Asymmetrical three-DOFs rotational-translational parallel-kinematics mechanisms based on Lie group theory

European Journal of Mechanics A/Solids 25 (2006) 550–558

Asymmetrical three-DOFs rotational-translational parallel-kinematics mechanisms based on Lie group theory Sameh Refaat a , Jacques M. Hervé b,∗ , Saeid Nahavandi c , Hieu Trinh c a Simplex Pty Ltd, Melbourne, Victoria 3116, Australia b Ecole Centrale Paris, Recherches Mécaniques, 92295 Chatenay-Malabry, France c Deakin University, Geelong, Vic 3217, Australia

Received 21 February 2005; accepted 1 November 2005 Available online 15 December 2005

Abstract The paper introduces four families of three-DOFs translational-rotational Parallel-Kinematics Mechanisms (PKMs) as well as the mobility analysis of such families using Lie group theory. Two of these families are mechanisms with one-rotational two-translational degrees of freedom (DOFs) and each of the other two has one-translational two-rotational DOFs. Four novel mechanisms are presented and discussed as representatives of these four families. Although these mechanisms are asymmetric, the components used to realise them are very similar and, hence, there is no great departure from the favourable modularity of parallel-kinematics mechanisms. © 2005 Elsevier SAS. All rights reserved. Keywords: Overconstrained mechanisms; Asymmetric parallel-kinematics mechanisms; Rotational-translational mechanisms; Three-DOFs PKMs; Lie group theory

1. Introduction PKMs have many advantages (Tsai, 1999; Merlet, 2000). Six-DOFs PKMs might be sought of as PKMs in the extreme where the drawbacks of such PKMs are extremely pronounced. These drawbacks are mainly limited workspace and poor manipulability. To utilise the benefits of the concept of parallel kinematics while avoiding its drawbacks, there is a trend of relying on PKMs with less than six DOFs. Three-DOFs PKMs attract decent amount of interest for this reason. A large number of these three-DOFs PKMs have been proposed. See Tsai (1999), Merlet (2000) for an early survey. Synthesis and enumeration of possible threeDOFs PKMs that can provide either translational or rotational DOFs have been extensively studied based on screw theory (Kong and Gosselin, 2004a, 2004b) and Lie group theory (Hervé and Sparacino, 1991; Karouia and Hervé, 2000, 2002a, 2002b, 2004; Angeles, 2004). Callegari and Tarantini (2003), Callegari et al. (2004) and Gosselin and Angeles (1989) have proposed further kinematic analysis of some three-DOFs PKMs.

* Corresponding author.

E-mail addresses: [email protected] (S. Refaat), [email protected] (J.M. Hervé), [email protected] (S. Nahavandi). 0997-7538/$ – see front matter © 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2005.11.001

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Overconstrained PKMs (Hunt, 1978) are those that have limbs that provide similar constraint(s). That is, the motion constraint provided by one limb is also provided by other limbs. These overconstrained mechanisms do move despite the fact that Chebyshev–Grübler–Kutzbach criterion in its original form (Hunt, 1978; Tsai, 1999; Merlet, 2000) concludes that they should not, and they (overconstrained mechanisms) are mobile only when certain geometrical condition is satisfied. The main advantage of these overconstrained mechanisms is the fact that they use less joints and links, resulting in simpler mechanism. The price is the need for strict manufacturing tolerance and the excessive loads on some links and/or joints. PKMs with three DOFs that are a combination of rotational and translational DOFs have also been studied. These were symmetrical non-overconstrained, i.e. each limb has five-DOFs. A PKM that utilises three R-P-S (i.e. RevolutePrismatic-Spherical) limbs and provides one-translational and two-rotational DOFs have been proposed (Hunt, 1983; Lee and Shah, 1988). A similar PKM that utilises three P-R-S limbs have also been reported (Merlet, 1991) and all equivalent limbs are disclosed in Huynh and Hervé (2005). Replacing the prismatic joints by revolute ones would maintain the mechanism’s three-DOFs. The R-S-R PKMs has been implemented as well (Hui, 1995; Dunlop and Jones, 1997). An overconstrained asymmetrical PKM with one-rotational and two-translational DOFs has been proposed as well (Liu et al., 2002). Apart from these mechanism-specific efforts, screw theory was presented as a tool for synthesis of lower-mobility PKMs (Fang and Tsai, 2002; Huang and Li, 2002; 2003). There was no specific focus on three-DOFs rotationaltranslational PKMs though. Here, there is a specific focus on three-DOFs PKMs as they can offer useful solutions in many applications, for example, the problematic rotational DOFs in machine tools (Refaat, 2004). 2. Lie group theory Screw theory is increasingly utilised to synthesise PKMs. Many reviews of the basics of screw theory can be found in the literature (Tsai, 1999; Kong and Gosselin, 2004a, 2004b). However, screw theory natively deals with the instantaneous (or local) mobility of a mechanism. Lie group, on the other hand, is a useful tool for full-cycle or finite mobility. In other words, screw-theory represents the differential aspect of Lie group theory. Background about Lie group theory can be found in the literature (Hervé, 1999; Angeles, 2004). It may be worth recalling that a Lie group is a set endowed with the algebraic structure of a group together with the algebraic structure of an analytic manifold. In the special case of the 6-dimensional (6D) displacement Lie group, the product of displacements, which can be represented by a matrix product in any frame of reference, is the closed binary operation of the group. As a matter of fact, the product of two displacements is still a displacement; the identical transformation is the neutral displacement E; and any displacement D has an inverse denoted D −1 ; DD −1 = D −1 D = E. Moreover, differential calculation is allowed with displacements. An infinitesimal displacement is a transformation, which can be represented by the addition of the identity and a twist. The following is notations of Lie subgroups of displacements; G(x): Planar gliding perpendicular to unit vector x. R(N, x): Set of rotations around the axis having a frame of reference (N, x). T : Set of 3-DOF translations. T2 (⊥x): Set of 2-DOF translations perpendicular to x. T (x): Linear translations parallel to x. S(O): Spherical motion about the point O. X(x): Set of Schoenflies (Schönflies) motions of direction x (it is a 4D Lie subgroup). Vectors are bold-faced characters, unit vectors are bold lower-case characters, points are capital letters and calligraphic letters indicate subgroups. From Mozzi–Chasles theorem, any finite or infinitesimal displacement is a screw motion. Hence, the subgroups H (N, u, p) of helical motions of given axes (N, u) and pitches p cannot be ignored. So is the 3D subgroups Y(u, p). However, mechanisms implementing screw pairs are out of scope here though. The improper subgroups are E , which contains only the identity E, and D, which is the six-dimensional group of displacements. Arthur Schoenflies (also spelt Schönflies) is a mathematician who wrote a book chapter about a special 4-DOF motion type. A Schoenflies motion (X-motion) can be defined as the commutative product T R(N, x), the point N

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being any one. The X(x) subgroup contains an infinity of 1D subgroups R(N, x) of rotations, which have the same direction of axis. Hence, without loss of information, a X-motion can be called a three-translational-one-rotational motion that constrains two rotational DOFs. However, the constraint space is not endowed directly with a group algebraic structure; moreover the constraints are actually wrenches that are only locally defined through the expression of the power. The power is the invariant Klein form of general Lie’s theory. Any serial array of H, R, P pairs producing 4-DOF motion between the distal bodies of the chain, is a generator of X-motion, provided that the H or R pairs have parallel axes. The set of Schoenflies motion is endowed with the algebraic structure of a four-dimensional Lie group. Hence, the product of Schoenflies motion subsets is a Schoenflies motion subset because of the product closure in any subgroup. For instance, the serial array of two H pairs with parallel axes produces a 2D manifold of displacements included in a X-subgroup. This 4-DOF motion type generalises the 3-DOF planar motion. As a matter of fact, it can be obtained also as the commutative product of a planar motion subgroup by a linear translation subgroup, namely X(x) = G(x)T (x) = T (x)G(x). The set of twists of a displacement Lie subgroup is endowed with the algebraic structure of a Lie algebra and is called a Lie subalgebra. Algebra in its general meaning is an aspect of mathematics. In our context, an algebra is a particular algebraic structure. An algebra is a vector space endowed with a closed product. By the Lie bracket, a closed product in the twist space can be defined. Hunt (1978) who ignored Lie’s theory of continuous groups of transformations found the Lie subalgebras of twists as “screw systems that guarantee full-cycle mobility”. Moreover in Lie’s theory, the exponential map of a Lie subalgebra provides the corresponding Lie subgroup (Selig, 1996). Hence there is a one-to-one mapping between Lie subalgebras of twists and Lie subgroups of finite displacements. The main useful algebraic property is the closure of the product in any displacement subgroup. From a long time, everybody knows that the products of translations are translations, the products of planar displacements are planar displacements along the same plane, the products of spherical displacements are spherical displacements around the same point. Group theory formalises and generalises that intuitive approach. 3. Four structural asymmetrical 3-DOFs families of PKMs Symmetrical PKMs are those that have limbs of identical architectures. Symmetry represents one of the main advantages of PKMs that allows their modularity and reduces their cost. Although the mechanisms presented here are not symmetric the departure from modularity is very minor as the limbs utilised are manufactured from very similar components. In most cases, limbs actuators are identical too. Here, four families of rotational-translational DOFs PKMs are presented. Each of the first two families provides the platform with one-rotational-two-translational DOFs and each of the second two families provide the platform with one-translational-two-rotational DOFs. All the PKMs presented here have three limbs to allow the actuators to be placed on the machine base. The planar PKM is a commonplace one-rotational-two-translational DOFs PKM. This is not the one presented here though as this family is fairly well established (Tsai, 1999; Merlet, 2000). Instead the rotation axis is selected to be not normal to the translation plane. Fig. 1 depicts a PKM that provides one-rotational and two-translational DOFs. Using Lie group theory, one can demonstrate the full-cycle mobility of that mechanism as follows. The first limb generates the product G(x)R(O, y). O is a point of the axis of rotation and belongs to the moving platform. The second limb generates the same motion set. The third limb generates G(z)R(A, y)R(B, y). Using group algebraic properties of motions sets, one can show that G(z)R(A, y)R(B, y) is equal to G(z)G(y) by elimination of the redundancy of the square of G(z) ∩ G(y) = T (x) in the product G(z)G(y). For instance G(y) = T (x)R(A, y)R(B, y) can be proven at the level of motion types in a neighbourhood of the identity; as a matter of fact, the manifold T (x)R(A, y)R(B, y) has the dimension 3 and is contained in the 3D subgroup G(y). Moreover G(z)G(y) = G(z)T (x)R(A, y)R(B, y) = G(z)R(A, y)R(B, y) because G(z) ⊃ T (x) and G(z)T (x) = G(z), the product being closed in the subgroup G(z). Hence, the set of allowed motions for the moving platform is G(x)R(O, y) ∩ G(z)G(y). G(x) = T2 (⊥x)R(N, x),

∀N (for any point N ),

G(z)G(y) = T R(A, z)R(B, y),

∀A, ∀B,

G(x)R(O, y) ∩ G(z)G(y) = T2 (⊥x)R(N, x)R(O, y) ∩ T R(A, z)R(B, y)   = T2 (⊥x) R(N, x)R(O, y) ∩ T (x)R(A, z)R(B, y) ,

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Fig. 1. Member of the first one-rotational and two-translational DOFs asymmetrical PKM family.

R(N, x)R(O, y) ∩ T (x)R(A, z)R(B, y),

∀A, ∀B, ∀N leads to   R(N, x)R(O, y) ∩ T (x)R(A, z)R(O, y) = R(N, x) ∩ T (x)R(A, z) R(O, y) = R(O, y).

Therefore, the platform motions are represented by T2 (⊥x)R(O, y), which is a product with two-translational and one-rotational DOFs. The other members of the family are obtained by replacing the first two limbs by other limbs that generates the same 4D manifold of displacement G(x)R(O, y), and by replacing the third limb by limbs that generate the same 5D manifold of displacements G(z)G(y). G(x) is generated by any array of three pairs among R and P pairs, provided that the R axes are parallel to x and the P pair are perpendicular to x and eliminating the combination PPP that generates T2 (⊥x) ⊂ G(x). The enumeration of limbs that generate G(z)G(y) without passive mobility between the distal bodies is now available online (Lee and Hervé, 2005). One can readily verify that any serial arrangement of a 2-DOF planar chain and a 3-DOF planar chain generates a doubly planar 5-DOF kinematic bond. Fig. 2(a) shows a member of another family that also provides one-rotational and two-translational DOFs. Two limbs generate G(x)R(O, y); a third limb generates T (y)G(y) = X(y) (if the axes of the R pairs are not in a plane). Again, using Lie group theory one can demonstrate the full-cycle mobility of that mechanism. The set of allowed displacements of the moving platform is G(x)R(O, y) ∩ X(y). G(x) = T2 (⊥x)R(A, x),

∀A,

X(y) = T R(B, y), ∀B

⇒

X(y) = T R(O, y),

G(x)R(O, y) ∩ X(y) = T2 (⊥x)R(A, x)R(O, y) ∩ T R(O, y)   = T2 (⊥x) R(A, x) ∩ T (x) R(O, y) = T2 (⊥x)R(O, y). This set is a product of a subgroup of planar translations and a subgroup of rotations. However, three fixed motors cannot actuate the platform motion. As a matter of fact, if the fixed joints are locked, then a 1-DOF motion can always

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(a)

(b) Fig. 2. Member (and limb) of the second one-rotational and two-translational DOFs asymmetrical PKM family.

happen because we have a planar 4R sub-chain explained by R(O, y) ∩ G(y) = R(O, y). See Fig. 2(b) for a better view of that limb. This problem can be overcome by using another generator of X(y) without an unactuated planar subchain. Alternatively, one can exchange the location/order of the P and R joints or use a cylindrical joint. The other members of the family are obtained by replacing the generator RRR of G(x) by equivalent chains producing also G(x) and by replacing the generator PRRR of X(y) by other generators of the same Schoenflies motion. Fig. 3(a) shows a representative of a family of PKMs that can provide the platform with one-translational-tworotational DOFs. That mechanism uses one planar-spherical limb and two other limbs, each of which generates the

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(a)

(b) Fig. 3. Member of the first one-rotational and two-translational DOFs asymmetrical PKM family.

5-DOF product of G(x) and R(N, y) motions. Another example of planar-spherical limb is shown in Fig. 3(b). The equivalent generators of planar-spherical kinematic bonds are described online (Hervé, 2003). In this family of mechanisms, two limbs produce G(x)RN, y) and a third limb produces G(y)S(N ). One can readily show G(x)R(N, y) ∩ G(y)S(N ) = T (z)R(N, x)R(N, y).

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Fig. 4. One-translational and two-rotational DOFs asymmetrical PKM.

The last proposed family is represented here by the mechanism of Fig. 4. It uses two limbs that are similar to the ones used in Fig. 3(a) and the third limb generates T (z)S(O). Notice, however, that this mechanism cannot be actuated using the three P joints. Remarks. • Among the three limbs, two limbs have the same architecture and produce the same motion type. Only two limbs with distinct architectures are sufficient to realise the kinematic constraint of the platform motion in each of the PKMs proposed. However, three limbs are used to ensure that there is one actuator per limb. • The shape of links to be used can be further optimised for better space utilisation. • The whole issue of actuator selection has not been discussed in details in the current work. 4. Conclusions Using the equivalencies that are proven by the closure of the product in any algebraic subgroup, four families of PKMs have been conceived. For the first family; two limbs generate the product G(x)R(O, y) and a third limb generates planar-planar displacements G(z)G(y) (the enumeration of the equivalencies was recently published, Lee and Hervé, 2005). The intersection and hence the platform motion is T2 (⊥x)R(O, y). That is, two translations parallel to the plane normal to the x direction and one rotation around an axis having the y direction. In a possible related family, one limb generates G(x)R(O, y) and two limbs generate G(z)G(y). For the second family, two limbs generate the product of planar displacements G(x) and rotations R(O, y), a third limb generates Schoenflies displacements X(y). The intersection, again, is T2 (⊥x)R(O, y). A related family implements one limbs producing G(x)R(O, y) and two limbs producing X(y). The third family utilises two limbs that generate the product G(x)R(O, y) and the third limb generates planar spherical displacements G(y)S(O). The intersection is T (z)R(O, x)R(O, y). That is, two rotations around the (O, x)

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and (O, y) axes and translation about z direction. In a related family, there are one limb producing G(x)R(O, y) and two limbs producing G(y)S(O). Each member of the fourth and the last family relies on two limbs that generate G(x)R(O, y) and a third limb generates T (z)S(O). The intersection is T (z)R(O, x)R(O, y). However, in these mechanisms, the three actuators cannot be fixed. More limb architectures can be synthesised by replacing a prismatic pair P that produces linear translation by a planar hinged parallelogram coupling PA that produces circular translation. As a matter of fact, two opposite bars of a hinged parallelogram remain parallel and therefore generate relative translation. The set of relative displacements is a 1D manifold denoted TC (⊥x) included in the 2D subgroup of planar translations T2 (⊥x). Because of the product closure in the subgroup G(x), one can demonstrate, for example, the set equality G(x) = R(A, x)TC (⊥x)R(B, x), ∀A, ∀B. 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