1) - \alpha \displaystyle{{\partial e} \over {\partial c_j^i}} $$
   (33)$$b_j^i (k) = b_j^i (k - 1) - \alpha \displaystyle{{\partial e} \over {\partial b_j^i}} $$

The topological structure of an FNN is designed from small processing units (neurons) that are connected within the network using weighted chains. Generally speaking, the basic model of the neural network includes three primary elements: (a) weight chains, (b) a calculator for summing the input data that are weighted by respective synapses of the neuron; and (c) an activation function for restricting the amplitude of the neural network output and the last output (Chiang et al., Reference Chiang, Chang, Li and Huang2009).

4.2. GPS/INS/Odometer Integrated System Using Fuzzy Neural Network

A detailed block diagram of the proposed fuzzy neural network is shown in Figure 2. In order to progress the implementation of the FNN, a number of parameters should be set. These contain the training threshold contribution, the training momentum, the training rate and the number of hidden layers to be used (Petropoulos et al., Reference Petropoulos, Vadrevu, Xanthopoulos, Karantounias and Scholze2010). An ANN with an optimal structure is expected to realize the closest accuracy to the prediction model using the most suitable number of hidden neurons and hidden layers. There are many ways to decide on the most suitable number of hidden neurons, such as genetic algorithms (Kuo et al., Reference Kuo, Chen and Hwang2001), which is applied in this paper.

Figure 2. Three layer neural network of proposed integrated system.

A genetic algorithm (GA) process requires three most important aspects: (1) the genetic representation of the solution domain, (2) the genetic operators of the solution domain and (3) an objective function to evaluate the solution domain. Steps of a general GA process are as follows:

1. (1) Initial: generate initial parent population and define the crossover and mutation probability;

2. (2) Selection: evaluate the objective function and select chromosomes for reproduction;

3. (3) Crossover and Mutation: create offspring using reproduction operators such as crossover and mutation;

4. (4) Termination: repeat the generational process until a termination condition has been reached.

The construction of objective function is key to the GA process. In the FNN of our GPS/INS/odometer integrated system, the velocity difference between the odometer and GPS is the output. Based on this, the objective function used in the GA to choose the FNN parameters is constructed as:

(34)$$\varphi (\lambda ) = (\delta v'_{ON} - \delta v_{ON} )^2 + (\delta v'_{OE} - \delta v_{OE} )^2 $$

where λ are the FNN parameters (number of hidden neurons), δv _ON and δv _OE are the difference between GPS and odometer (odometer velocity correction) in north and east directions, δv′ _ON and δv′ _OE are the difference velocity predicted by FNN. When the function φ trends to zero, the most appropriate parameters will be chosen.

As is well known, one important component of the FNN is to acquire the training data which are required to train the neurons to accurately estimate the system (Chiang and Chang, Reference Chiang and Chang2010). In the FNN of our GPS/INS/odometer integrated system, the training phase records the IMU observation and odometer observation in the NED frame as the input. The output, velocity difference between the odometer and GPS, is a 2-D vector with north velocity and east velocity as components. The odometer is only able to sense the displacement in a plane, so it is difficult to record the three-dimensional motion. In land navigation applications, the vehicle always remains on the Earth's surface, so the vehicle positions have no large changes in the vertical direction (Han and Wang, Reference Han and Wang2010). It is reasonable to assume that the vehicle moves in an approximately horizontal plane in most cases. Thus the ability of the odometer to sense the velocity in the down direction is weak (Yan, Reference Yan2006), so the output of the FNN excludes it. The reason for using the velocity differences instead of the velocity component itself as output is to simplify the learning and training process.

Figure 3 illustrates the FNN system configuration and training strategy. During the training phase, the velocity difference between GPS and the odometer is selected as the target for the network training during signal availability. Thus the navigation knowledge can be learnt, stored and accumulated (Wang et al., Reference Wang, Wang, Sinclair and Watts2006). On the other hand, if the GPS signal is unavailable and the network is well trained, the proposed FNN model receives the acceleration and rotation rates from INS and velocity from the odometer to generate the velocity difference between GPS and the odometer (odometer velocity correction). Based on that, the estimate velocity can be computed by summing the odometer velocity and FNN output during GPS outages. IMU data participates in the INS computation and FNN process. The weight of IMU data in FNN prediction is low so the possibility of causing divergence of the navigation solutions is limited by double counting of INS computation and FNN prediction.

Figure 3. FNN training architecture of proposed integrated system.

Figure 4 shows a loosely coupled GPS/INS/odometer integration structure with closed loop using fuzzy neural network, which is proposed in this paper. The observation input of the Kalman filter is changed to the odometer measurement corrected by FNN in a GPS outage.

Figure 4. A proposed loosely coupled GPS/INS/odometer integration architecture.