Proceedings of the ASME 2012 International Design Engineering

Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2012 August 12 -15, 2012, Chicago, IL, USA DETC 2012 -71289 Mobility Determination Of Mechanisms Based On Rigidity Theory Slavutin Michael School of Mechanical Engineering Faculty of Engineering Tel Aviv University Tel Aviv, Israel Offer Shai School of Mechanical Engineering Faculty of Engineering Tel Aviv University Tel Aviv, Israel Andreas Müller Institute of Mechatronics Technical University Chemnitz Germany

Rigidity theory deals mostly with the topological computation. Mechanism theory is mainly concerned with the geometrical analysis and generic statements. In rigidity theory there are two main types of representations: 1. Bar-joint representation. 2. Body-bar representation.

Bar and joint representation Between any two joints (2 d-revolute, 3 d-spherical) there exists at most one bar/constraint. Binary Link/one constraint 2 d – revolute joint 3 d – spherical joint

Body-bar Representation Bodies connected by bars (constraints). 1 Lower kinematic pair Higher kinematic pair 1 2 2 1 2 Hinge Spherical joint 1 2

Representing a mechanism by Bar-joint and Body-bar representations Body-bar: Two bodies and three bars. Bar-joint: Nine bars and seven joints.

Definition of a Body-Bar Assur Graph G is a 2 d/3 d Body-Bar Assur Graph IFF 1. G has 3/6 DOF 2. G does not contain any sub-graph (of more than one element) which also has 3/6 DOF. 3. There at most 2/5 constraints between any two bodies.

2 d Body-bar Assur Graph 1 2 (a) Body-bar Assur Graph 3 (b) Not a body-bar Assur Graph

Analysis mechanisms through body-bar graphs Rigidity theory + Projective Geometry

Analysis mechanisms through body-bar graphs Property 1: The length of the projection of the velocity of any point P, onto a line (not at infinity) with coordinates, denoted by , is equal to the scalar productof Z, with the line coordinate vector:

Body-bar rigidity matrix D 2 E C A 1 B F Each joint of a boy imposes constraints on its motion. 1 yb-y. C x. C-x. By. C - x. Cy. B y. C-y. B x. B-x. Cy. B - x. By. C 2 y. D-y. E x. E-x. Dy. E - x. Ey. D y. E-y. D x. D-x. Ey. D - x. Dy. E Fx 0 0 1 -y. F -1 x. F Fy 0 0 -y. A Ax 0 1 0 0 0 x. A 0 0 0 Ay -1 0 x. Bl y Bl Bl x. Bll y Bll = 0 0 v. A x y

The spatial body-bar rigidity matrix for mechanisms Property 2: For the constraint between the two bodies there will be at most six entries written according to the following equation: From the latter equation it is possible to derive the projection of each point of body A, let it be Ai, on the line of the bar/constraint connected to it, li, as follows:

Example: Body-Bar Rigidity Matrix of Stewart Platform The schematic description of the Stewart Platform.

Pebble game Algorithm 1. Determines the generic mobility of the system and each element. 2. Reveals redundancies. 3. Decomposes the system into Assur Graphs (body-bar or bar-joint).

The main rule/theorem underlying pebble game: The sum of the pebbles on both two end elements before assigning a pebble to a constraint should be at least: dof of a rigid body + 1. 2 d=3+1 Joint Four (2, 2) Body Four (1, 3), (2, 2), (2, 3), (3, 3) 3 d=6+1 Six (3, 3)? Seven (1, 6), (2, 5), (3, 4)……

Assigning pebbled to the elements in dimension d Element can be body or joint. To each element we assign pebbles according to its dof. 2 d 3 d Joint Two Three Body Three Six

Moving pebbles to the constraints from the elements Each constraint reduces one dof thus we move one pebble from the end element to the constraint. Element i Element j Element j The directed constraint is directed from the element from which the pebble was taken.

6 8 5 4 2 3 4 2 5 7 6 8 3 0 1 2 1 1 3 0

7 6 8 5 Mechanical system 4 2 2 1 3 4 2 5 7 6 8 3 0 1 Body-bar Graph 1 3 0