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To find the situation where singularity occurs, the determinant of the Jacobian is used.

RRR Planar robot

For the RRR planar robot with angles $q_1$, $q_2$, and $q_3$ and the length $L_1$, $L_2$, and $L_3$ as shown in the picture, the determinant of the Jacobian is as follows:

$$det(J) = L_1*L_2*sin(q_2)$$

$$L_1*L_2*sin(q_2) = 0$$

Therefore, singularity occurs when $q_2 = 0°$ or $q_2 = 180°$

I understand that singularity happens when the link is fully stretched or retracted. However, from the calculation of determinant of the Jacobian, it shows that the singularity will only occur when $q_2$ is 0 or 180 degrees.

My question is that when $q_3 = 0°$ or $180°$, why does the singularity not occur?

I just need to understand the physical meaning regarding the mathematical calculation.

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  • $\begingroup$ What is your Jacobain matrix? $\endgroup$
    – CroCo
    Jan 5 at 18:11
  • $\begingroup$ I understand that singularity happens when the link is fully stretched or retracted., singularities do not necessarily happen in this manner. A singularity occurs when the Jacobian matrix's rank degenerates. $\endgroup$
    – CroCo
    Jan 5 at 18:22
  • $\begingroup$ According to the paper in this link (ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1041568), the Jacobian matrix is shown on page 2 in Section 2.1. After the inverse of the Jacobian, J^-1 = L1*L2*sin(q2) $\endgroup$
    – Boonboy
    Jan 5 at 20:26
  • $\begingroup$ IEEE requires membership to access its resources, so please post it. $\endgroup$
    – CroCo
    Jan 7 at 13:35

1 Answer 1

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Sometimes I seem to equate planar and planar positioning, this would be one of those times. The workspace of the robot you are analyzing is the planar position and orientation workspace - i.e. $x$ and $y$ location of the end-effector and additionally some $\theta$ angle of the end-effector. In this case, the determinant of the Jacobian exists, and I'll take the papers word for the given formula being correct. It looks like what you're saying about the fully-extended and retracted states of the arm comes from the paper, so let's dig into that.

The fully-extended state of the arm refers to the first two links being parallel to each other, where their total length is the sum of each link's length, and the retracted state is the configuration where the two links are parallel but their length is the difference of their individual link length's. So, what about this robot guarantees that these configurations will always be singular?

Intuitively, I'm not quite sure. However, it appears that in these cases, we get something like $j_{1,1} = j_{1,2} + j_{1,3}$ and likewise $j_{2,1} = j_{2,2} + j_{2,3}$, where $j_{i,k}$ represents the $i^{\text{th}}$ row and the $k^{\text{th}}$ column of the Jacobian. Thus, we lose the ability to move the arm independently in some direction as the Jacobian has linearly dependent columns.

I can write out more of the math if you would like, but its really just a couple steps of work from the paper you listed.

As an aside, when the workspace of a robot has more than 2 DoFs (especially for orienting robots), the physical intuition of singularities is harder to visualize. As CroCo mentioned, it won't do much good to look into the matter more than the Jacobian loses rank, or likewise has linearly dependent columns.

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  • $\begingroup$ it is 2×3, and ..., unless the OP provides more details, we cannot be certain. The Jocabain matrix comes in different forms, so the size may vary even for the same manipulator. $\endgroup$
    – CroCo
    Jan 5 at 18:17
  • $\begingroup$ Considering this is a planar 3R robot, the only workspace the presented analysis would make sense in is the 2D positional workspace. In this case, there is only one form of the Jacobian - as others are defined relative to different orientation representations. Also, yes, I have edited the answer to reflect that the Jacobian is $2 \times 3$, thanks! $\endgroup$
    – domo_arigato
    Jan 5 at 18:22
  • $\begingroup$ the only workspace the presented analysis would make sense again the Jacobian matrix is a linear mapping between two entities. In the absence of a description of the task, a priori guess of its size is really difficult. $\endgroup$
    – CroCo
    Jan 5 at 18:29
  • $\begingroup$ Yes, that is what a matrix is. The statement made about singularities happening when the arm is retracted and my own personal experience with 3R analysis would leave me to believe the task space is the 2D position space. It is a guess, yes, but sometimes we can make informed decisions based on limited information. Not to mention that the arm is referred to as a "planar" arm. $\endgroup$
    – domo_arigato
    Jan 5 at 18:39
  • $\begingroup$ It would appear my "informed decision" was wrong :/ $\endgroup$
    – domo_arigato
    Jan 6 at 16:32

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