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The mathematical modelling of variation propagation in multistage machining processes helps to perform a quick analysis and diagnosis of the processes. The models for part quality prediction, such as Stream of Variation, include homogeneous transformations of the vectorial representations of parts and fixtures. However, these prediction models are complex when considering fixtures with locating surfaces and the associated matrix size is large. Towards mitigating the mathematical complexity, dual quaternions are proposed in representing and transforming a virtual part and fixture. To achieve this, the primary feature datum is assembled to the primary locating surface, followed by sliding the part to secondary and tertiary locating surfaces by reducing the distance between the vertices of the part and the locating surface. The prediction following the proposed approach gave a result within 0.36 % of the prediction made using CAD/CAM models and maintained the largest matrix size of 9 by 8 for a part with 9 features., QC 20220913

All manufacturing processes inevitably induce variations into manufactured parts that may result in nonconformance. Nonconforming parts incur costs due to the additional process required for rework or scrap loss. Hence, methodical efforts to reduce these variations are necessary for competitive manufacturing. To achieve this, effective variation reduction strategies have to be in place. In a multistage machining context, this could mean robust, rapid, and accurate approaches for representation and prediction of variations, change detection, variations source identification, and compensation. Moreover, the approaches used should be capable of handling all forms of errors contributing to the propagation of variations and nonconformance. Existing part variation and variation propagation analysis methods for multistage machining are limited to orientation and position errors, neglecting form errors. Form errors can be captured by utilizing the concept of Skin Models Shapes (SMSs). The application of SMSs for multistage machining and variation reduction strategies has been limited and not established yet. This thesis contributes to developing and demonstrating the use of SMSs for part quality prediction and variation reduction in multistage machining processes. The specific contribution of the thesis can be summarized as (i) the derivations of variation propagation models using dual quaternions; (ii) part quality prediction considering fixtures with locating surfaces, 3-2-1, and N-2-1 (N>3) locators; (iii) Octrees based method for performing statistical shape analysis; (iv) change and anomaly detection using machine learning classifiers; (v) variation source identification using pattern matching technique; (vi) and estimation of variation compensation values using dual quaternions., Alla tillverkningsprocesser orsaker oundvikligen variationer på den tillverkade detaljen. Detta medför kostnader i form av kassationer eller att artikeln måste ombearbetas. För att minska bearbetningsvariationerna, erfordras ett metodiskt arbetssätt för att kunna erhålla en konkurrenskraftig tillverkning. För att uppnå detta måste det finnas effektiva strategier så att variationerna kan reduceras till ett minimum. I en flerstegsbearbetningskontext måste robusta och exakta metoder finnas, så att detektering av avvikelser och variationer kan korrigeras och kompenseras. Befintliga analysmetoder som idag används för att behandla spridningsvariation av form och geometrifel i flerstegsbearbetningsoperationer, är begränsade till orienterings- och positionsfel, och behandlar inte formfel. Formfel kan behandlas med Skin Model Shapes (SMSs) konceptet. Tillämpning av SMSs för flerstegsbearbetning har hitintills varit begränsad, och metoder och strategier för reduktion av variationerna, är inte färdigutvecklat. Bara ett fåtal forskningsarbeten rapporterar studier inom området. Avhandling bidrar till att skapa ny kunskap och utveckla användningen av SMSs för och flerstegsbearbetningsprocesser och kan sammanfattas som: i. härledning till variationerna i propageringsmodeller med dual quaternions, ii. variation förutsägelse med beaktande av fixturer med lokaliseringsytor, 3-2-1 och N-2-1 (N> 3) lokalisatorer, iii. Octree-baserad metod för att utföra statistisk formanalys. iv. detektering och analys av förändringar och anomalier med hjälp av maskininlärning klassificering, v. identifiering av variationskällor med hjälp av mönstermatchningsteknik, vi. uppskattning av variationskompensationsvärden med dual quaternions.

All manufacturing processes inevitably induce variations into manufactured parts that may result in nonconformance. Nonconforming parts incur costs due to the additional process required for rework or scrap loss. Hence, methodical efforts to reduce these variations are necessary for competitive manufacturing. To achieve this, effective variation reduction strategies have to be in place. In a multistage machining context, this could mean robust, rapid, and accurate approaches for representation and prediction of variations, change detection, variations source identification, and compensation. Moreover, the approaches used should be capable of handling all forms of errors contributing to the propagation of variations and nonconformance. Existing part variation and variation propagation analysis methods for multistage machining are limited to orientation and position errors, neglecting form errors. Form errors can be captured by utilizing the concept of Skin Models Shapes (SMSs). The application of SMSs for multistage machining and variation reduction strategies has been limited and not established yet. This thesis contributes to developing and demonstrating the use of SMSs for part quality prediction and variation reduction in multistage machining processes. The specific contribution of the thesis can be summarized as (i) the derivations of variation propagation models using dual quaternions; (ii) part quality prediction considering fixtures with locating surfaces, 3-2-1, and N-2-1 (N>3) locators; (iii) Octrees based method for performing statistical shape analysis; (iv) change and anomaly detection using machine learning classifiers; (v) variation source identification using pattern matching technique; (vi) and estimation of variation compensation values using dual quaternions., Alla tillverkningsprocesser orsaker oundvikligen variationer på den tillverkade detaljen. Detta medför kostnader i form av kassationer eller att artikeln måste ombearbetas. För att minska bearbetningsvariationerna, erfordras ett metodiskt arbetssätt för att kunna erhålla en konkurrenskraftig tillverkning. För att uppnå detta måste det finnas effektiva strategier så att variationerna kan reduceras till ett minimum. I en flerstegsbearbetningskontext måste robusta och exakta metoder finnas, så att detektering av avvikelser och variationer kan korrigeras och kompenseras. Befintliga analysmetoder som idag används för att behandla spridningsvariation av form och geometrifel i flerstegsbearbetningsoperationer, är begränsade till orienterings- och positionsfel, och behandlar inte formfel. Formfel kan behandlas med Skin Model Shapes (SMSs) konceptet. Tillämpning av SMSs för flerstegsbearbetning har hitintills varit begränsad, och metoder och strategier för reduktion av variationerna, är inte färdigutvecklat. Bara ett fåtal forskningsarbeten rapporterar studier inom området. Avhandling bidrar till att skapa ny kunskap och utveckla användningen av SMSs för och flerstegsbearbetningsprocesser och kan sammanfattas som: i. härledning till variationerna i propageringsmodeller med dual quaternions, ii. variation förutsägelse med beaktande av fixturer med lokaliseringsytor, 3-2-1 och N-2-1 (N> 3) lokalisatorer, iii. Octree-baserad metod för att utföra statistisk formanalys. iv. detektering och analys av förändringar och anomalier med hjälp av maskininlärning klassificering, v. identifiering av variationskällor med hjälp av mönstermatchningsteknik, vi. uppskattning av variationskompensationsvärden med dual quaternions.

All manufacturing processes inevitably induce variations into manufactured parts that may result in nonconformance. Nonconforming parts incur costs due to the additional process required for rework or scrap loss. Hence, methodical efforts to reduce these variations are necessary for competitive manufacturing. To achieve this, effective variation reduction strategies have to be in place. In a multistage machining context, this could mean robust, rapid, and accurate approaches for representation and prediction of variations, change detection, variations source identification, and compensation. Moreover, the approaches used should be capable of handling all forms of errors contributing to the propagation of variations and nonconformance. Existing part variation and variation propagation analysis methods for multistage machining are limited to orientation and position errors, neglecting form errors. Form errors can be captured by utilizing the concept of Skin Models Shapes (SMSs). The application of SMSs for multistage machining and variation reduction strategies has been limited and not established yet. This thesis contributes to developing and demonstrating the use of SMSs for part quality prediction and variation reduction in multistage machining processes. The specific contribution of the thesis can be summarized as (i) the derivations of variation propagation models using dual quaternions; (ii) part quality prediction considering fixtures with locating surfaces, 3-2-1, and N-2-1 (N>3) locators; (iii) Octrees based method for performing statistical shape analysis; (iv) change and anomaly detection using machine learning classifiers; (v) variation source identification using pattern matching technique; (vi) and estimation of variation compensation values using dual quaternions., Alla tillverkningsprocesser orsaker oundvikligen variationer på den tillverkade detaljen. Detta medför kostnader i form av kassationer eller att artikeln måste ombearbetas. För att minska bearbetningsvariationerna, erfordras ett metodiskt arbetssätt för att kunna erhålla en konkurrenskraftig tillverkning. För att uppnå detta måste det finnas effektiva strategier så att variationerna kan reduceras till ett minimum. I en flerstegsbearbetningskontext måste robusta och exakta metoder finnas, så att detektering av avvikelser och variationer kan korrigeras och kompenseras. Befintliga analysmetoder som idag används för att behandla spridningsvariation av form och geometrifel i flerstegsbearbetningsoperationer, är begränsade till orienterings- och positionsfel, och behandlar inte formfel. Formfel kan behandlas med Skin Model Shapes (SMSs) konceptet. Tillämpning av SMSs för flerstegsbearbetning har hitintills varit begränsad, och metoder och strategier för reduktion av variationerna, är inte färdigutvecklat. Bara ett fåtal forskningsarbeten rapporterar studier inom området. Avhandling bidrar till att skapa ny kunskap och utveckla användningen av SMSs för och flerstegsbearbetningsprocesser och kan sammanfattas som: i. härledning till variationerna i propageringsmodeller med dual quaternions, ii. variation förutsägelse med beaktande av fixturer med lokaliseringsytor, 3-2-1 och N-2-1 (N> 3) lokalisatorer, iii. Octree-baserad metod för att utföra statistisk formanalys. iv. detektering och analys av förändringar och anomalier med hjälp av maskininlärning klassificering, v. identifiering av variationskällor med hjälp av mönstermatchningsteknik, vi. uppskattning av variationskompensationsvärden med dual quaternions.

All manufacturing processes inevitably induce variations into manufactured parts that may result in nonconformance. Nonconforming parts incur costs due to the additional process required for rework or scrap loss. Hence, methodical efforts to reduce these variations are necessary for competitive manufacturing. To achieve this, effective variation reduction strategies have to be in place. In a multistage machining context, this could mean robust, rapid, and accurate approaches for representation and prediction of variations, change detection, variations source identification, and compensation. Moreover, the approaches used should be capable of handling all forms of errors contributing to the propagation of variations and nonconformance. Existing part variation and variation propagation analysis methods for multistage machining are limited to orientation and position errors, neglecting form errors. Form errors can be captured by utilizing the concept of Skin Models Shapes (SMSs). The application of SMSs for multistage machining and variation reduction strategies has been limited and not established yet. This thesis contributes to developing and demonstrating the use of SMSs for part quality prediction and variation reduction in multistage machining processes. The specific contribution of the thesis can be summarized as (i) the derivations of variation propagation models using dual quaternions; (ii) part quality prediction considering fixtures with locating surfaces, 3-2-1, and N-2-1 (N>3) locators; (iii) Octrees based method for performing statistical shape analysis; (iv) change and anomaly detection using machine learning classifiers; (v) variation source identification using pattern matching technique; (vi) and estimation of variation compensation values using dual quaternions., Alla tillverkningsprocesser orsaker oundvikligen variationer på den tillverkade detaljen. Detta medför kostnader i form av kassationer eller att artikeln måste ombearbetas. För att minska bearbetningsvariationerna, erfordras ett metodiskt arbetssätt för att kunna erhålla en konkurrenskraftig tillverkning. För att uppnå detta måste det finnas effektiva strategier så att variationerna kan reduceras till ett minimum. I en flerstegsbearbetningskontext måste robusta och exakta metoder finnas, så att detektering av avvikelser och variationer kan korrigeras och kompenseras. Befintliga analysmetoder som idag används för att behandla spridningsvariation av form och geometrifel i flerstegsbearbetningsoperationer, är begränsade till orienterings- och positionsfel, och behandlar inte formfel. Formfel kan behandlas med Skin Model Shapes (SMSs) konceptet. Tillämpning av SMSs för flerstegsbearbetning har hitintills varit begränsad, och metoder och strategier för reduktion av variationerna, är inte färdigutvecklat. Bara ett fåtal forskningsarbeten rapporterar studier inom området. Avhandling bidrar till att skapa ny kunskap och utveckla användningen av SMSs för och flerstegsbearbetningsprocesser och kan sammanfattas som: i. härledning till variationerna i propageringsmodeller med dual quaternions, ii. variation förutsägelse med beaktande av fixturer med lokaliseringsytor, 3-2-1 och N-2-1 (N> 3) lokalisatorer, iii. Octree-baserad metod för att utföra statistisk formanalys. iv. detektering och analys av förändringar och anomalier med hjälp av maskininlärning klassificering, v. identifiering av variationskällor med hjälp av mönstermatchningsteknik, vi. uppskattning av variationskompensationsvärden med dual quaternions.

All manufacturing processes inevitably induce variations into manufactured parts that may result in nonconformance. Nonconforming parts incur costs due to the additional process required for rework or scrap loss. Hence, methodical efforts to reduce these variations are necessary for competitive manufacturing. To achieve this, effective variation reduction strategies have to be in place. In a multistage machining context, this could mean robust, rapid, and accurate approaches for representation and prediction of variations, change detection, variations source identification, and compensation. Moreover, the approaches used should be capable of handling all forms of errors contributing to the propagation of variations and nonconformance. Existing part variation and variation propagation analysis methods for multistage machining are limited to orientation and position errors, neglecting form errors. Form errors can be captured by utilizing the concept of Skin Models Shapes (SMSs). The application of SMSs for multistage machining and variation reduction strategies has been limited and not established yet. This thesis contributes to developing and demonstrating the use of SMSs for part quality prediction and variation reduction in multistage machining processes. The specific contribution of the thesis can be summarized as (i) the derivations of variation propagation models using dual quaternions; (ii) part quality prediction considering fixtures with locating surfaces, 3-2-1, and N-2-1 (N>3) locators; (iii) Octrees based method for performing statistical shape analysis; (iv) change and anomaly detection using machine learning classifiers; (v) variation source identification using pattern matching technique; (vi) and estimation of variation compensation values using dual quaternions., Alla tillverkningsprocesser orsaker oundvikligen variationer på den tillverkade detaljen. Detta medför kostnader i form av kassationer eller att artikeln måste ombearbetas. För att minska bearbetningsvariationerna, erfordras ett metodiskt arbetssätt för att kunna erhålla en konkurrenskraftig tillverkning. För att uppnå detta måste det finnas effektiva strategier så att variationerna kan reduceras till ett minimum. I en flerstegsbearbetningskontext måste robusta och exakta metoder finnas, så att detektering av avvikelser och variationer kan korrigeras och kompenseras. Befintliga analysmetoder som idag används för att behandla spridningsvariation av form och geometrifel i flerstegsbearbetningsoperationer, är begränsade till orienterings- och positionsfel, och behandlar inte formfel. Formfel kan behandlas med Skin Model Shapes (SMSs) konceptet. Tillämpning av SMSs för flerstegsbearbetning har hitintills varit begränsad, och metoder och strategier för reduktion av variationerna, är inte färdigutvecklat. Bara ett fåtal forskningsarbeten rapporterar studier inom området. Avhandling bidrar till att skapa ny kunskap och utveckla användningen av SMSs för och flerstegsbearbetningsprocesser och kan sammanfattas som: i. härledning till variationerna i propageringsmodeller med dual quaternions, ii. variation förutsägelse med beaktande av fixturer med lokaliseringsytor, 3-2-1 och N-2-1 (N> 3) lokalisatorer, iii. Octree-baserad metod för att utföra statistisk formanalys. iv. detektering och analys av förändringar och anomalier med hjälp av maskininlärning klassificering, v. identifiering av variationskällor med hjälp av mönstermatchningsteknik, vi. uppskattning av variationskompensationsvärden med dual quaternions.

With the onset of the age of space travel, asteroid missions have been steadily gaining interest. The pristine nature of asteroids due to their preserved state since the formation of the Solar System is an opportunity to unravel many mysteries about the Solar System. Also with the ever-growing need for resources, asteroids prove to be a plentiful source. With the discovery of asteroids in the close vicinity of our planet and a probable threat to the preservation of life, a need for defence missions has also arisen. The ever-growing need for better computational speed and accuracy has led to the development of new representations for attitude and position of the spacecraft in the past. The usual methods of representations of position and attitude (pose) are the Cartesian coordinates and quaternions. A recent development is the simultaneous representation of the pose of the spacecraft using dual quaternions which are eight-dimensional vectors. In this thesis, an attempt to incorporate these state-of-the-art guidance technologies to asteroid missions has been made. The novelty in this thesis is, using dual quaternions for SC pose and attitude representation to autonomously guide the spacecraft for mapping an asteroid and perform a touch and go descent using sampling-based motion predictive optimisation and successive convexification respectively. After the development and verification of these algorithms, different scenarios have been designed to find out the robustness of the dynamic successive convexification method. The scenarios have been run by both the Cartesian quaternion and dual quaternion based algorithms. They are found to behave similarly in their results, besides the latter being computationally more expensive as proved by different theses so far. The algorithm performs better than the one developed by Szmuk et al. (2017), but faces difficulties with badly scaled problems as is the nature of missions to asteroids. The software used for the thesis is ECOS, Aerospace Engineering

With the onset of the age of space travel, asteroid missions have been steadily gaining interest. The pristine nature of asteroids due to their preserved state since the formation of the Solar System is an opportunity to unravel many mysteries about the Solar System. Also with the ever-growing need for resources, asteroids prove to be a plentiful source. With the discovery of asteroids in the close vicinity of our planet and a probable threat to the preservation of life, a need for defence missions has also arisen. The ever-growing need for better computational speed and accuracy has led to the development of new representations for attitude and position of the spacecraft in the past. The usual methods of representations of position and attitude (pose) are the Cartesian coordinates and quaternions. A recent development is the simultaneous representation of the pose of the spacecraft using dual quaternions which are eight-dimensional vectors. In this thesis, an attempt to incorporate these state-of-the-art guidance technologies to asteroid missions has been made. The novelty in this thesis is, using dual quaternions for SC pose and attitude representation to autonomously guide the spacecraft for mapping an asteroid and perform a touch and go descent using sampling-based motion predictive optimisation and successive convexification respectively. After the development and verification of these algorithms, different scenarios have been designed to find out the robustness of the dynamic successive convexification method. The scenarios have been run by both the Cartesian quaternion and dual quaternion based algorithms. They are found to behave similarly in their results, besides the latter being computationally more expensive as proved by different theses so far. The algorithm performs better than the one developed by Szmuk et al. (2017), but faces difficulties with badly scaled problems as is the nature of missions to asteroids. The software used for the thesis is ECOS, Aerospace Engineering

Formulisanje odgovarajućih i efikasnih algoritama kinematike robota je od suštinskog značaja za analizu i razvoj serijskih manipulatora. Kinematičko modelovanje manipulatora se najčešće vrši u Dekartovom prostoru. Međutim, usled nedostataka najzastupljenijih matematičkih operatora za definisanje orijentacije kao što su Ojlerovi uglovi i rotacione matrice, nameće se potreba za jednoznačnim, kompaktnim, računski efikasnim metodom za određivanje orijentacije. Kao rešenje ovog problema predlažu se jedinični kvaternioni kao i razvoj kinematičkih modela u prostoru dualnih kvaterniona. U ovom radu je dat pregled geometrijskih opisa i transformacija koje se mogu primeniti u okviru navedenih prostora kako bi se rešili problemi kinematike robota. Poseban akcenat je na različitim matematičkim formalizmima koji se koriste za definisanje orijentacije krutog tela, kao što su rotacione matrice, Ojlerovi uglovi, osa i ugao rotacije, jedinični kvaternioni, kao i na njihovoj uzajamnoj vezi. Prednosti kinematičkog modeliranja u prostoru kvaterniona su istaknute. Osobine jediničnih i dualnih kvaterniona se analiziraju sa stanovišta robotike. Takođe, dat je novi algoritam direktne kinematike robota u prostoru dualnih kvaterniona. Ovaj algoritam je primenjen na humanoj centrifugi koja je modelirana kao troosni manipulator., Formulation of proper and efficient algorithms for robot kinematics is essential for the analysis and design of serial manipulators. Kinematic modeling of manipulators is most often performed in Cartesian space. However, due to disadvantages of most widely used mathematical constructs for description of orientation such as Euler angles and rotational matrices, a need for unambiguous, compact, singularity free, computationally efficient method for representing rotational information is imposed. As a solution, unit quaternions are proposed and kinematic modeling in dual quaternion space arose. In this paper, an overview of spatial descriptions and transformations that can be applied together within these spaces in order to solve kinematic problems is presented. Special emphasis is on a different mathematical formalisms used to represent attitude of a rigid body such as rotation matrix, Euler angles, axis-angle representation, unit quaternions, and their mutual relation. Benefits of kinematic modeling in quaternion space are presented. New direct kinematics algorithm in dual quaternion space pertaining to a particular manipulator is given. These constructs and algorithms are demonstrated on the human centrifuge as 3 DoF robot manipulator.

We consider a finite position synthesis and branching avoidance of four-bar linkages based on kinematic theory derived from dual quaternions. The branching defect is defined as a situation where all specified task positions are not connected by a continuous motion of the linkage end effector. The workspace of a four-bar linkage is derived to be the intersection of two hyperboloids in the kinematic image space. This is used to develop a fast method to determine branching of a linkage and to find an explicit solution to the end effector trajectory. A new synthesis method is developed where one task position is given a lower priority. Using the new branching analysis method and by synthesizing linkages with various values of one parameter of the linkage, we can determine a range of values for the constrained dimension which will give useful linkages for the four most prioritized task positions. Finally a method to find the closest useful linkage to the last task position is derived., Vi betraktar syntes och grendefekten av fyrlänks-mekanismer utifrån ett bergänsat antal task-positioner, baserat på kinematisk teori härledd från duala kvarternioner. Grendefekten är definierad som situationen där alla specificerade task-positionerna inte är sammankopplade av en kontinuerlig rörelse av mekanismens end-effector. Arbetsområ det av en fyrlänks-mekanism är härlett som snittet mellan två hyperboloider i det kinematiska bildrummet. Detta används för att utveckla en snabb metod för att avgöra om grendefekten föreligger för mekanismen, och för att härleda en explicit lösning för mekanismens end effector trajektoria. En ny syntes-metod är utvecklad där en av task-positionerna är given en lägre prioritet. Genom att använda den nya grendefektsanalys-metoden och beräkna länk-mekanismer med varierande värden av mekanismens parametrar, kan vi hitta ett intervall av värden på parametern som ger en användbar mekanism för de första fyra task-positionerna. Slutligen härleder vi en metod för att avgöra vilket värde på parametern som ger den länk-mekanism som når närmast den lägre prioriterade task-positionen.

We consider a finite position synthesis and branching avoidance of four-bar linkages based on kinematic theory derived from dual quaternions. The branching defect is defined as a situation where all specified task positions are not connected by a continuous motion of the linkage end effector. The workspace of a four-bar linkage is derived to be the intersection of two hyperboloids in the kinematic image space. This is used to develop a fast method to determine branching of a linkage and to find an explicit solution to the end effector trajectory. A new synthesis method is developed where one task position is given a lower priority. Using the new branching analysis method and by synthesizing linkages with various values of one parameter of the linkage, we can determine a range of values for the constrained dimension which will give useful linkages for the four most prioritized task positions. Finally a method to find the closest useful linkage to the last task position is derived., Vi betraktar syntes och grendefekten av fyrlänks-mekanismer utifrån ett bergänsat antal task-positioner, baserat på kinematisk teori härledd från duala kvarternioner. Grendefekten är definierad som situationen där alla specificerade task-positionerna inte är sammankopplade av en kontinuerlig rörelse av mekanismens end-effector. Arbetsområ det av en fyrlänks-mekanism är härlett som snittet mellan två hyperboloider i det kinematiska bildrummet. Detta används för att utveckla en snabb metod för att avgöra om grendefekten föreligger för mekanismen, och för att härleda en explicit lösning för mekanismens end effector trajektoria. En ny syntes-metod är utvecklad där en av task-positionerna är given en lägre prioritet. Genom att använda den nya grendefektsanalys-metoden och beräkna länk-mekanismer med varierande värden av mekanismens parametrar, kan vi hitta ett intervall av värden på parametern som ger en användbar mekanism för de första fyra task-positionerna. Slutligen härleder vi en metod för att avgöra vilket värde på parametern som ger den länk-mekanism som når närmast den lägre prioriterade task-positionen.

Dual quaternion is a more efficient tool without singularity to describe rigid-body transformation than other tools, such as homogeneous transformation matrix. In this paper, we propose a new variable structure control (VSC) based on unit dual quaternion Lie-group for the transformation of rigid-bodies in configuration space. We set up the dynamic model of the rigid-body motion represented by unit dual quaternion after the geometric structure of dual quaternion is revealed, and then propose a new unit-dual-quaternion-based VSC controller, including a new switching function and a new reaching law. At last, the proposed controller is specified into the configuration space of planar robot and applied with the dynamic model of the NuBot omni-directional mobile robot in simulation to show its effectiveness.

This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method. © 2008 Institute of Automation, Chinese Academy of Sciences and Springer-Verlag GmbH.

This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method. © 2008 Institute of Automation, Chinese Academy of Sciences and Springer-Verlag GmbH.

This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method. © 2008 Institute of Automation, Chinese Academy of Sciences and Springer-Verlag GmbH.

Das Ziel dieser Arbeit ist es, die Faktorisierbarkeit von Polynomen mit Koeffizienten in Quaternion-Algebren zu untersuchen. Wir setzen unseren Schwerpunkt dabei auf Split-Quaternion-Polynome. Der Hauptteil der Arbeit besteht aus einer Zusammenstellung von vier Artikeln, die bereits in Zeitschriften oder als Buchkapitel veröffentlicht wurden. Die präsentierten Publikationen bauen aufeinander auf und sind in chronologischer Reihenfolge angeordnet. Im ersten Artikel diskutieren wir die Existenz von Faktorisierungen für Polynome mit Koeffizienten im Schiefkörper der hamiltonschen Quaternionen, im Ringe der dualen Quaternionen und im Ring der Split-Quaternionen. Wir präsentieren den aktuellen Stand der Forschung, einschließlich der Beziehungen zwischen Faktorisierungstheorie und Kinematik. Anhand von Beispielen veranschaulichen wir die Herausforderungen in der Faktorisierungtheorie für Polynome mit Koeffizienten in Ringen mit Nullteilern. Der zweite Artikel konzentriert sich rein auf Split-Quaternion-Polynome und erweitert die sogenannte „Bond“-Theorie auf die hyperbolische Kinematik. Wir zeigen reelle und endliche Visualisierungen des Verhaltens von Mechanismen in „Bond“-Punkten. Dies ist im Rahmen der euklidischen Kinematik nicht möglich, da die entsprechenden Posen nur für imaginäre oder „unendliche“ Parameterwerte erreicht werden. Im dritten und vierten Artikel präsentieren wir eine strukturierte Charakterisierung von faktorisierbaren Split-Quaternion Polynomen. Der quadratische Fall kann in allen Einzelheiten behandelt werden, einschließlich Nullteilerpolynome. Zusätzlich zur Charakterisierung mittels Ungleichheitsbedingungen interpretieren wir unsere Ergebnisse geometrisch. Durch weitere geometrische Untersuchungen können wir auch schließlich Split-Quaternion-Polynome beliebigen Grades behandeln. Unsere Analyse führt zu einem Algorithmus, der nachweislich alle möglichen Faktorisierungen in lineare Faktoren findet. Die Resultate für Split-Quaternion-Polynome können, The goal of this thesis is to investigate factorizability of polynomials over quaternion algebras with a focus on split quaternion polynomials. The main content of the thesis is a compilation of four papers which are already published in journals or as chapters in books. The presented publications are arranged in chronological order and build up on each other. In the first publication, we discuss existence of factorizations for polynomials over the skew field of Hamiltonian quaternions, the ring of dual quaternions and the ring of split quaternions. We present the current state of the art including relations between factorization theory and mechanism science and illustrate the challenges of factorizing polynomials in rings with zero divisors by means of representative examples. The second publication solely deals with split quaternion polynomials and extends the theory of bonds to hyperbolic kinematics. We provide real and finite visualizations of the behavior of mechanisms at bond points. This is not possible in Euclidean kinematics since the according poses are only attained for imaginary or “infinite” parameter values. In the third and fourth publication we present a structured characterization of factorizable split quaternion polynomials. The quadratic case can be dealt with in all details, including polynomials being zero divisors. We provide inequality conditions as well as geometric interpretations for existence of factorizations. By further geometric investigations we can treat split quaternion polynomials of arbitrary degree as well. Our analysis results in an algorithm that provably finds all possible factorizations into linear factors. The developed techniques for split quaternion polynomials can be transferred back to the Euclidean theory. In doing so, we compute new factorizations of certain motion polynomials which could not be found by existing algorithms. In the appendix (a fifth publication), we investigate rational motions of exceptionally low degr, Dipl.-Ing. Daniel F. Scharler BSc, Dissertation Universität Innsbruck 2021