# Kinematic Synthesis of Tree Topologies

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Kinematic synthesis consists of the theoretical design of robots to comply with a given task. In this project we focus on finite point kinematic synthesis, that is, given a specific robotic topology and a task defined by spatial positions, we design a robot with that topology that complies with the task.

Tree topologies consist of loop-free structures where there can be many end-effectors. A characteristic of these topologies is that there are many shared joints. This allows some structures that may seem redundant to not actually be redundant when considering all the end-effectors at once. The main focus of this work is the design of grippers that have topologies similar to that of the human hand, which can be seen as a tree topology.

This research was developed in collaboration with Dr. Alba Perez-Gracia.

## Overview

In this research we investigated the kinematic synthesis of tree articulated structures. In particular these structures are interesting as they are found in natural life forms. A straight-forward example would be the case of human hands that can be modelled using revolute joints. In order to be able to perform kinematic synthesis of these structures, we must understand the entire process in depth and establish criteria in which the system of equations does not degenerate. For this purpose we have presented both practical applications of this work and the more theoretical aspects of these structures.

We start by defining the system of equations needed to solve for tree topologies using Clifford algebra. An approach to solve the equations numerically using a hybrid optimizer based on Levenberg-Marquadt local optimizer and genetic algorithms was presented. We show that this approach is able to consistently find solutions despite the complexity of these equations that make them unsolvable by traditional methods.

We later extend this formulation to take into account velocities, accelerations and further derivatives using Lie algebra. We show that this simplifies the equation system and allows more control in the definition of tasks. By defining the proper velocity and derivatives at points in space it is possible to approximate arbitrary curves. Additionally the same optimization framework can be used.

Finally we do a more in-depth theoretical analysis of the conditions needed in order to be able to perform finite point kinematic synthesis on tree topologies. This is not as straight-forward as it seems as there may be sub-topologies that become overdetermined for certain tasks. This causes the equations to generate and not have any solutions. In order to avoid this problem, the topology must be analyzed in depth. We propose representing the topology as a compacted tree graph, and present the necessary criteria for the design equations to be solvable for a given task.

Full source code for all the publications is available.

## Publications

### 2014

• • Kinematic Synthesis using Tree Topologies
• Edgar Simo-Serra, Alba Perez-Gracia
• Mechanism and Machine Theory (MAMT) 72:94-113, 2014

### 2012

• • Design of Multi-fingered Robotic Hands for Finite and Infinitesimal Tasks using Kinematic Synthesis
• Edgar Simo-Serra, Alba Perez-Gracia, Hyosang Moon, Nina Robson
• Advances in Robot Kinematics (ARK), 2012

### 2011

• • Design of Non-Anthropomorphic Robotic Hands for Anthropomorphic Tasks
• Edgar Simo-Serra, Francesc Moreno-Noguer, Alba Perez-Gracia
• ASME International Design Engineering Technical Conferences (IDETC), 2011
• • Kinematic Model of the Hand using Computer Vision
• Edgar Simo-Serra
• Degree Thesis, 2011

## Source Code

• • libdq, 2.2 (Feb, 2013)
• Library for using and manipulating unit dual quaternions to describe rigid body motions
• E. Simo-Serra
• • ArtTreeKS, 1.0 (Mar, 2012)
• Kinematic synthesis solver for tree topologies
• E. Simo-Serra