## 3 Methodology
### Impedance Control  
The forward kinematics of a robotic manipulator is written as [23]:  
\[x(t)=f(q) \tag{1}\]  
where \(x(t)\in\Re^{n}\) and \(q\in\Re^{n}\) are the pose (i.e., position/orientation) of the end-effector in the Cartesian space and the joint angle coordinates in the joint space, respectively. Differential kinematics is obtained by deriving (1) with respect to time:  
\[\dot{x}(t)=J(q)\dot{q} \tag{2}\]  
where \(J(q)\in\Re^{n\times n}\) is the Jacobian matrix. Differentiating (2) again results in the acceleration of the end-effector:  
\[\ddot{x}(t)=\dot{J}(q)q+J(q)\ddot{q} \tag{3}\]  
The robot arm dynamics in the joint space is given by:  
\[M(q)\ddot{q}+C(q,q)\dot{q}+G(q)=\tau-J^{T}(q)F \tag{4}\]  
Figure 1: Impedance controller diagram with amplification module for the force amplification strategy. Figure 2: Impedance controller diagram with intent prediction module for intent prediction strategy. where \(M(q)\in\Re^{n\times n}\) is the symmetric positive-definite inertia matrix; \(C(q,\dot{q})\dot{q}\in\Re^{n}\) is the Coriolis and Centrifugal forces; \(G(q)\in\Re^{n}\) is the gravitational force; \(\tau\in\Re^{n}\) is the vector of control input; \(F\in\Re^{n}\) denotes the net force exerted by the robot on the environment at the end-effector (\(F=F_{r}-F_{ext}\)). Otherwise, \(F_{r}\in\Re^{n}\) is the force exerted by the robot on the environment at the end-effector while \(F_{ext}\in\Re^{n}\) is the external force exerted by the environment on the robot at the end-effector. In our pHRI task, the environment is the human, specifically the human hand that is in contact with the robot. The robot dynamics can be written in the Cartesian space as:  
\[M_{x}(q)\ddot{x}+C_{x}(q,\dot{q})\dot{x}+G_{x}(q)=u-F \tag{5}\]  
where  
\[ \begin{array}{c}M_{x}(q)=J^{-T}(q)M(q)J^{-1}(q),\\ C_{x}(q,\dot{q})=J^{-T}(q)(C(q,\dot{q})-M(q)J^{-1}(q)J(q))J^{-1}(q),\\ G_{x}(q)=J^{-T}(q)G(q),\,u=J^{-T}(q)\tau,F=F_{r}-F_{ext}\end{array} \]  
As explained in Section 3.1, the most common interaction controllers for pHRI are admittance and impedance control. Both controllers are based on a target impedance model for the robot and they only differ in terms of input and output (see Figures 3 and 4). Therefore, the target robot impedance model can be represented as a mass-damper-spring system:  
\[M_{d}(\ddot{x}_{d}-\ddot{x})+B_{d}(\dot{x}_{d}-\dot{x})+K_{d}(x_{d}-x)=F_{d}=F _{r} \tag{6}\]  
where \(M_{d},B_{d},K_{d}\) are the virtual inertia, damping and stiffness of the robot, respectively. \(F_{d}\) is the desired force and \(x_{d}\) can be interpreted as the rest position of this virtual mass-damper-spring system. For pHRI tasks in which the human touches the robot, the human limb (arm+hand) can also be modeled as a mass-damper-spring system:  
\[M_{h}\ddot{x}_{h}+B_{h}\dot{x}_{h}+K_{h}(x_{h}-x_{hd})=F_{h} \tag{7}\]  
where \(M_{h},B_{h},K_{h}\) are the limb inertia, damping and stiffness respectively and \(F_{h}\) is the force applied by the human to the robot. The limb impedance values are not fixed and depend from person to person, as well as on the task being carried out. \(x_{h}\) is the position of the human wrist in the robot frame and \(x_{hd}\) can be interpreted as the desired target position. Discussion on the limb impedance parameters or their calculation are not in the purview of this paper. In fact, the investigation of the limb impedance parameters would have been necessitated if an admittance controller was used, to tackle the potential controller stability issues. Since we have used a torque-based impedance controller for our experiments, it does not face such issues. While real time measurements of limb impedance values could be used for interpreting the intent of the human operator during the operation, this would have required attaching EMG sensors on the arms of the human operator, such as in [24], thereby reducing the practicality of this research for an industrial meat cutting scenario.