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Classification of robot-manipulators with only singular configurations
Mech. Mach. TheoryVol. 30, No. 5, pp. 727-736, 1995 Copyright© 1995ElsevierScienceLtd 0094-114X(94)00068-9 Printed in Great Britain. All rights reserved 0094-114X/95$9.50+ 0.00
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C L A S S I F I C A T I O N OF R O B O T - M A N I P U L A T O R S W I T H O N L Y SINGULAR CONFIGURATIONS ADOLF KARGER Department of Mathematics, Charles University, Praha, Czech Republic
(Received 29 April 1992; in revisedform 29 November 1994; receivedfor publication 6 December 1994)
1. I N T R O D U C T I O N In this paper we shall find aU serial robot-manipulators which have all configurations singular. Such r o b o t - m a n i p u l a t o r s can be characterized by the condition that their Jacobian matrix has rank smaller than maximal for all locations o f the end effector. F o r simplicity we shall consider r o b o t - m a n i p u l a t o r s with only rotational axes.
2. N E C E S S A R Y
FORMALISM
AND DEFINITIONS
At the beginning we shall present a short review o f the notation used in Sections 3 and 4 o f the paper. Detailed exposition o f this material can be found for instance in [1], where one an find also m a n y references concerning this topic. Let p = A + 2v, 2 e R, be a straight line in the Euclidean space E 3, Ro = {0; e~, e2, e3} be a fixed o r t h o n o r m a l frame in E3. Let us denote by (v~, v2) the usual scalar product in E3, v~ x v2 be the usual vector (cross) p r o d u c t o f arbitrary vectors Vl, v2 in E3, let (v, v ) = 1. Let us depote by X = (v; A x v) the Pliicker-coordinates o f the straight line p in the frame Ro, g(u) be the rotation a r o u n d p with the angle o f rotation u. g(u) can be written as
where g(u) is a 4 × 4 matrix, T(u) is a 3-column and 2:(u) is a 3 x 3 matrix, y ( u ) e S O ( 3 ) . Let us write .~ = g'(u)g-~(u), where the prime denotes the derivative with respect to u. Then ,~ is a 4 x 4 matrix o f the form
where z is a 3-column and x is a 3 x 3 skew-symmetric matrix. Let us define the following correspondence i between skew-symmetric 3 × 3 matrices and 3-columns:
i
(0 "oX)(x / x3
0
\ -- x2
xj
-xl
=
x2 •
(1)
x3
The correspondence i has several useful properties: Let x, y be any two skew-symmetric 3 x 3 matrices, y ~ S0(3). Then
i(xy - - y x ) = i(x) x i(y),
x ' i ( y ) = i(x) x i(y), 727
i(),xy - I ) = ~/g(x).
(2)
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Adolf Karger
The fundamental correspondence between Pliicker-coordinates X = (v; A x v) of the straight line p and the matrix
z o) of the rotation
g(u) around p is the following: v=i(x),
A ×v=z.
This shows that each straight line p in E3 uniquely defines both X and ,~; in what follows we shall identify both of them with p whenever it will be convenient and by the pair X = (x; z), where (x, x) = 1, (x, z) = 0 we shall understand the appropriate one of the three mentioned objects. Pairs of vectors of the form X = (x; z), where x and z are arbitrary 3-columns, are called screws. The vector space of all screws will be denoted by L. L can be equipped with a structure very similar to the structure of the vector space of ordinary vectors in E3 by using dual numbers. Let us denote by D the set of all pairs ~t = a + eb, where a, b ~ R, e 2 = 0. D is the ring of dual numbers. The pair X = (x; y) from L can be written as a dual vector X d = x + ez, it is a 3-column of dual numbers, L d be the space of all screws written in the form of dual vectors. The ordinary scalar and vector products in E 3 can be extended to L d by using formally the same definitions. Let
(a~) X d = a2 , \a3/
yd
(b~) = b2 b3
be two dual vectors from L d. We define the scalar product by (X a, yd) = y~= ~a~bi and the vector product as X d x y d = (ci), where ci = E3k= t~ijkajbk, i = 1, 2, 3, and e~jk is the usual sign of the permutation tensor used for the definition of the usual vector (cross) product. These two operations in L d induce immediately new operations in L. Let X d = x + ez and yd = y + et be arbitrary screws in L d. Then
(X a, ra) = (x, y) + e[(x, t) + (y, z)].
(3)
Let X ~ be the dual form of a screw X = (x; z) which corresponds to a straight line p. Then (X d, X a) = (x, x) + 2e(x, z) = 1. This shows that screws corresponding to straight lines are unit vectors in L a. Let X d and Ya be two unit vectors in L d, which correspond to straight lines p and q, respectively. We can write (X ~, ya) = cos tp + ed sin ~o,
(4)
where ~0 is the angle of p and q and d is their distance (both oriented). Each screw can be written as the sum of two special screws, X = X~ + X2, where x = (x; z ) = (x; z - (x,
z)x) + (0; (x, z)x),
(5)
where we for simplicity suppose that (x, x ) = 1. XI = (x; z - ( x , z)x) in (5) corresponds to the rotational part of the screw X, X2 = (0; (x, z)x) is its translational part; (x, y) is the pitch, XI yields the Pliicker-coordinates of the axis of the corresponding screw motion. The vector (cross) product in L d induces a vector product in L which is expressed as follows: Let .~d, y d E L d, z d = x ' d × yd, X = ( x , z ) , Y = ( y ; t). Then Z=(x
x y;x x t+z xy).
(6)
If X and Y correspond to straight lines p and q, respectively, then the rotational part of Z corresponds to the axis (common perpendicular) of p and q. The correspondence i between skew-symmetric 3 x 3 matrices and ordinary vectors in E 3 can be extended to dual vectors as well.
Robot-manipulators with singular configurations
729
Let us denote by C6 the group of all orientation preserving congruences of the Euclidean space E3. Let
where T is the 3-column of the translation, 7 e S O ( 3 ) is the 3 x 3 orthogonal matrix of the rotational part. The group SO(3) can be extended to dual numbers by the formula gd = (E + i-~ (T)e)7, where E is the unit matrix. Elements gd form a group denoted by S 0 ( 3 , D). It is the special orthogonal group over dual numbers, it is defined by the condition gd.(gd)T= E and it is easy to show that the described correspondence between C6 and S 0 ( 3 , 1)) is an isomorphism (see [1] for instance). Let us describe now the action of groups C6 and S 0 ( 3 , D) in L and L d, respectively. Let p = A + 2v be a straight line in E3,
be a congruence of E3. Let us apply g to p. Then the corresponding straight line g(p) will be given by g ( p ) = T + T A +27v. Let X = ( x ; z ) be the Plficker-coordinates of p, x = v , z = A x v. The Pliicker-coordinates of g(p) are given by g(X), where g(X) = (~,x; T x 7x + 7z). On the other side the multiplication of the vector X d = x + ez by the dual matrix g d = (E + i-J(T)e)7 yields gd'Xd = (E + i-'(T)e)7(x + ez) = 7z + e(Tz + T x 7x) = [g(X)] d. We shall call the action of the group (76 in the screw space L determined by the natural action of C 6 in E 3 the adjoint action of C6 in L, Ad(g)X. The adjoint action is expressed either by multiplication of dual vectors by dual matrices from S 0 ( 3 , D) as we have shown above or by the formula A d ( g ) X = gXg -~ in the representation of screws by 4 x 4 matrices with help of (1) as we can see from (2). Properties of the ordinary cross product remain valid also for the vector product of dual vectors, we shall use the property u × (v × w ) = (u, w)v - (u, v)w,
which is true for ordinary vectors as well as for dual ones. Let X ~ , . . . , Xp be straight lines in E3 such that X~ # Xi+~, let gi(u~) be the rotation around the axis Xi, ui be the angle of rotation. The motion of the p-parametric robot-manipulator defined by axes Xj . . . . ,Xp is the motion of /?3 determined by the matrix function g(u~ . . . . . up) = gz (u~) . . . . gp(U.). The instantaneous position of axes Xj . . . . . Xp is determined by straight lines with Pliicker-coordinates Y~ . . . . . Yp, where (see for instance [2]) Y, = X,,
Y2 = Ad(g2)X, . . . . . Y. = Ad(g). . .g,_,)Xp.
This leads to the following Definition
A configuration Y~ . . . . , Yp of axes of the robot-manipulator defined by axes X~ . . . . . Xp is called singular iff the dimension of the vector space generated by vectors Y~ . . . . . Yp in L is less than p. A robot-manipulator will be called singular iff it has only singular configurations of its axes. The rank of a singular robot-manipulator is the maximum of all dimensions of spaces generated by r, . . . . . yp. 3. S T A T E M E N T OF THE C L A S S I F I C A T I O N T H E O R E M
Singular robot-manipulators with rotational axes only are the following ones: (1) Rank 3: (a) Xj . . . . . X r pass through one point, p > 3, (b) Xz . . . . , Xp are parallel, p > 3. (2) Rank 4: Robot-manipulator of rank 3 with one more arbitrary axis at the beginning or at the end. (3) Rank 5: (a) Robot-manipulators of rank 3 with two more arbitrary axes at the beginning or at the end or with one at the beginning and one at the end. (b) Robot-manipulators defined by
Adolf Karger
730
////
p=4
\\\\
p=5
X I or X 5
/ X 1 or X 5
////
p=6
Fig. 1.
axes X ~ , . . . , Xm, X,, + t . . . . , Xm +,, where m >~ 3, n >/3 and all axes in the group X~ . . . . , Xm and in the group Xm +~ . . . . . Xm +, are either parallel or pass through one point. Figure 1 shows axes of singular robot manipulators for p = 4, p = 5 and p = 6. Remark. Let us denote by P, the robot-manipulator defined by r parallel axes (the planar robot-manipulator), by S, the robot-manipulator defined by r axes which pass through one fixed point (the spherical robot-manipulator), R denotes as usually one rotational axis. Then the singular robot-manipulators are described as follows: (1) S,: P,, r > 3,
(2) RS,, SrR; RP,, P,R, r > 3, (3a) RS, R, RRSr, SrRR; RPrR, RRPr, PrRR; r > 3, (3b) S=S.; S=P,; ProS,; PmPn; m >/ 3, n >~ 3. 4. P R O O F OF T H E T H E O R E M At first we shall show that it is enough to prove the classification theorem for 6-parametric robot-manipulators (p = 6). If axes Xt . . . . . Xp determine a singular robot-manipulator, then the robot-manipulator with one more arbitrary axis at the beginning or at the end remains singular. This means that singular robot-manipulators with p < 6 are partial robot-manipulators of 6-parametric robot-manipulators. On the other side, if p > 6, then each six axes from X ~ , . . . , Xp must determine a singular robot-manipulator as well, because by adding new axes we cannot decrease the rank of the robot-manipulator. This means that we can restrict ourselves to 6-parametric robot-manipulators. Let X1 . . . . . X6 be the initial position of axes of such a robot-manipulator. Then the instantaneous configuration of axes of this robot-manipulator is given by straight lines ~1 . . . . . ~'~ with Pliicker-coordinates
L = x,,
72 = Ad(g~(ui))X2 . . . . .
?, =
Aa(~,(u~)...g~(u~))X6,
where (gi(ui)) is the rotation around the axis Xi with the rotation angle ui. The equation for singular robot-manipulators is det(~71 . . . . . Y6)= II?~. . . . . Y61 = 0 .
(7)
Robot-manipulators with singular configurations
731
We can write equation (7) in more symmetrical form--the configuration of axes of the robot-manipulator is independent of space congruences and therefore we can apply the congruence given by the matrix gzt(uz)g~J(u~)g~(u3) to vectors 17~. . . . . Y6. We obtain vectors Y~. . . . . Y6, where Y, = Ad(g2g3)-'Xj, Y5 = Ad(g4)Xs,
Y2= Ad(g~')X2,
}:3=)(3,
Y4=X4,
]:6 = Ad(g4gs)X6,
(8)
and the equation for singular robot-manipulators reads as l Y e , . . . , Y6I = 0
for all [ u l , . . - , u 6 ] ~ R 6-
(9)
Equation (6) is very complex one in the expanded form and not suitable for further computations even if we use computer algebra. Its expanded form has more than 1000 terms. To solve it, we shall simplify it at first by using some theoretical considerations. We shall show that it implies another equation, which is simpler and which can be explicitly solved. Lemma. Let X = (x; z), Y = (y; t) be Pliicker-coordinates of two straight lines. Then X × (X × Y) = cos ~pX - Y + ed sin ~pX, where e(x; z ) = (0;x), q~ is the angle of X and Y, d is the distance of X and Y. Proof. We have X × ( X x Y) = (X, Y ) X - (X, X) Y = (cos ~o + ed sin ~o)X - Y. Lemma. ~/~u5 (Y6) = Y5 × Y6. Proof. Y6 = g a g s ] ( 6 , O/Ous(Y6) = g4g'sX6 = g,g'sgs-' "gsX6 = g4{X~ × (gsX6) d} = g4X~ × g, gsX d = Y5 × Y6, where we use the representation by dual vectors and matrices. Lemma. ~2/(Ous)ZlYi . . . . . Y61 = IYI . . . . . Ys, eYsla5 sin ~5, where a5 and ot5 is the distance and angle o f X5 and :(6, respecthJely. From the last Lemma we see that (9) implies a5 sin 75[YI . . . . . Ys,eYs[ = 0 ,
(lO)
and by repeating the argument also a~ sin ~ as sin ~t~[e Y2, Y2 . . . . . Ys , e Ys I = 0 .
(ll)
Before we start with detailed computations, we shall simplify our denotations in the standard way (by using Denavit-Hartenberg parameters): Let us denote Ci = cos ct~, S~ = sin cti, where ~t~ is the angle from X~ to X~+ t, ai is the distance from Xi to ~ + t, d~ denotes the distance from the axis of X,_ t and X~ to the axis of X~ and Xi+ ~ (the offset). (The axis of two parallel straight lines is any straight line which intersects both of them perpendicularly.) Ci ~
COS
lli,
Si
= sin/,/i.
Remark. We note also that the inverse robot-manipulator to a singular robot-manipulator is also singular, because the configuration of axes of the inverse robot-manipulator is the same. The inverse robot-manipulator is obtained by interchanging the role of the base and end-effector of a robot-manipulator. For the initial position of the robot-manipulator we choose such a position (configuration of axes), in which all axes are parallel to a given fixed plane. For the basic orthonormal frame Ro = {O; e~, e2, e3 } in the base space E3 we choose the symmetry frame of axes )(3 and )(4. This means that the origin O lies on the axis of ,t"3 and X4 at the same distance from both of them, vectors e~ and e2 cut the angle between :(3 and )(4 by half. e3 has the direction of the axis of )(3 and ):4.
732
Adolf Karger
Let us have a straight line p = A + 2):, 2 e R, where y = (cos ct, sin 0~, 0), A is a point o f p, z = A x y. X = (y; z) ~ L be the corresponding vector from L. Let us denote again
.u,(; the rotation a r o u n d the straight line p with the angle o f rotation u, where ~ e SO(3), T e R 3. Let further
sin m(0~) =
si 0
r(u)=
COS0 Gt
cosu
,
-sinu
sin u
•
cos u /
We obtain immediately
= m(o~)r(u)m(--oQ. Because g(u) preserves the point A, we have
which yields T = (E - y)A. N o w we are prepared to c o m p u t e the expressions for vectors
Y~,..., r'6. Let us have Xi = (Yi; zi), where Ai and yi = rn(0Qel are known. Then
Ai+ j = Ai + aie3 + diy~, Yi+~ = m(~i)yi. Let us find at first the vector X :+ ~ = (y~; z:) = A d(g~(u~))X~+ ~ where z: is determined by the point A ~+ 7. Then
Y :+ ~ = ~:iYi+1,
.4 :+ l -~ Ai "q- aiYie3 q" diYi = Ai q- aim(ct )r(ui)e3 -k- dim(~ )el,
where Yi = m(~t)r(ui)m(-ct). This yields
y: = m(~)r(ui)m(-o~)rn(~i)m(~)el. --
I
t
t
F o r I:5 we have ~ - ~g3, A, = ½a3e3 and for Y5 = (Y~; z~), where z~ is determined by As, we have:
Y'5 = rn(½0~3)r(u,)rn(~4)e,,
t
I
A~ = 5a3e3 + a.m(½g3)r(u4)e3 + d4m(½g3)e,.
C o m p u t a t i o n yields //£ C4 -- 0"C4S 4 ;
I:5 = ~ 0C4 -~ Kc4S4
\
s.S4
xG4 + trH4\
-
I -
-
(12)
o-G4 - ~;H4/, R2
/
where Ga = S4(a4 + l a 3 c 4 ) , i g = COS ~0~3,
/-/4 = s4S4d4-- G(½a3 + a, c4),
R2 = d4c4S, + a, s4G,
O" = sin ½~3.
Analogically we obtain
y'6=75y6,
A;=As+as•se3+dsys,
y6=m(½0c3+oq+ots)el,
y'6 = rn(½~ 3 + ~4)r(us)rn(~q)el,
Y'6' = Y4Y'6 = m (½~t3)r (u4)m (oe,)r (us)m (or5)el = a" b . m (oc5)el,
As=(½a3+a4)e3+d4y4,
Robot-manipulators with singular configurations
733
where a = m(½o~3)r(u4), b = m(o~4)r(us),
A; = (½a, + a4)e3 + d4m(½~3)e, + asm(½~q + ~4)r(us)e3 + dsm(½0t3 + ot4)e,, A: = T4 + ?4A~ = ½a3e3 + a(a4e3 + d4el) + ab(ase3 + dsem). Computation yields y~ =
- ~Ls - tr(c, Ms - s, Fs) 7 - e L 5 + x(c4Ms - s4Fs)l S 4 Ms + c4Fs J
I
,
Ix[Bs - 2 (c, Ms - s, Fs)] - tr(c4A, - s4Ps - 2 Ls) t¢
tr[Bs -
26
(13)
(c4M5 - s4Fs)] + x(c4A5 - s4Ps -- -~ Ls), s4A s q- c4P s
where
Ms=G&c,+S4G,
N s = S , Gcs+C4Ss,
Ls= S4Sscs- C4Cs, Fs= ssSs, As = -a4Ls + asKs -
F4(d4 +
Ks=GGcs-S4&,
Bs = - a 4 M s - a s N s + dsS4Fs,
C4ds), P~ = asCsss + dsSsc4 + d4Ms.
We have r/.
It
Y6 = (y6, z6).
Finally
1:4= X4 =
a3V ½a~
•
(14)
Expressions for Y~, Y2, Y3 are obtained by the substitution ai--+--ot6_i,
ai---*--a6_i,
ui---~--u7_i,
(15)
di--+d7_i.
Now we are ready to write down equation (9) explicitly. The determinant IY~. . . . . I:6[ has its columns composed from components of vectors Y~. . . . . I:6. Vectors I:4, Ys and Y6 are given by expressions (13), (11) and (12), respectively. Expressions for It, Y2 and Y3 are obtained from Y6, I"5 and I"4 by the correspondence (15), M:, N2, K2, L2, F2, B2, A2, P2 are also defined by (15). The whole equation (9) is not presented here, because it is too complicated. It can be changed to more readable form by forming linear combinations of rows. As a result we obtain
det
T2
U~
- B s "k a 3 ( c 4 M 5 - s4F5)
- CaB 2 - $3(c3A 2 -t- s3P2)
U2
Zl
-CHBs't- S3(c4As-s4Ps)
-B2-a3(c3g2"k-s3F2)
1:2
Vl
s4M5
¢4F5
-s3M2 -k c3F2
R2
RI
s4-45 q- c4P5
-s3.42 -k- c3P2
+
=0,
(16)
where
R,=d~c3&+s3a2G,
V,=s3&,
Tl=&(a2+a:3),
U,=a=(G&+c~G&)-d:3&S3,
R2 = d4c4S4 + ha4C4,
V2 = s4S4,
r2 = S4(a4 + a3¢4),
U~ = a4(C3S4 + c4C4S3)- d4s4S3S4.
To solve equation (16) we have to consider various possibilities as follows:
l.S3 ~ O. 1. S~ = $ 4 = 0 .
734
Adolf Karger
We substitute c4 = 0 into (16) and obtain the equation d e t ( - C3B5 - $3s4P5
\
- B2 - a3(c3M2 -~- $3 F2)X) --- 0. - s 3 M z + c3F2 /I
s4M5
Terms at $4c3 and $4s3 yield $3 P5 F2 - a3 M5 M2 = 0,
a3 M5 Fz + s3 P5 M2 = 0.
The determinant of this system is equal to 6 = S~ Ps2 + a3M25 : • Let 6 = 0. Then P5 = M5 = 0, which yields $5 = a5 C5 = 0, which is impossible. Therefore we have F2 = M2 = 0, so St = 0, similarly we obtain $5--- 0. We have obtained the singular robot-manipulator of the type P3P3. 2. $ 2 = 0 , $4=~0. We substitute s3 = 0 into (16) and obtain
V2 R2
det
s4M5 + c4Fs s4 As + c4P5
c3F2 c3P2
= O.
a] Let B2 = 0. Then a2 C2S~ = 0, therefore S~ = 0 and we have ~ = M2 = 0, P2 ¢ 0. Let $5 = 0 and let us compute the component into s2. We obtain $4 = 0, which is impossible. Therefore Ss ~ 0. Let us substitute s4 = 0. Then F s • U2 = 0 which yields a 4C 3 -- a4 6"4 = 0. The remaining equation is
- C 3 B s 4+MS3(c4As , + ()4Fs- s4P5)) =
d e t ( -d4S3l
Components yield d4Ms - Ps = O, daFs + A5 = O. The solution is S, = $2 = a4 = as-----d5 = 0, which is the robot-manipulator P3S3. b] Let B2 ¢ 0. We obtain the equation d. [
s4S4
s4Ms + c4Fs~ = 0
et~d4c4S4+ s4a4C4
s4As d-c4P5/I
"
It yields d4Fs = O. Let S s = 0. The component to s2 leads to $4P5 = 0, which is impossible and therefore we have Ss :/: 0 and d4 = 0. Remaining equations are
S4A s - a 4 C 4 M s=O,
SaP s - a 4 C 4 F s=O.
They give d5 = a4 = as = 0. We substitute this result together with s 5 = 0 into (16) and obtain a simple equation a3B2Fs = 0 which yields a 3 -----0 and we obtain the robot-manipulator R R S4.
3. S2S4 :/: 0. A) ai $1 as :/: 0. We consider equation (1 1), which reads as T2
I
Ut
U2 Ti
det
- C4
- C3 C2 + c3 $2 $3"
-- C3 64 31- c4 S3 S4
- C2
V2
V~
0
0
R2
Rt
V2
VI
= 0.
We substitute s4 = 0 and obtain d4 = a3 = a4 = 0, by symmetry we have also a2 = d3 = 0 and the result is the robot-manipulator R S4 R.
B) a,S, = 0 , asSs S 0 . We consider equation (10) which reads as
det
T2
U1
-C 4
us
Tt
-C C4 + c, S3S,
V2
Vt
0
R2
R,
V2
-- C3B 2 -- $3 (c3A2 + s3P2)] - n 2 - a 3 ( c 3 n 2 q- s3F2) [ =0. - s 3 M 2 + c3F2 | --s3A2+c3P2 ]
(17)
Robot-manipulators with singular configurations
735
We substitute s4 = 0 into (! 7) and obtain
det(:
-c,
- C~C~ + c3S3S~ 0
- C3 B~ - S~ (c3 A ~ + s~ P 2 ) \
- B ~ - a~(c3M2 + s~F2) J = 0 . - s ~ A , + c~P~ /
a] L e t S ~ = 0 . Let us c o m p u t e the c o m p o n e n t to s2. It yields a 3 ----- a 4 = d 4 = 0. Substitution into (17) yields for the s2 c o m p o n e n t a2 = d~---0 and we obtain the r o b o t - m a n i p u l a t o r R $4 R. b] Let aj = 0, S~ #- 0. We substitute s 3 = 0 and consider the c o m p o n e n t to c2. We obtain the equation
d J'a2S3C2 - a3S2C eta, $2 (a2 + a3 c3)
$3A2 - a3C3M2~ B2 + a3 c3 M2 ,} = 0,
which yields az = O. bl] Let d2 4= O. We consider the c o m p o n e n t o f c2 in (17) and substitute c3 = 0. It yields d 4 = a 4 C 3 = O. Substitution o f s4 = 0 yields further a~ = a4 --- 0. The c o m p o n e n t to s2 in (17) leads to 6 = 0 and to the r o b o t - m a n i p u l a t o r R $4 R. b2] Let d2 = O. We substitute s3 = 0 into (17) and c o m p u t e the c o m p o n e n t to s2. The coefficient at c4s4 yields S](a~ - d~$23) = 0, the constant term yields a3d3 = 0 and we obtain the r o b o t - m a n i p u l a t o r $4 R R. C ) a i S l a s S 5 = O.
a] L e t S r = S s = 0 . The c o m p o n e n t to S2SsS3C4S4 in (16) yields a2 = 0, similarly we obtain a 4 = 0. At s2css4 we obtain a3 = 0, c2sss3cas4 yields d3 = 0, similarly we have d4 = 0 and we have a R $4 R robot-manipulator. b] Let a l = a s = 0 . We consider the coefficient at c2sss3c4 in (16) and we obtain a2a4 = O. i) Let az = O, a4 # O. The coefficient at css2s3s 4 yields d2 = 0 . The coefficient at c5s2 is -S4a24a3 + S2S3S4a4d3ds, the coefficient at css2s4 is S2S4a3a4d5 + S2S3a]d3. We obtain the following system o f equations:
-a4"a3 + S4ds" S3d3 = O,
S4ds"a3 + a4" S3d3 = O.
Because a42+ $4d5 2 2 # O, we have a 3 = d 3 = 0 and we obtain a $4 R R robot-manipulator. ii) Let a2 = a4 = 0. The coefficient at c2css3s4cn yields a3d2d5 = 0, the coefficient at c2csc3s~ leads to the equation - a ~ d 2 + S~dzd4 + C a S ~ d 2 d 4 d 5 = 0 . The coefficient at c2css4ca yields d2d4d5 = 0, similarly d2d3ds=O. If a3 = 0, then d2d4 = 0 and d3d s = 0, which gives a $4 or $3S3 robot-manipulator. If a 3 :# 0, then d2 d5 = 0. d2 = d5 = 0 leads to a $3 $3 robot-manipulator. Let d2 = O, d5 # O. Then d3 = 0, the coefficient at c2c5c4s 2 yields - a ~ d 5 + S]dsd~ = 0, therefore a~ = 0, a contradiction. c] Let S ~ = a s = 0 . The coefficient at s 2 c5s3 c3 c4 yields a4 = 0, at s2s5 we obtain a3 = 0. The coefficient at s,_sss~ leads to d4 = 0. If d5 = 0, we have a R R $4 robot-manipulator, therefore let d5 # 0. The coefficient at c2css3s4c4 yields d 3 = 0 , the coefficient at s2css3s4c 4 yields a 2 = 0 and we obtain a R S4R robot-manipulator. lI. $3 = O, a 3 ~ O. A n easy c o m p u t a t i o n shows that (9) changes to (16) similarly as in the case I. We have to consider the following cases: a] Let S: $4 :/: 0. al) Let St $5 # 0. The coefficient at s2sss~ yields a 5 = 0, similarly a~ = 0. The coefficient at c2c5s~ leads to d2 = 0, similarly we obtain d5 = 0. At c2c5s3c 3 we obtain a2 = 0, similarly we have a4 = 0 and we have obtained a Sa $3 robot-manipulator. a2) Let St = 0, 5'5 :# 0.
736
Adolf Karger
At S2SsS~ we obtain a 3 = 0, a contradiction. a3) Let St = $5 = 0. At c2sss3 we obtain alas = 0, a contradiction. b] Let $26=0, $ 4 = 0 . $5 = 0 leads to a P4 robot-manipulator, so we m a y suppose $5 =~ 0. b l ) Let $1 # 0. At c2css3s4c4 we obtain ala4 = 0, a contradiction. b2) Let S~ # 0. The coefficient at s2s4 yields a 4 = 0, a contradiction. c] Let $ 2 = $ 4 = 0 . In this case we have a P4 robot-manipulator and the consideration of all possible cases is finished. Remark. The definition of singular robot manipulator can be extended to an arbitrary Lie group in the following way: Let G be a Lie group of dimension n, g be its Lie algebra. Let X1 . . . . . Arm be elements from g such that X~ and X~+ ~ are linearly independent for i = 1. . . . . m - 1. The mapping r: g --* G given by the formula r(ul . . . . . u,,) = exp(ul Xl ) . . . . . exp(u,,X,.)
is called singular iff its differential is a singular mapping for all values of [u~ . . . . . u~.] e R m. Similarly as in the special case of singular robot-manipulator we would like to find all possible m-tuples of vectors in g, which define a singular mapping r. According to the p r o o f of the classification theorem given above, we see that we always have the following two types o f solutions: a) We have a subsequence X ~ , . . . , X o + p in X~ . . . . . X~ such that vectors Xa . . . . . Xa+p are linearly dependent and they generate some subalgebra of g. (This corresponds to S 4 and P4 robot-manipulators.) b) We have two subsequences xl = Xa . . . . , X,+p and xz = Xa+p+l . . . . . Xa+p+q o f x I . . . . . X,, in g such that there exist subalgebras LI and L 2 of g with the following properties: xt generates Lt, x2 generates L2, the dimension of the intersection of Lt with L2 is at least one. (This corresponds to S 3 S 3, S3P 3, P3 S3, S3 $3 robot-manipulators.) In the case of robot manipulators these two types give all solutions for singular m a p r, it seems to be reasonable to expect similar situation also in the general case, but to my knowledge no results in this direction are known.
1. 2. 3. 4.
REFERENCES A. Karger and J. Novak, Space Kinematics and Lie Groups. Gordon & Breach, NY (1985). A. Karger, Manuscr. Math. 65, 311-328 (1989). A. Karger, Manuscr. Math. 62, 115-162 (1988). J. J. Craig, Introduction to Robotics. Addison-Wesley (1986).
Pc, graph
C L A S S I F I C A T I O N OF R O B O T - M A N I P U L A T O R S W I T H O N L Y SINGULAR CONFIGURATIONS ADOLF KARGER Department of Mathematics, Charles University, Praha, Czech Republic
(Received 29 April 1992; in revisedform 29 November 1994; receivedfor publication 6 December 1994)
1. I N T R O D U C T I O N In this paper we shall find aU serial robot-manipulators which have all configurations singular. Such r o b o t - m a n i p u l a t o r s can be characterized by the condition that their Jacobian matrix has rank smaller than maximal for all locations o f the end effector. F o r simplicity we shall consider r o b o t - m a n i p u l a t o r s with only rotational axes.
2. N E C E S S A R Y
FORMALISM
AND DEFINITIONS
At the beginning we shall present a short review o f the notation used in Sections 3 and 4 o f the paper. Detailed exposition o f this material can be found for instance in [1], where one an find also m a n y references concerning this topic. Let p = A + 2v, 2 e R, be a straight line in the Euclidean space E 3, Ro = {0; e~, e2, e3} be a fixed o r t h o n o r m a l frame in E3. Let us denote by (v~, v2) the usual scalar product in E3, v~ x v2 be the usual vector (cross) p r o d u c t o f arbitrary vectors Vl, v2 in E3, let (v, v ) = 1. Let us depote by X = (v; A x v) the Pliicker-coordinates o f the straight line p in the frame Ro, g(u) be the rotation a r o u n d p with the angle o f rotation u. g(u) can be written as
where g(u) is a 4 × 4 matrix, T(u) is a 3-column and 2:(u) is a 3 x 3 matrix, y ( u ) e S O ( 3 ) . Let us write .~ = g'(u)g-~(u), where the prime denotes the derivative with respect to u. Then ,~ is a 4 x 4 matrix o f the form
where z is a 3-column and x is a 3 x 3 skew-symmetric matrix. Let us define the following correspondence i between skew-symmetric 3 × 3 matrices and 3-columns:
i
(0 "oX)(x / x3
0
\ -- x2
xj
-xl
=
x2 •
(1)
x3
The correspondence i has several useful properties: Let x, y be any two skew-symmetric 3 x 3 matrices, y ~ S0(3). Then
i(xy - - y x ) = i(x) x i(y),
x ' i ( y ) = i(x) x i(y), 727
i(),xy - I ) = ~/g(x).
(2)
728
Adolf Karger
The fundamental correspondence between Pliicker-coordinates X = (v; A x v) of the straight line p and the matrix
z o) of the rotation
g(u) around p is the following: v=i(x),
A ×v=z.
This shows that each straight line p in E3 uniquely defines both X and ,~; in what follows we shall identify both of them with p whenever it will be convenient and by the pair X = (x; z), where (x, x) = 1, (x, z) = 0 we shall understand the appropriate one of the three mentioned objects. Pairs of vectors of the form X = (x; z), where x and z are arbitrary 3-columns, are called screws. The vector space of all screws will be denoted by L. L can be equipped with a structure very similar to the structure of the vector space of ordinary vectors in E3 by using dual numbers. Let us denote by D the set of all pairs ~t = a + eb, where a, b ~ R, e 2 = 0. D is the ring of dual numbers. The pair X = (x; y) from L can be written as a dual vector X d = x + ez, it is a 3-column of dual numbers, L d be the space of all screws written in the form of dual vectors. The ordinary scalar and vector products in E 3 can be extended to L d by using formally the same definitions. Let
(a~) X d = a2 , \a3/
yd
(b~) = b2 b3
be two dual vectors from L d. We define the scalar product by (X a, yd) = y~= ~a~bi and the vector product as X d x y d = (ci), where ci = E3k= t~ijkajbk, i = 1, 2, 3, and e~jk is the usual sign of the permutation tensor used for the definition of the usual vector (cross) product. These two operations in L d induce immediately new operations in L. Let X d = x + ez and yd = y + et be arbitrary screws in L d. Then
(X a, ra) = (x, y) + e[(x, t) + (y, z)].
(3)
Let X ~ be the dual form of a screw X = (x; z) which corresponds to a straight line p. Then (X d, X a) = (x, x) + 2e(x, z) = 1. This shows that screws corresponding to straight lines are unit vectors in L a. Let X d and Ya be two unit vectors in L d, which correspond to straight lines p and q, respectively. We can write (X ~, ya) = cos tp + ed sin ~o,
(4)
where ~0 is the angle of p and q and d is their distance (both oriented). Each screw can be written as the sum of two special screws, X = X~ + X2, where x = (x; z ) = (x; z - (x,
z)x) + (0; (x, z)x),
(5)
where we for simplicity suppose that (x, x ) = 1. XI = (x; z - ( x , z)x) in (5) corresponds to the rotational part of the screw X, X2 = (0; (x, z)x) is its translational part; (x, y) is the pitch, XI yields the Pliicker-coordinates of the axis of the corresponding screw motion. The vector (cross) product in L d induces a vector product in L which is expressed as follows: Let .~d, y d E L d, z d = x ' d × yd, X = ( x , z ) , Y = ( y ; t). Then Z=(x
x y;x x t+z xy).
(6)
If X and Y correspond to straight lines p and q, respectively, then the rotational part of Z corresponds to the axis (common perpendicular) of p and q. The correspondence i between skew-symmetric 3 x 3 matrices and ordinary vectors in E 3 can be extended to dual vectors as well.
Robot-manipulators with singular configurations
729
Let us denote by C6 the group of all orientation preserving congruences of the Euclidean space E3. Let
where T is the 3-column of the translation, 7 e S O ( 3 ) is the 3 x 3 orthogonal matrix of the rotational part. The group SO(3) can be extended to dual numbers by the formula gd = (E + i-~ (T)e)7, where E is the unit matrix. Elements gd form a group denoted by S 0 ( 3 , D). It is the special orthogonal group over dual numbers, it is defined by the condition gd.(gd)T= E and it is easy to show that the described correspondence between C6 and S 0 ( 3 , 1)) is an isomorphism (see [1] for instance). Let us describe now the action of groups C6 and S 0 ( 3 , D) in L and L d, respectively. Let p = A + 2v be a straight line in E3,
be a congruence of E3. Let us apply g to p. Then the corresponding straight line g(p) will be given by g ( p ) = T + T A +27v. Let X = ( x ; z ) be the Plficker-coordinates of p, x = v , z = A x v. The Pliicker-coordinates of g(p) are given by g(X), where g(X) = (~,x; T x 7x + 7z). On the other side the multiplication of the vector X d = x + ez by the dual matrix g d = (E + i-J(T)e)7 yields gd'Xd = (E + i-'(T)e)7(x + ez) = 7z + e(Tz + T x 7x) = [g(X)] d. We shall call the action of the group (76 in the screw space L determined by the natural action of C 6 in E 3 the adjoint action of C6 in L, Ad(g)X. The adjoint action is expressed either by multiplication of dual vectors by dual matrices from S 0 ( 3 , D) as we have shown above or by the formula A d ( g ) X = gXg -~ in the representation of screws by 4 x 4 matrices with help of (1) as we can see from (2). Properties of the ordinary cross product remain valid also for the vector product of dual vectors, we shall use the property u × (v × w ) = (u, w)v - (u, v)w,
which is true for ordinary vectors as well as for dual ones. Let X ~ , . . . , Xp be straight lines in E3 such that X~ # Xi+~, let gi(u~) be the rotation around the axis Xi, ui be the angle of rotation. The motion of the p-parametric robot-manipulator defined by axes Xj . . . . ,Xp is the motion of /?3 determined by the matrix function g(u~ . . . . . up) = gz (u~) . . . . gp(U.). The instantaneous position of axes Xj . . . . . Xp is determined by straight lines with Pliicker-coordinates Y~ . . . . . Yp, where (see for instance [2]) Y, = X,,
Y2 = Ad(g2)X, . . . . . Y. = Ad(g). . .g,_,)Xp.
This leads to the following Definition
A configuration Y~ . . . . , Yp of axes of the robot-manipulator defined by axes X~ . . . . . Xp is called singular iff the dimension of the vector space generated by vectors Y~ . . . . . Yp in L is less than p. A robot-manipulator will be called singular iff it has only singular configurations of its axes. The rank of a singular robot-manipulator is the maximum of all dimensions of spaces generated by r, . . . . . yp. 3. S T A T E M E N T OF THE C L A S S I F I C A T I O N T H E O R E M
Singular robot-manipulators with rotational axes only are the following ones: (1) Rank 3: (a) Xj . . . . . X r pass through one point, p > 3, (b) Xz . . . . , Xp are parallel, p > 3. (2) Rank 4: Robot-manipulator of rank 3 with one more arbitrary axis at the beginning or at the end. (3) Rank 5: (a) Robot-manipulators of rank 3 with two more arbitrary axes at the beginning or at the end or with one at the beginning and one at the end. (b) Robot-manipulators defined by
Adolf Karger
730
////
p=4
\\\\
p=5
X I or X 5
/ X 1 or X 5
////
p=6
Fig. 1.
axes X ~ , . . . , Xm, X,, + t . . . . , Xm +,, where m >~ 3, n >/3 and all axes in the group X~ . . . . , Xm and in the group Xm +~ . . . . . Xm +, are either parallel or pass through one point. Figure 1 shows axes of singular robot manipulators for p = 4, p = 5 and p = 6. Remark. Let us denote by P, the robot-manipulator defined by r parallel axes (the planar robot-manipulator), by S, the robot-manipulator defined by r axes which pass through one fixed point (the spherical robot-manipulator), R denotes as usually one rotational axis. Then the singular robot-manipulators are described as follows: (1) S,: P,, r > 3,
(2) RS,, SrR; RP,, P,R, r > 3, (3a) RS, R, RRSr, SrRR; RPrR, RRPr, PrRR; r > 3, (3b) S=S.; S=P,; ProS,; PmPn; m >/ 3, n >~ 3. 4. P R O O F OF T H E T H E O R E M At first we shall show that it is enough to prove the classification theorem for 6-parametric robot-manipulators (p = 6). If axes Xt . . . . . Xp determine a singular robot-manipulator, then the robot-manipulator with one more arbitrary axis at the beginning or at the end remains singular. This means that singular robot-manipulators with p < 6 are partial robot-manipulators of 6-parametric robot-manipulators. On the other side, if p > 6, then each six axes from X ~ , . . . , Xp must determine a singular robot-manipulator as well, because by adding new axes we cannot decrease the rank of the robot-manipulator. This means that we can restrict ourselves to 6-parametric robot-manipulators. Let X1 . . . . . X6 be the initial position of axes of such a robot-manipulator. Then the instantaneous configuration of axes of this robot-manipulator is given by straight lines ~1 . . . . . ~'~ with Pliicker-coordinates
L = x,,
72 = Ad(g~(ui))X2 . . . . .
?, =
Aa(~,(u~)...g~(u~))X6,
where (gi(ui)) is the rotation around the axis Xi with the rotation angle ui. The equation for singular robot-manipulators is det(~71 . . . . . Y6)= II?~. . . . . Y61 = 0 .
(7)
Robot-manipulators with singular configurations
731
We can write equation (7) in more symmetrical form--the configuration of axes of the robot-manipulator is independent of space congruences and therefore we can apply the congruence given by the matrix gzt(uz)g~J(u~)g~(u3) to vectors 17~. . . . . Y6. We obtain vectors Y~. . . . . Y6, where Y, = Ad(g2g3)-'Xj, Y5 = Ad(g4)Xs,
Y2= Ad(g~')X2,
}:3=)(3,
Y4=X4,
]:6 = Ad(g4gs)X6,
(8)
and the equation for singular robot-manipulators reads as l Y e , . . . , Y6I = 0
for all [ u l , . . - , u 6 ] ~ R 6-
(9)
Equation (6) is very complex one in the expanded form and not suitable for further computations even if we use computer algebra. Its expanded form has more than 1000 terms. To solve it, we shall simplify it at first by using some theoretical considerations. We shall show that it implies another equation, which is simpler and which can be explicitly solved. Lemma. Let X = (x; z), Y = (y; t) be Pliicker-coordinates of two straight lines. Then X × (X × Y) = cos ~pX - Y + ed sin ~pX, where e(x; z ) = (0;x), q~ is the angle of X and Y, d is the distance of X and Y. Proof. We have X × ( X x Y) = (X, Y ) X - (X, X) Y = (cos ~o + ed sin ~o)X - Y. Lemma. ~/~u5 (Y6) = Y5 × Y6. Proof. Y6 = g a g s ] ( 6 , O/Ous(Y6) = g4g'sX6 = g,g'sgs-' "gsX6 = g4{X~ × (gsX6) d} = g4X~ × g, gsX d = Y5 × Y6, where we use the representation by dual vectors and matrices. Lemma. ~2/(Ous)ZlYi . . . . . Y61 = IYI . . . . . Ys, eYsla5 sin ~5, where a5 and ot5 is the distance and angle o f X5 and :(6, respecthJely. From the last Lemma we see that (9) implies a5 sin 75[YI . . . . . Ys,eYs[ = 0 ,
(lO)
and by repeating the argument also a~ sin ~ as sin ~t~[e Y2, Y2 . . . . . Ys , e Ys I = 0 .
(ll)
Before we start with detailed computations, we shall simplify our denotations in the standard way (by using Denavit-Hartenberg parameters): Let us denote Ci = cos ct~, S~ = sin cti, where ~t~ is the angle from X~ to X~+ t, ai is the distance from Xi to ~ + t, d~ denotes the distance from the axis of X,_ t and X~ to the axis of X~ and Xi+ ~ (the offset). (The axis of two parallel straight lines is any straight line which intersects both of them perpendicularly.) Ci ~
COS
lli,
Si
= sin/,/i.
Remark. We note also that the inverse robot-manipulator to a singular robot-manipulator is also singular, because the configuration of axes of the inverse robot-manipulator is the same. The inverse robot-manipulator is obtained by interchanging the role of the base and end-effector of a robot-manipulator. For the initial position of the robot-manipulator we choose such a position (configuration of axes), in which all axes are parallel to a given fixed plane. For the basic orthonormal frame Ro = {O; e~, e2, e3 } in the base space E3 we choose the symmetry frame of axes )(3 and )(4. This means that the origin O lies on the axis of ,t"3 and X4 at the same distance from both of them, vectors e~ and e2 cut the angle between :(3 and )(4 by half. e3 has the direction of the axis of )(3 and ):4.
732
Adolf Karger
Let us have a straight line p = A + 2):, 2 e R, where y = (cos ct, sin 0~, 0), A is a point o f p, z = A x y. X = (y; z) ~ L be the corresponding vector from L. Let us denote again
.u,(; the rotation a r o u n d the straight line p with the angle o f rotation u, where ~ e SO(3), T e R 3. Let further
sin m(0~) =
si 0
r(u)=
COS0 Gt
cosu
,
-sinu
sin u
•
cos u /
We obtain immediately
= m(o~)r(u)m(--oQ. Because g(u) preserves the point A, we have
which yields T = (E - y)A. N o w we are prepared to c o m p u t e the expressions for vectors
Y~,..., r'6. Let us have Xi = (Yi; zi), where Ai and yi = rn(0Qel are known. Then
Ai+ j = Ai + aie3 + diy~, Yi+~ = m(~i)yi. Let us find at first the vector X :+ ~ = (y~; z:) = A d(g~(u~))X~+ ~ where z: is determined by the point A ~+ 7. Then
Y :+ ~ = ~:iYi+1,
.4 :+ l -~ Ai "q- aiYie3 q" diYi = Ai q- aim(ct )r(ui)e3 -k- dim(~ )el,
where Yi = m(~t)r(ui)m(-ct). This yields
y: = m(~)r(ui)m(-o~)rn(~i)m(~)el. --
I
t
t
F o r I:5 we have ~ - ~g3, A, = ½a3e3 and for Y5 = (Y~; z~), where z~ is determined by As, we have:
Y'5 = rn(½0~3)r(u,)rn(~4)e,,
t
I
A~ = 5a3e3 + a.m(½g3)r(u4)e3 + d4m(½g3)e,.
C o m p u t a t i o n yields //£ C4 -- 0"C4S 4 ;
I:5 = ~ 0C4 -~ Kc4S4
\
s.S4
xG4 + trH4\
-
I -
-
(12)
o-G4 - ~;H4/, R2
/
where Ga = S4(a4 + l a 3 c 4 ) , i g = COS ~0~3,
/-/4 = s4S4d4-- G(½a3 + a, c4),
R2 = d4c4S, + a, s4G,
O" = sin ½~3.
Analogically we obtain
y'6=75y6,
A;=As+as•se3+dsys,
y6=m(½0c3+oq+ots)el,
y'6 = rn(½~ 3 + ~4)r(us)rn(~q)el,
Y'6' = Y4Y'6 = m (½~t3)r (u4)m (oe,)r (us)m (or5)el = a" b . m (oc5)el,
As=(½a3+a4)e3+d4y4,
Robot-manipulators with singular configurations
733
where a = m(½o~3)r(u4), b = m(o~4)r(us),
A; = (½a, + a4)e3 + d4m(½~3)e, + asm(½~q + ~4)r(us)e3 + dsm(½0t3 + ot4)e,, A: = T4 + ?4A~ = ½a3e3 + a(a4e3 + d4el) + ab(ase3 + dsem). Computation yields y~ =
- ~Ls - tr(c, Ms - s, Fs) 7 - e L 5 + x(c4Ms - s4Fs)l S 4 Ms + c4Fs J
I
,
Ix[Bs - 2 (c, Ms - s, Fs)] - tr(c4A, - s4Ps - 2 Ls) t¢
tr[Bs -
26
(13)
(c4M5 - s4Fs)] + x(c4A5 - s4Ps -- -~ Ls), s4A s q- c4P s
where
Ms=G&c,+S4G,
N s = S , Gcs+C4Ss,
Ls= S4Sscs- C4Cs, Fs= ssSs, As = -a4Ls + asKs -
F4(d4 +
Ks=GGcs-S4&,
Bs = - a 4 M s - a s N s + dsS4Fs,
C4ds), P~ = asCsss + dsSsc4 + d4Ms.
We have r/.
It
Y6 = (y6, z6).
Finally
1:4= X4 =
a3V ½a~
•
(14)
Expressions for Y~, Y2, Y3 are obtained by the substitution ai--+--ot6_i,
ai---*--a6_i,
ui---~--u7_i,
(15)
di--+d7_i.
Now we are ready to write down equation (9) explicitly. The determinant IY~. . . . . I:6[ has its columns composed from components of vectors Y~. . . . . I:6. Vectors I:4, Ys and Y6 are given by expressions (13), (11) and (12), respectively. Expressions for It, Y2 and Y3 are obtained from Y6, I"5 and I"4 by the correspondence (15), M:, N2, K2, L2, F2, B2, A2, P2 are also defined by (15). The whole equation (9) is not presented here, because it is too complicated. It can be changed to more readable form by forming linear combinations of rows. As a result we obtain
det
T2
U~
- B s "k a 3 ( c 4 M 5 - s4F5)
- CaB 2 - $3(c3A 2 -t- s3P2)
U2
Zl
-CHBs't- S3(c4As-s4Ps)
-B2-a3(c3g2"k-s3F2)
1:2
Vl
s4M5
¢4F5
-s3M2 -k c3F2
R2
RI
s4-45 q- c4P5
-s3.42 -k- c3P2
+
=0,
(16)
where
R,=d~c3&+s3a2G,
V,=s3&,
Tl=&(a2+a:3),
U,=a=(G&+c~G&)-d:3&S3,
R2 = d4c4S4 + ha4C4,
V2 = s4S4,
r2 = S4(a4 + a3¢4),
U~ = a4(C3S4 + c4C4S3)- d4s4S3S4.
To solve equation (16) we have to consider various possibilities as follows:
l.S3 ~ O. 1. S~ = $ 4 = 0 .
734
Adolf Karger
We substitute c4 = 0 into (16) and obtain the equation d e t ( - C3B5 - $3s4P5
\
- B2 - a3(c3M2 -~- $3 F2)X) --- 0. - s 3 M z + c3F2 /I
s4M5
Terms at $4c3 and $4s3 yield $3 P5 F2 - a3 M5 M2 = 0,
a3 M5 Fz + s3 P5 M2 = 0.
The determinant of this system is equal to 6 = S~ Ps2 + a3M25 : • Let 6 = 0. Then P5 = M5 = 0, which yields $5 = a5 C5 = 0, which is impossible. Therefore we have F2 = M2 = 0, so St = 0, similarly we obtain $5--- 0. We have obtained the singular robot-manipulator of the type P3P3. 2. $ 2 = 0 , $4=~0. We substitute s3 = 0 into (16) and obtain
V2 R2
det
s4M5 + c4Fs s4 As + c4P5
c3F2 c3P2
= O.
a] Let B2 = 0. Then a2 C2S~ = 0, therefore S~ = 0 and we have ~ = M2 = 0, P2 ¢ 0. Let $5 = 0 and let us compute the component into s2. We obtain $4 = 0, which is impossible. Therefore Ss ~ 0. Let us substitute s4 = 0. Then F s • U2 = 0 which yields a 4C 3 -- a4 6"4 = 0. The remaining equation is
- C 3 B s 4+MS3(c4As , + ()4Fs- s4P5)) =
d e t ( -d4S3l
Components yield d4Ms - Ps = O, daFs + A5 = O. The solution is S, = $2 = a4 = as-----d5 = 0, which is the robot-manipulator P3S3. b] Let B2 ¢ 0. We obtain the equation d. [
s4S4
s4Ms + c4Fs~ = 0
et~d4c4S4+ s4a4C4
s4As d-c4P5/I
"
It yields d4Fs = O. Let S s = 0. The component to s2 leads to $4P5 = 0, which is impossible and therefore we have Ss :/: 0 and d4 = 0. Remaining equations are
S4A s - a 4 C 4 M s=O,
SaP s - a 4 C 4 F s=O.
They give d5 = a4 = as = 0. We substitute this result together with s 5 = 0 into (16) and obtain a simple equation a3B2Fs = 0 which yields a 3 -----0 and we obtain the robot-manipulator R R S4.
3. S2S4 :/: 0. A) ai $1 as :/: 0. We consider equation (1 1), which reads as T2
I
Ut
U2 Ti
det
- C4
- C3 C2 + c3 $2 $3"
-- C3 64 31- c4 S3 S4
- C2
V2
V~
0
0
R2
Rt
V2
VI
= 0.
We substitute s4 = 0 and obtain d4 = a3 = a4 = 0, by symmetry we have also a2 = d3 = 0 and the result is the robot-manipulator R S4 R.
B) a,S, = 0 , asSs S 0 . We consider equation (10) which reads as
det
T2
U1
-C 4
us
Tt
-C C4 + c, S3S,
V2
Vt
0
R2
R,
V2
-- C3B 2 -- $3 (c3A2 + s3P2)] - n 2 - a 3 ( c 3 n 2 q- s3F2) [ =0. - s 3 M 2 + c3F2 | --s3A2+c3P2 ]
(17)
Robot-manipulators with singular configurations
735
We substitute s4 = 0 into (! 7) and obtain
det(:
-c,
- C~C~ + c3S3S~ 0
- C3 B~ - S~ (c3 A ~ + s~ P 2 ) \
- B ~ - a~(c3M2 + s~F2) J = 0 . - s ~ A , + c~P~ /
a] L e t S ~ = 0 . Let us c o m p u t e the c o m p o n e n t to s2. It yields a 3 ----- a 4 = d 4 = 0. Substitution into (17) yields for the s2 c o m p o n e n t a2 = d~---0 and we obtain the r o b o t - m a n i p u l a t o r R $4 R. b] Let aj = 0, S~ #- 0. We substitute s 3 = 0 and consider the c o m p o n e n t to c2. We obtain the equation
d J'a2S3C2 - a3S2C eta, $2 (a2 + a3 c3)
$3A2 - a3C3M2~ B2 + a3 c3 M2 ,} = 0,
which yields az = O. bl] Let d2 4= O. We consider the c o m p o n e n t o f c2 in (17) and substitute c3 = 0. It yields d 4 = a 4 C 3 = O. Substitution o f s4 = 0 yields further a~ = a4 --- 0. The c o m p o n e n t to s2 in (17) leads to 6 = 0 and to the r o b o t - m a n i p u l a t o r R $4 R. b2] Let d2 = O. We substitute s3 = 0 into (17) and c o m p u t e the c o m p o n e n t to s2. The coefficient at c4s4 yields S](a~ - d~$23) = 0, the constant term yields a3d3 = 0 and we obtain the r o b o t - m a n i p u l a t o r $4 R R. C ) a i S l a s S 5 = O.
a] L e t S r = S s = 0 . The c o m p o n e n t to S2SsS3C4S4 in (16) yields a2 = 0, similarly we obtain a 4 = 0. At s2css4 we obtain a3 = 0, c2sss3cas4 yields d3 = 0, similarly we have d4 = 0 and we have a R $4 R robot-manipulator. b] Let a l = a s = 0 . We consider the coefficient at c2sss3c4 in (16) and we obtain a2a4 = O. i) Let az = O, a4 # O. The coefficient at css2s3s 4 yields d2 = 0 . The coefficient at c5s2 is -S4a24a3 + S2S3S4a4d3ds, the coefficient at css2s4 is S2S4a3a4d5 + S2S3a]d3. We obtain the following system o f equations:
-a4"a3 + S4ds" S3d3 = O,
S4ds"a3 + a4" S3d3 = O.
Because a42+ $4d5 2 2 # O, we have a 3 = d 3 = 0 and we obtain a $4 R R robot-manipulator. ii) Let a2 = a4 = 0. The coefficient at c2css3s4cn yields a3d2d5 = 0, the coefficient at c2csc3s~ leads to the equation - a ~ d 2 + S~dzd4 + C a S ~ d 2 d 4 d 5 = 0 . The coefficient at c2css4ca yields d2d4d5 = 0, similarly d2d3ds=O. If a3 = 0, then d2d4 = 0 and d3d s = 0, which gives a $4 or $3S3 robot-manipulator. If a 3 :# 0, then d2 d5 = 0. d2 = d5 = 0 leads to a $3 $3 robot-manipulator. Let d2 = O, d5 # O. Then d3 = 0, the coefficient at c2c5c4s 2 yields - a ~ d 5 + S]dsd~ = 0, therefore a~ = 0, a contradiction. c] Let S ~ = a s = 0 . The coefficient at s 2 c5s3 c3 c4 yields a4 = 0, at s2s5 we obtain a3 = 0. The coefficient at s,_sss~ leads to d4 = 0. If d5 = 0, we have a R R $4 robot-manipulator, therefore let d5 # 0. The coefficient at c2css3s4c4 yields d 3 = 0 , the coefficient at s2css3s4c 4 yields a 2 = 0 and we obtain a R S4R robot-manipulator. lI. $3 = O, a 3 ~ O. A n easy c o m p u t a t i o n shows that (9) changes to (16) similarly as in the case I. We have to consider the following cases: a] Let S: $4 :/: 0. al) Let St $5 # 0. The coefficient at s2sss~ yields a 5 = 0, similarly a~ = 0. The coefficient at c2c5s~ leads to d2 = 0, similarly we obtain d5 = 0. At c2c5s3c 3 we obtain a2 = 0, similarly we have a4 = 0 and we have obtained a Sa $3 robot-manipulator. a2) Let St = 0, 5'5 :# 0.
736
Adolf Karger
At S2SsS~ we obtain a 3 = 0, a contradiction. a3) Let St = $5 = 0. At c2sss3 we obtain alas = 0, a contradiction. b] Let $26=0, $ 4 = 0 . $5 = 0 leads to a P4 robot-manipulator, so we m a y suppose $5 =~ 0. b l ) Let $1 # 0. At c2css3s4c4 we obtain ala4 = 0, a contradiction. b2) Let S~ # 0. The coefficient at s2s4 yields a 4 = 0, a contradiction. c] Let $ 2 = $ 4 = 0 . In this case we have a P4 robot-manipulator and the consideration of all possible cases is finished. Remark. The definition of singular robot manipulator can be extended to an arbitrary Lie group in the following way: Let G be a Lie group of dimension n, g be its Lie algebra. Let X1 . . . . . Arm be elements from g such that X~ and X~+ ~ are linearly independent for i = 1. . . . . m - 1. The mapping r: g --* G given by the formula r(ul . . . . . u,,) = exp(ul Xl ) . . . . . exp(u,,X,.)
is called singular iff its differential is a singular mapping for all values of [u~ . . . . . u~.] e R m. Similarly as in the special case of singular robot-manipulator we would like to find all possible m-tuples of vectors in g, which define a singular mapping r. According to the p r o o f of the classification theorem given above, we see that we always have the following two types o f solutions: a) We have a subsequence X ~ , . . . , X o + p in X~ . . . . . X~ such that vectors Xa . . . . . Xa+p are linearly dependent and they generate some subalgebra of g. (This corresponds to S 4 and P4 robot-manipulators.) b) We have two subsequences xl = Xa . . . . , X,+p and xz = Xa+p+l . . . . . Xa+p+q o f x I . . . . . X,, in g such that there exist subalgebras LI and L 2 of g with the following properties: xt generates Lt, x2 generates L2, the dimension of the intersection of Lt with L2 is at least one. (This corresponds to S 3 S 3, S3P 3, P3 S3, S3 $3 robot-manipulators.) In the case of robot manipulators these two types give all solutions for singular m a p r, it seems to be reasonable to expect similar situation also in the general case, but to my knowledge no results in this direction are known.
1. 2. 3. 4.
REFERENCES A. Karger and J. Novak, Space Kinematics and Lie Groups. Gordon & Breach, NY (1985). A. Karger, Manuscr. Math. 65, 311-328 (1989). A. Karger, Manuscr. Math. 62, 115-162 (1988). J. J. Craig, Introduction to Robotics. Addison-Wesley (1986).