Technion Israel Institute of Technology Department of

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 2 q-e wedge of two dimensional discs with common center V, where q is the number of aligned branches, and:

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 c 1, c 2 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 c 1 , c 2 of a star-shaped manipulator are in the same component iff 1) f(c 1) , f(c 2) 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 C 1 and C 2 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: d. F 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.

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