## 3 The Identification Method of Impact Force
### Impact Force Modeling  
The modeling process of [PERSON]'s normal impact force is as follows [9]. The internal pressure \(p\) of the impact end-effector is assumed to be constant during the impact time. Thus, the magnitude of the normal impact force \(F_{n}\) can be geometrically derived using the amount of depression of the inflatable ball \(\delta\), as follows:  
\[F_{n}=PS=P\pi\Big{\{}2R_{c}\delta-\delta^{2}\Big{\}}, \tag{1}\]  
where \(R_{c}\) is the radius of the model. In fact, the impact end-effector is deformed during impact, which causes the internal pressure \(P^{\prime}\) of the impact end-effector to increase. The normal impact force should be expressed as \(F_{n}{}^{\prime}=P^{\prime}S\). Because \(P^{\prime}\) is greater than \(P\), \(F_{n}{}^{\prime}\) is greater than \(F_{n}\), which causes the normal impact force calculated by [PERSON]'s model to be smaller than the actual value. The modeling process after considering the internal pressure change of the flexible end-effector during impact is as follows. The object of this study is large space debris. The size of the flexible end-effector is much smaller than the size of the space debris, so the contact surface of the space debris is simplified to a plane. The impact force model is shown in Figure 3, where \(k_{n}\) is the equivalent normal stiffness, \(c_{n}\) is the equivalent normal damping, \(k_{t}\) is the equivalent tangential stiffness, \(c_{t}\) is the equivalent tangential damping, \(p\) is the initial pressure, \(P^{\prime}\) is the pressure after the impact, \(v_{1}\) represents the initial impact velocity of the end-effector, and \(v_{2}\) is the component velocity of space debris. It is assumed that the space debris is rigid and that the flexible end-effector is only elastically deformed. Assumptions are made, wherein (1) the gas is isothermal and (2) the perimeter of the cross-section of the membrane is constant. The cross-section is changed from a circular shape to a racetrack shape, as shown in Figure 4.  
It is assumed that the amount of depression of the inflatable ball under static load is \(\delta\), the height of the section after compression is \(d=D\)-\(\delta\), and the radius of the contact surface is \(b\). \(D\) is the diameter of the elastic ball before deformation. \(D^{\prime}\) is the diameter of the elastic ball in the direction perpendicular to the amount of depression after deformation and is also the diameter of the elastic ball after deformation. \(d\) represents the height of the elastic ball in the direction of the depression amount after deformation. According to the constant section perimeter, the deformed volume is as follows:  
\[V^{\prime}=2\pi\int_{0}^{d/2}\biggl{(}b+\sqrt{(d/2)^{2}-y^{2}}\biggr{)}^{2}dy+ \pi b^{2}d. \tag{2}\]  
Based on the isothermal assumption, the internal pressure after deformation is as follows:  
\[P^{\prime}=\frac{P_{0}V_{0}}{V^{\prime}}=\frac{P_{0}D^{3}/6}{2\int_{0}^{d/2} \biggl{(}b+\sqrt{(d/2)^{2}-y^{2}}\biggr{)}^{2}dy+b^{2}d}. \tag{3}\]  
According to the force-balance relationship between the pressure plate and the contact part of the inflatable ball, the elastic force \(F_{k}\) is as follows:  
\[F_{k}=P^{\prime}\pi b^{2}. \tag{4}\]  
Figure 4: Section structure change. Figure 3: Impact force model. According to [PERSON]'s damping model [13; 14], the damping force \(F_{c}\) is as follows:  
\[F_{c}=-c_{n}F_{k}\dot{\delta}_{n}. \tag{5}\]  
The damping coefficient is as follows:  
\[c_{n}=\frac{3\left(1-e^{2}\right)}{4v_{1}}, \tag{6}\]  
where \(e\) is the recovery coefficient of the material and \(v_{1}\) is the initial relative velocity of the impact. In summary, based on Hertzian contact theory [13; 14], the normal impact force \(F_{n}\) is as follows:  
\[F_{n}=F_{k}+F_{c}=\frac{\pi\delta^{2}P_{0}D^{3}/6}{32\int_{0}^{d/2}\!\!\left(b+ \sqrt{\left(d/2\right)^{2}-y^{2}}\right)^{2}dy+16b^{2}d}-c_{n}F_{k}\dot{\delta }_{n} \tag{7}\]  
Different from the friction behavior of metal, the friction coefficient of rubber in the process of friction is not constant.