The work presented in this thesis describes the development of a novel strategy for the steady state tip position control of a single link flexible robot arm. Control is based upon a master/slave relationship. Arm trajectory is defined by through 'master' positioning head which moves a laser through a programmed path. Tip position is detected by an optical system which produces an error signal proportional to the displacement of the tip from the demand laser spot position. The error signal and its derivative form inputs to the arm 'slave' controller so enabling direct tip control with simultaneous correction for arm bending. Trajectory definition is not model-based as it is defined optically through movement of the positioning head alone. A critical investigation of vacuum tube and solid state sensing methods is undertaken leading to the development of a photodiode quadrant detector beam tracking system. The effect of varying the incident light parameters on the beam tracker performance are examined from which the optimum illumination characteristics are determined. Operational testing of the system on a dual-axis prototype robot using the purpose-built beam tracker has shown that successful steady state tip control can be achieved through a PD based slave controller. Errors of less than 0.05 mm and settling times of 0.2 s are obtained. These results compare favourably with those for the model-based tip position correction strategies where tracking errors of ± 0.6 mm are recorded.

Flexible link manipulators (FLM) are well-known for their light mass and small energy consumption compared to rigid link manipulators (RLM). These advantages of FLM are even of greater importance in applications where energy efficiency is crucial, such as in space applications. However, RLM are still preferred over FLM for industrial applications. This is due to the fact that the reliability and predictability of the performance of FLM are not yet as good as those of RLM. The major cause for these drawbacks is link flexibility, which not only makes the dynamic modeling of FLM very challenging, but also turns its end-effector trajectory tracking (EETT) into a complicated control problem. <p>The major objectives of the research undertaken in this project were to develop a dynamic model for a FLM and model-based controllers for the EETT. Therefore, the dynamic model of FLM was first derived. This dynamic model was then used to develop the EETT controllers. <p>A dynamic model of a FLM was derived by means of a novel method using the dynamic model of a single flexible link manipulator on a moving base (SFLMB). The computational efficiency of this method is among its novelties. To obtain the dynamic model, the Lagrange method was adopted. Derivation of the kinetic energy and the calculation of the corresponding derivatives, which are required in the Lagrange method, are complex for the FLM. The new method introduced in this thesis alleviated these complexities by calculating the kinetic energy and the required derivatives only for a SFLMB, which were much simpler than those of the FLM. To verify the derived dynamic model the simulation results for a two-link manipulator, with both links being flexible, were compared with those of full nonlinear finite element analysis. These comparisons showed sound agreement. <p>A new controller for EETT of FLM, which used the singularly perturbed form of the dynamic model and the integral manifold concept, was developed. By using the integral manifold concept the links lateral deflections were approximately represented in terms of the rotations of the links and input torques. Therefore the end-effector displacement, which was composed of the rotations of the links and links lateral deflections, was expressed in terms of the rotations of the links and input torques. The input torques were then selected to reduce the EETT error. The originalities of this controller, which was based on the singularly perturbed form of the dynamic model of FLM, are: (1) it is easy and computationally efficient to implement, and (2) it does not require the time derivative of links lateral deflections, which are impractical to measure. The ease and computational efficiency of the new controller were due to the use of the several properties of the dynamic model of the FLM. This controller was first employed for the EETT of a single flexible link manipulator (SFLM) with a linear model. The novel controller was then extended for the EETT of a class of flexible link manipulators, which were composed of a chain of rigid links with only a flexible end-link (CRFE). Finally it was used for the EETT of a FLM with all links being flexible. The simulation results showed the effectiveness of the new controller. These simulations were conducted on a SFLM, a CRFE (with the first link being rigid and second link being flexible) and finally a two-link manipulator, with both links being flexible. Moreover, the feasibility of the new controller proposed in this thesis was verified by experimental studies carried out using the equipment available in the newly established Robotic Laboratory at the University of Saskatchewan. The experimental verifications were performed on a SFLM and a two-link manipulator, with first link being rigid and second link being flexible.<p>Another new controller was also introduced in this thesis for the EETT of single flexible link manipulators with the linear dynamic model. This controller combined the feedforward torque, which was required to move the end-effector along the desired path, with a feedback controller. The novelty of this EETT controller was in developing a new method for the derivation of the feedforward torque. The feedforward torque was obtained by redefining the desired end-effector trajectory. For the end-effector trajectory redefinition, the summation of the stable exponential functions was used. Simulation studies showed the effectiveness of this new controller. Its feasibility was also proven by experimental verification carried out in the Robotic Laboratory at the University of Saskatchewan.

Flexible link manipulators (FLM) are well-known for their light mass and small energy consumption compared to rigid link manipulators (RLM). These advantages of FLM are even of greater importance in applications where energy efficiency is crucial, such as in space applications. However, RLM are still preferred over FLM for industrial applications. This is due to the fact that the reliability and predictability of the performance of FLM are not yet as good as those of RLM. The major cause for these drawbacks is link flexibility, which not only makes the dynamic modeling of FLM very challenging, but also turns its end-effector trajectory tracking (EETT) into a complicated control problem. <p>The major objectives of the research undertaken in this project were to develop a dynamic model for a FLM and model-based controllers for the EETT. Therefore, the dynamic model of FLM was first derived. This dynamic model was then used to develop the EETT controllers. <p>A dynamic model of a FLM was derived by means of a novel method using the dynamic model of a single flexible link manipulator on a moving base (SFLMB). The computational efficiency of this method is among its novelties. To obtain the dynamic model, the Lagrange method was adopted. Derivation of the kinetic energy and the calculation of the corresponding derivatives, which are required in the Lagrange method, are complex for the FLM. The new method introduced in this thesis alleviated these complexities by calculating the kinetic energy and the required derivatives only for a SFLMB, which were much simpler than those of the FLM. To verify the derived dynamic model the simulation results for a two-link manipulator, with both links being flexible, were compared with those of full nonlinear finite element analysis. These comparisons showed sound agreement. <p>A new controller for EETT of FLM, which used the singularly perturbed form of the dynamic model and the integral manifold concept, was developed. By using the integral manifold concept the links lateral deflections were approximately represented in terms of the rotations of the links and input torques. Therefore the end-effector displacement, which was composed of the rotations of the links and links lateral deflections, was expressed in terms of the rotations of the links and input torques. The input torques were then selected to reduce the EETT error. The originalities of this controller, which was based on the singularly perturbed form of the dynamic model of FLM, are: (1) it is easy and computationally efficient to implement, and (2) it does not require the time derivative of links lateral deflections, which are impractical to measure. The ease and computational efficiency of the new controller were due to the use of the several properties of the dynamic model of the FLM. This controller was first employed for the EETT of a single flexible link manipulator (SFLM) with a linear model. The novel controller was then extended for the EETT of a class of flexible link manipulators, which were composed of a chain of rigid links with only a flexible end-link (CRFE). Finally it was used for the EETT of a FLM with all links being flexible. The simulation results showed the effectiveness of the new controller. These simulations were conducted on a SFLM, a CRFE (with the first link being rigid and second link being flexible) and finally a two-link manipulator, with both links being flexible. Moreover, the feasibility of the new controller proposed in this thesis was verified by experimental studies carried out using the equipment available in the newly established Robotic Laboratory at the University of Saskatchewan. The experimental verifications were performed on a SFLM and a two-link manipulator, with first link being rigid and second link being flexible.<p>Another new controller was also introduced in this thesis for the EETT of single flexible link manipulators with the linear dynamic model. This controller combined the feedforward torque, which was required to move the end-effector along the desired path, with a feedback controller. The novelty of this EETT controller was in developing a new method for the derivation of the feedforward torque. The feedforward torque was obtained by redefining the desired end-effector trajectory. For the end-effector trajectory redefinition, the summation of the stable exponential functions was used. Simulation studies showed the effectiveness of this new controller. Its feasibility was also proven by experimental verification carried out in the Robotic Laboratory at the University of Saskatchewan.

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[pt] Recentemente, grande interesse tem se voltado na robótica para o conceito de manipuladores flexíveis. Estes dispositivos apresentam uma coluna vertebral deformável continuamente, em oposição aos tradicionais manipuladores robóticos elo/junta/elo com elos rígidos. Sistemas flexíveis oferecem um aumento em potencial da capacidade de interação com o ambiente, estando aptos a se ajustarem às limitações do meio através de sua deformação. Robôs flexíveis oferecem possibilidades atrativas para o uso em diversas aplicações, como em posicionamento em ambientes complexos com obstáculos, endoscópios ativos, e manuseamento de materiais frágeis. O uso de polímeros, em particular elastômeros, tem sido explorado nestes manipuladores para promover simplicidade de operação e menor rigidez, necessária para uma interação homemmáquina com maior segurança. Usufruindo-se destes conceitos, esta dissertação aborda o desenvolvimento de um manipulador pneumático flexível de longo alcance. O manipulador é composto por uma estrutura modular, formada por vários elos ligados serialmente, permitindo que em sua extremidade sejam acopladas ferramentas que auxiliem a execução de diferentes tarefas. O sistema é baseado em um atuador pneumático denominado Músculo Artificial Pneumático (Pneumatic Artificial Muscles, PAM). Ao serem pressurizados, estes dispositivos se contraem, exercendo uma força em sua extremidade proporcional à pressão aplicada. A movimentação do manipulador desenvolvido é obtida a partir da diferença de pressão entre câmaras independentes localizadas em seu interior. Modelos analíticos dos sistemas desenvolvidos foram elaborados. O controle do manipulador é feito a partir de servoválvulas pneumáticas controladas por computador. Experimentos foram realizados para verificar os modelos desenvolvidos. O sistema desenvolvido pode ser aplicado à tarefa de inspeção interna de reservatórios de combustíveis. Inspeções internas atualmente requerem um completo esvaziamento do reservatório, se tornando muito trabalhosas e resultando em altos custos. Uma versão do manipulador é adaptada para executar esta tarefa sem a necessidade de esvaziar os tanques, devido à segurança intrínseca do sistema pneumático. / [en] Recently, the concept of flexible manipulators has attract great interest. These devices present a continuously deforming vertebral column, in opposition to the traditional robotic manipulators link/joint/link with rigid links. Flexible systems offer a potential increase in the capacity of interaction with the environment, being apt to adjust itself to the constrants through its deformation. Flexible robots offer attractive possibilities for usage in many applications, as complex environments positioning with active obstacles, endoscopies, and manipulating fragile materials. Polymers, specially elastomers, have been explored in these manipulators to guarantee simple operation and minor rigidity, necessary for a higher security man-machine interaction. Making a good use of these conceptions, this dissertation presents the development of a long-reach flexible pneumatic manipulator. The manipulator is composed of a modular structure, formed by links attached serially, allowing tools to be connected in its end-point for assistence in different tasks. The system is based on a pneumatic actuator called Pneumatic Artificial Muscle (PAM). When pressurized, these devices contract themselves, exerting a proportional force in its end-points proportional to the applied pressure. The manipulator´s motion is obtained from the pressure difference between the independent chambers located in its interior. Developed systems analytical models have been elaborated. Pneumatic valves, commanded by computer, control the manipulator. Experiments have been carried through to test the developed models. The developed system can be applied to internal inspection of fuel tanks. Internal inspections currently require a complete tank ullage, becoming very laborious and resulting in high costs. A manipulator´s version is adapted to execute this task in a full fueled tank, due to intrinsic security of the pneumatic system.

This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

De nos jours, les exigences en productivité dans le monde industriel imposent aux robots un comportement optimal en termes de précision géométrique et dynamique, et en termes de temps de réponse. Ainsi, la présence des flexibilités dans les liaisons pivots des structures mécaniques légères se déplaçant à grande vitesse et sous charges importantes peut limiter dynamiquement la précision et le temps de stabilisation sur la pose finale du robot. La problématique traitée dans ces travaux concerne la maîtrise des vibrations des robots sériels à liaisons flexibles durant les opérations de prise et dépose (Pick and Place).Dans ces travaux, nous effectuons une modélisation et une identification expérimentale des paramètres géométriques et dynamique d’un robot à liaisons flexible. Ce modèle sera utilisé dans la synthèse d’une loi de commande basée modèle dédiée aux robots à flexibilité articulaire. Cette stratégie permet de réduire les vibrations lors des phases exigeantes dynamiquement. Des simulations sur un robot Scara sont alors conduites pour valider la pertinence de cette loi de commande qui intègre un modèle des flexibilités présentes dans les liaisons pivots dans le schéma de commande. Nous appliquons sur le même simulateur du robot à liaisons flexibles trois autres stratégies de commande afin de faire une comparaison (commande PD, commande dédiée aux robots rigides et commande ne considérant pas les amortissements). Le schéma de la loi de commande basée modèle permet de respecter la précision de pose finale avec une diminution du temps de stabilisation. Finalement, Le calcul de l’erreur d’asservissement nous a permis de constater l’influence des erreurs de modélisation de la flexibilité sur la précision de la tâche. Dans ce contexte, une analyse de sensibilité aux paramètres influents est établie. / Nowadays, the demand of productivity in the industrial world of robotics require robots to behave optimally in terms of geometric and dynamic accuracy and response time. Thus, the presence of flexibilities in rotational joints can dynamically limit the position control of manipulators having lighter arms, higher payload-to-weight ratio and doing tasks at high speed. The problem addressed in this work concerns the vibration control of serial robots with flexible joints performing Pick and Place tasks. In this work, we carry out modelling and experimental identification of the geometric and dynamic parameters of a robot with flexible joints. This model is then used in the synthesis of a model-based control law dedicated to manipulators with flexible joints. This strategy reduces vibrations resulting from joints sensitivity during dynamically demanding phases. Simulations on a Scara robot are then conducted to validate the relevance of the proposed control law which integrates joint flexibilities in the form of a feedback loop in the control diagram. To this end, three other control strategies (PD control, control dedicated to rigid structures and control not considering damping) are applied to the same simulator in order to make a comparative analysis. The diagram of the model-based control law allows to respect the set point with a reduction in the stabilization time.Finally, the calculation of the servo error allowed us to see the influence of flexibility modeling errors on the accuracy of the task. In this context, the sensitivity of this control law is evaluated through a sensitivity analysis.

Ο σκοπός αυτής της διπλωματικής είναι η ανάπτυξη ενός προσαρμοστικού ελεγκτή για έναν εύκαμπτο ρομποτικό βραχίονα. Οι μετρήσεις του συστήματος θεωρούνται πως παρεμβάλλονται από θόρυβο, του οποίου τα όρια είναι γνωστά εξ’αρχής. Ένας Set Memebership εκτιμητής υπολογίζει το δυνατό set (ορθότοπο) μέσα στο οποίο βρίσκονται οι τιμές του διανύσματος των παραμέτρων. Από τις ακμές του ορθοτόπου αυτού προκύπτουν τα όρια μέσα στα οποία βρίσκονται οι παράμετροι του συστήματος, τα οποία χρησιμοποιούνται για τον υπολογισμό της αβεβαιότητας της εκτίμησης της εξόδου του συστήματος. Ο ελεγκτής καθορίζει τα κέρδη του μέσα σε μια online βελτιστοποίηση ενός κόστους, το οποίο βάζει κάποια βάρη στην προσπάθεια του ελέγχου (control effort), στην προκλημένη αβεβαιότητα στην έξοδο του συστήματος αλλά και στο σφάλμα παρακολούθησης της εξόδου με ένα σήμα αναφοράς. Μετά την εφαρμογή του ελεγκτή, ελέγχεται η ευστάθεια των οριακών κλειστών συστημάτων που προκύπτουν από την εφαρμογή κάθε πιθανού νόμου ελέγχου. Εξετάζεται επίσης η συμπεριφορά του Set Memebership εκτιμητή σε περίπτωση σφάλματος, δηλαδή στην περίπτωση που το σύστημά μας αλλάζει καθώς δουλεύει ο έλεγχος. / The development of an adaptive controller for a flexible link manipulator is the subject of this diploma thesis. The system’s measurements are assumed to be corrupted with noise of a priori known bounds. A Set Membership Identifier computes the feasible set (orthotope) within which the parameter vector resides. The orthotope’s vertices provide the parameter-vector’s bounds, which are used to compute the predicted system-output uncertainty. The controller tunes its gains through an on-line minimization of a cost that penalizes the control effort, the induced uncertainty on the system output, and the tracking error. After the application of the controller, the stability of the ‘extreme’ closed loop systems, derived from every possible control law, is checked. The behavior of the Set Membership Identifier is checked in the case where a fault occurs, which means that there is a change in our system’s structure while the controller is functioning.

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