Problems in understanding kinematics

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The discussion centers on difficulties in understanding kinematics related to an omnidirectional robot, particularly concerning translational and tangential velocities. There is confusion regarding the derivation of equation (8) from the first paper and its implications for the robot's movement, especially regarding the multiplication by R(theta). Participants seek clarification on the rotation matrix and the equation for v_trans,i, as well as the derivation of the pure rolling constraint. The conversation highlights the need for a deeper understanding of the relationships between drive velocity, wheel direction, and potential sliding motion. Overall, the thread emphasizes the complexities of kinematic equations in robotic applications.
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wheeled robot kinematics and constraints
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).
66815456-dcb60900-ef6a-11e9-966b-2a5821ebff7e.jpg

66815457-dd4e9f80-ef6a-11e9-9469-29b2c29bf48d.jpg
66815458-dd4e9f80-ef6a-11e9-848a-4c41f34c2c68.jpg

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.
66815705-5948e780-ef6b-11e9-976c-843893a73e4d.jpg

At last, there is another lecture about kinematics. The problem is still about the constraints. How can I derive this pure rolling constraint.
QQ截图20191016114206.jpg
 
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It looks to me as though equation (8) in the first paper is the dot product of the drive velocity with the direction vector of the drive's wheel, i.e. the component of the velocity in the direction of the wheel spins. I guess that would imply that the wheel is also sliding sideways. If ##\theta## is a function of time, then the extra multiplication by ##\mathbf R(\theta(t))## would give the direction of the wheel at time ##t##.
 
tnich said:
It looks to me as though equation (8) in the first paper is the dot product of the drive velocity with the direction vector of the drive's wheel, i.e. the component of the velocity in the direction of the wheel spins. I guess that would imply that the wheel is also sliding sideways. If ##\theta## is a function of time, then the extra multiplication by ##\mathbf R(\theta(t))## would give the direction of the wheel at time ##t##.
Thank you, tnich. How can we get the equation of the rolling constraint if we focus on the 3rd approach.
 
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