, by the European Union through the Regional program competitiveness and employment 2007-2013 (ERDF -Auvergne region), by the Auvergne region and by French Institute for Advanced Mechanics.

A special thanks to my family. Words cannot express how grateful I am to my mother and father for all of the sacrifices that you've made on my behalf. Your prayer for me was what sustained me so far. I thank my mother-in law and father-in-law for all the support throughout my thesis.

I would also like to thank all of my colleagues and friends who supported me in writing and incented me to strive towards my goal. A special mention to my best friend Mohamed Mohaideen, Ashok, Hemaanand, Karthik, Nadege, Lazher, Adel, Khaled, Jun, Said, Djily, Lam, Anh-Vu, Oleksii, Bassem, Seifeddine and to name a few whose support was priceless on many occasions. I would like to thank Montassar for the perfect design of the robot with which we were able to realize a real robot and perform experimentations on it. Above all, I would like to express appreciation and heartfelt thanks to my beloved husband Raghunath RAJAVELU who spent sleepless nights with me and was always my support and strength in the moments when there was no one. I dedicate my thesis to Raghu, my mom and to my dad.

Table of contents

Introduction Motivation

Modern robotic world is seeking for robots which are able to adapt and reconfigure according to the environment and the task that has to be performed. With the development of science and technology and especially the progress of deep space and undersea exploration, the traditional mechanisms with limited adaptability and flexibility are facing challenges.

Reconfigurable and deployable mechanical systems, being capable of adapting themselves to variable tasks, have been attracting great attentions in both academia and industry.

Reconfigurable mechanisms are well suited for applications in domestic, hazardous, deep space, deep sea, and complex manufacturing environments. Deployable structures are also widely used in a variety of spacecrafts and planetary detectors, with increasing requirements upon transformation, dexterity, modularization, reconfigurability, and reliability. Research and development of reconfigurable and deployable mechanisms face a series of challenges both in theoretical and technical aspects. These robots are high in demand because of their beneficiary property of performing multiple tasks instead of multiple robots that perform only one particular task [START_REF] Aimedee | Systematization of Morphing in Reconfigurable Mechanisms[END_REF].

In this thesis, I take into account a strict constraint criterion for systematizing the reconfigurable mechanisms, i.e. we only consider mechanisms in which reconfigurability is achieved without assembling and disassembling of the parts. Henceforth, mechanisms presented in the following sections are those which are able to reconfigure itself by selflocking, changing relative position of joint axes, superposing or aligning two or more links, superposing the axis of two or more joints and so on.

Thesis goals

This thesis mainly addresses three major parts each of which are discussed in the following subsections. Also for the conceptual and methodological developments, we consider the following scientific issues as listed below:

Systematization and structural analysis

This part of the thesis is dedicated to the development of a systematization approach for reconfigurable mechanisms with respect to their structural parameters such as mobility, connectivity, redundancy and number of overconstraints. IFToMM terminology defines the mobility or degree of freedom as the number of independent coordinates required to define the configuration of a kinematic chain or mechanism [START_REF] Ionescu | Terminology for mechanisms and machine science[END_REF]. The connectivity between two links of a mechanism represents the number of independent finite and/or infinitesimal displacements allowed by the mechanism between the two links. In free-ofsingularity branches, full-branch connectivity is defined by the number of finite displacements [START_REF] Gogu | Branching singularities in kinematotropic parallel mechanisms[END_REF], [START_REF] Dai | Finite displacement screw operators with embedded chasles motion[END_REF]. In singular configurations, instantaneous connectivity can be defined by the number of finite and/or infinitesimal displacements. The various types of motions associated with the connectivity are defined in terms of translational or rotational velocities. In this way, both finite and infinitesimal motion derivatives with respect to time could be integrated using a linear transformation method. The number of over constraints of a mechanism is given by the difference between the maximum number of joint parameters that could lose their independence in the closed loops, and the number of joint parameters that actually lose their independence in the closed loops [Gogu, 2008a]. The redundancy is given by the difference between the mobility of the parallel mechanism and the connectivity between the moving platform and the fixed base. Redundancy introduces internal mobilities in the limbs [Gogu, 2008a].

The mechanisms are analyzed structurally using the formulae of structural parameters proposed by Gogu [Gogu, 2008a]. These parameters help us to understand the mechanism and to systematize it according to type of mechanism, whether the mechanism is overconstraint or non-overconstraint, redundant or non-redundant, with/without internal mobilites, etc.

Geometric and kinematic formulation

To resolve the practical problems of modeling, control, simulation and development of the robot, the structural parameters are required. Various types of singularities are also systematized and analyzed by taking into account the structural parameters. Further to know the location of the robot links relative to each other, we need to compute the geometric model. We use Travel Coordinate System [START_REF] Gogu | Mathématiques pour la robotique : Représentation des mouvements des corps solides[END_REF], [START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]] to determine the position and orientation of the joint axes at each instant. To find out the linear and angular velocities of each joint variable, we need to formulate the kinematic equations for the robot under consideration.

Control strategies

This part is dedicated to the development of trajectory generation and control strategy, based on actuation redundancy. The challenging task in this control part is to develop an advanced control law in order to synchronize several actuators to have a smooth transition from one assembly mode to another without causing wear and tear to the robot. Choice of actuated joints also plays a vital role in ensuring high performance and controllability of the mechanism.

In this research we focus on the 8-bar single loop mechanism to illustrate the developments achieved in the three parts mentioned above. As it will be shown, this mechanism exhibits an interesting capacity to reconfigure. It has two degrees of mobility in a general configuration but needs at least five motors to be fully controlled in all singular configurations.

Thesis Structure

The present thesis is divided into four chapters. Chapter 1 discusses the related literature review in detail while chapter 2 discusses the structural analysis and geometric modelling of an eight-bar linkage. The singular configurations and their bifurcated branches are discussed in chapter 3. Control strategies of the reconfigurable eight bar mechanism, tested by simulation on a prototype with five actuated motors, are presented in chapter 4.

 In chapter 1, a bibliographic review has been done, taking into consideration most of the reconfigurable mechanisms studied in the literature. Mechanisms are reconsidered and studied in detail by taking into account the way these mechanisms achieve reconfigurability. Furthermore we systematized them according to their types of morphing, types of motions and their applications. The proposed systemization is useful for the upcoming researches and helps to have a clear idea on the concept of reconfigurable mechanisms.

 In chapter 2, we take into account a particular design of the reconfigurable mechanism namely the spatial eight-bar single loop linkage. The structural parameters of this mechanism are analyzed for a general configuration. Structural analysis has shown that the eight bar linkage have two degrees of freedom in a general non-singular configuration. Further various assembly modes of the eight bar linkage were identified. The singular configurations of this particular mechanism have been identified and classified according to the types of singularities (constraint singularity, redundant singularity and constraint-redundant singularity). The properties of redundant and constraint-redundant singularities are defined for the first time in connection with the four main structural parameters: mobility, connectivity, number of overconstraints and structural redundancy. Therefore, we have defined and identified the different branches of the mechanism which can be singular or not. We found out that this mechanism has an interesting property of continuously transiting from one type of singularity to another by always remaining in a singular configuration. This property enhances the reconfiguration capability of the mechanism.

 The chapter 3 of thesis is dedicated to the analysis of singular configurations of the 8bar reconfigurable mechanism. The mechanism has a large number of branches and can switch from one to another by crossing a singular configuration. The geometric models corresponding to each branch have been derived. They give all possible relations between joint variables and imposed by joint kinematic constraints. These models are useful for mechanism control, especially for configuration changes at singular positions where the actuation redundancy is used.

 In chapter 4, the dynamic model of the mechanism is built in ADAMS software with complete parameterization of length and angle between each joint. The challenge is to control the eight bar linkage by synchronizing all the five motors in such a way to avoid the lock of the robot. This is done by using the geometric model relations to control the mechanism for a planned trajectory. The implementation of the numerical results and simulated results on a real prototype has been carried out. The performance of the robot is monitored.

1 Chapter 1: Types of morphing and systematization of reconfigurable mechanisms

Metamorphic mechanisms

The development in the second part of 1990s raised much interest in reconfigurable mechanisms with researchers generating numerous types of such mechanisms. The change of states, configurations and/or functional modes is the transformation process that we called morphing in the sense of image processing or metamorphosis in the sense of biology. In the time of rigorous development of science and technology, the cross-disciplinary concept brings thriving progress of the technology and thus results in development of reconfigurable mechanisms characterized by a certain ability of morphing [START_REF] Aimedee | Systematization of Morphing in Reconfigurable Mechanisms[END_REF]. Metamorphic mechanisms originated by metamorphosis involving change in form, topology and configuration of a mechanism [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF], [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF] is an extension of the context of biology to mechanism topology change [Zhang et al., 2008 (b)] characterized by change of main structure parameters of a mechanism (mobility, connectivity, overconstrained and redundancy) in a number of topological configurations.

The initial concept of "metamorphic mechanisms" was originated by metamorphosis, which depicts the change in form, topology, and configuration. Metamorphic robots were presented as a class of serial reconfigurable manipulators allowing the creation of structures presenting a large number of different anatomies each, many of which are outside the current line of design practice for robots [START_REF] Charalampos | Index based optimal anatomy of a metamorphic manipulator for a given task[END_REF].

Kinematotropic mechanisms

In the 1990s, two new types of mechanisms were identified as kinematotropic linkages [START_REF] Wohlhart | Kinematotropic linkages[END_REF] and metamorphic mechanisms [START_REF] Dai | Conceptual Design of the Dexterous Reconfigurable Assembly and Packaging System (D-RAPS)[END_REF], [START_REF] Dai | Mobility in metamorphic mechanisms of foldable/erectable Kinds[END_REF] and [Zhang et al., 2008 (a)]. Kinematotropic mechanisms use the bifurcation phenomenon and extend it to involving mobility change with change of relative orientations of joint axes [START_REF] Galletti | Single-loop kinematotropic mechanisms[END_REF][START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF].

We can see that both kinematotropic mechanisms and metamorphic mechanisms are complementary towards using mobility change.

Reconfigurable mechanisms

In the first decade of the 21 st century, much interest has been raised with a broad search for reconfigurable mechanisms. In 2000, Yan and Liu [START_REF] Yan | Finite-state-machine representations for mechanisms and chains with variable topologies[END_REF] investigated mechanisms and chains with variable topologies by using the finite-state-machine representation and in 2003 the work was extended to joint-code representation [START_REF] Yan | Joint-codes representations for mechanisms and chains with variable topologies[END_REF]. Variable kinematic joints were further investigated by Yan and Kuo [START_REF] Yan | Topological representations and characteristics of variable kinematic joints[END_REF] and [START_REF] Yan | On the mobility and configuration singularity of mechanisms with variable topologies[END_REF]. With the variable prismatic joint [START_REF] Yan | Topological representations and characteristics of variable kinematic joints[END_REF], a number of mechanisms with variable topologies were presented by Yan and Kang [START_REF] Yan | Structural analysis and configuration synthesis of mechanisms with variable topologies[END_REF].

In the same period, Lee and Hervé [START_REF] Lee | Discontinuous Mobility of One Family of Spatial 6R Mechanisms through the Group Algebraic Structure of Displacement Set[END_REF] revealed discontinuously movable mechanisms in 2002 by fully exploring the branching characteristics of mechanisms where one particular motion ends and switches to a different motion with a changed mobility. This resulted in a number of reconfigurable mechanisms with mobility change [START_REF] Lee | Discontinuously Movable Seven-link Mechanisms via Group-Algebraic Approach[END_REF], [START_REF] Lee | Discontinuously Movable 8R Mechanisms with an Infinity of Bifurcations, 12th IFToMM World Congress[END_REF] in 2005 and 2007. The branching characteristics entail much exploration and were investigated by Kong and Gosselin in developing multimode parallel mechanisms in 2007 [START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF].

(a.) (b.) Modular self-reconfigurable (MSR) robots are robots composed of a large number of repeated modules that can rearrange their connectedness to form a large variety of structures. An MSR system can change its shape to suit the task, whether it is climbing through a hole, rolling like a hoop, or assembling a complex structure with many arms. These systems have three promises namely,

 Versatility: The ability to reconfigure allows a robot to form morphologies that are well-suited for a variety of given tasks.  Robustness: Since the system is composed of many repeated parts which can be rearranged during operation, faulty parts can be discarded and replaced with an identical module on-the-fly, leading to self-repair.  Low cost: MSR systems lower the module costs since mass production of identical unit modules has an economic advantage that scales favorably. Also, a range of complex machines can be made from a set of modules saving the cost versus having multiple single-function machines for doing different tasks. The acm-r5 is based on the mechanics behind snakes. Powered by a lithium-ion battery, the ACM-R5 is a radio-controlled amphibious robot designed to move like its real world counterpart. It can slither or swim underwater for 30 minutes on a full charge. An intricate sensor system (attitude/torque), small-sized camera, and a 32bit micro controller are placed inside the robot. While this robot seems more like a single object based robot it is made up of self-similar parts that work together to accomplish changing geometrical demands.

In the development of reconfigurable mechanisms, various ways to achieve reconfiguration were investigated and developed. In the study of reconfigurable packaging in late 1990s and in 2000s, metamorphic mechanisms were associated with ability to change their geometrical structures [START_REF] Dai | Conceptual Design of the Dexterous Reconfigurable Assembly and Packaging System (D-RAPS)[END_REF], [START_REF] Dai | Kinematics and mobility analysis of origami-carton folds in packing manipulation based on the mechanism equivalent[END_REF]. Other approaches for reconfigurability and metamorphosis include change of number of links by link coincidence and self-locking [Zhang et al., 2008(a)], [START_REF] Leonesio | A general approach for self-locking analysis in closed kinematic chains[END_REF] and change of geometrical constraint to links and joints [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF], [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF], [START_REF] Zhang | A kirigami-inspired 8R linkage and its evolved overconstrained 6R linkages with the rotational symmetry of order two[END_REF], [Zhang et al., 2011 (b)], [START_REF] Zhang | Geometric constraint and mobility variation of Two 3SvPSv metamorphic parallel mechanisms[END_REF], [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF]. In this way, a metamorphic mechanism was considered as a mechanism set composed by multiple kinematic chains which have the ability to be transformed sequentially from one to another following specific rules in order to meet different requirements of tasks. In the paper presented by Charalampos et al [START_REF] Charalampos | Index based optimal anatomy of a metamorphic manipulator for a given task[END_REF], metamorphic manipulators were presented as a class of serial reconfigurable mechanisms allowing creation of a large number of different topologies.

Types of Morphing

In this section we will review various reconfigurable mechanisms in their essence in morphing or metamorphosis to present a systematic classification of morphing in their use to reconfigure a mechanism. Therefore, three types of morphing are classified as topological morphing, geometrical morphing and multi-furcating morphing and further classifications are made in terms of their intrinsic characteristics. These morphing types are then related to types of mechanisms and motion to demonstrate the morphing process in reconfigurable mechanisms [START_REF] Aimedee | Systematization of Morphing in Reconfigurable Mechanisms[END_REF]

Topological morphing

From a mathematical point of view, "topology" refers to continuous deformations including stretching and bending. This includes such properties as connectedness, continuity and boundary. In the field of mechanism and machine science, the concept of "topology" has been employed and extended to studying structural characteristics of mechanisms and machines. In general terms, "topology" is defined as a way that parts of a system are organized or connected. Topological morphing is defined by changing between the physical topology to an equivalent topology with fewer number of joints and links without assembly and disassembly. This change to the equivalent topology could be achieved by joint-motion range limits and by internal forces as shown in figure 1.3. The equivalent topology is associated with a reduced number of joints and links with respect to the initial topology. The way to interconnect links and joints could also change. This interconnection could be represented by the topological graph [Gogu, 2008a] and connectivity matrix [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF], [START_REF] Dai | Matrix representation of topological configuration transformation of metamorphic mechanisms[END_REF]. The mechanism state matrices were discussed by [START_REF] Brain | Mechanism State Matrices for Planar Reconfigurable Mechanisms[END_REF]. It deals with the state matrices which enable to represent the topological characteristics of planar reconfigurable mechanisms. These matrices also help calculating the degrees of freedom of the planar mechanisms with one DOF joints. The state matrices of the spatial reconfigurable mechanisms were also discussed by Philip et al [START_REF] Philip | Mechanism State Matrices for Spatial Reconfigurable Mechanisms[END_REF].

Figure 1.3 Classification of topological morphing

Topological morphing

By limiting joint motion range By using internal forces

Morphing by limiting joint motion range

Joint-motion range limit is used in some of the mechanisms presented by Zhang and Dai [Zhang et al., 2008(a)]. Different working phases were generated from a source metamorphic mechanism by using a range limit with an orientation restriction. A spherical joint in this approach is degenerated into a revolute joint and a cylindrical joint is degenerated into a prismatic joint to achieve reconfigurability while a mechanism falls into a motion range that has a range limit. Therefore a physical topology can be reconfigured to an equivalent topology with various states and phases during the process of metamorphosis. The motion limit of the variable prismatic joint was used by Yan and Kang [START_REF] Yan | Structural analysis and configuration synthesis of mechanisms with variable topologies[END_REF] to construct mechanisms with variable topologies. A motion range limit was further used in a mechanism presented by Zhang et al [Zhang et al., 2011(b)] on changing joint-motion direction when a joint reaches a certain geometry that guides the joint to rotate in a different direction. In figure 1.4, OABC, O C and O C represent the corresponding mechanism motion states such as translation, oscillating and ceasing, respectively. With this approach, these operations such as translating, oscillating and ceasing can be achieved by morphing.

Morphing by using internal forces

Using an internal force while limiting joint motion, a mechanism could change its configuration under this morphing. An example in figure 1.5 could be illustrated in a mechanism proposed by Zhang et al [Zhang et al., 2011(b)] and by [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF] whose reconfiguration is achieved by using a potential force from a spring with the joint motion range limit to create branching configurations. In this morphing, a metamorphic mechanism converts into a lower mobility mechanism by self-locking [START_REF] Leonesio | A general approach for self-locking analysis in closed kinematic chains[END_REF] a joint using the geometric limit together with a potential force that creates branching configurations where a slider acts as a mechanical element sprung back by a spring, leading to a different configuration. The mechanisms with variable structure and geometry were presented by Sarkissyan et al [START_REF] Sarkissyan | Synthesis of mechanisms with variable structure and geometry for reconfigurable manipulation systems[END_REF] illustrating three types of reconfigurable mechanisms.

Reconfiguration is implemented by locking joints and also by combining one or two subchains with axes of the kinematic pairs. It is ensured that the mechanisms chosen have the minimum degrees of freedom to obtain optimal solutions to the given tasks.

A new class of reconfigurable mechanism from a 3-CPS mechanism was discussed by Carbonari et al [START_REF] Carbonari | A new class of reconfigurable parallel kinematic machines[END_REF]. The mechanism is composed of 3 legs in which each leg is composed by a cylindrical, prismatic and a spherical joint. Reconfiguration is achieved by instantaneously locking one at a time, the revolute joints in the spherical joint. By doing this we obtain pure rotation and pure translation in the mechanism.

Further to this, a class of mechanisms called as parallel robot with enhanced stiffness (PRES) has been studied in Moosavian and Xi [START_REF] Moosavian | Design and analysis of reconfigurable parallel robots with enhanced stiffness[END_REF], to enhance the static and stiffness characteristics by varying their topology without a need of actuation redundancy. This was completed by locking passive joints which are used in reconfiguring the mechanisms.

According to Finistauri et al., group morphing is a combination of geometric and topological reconfiguration and is seen in advanced systems such as variable geometry truss mechanisms (VGTM). One such example is the modular wing truss capable of multiple level of reconfiguration to achieve unique wing shapes [START_REF] Finistauri | Type synthesis and kinematics of a modular variable geometry truss mechanism for aircraft wing morphing[END_REF]. The topological morphing is also coupled with other morphing as introduced by Xi and Finistauri [START_REF] Xi | Reconfiguration analysis of a fully reconfigurable parallel robot[END_REF] in group morphing.

Geometrical morphing

After a critical review, we define geometrical morphing as the transformation of a particular geometrical position and orientation of relative joint axes and links achieved by geometrical constraints. This morphing can be achieved in different ways as represented in figure 1.6 and is discussed in the following subsections.

Morphing by changing relative position of joint axes

In changing of relative positions and orientations of joint axes without superposition or additional means, kinematotropic mechanisms proposed in 1996 entail a mobility change. Particular cases can be seen in Wunderlich mechanism, and Wren platform illustrated and the Queer-square linkage proposed by Wohlhart [START_REF] Wohlhart | Kinematotropic linkages[END_REF], and the derivative Queersquare mechanism presented by Qin et al [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF]. Using special joint property and link superposition to change relative positions and orientations of joint axes with superposition, multi-loop metamorphic mechanisms entail the mobility change as shown in figure 1.7 [START_REF] Dai | Mobility in metamorphic mechanisms of foldable/erectable Kinds[END_REF]. Both mechanisms complement the mobility change through change of relative positions of axes using different ways. In these mechanisms, a joint axis changes its relative position with respect to other joint axes following particular reconfigurations. These changes subsequently produce a change in the geometrical structure of a mechanism. By this way, kinematotropic mechanisms and a particular type of metamorphic mechanisms can be characterized under geometrical morphing.

Superposing joint axes

Superposing joint axes is a way of morphing. This was implemented by the vA joint. A vA joint is the variable-axis joint which decomposes the spherical motion of a conventional Geometrical morphing

Morphing by changing relative position of joint axes

Morphing by superposing or aligning two/more links Morphing by using plane symmetry Morphing using internal mobilities spherical joint [START_REF] Zhang | Geometric constraint and mobility variation of Two 3SvPSv metamorphic parallel mechanisms[END_REF]. Different rotation phases about three intersecting axes are produced by the vA joint. The three stable working phases are derived from the kinematic joint and the constraints supplied by this kinematic joint are changeable following the variation of the rotational axis. The vA joint is formed by three individual joints with their axes intersecting each other. The second-joint axis is the reconfigurable axis that intersects the first joint axis in 45 degree. This joint rotates the third joint axis either out of or into the plane formed by the first and the second joint axes. The former, while the third joint axis is out of the plane constitutes a spherical joint, the latter, while the third joint axis is in the plane forms a precondition for constituting both U-joint and R-joint. In the latter case when all joint axes are coplanar, there are two subcases. The first subcase occurs when the third joint axis is perpendicular to the first joint axis; this constitutes a U-joint circumstance. The second case occurs when the thirdjoint axes is in line with the first joint axis, this forms a R-joint circumstance. The transition from the U-joint case to the R-joint case is completed by the reconfigurable joint that rotates the third-joint axis out of the plane by 180 degree about the reconfigurable joint axis and to be back to the plane in alignment with the first-joint axis. This gives the R-joint case.

Therefore the vA joint is a variable-axis joint where the relative orientations of three intersecting axes are changing to achieve three different working phases of joint configurations.

This superposition of joint axes results in change of joint mobility and change of structural parameters of a parallel mechanism as in the mechanism presented by Zhang et al [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF], [START_REF] Zhang | Geometric constraint and mobility variation of Two 3SvPSv metamorphic parallel mechanisms[END_REF]. Using this joint property to change the relative positions of joint axes is a type of metamorphic mechanisms. In this process, the source metamorphic phase makes a transition to various phases as shown in figure 1.8.

Enabling joint axes to be coplanar

Enabling joint axes to be coplanar is a way of morphing. While doing this enabling, various geometrical properties such as co-linearity, parallelism and perpendicularity of joint axes can be achieved. This creates a set of metamorphic parallel mechanisms with reconfigurability in geometric morphing.

This can be demonstrated in the vA joint whose joint axes are made to lie on a plane from their initial state of spatial arrangement [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF] to enable the vA joint to change to a Hooke joint from its initial spherical joint state.

In the mechanism presented by Zhang et al [START_REF] Zhang | A new metamorphic mechanism with ability for platform orientation switch and mobility change[END_REF], a metamorphic mechanism with articulated links is illustrated. By aligning links, a union of two links can be obtained acting as a single link. By doing so, axes of two revolute joints become collinear. Assembling four L-shaped flat cards inspired chains, a new metamorphic mechanism was obtained to change relative positions of joint axes. This mechanism performs circular translational motions.

Superposing a joint axis with a link

This morphing can be executed by the rT (Fig. 1.9) joint where one axis of the joint could be aligned with a limb of a parallel mechanism to initiate an idle mobility and subsequently reduce the mobility of the platform as presented by Gan et al [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF]. The examples presented in the paper are multi-loop mechanisms.

The rT joint is a reconfigurable Hooke joint in which an axis of a traditional T joint could rotate and change the initial orientation with respect to a link it connects. Usually a Hooke joint consists of two revolute joints with axes intersecting at the right angle to form a Tshaped connector of mobility two. Based on metamorphosis, it is expected that the direction of a rotation axis be altered to enable change of axis orientations. This leads to a new design that varies [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF] the rotation axis to realize the reconfiguration of the joint. This ability of changing the rotation axis is initiated by rotating one axis along the groove of a ring. This subsequently creates the rT joint, where T is commonly used for a Hooke joint and 'r' stands for reconfigurable with the added revolute joint.

In the metamorphic parallel mechanism, two prominent phases of the rT joint [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF] are used. With the relative position between the radial axis in the grooved ring and the limb it installs, two phases are presented. Phase 1 is achieved by fixing the radial axis perpendicular to a limb that the limb has two rotations in pitch motion about the radial axis and yaw motion about the other axis. Phase 2 is achieved by fixing the radial axis in line with the limb where the radial axis rotation becomes an internal mobility and the limb it installs will have only one rotation being exerted. These two phases stemming from change of the reconfigurable Hooke joint exert an alterable number of rotations to a mechanism. Subsequently, a family of metamorphic parallel mechanisms were created by Gan et al [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF] with a way of installing the reconfigurable Hooke joint rT as shown in figure 1.9. In these parallel mechanisms, three rT joints are installed between limbs and the platform, and other three rT joints are installed between limbs and a basis, together with three prismatic joints on limbs. All axes of the rT joints intersect at a single point to create the pure rotation. With the change of the rT joints by superposing one axis aligned with a limb, the platform mobility is changing.

With a combination of rT joints installed in a mechanism, more metamorphic parallel mechanisms can be identified. Changing the arrangement of joint axes of the rT joints enables change of a motion type of a parallel mechanism. By making all rT joint axes in the metamorphic parallel mechanisms [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF] in parallel to a base, the mechanism generates the translation-only mode to achieve the morphing from a pure rotation to a pure translation.

1.4.2.1.5 Using the joint-motion switch Switching joint-motion direction proposed by Yan and Kuo [START_REF] Yan | On the mobility and configuration singularity of mechanisms with variable topologies[END_REF] using a joint coined as the sA joint which is the switch-axis joint, a mechanism can reconfigure itself to achieve various mobility and connectivity. This shift axis of rotation to redirect the joint axis to a different orientation and to change the link orientation is used in a mechanism presented by Zhang et al [Zhang et al., 2012(d)] as shown in figure 1.10. To achieve the reconfiguration, the mechanism has four sA joints together with four conventional joints. By changing the orientation of joint axes, reconfigurability in geometrical morphing is elegantly achieved. A further single loop mechanism example proposed by Zhang and Ding [Zhang et al., 2012(b)] achieves morphing by utilizing a motion-type switch to change a mechanism from a planar mechanism to a spatial one as illustrated in figure 1.11 [START_REF] Li | Augmented adjacency matrix for topological configuration of the metamorphic mechanisms[END_REF]. With change of the axis orientation based on a metamorphosis process, Li et al [START_REF] Li | Topological Representation and Operation of Motion Space Exchange Reconfiguration of Metamorphic Mechanisms Computational Kinematics[END_REF] proposed a mechanism of a multiple look that can be reconfigured from the planar motion to the spatial motion. In the study, a joint-axis matrix and an augmented adjacent matrix of kinematic chains were proposed to model the metamorphosis process. Military domains require more manpower which seems to be risky. This manpower can be replaced by reconfigurable robots. One such robot which can be used for this application having three uniform modules, each forming a reconfigurable mechanism, was studied by Wang et al [START_REF] Zhang | Design and Realization of a Novel Reconfigurable Robot with Serial and Parallel Mechanisms[END_REF]. This JL-I robot achieves highly adaptive locomotion to drive serial and parallel mechanisms to form an active joint. This generation of active joint helps change shape and enhance motion on rough terrains. The yaw and pitch motions are achieved by the parallel mechanisms whereas the third rotation is achieved by a serial mechanism.

Therefore by switching between motions, reconfigurability is achieved.

Morphing by superposing or aligning two or more links

The metamorphic palm of a novel multi-fingered robotic hand presented by Dai and Wang [START_REF] Dai | Geometric Analysis and Synthesis of the Metamorphic Robotic Hand[END_REF], [Dai et al., 2009 (b)] as illustrated in figure 1.12 and by Cui and Dai [START_REF] Cui | Posture, workspace, and manipulability of the metamorphic multifingered hand with an articulated palm[END_REF] is a spherical five-bar linkage having a characteristic of metamorphosis. There are two drivers on the palm which adjusts the position and orientation of the palm mechanism. When one of the drivers is fixed and the crank link of the palm overlaps the base link, two links are aligned, resulting in the palm to operate as a reconfigurable spherical four-bar linkage with one degree of freedom.

This mechanism was used in the creation of a novel multi-fingered hand as a metamorphic palm done by Dai et al [START_REF] Dai | Geometric Analysis and Synthesis of the Metamorphic Robotic Hand[END_REF], [Dai et al., 2009], [START_REF] Cui | Posture, workspace, and manipulability of the metamorphic multifingered hand with an articulated palm[END_REF] and by Wei et al [START_REF] Wei | Kinematic analysis and prototype of a metamorphic anthropomorphic hand with a reconfigurable palm[END_REF] as shown in figure 1.12.

The mechanism in an anthropomorphic hand presented by Wei et al [START_REF] Wei | Kinematic analysis and prototype of a metamorphic anthropomorphic hand with a reconfigurable palm[END_REF] illustrates a folding palm of the metamorphic hand where each finger consists of three revolute joints. Various tasks can be completed using the metamorphic hand as illustrated in figure 1.13. By aligning two links at one occasion, various palm configurations achieve, resulting in stretching and folding the palm with an in-hand manipulation. The process of aligning links leads to systematizing this mechanism under geometrical morphing. Further example on aligning links can be seen in a spatial 10-bar mechanism derived from the augmented Assur group proposed by Li and Dai [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF], [START_REF] Li | Augmented adjacency matrix for topological configuration of the metamorphic mechanisms[END_REF]. Design of a metamorphic mechanism with three configurations was presented by Zhang et al [Zhang et al., 2011(b)]. This mechanism has a transformation sequence from one configuration to another by aligning two links using the force or geometrical limits. The mechanism was used in a door-opening structure of a space shuttle as a crank-slider metamorphic mechanism via stiffness variation. This mechanism presents as an example of geometrical morphing.

A planar lift mechanism with a number of revolute joints was proposed by Zhao et al [START_REF] Zhao | A reconfigurable linkage and its application in lift mechanism[END_REF]. This is an interesting mechanism portable by means of the zigzag arrangement of joints and by forming a triangular closed loop to operate as a lift. The lift mechanism is constructed from scissor-like linkages as a deployable unit. During the operation, the links of the mechanism get aligned in a single line and are hence compacted to complete the lift mechanism.

Morphing using plane symmetry

For a reconfigurable mechanism with a feature of plane symmetry and metamorphosis, its plane symmetry is used. This is in particular presented by kinematotropic mechanisms [START_REF] Wohlhart | Kinematotropic linkages[END_REF], [START_REF] Galletti | Single-loop kinematotropic mechanisms[END_REF], [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF], [START_REF] Fanghella | Stability of Branches of a Kinematotropic Mechanism[END_REF].

The case can also be seen in a reconfigurable mechanism proposed by Zhang et al [Zhang et al., 2012 (d)]. Eight links are connected by four sA joints and four revolute joints. At a particular instant of plane symmetry, changing relative positions of two joints with respect to each other, reconfigurability using geometric morphing is achieved. A mechanism of an origami presented in figure 1.14 by Dai and Rees Jones [START_REF] Dai | Mobility in metamorphic mechanisms of foldable/erectable Kinds[END_REF], [START_REF] Dai | Kinematics and mobility analysis of origami-carton folds in packing manipulation based on the mechanism equivalent[END_REF], demonstrates a commonly found metamorphic mechanism. Taking panel sections and creases as links and revolute joints respectively, an origami mechanism with ten revolute joints was illustrated and reconfigurability was achieved by using the plane symmetry with five revolute joints in one side and other five revolute joints in the other side.

This reconfigurable mechanism is used in the industry for origami folding in the confectionary industry to adapt to various shapes of origami [START_REF] Dai | Mobility in metamorphic mechanisms of foldable/erectable Kinds[END_REF][START_REF] Yao | Automatic folding of cartons using reconfigurable robotic system[END_REF]. A further example of carton erection is dealt by Dai et al [Dai et al., 2009 (a)], [START_REF] Dubey | Complex Carton Packaging with Dexterous Robot Hands[END_REF], where various motions in erecting origami-cartons are used to reconfigure the machine to adapt for different origami-carton sizes and styles.

Morphing using internal mobilities

This type of morphing can be achieved when internal mobilities are produced while replacing an existing joint with a reconfigurable joint. This is mostly done by using the rT joint or vA joint. Morphing of this type can be illustrated in examples presented by Gan et al [START_REF] Gan | Mobility Change in Two Types of Metamorphic Parallel Mechanisms[END_REF]. In a 3-UPU mechanism, replacing all U joints by rT joints, a mechanism with variable mobility changes motion from a pure rotation to a pure translation. In this transition, the mechanism generates internal mobilities and forms a reconfigured mechanism.

A further example of geometrical morphing was demonstrated in figure 1.15 by Gan et al [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF]. By altering the rT joint one by one in all three limbs of a metamorphic parallel mechanism, resulting in switching a global mobility to a local mobility gradually, the mobility changes from 6 to 1 consecutively. Therefore, by changing the internal mobilities, reconfigurability is achieved through alteration of the rT joints. The mechanisms presented are double loop mechanisms.

A type-changeable kinematic pair with various phases has been discussed by Zhang et al [Zhang et al., 2012 (c)]. The platform of a reconfigurable parallel mechanism has the ability to change the phases of the kinematic pair, leading to mobility change from 6 DOFs to 3DOFs. The type-changeable kinematic pair changes its topological configuration, resorting to link annex. By annexing links, reconfiguration is achieved in this parallel mechanism.

A systematic way of synthesizing metamorphic parallel mechanisms, using screw theory based on the internal mobility principle in co-line, co-plane and co-sphere with internal mobilities, was proposed by Gan et al [START_REF] Gan | Constraint-Based Limb Synthesis and Mobility-Change Aimed Mechanism Construction[END_REF] and some new metamorphic parallel mechanisms were constructed. The co-line principle was then extended for a new family of metamorphic parallel mechanisms based on point-plan geometric constraint in [START_REF] Gan | Constraint-Plane-Based Synthesis and Topology Variation of A Class of Metamorphic Parallel Mechanisms[END_REF]. The connectivity morphing covers several morphing branches where the degree of mobility is constant but connectivity changes. The mechanism provided by [START_REF] Galletti | Single-loop kinematotropic mechanisms[END_REF]] is a single loop planar four-bar linkage with one degree of freedom. At an initial stage, the mechanism has two prismatic joints with parallel axis and at this point, the mobility is one and the connectivity of the prismatic pair is equal to one with a zero-mobility revolute joint. The mechanism then makes a transition and moves to a singular position and finally comes out of this position with coincident revolute joint axes as illustrated in figure 1.16. At this instant the mobility remains the same but the connectivity changes with the connectivity of the revolute pair as one and that of the prismatic as zero. This is an example in which morphing can be achieved. Mobility morphing mostly occurs in bifurcation in kinematotropic mechanisms [START_REF] Wohlhart | Kinematotropic linkages[END_REF], [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF], [START_REF] Fanghella | Stability of Branches of a Kinematotropic Mechanism[END_REF] that results in change of mobility. In 2013, a novel spatial-spherical metamorphic mechanism with bifurcated motion was presented by Gan and Dai [Gan et al., 2013 (a)]. This presents a class of 3-PUP double loop mechanisms with various configurations. When the mechanism is in a constraint singular configuration, it bifurcates hence the mobility changes from 3 to 2 with one translation and one rotation and the connectivity in the sense of motion type changes. Plitea et al also illustrated a robot with 6 degrees of freedom which enable its reconfiguration to achieve mechanism variation from 6 to 2 degrees of freedom [START_REF] Plitea | Structural design and kinematics of a new parallel reconfigurable robot[END_REF]] Furcating morphing generates different branches in mechanism reconfiguration leading to classification of various numbers of branches in morphing. Bifurcation takes place in a constraint singularity position in which connectivity between moving and fixed platforms increases instantaneously with no change in limb connectivity. When branching occurs in a constraint singularity, a mechanism reaches different configurations as branches as shown in figure 1.17 and can have different independent motions of a moving platform [START_REF] Gogu | T2R1-type parallel manipulators with bifurcated planar-spatial motion[END_REF] leading to change in connectivity and/or mobility.

Several investigations have been made for bifurcation by Gan et al [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF]; Gan and Dai [Gan et al., 2013 (a)]; Zhang et al [START_REF] Zhang | Geometric constraint and mobility variation of Two 3SvPSv metamorphic parallel mechanisms[END_REF], [Zhang et al., 2012 (a)]; Lee and Hervé [START_REF] Lee | A Discontinuously Movable Constant Velocity Shaft Coupling of Koenigs Joint Type[END_REF]; Gogu [START_REF] Gogu | T2R1-type parallel manipulators with bifurcated planar-spatial motion[END_REF]; and multi-furcating morphing by Qin et al [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF]. When a metamorphic mechanism or a kinematotropic mechanism passes through a constraint singularity then bifurcates by influence of geometrical constraints such as parallelism or perpendicularity, morphing is achieved.

Branching in constraint singularities

In the example presented by Gogu [START_REF] Gogu | Branching singularities in kinematotropic parallel mechanisms[END_REF] four types of branching singularities are identified as type A (BS-A), type B (BS-B), type C (BS-C) and type D (BS-D) leading to furcating morphing. Each of them has their own properties of structural parameters.

A typical bifurcating morphing occurs in the parallel mechanisms with multiple modes proposed by [START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF], [START_REF] Kong | Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method[END_REF].

In 2009, a group of kinematotropic mechanisms with bifurcated motion for each type of branching in constraint singularities has been presented by Gogu [START_REF] Gogu | Branching singularities in kinematotropic parallel mechanisms[END_REF]. During the process of bifurcation, axes of the last revolute joint of the first limb and the first revolute pair of the second limb overlap leading to a morphing process. A family of mechanisms presented by Gogu [START_REF] Gogu | Maximally regular T2R1-type parallel manipulators with bifurcated spatial motion[END_REF] also illustrates bifurcation morphing.

The multi-loop mechanism presented in the paper by Zhang et al [START_REF] Zhang | A new metamorphic mechanism with ability for platform orientation switch and mobility change[END_REF] is an example of bifurcating morphing where one instantaneous rotation and a translation are exchanged during the motion.

A metamorphosis that results in reconfiguring a mechanism from generating the pure translation to the pure rotation can be illustrated in a metamorphic mechanism proposed by Zhang et al in figure 1.18 [START_REF] Zhang | Topology and constraint analysis of phase change in the metamorphic chain and its evolved mechanism[END_REF] where the platform implements a translational motion from the transitory position, passing through a constraint singularity position. This produces a bifurcated motion branch with one translation. When the platform implements a rotation motion and passes to the transitory position, the second motion branch can be realized. Therefore, two bifurcating motions i.e. a twisting motion between the platform and the base and a curvilinear motion are implemented. Hence reconfigurability is achieved using the bifurcating morphing.

The 3-PUP parallel mechanism presented by Gan and Dai [Gan et al., 2013 (a)] consists of three identical PUP limbs, in which axes of three U joints are parallel correspondingly to each other and the P joints on the base are perpendicular to the base plane. When three limbs have the same length, the home position configuration is when the platform is parallel to the base, acting as the constraint singularity configuration. The bifurcation is trigged at this position enabling the mechanism to possess furcating morphing.

Instantaneous locking bifurcation

By instantaneous locking of joints, a mechanism can branch into a non-singular configuration. This is called as the instantaneous locking bifurcation.

This type of bifurcation occurs in double loop mechanisms dealt by Gogu [START_REF] Gogu | T2R1-type parallel manipulators with bifurcated planar-spatial motion[END_REF]. These mechanisms require two independent planar translations and one independent rotation of the mobile platform around an axis perpendicular to the plane of translations or parallel to this plane. In the first case, the moving platform has a planar motion, and in the second case, it has a spatial motion. By instantaneously locking a revolute joint, two bifurcating motions occur switching from planar to spatial motion.

Types of furcating morphing

The following types of furcating morphing are identified in figure 1 Another configuration is obtained by rotating pairs of two planar sets with non-coincident axes of the normal revolute joints hence making this pair as locked revolute joints. At this configuration the mechanism has 3 degrees of freedom. The mechanism uses the kinematotropic bifurcation morphing.

Furcating morphing in metamorphic parallel mechanisms

Furcating morphing in metamorphic parallel mechanisms entails mobility change by using furcation with change of joint property. They perform bifurcation and hence possess the property of furcating morphing.

At the home position configuration, mobility changes from 6 to 1 by altering the rT joints from one phase to another in an example proposed by Gan et al [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF]. It is found

Furcating morphing

Kinematotropic bifurcating morphing

Furcating morphing in metamorphic parallel mechanism

Bifurcation in discontinously movable parallel mechanisms

Multifurcating morphing

Actuation morphing that the 3 (rT)C(rT) metamorphic parallel mechanisms has the bifurcation point at the home position when it is evolved into a configuration with mobility fewer than 3. The home position presents the bifurcation point with two branch motions each of which has mobility one. In the last topological configuration, two branches have neither the same mobility nor the same motion. Hence this mechanism can be characterized under furcating morphing.

In the mobility 2 phase of the 4rTPS metamorphic parallel mechanism, two rotation branches bifurcate at the home position along two perpendicular directions while a common translation exists all the time as shown by Gan et al [Gan et al., 2013 (b)]. A further example with the reconfigurable Hooke joint is used in the mechanism presented by Gan et al [START_REF] Gan | Reconfiguration and Actuation Scheme of 3rTPS Metamorphic Parallel Mechanisms with Parallel Constraint Screws[END_REF], where the main characteristic is the mobility change from full mobility to lower mobility with constraint forces. This mechanism exhibits the property of reconfigurability by changing the phases of the rTPS limbs.

Bifurcation in discontinuous movable mechanisms

Discontinuously movable mechanisms proposed by Lee and Hervé shown in figure 1.21 [START_REF] Lee | A Discontinuously Movable Constant Velocity Shaft Coupling of Koenigs Joint Type[END_REF] uses the bifurcation principle in generating different mobility in different motion branches where a motion would be ended at the constraint singularity discontinuously and switched to a different motion with different mobility. This reflects particular characteristics of this type of mechanisms by fully utilizing their branching behaviors. Subsequently, a number of these types of mechanisms [START_REF] Lee | Discontinuous Mobility of One Family of Spatial 6R Mechanisms through the Group Algebraic Structure of Displacement Set[END_REF], [START_REF] Lee | Discontinuously Movable Seven-link Mechanisms via Group-Algebraic Approach[END_REF], [START_REF] Lee | Discontinuously Movable 8R Mechanisms with an Infinity of Bifurcations, 12th IFToMM World Congress[END_REF], [START_REF] Lee | A novel discontinuously movable six-revolute mechanism, Reconfigurable Mechanisms and Robots[END_REF] were generated with their various branch behaviors.

Multifurcating morphing

Multifurcating morphing is the extent of furcating morphing. The feature of it can be illustrated by parallel mechanisms with multiple operating modes.

Kong et al [START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF] presented a family of parallel mechanisms with multiple operating modes. These mechanisms are of 3 degrees of freedom and produce both spherical motion and spatial translational motion systematically. By entailing the characteristics of branches as modes, multiple modes with both spherical and translational motions are generated. The main particularity of these modes is that they have the same mobility but generate different motions; we call this reconfiguration as multifurcating morphing.

Figure 1.22 is the derivative queer-square mechanism presented by Qin et al [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF]. It clearly gives the process of multifurcation morphing where two phenomenon and fourteen states were discussed. Phenomenon one takes place when the mechanism passes through a constraint singular position leading to mobility changes. The mechanism changes its motion permanently during this phenomenon. Whereas, in phenomenon two, mobility change occurs as a result of change of orientations of the joint axes by passing through a constraint singularity. With the help of this phenomenon, multifurcation is achieved with different motions in four categories and fourteen states in the derivative queer-square mechanism. A family of reconfigurable parallel mechanisms with three identical kinematic reconfigurable limbs connecting the platform with the base is dealt by Ye et al [START_REF] Ye | A new family of reconfigurable parallel mechanisms with diamond kinematotropic chain[END_REF]. This reconfigurable mechanism uses a four-bar linkage with equal dimensions making it as a diamond shaped kinematotropic mechanism. By applying a constraint force or couple on the reconfigurable limb, the mechanism is able to switch from one branch to another and also come out of the singular configuration allowing the mechanism to perform different kinds of motion. Due to its ability to generate different motions, we systematize this mechanism under multifurcating morphing.

Actuation morphing

By using different combinations of actuations, mechanism morphing can be achieved in different branches with different operating modes.

The mechanism presented by Kong et al [START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF], uses different numbers of actuators for different modes. In parallel mechanisms, reduced number of actuators can be used to generate different motion patterns. In this, initially six actuators are used to perform spherical motions and the number of actuators is reduced to four to switch to various modes.

Joint-Motion morphing

The joint-motion morphing occurs when a mechanism generates an idle mobility or when the mechanism switches between active and idle mobility. This type of morphing can be achieved by various ways as illustrated in figure 1.23 and is discussed in the following. Switching between idle and active joints to achieve morphing can be illustrated in a metamorphic mechanism [START_REF] Gan | Constraint analysis on mobility change of a novel metamorphic parallel mechanism[END_REF]. With presence of a source generator and by operation of metamorphosis, pure rotation and pure translation are achieved respectively in subphases 1 and 2. From this, by reconfiguring the axis of rotation of the rT joint, the joint motion switch between idle and active joints is achieved.

Further example with idle-active switch morphing is the pin-in-slot joint presented by Yan and Kuo [START_REF] Yan | Topological representations and characteristics of variable kinematic joints[END_REF]. At the initial configuration, the mechanism performs both rotation and translation with 2 degrees of freedom. During this motion, when the pin reaches the end of the slot, a mobility produced by translation is restricted whereas it continues to rotate. This changes the property of a joint and leads to changing the number of degrees of freedom from 2 to 1.

Figure 1.24 is an application of a metamorphic mechanism to a steel-ingot cutting machine proposed by Li and Dai [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF] which illustrates two working stages of the machine.

In the first working stage the movement of the lower cutter is blocked by a spring and the upper cutter moves downwards. By this, one movement and hence one mobility produced by

Joint motion morphing

Idle-active switching morphing Idle joint morphing the lower cutter becomes idle. When the mechanism reaches the second working stage, the upper cutter remains static and the lower cutter moves upward. By this operation, the mobility of the upper cutter becomes idle and that of the lower cutter becomes active. This single loop mechanism clearly illustrates switching between idle-active morphing.

Idle-joint morphing

Changing an active joint to an idle joint is the idle-joint morphing. In 2009, Zhang et al [START_REF] Zhang | A new metamorphic mechanism with ability for platform orientation switch and mobility change[END_REF] presents a mechanism where two panels are folded together and made as a single link. By doing this with the mechanism equivalence principle [START_REF] Dai | Mobility in metamorphic mechanisms of foldable/erectable Kinds[END_REF], two revolute joints become perpendicular and two independent rotational degrees of freedom are produced. Hence by annexing, an idle revolute joint is generated leading to idle-joint motion morphing.

In the metamorphic robotic hand [START_REF] Dai | Geometric Analysis and Synthesis of the Metamorphic Robotic Hand[END_REF] shown in figure 1.25, [Dai et al., 2009 (b)] and a spatial hybrid mechanism [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF], an idle joint was generated, by superposing and by annexing two links respectively. The morphing in these mechanisms is also coupled with the geometrical morphing. By superposing two revolute joints in a 5R spherical metamorphic mechanism presented by Li and Dai [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF] a degree of freedom is lost and hence an idle joint is generated.

Reconfigurability is achieved in this mechanism by using the idle-joint morphing which is implemented in the steel ingot cutting machine. These steel ingots were used in a number of rail-craft systems including steel tools, rails, bore heads and indirectly in the tunnel bore via the production of steel blocks.

In each of the double loop mechanisms presented in the paper by Zhang et al [START_REF] Zhang | Geometric constraint and mobility variation of Two 3SvPSv metamorphic parallel mechanisms[END_REF], an idle mobility was generated by using the principle of metamorphosis of the vA joints. This spatial-spherical metamorphic parallel mechanism evolves themselves into different mobility configurations.

Note:

Some metamorphic mechanisms can be grouped under two or more types of morphing. For instance the mechanisms proposed by Dai and Wang [START_REF] Dai | Geometric Analysis and Synthesis of the Metamorphic Robotic Hand[END_REF] and Li and Dai [START_REF] Li | Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on Augmented Assur Group[END_REF] possess both geometrical as well as idle-joint morphing. Also, the crank-slider mechanism presented by Zhang et al [Zhang et al., 2011(b)] uses both the motion range limit (topological morphing) as well as aligning two links using force or geometrical limits (geometrical morphing). Hence there is a common property between these two morphing types.

Applications

The reconfigurable eight-bar linkage has various interesting applications in the day to day life and in industrial fields. One of the brainstorming applications of the eight-bar linkage is the reconfigurable cube mechanism which is commonly called as RCM [Kuo et al., 2014]. It is equivalent to a single-loop 8R spatial linkage possessing eight different topological configurations during reconfiguration. The reconfigurable cube mechanism is a foldable puzzle mechanism that has the ability to manipulate, alter its topological configurations followed by changes in mobility. This RCM can be viewed as a reconfigurable mechanism, variable topology mechanism, kinematotropic linkage or discontinuous mobility mechanism.  Metamorphic palm illustrated in figure 1.12 is capable of generating reconfigurable motions with the help of its foldable and flexible palm.

 The Metamorphic hand presented in figure 1.29 plays an important role in industrial automation and space technology. The applications presented above are largely used in the robotic industry due to its versatile reconfiguration motions and configurations.

Conclusion

This chapter has presented a comprehensive review of morphing techniques in reconfigurable mechanisms and has examined their relationship and existence in the mechanisms to help achieve reconfiguration. Based on mobility and connectivity, the morphing process has been analyzed in terms of topological, geometrical and furcating morphing, each of which has been revealed through literature in its broadness in the field of reconfigurable mechanisms and robots. These morphing techniques have been further related to mechanism type and motion type, and their applications have been identified. Typical reconfigurable joints such as vA joints and rT joints have been illustrated through the review in achieving mechanism reconfiguration. This chapter on bibliographic review has presented a foundation for the study of morphing techniques and metamorphosis in reconfigurable mechanisms.

In this thesis, we focus on furcation morphing. Furcation morphing does not require joint activation or locking. Nevertheless, we show in the following chapters that actuation redundancy is used to control the mechanism to perform multifurcation and switch between several branches.

Chapter 2: Structural analysis and geometric modeling of reconfigurable robots: Application to the single-loop eight bar linkage

This second chapter is devoted to the structural synthesis and modeling of reconfigurable robots. The first section presents the objectives and the general perspectives of structural analysis and synthesis. We then recall the formulation and parameters based on TCS method [START_REF] Gogu | Mathématiques pour la robotique : Représentation des mouvements des corps solides[END_REF] in order to obtain an expression of the geometric models. The rest of the chapter deals with the application of these methods to a single-loop eight-bar mechanism for its general non-singular configurations.

Structural analysis of parallel manipulators

Recent advances in research on parallel robots have contributed mainly to expand their potential use to both terrestrial and space applications including areas such as high speed manipulation, material handling, motion platforms, machine tools, medical fields, planetary and underwater exploration [Gogu, 2008a]. Therefore, the need for methodologies devoted to the systematic design of highly performing parallel robots is continually increasing. Structural synthesis is directly related to the preliminary phase of robot design, and represents one of the highly challenging subjects in recent robotics research. [START_REF] Patel | Parallel Manipulators Applications -A Survey[END_REF], [START_REF] Kong | Type synthesis of parallel mechanisms with multiple operation modes[END_REF], [Gogu, 2008a], [START_REF] Gogu | Branching singularities in kinematotropic parallel mechanisms[END_REF], [START_REF] Gogu | Structural synthesis of parallel robots, Part 3: Topologies with planar motion of the moving platform[END_REF], [START_REF] Gogu | T2R1-type parallel manipulators with bifurcated planar-spatial motion[END_REF], [START_REF] Qin | Multi-furcation in a derivative queer-square mechanism[END_REF].

Introduction

Mobility is the main structural parameter of a mechanism and also one of the most fundamental concepts in the kinematic and the dynamic modelling of mechanisms. Mobility is used to verify the existence of a mechanism, to indicate the number of independent parameters in the both kinematic and the dynamic models and to determine the number of inputs needed to drive the mechanism. The state of the art in the development of mobility formulae was largely discussed in [Gogu, 2005a]. Earlier works on the mobility of mechanisms go back to the second half of the XIX century to Chebychev [Chebychev, 1869], Sylvester [START_REF] Sylvester | On recent discoveries in mechanical conversion of motion[END_REF], Grübler [Grübler 1883,1885], Somov [Somov ,1887] and Hochman [START_REF] Hochman | Kinematics of machinery[END_REF]]. During the XX century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulas and approaches were derived and presented in the literature by Koenigs [START_REF] Koenigs | Introduction à Une Théorie Nouvelle Des Mécanismes[END_REF], Grübler [Grübler 1916[Grübler , 1917]], Malytsheff [START_REF] Malytsheff | Analysis and synthesis of mechanisms with a viewpoint of their structure[END_REF], Kutzbach [START_REF] Kutzbach | Mechanische Leitungsverzweigung, ihre Gesetze und Anwendungen[END_REF], Dobrovolski [Dobrovolski 1949[START_REF] Dobrovolski | Theory of mechanisms[END_REF], Artobolevski [Artobolevski , 1953], Moroskine [START_REF] Moroskine | General analysis of the theory of mechanisms[END_REF][START_REF] Moroskine | On the geometry of complex kinematic chains[END_REF] Voinea and Atanasiu [START_REF] Voinea | Contribution à l'étude de la structure des chaines cinématiques[END_REF], Kolchin [START_REF] Kolchin | Experiment in the construction of an expanded structural classification of mechanisms and a structural table based on it[END_REF], Rossner [START_REF] Rossner | Zur strukturellen Ordnung der Getriebe[END_REF], Boden [START_REF] Boden | Zum Zwanglauf gemischt ra¨umlich-ebener Getriebe[END_REF], Manolescu and Manafu [START_REF] Manolescu | Sur la détermination du degré de mobilité des mécanismes[END_REF], Ozol [START_REF] Ozol | On a new structural formula of mechanisms, Izv.-Vuz-ov[END_REF], Hunt and Phillips [START_REF] Hunt | Zur Kinematic mechanischer Verbindung fu¨r ra¨umliche Bewegung[END_REF], Waldron [START_REF] Waldron | The constraint analysis of mechanisms[END_REF], Manolescu [START_REF] Manolescu | For a united point of view in the study of the structural analysis of kinematic chains and mechanisms[END_REF], Bagci [Bagci, 1971], Antonescu [START_REF] Antonescu | Extending of structural formula of Dobrovolski to the complex mechanisms with apparent family[END_REF][START_REF] Antonescu | General formula for the d.o.f. of complex structure manipulators and robots[END_REF], Freudenstein and Alizade [START_REF] Freudenstein | On the degree-of-freedom of mechanisms with variable general constraint[END_REF], Hunt [START_REF] Hunt | Kinematic Geometry of Mechanisms[END_REF], Hervé [Hervé 1978a[Hervé , 1978b]], Baker [START_REF] Baker | On relative freedom between links in kinematic chains with cross-jointing[END_REF][Baker ,1981]], [START_REF] Gronowicz | Identifizierungs-Methode der Zwanglaufbedingungen von kinematischen ketten[END_REF] , Davies [Davies 1983a[Davies , 1983b[Davies , 1983c], Agrawal and Rao [Agrawal et al., 1987a[Agrawal et al., , 1987b[Agrawal et al., , 1987c]], Angeles and [START_REF] Gosselin | Détermination du degré de liberté des chaines cinématiques[END_REF], Dudita and Diaconescu [START_REF] Dudita | Optimizarea structurala a mecanismelor[END_REF], Fanghella and Galletti [Fanghella et al., 1988[START_REF] Fanghella | Mobility analysis of single-loop kinematic chains: an algorithmic approach based on displacement groups[END_REF], Fayet [Fayet 1995a[Fayet , 1995b[Fayet , 1995c] Tsai [START_REF] Tsai | Robot Analysis: the Mechanics of Serial and Parallel Manipulators[END_REF], [START_REF] Mccarthy | Geometric Design of Linkages[END_REF].

Contributions have continued to emerge in the last years: Huang et al. [START_REF] Huang | Type synthesis of symmetrical lower-mobility parallel mechanisms using the constraint-synthesis method[END_REF], Rico and Ravani [Rico et al., 2003a], Rico, Gallardo and Ravani [Rico et al., 2003b].

The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. For this reason, the real and practical value of these approaches is very limited in spite of their valuable theoretical foundations. The challenging and difficult objective of structural synthesis is to find a method to set up the mechanical architecture achieving the required structural parameters. The mechanical architecture is defined by number, type and relative position of the joint axes in the parallel robot. The structural parameters are mobility, connectivity, redundancy and degrees of freedom and the motion-type of the moving platform. Formula for a quick calculation of mobility and structural parameters was proposed by Gogu in 2005 [Gogu, 2005b]. Usually, these structural parameters are easily determined by inspection without need to develop the set of kinematic constraint equations.

In general, parallel manipulators performances are highly dependent on their mechanical architecture, so that structural synthesis becomes the central problem in the preliminary design phase. To synthesize new reconfigurable robots, it is necessary to investigate structural properties of particular mechanisms. Indeed, the mechanism reconfigurability can be highlighted by the changes of its structural parameters. These parameters are determined by using recently developed methods and formulae [Gogu, 2008a].

Structural parameters of a mechanism

A parallel mechanism is a mechanism in which an end-effector is connected to a reference link by k≥2 structurally independent kinematic chains called limbs or arms [Gogu, 2008a]. In a parallel mechanism, the end-effector and the reference link are usually called moving and reference platforms. They represent the distal links of the parallel mechanism and of each limb as well. The limb can be simple or complex kinematic chain. In a simple limb just monary and binary links exist. They are connected in the kinematic chain by one or two joints. At least one polinary link is combined with monary and binary links in a complex limb in which at least one closed loop exists.

The main structural parameters of a parallel mechanism are associated with mobility, connectivity, redundancy and overconstraint. We recall briefly the meaning of these parameters. More details can be found in [Gogu, 2008a].

IFToMM terminology defines the mobility or the degree of freedom as the number of independent coordinates required to define the configuration of a kinematic chain or mechanism. We note that the classical formulae for a quick calculation of mobility, known as Chebychev-Grübler-Kutzbach formulae do not fit many classical mechanisms, recent parallel robots and kinematotropic mechanisms. These formulae have been recently reviewed and their limits have been set up [Gogu, 2005a]. New formulae for quick calculation of the mobility have been proposed and demonstrated recently via the theory of linear transformations [Gogu 2005b[Gogu , 2008a]].

The connectivity between two links of a mechanism represents the number of independent finite and/or infinitesimal displacements allowed by the mechanism between the two links. Limb connectivity is given by the connectivity between the distal links of the limb. The connectivity of a parallel mechanism is given by the connectivity between the characteristic link (end-effector) and the fixed base.

The number of overconstraints of a mechanism is given by the difference between the maximum number of joint parameters that could lose their independence in the closed loops, and the number of joint parameters that actually lose their independence in the closed loops.

The structural redundancy is given by the difference between the mobility of the parallel mechanism and the connectivity of the end-effector. Redundancy introduces internal mobilities in the limbs. The internal mobilities in a limb are given by the difference between limb mobility and connectivity.

The following formulae have been proposed in [Gogu, 2008a] for the calculation of the structural parameters of a parallel mechanism

1 2 k F G -G -...-G 
in which the mobile platform by k simple and/or complex limbs G

j (1 Gj - 2 Gj -…-n Gj ), j=1,2,…,k.: p F j F j1 M f r    , (2.1) N F =6q-r F , (2.2) T F =M F -S F , (2.3)
where, (2.9)

Gj Gj S dim( R )  , (2.4) F F G1 G2 Gk S dim( R ) dim( R R ... R )      , (2.5) k F Gj F l j1 r S S r      , ( 2
The following notations are used in these formulae:

1 2 k F G -G -...-G  ,
p Gjnumber of joints of G j -limb, p -total number of joints of parallel mechanism F, m -total number of links in mechanism F including the moving and reference platforms, q -total number of independent closed loops in the sense of graph theory, f j -mobility of the jth joint, r F =r p +r l -total number of joint parameters that lose their independence in the closed loops of mechanism F, r p -total number of joint parameters that lose their independence in the closed loops formed between the limbs of mechanism F. r l -number of joint parameters that lose their independence in the closed loops that may exist in the limbs of mechanism F. r Gj -number of joint parameters that lose their independence in the closed loops that may exist in the loops of limb G j , k=k 1 +k 2 -total number of limbs of the mechanism F, k 1 -number of simple limbs of mechanism F, k 2 -number of complex limbs of mechanisms F. Dim(R Gi ) -dimension of the basis of the vector space R Gi .

The key parameters used in Eqs. (2.1)-(2.6) are the basis and the dimension of R Gj . They can be easily obtained by inspection for each limb by identifying the independent motions between the distal link n Gj and 1 Gj in the kinematic chain G j disconnected from the parallel mechanism. An analytical method to compute these parameters has been developed in [Gogu, 2008a] just for verification and for a better understanding of the meaning of these parameters. In this method, the dimension of R Gj is given by the rank of the forward velocity Jacobian J Gj of G j -limb disconnected from the parallel mechanism, that is S Gj =dim(R Gj )=rank(J Gj ).

Note 1. Equations (2.1)-(2.9) are valid for any parallel mechanism with structurally independent limbs. Equation (2.7) gives the condition of existence of structurally independent limbs. It indicates that the limbs of the parallel mechanism F  G 1 -G 2 -…G k must be defined in such a way that each joint belongs to just one limb.

Note 2. The intersection in Eq. (2.5) is consistent if vector spaces R Gi are defined by the translational and rotational velocities of the same point belonging to the moving platform with respect to the same reference frame. This point, denoted by H, is called the characteristic point. It must be the point with the most restrictive motions of the moving platform.

Note 3. The connectivity S F of the moving platform n≡n Gj with respect to the reference platform 1≡1 Gj in the mechanism F  G 1 -G 2 -…G k is less than or equal to the mobility M F of mechanism F. In the same way, the connectivity S Gj of the moving platform n Gj with respect to the reference platform 1 Gj in limb G j disconnected from the mechanism F is less than or equal to the mobility M Gj of limb G j .

Note 4. The basis (R F ) of the vector space R F of relative velocities between the moving and reference platforms in the mechanism F  G 1 -G 2 -…G k must be valid for any point of the moving platform n≡n Gj .

Note 5. When there are various ways to choose the bases of the vector spaces R Gj in Eq. (2.4), the bases (R Gj ) are selected such that the minimum value of S F is obtained by Eq. (2.5). By this choice, the result of Eq. (2.1) fits in with the definition of general mobility as the minimum value of the instantaneous mobility. The various bases of the vector spaces R Gj have the same dimension.

We note that the bases of vector spaces R Gj and R F may contain up to 6 independent velocity vectors v x , v y , v z , ω x , ω y , and ω z . By v x , v y , v z we denote the independent linear velocities of the characteristic point H of the moving platform and by ω x , ω y , ω z the independent angular velocities of the moving platform.

(a) (b) For example the basis of vector space R Gj of a planar limb with three revolute joint is always (R Gj )=(v x ,v y ,ω z ) if the three joint axes are parallel to z-axis. This is illustrated in figure 2.1 (a) with three simple revolute joints (3R).

For the same dimension S Gj , the basis of vector space R Gj of certain kinematic chains may be defined by different combinations of velocity vectors v x , v y , v z , ω x , ω y , and ω z . For example, in a spatial limb with three revolute joints with orthogonal axes and non zero distance between the joint axes adjacent to the same link, vector space R Gj has always three dimensions, but the basis can be defined by various combination of three out of six vectors v x , v y , v z , ω x , ω y , and ω z . The rotational joint axes of three orthogonal revolute joints are illustrated in figure 2.1 (b).

In these cases, the bases of R Gj in Eq. (2.5) are selected such as the minimum value of S F is obtained. By this choice, the result of Eq. (2.5) fits in with general mobility definition as the minimum value of the instantaneous mobility in a free of singularity branch. The bases of R Gj giving larger values of S F are associated with robot singularities.

1 0 2 3 v x v y ω z x 0 z 0 y 0 O 0 H y 0 x 0 z 0 H v Z v X v Y ω y ω X ω Z 1 0 3 2 O 0 55

Conclusion

In this section, we have presented the formulae (2.1) -(2.9) to calculate the main structural parameters of parallel mechanisms. The structural parameters such as mobility, connectivity, structural redundancy, degrees of freedom and motion-type of the moving platform help in setting up the mechanical architecture defined by the number, the type and the relative position of joint axes in the parallel manipulator.

Geometric modelling of an open loop kinematic chain

This section is devoted to the geometric modeling of robotic open/closed loop kinematic chains. The geometric model helps to represent the configuration of the robot based on joint variables. This section presents a general geometric model for serial robots along with the open loop and closed loop geometric models.

Introduction

The principal methods proposed in the literature for geometric model parameterization and formulations of a robot are presented in [START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]]. They are Denavit-Hartenberg, Paul, Khalil-Kleifinger, Yih, Sheth-Uicker, Litvin, Megahed and Gogu-Coiffet-Barraco methods. In the context of the geometric modeling, the method of Denavit-Hartenberg [START_REF] Hartenberg | Kinematic synthesis of linkages[END_REF]] and its adaptations were most commonly used. However this method includes more number of operations which complicates the calculations. Hence, we have chosen the formalization of Gogu-Coiffet-Barraco which is commonly called as Travelling Coordinate System (TCS method) as its use allows simplification of the calculations and provides more flexible notation [START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]]. It is considered that initially the type of kinematic pairs (axis of rotation or translation) and the relative position of the different axes (perpendicular or parallel) are known.

We will first recall the Travelling Coordinate System method [START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]] and the notations used in the definition of the geometric and kinematic models.

Travelling Coordinate System

The objective of this method is to express at each moment, the relative positions and orientations of the links of an open kinematic chain according to joint variables. A coordinate system which travels over the kinematic chain of the robot starting from the reference link to the final link is considered. The coordinate systems attached to the links of the robot are the intermediate positions of the coordinate system travelling from the reference frame to the endeffector frame. The TCS glides over the links of the kinematic chain of the robot and carries out the relative displacement of rotation and of translation. For TCS method, a hypothesis is made i.e. the axes of the kinematic pairs of rotation or of translation are supposed to be orthogonal or parallel to each other. We fix on to this hypothesis which is valid for a majority of robots and most commonly for parallel robots. These are various criterions in the formulation of the travelling coordinate system. A detailed description with examples illustrating these criterions has been dealt in [START_REF] Gogu | Mathématiques pour la robotique : Représentation des mouvements des corps solides[END_REF] and [START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]].

The two dimensional parameters which are considered for each kinematic chain i are:

-a i called as eccentricity which is the distance defined on the axes of the joint (i-1, i) -b i called as the length of the element i, is the distance defined on the direction of the common normal to the axes of joints (i-1, i) and (i, i+1).

These parameters are illustrated with two revolute joints in figure 2.2 as shown below.

Note:

The indices nm, in this order, indicates the relative displacement of element n with respect to element m, produced by the kinematic pair (m-n).

The circular permutation rule helps us to define the joint axis. By making a product of the translational and rotation homogeneous matrices, we obtain the matrix describing the configuration of each characteristic frame of the mechanism relatively to the previous joint.

If the axes of the adjacent joints of the element i are parallel, then the two joints (i-1, i) and (i, i+1) will have the same axis such as x, y or z axis of TCS. We follow the circular permutation rule to denote the direction of the common normal to the joints as in figure 2.3.

Geometric modelling of a robot using TCS method

In the direct geometric model, the finite displacements of the joints are independent variables, the position and orientation of the robot elements are the dependent parameters. We need also to compute the geometric model to know the location of the robot links relative to each other. The solution of the direct geometric model is obtained by the multiplication of the homogeneous operators which modelize the elements and the finite displacements in the joints.

We can obtain the position and orientation of the final frame with respect to the reference frame, by the matrix product,

= (2.10)
where, is a 4x4 matrix which express the position and orientation of the frame i with respect to frame i-1. Each operator in its structure must have one and just one independent variable defining the relative displacement in the joint .

Link axis (i-1, i) Link axis (i, i+1) Common normal The solution for the inverse geometric model expresses the final displacements in the kinematic pairs in function of the position and the orientation of the final element of the robot. The mechanical architecture defined by number, type and relative position of joint axes of the robot and the geometrical dimensions of the kinematic elements are the inputs.

The solutions of the direct and inverse geometric model involving the angle of rotation of each joint and the dimensional parameters of each link will be implemented in the reconfigurable eight-bar mechanism. This application is followed in section 2.3.

Calculation of the forward velocity Jacobian of the open kinematic chains associated to simple limbs

Jacobian matrices are a useful tool, and commonly used throughout robotics and control theory. Basically, a Jacobian defines the kinematic relationship between two different representations of a system. For example, if we have a 2-link robotic arm, there are two obvious ways to describe its current position: 1.) the end-effector position and orientation, and 2.) as the set of joint angles (which we will denote q). The Jacobian for this system relates how movement of the elements of q causes movement of the elements of . , , , , ,

z y x z y x v v v   
Formally, a Jacobian is given by:

                    z y x z y x v v v    = J
This tells us that the end-effector velocity is equal to the Jacobian, J, multiplied by the joint angle velocity.

Mathematically, the forward geometric equations define a function between the cartesian positions and orientations and joint positions. The velocity relationships are then determined by the Jacobian of this function. The Jacobian is a matrix-valued function and can be thought of as the vector version of the ordinary derivative of a scalar function. This Jacobian or Jacobian matrix is one of the most important quantities in the analysis of the possible velocity bases , connectivity (S G ) and also for the control of robot motion. Also Jacobian matrices are used in every aspect of robotic manipulation: in the planning and execution of smooth trajectories, in the determination of singular configurations, in the derivation of the dynamic equations of motion, and in the transformation of forces and torques from the end-effector to the manipulator joints.

The kinematic model is formulated for the limbs associated with the eight bar reconfigurable single loop parallel mechanism using the joint angles and the homogeneous matrix. For the associated open chain, the relationship between the operational velocity, Jacobian matrix and joint velocities are as follows: The calculation of Jacobian is done using the formulation proposed by Gogu [START_REF] Gogu | Mathématiques pour la robotique : Représentation des mouvements des corps solides[END_REF][START_REF] Gogu | Mathématiques pour la robotique : Représentation des déplacements des robots[END_REF]. We use the formulation of TCS and the homogeneous matrix to calculate the Jacobian. We can express the Jacobian matrix in the system of coordinates e situated in any point of any element of the robot. This method implies symbolic calculation of the Jacobian matrix by using the components of the homogeneous matrix from the direct geometric model. One of the advantages of using this method is that the number of mathematical operators necessary for the symbolic calculation of the Jacobian matrix is much diminished when compared to the other existing methods in the literature.

                    z y x z y x v v v    =   h p J                            
The eight bar reconfigurable mechanism comprises two simple limbs. Using this method, the 6x4 Jacobian matrix is calculated for each limb and hence the rank of this matrix is calculated. For the general configuration of the mechanism, a rank of 4 for each calculated Jacobian is necessary. Henceforth, we select all the 4x4 Jacobian submatrices of rank 4 and systematize the operational vector bases corresponding to their velocities. We then calculate (R F ) which is the intersection between vector spaces of each limb. The choice of the vector space should respect a criterion presented previously in Note 5. While there are various ways to choose the basis for the operational spaces, the bases of are selected such as the minimum value of is obtained. By this choice, the calculation of mobility sets up a general mobility as the minimum value of the instantaneous mobility. Respecting this criterion, (R F ) is valid for a general eight-bar reconfigurable mechanism. From Table 2.1, we can see the various possible combinations of the vector spaces giving S F =2. The table below illustrates the direct kinematic model for one solution of the vector space for each limb.

Direct kinematic model of G 1

Direct kinematic model of G

2                              43 32 21 10 4 4 1            x G z y x z J v                              54 65 76 87 4 4 2           x G z y y x J v v

Analysis of an eight-bar linkage in a general configuration

A bibliographic study was made on several reconfigurable mechanisms discussed in the literature in chapter 1. Out of all those, we have chosen to focus on the single-loop eight-bar linkage. In fact, for a special design we propose for this mechanism, we have detected numerous remarkable kinematic properties in relation with reconfigurability. In the forthcoming sections, we undertake a complete analysis of this mechanism in order to exhibit all these properties. Gogu's formulae and the TCS method have been used to carry out the structural analysis and the geometric modeling of this special eight-bar mechanism in its general configuration.

Introduction and description of the special eight-bar linkage

A single-loop eight bar linkage has two degrees of freedom in its general configuration. A particular design of this mechanism with alternate orthogonal and parallel joint axes and symmetric lengths exhibits remarkable properties of reconfiguration [START_REF] Aimedee | Redundant versus constraint singularities in parallel mechanisms[END_REF]. The mechanism has concurrent axis where two revolute joints intersect each other making the square shape with four common points of intersection. Hence we can coin this concurrency and say the eight-bar mechanism to be a quadric-symmetric mechanism. This mechanism is a modified version of the eight-bar mechanisms presented by Wei and[START_REF] Wei | A spatial eight-bar linkage and its association with the deployable platonic mechanism[END_REF]Dai, 2014] and Zhang et al., 2011[Zhang et al., 2011a].

Structural analysis of the eight-bar linkage

In this section, we illustrate the application of the formulae (2.1)-(2.9) to calculate the global mobility in a non-singular configuration of the reconfigurable single-loop eight-bar mechanism presented in Figure 2.5. The revolute joints adjacent to the eight links of the mechanism have parallel or orthogonal axes. The mechanism can be considered as a parallel mechanism in which the distal link 5 is connected to the reference link 1 by two simple 4R limbs of type R  R||R  R, where R denotes the revolute joint,  and || the perpendicular and parallel position of joint axes. Links 1, 5, 3 a and 3 b have the same length. Links 2 a , 2 b , 4 a and 4 b have also the same geometric configuration and dimensions. The axes of the revolute joints adjacent to reference link 1 are parallel, and the axes of the revolute joints adjacent to the moving platform 5 are also parallel. No closed loops exist inside a limb, that is r l =0. This linkage is called single-loop eight-bar mechanism with orthogonal/parallel axes. To simplify the notations of the links e Gj (j=1, 2, and e=1,2,…5) by avoiding the double index in Figure 2.5 we have denoted by e a the links belonging to limb G 1 (e a ≡e G1 ) and by e b , the links of limb G 2 (e b ≡e G2 ). As we have mentioned, the distal links 1 and 5 belong to the two limbs (1≡1 a ≡1 b and 5≡5 a ≡5 b ).

In a non-singular configuration the eight-bar mechanism has the following structural parameters: The detailed calculation using Maple software is illustrated in the Appendix I. In a general configuration, we calculate six joint angles with respect to two independent angles. By using the TCS method, we define the frames and the joint angles. By taking the rules of the TCS method into account, we formulate the travelling coordinate system for the open kinematic chain associated with the eight-bar mechanism as follows:

R R R R R R R R x z z y y x x z     // // //
      10 01  z R        21 12  x R       2 ' 22 b T y        32 3 ' 2  x R        43 34  y R       4 ' 44 b T z        54 5 ' 4  y R        65 56  z R       6 ' 66 b T x        76 7 ' 6  z R        87 78  x R       8 ' 88 b T y
In order to determine the mechanism configuration, the geometric model of mechanism has been solved. This has been performed by formulating the six loop closure constraint equations while considering two independent input joints. In our calculation, we consider and as the independent variables and , , , , and as the dependent variables.

In this chapter we will make use of these constraint equations to solve kinematic problems, namely, the initial position or assembly problem, the finite displacement problem, and the x 6

velocity and acceleration analysis. The six kinematic equations are obtained by using the properties of perpendicularity and parallelism of the SSL8B mechanism.

Note:

SSL8B: special single-loop 8-bar The detailed calculations of these kinematic constraint equations and their complete solution are presented in Appendix II.

To summarize, the six kinematic constraint equations are as follows:

[where is obtained from ] ,

By solving these six equations, six dependent joint variables are determined in terms of and in a particular configuration of the mechanism. Multiple solutions of the geometric model were found corresponding to several assembly modes.

After solving the loop-closure equations, we obtain the values of dependent angles for a particular configuration as follows:

(2.11) )

(2.12) .9 shows an assembly mode which validates and satisfies the above set of equations. This is one of the singular configurations of the SSL8B mechanism where the axes are parallel to the plane

- (2.13) (2.14) (2.15) - (2.16)
i i i z y O
. This mechanism has a continuous mobility change of two to five degrees of freedom without assembling/ disassembling of the mechanism. Also the mechanism exhibits different bifurcation modes to get out of singularity. This leads to an interesting property of continually transiting from one assembly mode to another. The angular values and its detailed description will be dealt in chapter 3.

Conclusion

We applied the formulae of structural parameters to a reconfigurable eight-bar linkage. The mechanism is parameterized using the TCS method followed by the formulation of the geometric model. The structural analysis and the geometric modeling are performed for a general configuration of the mechanism. Various assembly modes and singularities will be analyzed in detail in the next chapter.

Conclusion

In this chapter, we presented the structural analysis and the geometric modeling based on the TCS method applied to the proposed SSL8B mechanism. Initially we recalled the structural parameters such as mobility, connectivity, overconstraint and redundancy of the parallel robots, and their calculation formulae. A structural analysis has been performed for general configuration of the mechanism showing all possible motions of the end-effector with respect to the reference frame. Thereafter, the TCS method has been implemented for the geometric modeling of the SSL8B mechanism. The mechanism kinematic constraints have been formulated in order to determine the mechanism configuration with respect to two independent joint variables. This resolution is required for mechanism control. One of the possible solutions of the geometric model was given for a non-singular configuration. The other solutions of the geometric model as well as those in singularity will be discussed in the following chapter. Also, possible singular configurations of this mechanism and the nature of these singularities will be discussed in Chapter 3.

Chapter 3: Singularity analysis of the single-loop eight bar reconfigurable mechanism

A parallel manipulator is naturally associated with a set of constraint functions defined by its closure constraints. The differential forms arising from these constraint functions completely characterize the kinematic properties of the manipulator. In this chapter, we provide a thorough geometric study on the various types of singularities of a parallel manipulator and their relations with the structural parameters. The role that redundant actuation plays in reshaping the singularities and improving the performance of the manipulator.

Parallel manipulators and its singularities

The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a mobile platform is connected to a reference element by at least two limbs that can be simple or complex [Gogu, 2008a]. Perhaps, the best known parallel manipulator is formed from six linear actuators that support a movable base for devices such as flight simulators. This device is called a Stewart platform or the Gough-Stewart platform in recognition of the engineers who first designed and used them [START_REF] Merlet | Parallel Robots[END_REF]. A drawback of parallel manipulators, in comparison to serial manipulators, is their limited workspace. As for serial manipulators, the workspace is restricted by joint limits and self collisions. The workspace is also limited by the existence of singularities, which are positions where, the variation of the actuated joint variables is infinitely smaller than the variation of the displacement of the moving platform. Conversely, at a singular position, an external force applied on the end-effector induce infinitely larger force or torque in the actuated joint, which may result in loss of control and a kind of "explosion" of the manipulator. The determination of the singular positions is difficult (for a general parallel manipulator, this is an open problem). This implies that it is dangerous to utilize the manipulator near singular configurations, which generally limits the effective workspace of the parallel manipulators to a small region where one knows that there is no singularity.

Another drawback of parallel manipulators is their nonlinear behavior: the command (or control) which is needed for getting a linear or a circular movement of the end-effector depends dramatically on the location in the workspace and does not vary linearly during the movement. The following sub-sections deals with the definitions of different types of singularities and the impacts of each on the parallel manipulator kinematics

Importance of singularities

Many research works have studied the singularities of serial and parallel manipulators.

Compared with its serial counterparts, a parallel manipulator (or a closed-chain mechanism) has a much more complex structure in terms of its kinematics, dynamics, trajectory planning and control. In particular, the configuration space of a parallel manipulator is not even explicitly known; it is implicitly defined by a set of constraint functions introduced by the manipulator's closure constraints [START_REF] Liu | Singularities of parallel manipulators: A geometric treatment[END_REF]. A parallel manipulator also has, in addition to the usual end-effector singularities, different types of singularities such as configuration space singularities and actuator singularities. Understanding the intrinsic nature of the various types of singularities and their relations with the kinematic parameters and the configuration spaces is of ultimate importance in design, trajectory or task planning and control of the system. In the upcoming sections we will discuss the major types of constraint and redundant singularities identified in the SSL8B mechanism.

Types of singularities

Unlike its serial counterparts, where there have been well established mathematical tools for their analysis, studies on singularities of parallel manipulators were confined to basic issues such as definition, classification and identification of singularities. Furthermore, the mathematical tools used in most studies were directly borrowed from that for serial manipulators and were applicable only to local analysis. The unique structures of parallel mechanisms were not fully explored. Gosselin and Angeles [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] were perhaps the first to define and study singularities of closed-loop kinematic chains. Based on some derived Jacobian relations, they introduced several notions of singularities which formed a basis of later research. Park and Kim [START_REF] Park | Singularity analysis of closed kinematic chains[END_REF] used differential geometric tools to study singularities of parallel mechanisms and provided a finer classification of singularities. In their later works, they proposed the use of redundant actuation as a means of eliminating actuator singularities and improving manipulator performances. Merlet and others [START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF], [START_REF] Hunt | Special configurations of robot-arm via screw theory (part 1)[END_REF], [START_REF] Ma | Architecture singularities of platform manipulators[END_REF], [START_REF] Kumar | Kinematics of redundantly actuated closed chains[END_REF] studied extensively singularities of the Stewart-Gough platform and several of its variants.

The determination of singularities for parallel manipulator is indeed very complex. An interesting example is offered by that of the Seoul National University (SNU) manipulator, a 3-DOF translational manipulator with the joints of its three subchains arranged in the order of universal-prismatic-universal (UPU). Zlatanov et al. [START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF] studied the rotational and translational singularity using screw theory and classified it as a constraint singularity. The same singularity was also identified by Joshi and Tsai [START_REF] Joshi | Jacobian analysis of limited-DOF parallel manipulators[END_REF], Simaan and Shoham [START_REF] Simaan | Singularity analysis of a class of composite serial in-parallel robots[END_REF], and Wolf et al. [START_REF] Wolf | Investigation of singularities and self-motions of the 3-UPU robot[END_REF] using an augmented Jacobian matrix which took the constraints into account.

In the following sections, we identify and describe all possible singular configurations of the SSL8B mechanism. We characterize the singular configurations using structural parameters.

We investigate the intrinsic nature of the various singularities of a parallel mechanism, their relations with the kinematic parameters and classify them accordingly under different types of singularities.

We propose to use redundant actuation for the SSL8B mechanism, which translates into more actuation options than required to perform a particular task in its general configuration. The role of redundant actuation plays in reshaping the singularities and improving the manipulator's performance. We present a detailed classification of parameterization singularities and identify those which are potentially dangerous and should be avoided or eliminated through design.

Singularities in connection with structural parameters

The aim of this chapter is to identify and make an analysis on the nature of the singularity behavior with the associated structural parameters by using the new formulae recently proposed in [Gogu 2008a]. Singular configurations of parallel robots are particular configurations at which the robot loses its natural rigidity. When a parallel manipulator reaches its singular configuration, the platform becomes uncontrollable. Also, singularities can damage the robot; hence measures should be taken to avoid singularities in the reachable space during designing the robot. The singular configurations can be determined by analyzing the rank of the matrix J or J -1 (inverse of the Jacobian matrix).

In a singular configuration at least one of the structural parameters in Eq. 2.6 is instantaneously altered. We denote by an anterior superior index i the instantaneous values of the structural parameters defined above. We note that, in general, any structural parameter used in Eqs. (2.1)-(2.9) can be affected by singular configurations excepting f j and q. Various singular configurations can be associated with the eight-bar mechanism analysed in this section. They can be defined by the following configurations of the rotation axes of the eight revolute joints: (1) coplanar, (2) intersecting a line, (3) parallel to a plane, and (4) parallel to a plane and in the same time intersecting a line. As we will illustrate in the next sections, the last configuration can be associated with a constraint singularity, the third configuration with a constraint or with a redundant singularity, the second with a redundant singularity and the first with a constraint-redundant singularity. In all singular or non-singular configurations of the eight-bar mechanism, the following structural parameters are invariants: m=8, p 1 =p 2 =4, p=8, q=1, k 1 =2, k 2 =0, k=2, r G1 =r G2 =0,M G1 =M G2 =4, r l =0.

Constraint Singularities

A constraint singularity is a configuration of the parallel manipulator in which both the connectivity of the moving platform with respect to the fixed base and the mobility of the parallel mechanism increase their instantaneous values with no change in limb connectivity and mobility. The following properties of the parallel mechanisms have been associated with the constraint singularities in [Gogu, 2008b].

Property 1: If the vector space of relative velocities between the distal links n Gj and 1 Gj in the kinematic chain G j disconnected from the mechanism F, accept various bases, the connectivity of the moving platform can increase instantaneously its value i S F >S F , with no instantaneous change in limb connectivity ( i S Gj =S Gj ).

This property results directly from Eq. 2.5 and shows that a constraint singularity may occur when the vector space of relative velocities between the distal links n Gj and 1 Gj in the kinematic chain G j disconnected from the mechanism F, accept various bases at least for one of the limbs.

Property 2: An instantaneous increase of the connectivity of the moving platform ( i S F -S F ) is accompanied by an identical increase of the mobility of the parallel mechanism ( i M F -M F ) when no instantaneous changes occur in limb connectivity and the number of joint parameters that lose their independence in the closed loops that may exist in limb G j i M F -M F = i S F -S F (3.1) when i S Gj =S Gj and i r l =r l . This property results directly from Eqs. 2.1 and 2.6.

Property 3: An instantaneous increase of the connectivity of the moving platform ( i S F -S F ) is accompanied by an identical increase of the degree of overconstraint of the parallel mechanism ( i N-N) when no instantaneous changes occur in limb connectivity and the number of joint parameters that lose their independence in the closed loops that may exist in limb G j i N-N= i S F -S F (3.2) when i S Gj =S Gj and i r l =r l . This property results directly from Eqs. 2.2 and 2.6. Property 4: An instantaneous increase of the connectivity of the moving platform ( i S F -S F ) accompanied by an identical increase of the mobility of the parallel mechanism ( i M F -M F ) does not affect the redundancy of the parallel mechanism.

This property results directly from Eq. 2.3.

The four properties characterizing the constraint singularities are illustrated in this section with respect to the single-loop eight-bar mechanism with orthogonal/parallel axes. Two constraint singularities can be associated with this mechanism as presented in Figure 3.2.

              b b b a a a a          0 12 b 
. Each limb has S Gi =4 and the same basis of the velocity vector space ( i R G1 )=( i R G2 )= (v y ,v z ,ω y ,ω z ). Each limb has S Gi =4 and the same basis of the velocity vector space ( i R G1 )=( i R G2 )= (v y ,v z ,ω y ,ω z ). For this singular configuration, Eqs. (2.1)-(2.9) give the following values for the structural parameters:

i M G1 = i M G2 =4, i r F =4, i M F =4, i S F =4, i N F =2, i T F =0. The basis of the vector space i R F is ( i R F )= (v y ,v z ,ω y ,ω z ).
The mechanism can get out of this constraint singularity in a non-singular branch with various bases of vector space R F , as indicated in section 2.1.2, or in the constraint singularity configuration presented in Figure 3.2b. A non-singular branch defines the connexe set of non-singular configurations of the mechanism in which the structural parameters keep the same value. The transition from one branch to another without disassembling the mechanism is always done passing through a singular configuration.

The vector spaces associated with each limb of the mechanism presented in Figure 3.2a are illustrated below by means of calculating the Jacobian matrix of each limb. . Each limb has

) , , , ( 4 )
( , 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) , , , ( 4 )
z y z y G z y z y G v v R J Rank b b J v v R J Rank b b J                                                      Table 3.
             b b a a a a            40 ,

S

Gi =4 and each vector space R Gi (i=1 and 2) can have one of the following bases: (v

x ,v z ,ω x ,ω z ), (v x ,v z ,ω y ,ω z ), (v y ,v z ,ω x ,ω z ) or (v y ,v z ,ω y ,ω z ).
Two different values can be obtained for the connectivity between the moving platform and the base. The minimum value i S F =3 is associated with the mechanism in Figure 3 where the moving platform have just three independent motions. For example, if we consider

( i R G1 )=(v x ,v z ,ω x ,ω z ) and ( i R G2 )=(v y ,v z ,ω x ,ω z ) we get ( i R F )=(v z ,ω x ,ω z ) and if ( i R G1 )=(v x ,v z ,ω y ,ω z ) and ( i R G2 )=(v y ,v z ,ω y ,ω z ) we get ( i R F )=(v z ,ω y ,ω z ). Eqs. 1-9 give: i M G1 = i M G2 =4, i r F =5, i M F =3, i S F =3, i N F =1, i T F =0, T G1 =T G2 =0
. The vector spaces associated with each limb of the mechanism presented in Figure 3.2b are illustrated below as: ) ( , The mechanism can get out of this constraint singularity in a non-singular branch with various bases of vector space R F , as indicated in section 2.1.2, or in a constraint-redundant singularity as will be present in the following section.

1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 2 85 85 1 04 04 z y z y z x z y z y z x z x z x G z y z y z x z y z y z x z x z x G v v or v v v v v v R J Rank b b b b b b J v v or v v v v v v R J Rank b b b b b b J                                                                     Table 3.2

Constraint-redundant singularities

A constraint-redundant singularity is a configuration of the parallel mechanism in which the following structural parameters increase instantaneously their values with respect to a nonsingular configuration [START_REF] Aimedee | Redundant versus constraint singularities in parallel mechanisms[END_REF]

i M Gj =M Gj ).
By analogy with the constraint singularity analysis, the following properties of parallel mechanisms can be associated with constraint-redundant singularities.

Property 5: Instantaneous mobility of the parallel mechanism is divided between the instantaneous mobility of the moving platform i S F and the instantaneous internal mobilities in the limbs i T F i M F = i S F + i T F .

(3.3)

This property results directly from Eq. (2.3).

Property 6: An instantaneous increase of mechanism redundancy ( i T F -T F ) is identical with the instantaneous increase of internal mobilities in the limbs disconnected from the parallel mechanism

kk ii F F Gj Gj j 1 j 1 T T T T      .
(3.4)

This property results directly from the definition of internal mobilities in connection with mechanism redundancy.

Property 7: An instantaneous increase of limb redundancy ( i T Gj -T Gj ) is identical with the instantaneous decrease of the connectivity between the moving platform and the reference link (S Gj -i S Gj ) in the limb disconnected from the parallel mechanism i T Gj -T Gj = S Gj -i S Gj .

(3.5)

This property results from Eq. ( 2.3) by taking into account that the mobility of the limb disconnected from the parallel mechanism is not affected by this singular configuration.

Property 8: An instantaneous increase of the mobility of the parallel mechanism ( i M F -M F ) is accompanied by an identical increase of the degree of overconstraint of the parallel mechanism

( i N F -N F ) i N F -N F = i M F -M F
(3.6) This property results directly from Eqs. 2.1 and 2.2.

The four properties characterizing the constraint-redundant singularities are illustrated in this section with respect to the configuration of the single-loop eight-bar mechanism with orthogonal/parallel axes presented in Figure 3.3. In this singular configuration, the axes of the eight revolute joints are coplanar and situated in plane O 1 x 1 z 1 .

Each limb has S Gi =3 and the same basis of the velocity vector space ( i R G1 )=( i R G2 )=(v y ,ω x ,ω z ). For this singular configuration, Eqs. (2.1)-(2.9) give the following values for the main structural parameters:

i M G1 = i M G2 =4, i r F =3, i M F =5, i S F =3, i N F =3, i T F =2, T G1 =T G2 =1. The basis of the vector space i R F is ( i R F )= (v y ,ω x ,ω z )
. Each limb has one internal mobility associated with an unlimited rotation of the sub chain 2-3-4 around the superposed rotational axis of the first and the last revolute joints of the limb. The mechanism can get out of this constraint singularity in a non-singular branch with various bases of vector space R F , as indicated in section 2.1.2, or in various redundant singularities as presented in the following section. The following values are associated with the joint rotation angles in the constraint-redundant singularity in Figure 3

              b b b a a a a          270 12 b  .
The vector spaces associated with each limb of the mechanism presented in Figure 3.3 are illustrated below as:

) , , (
3 ) ( , 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) , , (
3 ) ( , 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 85 85 1 04 04 z x y G z x y G v R J Rank b b b J v R J Rank b b b J                                                       Table 3
.3 Possible combinations of vector spaces with minimum S F for the configuration in Fig. 3.3 3

Redundant Singularities

A redundant singularity is a configuration of the parallel mechanism in which the following structural parameters change instantaneously their values with respect to a non-singular configuration [START_REF] Aimedee | Redundant versus constraint singularities in parallel mechanisms[END_REF]]: a) connectivity of the moving platform with respect to the fixed base in the parallel mechanism decreases, i S F <S F b) mobility of the parallel mechanism keep its value or increases, i M F ≥M F c) redundancy of the parallel mechanism, i T F >T F increases, with no change in limb mobility ( i M Gj =M Gj ).

The properties defined in the previous section for the constraint-redundant singularities are also applicable to the redundant singularities with a slight reformulation of property 8. A new specific property is also associated in this section to this type of singularities.

Property 8bis:

A possible instantaneous increase of the mobility of the parallel mechanism ( i M F -M F ) is accompanied by an identical increase of the degree of overconstraint of the parallel mechanism ( i N F -N F ). This property results directly from Eqs. 2.1 and 2.2.

Property 9: If the instantaneous value of mechanism mobility i M F is equal to the instantaneous value of mechanism redundancy i T F , the moving platform of the parallel mechanism is instantaneously blocked and the parallel mechanism has just internal mobilities. This property results from Eq. (3.3).

The properties characterizing the redundant singularities are illustrated in this section with respect to the configurations of the single-loop eight-bar mechanism with orthogonal/parallel axes presented in Figure 3.4.

In the singular configuration in Figure 3.4a the axes of the eight revolute joints intersect O 1 z 1 axis and the configuration in Figure 3.4b they intersect a line passing by point H and parallel with O 1 z 1 . We recall that two parallel axes intersect at infinity. In the singular configuration in Figure 3.4c, the axes of the eight revolute joints are parallel to the plane O 1 y 1 z 1 . We recall that two parallel axes intersect at infinity. In the singular configuration in Figure 4c, axes of the eight revolute joints are parallel to the plane O 1 y 1 z 1 . The following values are associated with the joint rotation angles in the redundant singularity: In Figure 3 The vector spaces associated with each limb of the mechanism presented in Figure 3.4a are illustrated below as:

              b b b a a a a          270 12 b  . (a) (b) (c)
) , , ( 3 ) ( ,
1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) , , ( ) , , ( ), , , ( 3 ) ( ,
1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 2 85 85 1 04 04 z x y G z z x z y z z y x G v R J Rank b b b J v v or v v R J Rank b b b b b J                                                          
              b b b a a a a          0 12 b  .
Table 3.5 Possible combinations of vector spaces with minimum S F for the configuration in Fig. 3

.4b ) 1 ) 1 ) 1 ) , , ( ), , , ( ), , , ( 3 )
( , 0 1 1 1 0 0 0 0 0 0 0 0 0 0 ) , , ( 3 )
z y z z y x z z x G z x y G v v v v R J Rank b b b b b J v R J Rank b b b J                                                            In Figure 3.4c : , 0 ,
              b b b a a a a          0 12 b  . ) , , ( ) , , ( ), , , ( ), , , ( ), , , ( ), , , ( 3 )
( , 0 1 1 1 0 0 0 0 0 0 0 0 0 0 ) , , ( ) , , ( ), , , ( ), , , ( 3 )
y y x z y x y z x z z x z y x z y x G y y x z x z y x z y x G v v or v v v v v v v v v v R J Rank b b b b b J v v or v v v v v v R J Rank b b b b b J                                                               
Table 3.6 Possible combinations of vector spaces with minimum S F for the configuration in Fig. 3.4a

1 1 1 1 1 1 1 1 1
The Jacobian matrix associated with each limb of the mechanism is presented in Annex III.

In the configurations presented in Figure 3.4, each limb has S Gi =3. The velocity vector spaces have the following bases:

( i R G1 )=(v x ,ω y ,ω z ), ( i R G2 )= (v y ,ω x ,ω z ), ( i R F )=(ω z ) for the configuration in Figure 3.4a, ( i R G1 )=(v y ,ω x ,ω z ), ( i R G2 )=(v x ,ω y ,ω z ), ( i R F )=(ω z ) for the configuration in Figure 3.4b, and ( i R G1 )=(v x ,v y ,v z ), ( i R G2 )=(v x ,ω y ,ω z ), ( i R F )=(v x )
for the configuration in Figure 3.4c. For these singular configurations, Eqs. (2.1)-(2.9) give the following values for the main structural parameters:

i M G1 = i M G2 =4, i r F =5, i M F =3, i S F =1, i N F =1, i T F =2, T G1 =T G2 =1.
As in the constraint-redundant singularity, each limb has one internal mobility associated with an unlimited rotation of the sub chain 2-3-4 around the superposed rotational axis of the first and the last revolute joints of the limb. The mechanism can get out of these constraint singularities in a non-singular branch with various bases of vector space R F , as indicated in section 2.1.2, or back in the constraint-redundant singularity.

The mechanism can transit continuously from one type of singularity to another by remaining always in a singular configuration. This is an interesting property of this mechanism enhancing its reconfiguration capability. For example, by locking two, three or four revolute joints with superposed axes in Figure 3.4c, the eight-bar mechanism can be reconfigured in six, five or four-bar mechanisms. The three mechanisms have an equivalent kinematics with a planar four four-bar parallelogram mechanism and different degrees of overconstraint.

The six-bar mechanism can be obtained by locking the first revolute joint of each limb of the eight-bar mechanism in Figure 3.4c. The resulting six-bar mechanism has the following structural parameters:

m=6, p 1 =p 2 =3, p=6, q=1, k 1 =2, k 2 =0, k=2, r l =0, S G1 =S G2 =3, r G1 =r G2 =0, (R G1 )=(v x ,v y ,v z ), (R G2 )= (v x ,ω y ,ω z ), (R F )=(v x ), M G1 =M G2 =3. Equations (2.1)-(2.6) give r F =5, M F =1, S F =1, N F =1, T F =0.
This six-bar mechanism has one degree of mobility and one degree of overconstraint.

Two five-bar mechanisms can be obtained by locking the first revolute joint of each limb and the last joint of just one limb in the eight-bar mechanism in Figure 3.4c. For example, if the joints between links 1 and 2a, 1 and 2b, and 4a and 5 are locked in Figure 3.4c, the resulting five-bar mechanism has the following structural parameters:

m=5, p 1 =2, p 2 =3, p=5, q=1, k 1 =2, k 2 =0, k=2, r l =0, S G1 =2, S G2 =3, r G1 =r G2 =0, (R G1 )=(v x ,v z ), (R G2 )= (v x ,ω y ,ω z ), (R F )=(v x ), M G1 =2, M G2 =3. Equations (2.1)-(2.6) give r F =4, M F =1, S F =1, N F =2, T F =0.
This five-bar mechanism has one degree of mobility and two degrees of overconstraint.

The four-bar planar parallelogram mechanisms can be obtained by locking the first and the last revolute joints of each limb of the eight-bar mechanism in Figure 3.4c. The resulting fourbar mechanism has the following structural parameters:

m=4, p 1 =2, p 2 =2, p=4, q=1, k 1 =2, k 2 =0, k=2, r l =0, S G1 =2, S G2 =2, r G1 =r G2 =0, (R G1 )=(v x ,v z ), (R G2 )= (v x ,ω y ), (R F )=(v x ), M G1 =2, M G2 =2. Equations (2.1)-(2.6) gives r F =3, M F =1, S F =1, N F =3, T F =0
. The mechanism has one degree of mobility and three degrees of overconstraint.

Transition from one singularity to another

In the field of parallel manipulators, the possibility of changing assembly mode without passing through a singular configuration is well known. These kinds of transitions have been studied and it is a common issue [START_REF] Pagis | Design of a Controller for Enlarging Parallel Robots Workspace through Type 2 Singularity Crossing[END_REF]. This section focuses on the SSL8M eight-bar reconfigurable mechanism for which we propose a continuously transiting from one singular assembly mode to another by using redundant actuation. In our work, we have identified six singularities as studied above from which the mechanism can bifurcate into several assembly modes.

This mechanism has an interesting property to continually transit from a constraint singularity to a constraint -redundant singularity by remaining always in a constraint singularity branch as shown in Figure 3.5. For example, the mechanism can pass from a constraint singularity as in Figure 3.2a to a constraint-redundant singularity as in Figure 3.3 by passing through various singular configurations of type Figure 3.2b.

The singular configurations in Figure 3.5 are defined by the following geometric features. The angles of rotation in the revolute joint between link 2a and link 1 denoted by (2a, 1) and between 2b and 1 denoted by (2b, 1) are identical and have opposite signs. The same is with the angles of rotation in the joint (5, 4a) and (5, 4b). In the constraint singularity branch, the rotational axis in the following joints (1, 2a), (5, 4a), (5, 4b) and (1,2b) are always parallel.

Conclusion

In this chapter, constraint, redundant and constraint-redundant singularities have been comparatively defined and analysed. The natures of these singularities have been explained and their properties have been formalized using the novel formulae of mobility, connectivity, overconstraint and redundancy of parallel robots, recently proposed in the literature. The common property of the three types of singularities resides in the instantaneous change of connectivity between the moving platform and the base which takes place with no change in limb mobility. The common properties of constraint and constraint-redundant singularities consist in the instantaneous increase of mechanism mobility and connectivity between the moving platform and the base. The difference between these two types of singularities consists in the fact that mechanism redundancy is not affected in constraint singularity but increases in constraint-redundant singularity. The properties of the constraint-redundant singularities are also applicable to redundant singularities. The main difference between these two types of singularities is associated with the fact that the connectivity of the moving platform with respect to the base in the parallel mechanism increases in a constraint-redundant singularity and it decreases in a redundant singularity. The analysis of these singularities associated with a reconfigurable eight-bar single loop mechanism with orthogonal/parallel axes has underlined the capacity of this mechanism to continually transit from one type of singularity to another by remaining always in a singular configuration. From these different singularities the mechanism can also get out at any time in a non-singular configuration. The singularity analysis presented in this chapter is useful for the study of reconfiguration capability of this mechanism which can take place in the various singular configurations illustrated in this chapter.

4 Chapter 4: Control strategies of the SSL8B mechanism reconfiguration: Simulation and experimental results

Introduction

Industrial robotics is booming and there are about 1.5 million industrial robots today active in the world [IFR, 2014]. Many of these robots are constituted by a serial architecture: the robot is characterized by an open kinematic chain, which implies that their links are mounted in series. These systems are relatively simple to model and feature a large workspace. However, their report on payload weight of the robot is very low. A parallel architecture is distinguished from a serial architecture where it has several kinematic chains connecting the base (fixed) to the mobile platform. These architectures have several advantages: each limb typically has a single motor which can significantly reduce the weight of the movable part of the robot. These architectures also allow improved stiffness and have a better dynamic behavior [START_REF] Tlusty | Fundamental comparison of the use of serial and parallel kinematics for machine tools[END_REF] (acceleration, report payload / total weight) as well as a better theoretical precision [START_REF] Briot | Are parallel robots more accurate than serial robots[END_REF]. From a theoretical point of view, parallel kinematic machines allow better dynamic performances than serial ones, in terms of speed, accuracy and stiffness [START_REF] Merlet | Parallel robots[END_REF]. Due to this, they seem perfectly suitable for industrial high-speed applications, such as pick and place or high speed machining. On the other hand, recent machines allow maximal acceleration, which is not achievable by the serial kinematic machines.

Despite all these advantages, the proportion of parallel robots in operation in the industry is widely lower than that of serial robots. This under-representation of parallel robots is due to two key points: parallel robots are more complex, which complicates their modeling and control. Moreover, workspace is, at equal size, lower than that of serial robots. We believe that the relative complexity of parallel robots is not a major obstacle to their industrial development. It is time being largely negligible compared to the operation time of the industrial machinery. It is mainly the small size of their workspace that now limits applications. This small size is generally due to the presence of singularities [START_REF] Arakelian | Design of planar 3-DOF 3-RRR reactionless parallel manipulators[END_REF][START_REF] Conconi | A new assessment of singularities of parallel kinematic chains[END_REF][START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF].

The control problem for the parallel robotic platform was rigorously analyzed in the robotic community. A great variety of control approaches have been proposed. Generally, the classical control strategies from serial robotics can be used for parallel kinematic machines. By cons, even if the conventional control approaches of the serial robots can be used for parallel robots, they can induce larger errors because of the coupling between the axes of the parallel machines. Several issues with accuracy and stiffness of parallel kinematic machines were studied in the literature. One major issue is the presence of numerous joints which causes kinematic model errors because of clearance and assembly defects [START_REF] Wang | On the accuracy of a Steward platform-part I: the effect of manufacturaing tolerances[END_REF].

Another issue is that, the actuators of the parallel kinematic machine tool do not apply a torque along the end-effector motion axis, contrary to a serial one [START_REF] Tlusty | Fundamental comparison of the use of serial and parallel kinematics for machine tools[END_REF]. This results in a decrease of stiffness leading to a lack of accuracy during machining process. In the following section, we are going to present all the control laws used in robotics that can be used to control parallel robots.

Control Schemes in Robotics

The knowledge on parallel robotics comes directly from the serial one. Therefore, parallel kinematic machines are mainly controlled with the same strategies as serial ones [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. Figure 4.1 illustrates the various control laws used for robotic manipulators. The most commonly used control law for industrial robots is a decentralized "proportional, integral, derivative" (PID) control which is also called as linear single-axis control. More sophisticated linear or nonlinear control schemes have been developed, such as computed torque control (also called as decoupling control) and passivity-based control. Others advanced control laws such as adaptive control and predictive controls are proposed in the literature to minimize large servo errors. In the following subsections we analyze these control schemes and thereafter make a choice of control for the SSL8B mechanism described in section 4.1.2.

Proportional, Integral and Derivative control

The PID control is most popular solution for the control of serial robots and processes. For most of today's industrial robots, a local decentralized PID control with constant gains is implemented for each joint. The advantages of this control are simplicity of implementation and the low computational cost. Also, the choice of the gains can be achieved according to the first natural frequency of the robot [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. However, for this decentralized control law, it is assumed that each axis is independent of the others. We can see that this is not a publicity to make it simple and obtain slow movements but this assumption is quickly becoming invalid on parallel robots with high dynamics.

PID control in joint space is implemented to control the joints of the mechanism with constant gains without using any model of the robot. Therefore, this kind of control law can be quickly implemented on the robot controller. However, by using PID control in joint space, the endeffector motion is not controlled directly and it is then difficult to compensate pose errors due to geometric calibration uncertainties (obstacle avoidance is performed in trajectory planning and / or active perception). As a consequence, if the motion of the end-effector has to be specified and controlled, PID control in task space will be preferred to PID control in joint space. However, since PID control in task space requires the knowledge of direct or / and inverse geometric model, PID control in task space is very difficult to implement for parallel robots. The block diagram of such a control scheme in the joint space is shown in figure 4.2. The control law for the corresponding scheme is given by:

        t t d I d d d p d q q K q q K q q K 0 ) ( ) ( ) (    (4.1)
where, ) (t q d and ) (t q d  denote the desired joint positions and velocities (computed by using trajectory generator as described in Sec.4.2.3), and

Computed torque control

Another most commonly used control strategy is the computed torque control. The control law is based on the formulation of inverse dynamic model of the robot:

    q q H q q A    ,    (4.2)
Where,   q A is the inertia matrix of the robot and   q q H  , is the vector of dynamic parameters such as inertial and frictional parameters.

By replacing q  in equation 4.2 by an adapted control signal u, an exact linearization of the dynamics is ensured. The following control signal is used:

e K e K q u d v d       (4.3)
This control strategy can be improved with friction and backlash compensation, like that of the linear single-axis control. This control does not cope very well with modeling errors [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF] and require the dynamic model of the robot. Modeling errors create perturbation on the tracking error behavior which may lead to a lack of stability and accuracy. Therefore this kind of control law can be coupled with an adaptive control algorithm (as described hereafter, see Fig. 4.3) in order to reduce the tracking error. Then, the main advantage of computed torque control for parallel robots is the fact that coupling between axis are taken into account. On the contrary, the main inconvenient is the fact that the control law requires a good dynamic model of the robot which is difficult to obtain for parallel robots.

Further analysis of the different types of controls applied to parallel mechanisms are proposed in [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF][START_REF] Spong | Robot modeling and control[END_REF][START_REF] Paccot | A review on the dynamic control of parallel kinematic machine : theory and experiments[END_REF].

Passivity-based control

In this section, we investigate another approach that used the property of passivity of the robot. These control laws modifies the natural energy of the robot in order to satisfy the desired position control. Hamiltonian formulations are used to calculate the dynamics of the robot.

The Hamiltonian gives the total energy of the robot: (4.4) where,

) , ( q q E  is the kinematic energy of the robot equal to

q q A q T   ) ( 2 1 ) (q U
is the potential energy of the robot ) (q A is the inertia matrix of the robot The passivity control can be classified as passivity-based position control, passivity-based tracking control and Lyapunov-based method.

Let us assume that we want to drive the robot to a desired position d q . Intuitively, this can be achieved by shifting the open-loop energy minimum from ) 0 , 0

(   q q  towards ) 0 , 0 (   e q 
for the closed-loop system, where

) ( q q e d  
is the position error. This shifting can be obtained by reshaping the potential energy of the system such that it attains the desired minimum at ) 0 (  e . Let us define the control law as:

v q q U q q U           ) ( ) ( (4.5)
where, v is the (nx1equation format) new input control vector and,

e K e U p T 2 1   (4.6)
After simplification, the control law in equation (4.5) becomes,

U E H   95 ) ( d d p q Q q K e K      (4.7)
Equation 4.7 represents gravity compensation and a linear state-feedback loop where ) (q Q is the vector of gravity torques.

Adaptive control

This control method is used to estimate or adjust on-line the parameter values used in the control law. For example, this kind of algorithm is used to adapt on-line the robot model parameters since it (kinematic, dynamic …) is not often exactly known. The adaptive control is further classified into adaptive feedback linearizing control and adaptive passivity-based control. [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF] Adaptive control differs from other types of control by the fact that the parameters of the model it uses varies over time. Figure 4.3 shows the block diagram of the adaptive control applied to the computed torque control law. Rather than calculating the control to be applied on a horizon prediction as done in predictive control, a law of adaptation modifying the parameters of the dynamic model is determined. This is particularly suitable for the control of systems whose parameters change over time.

Predictive control

Predictive control was proposed explicitly for the first time in 1960 [START_REF] Propoï | Use of linear programming methods for synthesizing sampled-data automatic systems[END_REF]. However, it was not discovered until 1987 that the first generalized predictive control has been formalized [START_REF] Clarke | Generalized predictive control. Chap I : The basic algorithm[END_REF]. The idea is to insert into the control algorithm a predictor of concerning the evolution of the process output from a model [Richalet 1993b]. The computer determines, at the present sampling instant, the control sequence to be applied over a period of prediction so that the output has the desired behavior over the horizon. This type of control is particularly suitable for mobile robotics [START_REF] Bouton | An active anti-rollover device based on Predictive Functional Control : application to an All-Terrain Vehicle[END_REF][START_REF] Lenain | Adaptive and predictive non linear control for sliding vehicle guidance : application to trajectory tracking of farm vehicles relying on a single RTK GPS[END_REF][START_REF] Lenain | Contribution à la modélisation et à la commande de robots mobiles en présence de glissement[END_REF] and in industry [Richalet 1993a], but is difficult to adapt to parallel mechanisms because of its models complexity. Indeed, predictive control algorithm requires a long computation time not very compatible with the parallel robot models.

Most applications in parallel robots with the predictive control robot are based on a principle similarly based on a compromise between modeling accuracy and performance of the order. This technique seems difficult to integrate to our robot.

Conclusion and choice of position control

The control laws presented in this section rely on the availability of joint positions and velocities. Out of all the methods presented above, we choose position control scheme to be implemented in the SSL8B mechanism. As with other control schemes like dynamic control, adaptive control, etc… it is difficult to control the motions of the 8-bar mechanism (since the model used to compute the control variable is too difficult to obtain), then the position control is desirable.

Control of the 8-bar mechanism

As discussed in section 4.1.1, there are many different control approaches for parallel manipulators. This section discusses the particular case of redundant actuation and the two position control strategies that are implemented on the 8-bar reconfigurable mechanism.

Parallel mechanisms frequently contain an unstable type of singularity that has no counterpart in serial mechanisms. The robot loses the ability to counteract external forces in certain directions, when the mechanism is at or near this type of singularity. The singularity can be modified or removed, by adding kinematic linkages which alters the mechanism. Another approach is to actuate certain unactuated degrees of freedom. The manipulability is guaranteed to improve over the original mechanism, but the mechanism is now over-actuated. The following section deals with the redundant actuation to overcome this problem.

Particular case of redundant actuation mechanism

By using 5 actuators, the 8-bar mechanism can be considered partially as a redundant actuated mechanism. These redundant robots are mechanisms with more actuators than required for doing the prescribed task in the task space. Redundant actuation is achieved by [Gogu, 2008a]:

 Actuating some of the passive joints within the existing limbs.  Introducing additional actuated limbs beyond the minimum necessary to actuate the manipulator.  Introducing some additional actuated joints within the limbs beyond the minimum necessary to actuate the manipulator.

Actuator redundancy does not affect the connectivity of the end-effector but only increases the number of actuators. In all case, only 'M' actuators can have independent motions. The other actuators have dependent motions. A selective choice of actuators with independent motion could have more advantages.

Redundancy in parallel manipulators is used to eliminate some singular configurations [START_REF] Wang | Kinematics analysis and design of kinematically redundant parallel Mechanisms[END_REF][START_REF] Kurtz | Mutiple-goal Kinematic optimisation of parallel spherical mechanisms With actuator redundancy[END_REF][START_REF] Merlet | Les Robots parallèles[END_REF]Firmani andPodhorodeski 2004, Alberich et al. 2006], to minimize the joint rates, to optimize the joint torques/forces [START_REF] Dasgupta | Force redundancy in parallel manipulators[END_REF]Bruckman et al 2006;[START_REF] Nokleby | Force capabilities of redundantly-Actuated parallel manipulators[END_REF] to increase dexterity workspace [Marquet et al 2001a,b], stiffness [START_REF] Chakarov | Study of the antagonistic stiffness of parallel manipulators, With actuation redundancy[END_REF]], eigen frequencies, kinematic and dynamic accuracy [START_REF] Valasek | The innovative potential of redundantly actuated PKM[END_REF], to improve velocity performances [START_REF] Krut | Velocity performance indices for parallel mechanisms With actuation redundancy[END_REF]] and both kinematic and dynamic control algorithms [START_REF] Liu | Analysis and control of redundant parallel manipulators[END_REF][START_REF] Cheng | Dynamics and control of redundantly actuated parallel Manipulators[END_REF], to develop large forces in micro electro-mechanical systems [START_REF] Mukherjee | Massively parallel binary manipulators[END_REF], decoupling the orientations and the translations [START_REF] Chen | Structure synthesis and singularity analysis of a parallel Manipulator based on selective actuation[END_REF][START_REF] Gogu | Fully-isotropic redundantly-actuated parallel manipulators with five degrees of Freedom[END_REF], to obtain reconfigurable platforms [START_REF] Mohamed | Design and analysis of kinematically redundant parallel Manipulators with configurable platforms[END_REF] and limbs [START_REF] Fischer | Design of a reconfigurable planar Parallel manipulator[END_REF] or combined advantages [START_REF] Nahon | Reducing the effect of shocks using redundant actuation[END_REF]Zanganach and Angeles 1994a,b;[START_REF] Kim | Operational quality analysis of parallel manipulators with actuation redundancy[END_REF]Kock andScumacher 1998, 2000;Mohamed 2003a,b].

New formulae for calculating the degree of structural redundancy of the parallel robots have been proposed by [START_REF] Gogu | Fully-isotropic redundantly-actuated parallel manipulators with five degrees of Freedom[END_REF][Gogu, , 2008a]]. Actuation redundancy is achieved by introducing additional actuated joint within the limbs, beyond the minimum necessary to actuate the manipulator. In this way, motion coupling in parallel manipulators can be reduced.

For our mechanism we have a minimum of two degrees of freedom in a general configuration and five degrees of freedom for a planar constraint-redundant configuration. Hence we introduce five motors to cross from one type of singularity to another. Redundant robot control techniques cannot be used in all cases. Therefore, we prefer using more advanced techniques (multi-model control) [START_REF] Pagis | Augmentation de la taille de l'espace de travail operationnel des robots paralleles en traversant les singularites de Type 2 : Generation de trajectoires optimales et commande avancee[END_REF] coupled with PID controller to reflect the local actuation redundancy of the robot. The following sections explain the various control strategies used for the SSL8B mechanism.

General principle of position control dedicated to SSL8B reconfigurable mechanism

This part aims to detail the two position control approaches that are implemented for the SSL8B mechanism, namely the synchronization of the actuators and modification of the degrees of actuation. Synchronization of actuators can be achieved by two ways: namely the PID controller method (by tuning the values of the gain) and by the conventional control of generating trajectories respecting the geometric model. The modification of the degrees of actuation is attained by the multi-model control strategy. In the following subsection, we will discuss in details about these approaches with some simulated results. These position control techniques can follow a path in the joint space. This trajectory is generated using conventional approaches as discussed in the next section.

Trajectory generation

Trajectory generation is obtaining a desired motion of the parallel manipulator in the joint space. The basic concept of trajectory generation is to move the robot manipulator from an initial joint position to some desired final joint position by using interpolation functions as described in [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF].

Several interpolation functions (denoted in the sequel) can provide a trajectory generation and are described below: Six constraints should be satisfied to obtain an interpolation of polynomial of fifth degree. These six constraints are as follows:

        f t t 4         f t t 5         f t t Bang-bang profile 2         f t t         f t t 2         f t t Position control
, , (0) = 0, ( ) = 0, , .

These constraints are used to calculate the coefficients of a fifth degree polynomial which gives us the interpolation function as: r(t) = 10 -15 + 6

The two approaches used for control of the 8-bar mechanism will be presented in detail in the following section and both of them uses the fifth degree polynomial for trajectory generation.

The first control strategy: Synchronization of the actuators

The robot has a property of transiting from one singular configuration to another. This is attained by synchronizing the actuators. Synchronization is achieved either by adjusting the values of the gains in the PID controller or by using the conventional control by generating a trajectory respecting the geometric model of the SSL8B mechanism.

PID control by adjusting the values of gains

Classical tuning of the gains K p , K d and K i allows to have a limited positioning error. The integral gain K i is generally increased to compensate for the dry frictions and gravity effects, while the derivative gain K d is generally decreased to cope with measurements noise.

Additional features such as friction, gravity and backlash compensation in the industrial controllers improve accuracy. Therefore, a PID control ensures an efficient compensation of the small dynamics and the behaviors of the stiff mechanical structure ensure a good static accuracy.

Conventional control with trajectory generation

Having obtained the geometric model of the SSL8B mechanism (chapter 2: section 2.3.4), we describe in mathematical form the relationship between the joint variables, position and orientation of the robot. This mathematical representation is obtained from the travelling coordinate system which involves the link between the rotation and translation of the adjacent links. As a conclusion, by using the geometric model of the robot for dependent actuated joints coupled with a trajectory generator for independent actuated joints, it is possible to synchronize on-line all the actuated joints motions.

The second control strategy: Modification of the degrees of actuation using multimodel control strategy

In this control strategy, we use a selection matrix which helps us to modify the degrees of actuation with respect to time. where, and where, is calculated in function of time from which the robot approaches a singular position. can be calculated with the conditioning of matrix J. The diagonal matrix 'S' can also be used with transition functions where the diagonal permits to deactivate one or more motors in a flexible manner to avoid the deterioration of the mechanism.

PID controller

+ - d/dt S K p K d 8-bar robot q d q d (t) + + + ε ʃ K I ʃε Multi-model
With, (t) = a 0 + a 1 t+ a 2 t 2 + a 3 t 3 (t= t 1 ) = 1 (t= t 2 ) = 0.

(t= t 1 ) = (t= t 2 ) = 0 (t 1 and t 2 are calculated in function of the conditioning of matrix J)

+ + + +                                            0 0 0 1 3 2 1 0 3 2 1 0 1 1 3 2 1 0 2 2 2 2 1 1 3 2 2 2 2 3 1 2 1 1 a a a a t t t t t t t t t t This is of the form B A X B AX 1    
. Hence we calculate the transition matrix 'S'.

In the following section, we are going to test these control laws on the robot models carried out by ADAMS to perform advanced simulations using Adams / Simulink. Henceforth, we present firstly the virtual models/prototype of the SSL8B reconfigurable robot.

t 1 t 2 t 1 A X B 102

Advanced simulation results of the SSL8B control law

Now, we implement the control strategies presented in section 4.2.2 to our SSL8B reconfigurable mechanism. Before that, we will make a study on the prototype modeled using ADAMS. Section 4.3.1 details the development of the prototype. The configuration shown in figure 4.16b is simulated from the initial position of the planar 8bar mechanism to this folded configuration by actuating four motors (we actuate always 5 motors even if the motion is null: the applied torque maintain the joint in a fixed position), two motors at the base and two motors of the two arms with synchronized rotational movement.

From this configuration (figure 4.16b) and controlling one motor from the base we obtain the following configuration: In this section, we are going to shows some simulations results by implementing the control strategies on the SSL8B mechanism.

Principle of the advanced simulations using Adams/Simulink

The model should be completed and should include all necessary geometry, constraints, forces, and measures in order to be controlled by a Once the Adams model block is obtained, the global dynamic model can be simulated with the control within the Simulink environment.

Tuning of gains using PID controller

The most common solution in robotics consists in adjusting the gains in order to obtain a negative real triple pole. This yields the fastest possible response without overshoot. Determining gains of PID controllers can be done with these three equations below from the inertia matrix 'A' obtained by the dynamic model [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF].

Where, is the frequency and corresponds to the maximum magnitude of the inertia matrix of the robot.

However for mechanisms whose architecture is complex, including parallel robots, reconfigurable, redundant ... etc. control becomes increasingly difficult.

In order to obtain a 4-bar mechanism from the 8-bar, we must perform a movement of +/-90 ° in the motors situated on the base of the robot. By doing this, we achieve the following 5 trajectories as shown in figure 4.19 using the fifth degree polynomial interpolation. The challenge now is to be able to adjust the gains of PID controllers. For the PID control scheme by gain tuning, we have found out the exact values of the gains at which the robot is able to go from the planar configuration to the four bar mechanism. The values we used are as follows: Tuning of the gains K p , K d and K i plays a prominent role to stabilize and synchronize the actuators. The following graph shows the position errors of the 8-bar mechanism by tuning the gains in the PID controller.

Trajectory generator respecting the geometric model

Generating movement for the robot can be modeled by a mathematical equation of a trajectory. Joint or operational parameters depending on the type of control follow this trajectory path to reach a desired final position. In this control strategy, we develop a trajectory respecting the geometric model of the 8-bar mechanism and by using the 5 th polynomial interpolation (see Section 4.3) in order to compute the trajectory of the independent joint.

Multi-model control law

The second control scheme implemented for the SSL8B mechanism is the multi-model system. Using multi-model control (See section 4.2.5), we deactivate some actuators by setting a threshold period. Few seconds before the robot goes to the position of four bars, we make two of the motors passive. This is illustrated in figure 4.5.

At the singular configuration, the robot is difficult to control. These extra degrees of actuation help us to cross singular positions and branch to other positions with lesser degrees of freedom.

The values of the gains are tuned in such a way that the robot is able to stabilize at its desired final position with negligible error. Hence, by tuning, the gain values we used in the PID controller to stabilize the robot with multi-model control law are as follows:

For Torque 1: K p =0.098, K d = 0.098s For Torque 2: K p = 15, K d =1s For Torque 3: K p = 15, K d =1s For Torque 4: K p = 15, K d =1s For Torque 5: K p =0.062, K d = 0.062s The graphs in figure 4.27 show the stabilization of the torques after a threshold period of 19.7 seconds. The robot is able to move from the planar configuration with 5 degrees of mobility to a 4-bar mechanism and then it is able to stabilize in this configuration of 4 bars. The graphs shown in figure 4.28 (a) illustrates the output position of the five controlled joints with a movement of 90 o and -90 o . In figure 4.28 (b), we can see that the mechanism is able to reach the desired position with a negligible error.

Note that the evolution of the two joint variables is synchronized and the two curves converge towards the final value of = and = . This is achieved by the synchronous movement of the two lateral arms of the 8-bar mechanism from the initial configuration to the 4-bar configuration.

Critical analysis of the control laws

From the above sections, we can summarize the control laws as follows: 4.3 illustrates the pros and cons of the different control strategies used for the 8-bar reconfigurable mechanism. From this table we can conclude that the multi-model control strategy is preferable to be implemented on the SSL8B mechanism as the robot is able to go from the planar configuration to the 4-bar mechanism with negligible error. However, this kind of control law is difficult to be implemented since the time to switch between one actuation states to another depends on the joints trajectories and the singular position of the robot.

Conclusion

In this chapter we discussed the interesting property of reconfigurability of an eight-bar mechanism. Reconfigurability is achieved by crossing singularities without disassembling and assembling of the robot. In order to cross singularities without losing the control of mechanism, we have proposed few control schemes in this chapter. From the two control strategies presented in this chapter, we can conclude that the multi-model control law shows better performances with negligible errors and the robot is able to stabilize at the final configuration. In the case of the single-loop 8 bar mechanism, 5 joints have to be actuated while it has only 2 DOFs in its general configuration. The configuration control with bifurcation from one configuration to another is also explained in this chapter. Also the mechanical design, modeling and simulation in Adams are illustrated and permit us to show the relevance of the proposed control algorithms by using advanced simulations.

The SSL8B mechanism is under fabrication and these simulated results will be checked with the real 8-bar mechanism. A perspective of the thesis will be to develop advanced accurate and robust control laws.

Conclusion

Synthesis and contributions

The contribution of this thesis can be explained in four parts as detailed from chapters 1 to 4. This thesis mainly addresses three major parts namely the systematization and structural analysis, geometric and kinematic formulation followed by the control strategies of the 8-bar reconfigurable mechanism.

The first chapter proposes a complete systematization of the types of morphing in the reconfigurable mechanisms and details the systematization of the structural parameters such as mobility, connectivity, redundancy and number of over-constraints. These morphing techniques have been further related to mechanism type and motion type, and their applications have been identified. This chapter on bibliographic review has presented a foundation for the study of morphing techniques and metamorphosis in reconfigurable mechanisms. Hence this chapter presents a comprehensive review of morphing techniques in reconfigurable mechanisms and their relationship and existence in the mechanisms to help achieve reconfiguration has been examined.

The second chapter presents the structural analysis and the geometric modeling of the eight bar reconfigurable mechanisms. We used the travelling coordinate system for the geometric formulation of the robot. The mechanism kinematic constraints have been formulated in order to determine the mechanism configuration with respect to two independent joint variables. We need these constraint equations for the mechanism control. The geometric relationship between the joint variables for the general non-singular configuration has been determined with the frames of the travelling coordinate system.

The third chapter comprises singularity analysis of the SSL8B mechanism. The solutions of the geometric model for all the singular configurations identified have been formulated. Also, various singular configurations of this mechanism and the nature of these singularities are discussed in Chapter 3. This mechanism possesses an interesting property of reconfigurability. Different types of singularities such as constraint singularity, redundant singularity and constraint-redundant singularities are identified for the SSL8B reconfigurable mechanism. From these different singularities the mechanism can also get out at any time in a non-singular configuration. The singularity analysis presented in this chapter is useful for the study of reconfiguration capability of this mechanism which can take place in the various singular configurations illustrated in this chapter.

The fourth chapter addresses the control of the SSL8b especially for bifurcation from a singular configuration to different branches of the mechanism. In order to control the mechanism in any configuration, we proposed the use of redundant actuation. Indeed, five joints are actuated, which correspond to the maximal number of instantaneous mobilities that the mechanism can reach in a constraint singularity, while it has only two mobilities in its general configuration. Following this, we present two control schemes implemented for the SSL8B mechanism namely the synchronization of the actuators and modification of the degrees of actuation. The robot is modeled using Adams software and the dynamic parameters are tabulated. The control schemes are applied for the 8-bar mechanism and the simulation results are illustrated. From the two control strategies presented in this chapter, we can conclude that the multi-model control law shows better performances with negligible error and the robot is able to stabilize at the final configuration. The chapter is concluded with a comparative analysis of the control strategies presented.

Perspectives

The future work of this thesis could concentrate on the following aspects: i) investigate the reconfigurability of other kind of reconfigurable mechanism to identify and characterize other types of singularities used during the reconfiguration phase, ii) formulate general rules for structural synthesis of reconfigurable mechanism starting from formalizing the singularity conditions; iii) control the real prototype to obtain optimized results by implementing the multi-model control law to obtain a negligible error; iv) generalize the multi-model control to be applicable to other types of reconfigurable mechanism; formulate general conditions for achieving reconfigurability of other mechanism with continuous transitions from one type of singularity to another. Solving the final simplified equations of ( 2) and ( 13), we obtain two values for two values for

Publications

The complete schema of all the solutions is illustrated in a tabulation in the Annex II.

The third kinematic constraint equation is obtained as follows,

We know that, // and  ( 14)

From ( 14) we have, 

 (Since ) are orthogonal Also, 

From equation ( 15) and ( 16 Hence we obtain the Adams block to be connected with the PID controller in Simulink. Now to connect the block with Simulink, we need to call the file generated by Adams in the command window of Matlab. In our case, we call the function by typing, "Controls_Plant" as enumerated in figure A.0.5 and then to obtain the block, we type "adams_sys".

  Introduction ...........................................................................................................................................

FIGURE 1

 1 FIGURE 1.8 (A) THE 3SVPSV-I (B) PHASE 3RVPSV OF THE METAMORPHIC PARALLEL MECHANISM 3SVPSV-I (C) PHASE 3UVPUV OF THE METAMORPHIC PARALLEL MECHANISM 3SVPSV-I (D) THE SVPSV-II METAMORPHIC PARALLEL MECHANISM (E) PHASE 3UVPUV OF THE METAMORPHIC PARALLEL MECHANISM 3SVPS [ZHANG ET AL., 2013] ................................................................. 26

FIGURE 4 .

 4 FIGURE 4.25 GRAPH ILLUSTRATING THE FILTERED INPUT TORQUES OF THE FIVE JOINTS ...................

FIGURE 4 .

 4 FIGURE 4.26 POSITION ERRORS FOR EACH JOINT .................................................................................

FIGURE 4 .

 4 FIGURE 4.27 GRAPH ILLUSTRATING THE INPUT TORQUES OF THE FIVE JOINTS ...................................

FIGURE 4 .

 4 FIGURE 4.28 GRAPH ILLUSTRATING (A) MEASURED POSITION OF THE JOINTS (B) POSITION ERROR ..

Figure

  Figure 1.1 (a.) M-TRAN III self-reconfigurable robot (b.) Metamorphosis of M-TRAN III [Kurokawa et al., 2006]

Figure 1 .

 1 Figure 1.1 illustrates one of the leading robots in the area of self-reconfigurable modular robotics, the M-TRAN III (Modular Transformer). These robots can change shape to walk, slither or crawl and climb over various obstacles on the way.

Figure 1 .

 1 Figure 1.2 ACM-R5 amphibious robot [Shumeiu et al., 2011]

Figure

  Figure 1.4 Workflow chart of the metamorphic mechanism in all configurations [Zhang et al., 2011(b)]

Figure

  Figure 1.5 (a) and (b) Schematic graphs of a planar five bar force limit metamorphic mechanism [Zhang et al.,2011(b)]

Figure 1 . 6

 16 Figure 1.6 Classification of geometrical morphing

Figure

  Figure 1.7 (a) A card box with a crash-lock base and (b) A special mechanism equivalent to the box [Dai et al., 1999]

Figure 1 .

 1 Figure 1.8 (a) The 3SvPSv-I (b) Phase 3RvPSv of the metamorphic parallel mechanism 3SvPSv-I (c) Phase 3UvPUv of the metamorphic parallel mechanism 3SvPSv-I (d) The SvPSv-II metamorphic parallel mechanism (e) Phase 3UvPUv of the metamorphic parallel mechanism 3SvPS [Zhang et al., 2013]

Figure 1

 1 Figure 1.9 (a) 3(rT)g1P(rT)g3-pure rotation (b) 3(rT)g1P(rT)g2-pure translation [Gan et al., 2009]

Figure 1 .

 1 Figure 1.10 A reconfigurable eight-bar linkage [Zhang et al., 2012(d)]

Figure

  Figure 1.11 Spatial hybrid mechanism when link 4 and link 5 have been annexed [Zhang et al., 2012(b)]

Figure 1 .

 1 Figure 1.12 (a) Palm workspace and the corresponding hand poses with minimum palm workspace (b) The twisting motion [Dai et al., 2009]

Figure 1 .

 1 Figure 1.13 Prehensile tests of the metamorphic anthropomorphic hand [Wei et al., 2011]

Figure 1 .

 1 Figure 1.14 (a) A card box with a crash-lock base (b) A special mechanism equivalent to the box (c) A flattened configuration of a card box (d) Equivalent mechanism of a whole box [Dai and Rees Jones, 1999]

Figure 1 .

 1 Figure 1.15 (a) The 3(rT)b1C(rT)b1 (b) The 2(rT)b1C(rT)b1-1(rT)b2C(rT)b1 (c) The 1(rT)b1C(rT)b1-2(rT)b2C(rT)b1 (d) The 3(rT)b2C(rT)b1 (e) The 2(rT)b2C(rT)b1-1(rT)b2C(rT)b2 [Gan et al.,2010] 1.4.3 Connectivity and Mobility change in furcating morphing

Figure 1 .

 1 Figure 1.16 A planar four-bar linkage R-P-R-P (and R-R-R-R)[START_REF] Galletti | Single-loop kinematotropic mechanisms[END_REF]

Figure 1 .

 1 Figure 1.17 T2R1-type parallel manipulator with uncoupled and bifurcated planar-spatial motion of the moving platform: constraint singularity (a) branch with planar motion (b) branchwith spatial motion (c) limb topology[START_REF] Gogu | T2R1-type parallel manipulators with bifurcated planar-spatial motion[END_REF]

Figure 1 .

 1 Figure 1.18 (a) Kinematic structure of the metamorphic mechanism (b) equivalent model with bifacial constraint forces [Zhang et al., 2010]

  .19 as: (a) kinematotropic bifurcating morphing (b) furcating morphing in metamorphic parallel mechanisms (c) bifurcation in the discontinuously movable parallel mechanisms (d) multifurcating morphing and (e) actuation morphing. Detailed explanations with its corresponding examples are described in the following sub-sections.

Figure

  Figure 1.19 Classification of furcating morphing 1.4.4.1 Kinematotropic bifurcation morphing

Figure 1 .

 1 Figure 1.20 A kinematotropic chain (from 2 to 3 degrees of freedom) [Galletti et al., 2001]

Figure 1 .

 1 Figure 1.21 One bifurcation configuration of DM RC -//-Rc CVSC. (a) A bifurcation at double point of top surface; (b) at a transition posture; (c) symmetric motion at a mode I ; (d) motion at a mode II [Lee and Herve,2012]

Figure 1 .

 1 Figure 1.22 (a) General structure and (b) Singular posture of the derivative queer-square mechanism [Qin et al., 2014]

Figure

  Figure 1.23 Classification of Joint motion morphing 1.4.5.1 Idle-active switch morphing

Figure 1 .

 1 Figure 1.24 The working stages of the metamorphic mechanism extracted from a steel-ingot cutting machine [Li and Dai, 2012]

Figure 1 .

 1 Figure 1.25 A metamorphic robotic hand with a spherical five-bar [Dai et al., 2007]

Figure 1 .

 1 Figure 1.26 The reconfigurable cube mechanism (RCM) [Kuo et al., 2014]

Figure 1 .

 1 Figure 1.28 A carton fold [Dai et al.,2008]

Figure 1 .

 1 Figure1.29 Metamorphic hand with its palm in a reconfigured position[START_REF] Wei | Kinematic analysis and prototype of a metamorphic anthropomorphic hand with a reconfigurable palm[END_REF]

Figure 1 .

 1 Figure 1.30 Schematic graph of a planar five bar force limit metamorphic mechanism [Zhang et al.,2011a]

Figure 1 .

 1 Figure 1.32 (a) Robotic finger for packaging (b) A demonstration [Dai and Caldwell, 2010]

MF

  the mobility (or degree of mobility) of the parallel mechanism F, N F -number of overconstraints (degree of overconstraint) of the parallel mechanism F, T Fnumber of structural redundancies (degree of redundancy) of parallel mechanism F, T Gjnumber of structural redundancies (degree of redundancy) of the kinematic chain G j disconnected from the parallel mechanism F, R Gj -vector space of relative velocities between the mobile and the reference platforms, n Gj and 1 Gj , in the kinematic chain G j disconnected from the parallel mechanism F, R F -vector space of relative velocities between the mobile and the reference platforms, between the mobile and the reference platforms, n Gj and 1 Gj , in the kinematic chain G j disconnected from the parallel mechanism F, S F -connectivity between the mobile and the reference platforms Gj nn  and Gj 11  , in the parallel mechanism

Figure 2 . 1

 21 Figure 2.1 Simple open kinematic chain with 3R joints (a) three joint axes are parallel to z-axis (b) three revolute joints with orthogonal axes

Figure 2 . 2

 22 Figure 2.2 Example of modeling associated with TCS method illustrating the lengths a i and b iTo explain the angle of rotation between the frames, let us consider the passage of TCS of two elements 'm' and 'n' with a kinematic pair. These elements undergo a finite displacement of rotation/translation around the axes of the TCS superposed by the axes of the pair . The finite displacement of rotation is represented by and the finite displacement of translation is denoted by .

Figure 2 . 3

 23 Figure 2.3 Joint axes (i-1, i) and (i, i+1) are parallelIf the axes of the adjacent joints are perpendicular, the axes of TCS carried out by the joint (i, i+1) is obtained by the circular permutation with respect to the axes of TCS carried out by (i-1, i). This rule is illustrated in figure2.4 along with the direction of the common normal.

Figure 2 . 4

 24 Figure 2.4 Joint axes (i-1, i) and (i, i+1) are perpendicular

Figure 2 . 5

 25 Figure 2.5 Single loop eight-bar mechanism with orthogonal/ parallel axes in a non-singular configuration

2. 3 . 3

 33 Parameterization of the eight-bar linkage using Travelling Coordinate SystemThe schematic diagram of the eight-bar linkage with its origin and the length of the elements of each link are represented in figure 2.6. One of the main characteristic of this mechanism is that the lengths b 2 = b 4 = b 6 = b 8. For the eight-bar mechanism, we use the TCS parameters to determine the position and orientation of the characteristic frames attached to each link. The notation below shows the axis of rotation for each revolute joint with parallel and orthogonal axes in the associated open loop kinematic chain obtained by splitting the reference link.

Figure

  Figure 2.6 Schematic diagram of eight bar linkage

Figure 2 . 7

 27 Figure 2.7 Modeling of the eight bar reconfigurable mechanism by TCS method

Figure 2 . 8

 28 Figure 2.8 Schema of the eight bar mechanism with its natural coordinates

Figure 2 . 9

 29 Figure 2.9 Single-loop eight-bar mechanism with orthogonal/ parallel axes Figure 2.9 shows an assembly mode which validates and satisfies the above set of equations. This is one of the singular configurations of the SSL8B mechanism where the axes are parallel to the plane

Figure

  Figure 3.1 Structure of Steward Platform[START_REF] Merlet | Parallel Robots[END_REF]

  Figure 3.2 SSL8B mechanism two distinct constraint singularities In the singular configuration in Figure 3.2a, the axes of the eight revolute joints are parallel to plane O 1 x 1 y 1 and in the same time intersect axis O 1 x 1 . The following values are associated with the joint rotation angles in the constraint singularity in Figure 3.2a: , 90 , 0 , 90 , 0 , 90 , 180 , 90 23 34 45 54 43 32 21

4

  In the singular configuration in Figure3.2b, the axes of the eight revolute joints are parallel to the plane defined by the rotation axes of the two first revolute joints of limb G 1 . The following values are associated with the joint rotation angles in the constraint singularity in Figure2b

  Possible combinations of vector spaces with minimum S F for the configuration in

Figure 3 .

 3 Figure 3.3 Single-loop eight-bar mechanism with orthogonal/ parallel axes in constraintredundant singularity

Figure 3 .

 3 Figure 3.4 Single-loop eight-bar mechanism with orthogonal/ parallel axes in three different redundant singularities

Figure 3 . 5

 35 Figure 3.5 Transition between constraint singularity and constraint -redundant singularity of an eight-bar mechanism

Figure 4 . 1

 41 Figure 4.1 Control methods for Robotic manipulators

Figure 4 . 2

 42 Figure 4.2 Block diagram of a PID control scheme in the joint space

  the proportional, derivative and integral gains of each axis.

Figure 4 . 3

 43 Figure 4.3 Principle of the adaptive control applied to computed torque control [Pagis, 2015]

  Figure 4.4 illustrates the classification of position control for the 8-bar mechanism.

Figure 4 . 4

 44 Figure 4.4 Control strategies for SSL8B mechanism

  joint coordinate vectors corresponding to the initial and final configurations. is the transfer time of the trajectory and D is the joint displacement corresponding to the difference between andIn our work, in order to ensure that we have continuity in acceleration as well as to avoid exciting the resonances in the mechanism, we use the fifth degree polynomial interpolation function with smooth position and velocity profiles. The general form of this polynomial is: = + By taking derivation of this equation, we get the joint velocity and acceleration as, = =

Figure 4 . 5

 45 Figure 4.5 Schema of the principle of the multi-model control law Figure 4.5 illustrates the control law of the multi-model control strategy. 'S' is a selection matrix which permits to activate or deactivate a motor in real time.

Figure 4 . 6

 46 Figure 4.6 Graph of σ Vs t

  of the prototype and development of the ADAMS model The prototype consists of 8 parts as shown in the figure 4.7.

Figure 4 . 7

 47 Figure 4.7 Front view of the 8 bar linkageThe base of the robot comprises an aluminium plate which serves to fix the robot on the support and to maintain the motors and other accessories. Two motors with their gearboxes are installed at the base of the robot. The four square shaped parts helps to rotate the side arms of the 8-bars about the axes of the base engine. The end-effector is composed of an aluminum plate with two revolute joints.

Figure 4 .

 4 Figure4.11 shows the real prototype of the SSL8B mechanism. Two motors are already mounted on the base. Three motors will be fixed on the robot arm. Once this is done, the robot is ready for experimentation.

Figure 4 .

 4 Figure 4.11 Real prototype of the SSL8B mechanism

Figure 4 .

 4 Figure4.12 shows the gearboxes which help to couple the prototype with the motors. They are used to transmit the rotational movement of the motors mounted on the arms (3 motors where two are on the left arm and one on the right arm) from the axis of rotation of these actuators. So these bevel gears will be coupled with three actuators. Coupling and transmission of rotational movement is by a wedge on the side as shown below in figure 4.12.

Figure 4 .

 4 Figure 4.16 Two solutions after bifurcation (a.) Solution with constraint-redundant singularity (b.) Solution with constraint singularity In figure 4.16, the SSL8B mechanism is able to bifurcate from a planar constraint-redundant singular configuration (figure 4.16a) to a constraint singular configuration (figure 4.16b).These configurations are identified as a constraint-redundant singularity and constraint singularity in Aimedee et al.[START_REF] Aimedee | Redundant versus constraint singularities in parallel mechanisms[END_REF]

Figure 4 .

 4 Figure 4.17 8-bar mechanism with folded and outstretched arm configuration 4.3.4 Advanced simulation results

  simple PID controller. Any changes in the model under ADAMS are automatically taken into account in the dynamic model under Simulink. The successful interface between ADAMS and MATLAB/Simulink yields a powerful tool to model the mechanism and to analyze the dynamic behavior of the model. The input torques, the output positions and velocities are defined as state variables and represent the different data exchange between Adams and Matlab as explained in the figure 4.18

Figure 4 .

 4 Figure 4.18 ADAMS/Controls step by step process

Figure 4 .

 4 Figure 4.19 Desired trajectories of the five joints

Figure 4 .

 4 Figure 4.20Graph illustrating the input torques of the five joints

Figure 4 .

 4 Figure 4.20, figure 4.21 and figure 4.22 shows the evolutions of the input torques (with and without filter) and joint positions with respect to time. From the graph in figure 4.23, we can say that by tuning the gain using the PID control law, joint 1 and joint 2 are controlled at t = 11.1s with a negligible error whereas other joints are able to stabilize itself to reach the final configuration of 4-bar with zero error. The PID control strategy is used for passing from the planar 8-bar configuration to a 4-bar configuration. To pass through other singular configurations, the multi-model control strategy is better suited as presented in section 4.3.4.4.

  114

Figure 4 .

 4 Figure 4.24 Measure position of the joints

Figure 4 .

 4 Figure 4.27 Graph illustrating the input torques of the five joints

  Redundant versus constraint singularities in parallel mechanisms, In: Proceedings of the Imeche, Part C: Journal of Mechanical Engineering Science, in press, accepted on September 27, 2015. Aimedee F., Dai J., Gogu G., Bouzgarrou B. C., Bouton N., Systematization of Morphing in Reconfigurable Mechanisms, Mechanism and Machine Theory 96 (2016) pp: 215x5, y5 and z5 are substituted in the above equation and the equation is simplified.After solving and simplifying the above equation with the trigonometric formulae we get, equation is obtained by various sets of equations involving parallelism and perpendicularity and thereafter making a common intersection between those equations as illustrated below. In the general configuration



  Figure A. 0.4 Generation of the control plant with input and output variables

Figure

  Figure A.0.5 Matlab command for obtaining the Adams block

  Table of figures: FIGURE 1.1 (A.) M-TRAN III SELF-RECONFIGURABLE ROBOT (B.) METAMORPHOSIS OF M-TRAN III [KUROKAWA ET AL., 2006] ............................................................................................................ 20 FIGURE 1.2 ACM-R5 AMPHIBIOUS ROBOT [SHUMEIU ET AL., 2011] ..................................................... 21 FIGURE 1.3 CLASSIFICATION OF TOPOLOGICAL MORPHING .................................................................. 22 FIGURE 1.4 WORKFLOW CHART OF THE METAMORPHIC MECHANISM IN ALL CONFIGURATIONS [ZHANG ET AL., 2011(B)] ............................................................................................................................. 23 FIGURE 1.5 (A) AND (B) SCHEMATIC GRAPHS OF A PLANAR FIVE BAR FORCE LIMIT METAMORPHIC MECHANISM [ZHANG ET AL.,2011(B)] ........................................................................................... 24

FIGURE 1.6 CLASSIFICATION OF GEOMETRICAL MORPHING .................................................................. 25 FIGURE 1.7 (A) A CARD BOX WITH A CRASH-LOCK BASE AND (B) A SPECIAL MECHANISM EQUIVALENT TO THE BOX [DAI ET AL., 1999] ..................................................................................................... 25

  ... 35 FIGURE 1.17 T2R1-TYPE PARALLEL MANIPULATOR WITH UNCOUPLED AND BIFURCATED PLANAR-SPATIAL MOTION OF THE MOVING PLATFORM: CONSTRAINT SINGULARITY (A) BRANCH WITH PLANAR MOTION (B) BRANCH WITH SPATIAL MOTION (C) LIMB TOPOLOGY [GOGU, 2012] .......... 36 FIGURE 1.18 (A) KINEMATIC STRUCTURE OF THE METAMORPHIC MECHANISM (B) EQUIVALENT MODEL WITH BIFACIAL CONSTRAINT FORCES [ZHANG ET AL., 2010] ........................................................ 38 FIGURE 1.19 CLASSIFICATION OF FURCATING MORPHING..................................................................... 39 FIGURE 1.20 A KINEMATOTROPIC CHAIN (FROM 2 TO 3 DEGREES OF FREEDOM) [GALLETTI ET AL., 2001] .............................................................................................................................................. 39 FIGURE 1.21 ONE BIFURCATION CONFIGURATION OF DM RC -//-RC CVSC. (A) A BIFURCATION AT DOUBLE POINT OF TOP SURFACE; (B) AT A TRANSITION POSTURE; (C) SYMMETRIC MOTION AT A MODE I ; (D) MOTION AT A MODE II [LEE AND HERVE,2012] ........................................................ 40 FIGURE 1.22 (A) GENERAL STRUCTURE AND (B) SINGULAR POSTURE OF THE DERIVATIVE QUEER-SQUARE MECHANISM [QIN ET AL..31 WORKFLOW CHART OF THE METAMORPHIC MECHANISM IN ALL CONFIGURATIONS [ZHANG ET AL., 2011B] ................................................................................................................. 47 FIGURE 1.32 (A) ROBOTIC FINGER FOR PACKAGING (B) A DEMONSTRATION [DAI AND CALDWELL, 2010] .............................................................................................................................................. 48 FIGURE 2.1 SIMPLE OPEN KINEMATIC CHAIN WITH 3R JOINTS (A) THREE JOINT AXES ARE PARALLEL TO Z-AXIS (B) THREE REVOLUTE JOINTS WITH ORTHOGONAL AXES ................................................... 54 FIGURE 2.2 EXAMPLE OF MODELING ASSOCIATED WITH TCS METHOD ILLUSTRATING THE LENGTHS A JOINT AXES (I-1, I) AND (I, I+1) ARE PERPENDICULAR ........................................................ FIGURE 2.5 SINGLE LOOP EIGHT-BAR MECHANISM WITH ORTHOGONAL/ PARALLEL AXES IN A NON-SINGULAR CONFIGURATION ........................................................................................................... FIGURE 2.6 SCHEMATIC DIAGRAM OF EIGHT BAR LINKAGE .................................................................. FIGURE 2.7 MODELING OF THE EIGHT BAR RECONFIGURABLE MECHANISM BY TCS METHOD ............. FIGURE 2.8 SCHEMA OF THE EIGHT BAR MECHANISM WITH ITS NATURAL COORDINATES .................... FIGURE 2.9 SINGLE-LOOP EIGHT-BAR MECHANISM WITH ORTHOGONAL/ PARALLEL AXES .................. FIGURE 3.1 STRUCTURE OF STEWARD PLATFORM [MERLET, 2008] ..................................................... FIGURE 3.2 SSL8B MECHANISM TWO DISTINCT CONSTRAINT SINGULARITIES ..................................... 4.6 GRAPH OF Σ VS T .............................................................................................................. FIGURE 4.7 FRONT VIEW OF THE 8 BAR LINKAGE ................................................................................ FIGURE 4.8 BASE OF THE ROBOT ......................................................................................................... FIGURE 4.9 SQUARE SHAPED PARTS OF 2, 4, 6 AND 8 IN REFERENCE TO FIGURE 4.7 ........................... FIGURE 4.10 ACCESSORIES USED IN RIGHT AND LEFT ARM ................................................................. 4.14 SSL8B MECHANISM AT THE 4-BAR SINGULAR CONFIGURATION .................................... FIGURE 4.15 OTHER 4-BAR CONFIGURATIONS (OPPOSED) .................................................................. FIGURE 4.16 TWO SOLUTIONS AFTER BIFURCATION (A.) SOLUTION WITH CONSTRAINT-REDUNDANT SINGULARITY (B.) SOLUTION WITH CONSTRAINT SINGULARITY ................................................. FIGURE 4.17 8-BAR MECHANISM WITH FOLDED AND OUTSTRETCHED ARM CONFIGURATION ........... FIGURE 4.18 ADAMS/CONTROLS STEP BY STEP PROCESS .................................................................. FIGURE 4.19 DESIRED TRAJECTORIES OF THE FIVE JOINTS .................................................................. FIGURE 4.20GRAPH ILLUSTRATING THE INPUT TORQUES OF THE FIVE JOINTS .................................... FIGURE 4.21 GRAPH ILLUSTRATING THE INPUT TORQUES OF THE FIVE JOINTS USING FILTER ............

, 2014] ....................................................................................... 41 FIGURE 1.23 CLASSIFICATION OF JOINT MOTION MORPHING ................................................................ 42 FIGURE 1.24 THE WORKING STAGES OF THE METAMORPHIC MECHANISM EXTRACTED FROM A STEEL-INGOT CUTTING MACHINE [LI AND DAI, 2012] .............................................................................. 43 FIGURE 1.25 A METAMORPHIC ROBOTIC HAND WITH A SPHERICAL FIVE-BAR [DAI ET AL., 2007] ....... 44 FIGURE 1.26 THE RECONFIGURABLE CUBE MECHANISM (RCM) [KUO ET AL., 2014] ........................... 45 FIGURE 1.27 THE EIGHT CONFIGURATIONS OF THE RCM [KUO ET AL., 2014] ...................................... 45 FIGURE 1.28 A CARTON FOLD [DAI ET AL.,2008] .................................................................................. 46 FIGURE 1.29 METAMORPHIC HAND WITH ITS PALM IN A RECONFIGURED POSITION [WEI ET AL.,2011] 46 FIGURE 1.30 SCHEMATIC GRAPH OF A PLANAR FIVE BAR FORCE LIMIT METAMORPHIC MECHANISM [ZHANG ET AL.,2011A] .................................................................................................................. 47 FIGURE 1I AND B I ............................................................................................................................................ 56 FIGURE 2.3 JOINT AXES (I-1, I) AND (I, I+1) ARE PARALLEL .................................................................. 57 FIGURE 2.4 FIGURE 3.3 SINGLE-LOOP EIGHT-BAR MECHANISM WITH ORTHOGONAL/ PARALLEL AXES IN CONSTRAINT-REDUNDANT SINGULARITY ...................................................................................... FIGURE 3.4 SINGLE-LOOP EIGHT-BAR MECHANISM WITH ORTHOGONAL/ PARALLEL AXES IN THREE DIFFERENT REDUNDANT SINGULARITIES ....................................................................................... FIGURE 3.5 TRANSITION BETWEEN CONSTRAINT SINGULARITY AND CONSTRAINT -REDUNDANT SINGULARITY OF AN EIGHT-BAR MECHANISM ............................................................................... FIGURE 4.1 CONTROL METHODS FOR ROBOTIC MANIPULATORS .......................................................... FIGURE 4.2 BLOCK DIAGRAM OF A PID CONTROL SCHEME IN THE JOINT SPACE .................................. FIGURE 4.3 PRINCIPLE OF THE ADAPTIVE CONTROL APPLIED TO COMPUTED TORQUE CONTROL [PAGIS, 2015] .............................................................................................................................................. FIGURE 4.4 CONTROL STRATEGIES FOR SSL8B MECHANISM ................................................................ FIGURE 4.5 SCHEMA OF THE PRINCIPLE OF THE MULTI-MODEL CONTROL LAW .................................. FIGURE FIGURE 4.11 REAL PROTOTYPE OF THE SSL8B MECHANISM .............................................................. FIGURE 4.12 THE THREE BEVEL GEARBOXES ACQUIRED ..................................................................... FIGURE 4.13 ADAMS MODEL OF THE EIGHT BAR MECHANISM .......................................................... FIGURE FIGURE 4.22 MEASURE POSITION OF THE JOINTS ................................................................................ FIGURE 4.23 POSITION ERRORS FOR EACH JOINT ................................................................................. FIGURE 4.24 MEASURE POSITION OF THE JOINTS ................................................................................

TABLE 2 .

 2 1 VARIOUS BASES OF VECTOR SPACE R F ................................................................................. 59 TABLE 2.2 DIRECT KINEMATIC MODEL FOR EACH LIMB ....................................................................... 62 TABLE 2.3 STRUCTURAL PARAMETERS OF THE EIGHT BAR RECONFIGURABLE MECHANISM: ............... 64 TABLE 3.1 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION IN FIG. 3.2A .................................................................................................................................... 78 TABLE 3.2 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION IN FIG. 3.2B .................................................................................................................................... 79 TABLE 3.3 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION

IN FIG. 3.3 ...................................................................................................................................... 82

TABLE 3.4 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION IN FIG. 3.4A .................................................................................................................................... 84 TABLE 3.5 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION IN FIG. 3.4B .................................................................................................................................... 84 TABLE 3.6 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F FOR THE CONFIGURATION IN FIG. 3.4A .................................................................................................................................... 86 TABLE 4.1 VARIOUS INTERPOLATION FUNCTIONS ................................................................................ 98 TABLE 4.2 MASS AND INERTIAL PARAMETERS OF THE 8-BAR MECHANISM ........................................ 103 TABLE 4.3 PROS AND CONS OF DIFFERENT CONTROL LAWS USED FOR THE SSL8B MECHANISM....... 117 TABLE A2.1 POSSIBLE BASES FOR EACH LIMB .................................................................................... 137 TABLE A3.1 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F PRESENTED IN FIGURE 3.4A .............................................................................................................................................. 153

TABLE A3 .

 A3 2 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F PRESENTED IN FIGURE 3.4B .............................................................................................................................................. 155TABLE A3.3 POSSIBLE COMBINATIONS OF VECTOR SPACES WITH MINIMUM S F PRESENTED IN FIGURE 3.4C .............................................................................................................................................. 157

Table 2 .1 Various bases of vector space R F

 2

	2
	2

Table 2 .2 Direct Kinematic model for each limb

 2

Table 2 .3 Structural parameters of the eight bar reconfigurable mechanism:

 2 The joint arrangement and structural parameters of the solutions presented in figure2.5 are systematized in Table2.2. From table 2.2, we can observe that the mechanism has only simple limbs (k 1 =1) with one degrees of freedom revolute joints. The mechanism has eight revolute joints in two simple limbs consisting of four revolute joints in each limb. The eight bar reconfigurable mechanism has two degrees of freedom in a general non-singular configuration with six joint parameters that lose their independence in the closed loops of mechanism. Since there are no closed loops existing in the simple limbs of the mechanism, r l =0. The basis of the vector space R F can be defined by any combination of two independent translational velocities and/or two independent rotational velocities as presented in Table2.1. The connectivity and the bases of the vector spaces presented in this table are determined by using the rank and the corresponding submatrices of the Jacobian matrices of the two open kinematic chains associated to the simple limbs of the parallel mechanism as presented in Annex II

	No. Structural parameter		Notations	Solution
	1	m	total number of links including the fixed base
	2	p 1	number of joints in limb 1
	3	p 2	number of joints in limb 2
	4	p	total number of joints in the parallel
			mechanism
	5	q	number of independent closed loops in the
			parallel mechanism
	6	k 1	number of simple limbs of mechanism
	7	k 2	number of complex limbs of mechanisms
	8	k	total number of limbs of mechanism
	9		vector space of relative velocities between the
			mobile and the reference platforms, n Ga and	See Table 2.1
			1 Ga , in the kinematic chain G a disconnected
			from the parallel mechanism
	10		vector space of relative velocities between the	See Table 2.1
			mobile and the reference platforms, n Gb and
			1 Gb , in the kinematic chain G b disconnected
			from the parallel mechanism
	11	S G1 =S G2	connectivity between the mobile and the
			reference platforms, n Gj and 1 Gj , in the
			kinematic chain G j disconnected from the
			parallel mechanism F,
	12	r G1 =r G2	number of joint parameters that lose their
			independence in the closed loops that may
			exist in the loops of limb G a and G b
	13	M G1 =M G2	mobility of kinematic chains associated with
			G a and G b
	14		vector space of relative velocities between the
			mobile and the reference platforms,	Gj nn 	See Table 2.1
			and	Gj 11  ,
	15		connectivity between the mobile and the
			reference platforms	Gj nn 	and	Gj 11  , in the
			parallel mechanism in the parallel mechanism
	16	r l	number of joint parameters that lose their
			independence in the closed loops that may
			exist in the limbs of mechanism
	17	r F	total number of joint parameters that lose
			their independence in the closed loops of
			mechanism
	18	M F	mobility (or degree of mobility) of parallel
			mechanism
	19	N F	number of overconstraints (degree of
			overconstraint) of parallel mechanism
	20	T G1 =T G2	number of structural redundancies (degree of
			redundancy) of the kinematic chains G a and
			G b disconnected from the parallel mechanism

Table 3 .4 Possible combinations of vector spaces with minimum S F for the configuration in Fig. 3.4a

 3

	1
	1
	1

Table 4 .1 Various interpolation functions

 4

	Interpolation functions	Equation
	Linear Interpolation
	Third degree polynomial
	Fifth degree polynomial
		3

Table 4 .3 Pros and Cons of different control laws used for the SSL8B mechanism

 4

	Control Laws	Pros			Cons
	PID control	 Simplicity	of	 Dynamic performance of
		implementation and low	the	robot	varies
		computational cost		according	to	its
				configuration
				 Poor dynamic accuracy
				when tracking a high
				velocity trajectory
	Conventional	 Simple way to describe the	 Use of interpolation is
	control using	desired input positions		difficult to exploit with
	trajectory generation	 Possible to synchronize the joints so that they	increasing the number of points
		reach their final points	 Increased number of
		simultaneously		points	increases	the
				computational burden of
				trajectory generation.
	Multi-model control	 Stabilize the movement of	 Difficult to be set.
		the robot
		 Minimal error
	Table

  There are two possible ways to obtain the joint angles for the fifth kinematic constraint equation. They are as follows, i. From figure 2.7 we can see that,

																(29)
	where,
	After solving and simplifying equation (21) we obtain,
			S	(  32			21	) C	(  54			43	)		0	(22)
	Solving equation (22), we get a set a solutions for 32  and 54  . The complete sets of solutions are illustrated in Annex II. Solving equation (29) we obtain, 0 87 765 543  S C C
					C	(  54				43	) C	(  76		65 	)	S	(  87	)		0	(30)
	The fourth kinematic constraint is quite simple and is directly obtained by homogeneous
	matrix as, By solving equation (30), we get a set of solutions for 43  , 54  , 65  and 76  .
	The sixth kinematic constraint equation is obtained from figure 2.7 by equating	(23) and
																		(31)
	Substituting the values of	and in equation (31) and after simplification we get,
					543 S C	 87 S 765			S	 543 C 87		0	(24)
	where,	) we get, is obtained from the matrix of (  ( ) ( 76 43 54	65	)		) // , i.e the x, y and z coordinates of 0 ) ( ) ( ) ( 87 43 54 87
	corresponds to		and												respectively.
	From figure 2.7, we can say that,								//						//
										 	(17)
	Substituting the values of	and	and simplifying the calculations we get,
	From equation (16) and (17) we get,					(		543 C C	765	)  0 	(18)
				  [ C	( 54  (				43	)][ C	) = 0 76   65  (	)]		0	(19) (25)
	Solving equation (25) we obtain two solutions for 54  and two solutions for 76  .
	Also from figure 2.7 we can say that, ii. From figure 2.7 we can see that,										//	and		(20)
	From equation (19) and (20) we get,														(26)
																		(21)
		Also,													//		//
															(27)
	From equation (26) and (27) we get,					(									) 	(28)

Acknowledgements

I would like to express my special appreciation and thanks to my director Professor Grigore GOGU, for being a tremendous mentor for me. I am extremely thankful and indebted to him for sharing expertise, sincere and valuable guidance and encouragement extended to me. I also place on record, my sense of gratitude to my cosupervisors Dr. Chedli BOUZGARROU and Dr. Nicolas BOUTON who gave me a number of ways to improve the presented techniques and played important role in the successful completion of my doctorate degree.

I would also like to thank my committee members, professor Jian DAI, professor Said ZEGHLOUL and Dr. Stephane CARO for serving as committee members. I also want to thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions, thanks to you. I thank the funders of this thesis. This work has been sponsored by the French government research program "Investissements d'avenir" through the IMobS 3 Laboratory of Excellence (ANR-10-LABX-16-01) and the RobotEx Equipment of Excellence (ANR-10-

The dynamic model represents the robot's motion equations and gives an idea about the robot's behavior when performing a specific task. This model allows establishing relationships between couples (forces) developed by the actuators and the positions, speeds and accelerations of the mechanism joints to control.

Parallel robots are single or multiple closed loop mechanisms. Complexity in the formulation of the dynamic model using the Lagrangian method increases with the number of joints and links as well as the loop-closer constraints. Developing the dynamic model for a parallel manipulator has been of great challenge. The ADAMS software helps to generate the dynamic model of a closed loop mechanism. Since obtaining the "dynamic model" of the 8-bar mechanism is complex, we use modeling robots with ADAMS/ Controls. This helps us to generate a robot block for the use in MATLAB/Simulink.

The mechanism can be controlled using ADAMS/Controls in connection with the MATLAB/Simulink environment. The ADAMS/Controls product is an add-on product to ADAMS. This product has been created for the ADAMS user to analyze the mechanism in either the control application environment or in the ADAMS environment. The model is used in the control scheme to predict the behavior of the system using a PID controller. The SSL8B is firstly modeled in ADAMS and is parameterized which facilitates model modification. The length of the bars and the orientation of the axes of rotation of each revolute join are parameterized. Figure 4.13 show the 5 actuators that are controlled for a planar 8-bar mechanism. and shown in figure 4.13 are the joint torques corresponding to the five joints to be controlled.

Application of the two control strategies on the SSL8B reconfiguration mechanism

In section 4.2.2 we have discussed the theoretical aspects on the two control strategies. In the former part of this section, we will induce few motions in the joints in ADAMS to show the various configurations of the mechanism. In the latter part of this section, we are going to apply control algorithms on the prototype of the 8-bar reconfigurable mechanism and show some simulated results using ADAMS coupled with Simulink.

By actuating the two motors at the base and by imposing a rotational movement of 90 °, the robot arrives to the 4-bar mechanism configuration as shown below: There are numerous assembly modes for this mechanism. We will consider some particular assembly modes to explain how the eight-bar linkage reconfigures.

Annex I

Geometric modeling of the eight-bar mechanism:

The expression of the homogeneous matrices defining the direct geometric model of the open chain associated with the 8-bar reconfigurable mechanism calculated using maple software are as follows, 'b i ' is the distance defined on the direction of the common normal to the axes of joints (i-1, i) and (i, i+1).

Annex II

The table below presents all possible combinations of the vector spaces of dimension 4 for each limb. Among these the ones with the minimum S F are selected as presented in Table 2.1 in Chapter 2. The other bases are associated with various types of singularities.

Formulating the six loop closure constraint equations while considering two independent input joints:

The first kinematic constraint equation corresponding to the natural coordinates as shown in figure 2.8 is obtained from the coordinates of and

(1) where,

Angular values for a particular configuration of an eight-bar reconfigurable mechanism

The tabulation presented below shows the complete set of joint values for each joint angle. The solutions high lightened are a complete set of joint angular values for a particular configuration.

Annex III

The vector spaces associated with each limb of the mechanism presented in Figure 3.4a are illustrated below by means of calculating the Jacobian matrix of each limb. Qcurrent=[q1,q2,q3,q4,q5]; end % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % if(0.5<u && u<=20); temps_recale=temps-0.5; % % Independent angles q1 = pi/2; q2 = 0; % Dependent on q1 and q2 q3 = 0; q4 = 0; q6 = 0; q5 = 0; q7 = 0; q8 = -pi/2; qi= [0,0,0,0,0]; qf= [q1,q2,q3,q7,q8]; Qcurrent=[q1,q2,q3,q4,q5]; end

Coupling between Adams / Simulink

Once the Adams model of the 8-bar linkage is ready, it is necessary to generate the input joint torques and the output positions of the Adams block. In our case, we have 5 input torques namely and with 5 output positions ( and . To generate these variables we do the following:

Step 1:

Go to Build System elements State variables New

Now we are free to create our state variables one by one. As shown in the figure A.0.1, to create the input torques, we go to the main toolbox and select forces and click Applied Force: Torque (Single-component), then we select the action and the reaction body and define the function as : VARVAL(Torque_1). We repeat the same for the remaining torques and . Step 2:

We now define the plant input and output for the control in Simulink. For this, we do the following:

Go to Build Data elements Plants Plant input New

For the plant input, we add the variable name as the Torque_1, Torque_2, Torque_3, Torque_4, Torque_5 as shown in figure A.0.3