Performance analysis plays a vital role in obtaining the fundamental tools to evaluate and carry out optimal designing of a mechanism. In literature, this has been generally done by analyzing certain workspace-related performance measures based on the kinematic properties of mechanisms. Hence, workspace analysis is a pertinent issue to address, for subsequent development of performance measures and design optimization. In general, workspace analysis consists of three main components, namely, (i) workspace definition, (ii) workspace evaluation approaches, and (iii) workspace generation. While the first two components have been addressed in the previous sections, this section will address the workspace representation and volume determination, especially for CDRs with orientation capabilities. Several performance measures based on workspace quantity and quality will also be presented to evaluate the design of CDRs.

Workspace Representation

In literature, there have been two main approaches for the workspace representation of CDRs, namely, (i) analytical and (ii) numerical approaches. The analytical approach derives the workspace from the closed-form solutions of the kinematic constraints, and the workspace boundary is generally characterized by the occurrence of the loss of force-closure condition. This is usually considered as a singularity pose and is based on the conventional singularity definition as a pose where the moving platform has uncontrollable DOFs instantaneously. Bosscher and Ebert-Uphoff (2004a) characterized the wrench-feasible workspace boundary based on geometric properties of the available net wrench set. Stump and Kumar (2004) derived the closed-form expression for the workspace boundary based on convex analysis and using tools from semi-definite programming. Verhoeven and Hiller (2000) developed a method to find closed-form expressions of the conservative bounds for the controllable workspace. Oh and Agrawal (2006a, b) proposed an interesting approach to analytically generate feasible workspace through the choice of a control law, which considered input constraints and disturbances. Based on a given initial condition, this method allowed recursive calculations of subsequent feasible domains, which is useful in extending the end-effector motion range. Although analytical methods have an explicit expression and high computational efficiency, they are more suited for lower DOF CDRs with simpler kinematic constraints and architectures. Such approaches become complex and computationally challenging when incorporating actual implementation issues (such as mechanical interferences, task-specific kinematic constraints, cable tension bounds, and CDR configurations with redundant cables) due to the increasing number of inequalities to consider. Hence numerical approaches are generally more effective in coping with such cases and are demonstrated in works by Ceccarelli (2004), Pusey et al. (2003), Alp and Agrawal (2002), Gosselin and Wang (2004), and Pham et al. (2005).
The numerical approach generally involves using of discretization algorithms to quantify the workspace, solving the inverse kinematics at each pose, and verifying the constraints that limit the workspace. For CDRs, this involves force-closure analysis. Such numerical approaches can be applied to any type of CDR architectures. However, this approach requires proper representation of the pose parameters in the workspace with an equi-volumetric finite partitioning scheme. Ceccarelli (2004) proposed such an equi-volumetric finite partition scheme based on a binary matrix formulation. This formulation divided the cross section of the workspace into small rectangles, represented by a feature point. An efficient workspace boundary determination algorithm was also developed, which is based on the geometry of the grid and the number of surrounding feature points.
There are two main categories of pose parameters, namely, translational and orientation pose parameters (Merlet 2000). Translational pose parameters can be easily visualized in 3-dimensional space, and the visualization in R3 represents the actual task space. However, is it not so trivial for the case of orientation pose parameters, especially the Special Orthogonal Group, i.e., SO(3) orientations. There are several conventions that have been developed to represent and parameterize rigid body rotations, such as the Cayley-Rodrigues parameters, axis-angle parameters, unit quaternions, Euler angles, T&T angles, and exponential coordinates (Gosselin and Wang 2004; Bonev and Ryu 2001; MacCarthy 1990; Park and Ravani 1997). Hence, there are two basic requirements to be considered in the selection of the appropriate parameterization method: (i) the parametric domain must have a closed boundary so as to make finite partition possible, and (ii) the parameterization needs at most three parameters so that the embodied orientation workspace can be readily visualized in three-dimensional space.
There are three parameterization methods (among the various mentioned previously) which fulfill these two requirements, i.e., the Euler angles, the T-&-T angles, and the exponential coordinates. The rigid body rotation group SO(3) can be represented as a rectangular parallelepiped (Chirikjian and Kyatkin 2000), a solid cylinder (Bonev and Ryu 2001), and a solid sphere (Park and Ravani 1997) when using the Euler angles, the T&T angles, and exponential parameterizations, respectively. Various partition schemes have been proposed for these three parameterization methods based on their geometries. Readers may refer to a related paper by the author (Yang et al. 2006) for details on the partition schemes proposed and the workspace analysis of the different parameterization methods. Nevertheless, these three partition schemes possess the same essential features, i.e., they are all equi-volumetric, parametric, proportional, and exact partitions of their corresponding geometric identities. However, as a result of the parameterization, integration measures have to be introduced when computing the workspace volume from its parametric domain. Consequently, the orientation workspace volume is numerically computed as a weighted sum of the equi-volumetric elements in which the weightages are the element-associated integration measures (Chirikjian and Kyatkin 2000). Following this approach, various global performance measures of the orientation workspace can also be readily computed.
The following subsections will present the workspace representation for two common orientation configuration of modular CDRs, i.e., 2-DOF and 3-DOF orientation.

CDR with 2-DOF Orientation

A CDR with 2-DOF orientation has a passive 2-DOF universal joint connected between the base and moving platform so as to limit the motion of the moving platform to the two rotating axes. The total orientation workspace of the 2-DOF CDR belongs to a subset of the Special Orthogonal Group SO(3) with a complex boundary. Thus, the workspace is represented using the parametric domains, θ1 and θ2, which are the angular displacements about the rotational axes of the 2-DOF CDR. In the parametric space, the orientation workspace of the 2-DOF CDR can be represented by a plane in R2, in which a one-to-one mapping between θ1 and θ2 and the orientation of the 2-DOF CDR can be established. The parametric domain is partitioned into finite elements with equal volume by dividing each axis in the parametric domain into h identical segments (see Fig. 20). There are h2 equi-volumetric elements in the planar workspace, and each element is specified by its feature point fij. These equi-volumetric elements have dimensions which are inversely proportional to the number of partitions h, implying that all the elements in this partition scheme converge uniformly when h is sufficiently large. When the elements are small, the local property of the entire element can be represented by fij anywhere inside the element (Ceccarelli 2004). The use of feature points will facilitate numerical calculation of the orientation workspace volume where each fij represents a volumetric element, |det(JI)|dθ1dθ2.

CDR with 3-DOF Orientation

These 3-DOF CDRs are in fact cable-driven spherical mechanisms. Hence, the SO(3) orientation workspace analysis is not as straightforward. While quaternion seems to be used as the standard method for representing motion in computer animations and has the advantages in terms of lack of gimbal lock and insensitivity to round off errors, but they suffer from problems of interpretation. Euler angles, on the other hand, allow some degree of intuition for the rotation angles, but suffer from singularities. Representations using equivalent angle-axis have also been proposed (Korein 1984), but the main issue is that interpretation of joint motions must have an intuitive feel while being mathematically tractable at the same time. Hence, there is a great debate as to the use of Euler angles with quaternion and equivalent angle-axis for workspace representations. For the 3-DOF CDR, the Tilt-&-Torsion angles parameterization is adopted to represent its workspace. The advantage of this representation is that it is able to separate the 3-DOF orientation into two components: the 2-DOF tilt and the 1-DOF twist. The tilt component (ϕ and θ) determines the direction of the axis, while the twist component (σ) determines the rotation about the axis itself, as shown in Fig. 21. This representation is intuitive, and the joint motion limit boundaries can be easily identified using this representation.

Workspace Volume

Workspace volume is normally employed as the quantitative measure for workspace analysis. However, for mechanisms with orientation capabilities, its orientation workspace volume is not equal to its geometric volume in its parametric space due to the nonlinear parameterization. Instead, the workspace volume can be numerically computed as a weighted volume sum of its constituent volumetric elements in its parametric space, in which the weightages are the element-associated integration measures. During the computation of the volume of SO(3), an integration measure has to be introduced in order for the integral of the function to be independent of the parameterization. This integration measure is the absolute value of the determinant of the Jacobian, JI, which relates the angular velocity, ω ∈ ℜ3, to the rate of change of the parameters θ ∈ ℜ2:

The following subsections will present the workspace volume determination for two common orientation configuration of modular CDRs, i.e., 2-DOF and 3-DOF orientation.

CDR with 2-DOF Orientation

The orientation workspace volume of the 2-DOF CDR is not the geometric volume in its parametric space due to nonlinear parameterization. The orientation workspace volume is obtained by defining a volumetric element, |det(JI)|dθ1dθ2, and finding the integral of the function. The term |det(JI)| is the integration measure, Iv. For the 2-DOF CDR, the Jacobian matrix JI ∈ ℜ3x2 is rectangular, but a matrix has to be square for the determinant to exist. Note that the rotation of the 2-DOF CDR can also be represented using the XYZ Euler angles parameterization, in which the rotation matrix can then be represented by

where sZ = [0 0 0 0 0 1]T and θ3 = 0. Using the XYZ Euler angles parameterization, the integration measure is derived in Appendix “Formulation of the Jacobian and Integration Measure Associated with 2-DOF CDR’s Parameterized Rotations” and is given as

The orientation workspace volume of the 2-DOF CDR, VSO(2), is the integration of the volumetric element over the parametric domain, defined by the two rotational angles, and is given as follows:

Employing the equi-volumetric partition scheme, the orientation workspace volume of the 2-DOF CDR is numerically computed by

where htotal is the total number of poses (i.e., feature point) and θ(2i) is the θ2 angle at the ith feature point.

CDR with 3-DOF Orientation

When quantifying the workspace volume of SO(3), VSO(3) is not equal to the geometric volume of the cylindrical workspace (i.e., VCartesian) due to the nonlinear parameterization. Hence, a volume-associated integration measure (Iv) has to be introduced to accurately determine VSO(3) from VCartesian. From the concept of metrics, Iv is defined as the absolute value of the Jacobian matrix which relates the rate of change of the orientation parameters to describe the rotation to the angular velocity of the rigid body. When computing the volume of SO(3) from the cylindrical coordinate representation of T&T angles, an equivalent Iv is determined as sin θ /θ by adopting a similar approach presented by Chirikjian and Kyatkin (2000) (refer to Appendix “Formulation of the Integration Measure for SO(3) Representation in Cylindrical Coordinates” for formulation details.). The volume for the discretized SO(3) workspace is now given as

where htotal is the total number of poses (i.e., feature point), θi is the θ angle at the ith feature point, and p is the number of circular bands, with respect to the cylindrical workspace as shown in Fig. 22.

Workspace Performance Measures

The most widely studied performance measures are based on workspace quantity (i.e., volume) and workspace quality (i.e., manipulability, condition index, dexterity, stiffness). Although majority of design optimization approaches utilize workspace volume as a performance measure, a mechanism designed for maximum workspace may lead to undesirable kinematic characteristics such as poor dexterity or manipulability (Tsai and Joshi 2002; Yang et al. 2003). Hence, it is also important to determine the quality of the workspace in addition to workspace quantization. A variety of workspace quality measures have been proposed such as manipulability, dexterity, condition index, and stiffness, but most of them are equivalent to one another as they all depend on the conditioning of the Jacobian matrix (Merlet 2000). A more commonly used workspace quality measure is the global condition index (GCI), which is a measure of the kinematic dexterity of the robot over the whole workspace (Gosselin and Angeles 1991). Large values for GCI (close to one) ensure good performance with respect to force and velocity transmission. Regardless of the quality measures used, a mechanism far from singularities will always have a high workspace quality measure. For CDRs, GCI has also been employed to study the workspace quality (Pusey et al. 2003; Fattah and Agrawal 2002). However, Jacobian-based measures do not reflect the stiffness of the cables. Hence, several other measures have been proposed. Bosscher and Ebert-Uphoff (2004b) proposed a slope-based measure to analyze the stability of an under-constrained CDR (i.e., no. of cables, m ≤ n-DOF), while Verhoeven and Hiller (2002) proposed a tension factor which reflects the tension distribution among the cables. Pham et al. (2006b) proposed a similar tension factor, but it utilized a linear programming approach.
The following subsections will present several performance measures based on the workspace volume and workspace quality (using the stiffness matrix).

Quantitative Measures

Given a specific task workspace, the main objective in the dimension optimization of a CDR is to achieve a workspace that closely matches the specified workspace. Hence, a performance evaluation index is required that will quantitatively determine how much workspace volume of the specified workspace has been matched by the designed CDR. This proposed performance quantity measure, known as the Workspace Matching Index (WMI), is described as follows:

where the maneuverable workspace refers to the poses satisfying the force-closure conditions, as well as other requirements such as minimum stiffness.
For the 2-DOF CDR, Eq. 55 is numerically given as

For the 3-DOF CDR, Eq. 55 is numerically given as

where ha is the number of poses achievable by the CDR and is a subset of htotal, while htotal is the total number of poses in the specified task workspace. WMI ranges from 0 to 1. A WMI value 1 (ideal) indicates that the CDR is able to achieve all the poses in its corresponding specified workspace.
The proposed performance quantity measure (WMI) is illustrated using a 3-DOF modular CDR. This modular CDR is part of a 7-DOF humanoid arm (Yang et al. 2011) whose objective is to achieve the same workspace as the human arm. In this illustration, the 3-DOF CDR represents the human shoulder joint with their typical range of motion (Hamilton and Luttgens 2002), and the approximated maximum range of the tilt-&-torsion angles are ϕ ϵ [- 180o, 180o], θ ϵ [0o, 140o], and σ ϵ [- 90o, 90o], respectively. Figure 23a–c show the workspace plot based on the shoulder range of motion, the optimized workspace plot, and the corresponding optimized 3-DOF CDR configuration, respectively (readers may refer to (Yang et al. 2011) for the detailed design optimization of the 7-DOF humanoid arm).

Qualitative Measures

Besides workspace quantity, it is also important to determine the quality of the workspace. Especially for CDRs, it utilizes flexible cables which greatly affect the moving platform’s stiffness. Hence it is very critical to maintain sufficient stiffness. As shown by Strang (1976), the condition number of a matrix can be used in numerical analysis to estimate the error generated in the solution of a linear system of equations by the error on the data. When applied to the Jacobian matrix, the condition number gives a measure of the accuracy of the Cartesian velocity and the static load acting on the end of the moving platform. Similarly, the condition number of the stiffness matrix K (determined in section “Stiffness Analysis”) can also be used to give a measure of accuracy for CDRs, which includes the effect of cable stiffness. The Stiffness Condition Index, i.e., SCI, (which is the reciprocal of the condition number) is given as the ratio of the minimum to the maximum singular value of K at a given pose. SCI ranges from 0 to 1. Large values for SCI (close to 1) ensure good stiffness with respect to force and velocity transmission, while SCI values close to 0 indicate poor stiffness.
Adopting a similar approach as the Global Conditioning Index proposed by Yang et al. (2003), the Global Stiffness Conditioning Index (GSCI) is proposed as a performance quality measure. GSCI is the integration of SCI over the whole workspace and is given by

where V represents the workspace volume of the CDR. GSCI ranges from 0 to 1. Large values for GSCI (close to 1) ensure good overall stiffness with respect to force and velocity transmission, while GSCI values close to 0 indicate poor overall stiffness.
The proposed performance quality measure GSCI is illustrated using the 3-DOF modular CDR. In this illustration, dimension optimization is carried out on symmetric six-cable 3-DOF CDRs with the topologies shown in Fig. 24, and Table 7 shows the optimization results based on both GSCI and WMI.