Introduction to Continuum Mechanics and FEM 

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Soft continuum robot applications

The characteristics of soft continuum manipulators, like their natural compli-ance, high power to weight ratio, and reduced dimensions compared to their  rigid counterparts, make them particularly suitable for applications in which the contact with humans is unavoidable or even desired. Applications such as skeletal trauma treatment [Wilkening 2017] [Alambeigi 2017], endoscopy [Conrad 2013] [Cianchetti 2013] [Fraś 2015] and minimally invasive surgery [Qu 2016] [Orekhov 2016] [Mahoney 2016] have proved the great potential of application of continuum robots in the medical field, as extensively reviewed in [Burgner-Kahrs 2015].
As suggested by the very first prototypes of continuum manipulators [Anderson 1967]. The slender shape and high dexterity of continuum ma-nipulators can be exploited in tasks such as minimally invasive inspection [Mehling 2006] [Tonapi 2014] and search and rescue [Bajo 2010] [Li 2017].
Continuum robots have been studied with the goal of exploiting their lo-comotive capabilities [Godage 2012] [Kang 2012] [Arienti 2013], although the review presented in the following of this manuscript is concerned mainly on manipulation.

Bio-inspiration

If one sees for the first time a continuum manipulator, without any previous knowledge on the concept behind it, one can immediately identify the re-markable morphological similarities that this type of devices have with some soft-bodied animals, particularly with the muscular hydrostats. Muscular hy-drostats, commonly found in elephant trunks, mammal tongues and octopus tentacles are soft muscular structures that can bend, extend and twist and provide the force required for movement and skeletal support to animals (or limbs) that lack a rigid skeleton, see Fig. 2.2.
Muscular hydrostats are typically composed by a fluid-filled cavity sur-rounded by a muscular wall reinforced with connective tissue fibres. The arrangement of the muscle fibres in a hydrostatic muscle include both circular and longitudinal muscle fibres. These two muscle fibres can antagonize one another to produce a variety of shape changes including elongation and bend-ing [Kier 1992]. The fluid inside the cavity of a hydrostatic limb is mainly a liquid which resists to volume change, thus, to create an elongation of the limb, the circular muscles contract to decrease the diameter while increasing the length to allow for a constant volume inside the cavity. The study of the biomechanics of hydrostatic structures have shown also additional fibres with more intricate configuration patterns that allow for more complex shape changes like twisting, present in octopus tentacles and mammals and reptile tongues [Kier 1985]. The complete replication of hydrostats is very complex, but the study of their underlying function principles have given roboticists an interesting insight and a solid starting point in the design of soft, continuum manipulators.

State of the art in soft, continuum manipu-lators

This section describes the research developed in the last 20 years, towards the design, modeling and control of soft, continuum manipulators. The review is divided into each of the subtopics mentioned in order to provide a clear overview of the field.

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Design of continuum manipulators

As stated in [Walker 2013a], the first prototype of a continuum robot reported in the literature was the Tensor Arm [Anderson 1967], designed by Anderson and Horn in 1967. Conceived to be used under water, the prototype was able to achieve a wide range of shapes; however, the relationship between the shapes and inputs was highly complex and challenging for the computational resources of that time. Based on extrinsic actuation, the robot used nylon filaments routed along the structure through spacer discs that apply torques directly to the backbone to produce bending, see Fig. 2.6. As is usual in tendon-based designs, the termination points of the cables define the bending sections. This prototype has inspired since then, a significant number of designs based on the same principle.

Table of contents :

1 Introduction 
1.1 General Introduction
1.2 Framework and context
1.3 Motivation
1.4 Chapter summary
1.5 List of Publications
2 State of the Art 
2.1 Introduction
2.2 Continuum manipulator definition
2.2.1 Soft continuum robot applications
2.2.2 Bio-inspiration
2.2.3 Classification
2.3 State of the art in soft, continuum manipulators
2.3.1 Design of continuum manipulators
2.3.2 Modeling of continuum robots
2.3.3 Dynamics and control of continuum robots
2.4 Work contextualization and contributions
3 FEM-based model of Continuum Manipulators 
3.1 Introduction
3.2 Continuum mechanics framework
3.2.1 Constitutive material law
3.2.2 Forces in the continuum manipulator
3.3 Finite Element Method
3.4 FEM-based kinematics of soft manipulators
3.4.1 Constraint for the end-effector
3.4.2 Actuator constraint model
3.4.3 Sensor constraint model
3.5 Reduced model in the constraint space
3.5.1 Reduced compliance on the constraint space
3.5.2 Coupled Kinematic Equations
3.5.3 Inverse kinematic model solution by convex optimization
3.6 Method implementation
3.6.1 Simulation framework
3.6.2 Corotational FEM
3.6.3 Mesh generation
3.6.4 Description of the Compact Bionic Handling Assistant
3.6.5 Simulation of the CBHA
3.7 Kinematic models
3.7.1 Forward kinematic models
3.7.2 Inverse kinematic model
3.8 Conclusion of the chapter
4 Closed loop control of soft, continuum manipulators 
4.1 Introduction
4.2 Feed-forward control of continuum manipulators
4.3 Closed-loop control of continuum manipulators
4.3.1 Closed-loop control law design
4.3.2 Robustness analysis
4.4 Conclusions of the chapter
5 Conclusion and Perspectives 
5.1 Summary of conclusions
5.2 The FeTCh manipulator
5.3 Perspectives
A Introduction to Continuum Mechanics and FEM 
A.1 Introduction
A.1.1 Continuum Mechanics
A.2 Finite Element Method for linear elastic bodies
A.2.1 Discretization of the domain
A.2.2 Element solution
A.2.3 Assembly
A.2.4 Displacement boundary conditions
A.2.5 Solution method
B Domain decomposition of continuum manipulators 
B.1 Domain decomposition of the CBHA
Bibliography

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