The control of nonholonomic wheeled mobile robots (WMRs) has gained a lot of attention in the field of robotics over the past few decades as WMRs provide an increased range of motion resulting in a larger workspace. This research focuses on the application of Model Predictive Control (MPC) for real-time trajectory tracking of a nonholonomic WMR. MPC is a control strategy in which the control law is designed based on optimizing a cost function. The input and output constraints that may arise in practical situations can be directly incorporated into the control system using MPC. Computation time is the biggest hurdle in adapting MPC strategies for trajectory tracking. This research applies a non-feasible active set MPC algorithm developed in [1] which is faster than the traditional active set methods (ASMs). A discrete-time linear model of a general WMR is used for the simulation. MATLAB simulations are performed for tracking circular as well as square trajectories using the discretized WMR model and the non-feasible ASM (NF-ASM). The performance of NF-ASM is compared to two other well-known traditional algorithms, i.e. Fletcher’s ASM and MATLAB’s Quadratic Programming algorithm. It is shown that, although all these algorithms are capable of providing satisfactory trajectory tracking performance, NF-ASM is a better choice in terms of the simulation time and required number of iterations for realtime trajectory tracking of any type as long as the constraints on the inputs stay active for a long period during the simulation. / UOIT

A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order.

A virtual holonomic constraint (VHC) for an Euler-Lagrange Control System is a smooth relation between the configuration variables that can be made invariant through application of suitable feedback. In this thesis we investigate the role played by VHCs in the synchronization of Euler-Lagrange systems. We focus on two problems. For $N$ underactuated cart-pendulums, we design a smooth feedback that fully synchronizes the cart-pendulums while simultaneously stabilizing a periodic orbit corresponding to a desired oscillation for the pendulums. A by-product of our results is the ability to simultaneously synchronize the pendulums and stabilize the unstable upright equilibrium. The second synchronization problem investigated in this thesis is bilateral teleoperation, whereby a master robot is operated by a human while a slave robot synchronizes to the master. For two identical planar manipulators, we develop a methodology to achieve teleoperation in the presence of a hard surface, with simultaneous force control.

A virtual holonomic constraint (VHC) for an Euler-Lagrange Control System is a smooth relation between the configuration variables that can be made invariant through application of suitable feedback. In this thesis we investigate the role played by VHCs in the synchronization of Euler-Lagrange systems. We focus on two problems. For $N$ underactuated cart-pendulums, we design a smooth feedback that fully synchronizes the cart-pendulums while simultaneously stabilizing a periodic orbit corresponding to a desired oscillation for the pendulums. A by-product of our results is the ability to simultaneously synchronize the pendulums and stabilize the unstable upright equilibrium. The second synchronization problem investigated in this thesis is bilateral teleoperation, whereby a master robot is operated by a human while a slave robot synchronizes to the master. For two identical planar manipulators, we develop a methodology to achieve teleoperation in the presence of a hard surface, with simultaneous force control.

A Lagrange--d'Alembert integrator is a geometric numerical method for finding numerical solutions to the Lagrange--d'Alembert equations for mechanical systems with nonholonomic constraints that are linear in the velocities. The integrator is developed from geometry and principles that are analogues of the continuous theory.<p>Using discrete analogues of the symplectic form and momentum map, the resulting methods are symplectic and momentum preserving whenever the continuous system is symplectic and momentum preserving. In addition, it is possible to, in principle, generate Lagrange--d'Alembert integrators of any method order.

Geometric and holonomic quantum gates perform transformations that only dependon the geometry of a loop covered by the parameters controlling the gate. Thesegates require adiabatic time evolution, which is achieved in the limit when the looptakes infinite time to complete. However, it is of interest to also know thetransformation properties of the gates for finite run times. It has been shown [Phys.Rev. A 73, 022327 (2006)] that some holonomic gates for a trapped ion system showrevival structures, i.e., for some finite run time the gate performs the sametransformation as it does in the adiabatic limit. The purpose of this thesis is to investigate if similar revival structures are shown alsofor geometric and holonomic quantum gates for spin systems. To study geometricquantum gates an NMR setup for spin-1/2 particles is used, while an NQR setup forspin-3/2 particles is used to study holonomic quantum gates. Furthermore, for thegeometric quantum gates the impact of some open system effects are examined byusing the quantum jump approach. The non-adiabatic time evolution operators of thesystems are calculated and compared to the corresponding adiabatic time evolutionoperators by computing their operator fidelity. The operator fidelity ranges between0 and 1, where 1 means that the gates are identical up to an unimportant phasefactor. All gates show an oscillating dependency on the run time, and some Abeliangates even show true revivals, i.e., the operator fidelity reaches 1.

Στόχος αυτής της διπλωματικής εργασίας είναι η κατασκευή και ο έλεγχος μιας ομάδας αυτόνομων και συνεργαζόμενων κινούμενων ρομπότ τα οποία θα είναι σε θέση να δημιουργήσουν ένα σύστημα επικοινωνίας με μεταβλητή εμβέλεια, κάνοντας χρήση των ίδιων των ρομπότ ως αναμεταδότες πληροφορίας. Τα δύο ρομπότ εκκινούν ως ένα και στη συνέχεια αποδεσμεύονται και ακολουθούν μια τροχιά σε σχηματισμό Leader-follower. Για την επίτευξη του στόχου αυτού κατασκευάστηκαν 2 όμοια μη ολονομικά οχήματα (ref μανεσης) με δύο κινητήριους και έναν παθητικό περιστρεφόμενο τροχούς (εικόνα) και υλοποιήθηκε αλγόριθμος μη γραμμικού ελέγχου των κινούμενων οχημάτων. Για την ασύρματη επικοινωνία μεταξύ των κόμβων του ηλεκτρονικού υπολογιστή και των 2 ρομπότ κατασκευάστηκε ένα ολοκληρωμένο κύκλωμα μετάδοσης της πληροφορίας και προγραμματίστηκε για χρήση με το πρωτόκολλο ZigBee Pro. Στο κεφάλαιο 1 παρατίθεται ένα εισαγωγικό κείμενο πάνω στις τεχνολογίες που χρησιμοποιήθηκαν για την παρούσα διπλωματική εργασία μαζί με μια περιορισμένη βιβλιογραφική ανασκόπηση. Στο κεφάλαιο 2 δίνεται η περιγραφή της πλατφόρμας Lego Mindstorms NXT που χρησιμοποιήθηκε για την κατασκευή των κινούμενων ρομπότ. Επίσης περιγράφεται η κατασκευή των ρομπότ, η διαδικασία απόσπασης των δύο μελών της ομάδας και δίνεται το κινοδυναμικό μοντέλο τους. Στο κεφάλαιο 3 γίνεται αναφορά στην ασύρματη δικτυακή επέκταση της πλατφόρμας Lego Mindstorms NXT. Αναφέρεται ο σχεδιασμός του PCB στο οποίο ενσωματώνεται η πλατφόρμα ασύρματης επικοινωνίας (Jennic 5148), η δικτύωση του PCB για την επικοινωνία του με το Lego μέσω πρωτοκόλλου I2C καθώς και πλεονεκτήματα και μειονεκτήματα της πλατφόρμας ασύρματης επικοινωνίας. Στο κεφάλαιο 4 παρουσιάζεται ο έλεγχος των συνεργαζόμενων δικτυωμένων ρομποτικών οχημάτων και δίνονται η περιγραφή του σεναρίου που υλοποιήθηκε, οι νόμοι ελέγχου και τα πειραματικά αποτελέσματα. Στο κεφάλαιο 5 βρίσκονται τα συμπεράσματα από το πείραμα μαζί με προτάσεις για μελλοντικές βελτιώσεις της διάταξης και χρήσης της ως εκπαιδευτικό μέσο για την εξοικείωση των φοιτητών με multi-agent προβλήματα ρομποτικής. Το κεφάλαιο 6 είναι η βιβλιογραφία, ενώ τα παραρτήματα Α και Β περιέχουν τον πηγαίο κώδικα σε C και LabView και τα schematics των PCBs. / Creation of non holonomic moving robots using Lego Mindstorms NXT, control and cooperation of two of those roobots and development of a wireless communication platform for the establishment of robotic networks.

The following dissertation investigates a problem related to the practice of quantum tomography, where one usually estimates the parameters associated with quantum states or measurements. In particular, the question answered is whether and how one could detect if states and measurements are correlated. A similar question answered is how one could detect state-preparation non-localities and measurement non-localities in multiqudit systems. The solution involves an analysis of certain matrix quantities called \emph{partial determinants}. Partial determinants are an application of the Born rule that can be interpreted as tomography over a loop in the space of state and measurement settings. From this perspective, the notion of state and observable become \emph{non-holonomic} | that is, state and observable parameters can be defined ``locally'' over each setting but not globally over all settings. As such, state and measurement parameters are not estimated because such estimated values don't exist in correlated systems, but rather the inability to estimate such values is quantified. Therefore, partial determinants are a measure of the amount of contradiction that would result from any claim of such a estimated values by propagating these estimates through a `tomography loop' of data collected by various experiments. Such measures of contradiction are generally known as \emph{holonomies}.

In this dissertation we introduce a new class of pursuit-evasion games; the herding problem. Unlike regular pursuit evasion games where the pursuer aims to hunt the evader the objective of the pursuer in this game is to drive the evader to a certain location on the x-y grid. The dissertation deals with this problem using two different methodologies. In the first, the problem is introduced in the continuous-time, continuous-space domain. The continuous time model of the problem is proposed, analyzed and we came up with an optimal control law for the pursuer is obtained so that the evader is driven to the desired destination position in the x-y grid following the local shortest path in the Euler Lagrange sense. Then, a non-holonomic realization of the two agents is proposed. In this and we show that the optimal control policy is in the form of a feedback control law that enables the pursuer to achieve the same objective using the shortest path. The second methodology deals with the discrete model representation of the problem. In this formulation, the system is represented by a finite di-graph. In this di-graph, each state of the system is represented by a node in the graph. Applying dynamic programming technique and shortest path algorithms over the finite graph representing the system, we come up with the optimal control policy that the pursuer should follow to achieve the desired goal. To study the robustness, we formulate the problem in a stochastic setting also. We analyze the stochastic model and derive an optimal control law in this setting. Finally, the case with active evader is considered, the optimal control law for this case is obtained through the application of dynamic programming technique. / Ph. D.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 &lt; &alpha; &lt; 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.