Technion – Israel Institute of Technology, Department of Mechanical Engineering 1/27 Configuration spaces of parallel mechanisms & Applications Nir Shvalb A part of Doctoral dissertation supervised by Prof. Moshe Shoham. Technion – Israel Institute of Technology, Department of Mechanical Engineering 2/27 General Outline • A short review: 1. Configuration spaces pursuit. 2. Applications. • Spider like mechanisms. • Motion planning – a short summery (?). • Uncertainty singularities. Technion – Israel Institute of Technology, Department of Mechanical Engineering 3/27 Up to date work: Configuration spaces Definition: The set of all feasible configurations = The Configuration space. Research issues the # connected components, Alg. Top. Groups and singularity • Kapovitch & Milson (1995-1999) • Hausmann & Knutson (1994) • Kamiyama & Tezuka (1996-2000) • Trinkle & Milgram (2000-2001) • Farber (2006) • Holcomb (2003) • Grihst & Abrahms (2003) Closed chain linkages Technion – Israel Institute of Technology, Department of Mechanical Engineering 4/27 Up to date work: Applications •Singularity • Angeles & Gosselin (1990) • Hunt (1978) • Zalatnov, Felton & Benhabib (1998) • Motion planning: 1. Explicit continuous planers. Trinkle (2003) 2. Theoretical assessment as to the minimal number of such planners. - Farber (2002) Technion – Israel Institute of Technology, Department of Mechanical Engineering 5/27 Spider like mechanism Definition: A spider like mechanism consists of k ‘free’ legs each having n(i) links and all legs meet at their End-effector. Theorem: The configuration space C of such mechanisms is a smooth manifold iff it does not contain aligned configurations. N. Shvalb, D. Blanc & M. Shoham “The Configuration space of arachnoid mechanisms”, Fund. Math. 17 (2005), 1023-1042 Technion – Israel Institute of Technology, Department of Mechanical Engineering 6/27 Spider like mechanism Notation: Elbow Up, Elbow Down From now on we focus on Spider like mechanisms having exactly 2 links in each leg: UU-U UD-U DU-U DD-U Technion – Israel Institute of Technology, Department of Mechanical Engineering 7/27 Spider like mechanism Let V be a node-configuration in C of planar spider like mechanism with k legs, each with two links. Theorem: V has a neighborhood in C which is a 2q-e wedge of two dimensional discs with common center V, where q is the number of aligned branches, and: 1 all aligned branches have common direction. e otherwise. 2 Technion – Israel Institute of Technology, Department of Mechanical Engineering 8/27 Spider like mechanism Defintion: Genus Theorem: By simply calculating the Euler characteristic one find the topological type: • W is one of the work-space connected components. • g is 1 if Conv(W) is a disc, and zero otherwise. • The #of annuli which wholly contain W is denoted by b. N. Shvalb, D. Blanc & M. Shoham “The Configuration space of arachnoid mechanisms”, Fund. Math. 17 (2005), 1023-1042 Technion – Israel Institute of Technology, Department of Mechanical Engineering 9/27 Motion planning implementation Given: planar tree shaped mechanism Task: given two configuration c1,c2 determine: (1) If they are in the same connected component of C. (2) Find a path in C connecting them. Define the kinematic map where and denote: Guanfeng Liu, J. C. Trinkle & N. Shvalb, "Motion Planning for a Class of Planar Closed-Chain • Manipulators" IEEE International Conference on Robotics and Automation, 2006.. N. Shvalb, G. Liu, M. Shoham & J. Trinkle “Motion Planning for a Class of Planar Closed-Chain Manipulators”, Submitted to The International Journal of Robotics Research. Technion – Israel Institute of Technology, Department of Mechanical Engineering 10/27 Motion planning implementation Technion – Israel Institute of Technology, Department of Mechanical Engineering 11/27 Motion planning implementation First Observation Technion – Israel Institute of Technology, Department of Mechanical Engineering 12/27 Motion planning implementation Second Observation Theorem: The C-space has two components iff has one components iff No other cardinality is possible. To implement this let us look at the following… Technion – Israel Institute of Technology, Department of Mechanical Engineering 13/27 Motion planning implementation Technion – Israel Institute of Technology, Department of Mechanical Engineering Motion planning implementation 14/27 Thus: Theorem: two configurations c1 ,c2 of a star-shaped manipulator are in the same component iff 1) f(c1) ,f(c2) are in the same component of the work space. 2) for each leg with three long links in all cells the elbow signs are the same for C1 and C2 A small lie here Guanfeng Liu, J. C. Trinkle & N. Shvalb, "Motion Planning for a Class of Planar Closed-Chain • Manipulators" IEEE International Conference on Robotics and Automation, 2006.. N. Shvalb, G. Liu, M. Shoham & J. Trinkle “Motion Planning for a Class of Planar Closed-Chain Manipulators”, Submitted to The International Journal of Robotics Research. Technion – Israel Institute of Technology, Department of Mechanical Engineering 15/27 Motion planning implementation Technion – Israel Institute of Technology, Department of Mechanical Engineering 16/27 Motion planning implementation Technion – Israel Institute of Technology, Department of Mechanical Engineering 17/27 Uncertainty singularity Given: parallel mechanism with polygonal platform. Task: Find conditions and characterize configuration space singularities (Uncertainty singularities). N. Shvalb, M. Shoham & D. Blanc “The Configuration space of parallel polygonal mechanism" Submitted to • Homotopy Homology and Applications. N. Shvalb, D. Blanc & M. Shoham“Uncertainty configuration singularities in parallel mechanisms" In preparation. Technion – Israel Institute of Technology, Department of Mechanical Engineering 18/27 Uncertainty singularity Notations: superscripts, subscripts, link vectors, leg vectors, string vectors. Recall: And consider the map: Technion – Israel Institute of Technology, Department of Mechanical Engineering 19/27 Uncertainty singularity A skip here Technion – Israel Institute of Technology, Department of Mechanical Engineering 20/27 Uncertainty singularity Carefully examining we conclude Lemma: dF has not maximal rank iff for all participant legs (1) (2) All are aligned. on the moving platform plane. (1) If 2 vectors are participants in the “game” this is not a “generic case”: (2) If 3 vectors are participants this is generic - (coupler curve intersection with a circle) !!! (3) If 4 vectors are participants this is not generic and we disallow this (or more then 4) So we need a “finer” handling with the three aligned legs case… Technion – Israel Institute of Technology, Department of Mechanical Engineering 21/27 Uncertainty singularity Recall the following theorem: Technion – Israel Institute of Technology, Department of Mechanical Engineering 22/27 Uncertainty singularity Assume there are three aligned legs and one plane and we prove by induction Denote the work map by Lastly define the spaces: Technion – Israel Institute of Technology, Department of Mechanical Engineering 23/27 Uncertainty singularity And consider the following diagram: Note that the configuration space is pre-image of under h Technion – Israel Institute of Technology, Department of Mechanical Engineering 24/27 Uncertainty singularity Technion – Israel Institute of Technology, Department of Mechanical Engineering 25/27 Uncertainty singularity Theorem: if no three aligned legs meet in one point and are on the same plane the configuration space is a smooth orientable manifold. A skip here Line dependence vs. our theorem N. Shvalb, M. Shoham & D. Blanc “The Configuration space of parallel polygonal mechanism" Submitted to • Homotopy Homology and Applications. N. Shvalb, D. Blanc & M. Shoham“Uncertainty configuration singularities in parallel mechanisms" In preparation. Technion – Israel Institute of Technology, Department of Mechanical Engineering 26/27 Uncertainty singularity Technion – Israel Institute of Technology, Department of Mechanical Engineering 27/27 N. Shvalb, D. Blanc & M. Shoham “The Configuration space of arachnoid mechanisms”, Fund. Math. 17 (2005), 1023-1042 N. Shvalb, G. Liu, M. Shoham & J. Trinkle “Motion Planning for a • Class of Planar Closed-Chain Manipulators”, Submitted to The International Journal of Robotics Research. Guanfeng Liu, J. C. Trinkle & N. Shvalb, "Motion Planning for a Class • of Planar Closed-Chain Manipulators" IEEE International Conference on Robotics and Automation, 2006.. N. Shvalb, M. Shoham & D. Blanc “The Configuration space of parallel • polygonal mechanism" Submitted to Homotopy Homology and Applications. N. Shvalb, D. Blanc & M. Shoham“Uncertainty configuration • singularities in parallel mechanisms" In preparation. Technion – Israel Institute of Technology, Department of Mechanical Engineering Push Me Pull You
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