Kinematics of Soft-continuum Manipulators using PH-Curves 

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Soft-continuum Robot Applications

Soft-continuum manipulators are becoming more popular nowadays. The very first prototype of the continuum manipulator [Anderson 1967] suggested their use in the inspection [Tonapi 2014], search and the rescue tasks [Li 2017] [Bajo 2010].
The characteristics of soft-continuum manipulators, their compact structure, compliant nature, light-weight structure, make them suitable for many applications. The profound use of soft-continuum manipulators is in the work environment where humans have to interact with the robots. This involves the application of continuum robots in the medical field. Applications, such as, endoscopy [Fras 2015] [Conrad 2013] [Cianchetti 2013], skeleton trauma treatment [Alambeigi 2017] [Wilkening 2017] and minimally invasion surgery [Orekhov 2016] [Qu 2016] [Mahoney 2016] have proved the use of continuum manipulators in the medical field [Burgner-Kahrs 2015].


Continuum manipulators can be broadly classified according to the type of backbone they possess. Therefore, they are classified as single or multibackbone manipulators. Single backbone manipulators (Fig. 2.4 (left)) use to have a central structure along the manipulator which supports the passage of the actuation system along the body of the manipulator [Burgner-Kahrs 2015]. Many single backbone manipulators have tendons along their structure, which are spaced by the discs attached to the backbone as a way of transmission. The end-points of the tendons define the length of the bending section. Fig. 2.4 (right) shows a multi-backbone continuum manipulator. These manipulators normally consist of a parallel arrangement of the elastic elements constrained in a way.

Shape Estimation of Soft-continuum Manipulators

Soft-continuum manipulators have enormous advantages which enable their use in many applications. Their ability to manipulate through confined spaces is extensively used in many applications like medical, military, and nuclear, etc. Many research works are devoted to model kinematics and dynamics of the soft-continuum manipulators and are continuously developing to enhance their accuracies, for the purpose of control strategy development. Shape estimation of soft-continuum manipulators still has very limited literature. The classification of the shape estimation approaches for the soft-continuum manipulators is as shown in Fig. 2.6.

Sensor Based Approaches

Most of the literature regarding shape estimation of the soft-continuum manipulators covers sensor based approaches. They need to install the specific sensors to measure and estimate the shape of the robots. Therefore, the placement and the dimension of these sensors is important according to the type of the robot. Also, these approaches request associated investments. Following existing works using different types of sensors are discussed as follows:

Fiber Bragg Gratings (FBGs)

FBG sensors are commonly used sensors in the field of shape estimation. A small sized sensor is introduced in [Araújo 2001] which can be embedded in any layer of a composite material. The configuration of this sensor is based on the intrinsic bend sensitivity of Bragg gratings written in D-type fibers. The sensor gives the information of the curvature of the manipulator at the points at which the sensors are embedded. Further, curvature information is used for the shape reconstruction of the soft-continuum manipulators. These sensors are suitable to use in smart structures due to their small size. For the first time, [MacPherson 2006] used the multiplexed Fiber Bragg grating sensors into a multi-core fiber for the shape estimation. This work is demonstrated
for the application to the structural monitoring process. In the medical field, [Yi 2007] proposed the shape estimation of the colonoscope while it is proceeding inside the colon. The design of the sensor is discussed along with its positioning. The sensor provides the information of the curvature and torsion, and further, the principle based on differential geometry is used for curve-based shape reconstruction. A Bragg gratings based optical fiber bend sensor is used in an eccentric core polymer optical fiber [Chen 2010]. This sensor provides high bend sensitivities and the wide range of curvature measurements. In medical, for the MRI process, 3-D shape of the needle and its deflection is estimated using FBG sensors. The needle is used with a fixture, and the fixture has the grooves to accommodate the optical fibers. The sensors provide the curved profile and the deflection of the tip of the needle once it is inserted into the tissues. Three FBG sensors are used, and the placement f the sensors is discussed. The work of [Roesthuis 2014] provides a prototype
of a continuum nitinol needle embedded with 12 FBG sensors. FBG sensors provide the axial strain in the needle which gives the curvature of the needle. Further, the 3-D shape of the needle is estimated using kinematic and mechanics based model. The shape estimation of Dexterous Continuum Manipulators (DCMs) is proposed by sensing their curvatures using FBGs in [Liu 2015]. Fig. 2.7 shows the shape estimation of the DCM used for minimally invasive surgery. While advancing in this direction, [Farvardin 2016]

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Table of contents :

1 Introduction 
1.1 General Introduction
1.2 Framework and Context of the Thesis
1.3 Thesis Objective
1.4 Research Problem Statement
1.5 Contribution Positioning in the Framework of the Group Activities .
1.6 Main Contributions
1.7 Disseminated Results
1.8 Manuscript Organization
2 State of the Art on Soft-continuum Manipulators: Shape & Kinematics
2.1 Introduction
2.2 Soft-continuum Manipulator
2.2.1 Definition
2.2.2 Soft-continuum Robot Applications
2.2.3 Classification
2.3 Shape Estimation of Soft-continuum Manipulators
2.3.1 Sensor Based Approaches
2.3.2 Mathematical Model-Based Approaches
2.3.3 Work Contextualization and Contributions
2.4 Kinematic Modeling of Soft-continuum Manipulators
2.4.1 Qualitative Approaches
2.4.2 Quantitative Approaches
2.4.3 Hybrid Approach
2.4.4 Work Contextualization and Contributions
2.5 Conclusion of the Chapter
3 Shape Reconstruction of Soft-continuum Manipulators 
3.1 Introduction
3.2 Geometrical Curves Representation
3.3 Non-Parametric Representation
3.3.1 Explicit Form
3.3.2 Implicit Form
3.3.3 Drawbacks of Non-parametric Representation to Model Softcontinuum Manipulators
3.4 Parametric Representation
3.4.1 Analytical Curves
3.4.2 Synthetic Curves
3.5 Synthetic Curves
3.5.1 Hermite
3.5.2 Bezier
3.5.3 B-Splines and NURBS
3.5.4 Pythagorean Hodograph
3.6 Synthetic Curves v/s Soft-continuum Manipulators
3.7 Shape Reconstruction of Soft-continuum Manipulators Based on Curves
3.7.1 Compact Bionic Handling Assistant Manipulator
3.7.2 Experimental Setup
3.7.3 Results and Discussions
3.8 Modified PH-Curves for Reconstructed Shape
3.8.1 Calibration of the Shape
3.8.2 Relationship between Quintic PH and Calibrated Curve
3.8.3 Results and Discussions
3.9 Conclusion of the Chapter
4 Kinematics of Soft-continuum Manipulators using PH-Curves 
4.1 Introduction
4.2 Inverse Kinematics of Soft-continuum Manipulators from PH-curves
4.2.1 Shape Reconstruction towards Inverse Kinematic Model
4.2.2 Application of IKM to the CBHA Manipulator
4.2.3 Experimental Validation
4.2.4 Results and Discussions
4.3 Performances Robustness of PH-curves
4.3.1 NN-based Approach for End-point Approximation
4.4 Forward Kinematics of Continuum Manipulators from PH-curves
4.4.1 Inverse Kinematic Model towards Forward Kinematics
4.4.2 Application of FKM to the CBHA Manipulator
4.4.3 Results and Discussions
4.5 Conclusion of the Chapter
5 Towards Reconstruction of Soft-continuum Closed Loop Kinematic Chain: PH-based Approach 
5.1 Introduction
5.2 Problem Statement
5.3 Concatenation of PH Curves
5.3.1 Case 1: C1 Continuity
5.3.2 Case 2: C0 Continuity
5.4 Experimental Validation
5.4.1 Experimental Setup
5.4.2 Results
5.5 Conclusion of the Chapter
6 Conclusion and Prospective 
6.1 Summary of Conclusions
6.2 Future Works
A Introduction to Quaternions 
A.1 Introduction
A.2 Quaternion Algebra
A.2.1 Addition and Multiplication
A.2.2 Conjugate
A.3 Solution of equation A~iA = ~c
B ANSYS Model 
B.1 FEM Model of CBHA in ANSYS
B.1.1 Contact information between two sections
B.1.2 Connection of three tubes
B.1.3 Material Properties
B.1.4 Mesh
B.1.5 Detailing of Mesh
B.1.6 Input pressure graph of P1
B.1.7 Results of Deformation


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