# Design and Stiffness Optimization of Bionic Docking Mechanism for Space Target Acquisition
## 3 Model of Bionic Docking Mechanism
### Kinematics Equation  
Establish the absolute inertial coordinate system \(OXYZ\), and establish the conjoined coordinate system \(O_{k}X_{k}Y_{k}Z_{k}\) at the end face of segment \(k\) (\(k=1,2,3,4,5\)). The linear velocity \(\boldsymbol{v}_{k}^{k-1}\) and angular velocity \(\boldsymbol{\omega}_{k}^{k-1}\) of the relative motion between two adjacent segments \(k\) and \((k-1)\) are selected as the generalized velocity \(\boldsymbol{y}_{l}\), and meanwhile the linear displacement \(\boldsymbol{s}_{k}^{k-1}\) and angular displacement \(\boldsymbol{\theta}_{k}^{k-1}\) are selected as the generalized coordinates \(\boldsymbol{x}_{l}\) (\(l=1,2\), \(3,\ldots,10\)), that is:  
\[ \begin{array}{l}\boldsymbol{\omega}_{1}^{0}=\left[\begin{array}{ccc} \boldsymbol{y}_{1}&\boldsymbol{y}_{2}&\boldsymbol{y}_{3}\end{array}\right]^{ \mathrm{T}}&\boldsymbol{\theta}_{1}^{0}=\left[\begin{array}{ccc}\boldsymbol {x}_{1}&\boldsymbol{x}_{2}&\boldsymbol{x}_{3}\end{array}\right]^{\mathrm{T}} \\ \boldsymbol{\omega}_{2}^{1}=\left[\begin{array}{ccc}\boldsymbol{y}_{4}&0&0 \end{array}\right]^{\mathrm{T}}&\boldsymbol{\theta}_{1}^{1}=\left[\begin{array} []{ccc}\boldsymbol{x}_{4}&0&0\end{array}\right]^{\mathrm{T}}\\ \boldsymbol{\omega}_{3}^{2}=\left[\begin{array}{ccc}0&\boldsymbol{y}_{5}&0 \end{array}\right]^{\mathrm{T}}&\boldsymbol{\theta}_{3}^{2}=\left[\begin{array} []{ccc}0&\boldsymbol{x}_{5}&0\end{array}\right]^{\mathrm{T}}\\ \boldsymbol{\omega}_{3}^{4}=\left[\begin{array}{ccc}0&0&\boldsymbol{y}_{6} \end{array}\right]^{\mathrm{T}}&\boldsymbol{\theta}_{3}^{4}=\left[\begin{array} []{ccc}0&0&\boldsymbol{x}_{6}\end{array}\right]^{\mathrm{T}}\\ \boldsymbol{v}_{1}^{0}=\left[\begin{array}{ccc}\boldsymbol{y}_{7}&\boldsymbol {y}_{8}&\boldsymbol{y}_{9}\end{array}\right]^{\mathrm{T}}&\boldsymbol{ \theta}_{1}^{0}=\left[\begin{array}{ccc}\boldsymbol{x}_{7}&\boldsymbol{x}_{8 }&\boldsymbol{x}_{9}\end{array}\right]^{\mathrm{T}}\\ \boldsymbol{v}_{5}^{4}=\left[\begin{array}{ccc}0&0&\boldsymbol{y}_{10} \end{array}\right]^{\mathrm{T}}&\boldsymbol{s}_{5}^{4}=\left[\begin{array}{ cccc}0&0&\boldsymbol{x}_{10}\end{array}\right]^{\mathrm{T}}\end{array}  \tag{1}\]  
where  
\(\boldsymbol{x}_{10\times 1}\) is used to store generalized coordinates;  
\(\boldsymbol{y}_{10\times 1}\) is used to store generalized velocity. Then the relative rotation matrix of the connected coordinate system in segment \(k\) and \((k-1)\) is:  
\[ \begin{array}{l}\boldsymbol{A}_{1}^{0}=\mathrm{Rot}(X,\boldsymbol{x}_{1}) \cdot\mathrm{Rot}(Y,\boldsymbol{x}_{2})\cdot\mathrm{Rot}(Z,\boldsymbol{x}_{3}) \\ \boldsymbol{A}_{2}^{1}=\mathrm{Rot}(X,\boldsymbol{x}_{4})&\boldsymbol{A}_{2}^{2} =\mathrm{Rot}(Y,\boldsymbol{x}_{5})\\ \boldsymbol{A}_{4}^{3}=\mathrm{Rot}(Z,\boldsymbol{x}_{6})&\boldsymbol{A}_{5}^{4} =I\end{array}  \tag{2}\]  
where  
\(\boldsymbol{I}\) is the \(3\times 3\) identity matrix. Then the absolute rotation matrix of the segment \(k\) connected coordinate system \(O_{k}X_{k}Y_{k}Z_{k}\) relative to the inertial system \(OXYZ\) is:  
\[\boldsymbol{A}_{k}^{0}=\boldsymbol{A}_{1}^{0}\cdot\boldsymbol{A}_{2}^{1}\cdots \boldsymbol{A}_{k}^{k-1} \tag{3}\]  
Since the base has three rotational DOFs, it is assumed that coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\) connected to the base first rotates around the \(OX\) axis of the inertial coordinate system \(OXYZ\) and the angular velocity is \(\dot{\boldsymbol{x}}_{1}\), then around the \(O_{1}Y_{1}\) axis of the coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\) connected to the base, the angular velocity is \(\dot{\boldsymbol{x}}_{2}\), and finally around the  
Figure 5: Model of the bionic docking mechanism in space docking. \(O_{1}Z_{1}\) axis of the coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\) connected to the rotated base, the angular velocity is \(\dot{\mathbf{x}}_{3}\), Then the angular velocity \(\mathbf{\omega}_{1}^{0}\) of the base in the inertial system \(OXYZ\) is:  
\[\mathbf{\omega}_{1}^{0}=\left( \begin{array}{c}\dot{\mathbf{x}}_{1}\\ 0\\ 0\end{array} \right)+\text{Rot}(X,\mathbf{x}_{1})\cdot\left(