This chapter focused on the performance analysis tools such as force closure, stiffness, workspace, and cable tension planning, required in the design of cable-driven robots (CDRs). CDRs are a special class of parallel mechanisms in which the end-effector is actuated by cables, instead of rigid-linked actuators. They are characterized by lightweight structures with low moving inertia and large workspace, due to the location of the cable winching actuators at the fixed base of the structure and thereby reducing the mass and inertia of the moving platform. However, unlike rigid links, cables exhibit a unilateral driving property, i.e., it can only exert a tensile force. Hence, analysis approaches obtained for rigid-linked robots cannot be directly applied, and such CDRs require additional force considerations. As such, force-closure algorithms which focused on determining if equilibrium can be achieved with a set of positive cable tension under the influence of external wrenches were presented for both single-bodied CDRs as well as multi-bodied CDRs. Case studies were presented to illustrate the application of the proposed algorithms.
In addition, CDRs also possess an intrinsically safe feature due to the cables’ flexibility, which allow CDRs to provide safe manipulation in close proximity to their human counterparts. On the other hand, this flexibility causes CDRs to have relatively low stiffness and thereby significantly affecting the positional accuracy. Furthermore, it is not possible to achieve significant stiffness regulation by manipulating the cable tensions for fully constrained CDRs. As such, the concept of CDR with variable stiffness characteristics (through the use of variable stiffness devices (VSDs) attached in series to cables) was introduced. A case study was presented to verify the ability of the CDR to significantly regulate its stiffness after the VSDs were employed.
Performance analysis plays a vital role in obtaining the fundamental tools to evaluate and carry out optimal designing of a CDR. In literature, this has been generally done by analyzing the workspace. Workspace representation techniques such as equi-volumetric workspace partitioning for CDRs with orientation capabilities were presented. For these CDRs, mathematically tractable workspace volumes were also presented as the orientation workspace volume was not equal to the geometric volume in the parametric space due to the nonlinear parameterization. Both quantitative and qualitative performance measures based on workspace volume and eigenvalues of the stiffness matrix, respectively, were presented together with case study examples to illustrate the application of the proposed performance measures.
Planning through optimization of the cable tension solution for a given pose is an important aspect for CDRs. This is due to the unilateral driving property of cables, which also makes CDRs to be redundantly actuated. Hence, there exist an infinite number of cable tension solutions for a particular pose. A quadratic programming-based tension optimization method was proposed to obtain adjustable tension solution for CDRs. The tension solution can be manipulated by selecting a desired cable tension, in which the tension optimization will aim to achieve. Adjustable cable tensions allow CDRs to regulate its stiffness and also enable the CDR to avoid tension limits and allow the stiffness of the CDR to be regulated at a particular pose. A case study was also presented to illustrate the application of the proposed cable tension planning algorithm.
In summary, various performance analysis tools for CDRs were presented in this chapter with the aim of allowing readers to get a quick overview and a head start in the design of CDRs.

Appendix 1: Formulation of the Jacobian and Integration Measure Associated with 2-DOF CDR’s Parameterized Rotations

Jacobian Associated with 2-DOF CDR’s Parameterized Rotations

Integration Measure Associated with 2-DOF CDR’s Parameterized Rotations

Appendix 2: Formulation of the Integration Measure for SO(3) Representation in Cylindrical Coordinates

It has been addressed in (Bonev and Gosselin 2005) that the entire rigid body group SO(3) can be visualized as a solid cylinder when the cylindrical coordinates are employed to represent the Tilt-&-Torsion angles, as shown in Fig. 22. However, integration measures need to be introduced when computing the volume of the orientation workspace of rigid body rotations from the T&T angles domain. It can be verified that if Cartesian coordinates are used to represent the T&T angles, the integration measure will be same as that of the Euler angle representation, which is given by sin θ. In this case, the volume of the entire rigid body rotation group is given as

Equation 71 has the same form as when the T&T angles are represented with Cartesian coordinates, i.e., x ≡ ϕ, y ≡ θ, and z ≡ σ. However, since the T&T angles are normally represented with cylindrical coordinates, i.e., x ≡ θ cos ϕ, y ≡ θ sin ϕ, and z ≡  σ, an additional integration measure needs to be included for the change of the coordinate representation. Geometrically, such a transformation of the coordinate representations maps the parametric domains of the T&T angles from a rectangular parallelepiped to a solid cylinder. It can be further verified that determinant of the Jacobian (i.e., the additional integration measure) for the transformation of the coordinate representations is given by 1/θ . The resultant integration measure becomes |sin θ/θ|. It follows that the volume of the entire SO(3) under cylindrical coordinate representation of the T&T angles is given by

where D2x R represents a solid cylinder. If the integration is computed using cylindrical coordinates, Eq. 72 can be rewritten as

Although Eqs. 71 and 73 are equivalent for the volume computation of SO(3), they possess different geometrical meanings. Equation 71 is associated with the Cartesian coordinate representation of the T&T angles, while Eq. 73 is associated with the cylindrical coordinate representation of the T&T angles. In Eq. 73, the terms sin θ/θ| and θdϕdθdσ represent the integration measure and the differential volume element, respectively.
Equations 72 and 73 also indicate that the integration measure becomes singular when θ approaches 0 or π. However, with strategic selection of the cylindrical coordinate representation for T&T angles, singularity point at θ = 0 can be avoided. This is a significant feature for the numerical volume computation of SO(3) through its parametric domains.
With the equivalent integration measure derived for the cylindrical coordinate representation of T-&-T angles, the integration or convolution of a rotation-dependent function f (R) over a set of rotations S ∈ SO(3) in the T&T angles domain is given by

where Qt denotes the parameter space of the T&T angles (with cylindrical coordinate representation), i.e., a subset of the solid cylinder. After the finite partition of the solid cylinder, the orientation workspace for a set of rotations S ∈ SO(3)  can be numerically computed as

where vt is the unit volume of the equi-volumetric partition scheme in the cylindrical coordinate  representation of the T&T angles. Consequently, Eq. 74 can be written as

where (ϕijk, θijk, σijk) ∈ Qt.

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