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\Copyright 1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish thismaterial for advertising or promotional purposes or for creating new collective works for resale or redistribution toservers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE."Accepted for publication in IEEE Transactions on Robotics and Automation, 1999Comments on \Closed Form Forward KinematicsSolution to a Class of Hexapod Robots" yHerman Bruyninckx, Joris De SchutterKatholieke Universiteit Leuven, Dept. Mechanical EngineeringCelestijnenlaan 300B, B-3001 Heverlee, BelgiumAbstractThe paper [8] by Yang and Geng presents a class of parallel manipulators that has an e cient closedform solution to its forward kinematics. This Correspondence points to some missing insights in that paper(and in many others), especially with respect to singular con gurations.KeywordsParallel manipulators, closed form solutions, singularities.1 IntroductionMost literature on parallel manipulators considers so-called Stewart-Gough platforms, i.e., a xed base and amoving end e ector are connected by six prismatic, actuated \legs" attached to base and end e ector by meansof universal and/or spherical joints. It is well known that the forward kinematics of arbitrary Stewart-Goughplatforms are di cult to solve. However, these kinematics become much easier for some special geometries. Thecited paper, [8], proposes ...

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• Publié par: Muchang
• Nombre de lectures: 51
• Langue: English

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“Copyright 1999 IEEE. Personal use of this material is permitted.However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.”

Accepted for publication inIEEE Transactions on Robotics and Automation, 1999 Comments on “Closed Form Forward Kinematics Solution to a Class of Hexapod Robots”

∗ † Herman Bruyninckx, Joris De Schutter Katholieke Universiteit Leuven, Dept.Mechanical Engineering Celestijnenlaan 300B, B-3001 Heverlee, Belgium

Abstract The paper [8] by Yang and Geng presents a class of parallel manipulators that has an efficient closed form solution to its forward kinematics.This Correspondence points to some missing insights in that paper (and in many others), especially with respect to singular configurations.

Keywords Parallel manipulators, closed form solutions, singularities.

1 Introduction Most literature on parallel manipulators considers so-called Stewart-Gough platforms, i.e., a fixed base and a moving end effector are connected by six prismatic, actuated “legs” attached to base and end effector by means of universal and/or spherical joints.It is well known that the forward kinematics of arbitrary Stewart-Gough platforms are difficult to solve.However, these kinematics become much easier for some special geometries.The cited paper, [8], proposes one such special geometry:ifA i andB i (i= 1, . . . ,5) denote the vectors (on end effector and base, respectively) between one (arbitrary chosen) “zeroth” attachment point and the five other attachment points on the platform, then the special geometry is calledlinearly relatedif there exists a 3×3 matrixMsuch that: A i =M B i ,fori= 1, . . . ,5.(1) The most difficult equation in the forward position kinematics is then a quartic in one variable.Only eight different solutions exist, four of those being mirror images of the four others with respect to the base.

2 Discussion Our first comment concerns some important missing references to earlier, original work:(i) the authors neglect the work on the so-called “321” geometry by Nanua and Waldron, [4, 5], which pre-dates theirs, (ii) the linearly related geometry has already been published in 1996 in [1], and (iii) a special case hereof (base and end effector are similar hexagons, i.e.,Min Eq. (1) is a multiple of the unity matrix) was presented by several authors, [2, 6, 7], also long before the paper under discussion. ∗ Post-Doctoral Researcher of the Fund for Scientific Research–Flanders in Belgium (F.W.O.).The financial support of the Belgian Inter-Universitary Attraction Poles (I.U.A.P.) is gratefully acknowledged. † Manuscript submitted: July 1998.

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Our second comment concerns the fact that an important geometric constraint has not been made explicit: the base and end effector must beplanar; the contrary is suggested by the definition (1) that seemingly involves arbitrary three-vectors. Our next two comments relate to a potentially dangerous lack of insight concerning the occurrence of singular manipulator configurations. First, the authors give a numerical example of their algorithm, usingM=diag(.5, .5,0) in Eq. (1).They motivate this choice by stating that “in most real world designs, symmetries are often desirable.”However, this isnotthe case, on the contrary:symmetries in the geometries of base and end effector increase the risk of coming up with “architecturally singular” parallel manipulators, [3], i.e., manipulators thatalwayshave a passive degree of freedom which cannot be blocked by the available actuators.This fact was already stated explicitly in, e.g., [1] and [2]. Secondly, our final and most serious comment concerns the fact that the described family of parallel manip- ulatorsis in a singular configuration whenever the end effector plate is parallel to the base platethat. Remark this is not just an isolated singular point, but a whole manifold of singularities! Mathematically speaking, the Jacobian matrixJbecomes rank-deficient in a singular configuration.This 6×6 matrixJdescribes the velocity transformation between joint space and Cartesian space: t=J q ˙ ,(2) withtthe instantaneous velocity (“twist”) of the end effector corresponding to the velocitiesq˙ j (j= 0, . . . ,5) of the actuated prismatic joints.Using the notations from [8], the transpose of thejth rowJ j of the Jacobian matrixJis (proportional to):   T L j J=.(3) j B j ×L j L j is the vector along thejth leg,B 0 = 0, andL 0 =p, i.e., the vector between the origins of the reference frames on both platforms.Taking into account Eq. (1), the following relationship connectsB i andL i : L i = (p+RM B i )−B i ,(4) withRthe rotation matrix between base and end effector reference frames.Hence, theith row ofJbecomes   T p+ (RM−I)B i J=.(5) i B i ×L i If the end effector is parallel to the base,Ris of the form   cos(φ)−sin(φ) 0   sin(φ) cos(φ) 0 .(6) 0 01 If both end effector and base are planar,Mis of the form   a b0   c d0 .(7) 0 0 1 Hence, (RM−I)B i is parallel to the base plate, andB i ×L i =B i ×pif base and end effector. Moreover, are planar, the vectorsB 3 ,B 4 andB 5 are linear combinations ofB 1 andB 2 (possibly after reordering of the indices).In the singularity analysis, we are interested in the vanishing of the determinant of the Jacobian matrix. Thisvanishing of the determinant does not change under row operations on the matrix.Hence, consider P 2 i ii ctαtimes the second row pl i =α(i= 4 1 usα 2 the following row operations:ifB j=1 j B j ,5,6), then subtra times the third row from theiThe transpose of the resulting matrix then looks liketh Jacobian row.   L 0 L 1 L 2 α 4 pα 5 pα 6 p ,(8) 0B 1 ×p B 2 ×p0 0 0 withα k The determinant of this matrix vanishes, and hence the manipulator is in a singularsome real scalars. configuration.

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3 Conclusions The importance and abundance of singularities is usually severely underestimated in the literature on parallel kinematic chains.As a rule of thumb, one should not strive to introduce symmetries and linear relationships in the geometry of base and end effector.

References [1] H.Bruyninckx and J. De Schutter.A class of fully parallel manipulators with closed-form forward position kinematics.InJ.LenarcˇicˇandV.Parenti-Castelli,editors,Recent Advances in Robot Kinematics, pages 411–419,Portoroˇz-Bernardin,Slovenia,1996. [2] H.-Y.Lee and B. Roth.A closed-form solution of the forward displacement analysis of a class of in-parallel mechanisms. InIEEE Int. Conf. Robotics and Automation, pages 720–724, Atlanta, GA, 1993. [3] O. Ma and J. Angeles.Optimum architecture design of platform manipulators.InInt. Conf. Advanced Robotics, pages 1130–1135, Pisa, Italy, 1991. [4] P.Nanua. Directkinematics of parallel mechanisms.Master’s thesis, Ohio State University, Ohio, 1988. [5] P.Nanua and K. J. Waldron. Direct kinematic solution of a special parallel robot structure. In A. Morecki, G. Bianchi, and K. Jaworek, editors,Proc. 8th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pages 134–142, Warsaw, Poland, 1990. [6] S.V. Sreenivasan, K. J. Waldron, and P. Nanua.Closed-form direct displacement analysis of a 6-6 Stewart platform.Mechanism and Machine Theory, 29(6):855–864, 1994. [7] G. Wang.Forward displacement analysis of a class of the 6-6 Stewart platforms.InProc. 1992 ASME Design Technical Conferences–22nd Biennal Mechanisms Conference, pages 113–117, Scottsdale, AZ, 1992. [8] J.Yang and Z. J. Geng. Closed form forward kinematics solution to a class of Hexapod robots.IEEE Trans. Rob. Automation, 14(3):503–508, 1998.

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