# Why in trajectory planning in joint space the manipulator never crosses a singularity?

I am studying robotics kinematics and I have studied that if we plan a trajectory in joint space it is impossible for the robot manipulator to cross a singularity during its motion, while in cartesian space this is possible.

But why in joint space the manipulator never meet a singularity?

The term "singularity" characterizes those configurations in the joint space where the Jacobian matrix loses rank and thus it is not directly invertible.

The Jacobian, in turn, is used to remap a trajectory from the Cartesian space to the joint space.

Therefore, if you plan the trajectory straight in the joint space, then you are not going to use the Jacobian and, as a consequence, you will not have to deal with singularities.

• Is this also true for parallel kinematic machines? Don't those have structural singularities? I know the question references manupulators, which are mostly serial..I am just asking for the sake of completeness of the answer. – 50k4 Jan 29 '20 at 12:58
• I believe that parallel structures are characterized by a constrained motion (i.e. the movement of a joint depends on other joints) in the joint space, which is technically different from singularities. For example, constrained motion does not imply that the joint velocities could go to infinity as the result of control actions, as it exactly happens in singular configurations instead. – Ugo Pattacini Jan 29 '20 at 15:01
• I have not studied parallel mechanisms, but I think I recall something about singularities of the parallel mechanism leading to a structural problem. This video may support that: youtu.be/RxIMTVVb3bs. I need to dig into the math. – SteveO Jan 30 '20 at 3:42
• I'm not an expert of parallel mechanisms either ;) Anyway, this resource is quite enlightening: fields.utoronto.ca/journalarchive/mics/3-9.pdf. Therein, singularities are defined in terms of the determinant of the matrix Q used in the forward kinematics case, which is then different from the Jacobian (differential case). To conclude, yes we may have singularities in the joint space for parallel mechanisms although the term "singular" needs to be correctly referred to. – Ugo Pattacini Jan 30 '20 at 13:00

Using quaternions for 3D rotations avoids the 'gimbal lock' problem, being one form of singularity in a manipulator. For a start to learning about quaternions in gaming and robotics, see https://www.3dgep.com/understanding-quaternions/

• This is not a correct answer. While you are true about the Clifford algebra / quarterian representation not resulting in mathematical singularities, the mathematical construct can do nothing about physical singularities. They exist in many manipulators regardless of the math. – SteveO Jan 30 '20 at 23:29
• Thanks, @SteveO for reminding me that there are physical situations that are also called singularities. I'd forgotten the overloading of the word with multiple concepts. Typically, those situations are where there are multiple solutions (just fix one dof, and re-solve) or none (unreachable). So, I'll update my answer. – user24605 Jan 31 '20 at 2:35
• You are welcome @RigTig. There is also the degenerate case where the manipulator does not have full rank - it does not span its workspace dimensions. For example, place the wrist center of an articulated manipulator arm right over the shoulder, and there is no way to move perpendicular to the plane of the arm. – SteveO Jan 31 '20 at 2:41
• Now that I think about it like described by @SteveO, isn't the question making an assumption that is incorrect: in general, a jointed manipulator might pass through any number of singularities. I just read it as I was thinking about use of quaternions, so that was the limit of my (non-)thinking. I am thinking now that my answer is almost off-topic!! – user24605 Jan 31 '20 at 2:55