can only As Hunt stated in Chapter 4 of his book (Hunt, 1978): Yet neither Kempe nor anyone else since has established a method for isolating the best, or the simplest, linkage for tracing a particular curve. In the history all feasible linkages with a small number of links for algebraic curves generation were invented by somegreatmasters using their geometrical intuitions (Please see the Appendix for details). Nev- ertheless geometrical intuitions are di may not guarantee all solutions for a synthesis problem be found. The above investigation raises a question: Are there any undiscovered 6-bar linkages for straight-line generation? This paper proposes a numerical approach to attack the problem. Notethatitispossibletoextendthe approach for nding spatial 6R single loop overconstrained mechanisms (see remarks at the end of Section 2.2). 2. Figure 1. Six arrangements of 6-bar linkages for a path generation, with the asterisk denoting the position of coupler-point Figure 1 illustrates 6 possible arrangements of 6-bar linkages for straight- line generation. As can be seen from Fig. 4 that the existing straight-line 6-bar linkages are either Watt-I1 linkages or Stephenson-I linkages. In- deed based on the principle of inversion and Robert’s cognate theorem, we can conclude that Stephenson-II2 linkages and Stephenson-III link- ages cannot generate a straight-line. Other arrangements should be 1 This is still an open problem. Smith, 1998 tried to prove it but failed. 2 According to (Artobolevskii, 1964), Alekseyev, 1939 discovered the dimensional relationships of the generalized linkage on 1939, but the authors are not able to find Alekseyev’s proof, while the short proof given in Artobolevskii’s book is indeed invalid. Z. Luo and J.S. Dai114 Searching for 6-bar Straight-line Linkages . find feasible linkages with a large number of links . fficult to be duplicated, and they to the derivation of coupler-curve equations of general planar linkages (see Primrose et al., 1967, Almadi, 1996, and Wampler, 1999). However it is important to develop problem-speci analysis (Dukkipati, 2001, Karger, 1998) or numerical analysis (Luo and Dai, 2005). 2.1 Synthesize Stephenson-I Linkages for Figure 2. Two representations for synthesizing Stephenson-I linkages. (a) is Alek- seyev’s representation (see Artobolevskii, 1964), and (b) is a new representation In Alekseyev’s representation, suppose a coupler-curve equation in (x Q ,y Q ) is obtained, the coupler curve is a straight line if and only if there exist (x 0 ,θ) which satisfy y Q ≡ tan θ (x Q − x 0 ). However since tan θ can vary from zerotoin nity,(x Q ,y Q ) should be parameterized. In our representation, we assume the straight-line is along the x-axis, thus y Q ≡ 0. We further specify x A = 0. As can be seen, there are 10 structural parameters (a, b, c, e, f, g, h, y A ,x D ,y D ). Alternatively we can use (a, b, c, d, e, f, g, h, y A ,θ 0 ). Refer to Fig. 2(b), we obtain the following three loop-closure equations: a cos θ 1 + b cos θ 2 − c cos θ 3 = x D − x A a sin θ 1 + b sin θ 2 − c sin θ 3 = y D − y A (1) (a + e)cosθ 1 + f cos θ 4 = x Q − x A (a + e)sinθ 1 + f sin θ 4 = y Q − y A (2) (c + h)cosθ 3 + g cos θ 5 = x Q − x D (c + h)sinθ 3 + g sin θ 5 = y Q − y D (3) Using classic resultant methods, it is not di i 1, ,5) and a 16 th degree bivariate polynomial in (x Q ,y Q ) is obtained. Searching for Undiscovered Planar Straight-line Linkages 115 examined individually. Synthesizing straight-line linkages is closely related a Straight-line Motion . fic methods based on symbolic fficult to eliminate θ (i = Since y Q ≡ 0, we obtain a univariate polynomial in x Q . Denote it as: P s1 = 16 i=0 a i x i Q =0 (4) Since the linkage can pass in s1 should be incidentally zero. Using a symbolic computing software such as Mathematica, we obtain that: a 16 =0; a 15 =0; a 14 = 65536 a 2 c 2 (ce − ah) 2 d 2 (5) It follows that a 13 = 0. Substitute c = ah/e into a 12 we obtain −a 2 b 2 + a(−2b 2 + d 2 )e +(−b 2 + d 2 )e 2 =0 (6) Solve the above equation yields e 1 = ab 2 /(d 2 − b 2 )ore 2 = −a.Onlye 1 is feasible. It follows that h = cb 2 /(d 2 − b 2 )anda 11 = 0. Substitute the above into coe 10 to a 7 yields a 10 = f 1 (a, b, c, d, f, g, θ 0 ) a 9 = f 2 (a, b, c, d, f, g, θ 0 ) a 8 = f 3 (a, b, c, d, f, g, y A ,θ 0 ) a 7 = f 4 (a, b, c, d, f, g, y A ,θ 0 ) (7) Notethatweusex D = x A + d cos θ 0 ,y D = y A + d sin θ 0 to simplify symbolic expressions. One may want to eliminate (y A ,θ 0 )from the above equations and then solve for (f, g). Unfortunately those equations are quite complicate to solve due to the “pyramidal e We then adapt Karger’s technique to the problem and try to obtain more information (see Karger, 1998 for more details). Karger’s Proposition:LetP (x)= n j=0 (a j + b j cos x)sin j x =0for all x.Thena j = b j =0(j =0, ,n) Now we eliminate θ i (i =2, ,5) and x Q using Resultant methods, this leads to P (θ 1 )= 7 j=0 (a j + b j cos θ 1 )sin j θ 1 =0 (8) Following the procedures in (Karger, 1998), we obtain the coe the two terms with the highest order in variables (cos θ 1 , sin θ 1 ). a 7 = g 1 (a, c, f, h, y A ,x D ,y D ) b 6 = g 2 (a, c, h, y A ,x D ,y D ) (9) Z. Luo and J.S. Dai116 finity many points along the x axis, all the coe Since link lengths can not be zero, we obtain ce = ah. fficients of P fficients a ffect” (Karger, 1998). fficients of . Incidentally b 7 =0,ande is substituted by ah/c.From the above equations, we obtain x D =0 or y D = 0 (10) When x D =0,wehaveθ 0 = π/2, Substitute θ 0 = π/2intoEq.(7),from f 1 we can obtain f 2 − g 2 = b 2 c 2 d 2 − a 2 b 2 d 2 (d 2 − b 2 ) 2 (11) Substitute the above equation into f 2 in Eq. (7), we obtain f = bcd d 2 − b 2 and g = adb d 2 − b 2 (12) It seemsthatevenforthesimplest case of 6-bar linkages, symbolic de- ductions are not quite straightforward. Indeed we have tried the above procedure to synthesize other generic 6-bar linkages but currently no analogous results have been obtained. However a supercomputer may help the symbolic computations. In contrast, we can use numerical algo- rithmstosolvetheaboveproblem conveniently. For example, given 10 points along the x-axis, we obtain a system of 10 polynomials (i.e. Eq. (4)) in 10 unknown variables (a, b, c, e, f, g, h, y A ,x D ,y D ). Together with tun- nelling techqniques, random restarts of Levenberg-Marquart method can 2.2 Synthesize Watt-I2 Linkages for a Straight-line Motion Symbolic Synthesis Equations. Consider a generic Watt-I2 mech- anism shown in Fig. 3, let’s call the illustrated pose the initial pose of the O Q A B C P D E Z 1 Z 4 Z 5 Z 8 Z 2 Z 3 Z 7 Z 6 5 2 1 3 7 Q ' Figure 3. Design parameters in the Watt-I2 mechanism Searching for Undiscovered Planar Straight-line Linkages 117 . When y D to get enough information using symbolic computation. find multiple solutions (see Luo and Dai, 2005for moreinformation). = 0, Eq. (7) still can’t be simplified. Currently we are not able cident with the coupler-point Q at the initial pose. There are 14 design ,y,x A ,y A ,x B ,y B ,x C ,y C ,x D ,y D ,x E ,y ,x P ,y P ). Alter- natively, we can use complex vectors Z For this problem, we prefer to derive the synthesis equations using com- plex numbers for compactness. Referring to Fig. 3, when Q is moved to a new position Q after a displacement of δ = x + iy, the following three loop-closure vector equations can be obtained Z 1 (e i∆θ 1 − 1) + Z 2 (e i∆θ 2 − 1) − Z 3 (e i∆θ 3 − 1) = 0 (13a) Z 3 (e i∆θ 3 − 1) + Z 4 (e i∆θ 2 − 1) + Z 5 (e i∆θ 5 − 1) = δ (13b) Z 6 (e i∆θ 3 − 1) + Z 7 (e i∆θ 7 − 1) + Z 8 (e i∆θ 5 − 1) = δ (13c) Rearrange Eqs. (13a) and (13b), one obtains: Z 1 e i∆θ 1 = Z 3 (e i∆θ 3 − 1) − Z 2 (e i∆θ 2 − 1) + Z 1 (14a) Z 5 e i∆θ 5 = δ − Z 3 (e i∆θ 3 − 1) − Z 4 (e i∆θ 2 − 1) + Z 5 (14b) The angles θ 1 and θ 5 can be eliminated by multiplying each side of Eqs. (14a) and (14b) with its complex conjugate. Expanding and rearranging the results yields p 1 e i∆θ 2 + p 2 e −i∆θ 2 + p 3 = 0 (15a) p 4 e i∆θ 2 + p 5 e −i∆θ 2 + p 6 = 0 (15b) where p i (i =1, ,6) are expressions in θ 3 and the 14 design variables. Note that Eqs. (15a) and (15b) are indeed two real number equations. Solve Eqs. (15a) and (15b) for e i∆θ 2 and e −i∆θ 2 by Cramer’s rule, and then apply the identity e i∆θ 2 e −i∆θ 2 = 1 leads to (p 1 p 6 − p 3 p 4 )(p 2 p 6 − p 3 p 5 )+(p 1 p 5 − p 2 p 4 ) 2 = 0 (16) It is easy to verify that Eq. (16) is also a real number equation. De- note e i∆θ 3 as θ 3 ,andmultiply the above equation by θ 3 3 , a sixth-order polynomial in θ 3 can be obtained as: m 6 θ 6 3 + m 5 θ 5 3 + m 4 θ 4 3 + m 3 θ 3 3 + m 2 θ 2 3 + m 1 θ 3 + m 0 = 0 (17) i Similarly, by manipulating Eqs. (13b) and (13c), one obtains another two equations (q 1 q 6 − q 3 q 4 )(q 2 q 6 − q 3 q 5 )+(q 1 q 5 − q 2 q 4 ) 2 = 0 (18) n 6 θ 6 3 + n 5 θ 5 3 + n 4 θ 4 3 + n 3 θ 3 3 + n 2 θ 2 3 + n 1 θ 3 + n 0 = 0 (19) Z. Luo and J.S. Dai118 variables (x E (i =1, ,7) as design variables. i mechanism. For simplicity, we set the the origin of the fixed frame coin- where the coefficients m (i = 0, , 6) are expressions in design variables. O O ∆ ∆ The necessary condition for Eqs. (17) and (19) to have a common solu- tion of θ 3 is that the determinant of their resultant matrix becomes zero. Here the Bezout resultant matrix will be used, which can be obtained using the Bezout-Cayley formulation (Almadi, 1996). B =[b ij ] 6×6 (20) Expand the determinant of the Bezout matrix, one obtains det(B)= r m=0 r n=0 a mn x m y n =0,m+ n ≤ r (21) where a mn are expressions in the aforementioned 14 design variables, while r is case dependent. In a generic case where Z 8 =0, Z 5 =0, r = 54; in case Z 8 =0,r = 16; while in case Z 5 =0,r =8. Itcanbe zero. Eq. (21) can be further factored since it always has a trivial factor: gcd(m 6 m 0 ,n 6 n 0 )=(x − x C ) 2 +(y − y C ) 2 (22) where gcd means the greatest common factor. Thus for a generic Watt- I2 linkage, its coupler curve equation is a bivariate polynomial of order 52, which in general has 1431 monomials. It is impractical to expand det(B) and collect coe cients of x as did in subsection 3.1. Numerical Approach and Analysis. In path generation synthesis, for each given precision point δ = x + iy, Eq. (21) is a polynomial in 14 design variables. Therefore if 14 precision points besides the origin are will be obtained. In other words, a Watt-I2 linkage generally can pass at maximum 15 precision points including the origin. Therefore if it can pass 16 precision points on a line, then theoretically it must contain a segment of that line. Note that in precision position synthesis problemstherearegenerally positive dimensional manifolds of extraneous solutions. Extraneous so- lutions arise when m 6 m 0 or n 6 n 0 is identically zero. It can be shown that the conditions for m 6 m 0 or n 6 n 0 to be identically zero are, Z 3 =0 or Z 2 + Z 4 =0 or Z 1 + Z 2 − Z 3 = 0 (23) Z 3 =0 or Z 6 =0 or Z 5 = Z 8 =0 or Z 5 Z 6 − Z 3 Z 8 = 0 (24) Some of the conditions correspond to degenerated linkages while other neous solutions is the tunnelling (de ation) method (Luo and Dai, 2005). Searching for Undiscovered Planar Straight-line Linkages 119 verified that the imaginary component of the determinant is identically specified, a determined system of 14 polynomials in 14 design variables are mathematical figments. An effective approach to exclude such extra- fl Although the above formulation is compact, numerical tests show that classic iterative methods normally can not converge within 1000 itera- should choose equations with less nonlinearity. Besides multi-precision arithmetic may be preferable for better accuracy and reliability. Cur- rently we use the following approach for better reliability. points (besides the origin) to be passed along the x-axis, there are the 14 structural variables (x O ,y O ,x A ,y A ,x B ,y B ,x C ,y C ,x D ,y D , x E ,y E ,x P ,y P ) and 15 incremental angular variables θ 3 k (k = 1, ,15). There are 30 equations in 29 variables. Multi-start of Levenberg-Marquart method is used to solve the system. 2 Once a converged point is obtained, we then assign small intervals to the 14 structural parameters of the converged point, and use interval arithmetic to evaluate the corresponding interval box. After a coupler of days of program running, we have got a large num- ber of converged approximate solutions. It is observed that most runs can converge to stationary points with a function residual smaller than 1.0e-10. However all the converged solutions are not exact solutions. more points to increase the reliability. However there is no obvious posi- that the instantaneous center of velocity at the initial pose should be The obtained interval boxes will then be used as the search domains of multi-start classic iterative methods to accelerate the process. The numerical approach can be extended to the synthesis of overcon- strained spatial single-loop mechanisms. It is well known that a spatial chain can reach 21 precision positions (Perez, 2003). Therefore give more constrained mechanisms can be found by precision position synthesis. nisms should be avoided using tunnelling techniques. 3. Conclusions In this paper, we have investigated the problem of searching for undis- covered straight-line linkages. The dimensional relationships in Hart’s Z. Luo and J.S. Dai120 on the y-axis. Meanwhile we are planning to run interval method use parallelized computers to identify potential interval boxes. tions when double-precision float-point arithmetic is used. Therefore we 1 Given 15 Most converged approximate solutions pass 14 precision points in differ- ent configurations and pass near a 15 th point. Later we have also added tive effect. Currently we are programming to include another constraint 6R manipulator has up to 16 configurations, while a spatial 5R open than 16 rotation angles about a fixed axis, spatial 6R single-loop over- Nevertheless similar numerical difficulties arise, e.g. planar 6R mecha- Two real equations Eq. (16) and Eq. (18) are used first. second straight-line linkage have been deduced using symbolic calcula- tions. A numerical approach is then proposed for solving more compli- cate cases. Although no new mechanisms have been found at the current stage, this research is a first step towards an automatic approach for dis- covering new overconstrained mechanisms. References Alekseyev, N.I., (1939), Hart’s straight line mechanism. Scientific Reports of the Moscow Hydro-improvement Institute, VI. Almadi, A.N., (1996), On new foundations of kinematics using classical and modern algebraic theory and homotopy. PhD thesis, University of Wisconsin-Milwaukee. Artobolevskii, I.I., (1964), Mechanisms for the Generation of Plane Curves. Trans- lated by Wills, R.D. & Johnson, W., Macmillan NY. Bricard, R., (1927), Lecons de Cin´ematique (2 volumes), Gauthier-Villar, Paris. Dai, J.S. and Rees Jones, J., (1999), Mobility in metamorphic mechanisms of fold- Dijksman, E., (1975), Kempe’s (focal) linkage generalized, particularly in connec- tion with hart’s second straight-line mechanism, Mechanism and Machine Theory, Dukkipati, R.V., (2001), Spatial Mechanisms, Analysis and Synthesis, Chapter 4.1 Existence Criteria of Mechanisms, Alpha Science Press. Gao, X.S., Zhu, C.C., Chou, S.C., and Ge, J.X., (2001), Automated generation of Kempe linkages for algebraic curves and surfaces. Mechanism and Machine Theory, Harry Hart, (1877), On some cases of parallel motion. Proc. London Math Soc. vol. 8, Hunt, K.H., (1978), Kinematic Geometry of Mechanisms, Oxford University Press. Kapovich, M., Millson, J., (2002), Universality theorem for configuration spaces of Karger, A., (1998), Classification of 5R closed kinematic chains with self mobility. Kempe, A.B., (1873), On the solution of equations by mechanical means, Cambridge Kempe, A.B., (1877), How to Draw a Straight Line, London: Macmillan and Co. Koenigs, G., (1897), Le¸cons de cin´ematique, Hermann, Paris. [4.4, 9.3, 14.4, 14.6 15.3] Luo, Z.J. and Dai, J.S., (2005), Pattern bootstrap: a new method which gives effi- ciency for some precision position synthesis problems, ASME J. Mechanical Design (Accepted). Peaucellier, C., (1873), Note sur une question de geometrie de compass, Nouvelles Annales der Mathematiques, vol. 12, pp. 71–81. Primrose, E.J.F., Freudenstein, F., Roth, B., (1967), Six-Bar Motion. Archive for Smith, W.D., 1998, Plane mechanisms and the “downhill principle”, in the series of “Computational power of machines made of rigid parts”, lectures given at Prince- Sylvester, J.J., (1875), History of the plagiograph, Nature, vol. 12, pp. 214–216. Searching for Undiscovered Planar Straight-line Linkages 121 able/erectable kinds, ASME J. of Mechanical Design, vol. 121, no. 3, pp. 375–382. vol. 10, no. 6, pp. 445–460. vol. 36, pp. 1019–1033. pp. 286–289. planar linkages, Topology, vol. 41, no. 6, pp. 1051–1107. Mechanism and Machine Theory, vol. 33, pp. 213-222. Messenger of Mathematics, vol. 2, pp. 51–52. Rational Mechanics and Analysis, vol. 24, pp. 22–41. ton University, pp. 1–26. Wampler, C.W., (1999), Solving the kinematics of planar mechanisms, ASME J. Me- chanical Design. vol. 121, pp. 387–391. Perez, A., (2003), Dual Quaternion Synthesis of Constrained Robotic Systems, PhD thesis, University of California, Irvine. Appendix: Existing 6-bar Straight-Line Linkages Figures 4 illustrates four known 6-bar straight-line linkages. Cases (a) is based on the principle of inversor (Hart, 1877). Case (b) is a generalized case of Case (a) discovered by Sylvester, 1875 and Kempe, 1877. Cases (c) and (d) were first invented by Hart, 1877 and Bricard, 1927 respectively. Later Dijksman, 1975 unified the two cases into a generalized Case (e). For all four cases, the coupler points drawing a straight-line are labelled as Q. Especially in case (c), Q 1 and Q 2 trace two perpendicular straight-lines, while any other point G on the same coupler traces an ellipse. In case (a), BD = CE, BE = CD, OC = BC, BO/BE = CP/CE = BP/BD, O P O = O P P . In case (b), BD = CE, BE = CD,∆OBE ∆QBD ∆PCE, and θ = ∠POQ In the generalized case of Cases (c) and (d), AB = a, BC = b, CD = c, AD = d, BE = e, CF = h, EQ = f,GQ = g, e = ab 2 /(d 2 − b 2 ),f = cdb/(d 2 − b 2 ),g = adb/(d 2 − b 2 ),h = cb 2 /(d 2 − b 2 ). Especially in case (c), AB = BC,OC = CB; and in case (d), AE = CF,EQ = FQ. E O B C D Q P O P A D B C E F Q 1 Q 2 G (a) (b) A B E D C F Q (c) (d) P O B E Q O P D C B A D E C F Q (e) Figure 4. Four known 6-bar linkages for a straight-line motion. Z. Luo and J . S . Da i 122 TYPE SYNTHESIS OF THREE-DOF UP-EQUIVALENT PARALLEL MANIPULATORS USING A VIRTUAL-CHAIN APPROACH Xianwen Kong D´epartement de G´enie M´ecanique, Universit´e Laval, Qu´ebec, Qu´ebec, Canada, G1K 7P4 xwkong@gmc.ulaval.ca Cl´ement M. Gosselin D´epartement de G´enie M´ecanique, Universit´e Laval, Qu´ebec, Qu´ebec, Canada, G1K 7P4 gosselin@gmc.ulaval.ca Abstract Three-DOF UP-equivalent parallel manipulators are the parallel coun- terparts of the 3-DOF UP serial manipulators, which are composed of one U (universal) and one P (prismatic) joint. Such parallel ma- nipulators can be used either independently or as modules of hybrid manipulators. Using the virtual-chain approach that we proposed else- where for the type synthesis of parallel manipulators, this paper deals with the type synthesis of this class of 3-DOF parallel manipulators. In addition to all the 3-DOF UP-equivalent parallel manipulators pro- posed in the literature, a number of new 3-DOF overconstrained or non-overconstrained UP-equivalent parallel manipulators are identified. Keywords: Three-DOF parallel manipulator, Type synthesis, Virtual chain, Screw Theory, Overconstrained mechanism 1. Introduction Three-DOF UP-equivalent parallel manipulators have a wide range of applications including assembly and machining. Such parallel manip- ulators can be used either independently or as modules of hybrid ma- nipulators. Two UP-equivalent parallel manipulators, which are used as modules in hybrid manipulators, have been proposed in [Neumann, 1988; Huang et al., 2005]. However, the systematic type synthesis of the UP-equivalent parallel manipulator is very difficult and has not been © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 123–132. 123 In order to provide alternatives to the currentinvestigated yet. [...]... Overconstrained 3-2 3-1 2-2 3-3 -2 3-3 -1 3-2 -2 2-2 -2 3-2 -0 3-1 -1 2-2 -0 2-1 -1 3-3 - 3-2 3-3 - 3-1 3-3 - 2-2 3-2 - 2-2 3-3 - 2-0 3-3 - 1-1 3-3 - 1-0 3-2 - 2-0 3-2 - 1-1 3-2 - 1-0 3-1 - 1-1 2-2 - 2-0 3-1 - 1-0 2-2 - 1-0 2-1 - 1-1 2-1 - 1-0 1-1 - 1-1 Moving platform Base (a) Non-overconstrained 3-0 2-1 3-0 -0 2-1 -0 1-1 -1 3-3 -0 2-2 -1 3-3 - 3-0 3-2 - 2-1 2-2 - 2-1 2-2 - 1-1 3-1 - 0-0 3-0 - 0-0 1-1 - 1-0 2-1 - 0-0 Moving platform Base (b) | ¨ ˇˇˇ Figure 4 Two UP-equivalent... UP-equivalent parallel kinematic chains, we can obtain a large number of UP-equivalent parallel kinematic chains By further applying the validity condition of actuated joints [Kong and Gosselin, 2005a], we 131 Three-DOF Up-equivalent Parallel Manipulators Table 3 Families of 3-DOF m-legged UP-equivalent parallel manipulators m 2 3 4 3-3 3-3 -3 3-2 -1 3-1 -0 3-3 - 3-3 3-3 - 2-1 2-2 - 2-2 3-3 - 0-0 3-2 - 0-0 2-2 - 0-0 ... system of a parallel kinematic chain is the linear combination of all of its leg-wrench systems in any configuration [Kumar et al., 2000], it is then concluded that the wrench system of any leg in a UP-equivalent parallel kinematic chain is a ci (0 ≤ ci ≤ 3 )- -system, including 2- 0 -1 - ∞ -system, 2- 0 -system, 1- 0 -1 - ∞ -system, 1- 0 -system, 1- ∞ -system and 0-system, in any general configuration... performing the type synthesis of 3-DOF single-loop kinematic chains and then constructing UP-equivalent parallel manipulators using the types of 3-DOF single-loop kinematic chains 3 Type Synthesis of 3-DOF Single-loop Kinematic Chains Involving a UP Virtual Chain In Section 2.2, the wrench systems of legs for UP-equivalent parallel manipulators have been determined Then, the number of 1-DOF joints of... 2 )- -system is equal to (6 − ci ) In the case of ci = 0, the associated single-loop kinematic chains are not overconstrained Such a single-loop kinematic chain is composed of the UP virtual chain and six R and P joints Many types of single-loop kinematic chains can be obtained Among these types, the types with simple structure, such as UPSV, PUSV and RUSV, are of practical interest In the following,... axes of R joints, the intersections of the R joints ˇ joints within the same within the same leg, the intersections of the R ¨ ¯ leg, and the intersection of the axes of the R joint and the R joint within ˝ the same leg determine a common line The axes of the R joints are parallel to the above common line For a better understanding of the notation used, a few single-loop kinematic chains involving a UP... manipulator shown in Fig 4(a) belongs to Family 2-1 -1 and is overconstrained The | ˝˝ ˇˇˇ RR(RRR)E -2 -RRR(RR)E shown in Fig 4(b) belongs to Family 1-1 -1 and is not overconstrained It is noted that the UP-equivalent parallel manipulators proposed in [Neumann, 1988; Huang et al., 20 05] belong respectively to Families 3-0 - 0-0 and 3-0 -0 listed in Table 3 X Kong and C M Gosselin 132 5 Conclusions The type... Composition of 3-DOF overconstrained single-loop kinematic chains with a UP virtual chain ci Leg-wrench system 3 2 2- 0 -1 - ∞ 1- 0 -1 - ∞ 2- 0 1- ∞ 1 Planar unit Spherical unit 1 1 1 1- 0 1 Composition Coaxial Codirectional unit unit 2 1 1 1 1 Parallelaxis unit 1 1 2 1 wrench system of each of these kinematic chains always includes a specified number of independent wrenches of zero-pitch or in nite-pitch... synthesis of overconstrained single-loop kinematic chains involving a UP virtual chain As pointed out in [Kong and Gosselin, 2005b], the types of over constrained single-loop kinematic chains can be constructed using seven compositional units A compositional unit is a serial kinematic chain with specific characteristics, namely: In any general configuration, the 127 Three-DOF Up-equivalent Parallel Manipulators... useful in the type synthesis of parallel manipulators 4 Construction of UP-equivalent Parallel Manipulators Now let us see how to construct UP-equivalent parallel manipulators from the 3-DOF single-loop kinematic chain involving a virtual chain By removing the virtual chain in a 3-DOF single-loop kinematic chain involving a virtual chain, one leg for UP-equivalent parallel manipulators can be obtained . 2-1 - 0-0 3-3 - 2-1 3-2 - 2-2 3-3 - 2-0 3-3 - 1-1 3-2 - 2-1 1-1 - 1-0 2-2 - 2-2 3-3 - 1-0 3-2 - 2-0 3-2 - 1-1 2-2 - 2-1 3-3 - 0-0 3-2 - 1-0 3-1 - 1-1 2-2 - 2-0 2-2 - 1-1 3-2 - 0-0 3-1 - 1-0 2-2 - 1-0 2-1 - 1-1 3-1 - 0-0 2-2 - 0-0 2-1 - 1-0 1-1 - 1-1 Moving. Non-overconstrained 2 3-3 3-2 3-1 2-2 3-0 2-1 3 3-3 -3 3-3 -2 3-3 -1 3-2 -2 3-3 -0 3-0 -0 2-1 -0 3-2 -1 2-2 -2 3-2 -0 3-1 -1 2-2 -1 1-1 -1 3-1 -0 2-2 -0 2-1 -1 4 3-3 - 3-3 3-3 - 3-2 3-3 - 3-1 3-3 - 2-2 3-3 - 3-0 3-0 - 0-0 2-1 - 0-0 3-3 - 2-1 . c i ≤ 3 )- -system, in- cluding 2- 0 -1 - ∞ -system, 2- 0 -system, 1- 0 -1 - ∞ -system, 1- 0 -system, 1- ∞ -system and 0-system, in any general configuration. 2.3 When we connect the base and