# Phasor-Like Interpretation of the Angular Velocity of the Wheels of Omnidirectional Mobile Robots

^{*}

## Abstract

**:**

## 1. Introduction

#### New Contribution

## 2. Materials and Methods

#### 2.1. Three-Wheeled Omni Mobile Robot (3WOMR)

#### 2.2. Four-Wheeled Mecanum Mobile Robot (4WMMR)

#### 2.3. Inverse Kinematics of a Three-Wheeled Omni Mobile Robot (3WOMR)

#### 2.4. Inverse Kinematics of a Four-Wheeled Mecanum Mobile Robot (4WMMR)

## 3. Representation of the Angular Velocity of the Wheels of a 3WOMR

#### 3.1. Example Trajectories for $M=\left(v=0.3m/s,{\alpha}_{i=1\dots 116}=22.5\xb0\xb7\left(i-1\right),\omega =0rad/s\right)$

#### 3.2. Example Trajectories for $M=\left(v=0.3m/s,{\alpha}_{i=1\dots 16}=22.5\xb0\xb7\left(i-1\right),\omega =0.1rad/s\right)$

## 4. Representation of the Angular Velocity of the Wheels of a 4WMMR

#### 4.1. Example Trajectories for $M=\left(v=0.3m/s,{\alpha}_{i=1\dots 16}=22.5\xb0\xb7\left(i-1\right),\omega =0rad/s\right)$

#### 4.2. Example Trajectories for $M=\left(v=0.3m/s,{\alpha}_{i=1\dots 16}=22.5\xb0\xb7\left(i-1\right),\omega =0.1rad/s\right)$

## 5. Phasor-Like Interpretation of the Angular Velocity of the Wheels of a 3WOMR

## 6. Phasor-Like Interpretation of the Angular Velocity of the Wheels of a 4WMMR

## 7. Implementation of Multi-Wheeled Omnidirectional Mobile Robots

#### 7.1. Asymmetric Three-Wheeled Omni Mobile Robot

#### 7.2. Symmetric Four-Wheeled Omni Mobile Robot

#### 7.3. Asymmetric Four-Wheeled Mecanum Mobile Robot

#### 7.4. Eight-Wheeled Mecanum Mobile Robot

#### 7.5. Hybrid Omnidirectional Mobile Robot

## 8. Discussion, Limitations, and Conclusions

#### 8.1. Discussion

#### 8.2. Limitations

#### 8.3. Conclusions and Future Scope

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Wheels frequently used in omnidirectional mobile robots: (

**a**) single omni wheel; (

**b**) optimal omni wheel; (

**c**) mecanum wheel.

**Figure 2.**Representation of the motion command, $M=\left(v,\alpha ,\omega \right)$, defined in the mobile robot frame $\left\{b\right\}$ of a mobile platform: (

**a**) top-view of a robot using three optimal omni wheels; (

**b**) top-view of a robot using four mecanum wheels. The free rollers of the wheels that are in contact with the floor are represented with wider lines.

**Figure 3.**APR-02 mobile robot: (

**a**) complete robot; (

**b**) top-view of its internal motion system based on three omni wheels.

**Figure 4.**Top-view representation of the parameters of a three-wheeled omni mobile robot: (

**a**) motion parameters and system frames; (

**b**) wheel parameters.

**Figure 5.**OSOYOO mecanum mobile robot: (

**a**) complete robot; (

**b**) top-view of its motion system based on four mecanum wheels.

**Figure 6.**Top-view representation of the parameters of a four-wheeled mecanum mobile robot: (

**a**) motion parameters and wheel frames; (

**b**) wheel parameters.

**Figure 7.**Simulation of the trajectories of the APR mobile robot obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\mathsf{\alpha}}_{\mathrm{i}=1\dots 16}=22.5\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0 rad/s in the case of a displacement during $t$ = 16.0 s.

**Figure 8.**Representation of the angular velocity of the omni wheels obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\alpha}_{i=1\dots 360}=1\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 9.**Simulation of the trajectories of the APR mobile robot obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\mathsf{\alpha}}_{\mathrm{i}=1\dots 16}=22.5\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0.1 rad/s in the case of a displacement during $t$ = 15.7 s.

**Figure 10.**Representation of the angular velocity of the omni wheels obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\alpha}_{i=1\dots 360}=1\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0.5 rad/s (used instead of $\omega $ = 0.1 rad/s in order to visually enhance the effect of the average shift caused by $\omega $).

**Figure 11.**Simulation of the trajectories of the mecanum mobile robot obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\mathsf{\alpha}}_{\mathrm{i}=1\dots 16}=22.5\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0 rad/s in the case of a displacement during $t$ = 16.0 s.

**Figure 12.**Representation of the angular velocity of the mecanum wheels obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\alpha}_{i=1\dots 360}=1\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 13.**Simulation of the trajectories of the mecanum mobile robot obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\mathsf{\alpha}}_{\mathrm{i}=1\dots 16}=22.5\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0.1 rad/s in the case of a displacement during $t$ = 15.7 s.

**Figure 14.**Representation of the angular velocity of the mecanum wheels obtained with a motion command $M=\left(v,{\alpha}_{i},\omega \right)$ with ${\alpha}_{i=1\dots 360}=1\xb0\xb7\left(i-1\right)$ when $v$ = 0.3 m/s and $\omega $ = 0.5 rad/s.

**Figure 15.**Asymmetric three-wheeled omni mobile robot: (

**a**) schematic top-view representation; (

**b**) profile of the angular velocities of the wheels for $v$ = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 16.**Symmetric four-wheeled omni mobile robot: (

**a**) schematic top-view representation; (

**b**) profile of the angular velocities of the wheels for $v$ = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 17.**Asymmetric four-wheeled mecanum mobile robot: (

**a**) schematic top-view representation; (

**b**) profile of the angular velocities of the wheels for $v$ = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 18.**Symmetric eight-wheeled mecanum mobile robot: (

**a**) schematic top-view representation; (

**b**) profile of the angular velocities of the wheels for v = 0.3 m/s and $\omega $ = 0 rad/s.

**Figure 19.**Hybrid six-wheeled omnidirectional mobile robot: (

**a**) schematic top-view representation; (

**b**) profile of the angular velocities of the wheels for v = 0.3 m/s and $\omega $ = 0 rad/s.

**Table 1.**Dimensional parameters of the APR-02 omnidirectional mobile robot [27].

Parameter | Symbol | Value (m) |
---|---|---|

Chassis radius | $-$ | 0.2790 |

Wheel radius | ${r}_{{w}_{i}=1\dots 3}$ | 0.1480 |

Wheel width | $-$ | 0.0465 |

Distance between the centroids of the robot and the wheels | ${d}_{i=1\dots 3}$ | 0.1950 |

Angle between ${x}_{b}$ and the line joining the centroids of the robot and the wheels | ${\delta}_{i=1\dots 3}$ | [60, 180, 300]° |

Angle between the rolling direction of the passive rollers and the wheels axis ${y}_{{w}_{i}}$ | ${\gamma}_{i=1\dots 3}$ | 0° |

**Table 2.**Dimensional parameters of the OSOYOO mecanum mobile robot [55].

Parameter | Symbol | Value (m) |
---|---|---|

Chassis width | $-$ | 0.2000 |

Chassis height | $-$ | 0.1550 |

Wheel radius | ${r}_{{w}_{i=1\dots 4}}$ | 0.0375 |

Wheel width | $-$ | 0.0350 |

Distance between the centroids of the robot and the wheels | ${d}_{i=1\dots 4}$ | 0.1163 |

Coordinates of the wheel centers relative to the robot frame | ${\left({x}_{i},{y}_{i}\right)}_{i=1\dots 4}$ | (0.05, 0.105)_{1} (−0.05, 0.105) _{2} (−0.05, −0.105) _{3} (0.05, −0.105) _{4} |

Unsigned representation of the wheel center coordinates | $\left({l}_{x},{l}_{y}\right)$ | (0.05, 0.105) |

Angle between ${x}_{b}$ and the line joining the centroids of the robot and the wheels | ${\delta}_{i=1\dots 4}$ | [64.5367, 115.4633, 244.5367, 295.4633] |

Angle between the rolling direction of the passive rollers and the wheels axis ${y}_{{w}_{i}}$ | ${\gamma}_{i=1\dots 4}$ | [−45 45 −45 45]° |

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**MDPI and ACS Style**

Palacín, J.; Rubies, E.; Bitriá, R.; Clotet, E.
Phasor-Like Interpretation of the Angular Velocity of the Wheels of Omnidirectional Mobile Robots. *Machines* **2023**, *11*, 698.
https://doi.org/10.3390/machines11070698

**AMA Style**

Palacín J, Rubies E, Bitriá R, Clotet E.
Phasor-Like Interpretation of the Angular Velocity of the Wheels of Omnidirectional Mobile Robots. *Machines*. 2023; 11(7):698.
https://doi.org/10.3390/machines11070698

**Chicago/Turabian Style**

Palacín, Jordi, Elena Rubies, Ricard Bitriá, and Eduard Clotet.
2023. "Phasor-Like Interpretation of the Angular Velocity of the Wheels of Omnidirectional Mobile Robots" *Machines* 11, no. 7: 698.
https://doi.org/10.3390/machines11070698