pro-Keywords: Three-DOF parallel manipulator, Type synthesis, Virtual chain, Screw Theory, Overconstrained mechanism Three-DOF UP-equivalent parallel manipulators have a wide range of ap
Trang 1can only As Huntstated 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 foralgebraic curves generation were invented by some great masters usingtheir 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 Theabove investigation raises a question: Are there any undiscovered 6-barlinkages for straight-line generation? This paper proposes a numericalapproach to attack the problem Note that it is possible to extend theapproach for nding spatial 6R single loop overconstrained mechanisms(see remarks at the end of Section 2.2)
straight-we can conclude that Stephenson-II2 linkages and Stephenson-III ages cannot generate a straight-line Other arrangements should be
link-1This is still an open problem Smith, 1998 tried to prove it but failed.
2According 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,
Z Luo and J.S Dai
114
Searching for 6-bar Straight-line Linkages
.find feasible linkages with a large number of links
fficult to be duplicated, and they
Trang 2to 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 andDai, 2005)
Figure 2. Two representations for synthesizing Stephenson-I linkages (a) is seyev’s representation (see Artobolevskii, 1964), and (b) is a new representation
Alek-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 (x0, θ) which satisfy y Q ≡ tan θ (xQ − x0) However since
tan θ can vary from zero to in 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 xA = 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 , θ0) Refer to Fig 2(b), we obtain the followingthree loop-closure equations:
a cos θ1+ b cos θ2− c cos θ3 = x D − xA
a sin θ1+ b sin θ2− c sin θ3 = y D − yA (1)(a + e) cos θ1+ f cos θ4= x Q − xA
(a + e) sin θ1+ f sin θ4 = y Q − yA (2)(c + h) cos θ3+ g cos θ5 = x Q − xD
(c + h) sin θ3+ g sin θ5= y Q − yD (3)
Using classic resultant methods, it is not di i
1, , 5) and a 16th 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 =
Trang 3Since y Q ≡ 0, we obtain a univariate polynomial in xQ Denote it as:
P s1 =
16
i=0
Since the linkage can pass in
s1 should be incidentally zero Using a symboliccomputing software such as Mathematica, we obtain that:
a16= 0; a15= 0; a14= 65536 a2c2(ce − ah)2d2 (5)
It follows that
a13= 0 Substitute c = ah/e into a12 we obtain
−a2b2+ a( −2b2+ d2)e + ( −b2+ d2)e2 = 0 (6)
Solve the above equation yields e1= ab2/(d2− b2) or e2 =−a Only e1
is feasible It follows that h = cb2/(d2− b2) and a11= 0 Substitute theabove into coe 10to a7 yields
Note that we use x D = x A + d cos θ0, y D = y A + d sin θ0 to simplify
symbolic expressions One may want to eliminate (y A , θ0) 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 obtainmore information (see Karger, 1998 for more details)
Karger’s Proposition: Let P (x) =n
j=0 (a j + b j cos x) sin j x = 0 for all x Then a j = b j = 0(j = 0, , n)
Now we eliminate θ i (i = 2, , 5) and x Q using Resultant methods, thisleads to
P (θ1) = 7
j=0 (a j + b j cos θ1) sinj θ1 = 0 (8)Following the procedures in (Karger, 1998), we obtain the coe
the two terms with the highest order in variables (cos θ1, sin θ1)
Trang 4Incidentally b7 = 0, and e is substituted by ah/c. From the aboveequations, we obtain
de-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 nelling techqniques, random restarts of Levenberg-Marquart method can
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, 2005 for more information)
= 0, Eq (7) still can’t be simplified Currently we are not able
Trang 5cident 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 ofδ = x + iy, the following three
loop-closure vector equations can be obtained
Z1(e i∆θ1 − 1) + Z2(e i∆θ2 − 1) − Z3(e i∆θ3 − 1) = 0 (13a)
Z3(e i∆θ3 − 1) + Z4(e i∆θ2 − 1) + Z5(e i∆θ5 − 1) = δ (13b)
Z6(e i∆θ3 − 1) + Z7(e i∆θ7 − 1) + Z8(e i∆θ5 − 1) = δ (13c)Rearrange Eqs (13a) and (13b), one obtains:
Z1e i∆θ1 = Z3(e i∆θ3 − 1) − Z2(e i∆θ2 − 1) + Z1 (14a)
Z5e i∆θ5 =δ − Z3(e i∆θ3− 1) − Z4(e i∆θ2− 1) + Z5 (14b)The angles θ1 and θ5 can be eliminated by multiplying each side
of Eqs (14a) and (14b) with its complex conjugate Expanding andrearranging the results yields
It is easy to verify that Eq (16) is also a real number equation
De-note e i∆θ3 as θ3, and multiply the above equation byθ3
3, a sixth-orderpolynomial inθ3 can be obtained as:
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.
Trang 6The necessary condition for Eqs (17) and (19) to have a common tion ofθ3 is that the determinant of their resultant matrix becomes zero.Here the Bezout resultant matrix will be used, which can be obtainedusing the Bezout-Cayley formulation (Almadi, 1996).
where a mn are expressions in the aforementioned 14 design variables,
while r is case dependent In a generic case where Z8 = 0, Z5 = 0,
r = 54; in case Z8 = 0, r = 16; while in case Z5 = 0, r = 8 It can be
zero Eq (21) can be further factored since it always has a trivial factor:
gcd(m6m0, n6n0) = (x − xC)2+ (y − yC)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 pointδ = x + iy, Eq (21) is a polynomial in 14
design variables Therefore if 14 precision points besides the origin arewill be obtained In other words, a Watt-I2 linkage generally can pass
at maximum 15 precision points including the origin Therefore if it canpass 16 precision points on a line, then theoretically it must contain asegment of that line
Note that in precision position synthesis problems there are generallypositive dimensional manifolds of extraneous solutions Extraneous so-
lutions arise when m6m0 or n6n0 is identically zero It can be shown
that the conditions for m6m0 or n6n0 to be identically zero are,
Z3 = 0 or Z2+ Z4= 0 or Z1+ Z2− Z3 = 0 (23)
Z3 = 0 or Z6 = 0 or Z5 = Z8= 0 or Z5Z6− Z3Z8 = 0 (24)Some of the conditions correspond to degenerated linkages while otherneous 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
Trang 7Although the above formulation is compact, numerical tests show thatclassic iterative methods normally can not converge within 1000 itera-should choose equations with less nonlinearity Besides multi-precisionarithmetic 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 θ3k (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 useinterval 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 runscan converge to stationary points with a function residual smaller than1.0e-10 However all the converged solutions are not exact solutions
more points to increase the reliability However there is no obvious that the instantaneous center of velocity at the initial pose should be
posi-The obtainedinterval boxes will then be used as the search domains of multi-startclassic iterative methods to accelerate the process
The numerical approach can be extended to the synthesis of strained spatial single-loop mechanisms It is well known that a spatialchain can reach 21 precision positions (Perez, 2003) Therefore give moreconstrained mechanisms can be found by precision position synthesis.nisms should be avoided using tunnelling techniques
differ-6R manipulator has up to 16 configurations, while a spatial 5R openthan 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
Trang 8second straight-line linkage have been deduced using symbolic tions A numerical approach is then proposed for solving more compli-cate cases Although no new mechanisms have been found at the currentstage, this research is a first step towards an automatic approach for dis-covering new overconstrained mechanisms.
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 Dijksman, E., (1975), Kempe’s (focal) linkage generalized, particularly in connec-
fold-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 Sylvester, J.J., (1875), History of the plagiograph, Nature, vol 12, pp 214–216.
Prince-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.
Trang 9Wampler, C.W., (1999), Solving the kinematics of planar mechanisms, ASME J chanical Design vol 121, pp 387–391.
Me-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 ageneralized 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 ageneralized Case (e) For all four cases, the coupler points drawing a
straight-line are labelled as Q Especially in case (c), Q1 and Q2 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 ∆P CE, and θ = ∠P OQ 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 = ab2/(d2 − b2), f = cdb/(d2− b2), g = adb/(d2− b2), h = cb2/(d2− b2) Especially in case
P O
Figure 4. Four known 6-bar linkages for a straight-line motion.
Z Luo and J.S Dai
122
Trang 10D´ 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 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.
ma-In addition to all the 3-DOF UP-equivalent parallel manipulators posed in the literature, a number of new 3-DOF overconstrained or non-overconstrained UP-equivalent parallel manipulators are identified.
pro-Keywords: Three-DOF parallel manipulator, Type synthesis, Virtual chain, Screw
Theory, Overconstrained mechanism
Three-DOF UP-equivalent parallel manipulators have a wide range
of applications including assembly and machining Such parallel ulators can be used either independently or as modules of hybrid ma-nipulators Two UP-equivalent parallel manipulators, which are used
manip-as modules in hybrid manipulators, have been proposed in [Neumann,1988; Huang et al., 2005] However, the systematic type synthesis of theUP-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
Trang 11the type synthesis of UP-equivalentparallel manipulators needs further investigation.
Using the virtual-chain approach proposed in [Kong and Gosselin,2005a]1, the type synthesis of UP-equivalent parallel manipulators isdealt with in this paper In Section 2, the virtual-chain approach for thetype synthesis of parallel manipulators is recalled The type synthesis
of 3-DOF single-loop kinematic chains is performed in Section 3 InSection 4, we discuss how to construct UP-equivalent parallel kinematicchains and UP-equivalent parallel manipulators using 3-DOF single-loopkinematic chains Two new UP-equivalent parallel manipulators are alsopresented Finally, conclusions are drawn
2.
2.1
As proposed in [Kong and Gosselin, 2005a], the motion pattern of
an f -DOF parallel manipulator can be represented by a virtual chain
which is the simplest serial or parallel kinematic chain that can expressthe motion pattern well
The virtual chain for the motion pattern of the 3-DOF PMs to besynthesized in this paper is the UP virtual chain shown in Fig 1(a)
In the UP virtual chain, the direction of the P (prismatic) joint is pendicular to the axis of its adjacent R (revolute) joint within the U(universal) joint
In this paper, we limit ourselves to non-redundant parallel
mani-The Virtual-chain Approach
The Virtual Chain
Representation of Instantaneous Constraints
UP-equivalent parallel manipulators,
Trang 122000; Davidson and Hunt, 2004] for example, provides an efficient tool
to address this issue
The instantaneous constraints exerted on the moving platform by thebase through the kinematic chain (virtual chain, leg of a parallel kine-matic chain or a parallel kinematic chain) is represented by a screw sys-tem which is called the wrench system of the kinematic chain (virtualchain, leg of a parallel kinematic chain or a parallel kinematic chain).For brevity, the wrench system of a leg is also called a leg-wrench system
Wrench system of UP-equivalent parallel kinematic chains.
In any general configuration, a UP-equivalent parallel kinematic chainand its corresponding UP virtual chain have the same wrench system.Finding the wrench system of the UP-equivalent parallel kinematic chain
is thus equivalent to finding the wrench system of the UP virtual chain[Fig 1(b)]
It can be found without difficulty that the wrench system of the equivalent parallel kinematic chain is a 2-ζ0-1-ζ ∞-system [see Fig 1(b)].Here, ζ0 and ζ ∞ denote, respectively, a wrench of zero pitch and awrench of infinite-pitch One base of the 2-ζ0-1-ζ ∞-system is composed
UP-of (a) two non-collinearζ0 whose axes pass through the center of the Ujoint and are perpendicular to the direction of the P joint and (b) aζ ∞
whose direction is perpendicular to the axes of the R joints within the
U joint
Leg-wrench system of UP-equivalent parallel kinematic chains.
As the wrench system of a parallel kinematic chain is the linear
com-UP-equivalent parallel kinematic chain is a c i(0≤ c i ≤ 3)-ζ-system,
in-cluding 2-ζ0-1-ζ ∞-system, 2-ζ0-system, 1-ζ0-1-ζ ∞-system, 1-ζ0-system,1-ζ ∞-system and 0-system, in any general configuration
2.3
When we connect the base and the moving platform of a parallelkinematic chain by an appropriate UP virtual chain, the function of theparallel kinematic chain is not affected (Fig 2) Any of its legs and the
UP virtual chain will constitute a 3-DOF single-loop kinematic chain.Thus, a parallel kinematic chain is a UP-equivalent parallel kinematicchain if it satisfies the following two conditions:
Three-DOF Up-equivalent Parallel Manipulators 125
Conditions for a UP-equivalent Parallel
Manipulator
et al., 2000], it is then concluded that the wrench system of any leg in abination of all of its leg-wrench systems in any configuration [Kumar
Trang 13Leg 1 Leg 2 Leg 3Base
Moving platform
(b) Leg 1 Leg 2 Leg 3Virtual chain
Base Moving platform
at least the UP-motion The second condition further guarantees thatthe degree of freedom of the moving platform is three
Based on the above conditions, the type synthesis of parallel lators can be performed by first performing the type synthesis of 3-DOFsingle-loop kinematic chains and then constructing UP-equivalent paral-lel manipulators using the types of 3-DOF single-loop kinematic chains
manipu-3.
In Section 2.2, the wrench systems of legs for UP-equivalent lel manipulators have been determined Then, the number of 1-DOF
paral-joints of a leg with a c i(0 ≤ c i ≤ 2)-ζ-system is equal to (6 − c i) In
the case of c i = 0, the associated single-loop kinematic chains are notoverconstrained Such a single-loop kinematic chain is composed of the
UP virtual chain and six R and P joints Many types of single-loop matic chains can be obtained Among these types, the types with simplestructure, such as UPSV, PUSV and RUSV, are of practical interest
kine-In the following, we will focus on the type synthesis of overconstrainedsingle-loop kinematic chains involving a UP virtual chain
compositional units A compositional unit is a serial kinematic chain
with specific characteristics, namely: In any general configuration, the
126
Type Synthesis of 3-DOF Single-loop
Chains Involving a UP Virtual Chain
constrained single-loop kinematic chains can be constructed using seven
As pointed out in [Kong and Gosselin, 2005b], the types of over
Kinematic
X Kong and C M Gosselin
Trang 14Table 1. Composition of 3-DOF overconstrained single-loop kinematic chains with a
UP virtual chain.
c i Leg-wrench
system
Composition Planar Spherical Coaxial Codirectional Parallelaxis
(a) Parallelaxis compositional units Serial kinematic chains composed
of at least one R joint and at least one P joint in which the axes
of all the R joints are parallel and not all the directions of the Pjoints are perpendicular to the axes of the R joints
(b) Planar compositional units Serial kinematic chains in which all
the links are moving along parallel planes A planar serial matic chain is denoted by ()E
kine-(c) Spherical compositional units Serial kinematic chains composed
of two or more concurrent R joints Each R joint of a sphericalserial kinematic chain is denoted by ˙R
(d) Coaxial compositional units Serial kinematic chains composed of
two coaxial R joints
(e) Codirectional compositional units Serial kinematic chains
com-posed of two P joints whose directions are parallel Each P joint
of a codirectional serial kinematic chain is denoted by P.
For each class of single-loop kinematic chains that has a UP virtualchain and a specified leg-wrench system, the specific types can be readily
Three-DOF Up-equivalent Parallel Manipulators 127
following five compositional units as shown in Table 1
Trang 15the type synthesis of single-loop mechanisms, a mechanism with a ial or codirectional compositional unit is regarded to be degeneratedand is therefore discarded In the type synthesis of parallel mechanisms,however, a single-loop kinematic chain that contains a coaxial or codi-rectional compositional unit should be used since one joint of the coaxial
coax-or codirectional compositional unit belongs to one leg of a parallel anism while the other joint belongs to the virtual chain
mech-In the representation of types of 3-DOF single-loop kinematic chainsinvolving a UP virtual chain, the following notations are used The jointswithin a ()|
E constitute a planar kinematic chain, whose associated plane
of relative motion is parallel to the direction of the P joint of the UP tual chain The joints within a ()
vir-E constitute a planar kinematic chain,whose associated plane of relative motion is parallel to the direction ofthe P joint of the UP virtual chain and perpendicular to the axis of thesecond R joint within the U joint of the UP virtual chain The P jointwhose direction is parallel to the direction of the P joint within the UPvirtual chain is denoted by P The R joints are represented by ˙R, ˇR, ¨R,
¯
R, ˝R and ´R due to the different geometric conditions that the R joints
˙point on the axis of the first R joint within the U joint of the UP virtualchain The axes of all the ˇR joints within a leg intersect at the center
of the U joint of the UP virtual chain ¨R ( ¯R) denotes an R joint that
is coaxial with the first (second) R joint within the U joint of the UPvirtual chain ˝R ( ´R) denotes an R joint whose axis is parallel to the axes
of the the first (second) R joint within the U joint of the virtual chain.Considering that each leg of the UP-equivalent parallel kinematicchain and the same UP virtual chain constitute a 3-DOF single-loopkinematic chain, the above notations can also be used to represent thetypes of UP-equivalent parallel kinematic chains, UP-equivalent paral-lel manipulators and their legs The geometric conditions for the UP-pulators and their legs can be obtained as follows
All the P joints are along the same direction. All the planes of
relative motion of the planar chains associated with ()
E are parallel.The above planes, the planes of relative motion of the planar chainsassociated with ()|
E as well as the direction of the P joints all parallel
to a common direction The axes of the ´R joints are parallel to a linethat is perpendicular to (a) the planes of relative motion of the planarchains associated with ()
E, (b) the intersection of the planes of relativemotion of the planar chains associated with ()|
E, and (c) the direction
128
obtained and shown in Table 2 It is noted that in the existing works on
satisfy The axes of all the R joints within a leg intersect at a common
parallel chains,
mani-parallel
X Kong and C M Gosselin