PR McAree

The International Journal of Robotics Research http://ijr.sagepub.com

An Explanation of Never-Special Assembly Changing Motions for 3–3 Parallel Manipulators P. R. McAree and R. W. Daniel The International Journal of Robotics Research 1999; 18; 556 DOI: 10.1177/02783649922066394 The online version of this article can be found at: http://ijr.sagepub.com/cgi/content/abstract/18/6/556

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P. R. McAree R. W. Daniel Department of Engineering Science The University of Oxford Oxford, OX1-3PJ, UK

Abstract When the leg rods of a fully in-parallel manipulator are fixed in their lengths, it is usual that the device can be assembled in several distinct ways. Sometimes it happens that motion between such assemblies can take place such that the linkage is never at a special configuration; that is, a configuration where the moving-platform body acquires uncontrollable freedom relative to the base. The possibility of such motion has implications for control. Focusing on 3–3 devices, we present a geometric explanation of how these motions arise, and give a sufficient condition for their existence. For the 3–3 planar-motion device, we show that never-special assembly changing motions can be excluded by making platform and base triangles similar, and we conjecture that appropriate, perhaps identical, specialization for the octahedral manipulator has the same effect.

KEY WORDS—parallel robots, octahedral manipulator, coalesced assemblies, assembly-changing motions, singularityfree evolutions

1. Introduction In spatial mechanism, especially in robotics, specialized geometries are preferred to the almost complete exclusion of general systems. The joint axes of practicable serial robots, for instance, are usually arranged in groups of twos and threes that are parallel or at right angles, some perhaps being concurrent in a spherical wrist. Such a design has among its advantages: • fewer inverse-kinematic solutions, viz. the PUMA, which has a maximum of 8 as against the general 6R, which has as many as 16; • a closed-form solution to the inverse kinematics if three consecutive axes are concurrent at a point (Pieper 1968); and The International Journal of Robotics Research Vol. 18, No. 6, June 1999, pp. 556-574, ©1999 Sage Publications, Inc.

An Explanation of Never-Special Assembly Changing Motions for 3–3 Parallel Manipulators

• a well-defined and easily understood singular behavior (Burdick 1988; McAree, et al. 1991). Together with simplicity of layout, these are good reasons for adopting a specialized structure over a more general one. However, a small perturbation in the design parameters of such a linkage usually results in changes to its mechanical behavior. Pai and Leu (1989) and Wenger (1998) have observed that such changes arise out of the nongenericity that comes with specialization. Think of the specialized designs as constituting a hypersurface in design parameter space, and connected components of the complement corresponding to generically different types of mechanical behavior. Among the behaviors that emerge for generic designs is the possibility for undertaking motion between closures or postures without encountering a special configuration. Burdick (1988) gives an example of such a motion for a 3-R linkage. From time to time it has been argued that these neverspecial (or never-singular) posture-changing motions are desirable on the grounds that there are advantages in being able negotiate special configurations. We hold that such motions are pernicious, because the mechanical behavior of a device able to undertake them is intrinsically more complex than that of a mechanism whose design is specialized to avoid them. Of course, every real device must exhibit generic behavior to a degree, if for no other reason than manufacturing inaccuracies perturb it from its kinematic reference. But so long as there exists an appropriate balance between sensor resolution, component tolerance, and desired end-effector accuracy, the kinematic reference is valid. We hold that the sensible practice is to design for the simplest possible mechanical behavior. This argues for specialization and against designs that deliberately aim for genericity; among other things, it precludes designs that seek to allow never-special posture-changing motion. It is well known that an in-parallel actuated device can usually be assembled in several ways for given fixed lengths of the leg rods and Innocenti and Parenti-Castelli (1998) have

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McAree and Daniel / Never-Special Assembly Changing Motions shown that motions can sometimes be found that move the device from one such assembly to another with the same legrod lengths without ever encountering a special configuration. They call such motions “singularity-free evolutions”; here we call them “never-special assembly changing motions” (or just “never-special motions”) to emphasize that the platform body remains fully constrained throughout the motion. Wenger and Chablat (1998) give an explanation of these never-special assembly changing motions via notions of characteristic surfaces and uniqueness domains. The purpose of this paper is to develop a deeper understanding of how never-special assembly changing motions arise. We focus on the octahedral or 3–3 manipulator, shown in Figure 1, because we see its fully triangulated form as the best starting point for practical parallel-manipulator design. But our development is given dually for the simpler 3–3 planar-motion device in Figure 2, and in Sections 7 and 9 we use this device to illustrate some key points. Much is already known about the octahedral manipulator. A detailed description of its geometry and special configurations have been given (Hunt and McAree 1998). Methods for solving the forward kinematics of this device are described by several authors, including Nanua, Waldron, and Murthy (1990), Charentus and Renaud (1989), Merlet (1992), and McAree and Daniel (1996), all of which either state or prove that, with the leg lengths fixed, there are as many as 16 different ways of assembling the linkage as a structure. Hunt and Primrose (1993) noted that when assemblies having the same leg-rod lengths coalesce, the platform acquires uncontrollable freedom; they concluded that coalesced assemblies are necessarily special configurations. But the story does not end at this. In particular, coalescence is inextricably linked to

Fig. 1. The octahedral manipulator, so called because its 12 triangular faces define an octahedron. The platform triangle B1 B2 B3 moves relative to a fixed-base triangle A1 A2 A3 . Double-ball-joint-ended rods between platform and base serve as six connecting “legs.”

557

Fig. 2. A planar motion device, actuated by three connecting leg rods. The platform B1 B2 B3 acquires first-order freedom when the three leg-rod lines intersect at a common point.

other aspects of behavior, and a main theme we pursue is that a sufficient condition for being able to command the linkage through a never-special assembly changing motion is coalescence of three assemblies. Wenger (1998), citing earlier work (Wenger and Omri 1995), points out that the same condition holds for serially actuated linkages. We aim (1) to show why three assemblies coalescing enables the platform to undertake never-special motions, (2) to explore sufficient conditions for triple coalescence, and (3) to consider whether the possibility of these motions might be “designed out” by appropriate specialization. Though we do not consider connections with other in-parallel actuated arrangements, several of our observations also have valid interpretations in the broader setting of those devices also. Our interest in these questions arises from our use of an adaptation of Figure 1 as the input device for a force-reflecting telerobotic system (McAree and Daniel 1996; Daniel and McAree 1998). In this role, the linkage functions as a six degree-of-freedom joystick that a human operator moves arbitrarily. A slave arm (typically a serial robot) connected to the input device by computer mimics motions made by the operator, enabling interaction with objects in a remote environment. If we are to track the moving platform unambiguously, using only leg-length sensors, the device must not undertake never-special motion. Until recently, we believed the symmetry imposed by the coalesced ball-socket joints of the octahedral manipulator was sufficient to ensure the device could not be made to undertake never-special motion. We stated this in a previous work (McAree and Daniel 1996), where we described a forwardkinematic solution for tracking the moving platform. We thank Jean-Pierre Merlet of the Institut National de Recherche


558

THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / June 1999

en Informatique en Automatique (INRIA) at Sophia Antipolis for a computer-generated counterexample that demonstrated the claim to be wrong. This stimulated our interest in the problem.

2. Configuration Space: Constraint Equations Suppose the locations of the double ball-socket joints at A1 , A2 , and A3 of Figure 1 are known. We let 8 be the product space of leg-rod lengths L ≡ (L1 , L2 , . . . , L6 ) and θ ≡ (θ1 , θ2 , θ3 ), where the θi are dihedral angles between (nonadjacent) leg-triangles A1 B2 A2 , A2 B3 A3 , and A3 B1 A1 , and the base-triangle A1 A2 A3 . Note that specifying a location of the platform determines the leg-rod lengths and the angles θi uniquely. But these nine parameters are not independent. The fixed distances between points Bi amount to three constraints that restrict physical assemblies to a (9 − 3 =) 6-D configuration space C in 8. Each point of C is identifiable, one-to-one, with a linkage configuration, and if the leg-rod lengths L are used as control parameters, the angles θ are necessarily dependent on them. We note that changing the distances between points Bi changes the set C in 8 such that all the possible designs having 4A1 A2 A3 as base constitute a three-parameter family of hypersufaces C that are like “onion skins” in their foliation of 8. Like the octahedral manipulator, for the planar motion device of Figure 2, the dependency between L and θ is embodied in the condition that the distances between B1 , B2 , and B3 are fixed. Each configuration of this linkage can thus be thought of as a point on (6 − 3 =) 3-D surface, C, in the 6-D space 8 = L × θ . Comments made about C in the setting of the octahedral manipulator apply here also. In particular, each point on C is in one-to-one correspondence with a configuration of the platform. The basis for similarity between Figures 1 and 2 lies in the following observations: both mechanisms have triangular platform and base (i.e., both are 3–3 devices); both have configuration spaces C defined by three constraint functions; and for both, the dependent variables are parameterized by three angles. Accordingly, constraint equations for both devices take the same form: 01 (L, θ ) = [B2 (L, θ ) − B1 (L, θ )]T [B2 (L, θ ) − B1 (L, θ )] − |B2 − B1 |2 = 0, 02 (L, θ ) = [B3 (L, θ ) − B2 (L, θ )]T [B3 (L, θ ) − B2 (L, θ )] − |B3 − B2 |2 = 0, 03 (L, θ ) = [B1 (L, θ ) − B3 (L, θ )]T [B1 (L, θ ) − B3 (L, θ )] − |B1 − B3 |2 = 0, (1) where the vector-valued function Bi = Bi (L, θ ) gives the coordinates of connection point Bi in terms of L and θ . Ex-

panding each 0i as a series about the nominal configuration (L, θ ) yields 10i (L, θ ) =   p 3 X X ∂ ∂  0i (L, θ )  1θj + 1Lj ∂θj ∂θj j =1

j =1

 2 p 3 X ∂ ∂  1 X 1θj + 1Lj 0i (L, θ ) + 2! ∂θj ∂θj j =1 j =1  n p 3 X 1 X ∂ ∂  + ... + 1θj + 1Lj 0i (L, θ ) n! ∂θj ∂Lj j =1

j =1

+ . . . = 0, (2) with the entire expression summing to zero because the platform is rigid. Here 1 L ≡ (1L1 , 1L2 . . . , 1Lp ) denotes an incremental change in the leg-rod lengths, p being the number of leg rods (p = 3 for the planar-motion device, p = 6 for the octahedral manipulator); 1θ ≡ (1θ1 , 1θ 2 , 1θ3 ) denotes an incremental change in the angles P3 Pp ∂ ∂ n θ ; and + j =1 1Lj is an operator j =1 1θj ∂θj ∂θj on the constraint function 0i (L, θ ). The second term, for instance, means 2  p 3 X X ∂ ∂  0i (L, θ )  1θj + 1Lj ∂θj ∂Lj j =1

=

j =1

3 3 X X

1θj 1θk

j =1 k=1

+2 +

p 3 X X

j =1 k=1 p p XX

∂ 2 0i (L, θ ) ∂θj ∂θk

(3)

∂ 2 0i (L, θ ) 1θj 1Lk ∂θj ∂Lk

1Lj 1Lk

j =1 k=1

∂ 2 0i (L, θ ) . ∂Lj ∂Lk

For eq. (2) to be valid, the three dihedral angles θi must be defined, and this requires the leg rods to take finite lengths. For the octahedral manipulator, it further requires that no triangular faces from the set {A1 B1 A2 , A2 B2 A3 , A3 B3 A1 } collapse. We exclude from consideration any configuration where one or more of the θs is undefined. We also exclude as being unrealistic any linkages where either the platform or base triangles have collapsed. Though the angles θ vary continuously with leg motions, in Sections 4 and 6 we find it useful to separate out the effects of 1 L and 1θ by applying the leg-rod displacement first, keeping the angles θ constant, and imagining the platform 0 distorting by an amount 1 0 = ∂0 ∂L 1 L. This gives a means


McAree and Daniel / Never-Special Assembly Changing Motions for visualizing the constraints imposed by the fixed distances between the points B1 , B2 , and B3 . Summarizing the first few terms of eq. (2) into matrix form gives 0 (L, θ ) 0 (L, θ ) ∂0 ∂0 1θ + 1L ∂θθ ∂L     2 2 θ) θ) 1 (L,θ 1θ 1θ T ∂ 0∂θ 1θ T ∂ 0∂θθ1 (L,θ ∂L 1 L θ2   1  T ∂ 2 02 (L,θθ )  +1θ T ∂ 2 02 (L,θθ ) 1 L +  1 θ 1θ 2     ∂θθ ∂L ∂θθ 2 ∂ 2 03 (L,θθ ) ∂ 2 03 (L,θθ ) T (4) T 1θ 1θ 1θ ∂θθ ∂L 1 L ∂θθ 2   2 (L,θ θ) 1 1L 1 LT ∂ 0∂L 2  1 2 (L,θ θ) 2 +  1L  1 LT ∂ 0∂L 2   + O(3) + . . . = 0. 2 2 θ) 3 (L,θ 1 LT ∂ 0∂L 1 LT 2

0= 10

Here the matrix

0 ∂0 ∂θθ encapsulates

first-order terms in θ . It is

0 ∂0 =2 ∂θθ  1 2 −(B1 − B2 )T ∂B (B1 − B2 )T ∂B ∂θ1 ∂θ2  ∂B  0 (B2 − B3 )T ∂θ22  1 −(B3 − B1 )T ∂B 0 ∂θ1



0



∂B3  ∂θ3 . ∂B3 T (B3 − B1 ) ∂θ3

−(B2 − B3 )T

(5) 0 Similarly, first-order terms in L are embodied in a matrix ∂0 ∂L . For the planar-motion device, this is a 3 × 3 matrix, and can be obtained by replacing the θi terms in eq. (5) with Li terms. For the octahedral manipulator, there are six, rather than three, 0 leg rods to consider, and ∂0 ∂L is a 3 × 6 matrix.

The second-order terms appear in matrices ∂ 2 0i , ∂L2

∂ 2 0i ∂ 2 0i , , and ∂θθ 2 ∂θθ ∂L

i = 1, 2, 3. The 3 × 3 matrix ∂∂θθ021 , for instance, can be found by differentiating the first row of the matrix of eq. (5) with respect to θ to give 2

∂ 2 01 =2 ∂θθ 2  ∂BT   

1 ∂B1 ∂θ1 ∂θ1

+ (B1 − B2 )T ∂BT ∂B1 ∂θ1

∂ 2 B2 ∂θ12

∂BT ∂B1 ∂θ1

− ∂θ22

(B1 − B2 )T

− ∂θ22

0

∂ 2 B2 ∂θ22

0

0



  0 . 0 (6)

0 In the same way, ∂∂θθ022 and ∂∂θθ023 follow from rows 2 and 3 of ∂0 ∂θθ . 2

559

3. Regular and Special Configurations The view we take is that kinematic behavior of the device is imprinted on the local structure of C by its projection into the space of leg lengths. This “imprinting” is revealed by asking, How many terms of the series expansion are needed to describe arbitrary local motion at a given configuration? Where first-order terms are sufficient, the configuration is said to be regular. For every regular configuration, an order O() change in the leg rods results in an order O() change in the angles θ ; i.e., 0 (L, θ ) 0 (L, θ ) ∂0 ∂0 1θ + 1L = 0. ∂θθ ∂L

(7)

All other configurations are said to be special, and occur when eq. (7) fails to describe the local relationship between 0 1 L and 1θ . The simplest possibility is for ∂0 ∂θθ to drop rank so that second-order terms are needed to describe the constraints. There have been several studies of these first-order degeneracies. McAree and Daniel (1996) discussed them from the perspective of eq. (5), and Hunt and McAree (1998) continued from that of line geometry. Inter alia: 0 • A drop in rank of ∂0 ∂θθ amounts to the loss of first-order platform constraint, and equally, the acquisition of firstorder platform freedom. The lost constraint can be 0 identified with the left kernel of ∂0 ∂θθ , denoted here by u, and the acquired freedom with the right kernel, denoted by v. 0 • When ∂0 ∂θθ drops rank, the planes defined by triangles A1 B2 A2 , A2 B3 A3 , and A3 B1 A1 have their common intersection Q in the plane of the platform-triangle B1 B2 B3 . The dimension of this intersection equals the number of first-order constraints lost (or first-order freedoms acquired).

• A loss of first-order constraint implies linear dependence among the leg-rod lines. This allows special configurations to be identified by a drop in rank of the 6 × 6 matrix J −1 , whose rows are the coordinates of the leg-rod lines. The corank of the matrix J −1 equals 0 the corank of ∂0 ∂θθ , and reveals the number of constraints lost. • The octahedral manipulator can lose as many as three first-order constraints (or gain as many as three firstorder freedoms).

2

and

• The loss of two first-order constraints is always brought about by two leg triangles collapsing.

i = 1, 2, 3 are similarly found. For the planar-motion device, all are 3 × 3 matrices. For the octahedral manipulator, ∂ 2 0i ∂ 2 0i ∂θθ ∂L are 3 × 6 matrices, while ∂L2 are 6 × 6 matrices.

• The loss of three first-order constraints is always the consequence of base and platform triangles becoming coplanar.

All three are symmetric but rank deficient. Matrices ∂ 2 0i , ∂L2

∂ 2 0i ∂θθ ∂L


560


0 −1 are given in the Several other observations about ∂0 ∂θθ and J Appendix. The planar-motion device of Figure 2 loses at most one first-order constraint. At such a configuration, the three legrod lines are concurrent at a finite or infinite point. This is self-evident when viewed as the requirement for linear dependence of the three actuator rods. We note that just as with the octahedral manipulator, when the leg-rod lines are linearly 0 dependent, the matrix ∂0 ∂θθ drops rank.

4. Coalesced Assemblies as Special Configuration

keeping the angles θ fixed so that the imagined platformtriangle distortion 1 0 occurs only in the direction u3 corresponding to no first-order constraint, and let the magnitude of this distortion be λ having order O( 2 ) or higher. The equilibrating motion 1θ = 1 v1 + 2 v2 + 3 v3 is of order O() in direction v3 , and O( 2 ) or higher in directions v1 and v2 . Substituting 1θ into eq. (2) and taking terms to the second order gives   p p 3 20 20 X X X 1 ∂ ∂ ∂0 i i   i+ l vj l + 1Lj 1Lk ∂Lk ∂θj ∂Lk 2 ∂Lj ∂Lk k=1

From eq. (4), we can understand why coalesced assemblies are also special configurations and vice versa. Start with the planar device, and place it in a special configuration so that the leg-rod lines have a point of mutual intersection. The dominant terms of eq. (4) for a motion v in the right kernel 0 of ∂0 ∂θθ satisfy      T ∂ 20 1 v T ∂ 20 1 2 v v 0 θ ∂L ∂θθ 2   ∂0 +  ∂θ . (8) ..  1 L = − 2  .. ∂L . . Provided the right-hand side of eq. (8) has full rank, the legrod motion 1 L resulting from the motion v can be written  −1    2 ∂ 20 1 vT ∂∂θθ021 v vT ∂θ 0 2  ∂0 θ ∂L  1L = − +  .   .. .. 2 ∂L .   T ∂ 20 1 ∂0 v ∂θθ 2 v 0 −1 2 2 2 ,  =− I − 3 + 3 − . . . .. 2 ∂L .  2  −1 v ∂ 01 0 ∂0  ∂θθ. ∂L  , where 3 = .. ∂L   ∂ 20 0 −1 vT ∂θθ 21 v 2 ∂0 .  ≈− .. 2 ∂L .

j,l=1

+

3 X j,k=1

0 ∂0 = ∂θθ

3 X i=1

σi ui viT ,

σ3 = 0,

σ1 ≥ σ2 .

(10)

(Note we use row-column notation to index elements of ui and vi : ui = (u1i , u2i , u3i ).) Suppose we alter the legs while

1 ∂0i k vj k + ∂θj 2

3 X

l m

l,m=1

vj l vkm

j,k=1

∂ 2 0i ∂θj ∂θk

+ O( ) = 0. 3

(11)

We note from eq. (10) (σ3 being zero) that 3 X

X ∂0i = j σj uij . ∂θj 2

k vj k

j,k=1

j =1

Then we let l m

3 X

X ∂ 2 0i = l m µlmj uij , ∂θj ∂θk 3

vj l vkm

j,k=1

j =1

with coefficients µlmj taken over all l and m summing to the imagined “platform distortion” along uj . Only terms in 32 are of order O( 2 ); all other terms, l m , l 6 = m 6 = 3, are order O( 3 ) or higher. Thus we can write p X ∂0i k=1

∂Lk

+ O() 1Lk +

2 X

j σj uj i

j =1 3 2 X + 3 µ33j uj i = 0, 2

(12)

j =1

or in terms of the left singular vectors

(9) For every motion v satisfying this expression, there exists a complementary motion −v in the opposite direction that also satisfies it. As goes to zero, these two assemblies, one for either side of the special configuration, come together. To generalize this conclusion to Figure 1, we start by ex0 pressing the rank-2 matrix ∂0 ∂θθ via its singular value decomposition; i.e.,

j =1

3 X

λu3 +

2 X i=1

1 X i σi ui + 32 µ33i ui + O( 3 ) = 0, 2 3

λ of order

i=1 2

(13)

O( ).

Equating terms, we obtain the following expressions to O( 2 ) for the coefficients of 1θ : 1 µ331 , 1 = − 32 2 σ1

1 µ332 2 = − 32 , 2 σ2

1 λ = − 32 µ333 . 2 (14)

For λ negative, the equation set has two real solutions cor1

responding to the two allowed values of 3 = ± µ2λ332 . As λ


3

McAree and Daniel / Never-Special Assembly Changing Motions

561

goes to zero, these two assemblies coalesce. Should λ become positive, coalesced real solutions become a complex conjugate pair which amounts to the vanishing of two solutions. As pointed out by Hunt and McAree (1998), the conclusion that special configurations coincide with coalesced assemblies also follows from Schoenflies’s work (1886, ch. 3, sect. 3), where the perspective is that of screw geometry. We leave it to the reader to reconcile one view with the other.

5. Geometric Interpretation of Coalescence: The Branch Locus Consider the projection of C onto the space of leg-lengths L, π : L × θ 7 → L, noting that pre-images of this map for given leg lengths are the solutions to the forward kinematics. Here we want to view the projection geometrically, as in Figure 3. The resulting bifurcation set corresponds to critical leg-rod lengths where the number of ways of assembling the platform changes or branches, either increasing or decreasing by two (or some multiple of two) as pairs of solutions go from being complex to real and vice versa, consistent with eq. (14). We call the locus of critical leg lengths the branch locus. Leg lengths on the branch locus have some or all of their assembles coalesced, and these coalesced assemblies are, by the arguments of Section 4, necessarily special configurations. The structure of C is locally linear at regular configurations, consistent with eq. (7). At first-order special configurations, it is locally quadratic, consistent with eq. (14), folding back on itself. For both devices these fold sets have codimension 1 in C, as does the depiction of the fold of Figure 3. It is standard to interpret a matrix determinant as the oriented volume defined by the rows of the matrix, with the sign giving orientation; see for instance the work of Strang (1988, 0 sect. 4.1). The row space of ∂0 ∂θθ is complementary to C, and gives the rate at which the platform triangle distorts as the angles θ are altered. On an upward-facing region, e.g., at P 0 , the 0 volume defined by the rows of ∂0 ∂θθ is directed upward, the determinant being positive. On a downward-facing region, e.g., 0 at P 00 , ∂0 ∂θθ is oriented downward, the determinant there being negative. And either side of a special configuration, that is, either side of a fold in C, the sign of the determinant is reversed. Sign reversal is often cited in connection with special configurations of serial arms. Innocenti and Parenti-Castelli (1998) asserted that in the setting of parallel manipulation, passage through a special configuration does not necessarily result in the determinant changing sign, and as we shortly see, there is a basis for this claim. However, two facts remain inescapable: at a special configuration the determinant is always zero, and if the motion passes through a doubly coalesced assembly, the determinant’s sign is always reversed.

Fig. 3. Projection of configuration space C onto the space of leg-lengths L, with C folding back on itself. The branch locus is the projection of the fold set. The figure depicts the family of configuration spaces foliating the space 8, and 0 ∂0 ∂θθ as an oriented volume (here a vector) pointing upward on an upward-facing region of C.

6. Second-Order Degeneracies: Triply Coalesced Configurations This discussion of special configurations, vis-á-vis the interpretation of them as “folds” in C under the projection into leg space, vexes consideration of further degeneracy of eq. (2) and the implications this has for the behavior of the linkage. In exploring these questions, we limit ourselves to corank-1 special configurations, with the objective of finding the conditions needed for the additional loss of second-order constraint. For the planar linkage of Figure 2, we lose nothing by 0 this restriction, because for this device ∂0 ∂θθ never drops rank by more than one (so long as we keep all leg rods finite and don’t concern ourselves with degenerate forms such as collapse of the platform triangle). For the octahedral manipulator, as noted in Section 3, there are corank-2 and -3 special configurations to consider. Higher-order degeneracy of these is intrinsically more subtle; it lies beyond the scope of this paper. So too does loss of higher than second-order constraint. Suppose then that either linkage is in a configuration such 0 that ∂0 ∂θθ has rank 2, and let {v1 , v2 , v3 } be the orthonormal basis for the space 1θ defined eq. (10). The vector v3 whose corresponding singular value is zero is then a unit vector in the 0 right kernel of ∂0 ∂θθ , while the associated left-singular vector u3 spans the left kernel and corresponds to the direction of the lost constraint.


562


From eq. (13) it follows that to lose both first- and secondorder constraints in direction u3 , the coefficient µ333 must equal zero. In matrix notation this condition can be written either as   2 v3T ∂∂θθ021 v3  ∂ 20  T  2 u3T  (15) v3 ∂θθ 2 v3  = 0, 2 ∂ 03 T v3 ∂θθ 2 v3 or as h i 2 2 2 v3T u13 ∂ 021 + u23 ∂ 022 + u33 ∂ 023 v3 = 0. ∂θθ ∂θθ ∂θθ

(16)

Special configurations satisfying eqs. (15) and (16) occur when (at least) three assemblies coalesce. The demonstration of this proceeds as an extension of the arguments used in Section 3. Place the linkage in a special configuration where eq. (15) is satisfied, and choose a motion 1 L of the leg rods so that the imagined platform-triangle distortion is in the direction u3 only. Let the magnitude of this distortion be λ. The necessary change in θ will satisfy 1θ = 1 v1 + 2 v2 + 3 v3 , with 3 of order O() and 1 and 2 of order O( 2 ) or higher. Taking terms in eq. (2) to order O( 3 ), we now have 3 X ∂0i + O() 1Lk + k vj k ∂Lk ∂θj

p X ∂0i k=1

1 + 2 +

l m

l,m=1

j,k=1

3 X

1 6

3 X

m n p

m,n,p=1

∂ 2 0i vj l vkm ∂θj ∂θk 3 X

vj m vkn vlp

j,k,l

1 1 1 σ1 + 32 µ331 + 1 3 µ131 + 2 3 µ231 + 33 ψ3331 = 0, 2 6 1 2 1 3 2 σ2 + 3 µ332 + 1 3 µ132 + 2 3 µ232 + 3 ψ3332 = 0, 2 6 1 1 3 µ133 + 2 3 µ233 + 33 ψ3333 = −λ. 6 (20) For small motions, the first two of the above equations are dominated by quadratic terms. Using these as order-O( 2 ) approximations to 1 and 2 , we get 1 = −32

2 = −32

and

(17) ∂ 3 0i ∂θj ∂θk ∂θl

µ332 . 2σ2

(21)

When substituted into the third equation, we obtain the following cubic in 3 : ψ3333 σ2 − 3µ332 µ233 σ1 − 3µ331 µ133 σ2 33 (22) + 6λσ1 σ2 = 0, and as λ tends to zero, this equation becomes (23)

confirming that three assemblies have coalesced. Of course this argument breaks down if third-order constraints also vanish, but it holds otherwise. A local solution to forward kinematics, valid to O( 3 ), is obtained by including the cubic terms discarded above. This results in a fifth-order polynomial 5 X

+ O( ) = 0. 4

p

βp 3 = 0,

(24)

p=0

If we set m n p

µ331 , 2σ1

33 = 0,

j,k=1

3 X

Equating coefficients for ui and setting each to zero consistent with the rigid-body constraint on the platform body results in the following three equations:

3 X j,k,l

X ∂ 3 0i vj m vkn vlp = m n p ψmnpj uij , ∂θj ∂θk ∂θl 3

j =1

(18) noting that in this expression only terms involving 33 have order O( 3 ) (all others necessarily being of order O( 4 ) or higher), the following expression is valid order O( 3 ): λu3 +

2 X i=1

+ 3

1 X i σi ui + 32 µ33i ui 2

2 X i,j =1

λ

2

i=1

1 X j µj 3i ui + 32 ψ333i ui = 0, 6 3

i=1

of order O( ). 3

(19)

with = −6λσ1 σ2 , = −6λ(µ232 σ1 − µ131 σ2 ), = +6λ(µ231 µ132 − µ131 µ232 ) − µ233 ψ3332 σ1 , = −σ1 σ2 ψ3333 − 3σ1 µ233 µ332 − 3σ2 µ133 µ331 + µ133 µ231 ψ3332 − µ233 µ131 ψ3332 , β4 = −ψ3333 µ232 σ1 − σ2 µ133 ψ3331 − 3µ232 µ133 µ331 + 3µ133 µ231 µ332 + 3µ233 µ132 µ331 − 3µ233 µ131 µ332 − ψ3333 µ131 σ2 , β5 = −(−µ232 µ133 + µ233 µ132 )ψ3331 + ψ3333 µ231 µ132 − ψ3333 µ131 µ232 .

β0 β1 β2 β3

As we are only taking terms to order O( 3 ), and since 3 is small, we discard coefficients β4 and β5 . The resulting cubic


McAree and Daniel / Never-Special Assembly Changing Motions equation has either three real solutions or one, the number of (real) solutions being the number of local assemblies. What physical interpretation are we to give to triply coalesced assemblies over and above those usually ascribed to a double coalescence? Where two assemblies coalesce, the platform body relies on second-order constraints to resist the component of a disturbance force that causes motion in the 0 right kernel of ∂0 ∂θθ ; the propensity to “wobble” at these config0 urations is quadratically determined in the kernel of ∂0 ∂θθ , but is otherwise linearly determined. Because of this the linkage is readily displaced in the direction of lost constraint, and this is identified with the acquisition of transitory freedom. Should three assemblies coalesce, the linkage relies on thirdorder constraints and the propensity to wobble, now cubic, is much greater. If double coalescence is intolerable because the linkage structure is too poorly conditioned to be useful, triple coalescence is more so. This argues that triply coalesced assemblies are pernicious and best avoided, or better still, “designed out” by judicious dimensioning. Figure 4 provides a more fundamental interpretation of triple coalescence and suggests why it gives a sufficient prerequisite for never-special motions. Here the fold of Figure 3 has been folded. When C (the configuration space) is projected into leg space, the branch locus cusps at the point P . Points inside the cusp, e.g., A0 , have three local pre-images (assemblies) in C; points outside it have one; and at P the three assemblies coalesce. This “folding of the fold set” is known as a cusp-type singularity (Poston and Stewart 1978; Bruce and Giblin 1992), and eq. (24) amounts to a crude form of the versal unfolding of this singularity. By diffeomorphism, eq. (24) could be brought into model form (for instance, see Bruce and Giblin’s work (1992)), although we do not pursue this here. We make the following observations based on Figure 4: • The triple coalescence at P allows the linkage to negotiate a special configuration in moving between two assemblies having the same leg lengths. A candidate motion is any that encircles P (e.g., that from A1 to A3 ). • A trajectory P passing though the special configuration 0 0 ∂0 = 0 at P . Note, however, that det has det ∂0 ∂θθ ∂θθ will take the same sign on either side of this special configuration if the trajectory remains on upward-facing regions. Innocenti and Parenti-Castelli (1998) report that they have observed this phenomenon in numerical investigation. • Hunt and Primrose (1993) describe an imagined procedure wherein an in-parallel actuated device is moved very close to a special configuration and the actuators then locked. Ill-conditioning of the structure together with compliance in the ball-socket joints and leg rods allows it to be “popped” through to the nearby assembly. Near a triply coalesced assembly it is possible to

563

Fig. 4. Cusp-type singularity and its projection into leg space. Three assemblies coalesce P . A candidate never-special motion is shown from A1 to A3 .

“pop” between the three nearby assemblies in the same imagined way, but we should not expect to be able to cycle through them, moving from one to the next in a cyclic sequence. One assembly will be central, e.g., A2 of Figure 4, so that in popping between the other two, here A1 and A3 , it will usually be visited. Objections to Figure 4 because it is dimensionally inconsistent with the spaces C for the linkages of Figures 1 and 2 are best reconciled by viewing doubly and triply coalesced assemblies via their codimension, rather than their dimension, making recourse to the accepted notion that objects with the same codimension have similar properties (see, e.g., Poston and Stewart (1978)). In Figure 4, C is a surface; the fold set corresponding to first-order degeneracy is of codimension 1; and the cusp at P , where there is second-order degeneracy, has codimension 2. While the configuration space C for the planar device has dimension 3, sets of interest (folds and cusps) have codimensions 1 and 2 consistent with Figure 4 (similarly for the octahedral manipulator). Provided we don’t go beyond corank 1, second-order degeneracy, Figure 4 is valid if we think of it as describing a two-dimensional slice of C where, say, all but two of the leg rods are fixed. Whitney (1955) showed that the projection of a surface to a plane always splits into folds and cusps, but nothing more complex. We expect then that for small perturbations of the fixed leg lengths, the topological structure of nearby branch-loci slices are, ordinarily,


564


preserved. A change in the structure of a slice of the branch locus indicates higher order degeneracies of eq. (2). If the branch locus in any such two-dimensional slice of C cusps, we can state categorically that the device will be able to undertake a never-special motion. Were we to locate the platform body so that three assemblies coalesce, then any two legs of the six could be actuated, keeping the others fixed, and the resulting slice of the branch locus would cusp. Slices of C and L, together with the imagery of Figures 3 and 4, not only explain never-special assembly-changing motions, but give a graphical construction that reveals the possibility of such motion. Extending the argument that folds of C are double coalescence, and cusps are triple coalescences, we might expect for the planar device, among perhaps some other possibilities, to find a finite number of configurations (i.e., a codimension-3 set) where four assemblies coalesce consistent with the loss of first-, second-, and third-order constraint. Ordinarily, however, we do not expect to find configurations where five (of more than five) assemblies coalesce. A full description of higher-order degeneracies of the octahedral manipulator goes well beyond anything we attempt here. But the possibilities will very likely include codimension-3 sets of configurations where four assemblies coalesce, and within these, codimension-4 sets where five assemblies coalesce, and so on, each set corresponding to the vanishing of successively higher-order terms in eq. (2). The coalescence of four, five, and more assemblies through degeneracy of third-, fourth-, and higher-order terms does not vitiate against undertaking never-special motions. If four assemblies coalesce somewhere within a set of triple coalescences, never-special motions are still possible, and other undesirable behavior might also show itself. This should not be confused as implying that coalescence of four assemblies allows never-special motions. The key to never-special motion lies in the simultaneous loss of first- and second-order constraint, possibly higher constraint also, in the same direction. We note that four assemblies can coalesce, without implying anything of the behavior of second- or higher-order terms, by intersection of two fold sets (double coalescences) in C.

7. An Example Focusing on the Planar-Motion Device In the work of Innocenti and Parenti-Castelli (1998), it is demonstrated that the planar-motion linkage can undertake never-special motion. We now give an explanation of this motion using the ideas developed in Section 6. If we fix the leg lengths, this linkage has at most six assemblies. This is demonstrated in several places (Hunt 1983; Merlet 1990; Gosselin, Sefrioui, and Richard 1992; Gosselin and Merlet 1994). We follow Hunt’s idea (1983) and use Bezout’s theorem, which states that the maximum number of

finite intersections of two curves having orders n1 and n2 and circularities p1 and p2 is n1 n2 − 2p1 p2 . We disconnect leg-rod 3 so that B3 is a coupler point of the hinged planar four-bar A1 B1 B2 A2 . We see that B3 traces a curve of degree 6, circularity 3 (Hunt 1990, sect. 7.3). It is convenient to call this the “coupler curve” of B3 ; it intersects the circle (degree 2, circularity 1) centered at A3 , radius L3 , which we call the “circle” of B3 , in at most (6 × 2 − 2 × 3 ×1 =) 6 finite points. Each intersection is identified with an assembly. In the coordinate frame of Figure 2, the locations of B1 , B2 , and B3 are given by B1 ≡ L1 cos(θ1 ) L1 sin(θ1 ) , B2 ≡ A2x + L2 cos(θ2 ) L2 sin(θ2 ) , B3 ≡ A3x + L3 cos(θ3 ) A3y + L3 sin(θ3 ) ,

from which it follows that the matrix 0 ∂0 = ∂θθ  L1 (A2x s1 − L2 s12 )   0   2    L1 (A3 s1 − L3 s31 x −A3y c1 )

0 ∂0 ∂θθ is

L2 (L1 s21 − A2x s2 )

0

−L2 [(A2x − A3x )s2 −L3 s23 + A3y c2 ] 0



 L3 [(A2x − A3x )s3   −L2 s23 + A3y c3 ]  ,   L3 (A3x s3 − L1 s31  −A3y c3 )

where si = sin(θi ), ci = cos(θi ), and sij = sin(θi − θj ). 0 Excluding the leg rods from having zero length, ∂0 ∂θθ drops rank when A2x sin(θ2 ) sin(θ3 − θ1 ) − A3x sin(θ3 ) − A3y cos(θ3 ) sin(θ1 − θ2 ) = 0. Nonzero rows and columns of ad 0 ∂0 ∂θθ .

0 ∂0 ∂θθ

(25)

are then, respectively,

The condition that must be satleft and right kernels for isfied, along with eq. (25), for three assemblies to coalesce is κ 1 κ2

κ3 κ4

−κ1 κ4

∂ 2 01 ∂ 2 02 ∂ 2 03 κ 1 κ4 − κ2 κ5 + κ 3 κ5 2 2 ∂θθ ∂θθ ∂θθ 2




 κ1 κ2  κ3 κ4  = 0, −κ1 κ4 (26)


565

where

  L1 (A2x c1 + L2 c21 ) −L1 L2 c21 0 ∂ 2 01 −L2 (A2x c2 − L1 c21 ) 0 , −L1 L2 c21 = 2 ∂θθ 2 0 0 0 

∂ 2 02 ∂θθ 2

0

 0  = 2   0

0

0

−L2 [(A2x − A3x )c2 −L3 c23 − A3y s2 ] −L2 L3 c23



L1 (A3x c1 + L3 c31  +A3y s1 )   ∂ 2 03 0 = 2  ∂θθ 2   L L c

   ,   L3 [(A2x − A3x )c3  +L2 c23 − A3y s3 ] −L2 L3 c23

0 0 0

1 3 31



L1 L3 c31



   , 0   L3 (L1 c31 − A3x c3  −A3y s3 )

and κ1 = L2 s2 A3x + 2L2 (L3 s23 − 2L2 A2x s2 − A3y c2 ), κ2 = −2L3 (L1 s13 − A3x s3 + A3y c3 ), κ3 = −2L3 ((A3x + A2x )s3 − L2 s23 + A3y c3 , κ4 = −2L2 ((A2x − A3x )s2 − L3 s23 + A3y c2 , κ5 = −2L2 (A2x s2 − L1 s12 ). The expansion of this expression is too complicated to yield any real insight; its main utility lies in providing a test for triple coalescence. We can affect a useful simplification of the problem by fixing the length of one of the actuated rods, say L1 , so that two parameters, rather than three, define a configuration of the device. We take angles θ1 and α in Figure 2, and we note that the projection of an (α, θ1 )-slice of C into the (L2 , L3 )-plane is consistent with the depiction of Figures 3 and 4. Specifically, where the branch locus in the (L2 , L3 )-plane cusps, three assemblies coalesce. We undertake never-special motion by the encircling of a cusp. For platform- and base-triangle dimensions, we take those used by Innocenti and Parenti-Castelli (1998): a1 a3 b1 b3

= |A2 − A1 | = 15.91, = |A1 − A3 | = 10.0, = |B2 − B1 | = 17.04, = |B3 − B1 | = 20.84,

a2 = |A3 − A2 | = 18.79, b2 = |B3 − B2 | = 16.54,

and fix L1 at 14.98. The locus of configurations satisfying eq. (25) on the (α, θ1 )-plane is shown in Figure 5. The dashed box indicates 0 − 2π extents of α and θ1 , with the locus repeating itself outside this region. This curve partitions the (α, θ1 )-surface (topologically a 2-torus) into two components,

0 Fig. 5. Zero-determinant contour of ∂0 ∂θθ of the planar platform with |A2 − A1 | = 15.91, |A3 − A2 | = 18.79, |A1 − A3 | = 10.0, |B2 − B1 | = 17.04, |B3 − B2 | = 16.54, |B3 − B1 | = 20.84, and L1 = 14.98.

one corresponding to configurations where the determinant of 0 ∂0 ∂θθ is positive, the other where it is negative. Figure 6 shows the branch locus resulting from the projection of the singular configuration set of Figure 5 into the (L2 , L3 )-plane, partitioning it into regions having 0, 2, 4, or 6 assemblies, as indicated. It is not possible to have an odd number of distinct assemblies, because assemblies must appear and disappear in pairs, consistent with the arguments of Section 4. Where the branch locus cusps, for instance at points A, B, C, D, and E, three assemblies coalesce consistent with the second-order terms of the constraint expansion degenerating in the singular direction. Platform configurations corresponding to these cusped leg lengths are given in Table 1, and the pre-images of each are as indicated by the appropriate letter in Figure 5. The apparent cusps where L2 = 0 and L3 = 0 do not, however, correspond to triply coalesced assemblies. They arise because the constraint functions are nondifferentiable for L2 = 0 and L3 = 0. (To make them differentiable, one can square the leg lengths.) Platform assemblies corresponding to each cusp are shown in Figure 7. Coupler curves of B3 for the hinged planar fourbar A1 B1 B2 A2 are traced out in each plot, as are the circles centered at A3 with radius L3 . As already noted, intersections between these curves can be identified with assemblies for the chosen leg-rod lengths. And because three assemblies have coalesced at B3 , we expect the coupler of B3 to osculate to first order with the circle of B3 . Inspection confirms this, with the coupler curve approaching the circle on one side, becoming tangent to it, and leaving on the opposite side. (At first-order special configurations, the coupler curve of B3 is tangent to this circle of B3 ; at regular configurations, it intersects it).


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Table 1. Triply Coalesced Configurations with L1 = 14.98a L1 L2 L3 A B C D E

14.9800 14.9800 14.9800 14.9800 14.9800

16.0305 17.9860 30.4276 31.2717 13.8533

29.5598 26.4320 26.6088 16.1766 6.2249

θ1

θ2

θ3

15.4752 −10.5962 133.6913 114.0472 177.4512

13.8244 179.7018 124.1355 −178.8453 −179.5421

19.6981 −92.2033 129.0776 −89.1941 99.3252

a. The L terms have units of length; angles θ are given in degrees.

Fig. 6. The bifurcation set associated with projection of the contour of Figure 5 onto the (L2 , L3 )-plane with L1 = 14.98. Other dimensions are given in the caption of Figure 5. Numbers correspond to the number of assembly modes for given leg-rod lengths. Cusps at A, B, C, D, and E indicate leg lengths having three coalesced assemblies. Other points on the branch locus have two of their assemblies coalesced. Cusps on L2 = 0 (L3 = 0) arise because θ2 (θ3 ) is undefined, the constraint equations being nondifferentiable.

This gives a second, arguably more intuitive, test: at a triply coalesced assembly, the coupler curve of B3 has its center of curvature at A3 and has radius of curvature equal to L3 . The choice of B3 as the coupler point is arbitrary. Just as validly, we could disconnect leg 1 (or leg 2) and use B1 (or B2 ), and the same osculating pattern would be evident. Using Aronhold’s construction for path curvature (Hunt 1990, sect. 5.5.1), a triply coalesced assembly can be constructed ad hoc, starting with any four-bar and an identified coupler point. The same construction confirms that A3 is the center of curvature for points B3 in Figure 7. Moreover, the argument can be extended to where four assemblies coalesce because the coupler curve and the circle now osculate to second order, B3 being a point of stationary curvature. Citing Müller (1903), Hunt (1990, sect. 7.5) reports that it is possible to build a hinged four-bar coupler curve that has

six infinitesimally separated points in common with a circle. Aronhold’s construction applied here would (presumably) yield a platform with its six assemblies all coalesced. We will not attempt to reconcile this possibility with the enumerative argument given in Section 6 that for corank-1 degeneracy no more than four assemblies are expected to be able to coalescence other than to remark that the singular behavior of such a platform would be considerably more degenerate than anything we deal with here. To complete the example, Figure 8a shows Innocenti and Parenti-Castelli’s trajectory superimposed on the branch locus given in Figure 6. The trajectory starts and ends at S; as expected, it encircles a cusp (the cusp at D). The point S is located in the region of the (L2 , L3 )-plane that has six assemblies, and these are labeled S1 to S6 in Figure 8b. Also shown are the pre-images of each assembly, along with this cusp-encircling trajectory. The never-special motion is that from assembly S1 to S2 or vice versa (depending on whether the encirclement is clockwise or counterclockwise). Other never-special motions can, of course, be constructed by other encirclements of D, or for that matter, any other cusp, all this being consistent with the discussion in Section 6.

8. Design Specialization: Coincidence Constraints We turn now to the idea raised in Section 1 that it might be possible to altogether avoid second and higher-order degeneracy of eq. (2) by specializing the design in one way or another. This fits well with common sense; in the mechanical world, orderly structure and regular form are almost always good things. So whatever form the putative specialization takes—this is not altogether clear for the octahedral manipulator, though we have a fairly good idea of what it means for the planar 3–3—it is expected to be recognizable in terms of symmetry in the design. Start by noting the octahedral manipulator has regular structure when compared to the general 6–6, the regularity coming through coalesced pairs of ball-socket joints (three in the base, three in the platform body). This effects a direct reduction in the number of possible assembly modes. With the leg lengths fixed, the 6–6 has as many as 40 distinct assemblies (Wampler 1996), while the octahedral manipulator



a

b

c

d

567

e Fig. 7. Triply coalesced assemblies for leg lengths corresponding to points A, B, C, D, and E of Figure 6. Also shown are coupler curves of B3 for hinged four-bars A1 B1 B2 A2 . Each coupler curve osculates to first order with a circle of radius L3 centered at A3 . Other intersections between the coupler curve and the circle correspond to the remaining (noncoalesced) assemblies.


568


a b Fig. 8. The specimen never-special motion given by Innocenti and Parenti-Castelli (1998) is superimposed on the branch locus of Figure 6 (a). The trajectory encircles cusp D; pre-images of the trajectory in leg-space are shown (b). Pre-images of the start point S are labeled S1 to S6 . The never-special motion reported by Innocenti and Parenti-Castelli is that from S1 to S2 , and is seen to come close to the pre-image of the cusp D.

has at most 16. Hunt and Primrose (1993) explore various patterns of coalescence, and reveal arrangements that have no more than 24, 16, and 8 assemblies, the number generally diminishing as more order or symmetry is imposed. However, coalescing the ball-socket joints is not, on its own, enough to design-out second and higher-order degeneracy. Further specialization is needed, and having settled on an octahedral design, this must come from the relative dimensions of the platform and base triangles.

9. Design Specialization for the Planar Platform Consider Figure 2, again with the aim of shedding light on the octahedral manipulator, and chose the platform and base triangles to be congruent but otherwise dimensioned arbitrarily. If we fix a coordinate system in the base, the position and orientation of the platform triangle in this system uniquely defines the leg lengths and their directions. Now, suppose a new coordinate system is defined so that in this new system, the position of the platform triangle is same as that of the base triangle in the original system. Swapping platform and base triangles ordinarily gives a second, different, assembly, the two assemblies being identified by superposition. All regular assemblies are paired; this pairing is depicted in Figure 9. Since every regular assembly is one of a pair, every regular configuration in the neighborhood of a special configuration is paired. The implied conclusion that special configurations must themselves be paired holds with two exceptions, these being when α = 0 and α = π , with α as defined in Figure 2.

Excepting these, all other special configurations are paired, the pairs again being identified by superposition. The corresponding components of the branch locus partitions leg space into regions differing by four assemblies. We note the possibility of transition from zero assemblies to four, but discard possible transitions from four assemblies to eight, because six assemblies is the maximum upper bound. When α = 0, straightforward construction reveals that three leg lengths must all be equal. The resulting linkage has full-cycle mobility; there are infinitely many assemblies. When α = π , paired configurations obtained by swapping between platform and base triangles are indistinguishable. Not only are they the same assembly, the assembly is necessarily special because it must be the coalescence of two neighboring, paired, regular configurations. Again this can be confirmed in a straightforward way by construction, the aim being to show the rays directed along the three leg lines always have a common intersection. The associated component of the branch locus marks out transitions from zero assemblies to two or from two assemblies to four. Transitions from two to six assemblies cannot arise, nor can transitions from four to six, because both possibilities are inconsistent with the observation that α = π implies coalescence of paired configurations. Phrased differently, the branch locus associated with α = π marks transition from leg lengths having two assemblies to a number differing from two by two; i.e., zero or four assemblies. For congruent platform and base triangles, there are then zero, two, or four assemblies for a given set of leg lengths, but there are never six. And every regular configuration is paired



Fig. 9. Paired configurations of Figure 2, with congruent platform and base triangles. The platform triangle is shaded. These two configurations are identical by superposition, but are distinguished by values θi , i = 1, 2, 3 once the platform and base are identified. with another regular configuration. If three assemblies could coalesce, the argument used to show special configurations are paired (α 6 = π) would lead us to conclude that triply coalesced assemblies are also paired, with the six putative assemblies lying in two groups of three. But we reject this because it conflicts with our upper limit of four. This leads us to conclude that never-special motions are not possible. This conclusion is not restricted to linkages where platform and base triangles are congruent. Similarity between base and platform triangles gives a similar pairing of configurations, and this leads us to the same conclusions, specifically: (1) there is an upper bound of four assemblies for any set of leg lengths; (2) all regular configurations are paired; and (3) excepting where α = 0, π , special configurations are also paired. When α = 0, the similar platform-base linkage is no longer full-cycle mobile; like assemblies where α = π, these are special but not paired. The upper bound of four assemblies, together with the pairing of special configurations, allows us to conclude that triple coalescence is impossible, because any pairing would necessarily imply six assemblies. By analytical means Gosselin and Merlet (1994) have demonstrated this upper bound of four. Our demonstration gives a

569

slightly stronger statement of the interaction between geometry and assemblies than that provided by algebraic manipulation alone. These conclusions fit the well-established geometry of four-bar coupler curves; below we briefly sketch the relationship. See the works of Hartenberg and Denavit (1964) and Hunt (1990) for essential background. To generate a coupler curve, disconnect a leg, say leg 3, and consider the resulting hinged planar four-bar, i.e., A1 B1 B2 A2 . The choice of B3 as the coupler point distinguishes three points in the fixed plane called singular foci that are the three real intersections of the six imaginary asymptotes of the curve traced out by B3 . Points A1 , A2 are two of the singular foci. The third, As , lies at the vertex of the 4A1 A2 As , similar to 4B1 B2 B3 . In the setting of the planar-motion device, As coincides with A3 when base and platform triangles are similar. The significance of the singular foci lies in the similar relationship that they hold to the coupler curve; A3 has equal footing with fixed centers A1 and A2 , and properties possessed by any of them with respect to the coupler curve are common to all three. Roberts (1879) used this observation as the basis of his demonstration that given a hinged four-bar, with the coupler point identified, two other four-bars can be found that generate the same coupler curve (the Roberts-Chebyshev theorem). Each of the three linkages takes two of the three of the singular foci as fixed pivots. Hartenberg and Denavit (1964) and Hunt (1990) explain some of the geometry of these cognate linkages. Excepting some degenerate cases of no interest to us here (e.g., B3 coincident with B1 or B2 ), the curve traced by B3 never osculates with circles (A1 , L1 ) and (A2 , L2 ). Nor, by the virtue of the relationship between A1 , A2 , and A3 can it osculate with the circle centered at (A3 , L3 ). This rules out the possibility of a triple coalescence, and therefore also the possibility of never-special motion. We conclude this section by remarking briefly on how having similar platform and base triangles is manifested in the formalism of Section 2. The constraint functions themselves do not explicitly encode the relationship between the platform and base triangles. Any specialization must be introduced through interdependencies between the various quantities. Let 4B1 B2 B3 = λ 4A1 A2 A3 , where λ is a scaling factor that when set to unity gives congruent platform and base triangles. This suggests four equalities: L1 cos(θ1 ) + λa1 cos(α) = a1 + L2 cos(θ2 ), L1 sin(θ1 ) + λa1 sin(α) = L2 sin(θ2 ), L1 cos(θ1 ) + λa3 cos(α + γ1 ) = L3 cos(θ3 ) + a3 cos(γ1 ), L1 sin(θ1 ) + λa3 sin(α + γ1 ) = L3 sin(θ3 ) + a3 sin(γ1 ), (27) with γ1 = 6 A2 A1 A3 = 6 B2 B1 B3 . When eq. (27) is substituted into eq. (25), we get the following expression for the 0 determinant of ∂0 ∂θθ :


570


0 ∂0 ∂θθ

= 8λ3 a12 a32 L1 sin(γ1 ) sin(α) λ(a3 sin(θ1 − α) − a1 sin(θ1 − γ1 − α))

− a3 sin(θ1 ) + L1 sin(γ1 ) + a1 sin(θ1 − γ1 ) . (28)

Note that eq. (25) has factored. One of the factors is sin(α), which is zero when α = 0, and π when accounting for one class of special configuration listed above. The other is Q = λ(a3 sin(θ1 − α) − a1 sin(θ1 − γ1 − α)) − a3 sin(θ1 ) + L1 sin(γ1 ) + a1 sin(θ1 − γ1 ). When Q = 0, the platform has two pairs of coalesced assemblies; i.e., four assemblies in total. The signs of the factors can be used to index configurations; i.e., ++, +−, −+, and −−. 0 We note further that the matrix ∂0 ∂L , largely unconsidered in the discussions so far, has a determinant given by

0 ∂0 det ∂L

=

8λ4 a13 a33 sin(γ1 ) [λ − cos(α)] L2 L3 λ(a3 sin(θ1 − α) − a1 sin(θ1 − γ1 − α))

−a3 sin(θ1 ) + L1 sin(γ1 ) + a1 sin(θ1 − γ1 ) , (29)

bearing close resemblance to eq. (28), the two sharing the common factor Q. While this may be of some importance, we are unable to offer any insight based on it at present. Others may see significance in it. An (α, θ1 )-slice of C is shown in Figure 10a; its projection into (L2 , L3 ) is given in Figure 10b. Note (1) the absence of cusps; (2) the partitioning of C factors α = 0, π , and Q = 0 into four distinct regions; and (3) the existence of no more than four assemblies for any leg-rod lengths. Other slices show the same essential structure, all of this being consistent with the elimination of never-special motion by specializing the design. We note further that at the intersection of the α = π and Q = 0 fold sets, four assemblies coalesce. We reiterate that this does not imply the linkage can undertake never-special motion, nor does it imply that linkage acquires two first-order freedoms consistent with losing two first-order constraints. Only one freedom is acquired.

10. Design Specialization of the Octahedral Manipulator: Some Speculative Remarks The analysis and discussion of the previous section give us some hope that appropriate specialization of platform and base triangles for the octahedral manipulator will marginalize never-special motions. We conjecture that, as with the

Fig. 10. An (α, θ1 )-slice of configuration space (a); and its projection into the (L2 , L3 )-plane (b). The platform and base √ triangle dimensions are |A2 − A1 | = 20, |A3 − A2 | = 20 2, |A1 − A3 | = 20, L1 = 15, and λ = 0.75. In (a), the singular set has three components α = 0, α = π , and Q = 0. At the intersection of Q = 0 and α = π , four assemblies coalesce with the loss of first-order constraint only.

planar-motion device, a sufficient condition arises by making platform and base triangles similar. At present, however, we are unable to lend any direct support to this claim, but we note the platform depicted by Lee, Duffy, and Hunt (1998) has this geometric structure. The symmetrical octahedral manipulator platform defies our efforts to understand it. The expressions that emerge when we set out to study it by analytical means are complex yet subtle. They are not easily worked through by hand, and essential detail becomes lost in the mass of terms when computer-algebra techniques are applied. Synthetic reasoning of the sort applied to the planar device fails us here because the zigzag leg pattern does not allow paired configurations to be identified by superposition. But this does not rule out synthetic argument per se. Indeed, we see progress as being most likely to come from this quarter. We can offer the following observations that may help guide the reader who wishes to take these ideas further. The theoretical developments of this work are as applicable to the octahedral manipulator as they are to the planar-motion linkage. Like the planar device, three assemblies coalescing is a sufficient condition for the octahedral manipulator to undertake never-special motion. If we disconnect a ball-socket connection, say that at B3 , the linkage yields an RSSR chain and the point B3 traces out a surface of order 16, circularity 8 (Hunt 1990; Hunt and Primrose 1993), this surface can be viewed as the spatial counterpart of the four-bar coupler curve. Coalescence of three assemblies is synonymous with osculation


McAree and Daniel / Never-Special Assembly Changing Motions between the spin surface of the coupler and the circle traced by the leg pair meeting at B3 . As with the planar-motion linkage, elimination of never-special motions amounts to elimination of osculation. The singular foci, so useful in the setting of the planar linkage, have no spatial counterpart. But this does not rule out the possibility of symmetry in the spin surface itself having the same effect. Indeed, regularity of one sort or another is expected of this surface, because the axes of two R-pairs of the generating linkage intersect at a vertex of the base triangle, giving it the geometry of a spherical four-bar; see for instance Hunt’s earlier work (1990). With the planar device, similar platform and base triangles resulted in reduction from six to four of the maximum number of assemblies. We expect a similar reduction for a symmetrically designed octahedral manipulator; however, we have been unable to show this either analytically or synthetically. Numerical experiments suggest this number is likely to be 8, or perhaps 12; it is unlikely to be less than 8.

11. Conclusion Our aim has been to relate the triple coalescence of assemblies to the possibility of undertaking continuous motion from one assembly to another having the same leg lengths without ever meeting a special configuration. Though we have not gone beyond second-order degeneracy of corank 1 special configurations, we claim enhancement of the understanding of fully-in-parallel robot kinematics, with direct implications for design and perhaps also the prospect of developing better control algorithms. These issues have not been dealt with previously. The key idea is that the projection of configuration space into leg space provides a complete description of the kinematics. The bifurcation set or branch locus under this projection reveals interesting points where the number of assembly modes changes. Crossings of the branch locus can be used to count the number of assemblies associated with a given region of leg space. It is the cusping of this projection that allows never-special motions. We have shown that for the planar 3– 3 linkage, similarity of platform and base triangles excludes the possibility of these motions. We use a counting argument plus an upper bound provided by Bezout’s theorem to prove there are at most four assemblies. We further conjecture that specialization of the octahedral manipulator will likewise exclude such motions. However, the symmetries here seem more subtle. Our analysis was deliberately initiated from a naive perspective, relying on the continuity between configurations and expanding the constraint functions as a series. We took this approach largely because we found existing literature on catastrophe theory deals almost exclusively with a single potential function, and hence avoids the subtle effect of the geometry of the constraints. Our approach, based on viewing

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the expansion through left- and right-singular vectors of the constraint Jacobian, derives more from physical insight than mathematical rigor. However, having arrived at it, the usual catastrophe machinery can be brought into action. The simplest degeneracy is the loss of second-order constraint from a corank-1 special configuration; we give sufficient detail to enable the reader to devise computer software that tests for such degeneracies. Our diagrams were computed using this approach. We have concentrated on triple coalescences of assemblies, and have shown that such a triple coalescence is a sufficient condition for the existence of a never-special assembly-changing motion. A key to the tractability of our approach is the use of the 3×3 constraint Jacobian instead of the more-usual 6×6 Jacobian whose rows are the leg-rod lines. The implied relationship between the two is a consequence of the implicit function theorem. However, there is also a strong geometric relationship between them arising from their being Schur complements of the full 9 × 9 configuration-space Jacobian (see the Appendix). We again see that the geometry of the configuration space imposes considerable structure on the problem of describing the kinematics of parallel manipulators. Indeed, it seems to us that our approach might be extended to the full 6–6 parallel manipulator.

Appendix: Connections between Matrices 0 ∂0 −1 for the Octahedral Manipulator ∂θθ and J The relationship between first-order platform velocity of a six-freedom, fully-in-parallel device and the rate of extension of the leg rods is given by      ωx s1 P1 Q1 R1 L1 M1 N1 P2 Q2 R2 L2 M2 N2  ωy  s2       P3 Q3 R3 L3 M3 N3   ωz  s3       P4 Q4 R4 L4 M4 N4   vx  = s4  . (30)      P5 Q5 R5 L5 M5 N5   vy  s5  P6 Q6 R6 L6 M6 N6 vz s6 The elements of row i are the line coordinates of the linear actuator axis of leg i, normalized so that L2i + Mi2 + Ni2 . ω , v) describes how the platform twists and The six-vector (ω the right-hand-side vector s gives the corresponding linear actuator speeds. Equation (30) solves the inverse-velocity problem. Adhering to notation that is commonly employed for serial robots, we call this 6 × 6 matrix the “inverse Jacobian,” and denote it by J −1 . Special configurations are those where it drops rank from six to five (or perhaps less than five); the lines of the linear actuators then being linearly dependent on one another. The platform consequently acquires one (or perhaps more) first-order freedoms, amounting to the same thing as losing one (or more) first-order constraints. The corank of J −1 describes both the number of linear dependencies among the


572


actuator lines and the number of freedoms acquired by the platform. As a consequence of the implicit function theorem, special 0 configurations are also signified by a drop of rank matrix ∂0 ∂θθ . The purpose of this Appendix is to explain the relationship 0 −1 in terms of the geometry of the linkage. between ∂0 ∂θθ and J Start by letting F (L, θ ) be the function of the leg-rod lengths L and angles θ that governs the octahedral manipulator’s forward kinematics. First-order constraints on the device must then satisfy   ! ! ω ∂F ∂F s  v  = ∂L ∂θθ (31) dθθ , ∂0 ∂0 dt ∂L ∂θθ 0 ∂F ∂0 ∂0 where ∂F ∂L is 6 × 6, ∂θθ is 6 × 3, ∂L is 3 × 6, and ∂θθ is 3 × 3. If the platform is not at a special configuration, that is, if det ∂0 ∂θθ 6 = 0, this expression can be reorganized as   ! ∂F ∂F ! ! ω s 0 I6 θ ∂L ∂θ v = −1 ∂0 ∂0 dθθ 0 ∂0 ∂L ∂θθ ∂θθ dt 0 (32) ! ! ∂F ∂F s ∂L ∂θθ = dθθ . ∂0 −1 ∂0 I 3 dt ∂θθ ∂L

Note that   ω  v  = I6 0 0 I = 6 0

  ω v 0 ∂F ∂F

− ∂F ∂θθ I3 − ∂θθ I3

∂L ∂0 −1 ∂0 ∂θθ ∂L

∂F ∂θθ

!

I3

s dθθ dt

! ,

which gives   ! ! ω ∂F ∂F ∂0 −1 ∂0 s − 0 ∂L ∂θθ ∂θθ ∂L v = dθθ ∂0 −1 ∂0 I3 dt 0 ∂θθ ∂L ! (33) ∂F ∂F ∂0 −1 ∂0 s I6 0 − 0 ∂L ∂θθ ∂θθ ∂L = dθθ . ∂0 −1 ∂0 0 ∂0 I3 ∂θθ dt ∂θθ

So

" ∂F ∂F ω − = v ∂L ∂θθ

∂L

∂0 −1 ∂0 ∂θθ ∂L

!#

s ,

i.e., the matrix J , the inverse of the 6 × 6 matrix J −1 given in eq. (30), is ∂F ∂0 −1 ∂0 ∂F − , J = ∂L ∂θθ ∂θθ ∂L which is the Schur complement of the full 9 × 9 constrain 0 matrix (eq. (31)) with respect to ∂0 ∂θθ ; see for example the work of Kailath (1980).

Moreover, from eq. (33), ∂0 det det (J ) = det ∂θθ

∂F ∂θθ ∂0 ∂θθ

∂F ∂L ∂0 ∂L

! .

To find the relationship between the determinants of J and 0 ∂0 ∂θθ , we start by noting that the velocities of the three vertices B1 , B2 , and B3 are  dB  1

dt  dB   2=  dt 



dB3 dt

[−B1 ∧]    [−B2 ∧]   [−B3 ∧]

I3 I3

B1 − B 2 2 |B1 − B2 | B2 − B 1 B2 − B3 2 |B2 − B1 | 2 |B2 − B3 |

I3

0



 ω  v  ,  d0 0

0 0 B3 − B2 2 |B3 − B2 |

B 1 − B3  2 |B1 − B3 |      B3 − B1  2 |B3 − B1 | (34)

dt

where [−Bi ∧] is the skew-symmetric matrix that acts as a cross-product operator on velocities; i.e., the matrix   −Biy 0 Biz 0 Bix  . [−Bi ∧] = −Biz Biy −Bix 0 We can rewrite the 9 × 9 matrix of eq. (34) as   0 B1 − B 3 [ − B1 ∧ ] I3 B1 − B2 0  P S = [ − B2 ∧ ] I3 B2 − B1 B2 − B3 0 B3 − B2 B3 − B1 [ − B3 ∧ ] I3   I6   1     2 |B1 − B2 |     1     2 |B2 − B3 |     1 2 |B3 − B1 | (35)

0

0

where I6 is the 6 × 6 identity matrix. The determinant of the first of these two matrices is det(P ) = 6 |(B3 − B1 ) ∧ (B2 − B1 )|2 h i (B3 − B1 ) · (B2 − B1 ) − |B3 − B1 |2 − |B2 − B1 |2 . (36) This determinant depends only on the platform dimensions, and therefore is fixed for a given design. The second matrix in eq. (35), S, has determinant det(S) =

1 , 8 |B2 − B1 | |B3 − B2 | |B3 − B1 |


(37)

McAree and Daniel / Never-Special Assembly Changing Motions and, like det(P ), depends only on platform dimensions. The relationship between the left-hand side of eq. (34) and dθθ T is given by the block Jacobian the vector s dt  ∂B1   dB1  0 0 ∂L1,2 ,θθ 1 dt s  0   dB2  ∂B2 0 = Q , (38)    θ dθ ∂L3,4 ,θθ 2 dt  dt ∂B3 dB 3 0 0 L5,6 ,θθ 3 dt

∂Bi where ∂L,θ θ are 3 × 3 Jacobians associated with a vertex for differential changes L and θ . A short calculation shows that the determinants of these 3 × 3 matrices are ∂B1 L1 L2 det =− , |A1 − A2 | ∂L1,2 , θ 1 L3 L4 ∂B2 =− , det (39) |A2 − A3 | ∂L3,4 , θ 2 L5 L6 ∂B3 =− . det |A3 − A1 | ∂L5,6 , θ 3

The matrix Q is a 9 × 9 permutation matrix, having precisely two row exchanges, and thus has the determinant +1. The determinant on the left-hand side of this expression is therefore Q6 i=1 Li det(JB ) = . (40) |A1 − A2 | |A2 − A3 | |A3 − A1 | 0 Equating all terms, we find that the determinants of ∂0 ∂θθ and −1 J are related by 4 Q6 Li |B1 − B2 | |B2 − B3 | |B3 − B1 | 0 ∂0 det( ) = 3 i=0 |A1 − A2 | |A2 − A3 | |A3 − A1 | ∂θθ 4 Q6 Li |B1 − B2 | |B2 − B3 | |B3 − B1 | × 3 i=0 |(B − B1 ) ∧ (B2 − B1 )|2 R 3 det J −1 ,

(41) where R = (B3 − B1 ) · (B2 − B1 ) − |B3 − B1 |2 − |B2 − B1 |2 . Further calculations show that this scaling factor is itself the product of the three perpendicular distances of the points Bi from the base lines Ai Ai+1 , i taken modulo 3. Denoting these distances by R1 , the scaling factor relating determinants of 0 ∂0 −1 is therefore ∂θθ and J

det J

−1

=

3 Y i=0

0 ∂0 Ri det ∂θθ

.

0 We conclude by noting the simplest way for ∂0 ∂θθ to drop rank is by various combinations of its elements becoming

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zero. All special configurations of this sort fall into one of the categories identified in an earlier work (Hunt and McAree 1 1998, sect. 13–20). For instance, if products (B1 − B2 )T ∂B ∂θ1

3 and (B2 − B3 )T ∂B ∂θ3 of eq. (34) are simultaneously zero, we have the condition for four coplanar vertices. Permutations of the various patterns follow either cyclically or by reference to the list of cognates given by Hunt and McAree (1998). The reader wishing to pursue this connection will be aided by Hunt and McAree’s figures, but should keep in mind that some of the depicted configurations have one or more of the dihedral angles θ undefined. In such instances, reconciliation is best achieved using a cognate of the linkage for which all θ s are defined.

Acknowledgments Our thanks to Dr. J.-P. Merlet of INRIA, Sophia Antipolis who convinced us that the octahedral manipulator is able to undertake never-special motion. We gratefully acknowledge the input of Professor K. H. Hunt of Monash University who, during a short visit to Oxford in September 1997, engaged freely in discussion on the problem of never-special motion, providing numerous valuable insights into the geometry of four-bar coupler curves. This work was funded by the United Kingdom Engineering and Physical Sciences Research Council under contract GR/L15005 in collaboration with UK Robotics Ltd. and BNFL PLC.

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