# Dynamic and Wrench-Feasible Workspace Analysis of a Cable-Driven Parallel Robot Considering a Nonlinear Cable Tension Model

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## Abstract

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## 1. Introduction

- The dynamics of the redundant cable-driven parallel robot (CDPR) and the influence of pulley kinematics are presented.
- A methodology for determining the wrench-feasible workspace (WFW) of a cable-driven parallel robot (CDPR) is proposed based on a nonlinear cable tension model.
- The dynamic elastic modulus of the polymer cable for the nonlinear tension model is obtained based on the dynamic mechanical analysis (DMA) method by changing the frequencies of the applied force on the cable and is presented for the first time in this article.
- An approach to obtaining the redundant tensions and ensuring that the cable tensions are positive, using a nonlinear constraint for optimal function, is proposed.

## 2. Modeling of the CDPR System

#### 2.1. Kinematics of the Redundant-Actuated CDPR System

_{comp}is a cable length compensation, L

_{c}is the free cable length from point A to point B, r is the pulley radius and θ is the wrapping angle of the cable. For the two right-angled triangles ADO and AOE shown in Figure 3, OA is the common side. Therefore, based on the Pythagorean theorem for the two right-angled triangles, the following can be obtained:

_{1}+ θ

_{2}at point A equals the sought complementary angle θ at point O [5,23,24]. Using the elementary trigonometric function, the cable wrapping angle around the pulley is given by Equation (4).

_{2}, y

_{2}, x

_{2}), at the anchor point of end-effector A(x

_{1}, y

_{1}, z

_{1}), and at the center of end-effector G(x

_{0}, y

_{0}, z

_{0}), then the values d

_{x}, d

_{y}, and d

_{z}are coordinates in the x, y, z directions, which are calculated by the equation: L(x, y, z) = O − A − G. d

_{z}represents coordinates of point E with respect to the frame point B coordinates. Finally, the equation for determining the corrected cable length, which takes into account pulley kinematics to solve the inverse kinematics, can be transformed as follows:

#### 2.2. Dynamic Modeling for the CDPR System

_{i}denotes the tension vector for the ith cable, m is the mass of the end-effector, g is the gravitational acceleration, and F

_{r}and M

_{r}indicate the resultant forces and moment vector, respectively. For convenience, we assumed that the origin geometrical center was located at the center of gravity. Therefore, the moment caused by gravity could be ignored. Combining F

_{r}and M

_{r}together, we obtained the wrench vector:

^{T}represents the transposed matrix of J, and $\widehat{{e}_{i}}$ and $\widehat{{L}_{i}}$ denote the unit cable length vector. T is the vector of cable tensions:

_{e}and M

_{e}denote the external force and moment, respectively; matrices m and I are a [3 × 3] diagonal matrix of the end-effector mass and tensor of inertia, respectively; $\ddot{X}$ is the acceleration of the end-effector; and ω denotes the angular velocity vector. These are defined as follows:

## 3. Nonlinear Tension Model of the DYNAMICA Cable

#### 3.1. Nonlinear Cable Tension Formulation

_{0}is the initial length of the cable, and T

_{0}is the initial cable tension. This equation is valid for highly dynamic systems, such as CDPR systems.

^{T}represents the transposed matrix of J. To find solutions for the cable tensions, by inverting Equation (19) then multiplying by the pseudoinverse matrix of J and adding the term of the homogeneous solution [11], the equation of tension can be determined according to the equation below.

^{P}represents the Moore–Penrose pseudoinverse Jacobian matrix and [I

_{8}] is the [8 × 8] identity matrix. Equation (20) combines two parts, with the term on the right-hand side representing a particular solution and the second term representing the homogeneous solution. Unlike rigid arm mechanisms, the CDPR robot uses flexible cables to hold the end-effector. Therefore, the vector {z} is an arbitrary vector and was chosen to ensure that all cable tensions were positive. To maintain the positive cable tension and resolve the problem of actuation redundancy in the CDPR workspace, many methods have been proposed, such as the closed-form force distribution method [26] and an improved force distribution algorithm [27]. In this study, the nonlinear constraint of the optimal method was used and the cable tension was obtained by using the objective function, as shown in Equations (21)–(23).

^{P}was calculated and the vector {z} represents an arbitrary vector that was chosen to ensure that all cable tensions were positive. In this step, the nonlinear constraint of the optimal method was used, as shown in Equations (21)–(23). Finally, the cable tension was obtained.

#### 3.2. Dynamic Modulus of Elasticity

_{max}and ε

_{max}are the maximum amplitude of the applied stress and the maximum amplitude of the strain response, respectively. The ratio of the maximum amplitude stress–strain is denoted as E* and called the absolute value of the dynamic modulus, and φ is the phase angle. ${E}^{\u2033}$ represents the energy dissipation, also called the imaginary modulus. It is calculated as follows:

## 4. Workspace Analysis

^{2}, and a geometric model was built by MATLAB simulation. The weight of the end-effector is 10 N and the maximum loading limit of the cable is 4000 N for a Dynamica cable with a diameter of 6 mm (breaking strength certificate). The step size for the MATLAB WFW simulation is 100 mm. According to the definition of feasible workspace, all of the cable tension at each position needed to be constrained to maintain positive tension and not exceed the maximum loading of the cable. Besides, each box was denoted with a color in the diagram; each color represented the maximum cable tension value at each acting position. Moreover, when the calculated cable tension values exceeded the limited value, the end-effector at these positions was eliminated from the workspace domain and appeared vacant in the figures.

^{2}and ~200 N at an acceleration of 100 m/s

^{2}. Moreover, the total maximum tensions increased significantly from 600 N to 1000 N when the acceleration varied from 10 m/s

^{2}to 100 m/s

^{2}. In addition, maximum tension values appeared at the vertices and edges of the workspace. The negative cable tension disappeared in the moving direction due to the constraint of the optimal method. Considering the case of X-direction acceleration (Figure 7a,b, and Figure 8a,b), the same directions between the front view and back view surface were compared, and the tension distribution of the back view surface was slightly higher than that in the opposite surface. The tension region had a C shape, with the edges created by the contiguous faces, and the surrounding area had a higher tension distribution than that of the front view surface. This tension distribution was similar for both cases of X-direction acceleration. In particular, the vertical edge was the edge created between the back view surface of the X-direction and the front view surface of the Y-direction, which had the highest tension distribution in this position because of the optimal method and the variable cable length; thus, the cable was expected to maintain a positive tension for the acceleration in the X-direction—the greater the acceleration, the greater the tension distribution in this region. This finding means that the CDPR system is restricted from working in this area and the tension must be decreased so that the wrench-feasible workspace can be improved.

^{2}, the tension distribution reached the maximum value; therefore, the CDPR system should be restricted from working in this region. Overall, neither tension distribution case exceeded the allowable tension of 4000 N. Figure 9a,b show the quarter section of the wrench-feasible workspace in the simulation results, and they provide detailed information on the internal workspace and show that both quarter sections were filled with blue boxes. The cable tension distribution in the interior of the workspace is shown to be relatively low, with an average value of approximately 100 N. Therefore, these results were desirable. Additionally, Figure 9c,d show the workspace results at Z = 0.2 m and Z = 0.4 m, respectively. It is obvious that the blue boxes comprised the bulk of the internal workspace and the tension was similar in these two diagrams. However, the high-tension distribution appeared at the corner of its slice and the vicinities of this position. Figure 10a–d show the wrench-feasible workspace of CDPR when extending the moving ranges in the X and Y directions from 0.6 m to 0.8 m and the Z-axis from 0.1 m to 0.8 m. The applied acceleration was 100 m/s

^{2}. In these figures, it can be seen that both faces of the back view in the X and Y directions had the highest cable tension and appeared in regions that were eliminated from the workspace because they exceeded the constraint in the range of 0 ≤ Ti ≤ 4000 N. This shows that the simulation results were suitable for the wrench-feasible workspace of the proposed CDPR system as well as the real system due to the necessity of having a safe working space to prevent collisions between the end-effector and the frame during work. Therefore, extensions of the moving range directions and the high-tension distribution, which occurred at the vertical edges of the workspace, should be considered. In addition, compared with results from previous related studies in [4,22,32], the research results of the paper show that under the same conditions of motion acceleration, the end-effector range of motion is smaller but the wrench-feasible workspace analysis results are significantly improved. This indicates the effectiveness of the proposed method.

## 5. Conclusions

^{2}. The workspace was satisfactory in the case of the end-effector moving in the X- and Y-axis limits from −0.3 to 0.3 m. When the axis ranges were increased from −0.4 to 0.4 m, the cable tensions exceeded the constrained condition, and these positions were eliminated from the workspace.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the CDPR system: (

**a**) diagram of 6–DOF CDPR; (

**b**) asymmetric pulley locations.

**Figure 6.**Sinusoidal force causing sinusoidal elongation at different frequencies: (

**a**) 0.1 Hz; (

**b**) 1 Hz; (

**c**) 2 Hz; (

**d**) 5 Hz.

**Figure 7.**Overall workspace simulation at the acceleration of 10 m/s

^{2}: (

**a**) front plane of the X–axis; (

**b**) back plane of the X–axis; (

**c**) front plane of the Y–axis; (

**d**) back plane of the Y–axis.

**Figure 8.**Overall workspace simulation at the acceleration of 100 m/s

^{2}: (

**a**) front plane of the X–axis; (

**b**) back plane of the X–axis; (

**c**) front plane of the Y–axis; (

**d**) back plane of the Y-axis.

**Figure 9.**Quarter section of overall workspace simulation at the acceleration of 100 m/s

^{2}: (

**a**) quarter section of X–direction; (

**b**) quarter section of Y–direction; (

**c**) section view of Z = 0.2 m; (

**d**) section view of Z = 0.4 m.

**Figure 10.**The wrench-feasible workspace simulation at the acceleration of 100 m/s

^{2}when extending the axis ranges: (

**a**) front plane of the X–axis; (

**b**) back plane of the X–axis; (

**c**) front plane of the Y–axis; (

**d**) back plane of the Y–axis.

Pulleys | X (m) | Y (m) | Z (m) |
---|---|---|---|

P1 | −0.455 | 0.540 | 0.884 |

P2 | 0.455 | 0.540 | 0.880 |

P3 | −0.445 | −0.540 | 0.860 |

P4 | 0.455 | −0.540 | 0.881 |

P5 | −0.540 | 0.465 | 0.075 |

P6 | 0.540 | 0.455 | 0.071 |

P7 | −0.540 | −0.455 | 0.073 |

P8 | 0.540 | −0.455 | 0.072 |

Freq (Hz) | σ_{max} (MPa) | ε_{max} | E* (GPA) | E′ (GPA) | φ (Deg) | η |
---|---|---|---|---|---|---|

0.1 | 53.155 | 5.6877 10^{−4} | 93.5 | 92.6 | 34.7 | 0.14 |

1 | 53.54 | 5.2031 10^{−4} | 102.9 | 75.5 | 7.03 | 0.93 |

2 | 53.474 | 5.2052 10^{−4} | 102.7 | 75.4 | 7.03 | 0.93 |

5 | 53.442 | 4.9898 10^{−4} | 107.1 | 78.2 | 16.4 | 0.94 |

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**MDPI and ACS Style**

Kieu, V.N.D.; Huang, S.-C.
Dynamic and Wrench-Feasible Workspace Analysis of a Cable-Driven Parallel Robot Considering a Nonlinear Cable Tension Model. *Appl. Sci.* **2022**, *12*, 244.
https://doi.org/10.3390/app12010244

**AMA Style**

Kieu VND, Huang S-C.
Dynamic and Wrench-Feasible Workspace Analysis of a Cable-Driven Parallel Robot Considering a Nonlinear Cable Tension Model. *Applied Sciences*. 2022; 12(1):244.
https://doi.org/10.3390/app12010244

**Chicago/Turabian Style**

Kieu, Vu N. D., and Shyh-Chour Huang.
2022. "Dynamic and Wrench-Feasible Workspace Analysis of a Cable-Driven Parallel Robot Considering a Nonlinear Cable Tension Model" *Applied Sciences* 12, no. 1: 244.
https://doi.org/10.3390/app12010244