## 4 Experimental Study and Discussion
### Model Learning and Input Choice  
Motion models were trained for each subject. The GP-NARX model was first trained using \(u_{k}\) as the inputs and \(x_{k}\) as the output. The model structure is illustrated in Figure 7. SE was chosen as the kernel function when the stationary nature and the smoothness assumption of this radial basis function regression model coincide with the joint angle variation. Specifically, the automatic relevance determination squared exponential (SEard) kernel was used to provide different hyperparameters for the three different inputs, including the extracted feature of EMG, the joint angle and the angular velocity. The model hyperparameters were optimized through minimizing the differentiable multivariate function using the conjugate gradients.  
Figure 6: Rear and front view of the human thigh illustrating the selected two muscles. In our application here, the electrodes were placed only on the RF.  
The BPNN included three layers (i.e., one input layer, one hidden layer, and one output layer). The input to the BPNN in the measurement model was the joint vector \(x_{k}\) including the joint angle and angular velocity, and the interactive force \(y_{k}\) was the model output. Thus, the dimension of the input vector was 2 and that of the output vector was 1. Sigmoid function (logsig) and linear function (purelin) were selected as the transfer functions of hidden and output layers, respectively, as illustrated in Figure 8. In order to decide the numbers of hidden layers and neurons, the correlation coefficient was used to evaluate the network performance. After many attempts, the number of hidden layers was chosen as 1 and the number of hidden neurons was chosen as 5 to achieve a tradeoff between computational efficiency and accuracy. The Levenberg-Marquardt method was used to optimize the net.  
In fact, the joint angle instead of the joint vector containing both the joint angle and the angular velocity was also tested as the input to the measurement model. However, the performance was far from satisfactory even when more hidden layers and hidden neurons were used. It proved, from one aspect, that the biomechanical interpretation of human-exoskeleton interaction did provide a good foundation for the development of the measurement model. With the trained GP-NARX model and BPNN, the state space model formulated in (26) can be obtained and developed using the UKF algorithm.  
The way in which system models learn the mapping relations from neural signals to human joint angle varies, but ANN is one of the most widely used approaches [1]. Thus, different from the BPNN for the measurement model of SSM, a traditional BPNN, which used the extracted EMG features and the interactive force as the inputs and the joint vector containing the joint angle and angular velocity as the output, was also built for performance comparison. In consequence, the dimensions of the input vector and output vector were both 2. It was also found through numerous attempts that the increase of the numbers of hidden neurons and hidden layers did not improve the performance much. Thus, one hidden layer with 5 hidden neurons was also adopted here. Its architecture is illustrated in Figure 9.  
Figure 8: The BPNN for the measurement model of SSM, where \(\omega_{1,m}\) and \(\omega_{2,m}\) are the weights, \(b_{1,m}\) and \(b_{2,m}\) are the bias values. The input vector \(x_{k}\) whose dimension is 2 contains the joint angle and angular velocity, and the output vector \(y_{k}\) whose dimension is 1 contains the interactive force, and 2, 5 and 1 are the numbers of the neurons.  
Figure 7: The GP-NARX model, where the extracted EMG feature \(u_{k}\) is the input and the joint vector \(x_{k+1}\) is the output