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Analytical Kinematics Models and Special Geometries of a Class of 4-DOF Parallel Mechanisms
Matteo Zoppi, Dimiter Zlatanov, and Clment M. Gosselin, Member, IEEE
AbstractThe paper discusses forward and inverse kinematics of a class of four-degree-of-freedom (DOF), four-legged parallel mechanisms providing three rotational and one translational DOFs. A fully parametric analytical form solution to the inverseposition problem is provided. All working modes of the mechanism are shown and discussed. The equations of the forward-position problem are obtained under different leg arrangements, and a numerical example is provided. New special geometries in the class are proposed, including one suitable for keyhole surgery. Index TermsFour-degree-of-freedom (DOF) mechanisms, kinematics, parallel mechanisms (PMs), surgical robot.
Fig. 1. Generic mechanism in the class.

I. INTRODUCTION

HE LARGE variety of specic tasks for which parallel mechanisms (PMs) are employed, and the dramatic change that the manufacturing and industrial paradigms are facing today, make strongly feature-oriented, high-dynamics PMs a valuable choice for exible automation. To satisfy specic task requirements, a device needs to have suitable mobility [number and type of degrees of freedom (DOFs)]. Therefore, the choice of an architecture and geometry, with the required motion capability of the platform mobility, is of key importance. The selection of a specic satisfactory geometry is a complex problem, with many interrelated aspects. Some of these considerations are the transmission of the actuation and constraint forces through the legs, the presence and type of singularities in the workspace, and the possibility of collisions between links. Analytical models of the mechanism kinematics can be very helpful to cope with all these aspects, and are also necessary to implement the control system, once the mechanism is used as a robot. This paper presents the analytical model of the inverse and forward kinematics of a class of parallel 4-DOF mechanisms with three rotational and one translational DOFs (Fig. 1). These were the rst known fully PMs with four (or ve) DOFs, described in [1] and further analyzed in [2][4]. (Some modications of the translational 3-DOF Delta PM had been proManuscript received January 14, 2005; revised May 23, 2005. This paper was recommended for publication by Associate Editor W. Y. Chung and Editor F. Park upon evaluation of the reviewers comments. M. Zoppi is with the University of Genova, Department of Mechanics and Machine Design, PMAR Laboratory of Robotics, 16145 Genova, Italy (e-mail: zoppi@dimec.unige.it). D. Zlatanov is with the Musashi Institute of Technology, Department of Mechanical Systems Engineering, 158-8557 Tokyo, Japan (e-mail: zlatanov@ sc.musashi-tech.ac.jp). C. M. Gosselin is with the Universit Laval, Dpartement de Gnie Mcanique, Qubec, QC G1K 7P4, Canada (e-mail: gosselin@gmc.ulaval.ca). Digital Object Identier 10.1109/TRO.2005.853494

posed for 4-DOF SCARA-type motion [5].) The type synthesis of 3-, 4-, and 5-DOF PMs related to this class is addressed in [6][8]. Other 4-DOF and 5-DOF mechanisms have been presented in [9]. In this paper, we discuss the kinematics in detail. Additional material can be found in [2], where the analytical model has been combined with a set of specially designed Maple graphical procedures to create a fully parametric mathematical mock-up of the mechanism class. This mock-up has been used to automatically generate all the pictures used in the paper. Furthermore, we present some new and nontrivial special geometries. In particular, we analyze in more detail one version, which is presently being considered for the design of a robot for minimally invasive laparoscopic surgery. With the help of the Maple mock-up, it has been possible to design a mechanism with a sufciently large singularity and collision-free workspace for the foreseen surgical application. We present an example of a satisfactory set of values for the geometrical parameters, and we illustrate the 4-D workspace of this particular geometry with different 2-D sections and 3-D visualizations. II. THE CLASS OF MECHANISMS The mechanisms in the class are composed of a base and an end-effector link, connected to each other by two or more 5R . All legs have serial legs, labeled with capital letters the same architecture. The joints of each leg are numbered from , and the corresponding the base. The joint axes are , where . The rst direction vectors three axes, , , and , intersect in the xed point (the rotation center), common to all the legs. The fourth and fth , are joints are parallel. All end-effector joint axes, parallel to the platform plane , chosen arbitrarily from all planes xed in the end-effector with the same normal, . Asare not all parallel, the platform has full rosuming that the tational mobility around , and translates orthogonally to

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Fig. 2.

One leg with its joint screws.

spherical subchains. The heave subchain of the mechanism is the end-effector with all leg-heave subchains. Each leg generates a 5-DOF motion. Out of singular congurations, the structural constraint applied by the leg to the end-effector is one pure force through in direction , as extensively discussed in [3]. (The structural constraint is the constraint that the leg applies with all its passive and actuated joints free.) For this reason, the minimum number are not parof legs to obtain four DOFs is two, provided their allel. Versions of the mechanism with more than two legs have lie in parallel planes; such mechanisms are four DOFs if the overconstrained. The four-leg mechanism can be controlled by actuating the base joints (see [3]). (The combined actuated constraints, applied to the platform when the base joints are locked, form a 6-system.) A mechanism in the class is rhombic, if its leg-heave sub, chains have the same geometry two by two ( , , ) and are arranged . In symmetrically, with rhombic mechanisms, and . A rect;a angular mechanism is a rhombic mechanism with square mechanism is a rectangular mechanism with four identical legs [2]. III. INVERSE KINEMATICS The pose of the end-effector (orientation matrix and extrusion ) is given, and the aim is to compute the entire conguration of the mechanism. First, we solve the heave subchain, then the spherical subchain. The analytical model proposed works for every mechanism geometry, including the special geometries. Unless noted otherwise, vectors are given in the end-effector frame. 1) Heave Subchain: Consider the heave subchain of leg (Fig. 3). The aim is to express as a function of the endis a virtual 1-DOF effector pose. The quadrilateral mechanism with parameter . The angle is square. and By trigonometric considerations, for , we obtain the relation (1) where of (1) are , . The solutions

Fig. 3. Heave subchain of leg L. (a) Vectors. (b) Parameters.

[1]. The amount of translation (extrusion) of the end-effector and the 0-extrusion plane through is the distance between and parallel to . To command the mechanism by actuating base joints only, it is necessary to choose a layout with more than three legs. In the following, we consider the simplest case . Further legs can be adopted to of four legs obtain actuation redundancy or to eliminate singular congurations. Part of the results presented in this paper can be extended to -legged mechanisms. , a rotating frame We use a base reference frame . and an end-effector frame at the projection of on . Consider leg (Fig. 2). The heave plane is the plane through orthogonal to . The joint axes and intersect , respectively, at the points and . The distance from to is ; is the distance between and and , where is the projection of on the plane through parallel to . The third link spans the constant angle between and [Fig. 3(b)]. The angles , , spanned by the rst two links, are between and . describes the geometry of leg The set standing alone. between and ; the distance from The distance to ; the angle between and ; and the tilt angle and azimuth angle placing in (such that ), describe how leg is placed in the mechanism. Each leg chain can be divided into a spherical chain (spherical subchain), formed by the rst two links, and a planar chain (heave subchain), formed by the third and fourth links. The spherical subchain of the mechanism is the base with the leg

(2) distinguishes between the two feasible working where (Fig. 4), and modes of the quadrilateral (3) with and . to be real, it is necessary that It is clear from (3) that for . The quadrangle takes symmetric congurations for positive . The transition conand negative values of . Consider gurations between the two working modes of the quadrilateral

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Fig. 4. Working modes (a), (b) of the leg L heave subchain and (c), (d) of . (b) " . (c)  . the leg L spherical subchain. (a) " . (d) 

= 01

= +1

= 01

= +1

are at maximal extrusion, with , and collinear, and ; at minimal extrusion (if allowed by the geom, and are collinear, etry of the whole mechanism), with between and ; . with To obtain in , we rotate about at an angle , . is computed analogously (4) is obtained with a rotation of an angle The vector about , , with (5)

, where and . is a result of the inverse 2) Spherical Subchain: Vector kinematics of the heave subchain, while is easily obtained by , from expressed a change of reference frame, in the base frame. . The axis of the second It remains to compute vector joint lies at the intersection of two right circular cones with and , , , respecvertex, axis, and half-angle , , tively. For to lie on both cones, must satisfy and , where . Then, is obtained as a linear combination of , and Finally, note that (12) The coefcients are , ,

(13) , , . The Boolean parameter distinguishes between the two working modes of the spherical subchain of the leg (Fig. 4). The rotation angles of the actuated (base) joints, which are the solution of the inverse kinematics problem, are easily computed . from the four in (13) is real) iff A leg conguration is feasible ( , where , i.e., . An expressions for the unit vector , orthogonal to the plane dened by and is (14) with

, one for each leg, distinguish between 16 The four different working modes of the combined mechanism heave subchain. along and We compute the components of (6) where , and , is provided by (4),

A noteworthy simplication appears when (7) (8) (13) becomes , and , . IV. FORWARD KINEMATICS

The distances the intersection of

and (between , , and ) are

and the point

at

(9)

(10)

An expression of the angle

between

and

is

(11)

The actuation angles of the base joints are known, thus, the are given. The unknown is the end-effector pose . four To solve the problem in the general case is very complicated , (see [1]). Here, we consider rhombic mechanisms with . For these mechanisms, we derive four equations in the four unknown angles . For rhombic mechanisms, three of these equations can be expressed as quadratic forms, while the fourth cannot. Then, we consider square mechanisms and show that, in this case, all four equations can be expressed as quadratic forms. Although it may seem appropriate to use and some rotation parameters as variables, the resulting formulation becomes too complex. This is why, to obtain relatively simple equations, we use as intermediate unknowns the angles ( , ) of the second joints of the legs. In leg , is the angle

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about from to . Since lies on the cone with axis and half angle , an expression of as a function of is (15) , , where and . The resulting system of four equations is discussed below. An analytical form solution is not known, however, it can be solved numerically (see [10]). Things simplify for square mechanisms. In this case, an analytical form solution should be achievable for particular geometries (see [11] and [12]) due to the structure of the equations, although the problem remains open. 1) Rhombic Mechanisms: Two equations state that the right rectangular pyramid with base edges along has equal opposite faces, i.e., and (16) (17)

The third equation describes the angle and tween

, be(18)

By developing (18), simplifying and substituting in (16) and (17), we obtain the following second-degree equations in and : (19) (20) (21) and, by means of (15), we write them as three quadratic forms (22) . The matrices , where , are as shown in (23)(25), respectively, at the bottom of the page. and satisfy (1) for the same Finally, we write that and , we value of . We solve (1) for ; since

(23)

(24)

(25)

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obtain: the fourth equation is

. Thus,

(26) After squaring twice, (26) becomes a second-degree equation in , . Since the mechanism is rhombic, , . Therefore, and . Equation as a function of . (15) allows obtaining Equations (22)(26) are the nal system of four equations for the forward kinematics of rhombic mechanisms. 2) Square Mechanisms: For square PMs, and (22) are homogeneous. Equation (26) becomes , stating that . From (15), we write this , where is fourth equation as a quadratic form shown in (27) at the bottom of the page. So, the forward kinematics of the square mechanisms is expressed by a system of four second-degree, homogeneous equa, . tions V. SPECIAL GEOMETRIES This section presents some families of nontrivial special geometries with particular arrangements of the base and platform joints. Fig. 1 shows a generic mechanism in the class. In Fig. 5(a), the axes of opposing platform joints coincide, in , pairs, along two skew lines ( ). We call double-compass a mechanism with this setup of the platform, which can be then used as an elongated end-ef. In particular, the axes of the end-effector joints fector along , allowing a more compact can intersect shape of the physical end-effector [Fig. 5(b)]. In both cases, the mobility of the mechanism does not change, and the analytical models of the kinematics, presented in the previous sections, are valid. The forward kinematics problem for these special mechanisms reduces to the placement of a double-compass tetrapod on four given circles. For rectangular mechanisms, the four circles belong to two concentric spheres; in square mechanisms, they are on the same sphere. Double-compass geometries can have interesting robotic applications as multi-DOF actuated joints or wrists with an additional local translational mobility. The mechanism in Fig. 6(a),

Fig. 5. Double-compass mechanism geometries. (a) Shifted. (b) Cross end-effector.

Fig. 6. Double-compass geometry. (a) For a multi-DOF actuated joint. (b) With a full rotational mobility about the collinear base joint axes.

with coaxial base joints, is an example of an actuated spherical joint allowing local translation along the end-effector. Fig. 6(b) shows another double-compass geometry with and base joint axes collinear on opposite . Since the whole mechsides of anism rotates about , the constant-torsion workspace has a symmetry of revolution about . To prevent the occurrence of link collisions, the radii of the rst two (spherical) links are different in each pair of opposing legs. For the same reason, the and mechanism has an assembly mode with . In the examples considered thus far, the center of rotation is inside the mechanism. For some geometries, it is possible

(27)

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tiny incision. In the following section, we discuss the possibility of using a mechanism with the geometry of Fig. 7 for minimally invasive surgical robotics. VI. SURGICAL GEOMETRY Minimally invasive surgery is performed through sets of small incisions, rather than the traditional single major incision in the patients body. Of particular interest is laparascopic surgery, performed inside the abdomen of the patient. The surgeon operates with long, thin instruments inserted into the abdomen through tiny holes. Robotic systems are used to expand the surgeons capability. Instead of directly handling the instruments, the surgeon teleoperates a robot which performs the actual motion. This eliminates hand tremor, while allowing motion scaling and very high precision. A survey of the design issues in surgical robotics is presented in [13]. Independently from the setup of the system (robot placed on the oor beside the patient, mounted to the operative table, xed to the ceiling over the operative table, or attached to the patients body), the mobility required for the mechanism carrying the instrument during a minimally invasive operation is 4-DOF. The tool pivots about the point at which it enters in the patients body (pan, tilt, and spin, or, as we prefer, tilt-and-torsion angles [15]) and translates along its axis (insertionretraction) [14], [16]. Therefore, the RCM is a central design feature of any robot for keyhole surgery. (Sometimes, at the tip, the instrument has additional freedoms, but they are independent from the main mobility of the robot carrying the instrument.) The tilt-and-torsion angles represent the rotation of the end, by one rotation (tilt) about effector, from the pose with an axis (of some azimuth) orthogonal to and another rotation (torsion) about the new [15]. It seems that this representation of the instrument rotation is preferable, since in many surgical applications, a special plane for the pan does not really exist. A main reason for using parallel rather than serial manipulators in surgical applications is the desire to achieve higher accuracy and stiffness. On the other hand, a large translational workspace (provided by serial arms) is not required during the operation. The technical literature proposes applications of 6-DOF hexapods (StewartGough platforms) [17][19]. The required 4-DOF mobility can be obtained by suitably controlling the mechanism. There are two compelling reasons to consider a 4-DOF architecture instead of a hexapod. The rst is safety: with a 4-DOF mechanism, there is no danger of injuring the patient with an accidental translation in a disallowed direction (e.g., due to control error). Second, the size of the workspace of a PM tends to be smaller for higher DOF devices. (In part, this is often due to the larger number of legs required for actuation.) During surgery, the size of the rotational workspace is signicantly more important than the translation range and, in particular, a good capability for torsion about the tool axis is quite useful for many procedures. Stewart platforms have notoriously small ranges of rotation, even when an internal point, rather than an RCM, is

Fig. 7. Surgical geometry from two points of view, (a) and (b), and the corresponding geometric parameters (angles in rad). The disk in the pictures represents the safety plane.

to have on the outside. That is, to assemble and move the mechanism with most physical links inside the base pyramid (Fig. 7). This is necessary in order to adapt the architecture for applications with a remote center-of-motion (RCM) [13], [14]. The geometry in Fig. 7 is an example. An end-effector and its tip point with a long linear handle passes through can move in a free region, where the other moving links never enter. A mechanism with an RCM can be used to operate in a bounded region of space accessible only through a small opening (a keyhole). This allows arbitrary rotation about the center of the opening. However, since the linear handle of the instrument must always pass through the hole, the only feasible translation is along the tool axis (along the tangent to the handle at the opening). If the center of rotation is made to coincide with the center of the opening, the 4-DOF mobility allowed by the keyhole matches exactly the end-effector motion capability of the discussed architectures. Such motion constraints exist in laparoscopic surgery, where the patient is operated on through a

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xed. The use of a partly spherical architecture, like the suggested 4-DOF mechanism, has the inherent promise of a better range of motion. Therefore, the discussed PMs are the natural choice for this application. (The only other basic parallel architecture with these DOFs uses a leg where the last three joints form a planar were parallel to and [6], [7]. However, a pair, e.g., if PM with four such legs cannot be actuated by the base joints.) Additional freedoms are necessary to make the RCM coincident with the hole in the patients body, but they are not used during the operation; we can imagine that the robot is put somehow with the RCM at the abdominal keyhole incision, e.g., by a recongurable frame (an interesting setup is the one in [14], with the surgical system directly xed to the patients body). The geometry in Fig. 7 has both the required DOFs (obtained mechanically) and an RCM. We now show that there exists such a set of geometric parameters which ensures a sufciently large singularity-free workspace. The mechanism workspace is a 4-D region in the space of the end-effector coordinates. We represent it by the sets of feasible locations of the tip point for constant values of the end-effector torsion (constant-torsion volumes). In the following, we consider a minimal workspace that we suppose required for the application of the mechanism to laparoscopic surgery. This workspace contains singularity-free, constant-torsion, spherical-truncated-cone volumes, with center, half-angle, and , and 100 mm, for values of the height, respectively, , (desired constant-torsion torsion in a range of amplitude volumes). (Note that the surgical instrument can be inserted in the patients body manually or by an additional translational actuator and then be xed to the end-effector of the mechanism. Thus, the 100-mm translation is just the one required to move the instrument in the operative region.) To nd a collision-free design for mechanisms with an RCM is more challenging than when the center of rotation can be internal to the mechanism as in Fig. 1. Indeed, during surgery, all links (with the exception of part of the end-effector) must be conned to a half-space (safety space) bounded by a plane through (safety plane). Hence, the space available to the rst and second links of the leg is limited. Moreover, this reduces the maximum angle between and , which, in turn, leads to larger oscillations of their common plane during a given end-effector tilt, and larger collision risk. However, solutions can be found. Consider the geometry in Fig. 7. The lengths are made dimensionless with respect to . Opposite legs have the same geometry ( and , and ) and their spherical subchains move on the same surface. To avoid link interference (otherwise likely for high end-effector torsion), the two spherical surfaces (of the two pairs of legs) are and made distinct. For the same reason, , where ap. proximates the maximal geometrically feasible value of The distance of from the platform plane (which is related . The maxto the length of the surgical instrument) is imal possible end-effector extrusion is almost equal to , about 120 mm (or a as discussed below. Therefore, making little longer) will enable a translation range adequate for many

Fig. 8. Constant-torsion workspace taken at six values of the torsion (indicated under the pictures).

surgical procedures, while keeping the mechanism size sufciently small in relation to the patients abdomen and the space available in the operation room. Fig. 8 shows six 3-D sections of the 4-D workspace of the mechanism, taken at six equidistant torsion angles in the range . Each section is a constant-torsion volume, obtained numerically by solving the inverse kinematics (2)(13) for point on an equally spaced 3-D grid. The working mode considered is the one characterized by , , , and . Some 3-D sections of the workspace contain cavities and singularities. To illustrate this, we use three sets of 2-D sections, each taken at xed torsion and (Fig. 9 and Tables IIII). These sections lie on spherical surfaces centered at , with radius (depending on the actual value of ); equal to the distance for better viewing, we map them in the plane using conformal transformation [20]. Each table gathers 12 sections (pictures) taken at the same , and 12 different values of the torsion, spaced of each other. The external circle in each picture represents the map of an equator of the sphere on which the section lies, see Fig. 9. For better visibility, the pictures of each set are enlarged to the size of the table cell and appear with the same size, although the diameter of the sphere increases from Tables IIII (see captions).

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TABLE II EXAMPLE GEOMETRY: CONFORMAL MAPS OF THE WORKSPACE 2-D SECTIONS AT h = (18=7)l . DIAMETER OF MAPPED SPHERE: (24=7)l

Fig. 9. 2-D section of the mechanism workspace at constant h and torsion, and its plane mapping. TABLE I EXAMPLE GEOMETRY: CONFORMAL MAPS OF THE WORKSPACE 2-D SECTIONS AT h = (15=7)l . DIAMETER OF MAPPED SPHERE: (30=7)l

The points colored white are either impossible to reach or are achieved by a singular conguration. The nonwhite areas in the plots are colored from dark gray to light gray, depending on the conditioning of the Jacobian [21], [22]. Note that since the entries of the Jacobian are not dimensionally homogeneous (the mechanism has three rotational and one translational freedoms), the use of the conditioning is only to determine the presence or absence of singularities.

The black frames surrounding some of the pictures in Tables IIII underline that for the indicated values of the extrusion and torsion, the region with tilt angle less than is not completely inside the workspace, or not completely free of singularities. In such cases, the singularity-free constant-torsion workspace of the mechanism does not contain the corresponding desired constant-torsion volume. As remarked before, for safety reasons, it is necessary that during the surgical operation, all links remain on one side of the . The gures surrounded by a gray frame (in safety plane Tables IIII) represent combinations of the extrusion and torsion such that the constant-torsion workspace of the mechanism contains the corresponding desired constant-torsion volume, and for every conguration belonging to this desired constant-torsion volume, all links are in the safety space. The tables show that for values of the extrusion between and and torsion in the range , the workspace contains the singularity-free, constant-torsion, spherical-truncated-cone volumes mentioned above, and we can conclude that the mechanism, in principle, could be used for a surgical robot. In fact, the three tables present only discrete sections of the workspace, and obviously cannot rigorously prove that no other singularities are present in the desired workspace. A large number of 2-D sections have been calculated. We veried with

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TABLE III EXAMPLE GEOMETRY: CONFORMAL MAPS OF THE WORKSPACE 2-D SECTIONS AT h = (21=7)l . DIAMETER OF MAPPED SPHERE: (18=7)l

This is a very strong indication that the desired singularityfree workspace exists. Establishing with mathematical certainty that a workspace region is free of singularities is an interesting and difcult problem. Some methods have been proposed [23] and implemented in certain cases [24], [25]. In the future, they may be applied to deal with the proposed mechanism. A fully parametric, 3-D virtual mockup for the architecture has been created using Maple. The analytical models of the kinematics have been implemented, together with a library of graphic and computational tools allowing visualizing the whole mechanism and its parts in different congurations, as well as performing singularity analysis. This mathematical mockup [2] allowed multiple design iterations in reasonable time, and enabled us to satisfy the tough requirements (RCM, safety plane, no singularity or link interference) imposed by the surgical application. VII. CONCLUSIONS The paper provides a fully parametric analytical model of the inverse kinematics of a class of 4-DOF PMs and a relatively simple formulation of the forward kinematics of an interesting mechanism subclass. Moreover, new special geometries are described, and possible applications of these models are outlined. The models proposed are a natural starting point for any further research on this mechanism class, such as workspace analysis or force transmission, as well as for the design and any practical implementation of a version of the architecture. As an example, we discuss the possible use of this mechanism for robotic surgery. The proposed special geometry with good workspace characteristics shows that, in principle, this application is feasible. The mechanism still poses analytical challenges, such as solving the direct kinematics and nding an exact bound on the number of possible solutions. For a special geometry, this open problem can be simply formulated as the placement of a double-compass tetrapod on four circles of a sphere. REFERENCES
[1] D. Zlatanov and C. Gosselin, A family of new parallel architectures with four degrees of freedom, in Proc. Workshop Comput. Kinematics, F. Park and C. Iurascu, Eds., May 2001, pp. 5766. [2] M. Zoppi, High dynamics parallel mechanisms: Contributions to force transmission and singularity analysis, Ph.D. dissertation, Univ. Genova, Genova, Italy, Apr. 2004. [3] D. Zlatanov, M. Zoppi, and C. Gosselin, Singularities and mobility of a class of 4-DOF mechanisms, in Advances in Robot Kinematics, J. Lenarcic and C. Galletti, Eds. Sestri Levante, Italy: Kluwer, Jun. 28Jul. 1 2004, pp. 105112. [4] M. Zoppi, D. Zlatanov, and C. Gosselin, Kinematics equations of a class of 4-DOF parallel wrists, in Advances in Robot Kinematics, J. Lenarcic and C. Galletti, Eds. Sestri Levante, Italy: Kluwer, Jun. 28Jul. 1 2004, pp. 321328. [5] L. Rolland, The Manta and the Kanuk: Novel 4-DOF parallel mechanisms for industrial handling, in Proc. ASME Int. Mech. Eng. Congr. Expo., Nashville, TN, Nov. 1419, 1999, pp. 831844. [6] Z. Huang and Q. Li, On the type synthesis of lower-mobility parallel manipulators, in Proc. Workshop Fundam. Issues Future Res. Directions Parallel Mech. Manipulators, Quebec City, QC, Canada, 2002, [CD-ROM]. [7] J. Herv. (2003, Jan.) The planar-spherical kinematic bond: Implementation in parallel mechanisms. Tech. Rep.. [Online]. Available: http://www.parallemic.org/-Reviews/-Review013.html

Fig. 10. Desired constant-torsion volumes and singularity surfaces for different values of the torsion in the range [0;  ].

a resolution of

for , 1/300 rad for the tilt angle , for the azimuth angle (providing a quite homogeneous angular distribution of the verication points), and rad for the torsion. In all of these cases, the desired workspace was achieved. Moreover, the singularity surfaces were traced for values of the torsion in with a resolution , and examined together with a representation of the desired constant-torsion volumes. The result is illustrated by Fig. 10, where the surfaces are shown . Part of the singularity plot for values of the tilt in has been removed to allow viewing the desired constant-torsion volumes inside. No volume intersects any singularity surface.

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Matteo Zoppi received the M.Eng. degree (summa cum laude) in mechanical engineering in 2000, and the Ph.D. degree in robotics in 2004, both from University of Genova, Genova, Italy. Currently, he is a Researcher with the Department of Mechanics and Machine Design, University of Genova, PMAR Laboratory of Robotics. His research subjects are kinematics, dynamics, control, and design of robotic systems, with a particular emphasis on industrial automation and kinematics and dynamics of parallel and complex mechanisms. He is involved in several European projects and in European networks. Dr. Zoppi is a Member of the Italian Association of Robotics and Automation SIRI and the ASME.

Dimiter Zlatanov obtained the Diploma in mathematics and mechanics from the University of Soa, Soa, Bulgaria, in 1989, and the Ph.D. degree in mechanical engineering from the University of Toronto, Toronto, ON, Canada, in 1998. He is currently a JSPS Fellow at the Musashi Institude of Technology, Tokyo, Japan. His research interests include the design, kinematics, dynamics and control of robotic mechanisms.

Clment Gosselin (S88M89) received the B.Eng. degree in mechanical engineering from the Universit de Sherbrooke, Qubec, QC, Canada, in 1985, and the Ph.D. degree from McGill University, Montral, QC, Canada, in 1988. In 1988, he accepted a postdoctoral fellowship from the French government in order to pursue work at INRIA, Sophia-Antipolis, France, for one year. In 1989, he was appointed by the Department of Mechanical Engineering, Universit Laval, where he has been a Full Professor since 1997. He has held the Canada Research Chair on Robotics and Mechatronics since January 2001. His research interests are kinematics, dynamics, and control of robotic mechanical systems with a particular emphasis on the mechanics of grasping and the kinematics and dynamics of parallel manipulators and complex mechanisms. His work in the aforementioned areas has been the subject of several publications in international conferences and journals. He is the French-language Editor for the international journal Mechanism and Machine Theory. Dr. Gosselin is a member of ASME and CCToMM. He was the recipient of the Gold Medal of the Governor General of Canada in 1985, and the D. W. Ambridge Award from McGill University for the best thesis of the year in Physical Sciences and Engineering in 1988. He was also the recipient, in 1993, of the I.
. Smith award from the Canadian Society of Mechanical Engineering, for creative engineering. In 1995, he received a fellowship from the Alexander von Humboldt Foundation which allowed him to spend six months as a Visiting Researcher with the Institut fr Getriebetechnik und Maschinendynamik of the Technische Hochschule, Aachen, Germany. In 1996, he spent three months at the University of Victoria, for which he received a fellowship from the BC Advanced Systems Institute.

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