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Symmetrical characteristics of the workspace for spatial parallel mechanisms with symmetric structure
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Mechanism and Machine Theory
Mechanism and Machine Theory 43 (2008) 427–444
www.elsevier.com/locate/mechmt
Symmetrical characteristics of the workspace for spatial parallel mechanisms with symmetric structure Jing-Shan Zhao *, Fulei Chu, Zhi-Jing Feng Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, PR China Received 12 June 2006; received in revised form 12 February 2007; accepted 10 April 2007 Available online 7 June 2007
Abstract The fact that symmetry of a spatial parallel mechanism can be directly mapped into the workspace of the end-effector is defined as a strengthened theorem and is proved by utilizing the group theory. If the order of the symmetry group of the mechanism structure is n, then the searching range can be reduced to 1n of the initial one. This can lead to significant reductions of the computation task and time because only a fraction of the workspace may need to be investigated for a symmetric mechanism, the merits of which are especially obvious if discretization algorithms are used. Firstly, the symmetry mapping from the mechanism structure into the workspace is generalized in a much more extension by a group theoretical proof. And then, examples are presented to illustrate the applications of this research result in reducing the searching tasks of the reachable workspace of an end-effector. The strengthened theorem herein will be adapted to all symmetry cases and will be most useful for the conceptual design of spatial parallel mechanisms, particularly for those with complicated symmetries. 2007 Elsevier Ltd. All rights reserved. Keywords: Workspace; Symmetry; Theorem; Spatial parallel mechanism
1. Introduction The reachable workspace of the end-effector of a mechanism, which will be denoted as ‘‘workspace’’ for short in this paper, is the reachable domain of its end-effector and is bounded in the three dimensional space [1]. The workspace of a mechanism has been defined in literature as the totality of positions that a particular identified point of the mechanism (end-effector) can reach [2–4]. For designing a new mechanism, it is almost always necessary to determine the boundaries of the workspaces [3]. Wang et al. [5] presented an algorithm, called the boundary search method, to determine the workspace of a parallel machine tool. Bonev and Ryu [6] presented a discretization method for the computation of the orientation workspace of 6-DoF (degree of freedom) parallel mechanisms, defined as the set of all attainable orientations of the mobile end-effector about a fixed point. Gosselin [7] addressed the determination of *
Corresponding author. Fax: +86 01062782351. E-mail addresses: [email protected], [email protected] (J.-S. Zhao).
0094-114X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2007.04.004
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the workspace of 6-DoF parallel manipulators based on geometrical properties of the workspace. Gosselin and Jean [8] investigated the determination of the workspace of planar parallel manipulators with joint limits. Yang et al. [9] analyzed the workspace boundary of serial manipulators with non-unilateral constraints. Wang and Chirikjian [10], Karim Abdel-Malek et al. [11] discussed the boundary determination of manipulator workspaces. Karim Abdel-Malek et al. [12] also discussed the interior and exterior boundaries to the workspace of mechanical manipulators. Lai and Menq [13] analyzed the dexterous workspace of simple manipulators. Li et al. [14,15] explored the maximal singularity-free zones in the workspace of planar parallel mechanism with three DoFs and the General Gough-Stewart platform. Gouttefarde and Gosselin [16] discussed the workspace determination of planar parallel cable-driven mechanisms. Bonev and Gosselin [17] analyzed the determination problems of the workspace of symmetrical spherical parallel mechanisms. Jesu´s Cervantes-Sa´nchez et al. [18] focused on the workspace and singularity characterization of a 5R spherical symmetric manipulator. However, the general symmetry mapping from the mechanism structure into the workspace shape is seldom systematically investigated. Therefore, this paper further investigates the analogous symmetry properties between the workspace and the mechanism structure based on the work of [19]. Although symmetries of the workspace of a spatial parallel mechanism were discussed in [19], the preconditions that each identical kinematic chain of a spatial parallel mechanism should always keep collinear can be relaxed to be general symmetries. As a result, the analogous symmetry properties between the workspace and the mechanism can be generalized to be any symmetry structure and symmetry cases. The present study characterizes the general geometry relationships between the workspace and the mechanism structure. Group theoretical analysis is used to generalize the three symmetries proposed in [19] to any symmetry operations. 2. Group theoretical analysis on the symmetries of the workspace The symmetries of a planar equilateral polygon can be interpreted as a group, and therefore, the symmetry operations can be considered as the actions of such a group on some sets. In order to develop the model of group theoretical analysis on the symmetries of the workspace, we first introduce the primary concepts of group theory [20]. 2.1. Primary concepts of group theory A group is a non-empty close set G equipped with a binary operation such that (1) The associative law holds for every a, b, c 2 G, a (b c) = (a b) c. (2) There is an element e 2 G, called the identity, with e a = a = a e for all a 2 G. (3) Every a 2 G has an inverse, there is b 2 G with a b = e = b a. A group is called Abelian if it satisfies the commutative law: a b = b a holds for every a, b 2 G. If G is a group and if a 2 G, define the powers an, for n P 1, inductively: a1 = a and an+1 = a an. Define a0 = e and, if n is a positive integer, define an = (a1)n. If G is a group, if a 2 G, and if m, n P 1, then am+n = am an and (am)n = amn. Let G be a group and let a 2 G. If ak = e for some k P 1, then the smallest such exponent k P 1 is called the order of a; if no such power exists, then one says that a has infinite order. If G is a finite group, then every a 2 G has finite order. A motion Pis a distance preserving bijection u : R3 ! R3. If V is a polygon in theP plane, then its symmetry group ðV Þ consists of all the motions u for which u(V) = V. The elements of ðV Þ are called symmetries of V. Definition. If Vn is a regular polygon with n vertices v1, v2, . . . , vn and center O, then the symmetry group P ðV n Þ is called the dihedral group with 2n elements, and it is denoted by D2n. The dihedral group D2n contains the n rotations qj about the center by 2jp n where 0 6 j 6 n 1. The description of the other n elements depends on the parity of n. If n is an odd, then the other n symmetries are
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v3
v4
v2 O
v5
v1
Fig. 1. The reflection lines in the symmetry group when n = 5 is an odd.
v4
m3
v3 m2
v5
v2
O m1 v6
v1
Fig. 2. The reflection lines in the symmetry group when n = 6 is an even.
reflections in the distinct lines Ovi, for i = 1, 2, . . . , n. For example, the five lines of symmetry of V5 are Ov1, Ov2, Ov3, Ov4, and Ov5, which are shown in Fig. 1. If n = 2p is an even, then each line Ovi coincides with the line Ovp+i, giving only p such reflections; the remaining p symmetries are reflections in the lines Omi for i = 1, 2, . . . , p, where mi is the midpoint of the edge vi vi+1. For example, the six lines of symmetry of V6 are Ov1, Ov2, and Ov3, and Om1, Om2, and Om3, which are shown in Fig. 2. Definition. If X is a set and G is a group, then G acts on X if there is a function G · X ! X, denoted by (g, x) # gx, such that (1) ðghÞx ¼ gðhxÞ for all g, h 2 G and x 2 X. (2) ex = x for all x 2 X, where e is the identity in G. We also call X a G-set if G acts on X. If a group G acts on a set X, then fixing the first variable, say g, gives a function ag : X ! X, namely, ag : x # agx. This function is a permutation of X, for its inverse is ag1 : ag ag1 ¼ ae ¼ eX ¼ ag1 ag : It is easy to see that a : g # ag, is a homomorphism. Conversely, given any homomorphism u : G ! SX, define gx = u(g)(x). Thus, an action of a group G on a set X is another way of viewing a homomorphism G ! S X. Let X = {v1, v2, . . . , vn}, where vi, (i = 1, 2, . . . , n) is the vertices of a polygon, and G be the dihedral group D2n acting on X. G ¼ fT ; T 2 ; . . . ; T n ; ST ; ST 2 ; . . . ; ST n g, where T denotes a 2p n -rotation about the origin of the coordinate system (center of the equilateral polygon) and Tn = e, S denotes the reflection about a certain line and S2 = e, and e denotes an identity mapping.
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For each vertex vi 2 X, there is some g 2 G with gv1 = vi ; therefore, the orbit of v1 is O(v1) = X and D2n acts transitively. It is not difficult to prove that for any vi, the stabilizer Gvi is a subgroup of order 2.
2.2. Analogous symmetrical theorem of workspace for spatial parallel mechanisms In this paper, the workspace of the end-effector of a mechanism is defined as the volume within which the identified point, say, the geometry center of the mechanism, can be reached [2,21], which can also be considered as a set of points that satisfy a certain restricted equations. Therefore, in mathematics language, the workspace of the end-effector of a mechanism, denoted by X, is a mapping of the three dimensional Euclidean spaces R3: F : R3 ! X
ð1Þ
In applications, most spatial parallel mechanisms have symmetric structure. The symmetries can be depicted by a subgroup of a certain symmetric group Sn [20], no matter what kinds of symmetries the mechanism structure has—about a certain plane, about one axis or about one specified point. In order to exploit the relationship between the symmetries of the workspace and the structural characteristics of the kinematic chains, we proposed a symmetrical workspace theorem for spatial parallel mechanism [19]: If each identical kinematic chain of a spatial parallel mechanism always keeps collinear, and they are structural symmetric about a plane, an axis, or a point; then the workspace will also be symmetric about the corresponding plane, axis, or point individually. In fact, the above theorem can be considerably strengthened to be If the symmetry group of the workspace is denoted by GW, and the symmetry group of the kinematic chain structure of the end-effector in a particular configuration is denoted by GM: then GM must be a subgroup of GW, namely, the following relationship will always hold: GM GW
ð2Þ
Prior to proving the validation of the theorem, we also make the following assumptions [19]: (1) The absolute coordinates of the geometric center of the mechanism are denoted by r ¼ ½ x y z T . (2) The total number of the symmetric kinematic chains (need not require collinear for each chain) of the spatial parallel mechanism is indicated by n(n P 2). (3) The mechanism’s ith (i = 1, 2, . . . ,n) vertex linked by the ith 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 base’s ith (i = 1, 2, . . . , n) vertex linked by the ith kinematic chain is denoted by Bi, its absolute coor dinates are denoted by xABi y ABi zABi , where the superscript A indicates ‘‘absolute’’. As a result, the absolute coordinates of the ith vertex, Mi, can be obtained rAM i ¼ r þ RrLM i
ð3Þ
where R is a transformation matrix from the local coordinate system to the absolute one, T T are the absolute coordinates ½x y z are the absolute coordinates of rAM i ¼ xAM i y AM i zAM i L of ML i, r ¼ T L L the geometry center of the mechanism, and rM i ¼ xM i y M i zM i are the local coordinates of Mi. Proof. For mathematically, the workspace of the end-effector of a mechanism, X, is a mapping of the three dimensional Euclidean spaces R3 F : R3 ! X The workspace X can therefore be denoted as X ¼ frjF ðrÞ ¼ 0g where F(Æ) are a set of constraint equations of the workspace boundary surfaces
ð4Þ
J.-S. Zhao et al. / Mechanism and Machine Theory 43 (2008) 427–444
2
krAM 1 rAB1 k l1
431
3
7 6 A 6 krM 2 rAB2 k l2 7 7 6 F ðrÞ ¼ 6 7 .. 7 6 . 5 4 krAM m rABm k lm where m denotes the number of constraint equations. According to Eqs. (3), (5) can be expanded to be 3 2 kr þ RrLM 1 rAB1 k l1 7 6 6 kr þ RrLM 2 rAB2 k l2 7 7 6 F ðrÞ ¼ 6 7 .. 7 6 . 5 4
ð5Þ
ð6Þ
kr þ RrLM m rABm k lm
Suppose the symmetry operations in a particular configuration form a group GM. Obviously, GM must be a subgroup of a symmetry group S(V), which is a distance-preserving group and consists of the motions ui for which ui(vj) = vj. Consider the actions of the symmetry group GM on the workspace X defined by Eq. (4). For any symmetry operation ui 2 G, i = 1, 2, . . . , n, there is 3 2 3 2 L 1 A 1 kui ðrÞ þ Rui u1 kui ðrÞ þ RrLM 1 rAB1 k l1 i ðrM 1 Þ ui ui ðrB1 Þk ui ui ðl1 Þ 7 6 7 6 L 1 A 1 6 kui ðrÞ þ RrLM 2 rAB2 k l2 7 6 kui ðrÞ þ Rui u1 i ðrM 2 Þ ui ui ðrB2 Þk ui ui ðl2 Þ 7 7 6 7 6 ð7Þ F ðui ðrÞÞ ¼ 6 7¼6 7 .. .. 7 6 7 6 . . 5 4 5 4 L 1 A 1 kui ðrÞ þ RrLM m rABm k lm kui ðrÞ þ Rui u1 ðr Þ u u ðr Þk u u ðl Þ m i i i i i Mm Bm 1 Because u1 denotes a different permutation, i 2 GM and the binary operation is closed on GM, and ui r(1)r(2) . . . r(m), of the number series (1, 2, . . . , m). In Eq. (7), the actions of ui and u1 on vectors r, rLM j i A and rBj , and on scalar lj denote the corresponding permutations (or symmetry operations), where j = 1, 2, . . . , m. As a result, Eq. (7) can be transformed into 3 2 3 2 L 1 A 1 kr þ RrLM rð1Þ rABrð1Þ k lrð1Þ kr þ Ru1 i ðrM 1 Þ ui ðrB1 Þk ui ðl1 Þ 7 6 7 6 6 kr þ RrLM rAB k lrð2Þ 7 6 kr þ Ru1 ðrLM 2 Þ u1 ðrAB2 Þk u1 ðl2 Þ 7 i i i 7 6 rð2Þ rð2Þ 7 6 7 ð8Þ F ðui ðrÞÞ ¼ ui 6 7 ¼ ui 6 7 6 . .. 7 6 . 7 6 . . 5 4 5 4 L 1 A 1 kr þ Ru1 kr þ RrLM rðmÞ rABrðmÞ k lrðmÞ i ðrM m Þ ui ðrBm Þk ui ðlm Þ
From Eq. (8), we can find that the constraints 3 2 kr þ RrLM rð1Þ rABrð1Þ k lrð1Þ 7 6 6 kr þ RrLM rAB k lrð2Þ 7 7 6 rð2Þ rð2Þ 7 6 7 6 .. 7 6 . 5 4 L A kr þ RrM rðmÞ rBrðmÞ k lrðmÞ are equivalent to Eq. (6) for the differences only result from the new permutations of the sequence numbers of the constraint functions. As a result, Eq. (8) can be simplified as F ðui ðrÞ ¼ ui ðF ðrÞÞ
ð9Þ
Therefore, if the kinematic chain structure of the end-effector is kept unaltered by certain symmetry operation ui, i = 1, 2, . . . , n, then the workspace X = {rjF(r) = 0} is also unaltered under the symmetry operation ui, and therefore unaltered under the symmetry group GM.
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Because GW characterizes all the symmetries of the workspace, there must be GM GW. As a matter of fact, the relationship depicted by Eq. (2) can also be generalized to dexterous workspace. That is, GM GD if GD denote the dexterous workspace of the end-effector, which can be similarly proved. By the way, it is not difficulty to prove that if the order of the symmetry group of the mechanism structure is1 n, then the searching range of angle can be reduced to 0; 2p n . That is, the searching range can be reduced to n of the initial one. h 3. Examples and discussions Next, we will present the workspaces of some spatial parallel mechanisms to demonstrate the applications of the theorem and illustrate the utilities in conceptual design and engineering estimations. 3.1. The workspace of a Gough-Stewart parallel mechanism A general Gough-Stewart parallel mechanism is shown in Fig. 3. Assume the origin of the absolute coordinate system oxyz shown in Fig. 4 is superimposed with the geometry center of the fixed base plane B1 B2 B3 B4 B5 B6, the x-axis is parallel to B1 B6, the y-axis is perpendicular to B1 B6, and the z-axis is perpendicular to the base plane B1 B2 B3 B4 B5 B6. In similar manner, we also presume that the origin of the local coordinate system oL xL yL zL shown in Fig. 4 is superimposed with the geometry center of the mobile end-effector
Fig. 3. Gough-Stewart parallel mechanism.
Fig. 4. Top view of Gough-Stewart mechanism.
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M1 M2 M3 M4 M5 M6, the xL-axis is parallel to M1 M6, the yL-axis is perpendicular to M1 M6, and the zL-axis is perpendicular to the mobile plane M1 M2M 3M4 M5 M6. In the absolute coordinate system, the vertexes of the fixed base are denoted by Bi, i = 1, 2, . . . , 6, and the geometry center is o. The vectors of these vertexes are represented by rBi , respectively. In the local coordinate system, the coordinates of vertexes Mi of the mobile mechanism are represented by rLM i , the geometry center of the mobile end-effector is denoted by C, and the absolute coordinates are ð x y z Þ. The structure of the spatial parallel mechanism shown in Fig. 3 has very interesting characteristics: the hexagons M1 M2M3 M4 M5 M6 and B1 B2 B3 B4B5B6 are similar polygons; the kinematic chains of the end-effector M1 M2 M3 M4 M5 M6 are identical SPS chains and have the same actuations; the intersections of M1 B1 and M6 B6, M2 B2 and M3 B3, M4 B4 and M5 B5 in the top view shown in Fig. 4 just form the three vertexes of an equilateral triangle v1 v2 v3. It is not difficult to find that the symmetry operations of the Gough-Stewart mechanism shown in Fig. 4 forms a symmetric group S3, the elements of which are the permutations of a set V = {v1, v2, v3}. The exact number of the permutations is 3! = 6. Therefore, S3 = {u1, u2, u3, u4, u5,u6}, where u1 = (1), u2 = (123), u3 = (132), u4 = (23), u5 = (13), u6 = (12). It is not difficult to prove that S3 is just a dihedral group D6, which describes all the symmetries of the mechanism structure shown in Fig. 4. The first three elements of S3 represent the three rotations 0, 2p , and 4p about z-axis; and the last three denote three symmetries about ov1, ov2, and 3 3 ov3, respectively. Now, let us consider the actions of the symmetric group S3 on the workspace of the end-effector. The workspace of the mechanism can be denoted as X ¼ frjF ðrÞ ¼ 0g ð10Þ where F(Æ) are a set of constraint equations of the workspace boundary surfaces 3 2 A krM 1 rAB1 k l1 6 krA rA k l 7 27 6 M2 B2 7 ð11Þ F ðrÞ ¼ 6 .. 7 6 5 4 . krAM 6 rAB6 k l6 According to Eqs. (3), (11) can be expanded to be 3 2 kr þ RrLM 1 rAB1 k l1 6 kr þ RrL rA k l2 7 7 6 M2 B2 7 F ðrÞ ¼ 6 ð12Þ .. 7 6 5 4 . kr þ RrLM 6 rAB6 k l6 In the following, we will demonstrate that the workspace of the end-effector contains the symmetries of the mechanism structure. That is, the actions of dihedral group of the mechanism on the workspace expressed in (10) will make the workspace superimpose with itself. (1) The action of u1 on X. Obviously, u1(X) = X holds. (2) The action of u2 on X. Because 3 2 3 2 L 1 A 1 ku2 ðrÞ þ Ru2 u1 ku2 ðrÞ þ RrLM 1 rAB1 k l1 2 ðrM 1 Þ u2 u2 ðrB1 Þk u2 u2 ðl1 Þ 7 6 7 6 L 1 A 1 6 ku2 ðrÞ þ RrLM 2 rAB2 k l2 7 6 ku2 ðrÞ þ Ru2 u1 2 ðrM 2 Þ u2 u2 ðrB2 Þk u2 u2 ðl2 Þ 7 7 6 7 6 6 ku ðrÞ þ RrL rA k l3 7 6 ku ðrÞ þ Ru u1 ðrL Þ u u1 ðrA Þk u u1 ðl3 Þ 7 2 2 2 2 2 2 M3 B3 M3 B3 7 6 2 7 6 2 F ðu2 ðrÞÞ ¼ 6 7¼6 7 L 1 A 1 7 6 ku2 ðrÞ þ RrLM 4 rAB4 k l4 7 6 ku2 ðrÞ þ Ru2 u1 ðr Þ u u ðr Þk u u ðl Þ 4 2 2 2 2 2 M4 B4 7 6 7 6 6 ku ðrÞ þ RrL rA k l 7 6 ku ðrÞ þ Ru u1 ðrL Þ u u1 ðrA Þk u u1 ðl Þ 7 5 5 5 5 4 2 4 2 2 2 2 M5 B5 2 M5 2 B5 2 L A 1 L 1 A 1 ku2 ðrÞ þ RrM 6 rB6 k l6 ku2 ðrÞ þ Ru2 u2 ðrM 6 Þ u2 u2 ðrB6 Þk u2 u2 ðl6 Þ ð13Þ
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while u2 = (123) and u1 2 ¼ ð132Þ ¼ u3 , so Eq. (13) equals to 3 2 2 kr þ Ru3 ðrLM 1 Þ u3 ðrAB1 Þk u3 ðl1 Þ kr þ RrLM 5 6 kr þ RrL 6 kr þ Ru ðrL Þ u ðrA Þk u ðl Þ 7 6 6 3 M2 3 B2 3 2 7 M6 7 6 6 6 kr þ RrL 6 kr þ Ru3 ðrL Þ u3 ðrA Þk u3 ðl3 Þ 7 M3 B3 M1 7 6 6 F ðu2 ðrÞÞ ¼ u2 6 7 ¼ u2 6 6 kr þ RrLM 6 kr þ Ru3 ðrLM Þ u3 ðrAB Þk u3 ðl4 Þ 7 4 4 2 7 6 6 7 6 6 4 kr þ RrLM 3 4 kr þ Ru3 ðrLM 5 Þ u3 ðrAB5 Þk u3 ðl5 Þ 5 kr þ Ru3 ðrLM 6 Þ u3 ðrAB6 Þk u3 ðl6 Þ
rAB5 k l5
3
rAB6 k l6 7 7 7 rAB1 k l1 7 7 7 ¼ u2 ðF ðrÞÞ rAB2 k l2 7 7 7 rAB3 k l3 5
ð14Þ
kr þ RrLM 4 rAB4 k l4
Therefore, the action of u2 on X makes X rotate 2p about the z-axis and superimpose with itself. The action of 3 u3 on X can be similarly found to be a 4p -rotation about the z-axis and superimpose with itself. 3 (3) The action of u4 on X. Because 2 ku4 ðrÞ þ RrLM 1 6 ku ðrÞ þ RrL 6 4 M2 6 6 ku ðrÞ þ RrL M3 6 4 F ðu4 ðrÞÞ ¼ 6 6 ku4 ðrÞ þ RrLM 4 6 6 L ku ðrÞ þ Rr 4 4 M5
rAB1 k l1
3
rAB2 k l2 7 7 7 rAB3 k l3 7 7 7 rAB4 k l4 7 7 7 rAB5 k l5 5
ð15Þ
ku4 ðrÞ þ RrLM 6 rAB6 k l6 while u4 is a symmetry operation of the mechanism and u1 4 ¼ ð23Þ ¼ u4 , Eq. (15) can be further transformed into 3 2 3 2 L 1 A 1 ku4 ðrÞ þ Ru4 u1 kr þ Ru4 ðrLM 1 Þ u4 ðrAB1 Þk u4 ðl1 Þ 4 ðrM 1 Þ u4 u4 ðrB1 Þk u4 u4 ðl1 Þ 7 6 6 kr þ Ru ðrL Þ u ðrA Þk u ðl Þ 7 L 1 A 1 7 6 ku4 ðrÞ þ Ru4 u1 6 4 M2 4 B2 4 2 7 4 ðrM 2 Þ u4 u4 ðrB2 Þk u4 u4 ðl2 Þ 7 6 7 6 7 6 1 L 1 A 1 L A 7 6 6 ku4 ðrÞ þ Ru4 u4 ðrM 3 Þ u4 u4 ðrB3 Þk u4 u4 ðl3 Þ 7 6 kr þ Ru4 ðrM 3 Þ u4 ðrB3 Þk u4 ðl3 Þ 7 7 6 ¼ u F ðu4 ðrÞÞ ¼ 6 7 6 4 7 1 L 1 A 1 6 kr þ Ru4 ðrLM Þ u4 ðrAB Þk u4 ðl4 Þ 7 6 ku4 ðrÞ þ Ru4 u4 ðrM 4 Þ u4 u4 ðrB4 Þk u4 u4 ðl4 Þ 7 4 4 7 6 7 6 7 6 L A 6 ku4 ðrÞ þ Ru4 u1 ðrL Þ u4 u1 ðrA Þk u4 u1 ðl5 Þ 7 kr þ Ru ðr Þ u ðr Þk u ðl Þ 5 4 5 4 4 4 M B 4 M5 4 B5 4 5 4 5 5 L A 1 L 1 A 1 kr þ Ru4 ðrM 6 Þ u4 ðrB6 Þk u4 ðl6 Þ ku4 ðrÞ þ Ru4 u4 ðrM 6 Þ u4 u4 ðrB6 Þk u4 u4 ðl6 Þ ð16Þ Therefore, Eq. (16) is equivalent to 3 2 kr þ RrLM 6 rAB6 k l6 6 kr þ RrL rA k l 7 6 57 M5 B5 7 6 6 kr þ RrL rA k l4 7 M4 B4 7 6 F ðu4 ðrÞÞ ¼ u4 6 7 ¼ u4 ðF ðrÞÞ 6 kr þ RrLM rAB k l3 7 3 3 7 6 7 6 4 kr þ RrLM 2 rAB2 k l2 5
ð17Þ
kr þ RrLM 1 rAB1 k l1 Consequently, the action of u4 on X makes X reflect about the yoz-plane. The actions of u5 and u6 on X can be similarly found to be reflections about the planes of v2oz and v3oz, individually. Therefore, the workspace of the Gough-Stewart mechanism has all the symmetries of the mechanism. The workspace can be investigated as follows. Suppose the coordinate transformations, R, from the mobile end-effector to the fixed base are 231-type, there is
J.-S. Zhao et al. / Mechanism and Machine Theory 43 (2008) 427–444
2 6 R¼4 2 6 ¼4
cos h
0
0
1
sin h
0
sin h
32
cos w
76 0 54 sin w cos h
0
sin w cos w 0
0
32
1
0
76 0 54 0
cos u
0
sin u
1
0
435
3
7 sin u 5 cos u
cos w cos h
sin w cos h cos u þ sin h sin u
sin w cos h sin u þ sin h cos u
3
sin w
cos w cos u
cos w sin u
7 5
cos w sin h
sin w sin h cos u þ cos h sin u
sin w sin h sin u þ cos h cos u
ð18Þ
Substituting Eq. (18) into Eq. (3) results in rAM i ¼ r þ RrLM i ; i ¼ 1; 2; . . . ; 6 ð19Þ 2 3 2 3 xM i x Assume rLM i ¼ 4 y M i 5; i ¼ 1; 2; . . . ; 6 and r ¼ 4 y 5, Eq. (19) can be expanded to yield z zM i 3 2 x þ xLM i cos wcos h þ y LM i ð sin wcos hcos u þ sin hsin uÞ þ zLM i ðsin wcos hsin u þ sin hcos uÞ 7 6 y þ xLM i sin w þ y LM i cos w cosu zLM i cos w sinu rAM i ¼ r þ ArLM i ¼ 4 5 L L L z xM i cos wsin h þ y M i ðsin w sinh cos u þ cos h sinuÞ þ zM i ð sin wsin hsin u þ cos hcos uÞ ð20Þ
Substituting Eq. (20) into Eq. (4) yields a set of constraint equations of the workspace of the end-effector. To obtain the workspace boundary surfaces, we can resort to programming at a computer. Therefore, suppose 3 2 3 2 f1 ðx; y; zÞ f1 6 f 7 6 f ðx; y; zÞ 7 7 6 27 6 2 7 6 7 6 6 f3 7 6 f3 ðx; y; zÞ 7 7 6 7¼6 ð21Þ 6 f 7 6 f ðx; y; zÞ 7 7 6 47 6 4 7 6 7 6 4 f5 5 4 f5 ðx; y; zÞ 5 f6 ðx; y; zÞ f6 where
f1 ðx; y;zÞ ¼ ½x þ xLM 1 cos w cos h þ y LM 1 ðsinw cos h cos u þ sinh sinuÞ þ zLM 1 ðsinw cos hsin u þ sinh cos uÞ xB1 2 þ ½y þ xLM 1 sinw þ y LM 1 cos w cos u zLM 1 cos w sinu y B1 2 þ ½z xLM 1 cos w sinh þ y LM 1 ðsinw sin hcos u þ cos h sinuÞ þ zLM 1 ð sinw sinhsin u þ cos h cos uÞ zB1 2 f2 ðx; y;zÞ ¼ ½x þ xLM 2 cos w cos h þ y LM 2 ðsinw cos h cos u þ sinh sinuÞ þ zLM 2 ðsinw cos hsin u þ sinh cos uÞ xB2 2 þ ½y þ xLM 2 sin w þ y LM 2 cos w cos u zLM 2 cos w sinu y B2 2 þ ½z xLM 2 cos w sin h þ y LM 2 ðsin w sinhcos u þ cos hsinuÞ þ zLM 2 ðsinw sin hsinu þ cos hcos uÞ zB2 2 f3 ðx; y;zÞ ¼ ½x þ xLM 3 cos w cos h þ y LM 3 ðsinw cos h cos u þ sinh sinuÞ þ zLM 3 ðsinw cos hsin u þ sinh cos uÞ xB3 2 þ ½y þ xLM 3 sinw þ y LM 3 cos w cos u zLM 3 cos w sinu y B3 2 ½z xLM 3 cos w sinh þ y LM 3 ðsinw sin hcos u þ cos h sinuÞ þ zLM 3 ð sinw sinh sinu þ cos h cos uÞ zB3 2 f4 ðx; y;zÞ ¼ ½x þ xLM 4 cos w cos h þ y LM 4 ðsinw cos h cos u þ sinh sinuÞ þ zLM 4 ðsinw cos hsin u þ sinh cos uÞ xB4 2 þ ½y þ xLM 4 sin w þ y LM 4 cos w cos u zLM 4 cos w sin u y B4 2 ½z xLM 4 cos w sinh þ y LM 4 ðsinw sin hcos u þ cos h sinuÞ þ zLM 4 ð sinw sinh sinu þ cos h cos uÞ zB4 2
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f5 ðx; y; zÞ ¼ ½x þ xLM 5 cos w cos h þ y LM 5 ðsinw cos hcos u þ sinhsin uÞ þ zLM 5 ðsin w cos hsinu þ sinhcos uÞ xB5 2 þ ½y LM 5 þ xLM 5 sin w þ y LM 5 cos w cos u zLM 5 cos w sin u y B5 2 ½z xLM 5 cos w sin h þ y LM 5 ðsinw sinhcos u þ cos hsinuÞ þ zLM 5 ðsinw sin hsinu þ cos hcos uÞ zB5 2 f6 ðx; y; zÞ ¼ ½x þ xLM 6 cos w cos h þ y LM 6 ðsinw cos hcos u þ sinhsin uÞ þ zLM 6 ðsin w cos hsinu þ sinhcos uÞ xB6 2 þ ½y þ xLM 6 sinw þ y LM 6 cos w cos u zLM 6 cos w sin u y B6 2 ½z xLM 6 cos w sin h þ y LM 6 ðsinw sinhcos u þ cos hsinuÞ þ zLM 6 ðsinw sin hsinu þ cos hcos uÞ zB6 2
Assume that the coordinates of Mi, i = 1, 2, . . . , 6 in the local coordinate system oLxLyLzL are a2 r sin 5p a2 0 ; M 1 r cos 5p 6 6 5p M 2 r cos 6 þ a2 r sin 5p þ a2 0 ; 6 M 3 r cos 3p a2 r sin 3p a2 0 ; 2 2 M 4 r cos 3p þ a2 r sin 3p þ a2 0 ; 2 2 M 5 r cos p6 a2 r sin p6 a2 0 ; p a p a M 6 r cos 6 þ 2 r sin 6 þ 2 0 : The coordinates of Bi, i = 1, 2, . . . , 6 in the absolute coordinate system oxyz are B1 R cosðp2 þ b2Þ R sinðp2 þ b2Þ 0 ; B2 R cosð7p b2Þ R sinð7p b2Þ 0 ; 6 6 B3 R cosð7p þ b2Þ R sinð7p þ b2Þ 0 ; 6 6 B4 R cosð p6 b2Þ R sinð p6 b2Þ 0 ; B5 R cosð p6 þ b2Þ R sinð p6 þ b2Þ 0 ; B6 R cosðp2 b2Þ R sinðp2 b2Þ 0 : Suppose the lengths of B1 M1, B2 M2, B3 M3, B4 M4, B5 M5, B6 M6 are variables subjected to li 2 [lmin, lmax], i = 1, 2, . . . , 6. Given the values of the parameters: a, b, r, R, lmin, lmax, we can obtain the boundary surfaces of the workspace of the end-effector with the searching procedure which can be found in [5,6,22,23] 1. Solve the lowest position ð xLow y Low zLow Þ of the identified point of the end-effector when each of its legs has shrunk to the least length; and solve the highest position xHig y Hig zHig of the identified point of the end-effector when each of its legs has stretched to the largest length. 2. Establish parametric equations of cylindrical coordinates of the point within the workspace 8 > < x ¼ q cosðiDcÞ > :
y ¼ q sinðiDcÞ
ð22Þ
z ¼ zLow þ jDz
2p , and where q is the searching polar radius at plane z = zLow + jDz with a polar angle iDc, and 0 6 i < Dc zHig zLow 0 < j < Dz , and Dc is the angular interval and Dz is the distance interval along z-axis, and i and j are all natural numbers. 3. Suppose the workspace is sliced with an interval Dz along z-axis into k 1 planes, which are all parallel to z z xoy-plane, and where k is the largest natural number that is less than or equal to Hig Dz Low . And therefore, 0 < j 6 k. 4. Search the boundary curve at the jth plane. Let q = q(jDz, iDc) increase from zero with an expected interval Dq, namely q (jDz, iDc) = qjDq, qj = 1, 2, . . . . The stop criterion for such increasing is that when Eq. (22) is substituted into Eq. (21), there is at least one fp, p = 1, 2, . . . , 6, such that fp > l2max ; or fp < l2min for a set of pose angles h, w, u. Select the largest parameter q(jDz, iDc) to plot the boundary curve in the jth plane with 2p Eq. (22) as i increase from zero to Dc .
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5. When the curves in all of the slices are obtained, the boundary surfaces of the reachable workspace of the end-effector are gained. Different workspace shapes of the Gough-Stewart type mechanism are shown in Figs. 5–7. Although the workspace shapes are different with different structure parameters, the symmetry relationships between the workspace and the mechanism structure are still governed by Eq. (2). Therefore, the searching processes can be reduced to search the range of a generating group of the symmetric group of the mechanism. For a symmetric mechanism, if the order of the symmetry group of the mechanism structure is n, then the searching 2p range of angle about the z-axis can be reduced to 0; , for an example, the searching range of angle about the n z-axis can be reduced to 0; p3 . And as a result, the searching task and time are greatly reduced. 3.2. The workspace of a spatial 4-UPU parallel mechanism A Schoenflies type parallel mechanism is shown in Fig. 8. The mechanism proposed herein is actuated by four identical UPU (universal joint + prismatic joint + universal joint) kinematic chains [24].
Fig. 5. Different workspaces of the end-effector of a Gough-Stewart mechanism.
Fig. 6. Top views of the workspaces of the end-effector corresponding to Fig. 5.
Fig. 7. The mushroom workspace entity covered by the three boundary surfaces of Fig. 5. Left is the isometric view of the workspace entity, the middle is the top view and the right is a bottom view.
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The top view of the Schoenflies type spatial parallel mechanism is shown in Fig. 9. Therefore, we can consider a symmetric group S4, the elements of which are the permutations of a set V = {B1, B2, B3, B4}. The exact number of the permutations is 4! = 24. However, not all of the 24 permutations can form real symmetries of the mechanism shown in Fig. 9. As a matter of fact, the possible symmetries of the mechanism shown in Fig. 9 1M2k 1 B2 k are eight at most. In order to avoid the similar singularity, we knew that kM 6¼ kB [24]. Therefore, the kM 1 M 4 k kB1 B4 k symmetry group of the kinematic chains of the end-effector is G = {u1,u2,u3,u4}, where u1 = (1), u2 = (13)(24), u3 = (12)(34), u4 = (14)(23) and, G is a subgroup of dihedral group D8. G describes all the symmetries of the mechanism structure shown in Fig. 9. The first two elements of G express the two rotations of angles 0 and p about z-axis; and the last two denote two symmetries about xoz- and yoz-planes, respectively. Now, let us consider the actions of group G on the workspace of the end-effector. The workspace of the mechanism can be denoted as 2 A 3 jrM 1 rAB1 k l1 6 A 7 6 jrM 2 rAB2 k l2 7 7 ð23Þ F ðrÞ ¼ 6 6 krA rA k l 7 35 4 M3 B3 krAM 4 rAB4 k l4 According to Eqs. (1) and (23) can be expanded to 2 3 kr þ RrLM 1 rAB1 k l1 6 7 6 kr þ RrLM 2 rAB2 k l2 7 6 7 F ðrÞ ¼ 6 7 L A 4 kr þ RrM 3 rB3 k l3 5 kr þ RrLM 4 rAB4 k l4 (1) The action of u1 on X. Obviously, u1(X) = X holds. (2) The action of u2 on X. Because 2 3 ku2 ðrÞ þ RrLM 1 rAB1 k l1 6 7 6 ku2 ðrÞ þ RrLM 2 rAB2 k l2 7 7 F ðu2 ðrÞÞ ¼ 6 6 ku ðrÞ þ RrL rA k l 7 35 4 2 M3 B3 ku2 ðrÞ þ RrLM 4 rAB4 k l4
ð24Þ
ð25Þ
while u2 is a symmetry operation of the mechanism and u1 2 ¼ ð13Þð24Þ ¼ u2 , Eq. (25) is further equivalent to
Fig. 8. A kind of Schoenflies type spatial parallel mechanism.
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Fig. 9. The top view of the Schoenflies type spatial parallel mechanism.
3 2 3 ku2 ðrÞ þ Ru2 u2 ðrLM 1 Þ u2 u2 ðrAB1 Þk u2 u2 ðl1 Þ kr þ RrLM 3 rAB3 k l3 6 ku ðrÞ þ Ru u ðrL Þ u u ðrA Þk u u ðl Þ 7 6 kr þ RrL rA k l 7 6 2 2 2 M2 2 2 B2 2 2 2 7 47 6 M4 B4 7 F ðu2 ðrÞÞ ¼ 6 7 ¼ u2 ðF ðrÞÞ L A 6 ku2 ðrÞ þ Ru2 u2 ðrLM 3 Þ u2 u2 ðrAB3 Þk u2 u2 ðl3 Þ 7 ¼ u2 6 4 kr þ Rr r k l 15 M1 B1 5 4 kr þ RrLM 2 rAB2 k l2 ku2 ðrÞ þ Ru2 u2 rLM 4 u2 u2 rAB4 k u2 u2 ðl4 Þ 2
ð26Þ Therefore, the action of u2 on X makes X rotate p about the z-axis and superimpose with itself. (3) the action of u3 on X. Because 2 ku3 ðrÞ þ RrLM 1 6 ku ðrÞ þ RrL 6 3 M2 F ðu3 ðrÞÞ ¼ 6 4 ku3 ðrÞ þ RrLM 3 ku3 ðrÞ þ RrLM 4
3 rAB1 k l1 rAB2 k l2 7 7 7 rAB3 k l3 5 rAB4 k l4
ð27Þ
while u3 is a symmetry operation of the mechanism and u1 3 ¼ ð12Þð34Þ ¼ u3 , Eq. (23) is further equivalent to 3 3 2 2 kr þ RrLM 2 rAB2 k l2 ku3 ðrÞ þ Ru3 u3 ðrLM 1 Þ u3 u3 ðrAB1 Þk u3 u3 ðl1 Þ 7 7 6 6 6 kr þ RrLM 1 rAB1 k l1 7 6 ku3 ðrÞ þ Ru3 u3 ðrLM 2 Þ u3 u3 ðrAB2 Þk u3 u3 ðl2 Þ 7 7 7 6 6 F ðu3 ðrÞÞ ¼ 6 7 ¼ u3 6 kr þ RrL rA k l 7 ¼ u3 ðF ðrÞÞ L A 45 4 4 ku3 ðrÞ þ Ru3 u3 ðrM 3 Þ u3 u3 ðrB3 Þk u3 u3 ðl3 Þ 5 M4 B4 kr þ RrLM 3 rAB3 k l3 ku3 ðrÞ þ Ru3 u3 ðrLM 4 Þ u3 u3 ðrAB4 Þk u3 u3 ðl4 Þ ð28Þ
Fig. 10. Different workspaces of a Schoenflies type parallel mechanism.
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Therefore, the action of u3 on X makes it reflect about the xoz-plane. The action of u4 on X can be similarly found to be a reflection about the plane of yoz. Therefore, the workspace of the Schoenflies type parallel mechanism shown in Fig. 10 contains all the symmetries of those of the mechanism. The workspace of the end-effector can be analyzed below. The end-effector has four DoFs (one rotational DoF and three orthogonal translational DoFs), which can be analyzed according to [25]. Correspondingly, we can only select four independent variables, three of which are the absolute coordinates of the geometry center of the end-effector, denoted by (x, y, z); and the rest is the rotational angle about z-axis, denoted by a. As a result, the transformation matrix from the local coordinate system to the absolute one can be expressed as 2 3 cos a sin a 0 6 7 R ¼ 4 sin a cos a 0 5 ð29Þ 0
0
1
Suppose the coordinates of the ith, (i = 1, 2, 3, 4) vertex of the end-effector in the local coordinate system are denoted by M i ðxLM i ; y LM i ; zLM i Þ, the absolute coordinates of M i ðxAM i ; y AM i ; zAM i Þ can be obtained 2 A 3 2 3 2 L 3 2 3 2 32 L 3 2 3 xp i xM i x þ xLM i cos a y LM i sin a xM i x x cos a sin a 0 6 yA 7 6 7 6 7 6 7 6 76 L 7 6 7 ð30Þ 4 M i 5 ¼ 4 y 5 þ R4 y LM i 5 ¼ 4 y 5 þ 4 sin a cos a 0 54 y pi 5 ¼ 4 y þ xLM i sin a þ y LM i cos a 5 zAM i
z
zLM i
0
z
0
zLpi
1
z þ zLM i
The actuation condition of the ith kinematic chain will be 2
kM i Bi k L2i ¼ 0
ð31Þ
Expanding the above equation yields ðx þ xLM i cos a y LM i sin a xABi Þ2 þ ðy þ xLM i sin a þ y LM i cos a y ABi Þ2 þ ðz þ zLM i zABi Þ2 L2i ¼ 0
ð32Þ
According to Eq. (5), we have 3 2 ðx þ xLM 1 cos a y LM 1 sin a xAB1 Þ2 þ ðy þ xLM 1 sin a þ y LM 1 cos a y AB1 Þ2 þ ðz þ zLM 1 zAB1 Þ2 L21 7 6 6 ðx þ xL cos a y L sin a xA Þ2 þ ðy þ xL sin a þ y L cos a y A Þ2 þ ðz þ zL zA Þ2 L2 7 M2 M2 B2 M2 M2 B2 M2 B2 27 6 F ðx; y; z; aÞ ¼ 6 7 6 ðx þ xL cos a y L sin a xA Þ2 þ ðy þ xL sin a þ y L cos a y A Þ2 þ ðz þ zL zA Þ2 L2 7 M3 M3 B3 M3 M3 B3 M3 B3 35 4 ðx þ xLM 4 cos a y LM 4 sin a xAB4 Þ2 þ ðy þ xLM 4 sin a þ y LM 4 cos a y AB4 Þ2 þ ðz þ zLM 4 zAB4 Þ2 L24 ð33Þ ð xLM i
y LM i
zLM i
ð xABi
y ABi
zABi
Given the values of the parameters: Þ, Þ and the actuations li 2 ½lmin ; lmax , we can get the workspace of the end-effector with a similar procedure of Section 3.1. However, the searching range of the angle a can be reduced to 0; p2 for the mechanism structure has symmetry planes xoz and yoz. The workspaces are different if the structure parameters are different, some examples of which are shown in Figs. 10–12.
Fig. 11. Top views of the workspaces corresponding to those of Fig. 10.
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Fig. 12. Workspace entities covered by the three boundary surfaces of Fig. 10.
3.3. The workspace of a 3-RUU spatial parallel mechanism A 3-RUU spatial parallel mechanism is shown in Fig. 13. The mechanism is actuated by three identical RUU (revolute joint + universal joint + universal joint) kinematic chains. It is not difficult to find that the end-effector has three translational DoFs within its workspace according to [25]. The top view of the spatial parallel mechanism is shown in Fig. 14.
Fig. 13. A 3-RUU spatial parallel mechanism.
Fig. 14. The top view of the 3-RUU spatial parallel mechanism.
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When the identified point of the end-effector (geometry center OE) is superimposed with that of the fixed base, OB, the configuration symmetries of the kinematic chains of the end-effector is depicted by a symmetric group S3, which has been discussed in Section 3.1. Therefore, the structure symmetries of the mechanism under such a configuration can be depicted by a dihedral group D6. The first three elements of S3 represent the three rotations 0, 2p , and 4p about z-axis; and the last three denote three symmetries about oBB3C3, oBB1C1, 3 3 and oBB2C2, respectively. The workspace shapes of the end-effector are shown in Figs. 15–17. The searching range can also be reduced to 16 of the initial one with their symmetry characteristics. Many more such examples can be found in a number of recently published literatures [6,7,22,23,26–32].
Fig. 15. Different workspaces of the end-effector of a 3-RUU spatial mechanism.
Fig. 16. Top views of the workspaces of the end-effector corresponding to Fig. 15.
Fig. 17. Workspace entities covered by the three boundary surfaces of Fig. 15.
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4. Conclusions This paper further probes the analogous symmetry characteristics of the workspace of a spatial parallel mechanism. Based on the work of [19], we extend the theorem to be that the symmetry group of the kinematic chain structure of the end-effector must be a subgroup of that of the workspace of the end-effector. Examples demonstrate its applications in investigating the geometry shapes of the workspace, especially for those with complicated symmetry properties. The theorem offers a concrete safeguard for reducing the searching tasks of the workspace boundary surfaces of any symmetric spatial parallel mechanisms. For a symmetric mechanism, if the order of the symmetry group of the mechanism structure is n, then the searching range can be reduced to 1 of the initial one. n Acknowledgements This research was supported by FANEDD, China Postdoctoral Science Foundation and the National Natural Science Foundation of China under Grant 50425516. The authors gratefully acknowledge these support agencies. The authors would also like to thank Dr. Simon Guest in the Department of Engineering at University of Cambridge for his invaluable suggestions and the anonymous reviewers for their precious comments. References [1] D. Sen, T.S. Mruthyunjaya, A centro-based characterization of singularities in the workspace of planar closed-loop mechanisms, Mechanism and Machine Theory 33 (8) (1998) 1091–1104. [2] J.-P. Merlet, C. Gosselin, N. Mouly, Workspaces of planar parallel manipulators, Mechanism and Machine Theory 33 (1) (1998) 7– 20. [3] S. Dibakar, T.S. Mruthyunjaya, A computational geometry approach for determination of boundary of workspaces of planar mechanisms with arbitrary topology, Mechanism and Machine Theory 34 (1) (1999) 149–169. [4] 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. [5] 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. [6] I.A. Bonev, J. Ryu, A new approach to orientation workspace analysis of 6-DOF parallel mechanisms, Mechanism and Machine Theory 36 (1) (2001) 15–28. [7] C. Gosselin, Determination of the workspace of 6-DoF parallel manipulators, Journal of Mechanical Design 112 (3) (1990) 331–336. [8] C.M. Gosselin, M. Jean, Determination of the workspace of planar parallel manipulators with joint limits, Robotics and Autonomous Systems 17 (3) (1996) 129–138. [9] J. Yang, Karim Abdel-Malek, Y. Zhang, On the workspace boundary determination of serial manipulators with non-unilateral constraints, robotics and computer integrated manufacturing, doi:10.1016/j.rcim.2006.06.005. [10] Y. Wang, G.S. Chirikjian, Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N), IEEE Transactions on Robotics 20 (3) (2004) 399–408. [11] Karim Abdel-Malek, Frederick Adkins, et al., On the determination of boundaries to manipulator workspaces, Robotics and Computer Integrated Manufacturing 13 (1) (1997) 63–72. [12] Karim Abdel-Malek, Harn-Jou Yeh, Saib Othman, Interior and exterior boundaries to the workspace of mechanical manipulators, Robotics and Computer Integrated Manufacturing 16 (5) (2000) 365–376. [13] Z.-C. Lai, C.-H. Menq, The dexterous workspace of simple manipulators, IEEE Transactions on Robotics 4 (1) (1988) 99–103. [14] H. Li, C.M. Gosselin, M.J. Richard, Determination of maximal singularity-free zones in the workspace of planar three-degree-offreedom parallel mechanisms, Mechanism and Machine Theory 41 (10) (2006) 1157–1167. [15] H. Li, C.M. Gosselin, M.J. Richard, Determination of maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform, Mechanism and Machine Theory 42 (4) (2007) 497–511. [16] M. Gouttefarde, C.M. Gosselin, Analysis of the Wrench-closure workspace of planar parallel cable-driven mechanisms, IEEE Transactions on Robotics 22 (3) (2006) 434–445. [17] I.A. Bonev, C.M. Gosselin, Analytical determination of the workspace of symmetrical spherical parallel mechanisms, IEEE Transactions on Robotics 22 (5) (2006) 1011–1017. [18] J. Jesu´s Cervantes-Sa´nchez, J. Ce´esar Herna´ndez-Rodrı´uez, et al., Symmetric manipulator: workspace and singularity characterization, Mechanism and Machine Theory 39 (4) (2004) 409–429. [19] J.-S. Zhao, M. Chen, et al., Workspace of mechanisms with symmetry identical kinematic chains, Mechanism and Machine Theory 41 (6) (2006) 632–645. [20] Joseph.J. Rotman, Advanced Modern Algebra, Pearson Education, Inc., Publishing as Prentice-Hall, 2002. [21] A. Kumar, K.J. Waldron, The workspaces of a mechanical mechanism, Journal of Mechanical Design 103 (1981) 665–672.
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Mechanism and Machine Theory
Mechanism and Machine Theory 43 (2008) 427–444
www.elsevier.com/locate/mechmt
Symmetrical characteristics of the workspace for spatial parallel mechanisms with symmetric structure Jing-Shan Zhao *, Fulei Chu, Zhi-Jing Feng Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, PR China Received 12 June 2006; received in revised form 12 February 2007; accepted 10 April 2007 Available online 7 June 2007
Abstract The fact that symmetry of a spatial parallel mechanism can be directly mapped into the workspace of the end-effector is defined as a strengthened theorem and is proved by utilizing the group theory. If the order of the symmetry group of the mechanism structure is n, then the searching range can be reduced to 1n of the initial one. This can lead to significant reductions of the computation task and time because only a fraction of the workspace may need to be investigated for a symmetric mechanism, the merits of which are especially obvious if discretization algorithms are used. Firstly, the symmetry mapping from the mechanism structure into the workspace is generalized in a much more extension by a group theoretical proof. And then, examples are presented to illustrate the applications of this research result in reducing the searching tasks of the reachable workspace of an end-effector. The strengthened theorem herein will be adapted to all symmetry cases and will be most useful for the conceptual design of spatial parallel mechanisms, particularly for those with complicated symmetries. 2007 Elsevier Ltd. All rights reserved. Keywords: Workspace; Symmetry; Theorem; Spatial parallel mechanism
1. Introduction The reachable workspace of the end-effector of a mechanism, which will be denoted as ‘‘workspace’’ for short in this paper, is the reachable domain of its end-effector and is bounded in the three dimensional space [1]. The workspace of a mechanism has been defined in literature as the totality of positions that a particular identified point of the mechanism (end-effector) can reach [2–4]. For designing a new mechanism, it is almost always necessary to determine the boundaries of the workspaces [3]. Wang et al. [5] presented an algorithm, called the boundary search method, to determine the workspace of a parallel machine tool. Bonev and Ryu [6] presented a discretization method for the computation of the orientation workspace of 6-DoF (degree of freedom) parallel mechanisms, defined as the set of all attainable orientations of the mobile end-effector about a fixed point. Gosselin [7] addressed the determination of *
Corresponding author. Fax: +86 01062782351. E-mail addresses: [email protected], [email protected] (J.-S. Zhao).
0094-114X/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2007.04.004
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the workspace of 6-DoF parallel manipulators based on geometrical properties of the workspace. Gosselin and Jean [8] investigated the determination of the workspace of planar parallel manipulators with joint limits. Yang et al. [9] analyzed the workspace boundary of serial manipulators with non-unilateral constraints. Wang and Chirikjian [10], Karim Abdel-Malek et al. [11] discussed the boundary determination of manipulator workspaces. Karim Abdel-Malek et al. [12] also discussed the interior and exterior boundaries to the workspace of mechanical manipulators. Lai and Menq [13] analyzed the dexterous workspace of simple manipulators. Li et al. [14,15] explored the maximal singularity-free zones in the workspace of planar parallel mechanism with three DoFs and the General Gough-Stewart platform. Gouttefarde and Gosselin [16] discussed the workspace determination of planar parallel cable-driven mechanisms. Bonev and Gosselin [17] analyzed the determination problems of the workspace of symmetrical spherical parallel mechanisms. Jesu´s Cervantes-Sa´nchez et al. [18] focused on the workspace and singularity characterization of a 5R spherical symmetric manipulator. However, the general symmetry mapping from the mechanism structure into the workspace shape is seldom systematically investigated. Therefore, this paper further investigates the analogous symmetry properties between the workspace and the mechanism structure based on the work of [19]. Although symmetries of the workspace of a spatial parallel mechanism were discussed in [19], the preconditions that each identical kinematic chain of a spatial parallel mechanism should always keep collinear can be relaxed to be general symmetries. As a result, the analogous symmetry properties between the workspace and the mechanism can be generalized to be any symmetry structure and symmetry cases. The present study characterizes the general geometry relationships between the workspace and the mechanism structure. Group theoretical analysis is used to generalize the three symmetries proposed in [19] to any symmetry operations. 2. Group theoretical analysis on the symmetries of the workspace The symmetries of a planar equilateral polygon can be interpreted as a group, and therefore, the symmetry operations can be considered as the actions of such a group on some sets. In order to develop the model of group theoretical analysis on the symmetries of the workspace, we first introduce the primary concepts of group theory [20]. 2.1. Primary concepts of group theory A group is a non-empty close set G equipped with a binary operation such that (1) The associative law holds for every a, b, c 2 G, a (b c) = (a b) c. (2) There is an element e 2 G, called the identity, with e a = a = a e for all a 2 G. (3) Every a 2 G has an inverse, there is b 2 G with a b = e = b a. A group is called Abelian if it satisfies the commutative law: a b = b a holds for every a, b 2 G. If G is a group and if a 2 G, define the powers an, for n P 1, inductively: a1 = a and an+1 = a an. Define a0 = e and, if n is a positive integer, define an = (a1)n. If G is a group, if a 2 G, and if m, n P 1, then am+n = am an and (am)n = amn. Let G be a group and let a 2 G. If ak = e for some k P 1, then the smallest such exponent k P 1 is called the order of a; if no such power exists, then one says that a has infinite order. If G is a finite group, then every a 2 G has finite order. A motion Pis a distance preserving bijection u : R3 ! R3. If V is a polygon in theP plane, then its symmetry group ðV Þ consists of all the motions u for which u(V) = V. The elements of ðV Þ are called symmetries of V. Definition. If Vn is a regular polygon with n vertices v1, v2, . . . , vn and center O, then the symmetry group P ðV n Þ is called the dihedral group with 2n elements, and it is denoted by D2n. The dihedral group D2n contains the n rotations qj about the center by 2jp n where 0 6 j 6 n 1. The description of the other n elements depends on the parity of n. If n is an odd, then the other n symmetries are
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v3
v4
v2 O
v5
v1
Fig. 1. The reflection lines in the symmetry group when n = 5 is an odd.
v4
m3
v3 m2
v5
v2
O m1 v6
v1
Fig. 2. The reflection lines in the symmetry group when n = 6 is an even.
reflections in the distinct lines Ovi, for i = 1, 2, . . . , n. For example, the five lines of symmetry of V5 are Ov1, Ov2, Ov3, Ov4, and Ov5, which are shown in Fig. 1. If n = 2p is an even, then each line Ovi coincides with the line Ovp+i, giving only p such reflections; the remaining p symmetries are reflections in the lines Omi for i = 1, 2, . . . , p, where mi is the midpoint of the edge vi vi+1. For example, the six lines of symmetry of V6 are Ov1, Ov2, and Ov3, and Om1, Om2, and Om3, which are shown in Fig. 2. Definition. If X is a set and G is a group, then G acts on X if there is a function G · X ! X, denoted by (g, x) # gx, such that (1) ðghÞx ¼ gðhxÞ for all g, h 2 G and x 2 X. (2) ex = x for all x 2 X, where e is the identity in G. We also call X a G-set if G acts on X. If a group G acts on a set X, then fixing the first variable, say g, gives a function ag : X ! X, namely, ag : x # agx. This function is a permutation of X, for its inverse is ag1 : ag ag1 ¼ ae ¼ eX ¼ ag1 ag : It is easy to see that a : g # ag, is a homomorphism. Conversely, given any homomorphism u : G ! SX, define gx = u(g)(x). Thus, an action of a group G on a set X is another way of viewing a homomorphism G ! S X. Let X = {v1, v2, . . . , vn}, where vi, (i = 1, 2, . . . , n) is the vertices of a polygon, and G be the dihedral group D2n acting on X. G ¼ fT ; T 2 ; . . . ; T n ; ST ; ST 2 ; . . . ; ST n g, where T denotes a 2p n -rotation about the origin of the coordinate system (center of the equilateral polygon) and Tn = e, S denotes the reflection about a certain line and S2 = e, and e denotes an identity mapping.
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For each vertex vi 2 X, there is some g 2 G with gv1 = vi ; therefore, the orbit of v1 is O(v1) = X and D2n acts transitively. It is not difficult to prove that for any vi, the stabilizer Gvi is a subgroup of order 2.
2.2. Analogous symmetrical theorem of workspace for spatial parallel mechanisms In this paper, the workspace of the end-effector of a mechanism is defined as the volume within which the identified point, say, the geometry center of the mechanism, can be reached [2,21], which can also be considered as a set of points that satisfy a certain restricted equations. Therefore, in mathematics language, the workspace of the end-effector of a mechanism, denoted by X, is a mapping of the three dimensional Euclidean spaces R3: F : R3 ! X
ð1Þ
In applications, most spatial parallel mechanisms have symmetric structure. The symmetries can be depicted by a subgroup of a certain symmetric group Sn [20], no matter what kinds of symmetries the mechanism structure has—about a certain plane, about one axis or about one specified point. In order to exploit the relationship between the symmetries of the workspace and the structural characteristics of the kinematic chains, we proposed a symmetrical workspace theorem for spatial parallel mechanism [19]: If each identical kinematic chain of a spatial parallel mechanism always keeps collinear, and they are structural symmetric about a plane, an axis, or a point; then the workspace will also be symmetric about the corresponding plane, axis, or point individually. In fact, the above theorem can be considerably strengthened to be If the symmetry group of the workspace is denoted by GW, and the symmetry group of the kinematic chain structure of the end-effector in a particular configuration is denoted by GM: then GM must be a subgroup of GW, namely, the following relationship will always hold: GM GW
ð2Þ
Prior to proving the validation of the theorem, we also make the following assumptions [19]: (1) The absolute coordinates of the geometric center of the mechanism are denoted by r ¼ ½ x y z T . (2) The total number of the symmetric kinematic chains (need not require collinear for each chain) of the spatial parallel mechanism is indicated by n(n P 2). (3) The mechanism’s ith (i = 1, 2, . . . ,n) vertex linked by the ith 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 base’s ith (i = 1, 2, . . . , n) vertex linked by the ith kinematic chain is denoted by Bi, its absolute coor dinates are denoted by xABi y ABi zABi , where the superscript A indicates ‘‘absolute’’. As a result, the absolute coordinates of the ith vertex, Mi, can be obtained rAM i ¼ r þ RrLM i
ð3Þ
where R is a transformation matrix from the local coordinate system to the absolute one, T T are the absolute coordinates ½x y z are the absolute coordinates of rAM i ¼ xAM i y AM i zAM i L of ML i, r ¼ T L L the geometry center of the mechanism, and rM i ¼ xM i y M i zM i are the local coordinates of Mi. Proof. For mathematically, the workspace of the end-effector of a mechanism, X, is a mapping of the three dimensional Euclidean spaces R3 F : R3 ! X The workspace X can therefore be denoted as X ¼ frjF ðrÞ ¼ 0g where F(Æ) are a set of constraint equations of the workspace boundary surfaces
ð4Þ
J.-S. Zhao et al. / Mechanism and Machine Theory 43 (2008) 427–444
2
krAM 1 rAB1 k l1
431
3
7 6 A 6 krM 2 rAB2 k l2 7 7 6 F ðrÞ ¼ 6 7 .. 7 6 . 5 4 krAM m rABm k lm where m denotes the number of constraint equations. According to Eqs. (3), (5) can be expanded to be 3 2 kr þ RrLM 1 rAB1 k l1 7 6 6 kr þ RrLM 2 rAB2 k l2 7 7 6 F ðrÞ ¼ 6 7 .. 7 6 . 5 4
ð5Þ
ð6Þ
kr þ RrLM m rABm k lm
Suppose the symmetry operations in a particular configuration form a group GM. Obviously, GM must be a subgroup of a symmetry group S(V), which is a distance-preserving group and consists of the motions ui for which ui(vj) = vj. Consider the actions of the symmetry group GM on the workspace X defined by Eq. (4). For any symmetry operation ui 2 G, i = 1, 2, . . . , n, there is 3 2 3 2 L 1 A 1 kui ðrÞ þ Rui u1 kui ðrÞ þ RrLM 1 rAB1 k l1 i ðrM 1 Þ ui ui ðrB1 Þk ui ui ðl1 Þ 7 6 7 6 L 1 A 1 6 kui ðrÞ þ RrLM 2 rAB2 k l2 7 6 kui ðrÞ þ Rui u1 i ðrM 2 Þ ui ui ðrB2 Þk ui ui ðl2 Þ 7 7 6 7 6 ð7Þ F ðui ðrÞÞ ¼ 6 7¼6 7 .. .. 7 6 7 6 . . 5 4 5 4 L 1 A 1 kui ðrÞ þ RrLM m rABm k lm kui ðrÞ þ Rui u1 ðr Þ u u ðr Þk u u ðl Þ m i i i i i Mm Bm 1 Because u1 denotes a different permutation, i 2 GM and the binary operation is closed on GM, and ui r(1)r(2) . . . r(m), of the number series (1, 2, . . . , m). In Eq. (7), the actions of ui and u1 on vectors r, rLM j i A and rBj , and on scalar lj denote the corresponding permutations (or symmetry operations), where j = 1, 2, . . . , m. As a result, Eq. (7) can be transformed into 3 2 3 2 L 1 A 1 kr þ RrLM rð1Þ rABrð1Þ k lrð1Þ kr þ Ru1 i ðrM 1 Þ ui ðrB1 Þk ui ðl1 Þ 7 6 7 6 6 kr þ RrLM rAB k lrð2Þ 7 6 kr þ Ru1 ðrLM 2 Þ u1 ðrAB2 Þk u1 ðl2 Þ 7 i i i 7 6 rð2Þ rð2Þ 7 6 7 ð8Þ F ðui ðrÞÞ ¼ ui 6 7 ¼ ui 6 7 6 . .. 7 6 . 7 6 . . 5 4 5 4 L 1 A 1 kr þ Ru1 kr þ RrLM rðmÞ rABrðmÞ k lrðmÞ i ðrM m Þ ui ðrBm Þk ui ðlm Þ
From Eq. (8), we can find that the constraints 3 2 kr þ RrLM rð1Þ rABrð1Þ k lrð1Þ 7 6 6 kr þ RrLM rAB k lrð2Þ 7 7 6 rð2Þ rð2Þ 7 6 7 6 .. 7 6 . 5 4 L A kr þ RrM rðmÞ rBrðmÞ k lrðmÞ are equivalent to Eq. (6) for the differences only result from the new permutations of the sequence numbers of the constraint functions. As a result, Eq. (8) can be simplified as F ðui ðrÞ ¼ ui ðF ðrÞÞ
ð9Þ
Therefore, if the kinematic chain structure of the end-effector is kept unaltered by certain symmetry operation ui, i = 1, 2, . . . , n, then the workspace X = {rjF(r) = 0} is also unaltered under the symmetry operation ui, and therefore unaltered under the symmetry group GM.
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Because GW characterizes all the symmetries of the workspace, there must be GM GW. As a matter of fact, the relationship depicted by Eq. (2) can also be generalized to dexterous workspace. That is, GM GD if GD denote the dexterous workspace of the end-effector, which can be similarly proved. By the way, it is not difficulty to prove that if the order of the symmetry group of the mechanism structure is1 n, then the searching range of angle can be reduced to 0; 2p n . That is, the searching range can be reduced to n of the initial one. h 3. Examples and discussions Next, we will present the workspaces of some spatial parallel mechanisms to demonstrate the applications of the theorem and illustrate the utilities in conceptual design and engineering estimations. 3.1. The workspace of a Gough-Stewart parallel mechanism A general Gough-Stewart parallel mechanism is shown in Fig. 3. Assume the origin of the absolute coordinate system oxyz shown in Fig. 4 is superimposed with the geometry center of the fixed base plane B1 B2 B3 B4 B5 B6, the x-axis is parallel to B1 B6, the y-axis is perpendicular to B1 B6, and the z-axis is perpendicular to the base plane B1 B2 B3 B4 B5 B6. In similar manner, we also presume that the origin of the local coordinate system oL xL yL zL shown in Fig. 4 is superimposed with the geometry center of the mobile end-effector
Fig. 3. Gough-Stewart parallel mechanism.
Fig. 4. Top view of Gough-Stewart mechanism.
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M1 M2 M3 M4 M5 M6, the xL-axis is parallel to M1 M6, the yL-axis is perpendicular to M1 M6, and the zL-axis is perpendicular to the mobile plane M1 M2M 3M4 M5 M6. In the absolute coordinate system, the vertexes of the fixed base are denoted by Bi, i = 1, 2, . . . , 6, and the geometry center is o. The vectors of these vertexes are represented by rBi , respectively. In the local coordinate system, the coordinates of vertexes Mi of the mobile mechanism are represented by rLM i , the geometry center of the mobile end-effector is denoted by C, and the absolute coordinates are ð x y z Þ. The structure of the spatial parallel mechanism shown in Fig. 3 has very interesting characteristics: the hexagons M1 M2M3 M4 M5 M6 and B1 B2 B3 B4B5B6 are similar polygons; the kinematic chains of the end-effector M1 M2 M3 M4 M5 M6 are identical SPS chains and have the same actuations; the intersections of M1 B1 and M6 B6, M2 B2 and M3 B3, M4 B4 and M5 B5 in the top view shown in Fig. 4 just form the three vertexes of an equilateral triangle v1 v2 v3. It is not difficult to find that the symmetry operations of the Gough-Stewart mechanism shown in Fig. 4 forms a symmetric group S3, the elements of which are the permutations of a set V = {v1, v2, v3}. The exact number of the permutations is 3! = 6. Therefore, S3 = {u1, u2, u3, u4, u5,u6}, where u1 = (1), u2 = (123), u3 = (132), u4 = (23), u5 = (13), u6 = (12). It is not difficult to prove that S3 is just a dihedral group D6, which describes all the symmetries of the mechanism structure shown in Fig. 4. The first three elements of S3 represent the three rotations 0, 2p , and 4p about z-axis; and the last three denote three symmetries about ov1, ov2, and 3 3 ov3, respectively. Now, let us consider the actions of the symmetric group S3 on the workspace of the end-effector. The workspace of the mechanism can be denoted as X ¼ frjF ðrÞ ¼ 0g ð10Þ where F(Æ) are a set of constraint equations of the workspace boundary surfaces 3 2 A krM 1 rAB1 k l1 6 krA rA k l 7 27 6 M2 B2 7 ð11Þ F ðrÞ ¼ 6 .. 7 6 5 4 . krAM 6 rAB6 k l6 According to Eqs. (3), (11) can be expanded to be 3 2 kr þ RrLM 1 rAB1 k l1 6 kr þ RrL rA k l2 7 7 6 M2 B2 7 F ðrÞ ¼ 6 ð12Þ .. 7 6 5 4 . kr þ RrLM 6 rAB6 k l6 In the following, we will demonstrate that the workspace of the end-effector contains the symmetries of the mechanism structure. That is, the actions of dihedral group of the mechanism on the workspace expressed in (10) will make the workspace superimpose with itself. (1) The action of u1 on X. Obviously, u1(X) = X holds. (2) The action of u2 on X. Because 3 2 3 2 L 1 A 1 ku2 ðrÞ þ Ru2 u1 ku2 ðrÞ þ RrLM 1 rAB1 k l1 2 ðrM 1 Þ u2 u2 ðrB1 Þk u2 u2 ðl1 Þ 7 6 7 6 L 1 A 1 6 ku2 ðrÞ þ RrLM 2 rAB2 k l2 7 6 ku2 ðrÞ þ Ru2 u1 2 ðrM 2 Þ u2 u2 ðrB2 Þk u2 u2 ðl2 Þ 7 7 6 7 6 6 ku ðrÞ þ RrL rA k l3 7 6 ku ðrÞ þ Ru u1 ðrL Þ u u1 ðrA Þk u u1 ðl3 Þ 7 2 2 2 2 2 2 M3 B3 M3 B3 7 6 2 7 6 2 F ðu2 ðrÞÞ ¼ 6 7¼6 7 L 1 A 1 7 6 ku2 ðrÞ þ RrLM 4 rAB4 k l4 7 6 ku2 ðrÞ þ Ru2 u1 ðr Þ u u ðr Þk u u ðl Þ 4 2 2 2 2 2 M4 B4 7 6 7 6 6 ku ðrÞ þ RrL rA k l 7 6 ku ðrÞ þ Ru u1 ðrL Þ u u1 ðrA Þk u u1 ðl Þ 7 5 5 5 5 4 2 4 2 2 2 2 M5 B5 2 M5 2 B5 2 L A 1 L 1 A 1 ku2 ðrÞ þ RrM 6 rB6 k l6 ku2 ðrÞ þ Ru2 u2 ðrM 6 Þ u2 u2 ðrB6 Þk u2 u2 ðl6 Þ ð13Þ
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while u2 = (123) and u1 2 ¼ ð132Þ ¼ u3 , so Eq. (13) equals to 3 2 2 kr þ Ru3 ðrLM 1 Þ u3 ðrAB1 Þk u3 ðl1 Þ kr þ RrLM 5 6 kr þ RrL 6 kr þ Ru ðrL Þ u ðrA Þk u ðl Þ 7 6 6 3 M2 3 B2 3 2 7 M6 7 6 6 6 kr þ RrL 6 kr þ Ru3 ðrL Þ u3 ðrA Þk u3 ðl3 Þ 7 M3 B3 M1 7 6 6 F ðu2 ðrÞÞ ¼ u2 6 7 ¼ u2 6 6 kr þ RrLM 6 kr þ Ru3 ðrLM Þ u3 ðrAB Þk u3 ðl4 Þ 7 4 4 2 7 6 6 7 6 6 4 kr þ RrLM 3 4 kr þ Ru3 ðrLM 5 Þ u3 ðrAB5 Þk u3 ðl5 Þ 5 kr þ Ru3 ðrLM 6 Þ u3 ðrAB6 Þk u3 ðl6 Þ
rAB5 k l5
3
rAB6 k l6 7 7 7 rAB1 k l1 7 7 7 ¼ u2 ðF ðrÞÞ rAB2 k l2 7 7 7 rAB3 k l3 5
ð14Þ
kr þ RrLM 4 rAB4 k l4
Therefore, the action of u2 on X makes X rotate 2p about the z-axis and superimpose with itself. The action of 3 u3 on X can be similarly found to be a 4p -rotation about the z-axis and superimpose with itself. 3 (3) The action of u4 on X. Because 2 ku4 ðrÞ þ RrLM 1 6 ku ðrÞ þ RrL 6 4 M2 6 6 ku ðrÞ þ RrL M3 6 4 F ðu4 ðrÞÞ ¼ 6 6 ku4 ðrÞ þ RrLM 4 6 6 L ku ðrÞ þ Rr 4 4 M5
rAB1 k l1
3
rAB2 k l2 7 7 7 rAB3 k l3 7 7 7 rAB4 k l4 7 7 7 rAB5 k l5 5
ð15Þ
ku4 ðrÞ þ RrLM 6 rAB6 k l6 while u4 is a symmetry operation of the mechanism and u1 4 ¼ ð23Þ ¼ u4 , Eq. (15) can be further transformed into 3 2 3 2 L 1 A 1 ku4 ðrÞ þ Ru4 u1 kr þ Ru4 ðrLM 1 Þ u4 ðrAB1 Þk u4 ðl1 Þ 4 ðrM 1 Þ u4 u4 ðrB1 Þk u4 u4 ðl1 Þ 7 6 6 kr þ Ru ðrL Þ u ðrA Þk u ðl Þ 7 L 1 A 1 7 6 ku4 ðrÞ þ Ru4 u1 6 4 M2 4 B2 4 2 7 4 ðrM 2 Þ u4 u4 ðrB2 Þk u4 u4 ðl2 Þ 7 6 7 6 7 6 1 L 1 A 1 L A 7 6 6 ku4 ðrÞ þ Ru4 u4 ðrM 3 Þ u4 u4 ðrB3 Þk u4 u4 ðl3 Þ 7 6 kr þ Ru4 ðrM 3 Þ u4 ðrB3 Þk u4 ðl3 Þ 7 7 6 ¼ u F ðu4 ðrÞÞ ¼ 6 7 6 4 7 1 L 1 A 1 6 kr þ Ru4 ðrLM Þ u4 ðrAB Þk u4 ðl4 Þ 7 6 ku4 ðrÞ þ Ru4 u4 ðrM 4 Þ u4 u4 ðrB4 Þk u4 u4 ðl4 Þ 7 4 4 7 6 7 6 7 6 L A 6 ku4 ðrÞ þ Ru4 u1 ðrL Þ u4 u1 ðrA Þk u4 u1 ðl5 Þ 7 kr þ Ru ðr Þ u ðr Þk u ðl Þ 5 4 5 4 4 4 M B 4 M5 4 B5 4 5 4 5 5 L A 1 L 1 A 1 kr þ Ru4 ðrM 6 Þ u4 ðrB6 Þk u4 ðl6 Þ ku4 ðrÞ þ Ru4 u4 ðrM 6 Þ u4 u4 ðrB6 Þk u4 u4 ðl6 Þ ð16Þ Therefore, Eq. (16) is equivalent to 3 2 kr þ RrLM 6 rAB6 k l6 6 kr þ RrL rA k l 7 6 57 M5 B5 7 6 6 kr þ RrL rA k l4 7 M4 B4 7 6 F ðu4 ðrÞÞ ¼ u4 6 7 ¼ u4 ðF ðrÞÞ 6 kr þ RrLM rAB k l3 7 3 3 7 6 7 6 4 kr þ RrLM 2 rAB2 k l2 5
ð17Þ
kr þ RrLM 1 rAB1 k l1 Consequently, the action of u4 on X makes X reflect about the yoz-plane. The actions of u5 and u6 on X can be similarly found to be reflections about the planes of v2oz and v3oz, individually. Therefore, the workspace of the Gough-Stewart mechanism has all the symmetries of the mechanism. The workspace can be investigated as follows. Suppose the coordinate transformations, R, from the mobile end-effector to the fixed base are 231-type, there is
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2 6 R¼4 2 6 ¼4
cos h
0
0
1
sin h
0
sin h
32
cos w
76 0 54 sin w cos h
0
sin w cos w 0
0
32
1
0
76 0 54 0
cos u
0
sin u
1
0
435
3
7 sin u 5 cos u
cos w cos h
sin w cos h cos u þ sin h sin u
sin w cos h sin u þ sin h cos u
3
sin w
cos w cos u
cos w sin u
7 5
cos w sin h
sin w sin h cos u þ cos h sin u
sin w sin h sin u þ cos h cos u
ð18Þ
Substituting Eq. (18) into Eq. (3) results in rAM i ¼ r þ RrLM i ; i ¼ 1; 2; . . . ; 6 ð19Þ 2 3 2 3 xM i x Assume rLM i ¼ 4 y M i 5; i ¼ 1; 2; . . . ; 6 and r ¼ 4 y 5, Eq. (19) can be expanded to yield z zM i 3 2 x þ xLM i cos wcos h þ y LM i ð sin wcos hcos u þ sin hsin uÞ þ zLM i ðsin wcos hsin u þ sin hcos uÞ 7 6 y þ xLM i sin w þ y LM i cos w cosu zLM i cos w sinu rAM i ¼ r þ ArLM i ¼ 4 5 L L L z xM i cos wsin h þ y M i ðsin w sinh cos u þ cos h sinuÞ þ zM i ð sin wsin hsin u þ cos hcos uÞ ð20Þ
Substituting Eq. (20) into Eq. (4) yields a set of constraint equations of the workspace of the end-effector. To obtain the workspace boundary surfaces, we can resort to programming at a computer. Therefore, suppose 3 2 3 2 f1 ðx; y; zÞ f1 6 f 7 6 f ðx; y; zÞ 7 7 6 27 6 2 7 6 7 6 6 f3 7 6 f3 ðx; y; zÞ 7 7 6 7¼6 ð21Þ 6 f 7 6 f ðx; y; zÞ 7 7 6 47 6 4 7 6 7 6 4 f5 5 4 f5 ðx; y; zÞ 5 f6 ðx; y; zÞ f6 where
f1 ðx; y;zÞ ¼ ½x þ xLM 1 cos w cos h þ y LM 1 ðsinw cos h cos u þ sinh sinuÞ þ zLM 1 ðsinw cos hsin u þ sinh cos uÞ xB1 2 þ ½y þ xLM 1 sinw þ y LM 1 cos w cos u zLM 1 cos w sinu y B1 2 þ ½z xLM 1 cos w sinh þ y LM 1 ðsinw sin hcos u þ cos h sinuÞ þ zLM 1 ð sinw sinhsin u þ cos h cos uÞ zB1 2 f2 ðx; y;zÞ ¼ ½x þ xLM 2 cos w cos h þ y LM 2 ðsinw cos h cos u þ sinh sinuÞ þ zLM 2 ðsinw cos hsin u þ sinh cos uÞ xB2 2 þ ½y þ xLM 2 sin w þ y LM 2 cos w cos u zLM 2 cos w sinu y B2 2 þ ½z xLM 2 cos w sin h þ y LM 2 ðsin w sinhcos u þ cos hsinuÞ þ zLM 2 ðsinw sin hsinu þ cos hcos uÞ zB2 2 f3 ðx; y;zÞ ¼ ½x þ xLM 3 cos w cos h þ y LM 3 ðsinw cos h cos u þ sinh sinuÞ þ zLM 3 ðsinw cos hsin u þ sinh cos uÞ xB3 2 þ ½y þ xLM 3 sinw þ y LM 3 cos w cos u zLM 3 cos w sinu y B3 2 ½z xLM 3 cos w sinh þ y LM 3 ðsinw sin hcos u þ cos h sinuÞ þ zLM 3 ð sinw sinh sinu þ cos h cos uÞ zB3 2 f4 ðx; y;zÞ ¼ ½x þ xLM 4 cos w cos h þ y LM 4 ðsinw cos h cos u þ sinh sinuÞ þ zLM 4 ðsinw cos hsin u þ sinh cos uÞ xB4 2 þ ½y þ xLM 4 sin w þ y LM 4 cos w cos u zLM 4 cos w sin u y B4 2 ½z xLM 4 cos w sinh þ y LM 4 ðsinw sin hcos u þ cos h sinuÞ þ zLM 4 ð sinw sinh sinu þ cos h cos uÞ zB4 2
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f5 ðx; y; zÞ ¼ ½x þ xLM 5 cos w cos h þ y LM 5 ðsinw cos hcos u þ sinhsin uÞ þ zLM 5 ðsin w cos hsinu þ sinhcos uÞ xB5 2 þ ½y LM 5 þ xLM 5 sin w þ y LM 5 cos w cos u zLM 5 cos w sin u y B5 2 ½z xLM 5 cos w sin h þ y LM 5 ðsinw sinhcos u þ cos hsinuÞ þ zLM 5 ðsinw sin hsinu þ cos hcos uÞ zB5 2 f6 ðx; y; zÞ ¼ ½x þ xLM 6 cos w cos h þ y LM 6 ðsinw cos hcos u þ sinhsin uÞ þ zLM 6 ðsin w cos hsinu þ sinhcos uÞ xB6 2 þ ½y þ xLM 6 sinw þ y LM 6 cos w cos u zLM 6 cos w sin u y B6 2 ½z xLM 6 cos w sin h þ y LM 6 ðsinw sinhcos u þ cos hsinuÞ þ zLM 6 ðsinw sin hsinu þ cos hcos uÞ zB6 2
Assume that the coordinates of Mi, i = 1, 2, . . . , 6 in the local coordinate system oLxLyLzL are a2 r sin 5p a2 0 ; M 1 r cos 5p 6 6 5p M 2 r cos 6 þ a2 r sin 5p þ a2 0 ; 6 M 3 r cos 3p a2 r sin 3p a2 0 ; 2 2 M 4 r cos 3p þ a2 r sin 3p þ a2 0 ; 2 2 M 5 r cos p6 a2 r sin p6 a2 0 ; p a p a M 6 r cos 6 þ 2 r sin 6 þ 2 0 : The coordinates of Bi, i = 1, 2, . . . , 6 in the absolute coordinate system oxyz are B1 R cosðp2 þ b2Þ R sinðp2 þ b2Þ 0 ; B2 R cosð7p b2Þ R sinð7p b2Þ 0 ; 6 6 B3 R cosð7p þ b2Þ R sinð7p þ b2Þ 0 ; 6 6 B4 R cosð p6 b2Þ R sinð p6 b2Þ 0 ; B5 R cosð p6 þ b2Þ R sinð p6 þ b2Þ 0 ; B6 R cosðp2 b2Þ R sinðp2 b2Þ 0 : Suppose the lengths of B1 M1, B2 M2, B3 M3, B4 M4, B5 M5, B6 M6 are variables subjected to li 2 [lmin, lmax], i = 1, 2, . . . , 6. Given the values of the parameters: a, b, r, R, lmin, lmax, we can obtain the boundary surfaces of the workspace of the end-effector with the searching procedure which can be found in [5,6,22,23] 1. Solve the lowest position ð xLow y Low zLow Þ of the identified point of the end-effector when each of its legs has shrunk to the least length; and solve the highest position xHig y Hig zHig of the identified point of the end-effector when each of its legs has stretched to the largest length. 2. Establish parametric equations of cylindrical coordinates of the point within the workspace 8 > < x ¼ q cosðiDcÞ > :
y ¼ q sinðiDcÞ
ð22Þ
z ¼ zLow þ jDz
2p , and where q is the searching polar radius at plane z = zLow + jDz with a polar angle iDc, and 0 6 i < Dc zHig zLow 0 < j < Dz , and Dc is the angular interval and Dz is the distance interval along z-axis, and i and j are all natural numbers. 3. Suppose the workspace is sliced with an interval Dz along z-axis into k 1 planes, which are all parallel to z z xoy-plane, and where k is the largest natural number that is less than or equal to Hig Dz Low . And therefore, 0 < j 6 k. 4. Search the boundary curve at the jth plane. Let q = q(jDz, iDc) increase from zero with an expected interval Dq, namely q (jDz, iDc) = qjDq, qj = 1, 2, . . . . The stop criterion for such increasing is that when Eq. (22) is substituted into Eq. (21), there is at least one fp, p = 1, 2, . . . , 6, such that fp > l2max ; or fp < l2min for a set of pose angles h, w, u. Select the largest parameter q(jDz, iDc) to plot the boundary curve in the jth plane with 2p Eq. (22) as i increase from zero to Dc .
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5. When the curves in all of the slices are obtained, the boundary surfaces of the reachable workspace of the end-effector are gained. Different workspace shapes of the Gough-Stewart type mechanism are shown in Figs. 5–7. Although the workspace shapes are different with different structure parameters, the symmetry relationships between the workspace and the mechanism structure are still governed by Eq. (2). Therefore, the searching processes can be reduced to search the range of a generating group of the symmetric group of the mechanism. For a symmetric mechanism, if the order of the symmetry group of the mechanism structure is n, then the searching 2p range of angle about the z-axis can be reduced to 0; , for an example, the searching range of angle about the n z-axis can be reduced to 0; p3 . And as a result, the searching task and time are greatly reduced. 3.2. The workspace of a spatial 4-UPU parallel mechanism A Schoenflies type parallel mechanism is shown in Fig. 8. The mechanism proposed herein is actuated by four identical UPU (universal joint + prismatic joint + universal joint) kinematic chains [24].
Fig. 5. Different workspaces of the end-effector of a Gough-Stewart mechanism.
Fig. 6. Top views of the workspaces of the end-effector corresponding to Fig. 5.
Fig. 7. The mushroom workspace entity covered by the three boundary surfaces of Fig. 5. Left is the isometric view of the workspace entity, the middle is the top view and the right is a bottom view.
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The top view of the Schoenflies type spatial parallel mechanism is shown in Fig. 9. Therefore, we can consider a symmetric group S4, the elements of which are the permutations of a set V = {B1, B2, B3, B4}. The exact number of the permutations is 4! = 24. However, not all of the 24 permutations can form real symmetries of the mechanism shown in Fig. 9. As a matter of fact, the possible symmetries of the mechanism shown in Fig. 9 1M2k 1 B2 k are eight at most. In order to avoid the similar singularity, we knew that kM 6¼ kB [24]. Therefore, the kM 1 M 4 k kB1 B4 k symmetry group of the kinematic chains of the end-effector is G = {u1,u2,u3,u4}, where u1 = (1), u2 = (13)(24), u3 = (12)(34), u4 = (14)(23) and, G is a subgroup of dihedral group D8. G describes all the symmetries of the mechanism structure shown in Fig. 9. The first two elements of G express the two rotations of angles 0 and p about z-axis; and the last two denote two symmetries about xoz- and yoz-planes, respectively. Now, let us consider the actions of group G on the workspace of the end-effector. The workspace of the mechanism can be denoted as 2 A 3 jrM 1 rAB1 k l1 6 A 7 6 jrM 2 rAB2 k l2 7 7 ð23Þ F ðrÞ ¼ 6 6 krA rA k l 7 35 4 M3 B3 krAM 4 rAB4 k l4 According to Eqs. (1) and (23) can be expanded to 2 3 kr þ RrLM 1 rAB1 k l1 6 7 6 kr þ RrLM 2 rAB2 k l2 7 6 7 F ðrÞ ¼ 6 7 L A 4 kr þ RrM 3 rB3 k l3 5 kr þ RrLM 4 rAB4 k l4 (1) The action of u1 on X. Obviously, u1(X) = X holds. (2) The action of u2 on X. Because 2 3 ku2 ðrÞ þ RrLM 1 rAB1 k l1 6 7 6 ku2 ðrÞ þ RrLM 2 rAB2 k l2 7 7 F ðu2 ðrÞÞ ¼ 6 6 ku ðrÞ þ RrL rA k l 7 35 4 2 M3 B3 ku2 ðrÞ þ RrLM 4 rAB4 k l4
ð24Þ
ð25Þ
while u2 is a symmetry operation of the mechanism and u1 2 ¼ ð13Þð24Þ ¼ u2 , Eq. (25) is further equivalent to
Fig. 8. A kind of Schoenflies type spatial parallel mechanism.
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Fig. 9. The top view of the Schoenflies type spatial parallel mechanism.
3 2 3 ku2 ðrÞ þ Ru2 u2 ðrLM 1 Þ u2 u2 ðrAB1 Þk u2 u2 ðl1 Þ kr þ RrLM 3 rAB3 k l3 6 ku ðrÞ þ Ru u ðrL Þ u u ðrA Þk u u ðl Þ 7 6 kr þ RrL rA k l 7 6 2 2 2 M2 2 2 B2 2 2 2 7 47 6 M4 B4 7 F ðu2 ðrÞÞ ¼ 6 7 ¼ u2 ðF ðrÞÞ L A 6 ku2 ðrÞ þ Ru2 u2 ðrLM 3 Þ u2 u2 ðrAB3 Þk u2 u2 ðl3 Þ 7 ¼ u2 6 4 kr þ Rr r k l 15 M1 B1 5 4 kr þ RrLM 2 rAB2 k l2 ku2 ðrÞ þ Ru2 u2 rLM 4 u2 u2 rAB4 k u2 u2 ðl4 Þ 2
ð26Þ Therefore, the action of u2 on X makes X rotate p about the z-axis and superimpose with itself. (3) the action of u3 on X. Because 2 ku3 ðrÞ þ RrLM 1 6 ku ðrÞ þ RrL 6 3 M2 F ðu3 ðrÞÞ ¼ 6 4 ku3 ðrÞ þ RrLM 3 ku3 ðrÞ þ RrLM 4
3 rAB1 k l1 rAB2 k l2 7 7 7 rAB3 k l3 5 rAB4 k l4
ð27Þ
while u3 is a symmetry operation of the mechanism and u1 3 ¼ ð12Þð34Þ ¼ u3 , Eq. (23) is further equivalent to 3 3 2 2 kr þ RrLM 2 rAB2 k l2 ku3 ðrÞ þ Ru3 u3 ðrLM 1 Þ u3 u3 ðrAB1 Þk u3 u3 ðl1 Þ 7 7 6 6 6 kr þ RrLM 1 rAB1 k l1 7 6 ku3 ðrÞ þ Ru3 u3 ðrLM 2 Þ u3 u3 ðrAB2 Þk u3 u3 ðl2 Þ 7 7 7 6 6 F ðu3 ðrÞÞ ¼ 6 7 ¼ u3 6 kr þ RrL rA k l 7 ¼ u3 ðF ðrÞÞ L A 45 4 4 ku3 ðrÞ þ Ru3 u3 ðrM 3 Þ u3 u3 ðrB3 Þk u3 u3 ðl3 Þ 5 M4 B4 kr þ RrLM 3 rAB3 k l3 ku3 ðrÞ þ Ru3 u3 ðrLM 4 Þ u3 u3 ðrAB4 Þk u3 u3 ðl4 Þ ð28Þ
Fig. 10. Different workspaces of a Schoenflies type parallel mechanism.
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Therefore, the action of u3 on X makes it reflect about the xoz-plane. The action of u4 on X can be similarly found to be a reflection about the plane of yoz. Therefore, the workspace of the Schoenflies type parallel mechanism shown in Fig. 10 contains all the symmetries of those of the mechanism. The workspace of the end-effector can be analyzed below. The end-effector has four DoFs (one rotational DoF and three orthogonal translational DoFs), which can be analyzed according to [25]. Correspondingly, we can only select four independent variables, three of which are the absolute coordinates of the geometry center of the end-effector, denoted by (x, y, z); and the rest is the rotational angle about z-axis, denoted by a. As a result, the transformation matrix from the local coordinate system to the absolute one can be expressed as 2 3 cos a sin a 0 6 7 R ¼ 4 sin a cos a 0 5 ð29Þ 0
0
1
Suppose the coordinates of the ith, (i = 1, 2, 3, 4) vertex of the end-effector in the local coordinate system are denoted by M i ðxLM i ; y LM i ; zLM i Þ, the absolute coordinates of M i ðxAM i ; y AM i ; zAM i Þ can be obtained 2 A 3 2 3 2 L 3 2 3 2 32 L 3 2 3 xp i xM i x þ xLM i cos a y LM i sin a xM i x x cos a sin a 0 6 yA 7 6 7 6 7 6 7 6 76 L 7 6 7 ð30Þ 4 M i 5 ¼ 4 y 5 þ R4 y LM i 5 ¼ 4 y 5 þ 4 sin a cos a 0 54 y pi 5 ¼ 4 y þ xLM i sin a þ y LM i cos a 5 zAM i
z
zLM i
0
z
0
zLpi
1
z þ zLM i
The actuation condition of the ith kinematic chain will be 2
kM i Bi k L2i ¼ 0
ð31Þ
Expanding the above equation yields ðx þ xLM i cos a y LM i sin a xABi Þ2 þ ðy þ xLM i sin a þ y LM i cos a y ABi Þ2 þ ðz þ zLM i zABi Þ2 L2i ¼ 0
ð32Þ
According to Eq. (5), we have 3 2 ðx þ xLM 1 cos a y LM 1 sin a xAB1 Þ2 þ ðy þ xLM 1 sin a þ y LM 1 cos a y AB1 Þ2 þ ðz þ zLM 1 zAB1 Þ2 L21 7 6 6 ðx þ xL cos a y L sin a xA Þ2 þ ðy þ xL sin a þ y L cos a y A Þ2 þ ðz þ zL zA Þ2 L2 7 M2 M2 B2 M2 M2 B2 M2 B2 27 6 F ðx; y; z; aÞ ¼ 6 7 6 ðx þ xL cos a y L sin a xA Þ2 þ ðy þ xL sin a þ y L cos a y A Þ2 þ ðz þ zL zA Þ2 L2 7 M3 M3 B3 M3 M3 B3 M3 B3 35 4 ðx þ xLM 4 cos a y LM 4 sin a xAB4 Þ2 þ ðy þ xLM 4 sin a þ y LM 4 cos a y AB4 Þ2 þ ðz þ zLM 4 zAB4 Þ2 L24 ð33Þ ð xLM i
y LM i
zLM i
ð xABi
y ABi
zABi
Given the values of the parameters: Þ, Þ and the actuations li 2 ½lmin ; lmax , we can get the workspace of the end-effector with a similar procedure of Section 3.1. However, the searching range of the angle a can be reduced to 0; p2 for the mechanism structure has symmetry planes xoz and yoz. The workspaces are different if the structure parameters are different, some examples of which are shown in Figs. 10–12.
Fig. 11. Top views of the workspaces corresponding to those of Fig. 10.
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Fig. 12. Workspace entities covered by the three boundary surfaces of Fig. 10.
3.3. The workspace of a 3-RUU spatial parallel mechanism A 3-RUU spatial parallel mechanism is shown in Fig. 13. The mechanism is actuated by three identical RUU (revolute joint + universal joint + universal joint) kinematic chains. It is not difficult to find that the end-effector has three translational DoFs within its workspace according to [25]. The top view of the spatial parallel mechanism is shown in Fig. 14.
Fig. 13. A 3-RUU spatial parallel mechanism.
Fig. 14. The top view of the 3-RUU spatial parallel mechanism.
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When the identified point of the end-effector (geometry center OE) is superimposed with that of the fixed base, OB, the configuration symmetries of the kinematic chains of the end-effector is depicted by a symmetric group S3, which has been discussed in Section 3.1. Therefore, the structure symmetries of the mechanism under such a configuration can be depicted by a dihedral group D6. The first three elements of S3 represent the three rotations 0, 2p , and 4p about z-axis; and the last three denote three symmetries about oBB3C3, oBB1C1, 3 3 and oBB2C2, respectively. The workspace shapes of the end-effector are shown in Figs. 15–17. The searching range can also be reduced to 16 of the initial one with their symmetry characteristics. Many more such examples can be found in a number of recently published literatures [6,7,22,23,26–32].
Fig. 15. Different workspaces of the end-effector of a 3-RUU spatial mechanism.
Fig. 16. Top views of the workspaces of the end-effector corresponding to Fig. 15.
Fig. 17. Workspace entities covered by the three boundary surfaces of Fig. 15.
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4. Conclusions This paper further probes the analogous symmetry characteristics of the workspace of a spatial parallel mechanism. Based on the work of [19], we extend the theorem to be that the symmetry group of the kinematic chain structure of the end-effector must be a subgroup of that of the workspace of the end-effector. Examples demonstrate its applications in investigating the geometry shapes of the workspace, especially for those with complicated symmetry properties. The theorem offers a concrete safeguard for reducing the searching tasks of the workspace boundary surfaces of any symmetric spatial parallel mechanisms. For a symmetric mechanism, if the order of the symmetry group of the mechanism structure is n, then the searching range can be reduced to 1 of the initial one. n Acknowledgements This research was supported by FANEDD, China Postdoctoral Science Foundation and the National Natural Science Foundation of China under Grant 50425516. The authors gratefully acknowledge these support agencies. The authors would also like to thank Dr. Simon Guest in the Department of Engineering at University of Cambridge for his invaluable suggestions and the anonymous reviewers for their precious comments. References [1] D. Sen, T.S. Mruthyunjaya, A centro-based characterization of singularities in the workspace of planar closed-loop mechanisms, Mechanism and Machine Theory 33 (8) (1998) 1091–1104. [2] J.-P. Merlet, C. Gosselin, N. Mouly, Workspaces of planar parallel manipulators, Mechanism and Machine Theory 33 (1) (1998) 7– 20. [3] S. Dibakar, T.S. Mruthyunjaya, A computational geometry approach for determination of boundary of workspaces of planar mechanisms with arbitrary topology, Mechanism and Machine Theory 34 (1) (1999) 149–169. [4] 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. [5] 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. [6] I.A. Bonev, J. Ryu, A new approach to orientation workspace analysis of 6-DOF parallel mechanisms, Mechanism and Machine Theory 36 (1) (2001) 15–28. [7] C. Gosselin, Determination of the workspace of 6-DoF parallel manipulators, Journal of Mechanical Design 112 (3) (1990) 331–336. [8] C.M. Gosselin, M. Jean, Determination of the workspace of planar parallel manipulators with joint limits, Robotics and Autonomous Systems 17 (3) (1996) 129–138. [9] J. Yang, Karim Abdel-Malek, Y. Zhang, On the workspace boundary determination of serial manipulators with non-unilateral constraints, robotics and computer integrated manufacturing, doi:10.1016/j.rcim.2006.06.005. [10] Y. Wang, G.S. Chirikjian, Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N), IEEE Transactions on Robotics 20 (3) (2004) 399–408. [11] Karim Abdel-Malek, Frederick Adkins, et al., On the determination of boundaries to manipulator workspaces, Robotics and Computer Integrated Manufacturing 13 (1) (1997) 63–72. [12] Karim Abdel-Malek, Harn-Jou Yeh, Saib Othman, Interior and exterior boundaries to the workspace of mechanical manipulators, Robotics and Computer Integrated Manufacturing 16 (5) (2000) 365–376. [13] Z.-C. Lai, C.-H. Menq, The dexterous workspace of simple manipulators, IEEE Transactions on Robotics 4 (1) (1988) 99–103. [14] H. Li, C.M. Gosselin, M.J. Richard, Determination of maximal singularity-free zones in the workspace of planar three-degree-offreedom parallel mechanisms, Mechanism and Machine Theory 41 (10) (2006) 1157–1167. [15] H. Li, C.M. Gosselin, M.J. Richard, Determination of maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform, Mechanism and Machine Theory 42 (4) (2007) 497–511. [16] M. Gouttefarde, C.M. Gosselin, Analysis of the Wrench-closure workspace of planar parallel cable-driven mechanisms, IEEE Transactions on Robotics 22 (3) (2006) 434–445. [17] I.A. Bonev, C.M. Gosselin, Analytical determination of the workspace of symmetrical spherical parallel mechanisms, IEEE Transactions on Robotics 22 (5) (2006) 1011–1017. [18] J. Jesu´s Cervantes-Sa´nchez, J. Ce´esar Herna´ndez-Rodrı´uez, et al., Symmetric manipulator: workspace and singularity characterization, Mechanism and Machine Theory 39 (4) (2004) 409–429. [19] J.-S. Zhao, M. Chen, et al., Workspace of mechanisms with symmetry identical kinematic chains, Mechanism and Machine Theory 41 (6) (2006) 632–645. [20] Joseph.J. Rotman, Advanced Modern Algebra, Pearson Education, Inc., Publishing as Prentice-Hall, 2002. [21] A. Kumar, K.J. Waldron, The workspaces of a mechanical mechanism, Journal of Mechanical Design 103 (1981) 665–672.
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