## 2 Configuration and Kinematics Modeling
### Forward and Inverse Kinematics Based on Geometrical Relationships
#### 2.2.2 Kinematics between Joint Angles and End-Effector Position  
Based on the coordinate system in the last section, the forward kinematics model from joint angles to end-effector position could be easily derived, similarly via coordinate transformation. The transform matrix from the world coordinate system \(\{W_{0}\}\) to the end-effector frame \(\{R_{N}\}\) could be derived as:  
\[T_{W}^{N}=\prod_{i=1}^{N}\left(T_{i-1}^{i}\cdot Trans(0,H,0)\right) \tag{9}\]  
where \(T_{i-1}^{i}\) is defined by Equation (1) and the definition of \(Trans(0,H,0)\) is similar to Equation (2).  
Due to the high redundancy of CHCM, the analytic solution of inverse kinematics does not exist and the numerical methods usually spend tremendous amounts of computational time based on the pseudoinverse of the Jacobian matrix. To improve the real-time performance of inverse kinematics computing, an efficient solution method based on geometric relationship of CHCM and some assumptions of its motion feature is introduced in this section.  
In confined working scenarios, the motion of the manipulator is restricted by both environment and mechanical structure, which means the joint modules are supposed to move less within a maximum limited angle. Thus, to simplify, some assumptions of the manipulator's motion are proposed.  
1. The position of the end-effector is mainly in an optional plane \(PTY\) perpendicular to the \(Y\)-axis.
2. The last tubular structure is parallel to the \(Y\)-axis.
3. The movement of the end-effector in the \(Y\)-axis direction is achieved by selecting different optional planes.  
To simplify the analysis, the joints are represented by dots and the tubular structures are represented by lines with a length of \(L=H+2\cdot D\), as shown in Figure 4. Suppose that the proximal end point of \(i\)th line is at \(A_{i}(x_{i},y_{i},z_{i})(i=1,2,\cdots,N)\), and the position of end-effector is \(A_{N+1}(x_{N+1},y_{N+1},z_{N+1})\), whose relationship could be derived as Equation (10).  
\[\left\{\begin{array}{c}x_{n+1}=x_{n}\\ y_{n+1}=y_{n}+L\\ z_{n+1}=z_{n}\end{array}\right. \tag{10}\]  
To calculate the desired position of each joint, a reference pose of CHCM is created where \(\varphi_{1}=\chi,\theta_{1}=0\) and \(\varphi_{i}=0,\theta_{i}=0(i=2,\cdots,N-3)\), thus, the equation of \(PTY\) is \(y=(N-1)\cdot L\cdot\cos(\chi)+L\). The subscripts \(r\) and \(d\) denote reference pose and desired pose, respectively. When the proximal end point of \(N\)th line \(A_{N,d}(x_{N,d},y_{N,d},z_{N,d})\) is determined by Equation (11), the movement of last \(N-k+1\) joint modules could achieve the desired position of the end-effector.  
\[\left\{\begin{array}{c}\left\|A_{N,d}-A_{k+1,r}\right\|\geq(N-k-1)\cdot L\\ \left\|A_{N,d}-A_{k,r}\right\|\leq(N-k)\cdot L\end{array}\right. \tag{11}\]  
As shown in Figure 5, the points \(A_{i,d}(i=k+1,N)\) in this state are located on a straight line; specifically, the angles \(\varphi_{i,d},\theta_{i,d}(i=k+2,N-1)\) stay zero. Then, according to the system geometry structure, the end point of \((k+1)\)th line \(A_{k+1,d}\) satisfies Equation (12).  
\[\left\{\begin{array}{c}\left\|A_{k,d}-A_{k+1,d}\right\|=L\\ \left\|A_{k+1,d}-A_{N,d}\right\|=(N-k-1)\cdot L\end{array}\right. \tag{12}\]  
As the end-effector moves in \(PTY\), we select the more accessible intersection point of the circle determined by Equation (11) and a fixed plane as the coordinate of \(A_{k+1,d}\), which is defined by \(A_{1,r}\), \(A_{n-2,r}\)\(A_{n,r}\) and expressed as Equation (12). Thus, \(A_{k+1,d}\) also satisfies Equation (13).  
\[f(A(x,y,z))=a\cdot x+b\cdot y+z=0 \tag{13}\]  
Since \(A_{k+1,d}\) and \(A_{N,d}\) are