ði = 1, ... , 4Þ. ð13Þ
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 , 麦克纳姆轮的机器人移动平台英文文献和中文翻译(4):http://www.youerw.com/fanyi/lunwen_79249.html