# Kinematic Analysis and Motion Planning of Cable-Driven Rehabilitation Robots
## 3 Kinematics Analysis of Rehabilitation Robot
\begin{array}{l}N_{1}=({c_{1}}^{2}+{c_{2}}^{2}{s_{1}}^{2})((2{c_{1}}^{2}-{s_ {1}}^{2}+s_{2})\dot{\theta}_{1}\dot{\theta}_{2}+s_{1}c_{1}c_{2}\dot{\theta}_{2 }^{2}+s_{1}c_{1}s_{2}\ddot{\theta}_{2}-c_{2}\ddot{\theta}_{1})\\ N_{2}=s_{1}(c_{2}\dot{\theta}_{1}-c_{1}s_{1}s_{2}\ddot{\theta}_{2})(s_{1}s_{2 }c_{2}\dot{\theta}_{1}+c_{1}s_{2}\ddot{\theta}_{1})\\ N_{3}=(s_{1}s_{2}(\dot{\theta}_{1}+\dot{\theta}_{2})-2c_{1}c_{2}\dot{\theta}_{1 }\dot{\theta}_{2})\sqrt{c_{1}^{2}{s_{2}}^{2}+c_{2}^{2}}\\ N_{4}=({c_{1}}^{2}{s_{2}}^{2}+c_{2}^{2})(s_{1}{c_{2}}^{2}\dot{\theta}_{2}^{2}+c_ {1}c_{2}s_{2}\dot{\theta}_{1}\dot{\theta}_{2}+s_{1}c_{2}s_{2}\ddot{\theta}_{2}+ s_{1}\dot{\theta}_{1}^{2}-s_{1}{s_{2}}^{2}\dot{\theta}_{2}^{2}-c_{1}\ddot{ \theta}_{1})\\ N_{5}=s_{2}(c_{1}\dot{\theta}_{1}-s_{1}s_{2}c_{2}\ddot{\theta}_{2})(s_{1}s_{2 }c_{1}\dot{\theta}_{1}+c_{2}s_{1}\dot{\theta}_{2})\\ N_{6}=(-4c_{1}c_{2}s_{1}s_{2}\dot{\theta}_{1}\dot{\theta}_{2}+(s_{2}^{4}-c_{2} s_{2})^{2})\dot{\theta}_{2}^{2}+s_{2}^{2}(1-c_{1}s_{1})\dot{\theta}_{1}^{2}-c_{ 2}{s_{2}}^{3}\ddot{\theta}_{2})\sqrt{c_{1}^{2}{s_{2}}^{2}+c_{2}^{2}}\\ N_{7}=(-s_{1}^{2}s_{2}c_{2}\dot{\theta}_{2}-s_{2}^{2}c_{1}s_{1}\dot{\theta}_{1 })d\\ \dot{d}=\frac{-s_{1}^{2}s_{2}c_{2}\dot{\theta}_{2}-s_{2}^{2}c_{1}s_{1}\dot{ \theta}_{1}}{\sqrt{c_{1}^{2}{s_{2}}^{2}+c_{2}^{2}}}\end{array} \right.\end{array} \tag{10}\]  
Assuming that the angular velocity vector of the end lower leg relative to the global coordinate system is \(\omega=\left[ \begin{array}{cc}\omega_{x}&\omega_{y}&\omega_{z}\end{array}  \right]^{T}\) and the angular acceleration vector is \(\varepsilon=\left[ \begin{array}{cc}\varepsilon_{x}&\varepsilon_{y}&\varepsilon _{z}\end{array} \right]^{T}\), the motion velocity vector is \(\dot{X}_{a}=\left[ \begin{array}{cc}v_{x}&v_{y}&v_{z}&\omega_{x}&\omega_{y}& \omega_{z}\end{array} \right]^{T}\), which can be obtained from the principle of kinematics  
\[\dot{X}_{a}=\left[ \begin{array}{c}v\\ \omega\end{array} \right]=\left[ \begin{array}{cc}I_{3\times 3}&0_{3\times 3}\\ 0_{3\times 3}&\Lambda\end{array} \right]\dot{X} \tag{11}\]  
where  
\[\Lambda=\left[ \begin{array}{ccc}\cos\beta\cos\gamma&-\sin\gamma&0\\ \cos\beta\sin\gamma&\cos\gamma&0\\ -\sin\beta&0&1\end{array} \right]\cdots\dot{X}=\left[ \begin{array}{c}v\\ \omega_{p}\end{array} \right] \tag{12}\]  
so there are \(\omega=E\omega_{p}\), further available  
\[\varepsilon=\dot{E}\omega_{p}+E\varepsilon_{p} \tag{13}\]  
Furthermore, the cable length vector can be obtained by the closed vector method,  
\[l_{i}=a_{i}-p_{op}-\frac{Q}{P}Rb_{i}, \tag{14}\]  
the length of the cable is  
\[q_{i}=\|l_{i}\|=\|a_{i}-p_{op}-\frac{Q}{P}Rb_{i}\|, \tag{15}\]  
and the quantity product of Equation (15) can be obtained as  
\[{q_{i}}^{2}=(l_{i})^{T}l_{i}=\Big{(}a_{i}-p_{op}-\frac{Q}{P}Rb_{i}\Big{)}^{T} \Big{(}a_{i}-p_{op}-\frac{Q}{P}Rb_{i}\Big{)} \tag{16}\]  
In the process of rehabilitation training, as cable winder \(A_{i}\) will move on the rail bracket, vector \(a_{i}\) will change, the derivation is not zero, point \(B_{i}\) is fixed relative to the local coordinate system, vector \(b_{i}\) is a fixed value, and its derivative is zero. The derivation of time on both sides of Equation (16) is simplified  
\[q_{i}\dot{q}_{i}=(l_{i})^{T}\Big{(}\dot{a}_{i}-\dot{p}_{op}-\frac{Q}{P}Rb_{i} \Big{)} \tag{17}\]due to \(\underset{p}{\overset{O}{R}}Rb_{i}=\omega\times\underset{p}{\overset{O}{R}}Rb_{i}\), Equation (17) can be written as  
\[q_{i}\dot{q}_{i}=(l_{i})^{T}\Big{(}\dot{a}_{i}-\dot{p}_{op}-\omega\times \underset{p}{\overset{O}{R}}Rb_{i}\Big{)} \tag{18}\]  
Equation (18) divides both sides by \(q_{i}\), and the cable length speed is  
\[ \begin{array}{l}\dot{q}_{i}=u_{i}^{T}\Big{(}\dot{a}_{i}-\dot{p}_{op}-\omega \times\underset{p}{\overset{O}{R}}Rb_{i}\Big{)}\\ =-u_{i}^{T}\dot{p}_{op}-(\underset{p}{\overset{O}{R}}Rb_{i}\times u_{i})^{T} \omega+u_{i}^{T}\dot{a}_{i}\end{array}