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An equilateral tetrahedral mechanism
Robotics and Autonomous Systems 9 (1992) 227-236 Elsevier
227
An equilateral tetrahedral mechanism Paul J. Zsombor-Murray and Alexander Hyder Dept. of Mechanical Engineering McRCIM, McGill University, 817 Sherbrooke St. W, Montreal, QC, Canada H3A 2K6
Abstract Zsombor-Murray, P.J. and Hyder, A., An equilateral tetrahedral mechanism, Robotics and Autonomous Systems, 9 (1992) 227-236. Input-output kinematics of a novel, exoskeletal manipulator mechanism has been analysed. The abstract model is two equilateral tetrahedral wire frames which move relatively so that their edges remain in intersecting, parallel or coincident pairs. A mechanical, rigid body model was constructed and a computer animation was performed.
Keywords: Mechanism; Kinematics; Mobility; Workspace; Parallel; Overconstrained.
1. Introduction
The mobility of two equilateral tetrahedral wire frames, assembled so that the edges are constrained to form six intersecting pairs, was discov-
. . . . . .
_
Paul J. Zsombor-Murray is Associate Professor in ME at McGill University. His three degrees, Ph.D. (71), were obtained there. He conducts research in the Robotic Mechanical Systems group of the McGill Research Centre for Intelligent Machines and holds membership in IEEE (SM), ASME, ASEE and l'Ordre des Ing6nieurs du Qu6bec (PE). His professional interests include machine design, computer graphics, automata and geometric modeling.
Alexander Hyder is Senior Engineer at McDonnell Douglas Space Systems Company in Houston where he worked on dynamic simulation and :~ control of space robotic systems. He now supports NASA's Mission Operations Directorate and contributes to ,~ planning of robotic operations on Space Station FREEDOM. Hyder obtained his B.Eng (85) and M.Eng (89) in ME from McGill University. ~; ~ This article is based on his Master's research. Before that, as research engineer in McGill's Biomechanics Lab, he designed and conducted in-vitro experiments on post-mortum human specimens to study the dynamics of musculoskeletal systems.
ered in 1982 by Louis Tompos, a student of Ern6 Rubik at the Hungarian Academy of Craft and Design. Tompos took the model to the Hungarian Institute for Building Science and showed it to Tibor Tarnai who together with E. Makai, a mathematician of the Hungarian Academy of Science, proceeded to demonstrate the nature of possible motions of the equilateral set [1,2] and, recently, of more general, similar sets [3,4,5]. Stachel [6], at the Institut fiir Geometrie, Technical University, Vienna, derived independently a geometric proof pertaining to finite motions of the movable frame. By way of illustration he mapped typical surfaces (loci) traced by (a) the centre point of an edge, and (b) by a vertex. Apparently, Stachel and the Hungarians regarded the tetrahedral pair only as a curious toy and a challenging exercise in mobility analysis. They did not consider any serious physical implementation or possible application of such a mechanism. In this work we take a mechanistic approach. (1) A perfectly equivalent mechanism, composed of rigid bodies and exhibiting much of the mobility of the theoretical wire frames was designed and built and will be described. (2) An analysis is carried out and a computer animation of the wire frame set is described. It should be noted that this algorithmic procedure would be equally well suited to commanding displacements of a pow-
0921-8890/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
228
t'..I. Zsombor-Murrav, A. t@der
ered mechanism model. (3) Certain limited motions are described in terms of the simplest possible geometric invariants and parameters so that direct and inverse kinematics, i.e., Jacobians, are easily obtained. (4) Some possible applications and avenues of further research are outlined. Hyder [7] treated (1) and (2), above, in detail.
2. Rigid body model mechanism The mobile rigid body assembly of two tetrahedral flames shown in Fig. 1 is believed to be the only existing one. It was built to help visualize the motions, investigate design, fabrication and assembly problems and to plan the ultimate motorization and control of this mechanism. In any physical implementation it is difficult to satisfy the requirement that the flames be exactly congruent but one be located an infinitessimat distance 'inside' the other. The device in Fig. 1 overcomes this problem. Eight aluminum vertex blocks can be seen there. The cylinder has three symmetrical flats milled on it such that the three normals make an angle of t a n - ~ (1/v~-) with the cylinder axis. A hole is drilled normal to each flat such that the three hole axes intersect on the cylinder axis. The edges consist of 1 × 1 / 8 T H K aluminum angle. Three are seen to converge on each vertex block. Also visible there, mounted at each end of every edge angle, in the vee concavity with a metallic tangent screw garter, is an edge axis pin. This 1• plastic cylinder has a portion of its length reduced to 3 / 8 Q which protrudes into one of the vertex block holes. Thus, when the garters are tightened upon assembly, vertex blocks are held together with three adjacent edges by the axis pins in a 'three fingered bowler's grip'. Edges are flee to rotate about vertex block h o l e / e d g e axis pin axes. These axes comprise the virtual edges of the abstract wire frame model. Finally, sliding at the six edge pair intersections is effected by assembling the pair of tetrahedra with inside/outside parity. The inside partner has its edge angles turned concavely outward while the outer tetrahedron's angles; vee faces in. Although not evident in Fig. 1, each edge pair intersection cusp embraces a 1Q plastic ball girdled by a plastic washer to keep the angle lips parallel to the plane of intersection of the virtual edge axes.
Fig. 1. Double tetrahedral mechanism.
Figs. 2a and 2b show somewhat more clearly than photographs the way in which 8 vertex blocks 12 structural angles 24 axis pins 24 garters 6 balls and 6 washers are assembled so as to constitute the mechanism. This is an easy-to-build design but it is by no means ideal. The 'bowler's grip' is not positive;
An equilateral tetrahedral mechanism
VERTEXBLOCK BALL _R ~AR ~TE ,~I ~
(b)
UTWARD ~/X~ O FACING INWARD "~ ANGLE FACING ~XIS PIN ANGLE
Fig. 2. Solid model of the assembly and details.
notice the guy wires wrapping the outer member's vertices to control axial play at each axis pin. Sliding, rather than rolling, at the plastic balls introduces undue friction; motion is stiff, not smooth. Furthermore the vertex block design precludes coincidence of vertex pairs; a condition required for some theoretically valid poses of this mechanism. Nevertheless it should be quite easy to adapt three tractive, i.e., tensile, and three rotary actuators on each vertex block so as to motorize the sliding balls and to turn the edge angles. The preliminary model admirably serves to help visualize motion and as an exploratory basis upon which to found better designs.
ners of a cube to form a stella octangula or 8-pointed 3-D star. All edges are bisected at intersection with their pair mate as shown in Fig. 3. Each edge occupies a cube face diagonal. Mobility can be reduced to three types of rotation of one frame, called the movable one, with respect to the other, arbitrarily fixed one. The rotations are classified according to axial direction as follows: (1) Perpendicular to a face of the conceptual cube (CC); there are three such directions, (2) Parallel to an edge or face diagonal; there are six such directions, and (3) Parallel to cubic body diagonal; there are four. Rotation of type 1 may proceed continuously through +90 ° until the assembly assumes one of six immediately adjacent singularities called 'coincident positions' (CP) [1,2]. There are twelve such singularities and the other six are adjacent to the 'counter-basic position' (CBP) [1,2]. This other stella octangular singularity occurs when a type 1 rotation continues, from BP, through + 180 °. During a type 1 rotation an opposite edge pair of each frame remains bisected though, obviously, not orthogonal. The frame centroids remain coincident. A type 2 rotation may proceed continuously to a maximum angle of t a n - l v ~ ---- 54.7 °, the angle between a tetrahedral edge and face. This results in a 'terminal vertex singularity' (TV) wherein a single pair of vertices coincide. There are 24 of these, 12 immediately attainable via similar rotations from BP, the other 12 similarly adjacent to CBP. During a type 2 rotation two edge pairs
-
'7.-
3. Mobility and singularity The description of mechanism mobility starts from a key symmetrical singular configuration called the 'basic position' (BP) [1,2]. H e r e the vertices of the two frames occupy the eight cor-
229
Fig. 3. Basic position/conceptual cube.
230
P.Z Zsombor-Murray, A. Hyder
remain orthogonal and one edge on each frame remains bisected. Frame centroids do not remain coincident. Type 3 rotation has a range of _+60 °, ending in one of eight 'terminal face singularities' (TF) adjacent to BP. Another eight occur at 60 ° from CBP. At TF three vertex pairs and a face pair become coincident and the assembly assumes an equilateral hexahedral form. The centroid of the moving frame translates in a direction parallel to the type 3 axis of rotation. At T F centroids are separated by 3 ~ / 2 where p is the edge length of CC, not the tetrahedral edge length which is 2 ~ . In the following analysis CC is normalized to p = 2. During type 3 rotation, no orthogonal or bisected edge pairs survive. The three rotation types appear to be motions of one degree of freedom, i.e., sustained by only one of the 24 (12 linear/tractive ball and 12 rotary edge angle) possible actuation modes. However, for type 1 motion at least two actuators must be locked to maintain a pair of intersecting edges in midpoint contact. Similarly for type 2 motion at least one actuator must be locked so that one of a pair of intersecting edges maintains midpoint contact with its partner. Although, in principle, one actuator can continue the frame rotation from BP or CBP to any immediately adjacent CP, TV or TF, a minimum of three (two for type 2) actuators are required to initiate an excursion from or through a basic singularity. To maintain any branch of type 1 motion at least one actuator must act and two must be locked. These are actually special three-(and two)-degree-offreedom cases discussed in Section 4.4.
i
1\;,1
//.:.
~ (1,1,1)
-. .\\
Q4 ( - 1 , - 1 , - 1 )
3
1,-1)
(1,-1,-~) Fig. 4. Base coordinates of frames P and Q.
4.2. Compatibility conditions For any valid displacement of Q, the six intersecting tetrahedral edge pairs are
PrP,QtQ,, for all subscript combinations r = 1, 2, 3, r < s < 4, r + s + t + u = 10 and t < u < 4. The intersection requirement is satisfied if the volume of the tetrahedron PrP,QtQ. vanishes. Since P is fixed, the six constraint equations, ( l a ) - ( l f ) , can be writfen in terms of the coordinates of Qt and Qu only. ( z , - 1 ) ( x 2 - y a ) - ( z 2 - 1 ) ( x , - y , ) = 0,
(la)
( x 2 - 1 ) ( y 3 - - Z 3 ) -- ( X 3 -- l ) ( Y 2 - - Z 2 ) = 0 ,
(lb)
(Y3- l)(z,-Xl)
-- ( y , -
1 ) ( z 3 - x 3 ) = O, ( l c )
( x 4 + t ) ( y , + z , ) - (x, + 1)(Y4 + z4) = 0, ( l d )
4. Analytic solution: Rigid body motion (Y4 + 1 ) ( Z 2 + X 2 )
4.1. Terminology and base coordinates Adopting Tarnai's excellent formulation [2], one begins with a BP and labels the eight corners of CC P1, /92, P3, 1°4 and QI, Q2, Q3, Q4. P is the fixed flame and Q is the movable one. The normalized CC has edges of length p = 2 and is centred at the origin of Cartesian coordinates whose axes are each parallel to four edges. This is shown in Fig. 4 along with the coordinates P and of Q, in this BP.
(Z4 -}- 1)(X3 +Y3)
(ae) lt) -- (2:3 q- ])(x4 +y4) =0. -- (Y2
+ 1)(z4 +x4) =0,
These six along with six others, which simply state that the edges of Q are of a fixed length 2v~- so as to preclude intersection external to the interval QtQ,,
(xt-x,)
2
2
2
+(Yt-Y,) +(Zt-Z,) =8
constitute a system of 12 equations in 12 unknowns, formulated by Tarnai [2] who was loath
A n equilateral tetrahedral mechanism
231
to face the tedium of an attempt at direct algebraic solution. Recently, Chen [8], by dint of great perseverance succeeded in this regard and produced results which describe any valid position of Q as rigid body rotation with translation.
Z
/
." I
4.3. Rotation and translation
//. /
q'=[R]q+b,
(2)
where qr
~
/
/
Chen's solution is of the form
I/'-A
|
-4-
I t I I
\\ N..
[ R ] = [R(O)]
(3)
b = sb*(O)
(4)
= angular displacement vector of Q, b* = translation vector for normalized CC described in Fig. 4, = scale factor, s = a/(2x/2), = actual length of tetrahedral edge. 0
4. 4. Rotational manifold and angular displacement 0 is a Cartesian vector whose components are 0~, 0y and 0~. These are shown in Fig. 5 and are used to represent any angular displacement of a rigid body, say frame Q. 0 = l el is the magnitude of 0 and 0 J 0 , 0y/0 and 0 J 0 are the direction cosines of the rotation axis in the sense of an advancing right hand screw. It is easy to see that unconstrained rotation of Q would result in a set of vectors, radiating from 0, such that 0 < 7r. I.e., the points at the tips of all 0 would pack a sphere of radius 7r. This sphere is the manifold of unconstrained rotations of a rigid body. Notice that the origin, chosen to coincide with the orientation of Q at BP in the cartesian coordinate system of Fig. 4, is singular. The direction cosines are ambiguous; they vanish. Any point on the sphere surface is also ambiguous because it maps to the surface point diametrically opposite, i.e., a rotation of + 7r produces the same angular displacement as a rotation of - ~ - .
\\L
/
G
i"-%\ \
l 'li A.
O
,,
"
I/ / //
=
displaced position vector of any point of rigid body Q, q position vector of that point in BP, [R] = a 3 × 3 rotation matrix, b = translation vector of rigid body Q to satisfy Eqs. ( l a ) - ( l l ) . Note that
: ~/
| / Z , ~ I:
/
,
t\
"
"l /
/
Fig. 5. A n g u l a r d i s p l a c e m e n t v e c t o r and c o m p o n e n t s .
Notwithstanding such complication, Tarnai [2] showed that the constrained rotational manifold of Q is limited to those 0 which are parallel to the principal planes of CC. We see that if 0i = 0, 0 ~ 0; ~ Ok ¢ 0 ~
Type I motion occurs
0 i = O, Oj = 0 =it= O k
~
Type 1 motion occurs
0 i = O, Oj = Ok ¢ 0
~
Type 2 motion occurs
0 i = O; = Ok 4:0
~
Type 3 motion occurs,
where i, j, k = x , y, z, i 4 : j ~ k ~ i . In Section 2 it was described how BP is related to adjacent singularities, i.e., BP goes to or from CP via type 1 rotation of 0 = + 7r/2, BP goes to or from T V via type 2 rotation of tan0 = _+ v ~ , BP goes to or from T F via type 3 rotation of 0 = _+ ~-/3. These rotation submanifolds are shown in Fig. 6. It must be emphasized that this topology of singularities is by no means complete. E.g., if a type 3 rotation continues through 0 = 27r/3, beyond the adjacent TF, it returns to a rotated basic position RBP which is neither the initial BP nor any CBP. This is because a cyclic exchange of Q1 ---'O4 -+ 03 ~ Q1 occurs while Q2 returns to its original position. A rotation of base vetices also occurs during a limiting case of type I motion, without any intervening movement of the
P.J. Zsombor-Murray. ,4. Hyder
232
I Z
Y
(a)
(b) /I
2
indeterminate form 0 / 0 one obtains Oii = ~-/2 if 0i=0or0j=0. Keeping in mind Fig. 7 and accompanying arguments concerning limited motion of three degrees-of-freedom, consider Fig. 8. Here one can see clearly three typical coplanar actuators required to initiate, from BP, a type 3 motion in Fig. 8a. The arrows indicate traction exerted on the intersection points. Fig. 8b shows one driven actuator and two locked ones, the crosses. They are on non-intersecting edges. This is necessary to initiate and maintain a type 1 motion. Locked actuators constitute a deliberate action and they provide a line, through the two fixed points, about which this pure rotation takes place. A quick check with Eq. (13), below, shows that under this condition the translation vector, b, vanishes. Fig. 8c shows actuation of a type 2 motion. The locked actuator's edge defines the direction of the rotation axis.
(c) Fig. 6. Manifolds of one degree-of-freedom.
non-exchanged vertex. This case is illustrated in Section 4.6. Topological adjacency relationships among all possible singular positions of Q, along with their interconnecting, two-degree-of-freedom continua, are currently under investigation. Nevertheless, the limit of the two-degree-offreedom continua adjacent to BP has been mapped as shown in Fig. 7. The limiting curve, Oij(Oi, Oj) i s shown in a typical quadrant of the three orthogonal planes of the rotational manifold of Q. Notice that the lines of the three axes belong to two of three orthogonal planes. Any two such planes, together, contain all three degrees of rotational freedom. Therefore their intesection represents a distinct, although infinitessimal, region of three-degrees-of-freedom. The maximum type I angular displacement, Oii, has been determined by Chen [8] as tan(0,J2) =
~ [o? ( 0 z +
lOOiOj) --o i -
o,]
y
8,t
/ 1 fJ ///
/ !
/
f
•
"\\
/
/
(b)
/ / /
\ \ \
\ \ \
%
I
i
I
t /
r /
\ \
l
/ \
/ \
(s) This can be solved, e.g., for 0 i = 0i to yield tan0 u = v~-. By applying l'Hopital's rule to resolve the
/ \\
/ \\
"~.~
/j
/
/
J
Fig. 7. Manifolds of two degrees-of-freedom.
/
/
/
An equilateral tetrahedral mechanism
233
R, is " i", "
/
~
~L ~ .
8,~ q~,
• ., ~ •
' t
~
R=
....
....
L(Oi/O)2+cos 0
LOiOi/O2
LOiOj/O2 -(Oj/O) sinO
L(Oj/O)2+cos 0 (Oi/O) sinO
(Oj/O) s i n
0
]
-(Oi/O ) sin 0 / ' cos0
J
(10) "~ ~
,'.,'
. ":,,,, ~ "~
where (b)
L=l-cosO
(11)
and T=
0 1
0, 0
(12)
which effects the appropriate 7r/2 rotations on R to account for the assignments i = x , y, z
4.5. Results in computational form Generalizing Chen's [8] results so as to apply to any of the three planes of the rotational manifold, i.e., for types 1, 2 and I motions, we adopt the following numerical equivalence among ijk, xyz and 123 to account for all right handed pairings of axes.
-
i12(3) [xy(z) 23(1) -= [yz(x) 31(2) [zx(y)
bij,
-NOj]
1,
where
M = 2LOiOJO 2,
(14)
N = M sin 0 / [ 0 ( 1 + cos 0)1.
(15)
Type 3 motion is a special case because all three components of the angular displacement vector, 0, exist albeit of identical magnitude. Now
- plane (perpendicular), qij = R i j q +
j=y, z,x
=1,2,3 =2,3,1 as described above. Note that T T is transpose (T). Note also that T 1 = T and T 3 = I, the 3 × 3 identity matrix. It is not so easy, in fact it took Chen [8] considerable effort, to show that
Fig. 8. S i m p l e m o t i o n a c t u a t i o n s c h e m e s .
ij(k)
and
(6)
where qij, Rij and bij are just q', [R] and b from Eq. (2), generalized to the three planes of Fig. 7. Now
Rij = TkRT Tk,
(7)
bi/= Tkb,
(8)
(16)
qijk = R i / k q + bijk,
where Lf2+cosO Rijk= L f x L + L s i n O
L f x f y - f z sinO
Lfzfx+fysinO]
LfyZ+cosO
L f y f z - f x s i n O l,
[LLfx-f~sino t 4 , L + f x s i n 0
Lfz2+c°sO I (17)
or
qij = Tk( RTTkq + b ).
(9)
It is easy to see that the generic rotation matrix,
and
2Lex/(3 cos 0) ] bijk = 2Ley/(3 cos 0) , [ 2Lez/(3 cos 0)
(18)
P.J. Zsombor-Murray, A. Hyder
234
P
P
4
3
Q$
Q
Fig. 11. Type 3 motion animation.
Fig. 9. Type 1 motion animation.
4.6. Animation
where
fi=ei/v/3,
ei=Oi/lOil
(19)
t h e sign o f t h e a n g u l a r d i s p l a c e m e n t v e c t o r component.
It is now a s t r a i g h t f o r w a r d task to w r i t e a p r o g r a m , b a s e d o n Eqs. (9) a n d (16) a n d subject
I
q l ~ . ,"
PS(~
~z
i
I-,
p
~2
•
: / q
' ~
b
Fig. 10. Type 2 motion animation.
P 3
3 1 Fig. 12. Type I motion animation
at
Oil,
An equilateral tetrahedral mechanism
to the angular displacement limitation specified by Eq. (5), to perform the following, and other, animations. Figs. 9, 10, 11 and 12 were produced with such a program. They illustrate, respectively, typical type 1 motion, type 2 motion, type 3 motion and a type I motion along a limiting trajectory where 0 = Oij. The inset on each figure shows the trajectory, of the successive images of Q in each case, mapped on the rotational manifold.
5. Conclusions Any attempt at design improvement must address the problem of devising a mechanism which can assume BP and CBP. Aside from the obvious problem to contrive virtual vertices which may coincide there is also the question of how to resolve the ambiguity of intersection between a pair of coincident, parallel lines. The practical consideration involves the redistribution of the plastic ball-and-washer joints, i.e., how to have two emerge from a vertex where only one is available and the complementary problem of disposing of an extra one from some vertices when the mechanism moves away from BP. Another interesting proposition to enhance the mechanism is the addition of two more degrees of freedom with a third frame.
235
As regards applications, the suitability of this device as a wrist or as a universal joint may be investigated. A possible, revolutionary 'vehicle', based on this model, might also be considered. A BC is vertex stable, i.e., in this position the assembly rests on any of three adjacent vertices. Its mass centre is symmetrical; monopolar. At the end of a type 3 rotation it assumes an octahedral T F configuration which is absolutely convex, hence face stable. The mass centres of the two frames are at extreme displacement, a dipolar distribution. It would appear that the assembly might be made to tumble in prescribed direction over irregular terrain in an alternating sequence of face and vertex support stances. So as to see how the double tetrahedron might be used as a robotic joint, not quite a complete wrist since it has only two degrees-of-freedom (maybe 21, counting type 3 motion), consider Fig. 13. Three edges of each tetrahedron, which meet at a common vertex, have been extended to support a platform. Each platform is represented by a large triangular plane and the 12 edges are shown as rods of square section. With a little imagination one may see how a chain of four platforms and three such interconnecting joints may be assembled to make up a light, robust six-degree-of-freedom mechanism. On a whimsical note, one might, with a single joint, construct a rather interesting, adjustable stool to sit on an inclined, irregular surface.
Acknowledgement This research is supported by grant #A4219 awarded by The Natural Sciences and Engineering Research Council of Canada. The authors extend their gratitude also to two anonymous referees whose constructive and encouraging advice has led to considerable revision and, it is believed, improvement upon the original manuscript.
References
Fig. 13. A robotic joint.
[1] T. Tarnai and E. Makai, Physically inadmissable motions of a pair of tetrahedra, Proc. 3rd Int. Conf. Eng. Graph. & Desc. Geom., Vol. 2, Vienna (1988) 264-271.
236
P.J. Zsombor-Murray, A. ttyder
[2] T. Tarnai and E. Makai, A moveable pair of tetrahedra, Proc. R. Soc. Lon., A. 423 (1989) 423-442. [3] T. Tarnai and E. Makai, Kinematical indeterminacy of a pair of tetrahedral frames, Acta Technica Acad. Sci. Hung. 102 (1-2) (1989) 123-145. [4] T. Tarnai and E. Makai, Moveability of a pair of tetrahedral frames (In preparation). [5] E. Makai and T. Tarnai, Generalized forms of an overconstrained sliding mechanism consisting of two equal tetrahedra (In preparation).
[6] H. Stachel, Ein bewegliches Tetrahederpaar, Elemente der Mathematik 43 (3) (1988) 65-75. [7] A. Hyder and P.J. Zsombor-Murray, Design, mobility analysis and animation of a double equilateral tetrahedral mechanism, CIM-89-15, McRCIM Int. Rpt. McGill University, Montreal (1989). [8] Huan-Wei Chen, Kinematics and introduction to dynamics of a movable pair of tetrahedra, M. Eng. Thesis, McGill University, Dept. of Mechanical Engineering (1901).
227
An equilateral tetrahedral mechanism Paul J. Zsombor-Murray and Alexander Hyder Dept. of Mechanical Engineering McRCIM, McGill University, 817 Sherbrooke St. W, Montreal, QC, Canada H3A 2K6
Abstract Zsombor-Murray, P.J. and Hyder, A., An equilateral tetrahedral mechanism, Robotics and Autonomous Systems, 9 (1992) 227-236. Input-output kinematics of a novel, exoskeletal manipulator mechanism has been analysed. The abstract model is two equilateral tetrahedral wire frames which move relatively so that their edges remain in intersecting, parallel or coincident pairs. A mechanical, rigid body model was constructed and a computer animation was performed.
Keywords: Mechanism; Kinematics; Mobility; Workspace; Parallel; Overconstrained.
1. Introduction
The mobility of two equilateral tetrahedral wire frames, assembled so that the edges are constrained to form six intersecting pairs, was discov-
. . . . . .
_
Paul J. Zsombor-Murray is Associate Professor in ME at McGill University. His three degrees, Ph.D. (71), were obtained there. He conducts research in the Robotic Mechanical Systems group of the McGill Research Centre for Intelligent Machines and holds membership in IEEE (SM), ASME, ASEE and l'Ordre des Ing6nieurs du Qu6bec (PE). His professional interests include machine design, computer graphics, automata and geometric modeling.
Alexander Hyder is Senior Engineer at McDonnell Douglas Space Systems Company in Houston where he worked on dynamic simulation and :~ control of space robotic systems. He now supports NASA's Mission Operations Directorate and contributes to ,~ planning of robotic operations on Space Station FREEDOM. Hyder obtained his B.Eng (85) and M.Eng (89) in ME from McGill University. ~; ~ This article is based on his Master's research. Before that, as research engineer in McGill's Biomechanics Lab, he designed and conducted in-vitro experiments on post-mortum human specimens to study the dynamics of musculoskeletal systems.
ered in 1982 by Louis Tompos, a student of Ern6 Rubik at the Hungarian Academy of Craft and Design. Tompos took the model to the Hungarian Institute for Building Science and showed it to Tibor Tarnai who together with E. Makai, a mathematician of the Hungarian Academy of Science, proceeded to demonstrate the nature of possible motions of the equilateral set [1,2] and, recently, of more general, similar sets [3,4,5]. Stachel [6], at the Institut fiir Geometrie, Technical University, Vienna, derived independently a geometric proof pertaining to finite motions of the movable frame. By way of illustration he mapped typical surfaces (loci) traced by (a) the centre point of an edge, and (b) by a vertex. Apparently, Stachel and the Hungarians regarded the tetrahedral pair only as a curious toy and a challenging exercise in mobility analysis. They did not consider any serious physical implementation or possible application of such a mechanism. In this work we take a mechanistic approach. (1) A perfectly equivalent mechanism, composed of rigid bodies and exhibiting much of the mobility of the theoretical wire frames was designed and built and will be described. (2) An analysis is carried out and a computer animation of the wire frame set is described. It should be noted that this algorithmic procedure would be equally well suited to commanding displacements of a pow-
0921-8890/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
228
t'..I. Zsombor-Murrav, A. t@der
ered mechanism model. (3) Certain limited motions are described in terms of the simplest possible geometric invariants and parameters so that direct and inverse kinematics, i.e., Jacobians, are easily obtained. (4) Some possible applications and avenues of further research are outlined. Hyder [7] treated (1) and (2), above, in detail.
2. Rigid body model mechanism The mobile rigid body assembly of two tetrahedral flames shown in Fig. 1 is believed to be the only existing one. It was built to help visualize the motions, investigate design, fabrication and assembly problems and to plan the ultimate motorization and control of this mechanism. In any physical implementation it is difficult to satisfy the requirement that the flames be exactly congruent but one be located an infinitessimat distance 'inside' the other. The device in Fig. 1 overcomes this problem. Eight aluminum vertex blocks can be seen there. The cylinder has three symmetrical flats milled on it such that the three normals make an angle of t a n - ~ (1/v~-) with the cylinder axis. A hole is drilled normal to each flat such that the three hole axes intersect on the cylinder axis. The edges consist of 1 × 1 / 8 T H K aluminum angle. Three are seen to converge on each vertex block. Also visible there, mounted at each end of every edge angle, in the vee concavity with a metallic tangent screw garter, is an edge axis pin. This 1• plastic cylinder has a portion of its length reduced to 3 / 8 Q which protrudes into one of the vertex block holes. Thus, when the garters are tightened upon assembly, vertex blocks are held together with three adjacent edges by the axis pins in a 'three fingered bowler's grip'. Edges are flee to rotate about vertex block h o l e / e d g e axis pin axes. These axes comprise the virtual edges of the abstract wire frame model. Finally, sliding at the six edge pair intersections is effected by assembling the pair of tetrahedra with inside/outside parity. The inside partner has its edge angles turned concavely outward while the outer tetrahedron's angles; vee faces in. Although not evident in Fig. 1, each edge pair intersection cusp embraces a 1Q plastic ball girdled by a plastic washer to keep the angle lips parallel to the plane of intersection of the virtual edge axes.
Fig. 1. Double tetrahedral mechanism.
Figs. 2a and 2b show somewhat more clearly than photographs the way in which 8 vertex blocks 12 structural angles 24 axis pins 24 garters 6 balls and 6 washers are assembled so as to constitute the mechanism. This is an easy-to-build design but it is by no means ideal. The 'bowler's grip' is not positive;
An equilateral tetrahedral mechanism
VERTEXBLOCK BALL _R ~AR ~TE ,~I ~
(b)
UTWARD ~/X~ O FACING INWARD "~ ANGLE FACING ~XIS PIN ANGLE
Fig. 2. Solid model of the assembly and details.
notice the guy wires wrapping the outer member's vertices to control axial play at each axis pin. Sliding, rather than rolling, at the plastic balls introduces undue friction; motion is stiff, not smooth. Furthermore the vertex block design precludes coincidence of vertex pairs; a condition required for some theoretically valid poses of this mechanism. Nevertheless it should be quite easy to adapt three tractive, i.e., tensile, and three rotary actuators on each vertex block so as to motorize the sliding balls and to turn the edge angles. The preliminary model admirably serves to help visualize motion and as an exploratory basis upon which to found better designs.
ners of a cube to form a stella octangula or 8-pointed 3-D star. All edges are bisected at intersection with their pair mate as shown in Fig. 3. Each edge occupies a cube face diagonal. Mobility can be reduced to three types of rotation of one frame, called the movable one, with respect to the other, arbitrarily fixed one. The rotations are classified according to axial direction as follows: (1) Perpendicular to a face of the conceptual cube (CC); there are three such directions, (2) Parallel to an edge or face diagonal; there are six such directions, and (3) Parallel to cubic body diagonal; there are four. Rotation of type 1 may proceed continuously through +90 ° until the assembly assumes one of six immediately adjacent singularities called 'coincident positions' (CP) [1,2]. There are twelve such singularities and the other six are adjacent to the 'counter-basic position' (CBP) [1,2]. This other stella octangular singularity occurs when a type 1 rotation continues, from BP, through + 180 °. During a type 1 rotation an opposite edge pair of each frame remains bisected though, obviously, not orthogonal. The frame centroids remain coincident. A type 2 rotation may proceed continuously to a maximum angle of t a n - l v ~ ---- 54.7 °, the angle between a tetrahedral edge and face. This results in a 'terminal vertex singularity' (TV) wherein a single pair of vertices coincide. There are 24 of these, 12 immediately attainable via similar rotations from BP, the other 12 similarly adjacent to CBP. During a type 2 rotation two edge pairs
-
'7.-
3. Mobility and singularity The description of mechanism mobility starts from a key symmetrical singular configuration called the 'basic position' (BP) [1,2]. H e r e the vertices of the two frames occupy the eight cor-
229
Fig. 3. Basic position/conceptual cube.
230
P.Z Zsombor-Murray, A. Hyder
remain orthogonal and one edge on each frame remains bisected. Frame centroids do not remain coincident. Type 3 rotation has a range of _+60 °, ending in one of eight 'terminal face singularities' (TF) adjacent to BP. Another eight occur at 60 ° from CBP. At TF three vertex pairs and a face pair become coincident and the assembly assumes an equilateral hexahedral form. The centroid of the moving frame translates in a direction parallel to the type 3 axis of rotation. At T F centroids are separated by 3 ~ / 2 where p is the edge length of CC, not the tetrahedral edge length which is 2 ~ . In the following analysis CC is normalized to p = 2. During type 3 rotation, no orthogonal or bisected edge pairs survive. The three rotation types appear to be motions of one degree of freedom, i.e., sustained by only one of the 24 (12 linear/tractive ball and 12 rotary edge angle) possible actuation modes. However, for type 1 motion at least two actuators must be locked to maintain a pair of intersecting edges in midpoint contact. Similarly for type 2 motion at least one actuator must be locked so that one of a pair of intersecting edges maintains midpoint contact with its partner. Although, in principle, one actuator can continue the frame rotation from BP or CBP to any immediately adjacent CP, TV or TF, a minimum of three (two for type 2) actuators are required to initiate an excursion from or through a basic singularity. To maintain any branch of type 1 motion at least one actuator must act and two must be locked. These are actually special three-(and two)-degree-offreedom cases discussed in Section 4.4.
i
1\;,1
//.:.
~ (1,1,1)
-. .\\
Q4 ( - 1 , - 1 , - 1 )
3
1,-1)
(1,-1,-~) Fig. 4. Base coordinates of frames P and Q.
4.2. Compatibility conditions For any valid displacement of Q, the six intersecting tetrahedral edge pairs are
PrP,QtQ,, for all subscript combinations r = 1, 2, 3, r < s < 4, r + s + t + u = 10 and t < u < 4. The intersection requirement is satisfied if the volume of the tetrahedron PrP,QtQ. vanishes. Since P is fixed, the six constraint equations, ( l a ) - ( l f ) , can be writfen in terms of the coordinates of Qt and Qu only. ( z , - 1 ) ( x 2 - y a ) - ( z 2 - 1 ) ( x , - y , ) = 0,
(la)
( x 2 - 1 ) ( y 3 - - Z 3 ) -- ( X 3 -- l ) ( Y 2 - - Z 2 ) = 0 ,
(lb)
(Y3- l)(z,-Xl)
-- ( y , -
1 ) ( z 3 - x 3 ) = O, ( l c )
( x 4 + t ) ( y , + z , ) - (x, + 1)(Y4 + z4) = 0, ( l d )
4. Analytic solution: Rigid body motion (Y4 + 1 ) ( Z 2 + X 2 )
4.1. Terminology and base coordinates Adopting Tarnai's excellent formulation [2], one begins with a BP and labels the eight corners of CC P1, /92, P3, 1°4 and QI, Q2, Q3, Q4. P is the fixed flame and Q is the movable one. The normalized CC has edges of length p = 2 and is centred at the origin of Cartesian coordinates whose axes are each parallel to four edges. This is shown in Fig. 4 along with the coordinates P and of Q, in this BP.
(Z4 -}- 1)(X3 +Y3)
(ae) lt) -- (2:3 q- ])(x4 +y4) =0. -- (Y2
+ 1)(z4 +x4) =0,
These six along with six others, which simply state that the edges of Q are of a fixed length 2v~- so as to preclude intersection external to the interval QtQ,,
(xt-x,)
2
2
2
+(Yt-Y,) +(Zt-Z,) =8
constitute a system of 12 equations in 12 unknowns, formulated by Tarnai [2] who was loath
A n equilateral tetrahedral mechanism
231
to face the tedium of an attempt at direct algebraic solution. Recently, Chen [8], by dint of great perseverance succeeded in this regard and produced results which describe any valid position of Q as rigid body rotation with translation.
Z
/
." I
4.3. Rotation and translation
//. /
q'=[R]q+b,
(2)
where qr
~
/
/
Chen's solution is of the form
I/'-A
|
-4-
I t I I
\\ N..
[ R ] = [R(O)]
(3)
b = sb*(O)
(4)
= angular displacement vector of Q, b* = translation vector for normalized CC described in Fig. 4, = scale factor, s = a/(2x/2), = actual length of tetrahedral edge. 0
4. 4. Rotational manifold and angular displacement 0 is a Cartesian vector whose components are 0~, 0y and 0~. These are shown in Fig. 5 and are used to represent any angular displacement of a rigid body, say frame Q. 0 = l el is the magnitude of 0 and 0 J 0 , 0y/0 and 0 J 0 are the direction cosines of the rotation axis in the sense of an advancing right hand screw. It is easy to see that unconstrained rotation of Q would result in a set of vectors, radiating from 0, such that 0 < 7r. I.e., the points at the tips of all 0 would pack a sphere of radius 7r. This sphere is the manifold of unconstrained rotations of a rigid body. Notice that the origin, chosen to coincide with the orientation of Q at BP in the cartesian coordinate system of Fig. 4, is singular. The direction cosines are ambiguous; they vanish. Any point on the sphere surface is also ambiguous because it maps to the surface point diametrically opposite, i.e., a rotation of + 7r produces the same angular displacement as a rotation of - ~ - .
\\L
/
G
i"-%\ \
l 'li A.
O
,,
"
I/ / //
=
displaced position vector of any point of rigid body Q, q position vector of that point in BP, [R] = a 3 × 3 rotation matrix, b = translation vector of rigid body Q to satisfy Eqs. ( l a ) - ( l l ) . Note that
: ~/
| / Z , ~ I:
/
,
t\
"
"l /
/
Fig. 5. A n g u l a r d i s p l a c e m e n t v e c t o r and c o m p o n e n t s .
Notwithstanding such complication, Tarnai [2] showed that the constrained rotational manifold of Q is limited to those 0 which are parallel to the principal planes of CC. We see that if 0i = 0, 0 ~ 0; ~ Ok ¢ 0 ~
Type I motion occurs
0 i = O, Oj = 0 =it= O k
~
Type 1 motion occurs
0 i = O, Oj = Ok ¢ 0
~
Type 2 motion occurs
0 i = O; = Ok 4:0
~
Type 3 motion occurs,
where i, j, k = x , y, z, i 4 : j ~ k ~ i . In Section 2 it was described how BP is related to adjacent singularities, i.e., BP goes to or from CP via type 1 rotation of 0 = + 7r/2, BP goes to or from T V via type 2 rotation of tan0 = _+ v ~ , BP goes to or from T F via type 3 rotation of 0 = _+ ~-/3. These rotation submanifolds are shown in Fig. 6. It must be emphasized that this topology of singularities is by no means complete. E.g., if a type 3 rotation continues through 0 = 27r/3, beyond the adjacent TF, it returns to a rotated basic position RBP which is neither the initial BP nor any CBP. This is because a cyclic exchange of Q1 ---'O4 -+ 03 ~ Q1 occurs while Q2 returns to its original position. A rotation of base vetices also occurs during a limiting case of type I motion, without any intervening movement of the
P.J. Zsombor-Murray. ,4. Hyder
232
I Z
Y
(a)
(b) /I
2
indeterminate form 0 / 0 one obtains Oii = ~-/2 if 0i=0or0j=0. Keeping in mind Fig. 7 and accompanying arguments concerning limited motion of three degrees-of-freedom, consider Fig. 8. Here one can see clearly three typical coplanar actuators required to initiate, from BP, a type 3 motion in Fig. 8a. The arrows indicate traction exerted on the intersection points. Fig. 8b shows one driven actuator and two locked ones, the crosses. They are on non-intersecting edges. This is necessary to initiate and maintain a type 1 motion. Locked actuators constitute a deliberate action and they provide a line, through the two fixed points, about which this pure rotation takes place. A quick check with Eq. (13), below, shows that under this condition the translation vector, b, vanishes. Fig. 8c shows actuation of a type 2 motion. The locked actuator's edge defines the direction of the rotation axis.
(c) Fig. 6. Manifolds of one degree-of-freedom.
non-exchanged vertex. This case is illustrated in Section 4.6. Topological adjacency relationships among all possible singular positions of Q, along with their interconnecting, two-degree-of-freedom continua, are currently under investigation. Nevertheless, the limit of the two-degree-offreedom continua adjacent to BP has been mapped as shown in Fig. 7. The limiting curve, Oij(Oi, Oj) i s shown in a typical quadrant of the three orthogonal planes of the rotational manifold of Q. Notice that the lines of the three axes belong to two of three orthogonal planes. Any two such planes, together, contain all three degrees of rotational freedom. Therefore their intesection represents a distinct, although infinitessimal, region of three-degrees-of-freedom. The maximum type I angular displacement, Oii, has been determined by Chen [8] as tan(0,J2) =
~ [o? ( 0 z +
lOOiOj) --o i -
o,]
y
8,t
/ 1 fJ ///
/ !
/
f
•
"\\
/
/
(b)
/ / /
\ \ \
\ \ \
%
I
i
I
t /
r /
\ \
l
/ \
/ \
(s) This can be solved, e.g., for 0 i = 0i to yield tan0 u = v~-. By applying l'Hopital's rule to resolve the
/ \\
/ \\
"~.~
/j
/
/
J
Fig. 7. Manifolds of two degrees-of-freedom.
/
/
/
An equilateral tetrahedral mechanism
233
R, is " i", "
/
~
~L ~ .
8,~ q~,
• ., ~ •
' t
~
R=
....
....
L(Oi/O)2+cos 0
LOiOi/O2
LOiOj/O2 -(Oj/O) sinO
L(Oj/O)2+cos 0 (Oi/O) sinO
(Oj/O) s i n
0
]
-(Oi/O ) sin 0 / ' cos0
J
(10) "~ ~
,'.,'
. ":,,,, ~ "~
where (b)
L=l-cosO
(11)
and T=
0 1
0, 0
(12)
which effects the appropriate 7r/2 rotations on R to account for the assignments i = x , y, z
4.5. Results in computational form Generalizing Chen's [8] results so as to apply to any of the three planes of the rotational manifold, i.e., for types 1, 2 and I motions, we adopt the following numerical equivalence among ijk, xyz and 123 to account for all right handed pairings of axes.
-
i12(3) [xy(z) 23(1) -= [yz(x) 31(2) [zx(y)
bij,
-NOj]
1,
where
M = 2LOiOJO 2,
(14)
N = M sin 0 / [ 0 ( 1 + cos 0)1.
(15)
Type 3 motion is a special case because all three components of the angular displacement vector, 0, exist albeit of identical magnitude. Now
- plane (perpendicular), qij = R i j q +
j=y, z,x
=1,2,3 =2,3,1 as described above. Note that T T is transpose (T). Note also that T 1 = T and T 3 = I, the 3 × 3 identity matrix. It is not so easy, in fact it took Chen [8] considerable effort, to show that
Fig. 8. S i m p l e m o t i o n a c t u a t i o n s c h e m e s .
ij(k)
and
(6)
where qij, Rij and bij are just q', [R] and b from Eq. (2), generalized to the three planes of Fig. 7. Now
Rij = TkRT Tk,
(7)
bi/= Tkb,
(8)
(16)
qijk = R i / k q + bijk,
where Lf2+cosO Rijk= L f x L + L s i n O
L f x f y - f z sinO
Lfzfx+fysinO]
LfyZ+cosO
L f y f z - f x s i n O l,
[LLfx-f~sino t 4 , L + f x s i n 0
Lfz2+c°sO I (17)
or
qij = Tk( RTTkq + b ).
(9)
It is easy to see that the generic rotation matrix,
and
2Lex/(3 cos 0) ] bijk = 2Ley/(3 cos 0) , [ 2Lez/(3 cos 0)
(18)
P.J. Zsombor-Murray, A. Hyder
234
P
P
4
3
Q$
Q
Fig. 11. Type 3 motion animation.
Fig. 9. Type 1 motion animation.
4.6. Animation
where
fi=ei/v/3,
ei=Oi/lOil
(19)
t h e sign o f t h e a n g u l a r d i s p l a c e m e n t v e c t o r component.
It is now a s t r a i g h t f o r w a r d task to w r i t e a p r o g r a m , b a s e d o n Eqs. (9) a n d (16) a n d subject
I
q l ~ . ,"
PS(~
~z
i
I-,
p
~2
•
: / q
' ~
b
Fig. 10. Type 2 motion animation.
P 3
3 1 Fig. 12. Type I motion animation
at
Oil,
An equilateral tetrahedral mechanism
to the angular displacement limitation specified by Eq. (5), to perform the following, and other, animations. Figs. 9, 10, 11 and 12 were produced with such a program. They illustrate, respectively, typical type 1 motion, type 2 motion, type 3 motion and a type I motion along a limiting trajectory where 0 = Oij. The inset on each figure shows the trajectory, of the successive images of Q in each case, mapped on the rotational manifold.
5. Conclusions Any attempt at design improvement must address the problem of devising a mechanism which can assume BP and CBP. Aside from the obvious problem to contrive virtual vertices which may coincide there is also the question of how to resolve the ambiguity of intersection between a pair of coincident, parallel lines. The practical consideration involves the redistribution of the plastic ball-and-washer joints, i.e., how to have two emerge from a vertex where only one is available and the complementary problem of disposing of an extra one from some vertices when the mechanism moves away from BP. Another interesting proposition to enhance the mechanism is the addition of two more degrees of freedom with a third frame.
235
As regards applications, the suitability of this device as a wrist or as a universal joint may be investigated. A possible, revolutionary 'vehicle', based on this model, might also be considered. A BC is vertex stable, i.e., in this position the assembly rests on any of three adjacent vertices. Its mass centre is symmetrical; monopolar. At the end of a type 3 rotation it assumes an octahedral T F configuration which is absolutely convex, hence face stable. The mass centres of the two frames are at extreme displacement, a dipolar distribution. It would appear that the assembly might be made to tumble in prescribed direction over irregular terrain in an alternating sequence of face and vertex support stances. So as to see how the double tetrahedron might be used as a robotic joint, not quite a complete wrist since it has only two degrees-of-freedom (maybe 21, counting type 3 motion), consider Fig. 13. Three edges of each tetrahedron, which meet at a common vertex, have been extended to support a platform. Each platform is represented by a large triangular plane and the 12 edges are shown as rods of square section. With a little imagination one may see how a chain of four platforms and three such interconnecting joints may be assembled to make up a light, robust six-degree-of-freedom mechanism. On a whimsical note, one might, with a single joint, construct a rather interesting, adjustable stool to sit on an inclined, irregular surface.
Acknowledgement This research is supported by grant #A4219 awarded by The Natural Sciences and Engineering Research Council of Canada. The authors extend their gratitude also to two anonymous referees whose constructive and encouraging advice has led to considerable revision and, it is believed, improvement upon the original manuscript.
References
Fig. 13. A robotic joint.
[1] T. Tarnai and E. Makai, Physically inadmissable motions of a pair of tetrahedra, Proc. 3rd Int. Conf. Eng. Graph. & Desc. Geom., Vol. 2, Vienna (1988) 264-271.
236
P.J. Zsombor-Murray, A. ttyder
[2] T. Tarnai and E. Makai, A moveable pair of tetrahedra, Proc. R. Soc. Lon., A. 423 (1989) 423-442. [3] T. Tarnai and E. Makai, Kinematical indeterminacy of a pair of tetrahedral frames, Acta Technica Acad. Sci. Hung. 102 (1-2) (1989) 123-145. [4] T. Tarnai and E. Makai, Moveability of a pair of tetrahedral frames (In preparation). [5] E. Makai and T. Tarnai, Generalized forms of an overconstrained sliding mechanism consisting of two equal tetrahedra (In preparation).
[6] H. Stachel, Ein bewegliches Tetrahederpaar, Elemente der Mathematik 43 (3) (1988) 65-75. [7] A. Hyder and P.J. Zsombor-Murray, Design, mobility analysis and animation of a double equilateral tetrahedral mechanism, CIM-89-15, McRCIM Int. Rpt. McGill University, Montreal (1989). [8] Huan-Wei Chen, Kinematics and introduction to dynamics of a movable pair of tetrahedra, M. Eng. Thesis, McGill University, Dept. of Mechanical Engineering (1901).