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problems in understanding kinematics

Hi, I tried to understand kinematics after having an omnidirectional roobt. Some problems stop me to go further. Here I upload some contents of different papers talking about kinematics. For the 1st three pictures, I don't know how equation (8) is from and I am little confused about translational and tangentianal velocities. I don't know why it mutiplys R(theta) again in equation (8). image description image description image description

For the 2nd paper, I cannot obtain the equation of v_trans,i. Also, I am not sure if the rotation matrix in this case is R(theta) = [cos(theta) -sin(theta); sin(theta) cos(theta)]. Can anyone please tell some details about the kinematics. image description