# 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

- Han, L.; Qian, H.; Xing, K.; Xu, Y. Heavy-Payload Omnidirectional Robot. In Proceedings of the IEEE International Conference on Real-time Computing and Robotics, Angkor Wat, Cambodia, 6–10 June 2016. [Google Scholar] [CrossRef]
- Tian, Y.; Zhang, S.; Liu, J.; Chen, F.; Li, L.; Xia, B. Research on a new omnidirectional mobile platform with heavy loading and flexible motion. Adv. Mech. Eng.
**2017**, 9, 1687814017726683. [Google Scholar] [CrossRef] - Gao, Z.Q.; Chen, H.B.; Du, Y.P.; Wei, L. Design and development of an omni-directional mobile robot for logistics. Appl. Mech. Mater
**2014**, 602–605, 1006–1010. [Google Scholar] [CrossRef] - Qian, J.; Zi, B.; Wang, D.; Ma, Y.; Zhang, D. The Design and Development of an Omni-Directional Mobile Robot Oriented to an Intelligent Manufacturing System. Sensors
**2017**, 17, 2073. [Google Scholar] [CrossRef][Green Version] - Galati, R.; Mantriota, G.; Reina, G. Mobile Robotics for Sustainable Development: Two Case Studies. In Proceedings of the I4SDG Workshop 2021, Online, 25–26 November 2021. [Google Scholar] [CrossRef]
- Galati, R.; Mantriota, G.; Reina, G. Adaptive heading correction for an industrial heavy-duty omnidirectional robot. Sci. Rep.
**2022**, 12, 19608. [Google Scholar] [CrossRef] - Dickerson, S.L.; Lapin, B.D. Control of an omni-directional robotic vehicle with Mecanum wheels. In Proceedings of the Telesystems Conference, Atlanta, GA, USA, 26–27 March 1991. [Google Scholar] [CrossRef]
- Pin, F.G.; Killough, S.M. A new family of omnidirectional and holonomic wheeled platform for mobile robots. IEEE Trans. Robot. Autom
**1994**, 10, 480–489. [Google Scholar] [CrossRef][Green Version] - Purwin, O.; D’Andrea, R. Trajectory generation and control for four wheeled omnidirectional vehicles. Robot. Auton. Syst.
**2006**, 54, 13–22. [Google Scholar] [CrossRef] - Kim, K.B.; Kim, B.K. Minimum-Time Trajectory for Three-Wheeled Omnidirectional Mobile Robots Following a Bounded-Curvature Path With a Referenced Heading Profile. IEEE Trans. Robot.
**2011**, 27, 800–808. [Google Scholar] [CrossRef] - Li, W.; Yang, C.; Jiang, Y.; Liu, X.; Su, C. Motion Planning for Omnidirectional Wheeled Mobile Robot by Potential Field Method. J. Adv. Transport.
**2017**, 2017, 4961383. [Google Scholar] [CrossRef][Green Version] - Wang, C.; Liu, X.; Yang, X.; Hu, F.; Jiang, A.; Yang, C. Trajectory Tracking of an Omni-Directional Wheeled Mobile Robot Using a Model Predictive Control Strategy. Appl. Sci.
**2018**, 8, 231. [Google Scholar] [CrossRef][Green Version] - Zhang, R.; Hu, H.; Fu, Y. Trajectory tracking for omnidirectional mecanum robot with longitudinal slipping. In Proceedings of the MATEC Web of Conferences, Wuhan, China, 10–12 November 2018. [Google Scholar] [CrossRef][Green Version]
- Palacín, J.; Rubies, E.; Clotet, E.; Martínez, D. Evaluation of the Path-Tracking Accuracy of a Three-Wheeled Omnidirectional Mobile Robot Designed as a Personal Assistant. Sensors
**2021**, 21, 7216. [Google Scholar] [CrossRef] [PubMed] - Tagliavini, L.; Colucci, G.; Botta, A.; Cavallone, P.; Baglieri, L.; Quaglia, G. Wheeled Mobile Robots: State of the Art Overview and Kinematic Comparison Among Three Omnidirectional Locomotion Strategies. J. Intell. Robot. Syst.
**2022**, 106, 57. [Google Scholar] [CrossRef] [PubMed] - Grabowiecki, J. Vehicle Wheel. US Patent 1305535A, 3 June 1919. [Google Scholar]
- Blumrich, J.F. Omnidirectional Wheel. US Patent 3789947, 5 February 1972. [Google Scholar]
- Palacín, J.; Martínez, D.; Rubies, E.; Clotet, E. Suboptimal Omnidirectional Wheel Design and Implementation. Sensors
**2021**, 21, 865. [Google Scholar] [CrossRef] - Indiveri, G. Swedish Wheeled Omnidirectional Mobile Robots: Kinematics Analysis and Control. IEEE Trans. Robot.
**2009**, 25, 164–171. [Google Scholar] [CrossRef] - Ilon, B.B. Wheels for a Course Stable Selfpropelling Vehicle Movable in Any Desired Direction on the Ground or Some Other Base. US Patent 3876255A, 8 April 1975. [Google Scholar]
- Diegel, O.; Badve, A.; Bright, G.; Potgieter, J.; Tlale, S. Improved mecanum wheel design for omni-directional robots. In Proceedings of the Australian Conference on Robotics and Automation, Auckland, Australia, 27–29 November 2002. [Google Scholar]
- Gao, X.; Wang, Y.; Zhou, D.; Kikuchi, K. Floor-cleaning robot using omni-directional wheels. Ind. Robot.
**2009**, 36, 157–164. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef][Green Version] - Racz, S.-G.; Crenganis, M.; Barsan, A.; Marosan, I.-A. Omnidirectional autonomous mobile robot with mecanum wheel. In Proceedings of the International Student Innovation and Scientific Research Exhibition, 11–13 April 2019. [Google Scholar]
- Almasri, E.; Uyguroğlu, M.K. Modeling and Trajectory Planning Optimization for the Symmetrical Multiwheeled Omnidirectional Mobile Robot. Symmetry
**2021**, 13, 1033. [Google Scholar] [CrossRef] - Mohanraj, A.P.; Elango, A.; Reddy, M.C. Front and back movement analysis of a triangle-structured three-wheeled omnidirectional mobile robot by varying the angles between two selected wheels. Sci. World J.
**2016**, 2016, 7612945. [Google Scholar] [CrossRef] [PubMed][Green Version] - Palacín, J.; Rubies, E.; Clotet, E. The Assistant Personal Robot Project: From the APR-01 to the APR-02 Mobile Robot Prototypes. Designs
**2022**, 6, 66. [Google Scholar] [CrossRef] - Guo, Y. A new kind of wheel-model all-directional moving mechanism. J. Harbin Inst. Technol.
**2001**, 33, 854–857. [Google Scholar] - Mohd Salih, J.E.; Rizon, M.J.M.; Yaacob, S.; Adom, A.H.; Mamat, M.R. Designing omni-directional mobile robot with mecanum wheel. Am. J. Appl. Sci.
**2006**, 3, 1831–1835. [Google Scholar] [CrossRef] - Hijikata, M.; Miyagusuku, R.; Ozaki, K. Omni Wheel Arrangement Evaluation Method Using Velocity Moments. Appl. Sci.
**2023**, 13, 1584. [Google Scholar] [CrossRef] - Moore, K.L.; Flann, N.S. A six-wheeled omnidirectional autonomous mobile robot. IEEE Control Syst. Mag.
**2000**, 20, 53–66. [Google Scholar] [CrossRef] - Moore, K.L.; Davidson, M.; Bahl, V.; Rich, S.; Jirgal, S. Modelling and control of a six-wheeled autonomous robot. In Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334), Chicago, IL, USA, 28–30 June 2000. [Google Scholar] [CrossRef]
- Huang, Y.; Meng, R.; Yu, J.; Zhao, Z.; Zhang, X. Practical Obstacle-Overcoming Robot with a Heterogeneous Sensing System: Design and Experiments. Machines
**2022**, 10, 289. [Google Scholar] [CrossRef] - Tian, P.; Zhang, Y.N.; Zhang, J.; Yan, N.M.; Zeng, W. Research on Simulation of Motion Compensation for 8×8 Omnidirectional Platform Based on Back Propagation Network. Appl. Mech. Mater.
**2013**, 299, 44–47. [Google Scholar] [CrossRef] - Fornarelli, L.; Young, J.; McKenna, T.; Koya, E.; Hedley, J. Stastaball: Design and Control of a Statically Stable Ball Robot. Robotics
**2023**, 12, 34. [Google Scholar] [CrossRef] - Schröder, K.; Garcia, G.; Chacón, R.; Montenegro, G.; Marroquín, A.; Farias, G.; Dormido-Canto, S.; Fabregas, E. Development and Control of a Real Spherical Robot. Sensors
**2023**, 23, 3895. [Google Scholar] [CrossRef] [PubMed] - Zhan, Q.; Cai, Y.; Yan, C. Design, analysis and experiments of an omni-directional spherical robot. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation, Shanghai, China, 9–13 May 2011. [Google Scholar] [CrossRef]
- Zhang, L.; Kim, J.; Sun, J. Energy modeling and experimental validation of four-wheel mecanum mobile robots for energy-optimal motion control. Symmetry
**2019**, 11, 1372. [Google Scholar] [CrossRef][Green Version] - Hou, L.; Zhang, L.; Kim, J. Energy Modeling and Power Measurement for Three-Wheeled Omnidirectional Mobile Robots for Path Planning. Electronics
**2019**, 8, 843. [Google Scholar] [CrossRef][Green Version] - Samani, H.A.; Abdollahi, A.; Ostadi, H.; Rad, S.Z. Design and Development of a Comprehensive Omni Directional Soccer Player Robot. Int. J. Adv. Robot. Syst.
**2004**, 1, 191–200. [Google Scholar] [CrossRef][Green Version] - Tianran Peng, J.Q.; Qian, J.; Zi, B.; Liu, J.; Wang, X. Mechanical Design and Control System of an Omni-directional Mobile Robot for Material Conveying. Procedia CIRP
**2016**, 56, 412–415. [Google Scholar] [CrossRef][Green Version] - Bae, J.J.; Kang, N. Design Optimization of a Mecanum Wheel to Reduce Vertical Vibrations by the Consideration of Equivalent Stiffness. Shock Vib.
**2016**, 2016, 5892784. [Google Scholar] [CrossRef][Green Version] - Yang, X.; Zhang, H.; Cheng, T.; Ni, X.; Wu, C.; Zong, H.; Lu, H.; Lu, Z.; Shen, Y. An Omnidirectional and Movable Palletizing Robot based on Computer Vision Positing. In Proceedings of the IEEE International Conference on Intelligence and Safety for Robotics (ISR), Shenyang, China, 24–27 August 2018. [Google Scholar] [CrossRef]
- Wang, Z.; Yang, G.; Su, X.; Schwager, M. OuijaBots: Omnidirectional Robots for Cooperative Object Transport with Rotation Control Using No Communication. In Distributed Autonomous Robotic Systems; Springer: Cham, Switzerland, 2018; pp. 117–131. [Google Scholar] [CrossRef]
- Omnidirectional Robots for Cooperative Object Transport. Available online: https://youtu.be/4nLMYjqUoJ4 (accessed on 1 April 2023).
- Li, Y.; Ge, S.; Dai, S.; Zhao, L.; Yan, X.; Zheng, Y.; Shi, Y. Kinematic Modeling of a Combined System of Multiple Mecanum-Wheeled Robots with Velocity Compensation. Sensors
**2020**, 20, 75. [Google Scholar] [CrossRef][Green Version] - Eirale, A.; Martini, M.; Tagliavini, L.; Gandini, D.; Chiaberge, M.; Quaglia, G. Marvin: An Innovative Omni-Directional Robotic Assistant for Domestic Environments. Sensors
**2022**, 22, 5261. [Google Scholar] [CrossRef] - Siradjuddin, I.; Azhar, G.A.; Wibowo, S.; Ronilaya, F.; Rahmad, C.; Rohadi, E. A General Inverse Kinematic Formulation and Control Schemes for Omnidirectional Robots. Eng. Lett.
**2021**, 29, 1344–1358. [Google Scholar] - Steinmetz, C.P.; Berg, E.J. Complex Quantities and Their Use in Electrical Engineering. In Proceedings of the International Electrical Congress, AIEE, Chicago, IL, USA, 21–25 August 1893. [Google Scholar]
- Steinmetz, C.P.; Berg, E.J. Theory and Calculation of Alternating Current Phenomena, 1st ed.; The W. J. Johnston Company: New York, NY, USA, 1897. [Google Scholar]
- Hijikata, M.; Miyagusuku, R.; Ozaki, K. Wheel Arrangement of Four Omni Wheel Mobile Robot for Compactness. Appl. Sci.
**2022**, 12, 5798. [Google Scholar] [CrossRef] - Palacín, J.; Clotet, E.; Martínez, D.; Martínez, D.; Moreno, J. Extending the Application of an Assistant Personal Robot as a Walk-Helper Tool. Robotics
**2019**, 8, 27. [Google Scholar] [CrossRef][Green Version] - Bitriá, R.; Palacín, J. Optimal PID Control of a Brushed DC Motor with an Embedded Low-Cost Magnetic Quadrature Encoder for Improved Step Overshoot and Undershoot Responses in a Mobile Robot Application. Sensors
**2022**, 22, 7817. [Google Scholar] [CrossRef] - Palacín, J.; Rubies, E.; Clotet, E. Systematic Odometry Error Evaluation and Correction in a Human-Sized Three-Wheeled Omnidirectional Mobile Robot Using Flower-Shaped Calibration Trajectories. Appl. Sci.
**2022**, 12, 2606. [Google Scholar] [CrossRef] - OSOYOO ZZ012318MC Mecanum Robotic Kit. Available online: https://osoyoo.com/2019/11/08/omni-direction-mecanum-wheel-robotic-kit-v1/ (accessed on 1 April 2023).
- Dhanasekar, R.; Kolachalama, N.; Mahmud, S.; Mishkevich, E.; Tan, A. Optimization of Four-Way Controlled Intersections with Autonomous and Human-Driven Vehicles. In Proceedings of the IEEE MIT Undergraduate Research Technology Conference (URTC), Cambridge, MA, USA, 5–7 October 2018. [Google Scholar] [CrossRef]
- Bajracharya, B.; Gondi, V.; Hua, D. IoT Education Using Learning Kits of IoT Devices. Inf. Syst. Educ. J.
**2021**, 19, 40–44. [Google Scholar] - Almassri, A.M.M.; Shirasawa, N.; Purev, A.; Uehara, K.; Oshiumi, W.; Mishima, S.; Wagatsuma, H. Artificial Neural Network Approach to Guarantee the Positioning Accuracy of Moving Robots by Using the Integration of IMU/UWB with Motion Capture System Data Fusion. Sensors
**2022**, 22, 5737. [Google Scholar] [CrossRef] - Georgeon, O.; Vidal, J.R.; Knockaert, T.; Robertson, P. Simultaneous Localization and Active Phenomenon Inference (SLAPI). In Proceedings of the Third International Workshop on Self-Supervised Learning, Reykjavik, Iceland, 28–29 July 2022. [Google Scholar]
- Dosoftei, C.C.; Popovici, A.T.; Sacaleanu, P.R.; Gherghel, P.M.; Budaciu, C. Hardware in the Loop Topology for an Omnidirectional Mobile Robot Using Matlab in a Robot Operating System Environment. Symmetry
**2021**, 13, 969. [Google Scholar] [CrossRef] - Popovici, A.-T.; Dosoftei, C.-C.; Budaciu, C. Kinematics Calibration and Validation Approach Using Indoor Positioning System for an Omnidirectional Mobile Robot. Sensors
**2022**, 22, 8590. [Google Scholar] [CrossRef] [PubMed] - Fourier, J.B.J. Theorie Analytique de la Chaleur; Didot: Paris, France, 1822; pp. 499–508, (Translated in The Analytical Theory of Heat; Freeman, A., Translator; Dover Publications: New York, NY, USA, 2003; ISBN 0-486-49531-0). [Google Scholar]
- Zygmund, A. Trigonometric Series; Cambridge University Press: Cambridge, UK, 1935. [Google Scholar]
- Rudin, W. Principles of Mathematical Analysis; McGraw-Hill: New York, USA, 1976. [Google Scholar]
- Baca, J.; Yerpes, A.; Ferre, M.; Escalera, J.A.; Aracil, R. Modelling of Modular Robot Configurations Using Graph Theory. In Hybrid Artificial Intelligence Systems. HAIS 2008; Corchado, E., Abraham, A., Pedrycz, W., Eds.; Springer: Berlin, Germany, 2008; Volume 5271. [Google Scholar] [CrossRef][Green Version]
- Pagala, P.; Ferre, M.; Armada, M. Design of Modular Robot System for Maintenance Tasks in Hazardous Facilities and Environments. In ROBOT2013: First Iberian Robotics Conference. Advances in Intelligent Systems and Computing; Armada, M., Sanfeliu, A., Ferre, M., Eds.; Springer: Cham, Switzerland, 2014; Volume 253. [Google Scholar] [CrossRef]

**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