# A MLP-Hedge-Algebras Admittance Controller for Physical Human-Robot Interaction
## 3 Hedge-Algebras-Based Admittance Controllers
### MLP-HA-Based Admittance Controller  
As presented in the fourth paragraph of the Introduction section, the MLP network is chosen for the semantic input-output relationship in the HA based on its properties. The proposed MLP-HA-based admittance solution is presented in Figure 4, including two main blocks: the _MLP semantic inference mechanism in HA_ and the _MLP-HA-based admittance system_. In the _MLP semantic inference mechanism in HA_, _HA-terms_ and _HA rule-base_ are defined based on requirements of engineering problems. In addition, in this paper, they are presented in Tables 3-5 to solve several issues of pHRI. _The semantic relationship_ is presented in Tables 6 and 7. These relationships are then described by the MLP network, which is trained using backpropagation. This MLP network will be used as an interpolation method for the input-output semantic relationship in the _HA-MLP-based admittance system_. In the _HA-MLP-based admittance system_, _human-robot interactive forces/torques_ are measured by using a real-time external force/torque sensor mounted on the end-effector of the manipulator (haptic). _Transmitted power_ is calculated by using Equation (2). Those values are then mapped into HA-terms and HA-semantic values. Semantic values of these linguistic variables play as inputs of the _pretrained MLP model_ to interpolate the semantic value of the output. Finally, the physical value of output is obtained by a simple map from [0, 1] to its physical domain. The MLP neural network is used to predict output semantic values, which will add significant benefits to HA-based admittance controllers than existing approaches, namely:  
* The semantic relationship between inputs and output is described adequately;
* The semantic real-super-surface in 3D space becomes smoother;
* Semantic values of output will be approximated in the global boundary;
* The interpolation accuracy is improved. Figure 4: The diagram of the HA-MLP system, including two main processes: (1) the MLP network is used to represent the input–output semantic relationship, (2) the pre-trained MLP model is used for the admittance system. The configuration of the MLP network is characterized by the number of hidden layers and the number of neurons in hidden layers, the number of samples propagated through the network for each gradient update, the number of trained epochs, and the learning rate. The studies based on the theorems of [PERSON], for example, Refs. [43; 44; 45], have pointed out that all of the continuous mappings from \([0,1]^{\text{P}}\) to \([0,1]^{\text{n}}\) can be approximated by a multilayer perceptron network of which input layer includes p neurons, output layer includes n neurons, and hidden layers include (2p + 1) neurons. However, the number of samples in the training data set is also an important factor in choosing the MLP configuration with well-fitting and capacity. To choose the configuration of the MLP network, the recommendation in [49; 50] is used:  
\[N_{h}=\frac{N_{s}}{\beta(N_{i}+N_{o})} \tag{24}\]  
Here, \(N_{h}\), \(N_{i}\), \(N_{o}\), \(N_{s}\) and \(\beta\) are the number of hidden neurons, the number of input neurons, the number of output neurons, the number of samples in the training data set, and the arbitrary scaling factor, respectively. Note, \(\beta\) usually belongs to [2; 10]. In this paper, the input of the MLP-HA-based admittance controller are semantic values in the range \([0,1]\), so the sigmoid is used as a transformation function since it well-handles the data in the range \([-1,1]\). \[g(\lambda)=\frac{1}{1+e^{-\lambda}} \tag{25}\]  
The MLP-backpropagation in this paper is briefly described as below:  
Step 1: Choosing the initial configuration of MLP network, learning rate \(\eta>0\), momentum parameter set \(\alpha\), desired error E\({}_{\text{max}}\), initial weights W. Setting the current error E = 0, and the current sample k = 1.