ADVANCEMENTS IN BIOMIMETIC UNDERWATER PROPULSION: BOUNDARY INTERACTIONS AND CROSSFLOW COMPENSATION

Biomimetic propulsion is an active area of research in the field of robotics. This work presents advancements to help improve the performance of vehicles employing oscillating foil propulsion. For vehicles operating near the sea floor, ground effect on rolling and pitching foils was investigated in Manuscript 1. The ‘ground effect’ was found to be present and to cause a repulsion force on the foil as well as an increased thrust force. It was also shown that ground effect can be activated by biasing the foil into the ground effect zone while the vehicle maintains a greater distance from the boundary. The effects on the foil due to ‘ground effect’ were compared to previous work done with heaving and pitching foils showing that ‘ground effect’ is an inherently 3 dimensional phenomena and studies using 2 dimensional constrained flow cannot be extrapolated into rolling and pitching foils. The second manuscript expands on this work, investigating the effect that free surface has on the oscillating foil. Similar to ‘ground effect’ it was found that the presence of the boundary creates a repulsive force pushing the foil away from the boundary. However, instead of increased thrust, the free surface causes a decrease in thrust forces on the foil. The third manuscript uses an analytical model to design a control system that uses force feedback from the foil in order to compensate and estimate a constant disturbance such as crossflow. The control system was able to compensate for a constant crossflow and was able to estimate the angle of the crossflow based on the force feedback to within 0.5 degrees. These advancements will help improve vehicles with oscillating foil propulsion that will open up new opportunities for investigation and exploration of complex oceanic environments.

ACKNOWLEDGMENTS I am extremely grateful for all of the support I have gotten over these many years. I would like to personally thank Stephen Licht for taking me on as a graduate student so many years ago. Over the past 5 years I have learned a lot from him and it was because of him that I've had this great opportunity to obtain my doctorate.  propeller and lifting surfaces for control. While effective in calm waters, they do not perform well in areas of more dynamic flow, nor in near bottom or complex environments. Access to these areas is essential for the furthering study of nearshore ocean dynamics, removal of unexploded ordinance and countless other applications. This has led to an increase in biomimetics and bio-inspired engineering in order to develop novel ideas for new generations of UUVs.

Motivation
Complex underwater environments are difficult to navigate through, hindering efforts of robotics to operate in these areas. Turtle-like robots emulate the locomotion of turtles with oscillating(flapping) foils inspired by the fins and motions of a sea turtle. Oscillating foils provide vehicles with advantages such as increased low-speed maneuverability and agility.
Vehicles equipped with this type of propulsion have a wide range of applications, both scientific and militaristic. I believe that vehicles with biomimetic propulsion are the next technological step in autonomous vehicles. There are many different styles of flapping foils, however, most vehicles use rolling and pitching coupled motions since they are easier to implement in an autonomous vehicle. For this reason my research aims to improve the understanding of this type of system. Vehicles with greater low-speed maneuverability will allow autonomous systems to investigate more complex environments. My research aims to shed light upon the performance of these oscillating foil systems in the presence of different types of boundaries and also in the presence of a crossflow. Understanding how these environments affect the forces on the foil will allow us to improve the performance of vehicles operating in near-surface and near-bottom conditions. Crossflows are also a common occurrence due to ocean currents, however, it could also be a crossflow caused by the vehicle traveling upwards or downwards while simultaneously moving forward. Furthermore, advanced spinning maneuvers would also create more complex flow about the fin. vii

Manuscript Contributions
This dissertation is made up of three separate but related manuscripts. Manuscript 1 studies an oscillating foil operating near a solid boundary such as the sea floor where there is expected to be a 'ground effect zone.' This is a zone close to the boundary that affects the lift and thrust on a wing or foil. The effect has been studied extensively in aerodynamics as well as with other types of oscillating foil systems. Manuscript 1 describes the appearance and magnitude of this effect as well as comparing the effect to that seen on other system configurations. Manuscript 2 extends the work of Manuscript 1 to the effect that free-surface such as the ocean surface has on the oscillating foil. The effect is detailed in this work as well as being compared to results from the work on 'ground effect.' Finally, Manuscript 3 employs a new analytic model to design a control system that effectively compensates for and estimates the angle of a crossflow using force feedback from the foil. All of this work will help improve the performance of these biomimetic vehicles, providing better platforms to investigate and explore the oceans.

Publication Status
Manuscript 1 has already been published in Bioinspriation and Biomimetics.
Manuscript 2 will be submitted for review in the future. Manuscript 3 may be submitted for review in the future.
viii MANUSCRIPT 1 Rolling and pitching oscillating foil propulsion in ground effect.
In this paper, we investigate the effect of operating near a solid boundary on the forces produced by harmonically oscillating thrust generating foils. A rolling and pitching foil was towed in a freshwater tank in a series of experiments with varying kinematics. Hydrodynamic forces and torques were measured in the freestream and at varying distances from a solid boundary, and changes in mean lift and thrust were found when the foil approached the boundary. The magnitude of this ground effect exhibited a strong nonlinear dependence on the distance between the foil and the boundary. Significant effects were found within three chord lengths of the boundary, and ground effect can be induced at greater distances from the boundary by biasing the tip of the foil toward the boundary. Lift coefficients changed by as much as 0.2 at the closest approach to the ground, with changes >0.05 for all cases across Strouhal number ranging from 0.3 to 0.6, and nominal maximum angle of attack ranging from 20 • to 40 • . The ubiquity of the ground effect in high thrust kinematics suggests that the ground effect can provide a passive obstacle avoidance capability for foil propelled vehicles. By comparison to previous experimental work, we find that the ground effect experienced by a high-aspect ratio rolling and pitching foil is a fully three-dimensional phenomenon, as it is not accurately predicted when two-dimensional flow and/or two-dimensional kinematics are enforced. While two dimensional foil kinematics are more easily modeled for numerical studies, three-dimensional foil kinematics may be more practical for real world implementation in underwater vehicles.

Introduction and Background
The complexity of operating efficiently and effectively underwater has driven researchers and engineers to search for biological examples to improve propulsion mechanisms in underwater vehicles. In this paper, we focus on the use of harmonically oscillating foils in the presence of obstacles. For discussion, we can categorize oscillating foils by the number of degrees of freedom in the driving motion of the foil. Three propulsive configurations for rigid foils have been extensively studied: one-dimensional (pitching or heaving only), two-dimensional (combined pitching and heaving motions), and three-dimensional (coupled rolling and pitching motions). The effect of operating near a solid bottom boundary, i.e.
in 'ground effect', has been investigated in the first two foil configurations. This paper is a first experimental study of three dimensional high aspect ratio thrust producing foils in ground effect. can produce thrust through an alternating vortex wake structure. The wake structure depends on Strouhal number and maximum angle of attack. The dynamics of both modes of operation in the absence of obstacles have since been extensively studied.
Numerical studies with heaving only foils in the presence of a solid boundary [3] predict that a foil can experience either net suction or repulsion force, depending on the distance between the foil and the boundary. [4] found that self-propelled rigid foils in the presence of a wall will yield increased cruising speed with a reduced efficiency if the kinematics are unchanged. Flexible foils however will experience both increased speed and efficiency.
Numerical simulations by [5] and [6] have also found increased thrust by propulsors near a wall. Flow tank experiments with pitching only foils find that rectangular foils undergoing pitch oscillations generate higher thrust when near a boundary [7] and that a suction region exists very near the wall with a repulsion region at moderate distances from the wall [8].
Using a very flexible self propelled foil actuated in pitch [9] showed that the wall affects the jet structure of the wake, enhancing the thrust and altering the lift force when compared to freestream operation.

2-D Combined Heaving and Pitching Foils
Combining heave and pitch motions while enforcing two-dimensional flow has allowed investigators to more closely approximate real animal mechanisms while limiting system complexity, particularly in measurement and analysis of fluid flow. A wide variety of kinematic and structural parameters have been investigated experimentally with heaving and pitching foils. [10] performed experimental studies of the forces as well as flow visualization of the wake. The test apparatus included restrictor plates attached to the sides of the foil to reduce spanwise flow around the tips, thus minimizing three dimensional effects. Wake structure behind the foil was parameterized based on maximum angle of attack and Strouhal number. [11] found that there was a plateau of 50%-60% propulsive efficiency in a range of Strouhal numbers from 0.05 to 0.6 with maximum angle ranging from 10 • to 50 • and that by adding a bias to the mean foil pitch angle, effective force vectoring is possible. [12] found that variations in chordwise flexibility can be used to increase propulsive efficiency.
In experiments where flow around the foil tip was allowed, [13] and [14], found that adding in-line motion with a power/feathering stroke style provides substantial improvements on force vectoring without reducing propulsive efficiency.
Operation of pitching and heaving foils near a boundary was investigated in [15] and [16] for a flexible undulating foil with flow around both ends. In [17] and [8], ground effect is treated as a 2 dimensional flow phenomenon, but [18] suggests that ground effect is inherently three dimensional as flow around the tip of a foil has a strong effect on the hydrodynamic forcing near the boundary. Out of plane flow around the tip of the foil caused significant differences in both the magnitude and direction of the ground effect, when compared to the case with purely two dimensional flow.

3-D Rolling and Pitching Foils
For marine animals, fin propulsion is an inherently three dimensional phenomenon, both in terms of the motion of the foil and the resulting fluid flow. We classify as three dimensional foils those that use a rotary oscillation about a shoulder joint, instead of linear heaving motion, coupled with oscillatory pitching. This configuration is more appropriate for integration into underwater vehicles as a propulsion method [19]. For three dimensional foils, [20] and [21] found similar dependencies of lift and thrust coefficients on Strouhal number and maximum angle of attack as in two dimensional foils. The influence of a solid boundary has been extensively studied for one and two dimensional foils. In the current work, we extend our investigation of oscillating foil ground effect to encompass fully three dimensional geometry and flow during operation near the boundary. To our knowledge this is the first experimental study of ground effect in rolling and pitching foils.

Methodology
A rolling and pitching foil was oscillated while towed in a freshwater tank, in a series of experiments with varying foil kinematics. The instantaneous fluid forces and mo-ments applied to the foil were recorded as the foil moved from a constant depth deep water ('freestream') section, through a 'transition' section in which the tank depth linearly decreased, into a constant depth shallow water ('ground effect') test section. The foil drive mechanism is mounted to the overhead towing carriage using a trio of faired struts, as shown in Figure 1.2.3 and described in [22]. The test apparatus is This motorized tow carriage travels the length of the tank by rolling on top of the tank walls. The tank is 3.65m wide, with a total length of 30m and maximum depth of 1.55m.

Experimental Apparatus
The distances between the fin apparatus and the boundaries of the tank are equal to or greater than that used in foundational work on flapping foils, with similar chord lengths and Reynolds number, by [11] and [21] using two dimensional and three dimensional foils, respectively. The tank depth profile is defined using seven adjustable 2.43m long bottom panels spanning the width of the tank. The depth profile used in this effort is shown to scale in Figure 1 The foil shaft bearings are mounted to a rigid platform supported by a 6-DOF dynamometer inside the pitch canister, as detailed in Figure 1.2.5.
bearing. This 'pitch' canister is machined with an internal pocket which contains a 15W motor with an 85:1 gearing for actuation of the pitching motion. Low level motor control is accomplished with a programmable two-axis motion control card (Galil 1425) and two PWM motor drives (Advanced Motion Devices) housed in the roll canister. Feedback from encoders, mounted on the drive motor shafts, to the dedicated motion control processor is used to ensure repeatable and accurate kinematics. The motion control processor is programmed to halt the motion if deviations of greater than two degrees from the desired kinematics occurs on either axis. The entire apparatus is a modified and updated version of a foil drive removed from the underwater vehicle described in [23]. The foil drive was augmented to enable direct measurement of the forces applied to the foil as described in [24]. The foil has a uniform NACA-0012 cross-section, with span of 0.398m and chord of 0.095m, and is nominally rigid, with a titanium frame and cast outer shape.

Sensor and Foil
The test foil is a nominally rigid rectangular planform foil with a constant NACA-0012 cross section. It is constructed of a titanium framework cast inside resin with hardness The foil attachment within the pitch canister is shown in cutaway detail in Figure 1.2.5.
The pitch motor drive system and the foil shaft bearings are mounted directly to a 6-axis strain-gauge dynamometer (ATI Gamma SI-65-5) installed inside the pitch canister. The foil is mounted directly to the sensor to avoid direct measurement of drag forces on the drive apparatus. Flow interaction between the drive apparatus and the foil is assumed to be a secondary effect. As the external geometry of the drive apparatus is identical to that of the free swimming vehicle in [23], it provides an accurate representation of a real world application. A floating dynamic o-ring seal, as detailed in [20], is used to minimize mechanical coupling between the canister and the foil shaft where it exits the housing.
The six analog outputs from the dynamometer strain gauges are amplified and then Roll position is used to synchronize roll foil motion with the force data; the analog potentiometer output is acquired using the same DAQ module that acquires the dynamometer output. Carriage position relative to the tank is acquired at 10Hz using a laser distance measurement device (Astech LDM42A) mounted to the carriage. All communication between the automated experimental script (implemented in LabView 7.0) on the operator workstation, and the sensors and actuators is routed through the carriage mounted chassis via a fiber optic link.

Foil Kinematics
where Φ 0 is the amplitude of the roll motion, θ 0 is the amplitude of the pitch motion, and Φ bias is a mean roll bias, and Φ and θ are defined to be zero where the foil planform is parallel to the ground plane. The resulting nominal angle of attack, α, of the foil relative to the stationary fluid is given by, where U is the carriage speed. By convention, α is calculated at a point that is 70% of the distance from the axis of rotation to the foil tip in the spanwise direction, i.e. at a point r 0.7 from the origin.
The relevant non-dimensional parameters for harmonically oscillating foils generating thrust in unobstructed flow are the Strouhal number, St = 2r0.7Φ0f U , the maximum angle of attack, α max , and the Reynolds number, Re = U c ν , where ν ≈ 10 −6 m 2 s −1 is the kinematic viscosity of water [21]. For operation near the boundary, we define the final independent experimental parameter as H * = h c , where d is the distance from the tank bottom to the pitch rotation axis in the ground effect test zone assuming Φ bias = 0, following [18].
Reynolds number was held constant throughout; the foil was towed at U = 0.5 m/s in all trials, resulting in Re = 57, 000.
With carriage speed, foil geometry, and roll amplitude all constant, the desired St was achieved by varying the oscillatory frequency, f . given the geometry of the foil apparatus. The parameter space was chosen to align with previous work in three dimensional flapping foils [21] and work with two dimensional foils by [18] to which we compare the data gathered in this experiment. At this height with a 12 • roll amplitude, the closest approach to the bottom by the foil tip is 1 cm. Φ 0 = 12 • was maintained throughout all trials.

Experimental Procedure
Before each trial, the carriage is positioned against a hard stop at the deep end of the tank. The 6-axis dynamometer is tared to compensate for the in water weight of the foil before the flapping motion is started. The foil motion is started, and after three flapping cycles are completed, the carriage is accelerated up to 0.5m/s and travels the length of the tank, starting in freestream zone and ending in the ground effect test zone. Each run lasts approximately 40 seconds. After the carriage is returned to its starting position, the tank is allowed to settle for eight minutes before the next trial.

Results
In this effort, we will focus on the mean lift and thrust produced by the oscillating foil, and the changes that occur when the foil moves from the freestream into proximity with the tank bottom. Results are reported in terms of non-dimensional lift and thrust coefficients, The instantaneous lift and thrust coefficients are calculated as follows, where ρ is the density of water, U is the carriage speed, and S and c are the foil span C L andC T denote mean lift and thrust coefficients averaged over a complete cycle of motion, and are calculated as follows, where T is the period of the motion and τ i is the starting time for the i th cycle.
In each trial, the force data is divided into successive complete cycles, with each cycle identified using the starting and ending location of the foil within the tank. The individual cycles can then be labeled as occurring wholly within the freestream, wholly or in part within the transition zone, or wholly within the ground effect test zone. Cycles where the carriage is at rest or accelerating are discarded.   As H * decreases and the foil apparatus approaches the boundary, the flow beneath the apparatus will become constricted. We do not control for the fact that the drive system is being moved farther from the ground along with the foil and therefore we cannot directly separate the direct effects on the foils from these secondary effects due to the change in the interaction between the flow and drive system. The foil protrudes 5 cm from the leading edge of the drive mechanism minimizing turbulence from boundary layer effects but this is also a possible avenue for future investigation of near ground flapping foil propulsion. Any vehicle or drive system will constrict the flow beneath the body and should be expected to have similar vertical extent given similar foil geometry, hence the results should be broadly applicable to other three dimensional flapping foil systems.   showing that for all kinematics tested, the distance to the free surface of the tow tank has relatively small effect (ranging from -0.03 to 0.02) compared to the systemic bias due to misalignment or the change in lift due to ground effect (> 0.1). Comparison between freestream and near ground effect within a single experiment (i.e. a single traversal of the towing tank), rather than in two separate experiments, was used to ensure that any systemic biases due to misalignment of foil or drive system are included in the forces experience in both zones, and we can therefore reasonably assume that the gross effect of the misalignment is canceled. The current experiments do not allow us to assess the direct effect of the misalignment on the total magnitude of the ground effect, and this is an area of potential future efforts. For the purposes of discussion, we assume that this is a secondary effect that can be neglected.  C T increases with St for all α max , consistent with previous research in both two dimensional [11] [18] and three dimensional foils [25] [20] [21]. Values ofC T reported by [21], for a

Comparison to change in lift with two dimensional foil kinematics.
One motivation for this investigation is to determine what qualitative and quantitative differences in observed ground effect occur between three dimensional rolling and pitching foils, and two dimensional heaving and pitching foils. To that end a comparison is made in Figure 1.3.7 between the results from the present effort, and experimental results produced using the experimental method detailed by [18], where a two dimensional heaving and pitching foil was tested in small tow tank. The apparatus is detailed in Figure 1   Results from [18] in 2D: heaving and pitching with no spanwise flow (c), and 2D+: heaving and pitching with spanwise flow (d) ,

Cross platform comparison across a range of kinematic parameters.
A comparison between two-dimensional and three-dimensional foils, Figure 1 case, compared to the other cases.

Discussion
The presence of a nearby solid boundary influences the hydrodynamic forcing on an oscillating foil used for propulsion. We highlight three significant conclusions that can be derived from the experimental results presented above: 1. Foils used to generate high thrust with rolling and pitching oscillations experience a 'ground effect' which repels them from solid boundaries.
2. Ground effect can be induced at greater distances by biasing the tip of the foil toward the boundary.
3. The ground effect experienced by a high-aspect ratio rolling and pitching foil is a fully three-dimensional phenomenon, which cannot be accurately predicted when twodimensional flow and/or two-dimensional kinematics are enforced.
1.4.1 Foils used to generate high thrust with rolling and pitching oscillations experience a 'ground effect' which repels them from solid boundaries.
The results presented in Figure 1 In steady flow, absent obstructions or boundaries, symmetric foil oscillations with zero mean pitch angle produce zero mean lift over each complete cycle of motion [21]. The magnitude of the opposing instantaneous lift peaks is typically several times the magnitude of the mean thrust [20]. Given the presence of these large instantaneous lift forces, it is thus not unexpected that large mean lift forces can be produced through asymmetries in the kinematics. These large lift forces have been illustrated in bench top experiments [11] [13] [25] and in practical application on a vehicle [23], where a non-zero mean pitch angle (pitch 'bias') is used to vector total force and control the trajectory of an underwater vehicle.
Absent ground effect, a free-swimming vehicle (or animal) seeking to avoid a boundary would need to actively sense that boundary and alter foil kinematics to vector the total force away from it. However, the ubiquity of the ground effect in high thrust kinematics suggests that the ground effect can provide a passive obstacle avoidance capability. With proper trajectory design, the closed loop sense/react control problem could be alleviated by taking advantage of the naturally occurring cushion created by the ground effect. 1.4.3 The ground effect experienced by a high-aspect ratio rolling and pitching foil is a fully three-dimensional phenomenon, which cannot be accurately predicted when two-dimensional flow and/or two-dimensional kinematics are enforced.
Previous experimental and numerical studies of flapping foils have enforced artificial limitations on the degrees of freedom of the foil motion and the fluid flow. However, the results of the present work show that there are significant qualitative differences between the forces produced by two dimensional and three dimensional foil and flow geometries. Based on this observation, we believe that naively extrapolating from the two dimensional to the three dimensional using empirical coefficients based on aspect ratio, such as those presented for stationary foils in [26], is likely to fail.
One rationale for investigation of two dimensional foil kinematics and flow (i.e. infinite aspect ratio foils) is to inform strip theory estimates of total force on finite aspect ratio foils.
This approach assumes that an effective 2D heave and pitch trajectory can be calculated  [33] are more appropriately modeled using results for very low aspect ratio foils with spanwise flow, e.g. from [7] Studies of steady wing in ground effect studies may provide insight in to the differences in the force production of two and three dimensional flapping foils. [34] splits fixed wing ground effect into two phenomena one being span dominated and pertaining to the interaction between ground and the tip vortices and a second being chord dominated 'ram effect' that deals with high pressure build up under the wing. In the former, the tip vortices are compressed by the presence of the solid boundary increasing the effective span and also decreasing the strength or amount of downwash created by the tip vortices, which in turn decreases the energy lost as drag. [35] also notes that the decreased downwash changes the streamlines in the flow, increasing the effective angle of attack of the foil increasing lift.
As the distance between the wing and the ground decreases ram effect increases as the air speed beneath the wing begins to stagnate increasing the pressure under the wing. However, with convex foils, at low angles of incidence, lift can actually be decreased as a venturi nozzle begins to form between the foil and the ground, creating suction between the two.
It should also be noted that the chord dominated ground effect is a two-dimensional flow effect, whereas the decreased downwash of the tip vortices is a three-dimensional effect. These

Conclusion
The present work demonstrates that it is critical to investigate ground effect using fully three dimensional geometry. Swimming animals and biologically inspired underwater vehicles use a wide variety of different oscillating foil mechanisms to propel themselves. In the face of this variety, it is natural to seek fundamental knowledge through experimental and numerical investigations which simplify the underlying hydrodynamic mechanisms. As a result, there is a large body of work into two dimensional foil and flow geometries, including numerous studies that attempt to elucidate ground effect with two dimensional numerical simulations. However, extrapolating two dimensional results through techniques such as strip theory, or empirical correction factors for aspect ratio, will fail to capture the qualitative and quantitative differences between the two dimensional and three dimensional cases, which are strongly dependent on nominal foil kinematics.
Productive directions for future efforts include flow visualization or numerical simulations to illuminate the underlying hydrodynamic mechanisms at work in ground effect; variation of foil geometry, particularly aspect ratio; variation of foil kinematics along with the inclusion of in-line motion similar to work by [13]; The surface effect develops similarly to ground effect as an oscillating foil approaches the boundary. This suggests that similar hydrodynamic mechanisms are at work in both cases.
Changes seen in mean thrust coefficients due to the free surface are found to be negative where as ground effect sees positive. The negative change can be explained by energy lost due to wave generation as the foil operates near the surface. propulsion method that has evolved from these studies. The goal of oscillating foils is to take inspiration from sea animals such as a sea turtle in order to inspire novel propulsion methods for underwater vehicles.

Oscillating Foils
In our work, we distinguish between different oscillating foil configurations by referring to the degrees of freedom in the motion of the foil. Three common configurations are one dimensional (pitching or heaving motions only), two dimensional (coupled heaving and pitching motions), and three dimensional (combined rolling and pitching motions). This paper focuses on three dimensional (3-D) rolling and pitching oscillating foils and is the first experimental study of 3-D high aspect ratio thrust producing foils near a free surface. One and two dimensional kinematics have been studied both numerically and experimentally.
[1] found experimentally that the structure of a wake behind a pitching only 1-D oscillating foil can be modified significantly by changing the amplitude, frequency, or shape of the oscillating motion. The foil was also found to produce greater thrust than predicted by linear inviscid theory. Similar results are found by [5] in their numerical work on low aspect ratio oscillating foils.
In the 3-D rolling and pitching foils, the foil uses a rotary oscillation about a shoulder joint with coupled oscillatory pitching. This configuration has been used as propulsion in vehicles akin to a robotic turtle, [6]. Studies by [7], [8], and [9] identify the Strouhal number, St, and maximum angle of attack, α max , as the primary non-dimensional parameters that govern the change in lift and thrust coefficients of the oscillating foils.

Effect of the boundary
Unmanned Underwater Vehicles (UUV) have increasing needs to access more complex areas including operating in near surface and near bottom conditions. The 'ground effect' resulting from operating near a solid boundary has been extensively studied for static foils(see [10] for review). More recently numerical and experimental studies have examined ground effect for oscillating foils (see [11] for a more in depth review). The presence of suction an repulsion forces, as well as increased thrust, is typically reported within less than 2 chord lengths of the wall. The magnitude and direction of the forces perpendicular to the wall are determined by the kinematics of the foil as well as the distance between the foil and the solid boundary. Numerical studies by [12] predict ground effect in 1-D heaving foils where the net force is dependent on the distance to a solid boundary. The foil experiences first a suction force and then repulsion as the separating distance decreases. Further investigation reported in [13] found that as the foil approaches the boundary, vortexes will start to be formed and shed from the solid boundary. The vortex pairs begin to stretch and rotate as the separation decreases causing the wake to deflect away from boundary. Experiments by [14] showed that in a flow tank, a pitching only 1-D foil will experience increased thrust while in the presence of a solid boundary with a region of attraction and repulsion. However,in this case the attraction region was located at greater distances from the wall than the repulsion region. Using a 2-D heaving and pitching foil configuration, [15] found that restricting the span-wise flow around the tip of the foil can have a significant effect on the magnitude and direction of forces experienced by the foil and supports the idea of ground effect being an inherently three dimensional phenomenon. In [11], the presence of a solid boundary was shown to effect forces on a 3-D rolling and pitching foil. A non-zero mean lift, indicating an asymmetry in the wake structure, was seen across the entire range of the kinematics tested. The results were compared to [15] and showed that ground effect is a fully three dimensional phenomenon and that extrapolation of 2-D foil studies into 3-D foils would be inherently inaccurate.
Recently, work has been conducted on 2-D heaving and pitching foils near the surface.
[16] found a significant effect due to proximity of the free surface. The numerical studies found a nearly linear relationship between thrust and pitch amplitude for a given Strouhal number as well as a decrease in thrust as the foil approaches the surface.

Contributions
Near-surface autonomous underwater vehicle (AUV) operations have been of interest to both the military and civilian communities [17]. These operations include oceanographic data collection, mine countermeasures, as well as intelligence, surveillance and reconnaissance(ISR) missions. In the present work we extend the investigation from [11] to operation of a 3-D rolling and pitching foil near the free surface. When operating near the surface, wave generation is bound to have an effect on the forces experienced by a flapping foil.
Quantitative and qualitative analysis of near surface operation of oscillating foils can pro-vide insight into modeling and improve performance of autonomous vehicles equipped with this type of propulsor.

Methodology
Fluid forces and moments on a foil at varying distance from the free surface were measured for a foil towed in a water tank. The experimental apparatus used in [11] to conduct ground effect tests in the University of Rhode Island tow tank was modified to allow for testing near the surface. Details of the apparatus from [11] are repeated here for completeness. The position of the foil was raised and lowered to allow the foil to flap in the free stream and with increasing proximity of the free surface of the tank.

Experimental Apparatus
The foil drive mechanism consists of a two canister oscillating mechanism mounted to an overhead towing carriage using a trio of faired struts, as shown in Figure 2

Foil Drive Mechanism
The test foil is a nominally rigid rectangular planform foil with a constant NACA-0012 cross section and an aspect ratio of 4.2 with a span of 0.398m and a chord of 0.095m (See     Sinusoidal pitch and roll trajectories with a 90 • phase offset were used for all experiments. The foil motion can be completely described by (1.1) and (1.2),

Foil Kinematics
where Φ 0 is the amplitude of the roll motion, and θ 0 is the amplitude of the pitch motion.
Φ and θ are defined to be zero where the foil planform is parallel to the surface plane. The resulting nominal angle of attack as a function of time, α, of the foil is given by, where U is the carriage speed. By convention, α is calculated at the 70% chord distance, i.e. at a point r 0.7 from the origin, however α will vary across the span of the foil due to differences in angular velocity.
The relevant non-dimensional parameters of thrust producing oscillating foils are Strouhal number, St = 2r0.7Φ0f U , maximum angle of attack, α max , and the Reynolds number, Re = U c ν , where ν ≈ 10 −6 m 2 s −1 is the kinematic viscosity of water [8]. Furthermore we define an additional independent variable, H * = h c , the non-dimensional distance from the surface boundary, where h is the distance from the free surface to the axis of pitch rotation, following [11] where h represented distance to the solid boundary.  parameter space was chosen to align with previous work in 3-D oscillating foils by [8] and [11] to which we compare the data gathered in this experiment.

Experimental Procedure
Before each trial, the carriage is reset to a position against one end of the tank. The 6-axis dynamometer is tared to compensate for the in water weight of the foil before any motion. The foil motion is started, and after three periods elapse, the carriage is accelerated up to 0.5 m s and travels the length of the tank before it is stopped. The foil motion is stopped, the foil is homed to return it to zero roll and zero pitch and finally the carriage is returned to its starting position. Each trial runs for approximately 40 seconds, after which the experiment is reset and the tank is allowed to settle for eight minutes before the next trial begins. After all kinematics are run, the foil is raised to a new depth and the testing continues.

Results
Results are reported in terms of non-dimensional lift and thrust coefficients, C L and C T , respectively, as defined in equations 1.5 and 1.4, where ρ is the density of water, U is the carriage speed, and S and c are the foil span and chord lengths.
where T is the period of the oscillating motion and τ i is the start of the i th cycle.

Change in mean lift
As the foil approaches the surface, there is a change in both the mean lift and mean thrust coefficients. The magnitude of this change is dependent on the kinematic parameters.
The magnitude of change, ∆C L , is shown in Figure 2

Changes in mean thrust
Contours of changes in mean thrust coefficient, ∆C T , at the closest approach to the surface are shown in Figure 2.3.4a. ∆C T is negative across all tested kinematics and ranges from magnitudes of 3% to 25% with the maximum found at low St and low α max and minimum at high St and low α max . For, comparison ∆C T for a foil approaching a solid boundary, [11], is shown in Figure 2.3.4b. In these tests, thrust was increased across all kinematics tested ranging from 2 to 7 percent which is opposite to the current work. The

Discussion
The forces on a oscillating foil propulsor are affected by the presence of the free surface.
We highlight the significant conclusions that can be taken from these results below: • Thrust generating rolling and pitching oscillating foils experience a repulsion force as they approach a free surface.
• Observed effect on lift forces of an oscillating foil are similar for free surface and solid boundaries.
• Thrust production difference can be accounted for by wave generation.

2.4.1
Thrust generating rolling and pitching oscillating foils experience a repulsion force as they approach a free surface. Ground effect can provide vehicles passive boundary avoidance capabilities [11]. This passive avoidance capability is also present as an oscillating foil approaches a free surface boundary. The repulsion force present could be used to detect the boundary and if coupled with an analysis of the thrust forces a vehicle could distinguish a free surface boundary condition from a solid boundary condition.

2.4.3
The difference between ground effect and surface effect on thrust production can be accounted for by wave generation.
It is likely that a large portion of the power lost as the oscillating foil approaches the surface is due to wave generation as is common with a body moving near or on the surface.
Assuming this to be the case, the power lost due to the surface effect is calculated here by multiplying the thrust difference between the freestream and the closest approach to the surface by the freestream speed (equation 2.8). Equation 2.9 is the power per meter of crest of a deep water wave according to linear wave theory, [20], where ρ is the density of water, g is gravity, H is wave height and f is wave frequency. For these calculations we assume a wave frequency equal to the oscillation frequency of the foil. The waves generated during the test were not measured but these estimated values are not unrealistic. It should be emphasized that this is a conservative over-estimation since it is unlikely that 100% of the power would transfer into waves and the wake created is more complex that a single wave with this wave height. Both gravity and transverse waves are likely to appear in a complex wake pattern with frequencies based on both the oscillation frequency as well as the freestream velocity of the apparatus. The presence of this wave generation at the surface is a large difference from an oscillating foil operating near a solid boundary and can cause the negative effect on thrust production. It is possible that the effect on thrust can be altered by creating constructive or destructive interference between the body waves created by the apparatus passing close to the surface with the wave generated by the flapping motion of the foil, but this has not yet been tested.

Conclusion
The hydrodynamic forces on a 3-D rolling and pitching oscillating foil were characterized at varying distances to the free surface. We conclude that there is a similar effect akin to the previously researched ground effect [11]  could be used to not only detect different boundaries but classify and estimate distances to them. As Computational Fluid Dynamics (CFD) improve, this data set will also provide reference data for numerical analyses of 3-D rolling and pitching oscillating foils. We hope that these experiments will provide useful data for further validations and investigations both numerical and experimental into force production of oscillating foils.

MANUSCRIPT 3
Combination of force feedback and an analytic flapping foil model to improve control of a flapping foils in the presence of crossflow In this paper we combine an analytical model to design a control system to use force feedback from a rolling and pitching oscillating foil to compensate for and estimate the angle of crossflow. An analytic model was used to produce predictive force estimates of multiple oscillating foil systems, across a range of kinematics, and in the presence of crossflow. These predicted forces were compared with experimental data in order to validate the implementation of the analytic model. The comparison proved the code developed was able to accurately replicate previously published results, accurately predict forces on oscillating foil configurations at NUWC-NPT and URI, predict mean force coefficients over a range of foil kinematics and accurately estimate the effect that crossflow will have on the forces experienced by an oscillating foil. Once the code was validated, it was used to design and simulate a simple control system. The control system was implemented on the foil system at URI and was able to compensate and coupled with information from the model, estimate the crossflow angle within a half of a degree.

Introduction and Background
The ocean is a complex environment and navigating it has an endless list of challenges to overcome. New vehicles are being designed that have biomimetic propulsion to improve the maneuverability and agility of these platforms. With these new types of propulsion, the vehicles will need new control systems to allow for accurate thrust and lift production while operating. Oscillating foils in particular have been implemented in underwater vehicles and their control systems are being researched to improve the performance of the foils.

Oscillating Foils
Self-propelled underwater vehicles using oscillating foils can be seen in [1] and [2]. These vehicles use coupled rolling and pitching motions to induce a flow across a foil providing lift which can be vectored to produce thrust on a vehicle. More in depth descriptions of different oscillating foil configurations can be found in [3].

Digital Control Theory
Control of a system starts with the creation of a representative model of the system.
A continuous time state space model or 'plant' can be described by the equations in (3.1) where matrices A, B, C, and D are derived from the system.
where k is the time index and T is the sampling period. Before designing a controller model must also analyzed to ensure it was controllable. To determine if a discrete model is controllable, first the continuous time model must be controllable. This can be checked for a second-order system by forming the controllability matrix as described in (3.4) [4].
If the rank of the controllability matrix is equal to the order of the system, than it is controllable. The rank of the controllability matrix was equal to the order of the system and therefore, it is controllable, or there exists a inputs signal that will bring the system from any initial condition to any final condition in a finite amount of time [4]. If the continuous system is controllable, the discrete time is controllable if where T is the sampling interval and β max is imaginary part of the pole with the greatest magnitude. There are many methods of calculating the the gain matrix, L, for a regulator. This paper proposes using pole placement through state feedback using an iterative algorithm.
The iterative algorithm will calculate the gain to make the poles of the closed loop system equal to normalized bessel poles as described [4] while ensuring that the eigenvectors are as orthogonal as possible. Moving the poles to these location should provide a desirable system response driving the states of the system to zero with minimal overshoot.

Contributions
As biomimetic underwater propulsion evolves, new tools are needed to help improve design and control of new systems. In the present work we replicate results of a previously derived model for forces on a flapping foil and expand on validation cases of the model. We then use the model to design a control system for use on a flapping foil. The control system is tested validating the model as a useful design tool for flapping foil control systems.

Useful Definitions
Results are reported in terms of non-dimensional coefficients as defined where ρ is the density of water, U is the carriage speed, α is the maximum angle of attack, and S and c are the foil span and chord lengths. F z is the measured force perpendicular to the span of the foil and incoming flow. F x is the thrust force in line with incoming flow.C L andC T denote mean lift and thrust coefficients averaged over a complete cycle of motion, and are calculated as follows, for each cycle of motion: where T is the period of the oscillating motion and τ i is the start of the i th cycle.

Model Implementation
An analytic model has recently been developed at the Massachusetts Institute of Technology (MIT) to predict forces on wings of finite aspect ratio. Detailed in [5], this model is suited for rapid modeling of forces on a flapping foil with the goal of improving on current methods such as computational fluid dynamics (CFD) or unsteady vortex lattice methods (UVLM). CFD simulations are physically accurate, however, due to their complexity, they are computationally slow. UVLM is faster than full CFD simulations, however, they grow slower as the entire wake history is recorded. Research into reduced order modeling has   [5] found that the dynamics of unsteady wings are highly complicated but simulation outputs such as lift, circulation, and downwash can be modeled more easily [5]. This model presents a fast and computationally cheap solution to model unsteady flapping motion with the goal of aiding active control of oscillating propulsive foils.
Commonly, the wake of an oscillating foil is modeled as a continuous vortex sheet without viscous effects and lacking leading edge vortex (LEV) shedding. The model presented in [5] proposes a novel apprach that approximates the wake as sum of horseshoe vortices and state-space systems of the local wake history. The model uses the indicial method to obtain the state-space representations of the blade element systems similarly to [6]. [5] proposes two options using 1 state per blade element and 6 states per blade element to record the independent lift and circulation. An additional small downwash correction is provided to consolidate the true 3-D unsteady wake with the previously mentioned quasi-steady and 2-D unsteady wakes.
Using this model, at the University of Rhode Island (URI) we were able to develop a code to calculate the forces on a foil. Following [5], each time step in the model follows this sequence: 1. The circulation is determined by solving for the trailers and circulation states that represent the unsteady wake.
2. The instantaneous downwash is determined from the previously found circulation and the incident velocities are modified.
3. The lift is found from a combination of the downwash and lift states. 4. The time rate of change of all states are calculated, then integrated to give an updated state.
After reviewing the paper we decided to use the proposed reduced order state-space system method which includes simplifications allowing for one state per blade element to track both the lift and circulation of that element. We did not add the small additional downwash correction since the correction had little affect on the resultant lift forces and the additional states would have added computational cost. Finally, our model simulations were done in MATLAB R2015 without the use of the Drake toolbox mentioned in [5] and any added mass forces were neglected. [5] provides multiple validation cases in order to compare the new model to earlier work.

Result Replication
The code developed at URI was used to replicate the results of two of these cases to validate that the code was working properly. The two cases were the impulsive start of a foil from rest and the case of coupled heaving and pitching motions of the same foil. The foil, shown in

Evaluating Model Accuracy
After the model code was completed, and the results from the paper were replicated, the model was validated against experimental force data taken from 3-D rolling and pitching foils. The data was taken from two different setups; one at the University of Rhode Island (URI) and another at the Naval Undersea Warefare Center-Newport (NUWC-NPT). The model was compared to force data from both of these apparatuses and under different conditions. At URI we were able to compare the accuracy of the model over a range of operating kinematics and at NUWC-NPT were able to the test the model against a foil in the presence of a crossflow and with non-zero pitch biases. The experimental apparatuses of each location are described in Section 3.3.1.

Experimental Apparatuses
This section details the oscillating foil configurations and experimental methods of the tests run at each location.

University of Rhode Island
The foil drive mechanism consists of a two canister oscillating mechanism mounted to an overhead towing carriage using a trio of faired struts, as shown in Figure 3  Sinusoidal pitch and roll trajectories with a 90 • phase offset were used for all experiments. The foil motion can be completely described by (3.14) and (  where Φ 0 is the amplitude of the roll motion, and θ 0 is the amplitude of the pitch motion. Φ and θ are defined to be zero where the foil planform is parallel to the surface   plane. The resulting nominal angle of attack as a function of time, α, of the foil is given by, The foil is oriented with the freestream velocity in the X axis, and the fin rolls and pitches about the X and Z axes respectively.
The onshore computer sends roll and pitch velocities to the motion control board to generate the oscillatory fin motion. The carriage to which the foil is attached travels the length of the tank where foil motion is stopped and then the carriage is returned to its

Phase averaged lift and thrust
The model was used to predict the forces on the oscillating foil system at URI. The  Next, the model was tested against the experimental apparatus at the NUWC-NPT. peaks that are flatter than that of the experimental data and the magnitude of one of the peaks is under predicted. It should also be noted that given symmetric sinusoidal motion, both peaks of the signals should be equal as is shown in the model but is not the case in the experimental signal. This could possibly be caused by slight asymmetries in the experimental setup. We can surmise that the model gives a good representation of the forces seen in both systems but there are still some higher order hydrodynamic effects that do not appear in the model.

Range of Kinematics
Since the model could predict the forces on different types of foils in normal operating conditions, the scope was expanded to include a foil flapping at different kinematics. The    error. The amplitudes of the thrust signals, however, can vary by up to 40 percent at low maximum angles of attack but can reach as low as zero at higher maximum angles of attack but Strouhal number has little effect on the accuracy of the thrust amplitude prediction.

Cross Flow
After validating the model over a range of kinematics, crossflow and non zero pitch biases were introduced into the system. The following data sets were taken using the NUWC-

Using Analytic Model as a Design Tool
This model of the forces on a foil undergoing oscillatory motion is a potentially powerful tool for vehicles using this form of propulsion. We propose to use this model to develop a control system for a rolling and pitching foil that will compensate for and estimate a crossflow. During normal cruising operations, a vehicle maintains a desired mean thrust coefficient and a zero mean lift coefficient. Using feedback from the embedded force sensor we aim to control the vehicle to obtain desired force coefficients. Given a desired mean thrust coefficient of C T des = 0.5 the control system will adjust the flapping frequency and pitch bias in order to obtain the desired thrust coefficient while driving the mean lift coefficient to zero. The pitch amplitude will be calculated to achieve a maximum angle of attack for the current Strouhal number assuming a zero crossflow. It is understood that in the presence of crossflow the maximum angle of attack will vary with crossflow, but this method is employed to simplify and standardize the calculation of the pitch amplitude for a given Strouhal number.

System Model
The states in the system to be controlled are chosen to be the error in the mean lift and thrust coefficients as defined in (3.17). The state space system matrices are also defined below.
C T des = 0.5 The controllability matrix was formed for each continuous system created across the entire range of operating parameters and each was verified that it was full rank. Furthermore, since there are no complex poles of the system, the discrete time systems are controllable for all sampling intervals. Since each is controllable, the feedback loop was closed creating the system in (3.21) where L is the feedback gain matrix. (3.21)

Gain Calculation and Simulation
With access to the working flapping foil force model, it was possible to pre-calculate the feedback gains for a range of operating conditions. Separate gains were found for a range of Strouhal numbers, maximum angles of attack, and pitch bias angles by calculating the gradients of mean lift and thrust coefficients provided by the analytical force model to create a system model for each operating point and using a pole placement method to calculate the feedback gain. These gains were stored so that during the control tests, the correct gain could be quickly obtained for the current operating conditions of each step. The feedback gains were calculated to place the poles each closed looped system at the roots of a 2nd order Bessel polynomial normalized by the desired settling time. The desired settling time was 3 times the flapping period derived from each Strouhal number. With the gains calculated, the control system could be simulated by feeding the control inputs back into the analytical model code. The results are shown in Figures 3.4.1 and 3.4.2.     In order to test the control system designed, the experimental apparatus at URI was modified to allow for the application of a crossflow by adding a hinge to the assembly, shown in Figure 3.4.3, enabling the foil canisters to be rotated an angle σ with respect the oncoming flow. Each experimental trial followed these prescribed steps: 1. The foil was run at an initial set of kinematics 2. The force data recorded was processed to determine the current mean lift and thrust coefficients and the error of each.
3. Using the error and the pre-calculated feedback gains, a change in St and θ bias were found.   St 0 , respectively. There is greater overshoot in the experimental implementation of the control system than seen in simulation but the states settle in the same amount of time.
The initial errors are also different than predicted from the model but this was expected due to the slight discrepancies seen between the analytical model and experiments described in

Discussion
From all of the results shown here, we would like to highlight three distinct points: • The code based on the analytical model provides accurate force prediction across a range of foil configurations, kinematics, and environmental factors.
• A simple control system combined with an analytical model can compensate for and estimate crossflow.
• This model is a valid design tool for designing control systems of 3-D rolling and pitching oscillating foils 3.5.1 The code based on the analytical model provides accurate force prediction across a range of foil configurations, kinematics, and environmental factors.
We were able to encode the analytical model into a working MATLAB script that can be used to predict forces on oscillating foils. The results from our code matched those provided by [5]. Upon creating a working tool, we moved to test it against a new range of conditions. The model performed well predicting forces for both the foil at NUWC-NPT and the foil at URI. The prediction of the URI foil was slightly better possibly due to some suspect error in the experimental setup at NUWC-NPT since symmetric kinematics did not provide symmetric force signals. In Figure 3.3.8 it can be seen that the magnitude of one stroke of the flapping cycle is greater than the other. This is likely due to the device not being completely aligned with the flow. The model was tested against a range of kinematics and performed well with some slight over prediction of thrust at high α max . The model was also able to accurately predict forces when a crossflow was added to the simulation. The model does start to fail as crossflow diverges too far from the pitch bias angle as seen in foil. It should be noted that the model was able to accurately predict the forces without any sort of tuning parameters and was able to be implemented for different systems by simply knowing the geometry and motion profiles of the foils.
3.5.2 A simple control system combined with an analytical model can compensate for and estimate crossflow.
The state feedback regulator designed in this paper was able to compensate for increasing angles of crossflow and was able to achieve the desired mean lift and thrust coefficients.
There is some overshoot especially when correcting for large angles of crossflow but still settles to the desired values. In this system both the lift and thrust are coupled, but it seems that the error in mean thrust which is more variable causes disturbances in the mean lift coefficient. This could be fixed by decoupling the two, where the change in Strouhal number is based solely on the error in mean thrust and the pitch bias angle is based solely on the error in mean lift. From the analytical model we were able to simulate how the control system would perform and also found that a zero mean lift coefficient would not occur when the pitch bias oriented the fin motion directly into the flow but at an angle approximately 91% of the crossflow angle. This is most likely due to the fact that while the pitch can be aligned with the flow, the roll motion is still constrained to be perpendicular to the canisters. This means that as the crossflow value increases the roll motion which is normally completely perpendicular to the flow has an increasing amount of motion inline with the flow which will affect the angle of attack of the foil through out its motion. More information on the effect of adding inline motion to oscillating foils can be found in [8].
The simulation was able to provide the 91%σ relationship that allowed for estimation of the crossflow angle to within half of a degree by dividing the pitch bias angle of zero mean lift by the 91%. This simple system would allow a vehicle to maintain course, rejecting the steady disturbance of a crossflow, while simultaneously estimating information about the surrounding environment.

This model is a valid design tool for designing control systems of 3-D rolling and pitching oscillating foils
The analytical model accurately predicts the forces on a foil and has been shown to perform well predicting lift and thurst forces under different operating conditions and in the presence of constant disturbances such as a crossflow. This paper describes a simple state feedback regulator that was designed and simulated using the model. The control system performed remarkably similar to simulated results and was able to compensate for a crossflow on the body. The model was also able to provide information for more accurate estimation of the crossflow angle. Being able to use the model to rapidly design and test control systems is a powerful tool and can be significantly more cost efficient than running full scale vehicle experiments. The model proved to be a useful tool for designing a control system and was able to accurately predict the performance of a control system without any need of tuning.

Conclusion
As vehicles with oscillating foil propulsion progress, their control systems will also need to perform in more difficult conditions. Having force feedback implemented on individual foils will allow for more complex vehicle maneuvers. Individual foil control can improve the agility of a vehicle allowing for maneuvers such as corkscrews, and barrel rolls. Station keeping and low-speed hovering or investigative maneuvers would also improve with better foil control. New tools for control system design will be needed to enable cheaper more efficient research into these areas. A code was developed to replicate the analytical model published in [5]. This code was validated against results in [5] and was also validated against experimental data of various foil configurations, kinematic parameters, and environmental conditions. The code was able to replicate previously published results, accurately predict forces on oscillating foil configurations at NUWC-NPT and URI, predict mean force coefficients over a range of foil kinematics and accurately estimate the effect that crossflow will have on the forces experienced by an oscillating foil. Once the code was validated, it was used to design and simulate a simple control system. The control system was implemented on the foil system at URI and was able to compensate and coupled with information from the model, estimate the crossflow angle within a half of a degree.

Future Work
The steps taken here open up various paths for further research. The code can be used to test the model against more complicated environmental conditions such as unsteady disturbances such as the presence of wave induced velocities. The control in this paper was done in discrete tests off-line post processing before new kinematics being input to the foil system. Modifications could be made to the oscillating foil system to allow for real time control. The control in this paper was done with mean lift and thrust coefficients, the model can output instantaneous lift forces which could be used for more complex motion control of the foil, including non-sinusoidal oscillating motions. In this paper we used a simple feedback regulator to drive the error of the system to zero. More complicated control systems could be simulated to improve robustness and response characteristics.