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Tree Topologies and Structural Synthesis

What is a tree topology?

A tree topology for a kinematic chain has a set of common joints spanning several chains, possibly in several stages, and ending in multiple end-effectors. The tree topology is represented as rooted a tree graph; for this we follow the approach of [Tsai 2001], the root vertex being fixed with respect to a reference system. In tree topologies, a vertex can be connected to several edges defining several branches.


Open hands, that is, hands not holding an object in the fingers, are kinematic chains with a tree or hybrid topology. For our synthesis formulation, the internal loops in the hand structure are substituted using a reduction process, so that the hand has a tree topology with links that are ternary or above.


Tree topologies are denoted as SerialChain-(Branch1,Branch2,…,Branchb), where SerialChain are the common joints and the dash indicates a branching, with the branches contained in the parenthesis, each branch Branchi characterized by its type and number of joints. The figure shows a 1-(1-(1,1),1-(1-(1,1),1-(1,1),1)) tree, assuming all edges have a single revolute joint.




For computational purposes, the tree topology is identified usign a parent-pointer array and a joint array. The parent-pointer array indicates the (unique) previous edge for the directed tree that starts at the root. The joint array contains the number and type of joints per edge.


Enter a tree topology below to see its reduced graph representation.


Structural synthesis and enumeration

In the design problem, we are interested in enumerating the hand topologies being solvable for a certain number of task positions per fingertip, and with certain structural constraints, such as the number of fingertips, number of edges in the graph and number of splits, or palms.


An algorithm has been created that yields the list of candidate topologies for a set of requirements.


Resources

The draft papers below explains when a tree topology is solvable for kinematic synthesis.


Draft: Solvable Multi-Fingered Hands for Exact Kinematic Synthesis

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