# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 2 Modeling of Spacecraft-Manipulator System with FDCDM
### Contact Force Model  
All capture tasks require the robot to interact with space objects, where big impact forces can be exerted. To solve this problem, it is desirable to know the contact force in the whole capture process so that effective compliant control can be designed.  
Currently, a lot of research indicate that the results of [PERSON]'s model are similar to those of actual contact tests at low speed. In this paper, a compound contact model combining an optimized [PERSON]'s model and a friction model is employed to estimate the contact force without additional sensors. The original [PERSON]'s model is established based on the theory of elasticity contact, which considers that there is no energy loss in the contact process; however, there are the following problems: at the beginning of the collision, the contact force is discontinuous, and at the end of the collision, the collision force makes the contact objects attract each other. So, it cannot accurately reflect the actual contact process.  
[PERSON] thinks that the energy loss in the contact process can be expressed as a function of the coefficient of restitution and the initial collision velocity of two contact objects. So, inspired by [16], the [PERSON] contact model based on the [PERSON] model and a hysteresis damping term can be written as:  
\[\mathbf{F}_{\mathrm{n}}=\mathbf{K}\delta^{n}+\mathbf{\xi}\mathbf{K}\delta^{p}\dot{\delta}^{q} \tag{19}\]  
where \(K\) is the contact stiffness coefficient, \(\delta\) is the penetration depth, \(\mathbf{\xi}\) is the damping coefficient, and \(\dot{\delta}\) is the relative impact velocity. Generally, the parameters \(p\) and \(q\) are set as \(p=n\) and \(q=1\), respectively.  
The hysteresis damping factor \(\mathbf{\xi}\) is designed as:  
\[\mathbf{\xi}=\frac{6(1-C_{r})\mathbf{K}}{3C_{r}\dot{\delta}_{0}^{n}} \tag{20}\]  
where \(C_{r}\) and \(\dot{\delta}_{0}\) denote the restitution coefficient and the initial impact velocity, respectively.  
When the grippers come into contact with the target surface, tangential friction \(\mathbf{F}_{\mathrm{f}}\) will be generated. It is generally considered that the tangential friction is orthogonal to normal pressure \(\mathbf{F}_{\mathrm{n}}\) and can be studied separately.  
Because of the obvious advantages of the dynamic friction model compared with the static friction model, the dynamic friction model can describe the friction process more comprehensively. Among a variety of dynamic models, the LuGre friction model is a famous model in control research; it describes the Coulomb friction, pre-sliding, and Stribeck friction with first order differential equations, which can be described by [38]  
\[\left\{\begin{array}{l}\dot{z}=v-(\sigma_{0}z/g(v))|v|\\ \|\mathbf{F}_{\mathrm{f}}\|=\sigma_{0}z+\sigma_{1}\dot{z}+\sigma_{2}v\end{array}\right. \tag{21}\]where \(\sigma_{0}\) is the stiffness, \(\sigma_{1}\) is the damping, and \(\sigma_{2}\) is the stickiness. \(v\) is the relative velocity between contact surfaces. \(g(v)=F_{c}+(F_{s}-F_{c})\exp\left(-(v/v_{s})^{2}\right)\) denotes the Stribeck model. \(F_{c}\) is the coulomb friction force, \(F_{s}\) is the maximum static friction, and \(v_{s}\) is an empirical parameter which expresses the Stribeck speed.  
In this section, the [PERSON] model is employed to estimate the normal pressure, and the tangential friction is obtained by the LuGre model. Then, we can obtain the dynamic contact force between the manipulator and the target during capturing, so the precise stabilization control can be designed based on the dynamic contact model.