# Topological Design Methods for Mecanum Wheel Configurations of an Omnidirectional Mobile Robot

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## Abstract

**:**

## 1. Introduction

## 2. Kinematics Model of an Omnidirectional Mobile Robot with n Mecanum Wheels

#### 2.1. Mecanum Wheel Configurations of the Single Omnidirectional Mobile Robot

#### 2.2. Kinematics Constraint Model of a Single Mecanum Wheel and Kinematics Model of an n-Mecanum-Wheel Robot

## 3. Bottom-Roller Axle Intersections Approach for Judging Robot’s Omnidirectional Mobility

#### 3.1. Conditions for Omnidirectional Motion of a Mecanum-Wheeled Mobile Robot System

#### 3.2. Relation Between the Roller Axle Intersection Points Number on Three Mecanum Wheels and the Column Rank of the Jacobian Matrix

#### 3.2.1. No Intersection of the Three Bottom-Rollers Axles

#### 3.2.2. The Axles of the Three Bottom-Rollers Intersect at One Point

#### 3.2.3. The Axles of the Three Bottom-Rollers Intersect at Two Points

#### 3.2.4. The Axles of the Three Bottom-Rollers Intersect at Three Points

#### 3.3. Three-Mecanum-Wheel Configurations of the Mobile Robot

## 4. Symmetrical Wheel Configurations of the Four-Mecanum-Wheel Mobile Robot

#### 4.1. Judging the Four-Mecanum-Wheel Configurations by a Bottom-Roller Axle Intersections Approach

#### 4.2. Theoretical Verification for the Symmetrical Rectangular Configurations with Four Mecanum Wheels

## 5. Design Method of Mecanum Wheel Configurations for the Omnidirectional Mobile Robot

#### 5.1. Sub-Configuration Judgment Method for Judging the Omnidirectional Motion Capacity of the Wheel Combination Configurations

#### 5.2. Analysis of Mecanum Wheel Configurations and Combination Configurations for Common Omnidirectional Mobile Robots

#### 5.3. Topological Design Methods of Multi-Mecanum Wheel Configuration for Omnidirectional Mobile Robot

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Wheel configurations of the single-Mecanum-wheeled robot: (

**a**) centripetal configuration; (

**b**) symmetrical rectangular configuration.

**Figure 2.**Kinematic constraints of a Mecanum wheel and the coordinate systems of a mobile system: (

**a**) structural principle of a Mecanum wheel; (

**b**) Kinematic constraints diagram of a Mecanum wheel on the robot using a vector method; (

**c**) Kinematic constraint diagram of a Mecanum wheel using a matrix transformation method; (

**d**) Location of the mobile robot in the global coordinate system and the relationship regarding position between two local coordinate systems.

**Figure 3.**Wheel configuration of the mobile robot with three Mecanum wheels whose bottom-roller axles are parallel to each other.

**Figure 4.**Wheel Configuration of the mobile robot with three Mecanum wheels whose bottom-roller axles intersect at one point.

**Figure 5.**Wheel configuration of the mobile robot with three Mecanum wheels whose roller axles intersect at two points.

**Figure 6.**Configuration of mobile robot with three Mecanum wheels whose roller axles intersect at three points.

**Figure 7.**Three-Mecanum-wheel configurations: (

**a**) centripetal circular array configuration; (

**b**) Non-centripetal circular array configuration; (

**c**) star-type circular array configuration; (

**d**) centripetal circular array configuration of orthogonal Mecanum wheels; (

**e**) T-configuration of the orthogonal Mecanum wheels; (

**f**) an orthogonal Mecanum wheel.

**Figure 8.**Four-Mecanum-wheel configurations: (

**a**)

**–**(

**j**) rectangular configurations of four wheels that are arranged at the corner and whose axles are parallel to the centerline of the robot; (

**k**)

**–**(

**n**) centripetal configurations of four Mecanum wheels; (

**o**) centripetal circular array configurations of four orthogonal Mecanum wheels.

**Figure 10.**A tandem configuration composed of two symmetrical four-Mecanum-wheel sub-configurations.

**Figure 11.**Common symmetrical rectangular wheel configurations and application examples: (

**a**) the typical four-Mecanum-wheel configuration W4; (

**b**) a 12-Mecanum-wheel configuration W12; (

**c**) a combination configuration W36 with three W12; (

**d**) a 24-Mecanum-wheel configuration W24; (

**e**) an eight-Mecanum-wheel configuration W8, the example: MC-Drive TP 200 of CLAAS; (

**f**) the combination configuration W16 with two W8, the example: the combination of two MC-Drive TP 200 [12]; (

**g**) an eight-Mecanum-wheel configuration W8, which can combine into a 16-Mecanum-wheel configuration [32]; (

**h**) a rectangular combination configuration consisting of four W6 [13]. The examples in (

**a**–

**d**,

**h**) are KUKA omniMove AGVs.

**Figure 12.**Topological design methods for wheel configurations based on a symmetrical four-Mecanum-wheel configuration: (

**a**) end-to-end connection configuration; (

**b**) side-by-side connection configuration; (

**c**) a front-back symmetric configuration; (

**d**) a front-back asymmetric configuration; (

**e**) a symmetric configuration with four coaxial wheels series; (

**f**) end-to-end combination configuration; (

**g**) rectangular combination configuration; (

**h**) side-by-side combination configuration; (

**i**) distributed combination configuration.

**Figure 13.**The examples of deducing new wheel configurations based on three basic Mecanum wheel configurations by using the topological method: (

**a**) new configurations deduced from a centripetal circular array configuration of three Mecanum wheels in Figure 7a; (

**b**) new configurations deduced from centripetal circular array configuration of four Mecanum wheels in Figure 8n; (

**c**) new configurations deduced from centripetal circular array configurations of four orthogonal Mecanum wheels in Figure 8o.

**Table 1.**Relationship between the parameters of the three Mecanum wheels in Figure 3.

Serial Number | $\left|{\mathit{\alpha}}_{\mathit{i}}\right|\in [0\xb0,360\xb0)$ | $\left|{\mathit{\beta}}_{\mathit{i}}\right|\in [0\xb0,180\xb0]$ | $\left|{\mathit{\gamma}}_{\mathit{i}}\right|\in (0\xb0,90\xb0)$ |
---|---|---|---|

1 | ${\alpha}_{1}$ | ${\beta}_{1}$ | ${\gamma}_{1}$ |

2 | ${\alpha}_{2}$ | ${\beta}_{2}$ | ${\gamma}_{2}$ |

3 | ${\alpha}_{3}$ | ${\beta}_{3}$ | ${\gamma}_{3}$ |

**Table 2.**Relationship between the parameters of the three Mecanum wheels in Figure 4.

Serial Number | $\left|{\mathit{\alpha}}_{\mathit{i}}\right|\in [0\xb0,360\xb0)$ | $\left|{\mathit{\beta}}_{\mathit{i}}\right|\in [0\xb0,180\xb0]$ | $\left|{\mathit{\gamma}}_{\mathit{i}}\right|\in (0\xb0,90\xb0)$ |
---|---|---|---|

1 | ${\alpha}_{1}$ | −${\gamma}_{1}$ | ${\gamma}_{1}$ |

2 | ${\alpha}_{2}$ | −180°−${\gamma}_{2}$ | ${\gamma}_{2}$ |

3 | ${\alpha}_{3}$ | −${\gamma}_{3}$ | ${\gamma}_{3}$ |

**Table 3.**Relationship between the parameters of the three Mecanum wheels in Figure 5.

Serial Number | $\left|{\mathit{\alpha}}_{\mathit{i}}\right|\in [0\xb0,360\xb0)$ | $\left|{\mathit{\beta}}_{\mathit{i}}\right|\in [0\xb0,180\xb0]$ | $\left|{\mathit{\gamma}}_{\mathit{i}}\right|\in (0\xb0,90\xb0)$ |
---|---|---|---|

1 | ${\alpha}_{1}$ | −${\gamma}_{1}$ | ${\gamma}_{1}$ |

2 | ${\alpha}_{2}$ | −180°−${\gamma}_{2}$ | ${\gamma}_{2}$ |

3 | ${\alpha}_{3}$ | ${\beta}_{3}$ | ${\gamma}_{3}$ |

**Table 4.**Relationship between the parameters of the three Mecanum wheels in Figure 6.

Serial Number | $\left|{\mathit{\alpha}}_{\mathit{i}}\right|\in [0\xb0,\text{}360\xb0)$ | $\left|{\mathit{\beta}}_{\mathit{i}}\right|\in [0\xb0,\text{}180\xb0]$ | $\left|{\mathit{\gamma}}_{\mathit{i}}\right|\in (0\xb0,\text{}90\xb0)$ |
---|---|---|---|

1 | ${\alpha}_{1}$ | −${\gamma}_{1}$ | ${\gamma}_{1}$ |

`2 | ${\alpha}_{2}$ | −180°−${\gamma}_{2}$ | ${\gamma}_{2}$ |

3 | ${\alpha}_{3}$ | ${\beta}_{3}$ | ${\gamma}_{3}$ |

Number of Intersection Points | Typical Configurations | Column Full Rank |
---|---|---|

0 | No | |

1 | No | |

2 | Yes | |

3 | Yes |

**Table 6.**Characteristics of the Mecanum wheel configurations in Figure 8.

Configurations in Figure 8 | a | b | c | d | e | f | g | h | i | J | k | l | m | n |

Intersections | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 |

Column rank | 2 | 3 | 3 | 3 | 3 | 2 | 3 | 3 | 3 | 3 | 2 | 3 | 3 | 3 |

Column full Rank | N | Y | Y | Y | Y | N | Y | Y | Y | Y/N | N | Y | Y | Y |

Omnidirectional motion capacity | n | B | B | B | G | n | B | B | B | G/n | n | B | B | G |

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**MDPI and ACS Style**

Li, Y.; Dai, S.; Zhao, L.; Yan, X.; Shi, Y.
Topological Design Methods for Mecanum Wheel Configurations of an Omnidirectional Mobile Robot. *Symmetry* **2019**, *11*, 1268.
https://doi.org/10.3390/sym11101268

**AMA Style**

Li Y, Dai S, Zhao L, Yan X, Shi Y.
Topological Design Methods for Mecanum Wheel Configurations of an Omnidirectional Mobile Robot. *Symmetry*. 2019; 11(10):1268.
https://doi.org/10.3390/sym11101268

**Chicago/Turabian Style**

Li, Yunwang, Sumei Dai, Lala Zhao, Xucong Yan, and Yong Shi.
2019. "Topological Design Methods for Mecanum Wheel Configurations of an Omnidirectional Mobile Robot" *Symmetry* 11, no. 10: 1268.
https://doi.org/10.3390/sym11101268