## 3 Kinematic Analysis
### Inverse Velocity Solving and Jacobian
#### 3.3.1 Velocity Analysis  
Equation (9) can be derived easily according to Equations (6) and (7),  
\[(\mathbf{L}+\mathbf{l}_{i})^{A}\mathbf{u}_{i}={}^{A}\mathbf{B}_{i}-\mathbf{A}_{i} \tag{9}\]  
Conduct a dot multiply by itself operation for both sides of Equation (9),  
\[(\mathbf{L}+\mathbf{l}_{i})^{2}=\left({}^{A}\mathbf{B}_{i}-\mathbf{A}_{i}\right)^{T}\left({}^{ A}\mathbf{B}_{i}-\mathbf{A}_{i}\right) \tag{10}\]  
Differentiate t on both sides of Equation (10),  
\[(\mathbf{L}+\mathbf{l}_{i})^{\bullet}_{l_{i}}=\left({}^{A}\mathbf{B}_{i}-\mathbf{A}_{i}\right) ^{T}\mathbf{\frac{d_{B}^{A}T}{dt}}\mathbf{B}_{i} \tag{11}\]  
where  
\[\frac{\mathbf{d_{B}^{A}T}}{dt}=\left[ \begin{array}{cc}\frac{d_{B}^{A}\mathbf{R}}{dt} &\frac{d\mathbf{p}}{dt}\\ \mathbf{0}_{1\times 3}&0\end{array} \right] \tag{12}\]  
\[\frac{d\mathbf{P}}{dt}=\mathbf{v} \tag{13}\]  
\[\frac{d_{B}^{A}\mathbf{R}}{dt}=\mathbf{\omega}\times_{B}^{A}\mathbf{R} \tag{14}\]  
where \(\mathbf{v}=\left[v_{x},v_{y},v_{z}\right]^{T}\) and \(\mathbf{\omega}=\left[\omega_{x},\omega_{y},\omega_{z}\right]^{T}\) denote the velocity vector and angular velocity vector of shield machine relative to the global coordinate frame \(\{A\}\), respectively. Therefore, Equation (14) can be derived by substituting Equations (12)-(14) to Equation (11). \[\overset{\bullet}{l}_{i}={}^{A}\mathbf{u}_{i}{}^{T}\mathbf{\omega}\times_{B}^{A}\mathbf{R} \mathbf{B}_{i}+{}^{A}\mathbf{u}_{i}{}^{T}\mathbf{v} \tag{15}\]  
According to the Lagrange formula,  
\[{}^{A}\mathbf{u}_{i}{}^{T}\mathbf{\omega}\times_{B}^{A}\mathbf{R}\mathbf{B}_{i}=\left({}^{A} \mathbf{R}\mathbf{B}\mathbf{B}_{i}\times{}^{A}\mathbf{u}_{i}\right)^{T}\mathbf{\omega} \tag{16}\]  
Equation (15) can be further described as  
\[\overset{\bullet}{l}_{i}={}^{A}\mathbf{u}_{i}{}^{T}\mathbf{v}+\left({}^{A}\mathbf{R}\mathbf{B} _{i}\times{}^{A}\mathbf{u}_{i}\right)^{T}\mathbf{\omega} \tag{17}\]  
Equation (17) can be written in matrix form as  
\[\overset{\bullet}{l}_{i}=\mathbf{J}_{i}\left[ \begin{array}{c}\mathbf{v}\\ \mathbf{\omega}\end{array} \right]_{6\times 1} \tag{18}\]  
where  
\[\mathbf{J}_{i}=\left[ \begin{array}{cc}{}^{A}\mathbf{u}_{i}{}^{T}&\left({}^{A}\mathbf{R} \mathbf{B}\mathbf{B}_{i}\times{}^{A}\mathbf{u}_{i}\right)^{T}\\ \end{array} \right]_{1\times 6} \tag{19}\]The relationship between the extension velocity of cylinders and the generalized velocity of the shield machine can be described explicitly as  
\[\overset{\bullet}{l}=\mathbf{j}\overset{\bullet}{x} \tag{20}\]  
where \(\overset{\bullet}{l}\) is an n-dimensional column vector composed of the extension velocity of each cylinder; \(\overset{\bullet}{x}\) is a 6-dimensional column vector composed of the velocity vector, \(\mathbf{v}\), and angular velocity vector, \(\mathbf{\omega}\); \(J\) is the Jacobian matrix for the parallel mechanism of shield machine, and they can be expressed as:  
\[\overset{\bullet}{l}=\left[\overset{\bullet}{l_{1}},\overset{\bullet}{l_{2}}, \overset{\bullet}{l_{3}}\cdots\cdots\overset{\bullet}{l_{n}}\right]^{T}\overset{ \bullet}{x}=\left[ \begin{array}{c}\mathbf{v}\\ \mathbf{\omega}\end{array} \right]=\left[v_{x},v_{y},v_{z},\omega_{x},\omega_{y}, \omega_{z}\right]^{T} \tag{21}\]  
\[\mathbf{J}=\left[ \begin{array}{cc}A_{\mathbf{u}_{1}}T&\left(\overset{A}{{}_{B}}\mathbf{R }\mathbf{B}_{1}\times\overset{A}{{}_{\mathbf{u}_{1}}}\right)^{T}\\ \overset{A}{{}_{B}}\mathbf{R}\mathbf{B}_{2}\times\overset{A}{{}_{\mathbf{u}_{2}}}\right)^ {T}\\ \vdots&\vdots\\ A_{\mathbf{u}_{n}}\overset{T}{\left(\overset{A}{{}_{B}}\mathbf{R}\mathbf{B}_{n}\times \overset{A}{{}_{\mathbf{u}_{n}}}\right)^{T}}\end{array} \right]_{\mathbf{n}\times 6} \tag{22}\]  
The Jacobian matrix, \(\mathbf{J}\), not only shows the mapping relationship between the joint space of each branch chain (thrust cylinder) and the workspace of the shield machine, but also reveals the linear mapping relationship between the extension velocity of cylinders and the generalized velocity of the shield machine.