# A MLP-Hedge-Algebras Admittance Controller for Physical Human-Robot Interaction
## 4 Stability Considerations  
In physical human-robot interaction, the human factor is a part of the controller, which is extremely difficult to model and prove the stability of the system. Normally, the derivative equation of the admittance controller is the relationship between the external force and the end-effector's velocity. In other words, the linear/angular velocity of the end-effector only depends on the translation/rotation elements of the external wrench. Consequently, a large jerk during cooperation will appear if the external force changes dramatically. Experimental studies showed that the robot could present unstable behavior with very low virtual damping and high virtual inertia of stiff environment [54]. It is suggested that the human arm has a maximum impedance that occurs when the human increases the stiffness of their arm [55]. As analyzed in [9; 10], the stability of the conventional admittance controller is guaranteed by setting the virtual inertia as constant and equal to half the effective inertia of the manipulator in the directions of the motions during the experiments. The effective mass \(\mathbf{B_{t}}\) is expressed as:  
\[\mathbf{B_{t}}^{-1}(\mathbf{q})=\mathbf{J}(\mathbf{q})\mathbf{M}^{-1}( \mathbf{q})\mathbf{J}^{T}(\mathbf{q}) \tag{35}\]  
where \(\mathbf{M}\in\)\(\mathrm{R^{6}}\times\)\(\mathrm{R^{6}}\) is the mass matrix of the manipulator's configuration space, \(\mathbf{B_{t}}\in\)\(\mathrm{R^{6}}\times\)\(\mathrm{R^{6}}\) is the effective mass in the frame, and \(\mathbf{J}\) is the Jacobian matrix.  
By contrast, the stability and accuracy of fuzzy-based and HA-based controllers mainly depend on the expert knowledge-based rule base. As analyzed in [56], the fuzzy controller can be stable and ensure accuracy by choosing a good fuzzy rule and physical value domains of appropriate input and output variables. Furthermore, the mathematical foundation given by HA seems to form a new approach to solve fuzzy control problems, which is quite different from that based on the fuzzy sets. It has been shown that the HA-based controller causes smaller errors. Moreover, it can bring the controlled object to a stable state, while the controller based on other methods cannot. This point is clearly analyzed in [57; 58]. Another point should be mentioned for the stability of the proposed method, which is the stability of the MLP-based input-output semantic interpolation. As presented in [59], the weight dynamics of the neural network are described by a gradient system in most cases. Therefore, the stability of the weight dynamics is not a worry since it is well-known that the gradient system possesses a Lyapunov function.  
In addition, the proposed admittance controller uses the transmitted power as an additional input, which is the dot product of the external wrench vector and the actual velocity vector, whose purpose is to avoid the direct map from external forces/torques to linear/angular velocities. As presented, the natural human-robot interaction is also considered in this paper. Therefore, the velocity of end-effector and external forces include both linear and angular elements. Normally, the rotation elements have bigger effects on the stability and accuracy than translation elements since a small change of the rotation can lead to a big error. This observation raises the need to create a relationship between linear velocity and angular velocity to guarantee smooth cooperation. Based on this point, the linear velocity is calculated first based on translation elements of the external wrench and transmitted power, then the angular velocity is calculated based on the linear velocity and rotation elements of the external wrench. As shown in Equation (2), the value of transmitted power depends on \(\mathbf{V_{k-1}}\), \(\mathbf{W_{k-1}}\), \(\mathbf{F}\), and \(\mathbf{M}\). As a result, the value of the actual velocity will not change suddenly even when the external wrench changes sharply. By doing this, the jerk during cooperation is reduced.