# Hybrid-Compliant System for Soft Capture of Uncooperative Space Debris
## 5 Simulation Study and Results
### Debris-Mass and Compliance Relation  
As the masses of the servicer CubeSat, including the ADR system, were assumed to be known, the desired stiffness and the damping coefficients of the hybrid system's equivalent compliance must be selected. The selected parameters should ensure that the minimum contact time between the ADR system and the debris was achieved. An analytical solution for the optimal tuning of these coefficients is difficult to obtain since no analytical equation relates the contact time and the hybrid compliance coefficients. Because of this, an algorithm in MATLAB, consisting of a loop, was developed to search the successful cases (\(t_{c}>\) minimum contact time required) in a range of stiffness values, for a range of space debris masses and for a range of minimum contact time required to complete the capture. Specifically, the servicer CubeSat and the ADR system were simulated when approaching and coming into contact and the success in terms of the time of contact was noted. It was considered a successful case if it was greater than the contact time required for the successful capturing while not reaching the spring limit. Then, the corresponding spring's stiffness and the debris mass were stored. In this simulation study, for tuning the hybrid compliance of the system, the servicer CubeSat and the ADR system were modelled as a two-mass equivalent system, i.e., as two point masses connected by the hybrid compliance, as shown in Figure 14b. This compliance was considered hybrid since it consists of passive parts integrated into the PCU and the active part realized by the impedance-controlled linear actuator of the ACU, as shown in Figure 14a. The stiffness and damping coefficients to be altered during the search of the developed algorithm are denoted by \(k_{m}\) and \(b_{m}\), respectively. Once \(k_{m}\) is altered, by choosing critical damping, one can calculate \(b_{m}\) too, as follows  
\[b_{m}=2\sqrt{m_{\rm c}k_{m}}. \tag{19}\]  
The hybrid compliance's length \(l_{m}\) is equal to 0.05m since it is the sum of two lengths: \(l_{m}=l_{s}+l_{a}\); the length \(l_{s}\) of the passive physical compliance, equal to 0.025 m, and the maximum displacement \(l_{a}\) of the linear actuators' moving parts, equal to 0.025 m. The schematic of the system under simulation study as designed in Simscape is shown in Figure 15.  
Figure 15: Schematic of CubeSat-ADR system and debris in Simscape. The algorithm runs for the range of stiffness values \(k_{m}\) between [0.1-0.5] N/m, with a step of 0.1 N/m, for a range of space debris mass \(m_{d}\) between [1-100] kg, with a step of 1 kg, and for a range of minimum contact time \(t_{c}\) required for completion of the capturing between [4-12] s, with steps of 2 s.  
The resulting diagram is shown in Figure 16, providing the relation between these variables. More detailed visualization of the data of Figure 16 is provided for each contact time in the range of [4-12] s in Appendix A. The lines depicted in the 3D diagram correspond to points in 3D space, representing the successful cases obtained from the algorithm. Each point along the line represents three distinct values, namely: space debris mass, minimum contact time achieved, and hybrid compliance stiffness coefficient. Note that the [4-12] s range for this simulation for the minimum contact time required was selected since it was sufficient to showcase the necessity of tuning the system's compliance and, therefore, of the hybrid compliance concept. Simulation results for minimum contact time required greater than 12 s show that hybrid compliance is even more necessary if we want to target a wide range of debris between 0-100 kg. This can be easily shown by the trend shown in Figure 16: the minimum contact time required increases, and the range of debris masses to be targeted decreases. Moreover, a contact time of less than 4 s for successful capturing is considered unrealistically small.