1

Crossable Surfaces of Robotic Manipulatorswith Joint Limits

Karim Abdel-MalekHarn-Jou Yeh

Department of Mechanical EngineeringThe University of IowaIowa City, IA 52242

Tel. (319) 335-5676 Fax. (319) 335-5669amalek@engineering.uiowa.edu

AbstractA broadly applicable formulation for determining the crossability of singular surfaces and curvesin the workspace of serial robotic manipulators is presented. Singular surfaces and curves areanalytically determined using nullspace rank-deficiency criteria of the mechanism’s constraintposition Jacobian. Imposed joint limits in terms of inequality constraints are taken into accountin the formulation. Directions of admissible normal movements on a surface or curve areestablished from the analysis of the normal curvature of singular surfaces. The normal curvatureof a parametric surface used in this formulation, is determined from the first and secondfundamental forms adapted from differential geometry. Definiteness properties of a quadraticform developed from the acceleration analysis determine admissible normal movements. Forsingular surfaces resulting from active joint constraints, definiteness properties are not enoughand supplementary criteria are necessary. Two additional criteria are derived. This paper is acomplete treatment of the problem of determining whether a singular surface/curve in theworkspace is crossable. Planar and spatial numerical examples are presented to illustrate theformulation.

IntroductionThe problem of determining singular behavior of mechanical manipulators has receivedconsiderable attention. The reciprocal screw method for workspace generation, for example, isbased on the fact that when the hand reference point of the manipulator is on the workspaceboundary when the manipulator has a spherical wrist, all the joint axes of a manipulator arereciprocal to a zero pitch wrench axis (Sugimoto 1982). For each degree-of-freedom (DOF)lost, there exists one reciprocal screw that, if applied as a wrench to the end-effector, producesno virtual work for the manipulator joints. Wang and Waldron (1986), based upon earlier work(Waldron, et al. 1985), stated that as the Jacobian of the manipulator becomes singular, itscolumns, which are screw quantities, do not span the full rank of the matrix, therefore, theJacobian rank decreases by at least one. Proof of the nullspace criteria applied to the Jacobianwas presented by Spanos and Kohli (1985).

Other methods that are based on Jacobian singularity can be found in (Shu, et al. 1983, Litvin, etal. 1986a, and 1986b). An enumeration of singular configurations due to the vanishing of thedeterminant of the Jacobian and the Jacobian’s minors is presented by Lipkin and Pohl (1991).Shamir (1990) provided an analytical tool to determine if the singularities are avoidable for three

Abdel-Malek, K.and Yeh, H.J., (2000), "Crossable Surfaces of Robotic Manipulators withJoint Limits", ASME Journal of Mechanical Design, Vol. 122, No. 1, March 2000, pp. 52-61.

2

DOF manipulators. Gorla (1981) was able to get expressions for the set of singular points byassuming that link twists were multiple of π /2. Geometric approaches to the study of singularconfigurations of a manipulator arm were addressed by Lai and Yang (1986), Ahmad and Luo(1988), and Tourassis and Ang (1992). Other earlier important studies that discussedmanipulator singularities include Soylu and Duffy (1988) and Lai and Yang (1984).

Early studies that have addressed difficulties in the control of manipulators due to theappearance of interior curves and surfaces were reported by Waldron (1987) and Nielsen, et al.(1991). In the latter work, the controllability of a mechanical arm is discussed from a differentialgeometry point of view. Nielsen, et al. showed control difficulties based upon vector fields andtheir consecutive Lie brackets spanning a state space. A criterion to define possible motion (so-called feasible trajectory) from a singularity was presented by Chevallereau and Daya (1994) andChevallereau (1996).

The fundamental concept of crossable and noncrossable surfaces inside a manipulator’sworkspace was addressed by Oblak and Kohli (1988). Although their report has touched upon acrossable surface using the singularity criteria of the Jacobian, the paper does not present aunified method for identifying a crossable surface or curve, neither does it determine directionsof admissible movement. Further work on the issue of singularity determination and thestructure of singular regions inside the workspace was presented by Pai and Leu (1992),Burdick (1991; 1992), Karger (1995), and Merlet (1989).

Recently, Haug, et al. (1995; 1996) presented a numerical algorithm for identifying andanalyzing barriers to output control of manipulators using first and second-order Taylorapproximations of the output in selected directions. Haug, et al. showed that the outputvelocity in the direction normal to such curves and surfaces must be zero. The work wasextended to the investigation of the domains of mobility for a planar body moving amongobstacles (Haug, et al. 1998) and to the study the workspace (called operational envelope) ofthe Stewart platform (Adkins and Haug 1997). It should be noted that this method iscompletely numerical, yields only curves on the boundary that must be manually meshed todepict the envelope. Similar work based on the Euler-Rodrigues parameters and aimed atdetermining the workspace of serial and platform manipulators through the occurrence ofsingular behavior (Bulca, et al. 1999).

Recently, Abdel-Malek and Yeh (1997a) presented a general method by which all singularsurfaces and curves can be analytically determined for a 3DOF serial manipulator and henceobtained a closed form of the surface patches enveloping the workspace. The method andalgorithm were extended to the concept of swept volumes whereby the volume generated by themotion of a geometric entity in space is numerically determined (Abdel-Malek and Yeh 1997b)and was shown to apply to solid modeling. The concept of singular curves and surfaces wasthen used to determine feasible path trajectories where the end-effector can cross a given barrier(Abdel-Malek and Yeh 1997c). The formulation therein was compared with the results of thenumerical method by Haug, et al. (1996) in a joint paper (Abdel-Malek, et al. 1997). The sametheory implemented for the determination of the serial 5-DOF manipulator was presented by

3

Abdel-Malek, et al. (1999) and reported singular surfaces that are boundary to the workspacebut that can be further stratified.

We first present a rigorous formulation for identifying singular curves and surfaces internal andon the boundary to the workspace envelope. We then adapt methods from differential geometryto investigate properties associated with a proposed acceleration function on these curves andsurfaces. A number of examples are presented to demonstrate the theory and formulation.

FormulationIt has been shown (Abdel-Malek and Yeh 1997a) that singular surfaces/curves comprising theworkspace of serial robotic manipulators, can analytically be determined by studying the analyticJacobian of the mechanism. For a position vector x q= F( ) , where q R∈ n is the vector ofgeneralized coordinates of an n-DOF manipulator, the dimension of the nullspace of the

Jacobian Fq = ∂Φ ∂i jq is associated with singular behavior. Define the subvector of q as

p Rim= , where m n≤ − 1 such that q u p≡ ∪ i , where u R∈ −n m .

Generalized coordinates called singular sets, are determined from

S p q q u p pq= ≥iT

i iNull;dim ( ( )) ,F 1 = , for some > C (1)

where pi are joint variables that are either constant values of joints or defined as functions ofother variables. The set of analytic functions generated by a subset of a Euclidean space definedby zeros of a finite number of differentiable functions (sub-determinants) are called varieties(singular sets). The nullspace of the Jacobian corresponds to joint velocities that do not produceany end-effector velocity for a given manipulator configuration pi . Singular configurations pi

are obtained from S and from the reduced-order manipulator condition discussed in Abdel-Malek, et al. (1999), which occurs when joint constraints are active. It was shown thatsubstituting pi into F( )q yields singular surfaces and curves parameterized by Y( )u where u isthe new vector of generalized coordinates containing the remaining variables such that

F Y( ,u p ui ) ( )= (2)

A proof was presented showing that the basis of the null space of FqT is a normal vector to

singular surfaces/curves in the workspace. i.e., if a vector no evaluated at qo satisfies

Fq n 0Tο = (3)

where nο = γ γ γ1 2 3

T, it is the vector normal to singular surfaces/curves at qo . This

important result was used in the velocity analysis of the end-effector on singular surfaces/curvesto show that the end-effector on a singular surface may have only tangential velocities.Acceleration analysis of the end-effector revealed a quadratic form that can be used to establishpossible directions of admissible movements. The normal acceleration is derived as

an oT T

oT= =n x q n q

qq&& & &F (4)

where &&x is the acceleration of the end-effector, &q is the vector of joint velocities and two

subscripts indicate a second derivative of noTF . Definiteness properties of the quadratic form

4

in Eq. (4) predict the direction of admissible normal acceleration, but do not guarantee whetherthe end-effector can admit movement across a singular surface/curve (crossability analysis).

Consider the end-effector at a point on a singular surface with radius of curvature ρ o , with aspecified normal acceleration an , and a tangential velocity vt . This manipulator will admitmotion in one normal direction or another subject to

av

n

t

o

−2

ρ(5)

where vt = =& &x qqF is the tangential velocity and 1 ρo is the normal curvature of the singular

surface with respect to the tangent direction of vt ( ρ o is the radius of curvature). The sign of(5) above establishes the admissible direction of motion.

From the theory of differential geometry (Farin 1993), the First and Second FundamentalForms (denoted by I p and II p , respectively) of a parametric geometric entity Y( )u , where

u = u vT

, is defined as

I u uu upT T≡ δ δY Y (6)

where Y Yu u= ∂ ∂ and Y Yv v= ∂ ∂ , and

II u n uuup

T T≡ δ δY (7)

or expanded asII n n np

Tuu

Tuv

Tvvdu du dv dv= + +Y Y Y

2 22 (8)

where n is the normal vector to the singular surface and Y Yuv u v= ∂ ∂ ∂2 , Y Yvv v v= ∂ ∂ ∂2 , and

Y Yuu u u= ∂ ∂ ∂2 . The Normal Curvature Ko of a parametric singular surface at a configurationqo , can then be defined as the ratio

Koo

p

p

= =1

ρII

I(9)

Define the time-Modified First and Second Fundamental Forms as′ ≡I u uu up

T T& &Y Y (10a)

II u n uuu

′ ≡pT T

& &Y (10b)

such that the normal curvature can still be defined as

Koo

p

p

p

p

= = =′′

1

ρII

I

II

I(11)

The tangent velocity in terms of Y or F can be written asY Fu qu q& &= (12)

Hence, it can be shown that the time-Modified First Fundamental Form is indeed the square ofthe tangential velocity

′ =I p tv2

(13)

Therefore, the difference in acceleration components can be written as

5

av

a vnt

on t

p

p

− = −′′

22

ρII

I

= − ′an pII (14)

Since an is in terms of &q and to express II′p in terms of &q , it was shown that the velocity

vector on a singular surface can be written as

& &u E E qu q= −Y F

1(15)

where

E =�!

"$#

1 0 0

0 1 0 if the first and second rows of Yu are independent (16A)

E =�!

"$#

1 0 0

0 0 1 if the first and third rows of Yu are independent (16b)

E =�!

"$#

0 1 0

0 0 1 if the second and third rows of Yu are independent (16c)

For an end-effector on a singular surface, the crossability criteria was then expanded into aquadratic form written as

av

an

t

on p

T− = − ′ =2

ρII q Qq& & (17a)

where

Q n B n Bqq q uu q= −T T T T

F F Y F (17b)

and B is a generalized inverse of Yu defined in Eq. (15) by

B E Eu≡ −Y

1(18)

Definiteness properties of the quadratic form in Eq. (17) indicate the crossability of a singularsurface/curve. Since singular surfaces/curves that are non-crossable are impediments to motion,controllability issues are directly related to this formulation.

In this paper, we extend the formulation to manipulators with joint limits. The formulationincludes joint parameters that are subject to mobility constraints. Since additional singularsurfaces/curves may appear in the workspace due to active joint constraints, velocities of thejoints that reached their limits on those surfaces are nulled. The nullification of joint velocitieswill give rise to an apparent loss of information from the matrix of the quadratic form (Eq. 17).Therefore, additional (supplementary) criteria are developed to determine admissible normalmovements on those surfaces/curves.

Extending the Formulation to Include Joint LimitsJoint limits imposed in the form of inequality constraints such as q q qi i i

min max≤ ≤ , areparameterized into an equality using the following transformation such that new generalizedcoordinates λ i are introduced

q a bi i i i= + sin λ (19)

6

where ( )a q qi i i= +max min 2 and ( )b q qi i i= −max min 2 are the mid point and half range of the

inequality constraint (Haug, et al. 1994). The constraint function is then written in terms of the

extended vector s = λ λ λ1 2 ... n

T such that

x s q= F F( ) ( ( ))= λ i (20)

Differentiating with respect to time and using the chain rule, the velocity of the end-effector is& &x q sq s= F (21)

where qs = ∂ ∂λqi j and &s s= d dt . On a singular surface, the term Fq s q sq

o o, is rank-deficient.

Therefore, the rows of Fq sq are dependent and there exists a set of constants

no

Tn n n= 1 2 3 that satisfy

Fq sq n 0T

o = (22)

where no is the vector normal to a singular surface at ( qo , so ). In addition, the normalacceleration of the end-effector can be written as:

&& ( ) ( ) & ( ) ( ) & ( )( )&&xd

dt q

q

q

d

dt

q

q

qi

i

j

j

jj

i

j

j

jj

i

j

j

jj

j

n

=�!

"$##

∂∂

+∂∂

�!

"$##

+∂∂

%&K'K

()K*K=

∑ ∂Φ∂ λ

λ ∂Φ∂ λ

λ ∂Φ∂ λ

λ1

(23)

where d

dtq q q qi j i j( ) ( )( ) &∂Φ ∂ ∂ ∂ ∂ ∂ ∂λ λ≡ 2Φ

l l l l

where l=1,...,n and d

dtq qj j j j( ) ( ) &∂ ∂λ ∂ ∂λ ∂λ λ�

2l l

Expanding terms in the form of joint rates and collecting on similar terms yields

&&

& ( )( )( ) ( )( ) & ( )( )&&xdq

d q q

q

q

q

q

qi k

k

k

i

k j

j

k

i

j

j

k jj

i

j

j

jj

j

n

= ∂∂ ∂

∂∂

+ ∂∂

∂∂ ∂

�!

"$##

+ ∂∂

∂∂

%&K'K

()K*K=

∑ λλ λ λ λ

λλ

λ2 2

1

Φ Φ Φ(24)

where k n=1,..., . Written in matrix form, the acceleration is

&&

& & & & &&x q q qqq qi

T T

iT i

ii

i

n

i

d

dqq=

���

���=

∑l l l ll l ll l

Φ Φ Φλ + +1

(25)

where [ ]

...

. . .

...

ΦΦ Φ

Φ Φi

i i n

i n i n n

q q q q

q q q qqq =

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

�

!

"

$###

1 1 1

1

and [ ]

...

. . .

...

q

q q

q qi

i i n

i n i n n

ll=

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

�

!

"

$###

λ λ λ λ

λ λ λ λ

1 1 1

1

.

To obtain the normal acceleration, &&x is projected onto the normal no . Multiplying both sides of

Eq. (25) by the vector noT eliminates the last term of the right hand side (definition of the normal

as the basis of the null space in Eq. 22).

The component of the normal acceleration is thenan o

T T= =n x s H s&& & &

* (26)where

H q s q n qn

s qq s ss*( , )

(o o

ToT o

T

ii

i

n d

dqq= + ⋅

=∑F

F)1

(27)

7

In the formulation above (not considering joint limits), definiteness properties of Eq. (17)depended on the Q matrix. In this extended formulation (considering joint limits), the quadraticform is written in terms of the Q* matrix as

anT− ′ =II s Q sp & &

* (28)

where Q H q B n B qs q uu q s* = −∗ T T T T

F Y F (29)

Definiteness properties of the quadratic form of Eq. (28) defines the crossability of a singularsurface/curve.

Singular Surfaces Due to Active Joint ConstraintsFor a singular surface that is due to singularities that are at the upper or lower bounds of aninequality constraint, the surface will definitely be crossable if Q* has both positive and negative

eigenvalues. However, when Q* is either positive semi-definite or negative semi-definite, thesingular surface/curve may still be crossable. On those surfaces, the matrix of the quadraticform loses some terms because joint velocities are zero. Joint velocities at a singular

configuration derived from Eq. (19) (i.e., & cosmax minq q qi i i i= − 2 λ ) are zero at an upper or

lower bound ( λ π πi = − 2 2 or , i.e., q qi i= max or q qi i= min ). Therefore, some terms in Q*

are lost. Supplementary criteria are necessary. To motivate the discussion, consider the twoDOF (one prismatic and one revolute) planar manipulator shown in Fig. (1a) with joint limitsgiven by 0 41≤ ≤q and ( / )− ≤ ≤π π4 2q .

q1

q2

n1

n2

Ψ2

p2

p1

Ψ1

(a) (b)

Revolutejoint

Prismaticjoint

(c) (d)

Ψ3

Ψ6

Ψ4

Ψ1Ψ5

Ψ2

Figure 1. (a) A two DOF manipulator and (b) Some singular curves due to joint limits (c) Allsingular curves (d) Workspace envelope

Since q2 has its lower and upper bounds as − π 4 and π , the lines Y1 and Y2 are the twocorresponding singular curves. For the two cases when the end-effector is at points p1 and p2

as shown in Fig. (1b), the joint velocity of q2 will always be zero (i.e., & cos( )q2 5 8 2= −π π1 6 ).

Therefore, there will not be any component of normal acceleration at point p1 nor at p2 .

Definiteness properties of Q* are, therefore, not sufficient to determine all possible directions ofmotion for these two cases. It should be noted that determination of the workspace is readilycarried out first by plotting all singular curves (Fig. 1c) and identifying the boundary (Fig. 1d).

8

To address this case, we propose the projection of a variational movement δ δx x= q iiq due to

δqi onto the normal direction n such that the normal component

σ δ= n xTq ii

q (30)

determines admissible normal movement, where δqi is given a magnitude of ±1 as follows

δqq

qii

i

=+−%&'

1

1

if is at lower bound

if is at upper bound(31)

Positive values of σ in Eq. (30) indicate that the end-effector can admit movement in thepositive direction of n.

Thus, for a singular surface that is due to an active joint constraint, Eq. (30) providessupplementary information in lieu of terms lost due to the enforced zero joint velocity. Forexample, consider the case at point p1 again. Since Q* at this point has only zero eigenvalues,it is necessary to use Eq. (30) with ∂q2 1= + since joint 2 is at the lower limit. Equation (30)yields a positive value, which indicates that the end-effector admits motion into the positivedirection of n1 .

Another situation arises when the normal vector n is perpendicular to xqi. In this case, σ in

Eq. (30) evaluates to zero. Therefore, the direction of normal movement cannot be determined.The normal curvature of the singular surface with respect to the tangent direction of xq ii

qδ is

compared with the normal curvature of the trajectory curve when only qi is varying (other jointvariables are held constant).

To determine the normal direction of admissible movement, it is necessary to evaluate the signof the difference in the normal curvature. Substituting for the first and second fundamental forms(Eqs. 6 and 7) into the normal curvature (Eq. 9) yields

Κ Κ2 1− = −δ δ

δ δδ δ

δ δ

q q

q q

iT

q q i

i qT

q i

T T

T Ti i

i i

n x

x x

u n u

u uuu

u u

Y

Y Y(32)

where Κ1 is the normal curvature of Y2 with respect to the tangent direction xq iiq∂ , and Κ 2 is

the normal curvature of the trajectory curve when qi is varying while other joints are heldconstant.

The case at point p2 in Fig. (1b), for example, exhibits a zero normal curvature of Y2 . Thecurvature of the curve due to varying only q2 is positive with respect to the normal vector n2 .Therefore, the end-effector can move into the side of positive direction of n2 by decreasing q2 .

Both curves of the right-hand side of Eq. (32) must have the same tangent vector given by Eq.(12) such that

Yu u xδ δ= q iiq (33)

9

where u is the vector of parametric coordinates for the corresponding singular surface/curve.To simplify the calculation, δqi can be determined from Eq. (31).

Rearranging Eq. (33) and using the inverse defined in Eq. (18), the vector δu can be written asδ δu Bx= q ii

q (34)

Substituting Eq. (34) into Eq. (32) and simplifying yields

Κ Κ2 1− =−δ δ δ δ

δ δ

q q q q

q q

iT

q q i i qT T T

q i

iT

qT

q i

i ii i

i i

n x x B n x Bx

x xuu (35)

Since the denominator of the right-hand side of Eq. (35) is always positive, the sign is dependenton the numerator. Define the numerator as µ where

µ δ δ δ δ= −q q q qiT

q q i i qT T T

q ii i

i in x x B n x Bx

uu(36)

which can be written as

µ δ= −n x x B n x Bxuu

T

q q qT T T

q ii i

i iqJ L 2 (37)

Since δqi2 is always positive, the sign of Κ Κ1 2− depends on the term inside the brackets.

Define

µ δ= K qi2 (38)

where

K qT T T

qT

q qi ii i

= −x B n x Bx n xuu

(39)

Therefore, if K > 0 , the end-effector can admit movement into the positive direction of n . If

K < 0 , the end-effector admits movement into the negative direction of n .

From the above discussion, it can be seen that Eq. (30) and (38) are used to supplement thecriteria of the definiteness properties of Q* to determine crossability of a singular surface withjoint limits. Crossability criteria are summarized as follows.

Crossability Criteria(1) If Q* is indefinite (has both positive and negative eigenvalues), the singular surface/curve

is crossable for each joint qi which is at its limit.

(2) If Q* is either positive semi-definite or negative semi-definite, the following additionalcriteria must be evaluated.(a) If no is not perpendicular to xqi

, then σ in Eq. (30) must be evaluated

(b) If no is perpendicular to xqi, then K in Eq. (38) must be evaluated.

(i) If any of σ or K has a different sign than the nonzero eigenvalues of Q* , thesingular surface/curve is crossable.

(ii) If σ and K have the same sign as the nonzero eigenvalues of Q* , the singularsurface/curve is non-crossable.

The direction along which the end-effector can move, either in the positive or negative

direction of n, will be in the same sense of signs of σ and K .

10

Example 1: A Planar 3-bar LinkageThis example is presented to demonstrate the implementation of the presented formulation andto validate the results in comparison with those reported by Haug, et al. (1994) for the planar

three-bar linkage shown in Fig. 2 with the following imposed limits − ≤ ≤π π3 3

qi ; i = 1 3... .

x

y

4

21

q1

q2q3

Figure 2. A planar three DOF manipulatorThe position vector of a point on the end-effector is defined by

F( )cos cos( ) cos( )

sin sin( ) sin( )q =

+ + + + ++ + + + +

�!

"$#

4 2

4 21 1 2 1 2 3

1 1 2 1 2 3

q q q q q q

q q q q q q(40)

The ( )2 3× analytic Jacobian matrix is computed as

Fq =− − + − + + − + − + + − + +

+ + + + + + + + + + +�!

"$#

4 2 2

4 2 21 1 2 1 2 3 1 2 1 2 3 1 2 3

1 1 2 1 2 3 1 2 1 2 3 1 2 3

sin sin( ) sin( ) sin( ) sin( ) sin( )

cos cos( ) cos( ) cos( ) cos( ) cos( )

q q q q q q q q q q q q q q

q q q q q q q q q q q q q q (41)

and qs =

�

!

"

$###

( ) cos

( ) cos

( ) cos

2 6 0 0

0 2 6 0

0 0 2 6

1

2

3

π λπ λ

π λ

.

Only one singular set results from the dim[ ( ( ))]Null TFq q ≥ 1 defined by p1 2 30 0= = ={ , }q q .

Singularities resulting from the reduced order manipulator (Abdel-Malek and Yeh 1997a) areS ii2 2 7= =p , ,...,; @ where p2 1 33 0= = − ={ , }q qπ , p3 1 33 0= = ={ , }q qπ ,

p4 2 313 3 2= = − = −{ , tan ( )}q qπ , p5 2 3

13 3 2= = = − −{ , tan ( )}q qπ ,

p6 3 213 3 5= = − = −{ , tan ( )}q qπ , p7 3 2

13 3 5= = = − −{ , tan ( )}q qπ . Singularity sets resulting

from a combination of joint limits are pi io

joq q i j i j= ≠ =, , , , ...= B> C 8 19 where qi

o and q jo are

the limits of the inequality constraints. Singular curves are shown in Fig. (3a).

11

x

Y

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1.02.03.04.0

Figure 3. (a) Singular curves of the planar 3-bar linkageSubstituting pi ’s into Eq. (40) yields singular curves Yi part of which are shown in Fig. (3b).

Ψ1

Ψ3

Ψ9Ψ11 Ψ18

Ψ5

Ψ16

Ψ19

Ψ10

Ψ4

Ψ17Ψ6

x1.0 2.0 3.0 4.0 5.0 6.0 7.0

1.0

2.0

3.0

4.0

y

Figure 3. (b) Identifying singular curvesTo determine crossable curves, it is necessary to evaluate the criteria above for each independentsegment, i.e., curves that cross other curves are segmented. For example, consider a point p1

on curve Y19 at q1 4= π , q2 3= π , and q3 3= π . Evaluating the normal n from the basis ofthe null space of

Fq sqT

=−�

! "$#

3 4765 0 0

0 9315 0 0

.

.

at p1 yields the normal n = 0 2588 0 9659. .T

. Evaluating the matrix H* of Eq. (27) yields

H*

.

.

.

=−�

!

"

$###

2 4930 0 0

0 2 0944 0

0 0 10472

12

Since this is a planar example, the singular curve is parameterized by only one joint variable (i.e.,Y19 1( )q , therefore, xq1

is a ( )2 1× matrix. The matrix E is therefore a E = 1 0 vector and

B E Eu= = −−Y

101992 0. . The matrix Q* of the quadratic form is evaluated as

Q* .

.

=�

!

"

$###

0 0 0

0 2 094 0

0 0 1047

The eigenvalues of Q* are evaluated 0 1047 2 094, . , . and ; @ , which indicates a positive semi-

definite quadratic form. Since the constraints of joints 2 and 3 are active (upper bounds) andxq2

and xq3 are not perpendicular to n, the additional value of σ is evaluated. For q2 at its

upper limit, δq2 1= − , and σ δ= =n xTq q

2 2 2 which is positive. For q3 , choose δq3 1= − and

evaluate σ δ= =n xTq q

3 3 1 which is also positive. Both values for σ are positive (same signs as

the eigenvalues of Q* ). Therefore, surface Y19 only admits movement into the positivedirection of the vector normal n (i.e., only towards the internal of the workspace) as expected.

To further demonstrate, consider point p2 on Y6 at q1 8= π , q21 3 5= −tan ( ) , and

q3 3= −π . The matrix Q* of the quadratic form is indefinite (both negative and positiveeigenvalues) indicating a crossable curve at p2 . Similar analysis is performed for all curvesegments. For example, the curve Y8 is crossable for the segment of 0 7137 33. ≤ ≤q π ; non-crossable for the segment 0 0 71373≤ ≤q . ; and crossable for − ≤ ≤π 3 03q . The curve Y9 isnon-crossable for the segment of 0 33≤ ≤q π ; crossable for the segment − ≤ ≤0 7137 03. q ; andnon-crossable for − ≤ ≤ −π 3 0 71373q . . Crossable curves are shown dotted in Fig. (4a) whilenon-crossable lines are shown solid. Directions of admissible motion are shown by arrows oneach curve segment in Fig. (4b).

x

y

x

y

Figure 4. (a) Singular curves (b) Admissible normal movement directions

13

Example 2: A Four DOF ManipulatorThe manipulator shown in Fig. 5 comprises two revolute and two prismatic joints. Joint limitsare imposed as 0 21≤ ≤q π , 20 502≤ ≤q , ( / )− ≤ ≤π π4 3q , and 10 204≤ ≤q .

x0, x1

z1

z3

z0

q1

q2q3

z4

q4

Figure 5. A four DOF manipulator with prismatic joints

For a point on the end-effector, the position vector is

F( ) cos cos cos sin cos sin sinq = + + +q q q q q q q q q q qT

4 1 3 1 4 1 3 1 4 3 230 30 (42)

Inequality constraints are parameterized as q1 1= +π π λsin , q2 235 15= + sin λ ,

q3 3

3

8

5

8= +π π λsin , and q4 415 5= + sin λ . The Jacobian is derived as

Fq =− − −

+�

!

"

$###

q q q q q q q q q

q q q q q q q q q

q q q

4 1 3 1 4 1 3 1 3

4 1 3 1 4 1 3 1 3

4 3 3

30 0

30 0

0 1

sin cos sin cos sin cos cos

cos cos cos sin sin sin cos

cos sin

(43)

The rank-deficiency criteria of the Jacobian yields no singular sets since singularities obtainedfrom the determinants of the sub-Jacobians do not satisfy the inequality constraints. Thereduced-order manipulator set (substituting a joint limit value) yields four singular sets asp1 3 40 10= = ={ , }q q , p2 3 4 10= = ={ , }q qπ , p3 3 40 20= = ={ , }q q , p4 3 4 20= = ={ , }q qπ .

Other singular sets result from active joint constraints: p5 2 320 4= = = −{ , / }q q π ,

p6 2 320= = ={ , }q q π , p7 2 420 10= = ={ , }q q , p8 2 420 20= = ={ , }q q p9 2 350 4= = = −{ , / }q q π ,

p10 2 350= = ={ , }q q π , p11 2 450 10= = ={ , }q q , p12 2 450 20= = ={ , }q q , p13 3 44 10= = − ={ / , }q qπ , and

p14 3 44 20= = − ={ / , }q qπ . Substituting each singularity set into Eq. (42) yields parametric

equations of singular surfaces in R 3 . Figure 6 is a cross-section of the workspace volumedepicting all singular surfaces.

14

-50

-25

0

25

500

20

40

0

20

40

60

0

20

40

0

20

40

60

Figure 6. A cross-section of the workspace depicting all singular surfacesFigure 7a depicts a cross section (a longitudinal slice) due to q1 2= π / . To study thecrossability of singular surfaces, consider surface Y11 . Let p1 be a point on Y11 at q1 2= π / ,q2 50= , q3 3 4= π / , and q4 10= . The vector normal at p1 is calculated using Eq. (22) as

n = − −0 2 2 2 2/ /T

. The Eigenvalues of Q* are 0 0 10 6 50, , . , . −; @ . Since the

quadratic form is indefinite, the singular surface is crossable at this point.For another point p2 on surface Y11 at q1 2= π / , q2 50= , q3 6= −π / , and q4 10= . The

vector normal is n = −0 3 2 1 2/ /T

. Eigenvalues of Q* are 0 0 7 5 50, , . , . -−; @ . Since

the quadratic form is negative semidefinite, the additional criteria should be evaluated. For thissurface, the joint variables q2 and q4 are at their limits. The additional criteria are also negativeindicating a non-crossable surface at this point. The end-effector can move only in the directionof the negative normal vector. In fact, the region of surface Y11 for which − ≤ ≤π / 4 03q , isnon-crossable. The region for which 0 3≤ ≤q π is crossable. Similarly, for surface Y12 , theregion for which − ≤ ≤π / 4 03q is crossable and for which 0 3≤ ≤q π is non-crossable. Figure7b depicts crossable (solid) and noncrossable (dotted) traces of surfaces of a cross-section of theworkspace. Arrows indicating admissible normal movements are shown in Fig. (7c). An arrowpointing in one direction indicates the admissible motion of the end-effector in that direction.

15

Ψ12

Ψ7

Ψ8Ψ2Ψ4

Ψ10

Ψ11

Ψ3

Ψ9

Ψ6

Ψ13

Ψ1

Ψ5

Ψ14

Z0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

10.0 20.0 30.0 40.0 50.0

Y0

Z0

10.0

20.0

30.0

40.0

50.0

60.0

10.0 20.0 30.0 40.0 50.0

Y0

10.0 20.0 30.0 40.0 50.0

Y0

Z0

10.0

20.0

30.0

40.0

50.0

60.0

Figure 7. (a) A cross-section of the workspace at q1 2= π / , (b) crossable and noncrossablesurfaces, and (c) Admissible directions of normal movement

ConclusionsThe formulation presented in this paper and illustrated using planar and spatial manipulators withjoint limits, demonstrates a complete solution to the problem of determining singular behavior inthe workspace of open-chain mechanisms. New results regarding the admissibility of movementof the end-effector across singular surfaces are presented. Validation of these

16

It was shown that the crossability of a singular surface can be determined from the definitenessproperties of a quadratic form based on acceleration analysis. It was also shown that singularsurfaces that are due to active joint constraints require additional criteria to determine theircrossability. The additional criterion is needed because of the loss of information in thequadratic term due to zero enforced joint velocity on the singular surface. Two supplementarycriteria were derived based on the direction of the normal curvature for surfaces that have activeconstraints. Two conditions were shown to exist. The first is based upon the projection of adifferential motion on the normal to the singular surface. The second method is based uponcomparing the normal curvatures of the singular surface with that of a trajectory due to thevariation of the joint that reached the limit.

The planar and spatial examples introduced represent the first complete treatment of crossabilityanalysis of singular surfaces in an analytical formulation. Although control problems andmanipulability measures are a natural extension of this work, they have not been addressed inthis paper, but are the subject of current studies.

AcknowledgmentsThe authors gratefully acknowledge the support of the Department of Defense under grantnumber DAAE07-94-C-R094 and the Old Gold Award from the University of Iowa.

ReferencesAbdel-Malek, K and Yeh, HJ, 1997a, “Analytical Boundary of the Workspace for General Three

Degree-of-Freedom Mechanisms,” International Journal of Robotics Research. Vol. 16,No. 2, pp. 198-213.

Abdel-Malek, K and Yeh, HJ, 1997b, “Geometric Representation of the Swept Volume UsingJacobian Rank-Deficiency Conditions,” Computer-Aided Design, Vol. 29, No. 6, pp. 457-468.

Abdel-Malek, K, Adkins, F, Yeh, HJ, and Haug, EJ, 1997, “On the Determination of Boundariesto Manipulator Workspaces,” Robotics and Computer-Integrated Manufacturing, Vol. 13,No. 1, pp.63-72.

Abdel-Malek, K, and Yeh, HJ, 1997c, “Path Trajectory Verification for Robot Manipulators in aManufacturing Environment” IMechE Journal of Engineering Manufacture, Vol. 211,part B, pp. 547-556.

Abdel-Malek, K, Yeh, HJ, and Khairallah, N, 1999 “Workspace, Void, and VolumeDetermination of the General 5DOF Manipulator, Mechanics of Structures and Machines,Vol. 27, No. 1, pp. 91-117.

Ahmad, S. and Luo, S. 1988. Analysis of kinematic singularities for robot manipulators inCartesian coordinate parameters. Proceedings of the IEEE Int. Conf. on Rob. and Autom.,Philadelphia, PA, 840-845.

Allgower, E. L. and Georg, K. 1990. Numerical Continuation Methods: An Introduction.Springer-Verlag, Berlin.

Bulca, F.; Angeles, J.; Zsombor-Murray, P.J., 1999, “On the workspace determination ofspherical serial and platform mechanisms”, Mechanism & Machine Theory, v34 n3, pp.497-512.

17

Burdick, J. W. 1995. Recursice method for finding revolute-jointed manipulator singularities.ASME J. Mech. Des., 117:55-63.

Burdick, J.W., 1991, “A Classification of 3r regional manipulator singularities and geometries”,Proceedings of the IEEE International Conference on Robotics and Automation,Sacramento, CA, pp. 2670-2675.

Burdick, J.W., 1992, “A recursive method for finding revolute-jointed manipulatorsingularities”, Proceedings of the IEEE International Conference on Robotics andAutomation, Nice, France, pp. 448-453.

Chevallereau, C. and Daya, B. 1994, “A new method for robot control in singular configurationswith motion in any Cartesian direction”, Proceedings of the IEEE International Conferenceon Robotics and Automation, Vol. 4, San Diego, CA, pp. 2692-2697.

Chevallereau, C., 1996, “Feasible trajectories for a non redundant robot at a singularity” Proc.of IEEE International Conference on Robotics and Automation, Minneapolis, MN, pp.1871-1876.

Cugini, U. Radi, S. and Rizzi, C. 1994. A system for parametric surface intersection. inComputer-Aided Surface Geometry and Design, The Mathematics of Surfaces (A.Bowyer), Oxford University Press.

Denavit, J. and Hartenberg, R.S., 1955. A kinematic notation for lower-pair mechanisms basedon matrices. Journal of Applied Mechanics, 77:215-221.

Farin, G. 1993. Curves and Surfaces for Computer-Aided Geometric Design, Academic Press,Inc, San Diego, CA.

Fu, K. S. Gonzalez, R. C. and Lee, C. S. G. 1987. Robotics: Control, Sensing, Vision, andIntelligence, McGraw-Hill, Inc., New York.

Gorla, B. 1981. Influence of the control on the structure of a manipulator from a kinematic pointof view. Proc. 4th Symp. Thoery and Practice of Rob. Manipulators, Zaborow, Poland,30-46.

Haug, E. J. Adkins, F. A. Qiu, C. C. and Yen, J. 1995. Analysis of barriers to control ofmanipulators within accessible output sets. Proceedings of the 20th ASME DesignEngineering Technical Conference, Boston, MA, 82:697-704.

Haug, E. J. Luh, C. M. Adkins, F. A. and Wang, J. Y. 1994. Numerical algorithms for mappingboundaries of manipulator workspaces. Advances in Design Automation, ASME DE69(2):447-459.

Karger, A. 1995. Classification of robot-manipulators with only singular configurations.Mechanism and Machine Theory, 30: 727-736.

Kumar, A. and Waldron, K. J. 1981. The workspaces of a mechanical manipulator. ASME J. ofMech. Des., 103(3):665-672.

Lai, Z. C. and Yang, D. C. H. 1984. On the singularity analysis of simple six-link manipulatorsASME Paper No. 84-DET-220.

Lai, Z. C. and Yang, D. C. H. 1986. A new method for the singularity analysis of simple six-linkmanipulators. Int. J. of Rob. Res., 5(2):66-74.

Lipkin, H. and Pohl, E. 1991. Enumeration of singular configurations for robotic manipulators.ASME J. Mech., Trans., and Autom. in Des., 113:272-279.

Litvin, F. L. Fanghella, P. Tan, J. and Zhang, Y. 1986. Singularities in motion anddispplacement functions of spppatial linkages. ASME J. of Mech., Trans., and Autom. inDes., 108(4):516-523.

18

Litvin, F. L. Yi, Z. Castelli, V. P. and Innocenti, C. 1986. Singularities, configurations, anddisplacement functions for manipulators. International Journal of Robotics Research,5(2):52-65.

Merlet, J. P. 1989. Singular configurations of parallel manipulators and Grassman geometry.Int. J. of Rob. Res., 8:45-56.

Nakamura, Y., 1991, Advanced Robotics: Redundancy and Optimization, Addison-WesleyPublishing Company.

Nielsen, L. deWitt, C. C. and Hagander, P. 1991. Controllability issues of robots in singularconfigurations. Proceedings of IEEE Int. Conf. on Rob. and Autom., Sacramento, CA.

Oblak, D. and Kohli, D., 1988. Boundary surfaces, limit surfaces, crossable and noncrossablesurfaces in workspace of mechanical manipulators. ASME Journal of Mechanisms,Transmissions, and Automation in Design, (110):389-396.

Pai, D. K. and Leu, M. C. 1992, “Generecity and singularities of robot manipulators. IEEETransactions on Robotics and Automation, Vol. RA-8, pp. 545-559.

Pai, D. K. and Leu, M. C. 1992. Generecity and singularities of robot manipulators. IEEETrans. on Rob. Autom., 8:545-559.

Sciavicco, L. and Siciliano, B. 1996. Modeling and Control of Robot Manipulators. McGraw-Hill, Inc., New York.

Sefrioui, J. and Gosselin, C. M. 1994. Determination of the singularity loci of spherical three-degree-of-freedom parallel manipulators. Mechanism and Machine Theory, 30:559-579.

Sefrioui, J. and Gosselin, C. M. 1995. On the quadratic nature of the singularity curves ofplanar three-degree-of-freedom parallel manipulators. Mechanism and Machine Theory,30:533-551.

Shamir, T. 1990. The singularities of redundant robot arms. International Journal of RoboticsResearch, 2(1):113-121.

Shu, M. Kohli, D. and Dwivedi, S. H. 1986. Proceedings of the 6th World Congress on Theoryof Machines and Mechanisms, New Delhi, India, 988-993.

Soylu, R. and Duffy, J. 1988. Hypersurfaces of special configurations of serial manipulators andrelated concepts. Part II: Passive joints, configurations, component manifolds and someapplications. Journal of Robotic Systems, 5:31-53.

Spanos, J. and Kohli, D., 1985, “Workspace Analysis of Regional Sttructure of Manipulators,”ASME J. of Mech. Trans. and Aut. in Design, Vol. 107, pp. 219-225.

Sugimoto, K. Duffy, J. andHunt, K. H. 1982. Special configurations of spatial mechanisms androbot arms. Mechanism and Machine Theory, 117(2):119-132.

Tourassis, V. D. and Ang, M. H. 1992. Identification and analysis of robot manipulatorsingularities. Int. J. of Rob. Res., 11:248-259.

Waldron, K. J. 1987. Operating barriers within the workspace of manipulators. Proceedings ofthe Society of Manufacturing Engineers, Chicago, IL, Robots II/17th ISIR, 8:35-46.

Waldron, K. J. Wang, S. L. and Bolin, S. L. 1985. A study of the Jacobian matrix of serialmanipulators. ASME J. of Mech., Trans., and Autom. in Des., 107(2):230-238.

Wang, S. L. and Waldron, K. J. 1987. Study of the singular configurations of serialmanipulators. ASME J. of Mech., Trans., and Autom. in Des., 109(1):14-20.