# Real-Time Tension Distribution Design for Cable-Driven Parallel Robot
## 2 Problem Formulation
### Robot Dynamics Analysis  
By introducing the generalized coordinates vector \(q= \begin{bmatrix}p_{G}^{T},\dot{\phi}^{T},\dot{\psi}^{T}\end{bmatrix} \) for the pose of the end-effector, the robot dynamics can be expressed as follows [15]:  
\[M(q)\ddot{q}+C(q,q)\dot{q}+G(q)=A^{T}f \tag{6}\]  
where  
\[M(q)= \begin{bmatrix}m{l_{3\times 3}}&0_{3\times 3}\\ 0_{3\times 3}&\Theta^{T}{l_{p}}\Theta\end{bmatrix}  \tag{7}\]  
\[C(q,\dot{q})= \begin{bmatrix}0_{3\times 3}&0_{3\times 3}\\ 0_{3\times 3}&\Theta^{T}{l_{p}}\Theta+\Theta^{T}(\Omega)_{\times}({l_{p}} \Theta)\end{bmatrix}  \tag{8}\]  
\[G(q)= \begin{bmatrix}0\\ 0\\ -{m_{g}}\\ 0_{3\times 1}\end{bmatrix}  \tag{9}\]  
\(I_{p}\) denotes the inertia tensor of the end-effector about point \(P_{G}\) in \(\mathbb{C}_{o}\), and \(A\in\mathbb{R}^{m\times 6}\) is the Jacobian matrix, which is defined as follows:  
\[A=-\bar{J}\gamma \tag{10}\]  
with \(\gamma:= \begin{bmatrix}{l_{3\times 3}}&0_{3\times 3}\\ 0_{3\times 3}&\Theta\end{bmatrix} \). The condition that applies is that the tension of the \(i\)th cable must be \(f_{i}\in\begin{bmatrix}{f_{\texttt{min}}},{f_{\texttt{max}}}\  
approaches have limitations, such as a discontinuous solution or a complicated computation procedure. Real-time applications necessitate a technique that can both make the tension distribution problem continuous and reduce the amount of time spent computing the solution. Here, we present a method for determining the appropriate control input for a cable-driven robot that performs tension selection based on a tension function containing the hyperbolic tangent function. Because of the boundary constraint imposed by the hyperbolic tangent function, the proposed tension function achieves the desired tension limits. Adjusting the parameters within this tension function produces a change in tension that is suitable for the desired wrench. Compared to conventional optimization approaches, this strategy can determine the optimal tension distribution more rapidly and effectively while preserving the continuity of the tension's variation. 3 Tracking Control of Cable-Driven Robot with Hyperbolic Tangent Function-Based Tension Distribution  
Hyperbolic Tangent Function-Based Tension Distribution Method of Cable-Driven Robot with Known Jacobian Matrix  
We can select the desired control input of the robot system as follows in order to obtain the tracking control of a cable-driven robot [15]:  
\[A^{T}f=\tau_{d}=M\bar{q}_{r}+C\hat{q}+G \tag{11}\]  
where \(q_{r}\), the two-order time derivative of which is \(\bar{q}_{r}=k_{p}(q_{d}-q)+k_{d}(\hat{q}_{d}-\hat{q})+k_{i}\int_{0}^{\tau}(q_ {d}-q)\text{dt}\), represents the reference trajectory. From Equation (11), it is possible to achieve PID tracking control of a cable-driven robot with nonlinear compensation by properly adjusting the tension \(f\) in accordance with the unique Jacobian matrix \(J\) and desired wrench \(\tau_{d}\). We suggest a strategy for the distribution of continuous tension that utilizes a simple computation procedure. The value \(f\) is chosen based on the function defined below:  
\[f=f_{\texttt{min}}\mathbb{1}+\lambda[\mathbb{1}+\texttt{Tanh}(L)] \tag{12}\]  
where \(\mathbb{1}\in\mathbb{R}^{n}\) denotes a vector in which all elements are \(\mathbb{1}\), \(L=[L_{1},\ldots,L_{m}]^{f}\in\mathbb{R}^{n}\) represents a vector with an unknown value and that must be adjusted in accordance with the desired control input in wrench space, and \(\texttt{Tanh}(L)=[\texttt{tanh}(L_{1}),\ldots,\texttt{tanh}(L_{m})]^{T}\) and \(\lambda\) represent a positive scalar related to tension constraints, calculated as \(\lambda=\frac{f_{\texttt{max}}-f_{\texttt{min}}}{2}\). Consequently, considering the hyperbolic tangent function's boundary condition (\(-1\leqslant\texttt{tanh}x\leqslant 1\), with \(x\) denoting any scalar), any selection of \(L\) can satisfy the tension constraints (\(f_{i}\in[f_{\texttt{min}},f_{\texttt{max}}]\)).