## 4 Simulations
### Simulation Results  
It is noteworthy in this scope that we intensively focus on investigations of system behaviors and stability under contact-loss. The position tracking performance between the conventional and the one with adaptive law algorithms under the presence of uncertainties and disturbances in free-space, and an enhanced force tracking performance can be referred to previous works in [9; 10; 45]. The first simulation is given in a case of still constrained framework. We suppose the end-effector of the 6-DOF-KIRO robot moves toward along the z-direction as shown in Figure 5. The desired trajectory is set as \(\mathbf{x}_{d}=\left( \begin{array}{cc}2300&0&200-20t\end{array} \right)^{T} (\text{mm})\) and the location of the end-effector is initialized at \(\mathbf{x}_{initial}=\left( \begin{array}{cc}2000&0&150\end{array} \right)^{T} (\text{mm})\). To mimic the contact-loss in two case studies, the environment profile is set as a step function as  
\[x_{set}=\left\{ \begin{array}{ll}2200\ (mm)\ if\ t\ \leq\ 7.62s\\ 2530\ (mm)\ if\ t\ \geq\ 7.62s\end{array} \right. \tag{50}\]  
As can be seen in Equation (50), we suppose that the contact loss happens at the time of \(t=7.62\) s. Indeed, the contact-loss can suddenly occur if the position along the X-direction suddenly changes. Besides, the end-effector of the KIRO robot must apply a constant force of 100 N to the wall. The desired force induced from contacting command with the environment and moments induced from the command motions such as twisting, bending, etc., at the end-effector. Then, the desired force is set as \(\mathbf{F}_{d}=\left( \begin{array}{cccccc}F_{dx}&F_{dy}&F_{dz}&M_{dx}&M_{dy}& M_{dz}\end{array} \right)^{T}=\left( \begin{array}{cccccc}100&0&0&0&0&0 \end{array} \right)^{T}\). The manipulator behavior in case of treating impedance-contact-loss based on the virtual energy tank concept is compared with that of conventional impedance control. We suppose that the end-effector is commanded to work along the X-direction. Position tracking performances of conventional impedance and impedance contact-loss in X, Y, and Z directions are displayed in Figures 6-8, respectively. The continuous black line represents desired trajectory \(\left(\bullet\right)_{desired}\), the black dot-line represents the arbitrary environment \(\left(\bullet\right)_{env}\), the blue dot-dash line represents the end-effector position of conventional impedance response \(\left(\bullet\right)_{imp}\), and the dash red line represents the end-effector trajectory of impedance contact-loss \(\left(\bullet\right)_{imp-cl}\). For the shake of simplicity in the case of force responses, the desired force (\(F_{desired}\)), external impedance force (\(F_{ext-imp}\)), and external impedance contact-loss (\(F_{ext-imp-cl}\)) are presented by italic symbols hereafter since these values are investigated in only X-direction. Figure 5: Working scenario of the 6-DOF KIRO robot and prescribed expected trajectory after contact-loss happens. At the time of approximately 7.62 s, the sudden change in the environment framework causes contact-loss. At that time, the interacting force immediately decreases to zero, and the tank variable increases until its energy reaches the upper limit \(T^{u}\) at the time of approximately 7.63 s. Then, the parameters \(\beta\) and \(\varphi\) simultaneously switch their values as shown in Figure 9. Due to using the energy tank, the end-effector can generally maintain the same level with a very small position shifted. This is because the force control signal is deactivated, and the energy stored in a virtual spring-damper of each joint is dissipated. In the case of conventional impedance control, the end-effector moves until it reaches the constrained framework again at the time of 8.2 s as the requirement of force tracking effort.