I. Introduction
Cable-driven parallel manipulator (CDPM) has become a more active topic in robotics since the past three decades. CDPM is generally recognized as one type of parallel manipulators, in which the rigid links are replaced by cables. A common CDPM consists of a rigid body serving as an end-effector connecting with the fixed frame by number of cables. Due to its reconfigure-friendly mechanism, different applications and designs of CDPMs were brought to the industry. Its attention can be attributed to a fruitful amount of research on CDPM. Similar to the conventional parallel manipulators, one of the common issues that researchers would have encountered in a CDPM is the interference, especially for those complicated CDPMs formed by multibodies [1], [2]. The occurrence of interference of a CDPM could result in limited motion within a region of its usable workspace, depending on the initial region that it lies in [3]. Therefore, handling an interference in a CDPM usually requires us to answer the following questions: when and where does an interference occurs? What is the interference and how can it be handled? Most work in detection and localization of interference in CDPM have been addressed in [4]–[7]. Their work suggested some effective solutions in order to answer the when and where questions by providing techniques/algorithms for the detection of interference/collision in their specific CDPMs. Indeed, the first two questions are usually combined as the study on the interference detection. Techniques used in collision detection in computer graphics were also considered but the drawbacks of using those techniques in the real world were also addressed in a comprehensive survey [8]. Nevertheless, the methods introduced in the survey can be the starting points for investigation and development of feasible algorithms, which are able to detect collisions in physical systems. Some of the approaches introduced by Merlet et al in [3], which aimed at checking the interference among cylindrical legs of parallel robots over a given workspace or trajectory, can also be adopted and modified for the use of interference detection in CDPMs. Furthermore, the what question can be answered by classifying the objects involved in an interference. An interference can be classified as external interference or internal interference depending on the objects involved. External interference usually refers to the interference between any part of the CDPM (particularly the cable) and the environment. The internal interference herein refers to the self-interference of a CDPM due to cable and cable or cable and link/end-effector in the CDPM. The influence of self-interference were studied in [4]. It proposed some algorithms to determine the regions in the workspace where interference occurs with a fixed orientation assumption on the end-effector. The study in [5] extended the work in [4] by proposing an algorithm to determine the regions of self-interference with less computational time. Both of them can be used for trajectory planning within an interference-free workspace. Last but not least, ignoring the collisions [9] or permitting the collisions [10] were commonly found as the answers to the how question. In other words, they were not particularly handled or taken into account when there was cable-to-link interference [11]. Yet, there are a few methods introduced and developed so far in order to handle the relevant problems [6], [7]. However, interference sometimes can be useful. For instance, the cables involved in the cable-to-link interference can also be used as a means of actuation similar to those cables without interference occurred. Many natural phenomena, including human motions, in fact rely on the wrapping phenomenon in the alignments of muscles and bones. One example can be found from the glenohumeral joint (a.k.a shoulder joint). In which the muscle-bone alignment enables upper arm to perform internal and external rotation. Similarly, when a cable collides with a rigid body, the situation appears to be a cable wrapping on a rigid body phenomenon. When a cable wraps on a rigid-link, instead of putting a boundary in the workspace, continuously pulling the cable can produce a moment exerting on the link. To this end, the motivation of this work is to introduce the wrapping phenomenon in the modelling of a CDPM such that the cable force can be effectively utilized and some of the interference regions in the usable workspace can be released simultaneously. Some preliminary work has demonstrated that a wrappable cable is no longer a straight line but a 3-dimensional curve in an orthogonal coordinate system. With the “no slack” assumption on the cable, the parametric expression of the cable can be solved by using Lagrangian equation [12]. An illustrative example with comprehensive modelling on multicable wrapping on a cylindrical body was given in [13]. The goal of this paper is to demonstrate the approach and the feasibility of incorporating the wrapping phenomenon in the modelling of a CDPM. Through the study on the inverse dynamics of the CDPM, the result not only shows the resolution of singularity once cable is allowed to wrap but also indicates the possibility to release the usable workspace from the interference. In fact, the incorporation of cable wrapping can produce a more accurate model of a CDPM. It appears to be a promising answer to the how question which provides a novel perspective to the trajectory planning and workspace analysis.