# Hybrid-Compliant System for Soft Capture of Uncooperative Space Debris
## 5 Simulation Study and Results
### Evaluation of Hybrid Compliance  
Two versions of the ADR system were simulated to further demonstrate the importance of tuning the ADR system's compliance by adding an active compliant unit and to showcase the application of the proposed impedance controller. Subsequently, the responses were measured and the corresponding contact times were calculated and compared.  
The first system is a passive system, denoted as PCS, composed only of passive physical compliance. In this case, the ACU is inactive; thus, its prismatic joints are locked, see Figure 23a. The second system is a hybrid-compliant system, denoted as HCS and shown in Figure 23b, composed of passive and active compliance; the linear actuators apply forces \(F_{a}\) driven by the proposed impedance controller presented in Section 5.2 and given by Equation (7).  
The passive compliance's stiffness \(k_{s}\) equals 0.5 N/m following the use case presented in Section 5.2 and its physical length \(l_{s}\) is equal to 0.025 m. The linear actuator's space limit, i.e., the maximum displacement \(l_{a}=x_{e}-x_{s}\) allowed, is also equal to 0.025 m.  
Figure 24 shows the impact force generated at the tips of the PCS system (magenta line) and the HCS system (blue line). Figure 25 shows the actuator force commanded by the impedance controller to be less than 0.15 N.  
Figure 23: Comparison of ADR systems; (**a**) PCS: system with only passive compliance (inactive linear actuators); (**b**) HCS: hybrid system with both passive and active compliance.  
Observing Figure 24 and comparing the contact times, one can see that using the proposed impedance controller significantly increases the contact time from 7 s to 11.65 s. Hence, the applied controller adjusted the contact time between the ADR system's tip and the debris and ensured the minimum required; in this example, equal to 10 s. The desired contact time of the ADR system was ensured by the appropriate tuning of the mass, spring, and damper impedance parameters \(m_{f}\), \(k_{f}\), and \(b_{f}\) of Equation (5), respectively, and therefore of the IC gains.  
To find the appropriate impedance parameters \(m_{f}\), \(k_{f}\), and \(b_{f}\), the simulation results provided in Section 5.2 for the use case were employed. Specifically, to capture debris with a mass of 4 kg, the required hybrid compliance's stiffness and damping coefficients, \(k_{m}\) and \(b_{m}\), were found to be equal to 0.2 N/m and 0.17 Ns/m, respectively. Thus, the desired spring and damper parameters of the impedance filter \(k_{f}\) and \(b_{f}\) can be calculated based on Equation (16) as  
\[k_{f}=\frac{m_{f}}{\mu_{ef}}k_{m} \tag{21}\]  
and,  
\[b_{f}=\frac{m_{f}}{\mu_{ef}}b_{m}, \tag{22}\]  
to be equal to 0.2003 N/m and 0.1702 Ns/m, respectively. The desired mass parameter \(m_{f}\) is equal to \(m_{c}\). Using the desired impedance parameters \(m_{f}\), \(b_{f}\), and \(k_{f}\) derived, the IC gains \(k_{p}\) and \(k_{d}\) were calculated based on Equations (8) and (9) to equal 150 N/m and 128 Ns/m, respectively. The IC command \(F_{a}\), which drove the ACU's linear actuators, was calculated by Equation (7).  
Figure 24: Comparison of impact forces of passive and impedance controlled systems.  
Figure 25: Command force by the impedance-controlled linear actuator.  
The application of the IC implemented active (virtual) compliance into the ADR system, rendering it a hybrid-compliant system. Comparing the PCS and HCS, the necessity of the IC and, subsequently, of a hybrid-compliant ADR system for the successful capturing of debris, was validated.