A Unified Concept for the Graph Representation of Constraints in Mechanisms Andreas Müller University of Michigan – Shanghai Jiao Tong University Joint Institute Mechanical Engineering School Tel-Aviv University Offer Shai

The outline of this talk 1. The constraint graphs – Body-bar (BB) Bar-joint (BJ) Mixed 2. Their uses : Combinatorial analysis – causality between blocks. Decomposition into Assur Graphs Combinatorial algorithms for determining the topological mobility. Combinatorial synthesis Revealing the pseudo mechanisms Deriving the correct general mobility equation

1.1 Body-Bar Constraint Graph G=(V,E) is a body-bar graph iff: A vertex stands for a rigid body. Thus, a vertex possesses three DOFs (in 2D) or six DOFs in (3D). - Edges stand for kinematic pairs (higher or lower pairs). - An edge is between exactly two bodies. - Between two vertices there can be several edges. Body-bar constraint graph

1.2 Bar- Joint Constraint Graph G=(V,E) is a bar-joint graph iff: - A vertex represents a point where binary links interconnect through only lower kinematic pairs (two constraints). Since the vertex stands for a point it possesses two DOFs (in 2D) or three DOFs (in 3D) - Between two vertices there can be at most one edge. Bar-Joint constraint graph (a) (b)

The main problem: in mechanisms several rigid bodies can be connected through one joint (called “multiple joint”). For mechanisms with multiple joints there is no unique body-bar graph causing difficulties.

The problem of Multiple Joints and Body-Bar Graphs C 1 2 3 4 5 Different Body-Bar Graphs for the same mechanism B1 B2 B3 B4 B5 B1 B2 B3 B4 B5 Kinematic pairs: (B2,B4), (B3,B4) Kinematic pairs: (B2,B3), (B3,B4) Different structural representations 1 2 3 4 5 1 2 3 4 5

Different Body-Bar graphs might result with even wrong conclusions 1 2 3 4 5 6 8 7 1 4 5 6 7 8 3 2   Structure – 0 DOF Mechanism – 1 DOf (B1,B2), (B1,B7), (B1, B4) (B1,B2), (B2,B7), (B7, B4)

1.3 Mixed Constraint Graph (new) G=(VB  VJ, E) is a mixed graph iff: - Every vertex can stand for a body or a point. If the vertex corresponds to a body, v  VB, it possesses three DOFs (in 2D) or six DOFs (in 3D). - If the vertex corresponds to a point (i.e. the location of a joint as in the bar-joint), v  VJ, it possesses two DOFs (in 2D) or three DOFs (in 3D).

bodies are represented by RFR vertices 𝑣∈ 𝑉 𝐵 , and Constraint Graph Definition: A mixed constraint graph with characteristic 𝑔 is a graph 𝐺 𝑔 =( 𝑉 𝐵 ∪ 𝑉 𝐽 ,𝐸, 𝑑) where bodies are represented by RFR vertices 𝑣∈ 𝑉 𝐵 , and (multiple-) joints are represented by JFR vertices 𝑣∈ 𝑉 𝐽 10 K G H F L D 2 B I 4 8 1 3 5 6 C 7 9 E M A J 1 2 3 5 4 7 10 8 9 A B 6 C D E F G H I J K L M

Mixed constrained graph Application to Gears Simpson gear system Mixed constrained graph 4 5 6 1 2 3 1 2 3 4 5 6 A

2. Deriving the correct general mobility equation 2.1 The correct topological mobility. 2.2 In BJ there exist pseudo mechanism. 2.3 The general equation for mobility.

𝐓𝐡𝐞 𝐜𝐨𝐫𝐫𝐞𝐜𝐭 𝐭𝐨𝐩𝐥𝐨𝐠𝐢𝐜𝐚𝐥 𝐦𝐨𝐛𝐢𝐥𝐢𝐭𝐲 equation 𝛿 top = 𝛿 str 𝐵,𝐽,𝑓,𝑔 + 𝜌 top 𝛤 𝛿 top - Topological mobility 𝛿 str 𝐵,𝐽,𝑓,𝑔 - Structural mobility 𝜌 top 𝛤 - Topological correction number (determined by combinatorial algorithms)

𝛿 top = 𝛿 str 𝐵,𝐽,𝑓,𝑔 + 𝜌 top 𝛤 𝛿=0 𝛿 str =0 𝛿 top =0 𝛿=1 𝛿 str =0

𝐓𝐡𝐞 𝐠𝐞𝐧𝐞𝐫𝐚𝐥 𝐠𝐞𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐦𝐨𝐛𝐢𝐥𝐢𝐭𝐲 equation 𝛿 loc 𝐪 = 𝛿 str (𝐵,𝐽,𝑓,𝑔) + 𝜌 top (𝛤)+ 𝜌 geo 𝐪 𝛿 loc 𝐪 - Geometric mobility in configuration q 𝜌 geo 𝐪 − Geometric correction number (number of overconstraints) 𝐓𝐡𝐞 𝐠𝐞𝐧𝐞𝐫𝐚𝐥 𝐠𝐞𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐦𝐨𝐛𝐢𝐥𝐢𝐭𝐲 equation

In 3D BJ there exist Pseudo Mechanisms M is a pseudo mechanism ↔ 1. Its topological mobility is less or equal to zero - δtop(M)≤0. 2. There is no topological redundancy - ρtop(M)=0. 3. For any geometric realization it has a finite motion.   3D BJ Assur Graph – Triad, 3/6 Stewart manipulator , in a topological singularity. 3D BJ Assur Graph – Heptad, in a topological singularity.

Properties of pseudo mechanisms 1. There are links that can be removed without affecting the finite motion.  

Properties of pseudo mechanisms 2. Changing the connection (topology) of at least one ground link can result in a rigid structure.  

Examples of floating pseudo mechanisms 2 Double Triad c-Triad + Tetrad (c-triad) c-Triad + c-Tetrad c-Triad + Hexad(c-triad+c-triad)

The general equation for mobility 𝛿 loc 𝐪 = 𝛿 str (𝐵,𝐽,𝑓,𝑔) + 𝜌 top (𝛤)+ 𝜌 geo 𝐪 + 𝜌 pse (𝛤) 𝜌 pse (𝛤) – Pseudo correction number.

Thanks!