麦克纳姆轮的机器人移动平台英文文献和中文翻译(5)

菜单

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

+ 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

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.