# A MLP-Hedge-Algebras Admittance Controller for Physical Human-Robot Interaction
## 2 Fuzzy-Based Admittance Controller  
This section presents an admittance controller based on the inference mechanism of fuzzy logic, in which the velocity of the end-effector will be adjusted directly via the external wrench and the transmitted power without the IIDM. To guarantee safety, constraints of ISO10218 standards are considered to formulate the controller:  
\[\left|\mathbf{F}\right|\leq F_{M},\ \left|\mathbf{V}\right|\leq V_{M},\ \left| \mathbf{M}\right|\leq F_{M}L,\ \left|\mathbf{W}\right|\leq\frac{V_{M}}{L} \tag{1}\]  
\[P=\left\langle{}^{B}\mathbf{T}_{T}\mathbf{H},\dot{\mathbf{x}}\right\rangle\leq P _{M} \tag{2}\]  
\[{}^{B}\mathbf{T}_{T}=\left[ \begin{array}{cc}{}^{B}\mathbf{R}_{T}&0_{3}\\ 0_{3}&{}^{B}\mathbf{R}_{T}\end{array} \right] \tag{3}\]Here, **F** and **M** are external force/torque, and \(\textbf{H}=[\textbf{F}^{T}\textbf{M}^{T}]\). **V** and **W** are linear and angular velocities of the end-effector. \(F_{M}\), \(V_{M}\) and \(P_{M}\) are the maximum external force, linear velocity, and transmitted power allowed, respectively. \(\dot{x}\) is the measured velocity. \(\dot{B}\textbf{R}_{T}\) is the rotation matrix between the TCP (tool center point) frame and the base frame, and \(L\) is the length of the tool in the TCP frame. In the sequel, \(|\textbf{u}|\) it denotes the Euclidean norm of a vector **u** and \(\langle\textbf{u},\textbf{v}\rangle\) represents the dot product of vectors **u** and **v**. To eliminate the IIDM, the fuzzy-based admittance controller is designed based on constraints Equations (1)-(3) of the safety standard, without the dynamic admittance model. \(F_{M}\), \(V_{M}\), and \(P_{M}\) are used to determine physical domains demonstrating linguistic variables. The output is the velocity of the end-effector, including **V** and **W**. The inputs are the external force **F**, external torque **M** and the transmitted power \(P\). Based on Equations (1)-(3), it is clear that \(|\textbf{F}|\), \(|\textbf{M}|\), \(|\textbf{V}|\), \(|\textbf{W}|\), and \(P\) are used during data processing. Based on [46], the five triangular type membership function is used to describe \(|\textbf{F}|\), \(|\textbf{M}|\), \(|\textbf{V}|\), \(|\textbf{W}|\), and \(P\) by linguistic values, and their physical domains are \([0,F_{M}]\), \([0,F_{M}L]\), \([0,V_{M}]\), \([0,\frac{V_{M}}{L}]\), and \([0,P_{M}]\), respectively. In the ISO10218 safety standard, \(F_{M}\) and \(V_{M}\) are two main constraints, which should be considered. Other allowed parameters \(P_{M}\), \(M_{M}\), and \(W_{M}\) should be obtained based on Equations (1) and (2). Normally, these parameters must be chosen following the safety requirements of specific technological processes. In other words, these parameters are flexible, and the control rule-base should generate the output adaptively depending on different values of safety constraints. In this paper, the rule-based admittance controller adapts all values of desired user safety constraints since physical subdomains of linguistic values are adaptively adjusted based on their predefined proportion in the linguistic variables. This mechanism is available for all of the controllers in this paper, including fuzzy-based admittance controllers and hedge-algebra-based admittance controllers. However, it is noticed that the human effort will be proportional to \(\delta\) (here, \(\delta=\frac{V_{M}}{V_{M}}\)). The fuzzy rule base in Table 1 presents the relationship between \(|\textbf{V}|\), \(|\textbf{F}|\) and \(P\), where \(|\textbf{V}|\) is inferred based on \(|\textbf{F}|\) and \(P\). Then, \(|\textbf{W}|\) is inferred via \(|\textbf{V}|\) and \(|\textbf{M}|\) using the fuzzy rule base in Table 2. Here, \(Z\), \(S\), \(M\), \(QB\) and \(B\) stand for _zero_, _small_, _medium_, _quite big_, and _big_, respectively. The max-min composition is used to infer the fuzziness of outputs based on the fuzzy relations.