# Real-Time Tension Distribution Design for Cable-Driven Parallel Robot
## 1 Introduction
In addition to the programming-based tension distribution approach, there exist other studies on this issue that solve the problem by examining the relationship between the structural equation's solution space and the tension constraint's hypercube. [PERSON] [12;13] introduced the iterative Dykstra method, which computes tension distribution by locating the intersection between the solution space of the structural equation and the hypercube of the tension constraint. However, this method cannot compute force distributions that are continuous throughout a trajectory, and the rate of convergence is slow. In order to solve the problems of computation speed and continuous solution, [PERSON] suggested a closed-form method in [14], followed by an improved version in [11]. Furthermore, the Barycentric force distribution method and its improved version, which can realize real-time calculation for cable-driven robots with two redundant cables, are provided in [15; 16; 17]. The puncture method [18] is another real-time approach that does not require a certain redundancy level. Although these algorithms calculate a continuous sequence of tension distributions, they only consider a certain minimum, maximum, or medium value, which may not be sufficient for certain applications, for example, those that require force control of the end-effector through stiffness control (where stiffness is affected by the tension in each cable). In addition, they only seek to solve the continuous problem by searching for solutions in the kernel space of the structure matrix, neglecting the possible discontinuity in the computed control wrench. In this paper, we present a straightforward tracking control strategy for a cable-driven robot with a hyperbolic tangent function for force distribution. The tension is expressed here as a function with respect to a vector \(L\in\mathbb{R}^{m}\) with a direction that is not specified. Utilizing the bounding attribute of the hyperbolic tangent function, this function satisfies tension requirements with user-selectable upper and lower bounds; \(L\) is chosen in relation to the computed PID control input in the wrench space. In addition, we propose a method for selecting an optimal tension distribution in relation to the computed wrench in tension space in order to carry out the tension distribution optimally and thereby achieve certain objectives, such as the minimization or maximization of the cable's tension vector norm. The proposed method may address the issue of potential discontinuity in the computed control wrench. This characteristic is highlighted in the comparison of our method with existing methods in the discussion section. Based on the proposed tension distribution method, we provide an algorithm that adaptively controls a cable-driven robot with physical parameters, such as its gravity center, that are unknown. Appropriate parameter selection in our tension distribution approach can manage adaptive tuning parameter-induced control input vibration, resulting in continuous tension. The rest of our paper is organized as follows: first, we introduce the problem formulation in Section 2, followed by the proposed method for tension distribution to track robot control in Section 3. A version of this method with consideration of optimization is proposed in Section 4. In Section 5, we propose an adaptive tracking control law for cable-driven robots to account for uncertain parameter of the structure matrix of cable-driven robots. The efficiency of our strategy is demonstrated by the simulation results presented in Section 6. We provide a discussion of comparisons of our method and previous methods in Section 7. Lastly, we conclude the paper in Section 8.