the international journal of robotics research-1988-paden-43-61

43

Optimal KinematicDesign of 6RManipulators

Brad Paden*Department of Mechanical and EnvironmentalEngineering and the NSF Center for Robotic Systemsin Microelectronics

University of California, Santa BarbaraSanta Barbara, California 93106

Shankar SastryDepartment of Electrical Engineering and ComputerScience

University of California, BerkeleyBerkeley, California 94720

* Research funded by the Semiconductor Research Corporationunder grant number SRC 82-11-008. This research was conductedwhile the first author was with the Department of Electrical Engi-neering and Computer Science, University of California, Berkeley.

Abstract

A fundamental theorem for the kinematic design of robotmanipulators is formulated and proved. Roughly speaking,the theorem states that a manipulator having six revolutejoints is optimal if and only if the manipulator or its kine-matic inverse is an elbow manipulator. By "optimal" wemean a manipulator that has the properties of (i) maximalwork-volume subject to a constraint on its length and (ii)well-connected workspace—that is, the ability to reach allpositions in its workspace in each configuration. The notionof work-volume we use is that derived from the translation-invariant volume form on the group of rigid motions. Thisnotion of volume is intermediate between those of "reach-able" and "dextrous" workspace and appears to be morenatural in that it leads to simple analytical results.

1. Introduction

This paper develops a tight relationship between thekinematic performance and design of manipulatorshaving six revolute joints. In order to state preciselythis relationship, we develop the notions of length of a

manipulator, work-volume, maximal work-volume,well-connected workspace, and kinematic inverse.With these notions defined, a relationship between thedesign and performance of a manipulator having sixrevolute joints (6R) is summarized in the theorem: A6R manipulator has maximal work-volume and awell-connected workspace if and only if the manipulatoror its kinematic inverse is an elbow manipulator. Webegin our discussion of previous work with a review ofvarious notions of manipulator workspaces.A robot manipulator defines a map from its config-

uration space J (called jointspace) to the configurationspace of its hand, which is typically the group G ofproper rigid motions in R3 (our nomenclature is listedafter the appendix). For example, the manipulatorsketched in Fig. 1 defines a map from ~1 X T5 to G;for each set of joints displacements (a linear variableand five angular variables) we can specify the configu-ration of the hand. We write f: J ~ G for the forwardkinematic map of a manipulator. For a given manipu-lator, this map depends on which point in jointspacewe call zero, a detail which we resolve later. The imageof J under f, f(J) c G, is called the workspace of themanipulator.The thrust of the recent studies of manipulator

workspaces has been on the generation of projectionsof f(J) onto R3 and characterizing these projections.There are two important projections of f(J). They arecalled the reachable workspace and dextrous work-space by Kumar and Waldron (1981). These two pro-jections are obtained as follows. Each g E G can bewritten as a rotation about a point P followed by atranslation. That is, there is a natural diffeomorphism,gp, of G onto Gp X R3, where Gp is the subgroup of Gwhose elements leave P fixed. Let 7~3 be the natural

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44

Fig. 1. Schematic diagramof a P5R manipulator.

projection of Gp X IF83 onto its 1183 component. In termsof nR, and Op,

is called the reachable workspace (of P), and

is called the dextrous workspace (of P). The point Prepresents some significant point attached to the handof the manipulator. Often P is chosen to be a pointbetween the fingers of a manipulator hand or the pointof intersection of the wrist axes. In words, the reach-able workspace is the set of points that can be reachedby P, and the dextrous workspace is the set of pointsthat can be reached by P with arbitrary orientation ofthe hand.Bounds on WR(P) were obtained numerically by

Kumar and Waldron ( 1981 ). They observed that formanipulators with only revolute joints the boundary ofWR(P) consists of critical values’ of nR3 O gp O g Bygenerating a plot of these critical values, Kumar andWaldron were able to obtain graphical bounds onWR(P). Additional work on the shape of WD(P) andWR(P) is contained in Gupta and Roth ( 1982), where

they classify holes and voids in the workspace projec-tions and give conditions for the existence of holesand voids. The basic difference between a hole andvoid can be demonstrated with a bakery doughnut. Ifthe surface of a doughnut represents the reachableworkspace of a manipulator, then the doughnut holeis a hole in this workspace and the doughnut dough isa void. Gupta and Roth also discuss the behavior ofYYD(P) and WR(P) as functions of P. For a manipulatorhaving its final three axes intersecting at a wrist pointPw, the reachable workspace increases and the dex-trous workspace decreases with increasing 11 P - P wll.Further analytical work by Freudenstein and Primrose(1984) in this area develop precise relationships be-tween kinematic parameters and workspace parame-ters. This work was followed by an application toworkspace optimization for three-joint manipulators(Lin and Freudenstein 1986).Numerical studies of manipulator workspaces have

been carried out by Yang and Lee ( 1983}, Tsai andSoni (1985), and Hansen, Gupta, and Kazerounian(1983). These studies rely on numerical methods togenerate projections of the manipulator workspace.The advantage of these schemes over analytical ap-proaches is that mechanical constraints can be easilyincluded; a disadvantage is that general design princi-ples are hard to obtain.Work that is closely related to ours is that of Vijay-

kumar, Tsai, and Waldron (1986). These researchersdevelop design criteria for optimal manipulator per-formance, which is measured primarily in terms ofdextrous workspace. Vijaykumar, Tsai, and Waldronargue as we do that the elbow manipulator is an opti-mal design. Our work extends these results by elimi-nating the a priori assumptions that the manipulatorhas a positioning component and an orientation com-ponent and that the Hartenberg-Denavit parameterssatisfy certain constraints. We also extend these resultsto show that the kinematic inverse of an elbow manip-ulator is an optimal design as well. For a clear com-parison of our theorem (stated above and rigorouslydeveloped in Section 4) with the recent theory devel-oped by Vijaykumar, Tsai, and Waldron, we statetheir results in the form of a theorem: If a 6R manipu-lator M has maximal dextrous work-volume and satis-fies the assumptions that ( 1 ) the fourth, fifth, and sixthjoint axes intersect, (2) the Hartenberg-Denavit twist

1. Critical points of a map fare those points in its domain where Tffails to be surjective. Critical values are values in the range of f thatare the images of critical points. Regular points are noncriticalpoints, and regular values are noncritical values. See Abraham,Marsden, and Ratiu (1983).



45

angles are 0 or 90° for all links, and (3) either the linklength aj or the joint offset ri is zero for each link i,then M is an elbow manipulator.The approach taken in this paper to derive relation-

ships between design and performance of 6R manipu-lators differs in several ways from those mentionedabove: (1) Rather than projecting f(J) onto f1~3, weconsider the volume’ of f(J) as a subset of the groupof rigid motions as a performance measure of a kine-matic design. (2) We find all designs that optimize ourperformance measures. Our techniques differ as well.In our proof we find it essential to focus on the intrin-sic geometric objects that affect the kinematic per-formance of a 6R manipulator. Thus, we avoid, asmuch as possible, particular parameterizations of amanipulator’s geometry. For example, we do not useHartenberg-Denavit parameters. Although they pro-vide a good parameterization of a manipulator’s ge-ometry, they are by no means canonical. The impor-tant geometric objects are the joint axes, especially thepositions of the joints’ axes in the &dquo;zero configura-tion&dquo; (see Gupta 1986). We make the significance ofthe joint axes as clear as possible by expressing theforward kinematic map in terms of the elements in theLie algebra of the group of rigid motions that generatethe rotations about the joint axes. These generators arecommonly known as infinitesimal (or differential)twists about a screw. In addition to the benefit of usinga geometric notation, we are inclined to take advan-tage of the Lie group structure of the group of rigidmotions; for example, we can express rotations asexponentials of twists, and use the unique translation-invariant volume form on the group of rigid motionsto define the work-volume of a robot.The format of this paper is as follows: Section 2

describes the notation we use for elements in G andtheir expression as exponentials of elements of the Liealgebra of G. Section 3 contains the mathematicalframework we use for analyzing 6R manipulators.Here we define the notions of length, work-volume,maximal work-volume, and well-connected work-space. The definition of the kinematic inverse of a ma-nipulator is also formally stated. Section 4 centers on

the design theorem and its proof. Our conclusions aremade in Section 5.

2. Proper Rigid Motions

The group of proper rigid motions on 1~3 plays a cen-tral role in the study of rigid-link robot manipulators.It is the configuration space of the links of such ma-nipulators, the most important of which is the finallink or hand. This section describes the representationof rigid motions as exponentials of differential twistsabout a screw (Brockett 1983; Paden 1985). This is aspecial case of the more general notion of an exponen-tial map from a Lie algebra into a Lie group (Boothby1975). We review the representation of proper rigidmotions as a subgroup (homogeneous transforma-tions) of the group of invertible matrices. First, the no-tation used for points and vectors is introduced.

Identify points in physical three-space with points inR3 via an orthonormal right-handed coordinate systemas usual. Thus, a point P can be represented by anelement of R3 written

where the 1 in the last row allows for the matrix repre-sentation of rigid motions by homogeneous transfor-mations. If O is the origin of the coordinate system, then

Identifying a point with its coordinates in Eq. (3)should not cause any confusion, since only one coor-dinate system, fixed relative to the base of the manip-ulator, will be used in developing manipulator kine-matics. The reader will note that it is not necessary to

specify the coordinate system. This follows from the2. This notion of volume is directly related to Roth’s service coeffi-cient (Roth 1976) for manipulators whose last axis is revolute.



46

fact that objects we consider are intrinsic to three-space, and anything said about them is independent ofthe choice of coordinate system.Vectors have the form

For points P and Q, the displacement from Q to P,P - Q, is a vector since the fourth component &dquo; 1’s&dquo; inP and Q cancel. Also, if v is a vector, Q + v is thepoint Q translated by v. Both points and vectors willbe considered as elements in R3; their physical inter-pretations are completely different, however. The 0and 1 in the fourth component of vectors and points,respectively, will remind us of this and enforce a fewrules of syntax. For example, it is meaningless geo-metrically to add two points.The vector cross and dot products are computed in

a natural way. If v = (v,, v2, v3, 0)T and u = (u,, u2,u3, 0)’, then

It is well known that when points are represented as in(3), rigid motions can be identified with a 4 X 4 ho-mogeneous transformation of the form

where A E SO (3), the group of 3 X 3 orthogonal ma-trices with determinant 1, or simply the rotationgroup, b E R 3, and 0 is the zero row-vector in 1R3. LetG be the group (under matrix multiplication) of all4 X 4 matrices of the form (6). G is called the group ofproper rigid motions in R3. By Chasle’s theorem, anyelement in G can be decomposed into a rotation aboutan axis and a tr.anslation along the same axis; that is,any g E G can be thought of as a helical or &dquo;screw&dquo;

motion, which is why an element of G is called a twistabout a screw.

If we identify the identity element I E G with somenominal configuration of a rigid body, then any otherconfiguration of the rigid body is identified with therigid motion that translates the body from the identityconfiguration to the given configuration. With thisidentification, a curve in G defines a trajectory of arigid body such as the moving hand of a manipulator.

If g(t) is a C curve in G representing the trajectoryof a rigid body, then the generalized velocity of therigid body is given by dg(t)/dt E Tg(t)G, where T,,G isthe tangent space of G at h (Abraham, Marsden, andRatiu, 1983). The trajectory g(t) can be written

where P(t) is a C curve of points and AU) is a C’ I

curve in SO(3) that necessarily satisfies

Evaluating the derivatives of (7) and (8) with respectto t at t = 0 for an arbitrary curve g(t) through

yields

There is an isomorphism of TgG onto TIG given byright multiplication by g-’ . That is, r~g- j E TI G ’t/ 1] ETBG. Thus, the velocity of a rigid body can be identi-fied with an element of TIG. The advantage of doingthis is that the element of TIG that represents the ve-locity of a rigid body is independent of the choice ofthe identity position of the rigid body.

Let co = (WI’ úJ2, ~3, 0)T and define the cross-prod-uct operator constructed from co by



47

(Observe that S(co)b = <u X b V vectors co and b.)With this, ~ E TIG can be expressed as

for some to and v. This follows from Eq. (9) with A setequal to the 3 X 3 identity matrix.The vector space TIG is the Lie algebra of G and

has the same dimension (= six) as the manifold G.Elements in TIG are called (differential or infinitesi-mal) twists about a screw or simply twists. (Ball 1900;Roth 1984). This definition is motivated by the differ-ential version of Chasle’s theorem: the linearized mo-tion of a rigid body is the linearization of some helicalor &dquo;screw motion.&dquo; The position and the pitch of thetwist changes with time, of course. It is common prac-tice to identify TIG with R6. The six components in1~6 are called twist coordinates and are defined by thelinear invertible map tc: TI G - 11~6.

For notational convenience a map that extracts theangular velocity and the linear velocity vectors from atwist is defined similarly.

(Here is a case where we pay a notational price forusing the four component representation of points andvectors.)A twist is given the attributes of a pitch, an ampli-

tude, and a screw axis, where the notions of pitch andaxis derive their meaning from an ordinary machinescrew. They are defined as follows. Let

Then the pitch of ~ is defined by

The amplitude3 3of ~ is defined by

The amplitude of ~ is not a norm on TIG but is usefulin describing zero and infinite-pitch differential twistsparticularly. We define a screw aacis4 of ç E TIG to bea directed line lu (line I with nonzero direction vectoru) having the properties that (i) u X ~Q = 0 for all Qon lu and (ii) u has the same direction’ as w (v resp.) ifto =A 0 ((o = 0 resp.). It is well known that (1) every

3. Our definition of the amplitude of a differential twist about ascrew is unconventional in that we define an amplitude of an infi-nite-pitch differential twist about a screw.

4. Some authors define a screw axis to be an undirected line and

then by abuse of notation draw screw axes with a direction. Our def-inition includes this directedness. This is useful, since we like tospecify the sense of positive rotation.

5. The zero vector has arbitrary direction.



48

;

twist _tc-1 v has an axis (if o ~ 0, then the axis isunique), and (2) for each axis, pitch h c (-00, andamplitude M E (- 00, 00) there exists a unique twisthaving these attributes. A screw is a pair consisting ofa line (called the screw axis4) together with an elementof (-00, 00] (called the pitch). Thus, there is nearly aone-to-one relationship between twists and screws,and this is why screws are used to graphically representtwists.The following proposition allows us to relate the

rigid motion of an element of TIG and the change inits axis.

PROPOSITION 1 (Rigid motion of a differential twistabout a screw) Let ~ E TIG with pitch h, amplitudeM, and axis l u . If g E G, then g~g- E TI G with pitchh, amplitude M, and axis (gl}~.

It follows that g~g-1 has the same amplitude and pitchas ~. Now u X <~ = 0 V0 <=/~=> ~M X ~’~ =0 ‘dR E (gl)~. Finally, u has direction cv (v resp.) ifw ~ 0 (co = 0 resp.) implies gu has direction gco (gvresp.) if gcv ~ 0 (gco = 0 resp.). m

In the next section we will see that exponentials ofdifferential twists are very useful in the study of ma-nipulator kinematics. Elements of TIG are squarematrices, so their exponential is well defined. There isalso an important geometric interpretation given bythe following propositions and discussion. The firstproposition is well known.

PROPOSITION 2 Let co be a vector and define co ~ <~/II w when t.~ # 0. Then

Figure 2 gives the geometric interpretation of es~~’~as a rotation by 11 co II about the axis through the origin

Fig. 2. Interpretation of e~w>.

O having direction co. The following proposition tellsus how to compute the exponential of a general twist.

PROPOSITION 3 Let ( = [£-1 v , Q be a point on10)], _

an axis of ~, and P be an arbitrary point. Then

Proof: This follows from the preceding proposition(see Paden 1985 for a complete proof)..

Figure 3 gives the geometric interpretation of e~ as arotation about the axis of ~ by 11 co II followed by atranslation along the axis by the pitch times to. Notethat when the pitch of ~ is zero, e~ is a pure rotationabout the axis of ~ by II w II . When the pitch of ~ isinfinity, e~ is a pure translation by v.

3. Mathematical Framework

A formal theory of manipulator kinematics requires,first of all, a mathematical representation of manipu-lators. The amount of information contained in the

representation depends on what we are trying to ac-



49

Fig. 3. Interpretation of e~.

complish. In dynamics, for example, a representationmust contain information on inertias, etc. For thepurposes of this paper, the only significant objects on6R manipulators are the joint axes. Thus we will rep-resent a 6R manipulator by an ordered set (orderedfrom base to gripper) of six zero-pitch unit-amplitudetwists whose screw axes are coincident with the ma-

nipulator axes for some configuration of the manipu-lator.6 Figure 4 depicts a manipulator with its repre-sentative screws. The ordered set of twists~ = {ço, ..., ~5), çj E TjG, is called a representativeof the manipulator M. Since there are many represent-atives corresponding to different &dquo;zero configurations&dquo;and different senses of positive rotation of the jointaxes, we say that C is equivalent to ~, and write C - ~,if ( and ( are representatives of the same manipulator.Formally, C - ~ if there exists 0 E T’ such that

From Proposition 1 we have that ~~ ’, g E G, hasthe same pitch and amplitude as C, and that the axis ofg~g-’ is the axis of (translated by the rigid motion g.This gives the interpretation of &dquo;-&dquo; as ~ - ~ if the axisof each can be rotated successively about the pre-vious axes Ci-1, ... , (0 such that its axis is coinci-

Fig. 4. Representing a 6Rmanipulator by an orderedset of twists.

dent (possibly antiparallel) with that of çj. A manipu-lator is identified with an equivalence class generatedby -. If ~ is a representative of M, we write M = [(],and the brackets [.] ] are read &dquo;the equivalence classcontaining.&dquo;The length of a manipulator is important for ob-

taining an upper bound on its work-volume. Presently,we develop a notion of length for 6R manipulatorsthat has operational significance. Define the (non-empty) set of linking curves of ~ as the following set ofcurves that map the interval [0, 5] into 1~3.

Definition 1.

is called the set of linking curves of ~. ( We use thenotation c, to denote c evaluated at t and refer tothe line segments c~l,;+1~, i E (0, 1, 2, 3, 4), aslinks of ç.)

’

Thus, a linking curve is simply a curve that passesthrough the axes of the manipulator in order. Usingthe set of linking curves, we define the length of arepresentative ~ as the minimum of the lengths of alllinking curves.

Definition 2.

Define the length of ~, l~ by

On a physical manipulator, rotating the jointsclearly does not affect the lengths of the links; likewise,choosing a different representative of a manipulator

6. Since this paper deals exclusively with 6R manipulators, we willoften drop the 6R and simply write manipulator.



50

does not affect its length as defined in Definition 2.This is expressed formally in the following:PROPOSITION 4 If ( - ~, then l~ = IZ.

Let c E C~ and define the curve d by

Observe that d, is continuous and piecewise linear on[0, 5]. Also ci E Ci (i.e., ci is on the axis of ~i), so dj =e~~o ~ ~ ~ e~r-~~~-~~c; E çj. It follows that d E C~. Sincerigid motions preserve length, it is easy to see that

Thus, for every linking curve of ~, there exists a linkingcurve of ~ having the same length. By the symmetry ofthe equivalence relation, -~-, the opposite is also true.Thus, l~ = lt. 0

Proposition 4 tells us that each representative of amanipulator has the same length. Thus, the length of amanipulator is well defined in terms of its representa-tives.

Definitlon 3.

Let M = [(] . Then its length lM is

It is often the case that a property of a manipulatoris invariant under the reversal of the axis order. Re-

versing the order of the axes is equivalent to using thegripper end of the robot as the base and the base endas a gripper (clearly, this is not practical unless wereplace the final link with a gripping device and thefirst link with a mounting flange)! This is an importantsymmetry that is made precise by defining formallythe kinematic inverse of a manipulator.

Definition 4.

[{ Ç5, ... , ,~}] is called the kinematic inverse ofM. The representative of M-’ is denoted ~-l.

One property of a manipulator that is invariantunder kinematic inversion is its length:

PROPOSITION 5 Let M be a manipulator, then lM =

1M-I.

Proof: If c, is a linking curve for (, a representativeof M, then c5-, is a linking curve for ~-1, and viceversa. E

For manipulation we express the motion of thehand attached after the last joint in terms of the jointangles. If ~ represents the configuration of the manip-ulator axes when all the joint angles are set to zero,and if we define the resulting hand configuration to bethe identity configuration, then the configuration ofthe hand as a function of the joint angles is

This rigid motion of the hand relative to its identityconfiguration is a rotation about ~5 by 05 followed bya rotation about ~4 by ~4, etc. The map j~ defined by~: ~,~ ee~Q~,~, ... ees~5 is called the forward kine-matic map for [~] associated with ~. Thus, the forwardkinematic map is determined by the zero configura-tion of the manipulator’s axes (Gupta 1986).

It would be nice to be able to define a unique for-ward kinematic map; however, there does not seem tobe a natural way to do this. The following propositiondemonstrates the relationship between forward kine-matic maps associated with different representatives ofthe same manipulator.

PROPOSITION 6 If ~ --- ~, then 3 if) E T’ such that

for some choice of signs in the RHS of (24).



51

Thus, f~ = Rgh h, where h is an automorphism ofT6, and R, is a right translation by some g E G. Itfollows that f~T6) = Rg~~(T6) {i.e., a translated versionof,,~~{T6)}. If we define volume in G such that it is in-variant under translations by group elements, then wecan associate a unique volume to each manipulator.Let w be the unique translation invariant volume formon SO (3) (Boothby 1975) such that Jso{3~ w - 8n2.This normalization gives the units of orientation vol-ume in radians cubed. Define a volume form on R3 bydx 1B dy 1B dz, as usual. Then we construct the vol-ume form ~ ~ dx 1B dy 1B dz A w on R3 X SO (3).From Abraham, Marsden, and Ratiu ( 19$3), page 399,we have that dx A dy A dz induces a measure p (Le-besgue measure) on R3 and w induces a measure ~CZ onSO (3). The volume form S2 induces a measure onQ83 X SO (3), which is simply the product measure111 X ~2. This observation will simplify our calculationof volumes in 1~3 X ,SO (3), since we only calculatevolumes of rectangles of the form A X SO (3), A C 11~3.We make the identification of G with R3 X SO (3) andnote that is a translation-invariant volume form on G.

Intuitively, this notion of volume is quite simple.For example, consider an aircraft restricted to a cubeof airspace that is 1 km on a side. At each point in theairspace the aircraft can point itself anywhere in a 4~rsolid angle and roll 2n about the direction it is point-ing. Thus the orientation freedom or orientation vol-ume at this point is 4n X 2n = 8n2 rad3. Multiplyingby the positional volume, we obtain $n2 rad3km3 forthe volume of the free configuration space of the air-craft. This is the notion of work-volume we use for

manipulators. It has the advantage of being able totrade off orientation freedom for positional freedomsmoothly, unlike the popular notion of dextrous work-space.We define the work-volume of a manipulator as the

volume of the image of its jointspace under the for-ward kinematic map.

Definition 5.

The work-volume Vm of a manipulator M = [~] isgiven by

Remark The translation invariance of Q in (26)guarantees that UM is independent of the choice ofrepresentative ~. This property also tells us that thevolume is independent of hand size and depends onlyon the positions of the joint axes! This is a useful mea-sure for the robot manufacturer who is building ageneral-purpose robot and not designing the robot’shand. With the definitions above it is simple to obtainan upper bound on the work-volume of a manipulator.

PROPOSITION 7 For any manipulator M,

Proof: Let ~ be a representative of M, and let c E CZwith length Im. Then !!~(6’)cs - co ll ~ lM b’ 8 E_ T6,since we can construct a curve from co to f~(6)c5 oflength 1M for all 0, as was done in Proposition 4. Thus,

since cs is always &dquo;tethered&dquo; to Co by a curve of length1M, (The other ci are also tethered together by shortercurves.) It follows that



52

Thus,

This proposition says that the most we can expectfrom a manipulator in terms of work-volume is theupper bound given. Since large work-volume is adesirable property in general, we make the following:

Definition 6.

M has maximal work-volume (MWV) if VM =4~3TL(IM)3 ’ 8~.

In addition to large work-volume, we would like themanipulator to be able to reach all points in itsworkspace in each of its configurations. Thisguarantees that the manipulator can move betweenany two points in its workspace without changing itsconfiguration. We call this property the well-connectedworkspace property. To be precise, we define CZ to bethe set of critical points (Abraham, Marsden, andRatiu 1983) off, and we define well-connectedworkspace as follows.

Definition 7.

The manipulator [~] has well-connected workspaceif fj (B) =fZ(T 6)BfZ (CZ) for all connectedcomponents B (commonly called configurations)of T 6BCZ. (Since A and fc are related, for ( - C, bycompositions with diffeomorphisms this propertyis well defined for a manipulator.)

A manipulator having the well-connected-workspaceproperty has the ability (modulo obstacles andmechanical constraints) to move its gripper from oneregular value (Abraham, Marsden, and Ratiu 1983) toanother without passing through a critical value(singularity). This is a &dquo;nice&dquo; property ofmanipulators, and it guarantees that the manipulatorneed not change configurations (e.g., from elbow up toelbow down) in order to move from one regular valueof its forward kinematic map to another.

Kinematic inversion has no effect on the MWV oron the well-connected-workspace properties.

Fig. 5. Representative of anelbow manipulator.

PROPOSITION 8 Let M be a manipulator. Then~M - ~M- ~ -

Proof : Let ~ be a representative of M. Then

where I is the inversion map in the group G. That I isvolume preserving for translation-invariant volumeelements implies VM = V M-I’.

COROLLARY M has MWV if and only if M~ ’ hasMWV.

Using the fact that I is a diffeomorphism we canshow the following.PROPOSITION 9 M has well-connected workspace ifand only if M-1 has a well-connected workspace.

Proposition 8, its corollary, and Proposition 9 tell usthat the notion of a kinematic inverse is a fundamen-tal symmetry in the analysis of manipulators. We willuse this symmetry to simplify our proof of Theorem 1.When we exploit this symmetry, we say &dquo;by kinematicinversion....&dquo;We will see shortly that elbow manipulators are very

special 6R manipulators. First, a formal definition ofan elbow manipulator is given.

Definition 8.

M is an elbow manipulator if it has a representa-tive ~ satisfying the following (see Fig. 5): Thereexists a line segment COC5 with the properties that



53

5. ~2 is a perpendicular bisector of cocs.6. ~3n~4n~s=c5.7. ~3 -L ~4.8. Ç41.ÇS’

(The line segment coc5 is introduced in anticipationof a linking curve with endpoints co and cs .)PROPOSITION 10 Let M be an elbow manipulatorand cocs be as in the definition of the elbow manipula-tor. Then lM = ll c$ - co II .

Proof: Let be a representative of M that satisfiesDefinition 8. Since the last three axes of ~ intersectconsecutively orthogonally at cs, we can point jointaxis 5 arbitrarily. Thus we can rotate ~5 and Ç4 andthen about ~3 to a new position ~’ with ~’114. Thencoc5 is a mutual perpendicular of ~o and ~5 . There is nocurve shorter than II cs - co connecting ~o and ~, andtherefore l~ ~ II cs - co II . Now it is easy to construct alinking curve c, with clo, 5] = CoCs and whose length isII cs - coil. Therefore lM = II c5 - colI.8

This concludes the mathematical setup. In the nextsection a basic theorem is proved relating the conceptsof this section.

4. Optimality Theorem

It is generally accepted that &dquo;elbow&dquo; manipulatorshave large work-volumes. This is made precise, and aninteresting converse is proved in the following theorem.

Theorem 1: A 6R manipulator M has well-connectedworkspace and maximal work-volume if and only if Mor M-1 is an elbow manipulator.

The formal proof of this theorem contains manytechnical (yet important) details, so a heuristic proof isgiven first to give an overview of the basic geometrythat motivated the complete proof. Readers should si-tuate themselves such that one arm is free to do someof the &dquo;exercises.&dquo;

Heuristic Proof: (~==) This direction of the proof isrelatively easy. Elbow manipulators are modeled after

the human arm, which is capable, modulo mechanicalconstraints, to position and orient the hand anywherein a ball of radius arm’s length. Thus, the elbow ma-nipulator has MWV. This is also true for the kinematicinverse of an elbow manipulator by the kinematicinversion properties. The well-connected-workspaceproperty is left to the formal proof.(~) The basic idea of this part of the proof is to

construct a set of &dquo;exercises&dquo; that only a manipulatorwith maximal work-volume and well-connected work-

space could do and then successively conclude thegeometric features of such a manipulator.Exercise 1 (Shoulder and Wrist Swirl) A manipulatorwith MWV can reach any point on a sphere of arm’slength radius and any orientation on that sphere. Inother words, the manipulator has at least five degreesof freedom remaining out of six at full reach. Sincejoints that do not intersect the shoulder or the wrist arefrozen at full reach, there must be only one such joint(called the elbow).Exercise 2 (Armpit Scratcher) A manipulator withMWV must be able to reach its shoulder with its wrist.

Therefore, the upper arm and forearm have the samelength, and the axis of the elbow joint must be perpen-dicular to both the forearm and the upper arm.

Exercise 3 (Elbow Swirl) A manipulator with MWVmust be able to place its elbow anywhere on a sphereof radius half the arm’s length. For suppose there is aregion on this sphere that the elbow cannot reach.Then project this region onto the larger sphere ofradius arm’s length, and the wrist cannot reach thisregion, in contradiction to MWV. Since the elbowmust move with 2 DOF, there must be at least twojoints intersecting at the shoulder. By kinematic inver-sion there are at least two joints intersecting the wrist.So the manipulator must have two joints at one endand three at the other. We can assume without loss of

generality (by kinematic inversion) that the end withonly two joints is the shoulder. It is well known thatthese two joints must intersect orthogonally and mustbe orthogonal to the upper arm for the elbow to reachany point on a sphere.Exercise 4 ( Hand Raiser) Assume that the firstshoulder joint is vertical. Then by MWV the arm mustreach a configuration where the wrist is on the firstshoulder axis and above the shoulder by half the arm’s



54

length. In this configuration the elbow is 120 fromfull extension. Now the upper arm and lower arm andthe first shoulder axis lie in a plane. The elbow axis isperpendicular to both arm segments and is thereforeperpendicular to the plane. Also, the second shoulderjoint axis is perpendicular to the first joint axis andperpendicular to the upper arm, so the secondshoulder axis is perpendicular to the plane. It followsthat the second joint axis and the elbow axis are parallel.Exercise 5 (Elbow-Down Wrist Swirl) Choose a con-

figuration with the elbow pointing down. By MWVthe hand should reach all orientations about the cur-rent position of the wrist. If the three wrist axes areconsecutively orthogonal, then this is possible and wehave an elbow manipulator or its kinematic inverse. Ifnot, there is some orientation that is not possible inthe present elbow-down configuration. By MWV thisorientation is attainable by changing the elbow config-uration to another, say elbow-up, but this requires thatthe manipulator move through a singularity (full ex-tension or full retraction of the elbow), violating thewell-connected-workspace assumption. Therefore, theonly possibility is that the wrist axes are consecutivelyorthogonal.Thus, we have narrowed the possible manipulators

to the elbow manipulators and their kinematic in-verses. The formal proof now follows.

Proof: (~) By kinematic inversion, it is sufficient toprove this implication for M an elbow manipulator.Let ~ be a representative of an elbow manipulatorsatisfying the properties in the definition and the addi-tional (nonrestrictive) properties ~3 = ~5 andza (e5 - co) = II es - coll, where zo is a unit vector par-allel to ~o. (These added conditions allow us to write aclosed-form inverse kinematic solution fora.) Weprove the well-connectedness and maximal work-vol-ume of M by examining the solution to the kinematicequation7

where R E G is a desired rigid motion. Let z; be a unitvector parallel to çj, i E {0, ... , 5}. Also, C5 and coare as in Definition 8, and 1. = ilcs - coll is the lengthof the manipulator. The solution of Eq. (31 ) is wellunderstood, and a straightforward calculation yieldsthat 0 is a solution to (31) if and only if 0 satisfies(32-35):

(the upper and lower signs are matched in (33a) and(33b))

(the upper and lower signs are matched in (34a) and(34b))

Note that all quantities under the radical in (33-34)are nonnegative and that the domain of the ATAN2function is the entire plane. (The numerator and de-nominator are two distinct arguments of the ATAN2function. When they are both zero, the function isset-valued and the 8; may take any value in [0, 2n].)The only restrictions we have on R are given by (32).That is, there is a solution for R if and only if

Thus the work-volume of the elbow manipulator is4/3 n~lM~3 . gn2 and is therefore maximal.The singular configurations of the elbow manipula-

tor are those where the elbow is fully extended or re-

7. Note that we make the minor change in joint coordinates &thetas;1 →

&thetas;1 — &thetas;2/2. This is for convenience and has no effect on work-volumeor the well-connected-workspace property.



55

tracted, joints 3 and 5 are coincident, or the intersec-tion of the last three joints lies on joint axis zero. Acalculation of the Jacobian determinant of f~ (see Paul1981 for this type of calculation) yields that the set ofcritical points for fj in (31 ) is

It is easy to see that T6BC~ has eight connected com-ponents. We must verify that

for all connected components B ofT6BC{. Let R Ef~{T6}Bf~(C~). Then when we apply the solution pro-cedure (32-35) with such an R there must be nochoice of signs in (32-35) such that 0; E (0, 7r), i E(1, 2, 4) (otherwise R E C~). Thus, whenever wechoose a solution branch for 81, 62, and 0, (32, 33a,and 34a) there is a choice of 8j E (0, 7r) or 8j E (- n, 0),i E { l, 2, 4). It follows that we can find a solution ineach connected component of T6BC~ ~ (38) holds.This completes the proof of the &dquo;if &dquo; part of the theorem.

(~) Let M = [~] and c’ E Q with length 1M anddefine 1’ ~ maxõET611e8oCo ... eestsc~~ - c’o 11. If I’ <

1M, then V,,~ ~ 4I3 n(l’)3 ’ 8n2 < 4I3 7L(I,,~)3 ’ 8n2 and Mdoes not have MWV. Thus there must exist 6 suchthat II e8o{o ... e8sCscs - coll - 1M, Let ~ be a represent-ative corresponding to this configuration. As in Propo-sition 5, we can construct c E C~ from c’ such that chas length lM, co = co’, and c. = ee~~ ... ees~5ej~.Thus, there exists c, a linking curve for ~ with lengthlm = IIcS - coll. So clo, 5] = CoCs, a line segment. Since cis a linking curve, all ~i intersect cocs and the c, areordered on cac5. (Figure 6A shows a representativeconsistent with our knowledge of M at this point in theproof.) Let fi : T6 - G be the forward kinematic mapfor M associated with ~. Now f~{T6) is compact impliesthat for any x E fZ (T 6)1 , x has a neighborhood ofnonzero volume. Thus

(see proof of Proposition 8). Thusf~(T6) contains 0,defined by

Fig. 6. Pictorial outline ofthe proqf of the &dquo;only if &dquo; partof theorem 1.

and 0 is a 5-manifold.

Next, define

By definition of a linking curve, min > 0 and max < 5.

Also, max ~ min; otherwise there would be only onelink with nonzero length and MWV would not hold.



56

From (39), we have that

V0 in the inverse image fC (O). (41 )

Since co n ~i * ~1 b’i < min and cs n ~~ #= ø Vi > max,

(41 ) becomes

IICh - call, b’a, b E [0, 5]. Let a = min, b = 5. Then bya tethering argument similar to the first part of theproof of Proposition 8,

Since II c~n - co + II c 5 - cminll = 1M, the two terms inthe norm in (43) have the same direction and the firstterm attains the bound in (44):

or, in other words,

A similar argument with Cmax (kinematic inversion)yields

Let ?7 be a zero-pitch unit twist whose axis is coincidentwith CoCs. Then from (46) and (47) we have

The containment in (49) follows from (48). Since 0is a 5-manifold, the order of (00, ... , 8uùn- I’ 0,8max+1> ... , 85} must be 5 by Proposition 1.4 of theappendix. (Intuitively, we need 5 DOF to sweep out a5-manifold.) Thus, max - min C 1. (Figure 6B repre-sents approximately our knowledge of M at this pointin the proof-at most two of the ~i do not intersectco or cs). Suppose max - min = 1 and thatr~lBmirsSmineemaoYmaxl~ E f~ 1(O)} is a finite set. Then from(48) we have that there exists a finite set (~# i , ... ,~n} such that

and

Since the union is finite and each set in the union is

compact, it follows (Baire category theorem (Royden1968)) that one of the sets in the union has nonemptyinterior in O. Now 0 is a 5-manifold, so the order of(00, ... , 8min-l, 8max+l, ... , 65) must be 5 byProposition 1.4, but the order is at most 4 when max -min = 1. Thus we must have that (eem~~~~ee~ e~~ ~ 9 Ef ~-’ (B)} is infinite when max - min = 1. This and(46) imply that eemia~mineemas~maz~s = cs has an infinitenumber of solutions. This, in turn, implies that (seePaden 1985, page 24) çmax n Cs =1= ø or ~mi,, f1 c5 ~ ø.The definition of max then implies that ~~ n c5 = 0.By kinematic inversion ~min n co = O also. This is acontradiction, so max - min = 1 is impossible andmax = min. Thus, there is exactly one twist, Çmin,which intersects neither co nor c5. Next, we show that~min is a perpendicular bisector of coc5._M has maximal work-volume implies that the map0 lIe8oC:O ... eB5~5c5 - co is onto [0, lm]. Since(min is the only axis not intersecting c5 or co, f(6) _II eBmm~,~,~c5 _ co II , and by Proposition 1.3 we have that

Çmin is a perpendicular bisector of Co c 5 . (51)

So we have that part 5 of Definition 8 is satisfied if



57

min = 2. (We will see this shortly.) (Figure 6C is arepresentative consistent with our knowledge of M atthis point in the proof. The drawings are isometric.)

Next, we claim that MWV implies that d.- B H~ ~ ~ -

Suppose not. Then since d(T6) C II cmin - coIIS2, thereexists a unit vector u such that

which is a contradiction to MWV since A(T 6) closed(compact). Thus the claim is true.Now d is a smooth function onto a 2-manifold, so d

is a function of two or more of the 0; by Proposition1.4. Thus, min ~ 2. By kinematic inversion, min =max -- (5 - 2) = 3. Thus, there are two axes intersect-ing one end of coc5 and three the other. In otherwords, M or M-’ has min = 2. Since we are only try-ing to prove M or M-’ is an elbow manipulator, thereis no loss of generality in assuming M has the propertymin = 2. It follows from Proposition A.1 of the ap-pendix and the fact that d is onto 11 ci. - COllS2 that

At this point, parts 1, 2, 3, and 5 of Definition 8 aresatisfied. Next we show that ~, 11~2.

Recall that zo is the unit vector parallel to ~ . Thensince M has MWV, there exists 0 c T~ such that£(0)cs = co + (j~.rl2)2o (this point lies in B(co, 1M)), Let~’ be a representative corresponding to this configura-tion, and let c’ E C;: be the linking curve derivedfrom c (see the proof of Proposition 5). Figure 7 showsthe configuration of the manipulator represented byI’. Note that c§ , c2, and cs are not collinear and con-sider the plane containing them. Since ~o and eoc2 arein the plane, we have by (52) that çí is perpendicularto the plane. Now coc2 and cZCS are in the planeimply that Ç2 is perpendicular to the plane by (11),and so 2 1 This, in turn, implies8 that ~211~, and part4 is satisfied. (Figure 6D represents approximately theinformation we have at this point in the proof.)

Fig. 7. Showing that çJ II C;2’

Since min = max = 2, the last three twist axes inter-sect at ~c~, satisfying part 6 of the definition of theelbow manipulator. To show that these axes are conse-cutively orthogonal (7 and 8 of Definition 8), we write

Since the wrist axes of M are intersecting, the singu-larities of the manipulator decouple. That is, 0 is acritical point of f ~ if and only if (80, 01 , 02) is a criticalpoint of h’: (80, 6, , 82) H h(80, 01, 82)cs or (83, 04, 05)is a critical point of w : T3 ~ Gcs’ More concisely,

Since we know the relationship among ~o, ~1, andÇ2, it is straightforward to verify that h’ has the prop-erty that each connected component, Bl , of T3 B Ch,satisfies lt’(B~ ) = h’(T 3) B h’(Ch’) and T3 B Chi has fourconnected components. Also h’ is 1-1 when restrictedto a connected component of T3 B Cn, .

8. Angles between successive joints, and between joints and adjacentlinks are independent of choice of representative when the linkingcurves are related as in the proof of Proposition 5.



58

Suppose that ~3, ~4, and ~5 are not consecutivelyorthogonal. Then w is not onto Gcs by Proposition 1.2.Then w(T3) is a compact, proper subset of Gcs’ andw(T 3)C is a nonempty open set in Gcs’ In addition, let0 be a regular point of f~, and let (9ot, 81~, 8z;), i E( 1, 2, 3, 4), be the four solutions that satisfyh’(8oj, 81j, 02i) = h’00, 8¡, 02). Since the critical valuesof w have measure zero in G~5 (the measure on G~s isthat induced by a translation-invariant volume form),

has measure zero in Gcs’ Therefore w(T3)CB Dw is non-empty. Let

It follows that h(Bo, 8, , B2)R_’ is a regular value off¿and by the choice of R’, f~(8) ~ h(8o, 81, 02)RI. Themanipulator has maximal work-volume, so there exists0’ E T6 such that

connected components of T3 B C,,. implies that 0’ and~ are not in the same connected component ofT6 B C~ = T3 B Ch. >C T3 B Cw, thereby contradicting thewell-connected-workspace assumption. Thus

In other words, parts 7 and 8 are satisfied, and wehave that M or M-’ is an elbow manipulator.

Theorem 1 reinforces the generally accepted ideathat elbow manipulators are good kinematic designsand are optimal with respect to work-volume. It issomewhat surprising that the elbow manipulators andtheir kinematic inverses are the only designs that meetthe optimality criteria of maximal work-volume/well-connected workspace. This theorem should encouragespecial consideration of elbow manipulators and theirkinematic inverses in the study of path planning andcollision avoidance problems.

5. Conclusions

We have developed a formalism for describing 6Rmanipulators that allows us to define quantitativelythe properties of well-connected workspace and maxi-mal work-volume. The result of this formalism is afundamental theorem relating the design of 6R ma-nipulators to their performance.

Admittedly, the class of 6R manipulators is limited.However, 6 DOF is the minimum number requiredfor a manipulator’s work-volume to be nonzero, andthe extension of the theorem to 6 DOF manipulatorswith both R and P (prismatic) joints is feasible with anextension of the definition of the length of a manipu-lator. The length of a manipulator having both R andP joints is defined to be the length of the shortestcurve passing through the axes of the revolute joints(minimize over configurations also) in order plus thesum of the travel distances of the prismatic joints.When P joints are allowed, the class of optimal ma-nipulators enlarges to include the Stanford manipula-tor with zero shoulder offset and its kinematic inverse.Also note that the definitions of work-volume and

length of a manipulator extend directly to redundantnR manipulators. One can conclude from argumentsin the proof of Theorem I that nR redundant manip-ulators having maximal work-volume have their firsttwo and last two joint axes intersecting. Because re-dundancy supposedly increases the &dquo;connectivity&dquo; ofthe workspace, it will be interesting to see if there is auseful &dquo;very&dquo;-well-connectedness measure for redun-dant manipulators.

Appendix

Collected here for convenience are several subprob-lems that appear in the proof of Theorem 1.

Proposition 1.1: Let 4 and ~l be zero-pitch unittwists, O, P E ~3, O ~ P, and Ço m (i = O. If f(00, 9,) H eB~~eBe~P is onto 0 + IIP- OIISZ (thesphere of radius II P - D centered at O), then



59

Proof: Let zo and Zt be unit vectors parallel to ~o andÇl, respectively. Now

Premultiplying both sides by z,T

Thus

Next from (1.4) we have

Proposition 1.2: Let O E ~3 and ~o, ~, , ~2 be zero-pitch unit twists with (o m ~1 f~l Ç2 = O. If

Proof: f onto Go implies that

Let P # 0 be a point on Ç2, and let both sides of {I.12)act on P. Then ( I.12) implies

but GOP = O + 11 p - 01182, so the result follows fromProposition L 1.

Proposition 1.3: Let ~ be a zero-pitch unit twist and<9, ~ e ~ with <9 ~ P and (~ n <9P ~ 0. If/: 0-lIe8ÇP - O is onto [0, liP - 0 11 ], then ~ is a perpendic-ular bisector of OP.

but II e~~P - OII = 0 for some 0 implies that 11 P - QII =Il O - Q II, which implies that

~ n OP contains the midpoint of OP. (1.14)

Now, let z be a unit vector parallel to ~. Then

This together with (T.14} yields the result:

Proposition 1.4: Let M and N be m- and n-manifolds,respectively, and let f M-~ Nbe smooth. If f(M) hasnonempty interior then n2 = n.

Proof: Since the set of regular values of f is dense inN, there exists a regular value y E interior of f (M).Thus, there exists x E M such that Tf TM - TN issurjective implies that m = n. 0

Nomenclature ’ ; °

R Real numbers

II ~ li Euclidean norm in Ran

B(x, 6 ) The closed ball of radius,6 centered at x



60

SO(3) The group of proper rotations in If83 (the groupof 3 X 3 orthogonal matrices with determinantone)

G The group of proper rigid motions in ~~ (thegroup of 4 X 4 homogeneous transformations)

Go The isotropy group of G at Q E R3 (thoserigid motions that have the point Q as a fixedpoint)

si The unit circle (i.e., [0, 27r] with the endpointsidentified)

S2 The unit 2-sphereT&dquo; The n-torus

TM Tangent bundle of the manifold M

T,fM Tangent space of M at x

Tf Tangent map of fA Tensor wedge product- Equivalent to- Diffeomorphic to

~v Diffeomorphic to via a volume-preservingdiffeomorphism

II Parallel to

0, P Points in R3

OP The line segment connecting O and P1 Perpendicular to

9, ~ An ordered set or list of indexed objects (e.g.,8 = (80, ... , t95))

~ l The ordered set or list of objects whose ele-ments are those of ( with their order reversed

sc Complement of the set S

AB B The set difference A minus B

0 Empty setCr Continuously differentiable r times

CZ The set of linking curves of ~Cf The set of critical points of the function fATAN2 Two-argument arctangent functionmin The smallest element in a set

max The largest element in a set

For economy we make the following abuse of notation:A differential twist about a screw and its screw axis areoften identified so that for differential twists about

screws Ço and ~, , P E ~ n ~, means that the point P isin the intersection of the axes of (o and ~, . Also, adifferential twist about a screw will sometimes be re-ferred to as simply a twist.

References

Abraham, R., Marsden, J. E., and Ratiu T. 1983. Manifolds,Tensor Analysis, and Applications, Reading, Mass: Ad-dison-Wesley.

Ball, R. S. 1900. A Treatise on the Theory of Screws. Cam-bridge : Cambridge University Press.

Boothby, W. M. 1975. An Introduction to DifferentiableManifolds and Riemannian Geometry. New York: Aca-demic Press.

Brockett, R. W. 1983 (Beer Sheba, Israel). Robotic manipu-lators and the product of exponentials formula. Proc.MTNS-83 Int. Symp., pp. 120-129.

Freudenstein, F., and Primrose, E. 1984. On the analysisand synthesis of the workspace of a three-link, turning-pairconnected robot arm. ASME J. Mechanisms, Transmis-sions, and Automation in Design 106:365-370.

Gupta, K. C., and Roth, B. 1982. Design considerations formanipulator workspace. ASME J. Mechanical Design104:704-711.

Gupta, K. C. 1986. Kinematic analysis of manipulatorsusing the zero reference position description. Int. J. Ro-botics Res. 5(2):5 -13.

Hansen, J. A., Gupta, K. C., and Kazerounian, S. M. K.1983. Generation and evaluation of the workspace of amanipulator. Int. J. Robotics Res. 2(3):22-31.

Kumar, A., and Waldron, K. J. 1981. The workspaces of amechanical manipulator. ASME J. Mechanical Design103:6b5-672.

Lin, C.-C. D., and Freudenstein, F. 1986. Optimization ofthe workspace of a three-link turning-pair connected robotarm. Int. J. Robotics Res. 5(2):91-103.

Paden, B. E. 1985. Kinematics and control of robot manipu-lators. Ph.D. thesis, University of California, Berkeley,Department of Electrical Engineering and ComputerSciences.

Paul, R. P. 1981. Robot Manipulators: Mathematics, Pro-gramming, and Control. Cambridge, Mass.: MIT Press.

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Roth, B. 1984. Screws, motors, and wrenches that cannot be



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bought in a hardware store. In Robotics Research, eds. M.Brady and R. Paul, Cambridge, Mass.: MIT Press, pp.679-693.

Royden, H. L. 1968. Real Analysis. New York: Macmillan.Tsai, Y. C., and Soni, A. H. 1985. An algorithm for theworkspace of a general n-R robot. ASME J. Mechanisms,Transmissions, and Automation in Design 105:52-57.

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