## 1 Introduction  
Unlike conventional robots that work in structured environments, safe human-robot interactions have been a key element for the robots that work in unstructured and unpredictable environments. Inspired by the structure of human arms, a modular _cable-driven human-like robotic arm_ (CHRA) is developed for safe human-robot interaction, which employs cables to mimic the functionality of the human muscles. As shown in Figure 1, the CHRA consists of a shoulder joint, an elbow joint and a wrist joint in series, where the shoulder joint and the wrist joint have _three degrees of freedom_ (3-DOF) and the elbow joint has _one degree of freedom_ (1-DOF). The CHRA employs lightweight cables to drive the rigid links and the cables can be wound into winches mounted onto the base of the CHRA. A _variable-stiffness device_ (VSD) is designed and placed along with each driving cable to increase the flexibility of the cables. With these arrangements, the CHRA has flexibility advantages [1], low moving mass [2], large workspace [3] and high payload-to-weight ratio [4]. Due to these advantages, the CHRA is intrinsically safe for human-robot interactions. The proposed CHRA and its joint modules are one kind of _cable-driven mechanisms_ (CDMs). In the last decades, various CDMs have been designed for various applications, such as medical robots [5; 6; 7], rehabilitation robots [8; 9; 10; 11; 12], inspection and repair [13; 14; 15], and moving payloads [16; 17; 18; 19]. Unlike rigid links, the cables have the unilateral driving property (i.e., they can only pull, but can not push). Hence, the CHRA and its joint modules are redundantly actuated. For a given pose, a lot of cable tension solutions are feasible for the CHRA. Furthermore, the stiffness of the CHRA can be adjusted by regulating the cable tensions. The characteristic of variable stiffness increases the flexibility and safety of the CHRA. The problem of _stiffness-oriented cable tension distribution_ (SCTD) of a CDM aims at finding the optimal cable tensions to achieve a desired stiffness. However, in the last decades, most studies on cable tension distribution have been carried out to minimize the cable tensions to reduce the energy consumption [20; 21; 22; 23; 24; 25; 26; 27]. In [28], the SCTD problem of a CDM was studied by formulating it as an optimization model, and a gradient-projection-based algorithm was presented to solve the optimization problem. However, it utilized the determinant of the stiffness matrix as the objective function. With this method, the desired stiffness cannot be achieved accurately. In [29], the SCTD problem of a 3-DOF cable-driven spherical mechanism was studied and it established the optimization model with all entries of the stiffness matrix, not only the determinant of the stiffness matrix. However, it employed the conventional stiffness model of the CDM, which was derived by the conventional differential formula of the load to the displacement. Since the trajectory of the 3-DOF cable-driven spherical mechanism is a curve on Lie group SO(3) and SO(3) is nonlinear, the stiffness model based on the conventional differential formula is not exactly accurate on SO(3). Its stiffness should be evaluated by the variation of its load against its displacement on SO(3). In order to study the SCTD problem of the proposed CHRA, the SCTD problems of the 1-DOF and 3-DOF joint modules should be studied first. Due to the simple structure of the 1-DOF joint module, its SCTD problem can be solved easily. For the 3-DOF _joint module_ (3 DJM) of the CHRA, its trajectory is a curve on SO(3). We derive the stiffness model of the 3 DJM by the covariant derivative of the load to the displacement on SO(3), and call it the enhanced stiffness model of the 3 DJM. In this paper, we focus on investigating how the cable tension distribution problem oriented the enhanced stiffness of the 3 DJM. Due the complexity of the enhanced stiffness model, it is difficult to solve the cable tensions from the desired stiffness of the 3 DJM analytically.