# Design and Stiffness Optimization of Bionic Docking Mechanism for Space Target Acquisition
## 3 Model of Bionic Docking Mechanism
### Dynamic Equations  
For a tandem mechanism with \((N+1)\) segments, the Kane dynamic equations can be written as  
\[\begin{array}{l}\mathbf{F}_{l}+\mathbf{F}_{l}^{*}=\sum\limits_{k=1}^{5}\left(\mathbf{F}_ {k\cdot}\cdot\mathbf{V}_{kl}+\mathbf{M}_{k\cdot}\cdot\mathbf{W}_{kl}\right)\\ +\sum\limits_{k=1}^{5}\left(\mathbf{F}_{k\cdot}^{*}\cdot\mathbf{V}_{kl}+\mathbf{M}_{k \cdot}^{*}\cdot\mathbf{W}_{kl}\right)=0\end{array} \tag{16}\]  
where  
\(k=1,2,\ldots,5\);  
\(l=1,2,3,\ldots,10\).  
Substituting partial angular velocity (8), partial linear velocity (12), equivalent active force (torque) (14) and equivalent inertia force (torque) (15) into (16) and combining kinematics equation (5), the integrated nonlinear ordinary differential equations of the bionic docking mechanism in space docking are obtained:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}=\mathbf{P}_{10\times 10}\cdot\mathbf{y}_{10\times 1 }\\ \dot{\mathbf{y}}=\mathbf{I}_{10\times 10}^{-1}\cdot\mathbf{f}_{10\times 1}\end{array}\right. \tag{17}\]  
where  
\[\mathbf{P}_{1  
\[\begin{array}{l}\mathbf{L}_{d}=\sum\limits_{i=2}^{5}\left(\mathbf{I}_{i}\cdot\mathbf{\omega}_{i }^{0}+\mathbf{p}_{i}\times\mathbf{P}_{i}\right)\text{ is the angular momentum of the bionic docking mechanism;}\\ \text{nism;}\\ \mathbf{P}_{i}=m_{i}\mathbf{v}_{i}^{0}\text{ is the linear momentum of segment }i;\\ \mathbf{p}_{i}=\int\mathbf{v}_{i}^{0}\mathrm{d}t\text{ is the linear displacement of segment }i;\\ \mathbf{L}_{0}\text{ is the total angular momentum after collision.}\\ \text{Assuming that the initial states of the kinematic parameters are all zero, then the integral of the above formula can be obtained:}\\ \int\mathbf{L}_{b}\mathrm{d}t+\int\mathbf{L}_{d}\mathrm{d}t=\int\mathbf{L}_{0}\mathrm{d}t \text{ }\\ \text{where}\\ \int\mathbf{L}_{b}\mathrm{d}t=\mathbf{I}_{1}\cdot\mathbf{\theta}_{1}^{0}+\left(\int\mathbf{ p}_{1}\mathrm{d}t\right)\times\left(m_{1}\mathbf{v}_{1}^{0}\right);\\ \int\mathbf{L}_{d}\mathrm{d}t=\sum\limits_{i=2}^{5}\left[\mathbf{I}_{i}\cdot\mathbf{ \theta}_{i}^{0}+\left(\int\mathbf{p}_{i}\right)\times\left(m_{i}\mathbf{v}_{i}^{0} \right)\right];\\ \end{array} \tag{19}\]  
\(\mathbf{\theta}_{i}^{0}\) is the angular displacement of free-floating base;  
\(\mathbf{\theta}_{i}^{0}\) is the angular displacement of segment \(i\).  
The inequality can be expressed as:  
\[\int|\mathbf{L}_{b}|\mathrm{d}t+\int|\mathbf{L}_{d}|\mathrm{d}t\geq\int|\mathbf{L}_{0}| \mathrm{d}t \tag{20}\]  
where  
\[\int|\mathbf{L}_{b}|\mathrm{d}t\leq\mathbf{I}_{1}\cdot\left|\mathbf{\theta}_{1}^{0}\right| +\left(\int|\mathbf{p}_{1}|\mathrm{d}t\right)\times\left(m_{1}\left|\mathbf{\varphi}_ {1}^{0}\right|\right);\]  
\[\int|\mathbf{L}_{d}|\mathrm{d}t\leq\sum\limits_{i=2}^{5}\left[\mathbf{I}_{i}\cdot \left|\mathbf{\theta}_{i}^{0}\right|+\left(\int|\mathbf{p}_{i}|\mathrm{d}t\right) \times\left(m_{i}\left|\mathbf{v}_{i}^{0}\right|\right)\right]\text{.}\]  
Since \(\int|\mathbf{L}_{0}|\mathrm{d}t\), \(\mathbf{I}_{i}\), \(m_{i}\) are static values, the larger the \(\left|\mathbf{\theta}_{i}^{0}\right|\) and \(\left|\mathbf{p}_{i}\right|\), the smaller the \(\left|\mathbf{\theta}_{1}^{0}\right|\) and \(\left|\mathbf{p}_{1}\right|\), that is, the smaller the angular momentum of free-floating base, which is from space collision.  
Through the above analysis of angular momentum of the spaceborne capture mechanism after collision, minimizing the angular momentum of the base is the prerequisite to ensure its stability. Thus, the optimization strategy of the bionic docking mechanism is to increase vibration amplitudes of the Gyro-X joint, Gyro-Y joint, Gyro-Z joint and Slide-Z joint to avoid free-floating base instability. Obviously, the adjustment of vibration amplitude can be realized through the optimal design of spring stiffness coefficients.