Singularities in mechanisms II—trajectory tracking in the 3-axis wrist

Singularities in mechanisms IIÐtrajectory tracking in the3-axis wrist

Shiva Shankar

Department of Electrical Engineering, Indian Institute of Technology, Powai, Bombay 400076, India

Received 21 July 1997; received in revised form 23 March 1998

Abstract

This paper classi®es, up to a natural equivalence, 3-axis wrists given by rotations about some three®xed directions. Such wrists admit the 2-torus as a group of symmetries. These symmetries are useful indescribing the singularities of this mechanism, and in solving the trajectory tracking problem. # 1998Elsevier Science Ltd. All rights reserved.

1. Introduction

For the purposes of this paper, a mechanism is a smooth map f:X 4 Y, where X and Y aresmooth manifolds. In engineering parlance, Y, the workspace of the mechanism, is given apriori by the problem, whereas the jointspace X as well as the forward (kinematic) map f are tobe designed so that the mechanism can perform its stated function. In order that themechanism be able to reach every point in its workspace, f must be designed to be surjective, arequirement that will be in force throughout this paper.For instance, if the task of a mechanism is to orient a body, then the workspace Y is the

space of all orientations of 3-space, viz. SO(3). If the mechanism is to orient the body byrotating it about some n (®xed) directions, say u1, u2, . . . , un, then the jointspace of thismechanism is Tn, the n-torus, and the forward map is given by the product of these rotationsÐ

f : Tn4SO�3��y1; y2; . . . ; yn�4Run;yn . . .Ru2;y2Ru1;y1; �1�

where Rui,yiis rotation about the ui-axis by angle yi. [This mechanism, for surjective f, is an

example of an n-axis wrist. More generally, an n-axis wrist is any surjective f:Tn4SO(3).]A commonly required task of mechanism is trajectory tracking. Here, given a continuously

di�erentiable (i.e. C1) curve s(t) in the workspace of a mechanism, the problem is to ®nd a C1

Mechanism and Machine Theory 34 (1999) 513±526

0094-114X/99/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(98)00036-6

PERGAMON

curve t(t) in the joint space so that f[t(t)] = s(t). The traditional approach to solving theproblem was to try and ®nd an inverse function, i.e. a map g:Y 4 X such that f � g is theidentity map on Y. Then a curve t(t) in the joint space which tracked the given curve s(t) inthe workspace could be easily determined by t(t) = g[s(t)].This was the approach adopted to solve the trajectory tracking problem for the n-axis wrist.

However, in a fundamental paper, Gottlieb [1] showed by an elementary topological argument,the impossibility of constructing a (continuous) inverse function [the fundamental group, p1, ofSO(3) has torsion, while that of Tn is free, so that the identity map on SO(3) cannot factorthrough Tn]. Thus, the trajectory tracking problem for the n-axis wrist remained unsolved;indeed it was not even clear whether every curve in SO(3) could be tracked at all.As it turns out, not every trajectory in SO(3) can be tracked by an orthogonal 3-axis wrist

whose forward map is given as in Eq. (1), viz Ref. [2]. Such a wrist is pictured in Fig. 1; theadjective refers to the fact that the axes u1, u2, u3 are orthogonal. Such trajectories mustnecessarily pass through singularities of the mechanism [i.e. critical values of f:T34SO(3)]. Infact, the authors of Ref. [2] construct such trajectories by ®rst describing these singularities, butit turns out that these trajectories (that cannot be tracked) can however be tracked byreparametrizing them. Moreover, these reparametrizations might require the wrist to pause atthose singularities where it failed to track the original trajectory.Thus, the question arises as to whether this is the only obstruction to tracking a trajectory in

SO(3) by a 3-axis, not necessarily orthogonal, wrist. This paper answers this question in thea�rmativeÐevery trajectory in SO(3), or some reparametrization of it, can be tracked by any 3-axis wrist whose forward map is as in Eq. (1).While this is the immediate point of this paper, there is also a larger, more ambitious

purpose, which is to introduce the notation of a group of symmetries of a mechanism.

Fig. 1. Orthogonal 3-axis wrist.

S. Shankar /Mechanism and Machine Theory 34 (1999) 513±526514

Symmetries have long played a central role in mathematics and physics, and, in fact, this at®rst unconscious realization of symmetries in natural laws seems to have re¯ected itself inhuman activity from architecture to sculpture [3]. This is even more evident when we examinemechanisms designed by humans; cams, gears, pistons etc., and ®nd that they re¯ect geometricsymmetries which make their use, as well as their manufacture, that much more convenient.Thus, this paper solves the trajectory tracking problem by exploiting the symmetries of a 3-axiswrist; indeed, the point of view adopted is that the symmetries of a mechanism are offundamental importance in understanding its functioning. While this paper studies only thekinematics of a 3-axis wrist, other dynamical consequences of symmetries will appearelsewhere.Before the principal de®nition in this paper, a remark is perhaps in order. As noted above, a

trajectory can be tracked if it does not pass through any critical value of the forward map [2].Also, by Sard's Theorem, at least three axes are required to track trajectories in SO(3); thisfollows directly from Lemma 2 of Ref. [2]. Increasing the number of axes, however, increasesthe complexity of the mechanism; hence, this paper is concerned only with the 3-axis wrist,though many of these considerations apply in the more general situation.De®nition 1: A group of symmetries of a mechanism f:X 4 Y is a (Lie) group G together

with (smooth) actions c:G � X 4 X and f:G � Y4 Y with respect to which f is equivariant,i.e.

f�c�g;x�� f�g; f�x��for all g in G and x in X. Denote this by (G, c, f, f).Example 1: Let f:T34SO(3) be a 3-axis wrist given by rotations about u1, u2, u3, i.e. let f(y1,

y2, y3) = Ru3,y3Ru2,y2Ru1,y1 be a surjective map. Consider actions of T2 on T3 and SO(3) given by:

c : T2 � T34T3

��a;b�; �y1; y2; y3��4�a� y1; y2;b� y3�;and

f : T2 � SO�3�4SO�3��a; b�;A�4Ru3;bARu1;a:

Clearly, the forward map f is equivariant with respect to these actions, so that (T2, c, f, f) is agroup of symmetries of this 3-axis wrist. Later in this paper, the orbit structures of theseactions are studied in detail as a prelude to solving the trajectory tracking problem. While thisin itself should justify the notion of symmetries of a mechanism, there is here another reasonto consider the above actions of T2, which is brie¯y explained below. This is connected withthe fact that the jointspace and the workspace of a 3-axis wrist are Lie groups, and with theidea from physics that any departure from linearity is to be associated with a notion of force.The zero force situation on a Lie group is when the local law of evolution is given by left (or

right) invariant vector ®elds; then a particle would move along left (or right) translates of 1-parameter subgroups. Suppose f:G 4 H is a mechanism where G and H are Lie groups. If theforce of coupling G and H via the map f was required to be zero, then f should map left (orright) invariant vector ®elds to left (or right) invariant vector ®elds, i.e. f should be a surjective

S. Shankar /Mechanism and Machine Theory 34 (1999) 513±526 515

Lie morphism. This ideal situation is clearly not possible in the case of a 3-axis wrist (as T3 isabelian, whereas SO(3) is not), so what is desired is a surjective map f:T34SO(3) that is as``close'' to a Lie morphism as is possible in the following sense.Suppose f:G 4 H were a Lie morphism. Then f de®nes a left (or right) action f of G on H

by f(g, h) = f(g)h [or f(g, h) = hf(g)]. That these are actions is clearly equivalent to the factthat f is a Lie morphism. Consider also the natural action c of G on itself given by left (orright) multiplication. It follows that the morphism f is equivariant with respect to these actions.This is the zero coupling force situation. If this is not possible (as in the case of a 3-axis wrist),then what is required is a forward map that is equivariant with respect to actions of G, orsome maximal subgroup of it, on G and H. The forward map of the 3-axis wrist describedabove is one such. This ``minimal coupling force'' criterion will receive a more detailedtreatment elsewhere.

2. A class of 3-axis wrists

In the study of the trajectory tracking problem for a mechanism f:X 4 Y (where byrequirement f is a surjective map), and in particular in the local trajectory tracking problem(see Ref. [3]), the following questions are clearly relevant:

(1) Is every point of Y a regular value of f (so that if the dimension of X equals that of Y,then f is a local di�eomorphism)?If this is not so, then;

(2) Is there a regular point in the inverse image (under f) of every point in Y?If this is also not so, then:(3) Given any y in Y and any vector v tangent at y, is there a point in the inverse image of yand a tangent vector there which is mapped to v (by the derivative of f)?{In the case of a 3-axis wrist f:T34SO(3), the ®rst of these questions clearly admits a

negative answer [p1 of T3 is not a subgroup of p1 of SO(3)]}.

The answers to these questions are obviously invariant with respect to the followingequivalenceÐDe®nition 2: Two mechanisms f:X 4 Y and g:X 4 Y are equivalent if there are

di�eomorphisms h:X 4 X and k:Y 4 Y such that k � f = g � h.In the ®nal engineering solution to the trajectory tracking problem for a 3-axis wrist, an all

important question that ®rst needs to be answered is which of these forward maps f admit arealization in terms of basic engineering structures like cams, gears, joints, motors etc. In theabsence of a well-developed realization theory here (as opposed to circuit theory, say), thisseems to be a most di�cult question. The 3-axis wrist in the example above, where the forwardmap is given by the product of rotations about some three ®xed axes, can clearly be realized inthis engineering senseÐfor instance Fig. 1! Here rotations at joints a, b and c (which arealigned with the chosen three axes) are implemented using motors. Call such a mechanism a 3-axis wrist given by rotations. As it is not at all clear how to realize any other kind of wrist, thispaper con®nes itself to 3-axis wrists given by rotations, henceforth referred to, simply, as a 3-


axis wrist. The problem now is to answer questions (2) and (3) for such wrists. This is a verytractable problem because of the followingTheorem 2.1 There is, up to equivalence, only one 3-axis wrist (given by rotations).This theorem is a consequence of the following generalization of the classical ``Euler angles''.Proposition 2.1 Let u1, u2, u3 be three directions in R3. Then the map

f : T34SO�3��y1; y2; y3�4Ru3;y3Ru2;y2Ru1;y1 �2�

is surjective if and only if u1 and u3 are perpendicular to u2.Proof of Proposition: The idea of the proof is to show that under the stated conditions, and

only then, does the image of f act transitively on SO(3) under the action given by leftmultiplication. As no proper subset of SO(3) has this property, f must then be surjective. Thus,given arbitrary points A and B in SO(3), it needs to be shown that there is a point (y1, y2, y3)in T3 such that f(y1, y2, y3)A= B if and only if u1 and u3 are perpendicular to u2.Let U(S2) be the unit tangent bundle of the 2-sphere in R3 (with respect to the metric

induced on S2 from the euclidean metric on R3). Identify U(S2) with SO(3) by identifying apoint ( p, v) in U(S2) (where v is a unit vector tangent to S2 at p, so that p and v areperpendicular) with the unique positively oriented orthonormal basis that it completes to. Thenthe above transitivity statement translates to the followingÐgiven two points ( p, v) and (q, w)in U(S2), there is a point (y1, y2, y3) in T3 such that f(y1, y2, y3)( p, v) = (q, w). Here, if anelement A in SO(3) is considered as acting on S2 (isometrically), then the induced action onU(S2) by the derivative is identical with the left action of A on SO(3) via the identi®cationdescribed above.Let the directions u1, u2 and u3 be given by three points in S2, similarly denoted. Consider

now the orbit of u1 under the action of those elements in SO(3) that lie in f(T3). Every elementof f(T3) is a product of three rotationsÐ®rst a rotation about u1, then one about u2, and®nally a rotation about u3. The ®rst rotation ®xes u1 and the last ®xes u3. Thus, if an elementof f(T3) could move u1 to u3, that element must be a rotation about u2 alone. However, theorbit of u1 under rotations about u2 is a circle, say g, which must, therefore, contain u3. Hence,the points u1 and u3 must then make the same angle with u2. Suppose now that u1 is notperpendicular to u2. Then g is not a great circle and the orbit of this circle under rotationsabout u3 clearly cannot cover all of S2. Thus, f(T3) does not even act transitively on S2, letalone on U(S2). Hence u1, and, therefore, also u3, must be perpendicular to u2. In this case, theorbit g is a great circle, say G, perpendicular to u2.So suppose now that u1 and u3 are perpendicular to u2, in which case, given any point p in

S2, its orbit under rotations about u1 is a circle gp that intersects the great circle G (see Fig. 2).Then the orbit of p under rotations about u1 and u2, in that order, contains the great circle G.As G also contains u3, the orbit of G under rotations about u3 covers S2. Thus, by successivelyrotating about u1, u2 and u3, p can be moved to any other point on S2. As p was arbitrary, thismeans that f(T3) acts transitively on S2. The point, however, is to show that f(T3) actstransitively on U(S2) as well.So let q be any other point in S2, and let gq be its orbit under rotations about u3 (Fig. 2). As

u3 lies on the orbit G of u1 under rotations about u2, the orbit of gp under such rotations


intersects gq. Indeed, at a certain value of rotation about u2, say by y0, the circle Ru2,y0(gp)(which is gp rotated by y0 about u2) is tangent to gq. The tangents to these circles (orientedcounter-clockwise say) then make an angle of p radians [Fig. 3(a)]. As y increases beyond y0,the circles Ru2,y(gp) and gq intersect at two points [Fig. 3(b)] until they are tangent again[Fig. 3(c)]; now their tangents make an angle of 0 radians. Intermediate to these two (extreme)positions, when the circles intersect at two points, the angles between the tangents to the circlesat one set of intersection points decrease (continuously) from p to 0 radians, while at the otherset of intersection points they increase from p to 2p radians [Fig. 3(b)]. Thus, by appropriatelyrotating by a suitable angle y about u2, the oriented circles Ru2,y(gp) and gq can be made tointersect at any angle whatsoever.Suppose now that v and w are unit tangent vectors at p and q making angles a and b with gp

and gq, respectively. Clearly under any rotation about u1, ( p, v) is mapped to a point ( p1, v1)in U(S2), where p1 lies on gp and where v1 also makes the angle a with gp. Similarly underrotations about u3, (q, w) is mapped to some (q1, w1) where q1 is on gq and w1 makes the sameangle b with gq. Thus, to move ( p, v) to (q, w) by some element in f(T3), ®rst rotate about u1by y1 say, moving p to such a p1 which under a suitable rotation about u2 by say y2, moves tothat point of intersection of Ru2,y2(gp) and gq, where the tangents make the angle bÿ a (by theabove paragraph this is indeed possible). Then the image of v under Ru2,y2Ru1,y1, which is a unitvector v' at this point of intersection, makes an angle a with Ru2

,y2(gp) and hence, an angle bwith the circle gq. A ®nal rotation about u3 will then move v' to w.In the above argument it has been assumed that neither is p the point u1, nor is q the point u3

(so that gp and gq are both circles with positive radius). If, however, either of these is the case,then the situation is even simpler than the one just described. For instance, if p is u1, then rotateabout u1 so that the image v1 of v under this rotation (which is also tangent at u1) makes an anglep/2 + b with the great circle G (oriented from u1 to u3). Next rotate about u2 to move u1 along Gto the intersection, say p', of G with gq. This rotation moves v1 to a vector v' say, tangent at p'.

Fig. 2. GÐthe orbit of u1 under rotations about u2 is a great circle. gpÐthe orbit of p under rotations about u2.gqÐthe orbit of q under rotations about u3.


Clearly v' also makes the same angle p/2 + b with G. As G passes through u3, which is the centreof gq, the angle between G and gq at p' isÿp/2. Then v' makes an angle b with gq. The last rotationabout u3 ®nally moves v' to w. The case when q is the point u3 is similar.

Thus f(T3) acts transitively on U(S2). This implies that f is subjective. q

Proof of Theorem: Suppose f:T34SO(3) and g:T34SO(3) are two 3-axis wrists, given byrotations about (u1, u2, u3) and (v1, v2, v3), respectively. By the above proposition, u1 and u3 areperpendicular to u2 while v1 and v3 are perpendicular to v2. Let the angle between u1 and u3 bea and the angle between v1 and v2 be b (i.e. let a rotation of a radians about u2 move u1 to u3,and similarly for b).Let A be the (unique) element in SO(3) that moves (v2, v3) to (u2, u3); i.e. let A(v2) = u2,

A(v3) = u3. This rotation will, of course, not move v1 to u1 (unless angle a equals b); however,a rotation about u2 by the angle (bÿ a) will move Av1 to u1. Thus, rotating by y1, y2 and y3about u1, u2 and u3, respectively, is equivalent to rotating by y1, y2+(bÿ a) and y3 about Av1,Av2 and Av3. Composing next by Aÿ1 moves Av1, Av2 and Av3 back to v1, v2 and v3. Thus,Aÿ1f(y1, y2, y3)A equals g[y1, y2+(bÿ a), y3] for all (y1, y2, y3) in T3. However, rightmultiplication by A together with left multiplication by Aÿ1 is by de®nition the innerautomorphism on SO(3) de®ned by Aÿ1 (denoted here by IAÿ1). De®ne the di�eomorphismh:T34T3 by h(y1, y2, y3) = [y1, y2+(bÿ a), y3]. Then the diagram

T3 4fSO�3�

h # # IAÿ1T3 4

gSO�3�

commutes, which is to say that f and g are equivalent wrists. qThus, to study the trajectory tracking problem for the 3-axis wrist, it su�ces to consider any

one particular wrist.

Remark: It has been an implicit assumption from the very beginning that the points of thebody that is to be oriented by the wrist do not all lie on a straight line, so that the space of its

Fig. 3. Rotating by an appropriate angle about u2, gp can be made to intersect gq at any angle whatsoever.


orientations is SO(3). If, however, the points of the body are all co-linear, then the space of itsorientations is S2. As this is a 2-manifold, the question arises as to whether it is possible toorient such a body in every possible way, by two rotations, say about u1 and u2, i.e. by amechanism with just two axes. Call such a mechanism a 2-axis pointer; such a pointer ispictured in Fig. 4. It is easy to see that the forward map f:T24S2 of this mechanism is thecomposition

T24SO�3�4P S2

�y1; y2�4Ru2;y2Ru1;y14P�Ru2;y2Ru1;y1�where P:SO(3)4 S2 is the circle bundle over S2 obtained by identifying SO(3) with U(S2), asin the proof of the above proposition. Thus, if the body is initially pointing at x0 in S2, i.e. ifP(Id) equals x0, then the forward map is given by

f : T24S2

�y1; y2�4Ru2;y2Ru1;y1�x0�: �3�Consider the following actions of S1 on T2 and S2:

c : S1 � T24T2

�a; �y1; y2��4�y1; y2 � a�and

f : S1 � S24S2

�a; x�4Ru2;a�x�The forward map of the pointer is equivariant with respect to these actions, so that (S1, c, f,f) is a group of symmetries of this mechanism.

Fig. 4. 2-axis pointer.


The ®rst part of the proof of the above proposition shows that this forward map is surjectiveif and only if x0 and u2 are perpendicular to u1. An argument similar to the one in the theoremabove shows that there is, up to equivalence, only one 2-axis pointer. Also, as before, no mapf:T24S2 admits an inverse function. Thus, for a pointer too, there is the problem of trackingtrajectories, which does not admit a solution that an inverse function provides. However, as isevident, this is a more elementary situation than the case of the 3-axis wrist. Thus, this paperwill henceforth, except for a few remarks, deal only with the problem of orienting a body, notall whose points are co-linear.

3. Structure of the singularities of a 3-axis wrist

A singularity of a mechanism f:X 4 Y is, by de®nition, a critical point or critical value ofthe forward map f. As pointed out in the Introduction above, see also [2], trajectories in Y thatdo not pass through critical values of f can always be tracked. However, there are trajectories,which, of course, must necessarily pass through critical values of f, that cannot be tracked. Thequestion now is whether it is possible to track such trajectories after suitably reparametrizingthem. This question is suggested by the examples in Ref. [2], and this paper answers it for the3-axis wrist (given by rotations).Towards this end, it will ®rst be necessary to describe the structure of the singularities of a

3-axis wrist, and which by the above theorem is independent of the wrist. This description, forsay f:T34SO(3), is facilitated by its group of symmetries, viz. (T2, c, f, f) of Example 1; forobserve that by equivariance of f, the orbit of a critical point (or value) is a critical point (orvalue). Thus, as the action c of T2 on T3 is free, the set of critical points of any 3-axis wrist isthe disjoint union of copies of T2 embedded in T3. The description of the set of critical valuesof the wrist in the proposition below makes use of the following standard facts about SO(3).The Lie algebra so(3) of SO(3) consists of skew-symmetric 3 � 3 matrices and is identi®ed

with R3 via the map

so�3�4R3

X �0 ÿx3 x2

x3 0 ÿx1ÿx2 x1 0

0B@1CA4�x1; x2;x3� � x:

Then for this X in so(3), exptX is rotation about the axis determined by the corresponding x inR3 through the angle tvxv, where v � v is the euclidean norm on R3. Thus, d/dt(Rx,t)t = 0=x.Suppose G is a Lie group. Then the inner automorphism de®ned by g is the map

Ig : G4G

h4ghgÿ1:

The adjoint mapping de®ned by g, which is the derivative of Ig at the identity, maps the Liealgebra of G to itself and is denoted by Adg. In the case of SO(3), the adjoint mapping de®ned


by A is given by

AdA : R34R3

x4AdA�x� � Ax

(where so(3) has been identi®ed with R3 as above).

Finally, equip SO(3) with the left invariant inner product whose value at so(3) ishX ;Yi=ÿ 1/2 trace(XY). Via the above identi®cation, this is just the usual inner product onR3, i.e. ÿ1/2 trace(XY) = hx, yi= x1y1+x2y2+x3y3.

Proposition 3.1 The set of critical values of a 3-axis wrist is the disjoint union of two circles,both nonhomologous to zero.

Proof: Let f:T34SO(3) be a 3-axis wrist, given by rotations about some three points in S2,say u1, u2 and u3, where u1 and u3 are orthogonal to u2. Suppose y in T3 is a critical point of f.Then the image of the tangent space to T3 at y under the derivative of f is a proper subspaceof the tangent space to SO(3) at f(y). Under the derivative of Lf(y)ÿ1, i.e. of left multiplicationby f(y)ÿ1, this subspace is mapped to a proper subspace of the Lie algebra so(3). The followingcomputation detects this.

Let y = (y1, y2, y3) map to f(y1, y2, y3) = A, say. The tangent space to T3 at y is spanned bythe velocities of the curves (y1+t, y2, y3), (y1, y2+t, y3) and (y1, y2, y3+t) in T3 at t = 0. Theimages of these velocities in the Lie algebra under the derivative of Lf(y)ÿ1 � f are then given by:

d

dt�Aÿ1Ru3;y3Ru2;y2Ru1;y1�t�t�0 �

d

dt�Aÿ1ARu1;t�t�0 � u1;

d

dt�Aÿ1Ru3;y3Ru2;y2�tRu1;y1�t�0 �

d

dt�Ru1;ÿy1Ru2;tRu1;y1�t�0 � AdRu1;ÿy1 �u2� � Ru1;ÿy1�u2�;

and

d

dt�Aÿ1Ru3;y3�tRu2;y2Ru1;y1�t�0 �

d

dt�Aÿ1Ru3;tA�t�0 � AdAÿ1�u3� � Aÿ1�u3�:

(In these computations, the identi®cations described above have been employed.)

Thus, A = f(y1, y2, y3) is a critical value if and only if there is a nontrivial relation

a1u1 � a2Ru1;ÿy1�u2� � a3Aÿ1u3 � 0:

As a rotation about ui ®xes ui (i = 1, 2, 3), this is equivalent to

a1u1 � a2u2 � a3Ru2;ÿy2�u3� � 0:

However, a rotation about u2 moves u3 in the plane perpendicular to u2; the above relationthen implies that

a1u1 � a3Ru2;ÿy2�u3� � 0;

i.e. a2=0. Thus, A= f(y1, y2, y3) is a critical value of f if and only if y2 is such that either

Ru2;y2�u1� � u3 or Ru2;y2�u1� � ÿu3: �4�


As observed above, by equivariance of f, the orbits of these critical values under the T2-actionof Example 1 consists of critical values. The orbit through A= f(y1, y2, y3) is fRu3;bARu1;av(a,b) $ T2}, which is also fRu3;bRu2;y2Ru1;av(a, b) $ T2}. Now if Ru2;y2 (u1) = u3, then

Ru2;y2Ru1;a � Ru3;aRu2;y2 : �5�Thus, the above orbit is fRu2;y2Ru1;ava $ S1} and it includes all the critical values that satisfy the®rst part of Eq. (4). This is a circle in SO(3), clearly nonhomologous to zero [it in factgenerates p1 of SO(3)]. Also clearly, this orbit does not contain the critical values in Eq. (4)given by Ru2;y2 (u1) =ÿ u3. These critical values which similarly all lie on one orbit, constitute asecond circle, nonhomologous to zero, in SO(3). q

This description of the singularities of a wrist also answers question (2) above, viz., is there aregular point in the inverse image of a critical value? For suppose that A= f(y1, y2, y3) is acritical value, where Ru2;y2u1=u3, say. Then for all a in S1

f�y1�a; y2; y3 ÿ a� � Ru3;y3ÿaRu2;y2Ru1;y1�a � Ru2;y2Ru1;y3ÿaRu1;y1�a � Ru2;y2Ru1;y3Ru1;y1

� Ru3;y3Ru2;y2Ru1;y1 � f�y1; y2; y3� � A

(where the second and the fourth equalities follow from Eq. (5) above). Thus, the inverseimage of A is a circle in T3 embedded in the orbit of (y1, y2, y3), and which therefore are allcritical points. The orbit of (y1, y2, y3), which is a 2-torus, is then the union of the inverseimages of the points in the orbit of A. Thus, the set of critical points of a 3-axis wrist is thedisjoint union of two embedded copies of T2 in the jointspace, each of which is the inverseimage of one of the two circles of critical values described in the proposition above. In fact, itfollows from all this that each of these 2-tori is a circle bundle over the corresponding circle ofcritical values.Though the answers to questions (1) and (2) are negative for a 3-axis wrist, question (3),

however, admits a positive answer. This is essentially the content of the following proposition.Proposition 3.2 Let f:T34SO(3) be a 3-axis wrist. Let N be a connected component of critical

values of f, and let M = fÿ1(N). Then f maps a tubular neighbourhood of M onto a tubularneighbourhood of N.Proof: By the proposition above, N is an embedded circle in SO(3) and M an embedded

copy of T2 in T3. A standard result states that a compact submanifold of a Riemannianmanifold has a tubular neighbourhood which is equivalent to its normal bundle [4]. Thus, itsu�ces to show that the normal bundle of M in T3 is mapped, by the derivative of f, onto thenormal bundle of N in SO(3). Here T3 is given the product Riemannian structure(corresponding to the standard Riemannian metric on S1, say), while SO(3) is equipped withthe left invariant metric described in Propostion 3.1.Let A be any point in N, equal to f(y1, y2, y3) say, so that (y1, y2, y3) is in M. Without loss

of generality assume that Ru2;y2(u1) = u3 [viz. the ®rst part of Eq. (4)], where f is given byrotations about u1, u2, u3. As M is the orbit of (y1, y2, y3) under the T2-action of Example 1,i.e. M= {(a+ y1, y2, b+ y3)v(a, b) $ T2}, the normal bundle to M at (y1, y2, y3) is spanned bythe velocity of the curve (y1, y2+t, y3). The image of this vector under the derivative of f, lefttranslated to the Lie algebra of SO(3), now identi®ed with R3, is Ru1;ÿy1 (u2), (as computed inthe proof of the proposition above). The other points in M which are mapped to A are


{(y1+a,y2,y3ÿa)va $ S1}. The normals to M at these points are then mapped tofRu1;ÿ�y1�a�(u2)va $ S1}. The claim now is that these vectors span the 2-dimensional subspace ofthe tangent space at A normal to N.To show this, it will be ®rst necessary to calculate the tangent space to N at A. But as N is

the orbit of A under the T2-action of Example 1, it equals fRu3;bARu1;av(a, b) $ T2}, whichreduces by Eq. (5) to fARu1;ava $ S1}. Thus the tangent space to N at A is spanned by thevelocity of the curve ARu1;t at t = 0; its image (under left translation) in the Lie algebra istherefore u1.As u2 is orthogonal to u1, so are Ru1;ÿ�y1�a�(u2) for all a in S1. Hence, these vectors do span

the 2-dimensional subspace perpendicular to u1. Thus, every vector normal to N at A is indeedthe image of some vector normal to M. q

Remark: Similar results hold also for the 2-axis pointer f:T24S2, f(y1,y2,y3)=Ru2;y2Ru1;y1 (x0),where x0 and u2 are perpendicular to u1 (see the previous remark). Then the critical values of fare {u2, ÿu2}; the critical points, which are the inverse images of these two points, are twocircles in T2 (these are all orbits of the S1-action described before). As in the case of the 3-axiswrist, tubular neighbourhoods of each of these circles map onto tubular neighbourhoods (i.e.open neighbourhoods) of u1 and u2, respectively.It is the above proposition for the 3-axis wrist (and its analogue for the 2-axis pointer in the

remark) that allows trajectory tracking in these mechanisms. As the theorem below shows, thetrajectory, however, might have to be reparametrized before it can be tracked.Theorem 3.1 Every trajectory, or some reparametrization of it, can be tracked by a 3-axis wrist

(or a 2-axis pointer).Proof: Let f:T34SO(3) be a 3-axis wrist. Let S1 and S2 be the connected components of the

set of critical values of f (so that they are embedded circles), and let T1=fÿ1(S1) andT2=fÿ1(S2) be the connected components of the critical points (so that they are embedded 2-tori). As shown above Ti is a circle bundle over Si, i = 1, 2.Let s:[0, 1] 4 SO(3) be a �C1� trajectory in the workspace of f. Then sÿ1(S1[S2) is the

disjoint union of closed subintervals of [0, 1]. Let [t1, t2] be the ®rst such subinterval, i.e. let t1be the ®rst time that s intersects S1 or S2. As s([0, t1) does not pass through any critical valueof f, this part of the trajectory can be tracked; so let t1:[0, t1)4 T3 be a curve such that f �t1=s on [0, t1). Similarly, s can be tracked for times after t2 till it again intersects S1[S2, sayat time t3. Thus, let t2:(t2, t3)4 T3 be such that f � t2=s on (t2, t3).Let limt4t1t1(t) and limt4t2t2(t) be denoted by x1 and x2, respectively. That these limits exist

follows from the fact that T3 is compact. Clearly both x1 and x2 lie on the same 2-torus, T1 orT2; and equally clearly f(x1) = s(t1), f(x2) = s(t2).Suppose now that t1=t2. If x1 equals x2, then t1 and t2 patch up to yield a curve that tracks

s for all t in [0, t3). If not, let t' be a smooth curve in fÿ1[s(t1)] joining x1 and x2. This ispossible as fÿ1[s(t1)] is di�eomorphic to a circle. Now consider the concatenation t of thecurves t1, t' and t2 in T3. Reparametrize this curve so that t is C1. Then this curve tracks asimilarly reparametrized version of s([0, t3).If t1 does not equal t2, then choose a curve t':[t1, t2]4 Ti, i = 1, 2 as the case may be, such

that f(t') = s on [t1, t2]. This is possible as Ti is a circle bundle over Si. Then again thecorresponding concatenation tracks s on [0, t3).


The rest of the trajectory can be tracked in a similar fashion. The case of the 2-axis pointeris similar, if not even simpler. q

4. Conclusions

Shorn of technicalities, this paper has addressed the following questionÐCan a 3-axis wrist(2-axis pointer) track every trajectory in its workspace SO(3) (S2)? Given the counterexamplesin Ref. [2], the above question needs to be modi®ed toÐCan a 3-axis wrist (2-axis pointer)track every trajectory in its workspace perhaps after reparametrizing it? This paper establishesthat this is indeed possible. The main steps in arriving at this answer are the following:(1) If the forward map f in Eq. (2) [or (3)] is not surjective, then clearly there are trajectories

that cannot be tracked at all; just choose a trajectory that is not contained in the image of f.The ®rst step, therefore, is to ensure that the image of f is the entire workspace. Proposition2.1 (and the discussion in the remark following the proof of Theorem 2.1) give necessary andsu�cient conditions for this. Assume for the rest of this discussion that these conditions aresatis®ed.(2) From results in Ref. [2] it follows that trajectories cannot be tracked only because they

pass through singularities (i.e. critical values) of the forward map. Is it possible to track such atrajectory after reparametrizing it? To answer this question, clearly it is necessary tounderstand the geometry of the set of singularities.(3) This study for the 3-axis wrist (or the 2-axis pointer) is made easy by Theorem 2.1 (or

the remark following it) which states that up to equivalence (viz. De®nition 2), there is only onesuch mechanism. The notion of equivalence here ensures that the singularities of two 3-axiswrists (2-axis pointers) are di�eomorphic to one another. Thus, if after necessaryreparametrizations, trajectories can be tracked by one 3-axis wrist (2-axis pointer), then this isalso the case for any other such mechanism.(4) Having reduced the problem to one particular 3-axis wrist (2-axis pointer), the

description of the singularities of its forward map is reduced to elementary calculations (viz.Proposition 3.1), because these singularities are invariant under a group of symmetries. Thisfollows from the fact that the forward map is itself ``symmetric'', viz. De®nition 1.(5) It does not su�ce to just describe the topology of the set of singularities. One must also

study in®nitesimal neighbourhoods of the singular set. The reason for this is the following. Atrajectory enters the singular set at some point and exits it at another. Suppose that thistrajectory cannot be tracked. As the forward map is surjective the reason for this must be thatthe velocity at some point p on the singular set cannot be realized with a certain con®guration,say c1, of the mechanism. However, there might be some other con®guration, c2 say, (with theend e�ector at the same point p) where this velocity can be realized. Proposition 3.2 (and theremark following it) assert that such is indeed the case for the 3-axis wrist (2-axis pointer).(6) Finally, can one move from con®guration c1 to c2 with the end e�ector stationary? If so,

then the original trajectory can indeed by tracked after a suitable reparametrization. That thiscan be done is the content of Theorem 3.1.Apart from this solution to the trajectory tracking problem for the 3-axis wrist or the 2-axis

pointer, the principal contribution of this paper is to point out the role that groups of


symmetries can play in the study of mechanisms. The study of groups of symmetries haveplayed a central role in physics and mathematics, and in particular mechanics wheresymmetries correspond to constants of motion, viz. E. Noether's Theorem [5]. This paperintroduces such techniques in the solution of the important problem of trajectory tracking inmechanisms.

Acknowledgements

I would like to thank M. Sohoni for many useful conversations.

References

[1] D. Gottlieb, Robots and topology, in: Proceedings of the IEEE Conference on Robotics and Automation,

Utilitas Mathematica, Winnipeg, 1986, pp. 1689±1691.[2] S. Shankar, A. Saraf, Singularities in mechanisms 1, Mechanisms and Machine Theory 30 (1995) 1139±1148.[3] H. Weyl, Symmetry, Princeton University Press, NJ, 1952.

[4] M. Spivak, A Comprehensive Introduction to Di�erential Geometry, Publish or Perish, Boston, MA, 1979.[5] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1980.


Singularities in mechanisms II—trajectory tracking in the 3-axis wrist

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