IN RECENT years, several authors proposed new parallel

56 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006 Uncoupled Actuation of Pan-Tilt Wrists J. M. Hervé Abstract Using algebraic properties of displacement (or rigid-body motion) subsets, the paper introduces new two-degree-of-freedom (2-DOF) nonoverconstrained orientation mechanisms. The angles of pan and tilt are also referred to as the angles of precession and nutation, respectively, employing the standard terminology of Euler angles. A serial array of two revolute pairs provides the kinematic constraint of an end-effector. For instance, the first axis of rotation is fixed vertically, and the second axis rotates around the first axis, remaining parallel to the horizontal plane. Such a 2-DOF wrist is fit for orienting various devices like telescopes, cameras, antennas, etc. The first axis is fixed, and can be actuated by any powerful heavy servomotor. The paper discloses new mechanisms that allow the actuation of the second movable axis by a fixed servomotor. Moreover, the actuation of the movable axis is not a function of the pan.in other words, the pan and the tilt are controlled independently or in a fully uncoupled manner. Index Terms Orientation manipulators, parallel mechanisms, type synthesis, uncoupled motion, wrists. I. INTRODUCTION IN RECENT years, several authors proposed new parallel wrists [1] [9]. These wrists are also termed orientation parallel mechanisms. The set of all rotations around axes passing through a given point is denoted. In this paper, curly brackets are employed to designate displacement subsets. The elements of are called spherical motions. Any spherical joint of center generates a displacement subset that is. The set is endowed with the algebraic structure of a 3-D Lie group. In any Cartesian frame of reference having its origin at, the Lie group of geometrical point transformations is represented by the isomorphic matrix Lie group usually denoted SO(3) [10]. The elements of SO(3) are special (proper) orthogonal matrices, and these matrices act on vector arrays of Cartesian coordinates. The notation SO(3) comes from the classical mathematical theory of the general linear group GL(n), which acts on elements of. Unfortunately, the notation SO(3) ignores the location of the center of the spherical displacements. As a consequence, this classical notation is not effective for geometric reasoning. The Lie subgroups of are the 1-D Lie subgroups of rotations around axes passing through the given point and parallel to the unit vector. A revolute pair of axis generates the displacement subset.if is a Cartesian frame of reference, then the first two Manuscript received February 22, 2005; revised July 6, 2005. This paper was recommended for publication by Associate Editor J. Angeles and Editor H. Arai upon evaluation of the reviewers comments. The author is with the Ecole Centrale Paris, Grande voie des vignes, Chatenay-Malabry 92295, France (e-mail: jherve@ecp.fr). Digital Object Identifier 10.1109/TRO.2005.858859 coordinates of any generic point are transformed through a product by a matrix of the Lie subgroup SO(2) [10]. However, the classical notation SO(2) gives no information about the axis, which is essential for any geometric analysis. If two revolute pairs with axes intersecting at point generate two subgroups and, then the kinematic chain of the serial array of these pairs generates a displacement subset, which is the product represented as. In any algebraic group, the product of two subsets is the set of the element products. and are included in. Because of the product closure in the group, the product is also included in. If, then the intersection is the set that contains only the identity transform. is the improper displacement subgroup of dimension zero. The product is a 2-D manifold included in the 3-D subgroup, but it is not a Lie subgroup of. By the same token, one can show that if are linearly independent vectors, then is a 3-D manifold included in the 3-D Lie group. The identity transform belongs to both and. Hence, is a 3-D neighborhood of the identity in the group ; when dealing with motion type, one can ignore the boundaries of the neighborhood and, therefore,. Neglecting the possible difference in the amplitude of the displacements allowed, the serial array of three revolute pairs with axes intersecting at generates the subgroup of spherical rotations around. Hence, there are multiple ways to generate, namely, the spherical joint of center and any serial layout of three revolute pairs (RRR), provided that the axes intersect at and are not coplanar. A spherical parallel mechanism (or parallel wrist) can be constructed, implementing three distinct legs that generate the same subgroup [1], [2]. These mechanisms are overconstrained, because one leg is enough to produce the spherical motion and the other two legs are redundantly compatible with the first given leg. By addition of mobility in the legs, nonoverconstrained parallel wrists were devised [3] [7]. In these new parallel wrists, each leg generates a 5-D manifold of displacements, which contains the subgroup. The product of two Lie subgroups of rotation around two intersecting axes is not a Lie subgroup of displacements, but rather a 2-D manifold included in. However, this motion type is often useful in many applications for orienting objects that have axial symmetry. In most cases, is vertical and is horizontal. Employing the standard Euler terminology, the angle of rotation about the vertical axis is called 1552-3098/$20.00 2006 IEEE

HERVÉ: UNCOUPLED ACTUATION OF PAN-TILT WRISTS 57 Fig. 1. Fig. 2. (a) Serial pan-tilt wrist. (b) Parallel pan-tilt wrist. (a) Spherical pan-tilt wrist. (b) Nonoverconstrained 2-DOF wrist. the precession, and the angle of rotation about the horizontal axis is the nutation. In other terminology, these angles are named azimuth and elevation, or also pan and tilt; in this paper, the words pan and tilt will be preferred. That motion type is generated by the series array of two revolute pairs that is sketched in Fig. 1(a). One cannot use the closure of the product in any subgroup to establish that would be equal to another product of two factors. However, a fully parallel two-degree-offreedom (2-DOF) wrist with two limbs is depicted in Fig. 1(b). The two servomotors cannot be mounted on the frame. The pan motor can be on a first limb, and the tilt motor can be on a second limb, but that tilt motor is not fixed. In order to ease the understanding, actuated pairs are gray-colored and the limb that is used to actuate the tilt is drawn with gray lines. Gosselin and Caron [8] proposed a parallel actuation of a 2-DOF wrist, Fig. 2(a). The whole mechanism is a spherical single loop kinematic chain and all relative displacements are spherical motions around a center. This mechanism is overconstrained and requires five intersecting axes; else, the mechanism cannot work. Carricato and Parenti-Castelli [9] proposed a novel 2-DOF wrist. In this mechanism type, a fixed motor can actuate the tilt independently of the pan. The nonoverconstrained version of this wrist will be demonstrated and depicted, among others, in Section III, using group theory. II. GROUP THEORY IN KINEMATICS Notwithstanding relevant recent papers, for instance, [11], most of present-day authors dealing with robot kinematics and type synthesis of parallel manipulators do not usually employ group theory. Hence, it may be worth recalling some fundamental aspects of what can be found in the literature. Moreover, a clarified presentation of basic concepts seems to be necessary. Actually, the group-algebraic property of the Euclidean displacement set formalizes and generalizes properties that are generally considered as obvious. For instance, the product of two translations is still a translation, the product of two spherical motions around a point is a spherical motion around the same point, etc. Using the terminology of algebra, one will say that the product of translations is a closed product in the subset of translations, etc. Roughly speaking, an algebraic group is a set endowed with a closed product having the properties of a multiplication that is not always commutative. The set of rigid-body displacements is endowed with the algebraic structure of a group. More precisely, this group is a 6-D Lie group, which is also endowed with the algebraic structure of a smooth 6-D manifold. The author, who introduced a geometric notation for the Lie subgroups of displacements, discovered early on the importance of the algebraic structure of a group in the analysis and synthesis of mechanisms [12]. Further contributions were disclosed in [13]. The book of Karger and Novák [14], first published in Czech, and later translated into English, laid the mathematical foundations for the application of the Lie theory of groups to the kinematics of rigid-body systems. However, the key role of Lie subgroups is ignored in that book. Later, Selig [10] published a full book, which is an extensive and valuable contribution to the Lie theory of groups in kinematics and robotics. Nevertheless, Selig s work is a special development of the matrix subgroup SE(3) of the more general theory of the matrix group GL(n) acting on -dimensional vectors. Selig [15] became aware of some incompleteness in the matrix notation. The matrix Lie subgroups of SE(3) actually represent conjugacy (or conjugation) classes of subgroups, instead of subgroups of geometrical transformations. As a matter of fact, the matrix subgroups have to be associated with a frame of reference in order to become geometric subgroups. That is, matrix notation gives no information on the frame of reference and, therefore, is not adequate for reasoning with geometric entities. Most of the displacement Lie subgroups have a usual name, the initial of which is recalled in the author s notation. For example, R means rotation, T translation, S spherical displacement, and so on. The invariant geometric entity, which determines a particular subgroup among all the equivalent subgroups of its conjugacy class, is also included in the author s notation. For example, an axis can be determined by a frame of reference, and a subgroup of rotations around the axis is consequently denoted. A property of the subgroup is to be represented by a matrix subgroup SO(2) in any frame of reference. The required elements of the frame of reference are underlined in the foregoing example, as well as in Table I. Notation

58 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006 TABLE I SUBGROUP NOTATION TABLE III PRODUCTS OF DEPENDENT SUBGROUPS [12] TABLE II GRAPH OF BINARY RELATION TO BE A LIE SUBGROUP OF A LIE SUBGROUP SO(2) could be used instead of. Table I shows the connections between the geometric subgroups [12], [13] and the matrix subgroups, as explained in [10]. The inclusion of a Lie subgroup in a greater-dimensional Lie subgroup defines a binary relation of partial order between all the Lie subgroups. Most of the inclusions are subject to geometric conditions that are recalled in Table II. It may be worth recalling that in any group, the intersection of two subgroups is always a subgroup. If the intersection of two subgroups is the improper subgroup, which contains only one element, the identity transform, then the subgroups are said independent; else, subgroups are dependent. The product of two dependent subgroups contains twice the subgroups of their intersection. The redundancy in that intersection can be eliminated, thus yielding a regular representation of the product, i.e., without superfluous parameters. Detailed explanation of such an elimination is done in the paper for the cases and. The table of products of dependent subgroups that was published in [12] is reproduced here as Table III for completeness. Gray coloring indicates the cases that are used in the paper. The other cases are not relevant to our particular problem because the product contains the translation set, that has to be avoided. In a given kinematic chain, the set of feasible relative displacements of a rigid body with respect to a second body is called kinematic bond between these two bodies. Generally, a kinematic bond is a manifold in, which has a dimension. The integer is called dimension or degree of freedom of the bond. However, in very special closed-loop chains, a kinematic bond has a bifurcation and is not a manifold. In these singular cases, the DOF is not well defined. A

HERVÉ: UNCOUPLED ACTUATION OF PAN-TILT WRISTS 59 Fig. 3. (a) Coupled actuation of pan and tilt. (b) Special arrangement with two coaxial R pairs. kinematic chain that produces a given bond between two of its bodies is called a generator of the bond. A given bond generally has several generators that can be considered as kinematic equivalencies. III. GENERAL GEOMETRY OF PAN-TILT WRISTS In what follows, numerous novel pan-tilt wrists are synthesized with three advantageous properties: the mechanism is not overconstrained and, therefore, can work even in the presence of manufacturing and assembly errors; the actuation of the angle of tilt is not a function of the angle of pan; the tilt is actuated by a fixed servomotor, which may be linear or rotational (actuation by a helical pair is not considered). A general nonoverconstrained 2-DOF wrist mechanism is shown in Fig. 2(b). The fixed base is connected to the oriented body by two limbs. One limb is an array that produces the geometric constraint of the desired motion; the second limb is any kinematic chain that generates 6-DOF motions. In other words, the second limb is a generator of the displacement group. The 6-D Lie group is represented in any Cartesian frame of reference by the matrix group usually called special Euclidean group and denoted SE(3). A possible second limb architecture is, which is a serial chain of two revolute pairs, one prismatic pair, and a spherical pair. Such a mechanism is shown in Fig. 3(a). The pair can be chosen for the actuation of the tilt. However, the rotation around the vertical axis (pan) generally will modify the tilt for any given value of the translation in the actuated pair. In other words, the pan and the tilt are coupled. A more special second limb can be made of a serial arrangement of a 5-DOF kinematic chain and a revolute pair, which is coaxial with the fixed revolute pair producing the pan, Fig. 3(b). The kinematic chain is closed, including two adjacent coaxial pairs, which produce a passive rotation in the loop. Consequently, the chain works as an equivalent chain, which is a general spatial chain that is movable with one DOF, and the whole chain can rotate around the fixed axis. The mechanism can be Fig. 4. (a) General uncoupled actuation. (b) Special architecture. considered as equivalent to a rotating closed loop. An example of a mechanism with uncoupled actuation of the pan and the tilt is shown in Fig. 4(a). In this example, a moving prismatic (gray) pair actuates the angle without affecting the value of. In a special case of a possible kinematic chain shown in Fig. 4(b) for the independent actuation of, an actuated pair is adjacent and parallel to the pair that is coaxial with the actuated pair of the pan. Then the array is equivalent to a cylindrical pair that generates the 2-D Lie subgroup of displacements. is an Abelian or commutative group: the product of two transformations that belong to does not depend on the order of the two factors. One can also write being called a direct product of and in group theory. Hence, the order of the two kinematic pairs in can be changed without modifying the values of the rotation in and the translation in. The resulting mechanism of Fig. 5(a) is equivalent to the mechanism of Fig. 4(b). In the new layout, the actuated pair becomes fixed. A family of nonoverconstrained 2-DOF orientation mechanisms with a tilt actuation that is not coupled with the pan actuation has the architecture type proposed by Fig. 5(b). The oriented object and the fixed actuated pair are connected by a kinematic chain, which generates a 5-D manifold included in the 6-D Lie group of displacements. It is worth noticing that the notation does not characterize a precise manifold but only a manifold type, whereas the Lie subgroups are fully determined by the author s notation. Suitable manifolds have to obey two conditions. The manifold must contain the subgroup of rotations around the fixed axis, and must not contain the subgroup. In other words and The last condition is required to obtain the necessary six dimensions of the kinematic bond between the fixed base and the oriented body (see Fig. 5).

60 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006 Fig. 5. (a) Fixed actuation of an uncoupled pan-tilt wrist. (b) Its generic type. Fig. 6. family. (a) Nonmovable chain. (b) Its integration in a general wrist of the first A general possible solution is that is illustrated by the example of Fig. 5(a). can be any 4-D manifold that verifies. A less obvious solution is obtained if. is a displacement Lie subgroup of dimension. contains and is a manifold of dimension 5-. The condition movable chain, Fig. 6(a), which establishes the required property. Hence, the limb can be used to transmit the actuation in the fixed joint in a manner that is uncoupled with the pan, Fig. 6(b). However, particular geometric conditions have to be avoided, as explained in Section VI. There are many ways to generate an adequate manifold. A special way is the use of a product of two dependent subgroups. A simple example is given by with implies necessarily but not always sufficiently that and Hence, despite cannot be the 4-D subgroup of Schönflies (often spelt Schoenflies) motions, because always contains the 3-D subgroup of spatial translation, which intersects with the linear translation. As a matter of fact, one can write Likewise, though cannot be because is included in Hence, only two subgroups containing can be the subgroup, namely, the subgroup of spherical motions around the point Q with axis, and the subgroup or of planar displacements along a plane Pl that is perpendicular to the unit vector. IV. FIRST FAMILY OF WRISTS: An serial chain is chosen as an example of a limb that generates an adequate manifold. In this example, the adequate manifold is Because of the product closure in the subgroup can be decomposed into the product of three rotation subgroups around three axes intersecting at Q, and therefore, one can write provided that are linearly independent. It is worth recalling that the equality is valid for finite displacements only in a neighborhood of the identity. One can also write for any vector base can be equated to, and the product However, and designate the same axis, because belongs to the axis ; therefore, and are equal. The square of is equal to because of the product closure in the subgroup. The product is the 5-D manifold provided that the spherical pair center Q lies on the axis. The closed loop of structural type is generally a non- which is also equal to

HERVÉ: UNCOUPLED ACTUATION OF PAN-TILT WRISTS 61 Fig. 7. (a) Wrist with an SS arrayof Carricato and Parenti-Castelli and (b) related wrist without passive motion. Fig. 9. motion. (a) Wrist with an SG array, and (b) related wrist without passive Fig. 8. (a) Kinematic equivalences of an SS array, and (b) of an SG array. Fig. 10. (a) Nonmovable subchain and (b) its integration in a general wrist of the second family. These set equalities have a practical application. A 5-DOF serial array of two spherical pairs can be employed, thus leading to the sketch of Fig. 7(a). This mechanism is the actuator of the tilt in the pointing device of Carricato and Parenti-Castelli [9]. The free rotation around the axis determined by the two sphere centers is passive, with respect to the kinematic bond generated by. This passive motion can be eliminated. If stands for two pairs with axes intersecting at the center of the first pair, any open chain can replace the array, provided that the axes and the line of the two sphere centers make up a frame of reference, Fig. 8(a). Hence, the new mechanism of Fig. 7( b) without two coaxial revolute pairs is derived. Referring to Table III of dependent subgroups (Section II), a manifold can be also the result of. The intersection is the subgroup, where is the unit vector that is perpendicular to the plane direction Pl. can also be denoted. By elimination of the redundancy of the square of can be equated to the 5-D manifold. The condition implies that the plane Pl has to be nonparallel to. The optimal situation for this condition is obtained for. Fig. 8(b) shows a possible elimination of the passive rotation in the chain, which is equivalent to an chain. In Fig. 9(a), an array of a spherical pair and a planar pair generates the manifold. The implementation of this chain in a pan-tilt wrist is shown in Fig. 9(b), where the pair is replaced by an equivalent generator of planar motion (or planar gliding). Obviously, other generators, and of planar motion can also be implemented. In the symbolic notation of pair arrays, the underline indicates planar chains. V. SECOND FAMILY OF WRISTS: The manifold can also be the subgroup of planar displacements if and only if the plane is perpendicular to. A corresponding embodiment of the limb for the tilt actuation has the architecture, provided that the pair axes are not parallel to, Fig. 10(b). The closed loop of Fig. 10(a) generally is not movable (exceptions are explained in Section VI), which means The planar pair generates the subgroup of planar displacements, but its kinematic equivalencies also do. The equivalency

62 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006 Fig. 12. (a) Wrist with a GS array and (b) equivalent wrist with a planar-spherical subchain, RR-S, (c) PR-S, (d) PP-S. Fig. 11. Wrist with a subchain of type G. (a) RR : RRR-RR, (b) RRP -RR, (c) RP P -RR, (d) RRR-U. can be a series of three revolute pairs with axes perpendicular to the equivalent plane. As a matter of fact, the rotations that are produced by the pairs are included in the planar motion of, and the product of the three independent rotations is a closed product in the 3-D subgroup. That way, the mechanism of Fig. 11(a) is readily obtained. In this new system, there is no real pair that is coaxial with the precession pair. Other generators of planar motion, namely can replace the planar chain. Fig. 11(b) and (c) show examples with one and two pairs for the generation of.itis noteworthy that in any limb of the family type, the two pairs can have intersecting axes, thus making up a universal joint, and Fig. 11(d) depicts a simple mechanism that may have some practical interest. In the joint, the direction of an axis can be chosen almost freely, but must not be perpendicular to the plane of ; else, the serial array is singular. The adequate manifold can also be obtained by means of other kinematic chains. Some products of two dependent subgroups can be equated to,as explained above in the case of the family with. The numerous equivalencies of the serial layout of a generator of planar displacements and a generator of spherical displacements can be used once more. Fig. 12(a) shows the use of a array of a planar pair and a spherical pair to generate an adequate 5-D manifold. Fig. 12(b), (c), and (d) illustrate the implementation of some of the equivalencies of that are obtained by elimination of Fig. 13. Wrist with a planar-cylindrical subchain of type (a) RRC, (b) RP C. the superfluous 1-DOF rotation in. All the numerous equivalencies of the planar-spherical bond are discussed in [16], and are also disclosed as limbs of parallel 5-DOF manipulators by Li et al. [17]. In Fig. 11(b), a subchain can be replaced by, provided that the pair axis in is chosen parallel to the pair, thereby obtaining the device of Fig. 13. By the same token, the mechanism of Fig. 11(c) can become the special system of Fig. 13(b). VI. INADEQUATE LIMBS The limb that transmits the actuation in the fixed joint to the tilt generates 5-D manifolds of displacements, which have to be independent of the vertical translation or. In other words, must not contain. It is not quite simple to verify that condition. Many 5-D manifolds contain

HERVÉ: UNCOUPLED ACTUATION OF PAN-TILT WRISTS 63 In Fig. 15, two pairs are parallel in the limb producing the 5-D manifold, where is the unit vector in the direction of the two parallel pairs. Vector may be perpendicular to or not. The 2-D manifold is included in the 3-D subgroup. Hence, is included with broad meaning in, which is a product of two dependent subgroups. The intersection is equal to being perpendicular to both and. We can write Fig. 14. (a) Wrong pan-tilt wrist and (b) its movable subchain. because of the product closure in the subgroup. Hence, is proven. By other possible ways of elimination of the redundant subgroup, the set equality is in- can be established (see Table III), which proves that cluded in Consequently, the closed chain of Fig. 15(b) is movable with one DOF, and the mechanism of Fig. 15(a) does not work as a 2-DOF wrist. Fig. 15. (a) Wrong pan-tilt wrist and (b) its movable subchain. translational displacement subsets. The 5-D manifolds that contain must be rejected because. The 5-DOF limbs of translational parallel manipulators [18] produce 5-D manifolds that contain the 3-D subgroup. The manifolds containing a subgroup of planar translations must be rejected only if the plane is parallel to. A comprehensive discussion on these numerous 5-D manifolds, which must be avoided for obtaining adequate manifolds, lies outside of the scope of the paper. In what follows, two examples show that by adding special geometric conditions, a 5-D manifold may become inadequate. Fig. 14(a) is a particular geometry of the mechanism of Fig. 6(b). Two pairs have parallel axes, namely and. The manifold is. This manifold can be equated to a product of two dependent subgroups (see Table III) Hence, the manifold contains. If is perpendicular to, then contains. The closed chain of Fig. 14(b) is movable as a 1-DOF planar chain, and therefore, the device of Fig. 14(a) is not a 2-DOF wrist. Fig. 15(a) shows a special case of the mechanism of Fig. 10(b). VII. CONCLUSION Many new pan-tilt devices were disclosed in addition to the mechanical systems introduced by Carricato and Parenti-Castelli. In these parallel wrists, the two actuators can be mounted directly on the base and, hence, can be heavy and bulky without adding to the inertial forces and, consequently, without compromising the capability of a very fast pan-tilt wrist. Such a statement seems to be effective for the 2-DOF agile eye of the Robotics Laboratory of Laval University, Quebec, QC, Canada. However, the use of a limb also implies moving masses. In practice, the kinetic energy of the limb should be compared with the kinetic energy of a moving servomotor in order to choose the best design for a 2-DOF wrist. Moreover, in parallel 2-DOF wrists, the collision-free and singularity-free workspace is reduced and, moreover, the tolerance in the joints may degrade the performance. The discussion of such practical concerns lies outside of the scope of the paper that is essentially theoretical. Nevertheless, the new 2-DOF parallel wrists that are introduced in the paper may have advantageous features depending on specific applications. The actuation of the tilt is fully uncoupled with the actuation of the pan, thereby providing a direct control of these two angles. These angles can be actuated intuitively. The rotations are referred to the vertical direction and, therefore, can account for the gravity effect. The possible incorporation of a weight-balancing fixed device can be envisioned. Hence, the simplest mechanisms described here seem to be well-suited for orienting an object like a camera, an antenna, a laser beam, etc., and maybe also for adjusting the pitch of windmill wings or helicopter blades.

64 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 1, FEBRUARY 2006 The algebraic properties of displacement subsets constitute the cornerstone of the method used. The paper also contributes to the advancement of the science of mechanisms and machines. REFERENCES [1] H. Asada and J. A. Cro Granito, Kinematic and static characterization of wrist joints and their optimal design, in Proc. IEEE Int. Conf. Robot. Autom., St. Louis, MO, 1985, pp. 244 250. [2] C. Gosselin and J. Angeles, The optimum kinematic design of a spherical three-degree-of-freedom parallel manipulator, ASME J. Mech., Transmission, Autom. Des., vol. 111, no. 2, pp. 202 207, 1989. [3] M. Karouia and J. M. Hervé, A three-dof tripod for generating spherical rotation, in Advances in Robot Kinematics. Dordrecht, The Netherlands: Kluwer, 2000, pp. 395 402. [4], A family of novel orientational 3-DOF parallel robots, in Romansy. Vienna, Austria: Springer Wien, 2002, vol. 14, pp. 359 368. [5], New parallel wrists: Special limbs with motion dependency, in On Advances in Robot Kinematics. Dordrecht, The Netherlands: Kluwer, 2004, pp. 371 380. [6] X.-W. Kong and C. M. Gosselin, Type synthesis of 3-DOF spherical parallel manipulators based on screw theory, in Proc. ASME DETC, Montréal, QC, Canada, Sep. Oct. 2002, Paper DETC2002/MECH-21152. [7], Type synthesis of three-degree-of-freedom spherical parallel manipulators, Int. J. Robot. Res., vol. 23, no. 3, pp. 237 246, 2004. [8] C. Gosselin and F. Caron, Two-Degree-of-Freedom Spherical Orienting Device, U.S. Patent 5,966,991, Oct. 19, 1999. [9] M. Carricato and V. Parenti-Castelli, A novel fully decoupled two-degrees-of-freedom parallel wrist, Int. J. Robot. Res., vol. 23, no. 6, pp. 661 667, 2004. [10] J. M. Selig, Geometrical Methods in Robotics. New York: Springer, 1996. [11] J. Angeles, The qualitative synthesis of parallel manipulators, ASME J. Mech. Des., vol. 126, no. 4, pp. 617 624, 2004. [12] J. M. Hervé, Analyze structurelle des mécanismes par groupe des déplacements, Mech. Mach. Theory, vol. 13, no. 4, pp. 437 450, 1978. [13], The Lie group of rigid body displacements, a fundamental tool for mechanism design, Mech. Mach. Theory, vol. 34, no. 5, pp. 719 730, Jul. 1999. [14] A. Karger and J. Novák, Space Kinematics and Lie Groups. New York: Gordon and Breach, 1985. [15] J. M. Selig, Geometrical Foundations of Robotics, Singapore: World Scientific, 2000. [16] J. M. Hervé and I. Bonev. (2003) The planar-spherical bond, implementation in parallel mechanisms. [Online]. Available: http://www.parallemic.org/reviews/review013.html [17] Q.-C. Li, Z. Huang, and J. M. Hervé, Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements, IEEE Trans. Robot. Autom., vol. 20, no. 2, pp. 173 180, Apr. 2004. [18] A. Frisoli, D. Checcacci, F. Salsedo, and M. Bergamasco, Synthesis by screw algebra of translating in-parallel actuated mechanisms, in Adv. Robot Kinematics. Dordrecht, The Netherlands: Kluwer, 2000, pp. 433 440. science. Jacques M. Hervé was born in France in 1944. He received the Dipl.Ing. degree from Ecole Centrale Paris, Paris, France, in 1968, and the Ph.D. degree in 1976 from the University of Paris 6, Paris, France. He began an academic career in 1968, and in 1983, he was appointed Professor and became responsible for a research team in mechanical design at Ecole Centrale Paris. He has been an Invited Researcher in the U.S., Canada, and Japan, and is also a consultant for several companies. His professional interest is teaching and research in mechanism and machine