# 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

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**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