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## Über dieses Buch

“The mathematical investigations referred to bring the whole apparatus of a great science to the examination of the properties of a given mechanism, and have accumulated in this direction rich material, of enduring and increasing value. What is left unexamined is however the other, immensely deeper part of the problem, the question: How did the mechanism, or the elements of which it is composed, originate? What laws govern its building up? Is it indeed formed according to any laws whatever? Or have we simply to accept as data what invention gives us, the analysis of what is thus obtained being the only scientific problem left – as in the case of natural history?” Reuleaux, F., Theoretische Kinematik, Braunschweig: Vieweg, 1875 Reuleaux, F., The Kinematics of Machinery, London: Macmillan, 1876 and New York: Dover, 1963 (translated by A.B.W. Kennedy) This book represents the second part of a larger work dedicated to the structural synthesis of parallel robots. Part 1 already published in 2008 (Gogu 2008a) has presented the methodology proposed for structural synthesis. This book focuses on various topologies of translational parallel robots systematically generated by using the structural synthesis approach proposed in Part 1. The originality of this work resides in the fact that it combines the new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel mechanisms.

## Inhaltsverzeichnis

### 1. Introduction

This book represents Part 2 of a larger work on the structural synthesis of parallel robots. The originality of this work resides in combining new formulae for the structural parameters and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel robots. Part 1 (Gogu 2008a) presented the methodology of structural synthesis and the systematisation of structural solutions of simple and complex limbs with two to six degrees of connectivity systematically generated by the structural synthesis approach. Part 2 of this work focuses on the structural solutions of translational parallel robotic manipulators (TPMs) with two and three degrees of mobility.
This section recalls the terminology, the new formulae for the main structural parameters of parallel robots (mobility, connectivity, redundancy and overconstraint) and the main features of the methodology of structural synthesis based on the evolutionary morphology presented in Part 1.

### 2. Translational parallel robots with two degrees of freedom

The translational parallel robots with two degrees of freedom can be actuated by linear and/or rotating actuators. Topologies with coupled, decoupled and uncoupled motions along with maximally regular solutions are presented in this section. They give rise to two independent translations along with a constant orientation of the moving platform.

### 3. Overconstrained T3-type TPMs with coupled motions

T3-type TPMs are translational parallel robotic manipulators with three degrees of connectivity between the moving and fixed platforms S F = 3. They give three translational velocities v 1 , v 2 and v 3 in the basis of the operational velocity vector space (R F ) = (v 1 ,v 2 ,v 3 ) along with a constant orientation of the moving platform.

### 4. Non overconstrained T3-type TPMs with coupled motions

Equation (1.15) indicates that non overconstrained solutions of T3-type TPMs with coupled motions and q independent loops meet the condition $$\sum\nolimits_1^p {f_i } \, = \,3\, + \,6q\,.$$ Various solutions fulfil this condition along with S F = 3, (R F ) = (v 1 ,(v 2 ,v 3 ) and N F = 0. They can have identical limbs or limbs with different structures and may be actuated by linear or rotating motors.

### 5. Overconstrained T3-type TPMs with uncoupled motions

T3-type translational parallel robots with uncoupled motions with various degrees of overconstraint may be obtained by using three simple or complex limbs. In these solutions, each operational velocity given by Eq. (1.19) depends, in the general case, on just one actuated joint velocity: v i = v i (q i ), i = 1,2,3. The Jacobian matrix in Eq. (1.19) is a diagonal matrix.

### 6. Non overconstrained T3-type TPMs with uncoupled motions

Equation (1.15) indicates that non overconstrained solutions of T3-type TPMs with uncoupled motions and q independent loops meet the condition $$\sum\nolimits_1^p {f_i } \, = \,3\, + \,6q\,$$ along with S F = 3, (R F ) = (v 1 ,v 2 ,v 3 ) and N F = 0. They could have identical limbs or limbs with different structures and could be actuated by linear or rotating motors. Each operational velocity given by Eq. (1.19) depends on just one actuated joint velocity: v i = v i (q i ), i = 1,2,3. The Jacobian matrix in Eq. (1.19) is a diagonal matrix.
They can be actuated by linear or rotating actuators which can be mounted on the fixed base or on a moving link. In the solutions presented in this section, the actuators are associated with a revolute joint mounted on the fixed base.

### 7. Maximally regular T3-type translational parallel robots

Maximally regular T3-type translational parallel robots are actuated by linear motors and can have various degrees of overconstraint. In these solutions, the three operational velocities are equal to their corresponding actuated joint velocities: v 1 = q 1 , v 2 = q 2 and v 3 = q 3 . The Jacobian matrix in Eq. (1.19) is the identity matrix. We call Isoglide3-T3 the translational parallel mechanisms of this family.

### Backmatter

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