# Defining the Consistent Velocity of Omnidirectional Mobile Platforms

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## Abstract

**:**

## 1. Introduction

#### 1.1. New Contribution

#### 1.2. Structure of the Paper

## 2. Materials and Methods

#### 2.1. Platform Configurations Assessed

#### 2.2. Kinematic Model of an Omnidirectional Platform

## 3. Problem Definition

#### 3.1. Maximum Translational Velocity of a Mobile Platform

#### 3.2. Problem Caused by the Maximum Velocity of the Motors

#### 3.3. Profile of the Angular Velocities of the Wheels

## 4. Consistent Velocity

#### 4.1. Consistent Velocity Definition

#### 4.2. Consistent Velocity Computation

## 5. Results

#### 5.1. Consistent Velocity of the 3A Platform

#### 5.1.1. Mobile Platform 3A: Consistent Velocity for $\omega =0\mathrm{rad}/\mathrm{s}$

#### 5.1.2. Mobile Platform 3A: Consistent Velocity for $\omega =1\mathrm{rad}/\mathrm{s}$

#### 5.1.3. Mobile Platform 3A: Consistent Velocity as a Function of $\omega $

#### 5.2. Consistent Velocity of the Set of Mobile Platforms

#### 5.2.1. Consistent Velocity for $\omega =0\mathrm{rad}/\mathrm{s}$

#### 5.2.2. Consistent Velocity for $\omega \ne 0\mathrm{rad}/\mathrm{s}$

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Representation of (

**a**) the parameters of a three-wheeled omnidirectional mobile platform, and detail of ${\gamma}_{k}$ for (

**b**) an omni wheel and (

**c**) a mecanum wheel.

**Figure 2.**Example of a mobile robot with the 3A omnidirectional platform [21]: (

**a**) the entire structure of the robot; (

**b**) detail of its omni wheels.

**Figure 3.**Top-view trajectory of the robot implementing a motion command $M$ for 2 s: expected trajectory (cyan, solid line) and ground truth trajectory (red, dashed line) caused by the maximum angular velocity reachable by its motors.

**Figure 4.**Representation of the evolution of the angular velocity of wheel 1 (${\omega}_{k=1}$) of the 3A mobile platform [21,25] relative to the angular orientation of the motion $\alpha $. The solid line depicts the evolution obtained when $\omega =0$ rad/s, while the dashed line depicts the evolution when $\omega =1$ rad/s.

**Figure 6.**Top-view of the schematic representation of the 3A platform and polar representation of the maximum translational velocity in each direction (blue) and of the consistent velocity (red) when $\omega =0\mathrm{rad}/\mathrm{s}$.

**Figure 7.**Representation of the evolution of the angular velocities of the wheels (${\omega}_{k=\mathrm{1...3}}$) as a function of the direction $\alpha $ of the linear velocity of the 3A mobile platform when $\omega =0\mathrm{rad}/\mathrm{s}$: (

**a**) angular velocities required to move at the maximum linear velocities; (

**b**) angular velocities required to move at the consistent velocity.

**Figure 8.**Representation of the evolution of the angular velocities of the wheels (${\omega}_{k=\mathrm{1...3}}$) as a function of the direction $\alpha $ of the linear velocity of the 3A mobile platform when $\omega =1\mathrm{rad}/\mathrm{s}$: (

**a**) angular velocities required to move at the maximum linear velocity; (

**b**) angular velocities required to move at the consistent velocity; and (

**c**) polar representation of the maximum translational velocity in each direction (blue) and of the consistent velocity (red).

**Figure 9.**Polar charts showing the maximum linear velocity ${v}_{i,max}$ (blue) in each direction ${\alpha}_{i}$, for different values of $\omega $ for the 3A platform: (

**a**) $\omega =0\mathrm{rad}/\mathrm{s}$; (

**b**) $\omega =1\mathrm{rad}/\mathrm{s}$; (

**c**) $\omega =2\mathrm{rad}/\mathrm{s}$; (

**d**) $\omega =3\mathrm{rad}/\mathrm{s}$. The red circle represents the value of the consistent velocity.

**Figure 10.**Representation of the mobile platforms assessed, their polar charts showing the maximum linear velocity (blue arrow) ${v}_{i,max}$ in each direction ${\alpha}_{i}$ when $\omega =0\mathrm{rad}/\mathrm{s}$, and their consistent velocity (red circle).

**Figure 11.**Relationship between the consistent velocity and the angular velocity $\omega $ of the assessed mobile platforms.

**Table 1.**Mobile platform configurations assessed in this work: name, schematic representation, and angular orientation of the wheels (${\phi}_{k}$).

Name | 3A | 1A-2B | 1A-2C | 1A-1B-1C | 1A-1B-1D | 3E |

Diagram | ||||||

${\mathit{\phi}}_{\mathit{k}=\mathbf{1}\dots \mathbf{3}}$(°) | [0 0 0] | [−90 0 −90] | [30 0 −30] | [0 −90 −30] | [0 −90 60] | [49.37 49.37 49.37] |

Parameter | Case with $\mathit{\omega}=0\mathbf{rad}/\mathbf{s}$ | Case with $\mathit{\omega}=1\mathbf{rad}/\mathbf{s}$ | Case with $\mathit{\omega}=-1\mathbf{rad}/\mathbf{s}$ |
---|---|---|---|

Consistent velocity | 0.99 m/s | 0.80 m/s | 0.80 m/s |

Maximum of the maximum translational velocities | 1.15 m/s | 0.92 m/s | 0.92 m/s |

Minimum of the maximum translational velocities | 0.99 m/s | 0.80 m/s | 0.80 m/s |

$offset$ value | 0 rad/s | 1.32 rad/s | −1.32 rad/s |

Maximum velocity of the wheels (${\omega}_{1},{\omega}_{2},{\omega}_{3})$ | 6.70 rad/s | 6.70 rad/s | 4.07 rad/s |

Minimum velocity of the wheels (${\omega}_{1},{\omega}_{2},{\omega}_{3})$ | −6.70 rad/s | −4.07 rad/s | −6.70 rad/s |

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**MDPI and ACS Style**

Rubies, E.; Palacín, J.
Defining the Consistent Velocity of Omnidirectional Mobile Platforms. *Machines* **2024**, *12*, 397.
https://doi.org/10.3390/machines12060397

**AMA Style**

Rubies E, Palacín J.
Defining the Consistent Velocity of Omnidirectional Mobile Platforms. *Machines*. 2024; 12(6):397.
https://doi.org/10.3390/machines12060397

**Chicago/Turabian Style**

Rubies, Elena, and Jordi Palacín.
2024. "Defining the Consistent Velocity of Omnidirectional Mobile Platforms" *Machines* 12, no. 6: 397.
https://doi.org/10.3390/machines12060397