# Hybrid-Compliant System for Soft Capture of Uncooperative Space Debris
## 4 Impedance Controller
### System Modeling  
The servicer satellite, consisting of a main body (CubeSat), and the hybrid compliant system for soft capture, is modelled as a three-body equivalent system, as represented in Figure 10. The CubeSat, along with the ACU's F/T sensor and the fixed part of the ACU's linear actuators, are lumped into the first rigid body with mass \(m_{s}\). The moving part of the ACU's actuators, the plate that separates ACU and PCU, and the PCU's upper legs are lumped into a second rigid body with mass \(m_{e}\); while the lower legs and the gecko adhesive pads are lumped to a third rigid body having mass \(m_{c}\). The positions of the center of mass (CoM) of \(m_{s}\) and \(m_{e}\) are denoted by \(x_{s}\) and \(x_{e}\), and the position of the mechanism's tip is denoted by \(x_{c}\). The debris is modelled as a rigid body of mass \(m_{d}\), and the position of the point on the debris that comes into contact with the mechanism's tip is denoted by \(x_{d}\). Masses \(m_{s}\) and \(m_{e}\) are connected through linear actuators, allowing a translation degree of freedom to be controlled. The maximum displacement of the linear actuators' moving parts is denoted by \(l_{a}\). Masses \(m_{e}\) and \(m_{c}\) are connected through passive compliance, with stiffness \(k_{s}\) and damping \(b_{s}\), that models the compliance provided by the 6 torsional springs located in the revolute joints of PCU's legs shown in Figure 7. Figure 11 provides a simplified 2D view of the three-body system. Figure 10: Equivalent three-body system of the CubeSat and ADR system. Before the contact, the CubeSat-ADR system has a non-zero relative velocity with respect to the debris. Once the PCU's mass \(m_{c}\) arrives in contact with the flat surface of the debris mass \(m_{d}\) at the moment \(t_{i}\), \(m_{c}\) and \(m_{d}\) have the same position, i.e., \(x_{c}=x_{d}\), the passive compliance enters into motion instantly, and the impedance controller is activated. The aim of the simulation study was to showcase the importance of incorporating active and passive compliant components to dissipate impact energy, ensure contact time, and enhance the capture of a wider range of space debris masses. Therefore, the simulation study was based on the following assumptions. The motion synchronization between the servicer and the debris was established, resulting in a zero relative angular velocity. The desired contact point of the ADR system's tip on the debris was assumed to pass through the debris centre of mass, resulting in only a contact force and no external moment on the debris. The assumption was made that the centre of mass of the debris is known, supported by existing research on estimation techniques. Misalignments during realistic approach and contact were not considered and flat surfaces were assumed for both the ADR system and debris, generating contact force along the approach and contact axis. Based on these assumptions, a three-dimensional simulation of the equivalent three-body system yields single-axis motion for the servicer and the ADR system was performed, providing informative data along the approach and contact axis. Due to the absence of relative rotational motion, all motion occurs along this axis. The inclusion of the assumption of point masses in the simulation model, neglecting the moment of inertia, does not affect the study's conclusions. The paper presents the equations of motion for this equivalent system, focusing on the commonly employed central impact analysis of the single motion axis. Specifically, the system equations of motion for each of the three rigid bodies of the equivalent CubeSat-ADR system in Figure 11 with masses \(m_{s}\), \(m_{e}\), and \(m_{c}\), and for the space debris with mass \(m_{d}\), obtained during the contact between \(m_{c}\) and \(m_{d}\), are given by Equations (1), (2), (3) and (4), respectively.