Here, UiðtÞ is the voltage applied to the respective motor, cvφ̇iðtÞ is the torque due to the back EMF. The positive constant coefficients cu  and cv  are defined by the

magnitudes of the starting and nominal torques, the nominal angular velocity, and the nominal voltage. Substitute relations (13) into Eqs. (9)–(11) and eliminate the angular velocities of the wheels by using Eqs. (5)–(8) to   obtain

.. p

xC ðmR2 + 4J1Þ + 4ẏC ψ ̇J1 + 4ẋccv = Rcu     ffi2ffiffiðU1 sinðψ + π 4̸ Þ

+ U2 cosðψ + π 4̸   Þ + U3 cosðψ + π 4̸   Þ + U4 sinðψ + π 4̸   ÞÞ,

ð14Þ

.. .. p

yC ðmR2 + 4J1Þ yC − 4ẋCψ ̇J1 − 4ẏccv = − R   ffi2ffiffi U1 cosðψ + π 4̸ Þ

− U2 sinðψ + π 4̸   Þ − U3 sinðψ + π 4̸   Þ + U4 cosðψ + π 4̸   ÞÞ,  ð

15Þ

.JCR2 + 4J1ðl + dÞ2.ψ ̈ = − Rðl + dÞðM1 − M2 + M3 − M4Þ. ð16Þ

4 Optimization of Driving Torques

If the trajectory of the center of mass of the robot is specified, i.e., the time histories xC ðtÞ, yC ðtÞ and ψ ðtÞ are given, then Eqs. (9)–(11) form a system of three equations for four unknown torques MiðtÞ, i = 1, ... , 4. We can seek the solutions for which the sum of the squared torques applied to the  wheels,

P = M2ðtÞ + M2ðtÞ + M2ðtÞ + M2ðtÞ ð17Þ

1 2 3 4

is a minimum at each time instant. In other words, we seek the torques that min- imize the function PðM1, M1, M1, M4Þ. These torques are defined by

M1 min = pffi2ffiffi.xC

1Þ + 4ẏCψ ̇ J2. sin  ψ + π 4

̸ 4R

ðmR2 + 4J

1 ð Þ ð Þ

− pffi2ffiffi.yC

ðmR2 + 4J

1Þ − 4ẋCψ ̇ J1. cosðψ + π 4̸ Þ ð̸ 4RÞ

ð18Þ

− ψ ...JCR2 + 4J1ðl + dÞ2. ð̸ 4Rðl + dÞÞ ,

M2 min = pffi2ffiffi.xC ðmR2 + 4J1Þ + 4ẏCψ ̇ J1. cosðψ + π 4̸ Þ ð̸ 4RÞ

p  . .. .

− ffi2ffiffi yC ðmR2 + 4J1Þ − 4ẋCψ ̇ J1

..

sinðψ + π 4̸  Þ ð̸ 4RÞ

ð19Þ

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

M3 min = pffi2ffiffi.xC ðmR2 + 4J1Þ + 4ẏCψ ̇ J1. cosðψ + π 4̸ Þ ð̸ 4RÞ

p ..

− ffi2ffiffi.yC ðmR2 + 4J1Þ − 4ẋCψ ̇ J1. sinðψ + π 4̸ Þ ð̸ 4RÞ

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