This section presents the mathematical preliminaries for the modeling of CDRs. These preliminaries include the general kinetostatic analysis for both single-bodied (henceforth termed as “modular”) CDRs, as well as multi-bodied open chain CDRs (consisting of modular CDRs attached in series). Various theories in convex analysis and reciprocal screw are also presented to allow the readers to follow the proposed force-closure analyses in section “Mathematical Preliminaries.”

Kinetostatic Analysis for Modular CDRs

This subsection presents a brief overview of kinematic analysis for modular CDRs and highlights the relevant theories in convex analysis which will allow the readers to clearly follow the proposed force-closure analysis of CDRs in section “Force- Closure Analysis of Modular CDRs.” The kinematic diagram of an n-DOF modular CDR, controlled by m cables, is shown in Fig. 4. The moving platform is connected to the base through m cables. {KB} represents the fixed base coordinate frame, and {KS} represents the moving platform frame. All quantities are represented with respect to frame {KB} unless there is a trailing superscript “S” indicating that it is written in frame {KS}. The origin of frame {KS} with respect to frame {KB} is represented by position vector s, while the orientation of frame {KS} with respect to frame {KB} is represented by the rotation matrix R.

Kinetostatic Analysis for Multi-Bodied CDRs

This subsection presents a brief overview of kinematic analysis using reciprocal screw theory and highlights relevant theories which will allow the readers to clearly follow the proposed force-closure analysis of multi-bodied open chain CDRs (i.e., consisting of modular CDRs attached in series) in section “Force-Closure Analysis of Multi-Bodied CDRs.”