This section presents the algorithms developed to analyze force-closure for CDRs. These algorithms will include for both modular CDRs, as well as multi-bodied open chain configurations of the CDRs.

Force-Closure Analysis of Modular CDRs

A variety of force-closure algorithms have been proposed for checking the positive cable tension status of n-DOF fully constrained CDRs driven by m cables (where m ≥ n + 1). Ming and Higuchi (1994) showed that a pose of a CDR satisfies the force-closure condition when at least one solution in the null space of A is strictly positive. This method becomes complex for highly redundant CDRs. By checking the solutions in the null space of A, Ferraresi et al. (2007) derived a method to determine if a given pose is under force closure for CDRs with up to three redundant driving cables, i.e., n + 1 ≤ m ≤ n + 3. Gouttefarde et al. (2006) determined the force-closure workspace for a 6-DOF manipulator by delineating the workspace boundary. Murray et al. (1994), Stump and Kumar (2006), and Diao and Ma (2008) checked the existence of supporting/separating hyperplane to verify if a pose satisfied the force-closure condition. Diao and Ma (2008) modified Murray’s method (1994) by using the generalized cross product so that the method can be used for CDRs of up to six DOFs. Their methods had difficulties to deal with degeneracies, i.e., when the rank of the vectors, which were used to form the hyperplane, is not equal to n-1. However, Gouttefarde (2008) proved that the degeneracies can be ignored while checking for force-closure condition. Pham et al. (2006a) developed the recursive dimension reduction algorithm for forceclosure analysis. The algorithm was not computationally efficient as it had to decompose the n-dimensional space into numerous one-dimensional subspaces to check the tension status. Hassan and Khajepour (2011) employed Dykstra’s alternating projection algorithm to determine the workspace that is able to counter a particular wrench.
In order to address the abovementioned shortcomings, the authors (Lim et al. 2011) proposed an alternative approach for force-closure analysis that is systematic, easy to implement, and satisfies both the necessary and sufficient conditions for force closure.

Proposed Algorithm

Evaluation of Computational Complexity

The efficiency of the algorithm is verified by calculating the maximum number of operations required to check for force-closure condition at a moving-platform pose. In particular, the number of operations required by three generic force-closure algorithms, i.e., the proposed algorithm, the recursive dimension reduction algorithm (Pham et al. 2006a), and the method proposed in (Diao and Ma 2008), is determined and compared. In order to verify if the pose satisfies the force-closure condition, Diao and Ma (2008) checked the existence of supporting/separating hyperplane, while Pham et al. (2006a) checked the decomposed one-dimensional subspaces. The recursive dimension reduction algorithm and the method proposed in (Diao and Ma 2008) stop the computation once the pose is deemed not a force-closure pose. On the other hand, the proposed method stops the computation once the pose is proved to satisfy force-closure condition. Therefore, the recursive dimension reduction algorithm and the method proposed in (Diao and Ma 2008) are more efficient for poses that do not satisfy the force-closure condition, while the proposed method is more efficient for force-closure poses. When checking for the force-closure condition of a given pose, it is unknown whether the pose satisfies force-closure condition. Therefore, the maximum number of operations required by each algorithm is used to evaluate the algorithms’ efficiency so that the comparison does not favor any algorithm.
The recursive dimension reduction algorithm uses Gaussian elimination to reduce the dimension of matrix A. The method proposed in (Diao and Ma 2008) uses matrix determinants to generate the generalized cross product. The proposed method solves a system of linear equations. In fact, deriving matrix determinants and solving linear equations can all be done using Gaussian elimination. Strang (1976) provided the number of operations required to solve the matrix determinants and linear equations using Gaussian elimination. For Gaussian elimination, each division or each multiplication/subtraction is considered as an operation.

Simulation Studies and Results

The force-closure workspace consists of all the moving platform poses which satisfy force-closure condition. Figure 8 shows a 6-DOF spatial CDR driven by eight cables. The dimension of the base is 1 x 1 x 1 m, and the moving platform is a 0.3 x 0.2 x 0.1 m block. The Z-Y-X Euler angles, represented by α, β, and γ, respectively, are employed to define the orientation of the platform frame with respect to the base frame. Figures 9 and 10 show examples of the translational workspace (with fixed orientation) and the orientation workspace (with fixed position) generated, respectively. The translational workspace is computed with a step size of 0.01 m. The orientation workspace uses a step size of 5o to find the limits of the Euler angles. Within the limits, the workspace is computed with a step size of 0.5o.
The simulation result produced by the proposed algorithm is the same as those by (Pham et al. 2006a) and (Diao and Ma 2008). The algorithms are implemented in Matlab on an Intel P4 3.20 GHz with 512 MB RAM. The computational time for the workspace analysis is compared among the various algorithms (see Table 2). The table shows that the computation time required by the proposed algorithm is always less than the recursive dimension reduction algorithm (Pham et al. 2006a) but is comparable to the algorithm proposed in (Diao and Ma 2008). This algorithm can also be extended to generate the static workspace of under-constrained CDRs, and readers may refer to (Lim et al. 2011) for more details.

Force-Closure Analysis of Multi-Bodied CDRs

In the previous section, a force-closure algorithm was presented for generic n-DOF modular CDRs. It has also been proven that it requires a minimum of n + 1 cables with positive tension to fully constrain it. However, force-closure analysis of open chains driven by cables is still an open question. This subsection will present the force-closure algorithm for the case of an n-DOF multi-bodied open chain CDR.
As explained in section “Kinetostatic Analysis for Multi-Bodied CDRs,” the basic idea for the kinematic analysis of multi-bodied “proper” open chain CDRs is to express individual wrench screws as linear combinations of the reciprocal screws and determine the individual joint torques. The net required torques at each joint is then calculated based on the superposition of the scalar components in their respective reciprocal screws. The key distinction for force-closure analysis of cable-driven open chains is to treat external wrenches as requiring joint torques, while cable wrenches as providing joint torques (i.e., Proposition 2.5). In other words, under static equilibrium condition, the joint torques required by external wrenches must be equal to the joint torques provided by the cable forces. Force closure is achieved if the set of cable tensions is determined to be positive.

Proposed Algorithm

The proposed algorithm consists of five major steps (Mustafa and Agrawal 2012b):

1. Determine the various screws (i.e., twists, reciprocal wrenches, external wrenches, and cable wrenches).
2. Determine the joint torques required to sustain the external wrenches by expressing these external wrenches as linear combinations of the reciprocal wrenches.
3. Determine the joint torques provided by the cable forces by expressing these cable forces as linear combinations of the reciprocal wrenches.
4. Equilibrate the joint torques between the external wrenches and the cable forces.
5. Determine force closure at a particular pose by checking if there exists a set of positive cable tension.

As shown in Fig. 11, the following notations will be used in this section:

The proposed algorithm is now illustrated using a multi-bodied CDR with a “proper” open chain. As shown in Fig. 12, the spatial chain example is of an S-R-U configuration, driven by seven cables (Note: S, R, and U indicate a spherical, revolute, and universal joint, respectively).

Step A: Determining the Various Screw Representations

From Fig. 12, the unit screws for the various pure rotation twists (i.e., zero pitch) with respect to the inertial frame are described as follows:

Step B: Determining the Joint Torques Required to Sustain the External Wrenches

Step C: Determining the Joint Torques Provided by the Cable Wrenches

Step D: Equilibrating the Joint Torques

As mentioned in Proposition 2.5, external wrenches require joint torques, while cable wrenches provide joint torques. By equilibrating Eqs. 2126 with Eqs. 3237, respectively, the resultant equations can be combined and expressed in a matrix representation form as follows:

Step E: Determining Force Closure at a Particular Pose

A single rigid-bodied CDR is said to achieve force closure at a particular pose if and only if any arbitrary wrench applied to the moving platform can be sustained by a set of positive cable forces. For the case of multi-bodied open chain CDRs, external forces will act not only at its end-effector but also at its intermediate links. So force closure for open chain CDRs is achieved if and only if any arbitrary external wrench applied to its end-effector and its intermediate links can be sustained by a set of positive cable forces. This relationship between the external wrenches and cable forces acting on the spatial S-R-U cable-driven open chain is reflected in Eq. 38, and W represents the 6-D orthogonal space of force and moments, which is spanned by the column vectors ai. The force closure for the spatial case can be determined by rearranging Eq. 38 as follows:

Simulation Studies and Results

Figure 15 presents the geometric parameters and the external wrenches for the spatial S-R-U “proper” open chain CDR. The external wrenches are just the weight of the individual links, i.e., 1 kgf, acting from the center of each link. Figure 16 presents the various cable routing configurations of the spatial S-R-U open chain CDR.

Configurations S1–S4 were selected to represent several general cases one may encounter. For each pose of the various configurations, force closure was analyzed based on the linear inequality approach (Oh and Agrawal 2005). The lower and upper cable tension bounds were also set as 1 N and 300 N, respectively.
Table 3 presents the force-closure results of the spatial S-R-U open chain CDR. Randomly chosen poses (within the joint motion constraints) for Configurations S1, S2, and S3 were able to achieve force closure. However, for Configuration S4, it was unable to achieve force closure in its entire range of motion. This is intuitively expected as n cables attached to the n-DOF segments were definitely unable to provide the bilateral force/moment. This configuration was deliberately chosen to demonstrate the ability of the proposed methodology to analyze such infeasible configurations.