Why hasn’t anyone heard of cuspidal robots?

Achille Verheye
5 min readJan 15, 2021

In recent years we’ve seen a wealth of new robot arm designs driven by the massive growth in robotics and adoption into new markets. Most robot designers have never heard of cuspidality, a class of robot designs that come with the unique ability to avoid singularities (when the robot can’t move in some direction) in some cases but can be a nightmare for motion planning engineers. This article describes what they are and why they matter, while the follow-up tells you how to spot them and walks through an example.

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.

When the robot above switches from one IK solution to another (often called the elbow down and elbow up solutions), it must pass through a state in which it’s fully stretched out or folded in. In and around that state, it loses the ability to move in one direction, essentially losing a degree of freedom.

It’s widely believed by robot designers and engineers that a robot must always pass through such a singularity when moving from one IK solution to another. It turns out that’s not true for all robots: the so-called cuspidal robots allow avoiding such singular posture-changing moves. This isn’t something they teach you in school — most of the people in academia I spoke with including professors were surprised to hear of this.

As a result, several cuspidal robots exist and their designers don’t even know.

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.

In 1996 ABB launched the IRB 6400C (C for compact), which was very similar to the popular IRB 6400 and PUMA robot configurations, except a smart engineer realized that if he flipped the order of the first two joints, the robot would have a much smaller swept volume so you could put robots closer together and save on factory space.

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.

Only a year later they stopped commercialization of this robot— application engineers mentioned motion planning issues. We now know that the simple axis swap turned the robot cuspidal.

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.

If you’re using an unmodified industrial robot, it most likely won’t be cuspidal. Through trial and error robot manufacturers learned to stick to tried and tested designs. All other cases — beware!

With the exception of a few robots, most 6-axis robot designs that exist today are about the same, consisting of a spherical wrist on top of a shoulder that doesn’t have any offsets. As a result, you should really only worry about cuspidality if you’re seeing similar difficulties in motion planning, are considering using a new robot design, or are building a manipulator with a novel configuration of link lengths and offsets.

In those cases, it’s necessary to gain a more in-depth understanding of cuspidality.

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:



Achille Verheye

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