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Workspace of parallel manipulators with symmetric identical kinematic chains
Mechanism and Machine Theory 41 (2006) 632–645
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Workspace of parallel manipulators with symmetric identical kinematic chains Jing-Shan Zhao
a,*
, Min Chen b, Kai Zhou a, Jing-Xin Dong a, Zhi-Jing Feng
a
a
b
Department of Precision Instruments, Tsinghua University, Beijing 100084, PR China School of Engineering and Technology, China University of Geosciences, Beijing 100083, PR China
Received 15 December 2004; received in revised form 5 September 2005; accepted 14 September 2005 Available online 8 November 2005
Abstract In this paper, we propound an analogous symmetric theorem of workspace for spatial parallel manipulators with identical kinematic chains, and then present a strict mathematic proof by splitting the problem into planar symmetry, rotational axis symmetry and point symmetry (centrosymmetry). Examples demonstrate that this theorem can be utilized to investigate the geometry shapes of the workspace and to guide the conceptual design of spatial parallel manipulators, especially for those complicated spatial parallel manipulators with full degrees of freedom (DoF). The theorem offers a concrete safeguard for the workspace analysis of any symmetric spatial parallel manipulators. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Workspace; Theorem; Spatial parallel manipulator; Kinematic chain
1. Introduction Since workspace determination is generally an intermediate but critical step in analyzing and synthesizing manipulators, it is very important to have a theory to safeguard the estimation and conceptual design. The workspace of a manipulator is the domain of reach of its end-effector and is bounded in the 3D space [1]. The workspace of a manipulator has been defined in literature as the totality of positions that a particular identified point of the manipulator (end-effector) can reach [2,3]. The workspace boundary is the curve (in plane) or the surface (in space) that defines the extent of reach of the end-effector [2,4]. Many scholars have proposed algebraic method [4,5] and geometric method [6,7] to determine the workspace of a manipulator. For determination of different characteristics of the workspace, in order to compare different existing manipulators or design a new manipulator, it is almost always necessary to determine the boundaries of the workspaces [2]. Wang et al. [8] presented an algorithm, called the boundary search method,
*
Corresponding author. Fax: +86 01 06 278 2351. E-mail address: [email protected] (J.-S. Zhao).
0094-114X/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.09.007
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to determine the workspace of a parallel machine tool. They also analyzed the boundary workspace in order to expand its operating scope. Bhattacharya et al. [9] found that the singular manifolds of a platform type parallel manipulator always lie at the boundary separating two open diffeomorphic regions (branch modes) in the workspace of the device. Dai and Shah [10] investigated the orientation capability of a serial manipulator by examining the rotatability of its equivalent mechanism and by proposing the workspace region decomposition, based on the polynomial discriminant derived from the virtual angle analysis of the end-effector link of the equivalent mechanism. Bonev and Ryu [11] presented a discretization method for the computation of the orientation workspace of 6-DoF parallel manipulators, defined as the set of all attainable orientations of the mobile platform about a fixed point. The focuses of the above literature are mainly on finding the workspace of a manipulator, but seldom discuss the relationship between the shapes of the workspace and the geometry structure of the manipulator. There exists no theorem for spatial parallel manipulators which characterizes the geometry relationship between the reachable workspace of closed-loop manipulators and the structures of their kinematic chains. Therefore, the present study characterizes the workspace of spatial parallel manipulators with symmetric identical kinematic chains. We first establish a symmetric theorem for spatial parallel manipulators and then identify the structural symmetry with three cases: planar symmetry, rotational axis symmetry and point symmetry (centrosymmetry). For the sake of clarity, we also make some necessary assumptions and present a strict mathematic proof. 2. Analogous symmetric theorem of workspace for spatial parallel manipulators In this paper, the reachable workspace of a manipulator is defined as the volume within which the reference point, say, the geometry center of the manipulator, can be reached [12]. In applications, most spatial parallel manipulators have symmetric identical kinematic chains. These chains are symmetric about a certain plane, about one axis or about one specified point. For example, the manipulator shown in Fig. 1 has three identical kinematic chains, each one of which consists of two universal joints and one prismatic joint in series. The structural symmetric characteristics can be found in Fig. 2, where a1, a2 and a3 are the three symmetric planes and e is a symmetric axis that is intersected by these three symmetric planes. As to the point symmetry, we can find that the middle point of AD is the symmetric center of the planar inverse parallel 4-bar manipulator shown in Fig. 3.
Fig. 1. A spatial parallel manipulator made up of 3-PUU kinematic chains.
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Fig. 2. The structural symmetric of the spatial parallel manipulator.
Fig. 3. A planar inverse parallel 4-bar manipulator.
In order to exploit the relationship between the workspace shapes and the structural characteristics of the kinematic chains, we firstly propose a symmetric workspace theorem for spatial parallel manipulator: If each identical kinematic chain of a spatial parallel manipulator always keeps collinear, and they are structural symmetric about a plane, an axis, or a point; the reachable workspace of the manipulator will also be symmetric about the corresponding plane, axis, or point individually. However the converse theorem does not hold. Prior to demonstrating the validation of the theorem, we firstly make the following assumptions: (1) The absolute coordinates of the geometric center of the manipulator are denoted by ð x y z Þ. (2) The total number of the symmetric kinematic chains of the spatial parallel manipulator are indicated by n (n P 2). (3) The manipulators ith (i = 1, 2, . . . , n) vertex linked by theith collinear kinematic chain is denoted by Mi, and its local coordinates are denoted by xLM i y LM i zLM i , where the superscript L represents ‘‘local’’. (4) The bases ith (i = 1, 2, . . . , n) vertexA linked by the ith kinematic chain are denoted by Bi, its absolute y Bi zA coordinates are denoted by xA Bi Bi , where the superscript A indicates ‘‘absolute’’. As a result, the absolute coordinates of the ith vertex, Mi, can be obtained L rA M i ¼ r þ RrM i
ð1Þ
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635
where R is a transformation matrix from the local coordinate system to the absolute one, T A A A T x y z rA ¼ are the absolute coordinates Mi, r ¼ ð x y z Þ are the absolute coordinates of Mi Mi Mi Mi of T L L L L the geometry center of the manipulator, and rM i ¼ xM i y M i zM i are the local coordinates of Mi. Because each kinematic chain of the manipulator always keeps collinear, the constraint conditions of the kinematic chain will reduce to be one distance constraint equation no matter what kind of actuation of the chain is 2 A krA M i rBi k li ¼ 0
ð2Þ
where li is the length of the ith (i = 1, 2, . . . , n) kinematic chain, kÆk denotes the 2-norm of ‘‘Æ’’. Now, we will split the problem into three cases to demonstrate this theorem, namely, planar symmetric case, rotational axisymmetric case and centrosymmetric case. 2.1. The identical kinematic chains of a spatial parallel manipulator are symmetric about a certain plane Because the identical kinematic chains of a spatial parallel manipulator are symmetric about a certain plane, we can assume the xoz plane of the absolute coordinate system is superimposed with the symmetric plane and the y-axis will be naturally perpendicular to the plane according to right hand rule which can be shown in Fig. 2. Similarly, we also presume the xLoLzL plane of the local coordinate system is superimposed with the symmetric plane of the manipulator. Because the identical kinematic chains of a spatial parallel manipulator are symmetric about xoz plane, the symmetric plane xoz must pass through one kinematic chain if n is an odd number; and there must be two sets of equivalent number of kinematic chains if n is an even number. If n is an odd number and the sequence numbers of the joints are properly presumed relative to the absolute coordinate system (the nþ1 th kinematic chain is superimposed with xoz-plane) such that there exist 2 8 A A A A A xB1 ¼ xA > Bn ; y B1 ¼ y Bn ; zB1 ¼ zBn > > > > > A A A A A A > > > xB2 ¼ xBðn1Þ ; y B2 ¼ y Bðn1Þ ; zB2 ¼ zBðn1Þ > > > > > .. > > > <. ð3Þ A A A A A > xA Bj ¼ xBðnþ1jÞ ; y Bj ¼ y Bðnþ1jÞ ; zBj ¼ zBðnþ1jÞ > > > > > > > .. > > . > > > > > > A > : y B nþ1 ¼ 0 ð2Þ where j ¼ 1; 2; . . . ; n1 . 2 Similarly, in the local coordinate system, there will also be 8 xLM 1 ¼ xLM n ; y LM 1 ¼ y LM n ; zLM 1 ¼ zLM n > > > > > > L L L L L L > > > xM 2 ¼ xM ðn1Þ ; y M 2 ¼ y M ðn1Þ ; zM 2 ¼ zM ðn1Þ > > > > > .. > > > <. xLM j ¼ xLM ðnþ1jÞ ; y LM j ¼ y LM ðnþ1jÞ ; zLM j ¼ zLM ðnþ1jÞ > > > > > > > > .. > > . > > > > > > yL > : M nþ1 ¼ 0 ð2Þ
ð4Þ
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If n is an even number and the sequence numbers of the joints are also properly presumed relative to the coordinate system such that in the absolute coordinate system there exist 8 A A A A A xB1 ¼ xA > Bn ; y B1 ¼ y Bn ; zB1 ¼ zBn > > > A A A A A > xA > B2 ¼ xBðn1Þ ; y B2 ¼ y Bðn1Þ ; zB2 ¼ zBðn1Þ > > > > > > > ... < ð5Þ A A A A A xA > Bj ¼ xBðnþ1jÞ ; y Bj ¼ y Bðnþ1jÞ ; zBj ¼ zBðnþ1jÞ > > > > > .. > > . > > > > A A A > ; yA ; zA : xA B n ¼ xB n B n ¼ y B n B n ¼ zB n ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ Similarly, in the local coordinate system, there will also be 8 L xM ¼ xLM n ; y LM 1 ¼ y LM n ; zLM 1 ¼ zLM n > > > L1 > > xM 2 ¼ xLM ðn1Þ ; y LM 2 ¼ y LM ðn1Þ ; zLM 2 ¼ zLM ðn1Þ > > > > > > . > > < ..
ð6Þ
xLM j ¼ xLM ðnþ1jÞ ; y LM j ¼ y LM ðnþ1jÞ ; zLM j ¼ zLM ðnþ1jÞ > > > > > > .. > > . > > > > > : xLM n ¼ xLM n ; y LM n ¼ y LM n ; zLM n ¼ zLM n ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ ð2Þ ð2þ1Þ According to Eq. (1), there will be 2 A 3 2 2 L 3 3 xM j xM j x 6 A 7 6 6 L 7 7 6 y M 7 ¼ 4 y 5 þ R6 y M 7 j 5 j 5 4 4 zA Mj
z
ð7Þ
zLM j
Following, we only prove the case when n is an even number. With Eqs. (6), (7) can be transformed to be 2 A 3 2 2 L 3 3 xM ðnþ1j Þ xM j x 6 A 7 6 6 L 7 6 y M 7 ¼ 4 y 7 7 ð8Þ 5 þ R6 j 5 4 4 y M ðnþ1jÞ 5 z zA zLM ðnþ1jÞ Mj Define
2
A xA M 1 xB1
2
2 2 A A þ yA þ zA l21 M 1 y B1 M 1 zB 1
3
7 6 7 6 7 6 .. 7 6 7 6 . 7 6 !2 !2 !2 7 6 7 6 2 7 6 A A A A A A xM n xB n þ yM n yB n þ zM n zB n lðnÞ 7 6 2 7 6 ð 2Þ ð 2Þ ð 2Þ ð2Þ ð2Þ ð 2Þ 7 F ðx; y; zÞ ¼ 6 7 6 !2 !2 !2 7 6 7 6 A 2 A A A A A 7 6 xM x y z þ y þ z l n B n M n B n M n B n þ1 7 6 ð Þ nþ1 2 ð2 Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ 7 6 7 6 7 6 .. 7 6 7 6 . 5 4 A 2 2 A A 2 A A 2 xM n xA þ y y þ z z l Bn Mn Bn Mn Bi n
ð9Þ
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
Then, from Eq. (2), Eq. (9) can also be noted as: 3 2 2 A krA M 1 rB1 k l1 7 6 7 6 .. 7 6 7 6 . 7 6 7 6 7 6 A 2 A 7 6 rM rB l n ð n 7 6 ðn2Þ 2Þ ð Þ 2 7 6 7 F ðx; y; zÞ ¼ 6 7 6 7 6 A 2 A 7 6 rM l r nþ1 7 B n 6 ðnþ1Þ ð2 Þ ð2þ1Þ 2 7 6 7 6 7 6 .. 7 6 7 6 . 5 4 2 A krA r k l n Mn Bn
637
ð10Þ
If the lengths of the kinematic chains are changeable from the minimum value, lmin i , to the maximum value, max during motion, the lengths of li and ln+1i must satisfy that li ; lnþ1i 2 ½lmin ; l ; i ¼ 1; 2; . . . ; n2. Considi i min max ering the symmetric structure of the manipulator system, for any li 2 ½li ; li ; i ¼ 1; 2; . . . ; n2, we can find one max ln+1i such that lnþ1i 2 ½lmin and li = ln+1i. If the length of each chain is fixed, li = ln+1i of course holds. i ; li Hence, in order to simplify the discussion, we only utilize the symmetric condition
lmax , i
li ¼ lnþ1i
ð11Þ
According to Eq. (2), the reachable workspace of the manipulator should satisfy F ðx; y; zÞ ¼ 0 With Eqs. (5), (6) and (11), we can find that 3 2 02 2 L 31 3 xM 1 x 7 6 B6 6 7C 7 7 6 B6 y 7 þ R6 y LM 7C rA l2 7 6 1 B @4 4 1 5A 5 1 7 6 7 6 L z zM 1 7 6 7 6 7 6 7 6 .. 7 6 . 7 6 0 2 L 31 7 6 7 6 2 x M 3 n 7 6 B 6 ð 2Þ 7 C 7 6 B x 7C 6 7 6 B6 6 y L 7C 7 7 6 B6 2 A C 7 6 7 6 B4 y 5 þ R6 M ðnÞ 7C rB n lðnÞ 7 2 7 6 B 2 ð Þ C 7 6 2 7 6 @ 5A 4 L 7 6 z zM n 7 6 7 6 ð 2Þ 7 6 F ðx; y; zÞ ¼6 0 31 2 L 7 7 6 x M n 2 3 7 6 B x 6 ð2þ1Þ 7C 7 6 B 7C 6 7 6 B6 7C 7 6 yL 6 l2 n 7 B6 y 7 þ R6 M n 7C rA 7 6 B n B4 5 6 ð2þ1Þ 7C 6 ð 2Þ 7 ð2þ1Þ B 7C 6 7 6 @ z 5A 4 L 7 6 zM n 7 6 7 6 þ1 ð 2 Þ 7 6 7 6 7 6 . .. 7 6 7 6 02 3 2 L 31 7 6 7 6 x x Mn 7 6 B6 7 6 L 7C 7 6 B6 y 7 þ R6 y 7C rA l2 7 6 1 Bn @4 5 4 M n 5A 5 4 zLM n z
ð12Þ
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2 3 2 L 3 3 x xM n 6 7 6 6 7 7 2 4 y 5 þ R4 y LM n 5 rA 7 6 B n ln 7 6 L z 7 6 z Mn 7 6 7 6 .. 7 6 7 6 . 3 2 L 7 6 7 6 x M n 3 7 6 2 ð2þ1Þ 7 6 ! 7 6 x 7 6 L 7 6 7 7 6 6 y 7 þ R6 y M 2 A l r 6 5 6 4 nþ1 7 n B n ð2 Þ 7 ð2þ1Þ 7 6 þ1 7 6 ð 2 Þ 5 4 L 7 6 z 7 6 zM n 7 6 þ1 ð2 Þ 7 6 3 2 L ¼6 7 7 6 2 x M 3 n 7 6 ð 2Þ 7 6 ! 7 6 x 7 6 7 6 6 L 6 y M 7 2 A 6 4 y 7 rB n lðnÞ 7 5 þ R6 n 7 7 6 ð Þ 2 7 2 6 ð2Þ 7 6 5 4 L 7 6 z zM n 7 6 7 6 ð 2Þ 7 6 7 6 .. 7 6 . 7 6 2 3 2 L 3 7 6 x xM 1 7 6 7 6 6 7 6 y L 7 2 7 6 A 4 y 5 þ R4 M 1 5 ðrB1 Þ l1 5 4 L z zM 1 2
and therefore, we can obtain the following equation with (9): 3 2 A 2 A 2 A 2 xM n xA þ ð1Þ2 y A þ zA l2n Bn M n y Bn M n zBi 7 6 7 6 .. 7 6 7 6 . 7 6 !2 !2 !2 7 6 7 6 2 6 xA 2 A yA zA þ ð1Þ y A þ zA lðnþ1Þ 7 7 6 M n xB n M n B n M n B n 2 7 6 ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ 7 6 7 6 F ðx; y; zÞ ¼ 6 !2 !2 !2 7 7 6 2 2 A A A A A A 7 6 x y z x þ ð1Þ y þ z l n M n B n M n B n M n B n 7 6 ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ 7 6 7 6 7 6 7 6 .. 7 6 . 7 6 5 4 2 2 2 2 2 A A A A A A xM 1 xB1 þ ð1Þ y M 1 y B1 þ zM 1 zB1 l1 ð13Þ Consequently, from Eqs. (9) and (13), we can obtain F ðx; y; zÞ ¼ F ðx; y; zÞ
ð14Þ
Similarly, it is not difficult to prove that Eq. (14) is also valid for the case when n is an odd number. Therefore, the workspace is also symmetric about the symmetric plane of its kinematic chains—xoz-plane. 2.2. The identical kinematic chains of a spatial parallel manipulator are rotational symmetric about a certain axis Because the identical kinematic chains of a spatial parallel manipulator are rotational symmetric about an axis, we can assume the z-axis of the absolute coordinate system is superimposed with the symmetric axis shown in Fig. 2. Similarly, we also presume the zL-axis of the local coordinate system is superimposed with the symmetric axis of the manipulator.
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
Define a function T(h)(x, y, z) such that it satisfies that 2 32 3 cos h sin h 0 x 6 76 7 T ðhÞ ðx; y; zÞ ¼ 4 sin h cos h 0 54 y 5 0
0
1
639
ð15Þ
z
where T(h)(x, y, z) denotes the coordinates transformation from the new one to the old one when the fixed coordinate frame rotates an angle of h around its z-axis. If the n kinematic chains of the manipulator are rotational symmetric about z-axis, we can find that 2 32 3 x cos 2mp sin 2mp 0 n n 6 76 7 2mp T ðm.2pÞ ðx; y; zÞ ¼ 4 sin 2mp y ð16Þ cos 0 54 5 n n n z 0 0 1 where m = 1, 2, . . . , n. Because the identical kinematic chains of a spatial parallel manipulator are rotational symmetric about zaxis of the absolute coordinate system, the fixed joints must also be rotational symmetric about z-axis. And therefore the following relationships hold: yA zA yA zA ; i ¼ 1; 2; . . . ; n T ðm.2pÞ xA 2 B i xA ð17Þ Bj Bj Bj B B B i i i n yA zA where j = 1, 2, . . . , n, and Bi xA denote the absolute coordinates of point Bi, {Æ} denotes a set genBi Bi Bi erated by the element ‘‘Æ’’. Similarly, in the local coordinate system, there exist T ðm.2pÞ xLM j y LM j zLM j 2 M i xLM i y LM i zLM i ; i ¼ 1; 2; . . . ; n ð18Þ n L where M i xBi y LBi zLBi are the local coordinates of point Mi. As a result, F T ðm.2pÞ ð x y z Þ ¼ F ð x y z Þ ð19Þ n Consequently, the workspace is rotational symmetric about the same axis, z-axis. 2.3. The identical kinematic chains of a spatial parallel manipulator are centrosymmetric about a certain point Because the identical kinematic chains of a spatial parallel manipulator are centrosymmetric about one point, we can assume the origin of the absolute coordinate system is superimposed with the symmetric center. Define a function S(x, y, z) such that it satisfies 2 3 x 6 7 Sðx; y; zÞ ¼ 4 y 5 ð20Þ z Because the n kinematic chains of the manipulator are symmetric about the origin, according to Eq. (20) there must be 8 A A > y z < S xA yA zA 2 Bi xA Bj Bj Bj Bi Bi Bi ; i ¼ 1; 2; . . . ; n ð21Þ > : S xA yA zA yA zA 2 M i xA Mj Mj Mj Mi Mi M i ; i ¼ 1; 2; . . . ; n yA zA where Bi xA are the local coordinates of point Bi, M i xLBi y LBi zLBi are the local coordinates of Bi Bi Bi point Mi, {Æ} denotes a set generated by the element ‘‘Æ’’. Therefore, F ðSðx; y; zÞÞ ¼ F ðx; y; zÞ
ð22Þ
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Consequently, the workspace is centrosymmetric about the same point—the origin of the absolute coordinate system oxyz. From Eqs. (14), (19) and (22), we can find that the theorem is proved. 3. Examples and discussions We will present two kinds of parallel manipulators to demonstrate the applications of the theorem and illustrate the utilities in conceptual design and engineering estimations. 3.1. The workspace of a spatial 3-PUU parallel manipulator A spatial parallel 3-PUU (prismatic joint, universal joint, universal joint) parallel manipulator is shown in Fig. 1. The three identical kinematic chains of the manipulator are all made up of one prismatic joint and two universal joints in series. Because the manipulator plane M1M2M3 and the base plane B1B2B3 are all equilateral triangles, the manipulator has one rotational symmetric axis, denoted by e, and three symmetric planes, denoted by a1, a2 and a3 individually, which are shown in Fig. 2. Therefore, the three kinematic chains can be called both rotational symmetric and planar symmetric. Assume z-axis is superimposed with the rotational symmetric axis of the fixed base plane, and x-axis passes through prismatic joint B2. Similarly, presume zL-axis is superimposed with the symmetric axis of the moving manipulator plane, and xL-axis passes through M2 joint. According to [13,14], the DoF of the manipulator is three, which includes three translational movements along the three orthogonal directions at the ordinary cases. The constraint equations can be obtained according to Eq. (9): 2
3 2 2 2 þ y þ y LM 1 y A þ ðz þ zLM 1 zA B1 B1 Þ l 7 7 2 2 2 7 27 L A L A xA þ y þ y y þ z þ z z l 7 B2 M2 B2 M2 B2 7 5 2 2 2 2 A L A L A xB3 þ y þ y M 3 y B3 þ z þ zM 3 zB3 l
x þ xLM 1 xA B1
6 6 6 L F ðx; y; zÞ ¼ 6 6 x þ xM 2 6 4 x þ xLM 3
2
ð23Þ
where 2
xLM i
3
2
cos 2p ði 2Þ 3
3
6 7 6 L 7 6 y M 7 ¼ r6 sin 2p ði 2Þ 7; 4 5 4 i5 3 L zM i 0
i ¼ 1; 2; 3
ð24Þ
r is the radius of the circumcircle of equilateral triangle M1M2M3 2
xA Bi
3
2
cos 2p ði 2Þ 3
3
6 7 6 A7 6 y B 7 ¼ Ri 6 sin 2p ði 2Þ 7; 4 5 4 i5 3 A 0 zBi
i ¼ 1; 2; 3
ð25Þ
Ri, i = 1, 2, 3 are the displacement of the ith actuator, slider Bi, l1 6 Ri 6 l2. According to Eq. (16), n = 3. Let m = 1, there will be 2
cos
2p 3
6 2p T ð2pÞ ðx; y; zÞ ¼ 6 4 sin 3 3
0
sin 2p 3 2p cos 3 0
pffiffi 3 32 3 2 1 x 2 x 23 y 76 7 6 pffiffi 7 6 7 6 3 7 07 54 y 5 ¼ 4 2 x 12 y 5 z 1 z
0
ð26Þ
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
Therefore 8 L L L L > T ; y ; z x 2p > ð 3 Þ M 3 M 3 B3 ¼ rM 2 > > < T ð2pÞ xLM 2 ; y LM 2 ; zLM 2 ¼ T ð22pÞ xLM 3 ; y LM 3 ; zLM 3 ¼ rLM 1 3 3 > > > > : T 2p xL ; y L ; zL ¼ T 2p xL ; y L ; zL ¼ T 2p xL ; y L ; zL ¼ rL M3 ð 3 Þ M1 M1 M1 ð2 3 Þ M 2 M 2 M 2 ð3 3 Þ M 3 M 3 M 3
641
ð27Þ
where rLM i ; i ¼ 1; 2; 3 are the vector coordinates of Mi in the local coordinate system. 3 2 2 pffiffi 2 2 pffiffi 12 x 23 y þ xLM 1 xLB1 þ 23 x 12 y þ y LM 1 y LB1 þ z þ zLM 1 zLB1 l2 7 6 7 6 2 pffiffi 2 2 pffiffi 6 27 3 1 L L L L F T ð2pÞ ðx; y; zÞ ¼ 6 12 x 23 y þ xLM xLB 7 þ x y þ y y þ z þ z z l M1 B1 M1 B1 2 2 1 1 3 7 6 5 4 pffiffi pffiffi 2 2 2 12 x 23 y þ xLM 1 xLB1 þ 23 x 12 y þ y LM 1 y LB1 þ z þ zLM 1 zLB1 l2 ð28Þ Eq. (28) can be simplified as 3 2 h pffiffi i 2 3 2 1 L L L L L L x þ 2x 2 xM 1 xB1 þ 2 y M 1 y B1 þ xM 1 xB1 7 6 6 h pffiffi i 2 7 7 6 6 þy 2 þ 2y 3 xL xL 1 y L y L þ y LM 1 y LB1 7 M1 B1 M1 B1 7 6 2 2 7 6 7 6 2 7 6 2 L L 7 6 þ z þ zM 1 zB1 l 7 6 7 6 h i pffiffi 2 7 6 2 L L 7 6 x þ 2x 12 xLM xLB þ 23 y LM y LB x þ x M2 B2 2 2 2 2 7 6 6 h pffiffi i 2 7 7 6 3 1 2 L L L L L L 7 F T ð2pÞ ðx; y; zÞ ¼ 6 6 þy þ 2y 2 xM 2 xB2 2 y M 2 y B2 þ y M 2 y B2 7 3 7 6 7 6 2 7 6 7 6 þ z þ zLM 2 zLB2 l2 7 6 7 6 h i pffiffi 2 7 6 2 L L 7 6 x þ 2x 1 xLM xLB þ 3 y LM y LB x þ x M3 B3 2 2 3 3 3 3 7 6 6 h pffiffi i 2 7 7 6 6 þy 2 þ 2y 3 xL xL 1 y L y L þ y LM 3 y LB3 7 7 6 M3 B3 M3 B3 2 2 7 6 5 4 2 2 L L þ z þ zM 3 zB3 l Therefore, considering Ri, i = 1, 2, 3 have the same variable ranges, we can obtain from Eq. (29): 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 3 xA B3 M3 B3 M3 B3 7 6 7 6 2 2 2 6 27 L A L A F T ð2pÞ ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B1 M1 B1 M1 B1 1 3 7 6 5 4 2 2 2 x þ xLM 2 xA þ y þ y LM 2 y A þ z þ zLM 2 zA l2 B2 B2 B2 Similarly, we can deduce: 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 2 xA B M B M B 2 2 2 2 2 7 6 7 6 2 2 2 6 27 L A L A F T ð4pÞ ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B M B M B 3 3 3 3 3 3 3 7 6 5 4 2 2 2 2 L A L A L A x þ xM 1 xB1 þ y þ y M 1 y B1 þ z þ zM 1 zB1 l
ð29Þ
ð30Þ
ð31Þ
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Fig. 4. The reachable workspace of the manipulator.
Therefore, from Eqs. (23), (27), (30) and (31), we can find that F T ¼ F T ¼ 4p ðx; y; zÞ 2p ðx; y; zÞ ð3Þ ð3Þ F T ð6pÞ ðx; y; zÞ ¼ F ðx; y; zÞ. The reachable workspace of the manipulator can be obtained from Eq. (12). 3 The workspace of the manipulator is shown in Fig. 4 if we assign r = 300 mm, l1 = 300 mm, l2 = 1900 mm and l = 1600 mm. From Fig. 1, we can also find that the manipulator is also planar symmetric and the symmetric planes are a1, a2, and a3 shown in Fig. 2. If we consider the symmetric plane a2, which is superimposed by xoz, according to Eq. (13), we can gain 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 1 xA B M B M B 1 1 1 1 1 7 6 7 6 2 2 2 6 27 L A L A ð32Þ F ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B M B M B 2 2 2 2 2 2 7 6 5 4 2 2 2 x þ xLM 3 xA þ y þ y LM 3 y A þ z þ zLM 3 zA l2 B3 B3 B3 In the case shown in Fig. 2, there will be ( L y M 2 ¼ y LM 2 ¼ 0 y LM 1 ¼ y LM 3 and
(
A yA B2 ¼ y B2 ¼ 0 A yA B1 ¼ y B3
From Eqs. (32)–(34), there exist 3 2 2 h i2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 3 xA B3 M3 B3 M3 B3 7 6 7 6 h i 2 2 2 6 27 L A L A L A F ðx; y; zÞ ¼ 6 x þ xM xB 7 þ y þ y y þ z þ z z l M2 B2 M2 B2 2 2 7 6 5 4 2 h i2 2 2 L A L A L A x þ xM 1 xB1 þ y þ y M 1 y B1 þ z þ zM 1 zB 1 l 3 2 2 2 2 2 L A L A x þ xLM 3 xA þ y þ y y þ z þ z z l B3 M3 B3 M3 B3 7 6 7 6 2 2 2 6 27 L A L A L A ¼ 6 x þ xM xB 7 þ y þ y y þ z þ z z l M2 B2 M2 B2 2 2 7 6 5 4 2 2 2 2 L A L A x þ xLM 1 xA þ y þ y y þ z þ z z l B1 M1 B1 M1 B1
ð33Þ
ð34Þ
ð35Þ
From Eqs. (23) and (35), we can find that F(x, y, z) = F(x, y, z). If we assign r = 300 mm, l1 = 600 mm, l2 = 1900 mm and l = 1600 mm, the workspace of the manipulator is obtained as Fig. 5 shows. We can find that the workspace has the same corresponding symmetries as the manipulator has. It is symmetric about all the planes a1, a2, and a3 and rotational symmetric about the directional axis e (z-axis), which are shown in
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Fig. 5. The reachable workspace of the manipulator.
Fig. 6. The symmetric workspace corresponding to the structure of the manipulator.
Fig. 6. As a result, the reachable workspace of the manipulator shown in Fig. 1 has the same three symmetric planes and one symmetric axis as the manipulator has. 3.2. The workspaces of planar 4-bar parallel manipulators We now analyze the workspaces of the planar 4-bar parallel manipulators shown in Figs. 3 and 7. According to the theorem, the workspace should have the same symmetric characteristics as those of the manipulators kinematic chains. Given an axis passing through the middle point of line AD and perpendicular to line AD, we can find a configuration such that the two kinematic chains of the manipulator (BC) shown in Fig. 7 are symmetric about the axis. As a result, the reachable workspace (the possible locus) of the manipulator should be symmetric about this axis, too. As is well known, the reachable workspace of the parallel 4-bar manipulator is a circle, whose center is the middle point of line AD and the radius is the length of the bar AB. Just as we have pointed out that the converse theorem does not hold, though the workspace of the manipulator is symmetric about any line passing through the middle point of line AD, we cannot safeguard that the two kinematic chains of the manipulator (BC) are also symmetric about this line. Similarly, because the inverse parallel 4-bar manipulator shown in Fig. 3 is symmetric about the middle point of line AD, the reachable workspace should also be symmetric about the same point according to the theorem. The workspace (the possible locus) of this manipulator is shown in Fig. 8 if we set the lengths of AB and CD to be 200 mm, the lengths of link BC and AD to be 600 mm. Many more such examples can be found in a number of recently published literatures [15–19].
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Fig. 7. A planar parallel 4-bar manipulator.
Fig. 8. The workspace of the inverse parallel 4-bar manipulator.
4. Conclusion This paper proposes an analogous symmetric theorem of workspace for spatial parallel manipulators with symmetric identical kinematic chains. Through a strict mathematic proof, it reveals the analogous relationship between the workspace shape and the structure of the manipulator. Examples demonstrate that this theorem can be utilized to estimate the geometry characteristics of the workspace and to guide the conceptual design of spatial parallel manipulators, especially for those with full DoFs. The theorem offers a concrete safeguard for the workspace analysis of complicated spatial parallel manipulators. Acknowledgement This work was supported by China postdoctoral foundations. The authors would like to thank Professor Andre´s Kecskeme´thy and the anonymous refereeing scientists for their wise suggestions. References [1] D. Sen, T.S. Mruthyunjaya, A centro-based characterization of singularities in the workspace of planar closed-loop manipulators, Mechanism and Machine Theory 33 (8) (1998) 1091–1104. [2] S. Dibakar, T.S. Mruthyunjaya, A computational geometry approach for determination of boundary of workspaces of planar manipulators with arbitrary topology, Mechanism and Machine Theory 34 (1) (1999) 149–169. [3] J.K. Davidson, K.D. Chaney, A design procedure for RPR planar robotic workcells: an algebraic approach, Mechanism and Machine Theory 34 (2) (1999) 193–203. [4] M. Ceccarelli, A formulation for the workspace boundary of general N-revolute manipulators, Mechanism and Machine Theory 31 (5) (1996) 637–646. [5] M. Ceccarelli, C. Lanni, A multi-objective optimum design of general 3R manipulators for prescribed workspace limits, Mechanism and Machine Theory 39 (2) (2004) 119–132.
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[6] I.A. Bonev, J. Ryu, A geometrical method for computing the constant-orientation workspace of 6-PRRS parallel manipulators, Mechanism and Machine Theory 36 (1) (2001) 1–13. [7] C.M. Gosselin, M. Guillot, The synthesis of manipulators with prescribed workspace, Journal of Mechanical Design 113 (December) (1991) 451–455. [8] Z. Wang, Z. Wang, W. Liu, Y. Lei, A study on workspace, boundary workspace analysis and workpiece positioning for parallel machine tools, Mechanism and Machine Theory 36 (5) (2001) 605–622. [9] S. Bhattacharya, H. Hatwal, A. Ghosh, Comparison of an exact and an approximate method of singularity avoidance in platform type parallel manipulators, Mechanism and Machine Theory 33 (7) (1998) 965–974. [10] J.S. Dai, P. Shah, Orientation capability of planar manipulators using virtual joint angle analysis, Mechanism and Machine Theory 38 (3) (2003) 241–252. [11] I.A. Bonev, J. Ryu, A new approach to orientation workspace analysis of 6-DOF parallel manipulators, Mechanism and Machine Theory 36 (1) (2001) 15–28. [12] A. Kumar, K.J. Waldron, The workspaces of a mechanical manipulator, Journal of Mechanical Design 103 (July) (1981) 665–672. [13] J.S. Zhao, K. Zhou, et al., A new method to study the degree of freedom of spatial parallel mechanisms, The International Journal of Advanced Manufacturing Technology 23 (3–4) (2004) 288–294. [14] J.S. Zhao, K. Zhou, Z.J. Feng, A theory of degrees of freedom for mechanisms, Mechanism and Machine Theory 39 (6) (2004) 621– 643. [15] F. Xi, D. Zhang, C.M. Mechefske, S.Y.T. Lang, Global kinetostatic modelling of tripod-based parallel kinematic machine, Mechanism and Machine Theory 39 (4) (2004) 357–377. [16] J. Jesu´s Cervantes-Sa´nchez, J. Ce´sar Herna´ndez-Rodrı´guez, J. Angeles, On the kinematic design of the 5R planar, symmetric manipulator, Mechanism and Machine Theory 36 (11–12) (2001) 1301–1313. [17] F. Gao, X.J. Liu, X. Chen, The relationships between the shapes of the workspaces and the link lengths of 3-DoF symmetrical planar parallel manipulators, Mechanism and Machine Theory 36 (2) (2001) 205–220. [18] J. Jesu´s Cervantes-Sa´nchez, J. Cesar Herna´ndez-Rodrı´uez, J. Gabriel Rendo´n-Sa´nchez, On the workspace, assembly configurations and singularity curves of the RRRRR-type planar manipulator, Mechanism and Machine Theory 35 (8) (2000) 1117–1139. [19] J.J. Cervantes-Sa´nchez, J.G. Rendo´n-Sa´nchez, A simplified approach for obtaining the workspace of a class of 2-DoF planar parallel manipulators, Mechanism and Machine Theory 34 (7) (1999) 1057–1073.
Mechanism and Machine Theory www.elsevier.com/locate/mechmt
Workspace of parallel manipulators with symmetric identical kinematic chains Jing-Shan Zhao
a,*
, Min Chen b, Kai Zhou a, Jing-Xin Dong a, Zhi-Jing Feng
a
a
b
Department of Precision Instruments, Tsinghua University, Beijing 100084, PR China School of Engineering and Technology, China University of Geosciences, Beijing 100083, PR China
Received 15 December 2004; received in revised form 5 September 2005; accepted 14 September 2005 Available online 8 November 2005
Abstract In this paper, we propound an analogous symmetric theorem of workspace for spatial parallel manipulators with identical kinematic chains, and then present a strict mathematic proof by splitting the problem into planar symmetry, rotational axis symmetry and point symmetry (centrosymmetry). Examples demonstrate that this theorem can be utilized to investigate the geometry shapes of the workspace and to guide the conceptual design of spatial parallel manipulators, especially for those complicated spatial parallel manipulators with full degrees of freedom (DoF). The theorem offers a concrete safeguard for the workspace analysis of any symmetric spatial parallel manipulators. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Workspace; Theorem; Spatial parallel manipulator; Kinematic chain
1. Introduction Since workspace determination is generally an intermediate but critical step in analyzing and synthesizing manipulators, it is very important to have a theory to safeguard the estimation and conceptual design. The workspace of a manipulator is the domain of reach of its end-effector and is bounded in the 3D space [1]. The workspace of a manipulator has been defined in literature as the totality of positions that a particular identified point of the manipulator (end-effector) can reach [2,3]. The workspace boundary is the curve (in plane) or the surface (in space) that defines the extent of reach of the end-effector [2,4]. Many scholars have proposed algebraic method [4,5] and geometric method [6,7] to determine the workspace of a manipulator. For determination of different characteristics of the workspace, in order to compare different existing manipulators or design a new manipulator, it is almost always necessary to determine the boundaries of the workspaces [2]. Wang et al. [8] presented an algorithm, called the boundary search method,
*
Corresponding author. Fax: +86 01 06 278 2351. E-mail address: [email protected] (J.-S. Zhao).
0094-114X/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.09.007
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633
to determine the workspace of a parallel machine tool. They also analyzed the boundary workspace in order to expand its operating scope. Bhattacharya et al. [9] found that the singular manifolds of a platform type parallel manipulator always lie at the boundary separating two open diffeomorphic regions (branch modes) in the workspace of the device. Dai and Shah [10] investigated the orientation capability of a serial manipulator by examining the rotatability of its equivalent mechanism and by proposing the workspace region decomposition, based on the polynomial discriminant derived from the virtual angle analysis of the end-effector link of the equivalent mechanism. Bonev and Ryu [11] presented a discretization method for the computation of the orientation workspace of 6-DoF parallel manipulators, defined as the set of all attainable orientations of the mobile platform about a fixed point. The focuses of the above literature are mainly on finding the workspace of a manipulator, but seldom discuss the relationship between the shapes of the workspace and the geometry structure of the manipulator. There exists no theorem for spatial parallel manipulators which characterizes the geometry relationship between the reachable workspace of closed-loop manipulators and the structures of their kinematic chains. Therefore, the present study characterizes the workspace of spatial parallel manipulators with symmetric identical kinematic chains. We first establish a symmetric theorem for spatial parallel manipulators and then identify the structural symmetry with three cases: planar symmetry, rotational axis symmetry and point symmetry (centrosymmetry). For the sake of clarity, we also make some necessary assumptions and present a strict mathematic proof. 2. Analogous symmetric theorem of workspace for spatial parallel manipulators In this paper, the reachable workspace of a manipulator is defined as the volume within which the reference point, say, the geometry center of the manipulator, can be reached [12]. In applications, most spatial parallel manipulators have symmetric identical kinematic chains. These chains are symmetric about a certain plane, about one axis or about one specified point. For example, the manipulator shown in Fig. 1 has three identical kinematic chains, each one of which consists of two universal joints and one prismatic joint in series. The structural symmetric characteristics can be found in Fig. 2, where a1, a2 and a3 are the three symmetric planes and e is a symmetric axis that is intersected by these three symmetric planes. As to the point symmetry, we can find that the middle point of AD is the symmetric center of the planar inverse parallel 4-bar manipulator shown in Fig. 3.
Fig. 1. A spatial parallel manipulator made up of 3-PUU kinematic chains.
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Fig. 2. The structural symmetric of the spatial parallel manipulator.
Fig. 3. A planar inverse parallel 4-bar manipulator.
In order to exploit the relationship between the workspace shapes and the structural characteristics of the kinematic chains, we firstly propose a symmetric workspace theorem for spatial parallel manipulator: If each identical kinematic chain of a spatial parallel manipulator always keeps collinear, and they are structural symmetric about a plane, an axis, or a point; the reachable workspace of the manipulator will also be symmetric about the corresponding plane, axis, or point individually. However the converse theorem does not hold. Prior to demonstrating the validation of the theorem, we firstly make the following assumptions: (1) The absolute coordinates of the geometric center of the manipulator are denoted by ð x y z Þ. (2) The total number of the symmetric kinematic chains of the spatial parallel manipulator are indicated by n (n P 2). (3) The manipulators ith (i = 1, 2, . . . , n) vertex linked by theith collinear kinematic chain is denoted by Mi, and its local coordinates are denoted by xLM i y LM i zLM i , where the superscript L represents ‘‘local’’. (4) The bases ith (i = 1, 2, . . . , n) vertexA linked by the ith kinematic chain are denoted by Bi, its absolute y Bi zA coordinates are denoted by xA Bi Bi , where the superscript A indicates ‘‘absolute’’. As a result, the absolute coordinates of the ith vertex, Mi, can be obtained L rA M i ¼ r þ RrM i
ð1Þ
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635
where R is a transformation matrix from the local coordinate system to the absolute one, T A A A T x y z rA ¼ are the absolute coordinates Mi, r ¼ ð x y z Þ are the absolute coordinates of Mi Mi Mi Mi of T L L L L the geometry center of the manipulator, and rM i ¼ xM i y M i zM i are the local coordinates of Mi. Because each kinematic chain of the manipulator always keeps collinear, the constraint conditions of the kinematic chain will reduce to be one distance constraint equation no matter what kind of actuation of the chain is 2 A krA M i rBi k li ¼ 0
ð2Þ
where li is the length of the ith (i = 1, 2, . . . , n) kinematic chain, kÆk denotes the 2-norm of ‘‘Æ’’. Now, we will split the problem into three cases to demonstrate this theorem, namely, planar symmetric case, rotational axisymmetric case and centrosymmetric case. 2.1. The identical kinematic chains of a spatial parallel manipulator are symmetric about a certain plane Because the identical kinematic chains of a spatial parallel manipulator are symmetric about a certain plane, we can assume the xoz plane of the absolute coordinate system is superimposed with the symmetric plane and the y-axis will be naturally perpendicular to the plane according to right hand rule which can be shown in Fig. 2. Similarly, we also presume the xLoLzL plane of the local coordinate system is superimposed with the symmetric plane of the manipulator. Because the identical kinematic chains of a spatial parallel manipulator are symmetric about xoz plane, the symmetric plane xoz must pass through one kinematic chain if n is an odd number; and there must be two sets of equivalent number of kinematic chains if n is an even number. If n is an odd number and the sequence numbers of the joints are properly presumed relative to the absolute coordinate system (the nþ1 th kinematic chain is superimposed with xoz-plane) such that there exist 2 8 A A A A A xB1 ¼ xA > Bn ; y B1 ¼ y Bn ; zB1 ¼ zBn > > > > > A A A A A A > > > xB2 ¼ xBðn1Þ ; y B2 ¼ y Bðn1Þ ; zB2 ¼ zBðn1Þ > > > > > .. > > > <. ð3Þ A A A A A > xA Bj ¼ xBðnþ1jÞ ; y Bj ¼ y Bðnþ1jÞ ; zBj ¼ zBðnþ1jÞ > > > > > > > .. > > . > > > > > > A > : y B nþ1 ¼ 0 ð2Þ where j ¼ 1; 2; . . . ; n1 . 2 Similarly, in the local coordinate system, there will also be 8 xLM 1 ¼ xLM n ; y LM 1 ¼ y LM n ; zLM 1 ¼ zLM n > > > > > > L L L L L L > > > xM 2 ¼ xM ðn1Þ ; y M 2 ¼ y M ðn1Þ ; zM 2 ¼ zM ðn1Þ > > > > > .. > > > <. xLM j ¼ xLM ðnþ1jÞ ; y LM j ¼ y LM ðnþ1jÞ ; zLM j ¼ zLM ðnþ1jÞ > > > > > > > > .. > > . > > > > > > yL > : M nþ1 ¼ 0 ð2Þ
ð4Þ
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If n is an even number and the sequence numbers of the joints are also properly presumed relative to the coordinate system such that in the absolute coordinate system there exist 8 A A A A A xB1 ¼ xA > Bn ; y B1 ¼ y Bn ; zB1 ¼ zBn > > > A A A A A > xA > B2 ¼ xBðn1Þ ; y B2 ¼ y Bðn1Þ ; zB2 ¼ zBðn1Þ > > > > > > > ... < ð5Þ A A A A A xA > Bj ¼ xBðnþ1jÞ ; y Bj ¼ y Bðnþ1jÞ ; zBj ¼ zBðnþ1jÞ > > > > > .. > > . > > > > A A A > ; yA ; zA : xA B n ¼ xB n B n ¼ y B n B n ¼ zB n ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ Similarly, in the local coordinate system, there will also be 8 L xM ¼ xLM n ; y LM 1 ¼ y LM n ; zLM 1 ¼ zLM n > > > L1 > > xM 2 ¼ xLM ðn1Þ ; y LM 2 ¼ y LM ðn1Þ ; zLM 2 ¼ zLM ðn1Þ > > > > > > . > > < ..
ð6Þ
xLM j ¼ xLM ðnþ1jÞ ; y LM j ¼ y LM ðnþ1jÞ ; zLM j ¼ zLM ðnþ1jÞ > > > > > > .. > > . > > > > > : xLM n ¼ xLM n ; y LM n ¼ y LM n ; zLM n ¼ zLM n ð 2Þ ð2þ1Þ ð 2Þ ð2þ1Þ ð2Þ ð2þ1Þ According to Eq. (1), there will be 2 A 3 2 2 L 3 3 xM j xM j x 6 A 7 6 6 L 7 7 6 y M 7 ¼ 4 y 5 þ R6 y M 7 j 5 j 5 4 4 zA Mj
z
ð7Þ
zLM j
Following, we only prove the case when n is an even number. With Eqs. (6), (7) can be transformed to be 2 A 3 2 2 L 3 3 xM ðnþ1j Þ xM j x 6 A 7 6 6 L 7 6 y M 7 ¼ 4 y 7 7 ð8Þ 5 þ R6 j 5 4 4 y M ðnþ1jÞ 5 z zA zLM ðnþ1jÞ Mj Define
2
A xA M 1 xB1
2
2 2 A A þ yA þ zA l21 M 1 y B1 M 1 zB 1
3
7 6 7 6 7 6 .. 7 6 7 6 . 7 6 !2 !2 !2 7 6 7 6 2 7 6 A A A A A A xM n xB n þ yM n yB n þ zM n zB n lðnÞ 7 6 2 7 6 ð 2Þ ð 2Þ ð 2Þ ð2Þ ð2Þ ð 2Þ 7 F ðx; y; zÞ ¼ 6 7 6 !2 !2 !2 7 6 7 6 A 2 A A A A A 7 6 xM x y z þ y þ z l n B n M n B n M n B n þ1 7 6 ð Þ nþ1 2 ð2 Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ 7 6 7 6 7 6 .. 7 6 7 6 . 5 4 A 2 2 A A 2 A A 2 xM n xA þ y y þ z z l Bn Mn Bn Mn Bi n
ð9Þ
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Then, from Eq. (2), Eq. (9) can also be noted as: 3 2 2 A krA M 1 rB1 k l1 7 6 7 6 .. 7 6 7 6 . 7 6 7 6 7 6 A 2 A 7 6 rM rB l n ð n 7 6 ðn2Þ 2Þ ð Þ 2 7 6 7 F ðx; y; zÞ ¼ 6 7 6 7 6 A 2 A 7 6 rM l r nþ1 7 B n 6 ðnþ1Þ ð2 Þ ð2þ1Þ 2 7 6 7 6 7 6 .. 7 6 7 6 . 5 4 2 A krA r k l n Mn Bn
637
ð10Þ
If the lengths of the kinematic chains are changeable from the minimum value, lmin i , to the maximum value, max during motion, the lengths of li and ln+1i must satisfy that li ; lnþ1i 2 ½lmin ; l ; i ¼ 1; 2; . . . ; n2. Considi i min max ering the symmetric structure of the manipulator system, for any li 2 ½li ; li ; i ¼ 1; 2; . . . ; n2, we can find one max ln+1i such that lnþ1i 2 ½lmin and li = ln+1i. If the length of each chain is fixed, li = ln+1i of course holds. i ; li Hence, in order to simplify the discussion, we only utilize the symmetric condition
lmax , i
li ¼ lnþ1i
ð11Þ
According to Eq. (2), the reachable workspace of the manipulator should satisfy F ðx; y; zÞ ¼ 0 With Eqs. (5), (6) and (11), we can find that 3 2 02 2 L 31 3 xM 1 x 7 6 B6 6 7C 7 7 6 B6 y 7 þ R6 y LM 7C rA l2 7 6 1 B @4 4 1 5A 5 1 7 6 7 6 L z zM 1 7 6 7 6 7 6 7 6 .. 7 6 . 7 6 0 2 L 31 7 6 7 6 2 x M 3 n 7 6 B 6 ð 2Þ 7 C 7 6 B x 7C 6 7 6 B6 6 y L 7C 7 7 6 B6 2 A C 7 6 7 6 B4 y 5 þ R6 M ðnÞ 7C rB n lðnÞ 7 2 7 6 B 2 ð Þ C 7 6 2 7 6 @ 5A 4 L 7 6 z zM n 7 6 7 6 ð 2Þ 7 6 F ðx; y; zÞ ¼6 0 31 2 L 7 7 6 x M n 2 3 7 6 B x 6 ð2þ1Þ 7C 7 6 B 7C 6 7 6 B6 7C 7 6 yL 6 l2 n 7 B6 y 7 þ R6 M n 7C rA 7 6 B n B4 5 6 ð2þ1Þ 7C 6 ð 2Þ 7 ð2þ1Þ B 7C 6 7 6 @ z 5A 4 L 7 6 zM n 7 6 7 6 þ1 ð 2 Þ 7 6 7 6 7 6 . .. 7 6 7 6 02 3 2 L 31 7 6 7 6 x x Mn 7 6 B6 7 6 L 7C 7 6 B6 y 7 þ R6 y 7C rA l2 7 6 1 Bn @4 5 4 M n 5A 5 4 zLM n z
ð12Þ
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2 3 2 L 3 3 x xM n 6 7 6 6 7 7 2 4 y 5 þ R4 y LM n 5 rA 7 6 B n ln 7 6 L z 7 6 z Mn 7 6 7 6 .. 7 6 7 6 . 3 2 L 7 6 7 6 x M n 3 7 6 2 ð2þ1Þ 7 6 ! 7 6 x 7 6 L 7 6 7 7 6 6 y 7 þ R6 y M 2 A l r 6 5 6 4 nþ1 7 n B n ð2 Þ 7 ð2þ1Þ 7 6 þ1 7 6 ð 2 Þ 5 4 L 7 6 z 7 6 zM n 7 6 þ1 ð2 Þ 7 6 3 2 L ¼6 7 7 6 2 x M 3 n 7 6 ð 2Þ 7 6 ! 7 6 x 7 6 7 6 6 L 6 y M 7 2 A 6 4 y 7 rB n lðnÞ 7 5 þ R6 n 7 7 6 ð Þ 2 7 2 6 ð2Þ 7 6 5 4 L 7 6 z zM n 7 6 7 6 ð 2Þ 7 6 7 6 .. 7 6 . 7 6 2 3 2 L 3 7 6 x xM 1 7 6 7 6 6 7 6 y L 7 2 7 6 A 4 y 5 þ R4 M 1 5 ðrB1 Þ l1 5 4 L z zM 1 2
and therefore, we can obtain the following equation with (9): 3 2 A 2 A 2 A 2 xM n xA þ ð1Þ2 y A þ zA l2n Bn M n y Bn M n zBi 7 6 7 6 .. 7 6 7 6 . 7 6 !2 !2 !2 7 6 7 6 2 6 xA 2 A yA zA þ ð1Þ y A þ zA lðnþ1Þ 7 7 6 M n xB n M n B n M n B n 2 7 6 ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ ð2þ1Þ 7 6 7 6 F ðx; y; zÞ ¼ 6 !2 !2 !2 7 7 6 2 2 A A A A A A 7 6 x y z x þ ð1Þ y þ z l n M n B n M n B n M n B n 7 6 ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ ð 2Þ 7 6 7 6 7 6 7 6 .. 7 6 . 7 6 5 4 2 2 2 2 2 A A A A A A xM 1 xB1 þ ð1Þ y M 1 y B1 þ zM 1 zB1 l1 ð13Þ Consequently, from Eqs. (9) and (13), we can obtain F ðx; y; zÞ ¼ F ðx; y; zÞ
ð14Þ
Similarly, it is not difficult to prove that Eq. (14) is also valid for the case when n is an odd number. Therefore, the workspace is also symmetric about the symmetric plane of its kinematic chains—xoz-plane. 2.2. The identical kinematic chains of a spatial parallel manipulator are rotational symmetric about a certain axis Because the identical kinematic chains of a spatial parallel manipulator are rotational symmetric about an axis, we can assume the z-axis of the absolute coordinate system is superimposed with the symmetric axis shown in Fig. 2. Similarly, we also presume the zL-axis of the local coordinate system is superimposed with the symmetric axis of the manipulator.
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
Define a function T(h)(x, y, z) such that it satisfies that 2 32 3 cos h sin h 0 x 6 76 7 T ðhÞ ðx; y; zÞ ¼ 4 sin h cos h 0 54 y 5 0
0
1
639
ð15Þ
z
where T(h)(x, y, z) denotes the coordinates transformation from the new one to the old one when the fixed coordinate frame rotates an angle of h around its z-axis. If the n kinematic chains of the manipulator are rotational symmetric about z-axis, we can find that 2 32 3 x cos 2mp sin 2mp 0 n n 6 76 7 2mp T ðm.2pÞ ðx; y; zÞ ¼ 4 sin 2mp y ð16Þ cos 0 54 5 n n n z 0 0 1 where m = 1, 2, . . . , n. Because the identical kinematic chains of a spatial parallel manipulator are rotational symmetric about zaxis of the absolute coordinate system, the fixed joints must also be rotational symmetric about z-axis. And therefore the following relationships hold: yA zA yA zA ; i ¼ 1; 2; . . . ; n T ðm.2pÞ xA 2 B i xA ð17Þ Bj Bj Bj B B B i i i n yA zA where j = 1, 2, . . . , n, and Bi xA denote the absolute coordinates of point Bi, {Æ} denotes a set genBi Bi Bi erated by the element ‘‘Æ’’. Similarly, in the local coordinate system, there exist T ðm.2pÞ xLM j y LM j zLM j 2 M i xLM i y LM i zLM i ; i ¼ 1; 2; . . . ; n ð18Þ n L where M i xBi y LBi zLBi are the local coordinates of point Mi. As a result, F T ðm.2pÞ ð x y z Þ ¼ F ð x y z Þ ð19Þ n Consequently, the workspace is rotational symmetric about the same axis, z-axis. 2.3. The identical kinematic chains of a spatial parallel manipulator are centrosymmetric about a certain point Because the identical kinematic chains of a spatial parallel manipulator are centrosymmetric about one point, we can assume the origin of the absolute coordinate system is superimposed with the symmetric center. Define a function S(x, y, z) such that it satisfies 2 3 x 6 7 Sðx; y; zÞ ¼ 4 y 5 ð20Þ z Because the n kinematic chains of the manipulator are symmetric about the origin, according to Eq. (20) there must be 8 A A > y z < S xA yA zA 2 Bi xA Bj Bj Bj Bi Bi Bi ; i ¼ 1; 2; . . . ; n ð21Þ > : S xA yA zA yA zA 2 M i xA Mj Mj Mj Mi Mi M i ; i ¼ 1; 2; . . . ; n yA zA where Bi xA are the local coordinates of point Bi, M i xLBi y LBi zLBi are the local coordinates of Bi Bi Bi point Mi, {Æ} denotes a set generated by the element ‘‘Æ’’. Therefore, F ðSðx; y; zÞÞ ¼ F ðx; y; zÞ
ð22Þ
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Consequently, the workspace is centrosymmetric about the same point—the origin of the absolute coordinate system oxyz. From Eqs. (14), (19) and (22), we can find that the theorem is proved. 3. Examples and discussions We will present two kinds of parallel manipulators to demonstrate the applications of the theorem and illustrate the utilities in conceptual design and engineering estimations. 3.1. The workspace of a spatial 3-PUU parallel manipulator A spatial parallel 3-PUU (prismatic joint, universal joint, universal joint) parallel manipulator is shown in Fig. 1. The three identical kinematic chains of the manipulator are all made up of one prismatic joint and two universal joints in series. Because the manipulator plane M1M2M3 and the base plane B1B2B3 are all equilateral triangles, the manipulator has one rotational symmetric axis, denoted by e, and three symmetric planes, denoted by a1, a2 and a3 individually, which are shown in Fig. 2. Therefore, the three kinematic chains can be called both rotational symmetric and planar symmetric. Assume z-axis is superimposed with the rotational symmetric axis of the fixed base plane, and x-axis passes through prismatic joint B2. Similarly, presume zL-axis is superimposed with the symmetric axis of the moving manipulator plane, and xL-axis passes through M2 joint. According to [13,14], the DoF of the manipulator is three, which includes three translational movements along the three orthogonal directions at the ordinary cases. The constraint equations can be obtained according to Eq. (9): 2
3 2 2 2 þ y þ y LM 1 y A þ ðz þ zLM 1 zA B1 B1 Þ l 7 7 2 2 2 7 27 L A L A xA þ y þ y y þ z þ z z l 7 B2 M2 B2 M2 B2 7 5 2 2 2 2 A L A L A xB3 þ y þ y M 3 y B3 þ z þ zM 3 zB3 l
x þ xLM 1 xA B1
6 6 6 L F ðx; y; zÞ ¼ 6 6 x þ xM 2 6 4 x þ xLM 3
2
ð23Þ
where 2
xLM i
3
2
cos 2p ði 2Þ 3
3
6 7 6 L 7 6 y M 7 ¼ r6 sin 2p ði 2Þ 7; 4 5 4 i5 3 L zM i 0
i ¼ 1; 2; 3
ð24Þ
r is the radius of the circumcircle of equilateral triangle M1M2M3 2
xA Bi
3
2
cos 2p ði 2Þ 3
3
6 7 6 A7 6 y B 7 ¼ Ri 6 sin 2p ði 2Þ 7; 4 5 4 i5 3 A 0 zBi
i ¼ 1; 2; 3
ð25Þ
Ri, i = 1, 2, 3 are the displacement of the ith actuator, slider Bi, l1 6 Ri 6 l2. According to Eq. (16), n = 3. Let m = 1, there will be 2
cos
2p 3
6 2p T ð2pÞ ðx; y; zÞ ¼ 6 4 sin 3 3
0
sin 2p 3 2p cos 3 0
pffiffi 3 32 3 2 1 x 2 x 23 y 76 7 6 pffiffi 7 6 7 6 3 7 07 54 y 5 ¼ 4 2 x 12 y 5 z 1 z
0
ð26Þ
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
Therefore 8 L L L L > T ; y ; z x 2p > ð 3 Þ M 3 M 3 B3 ¼ rM 2 > > < T ð2pÞ xLM 2 ; y LM 2 ; zLM 2 ¼ T ð22pÞ xLM 3 ; y LM 3 ; zLM 3 ¼ rLM 1 3 3 > > > > : T 2p xL ; y L ; zL ¼ T 2p xL ; y L ; zL ¼ T 2p xL ; y L ; zL ¼ rL M3 ð 3 Þ M1 M1 M1 ð2 3 Þ M 2 M 2 M 2 ð3 3 Þ M 3 M 3 M 3
641
ð27Þ
where rLM i ; i ¼ 1; 2; 3 are the vector coordinates of Mi in the local coordinate system. 3 2 2 pffiffi 2 2 pffiffi 12 x 23 y þ xLM 1 xLB1 þ 23 x 12 y þ y LM 1 y LB1 þ z þ zLM 1 zLB1 l2 7 6 7 6 2 pffiffi 2 2 pffiffi 6 27 3 1 L L L L F T ð2pÞ ðx; y; zÞ ¼ 6 12 x 23 y þ xLM xLB 7 þ x y þ y y þ z þ z z l M1 B1 M1 B1 2 2 1 1 3 7 6 5 4 pffiffi pffiffi 2 2 2 12 x 23 y þ xLM 1 xLB1 þ 23 x 12 y þ y LM 1 y LB1 þ z þ zLM 1 zLB1 l2 ð28Þ Eq. (28) can be simplified as 3 2 h pffiffi i 2 3 2 1 L L L L L L x þ 2x 2 xM 1 xB1 þ 2 y M 1 y B1 þ xM 1 xB1 7 6 6 h pffiffi i 2 7 7 6 6 þy 2 þ 2y 3 xL xL 1 y L y L þ y LM 1 y LB1 7 M1 B1 M1 B1 7 6 2 2 7 6 7 6 2 7 6 2 L L 7 6 þ z þ zM 1 zB1 l 7 6 7 6 h i pffiffi 2 7 6 2 L L 7 6 x þ 2x 12 xLM xLB þ 23 y LM y LB x þ x M2 B2 2 2 2 2 7 6 6 h pffiffi i 2 7 7 6 3 1 2 L L L L L L 7 F T ð2pÞ ðx; y; zÞ ¼ 6 6 þy þ 2y 2 xM 2 xB2 2 y M 2 y B2 þ y M 2 y B2 7 3 7 6 7 6 2 7 6 7 6 þ z þ zLM 2 zLB2 l2 7 6 7 6 h i pffiffi 2 7 6 2 L L 7 6 x þ 2x 1 xLM xLB þ 3 y LM y LB x þ x M3 B3 2 2 3 3 3 3 7 6 6 h pffiffi i 2 7 7 6 6 þy 2 þ 2y 3 xL xL 1 y L y L þ y LM 3 y LB3 7 7 6 M3 B3 M3 B3 2 2 7 6 5 4 2 2 L L þ z þ zM 3 zB3 l Therefore, considering Ri, i = 1, 2, 3 have the same variable ranges, we can obtain from Eq. (29): 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 3 xA B3 M3 B3 M3 B3 7 6 7 6 2 2 2 6 27 L A L A F T ð2pÞ ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B1 M1 B1 M1 B1 1 3 7 6 5 4 2 2 2 x þ xLM 2 xA þ y þ y LM 2 y A þ z þ zLM 2 zA l2 B2 B2 B2 Similarly, we can deduce: 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 2 xA B M B M B 2 2 2 2 2 7 6 7 6 2 2 2 6 27 L A L A F T ð4pÞ ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B M B M B 3 3 3 3 3 3 3 7 6 5 4 2 2 2 2 L A L A L A x þ xM 1 xB1 þ y þ y M 1 y B1 þ z þ zM 1 zB1 l
ð29Þ
ð30Þ
ð31Þ
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Fig. 4. The reachable workspace of the manipulator.
Therefore, from Eqs. (23), (27), (30) and (31), we can find that F T ¼ F T ¼ 4p ðx; y; zÞ 2p ðx; y; zÞ ð3Þ ð3Þ F T ð6pÞ ðx; y; zÞ ¼ F ðx; y; zÞ. The reachable workspace of the manipulator can be obtained from Eq. (12). 3 The workspace of the manipulator is shown in Fig. 4 if we assign r = 300 mm, l1 = 300 mm, l2 = 1900 mm and l = 1600 mm. From Fig. 1, we can also find that the manipulator is also planar symmetric and the symmetric planes are a1, a2, and a3 shown in Fig. 2. If we consider the symmetric plane a2, which is superimposed by xoz, according to Eq. (13), we can gain 3 2 2 2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 1 xA B M B M B 1 1 1 1 1 7 6 7 6 2 2 2 6 27 L A L A ð32Þ F ðx; y; zÞ ¼ 6 x þ xLM xA 7 þ y þ y y þ z þ z z l B M B M B 2 2 2 2 2 2 7 6 5 4 2 2 2 x þ xLM 3 xA þ y þ y LM 3 y A þ z þ zLM 3 zA l2 B3 B3 B3 In the case shown in Fig. 2, there will be ( L y M 2 ¼ y LM 2 ¼ 0 y LM 1 ¼ y LM 3 and
(
A yA B2 ¼ y B2 ¼ 0 A yA B1 ¼ y B3
From Eqs. (32)–(34), there exist 3 2 2 h i2 2 2 L A L A þ y þ y y þ z þ z z l x þ xLM 3 xA B3 M3 B3 M3 B3 7 6 7 6 h i 2 2 2 6 27 L A L A L A F ðx; y; zÞ ¼ 6 x þ xM xB 7 þ y þ y y þ z þ z z l M2 B2 M2 B2 2 2 7 6 5 4 2 h i2 2 2 L A L A L A x þ xM 1 xB1 þ y þ y M 1 y B1 þ z þ zM 1 zB 1 l 3 2 2 2 2 2 L A L A x þ xLM 3 xA þ y þ y y þ z þ z z l B3 M3 B3 M3 B3 7 6 7 6 2 2 2 6 27 L A L A L A ¼ 6 x þ xM xB 7 þ y þ y y þ z þ z z l M2 B2 M2 B2 2 2 7 6 5 4 2 2 2 2 L A L A x þ xLM 1 xA þ y þ y y þ z þ z z l B1 M1 B1 M1 B1
ð33Þ
ð34Þ
ð35Þ
From Eqs. (23) and (35), we can find that F(x, y, z) = F(x, y, z). If we assign r = 300 mm, l1 = 600 mm, l2 = 1900 mm and l = 1600 mm, the workspace of the manipulator is obtained as Fig. 5 shows. We can find that the workspace has the same corresponding symmetries as the manipulator has. It is symmetric about all the planes a1, a2, and a3 and rotational symmetric about the directional axis e (z-axis), which are shown in
J.-S. Zhao et al. / Mechanism and Machine Theory 41 (2006) 632–645
643
Fig. 5. The reachable workspace of the manipulator.
Fig. 6. The symmetric workspace corresponding to the structure of the manipulator.
Fig. 6. As a result, the reachable workspace of the manipulator shown in Fig. 1 has the same three symmetric planes and one symmetric axis as the manipulator has. 3.2. The workspaces of planar 4-bar parallel manipulators We now analyze the workspaces of the planar 4-bar parallel manipulators shown in Figs. 3 and 7. According to the theorem, the workspace should have the same symmetric characteristics as those of the manipulators kinematic chains. Given an axis passing through the middle point of line AD and perpendicular to line AD, we can find a configuration such that the two kinematic chains of the manipulator (BC) shown in Fig. 7 are symmetric about the axis. As a result, the reachable workspace (the possible locus) of the manipulator should be symmetric about this axis, too. As is well known, the reachable workspace of the parallel 4-bar manipulator is a circle, whose center is the middle point of line AD and the radius is the length of the bar AB. Just as we have pointed out that the converse theorem does not hold, though the workspace of the manipulator is symmetric about any line passing through the middle point of line AD, we cannot safeguard that the two kinematic chains of the manipulator (BC) are also symmetric about this line. Similarly, because the inverse parallel 4-bar manipulator shown in Fig. 3 is symmetric about the middle point of line AD, the reachable workspace should also be symmetric about the same point according to the theorem. The workspace (the possible locus) of this manipulator is shown in Fig. 8 if we set the lengths of AB and CD to be 200 mm, the lengths of link BC and AD to be 600 mm. Many more such examples can be found in a number of recently published literatures [15–19].
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Fig. 7. A planar parallel 4-bar manipulator.
Fig. 8. The workspace of the inverse parallel 4-bar manipulator.
4. Conclusion This paper proposes an analogous symmetric theorem of workspace for spatial parallel manipulators with symmetric identical kinematic chains. Through a strict mathematic proof, it reveals the analogous relationship between the workspace shape and the structure of the manipulator. Examples demonstrate that this theorem can be utilized to estimate the geometry characteristics of the workspace and to guide the conceptual design of spatial parallel manipulators, especially for those with full DoFs. The theorem offers a concrete safeguard for the workspace analysis of complicated spatial parallel manipulators. Acknowledgement This work was supported by China postdoctoral foundations. The authors would like to thank Professor Andre´s Kecskeme´thy and the anonymous refereeing scientists for their wise suggestions. References [1] D. Sen, T.S. Mruthyunjaya, A centro-based characterization of singularities in the workspace of planar closed-loop manipulators, Mechanism and Machine Theory 33 (8) (1998) 1091–1104. [2] S. Dibakar, T.S. Mruthyunjaya, A computational geometry approach for determination of boundary of workspaces of planar manipulators with arbitrary topology, Mechanism and Machine Theory 34 (1) (1999) 149–169. [3] J.K. Davidson, K.D. Chaney, A design procedure for RPR planar robotic workcells: an algebraic approach, Mechanism and Machine Theory 34 (2) (1999) 193–203. [4] M. Ceccarelli, A formulation for the workspace boundary of general N-revolute manipulators, Mechanism and Machine Theory 31 (5) (1996) 637–646. [5] M. Ceccarelli, C. Lanni, A multi-objective optimum design of general 3R manipulators for prescribed workspace limits, Mechanism and Machine Theory 39 (2) (2004) 119–132.
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