# Spring Effects on Workspace and Stiffness of a Symmetrical Cable-Driven Hybrid Joint

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CDHJ Description and Kinematic Analysis

#### 2.1. CDHJ Description

#### 2.2. Kinematic Analysis

_{1 × 1}y

_{1}} is the global coordinate system, {O

_{2}x

_{2}y

_{2}} is a local coordinate frame, all the coordinate origins are at the center of the platform. The upper and lower platforms are thin round plates, their radii are b and a, respectively. Denote A

_{1}, B

_{1}, A

_{2}, and B

_{2}as the connecting points of cables 1 and 2, respectively; and the distance from O

_{1}to the rotating pair center as d. The spring is simplified and drawn as an arc.

_{1}, l

_{2}), the output is (y, θ). The kinematic relationship between the input and output can be obtained used the closed loop vector method:

## 3. Modeling of Spring Lateral Bending and Compression

**T**

_{1}and

**T**

_{2}acting on the upper plate are equivalent to generalized forces

**F**

_{1e},

**F**

_{2e}, and

**M**

_{e}. The frame {O

_{k}x

_{k}y

_{k}z

_{k}} is attached to the center of the helix section corresponding to the helix end point of each coil. The variable k is the index number of the active coils. Two Cartesian coordinate systems {O

_{0}x

_{0}y

_{0}z

_{0}} and {Oxyz} are fixed to the lower and upper platforms, respectively, with {O

_{0}x

_{0}y

_{0}z

_{0}} being the global coordinate system coincident on the center of helix section corresponding to helix initiation point of the first coil. The number of spring active coils is n. Each coil frame is rotated around its z-axes in equally, finally y

_{n}-axis and y-axis are tangent. In order to use Castigliano’s theorem, Figure 2b shows the infinitesimal elements defined in the helical spring as. The infinitesimal element angular position on k-th coil is defined as α on x

_{k}-z

_{k}plane. Two forces exert on each spring coil infinitesimal element. One is on tangential direction coincident on ɛ

_{z}, the other is on normal direction coincident on ɛ

_{x}.

_{k}x

_{k}y

_{k}z

_{k}}. Obviously, ${\mathbf{OP}}^{k}{=\mathbf{OO}}_{k}{}^{k}{+\mathbf{O}}_{k}{\mathbf{P}}^{k}$. ${\mathbf{O}}_{k}{\mathbf{P}}^{k}$ is denoted as the position of the infinitesimal element in the particular coil frame {O

_{k}x

_{k}y

_{k}z

_{k}}. In the deformed configurations of the spring bend, it is desired that it is the shape of a circular arc. Meanwhile, due the platform motion is symmetric, the analysis only discusses the clockwise bend as shown in Figure 1b. Hence, θ is negative. ${\mathbf{O}}_{k}{\mathbf{P}}^{k}{=[R\mathrm{cos}\alpha ,-R}_{\mathrm{c}}\theta \alpha /{2n\pi ,R\mathrm{sin}\alpha ]}^{T}$, where ${R}_{\mathrm{c}}=-y/\mathrm{sin}\theta $, R is radius of the helical spring, and n is the active coils number.

_{x}ε

_{y}ε

_{z}}. The moment vector ${\mathbf{M}}^{\mathsf{\epsilon}}{=\mathbf{Rot}}_{y}{\left(\alpha \right)\mathbf{M}}^{k}$.

**D**is a 3 × 3 matrix. Let

**K**be

_{p}**D**

^{−1}, then the important relationship becomes:

**K**and simplify, Equation (13) can become:

_{p}**K**is a more complex form of proportionality factor, which is determined by the spring configuration, the geometry and material properties and is not a constant. That is to say, the linear helical compression spring became a nonlinear stiffness spring under the bending and compression effects. So, CDHJ is a variable stiffness joint. Through translational motion, the stiffness of the spring is adjusted which determines the joint system stiffness.

## 4. Workspace and Stiffness Index of CDHJ

#### 4.1. Workspace Index

_{min}and θ

_{max}[52], is defined as:

_{w}, which is the area of joint workspace and is used to assess the size of the workspace. The boundary values in arbitrary units, corresponding to that showed in Equations (15)–(17) for the following numerical examples, are given in Table 1.

#### 4.2. CDHJ Stiffness Index

**r**

_{1}, and ${}_{{\mathrm{O}}_{2}}{}^{{\mathrm{O}}_{1}}\mathbf{R}{\mathbf{L}}_{{\mathrm{O}}_{2}{\mathrm{B}}_{m}}^{{\mathrm{O}}_{2}}\times {\widehat{\mathbf{l}}}_{2}$ as

**r**

_{2}; ${\mathbf{T}}_{2\times 1}=\left[\begin{array}{c}{T}_{1}\\ {T}_{2}\end{array}\right]$, T

_{1}and T

_{2}are the tension of cable1 and cable 2, respectively.

**K**also uses the above method to be homogenized. Thus, denote the translational stiffness index as ${k}_{\mathrm{t}}$. Similarly, denote the rotational stiffness index as ${k}_{\mathrm{r}}$. The index $f$ is used to evaluate the helical spring stiffness, and ${f=k}_{\mathrm{t}}{k}_{\mathrm{r}}$.

## 5. Numerical Simulation

#### 5.1. Cable Parameters

_{c}is diameter of steel wire, 2 mm. According to the method utilized in [42], the stiffness of the ith cable is formulated as:

_{C}denotes the modulus of elasticity of the cable, 68 Gpa; A

_{C}denotes the cross-sectional area of the cable, 5.28 mm

^{2}; l

_{cw}denotes the length of the actuating winch, which is assumed to be constant, 30 mm.

#### 5.2. Numerical Analysis of Spring Effects on CDHJ Workspace and Stiffness

#### 5.2.1. Spring Parameters (n, E/G) on CDHJ Workspace and Stiffness

_{w}is all at No. 5 material, the maximum of A

_{w}is all the same at n = 8 for No. 4 material and No. 6 material, n = 10 for No. 4 material and No. 6 material, n = 13 for No. 6 material, and n = 16 for No. 6 material. When CDHJ is the same material, A

_{w}is all the minimum at n = 5. Contrarily, when CDHJ is the same material, the minimum value of joint stiffness min(${f}_{\mathrm{J}}$) is the maximum at n = 5 as shown in Figure 4. If n = 5 is removed, min(${f}_{\mathrm{J}}$) under No. 1, No. 2, No. 3, No. 4, and No. 6 material increases with the increase of n.

#### 5.2.2. Variable Stiffness Spring Effects on the CDHJ Stiffness

#### 5.2.3. Spring Parameters (R, r) on CDHJ Workspace and Stiffness

_{w}decreases with the increase of R. Except R = 0.015 m, CDHJ workspace area A

_{w}increases with the increase of r. Whether how much r is, the maximum of A

_{w}is at R = 0.015 m. The A

_{w}maximum is at r = 0.0015 m, and 0.00175 m. A

_{w}approaches 0 at r = 0.00075 m, R = 0.0225 m, and r = 0.001 m, R = 0.0325 m. The minimum of A

_{w}is 0 at r = 0.00075 m, R = 0.025 m, 0.0275 m, and 0.0325 m.

_{w}approaches 0 at r = 0.00075 m, R = 0.0225 m, and r = 0.001 m, R = 0.0325 m, min(${f}_{\mathrm{J}}$) is relatively large. The difference is too large, therefore the value of min(${f}_{\mathrm{J}}$) approaches 0 at Figure 8a,b. In fact, they are 1.054 × 10

^{8}and 1.0432 × 10

^{8}and are relatively large compared with other cases.

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chung, C.S.; Wang, H.; Cooper, R.A. Functional assessment and performance evaluation for assistive robotic manipulators: Literature review. J. Spinal. Cord. Med.
**2013**, 36, 273–289. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hersh, M. Overcoming Barriers and Increasing Independence–Service Robots for Elderly and Disabled People. Int. J. Adv. Robot. Syst.
**2015**, 12, 1–33. [Google Scholar] [CrossRef] [Green Version] - Jiang, H.R.; Zhang, T.; Wachs, J.P.; Duerstock, B.S. Enhanced control of a wheelchair-mounted robotic manipulator using 3-D vision and multimodal interaction. Comput. Vis. Image Und.
**2016**, 149, 21–31. [Google Scholar] [CrossRef] - Bilyea, A.; Seth, N.; Nesathurai, S.; Abdullah, H.A. Robotic assistants in personal care: A scoping review. Med. Eng. Phys.
**2017**, 49, 1–6. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.; Cao, D.X.; Li, S.; Min, H.; Fan, F. Inverse kinematic tension analysis and optimal design of a cable-driven parallel-series hybrid joint towards wheelchair-mounted robotic manipulator. J. Eur. Syst. Autom.
**2018**, 51, 59–74. [Google Scholar] - Yang, K.S.; Yang, G.L.; Chen, S.L.; Wang, Y.; Zhang, C.; Fang, Z.J.; Zheng, T.J.; Wang, C.C. Study on stiffness-oriented cable tension distribution for a symmetrical cable-driven mechanism. Symmetry
**2019**, 11, 1158. [Google Scholar] [CrossRef] [Green Version] - Gagliardini, L.; Caro, S.; Gouttefarde, M.; Girin, A. Discrete reconfiguration planning for cable-driven parallel robots. Mech. Mach. Theory
**2016**, 100, 313–337. [Google Scholar] [CrossRef] [Green Version] - Zi, B.; Lin, J.; Qian, S. Localization, obstacle avoidance planning and control of a cooperative cable parallel robot for multiple mobile cranes. Robot. Comput. Int. Manuf.
**2015**, 34, 105–123. [Google Scholar] [CrossRef] - Kaluarachchi, M.M.; Ho, J.H.; Yahya, S. Design of a single motor, tendon driven redundant manipulator with reduced driving joint torques. Mech. Based Des. Struct. Mach.
**2018**, 46, 591–614. [Google Scholar] [CrossRef] - Chen, Q.; Zi, B.; Sun, Z.; Li, Y.; Xu, Q.S. Design and development of a new cable-driven parallel robot for waist rehabilitation. IEEE/ASME Trans. Mechatron.
**2019**, 24, 1497–1507. [Google Scholar] [CrossRef] - Liu, Y.J.; Wang, J.M.; Ben-Tzv, P. A cable length invariant robotic tail using a circular shape universal joint mechanism. J. Mech. Robot.
**2019**, 11, 051005. [Google Scholar] [CrossRef] - Christoforoua, E.G.; Phocasb, M.C.; Matheoub, M.; Müller, A. Experimental implementation of the ‘effective 4-bar method’ on a reconfigurable articulated structure. Structures
**2019**, 20, 157–165. [Google Scholar] [CrossRef] - Qiao, S.L.; Guo, H.W.; Liu, R.Q.; Deng, Z.Q. Self-adaptive grasp process and equilibrium configuration analysis of a 3-DOF UACT robotic finger. Mech. Mach. Theory
**2019**, 133, 250–266. [Google Scholar] [CrossRef] - Seriani, S.; Gallina, P.; Scalera, L.; Lughi, V. Development of n-DoF preloaded structures for impact mitigation in cobots. J. Mech. Robot.
**2018**, 10, 051009. [Google Scholar] [CrossRef] - Beckerle, P.; Salvietti, G.; Ünal, R.; Prattichizzo, D.; Rossi, S. A Human-robot interaction Perspective on Assistive and rehabilitation robotics. Front. Neurorobot.
**2017**, 11, 1–6. [Google Scholar] [CrossRef] [Green Version] - Azadi, M.; Behzadipour, S.; Faulkner, G. Antagonistic variable stiffness elements. Mech. Mach. Theory
**2009**, 44, 1746–1758. [Google Scholar] [CrossRef] - Boehler, Q.; Vedrines, M.; Abdelaziz, S.; Poignet, P.; Renaud, P. Synthesis method for the design of variable stiffness components using prestressed singular elastic systems. Mech. Mach. Theory
**2018**, 121, 598–612. [Google Scholar] [CrossRef] - Nam, K.H.; Kim, B.S.; Song, J.B. Compliant actuation of parallel-type variable stiffness actuator based on antagonistic actuation. J. Mech. Sci. Technol.
**2010**, 24, 2315–2321. [Google Scholar] [CrossRef] - Petit, F.; Friedl, W.; Hoppner, H.; Grebenstein, M. Analysis and synthesis of the bidirectional antagonistic variable stiffness mechanism. IEEE/ASME Trans. Mechatron.
**2015**, 20, 684–695. [Google Scholar] [CrossRef] - Yeo, S.H.; Yang, G.L.; Lim, W.B. Design and analysis of cable-driven manipulators with variable stiffness. Mech. Mach. Theory
**2013**, 69, 230–244. [Google Scholar] [CrossRef] - Xu, F.; Wang, H.; Au, K.W.S.; Chen, W.; Miao, Y. Underwater dynamic modeling for a cable-driven soft robot arm. IEEE/ASME Trans. Mechatron.
**2018**, 23, 2726–2738. [Google Scholar] [CrossRef] - Liu, F.; Xu, W.F.; Huang, H.L.; Ning, Y.H. Design and analysis of a high payload manipulator based on a cable-driven serial-parallel mechanism. J. Mech. Robot.
**2019**, 11, 051006. [Google Scholar] [CrossRef] - Scalera, L.; Palomba, I.; Wehrle, E.; Gasparetto, A.; Vidoni, R. Natural motion for energy saving in robotic and mechatronic systemsand. Appl. Sci.
**2019**, 9, 3516. [Google Scholar] [CrossRef] [Green Version] - Ham, R.V.; Sugar, T.G.; Vanderborght, B.; Hollander, K.W.; Lefeber, D. Compliant actuator designs. IEEE Robot. Autom. Mag.
**2009**, 16, 81–94. [Google Scholar] [CrossRef] - Vanderborght, B.; Albu-Schaeffer, A.; Bicchi, A.; Burdet, E.; Caldwell, D.G.; Carloni, R.; Catalano, M.; Eiberger, O.; Friedl, W.; Ganesh, G.; et al. Variable impedance actuators: A review. Robot. Auton. Syst.
**2013**, 61, 1601–1614. [Google Scholar] [CrossRef] [Green Version] - Wu, Y.S.; Lan, C.C. Linear variable-stiffness mechanisms based on preloaded curved beams. J. Mech. Des.
**2014**, 136, 122302. [Google Scholar] [CrossRef] - López-Martínez, J.; Blanco-Claraco, J.L.; García-Vallejo, D.; Giménez-Fernández, A. Design and analysis of a flexible linkage for robot safe operation in collaborative scenarios. Mech. Mach. Theory
**2015**, 92, 1–16. [Google Scholar] [CrossRef] - Lau, D.; Oetomo, D.; Halgamuge, S. Wrench-Closure Workspace Generation for Cable Driven Parallel Manipulators using a Hybrid Analytical-Numerical Approach. ASME J. Mech. Des.
**2011**, 133, 071004. [Google Scholar] [CrossRef] - Duan, Q.J.; Duan, X.C. Workspace Classification and Quantification Calculations of Cable-Driven Parallel Robots. Adv. Mech. Eng.
**2014**, 6, 358727. [Google Scholar] [CrossRef] [Green Version] - Phama, C.B.; Yeo, H.S.; Yang, G.L.; Chen, M. Workspace analysis of fully restrained cable-driven manipulators. Robot. Auton. Syst.
**2009**, 57, 901–912. [Google Scholar] [CrossRef] - Bosscher, P.; Riechel, A.; Ebert-Uphoff, I. Wrench-feasible workspace generation for cable-driven robots. IEEE Trans. Robot.
**2006**, 22, 890–902. [Google Scholar] [CrossRef] - Yuan, H.; Courteille, E.; Deblaise, D. Static and dynamic stiffness analyses of cable-driven parallel robots with non-negligible cable mass and elasticity. Mech. Mach. Theory
**2015**, 85, 64–81. [Google Scholar] [CrossRef] - Amare, Z.; Zi, B.; Qian, S.; Du, J.L.; Ge, Q.J. Three-dimensional static and dynamic stiffness analyses of the cable driven parallel robot with non-negligible cable mass and elasticity. Mech. Based Des. Struct.
**2017**, 46, 455–482. [Google Scholar] [CrossRef] - Nguyen, D.Q.; Gouttefarde, M.; Company, O.; Pierrot, F. On the Simplifications of Cable Model in Static Analysis of Large-Dimension Cable-Driven Parallel Robots. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Tokyo, Japan, 3–7 November 2013; pp. 928–934. [Google Scholar]
- Rogier, D.R.; Mitchell, R.; Amir, K. Out-of-plane vibration control of a planar cable-driven parallel robot using a multi-axis reaction system. IEEE/ASME Trans. Mech.
**2018**, 23, 1684–1692. [Google Scholar] - Zhang, S.; Cao, D.X.; Hou, B.; Li, S.; Min, H.; Zhang, X.L. Analysis on variable stiffness of a cable-driven parallel-series hybrid joint toward wheelchair-mounted robotic manipulator. Adv. Mech. Eng.
**2019**, 11, 1–12. [Google Scholar] [CrossRef] - Nguyen, D.Q.; Gouttefarde, M. Stiffness Matrix of 6-DOF Cable-Driven Driven Parallel Robots and Its Homogenization Advances in Robot Kinematics. In Advances in Robot Kinematics; Springer: Cham, Switzerland, 2014; pp. 181–191. [Google Scholar]
- Khalilpour, S.A.; Taghirad, H.D.; Habibi, H. Wave-Based Control of Suspended Cable Driven Parallel Manipulators. In Proceedings of the 2017 5th International Conference on Control Instrumentation and Automation, Shiraz, Iran, 21–23 November 2017; pp. 173–178. [Google Scholar]
- Behzadipour, S.; Khajepour, A. Stiffness of cable-based parallel manipulators with application to stability analysis. ASME J. Mech. Des.
**2006**, 128, 303–310. [Google Scholar] [CrossRef] - Wu, G. Stiffness analysis and optimization of a co-axial spherical parallel manipulator. Model. Ident. Control
**2014**, 35, 21–30. [Google Scholar] [CrossRef] [Green Version] - Hoevenaars, A.G.L.; Lambert, P.; Herder, J.L. Jacobian-based stiffness analysis method for parallel manipulators with non-redundant legs. J. Mech. Eng.
**2016**, 230, 341–352. [Google Scholar] [CrossRef] - Anson, M.; Alamdari, A.; Krovi, V. Orientation workspace and stiffness optimization of cable-driven parallel manipulators with base mobility. J. Mech. Robot.
**2017**, 9, 031011. [Google Scholar] [CrossRef] - Yuan, H.; Courteille, E.; Gouttefarde, M.; Hervé, P. Vibration analysis of cable-driven parallel robots based on the dynamic stiffness matrix method. J. Sound. Vib.
**2017**, 394, 527–544. [Google Scholar] [CrossRef] - Duan, Q.J.; Vashista, V.; Agrawal, S.K. Effect on wrench-feasible workspace of cable-driven parallel robots by adding springs. Mech. Mach. Theory
**2015**, 86, 201–210. [Google Scholar] [CrossRef] - Mustafa, S.K.; Agrawal, S.K. Force-Closure of Spring-Loaded Cable-Driven Open Chains: Minimum Number of Cables Required & Influence of Spring Placements. In Proceedings of the 2012 IEEE International Conference on Robotics and Automation, Saint Paul, MN, USA, 14–18 May 2012; pp. 1482–1487. [Google Scholar]
- Taghavi, A.; Behzadipour, S.; Khalilinasab, N.; Zohoor, H. Workspace Improvement of Two-Link Cable-Driven Mechanisms with Spring Cable. In Cable-Driven Parallel Robots; Springer: Berlin, Germany, 2013; pp. 201–213. [Google Scholar]
- Gao, B.T.; Song, H.G.; Zhao, J.G.; Guo, S.X.; Sun, L.X.; Tang, Y. Inverse kinematics and workspace analysis of a cable-driven parallel robot with a spring spine. Mech. Mach. Theory
**2014**, 76, 56–69. [Google Scholar] [CrossRef] - Yigit, C.B.; Boyraz, P. Design and modeling of a cable-driven parallel-series hybrid variable stiffness joint mechanism for robotics. Mech. Sci.
**2017**, 8, 65–77. [Google Scholar] [CrossRef] [Green Version] - Krużelecki, J.; Życzkowski, M. On the concept of an equivalent column in the stability problem of compressed helical springs. Arch. Appl. Mech.
**1990**, 60, 367–377. [Google Scholar] [CrossRef] - Leech, A.R. A Study of the Deformation of Helical Springs under Eccentric Loading. Ph.D. Thesis, Naval Postgraduate School, Monterey, CA, USA, 1994. [Google Scholar]
- Timoshenko, S.; Gere, J. Theory of Elastic Stability; McGraw-Hill: New York, NY, USA, 1961; p. 142. [Google Scholar]
- Hay, A.M.; Snyman, J.A. Optimization of a planar tendon-driven parallel manipulator for a maximal dextrous workspace. Eng. Optimiz.
**2005**, 37, 1–20. [Google Scholar] [CrossRef] - Bolboli, J.; Khosravi, M.A.; Abdollahi, F. Stiffness feasible workspace of cable-driven parallel robots with application to optimal design of a planar cable robot. Mech. Mach. Theory
**2019**, 114, 19–28. [Google Scholar] [CrossRef] - Raiss, P.; Rettig, O.; Wolf, S.; Loew, M.; Kasten, P. Range of Motion of Shoulder and Elbow in Activities of Daily Life in 3D Motion Analysis. Z. Orthop. Unfall.
**2007**, 145, 493–498. [Google Scholar] [CrossRef]

**Figure 1.**Diagram of the cable-driven hybrid joint (CDHJ). (

**a**) 3-D joint mechanism. (

**b**) 2-D joint diagram.

**Figure 2.**Helical spring lateral bending and compression model diagram. (

**a**) Spring force analysis diagram on x-y plane. (

**b**) Infinitesimal element on x-z plane from top view.

**Figure 3.**Spring parameters (n, E/G) on CDHJ workspace. (

**a**) A

_{w}at n = 5; (

**b**) A

_{w}at n = 8; (

**c**) A

_{w}at n = 10; (

**d**) A

_{w}at n = 13; (

**e**) A

_{w}at n = 16; (

**f**) A

_{w}at n = 20; (

**g**) A

_{w}at n = 25.

**Figure 4.**Spring parameters (n, E/G) on CDHJ stiffness. (

**a**) min(${f}_{\mathrm{J}}$) at n = 5; (

**b**) min(${f}_{\mathrm{J}}$) at n = 8; (

**c**) min(${f}_{\mathrm{J}}$) at n = 10; (

**d**) min(${f}_{\mathrm{J}}$) at n = 13; (

**e**) min(${f}_{\mathrm{J}}$) at n = 16; (

**f**) min(${f}_{\mathrm{J}}$) at n = 20; (

**g**) min(${f}_{\mathrm{J}}$) at n = 25.

**Figure 6.**Spring stiffness on CDHJ stiffness and cable tension at θ = −36°. (

**a**) ${f}_{\mathrm{J}}$ and $f$ at n = 5 for No. 1 material; (

**b**) ${T}_{1}$ and ${T}_{2}$ at n = 5 for No. 1 material; (

**c**) ${f}_{\mathrm{J}}$ and $f$ at n = 10 for No. 4 material; (

**d**) ${T}_{1}$ and ${T}_{2}$ at n = 10 for No. 4 material.

**Figure 7.**Spring parameters (R, r) on CDHJ workspace. (

**a**) A

_{w}at r = 0.00075 m; (

**b**) A

_{w}at r = 0.001 m; (

**c**) A

_{w}at r = 0.00125 m; (

**d**) A

_{w}at r = 0.0015 m; (

**e**) A

_{w}at r = 0.00175 m; (

**f**) A

_{w}at r = 0.002 m.

**Figure 8.**Spring parameters (R, r) on CDHJ stiffness. (

**a**) min(${f}_{\mathrm{J}}$) at r = 0.00075 m; (

**b**) min(${f}_{\mathrm{J}}$) at r = 0.001 m; (

**c**) min(${f}_{\mathrm{J}}$) at r = 0.00125 m; (

**d**) min(${f}_{\mathrm{J}}$) at r = 0.0015 m; (

**e**) min(${f}_{\mathrm{J}}$) at r = 0.00175 m; (

**f**) min(${f}_{\mathrm{J}}$) at r = 0.002 m

Parameters | Value |
---|---|

T_{min} (N) | 1 |

T_{max} (N) | 300 |

l_{min} (m) | 0.01 |

x_{min} (m) | 0 |

x_{max} (m) | 0.08 |

θ_{min} (rad) | −1.48 |

θ_{max} (rad) | 0 |

y_{min} (m) | 0.035 |

y_{max} (m) | 0.095 |

Spring Materials | Mark | E (Gpa) | G (Gpa) |
---|---|---|---|

carbon spring steel wire, oil quenched-tempered spring steel wire, alloy spring steel wire, spring steel | 1 | 206 | 78.5 |

stainless steel wire for spring (A) | 2 | 185 | 70 |

stainless steel wire for spring (B), (C) | 3 | 195 | 73 |

copper and copper alloy wire | 4 | 93.1 | 40.2 |

beryllium bronze wire | 5 | 129.4 | 42.1 |

spring-tempered steel | 6 | 195 | 81.5 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, S.; Sun, Z.; Lu, J.; Li, L.; Yu, C.; Cao, D.
Spring Effects on Workspace and Stiffness of a Symmetrical Cable-Driven Hybrid Joint. *Symmetry* **2020**, *12*, 101.
https://doi.org/10.3390/sym12010101

**AMA Style**

Zhang S, Sun Z, Lu J, Li L, Yu C, Cao D.
Spring Effects on Workspace and Stiffness of a Symmetrical Cable-Driven Hybrid Joint. *Symmetry*. 2020; 12(1):101.
https://doi.org/10.3390/sym12010101

**Chicago/Turabian Style**

Zhang, Shan, Zheng Sun, Jili Lu, Lei Li, Chunlei Yu, and Dongxing Cao.
2020. "Spring Effects on Workspace and Stiffness of a Symmetrical Cable-Driven Hybrid Joint" *Symmetry* 12, no. 1: 101.
https://doi.org/10.3390/sym12010101