Singularities

During this last term we found the singularities of our robot. We have an RPR robot which we already had the Jacobian matrix. So now, we focused on solving the singularities to know where our robot will not be able to generate any velocity; this was faster with Matlab. We know that singularities occur whenever the determinant of the Jacobian is equal to zero. Therefore, since our robot is a 3dof robot, we checked the determinant of the linear velocity of all 3 axis. The following was the determinant:

If we factorize and simplify, we get to the point in which we know, there will be a singularity when t3 is equal to zero or any multiples of pi. This means that when the robot is in home position, the end effector’s link won’t be able to generate velocity in the direction that it is pointing. It also has a singularity when it is at pi radians, however, the motor cannot go to that position, so that won’t be a problem in our robot.

There is another singularity in our robot when cos(t3)=-L1/L3. This singularity wouldn’t be a problem if L1 was larger than L3, which will make the cosine bigger than 1; this is not possible, so it would never occur. However, in our design, it’s the other way around; L3 is larger than L1. In our interface, we’ll have to be careful when than happens.