# An Enhanced Fusion Strategy for Reliable Attitude Measurement Utilizing Vision and Inertial Sensors
## 3 The Enhanced Fusion Strategy Based on RBF and CKF
### RBF Algorithm  
The radial basis function neural network is one of the extensively employed deep learning methods, which has good approximation, fast learning convergence and simple structure of feed-forward neural network. There are usually three layers of network structure, as shown in Figure 3 where \(x_{i}\) denotes the \(i\)th input element. The hidden layer contains m radial basis neurons, whose basis function responds locally to the input. \(\varphi_{i}(x)\) is the activation function in hidden layer. When the input is close to the basis function, a large number of outputs will be generated. \(W_{i}\) indicates the weight between the hidden layer and output layer. Y is the linear weighted sum of the output of hidden layer neurons.  
The input layer includes a set of sensory units, which represents the input of signal source nodes. The hidden layer has the sufficient dimension, which uses the activation functions to map a low dimensional nonlinear separable input to a high dimensional linear separable space. The output layer expresses a linear transformation from the hidden node to the output space. The output can be given by:  
\[y(x)=\sum W_{i}\varphi_{i}(x)+w_{0} \tag{17}\]  
where \(w_{0}\) is the bias term and \(W_{i}\) is the weight between the hidden layer and output layer. The Gaussian basis function is the most frequently used radial basis function for \(\varphi_{i}(x)\) with the following function.  
\[\varphi(x)=e^{-\frac{\left\lVert x-\mu\right\rVert^{2}}{2\sigma^{2}}} \tag{18}\]  
where \(x\) is the input of the network; \(\left\lVert\cdot\right\rVert\) represents the Euclidean distance; \(\mu\) and \(\sigma\) are the center and width of the Gaussian function, and the two factors is used to adjust the deviation. By analyzing the structure characteristics, the center nodes \(\mu\) in hidden layer determine the performance of RBF. An appropriate center node in the hidden layer will enhance the learning effect. Furthermore, the value of \(\mu\) can be determined by different methods. The commonly used method is the k-means [28] clustering algorithm to obtain a subset of the data points as the RBF centers, which can adjust the deviation of the RBF [29; 30]. The specific steps of k-means clustering algorithm are as follows:  
Step 1: Preset the output precision E of the whole neural network. k nodes in hidden layer are selected arbitrarily from the input sample X.  
Step 2: M samples among the X and the width of Gauss function are randomly selected as the center of gauss function in hidden layer and the width of gauss function, respectively.  
Step 3: K-means clustering algorithm is employed to average Euclidean distance from each simple to the center node for updating the center nodes and function width.  
Step 4: RBF is constructed by using the updated central data and function width, and the weights between the hidden layer and output layer. The output Y is obtained.  
Step 5: Calculate the sums of squared errors \(E(M)\) by comparing the practical output Y and theoretical output Y. If \(E(M)<E\), proceed to step 6. Otherwise, \(M=M+1\) and return to step 2.  
Step 6: Compare \(E(M)\) and \(E(M-1)\). If they both dissatisfy the accuracy requirement, \(M=M-1\) and repeat steps 2 to 5. Otherwise, if \(E(M-1)<E\), the M is set as the number of center nodes and end loop.  
Using the above-described procedures, the center nodes of hidden layer can be determined by the k-means clustering algorithm.  
Figure 3: Structure of RBF.