# Kinematic Analysis and Motion Planning of Cable-Driven Rehabilitation Robots
## 2 Structure of Rehabilitation Robot and Human Lower Limb Model
### Human Lower Limb Model  
During rehabilitation training, the patient's body is in a horizontal posture of pitching, and the hip joints of the lower limbs are fixed in the middle of the bed. Considering the movement of the lower limbs, the structure of the lower limbs is analyzed, and a coordinate system in the lower limbs of the human body is established [33], as shown in Figure 2. \(O-xyz\) is the global coordinate system, \(O_{0}\) is the hip position, which is used as the base reference system \(O_{0}-x_{0}y_{0}z_{0}\), \(O_{p}\) is the position of the end ankle and is fixedly connected to the end ankle. The zero connecting rod is used to adjust the height of the hip joint and without causing rotation. The intersection of joint 1, joint 2, and joint 3 intersect at one point, which is the hip joint; joint 4 and joint 5 intersect at one point, which is the knee joint; and joint 6 is the ankle joint, its rotation will not affect the end trajectory; \(d_{1}\) is the length of the thigh; and \(L_{2}\) is the length of the lower leg. Table 1 shows the Denavit-Hartenberg (D-H) parameter, and the comfortable range of motion of the human hip joint is determined according to the maximum range of motion of the human hip joint [34]. In light of the principle of kinematics [35; 36], the transformation matrix of the lower limb \({}^{p}_{o}T\) is  
\[{}^{p}_{o}T=\left[ \begin{array}{cc}R_{3\times 3}&P_{3\times 1}\\ 0&1\end{array} \right] \tag{1}\]  
where \(R_{3\times 3}\) is the rotation matrix and \(P_{3\times 1}\) is the spatial position, which represents the orientation and position of the end ankle relative to the base reference coordinate system, respectively. The posture changes of the end ankle can be obtained by reasonably planning the angles of each joint. In the process of rehabilitation, the posture of the ankle joint is not considered, and the position of the end lower leg is defined as the ankle joint. The position expression in the base frame is given, as shown in Formula (2). \[\left\{ \begin{array}{l}p_{x}=d_{1}c_{1}s_{2}+L_{2}(c_{5}(s_{1}s_{34}+c_{1}c_ {2}c_{34})+c_{1}s_{2}s_{5})\\ p_{y}=d_{1}c_{2}+l_{2}(c_{2}s_{5}-s_{2}c_{5}c_{34})\\ p_{z}=d_{1}s_{1}s_{2}-L_{2}(c_{5}(c_{1}s_{34}-c_{1}c_{2}c_{34})-s_{1}s_{2}c_{5} )\\ \end{array} \right. \tag{2}\]  
where \(s_{i}=\sin\theta_{i}\), \(c_{i}=\cos\theta_{i}\), \(s_{ij}=\sin(\theta_{i}+\theta_{j})\), and \(c_{ij}=\cos(\theta_{i}+\theta_{j})\)\((i,j=1,2,3,4,5,6)\). Assume that the position vector of the hip joint is \(p_{o}=\left[ \begin{array}{cc}x_{o}&y_{o}&z_{o}\end{array} \right]^{T}\) in the global coordinate system, when the knee joint is not bent, only the hip joint is subject to rehabilitation. Figure 1: (**a**) Overall design of cable-driven rehabilitation robot. (**b**) Schematic diagram of the mechanism of DC motor and servo motor. tion training, that is, \(\theta_{3}=-90\), \(\theta_{4}=0\), \(\theta_{5}=90\), \(\theta_{6}=0\), let \(d_{1}+L_{2}=b\), at this time, the position \(p_{op}\) and posture \(R_{op}\) of the end lower leg are  
\[\left\{ \begin{array}{c}p_{op}=\left[\begin{array}{c}p_{ox}\\ p_{oy}\\ p_{oz}\end{array}\right]=\left[\begin{array}{c}bc_{1}s_{2}\\ bc_{2}\\ bs_{1}s_{2}\end{array}\right]+\left[\begin{array}{c}x_{o}\\ y_{o}\\ z_{o}\end{array}\right]\\ R_{op}=\left[\begin{array}{cc}c_{1}s_{2}&-c_{1}c_{2}&-s_{1}\\ c_{2}&s_{2}&0\\ s_{1}s_{2}&-s_{1}c_{2}&c_{1}\end{array}\right]\end{array} \right. \tag{3}\]  
By deriving both sides of the position expression in Equation (3), it can be found that the velocity \(v_{op}\) of the lower leg of the human body is  
\[v_{op}=\left[ \begin{array}{c}\dot{p}_{ox}\\ \dot{p}_{oy}\\ \dot{p}_{oz}\end{array} \right]=\left[ \begin{array}{c}v_{ox}\\ v_{oy}\\ v_{oz}\end{array} \right]=\left[ \begin{array}{cc}-bs_{1}s_{2}&bc_{1}s_{2}\\ 0&-bs_{2}\\ bc_{1}s_{2}&bs_{1}c_{2}\end{array} \right]\left[ \begin{array}{c}\dot{\theta}_{ 1}\\ \dot{\theta}_{2}\end{array}