ð20Þ

M4 min = pffi2ffiffi.xC

− pffi2ffiffi.yC

ðmR2 + 4J1Þ + 4ẏCψ ̇ J1. sinðψ + π 4̸ Þ ð̸ 4RÞ

ðmR2 + 4J1Þ − 4ẋCψ ̇ J2. cosðψ + π 4̸ Þ ð̸ 4RÞ

ð21Þ

1

..

+ ψ .JCR2 + 4J1ðl2 + d2Þ. ð̸ 4Rðl + dÞÞ .

By applying a similar criterion (the minimum of the sum of the squares) to the voltages defined by Eqs. (14)–(16), we  obtain

U1 min = pffi2ffiffi.xC

p

ðmR2 + 4J

..

1Þ + 4ẏCψ ̇ J1 + 4ẋccv. sinðψ + π 4̸ Þ ð̸ 4RcuÞ

− ffi2ffiffi.yC ðmR2 + 4J1Þ − 4ẋCψ ̇ J1 + 4ẏccv. cosðψ + π 4̸ Þ ð̸ 4RcuÞ

− ψ ðJCR2 + 4J1ðl + dÞ2Þ ð̸ 4Rðl + dÞcuÞ − ψ ̇ðl + dÞcv ð̸ RcuÞ ,

U2 min = pffi2ffiffiðxC ðm + 4J1Þ + 4ẏCψ ̇ J1 + 4ẋccvÞ cosðψ + π 4̸ Þ ð̸ 4RcuÞ

+    2  yC ðm + 4J2Þ − 4ẋCψ ̇ J2 + 4ẏccv    sinðψ + π 4̸ Þ ð̸ 4RcuÞ

ð22Þ

ð23Þ

1 1

..

+ ψ ðJCR2 + 4J1ðl + dÞ2Þ ð̸ 4Rðl + dÞcuÞ + ψ ̇ðl + dÞcv ð̸ RcuÞ ,

Fig. 3   Mobile robot based

on the 4 WD Mecanum wheel mobile kit from NEXUSrobot and controlled by a plugin based software “RobotController”

..

ð    ðm + 4J1Þ + 4ẏCψ ̇ J1 + 4ẋccvÞ cosðψ + π 4̸ Þ ð̸ 4RcuÞ

+ pffi2ffiffi.yC

..

ðm + 4J2

Þ − 4ẋCψ ̇ J2

+ 4ẏccv. sinðψ + π 4̸  Þ ð̸ 4RcuÞ

ð24Þ

− ψ ðJCR2 + 4J1ðl + dÞ2Þ ð̸ 4Rðl + dÞcuÞ − ψ ̇ðl + dÞcv ð̸ RcuÞ ,

U4 min = pffi2ffiffi.xC

ðmR2 + 4J

1Þ + 4ẏCψ ̇ J1 + 4ẋccv. sinðψ + π 4̸ Þ ð̸ 4RcuÞ

− pffi2ffiffi.yC

ðmR2 + 4J1Þ − 4ẋCψ ̇ J1 + 4ẏccv.

cosðψ + π 4̸  Þ ð̸  4RcuÞ

ð25Þ

+ ψ ðJCR2 + 4J1ðl + dÞ2Þ ð̸ 4Rðl + dÞcuÞ + ψ ̇ðl + dÞcv ð̸ RcuÞ .

Constructing the optimal modes on the basis of integral criteria is reduced, as a rule, to numerical optimization procedures.

5 Conclusion

The kinematic and dynamic equations are derived for a four-wheeled robot with Mecanum wheels, subject to non-holonomic constraints (rolling without slipping). The optimal torques to be applied to the wheels and the voltages to be applied to the motors in order to provide a prescribed trajectory for the robot’s center of mass are found. To evaluate the theoretical results, a prototype of a mobile platform with four Mecanum wheels that implements the principles presented in the paper was used (see Fig. 3). The experimental data agree with the theoretical  predictions.

Acknowledgments This study was supported by the Development Bank of Thuringia and the Thuringian Ministry of Economic Affairs with funds of the European Social Fund (ESF) under grant 2011 FGR 0127.

References

Campion, G., Bastin, G., & D’Andrea-Novel, B. (1996). Structural properties and classification of kinematic and dynamic models of wheeled mobile robots. IEEE Transactions on Robotics and Automation, 12(1), 47–62.