## 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-(Branch_{1},Branch_{2},…,Branch_{b}), where SerialChain are the common joints and the dash indicates a branching, with the branches contained in the parenthesis, each branch Branch_{i} 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.