# Velocity and Singularity Analysis of a 5-DOF (3T2R) Parallel-Serial (Hybrid) Manipulator

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## Abstract

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## 1. Introduction

## 2. Manipulator Design

## 3. Position Analysis

- $OXYZ$ is a stationary reference frame attached to the base arbitrarily.
- $S{X}_{S}{Y}_{S}{Z}_{S}$ is a reference frame attached to the end-effector such that axis ${Z}_{S}$ is directed along the tool (unit vector $\widehat{n}$) and the remaining axes (${X}_{S}$ and ${Y}_{S}$) have an arbitrary direction; vector ${p}_{S}$ and rotation matrix ${R}_{S}$ define the position and orientation of $S{X}_{S}{Y}_{S}{Z}_{S}$ relative to $OXYZ$.
- $q={\left[\begin{array}{ccc}{q}_{1}& \dots & {q}_{5}\end{array}\right]}^{\mathrm{T}}$ are the actuated coordinates according to the previous section; all these coordinates are measured about the axes defined by unit vectors ${\widehat{s}}_{1}\dots {\widehat{s}}_{5}$ (in $OXYZ$).
- $\mathsf{\alpha}$ and $\mathsf{\beta}$ are the angles in the universal joint measured about its axes defined by unit vectors ${\widehat{u}}_{1}$ and ${\widehat{u}}_{2}$ (in $OXYZ$).

- As the end-effector connects with the platform by two prismatic joints, vector $\widehat{n}$ uniquely defines the orientation of the latter. The platform orientation, on the other hand, depends only on two angles $\mathsf{\alpha}$ and $\mathsf{\beta}$ in the universal joint. This condition allows us to express $\widehat{n}$ as a function of $\mathsf{\alpha}$ and $\mathsf{\beta}$ and find these angles from the corresponding equations.
- Having found $\mathsf{\alpha}$ and $\mathsf{\beta}$, we can write a vector loop equation for ${p}_{S}$. This vector depends only on parameters ${q}_{1}$, ${q}_{4}$, and ${q}_{5}$, which we find from the obtained equation.
- Having found ${q}_{1}$, we know the platform configuration relative to $OXYZ$. Hence, we know the coordinates of platform spherical joints ${A}_{2}$ and ${A}_{3}$ relative to the same frame. We can also write coordinates of spherical joints ${C}_{2}$ and ${C}_{3}$ as functions of ${q}_{2}$ and ${q}_{3}$. Coordinates of the spherical joints in each kinematic chain are connected by a known and constant distance between the joints. This allows us to form corresponding equations and find parameters ${q}_{2}$ and ${q}_{3}$.

## 4. Velocity Analysis

#### 4.1. Theory

#### 4.2. Numerical Example

## 5. Singularity Analysis

#### 5.1. Serial Singularities

#### 5.2. Parallel Singularities

- ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ are collinear (Figure 8a): ${\mathsf{\zeta}}_{2}$ and ${\mathsf{\zeta}}_{3}$ are linearly dependent, and their moment parts are collinear.
- ${\widehat{w}}_{2}$ (or ${\widehat{w}}_{3}$) is on a line passing through point ${C}_{1}$ (Figure 8b): ${\mathsf{\zeta}}_{2}$ (or ${\mathsf{\zeta}}_{3}$) has a zero moment part.
- ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ lie in a plane passing through point ${C}_{1}$ (Figure 8c): the moment parts of ${\mathsf{\zeta}}_{2}$ and ${\mathsf{\zeta}}_{3}$ are collinear.
- ${\widehat{w}}_{2}$ (or ${\widehat{w}}_{3}$) lie in the spider plane, spanned by vectors ${\widehat{u}}_{1}$ and ${\widehat{u}}_{2}$ (Figure 8d): the moment part of ${\mathsf{\zeta}}_{2}$ (or ${\mathsf{\zeta}}_{3}$) is collinear with the axis of ${\mathsf{\zeta}}_{c}$.
- ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ are parallel and lie in a plane parallel to the spider plane (Figure 8e): the moment parts of ${\mathsf{\zeta}}_{2}$ and ${\mathsf{\zeta}}_{3}$ and the axis of ${\mathsf{\zeta}}_{c}$ lie in a common plane.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Joint (unit) twists of the manipulator. Blue arrows designate zero-pitch twists; red arrows designate infinite-pitch twists.

**Figure 5.**Spiral-helical trajectory for velocity analysis. Green arrows show end-effector orientation.

**Figure 6.**Results of velocity analysis: blue lines—$q$; red lines—$\dot{q}$ according to the proposed algorithm; green dots—$\dot{q}$ by numerical differentiation of $q$.

**Figure 7.**Serial singularities: (

**a**) parallelepiped formed by vectors ${\widehat{s}}_{1}$, ${\widehat{s}}_{4}$, and ${\widehat{s}}_{5}$ degenerates; (

**b**) ${\widehat{w}}_{2}$ is othogonal to ${\widehat{s}}_{2}$.

**Figure 8.**Parallel singularities: (

**a**) ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ are collinear; (

**b**) ${\widehat{w}}_{2}$ passes through ${C}_{1}$; (

**c**) ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ lie in a plane passing through ${C}_{1}$; (

**d**) ${\widehat{w}}_{2}$ lie in the spider plane, spanned by ${\widehat{u}}_{1}$ and ${\widehat{u}}_{2}$; (

**e**) ${\widehat{w}}_{2}$ and ${\widehat{w}}_{3}$ are parallel and lie in a plane parallel to the spider plane.

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**MDPI and ACS Style**

Laryushkin, P.; Antonov, A.; Fomin, A.; Essomba, T. Velocity and Singularity Analysis of a 5-DOF (3T2R) Parallel-Serial (Hybrid) Manipulator. *Machines* **2022**, *10*, 276.
https://doi.org/10.3390/machines10040276

**AMA Style**

Laryushkin P, Antonov A, Fomin A, Essomba T. Velocity and Singularity Analysis of a 5-DOF (3T2R) Parallel-Serial (Hybrid) Manipulator. *Machines*. 2022; 10(4):276.
https://doi.org/10.3390/machines10040276

**Chicago/Turabian Style**

Laryushkin, Pavel, Anton Antonov, Alexey Fomin, and Terence Essomba. 2022. "Velocity and Singularity Analysis of a 5-DOF (3T2R) Parallel-Serial (Hybrid) Manipulator" *Machines* 10, no. 4: 276.
https://doi.org/10.3390/machines10040276