Why hasn’t anyone heard of cuspidal robots?

A cuspidal (ˈkəspədᵊl ; having cusps) robot changing postures without passing through a singularity. That unique ability comes with tricky motion planning.

A familiar motion planning problem

Afew years ago, I was using the light-weight Kinova MICO robot arm to clean bathrooms. The robot was programmed to follow a circular path to clean a sink but would suddenly start spinning around itself or leave the path and collide. We began to spend more and more resources creating advanced motion planning capabilities to deal with these odd motions but eventually had to give up and switched to a different robot arm. Only later did I realize that I was likely dealing with a cuspidal manipulator.

The Kinova Mico had trouble executing simple motions without leaving the intended path, which could cause collisions. These movements are different from instabilities arising from passing through a singularity when using the Jacobian transpose.

The class of robots you haven’t heard of

All robot arms have a number of singularities which are a set of joint angles at which the robot loses the ability to move in a certain direction. These singularities are avoided because they can cause dangerous instabilities and unexpected motions, which can for example ruin a sanding or deburring operation. The hand of a robot arm can also reach the same point in space (x, y, z) with different sets of joint angles, which we call postures or inverse kinematics (IK) solutions. Press your hand against a flat surface and you can still move your elbow around (different postures) while keeping your hand in the same position. (Note: your arm has at least 7 degrees of freedom (axes) and thus can have infinite IK solutions)

Simple robot arm moving from the elbow up posture (IK solution) to elbow down. It must pass through a singularity (middle). Unlike your arm, this robot can’t rotate its shoulder, however it can ‘superflex’ its elbow. The manipulability ellipsoid (oval shape at the hand) is flat at the singularity.

Small changes, large impact

Should you build or buy a cuspidal robot? The answer is most likely not. The ability to make such singularity-avoiding moves in reality isn’t much of an advantage and it means that these robots cannot smoothly execute some simple moves.

Announcement of the IRB 6400C in ABB Review 3/1996. I spoke with a (now retired) engineer who oversaw the project. Engineers working on the project weren’t familiar with the discovery of cuspidal robots.

Modifying an industrial robot can make it cuspidal

More recently, a Freedom Robotics client reached out to me for advice on motion planning for their robot. The robot’s arm would make circling motions which they called ‘flourishes’ and they couldn’t find all IK solutions. It was a commonly used non-cuspidal industrial robot, but they had added an offset to the last axis such that they could feed a tube through to the end-effector. This time around I realized sooner that they had unknowingly turned their robot into a cuspidal one.

A simple cuspidal robot that has issues following a straight path. White lines in the middle and right graphs are singularity surfaces. For an in-depth explanation, see part 2 of cuspidal robots.

Is my robot cuspidal?

While a few rules of thumb exist, the general case requires solving a tough mathematical problem and thus remains unsolved.

Are there any upsides?

Burning their fingers on novel and potentially cuspidal configurations, robot designers likely stuck to the same old designs. Simply tuning link lengths can make a robot non-cuspidal, so if they are armed with new understanding, could we see innovative robot designs come to life? That way, designs with certain advantages like with the ABB IRB6400C could be tweaked to get the best of both worlds. More research is needed to turn the downsides of cuspidal robots into an advantage and a possibly interesting differentiator in robot sales. Learn more in the follow-up article:

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Achille Verheye

Achille Verheye

Roboticist, geometer, plant-person, Belgian living in the Bay Area