This thesis presents both the control design used to stabilize a small rover on two wheels as a mobile inverted pendulum, as well as the steering design used to achieve a series of unique stable driving modes. Additionally, the rover is assessed as an educational platform for use in teaching various STEM topics at both the high school and undergraduate university levels. The vehicle, termed BeagleRover, uses four DC motors and four servo motors to independently steer each wheel and achieve a total of six different driving modes, four of which are inherently stable four-wheel drive modes while two are unstable two-wheel drives modes. Two of the four stable drive modes implement Ackermann steering geometry to reduce side slip when driving around a turn. Experimental results using measurements from an onboard gyroscope suggest that side slip is indeed reduced by this method. Stability in the two unstable drive modes is achieved through classical control methods including lead and lag control. Complementary filtering of gyroscope and accelerometer measurements is used to derive accurate body position data for use in feedback. In support of current STEM education trends, this thesis provides a detailed solution set to the steering, control and filtering problems for potential use in STEM course material.

This work presents a new trajectory planning formulation that aims to maximize the probability of finding a target inside a bound domain using a robot over a specified time interval. A preliminary algorithm is developed to detect stationary targets, which is further extended to detect moving targets. Values are assigned at every grid point in the domain based on its distance from the robot; each value represents the probability of not finding the target if it is at that location. A cost function is formulated that computes the likelihood of not finding the target for any given path provided the target is inside the domain.

This cost function is minimized with a set of bound constraints on inputs to obtain an optimal path to find the target. The algorithm incorporates an adjoint-based gradient method to link the input parameters to the cost function. The cost function is nonlinear which makes it hard for most commercial off-the-shelf (COTS) optimization packages to solve it.

For faster convergence the recently developed low storage reduced Hessian box constraint optimization method (LRH-B) was used.

Results show our algorithm outperforms other optimization algorithms, and also explain how this framework is beneficial in terms of application.

The accurate space-time discretization of the partial differential equations (PDEs) governing the dynamic behavior of complex physical phenomena is a core challenge in the field of Computational Fluid Dynamics, and in the simulation of turbulence in particular. However, it appears that over the last 30 years a disproportionate amount of attention has been addressed toward the improvement of spatial discretization techniques, while temporal discretization has relied, in most cases, on old consolidated approaches. Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) of the incompressible Navier-Stokes Equation (NSE) today often use a mixed implicit/explicit (IMEX) time integration approach developed in the mid 1980s, which combines the second-order implicit Crank-Nicolson (CN) method for the integration of the linear stiff terms and a third-order explicit low-storage Runge-Kutta-Wray (RKW3) method for the nonlinear terms. This hybrid approach, dubbed CN/RKW3, guarantees overall second-order accuracy for the time integration, while allowing an efficient storage implementation.

Our work focuses on the development of new mixed implicit/explicit time integration schemes of the Runge-Kutta type for the simulation of high-dimensional stiff ODEs, with particular attention to the simulation of the NSE. Compared with the venerable CN/RKW3 method, our numerical schemes have better accuracy, improved stability properties, and require the same or slightly increased storage.

We have also developed new relaxation schemes for the iterative solution of linear and, with some modification, nonlinear systems arising from the discretization of PDEs. These schemes prove especially advantageous when applied as the smoothing step in the multigrid solution of elliptic PDEs over stretched grids. A noteworthy application is the iterative solution of the pressure Poisson equation arising when imposing the diverge-free constraint during the simulation of the incompressible NSE using a fractional step method. Compared with the standard approach, our schemes require significantly less computation, while providing comparable converge rates.

We then discuss the implementation of an Ensemble Kalman Filter (EnKF) for the short-term prediction of ocean waves. The approach leverages one of our low-storage IMEX Runge-Kutta schemes for the highly-resolved simulation of the nonlinear equations used to describe wave propagation. We found that, using EnKF for data assimilation of wave measurement data, it is possible to perform accurate wave forecasting up to thirty seconds into the future, provided a sufficient number of ensemble members is employed.

Finally, we introduce a new direct multiple shooting algorithm for Nonlinear Model Predictive Control (NMPC). The new approach allows analytic calculation of the discretized trajectories and associated gradients, which are required when solving the nonlinear programming problem arising within the NMPC formulation. For the discretization of the trajectories, two solutions are proposed: one based on a Runge-Kutta discretization of the continuous-time model, and one leveraging a nonlinear discrete-time model based on Taylor-Lie algebra. This algorithm is then applied to the optimization of the power take-off of a point-absorber wave energy converter (WEC). Results have shown that NMPC improves the WEC power take-off with respect to linear MPC, since the nonlinear viscous forces affecting WEC dynamics are better accounted for. Moreover, the nonlinear formulation also allows the investigation of more complex configurations, such as one-way power flow.

A new class of derivative-free optimization algorithms is developed to solve, with remarkable efficiency, a range of practical nonconvex optimization problems whose function evaluations are computationally (or experimentally) expensive. These new algorithms, which are provably convergent under the appropriate assumptions, are response surface methods which iteratively minimize metrics based on an interpolation of existing datapoints and a synthetic model of the uncertainty of this interpolant, which itself is built on the framework of a Delaunay triangulation over the existing datapoints. Unlike other response surface methods, our algorithms can employ any well-behaved interpolation strategy.

An important subproblem in $\alpha$-DOGS is the estimation of the averaging process in stationary ergodic processes. A new approach for determining this quantity is also presented which is mathematically rigorous and numerically accurate.

There are six main algorithms developed in this class thus far (only four of them are presented in this thesis) which address a wide range of practical optimization problems. The first algorithm, dubbed $\Delta$-DOGS, efficiently minimizes expensive objective functions inside linearly-constrained feasible domains. The second algorithm, $\Delta$-DOGS(C), extends $\Delta$-DOGS to handle efficiently more general convex search domains. The third algorithm incorporates a grid into $\Delta$-DOGS to achieve better convergence by performing fewer function evaluations at the boundary of feasibility. The fourth algorithm, dubbed $\alpha$-DOGS, efficiently minimizes objective functions which are noisy, generally derived by taking the statistics of stationary and ergodic random processes. An important subproblem in $\alpha$-DOGS is the estimation of the averaging process in stationary ergodic processes. A new approach for determining this quantity is also presented which is mathematically rigorous and numerically accurate.

Rigorous convergence analyses of all of the algorithms proposed are presented, and necessary conditions to guarantee convergence to the global minimum are provided. For validation, the algorithms proposed have been tested on both well-known benchmark optimization problems as well as some new application-oriented optimization problems in ship design; some illustrative results will be shown.

A ball-balancing robot (BBR) is a robot that balances itself on top of a ball, typically using three omni-directional wheels. This class of robot features a highly coupled 3D nonlinear dynamics that is capable of natural and holonomic motion. This dissertation presents a new mechanical design, a tractable dynamic model that is accurate at high yaw rates, and an effective estimation and control strategy for a micro ball-balancing robot (MBBR). The miniaturization and the low-cost components used in this design add significant control challenges, which manifest in the form of reduced durability and high amounts of noise, friction, and vibration, making the design of effective state estimation and control strategies for it very difficult, especially under high yaw rates. This motivates the design and use of a reduced nonlinear model which captures the important high yaw-rate dynamics well, and the design of an effective model-based observer and controller based on this reduced nonlinear model. The primary contributions of this dissertation in the general area of ball-balancing robotics include: (1) the novel (and, now, patented) midlatitude and orthogonal-omniwheel orientation of the design, (2) a reduced (minimum complexity) nonlinear BBR dynamic model which well captures its high yaw-rate behavior, and (3) the development of an effective model-based estimator (Extended Kalman Filter) and controller which are capable of achieving remarkable performance of this delicate system under high yaw rates.

The novel omniwheel placement minimizes coupling and increases the normal force acting on the omniwheels, which helps to reduce the slipping caused by its very light body. Another contribution of this dissertation is: (4) the modeling and real-time implementation of drive/coast motor drivers, which is also used in the MBBR. Drive/coast motor drivers exhibit highly nonlinear behavior, which makes using them in a model-based controller difficult, but they allow for zero torque dynamics which can be quite useful for many wheeled robots. The drive/coast motor model has been implemented in our linear feedback controller, and has achieved remarkably good position tracking even under high yaw-rates. The performance of the observer and controller were verified with a motion capture system.

Slip rings are often used to distribute power to rotating elements, but have the disadvantages of adding friction to the system and wear over time. An alternative to this method is power by inductance through coils. Power can be transferred over an air gap by changing the magnetic flux in a coil thereby inducing voltage. In this project we will transfer power from a primary to a secondary coil to power a custom made PCB which synchronizes a row of LEDs. As the PCB rotates, it creates a perception of 2D images. The main purpose of this prototype is to create a low cost LED display platform by rotating a motor at 40Hz+ and powering it through inductive coupling.

The mission of an unmanned air vehicle (UAV) tethered to a small unmanned surface vehicle (USV) is considered. The tether doubles as a power umbilical and communications link, providing unlimited flight duration and secure data transfer while limiting mobility. Contrary to the majority of existing tethered UAV work which assumes a taut tether for dynamic stability, this dissertation addresses the challenge of tether management for a slack, hanging tether in a dynamic ocean environment up to sea state 4 on the Douglas scale. For controlled laboratory experimentation, a novel wave and boat motion replication mechanism is developed capable of replicating motion up to 2.2 m heave, 32◦ roll, and 35◦ pitch. A reference hanging tether model maximizes heave robustness, providing a target tether length, departure angle, and tension. An effective estimation and control strategy is presented and validated indoor through motion capture experimentation and outdoor using a differential global positioning system solution. Finally,the foundation toward a simulation model of the complete tethered UAV - USV team is developed. The dynamic partial differential equations of motion are derived by treating the tether as a continuous body using Hamilton’s principle of least action. A simulation model is then developed, discretizing the elastic tether with linear and quadratic shape functions. Finally, the simulation results are experimentally validated.

This paper explores 3D path planning for unmanned aerial vehicles (UAVs) in 3D point cloud environments. Derivative maps such as dense point clouds, mesh maps, octomaps, etc. are frequently used for path planning purposes. A target-oriented 3D path planning algorithm, directly using point clouds to compute optimized trajectories for an UAV, is presented in this article. This approach searches for obstacle-free, low computational cost, smooth, and dynamically feasible paths by analyzing a point cloud of the target environment, using a modified connect RRT-based path planning algorithm, with a k-d tree based obstacle avoidance strategy and three-step optimization. This presented approach bypasses the common 3D map discretization, directly leveraging point cloud data. Following trajectory generation, the algorithm creates way-point based, closed loop quadrotor controls for pitch, roll, and yaw attitude angle as well as dynamics commands for the UAV. Simulations of UAV 3D path planning based on different target points in the point cloud map are presented, showing the effectiveness and feasibility of this approach.

Alongside derivative-based methods, which scale better to higher-dimensional problems, derivative-free methods play an essential role in the optimization of many practical engineering systems, especially those in which function evaluations are determined by statistical averaging, and those for which the function of interest is nonconvex in the adjustable parameters.

This work focuses on the development of a new family of surrogate-based derivative-free optimization schemes, namely $\Delta$-DOGS schemes. The idea unifying this efficient and (under the appropriate assumptions) provably-globally-convergent family of schemes is the minimization of a search function which linearly combines a computationally inexpensive ''surrogate`` (that is, an interpolation, or in some cases a regression, of recent function evaluations - we generally favor some variant of polyharmonic splines for this purpose), to summarize the trends evident in the data available thus far, with a synthetic piecewise-quadratic ''uncertainty function`` (built on the framework of a Delaunay triangulation of existing datapoints), to characterize the reliability of the surrogate by quantifying the distance of any given point in parameter space to the nearest function evaluations.

This thesis introduces a handful of new schemes in the $\Delta$-DOGS family:

(a) $\Delta$-DOGS($\Omega$) designs for nonconvex (even, disconnected) feasible domains defined by computable constraint functions within a bound search domain.

(b) $\Delta$-DOGS($\Omega_Z$) accelerates the convergence of $\Delta$-DOGS($\Omega$) by restricting function evaluation coordinations at each iteration to lie on a Cartesian grid in a bounded search domain. The grid is successively refined as convergence is approached.

(c) gradient-based acceleration of $\Delta$-DOGS combines derivative-free global exploration with derivative-based local refinement.

(d) $\Delta$-DOGS($\Lambda$) modifies the idea of restricting function evaluations at each iteration to lie on a dense lattice (derived from an n-dimensional sphere packing) instead of a Cartesian grid. Moreover, it handles the linear constraint domain.

(e) $\alpha$-DOGSX designs to simultaneously increase the sampling time, and refine the numerical approximation, as convergence is approached.

This work also introduces a method to scale the parameter domain under consideration based on adaptive variation of the seen data in the optimization process, thereby obtaining a significantly smoother surrogate. This method is called the Multivariate Adaptive Polyharmonic Splines (MAPS) surrogate model.

Practical applications of these algorithms are explored that include

(A) the design of low-storage implicit/explicit Runge-Kutta (IMEXRK) schemes for high performance computing (HPC) problems such as the numerical simulation of turbulence flows, and (B) the design of airfoils and hydrofoils.

These algorithms have been compared with existing state-of-the-art algorithms, particularly the Surrogate Management Framework (SMF) using the Kriging model and Mesh Adaptive Direct Search (MADS), on both standard synthetic and computer-aided shape designs. We showed that in most cases, the new $\Delta$-DOGS algorithms outperform the existing ones.