# Novel Reconfigurable Spherical Parallel Mechanisms with a Circular Rail

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Original Mechanism

_{i}, B

_{i}, and C

_{i}) are parallel to each other and orthogonal to a plane, tilted to the circular rail plane by a certain angle; this angle remains the same for any chain configuration. The axis of the fifth joint (point D

_{i}), attached to the moving plate, intersects the axis of the first joint at point F

_{i}; this intersection is also preserved for any chain configuration. Note that point F

_{i}does not necessarily lie on the moving plate itself, i.e., EF

_{i}≠ 0, where point E corresponds to the moving plate center. Finally, by the mechanism design, the axes of the moving plate joints intersect at common point F, i.e., points F

_{i}coincide for all three kinematic chains.

_{i}, B

_{i}, and C

_{i}, respectively.

_{i}, B

_{i}, and C

_{i}are parallel to each other.

_{i}, B

_{i}, and C

_{i}.

## 3. Modified Mechanism

_{i}B

_{i}C

_{i}vertically (Figure 2a). In this case, the adjacent joint axes in each chain are either parallel or orthogonal to each other (Figure 2b); it is much easier to provide such a design in a physical system. In this arrangement, however, vectors ${\widehat{s}}_{Ai}$ of all the chains will be parallel to the horizontal plane—the space of constraint wrenches ${\mathsf{\zeta}}_{ci}$, given in Equation (2), will be two-dimensional (Figure 2c). Therefore, the moving plate will gain an additional DOF: a translational motion along axis Fz. Since only three actuators drive the carriages, the space of constraint wrenches ${\mathsf{\zeta}}_{ci}$ and actuation wrenches ${\mathsf{\zeta}}_{ai}$ will be five-dimensional. We have an uncontrolled translational motion of the moving plate along axis Fz, defined by twist $\mathsf{\xi}$ reciprocal to all ${\mathsf{\zeta}}_{ci}$ and ${\mathsf{\zeta}}_{ai}$:

- actuating a passive joint in at least one leg;
- introducing an additional (actuated) kinematic chain between the base and the output link.

_{i}, B

_{i}, or C

_{i}to be actuated in at least one leg. In this case, we can determine an additional actuation wrench, depending on the selected joint; for a general mechanism configuration, this wrench will be independent of other constraint and actuation wrenches. Therefore, the plate wrench system will be six-dimensional, the plate vertical motion will become controlled, and the mechanism will have four (controlled) DOFs. This approach preserves the advantages of the original topology but may lead to load distribution problems, requiring a specific design for a chain with the auxiliary actuator. On the other hand, we can introduce additional actuators in each leg, making the mechanism redundantly actuated. This redundancy causes new challenges, specifically regarding the mechanism control, but may be beneficial for certain mechanisms and help to enlarge the workspace or avoid singularities. For the discussed mechanism, however, these benefits are questionable.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Seeßle, J.; Waterboer, T.; Hippchen, T.; Simon, J.; Kirchner, M.; Lim, L.; Müller, B.; Merle, U. Persistent symptoms in adult patients one year after coronavirus disease 2019 (COVID-19): A prospective cohort study. Clin. Infect. Dis.
**2021**, ciab611. [Google Scholar] [CrossRef] - Gemelli against COVID-19 Post-Acute Care Study Group. Post-COVID-19 global health strategies: The need for an interdisciplinary approach. Aging Clin. Exp. Res.
**2020**, 32, 1613–1620. [Google Scholar] [CrossRef] - Merlet, J.-P. Parallel Robots, 2nd ed.; Springer: Dordrecht, The Netherlands, 2006; 402p. [Google Scholar] [CrossRef]
- Ceccarelli, M. Fundamentals of Mechanics of Robotic Manipulation; Springer: Dordrecht, The Netherlands, 2004; 312p. [Google Scholar] [CrossRef]
- Kong, X.; Gosselin, C.M. Type Synthesis of Parallel Mechanisms; Springer: Dordrecht, The Netherlands, 2007; 276p. [Google Scholar] [CrossRef]
- Saglia, J.A.; Tsagarakis, N.G.; Dai, J.S.; Caldwell, D.G. A high-performance redundantly actuated parallel mechanism for ankle rehabilitation. Int. J. Rob. Res.
**2009**, 28, 1216–1227. [Google Scholar] [CrossRef] - Wang, C.; Fang, Y.; Guo, S.; Chen, Y. Design and kinematical performance analysis of a 3-RUS/RRR redundantly actuated parallel mechanism for ankle rehabilitation. J. Mech. Robot.
**2013**, 5, 041003. [Google Scholar] [CrossRef] - Chen, Q.; Zi, B.; Sun, Z.; Li, Y.; Xu, Q. Design and development of a new cable-driven parallel robot for waist rehabilitation. IEEE/ASME Trans. Mechatron.
**2019**, 24, 1497–1507. [Google Scholar] [CrossRef] - Ben Hamida, I.; Laribi, M.A.; Mlika, A.; Romdhane, L.; Zeghloul, S.; Carbone, G. Multi-objective optimal design of a cable driven parallel robot for rehabilitation tasks. Mech. Mach. Theory
**2021**, 156, 104141. [Google Scholar] [CrossRef] - Pisla, D.; Nadas, I.; Tucan, P.; Albert, S.; Carbone, G.; Antal, T.; Banica, A.; Gherman, B. Development of a control system and functional validation of a parallel robot for lower limb rehabilitation. Actuators
**2021**, 10, 277. [Google Scholar] [CrossRef] - Aginaga, J.; Iriarte, X.; Plaza, A.; Mata, V. Kinematic design of a new four degree-of-freedom parallel robot for knee rehabilitation. J. Mech. Des.
**2018**, 140, 092304. [Google Scholar] [CrossRef] - Rastegarpanah, A.; Rakhodaei, H.; Saadat, M.; Rastegarpanah, M.; Marturi, N.; Borboni, A.; Loureiro, R.C. Path-planning of a hybrid parallel robot using stiffness and workspace for foot rehabilitation. Adv. Mech. Eng.
**2018**, 10, 1–10. [Google Scholar] [CrossRef][Green Version] - Pisla, D.; Gherman, B.; Vaida, C.; Suciu, M.; Plitea, N. An active hybrid parallel robot for minimally invasive surgery. Robot. Comput. Integr. Manuf.
**2013**, 29, 203–221. [Google Scholar] [CrossRef] - Dalvand, M.M.; Shirinzadeh, B. Motion control analysis of a parallel robot assisted minimally invasive surgery/microsurgery system (PRAMiSS). Robot. Comput. Integr. Manuf.
**2013**, 29, 318–327. [Google Scholar] [CrossRef] - Tanev, T.K. Minimally-invasive-surgery parallel robot with non-identical limbs. In Proceedings of the IEEE/ASME 10th International Conference on Mechatronic and Embedded Systems and Applications (MESA), Ancona, Italy, 10–14 September 2014; pp. 1–6. [Google Scholar] [CrossRef]
- Kuo, C.; Dai, J.S. Kinematics of a fully-decoupled remote center-of-motion parallel manipulator for minimally invasive surgery. J. Med. Devices
**2012**, 6, 021008. [Google Scholar] [CrossRef] - Cao, W.; Xu, S.; Rao, K.; Ding, T. Kinematic design of a novel two degree-of-freedom parallel mechanism for minimally invasive surgery. J. Mech. Des.
**2019**, 141, 104501. [Google Scholar] [CrossRef] - Lum, M.J.H.; Rosen, J.; Sinanan, M.N.; Hannaford, B. Optimization of a spherical mechanism for a minimally invasive surgical robot: Theoretical and experimental approaches. IEEE Trans. Biomed. Eng.
**2006**, 53, 1440–1445. [Google Scholar] [CrossRef] - Hwang, Y.-H.; Kang, S.-R.; Cha, S.-W.; Choi, S.-B. An electrorheological spherical joint actuator for a haptic master with application to robot-assisted cutting surgery. Sens. Actuators A Phys.
**2016**, 249, 163–171. [Google Scholar] [CrossRef] - Ottoboni, A.; Parenti-Castelli, V.; Sancisi, N.; Belvedere, C.; Leardini, A. Articular surface approximation in equivalent spatial parallel mechanism models of the human knee joint: An experiment-based assessment. Proc. Inst. Mech. Eng. H
**2010**, 224, 1121–1132. [Google Scholar] [CrossRef] - Lessard, S.; Bigras, P.; Bonev, I.A. A new medical parallel robot and its static balancing optimization. J. Med. Devices
**2007**, 1, 272–278. [Google Scholar] [CrossRef] - Essomba, T.; Laribi, M.; Zeghloul, S.; Poisson, G. Optimal synthesis of a spherical parallel mechanism for medical application. Robotica
**2016**, 34, 671–686. [Google Scholar] [CrossRef] - Li, Q.; Chen, Q.; Wu, C.; Hu, X. Two novel spherical 3-DOF parallel manipulators with circular prismatic pairs. In Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Philadelphia, PA, USA, 10–13 September 2006; pp. 325–328. [Google Scholar] [CrossRef]
- Veliev, E.I.; Ganiev, R.F.; Glazunov, V.A.; Filippov, G.S. Parallel and sequential structures of manipulators in robotic surgery. Dokl. Phys.
**2019**, 64, 106–109. [Google Scholar] [CrossRef] - Fang, H.R.; Chen, Z.H.; Fang, Y.F. A novel spherical parallel manipulator with circular guide. Appl. Mech. Mater.
**2013**, 325–326, 1014–1018. [Google Scholar] [CrossRef] - Laryushkin, P.A.; Zakharov, M.N.; Erastova, K.G.; Glazunov, V.A. Spherical manipulator with parallel structure. Russ. Eng. Res.
**2017**, 37, 585–588. [Google Scholar] [CrossRef] - Zhao, J.; Feng, Z.; Chu, F.; Ma, N. Kinematic synthesis of spatial mechanisms. In Advanced Theory of Constraint and Motion Analysis for Robot Mechanisms; Academic Press: Waltham, MA, USA, 2014; pp. 429–469. [Google Scholar] [CrossRef]
- Xu, C.C.; Xue, C.; Duan, X.C. A novel 2R parallel mechanism for alt-azimuth pedestal. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 428, 012053. [Google Scholar] [CrossRef][Green Version] - Wu, G.; Bai, S. Design and kinematic analysis of a 3-RRR spherical parallel manipulator reconfigured with four–bar linkages. Robot. Comput. Integr. Manuf.
**2019**, 56, 55–65. [Google Scholar] [CrossRef] - Huang, Z.; Liu, J.; Zeng, D. A general methodology for mobility analysis of mechanisms based on constraint screw theory. Sci. China Ser. E-Technol. Sci.
**2009**, 52, 1337–1347. [Google Scholar] [CrossRef] - Dai, J.S.; Sun, J. Geometrical revelation of correlated characteristics of the ray and axis order of the Plücker coordinates in line geometry. Mech. Mach. Theory
**2020**, 153, 103983. [Google Scholar] [CrossRef] - Zhao, J.; Li, B.; Yang, X.; Yu, H. Geometrical method to determine the reciprocal screws and applications to parallel manipulators. Robotica
**2009**, 27, 929–940. [Google Scholar] [CrossRef][Green Version] - Conconi, M.; Carricato, M. A new assessment of singularities of parallel kinematic chains. IEEE Trans. Rob.
**2009**, 25, 757–770. [Google Scholar] [CrossRef] - Kuo, C.-H.; Dai, J.S. Task-oriented structure synthesis of a class of parallel manipulators using motion constraint generator. Mech. Mach. Theory
**2013**, 70, 394–406. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Original spherical mechanism; (

**b**) one kinematic chain with corresponding twists and wrenches; (

**c**) constraint and actuation wrenches of the entire mechanism.

**Figure 2.**(

**a**) Modified spherical mechanism with an uncontrolled vertical translation; (

**b**) one kinematic chain with corresponding twists and wrenches; (

**c**) constraint and actuation wrenches of the entire mechanism.

**Figure 3.**(

**a**) Modified spherical mechanism with a passive spherical joint in point F; (

**b**) one kinematic chain with corresponding twists and wrenches; (

**c**) constraint and actuation wrenches of the entire mechanism.

**Figure 4.**(

**a**) Modified spherical mechanism with a PS chain and decoupled vertical motion; (

**b**) one kinematic chain with corresponding twists and wrenches; (

**c**) constraint and actuation wrenches of the entire mechanism.

**Figure 5.**(

**a**) Modified spherical mechanism with an SPS chain; (

**b**) one kinematic chain with corresponding twists and wrenches; (

**c**) constraint and actuation wrenches of the entire mechanism.

**Figure 6.**Modified spherical mechanism with an SPS chain and decoupled vertical motion; the spherical joint of the fourth chain coincides with the spherical motion center.

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**MDPI and ACS Style**

Laryushkin, P.; Antonov, A.; Fomin, A.; Glazunov, V. Novel Reconfigurable Spherical Parallel Mechanisms with a Circular Rail. *Robotics* **2022**, *11*, 30.
https://doi.org/10.3390/robotics11020030

**AMA Style**

Laryushkin P, Antonov A, Fomin A, Glazunov V. Novel Reconfigurable Spherical Parallel Mechanisms with a Circular Rail. *Robotics*. 2022; 11(2):30.
https://doi.org/10.3390/robotics11020030

**Chicago/Turabian Style**

Laryushkin, Pavel, Anton Antonov, Alexey Fomin, and Victor Glazunov. 2022. "Novel Reconfigurable Spherical Parallel Mechanisms with a Circular Rail" *Robotics* 11, no. 2: 30.
https://doi.org/10.3390/robotics11020030