Solutions to trajectory optimization problems are carried out by the direct and indirect methods. Under broad heading of these methods, numerous algorithms such as collocation, direct, indirect and multiple shooting methods have been developed and reported in the literature. Each of these algorithms has certain advantages and limitations. For example, direct shooting technique is not suitable when the number of nonlinear programming variables is large. Indirect shooting method requires analytical derivatives of the control and co-states function and a poorly guessed initial condition can result in numerical unstable values of the adjoint variable. Multiple shooting techniques can alleviate some of these difficulties by breaking down the trajectory into several segments that help in reducing the non-linearity effects of early control on later parts of the trajectory. However, multiple shooting methods then have to handle more number of variables and constraints to satisfy the defects at the segment joints. The sie of the nonlinear programming problem in the collocation method is also large and proper locations of grid points are necessary to satisfy all the path constraints. Stochastic methods such as Genetic algorithms, on the other hand, also require large number of function evaluations before convergence. To overcome some of the limitations of the conventional methods, improved solution techniques are developed. Three improved methods are proposed for the solution of trajectory optimization problems. They are • a genetic algorithm employing dominance and diploidy concept. • a collocation method using chebyshev polynomials , and • a hybrid method that combines collocation and direct shooting technique A conventional binary-coded genetic algorithm uses a haploid chromosome, where a single string contains all the variable information in the coded from. A diploid, as the name suggests, uses pair of chromosomes to store the same characteristic feature. The diploid genetic algorithm uses a dominant map for decoding genotype into a stable, consistent phenotype. In dominance, one allele takes precedence over another. Diploidy and dominance helps in retaining the previous best solution discovered and shields them from harmful selection in a changing environment. Hence, diploid and dominance affect a king of long-term memory in the genetic algorithm. They allow alternate solutions to co-exist. One solution is expressed and the other is held in abeyance. In the improved diploid genetic algorithm, dominant and recessive genes are defined based on the fitness evaluation of each string. The genotype of fittest string is declared as the dominant map. The dominant map is dynamic in nature as it is replaced with a better individual in future generations. The concept of diploidy and dominance in the improved method mimics closer to the principles used in human genetics as compared to any such algorithms reported in the literature. It is observed that the improved diploid genetic algorithm is able to locate the optima for a given trajectory optimization problem with 10% lower computational time as compared to the haploid genetic algorithm. A parameter optimization problem arising from an optimal control problem where states and control are approximated by piecewise Chebyshev polynomials is well known. These polynomials are more accurate than the interpolating segments involving equal spaced data. In the collocation method involving Chebyshev polynomials, derivatives of two neighboring polynomials are matched with the dynamics at the nodal points. This leads to a large number of equality constraints in the optimization problem. In the improved method, derivative of the polynomial is also matched with the dynamics at the center of segments. Though is appears the problem size is merely increased, the additional computations improve the accuracy of the polynomial for a larger segment. The implicit integration step size is enhanced and overall size of the problem is brought down to one-fourth of the problem size defined with a conventional collocation method using Chebyshev polynomials. Hybrid method uses both collocation and direct shooting techniques. Advantages of both the methods are combined to give more synergy. Collocation method is used in the starting phase of the hybrid method. The disadvantage of standalone collocation method is that tuning of grid points is required to satisfy the path constraints. Nevertheless, collocation method does give a good guess required for the terminal phase of the hybrid method, which uses a direct shooting approach. Results show nearly 30% reduction in computation time for the hybrid approach as compared to a method in which direct shooting alone is used, for the same initial guess of control. The solutions obtained from the three improved methods are compared with an indirect method. The indirect method requires derivations of the control and adjoint equations, which are difficult and problem specific. Due to sensitivity of the costate variables, it is often difficult to find a solution through the indirect method. Nevertheless, these methods do provide an accurate result, which defines a benchmark for comparing the solutions obtained through the improved methods. Trajectory design and optimization of a RLV(Reusable Launch Vehicle) Demonstrator mission is considered as a test problem for evaluating the performance of the improved methods. The optimization problem is difficult than a conventional launch vehicle trajectory optimization problem because of the following two reasons. • aerodynamic lift forces in the RLV add one more dimension to the already complex launch vehicle optimization problem. • as RLV performs a sub orbital flight, the ascent phase trajectory influences the re-entry trajectory. Both the ascent and re-entry optimization problem of the RLV mission is addressed. It is observed that the hybrid method gives accurate results with least computational effort, as compared with other improved techniques for the trajectory optimization problem of RLV during its ascent flight. Hybrid method is then successfully used during the re-entry phase and in designing the feasible optimal trajectories under the dispersion conditions. Analytical solutions obtained from literature are used to compare the optimized trajectory during the re-entry phase. Trajectory optimization studies are also carried out for the off-nominal performances. Being a thrusting phase, the ascent trajectory is subjected to significant deviations, mainly arising out of solid booster performance dispersions. The performance index during rhe ascent phase is modified in a novel way for handling dispersions. It minimizes the state errors in a least square sense, defined at the burnout conditions ensure possibilities of safe re-entry trajectories. The optimal trajectories under dispersion conditions serve as a benchmark for validating the closed-loop guidance algorithm that is developed for the ascent phase flight. Finally, an on-line trajectory command-reshaping algorithm is developed which meets the flight objectives under the dispersion conditions. The guidance algorithm uses a pre-computed trajectory database along with some real-time measured parameters in generating the optimal steering profiles. The flight objectives are met under the dispersion conditions and the guidance generated steering profiles matches closely with the optimal trajectories.

A spacecraft tour trajectory is a trajectory in which a spacecraft visits a number of objects in sequence. The target objects may consist of satellites, moons, planets or any other body in orbit, and the spacecraft may visit these in a variety of ways, for example flying by or rendezvousing with them. The key characteristic is the target object sequence which can be represented as a discrete set of decisions that must be made along the trajectory. When this sequence is free to be chosen, the result is a hybrid discrete-continuous optimization problem that combines the challenges of discrete and combinatorial optimization with continuous optimization. The problem can be viewed as a generalization of the traveling salesman problem; such problems are NP-hard and their computational complexity grows exponentially with the problem size. The focus of this dissertation is the development of a novel methodology for the solution of spacecraft tour trajectory optimization problems. A general model for spacecraft tour trajectories is first developed which defines the parameterization and decision variables for use in the rest of the work. A global search methodology based on the tabu search metaheuristic is then developed. The tabu search approach is extended to operate on a tree-based solution representation and neighborhood structure, which is shown to be especially efficient for problems with expensive solution evaluations. Concepts of tabu search including recency-based tabu memory and strategic intensification and diversification are then applied to ensure a diverse exploration of the search space. The result is an automated, adaptive and efficient search algorithm for spacecraft tour trajectory optimization problems. The algorithm is deterministic, and results in a diverse population of feasible solutions upon termination. A novel numerical search space pruning approach is then developed, based on computing upper bounds to the reachable domain of the spacecraft, to accelerate the search. Finally, the overall methodology is applied to the fourth annual Global Trajectory Optimization Competition (GTOC4), resulting in previously unknown solutions to the problem, including one exceeding the best known in the literature. / text

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Approved for public release; distribution is unlimited / This thesis considers the problem of generating optimal entry trajectories for a reusable launch vehicle following a control surface failure. The thesis builds upon the work of Dr. David Doman, Dr. Michael Oppenheimer and Dr. Michael Bolender of the Air Vehicles Directorate, Air Force Research Lab Dayton Ohio. The primary focus of this work is to demonstrate the feasibility of inner loop reconfiguration and outer loop trajectory retargeting and replanning for the X-33 reusable launch vehicle (RLV) following the imposition of a control surface failure. The trajectory generation model employs path constraints generated by an AFRL trim deficiency algorithm coupled with an inner loop control allocator and aerodynamic database that captures the full 6-DOF vehicle aerodynamic effects while utilizing an outer loop 3-DOF model. The resulting optimal trajectory does not violate the trim deficiency constraints and provides additional margins for trajectories flown during failure conditions. The footprints generated by the thesis show that contemporary footprint analysis for vehicles experiencing control surface failures are overly optimistic when compared to those footprints that consider vehicle aerodynamic stability and realistic landable attitudes at the threshold of the landing runway. The results of the thesis also show the performance reductions resulting from decoupling the inner and outer loop and that trajectories can be generated to the landing runway without using a region of terminal area energy management. / Commander, United States Navy

Modern GIS applications for transportation and defense often require the ability to store the evolving positions of a large number of objects as they are observed in motion, and to support queries on this spatiotemporal data in real time. Because the M-Tree has been proven as an index for spatial network databases, we have selected it to be enhanced as a spatiotemporal index. We present modifications to the tree which allow trajectory reconstruction with fast insert performance and modifications which allow the tree to be built with awareness of the spatial locality of reference in spatiotemporal data.

The use of autonomous UAV’s is a progressively expanding industry. This thesisfocuses on the landing procedure with the main goal to be independent of visualaids. That means that the landing site can be hidden from the air, the landingcan be done in bad weather conditions and in the dark. In this thesis the use ofradio signals is investigated as an alternative to the visual sensor based systems.A localization system is needed to perform the landing without knowing wherethe landing site is. In this thesis an Extended Kalman Filter (EKF) is derived andused for the localization, based on the received signal strength from a radio beaconat the landing site. There are two main goals that are included in the landing,to land as accurate and as fast as possible. To combine these two goals a simultaneoustrajectory optimization and target estimation problem is set up that can bepartially solved while flying. The optimal solution to this problem produces thepath that the UAV will travel to get the best target localization while still reachingthe target. It is shown that trying to move directly towards the estimated landingsite is not the best strategy. Instead, the optimal trajectory is a spiral that jointlyoptimizes the information from the sensors and minimizes the arrival time.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015. / I would like to acknowledge a number of people who contributed towards the completion of this thesis. Firstly, I thank my supervisor Professor Montaz Ali for his patience, enthusiasm, guidance and teachings. The skills I have acquired during this process have infiltrated every aspect of my life. I remain forever grateful. Secondly, I would like to say a special thank you to Professor Jan Snyman for his assistance, which contributed immensely towards this thesis. I would also like to thank Professor Dominque Orban for his willingness to assist me for countless hours with the installation of CUTEr, as well as Professor Jose Mario Martinez for his email correspondence. A heartfelt thanks goes out to my family and friends at large, for their prayers, support and faith in me when I had little faith in myself. Thank you also to my colleagues who kept me sane and motivated, as well as all the support staff who played a pivotal roll in this process. Above all, I would like to thank God, without whom none of this would have been possible.

Significant research has been conducted on the physics of ball and bat collisions in an effort to model and understand real-world conditions. This thesis expands upon previous research to determine the maximum distance a ball can travel under ideal circumstances. Bat mass, bat speed, pitch speed and pitch spin were controlled values. These values were selected based on the highest recorded MLB values for their respective category. Specifically these are: Babe Ruth’s largest bat, Giancarlo Stanton’s recorded swing speed and Aroldis Chapman’s fastest fastball. A model was developed for a planar collision between a bat and ball using conservation laws in order to achieve the maximum exit velocity of the ball during a head-on collision. However, this thesis is focused on home runs and long fly-balls that occur from oblique collisions rather than the line drives produced by head-on collisions. The planar collision model results were adjusted to oblique collisions based on data from previous experimental research. The ball and bat were assumed to be moving in opposite directions parallel to one another at the point of impact with the ball slightly elevated above the bat. The post-collision results for the launch angle, spin and final exit velocity of the ball were calculated as functions of the perpendicular distance from the centerline of the bat to the centerline of the ball. Trajectories of the ball were calculated using a flight model that measured the final distance of the ball based on lift and drag forces. The results indicate that the optimum pre-collision parameters described above will maximize the distance traveled by the ball well beyond the farthest recorded home run distance. Experimentally determined factors such as the drag coefficient and coefficient of restitution have a significant impact on the flight of the ball. Implications of the results are discussed.

In most robot applications, the control of the manipulators end-effector along a specified desired trajectory is the main concern. In these applications, the end-effector (tip) of the manipulator is required to follow a given trajectory. Several methods have been so far proposed for the motion control of robot manipulators. However, most of these control methods ignore either joint friction or joint elasticity which can be caused by the transmission systems (e.g. belts and gearboxes). This study aims at development of a comprehensive control strategy for the tip-trajectory tracking of flexible-joint robot manipulators. While the proposed control strategy takes into account the effect of the friction and the elasticity in the joints, it also provides a highly accurate motion for the manipulators end-effector. During this study several approaches have been developed, implemented and verified experimentally/numerically for the tip trajectory tracking of robot manipulators. To compensate for the elasticity of the joints two methods have been proposed; they are a composite controller whose design is based on the singular perturbation theory and integral manifold concept, and a swarm controller which is a novel biologically-inspired controller and its concept is inspired by the movement of real biological systems such as flocks of birds and schools of fishes. To compensate for the friction in the joints two new approaches have been also introduced. They are a composite compensation strategy which consists of the non-linear dynamic LuGre model and a Proportional-Derivative (PD) compensator, and a novel friction compensation method whose design is based on the Work-Energy principle. Each of these proposed controllers has some advantages and drawbacks, and hence, depending on the application of the robot manipulator, they can be employed. For instance, the Work-Energy method has a simpler form than the LuGre-PD compensator and can be easily implemented in industrial applications, yet it provides less accuracy in friction compensation. In addition to design and develop new controllers for flexible-joint manipulators, another contribution of this work lays in the experimental verification of the proposed control strategies. For this purpose, experimental setups of a two-rigid-link flexible-joint and a single-rigid-link flexible-joint manipulators have been employed. The proposed controllers have been experimentally tested for different trajectories, velocities and several flexibilities of the joints. This ensures that the controllers are able to perform effectively at different trajectories and speeds. Besides developing control strategies for the flexible-joint manipulators, dynamic modeling and vibration suppression of flexible-link manipulators are other parts of this study. To derive dynamic equations for the flexible-link flexible-joint manipulators, the Lagrange method is used. The simulation results from Lagrange method are then confirmed by the finite element analysis (FEA) for different trajectories. To suppress the vibration of flexible manipulators during the manoeuvre, a collocated sensor-actuator is utilized, and a proportional control method is employed to adjust the voltage applied to the piezoelectric actuator. Based on the controllability of the states and using FEA, the optimum location of the piezoelectric along the manipulator is found. The effect of the controllers gain and the delay between the input and output of the controller are also analyzed through a stability analysis.

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.