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Over the centuries, humanity has created ever more ingenious systems to traverse the oceans and skies of our planet. Modern ships and planes operate with powerful engines that require substantial amounts of fuel, leading to high operating costs. However, this approach becomes impractical for applications that require extended periods of autonomous operation without the possibility of refueling. This dissertation starts with the idea of operating systems by going with the flow: harnessing the wind and ocean currents by letting the system drift in favorable directions and strategically using a low-power engine to change flows when this is beneficial. As the power to counteract drag forces scales cubically with the relative velocity of the system, this new paradigm reduces the power required for operation by 2-3 orders of magnitude, thereby significantly reducing the system and operating costs. This could enable a host of novel applications that require low-cost and long-term operations, such as active environmental monitoring of the oceans and atmosphere or floating solar platforms. The primary case study used throughout this work is autonomous seaweed farms that roam the oceans while rapidly growing biomass for biofuel, bioplastic, or to sink it for carbon removal.In this dissertation, we systematically develop control techniques to tackle the four key challenges of operating by going with the flow: First, the system is severely underactuated with its own propulsion often being less than 1/10th of the magnitude of the surrounding flows. Second, to make strategic control decisions when to change flows, only coarse, deterministic forecasts are available. Third, the forecasts have a limited time horizon of 5-10 days, but realistic control objectives extend over weeks to months. Lastly, the forecast error defined as the difference between the forecasted and the true flows often exceeds the propulsion capabilities of the system, hence robust control is infeasible.We start by introducing techniques for continuous-time optimal control when the complex flows are known. We use dynamic programming for the objectives of navigation and maximizing seaweed growth. Next, we turn towards the challenge of operating with imperfect and short-term forecasts. Our insight is that the value functions obtained by the previously developed optimal control methods can be used as closed-loop control policies, which are equivalent to replanning on the forecast at every step. Through extensive simulation studies in realistic ocean conditions, we demonstrate that such frequent replanning allows for reliable operation despite significant forecast errors. To enable reasoning beyond the forecast horizon, we derive a discounted optimal control formulation and demonstrate how the value function can be extended by estimating the cost-to-go using historical flow averages. In the last part of this dissertation, we focus on how to handle constraints in these challenging environments. For that, we integrate time-varying obstacles into our value function and show empirically that this almost eliminates the risk of stranding. Moreover, we develop a hierarchical control approach to operate a fleet of underactuated autonomous systems while avoiding collisions and ensuring connectivity across the fleet.At the end of this dissertation, we summarize our techniques for operating by going with the flow which could enable a host of new applications of low-power autonomous systems in the oceans and skies. We also discuss promising ongoing and future research directions towards further improving the performance of underactuated robotic systems operating in flows.

An underactuated exoskeleton design for walking assistance is presented and evaluated. The exoskeleton uses one motor per leg and makes use of a pantograph to reduce the overall profile and allow the exoskeleton to closely follow the shape of the user's leg. Support is provided between the ball of the user's foot and their waist by compressing a spring in parallel with the user's leg during Stance Phase. The exoskeleton has a mass of 14.0 kg (30.8 lbs) and was tested up to a supplied spring force of 323.6 N (72.75 lbf) which equates to around 161.8 N (36.38 lbf) of assistive force at the waist. Range of motion tests showed minimal restriction at the knee and ankle, but some restriction of the hip. Human subject experiments using a simple gait detection method based on GRF at walking speeds from 0.45 m/s to 1.12 m/s (1.0 mph to 2.5 mph) were performed and showed an increase in the time between actual heel strike and predicted heel strike of approximately 0.05 seconds to 0.1 seconds. Lastly, calculations are presented examining the effect of exoskeleton assistance on the biological joint moments and optimizing the actuator design to reduce power consumption. The actual performance of the exoskeleton is compared with the calculations based on the joint angles during a typical walking cycle.

The variable length pendulum (VLP) is used as a platform to demonstrate that Lyapunov stabilizing control is achievable for underactuated systems in the presence of energy dissipation. The necessary background to describe the system is developed from elementary assumptions and Lagrangian mechanics. Phase portraits and limit cycles, whose discussion is essential to the development of the controller and to the presentation of results, are introduced. An existing Lyapunov control is adapted from the literature and extended to dissipative models and a discrete control. Nondimensionalization as described may be used for front-end engineering of model realizations. Results for control of both the VLP and the closely related skateboard model are then presented in a way that simplifies qualitative interpretations of the interactions of state variables.

The following study focuses on the design and validation of an underactuated robotic gripper built for the Tactical Hazardous Operations Robot (THOR). THOR is a humanoid robot designed for use in the DARPA Robotics Challenge (DRC) and the Shipboard Autonomous Fire Fighting Robot (SAFFiR) project, both of which pertain to completing tasks associated with disaster response. The gripper was designed to accomplish a list of specific tasks outlined by the DRC and SAFFiR project. Underactuation was utilized in the design of the gripper to keep its complexity low while acquiring the level of dexterity needed to complete the required tasks. The final gripper contains two actuators, two underactuated fingers and a fixed finger resulting in four total degrees of freedom (DOF). The gripper weighs 0.68 kg and is capable of producing up to 38 N and 62 N on its proximal and distal phalanges, respectively. The gripper was put through a series of tests to validate its performance pertaining to the specific list of tasks it was designed to complete. The results of these tests show the gripper is in fact capable of completing all the necessary actions but does so within some limitations.

This work develops feedback controllers for bipedal walking in 3D on level ground, both in simulation and experimentally. MARLO is a new robot that has been designed for the study of 3D-bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having six actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits in a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (cmt), a dimensionless measure of energetic efficiency. A feedback controller is designed that balances the robot during the quiet standing mode, and another feedback policy is developed such that the robot can take a transition step from quiet standing to walking. A feedback controller is designed that stabilizes steady-state 3D walking gaits, despite the high degree of underactuation of the robot. These controllers are combined through a state machine that handles switching among the three controllers controllers. In experiments on planarized (2D) and untethered (3D) versions of the robot with point feet and passive feet (prosthetic feet) walking over flat ground or on a ramp with a shallow slope, the adaptability of the designed controller to the environment (planar or untethered, flat ground or ramp), and to the morphology of the robot (point feet or passive feet), is demonstrated. In experiments on a planarized version of the robot with passive feet, the controller yielded stable walking after starting from quiet standing, autonomously and without any intervention from the operator. In experiments on an untethered (3D) version of the robot, the controller was able to balance the robot over flat ground or on a shallow ramp during the quiet standing mode. In addition, the controller yielded six-untethered ``human-like'' steps after starting from quiet standing, autonomously without any intervention from the operator.

In this work, a method for controlling a nonlinear underactuated system using augmented sliding mode control (SMC) is proposed. SMC requires inversion of the input influence matrix to derive the desired control law. In under or over actuated systems this matrix is nonsquare therefore a true inverse does not exist. The proposed control approach demonstrated in this work involves introducing a transformation matrix mapping the systems input influence matrix to a transformed system that is square and thus invertible. The proposed approach is shown to control selectable states with proper choice of the transformation matrix yielding good control performance. The methodology is applied to an underactuated nonlinear fuel cell system to show its viability in a real world application. A sliding mode controller is derived for the full nonlinear system with a switching gain accounting for modeling errors and uncertainties. Simulation results indicate the viability of the proposed control law and demonstrate the robust nature of the control law in the presence of significant modeling errors while maintaining tracking stability. Finally, the augmented SMC is compared to a traditional linear control architecture illustrating the effectiveness and advantages in tracking performance and control effort over traditional methods.

In this thesis, a novel method for control of non-square dynamical systems using a model following approach is developed. Control methodologies such as dynamic inversion and sliding mode control require an inversion of the input influence matrix. However, if the system input influence matrix is non-square direct inversion is not possible. Pseudo inversion of the input influence matrix may be performed for control allocation. However, pseudo inversion limits the control to states where the controller is directly applied. The pseudoinverse method does not permit the engineer to designate a particular state to control or track. When accurate tracking of states that are not directly controlled (“remaining states”) is required the pseudo inversion method is not useful. Current methods such as dynamic extension can be used to generate a square input influence matrix, essentially, creating an input influence matrix that is invertible. However, this method is tedious for large systems. In this work, a new transformation is applied to the original dynamical system model to develop an input influence matrix that is square. Assuming the system is controllable, the proposed transformation allows for accurate tracking of selectable states. Selection of the new transformation matrix is used to develop accurate tracking of certain states compared to the remaining states. A method based on optimal control theory is used to define the transformation matrix. The new approach is first applied to control a two mass system with simulation results presented showing the advantage of the proposed new control strategy. Finally, simulation results are presented for longitudinal control of an aircraft using one control input.

This dissertation studies the dynamics and control of multibody systems, and their numerical simulation, in a central gravitational field. Initially, the dynamics of multibody systems moving in a plane in a central gravity is studied. There is a cyclic orbital coordinate, and the corresponding conjugate momentum is conserved. The dynamics is reduced to eliminate this cyclic variable. A general development for analyzing stability of relative equilibria of the full dynamics, corresponding to equilibria of the reduced dynamics, is obtained. This is applied to some examples of multibody spacecraft in planar motion. A control scheme, based on averaging theory, is developed for orbit transfer from one relative equilibrium to another. This scheme is applied to a planar dumbbell-shaped rigid body in central gravity. The dynamics of such systems in three-dimensional motion also has a cyclic coordinate, and the associated conjugate momentum is conserved. The dynamics is reduced to eliminate this degree of freedom. Stability analysis of the relative equilibria of a rigid dumbbell-shaped body is carried out. Potential shaping with attitude feedback is used to stabilize the unstable relative equilibria of the dumbbell body. For numerical simulations of the dynamics of free and controlled multibody systems in central gravity, numerical integration algorithms obtained from discrete variational mechanics are used. These algorithms exactly preserve the symplectic form and conserved momenta of such systems. They also nearly preserve the total energy of conservative systems over long simulation times. These properties are usually not present in other numerical integration algorithms. Variational integration algorithms for the full and reduced dynamics of multibody systems in a potential field are obtained, and applied to the specific examples treated in this dissertation. Comparison with a standard Runge-Kutta fourth order integrator, for an example problem, shows the better performance of the variational integrators in long time simulations. These numerical simulations also confirm stability results obtained analytically. The controlled motion of the planar dumbbell model with averaging feedback control is simulated with a variational integrator to confirm analytically obtained results for this control scheme.

In this dissertation, we develop equations of motion for a class of multibody spacecraft consisting of a rigid base body and multiple rigid appendages connected to the base body. There has been much prior research on this topic; however, much of this research is not appropriate for nonlinear control design purposes. The motion of a multibody spacecraft is described by the position and attitude of a base body in an inertial frame and by the relative position and attitude of the attached bodies with respect to the base body; these latter quantities define the shape of the multibody spacecraft. Our aim is to develop equations of motion that reveal important nonlinear coupling effects between the translation, rotation and shape dynamics, but are simple enough for control design purposes. A rotation matrix is used to represent the attitude of the spacecraft. This allows us to avoid complexity related to the use of parameter representations such as Euler angles. Hamilton's variational principle gives three sets of nonlinear equations of motion. The latter part of this dissertation presents results of control problems for several underactuated multibody spacecraft examples. These include spacecraft with an unactuated internal sliding mass, spacecraft with unactuated fuel slosh dynamics, tethered spacecraft with attachment point actuation and the triaxial attitude control testbed with two proof mass actuation devices. These examples illustrate important features related to the dynamics and control of various underactuated multibody spacecraft. Differences in geometries of the spacecraft and gravitational assumptions require adoption of different types of control schemes. We use the multibody equations in this dissertation to formulate control equations for the models and to construct feedback controllers that achieves asymptotic stability (or convergence) to the desired (relative) equilibrium manifolds. Computer simulations demonstrate the effectiveness of the controllers.