Screw Theory and Reciprocity

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1 1 crew Theory and Reciprocity Latifah Nurahmi Definition of crew A spatial displacement of a rigid body can be expressed as a combination of a rotation about a line and a translation along the same line. This combined motion is called screw displacement. A unit screw $ is defined as: s s r s $

2 Definition of crew A unit screw $ is defined as: s $ r s s s = a unit vector along the axis of the screw $ r = a position vector of any point on the screw axis of $ λ = pitch Definition of crew A unit screw : (λ = 0), $ 0 s r s s $ r s s Zero-pitch screw 0 (λ = ), $ s Infinite-pitch screw s (λ, λ 0), $ h Finite-pitch screw r s s 2

3 crew ystems A screw system of order n (0 n 6) comprises all the screws that are linearly dependent on n given linearly independent screw. crew ystems 3

4 Definition of Twist and Wrench Twist A twist represents an instantaneous motion of a rigid body. Zero-pitch twist Infinite-pitch twist Wrench 0 pure rotation. pure translation. A wrench represents a system of forces and moments acting on a rigid body. Zero-pitch wrench F pure force Infinite-pitch wrench M pure moment 3 0 system Relations to Kinematic Joints 1 0 system system 1 system system 4

5 Relations to erial Kinematic Chain A rigid body constrained to move in the plane Available twists The end effector is constrained to rotate and translate in the plane. The screw system associated with the available twists is of order 3. Relations to erial Kinematic Chain Two rigid bodies connected by three prismatic joints Body 1 Body 2 The screw system describing all the available twists consists of all screws with infinite pitch. All translatory motions are instantaneously (in this particular example, also finite translatory motions) possible. The set of all twists is a third order screw system. 5

6 Definition of Reciprocity Work done by a wrench on a twist The rate of work done by a wrench w = [f T, m T ] T on a twist t = [w T, v T ] T is given by P f v mw Alternatively, f P v w m T t w 0 I I Reciprocity Two screws are said to be reciprocal if a wrench applied about one does no work on a twist about the other. Reciprocal crews f m v w f d 1 2 Two screws 1 (pitch h 1 ) 2 (pitch h 2 ) are reciprocal if and only if (h 1 + h 2 ) cos f - d sin f = 0 Remarks A wrench applied about 1 does no work on a twist about 2. A wrench applied about 2 does no work on a twist about 1. The condition for reciprocity is a purely geometric relationship 6

7 Reciprocal crews (continued) y f m x v w f z d 1 2 Derivation 1 cosf 0 sinf 0 0 1, 2 h 1 h2 cosf d sinf 0 h2 sinf d cosf cosf 0 sinf f w t m 0 w v 0 f, w h h d 1 2 cosf sinf 0 h2 sinf d cosf 0 0 f v mw 0 h1 h2 cosf d sinf 0 Examples crews reciprocal to a zero pitch screw 1 q f m f d 2 (h 1 + h 2 ) cos f - d sin f = 0 A wrench acting on a rigid body free to rotate about a revolute joint does no work on the rigid body if one of the following is true The wrench is of zero pitch and the axis intersects the axis of rotation The wrench of infinite pitch is perpendicular to the axis of rotation The pitch is non zero but equal to d tan f Note: If m cosf f d sin f, the wrench does no work. 7

8 Examples (continued) 1 f m crews reciprocal to an infinite pitch screw r f d f v mw 0 2 A wrench acting on a rigid body free to translate along a prismatic joint does no work on the rigid body if one of the following is true The wrench is of infinite pitch. The pitch is zero or finite, but the axis is perpendicular to the axis of the prismatic joint. Reciprocal crew ystems Definition of a screw system The vector space of all screws generated by taking all possible linear combinations of a finite number of screws. Reciprocal screw systems Two screw systems are reciprocal if every screw in the first screw system is reciprocal to all screws in the second screw system. Important Property Given a screw system, the set of all screws reciprocal to every screw in the given screw system is another screw system. 8

9 Example The screw system consisting of all screws reciprocal to a given zero pitch screw f m Given screw system is a first order screw system defined by a zero pitch screw. w v f The reciprocal screw system consists of All screws of zero pitch such that the screw axes intersect the axis of the given zero pitch screw crews of pitch equal to d tan f d 1 2 Properties Given a screw system, the set of all screws reciprocal to every screw in the given screw system is another screw system. The screw system reciprocal to a nth order screw system is of order (6-n) A rigid body subject to constraints has the following property: Available twists dim = n Reciprocity Constraint wrenches dim = 6-n 9

10 Another view point A pure force F perpendicular to t, cannot produce any translation along t Twist and wrench systems for a kinematic joint The set of all twists allowed by a joint is reciprocal to the set of all wrenches that can be resisted (passively) by the joint. Available twists dim = n A n degree-of-freedom joint has a screw system of order n associated with the available twists and a screw system of order (6-n) associated with the wrenches that can be transmitted by the joint. Reciprocity Constraint wrenches dim = 6-n 10

11 Reciprocal Twist and Wrench in Kinematic Joints Example 1 Two rigid bodies connected by a spherical joint Fixed rigid body Moving rigid body The set of available twists for the moving rigid body is described by a screw system consisting of all screws with zero pitch passing through the center of the spherical joint. The set of all twists is a three-dimensional vector space. The screw system is order 3. The set of all wrenches that do no work is described by the reciprocal screw system The reciprocal screw system is of order (6-3=) 3 It consists of all pure forces passing through the center of the spherical joint. 11

12 Example 2 A serial kinematic chain, composed of P and R joints connecting link a and b. If link a is fixed and link b is moving, the twist of link b is: The screw system associated with the available twists is of order 2. The screw system associated with the constraint wrenches (the wrenches that can do no work on the end effector) is of order 4. Example 3 Two rigid bodies connected by three prismatic joints Body 1 Body 2 The screw system describing all the available twists consists of all screws with infinite pitch. All translatory motions are instantaneously (in this particular example, also finite translatory motions) possible. The set of all twists is a third order screw system. The reciprocal screw system describing all the wrenches that do no work on the constrained rigid body (Body 2) consists of all infinite pitch screws. No couple can do work on Body 2. The set of all constraint wrenches is a third order screw system. 12

13 Example 4 A rigid body constrained to move in the plane Constraint wrenches Available twists The end effector is constrained to rotate and translate in the plane. The screw system associated with the available twists is of order 3. The screw system associated with the constraint wrenches (the wrenches that can do no work on the end effector) is also of order 3. Rangkuman Twist Wrench 13

14 Type ynthesis Type synthesis adalah salah satu proses desain untuk mensintesa jenis-jenis atau tipe-tipe kaki robot. Type synthesis dapat dilakukan berdasarkan crew theory dan Virtual chain Virtual Chain adalah sebuah serial kinematic chain yang mempunyai tipe gerakan tertentu untuk menggambarkan jenis gerakan pada parallel manipulator. Virtual chain diusulkan berdasarkan analisa wrench system dan virtual chain yang paling sederhana harus dipilih. Contoh 1 virtual chain pada parallel manipulator ebuah Parallel Manipulator (PM) mempunyai 3-DOF dengan 3 system Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga P-joints yang terhubung secara seri. PPP-Virtual Chain 14

15 Contoh 2 virtual chain pada parallel manipulator ebuah Parallel Manipulator (PM) mempunyai 3-DOF dengan 3 0 system Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga R-joints yang terhubung secara seri atau satu -joint. -Virtual Chain Contoh 3 virtual chain pada parallel manipulator ebuah Parallel Manipulator (PM) mempunyai 4-DOF dengan 2 system Virtual chain yang paling sederhana untuk menggambar PM tersebut adalah tiga P-joints dan satu R-joint yang terhubung secara seri. PPPR-Virtual Chain 15

16 Prosedur Type ynthesis Parallel Manipulator Prosedur Type ynthesis Parallel Manipulator Langkah 3 Merangkai kaki robot menjadi parallel manipulator yarat-syarat yang harus dipenuhi dalam merangkai kaki robot: 1. etiap kaki harus mempunyai dof yang sama dengan virtual chain 2. Wrench system dari PM harus sama dengan virtual chain Langkah 4 eleksi joint yang akan diaktuasi Dalam memilih joint yang akan diaktuasi (diberi motor), beberapa kriteri berikut harus diikuti: 1. Harus terdistribusi diantara semua kaki 2. Lebih disukai yang berada di bas 3. ebaiknya tidak ada P-joint yang tidak diaktuasi/pasif. 16

17 Type ynthesis chönflies Motion Parallel Manipulator Type ynthesis chönflies Motion Parallel Manipulator Langkah 1: Dekomposisi wrench system 2 system 1 system m c Δ c 1 c 2 c 3 c 4 c Kombinasi wrench system setiap kaki (2-5 kaki)

18 Type ynthesis chönflies Motion Parallel Manipulator Langkah 2: Type synthesis dari setiap kaki Kasus (1.) 2 system Menghitung jumlah joint di setiap kaki f F ( 6 c) 8 joints 8 joints termasuk PPPR Virtual chain Type ynthesis chönflies Motion Parallel Manipulator 18

19 Type ynthesis chönflies Motion Parallel Manipulator Langkah 2: Type synthesis dari setiap kaki Kasus (2.) 1 system Menghitung jumlah joint di setiap kaki f F ( 6 c) 9 joints 9 joints termasuk PPPR Virtual chain Type ynthesis chönflies Motion Parallel Manipulator Tabel 2. Tipe-tipe kaki c i Dof Class Type R-1P 2R-2P 1R-3P 5R 4R-1P 3R-2P 2R-3P Permutation PŔŔŔ Permutation PPŔŔ Permutation PPPŔ Permutation ŔŔŔȐȐ Permutation ŔŔȐȐȐ Permutation PŔŔŔȐ Permutation PŔŔȐȐ Permutation PŔȐȐȐ Permutation PPŔŔȐ Permutation PPŔȐȐ Permutation PPPŔȐ 19

20 Type ynthesis chönflies Motion Parallel Manipulator Langkah 3: Merangkai kaki m c Δ c 1 c 2 c 3 c 4 c Table 1 Table c i Dof Class Type R-1P 2R-2P 1R-3P 5R 4R-1P 3R-2P 2R-3P Permutation PŔŔŔ Permutation PPŔŔ Permutation PPPŔ Permutation ŔŔŔȐȐ Permutation ŔŔȐȐȐ Permutation PŔŔŔȐ Permutation PŔŔȐȐ Permutation PŔȐȐȐ Permutation PPŔŔȐ Permutation PPŔȐȐ Permutation PPPŔȐ Type ynthesis chönflies Motion Parallel Manipulator Langkah 4: election of the actuated joint Kriteria: 1. Harus terdistribusi diantara semua kaki 2. Lebih disukai yang berada di bas 3. ebaiknya tidak ada P-joint yang tidak diaktuasi/pasif. W a W c 6system Quadrupteron PM L i = P i Ȑ i 1Ȑ i 2Ȑ i 3Ŕ i i = 1, 2, 3 L 4 = PŔ 1 Ŕ 2 Ŕ 3, 20