A review of cuspidal serial and parallel manipulators Philippe Wenger and Damien Chablat

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A review of cuspidal serial and parallel

manipulators

Philippe Wenger and Damien Chablat

Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000 Nantes, France

ABSTRACT

Cuspidal robots can move from one inverse or direct kinematic solution to another without ever passing through

a singularity. These robots have remained unknown because almost all industrial robots do not have this feature.

However, in fact, industrial robots are the exceptions. Some robots appeared recently in the industrial market can

be shown to be cuspidal but, surprisingly, almost nobody knows it and robot users meet difﬁculties in planning

trajectories with these robots. This paper proposes a review on the fundamental and application aspects of cuspidal

robots. It addresses the important issues raised by these robots for the design and planning of trajectories. The

identiﬁcation of all cuspidal robots is still an open issue. This paper recalls in details the case of serial robots

with three joints but it also addresses robots with more complex architectures such as 6-revolute-jointed robot and

parallel robots. We hope that this paper will help disseminate more widely knowledge on cuspidal robots.

1 Introduction

When a new robot is to be implemented in a production site, its kinematic architecture must be chosen initially (serial

or parallel, choice of the number and types of axes, etc.). Then, the robot must be programmed and controlled so that the

tool it is handling can properly follow the trajectories deﬁned to carry out the required tasks. Most serial robots have the

ability to reach a pose in their workspace with several inverse kinematic solutions, e.g., "elbow up" and "elbow down". A

change of solution can be made to avoid a joint limit or a collision. It has long been assumed that any robot must necessarily

pass through a singular conﬁguration during a change of solution, as is the case for most industrial robots: the singular

conﬁguration "extended arm" must always be passed through to move between "elbow up" and "elbow down". This property

is indeed observed on robots that have geometrical simpliﬁcations, such as, parallel or intersecting axes, which is the case of

most industrial robots. Yet, these robots are exceptions. A cuspidal robot is deﬁned as a robot that can move from one solution

to another without ever passing through a singularity. The very ﬁrst mention of a non-singular change of solution was made

in 1988 by Innocenti and Parenti-Castelli from the University of Bologna, who demonstrated this phenomenon numerically

on two different serial robots with six revolute joints [1]. This revelation went unnoticed and was even sometimes considered

fanciful. In fact, the community was convinced that any robot must necessarily cross a singularity to change its solution,

as was the case for all industrial robots at that time. A mathematical proof - which later proved to be false - had even been

produced to conﬁrm this hypothesis [2]. Shortly after the Bologna researchers’ revelation, Burdick (Stanford University,

California) conﬁrmed the Italians’ revelations in his Ph.D thesis, this time on several serial robots with 3 revolute joints. It

was not until 1992 that a new formalism was proposed for serial robots capable of changing solution without passing through

a singularity [3], pointing out an error in the proof produced in [2]. It took many years before the scientiﬁc community began

to accept the existence of the so-called cuspidal robots. The term cuspidal was coined in 1995 with the demonstration of the

existence of a "cusp" point (see Fig. 7) on the locus of the singularities of cuspidal serial robots with three revolute joints [4].

Finally, an exhaustive classiﬁcation of 3R serial robots with mutually orthogonal joint axes was carried out in 2004 [5].

Studies on cuspidal parallel robots were initiated in 1998 by Innocenti and Parenti-Castelli [6] and have been limited to

planar robots [7], [8], [9], [10], [9], [11], [12], [13], [14], [15], [16], with the exception of a 6-degree-of-freedom decoupled

parallel robot [17].

If cuspidality might appear interesting at ﬁrst sight, it may, in fact, produce more difﬁculties than advantages. This

raises several crucial questions, both for the user and the designer. How do we know if a newly designed robot is cuspidal?

If yes, how do we know if the robot has changed its solution during motion? Which robots are cuspidal, or which robots

are noncuspidal? Any robot user should know if its robot is cuspidal or not and, more importantly, any robot manufacturer

should know how to select suitable geometric parameters to design a cuspidal or a noncuspidal robot. Unfortunately, this

arXiv:2210.05204v1 [cs.RO] 11 Oct 2022

Fig. 1: A 3R serial robot (left), a parallel robot with 3 telescopic legs (right).

topic has not attracted a lot of researchers. Several decades after the ﬁrst published work showing cuspidal robots [1], it

turns out that still almost nobody is aware of this feature, as conﬁrmed in a recent report [18]. However, we do think that

this topic is very important and the available knowledge should be disseminated more widely to the robotics community and

even taught in robotic courses. This has motivated us to write this review paper which, for the ﬁrst time, collects all available

results on cuspidal robots, both of the serial and parallel type.

This paper proposes a methodology to identify and classify cuspidal robots and describes in depth their solution changing

mechanism. Particular attention is given to a family of robots with three mutually orthogonal revolute joint axes. These

robots, which can be interesting alternatives to the usual robots, can be classiﬁed into cuspidal and noncuspidal robots on

the basis of the values of their geometric parameters. Robots with six revolute joints are more difﬁcult to analyze and are

still under research at the time of writing this paper. They are discussed through some examples of cuspidal robots, some of

which are used in the industry, such as painting or collaborative robots. The case of parallel robots, which is more delicate,

is also investigated. Examples of parallel planar, spherical and spatial 6-degree-of-freedom robots are analyzed, capable of

changing assembly mode without crossing a singularity.

2 Preliminaries

We recalled in this section some basic deﬁnitions and notions that are useful for the understanding of cuspidal robots.

2.1 Types of robots studied

Two main categories of robots are studied in this paper: serial robots (Fig. 1, left) and parallel robots (Fig. 1, right). A

large part of the paper (sections 3, 4, 5 and 6) is devoted to serial robots, and more particularly to 3R type robots, i.e. those

whose articulated chain is composed of three revolute joints as shown in Fig. 1, left. Besides, all the robots studied in this

paper are non-redundant, i.e. their number of degrees of freedom is equal to the number of coordinates describing the task.

This is the case for the two 3-degree-of-freedom robots of Fig. 1 where the task is deﬁned by three position coordinates in

space.

2.2 Postures and assembly-modes

For a serial robot, the word posture deﬁnes a joint conﬁguration that allows the end-effector to reach a given pose in its

workspace. A posture is thus associated with an inverse kinematic solution.

For common industrial robots, a posture can be easily identiﬁed. For an anthropomorphic robot such as the one in Fig. 2,

for example, these postures can be identiﬁed by the conﬁguration of the elbow (up or down), shoulder (right or left), and

wrist (ﬂip or noﬂip). The total number of combinations amounts to 23=8 distinct postures. Figure 2 shows the four postures

that can be identiﬁed by the elbow and shoulder conﬁgurations.

For a parallel robot, the term assembly mode deﬁnes a solution to the direct kinematics, i.e. a pose of the moving

platform of the robot associated to the given values of actuated joint variables.

Figure 3 shows the six assembly-modes of a planar parallel robot with three telescoping legs. For a given set of leg

lengths (Ai,Bi), the triangular platform (B1,B2,B3)can assume up to six different poses in the plane. This robot will be

analyzed in details in section 6.2.

Fig. 2: The four solutions of an anthropomorphic robot deﬁned by the elbow and shoulder conﬁgurations, while the wrist

conﬁguration is ﬁxed to one of its two conﬁguration. The remaining four postures are deﬁned by the other wrist conﬁguration.

Fig. 3: The six assembly-modes of a planar parallel robot with three telescoping legs.

2.3 Singularities

We brieﬂy recall here the concept of singularity, their role being essential for cuspidal robots. A singularity of a serial

robot deﬁnes a joint conﬁguration from which the robot cannot produce certain instantaneous motions. Moreover, at least

two inverse kinematic solutions coincide when the robot is on a singularity. Figure 4, left shows an anthropomorphic robot

(sketched without its wrist) in a fully outstretched arm singularity: locally, a movement along the direction of the arm is not

possible. In this singularity, the two inverse solutions associated with the solutions “elbow up” and “elbow down” shown

at the right of Fig. 4, coincide. Singularities generate boundaries in the workspace that cannot be crossed, because they are

located at the edge of the workspace. The fully extended arm singularity, for example, deﬁnes the robot’s reachability limit.

Singularity may also deﬁne internal boundaries that can be crossed or not, depending on the robot posture. This important

feature is investigated further in the paper.

We can see that for the robot in Fig. 4, the transition between the two solutions elbow up and elbow down is necessarily

done by crossing the fully outstretched arm singularity.

Parallel robots can have several types of singularities. In this paper, we will only consider parallel robots with "type 2"

singularities, also called "parallel singularities". On these singularities, the parallel robot has certain motions that cannot be

controlled anymore. Moreover, at least two of its assembly modes coincide.

2.4 Aspects

The notion of aspects was initially deﬁned for serial robots by Borrel [2]. We recall the following deﬁnition for non-

redundant serial robots.

Infeasible

motion

elbow up elbow down

Fig. 4: Anthropomorphic robot in a fully outstretched arm singularity (left). The two inverse kinematic solutions “elbow up”

and “elbow down” (right) coincide.

Let Dbe the robot joint space :

D={q|qimin ≤qi≤qimax,∀i∈[i,n]}(1)

where q= [q1,..,qn]Tis the joint vector, i.e. containing the nactuated joint variables.

Let X= [x1,..,xn]be a choice of output coordinates deﬁning the pose of the end-effector. These coordinates are linked to the

joint variables by the kinematic map fdeﬁned by X=f(q), that is, xi=fi(q),i=1,..,n.

The aspects are the largest sets in Ajin Dsuch that :

Ajis path-connected ;

∀q∈Aj,det(J)6=0 where J=h∂fi

∂qi(q)iis the Jacobian matrix of the robot.

Aspects are bounded by singularities and joint limits when they exist. For non-redundant serial robots, the aspects

deﬁned in this way are the largest domains Dfree of any singularity. For any noncuspidal robot, the aspects are the largest

uniqueness domains of the kinematic map f, i.e. there is only one inverse solution in each aspect.

Aspects were later generalized to parallel robots [19], see section 7.1.

As an illustrative example, let us consider a 3R robot as in Fig. 1 (left), whose geometric parameters are: d2=1, d3=2,

d4=1.5, r2=1, r3=0, α2=−90◦and α3=90◦. The dimensions are unit-less, without importance for the understanding

of the example. This robot is called orthogonal because its axes are mututally orthogonal. We also assume that the robot

has no joint limits. This robot will be used in several examples throughout the ﬁrst part of this paper. We show that the

determinant of the Jacobian matrix Jof this robot can be written as a product of two factors (see for example [20]) :

det(J) = (d3+cos(θ3)d4)(cos(θ2)(sin(θ3)d3−cos(θ3)r2) + sin(θ3)d2)

Note that the singularities, deﬁned by det(J) = 0, do not depend on θ1, which is always the case when the ﬁrst joint is

revolute. We can therefore represent the joint space in the (θ2,θ3)plane. The ﬁrst factor of det(J) cannot cancel here because

d4<d3. The second factor deﬁnes two curves of identical shape and translated one with respect to the other by πalong the

θ3axis. The singularities then divide the joint space into two aspects (in green and blue in Fig. 5, left). Indeed, as there are

no joint boundaries, it is necessary to identify the opposite sides of the joint space, which has the topology of a torus.

In the workspace, the singularities deﬁne boundary surfaces. The workspace is 3-dimensional but since the singularities

do not depend on θ1and there is no joint boundary, the workspace is symmetrical around the robot axis 1. It is therefore

sufﬁcient to represent only a half-section in a plane passing through this axis. Such a section, called cross section, can

be deﬁned in the plane (ρ,z)where ρ=px2+y2. The total workspace can be obtained by rotating this section by 360°

around axis 1 of the robot. For the robot under study, the singularities generate two closed curves in the cross section of the

workspace. One deﬁnes the outer boundary, the other deﬁnes the inner boundary. The inner boundary separates a central

region accessible with four solutions, from a peripheral region accessible with two solutions (Fig. 5, right).

Fig. 5: 3R cuspidal robot : the two aspects in the joint space (left, units in radians) and cross section of the workspace (right).

A non-singular solution changing trajectory is shown.

3 Cuspidal robot : deﬁnition, identiﬁcation and properties

3.1 Deﬁnition and example

A robot is said cuspidal if it can move from one solution to another without passing through a singularity. There are

thus several solutions in a single aspect. In other words, this aspect is no longer a uniqueness domain of the kinematic map f.

The name “cuspidal robot” was coined in connection with the existence condition of a particular singular point of

the workspace, called cusp point. We can show that if a cusp point exists, then the robot is cuspidal [4]. The exhaustive

classiﬁcation of orthogonal 3R robots that was done afterwards, shows that the existence of a cusp point is also a necessary

condition for these robots [5], [21]. In January 2022, a mathematical proof that the same is true for all 3R robots has been

established in the framework of a France-Austria project [22]. Note that, unfortunately, this necessary condition does not

hold for any robot, in particular when the robot is parallel (see §6.7).

Acusp is one of two stable singularities formed when a folded surface is projected onto a plane, the other singularity

being a "fold" [23]. These singularities are called stable because they do not disappear under the effect of a small perturbation

of the surface.

Let us take the case of the 3R robot from previous example (d2=1, d3=2, d4=1.5, r2=1, r3=0, α2=−90◦and

α3=90◦).

The point Xof coordinates x=2.5, y=0, z=0.5 in the reference frame (O,x,y,z)(see Fig. 5, right) is reachable with

four solutions corresponding to the following joint conﬁgurations (values in radians) :

q_1 = [−1.8,−2.8,1.9]t,q_2 = [−0.9,−0.7,2.5]t

q_3 = [−2.9,−3,−0.2]t,q_4 = [0.2,−0.3,−1.9]t

The four solutions are depicted in Fig. 6.

From Fig. 5, we see that the conﬁgurations q2and q3belong to the same aspect. Therefore, the passage from one to the

other can be realized without crossing singularities, for example through a linear trajectory (left). The trajectory connecting

q2and q3then generates a loop in the workspace (right).

We can show that three inverse kinematic solutions coincide when the robot is on a cusp point. This property is very

important because it allows us to search for the cusp points as triple roots of the characteristic polynomial of the inverse

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AreviewofcuspidalserialandparallelmanipulatorsPhilippeWengerandDamienChablatNantesUniversité,ÉcoleCentraleNantes,CNRS,LS2N,UMR6004,F-44000Nantes,FranceABSTRACTCuspidalrobotscanmovefromoneinverseordirectkinematicsolutiontoanotherwithouteverpassingthroughasingularity.Theserobotshaveremainedunknownbeca...

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