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1、 Robotics Research The International Journal of The online version of this article can be found at: DOI: 10.1177/02783640122067426 2001 20: 312The International Journal of Robotics Research Bruno Monsarrat and Clment M. Gosselin Grassmann Line Geometry Singularity Analysis of a Three-Leg Six-Degree-

2、of-Freedom Parallel Platform Mechanism Based on Published by: On behalf of: Multimedia Archives can be found at:The International Journal of Robotics ResearchAdditional services and information for Alerts: What is This? - Apr 1, 2001Version of Record at Tsinghua University on January 8, Downloaded f

3、rom Bruno Monsarrat Clment M. Gosselin Mechanical Engineering Department, Laval University Qubec, Qubec, Canada, G1K 7P4 gosselingmc.ulaval.ca Singularity Analysis of a Three-Leg Six-Degree-of-Freedom Parallel Platform Mechanism Based on Grassmann Line Geometry Abstract This paper addresses the dete

4、rmination of the singularity loci of a six-degree-of-freedom spatial parallel platform mechanism of a new type that can be statically balanced. The mechanism consists of a base and a mobile platform that are connected by three legs using fi ve-bar linkages. A general formulation of the Jacobian ma-

5、trix is fi rst derived that allows one to determine the Plcker vectors associated with the six input angles of the architecture. The lin- ear dependencies between the corresponding lines are studied using Grassmann line geometry, and the singular confi gurations are pre- sented using simple geometri

6、c rules. It is shown that most of the singular confi gurations of the three-leg six-degree-of-freedom par- allel manipulator can be reduced to the generation of a general linear complex. Expressions describing all the corresponding sin- gularitiesarethenobtainedinclosedform. Thus,itisshownthatfor a

7、given orientation of the mobile platform, the singularity locus cor- respondingtothegeneralcomplexisaquadraticsurface(i.e., either a hyperbolic, a parabolic, or an elliptic cylinder) oriented along the z-axis. Finally, three-dimensional representations that show the in- tersection between the singul

8、arity loci and the constant-orientation workspace of the mechanism are given. KEY WORDSstatic balancing, parallel manipulator, sin- gularities, Grassmann line geometry, singularity loci 1. Introduction In the context of manipulators and motion simulation mech- anisms, parallel architectures have par

9、ticularly aroused the The International Journal of Robotics Research Vol. 20, No. 4, April 2001, pp. 312-328, 2001 Sage Publications, Inc. interest of researchers over the past 30 years for their prop- erties of better structural rigidity, positioning accuracy, and dynamic performances. However, in

10、industrial applications of such mechanisms where displacements of heavy loads are involved, theoperatingcostsareincreasedsubstantially. This motivated the development of parallel architectures that can be statically balanced. A mechanism is statically balanced if its potential energy is constant for

11、 all possible confi gurations (i.e., zero actuator torques are required whenever the manip- ulator is at rest in any confi guration). To the knowledge of the authors, static balancing of spatial six-degree-of-freedom parallel manipulators was fi rst introduced by Streit (1991), Leblond and Gosselin

12、(1998), J. Wang (1998), Gosselin and Wang (2000), and Herder and Tuijthof (2000). Two static- balancing methods, namely, using counterweights and using springs, are used. The fi rst method, introducing additional masses into the system, tends to increase the inertia con- siderably. Because many comm

13、ercial applications involve large accelerations of the mobile platform, the use of coun- terweights is undesirable. On the other hand, the use of an alternative architecture for the legs, similar to what was used by Streit and Gilmore (1989), allows one to obtain an effi - cient static balancing usi

14、ng only springs. Moreover, Ebert- Uphoff, Gosselin, and Lalibert (2000) presented a class of spatial parallel platform mechanisms of novel architecture, with three legs or more using fi ve-bar linkages, that is suit- able for static balancing. A prototype with three legs was de- signed by Gosselin e

15、t al. (1999) in accordance with the static- balancing-relatedconstraints(seeFig.1). Akinematicanaly- sisofthatclassofmechanismwaspresentedbyEbert-Uphoff and Gosselin (1998). In the latter reference, the inverse kine- maticproblemwasresolvedandanexpressionoftheJacobian 312 at Tsinghua University on J

16、anuary 8, Downloaded from Monsarrat and Gosselin / Parallel Platform Mechanism313 Fig. 1.Prototype of the statically balanced three-leg six- degree-of-freedom parallel platform mechanism (as reported in Gosselin et al. 1999). matrix was obtained in a general form. Closed-form expres- sions of the co

17、rresponding singularity loci were determined only for the special case in which (i) the fi xed and mobile platforms are parallel and (ii) for each leg i, the rotation an- gle around the z-axis and the angle between the two proximal links of the parallelogram are actuated. Thispaperfollowsupwiththeco

18、mpletegeometricandan- alytical characterization of the singularity loci of the above- mentioned three-leg parallel manipulator. The determination of such confi gurations is a critical issue to be taken into ac- count early in the design process, thus maximizing the inher- ent performance of the syst

19、em during trajectory tracking. In practice, these degeneracies of the instantaneous kinematics lead to a change in the number of degrees of freedom of the mechanism and to a degradation of the stiffness properties that may lead to very high joint torques or forces. In both cases, the accuracy of the

20、 control will be critically affected. As shown by Gosselin and Angeles (1990), Gosselin and Sefrioui (1992), Gosselin and Wang (1997), and Mayer St- Onge and Gosselin (2000), an effi cient approach consists in obtaining the equations describing the singularity loci from the closed-form expressions o

21、f the determinant of the Jaco- bian matrix. The resulting singular confi gurations have been classifi ed in three main groups: type I, where the end effec- tor lies at the boundary of the Cartesian workspace; type II, where the output link gains one or more degrees of freedom (i.e., the end effector

22、 is movable when all the input joints are locked); and type III, where the chain can undergo fi nite mo- tions when the actuators are locked or a fi nite motion of the actuators produces no motion at the end effector. This clas- sifi cation was recently generalized by Zlatanov, Fenton, and Benhabib

23、(1995, 1998), who developed a unifi ed framework forthesingularityanalysisofnonredundantmechanisms. Six types of singularities refl ecting different possibilities for the occurrence of degeneracies of the instantaneous forward and inverse kinematics are defi ned. The determinant-based ap- proach was

24、 illustrated by Gosselin and Sefrioui (1992) and Collins and McCarthy (1998) with the analysis of planar 3-RPR parallel manipulators. The approach was also used by Gosselin and Wang (1997) and more recently by Bonev and Gosselin (2001) to determine the singularity loci of 3- RRR parallel manipulator

25、s. Moreover, the same procedure was implemented by G. Wang (1998) and St-Onge and Gos- selin (2000) to obtain the equations of the singularity loci of the well-known Gough-Stewart platform. However, the structure of parallel architectures requires in some cases that the closed-form expression of the

26、 determi- nant of the Jacobian matrix depend not only on the Cartesian but also on the joint coordinates. After substituting the ac- tive and passive joint coordinates with the Cartesian coordi- nates using the equations of the inverse kinematic problem, the closed-form expression of the correspondi

27、ng determinant is of a very complex form. Therefore, the determinant-based method is not adapted to the singularity analysis of the three- leg six-degree-of-freedom parallel mechanism that uses fi ve- bar-linkagesforacaseofactuationidenticaltotheoneshown in Figure 1. In this context, the approach us

28、ed here is the one intro- duced by Merlet (1988, 1989) and Mouly and Merlet (1992) in which the singularities of spatial 6-(RR)PS and 6-P(RR)S parallel manipulators were studied using Grassmann line ge- ometry. Theprocedureleadstoanexhaustivelistofgeometric conditions that correspond to singularitie

29、s. It is known that such an analysis allows the characterization of the degenera- cies of the Jacobian matrix that correspond to singularities of type II, following the classifi cation given by Gosselin and Angeles (1990). A recent work by Hao and McCarthy (1998) allowedonetospecifythedesignfeatures

30、ofparallelplatform mechanisms that guarantee that only line-based singularities exist. A complete classifi cation of linearly dependant sets of lines is also provided using Merlets notation. However, it was shown by Zlatanov, Fenton, and Ben- habib (1995) that the existence of an invertible 66 Jacob

31、ian matrix is not a suffi cient condition for nonsingularity unless thevelocityequationsbetweentheactiveandpassivejointve- locities are defi ned. Therefore, the occurrence of additional singular confi gurations is studied here through an analysis of the cases in which the 36 matrix relating the actu

32、ated joint velocities and the passive joint velocities is not defi ned. The organization of this paper is as follows. In Section 2, we will briefl y review the design of the prototype and in- troduce the corresponding notation. In Section 3, a general expression of the Jacobian matrix will be obtain

33、ed using the principle of virtual work, and the corresponding Plcker vec- tors associated with the six input angles will be derived. In Section4,thelineardependenciesbetweenthecorresponding at Tsinghua University on January 8, Downloaded from 314THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April

34、 2001 Fig. 2. Parallel platform mechanism with three legs using fi ve-bar linkages (CAD model by Jiegao Wang). Grassmann lines are studied and the singular confi gurations are described using simple geometric rules. The closed-form equations of the singularity loci are then obtained in Sec- tion 5.

35、It is shown that for a given orientation of the mobile platform, the singularity locus corresponding to the general linear complex is a quadratic surface (i.e., either a hyper- bolic, a parabolic, or an elliptic cylinder) oriented along the z-axis. In Section 5, additional singularities correspondin

36、g to thenonexistenceofthe36matrixdefi nedabovearestudied. To clearly illustrate the results, three-dimensional represen- tations that show the intersection between the singularity loci and the constant-orientation workspace of the mechanism are given. 2. Description of the Mechanism The type of mech

37、anism considered in this paper is shown in Figure 2. The architecture consists of a fi xed base and a mobile platform connected by three legs. The ith leg is attached to the base at point Pi0and to the mobile platform at point Pi5(see Fig. 3). The attachment points on the base and mobile platforms f

38、orm equilateral triangles. We defi ne a geometric parameter r that is used to describe the position of points Pi0, making the assumption that the attachment points ofthethreelegsareequallyspacedonacircleofradiusr with center at point O; that is, r = ? ?pi0?, i = 1,2,3. At point Pi0, the two links of

39、 the parallelogram are mounted using revolute joints. These provide two degrees of freedom, associated with angles i2and i3. In addition, the whole leg can rotate about the vertical axis at Pi0, which provides a third degree of freedom for each leg, associated with angle i1. This rotation includes t

40、he mounting points of Fig. 3. Kinematic parameters of one leg of the mechanism. the springs at the base, so that the springs always remain in the plane of the parallelogram. The upper end of the leg, Pi5, is attached to the mobile platform using a spherical joint. In practice, a universal joint comb

41、ined with an additional rev- olute joint has been used in the design of the prototype (see Fig. 2). If six of the joints are actuated, it results in a mecha- nism with six degrees of freedom. We defi ne a fi xed-base reference frame with its origin at point O and with axes x, y, and z such that the

42、base z-axis coincides with the axis of symmetry of the mechanism. A mobile frame is chosen fi xed to the mobile platform at point C with axes xp, yp, and zpsuch that the mobile zp-axis coin- cides with the axis of symmetry. The position of the mobile platform is described by the vector p = x,y,zT, w

43、hich denotes the coordinates of the point C in the reference frame. Glossary of Terms ij : angles describing the confi guration of the ith leg (j = 1,2,3) O: origin of the fi xed frame located at equal distance from the three points Pi0, i = 1,2,3 C: centroid of the mobile platform p: vector connect

44、ing the origin of the fi xed frame O to the centroid C (position vector of the mobile platform) Q: rotation matrix representing the orientation of the platform and defi ned by Euler angles (, , ) pi0 : vector connecting the origin of the fi xed frame to the attachment point Pi0of leg i at the base p

45、i5 : vector connecting the origin of the fi xed frame to the attachment point Pi5of leg i at the mobile platform bi: vector from centroid C to the attachment point Pi5of the ith leg with respect to the mobile frame at Tsinghua University on January 8, Downloaded from Monsarrat and Gosselin / Paralle

46、l Platform Mechanism315 b: distance between point C and the attachment points Pi5of the ith leg to the mobile platform; that is, b = ?bi?, i = 1,2,3 ?ir: length of the rth link of leg i. In the reference orientation of the mobile platform, the ori- entation of the mobile frame coincides with that of

47、 the base frame. This orientation will be represented by the standard Euleranglesthataredefi nedbyfi rstrotatingthemobileframe aboutthebasez-axisbyanangle, thenaboutthemobilexp- axis by an angle , and fi nally about the mobile yp-axis by an angle . For this choice of Euler angles, the rotation matri

48、x is defi ned as Q = Qz()Qx()Qy() = (cc sss)sc(cs+ ssc) (sc+ css)cc(ss csc) csscc , (1) where c= cos, s= sin, c= cos, s= sin, c= cos, and s=sin. In this context, (x,y,z,) denote the generalized coordinates of the three-leg six-degree-of-freedom parallel manipulator. The reader should refer to Figure

49、 3 and to the associated glossary of terms for a complete description of the mechanism and its confi guration. 3. Jacobian Matrix and Resulting Plcker Coordinates In this section, a general expression of the Jacobian matrix J for a manipulator with D legs, D = 3 to 6, is obtained using the principle

50、 of virtual work when the manipulator is in its equilibrium state. The following study will address the determination of the matrix J in the case where, for each leg, any arbitrary subset of the joints corresponding to i1, i2, i3, and ?i3is actuated. This particular formulation of the Jacobian matri

51、x will allow the determination of the Plcker vectors associated with the input angles of the manipulator. 3.1. General Expression of the Jacobian Matrix The Jacobian matrix derived here relates the velocities of the actuated joints and the platform velocity in the form = J ? p ? ,(2) where contains

52、the velocities of all actuated joints, p is the velocityofthecentroidC ofthemobileplatform, and isthe angular velocity vector corresponding to the skew-symmetric matrix QQT; that is, = vect(QQT). Let Cijbe the joint torque associated with the actuated angle ij, and let? ? ? be the corresponding six-

53、dimensional joint force vector. Let F be the force vector applied on the mobile platform, and let M be the torque vector acting on the centroid C of the mobile platform. If f denotes the generalized force vector (i.e., the external wrench acting on the mobile platform and defi ned at point C), we th

54、us obtain f = ? F M ? .(3) Let us defi ne the auxiliary vectors gir= 0?ir0T, i = 1,2,3, and the auxiliary rotation matrices Qi1= Rotz(i1) = cosi1sini10 sini1cosi10 001 (4) and (j = 2,3) Qij= Rotx?(ij) = 100 0 cosijsinij 0 sinijcosij . (5) The z- and x?- axes are shown in Figure 3. In the following,

55、vector si= sizsiysiz is the vector connecting point Pi0 the attachment point of leg i to the baseto point Pi5the attachmentpointoflegi totheplatform;thatis,si= pi5pi0. This equation can be rewritten in terms of angles i1, i2, and i3 in the fi xed frame R such that si= Qi1(Qi3gi1+ Qi2gi5) = cosi1sini

56、10 sini1cosi10 001 ?i1 0 cosi3 sini3 + ?i5 0 cosi2 sini2 .(6) We also defi ne the vector f ? i5 = f ? i5x f ? i5y f ? i5z T that denotes the force that the mobile platform exerts on leg i at the point Pi5expressed in the frame R ?. Then, the total virtual work of the ith leg can be described by W =

57、Ci1i1+ Ci2i2+ Ci3i3+ f ? i5 s ? i5 (7) wheres ? i5 = QT i1sidenotesthevirtualdisplacementofpoint Pi5in frame R ? such that s ? i5 = QT i1 dQi1 di1 (Qi3gi1+Qi2gi5)i1 +QT i1Qi1 ? dQi3 di3 gi1i3+ dQi2 di2 gi5i2 ? (8) Given that QT i1Qi1is the 33 identity matrix, we can rewrite the expression of vector

58、s ? i5 in the form s ? i5= 0 1 0 100 000 ?i1 0 cosi3 sini3 + ?i5 0 cosi2 sini2 i1 + ?i1 0 sini3 cosi3 i3+ ?i5 0 sini2 cosi2 i2 . (9) at Tsinghua University on January 8, Downloaded from 316THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2001 According to the principle of virtual work, the tot

59、al virtual work of the system must vanish; that is, W = 0. This allows us to rewrite eq. (7) in the compact form (?i1cosi3+ ?i5cosi2)f ? i5x + Ci1 ?i5sini2f ? i5y + ?i5cosi2f ? i5z + Ci2 ?i1sini3f ? i5y + ?i1cosi3f ? i5z + Ci3 T i1 i2 i3 = 0. (10) Solving eq. (10) for f ? i5x, f ? i5y, and f ? i5z , one can fi nd the expression of vector f ? i5 as a function of the joint torques Ci1, Ci2, and Ci3with respect to frame R ?: f ? i5 = 1 ?i1?i5sin