Results on Planar Parallel Manipulators with Cylindrical Singularity Surface

Transcription

1 Results on Planar Parallel Manipulators with Cylindrical Singularity Surface Georg Nawratil Institute of Discrete Mathematics and Geometry Differential Geometry and Geometric Structures 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France

2 Table of Contents [1] Introduction [2] Preliminary considerations [3] The Main Theorem [4] Preparatory work for the proof [5] Sketch of the proof [6] A further example [7] Remark & Conclusion [8] References 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 1

3 [1] Singular configurations of SGP The geometry of a Stewart Gough Platform is given by the six base anchor points M i := (A i, B i, C i ) T in the fixed space Σ 0 and by the six platform anchor points m i := (a i, b i, c i ) T in the moving space Σ. Σ m i Theorem Merlet [1992] A SGP is singular iff the carrier lines L i of L i the six legs belong to a linear line complex. Σ 0 M i Σ 0 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 2

4 [1] Analytical condition Plücker coordinates of L i can be written as (l i, l i ) := (R m i + t KM i,m i l i ) with R := (r ij ) = e2 0 + e 2 1 e 2 2 e 2 3 2(e 1 e 2 + e 0 e 3 ) 2(e 1 e 3 e 0 e 2 ) 2(e 1 e 2 e 0 e 3 ) e 2 0 e e 2 2 e 2 3 2(e 2 e 3 + e 0 e 1 ), 2(e 1 e 3 + e 0 e 2 ) 2(e 2 e 3 e 0 e 1 ) e 2 0 e 2 1 e e 2 3 t := (cosϕt 1 sinϕt 2,sin ϕt 1 + cosϕt 2, t 3 ) T and K := e e e e 2 3. Remark: The group SO 3 is parametrized by Euler Parameters (e 0,e 1, e 2, e 3 ). L i belong to a linear line complex Q := det(q) = 0 with Q := b l1 l l 6 b l6 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 3

5 [2] Preliminary considerations Definition SGP with Cylindrical Singularity Surface The manipulators singularity set is for any orientation of the platform a cylindrical surface with rulings parallel to a given fixed direction p in the space of translations. The set of SGPs with a cylindrical singularity surface contains the set of architecture singular SGPs. These two sets are distinct due to: Example [see Figure] m 1 = m 2, m 3 = m 4, m 5 = m 6 M 1 M 2 M 3 M 4 M 5 M 6 p m 5 M 5 M 3 m 2 M 1 m 1 m 6 m 3 m 4 M 1,...,M 6 can be coplanar M 6 M 4 M 2 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 4

6 [2] Preliminary considerations This manipulator is only in a singular configuration iff the three planes [M 1,M 2,m 1 ], [M 3,M 4,m 3 ] and [M 5,M 6,m 5 ] have a common intersection line. The singularity surface is a quadratic cylinder. m 1 m 2 Is this the only SGP with this property? We distinguish between planar and non-planar SGPs because the structure of architecturally singular SGPs depends on the planarity of the platform and the base; cf. Karger [2003,2008]. m 5 m 6 M 5 M 6 M 3 M 4 M 1 M 2 m 3 In this paper we only deal with planar SGPs. m 4 Projection direction is p. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 5

7 [3] The Main Theorem Main Theorem The set of planar parallel manipulators with no four anchor points on a line which possess a cylindrical singularity surface with rulings parallel to a given fixed direction p for any orientation of the platform equals the set of planar architecture singular manipulators (with no four anchor points on a line). Idea of the proof We choose an Cartesian frame with one axis t i p. Then Q := det(q) = 0 must be independent of t i for all e 0,...,e 3, t j, t k. The analytical proof is based on the resulting equations. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 6

8 [4] Preparatory work for the proof [A] Choose of Cartesian frames in the fixed space and the moving space As we consider only SGPs with planar platform we set c i = 0 for i = 1,...,6. We set up the planar base in a more general position as C 1 = 0, C i = [C 2 (B 3 A i A 3 B i ) + A 2 C 3 B i ] /(A 2 B 3 ) for i = 4, 5,6. Lemma of Karger [2003] For planar parallel manipulators with no four points on a line we can assume A 1 = B 1 = B 2 = a 1 = b 1 = b 2 = 0 A 2 B 3 B 4 B 5 a 2 (a 4 a 3 )coll(3, 4,5) 0 and with coll(i, j,k) := a i (b j b k ) + a j (b k b i ) + a k (b i b j ). 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 7

9 [4] Preparatory work for the proof [B] Algebraic characterization of the subset of architecture singular SGPs We perform the same elementary row operations with the matrix Q as described by Karger [2003]. Then the last row of Q is of the form (r 11 K 1 + r 12 A 2 K 2, r 21 K 1 + r 22 A 2 K 2,r 31 K 1 + r 32 A 2 K 2,r 21 C 2 K 3 + r 22 C 2 K 4, r 31 A 2 K 3 + r 32 A 2 K 4 r 11 C 2 K 3 r 12 C 2 K 4, r 21 A 2 K 3 r 22 A 2 K 4 )D 1 with D := A 2 B 3 B 4 B 5 coll(3, 4,5) and r ij the entries of the rotary matrix R. Theorem of Karger [2003] K 1 = K 2 = K 3 = K 4 = 0 are the four conditions which are satisfied iff a planar parallel manipulator with no four points on a line is architecturally singular. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 8

10 [5] Sketch of the proof I) Base is not parallel to p (i) Base is orthogonal to p (ii) Base is not orthogonal to p II) Base is parallel to p (i) M 1 M 2 is parallel to p (ii) M 1 M 2 is not parallel to p (a) M 1 M 2 is orthogonal to p (b) M 1 M 2 is not orthogonal to p 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 9

11 I) Base is not parallel to p (i) Base is orthogonal to p (C 2 = C 3 = 0) The proof of this case is hidden in the proof of the Theorem of Karger [2003]. Karger sets t 1 = t 2 = 0 and eliminates t 3 from Q. He proves in four steps (a),...,(d) that the resulting equations can only vanish for K 1 =... = K 4 = 0. (ii) Base is not orthogonal to p We start such as Karger by setting t 1 = t 2 = 0. Now Q can be written as Q = A 2 2(r 11 r 22 r 12 r 21 )Q 3 t A 2 B 3 Q 2 t Q 1 t 3 + Q 0. With the coefficients Q 1, Q 2,Q 3 the steps (a) and (b) can be done one by one. The steps (c) and (d) are different and therefore given in Nawratil [2008,A]. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 10

12 II) Base is parallel to p (C 2 = C 3 = 0) We eliminate t 1 from Q. We denote the coefficients of t i 1t j 2 tk 3 from Q by Q ijk. (i) M 1 M 2 is parallel to p (ϕ = 0) From Q 101 we can factor out K and from Q 100 we can even factor out K 2. Finally, we denote the coefficient of e a 0e b 1e c 2e d 3 of Q ijk by P ijk abcd and compute: P P P P = K 1 B 3 B 4 B 5 coll(3, 4, 5) = K 1 = 0 P P P P = K 2 A 2 B 3 B 4 B 5 coll(3, 4, 5) = K 2 = 0 P P P P = K 3 a 2 B 3 B 4 B 5 coll(3, 4, 5) = K 3 = 0 P P P P = K 4 a 2 B 3 B 4 B 5 coll(3, 4, 5) = K 4 = 0 Remark: This is the shortest possible analytical proof of the Theorem of Karger. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 11

13 II) Base is parallel to p (C 2 = C 3 = 0) (ii) M 1 M 2 is not parallel to p (a) M 1 M 2 is orthogonal to p (ϕ = π/2) From Q ijk (i > 0) we can factor out K. From Q 100 we can even factor out K 2. We factor out (e 0 e 1 e 2 e 3 ) of Q 2jk and compute the following 15 polynomials: P 1 [18] := P P 2 [42] := P P 3 [12] := P P 4 [42] := P P 5 [72] := P P 6 [36] := P P P P P 7 [42] := P P P P P 8 [30] := P P P P P 9 [30] := P P P P P 10 [18] := P P P 11 [42] := P P P 12 [36] := P P P 13 [24] := P P P 14 [12] := P P P 15 [24] := P P th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 12

14 (ii) M 1 M 2 is not parallel to p (b) M 1 M 2 is not orthogonal to p (sinϕcosϕ 0) In this case we compute the following 20 polynomials: P 1 [12] := P P P 2 [36] := P P P 3 [78] := P P P 4 [66] := P P P 5 [30] := P P P 6 [66] := P P P 7 [36] := P P P 8 [42] := P P P 9 [18] := P P P 10 [18] := P P P 11 [108] := P P P 12 [102] := P P P 13 [24] := P P P P P 14 [42] := P P P P P 15 [48] := P P P P P 16 [36] := P P P P P 17 [66] := P P P P P 18 [54] := P P P P P 19 [48] := P P P P P 20 [150] := P P P P th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 13

15 ad (ii) (a,b) In both cases we proceed as given in Nawratil [2008,B] K 2 = 0: All 15 resp. 20 polynomials can only vanish for architecture singular manipulators, i.e. K 1 = K 3 = K 4 = 0 = K 2 0 K 2 0: All coefficients of t 1 can only vanish for the following two solutions: S 1 : A i = B i cotϕ, A j = B j cotϕ, A k = A 2 + B k cotϕ, b k = 0, a 2 = a k, a i = K 1 b i /(K 2 A 2 ), a j = K 1 b j /(K 2 A 2 ), K 3 = 0 and K 4 = 0 ( ) S 2 : A i = A 2 + B i cotϕ, A j = A 2 + B j cotϕ, A k = B k cotϕ, a i = a 2 + b i K 3 /K 4, a j = a 2 + b j K 3 /K 4, a k = b k = 0, A 2 K 2 + K 4 = 0 and K 1 + K 3 = 0 ( ) for i, j, k {3,4, 5} and i j k i without contradicting A 2 B 3 B 4 B 5 a 2 (a 4 a 3 )coll(3, 4, 5) 0. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 14

16 The close of the proof We show that both solutions S i imply contradictions for the choice of M 6 and m 6 : If we set A 2 = 1 and replace K i in ( ) and ( ) by the explicit expressions we get: ( ) K 3 = (A 6 B 6 cotϕ)(a k a 6 ) K 4 = (A 6 B 6 cotϕ)b 6 a 6 = a k, b 6 = 0 = K 2 = 0 A 6 = B 6 cotϕ = M 1,M i,m j,m 6 are collinear ( ) K 1 + K 3 = (1 A 6 + B 6 cotϕ)a 6 K 2 + K 4 = (1 A 6 + B 6 cotϕ)b 6 a 6 = 0 and b 6 = 0 = K 2 = 0 A 6 = 1 + B 6 cotϕ = M 2,M i,m j,m 6 are collinear 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 15

17 [6] A further example S 1 and S 2 imply a further example for a planar SGP with cylindrical singularity surface. For the computation see Nawratil [2008,A]. m 6 m5 m 4 m 3 m 2 m 1 M 1,M 2,M 3,M 4 are collinear, m 1,m 2,m 3,m 4 are collinear, M 5 M 6 M 1 M 2 p, and m 5 = m 6. SGP is in a singular position iff m 5 = m 6 lies in the base or L 1, L 2, L 3, L 4 are coplanar = singularity surface splits into two planes M 4 M 3 M2 M 6 M 5 M 1 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 16

18 [7] Remark Theorem Röschel and Mick [1998] Planar SGPs are architecturally singular iff {M i,m i } for (i = 1,...,6) are fourfold conjugate pairs of points with respect to a 3-dimensional linear manifold of correlations or one of the two sets {M i } and {m i } is situated on a line. Therefore the given main theorem can be reformulated as follows: Main Theorem Planar SGPs with no four points on a line and a cylindrical singularity surface must consist of four-fold conjugate pairs of anchor points with respect to a 3-dimensional linear manifold of correlations. It would be nice to have a geometric proof for the main theorem similar to the one presented by Röschel and Mick [1998]. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 17

19 [7] Conclusion We proved that there do not exist non-architecturally singular planar SGPs and no four anchor points collinear which possess a cylindrical singularity surface. We gave the shortest possible analytical proof for the Theorem of Karger. Moreover, we presented two examples of planar manipulators with cylindrical singularity surface. A complete list of planar SGPs with a cylindrical singularity surface is in preparation. Nawratil, G., All Planar Parallel Manipulators with Cylindrical Singularity Surface, in preparation. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 18

20 [8] References Karger A. [2003] Architecture singular planar parallel manipulators, Mechanism and Machine Theory 38 (11) Karger A. [2006] Stewart-Gough platforms with simple singularity surface, Advances in Robot Kinematics: Mechanisms and Motion (J. Lenarcic, B. Roth eds.), , Springer. Karger A. [2008] Architecturally singular non-planar parallel manipulators, Mechanism and Machine Theory 43 (3) Merlet J.-P. [1992] Singular Configurations of Parallel Manipulators and Grassmann Geometry, International Journal of Robotics Research 8 (5) Nawratil G. [2008,A] Results on planar parallel manipulators with cylindrical singularity surface, Advances in Robot Kinematics: Analysis & Design (J. Lenarcic, P. Wenger eds.), Springer. Nawratil G. [2008,B] Main theorem on planar parallel manipulators with cylindrical singularity surface, In Proc. of 33rd South German Colloquium on Differential Geometry, TU Vienna. Röschel O. and Mick S. [1998] Characterisation of architecturally shaky platforms, Advances in Robot Kinematics: Analysis & Control (J. Lenarcic, M.L. Husty eds.), , Kluwer. 11th International Symposium on Advances in Robot Kinematics, Batz-sur-Mer June , France 19