PASSIVE OBJECT DETECTION VIA AN ARTIFICIAL LATERAL LINE AND A 2-D VISCOUS FLOW MODEL

In the field of ocean exploration and marine vehicle design there is a heavy reliance on the use of acoustic and visual sensors for mapping, maneuvering, and obstacle avoidance. In low light regions, constricted spaces, or areas where water is turbid, however, passive systems similar to the lateral line of fish can be advantageous. Using a set of measurements like these, a digital twin could be constructed such that it represents the true physical system for use in mapping, control system design, or tracking the loading of a structure over time. For moving bodies in the presence of solid boundaries such as a vehicle operating at the ocean floor or a vessel in shallow waters, it is critically important to properly estimate the local boundary conditions for use with a digital twin. The following study implements a viscous numerical simulation as a model for object identification. The presented methods include a classification schema via machine learning for a variety of wall shapes and sizes, as well as a general boundary estimation method using basis splines and an Unscented Kalman filter. At the cost of generality, the classifier is shown to be capable of identifying shape and size with relatively high accuracy especially as the number of available classes increases. The basis spline method uses an Unscented Kalman filter and modeled pressure as a proxy to determine the locations of control points which govern boundary conditions. The method is shown to work for a single and multiple protrusions from a wall, with a weighted estimate approach developed which outperforms the standard formulation of the Unscented Filter when accounting for multiple measurement locations along the length of the foil body. Performance of the method as well as the sensitivity to the design of the system are discussed, and future work is presented.

Personal validation of the classifier using pressure signatures for all features of interest. Plus signs signify flat walls, triangles are shown with triangle markers, half-circles with circle markers, and rounded squares with square markers. A blue marker represents a correct classification, purple represents a misclassified feature as another feature, and red is a misclassified feature as a flat wall. 24 15 How using different numbers of points changes the accuracy of the classifier for all wall features of interest with size. . . . . . . 26 16 The confusion matrix from the classification learner in MATLAB for shape and size. The letter before each shape corresponds to its relative size. . x 18 Accuracy of the classifier using p 4 , p 7 , p 11 as specified by MATLAB. 30 19 Validation of the classifier for 9 discrete sizes and 3 shapes. Plus signs signify flat walls, triangles are shown with triangle markers, half-circles with circle markers, and rounded squares with square markers. A blue marker represents a correct classification, purple represents a misclassified feature as another feature, and red is a misclassified feature as a flat wall. The size of the marker corresponds directly to the size of the feature, with bold markers showing misclassified size. The background color corresponds to The wall shape to be estimated using the B-spline UKF algorithm. Pressure is measured at the shown locations along the foil body. 46 23 The standard UKF.  Error metrics for both multi-shape cases. Height, width, and distance are in units of chord lengths, and the center is defined by the physical location of the center referenced from the start position of the simulations. Each feature is presented separately, while RMSE T was common between both features. The characteristics of the true wall are also presented for comparison. . . . 69 In addition, there has been a surge of interest in the development of digital twins for ocean vehicles. A digital twin is a model representation of a physical system which properly captures the physical surroundings and the local dynamics in order to provide an artificial representation of a system [8]. Given proper design and a set of inputs, a digital twin can give estimates of product life-cycle, fatigue life, fatigue points, and even optimize design [8,9]. Digital twins have been implemented for product design and manufacturing [10,11], for vehicle "health" monitoring [12], improving fuel economy [13], and floating production system monitoring [14].
While digital twins have been combined with dynamic systems in the past, they have been predominantly used for some type of optimization or fatigue life analysis.
Digital twins can be particularly useful for marine vehicles and control systems as they may be used in real time control or decision making. In order to inform the model of the physical surroundings, measurements must be assimilated into the twin via some filtering or ensemble method. This can be a particular challenge in situations such as a streamlined body passing by an object in fluid. In this flow field, the boundary conditions are constantly changing as the body moves through the medium, and those that represent the object are unknown. This becomes even more complex due to the presence of a fluid wake, which has memory and changes in time.
The following work describes a method for determining general boundary conditions in a simple 2D viscous flow model of a wing passing by an object on a wall. The presented method is a first step towards realizing a real time computational model integrated with the operation of a moving body in a fluid providing updated boundary conditions. The estimated boundary conditions provide a model which represents the local surroundings and could be implemented with a digital twin.

Biological Analog
Measurements of pressure on the surface of a foil were used as the input for determining local boundary conditions. A biological analog for this type of measurement is the mechanosensory lateral line of fishes and aquatic amphibians.
The lateral line is a system composed of sensory organs that run along the head and body ( Figure 1) which can sense changes in pressure and flow in the surrounding fluid [15]. Fish use this system for a variety of complex behavior such as schooling, predator avoidance, prey detection, communication, object detection and avoidance, and rheotaxis [16]. Due to the nature of adaptation across diverse environments, the lateral line has evolved for a variety of different tasks [3], including detecting flow conditions [17], encoding a Karman Vortex Street [1], and object identification [18]. The system is comprised of a network of sensors called neuromasts which exist both on the skin and inside a series of porous canals just below the skin ( Figure 2).
A neuromast is a bundle of hair cells covered by a gelatinous mass called a cupula, which provides an interface between the hair cells and surrounding fluid [19]. The bundle can contain anywhere from hundreds to thousands of hairs each with a ciliary bundle at the apical surface of the cell, whose function is closely related to that of hair cells found in the auditory systems of vertebrates. The bundles shown in Figure 3 are composed of stereocilia organized in order of their height with the tallest next to an elongated kinocilium [1,3].
There are two types of neuromasts: superficial neuromasts (SN) and canal neuromasts (CN) as seen in Figure 3. SNs are smaller receptors on the surface of the body that are sensitive to changes in fluid velocity. CNs, however, are larger pressure-gradient-sensitive receptors which appear to be able to respond to more rapid changes [20]. The difference in these forms of stimuli can be seen in Figure 4, Figure 2: A diagram of the Lateral Line Canal (llc) on the trunk, highlighting the canal neuromasts (n) and the posterior lateral line nerve (plln). Also shown are the pores, or openings, from the canal to the surrounding fluid (p), the scales of the fish (s), and the underlying muscular tissue (m) [2] where the SN is stimulated by an external flow of fluid while the CN is stimulated by pressure changes inside the canal due to movement of fluid along the pores of the canal.
Studies of behavior in fish have shown that blind cave fish are capable of detecting the presence of new obstacles in their environment and their ability to recognize both the size and shape of the object [21]. It has also been shown that the blind cave fish can be trained to make decisions based on object shape, they will make use of their lateral line canal (llc) by passing by the object prior to decision making [22]. The intriguing functionality of this sensory organ has lead to research in the field of artificial lateral line sensing for a variety of marine applications.

Previous Work on Artificial Lateral Lines
The field of artificial lateral line sensing and object recognition has been approached from a variety of angles from both the side of bio-mimicry and bioinspiration. Much of this research has been focused on the design of novel pressure and velocity sensors that mimic the lateral line system [23,24,25] with a variety of methods developed for interpreting these measurements. Sensing systems such as these could be used for mapping, navigation, or control system design for marine vehicles especially in turbid environments where traditional sensing systems become compromised.

Artificial Lateral Lines Sensor Development
Due to the types of stimuli to which it depends, engineers and scientists often considered the lateral line system analogous to an array of pressure sensors.
Fernandez et al. [26] developed a set of small pressure sensors that could contour to a wide variety of marine vehicles which could mimic this system. They tested a strain-gage pressure sensor with a diameter near 1 mm, and spaced them a few millimeters apart on a sensing body to mimic the lateral line. The resolution of the sensors was on the order of 1 Pa. Systems such as these have been built for a wide variety of investigations, including the detection of ground effect [27], hydrodynamic imaging [28], and object tracking [20].
Venturelli et al. [29] demonstrated that artificial lateral lines on fish-like robots could be used to distinguish between uniform fluid flows and a Karman vortex street. The Karman vortex street was distinguished from the uniform flow via use of the turbulence intensity on the surface of the body and a frequency spectrum analysis (FFT) to find the number of sensors with the same dominant frequency.
Pressure differences across the body were used to determine orientation and position of the body, and cross-correlation among the sensors was used to find the vortex traveling speed. Using this information, an estimate of wake wavelength was also determined based on vortex traveling speed and shedding frequency [29].

Object Detection and Identification
The problem of object detection is of particular interest for marine vehicles in regions where the ocean floor is close. It was shown by Fernandez et al. [30] and Maertens et al. [31] that object detection and tracking could be accomplished using pressure sensors in both passive and active systems. Bouffanais et al. [32] showed that location, size, and general shape of an object could be determined from embedded pressure sensors. In their experiments, a potential flow panel method was used to model a foil and a generalized transform was used to describe a shape near the foil. The shape description allowed for simple definitions of characteristics like size, location, and orientation. Through inversion of the pressure measurements via an unscented Kalman Filter it was possible to separately determine these characteristics, though it was determined that the sensing body must be sufficiently close to the object in order to properly characterize the object (within one body length).
Maertens et al. [31] applied this method of detection to a variety of object shapes and multiple bodies. In these experiments, a NACA 0018 foil body with embedded pressure sensors was dragged past a stationary object and the method described by Bouffanais et al. [32] was used to model object characteristics. The method of detection was validated for regimes which closely match potential flow approximations, however estimation of shape parameters can break down when flow separation causes significant deviations in the pressure determined from potential flow models. Additionally, the definition of object shapes in the generalized transform limited object recognition to a set of predefined shapes due to the nature of the parameterization method.
It has been demonstrated that viscous effects are critically important and influence physical lateral line systems on both the boundary layer and receptor scale [33]. Maertens and Triantafyllou [34] established that in the presence of separation, the shape information derived from pressure sensing may be improved through inclusion of viscous effects. In these experiments it was first determined through use of viscous simulations and potential flow models that the boundary layer on the surface on a moving fish is significantly influenced by unsteady viscous effects.
Further, through linear stability analysis of the boundary layer of a cylindrical body, a correction was applied to potential flow modeled pressures based on the displacement thickness of the boundary layer. It was also seen in these experiments that the boundary layer can act as a signal amplifier, providing improved estimates of object characteristics. While this provided evidence that viscous effects are important in object recognition, the method was limited due to the fact that viscous information was determined a priori through learning algorithms and thus could not be applied in real time.
Further, Boulogne et al. [35] applied the techniques of machine learning and neural networking for object localization using an artificial lateral line. They implemented a variety of machine learning algorithms to simulated pressure time histories of a cylinder moving past a stationary lateral line array. The cylinder was allowed to move through an x,y-plane at an angle relative to the sensor array, and the corresponding pressure from a potential flow model was used to train the network to determine the horizontal and vertical position of the cylinder relative to the array at any point in time. It was shown here that it was possible to determine the position of the cylinder relative to the sensing array with low euclidean error, however the exclusion of viscous effects and the use of a stationary sensing system limited the applicability of this method to a moving body. As it is known that the viscous boundary layer is critically important to the function of the lateral line of fish [33], it is necessary to use a more accurate model of the fluid in order to make this method more applicable in a marine setting.

Statement of Purpose
To date, no experiments have been done to determine a general object identification method the uses a viscous fluid model. For the development of a general method, the problem of a streamlined body passing by a protrusion from a flat wall was considered. Akin to a fish swimming by an object, measurements of pressure on the surface of the body were used in conjuncture with the developed method to determine characteristics of the protrusion. The purpose of this investigation is to improve upon previous work in two key areas. First, a fully viscous fluid model will be applied to directly include viscous effects in an object detection algorithm.
Second, to characterize an object's general shape, two methods will be investigated for object detection. Object recognition is performed using classification via machine learning as well as shape definitions via standard spline definitions coupled with an Unscented Kalman filter. These modifications use a viscous flow model for boundary identification to enable boundary updating in real-time digital twin systems.

Document Organization
Chapter 2 introduces the 2-D viscous simulation used as the model for the object recognition method. This is both an introduction of the simulation and an explanation of the implementation of modifiable boundary conditions.
Chapter 3 provides a method of object recognition using machine learning and neural networking.
Chapter 4 demonstrates a more general method of object recognition using an Unscented Kalman filter.
Chapter 5 provides an investigation and discussion of the method as well as an analysis of the results.
Chapter 6 and Chapter 7 provide conclusions and recommendations for future work.  Figure 6 shows the influence of a wall feature on the pressure in the fluid at three instances in time in LilyPad. Here it can be seen that the change in pressure in the fluid is gradual when approaching and moving away from the wall feature, with an increase in the magnitude of the pressure when the body is in the proximity of the wall feature. This demonstrates that the formation of the wake is gradual as the body moves through the fluid, and that the presence of the body causes a measurable change in the pressure. In the development of the algorithm, all measurements of pressure were generated through simulations, and no physical measurements were used, although the algorithm will be tested with physical measurements in the future.

Simulation Definitions
The numerical model used to represent the system was an open source, 2-D, Cartesian-Grid, Navier-Stokes solver named LilyPad [36]. Lilypad is a numerical tool designed to perform complex fluid-structure simulations at low computational cost. It is of particular interest to researchers as it provides immediate visual feedback and is easily modified. It implements the boundary data immersion method (BDIM) [37], an immersed boundary technique which uses kernal functions to blend field equations and boundary conditions for fluids and solids near solid boundaries. BDIM has been demonstrated to achieve second order accuracy in force estimates and has been validated for a variety of fluid-structure interaction (c) k = 18 Figure 6: The pressure in the fluid as the foil body passes by an object on a wall at three discrete points in time. The pressure from LilyPad is shown with contours of color described by the color bar to the right of each plot.
Typically, it is necessary to test the convergence and validation of a simulation in order to determine what characteristics and scales must be matched to properly capture the physics of the problem. In this investigation, however, it was of interest to develop an algorithm that could determine the conditions of the system based on a set of measurements from a simulated system. Due to this, it was not critically important that the model capture the exact physics of the problem.  Figure 5) and pressure was determined as a function of time via interpolation from the nearest computational grid point.

Object Shape
In the development of the estimation methods, several standard shapes are also defined in order to investigate the use of pressure as an identifier of object shape. Three shapes were investigated initially: a triangle, a half-circle, and a rectangle (shown in Figure 7). Length scales in the simulations were determined in non-dimensional units of chord lengths of the foil such that the height h * w , length l * w , distance from the wall H * , and distance to the center of the feature x * w were all relative to the size of the body. See Table 1 for further explanation of these parameters.
Here it was hypothesized that for any given feature of a specific size and shape, there would be a unique pressure signature along the length of the foil. This can be seen in Figure 8 and Figure 9 where pressure signatures for the three shapes of interest with varying height and length are presented. Figure 8 demonstrates the effect of varying the height of the feature on the measured pressure while Figure 9 shows the effect of varying length. All simulations were performed at a constant value for H * , and the pressure is presented as a spatial history relative to the location of the foil as it moves past the feature.  Figure 7: A diagram of the design of each wall shape for simulations. h * w , l * w , H * , and x * refer to the height and length of the feature, the distance between the foil and the wall, and the distance between the foil and the center of the feature, all non-dimensionalized by the chord length of the foil.
From Figure 8 it is clear that as the height of the feature increases, the differences in the shapes of the pressure signatures becomes more pronounced.
Further it can be seen that while the front of the foil (left column of Figure 8) Table 1: List of geometric parameters of the wall feature passing by the foil body.
Length of the bump (length-to-chord) l * w Also: l/c or L/c Height of the bump (height-to-chord) h * w Also: h/c or H/c Distance between the foil and the flat wall H * Also: H Distance between the foil and the center of the feature x * Also: x/c shows relatively unique pressure signatures, the shapes of the pressures seen in the wake of the foil tend to be much closer in trend and shape (right column of Figure 8). Comparing to Figure 9, it can clearly be seen that the length of the feature is critically important for distinguishing between shapes. Observing both columns in Figure 9, it can be seen that the pressure signatures begin to widen and show differences between one another.
The noticeable difference in these pressure signatures shows that pressure in a wake contains unique information about the object that created the wake. It is expected that similarities in time histories could make distinguishing shapes difficult, so pressure signatures on the leading edge and wake of the foil are used to identify unique information about features. Using this information, a neural network is developed to investigate shape recognition similar to the experiment by Boulogne et al. [35]. x/c

CHAPTER 3 Classification with Machine Learning
Machine learning can provide subtle solutions, and make connections that otherwise would be impossible to model [38]. This can be both a blessing and a curse, as neural networks often provide accurate answers, but require massive training sets to give sufficiently generalized solutions. Using machine learning allows for the design of a classifier that can make decisions about what types of features exist in the local flow field based on pressure time histories. There exist two major processing types in machine learning that could be applied to this type of data: supervised learning, and unsupervised learning. Supervised learning is used when an answer is known and a set of inputs and corresponding outputs can be compiled such that the network can determine a relationship between inputs and outputs.
Unsupervised learning is typically used when recognizable patterns in the data are not already known and rules must be developed based solely on observed data [38].
Supervised learning is used in neural networking to train predictors based on known inputs and outputs. Under the category of supervised learning there exists a number of training models that can be implemented based on the structure of the data. Three important methods that can be used in a classification learner are linear/non-linear discriminant learning, support vector machine (SVM) learning, and k-Nearest Neighbors (kNN) learning. The discriminant learning method and the SVM share similar properties, however a discriminant learner attempts to separate inputs based on multi-degree lines and a SVM method seperates inputs based on multi-degree hyperplanes [39]. Both of these methods create separations between clusters to judge where inputs fall based on their distance to each separation, and uses these to make predictions about unknown inputs. The kNN method is different due to the fact that it bases clusters of inputs based on k similarities seen in the data set, meaning that as k increase so does the amount of overlap in classes leading to a decrease in accuracy [39].
Due to the nature of this problem it was determined that supervised learning was the appropriate avenue for designing a classifier, with pressure time histories as input for determining wall shapes.

Training
For classification, a significant amount of training data is necessary in order to make accurate predictions. In order to define a simple classifier, the distance from completed. This separation of data was done to assure that data used for the secondary validation had not been used to train the classifier, and thus the network could be tested with completely unknown inputs. The data was separated into a Matlab table, similar to Table 2, with columns representing pressure time histories from multiple locations along the length of the foil as inputs to the system and the final column corresponding to the known class. The table was designed such each row corresponded to a particular simulation run, meaning that in the training set there was a total of 2880 rows. This table was loaded into the classification learner toolbox in Matlab using an internal cross-validation method to avoid over-fitting the classifier. Cross-validation partitions the training data into a set number of "folds", and the accuracy of the classifier is determined over each fold before being globally compared. It was decided to use 30 folds, and to simultaneously train the classifier using the three methods mentioned above, such that the accuracy of each learning style could be compared.

Triangle Versus Flat Wall
For the development of a classifier, it is critically important that the training sets used provide information sufficient to distinguish between known classes. Six measurements of pressure along the length of the foil body, three near the leading edge and three in the wake, were used to provide information about the protrusions class. The data was then separated such that only simulations pertaining to flat walls or triangles were input into the training table, in order to determine the classifiers ability to distinguish a single feature from a flat wall.
A number of classifiers were trained using decrement methods, SVM methods, and a kNN method to determine which style of classification fit the data best. It was then relevant to remove the number of inputs until a reduction in accuracy was noticed. This was done by re-training the networks with fewer time histories and observing the accuracy of prediction. As can be seen below in Figure 10, the prediction ability of each model was relatively unaffected as the number of time histories was reduced. In Figure 11 it can be seen that classification between a flat wall and a triangular feature protruding from a wall is possible, and easily done by the kNN network.
Even the smallest triangular feature was distinguishable from a flat wall due to the differences in magnitude seen in Figure 8. For this reason, it was determined that complexity could be added to the network to test it's ability when more class options are available.

All Shapes
The classifier is next defined using four available classifications: flat wall, triangular wall, circular wall, and a rounded square wall. This was done in a similar manner as before where 1445 sets of time histories were generated for each shape.
Again, half of the data was used for training and half of the data was set aside for later validation of the exported model. The same six points on the foil were used as time histories in the classification method. Fitting of the data to a variety of classification methods is seen in Figure 12 for different combinations of pressure time history locations. It can again be seen that as long as a point at the front and a point in the wake are used, only two points are needed for classification. Interestingly, the cubic SVM outperforms the kNN model when more shapes are used, which can be seen by comparing Figure 12 and Figure 10. Before, the kNN only outperformed the SVM models by a 1-2%, but when the problem became more complex, the cubic SVM was 10% better at estimating the correct shape.
Further information can be gleaned about the performance of this classifier by observing the confusion matrix from the machine learning toolbox ( Figure 13). Here it can be seen that 2.2% of the flat wall features in the training of the neural network were misclassified as triangular walls, meaning that a protrusion was predicted when none existed. Further, the majority of misclassification was accidentally classifying a feature as a triangle when it was not. Most important to note is that a wall feature was never misclassified as a flat wall, meaning that while there may be confusion in shape, there is no confusion about the fact that a protrusion existed.  Figure 14: Personal validation of the classifier using pressure signatures for all features of interest. Plus signs signify flat walls, triangles are shown with triangle markers, half-circles with circle markers, and rounded squares with square markers. A blue marker represents a correct classification, purple represents a misclassified feature as another feature, and red is a misclassified feature as a flat wall.
From Figure 13 and Figure 14, it is clear that the classifier has difficulty when a shape of a certain size looks like a different feature of a different size, e.g. a long circle looking like a rounded square. This seems reasonable when considering Figure 8, where the general shape of the pressure measurements were the same.
This could lead to features of certain length scales being misclassified as features of other length scales. In order to account for the fact that size can affect the ability of the classifier to appropriately determine shape, a third classifier was developed to incorporate size distinctions.

Shape and Size
To incorporate size in the classifier, a definition was chosen to signify whether shapes were small, medium, or large. Small shapes were defined to be approximately 1 / 3 the maximum possible size or smaller, medium shapes were defined to be between 1 / 3 and 2 / 3 the maximum size, and large shapes were defined to be larger than 2 / 3 the max size. The relative size of the shape could then be determined using a size assigned to both the height and the length, for example a "small" shape could have small length and height, or one small characteristic and another medium. A full explanation of all the possible size combinations can be seen below in Table 3, where "S" is small, "M" is medium, and "L" is large. Now shapes may be classified by shape and size, e.g. a triangle with small length and small height would be classified as "sTriangle".
All shapes were then appended with sizes (s, m, or l) according to this schema and re-processed for training a new classifier using size and shape. This classifier was tested with the same methods as previous classifiers, in order to determine the number of pressure points necessary and which classification method worked best.
It is clear from Figure 15 that the accuracy does not decrease as the number of time histories was reduced to two and further that the cubic SVM was the most accurate model. As seen in Figure 16, a wall feature is rarely misclassified as a flat wall (0.1%) and the majority of misclassification comes from size rather than shape. It can further be seen that while the majority of predictive issues revolve around the size of the feature, there can also be issues of prediction when the shapes are very small. This is most likely due to the fact that this is at the extent of the sensing range for this type of problem, and thus pressure signatures may be less unique than if the body were closer. In order to compare the predictive ability of the network, a second validation step was performed with half of the data ( Figure 17) and statistics about classification ability were computed (Table 4). Figure 16: The confusion matrix from the classification learner in MATLAB for shape and size. The letter before each shape corresponds to its relative size. Table 4 shows that the majority of misclassification of shape came from issues distinguishing between a circular wall and other features. This is to be expected as the shape of a circular pressure time history (Figure 8) is similar to the rounded square and triangular features, where the primary difference is that of the total magnitude of the signal. It can further be seen from Figure 17 that size was generally misclassified in regions where the shape fell on artificially imposed class boundaries, and the majority of shape misclassification occurred when the shape was near the extents (small or large) of the test matrix. The greatest errors in prediction came from attempting to distinguish between a circular and a triangular feature, especially when very small or large. While this proves to make classification of exact features difficult, it is important to note that there were almost no flat wall misclassifications and percent errors were relatively low. Due to the fact that the majority of misclassification occurred along imposed boundaries and in regions where the shape was at the extremes of the text matrix, more complexity was added to the classifier. Previously, adding shape classification as an option made it possible for the classifier to make more refined predictions and removed ambiguity around shapes that have similar pressure signatures for distinctly different length scales e.g. a small circle being classified as a medium size triangle. This was done again in the hopes of further removing ambiguity in the solution by further refining the size.  Table 3.

More General Size Classifier
For the fourth iteration of the classifier sizes were broken into 9 different categories and assigned one of three shapes such that there were 28 total unique classes, 27 shape-size combinations and one flat wall class. his allowed for an extension of the previous size regimes to include both relative length and height of the features (Table 5). Sizes were assigned to each feature by length then height, followed by a shape class e.g. a triangle with medium length and small height would be classified as "MSTriangle". Issues of misclassificaiton were still expected near imposed boundaries and close to the extents of the testing range, though overall classification was expected to improve with this added complexity. Further, it was demonstrated in Figure 10, Figure 12, and Figure 15 that anywhere between two and six pressure measurements was sufficient for this type of prediction. Due to this, measurements were used at three locations along the span of the body. It can be seen from Figure 18 that the selection of the location of pressure measurements is relatively ambiguous, as the performance of this classifier was comparable to those presented previously.  Figure 18: Accuracy of the classifier using p 4 , p 7 , p 11 as specified by MATLAB.
Comparing Figure 18 and Figure 12 it can be seen that the prediction accuracy of the network is unchanged by the addition of more discrete sizes. This is expected, as neural networks can make connections in data that are not otherwise obvious, and should be able to make predictions so long as it has been trained with the right information.
Interestingly when comparing Table 4 and Table 6, it can be seen that adding complexity to the classifier did not significantly change the performance. In some cases, the error of classification actually increases (Triangle MCA a rounded square) however the overall error drops. This is a common issue is the design of neural networks where added complexity can only improve predictive abilities to a certain point, after which more available information does not significantly improve classification [40].  Figure 19 shows again that the predominate errors in classification are found on the artificially imposed borders of the regions of classification. There are also more errors in the region of small shapes than other regions, especially for triangular features. This is again due to the similarities between the pressure time history shapes. It can also be seen that the majority of misclassifications are of size instead of actual feature shape, confirmed by Table 6. Removing some of the ambiguity in the prediction of the size of the feature slightly improved the ability to predict the shape of the feature. (d) Figure 19: Validation of the classifier for 9 discrete sizes and 3 shapes. Plus signs signify flat walls, triangles are shown with triangle markers, half-circles with circle markers, and rounded squares with square markers. A blue marker represents a correct classification, purple represents a misclassified feature as another feature, and red is a misclassified feature as a flat wall. The size of the marker corresponds directly to the size of the feature, with bold markers showing misclassified size. The background color corresponds to Table 5.  Figure 20) it is easy to make an estimate of shape with almost no misclassification of the feature. This indicates that allowing the object to be close to the foil helps with identification. Allowing the foil to move vertically towards the wall could therefore help with classification but would require additional classes for distance from the wall.

Discussion
From the progression of the classifier, it can be seen that pressure is capable of measuring object information. Further, it appears that as the number of available classes in the training set increased, the classifier improved. The majority of the prediction errors were observed to be from misclassifying the size of the object.
This implies that if the exact size (e.g. the height and length in units rather than a relative size scale) and shape of each feature was used to classify the shapes, than the network would perform better overall.
While it is clear that this method works for object recognition of this type,

CHAPTER 4 B-spline UKF Estimator
In this chapter, the general B-spline representation of a shape is used with an Unscented Kalman filter (UKF ) in order to estimate a general wall shape over time.
For the classifier, a set of predefined wall protrusions were simulated in LilyPad and the pressure measured on the surface of the foil was used to infer class. Rather than design a wall with predefined shapes, the wall is described with a B-spline curve. B-splines, calculated using Equation 1, are 2-D line segments which are described by a set of control points multiplied by a set of weighted basis functions [41].
Where S m,t (x) is a particular spline segment of polynomial order m at point x relative to the knot vector t, P i are the D control points, and B i,m (x) is the basis function calculated relative to the knot vector (Equation 2 and 3).
The wall in the simulation is now defined by a set number of D control points which locally govern the shape of the solid boundary in LilyPad. Representing the wall with B-splines allows general shapes to be made by modification of control point locations, and provides a natural extension to three-dimensional NURBS surfaces for 3-D problems. For simplicity, the number of knots can be calculated based on the order of the polynomial and the number of desired control points such that t contains D + m + 1 knots, or weights.

The Unscented Kalman Filter
The UKF is an extension of the Linear Kalman Filter, which has the ability to account for highly non-linear state and measurement relationships by assuming Gaussian statistics for the unknown states [42].
where k is discrete time, P are the vertical position of N control points with distribution covariance C P , p is a time history of pressure modeled at defined locations on a foil that are the output of the non-linear process F . In this case

The Sigma Transformation
The sigma transformation is a method of selecting a set of statistically relevant weighted sample points to capture the statistics of a random variable distribution [42]. These points contain the necessary information about the true mean and covariance of the random variable by design of the transform. The number of sigma points (L) are determined through the number of states by the relationship L = 2N + 1, which allows for the addition of more estimated states with little additional computation needed. Wan and Van Der Merwe [42] show that each sigma point can be calculated based on the estimated mean state,P, and the estimated state covariance C P : where λ = α 2 (N + κ) − N is a scaling parameter corresponding to the mean and β ≥ 0 is used to statistically incorporate information of higher order moments of the distributions, and β = 2 is optimal for the standard normal distribution. Wan and Van Der Merwe [42] provide suggested values of α, κ, and β but it is important to note that they must be tuned for specific problems.

The Unscented Transform
The unscented transform is a method to estimate the mean and covariance of a transformed random variable distribution from the sigma points using the defined weights. Using the sigma points as determined above, the sigma points are where p F i represents the pressure time history from a numerical simulation run using the boundary conditions, X i , determined through the sigma transformation.
After calculating the mean of each random variable, it is possible to determine the covariance matrices corresponding to each distribution as in Equation 13 and Equation 14.
The unscented transform is designed such that it passes the first and second moment of the statistical distributions of the random variables that represent the system through the non-linear relationships between state and measurement [42].
This process captures the mean and covariance of the system random variables as determined by the sigma points, X i . When combined with the Kalman Filter, the estimates from the unscented transform can be corrected based on the error between the estimated measurement and the observed measurement. Combining the sigma and unscented transforms provides accurate estimates with fewer random variable realizations than would be necessary for statistically sampling a random variable. For example, the transform allows for only a handful of realizations to be performed to capture the random variable statistics rather than hundreds or thousands. This is important when running numerical simulations that require significant processing efforts.
From Wan and Van Der Merwe [42], the cross-variance between the state and measurement estimates must be calculated (Equation 15) to make a correction to the estimates from the unscented transform.
The gain of the filter, K, is then calculated from the cross-variance as in Equation 16 and used to correct the mean of the state (Equation 17) based on the measurement p meas and the covariance (Equation 18).

B-Spline Estimation Algorithm
In order to implement the UKF, a set of a priori decisions were made about the wall shape is only locally defined by m + 1 control points [41]. This implies that the curve does not need to be globally estimated, but could instead be locally estimated based on a subset of N control points near the foil body that shift in location as the body propagates forward in time.
To initialize the method, the wall was assumed to be flat such that the D Once an update to the current window of control points is made, the root mean square error (RMSE) is calculated between the measured value of pressure and the estimate from the UKF, µ F . This process is repeated using the current window of control points until a convergence criteria has been met based on the calculated RMSE. In this study, the convergence cutoff was defined as the iteration at which the percent error between the current iteration and the previous iteration changes by less than one percent. This is synonymous with the criteria that the current estimate of the states of the system are no longer being substantially modified using the current set of measurements. Here, n control points on the downstream side of the window are removed from the window and no longer allowed to change. The foil then moves forward in time and n new control points on the upstream side of the estimate are added to the window such that N control points are again being estimated (Algorithm 1 and Figure 21). For further explanation of the variables used in the formulation of the UKF, see Table A.1 in Appendix A.
An important aspect of fluids is that they contain memory, such that the pressure at a single time on the foil is dependent on the time history of the wake.
This means that while the true foil is moving forward in time, every simulation run in the unscented transform algorithm needs to be initialized from the same time in order to maintain the history of the wake. It is also necessary to keep a history of previously estimated states in order to maintain the wall shape downstream of the estimate window in each simulation. This causes the simulation time to increase as the number of sigma parameters being estimated increased. Once control points become locked in position, i.e. the window of N control points has moved so a local point is no longer allowed to move, the shape of the wall becomes fixed in that location. To reduce computational time, the simulations could be initialized to a start position later in time based on the fixed wall shape, however this is not implemented in the current version of the algorithm and will be an aspect of future work.

Importance of Spline Order
Due to the design of the method, it is necessary to keep the order, m, of the spline curve estimate low. This is due to the local definition of the spline curve

5:
Run simulation for each X i concurrently (Include previously estimated control points) return p F i for each X i 6: function Unscented(W m,c , p F ,Eq 5) return µ F , C y 7: procedure Gain Update(W c , X, µ P , C x , p F , µ F , C y ) returnP, C P 8: procedure Compare Error

10:
Calculate the percent different in RMSE 11: if Percent difference ≤ threshold then 12: Save P i → P n control points 13: Shift P n+1 to P 1 , add n control points 14: Shift the covariance matrix 15: Step k forward in time 16: Re-sample measurement 17: Return to step 2

Important Characteristics
In order to assess the performance of the estimation, a variety of error metrics are observed in the flat wall regions of the boundary estimates and in regions where features exist. Since this method is developed in the interest of digital wins as well as object detection, the performance is considered from both the point of object recognition as well as boundary identification. For this reason, error metrics are defined based on the estimation of the width, height, and center location of the feature. Further, it was of interest to determine the minimum distance between the foil body and boundary estimate as well as the total RMSE between the full shape estimate and the true wall shape.

CHAPTER 5
General Object Estimation Using UKF Algorithm

One Feature
We visit again the problem of a streamlined body passing by a protrusion from a flat wall. A general shape was defined with D = 124 control points (Figure 22), and ten control points in a local area modified to make a smooth, continuous protrusion from the wall. The protrusion was defined such that the total length of the feature, l * w , was less than one chord length, and the height, h * w , was chosen such that the body could be placed at a foil wall distance of H * = 0.75 without collision. This guaranteed that the flat wall regions were within the sensing range defined by Bouffanais et al. [32] for the entirety of each simulation. The important characteristics that define the feature are shown in Table 7 where H is the height of the feature, W is the width, dis is the minimum distance between the wall and the foil body, and cen is the location of the center of the feature in non-dimensional units (chord lengths).  [43] and Jauch et al. [44] in recursive spline estimation using Kalman Filters, it was decided that the noise level on the state would be set as Q = 0 and the noise level for the measurement was set proportional to the current LilyPad resolution, such that R = 0.5 × 10 −3 I. Further, following Wan and Van Der Merwe [42] and Julier [45], for this investigation it was decided to set the UKF constants to α = 1, β = 5, and κ = 1. The effect of these constants on the solution are evaluated later. Figure 22: The wall shape to be estimated using the B-spline UKF algorithm.
Pressure is measured at the shown locations along the foil body.

Standard b-Spline UKF
Initially, to estimate the B-spline we use a standard formulation of the UKF, where each pressure measurement at some given point in time is considered to be an observation of the system. Using this method the measurement array is constructed as a single column array, p meas : where    The fluctuations on the flat wall are partly due to the fact that the sensing body is relatively far from the flat wall, where sensitivity to differences in pressure will be small as discussed by Bouffanais et al. [32], however this does not explain the large error in shape observed on the back of the feature. Instead, since our estimate is based on a proxy of the wall shape through pressure measurements, errors in the wall shape that don't significantly alter the pressure (such as fluctuation in the wall where it should be flat) don't significantly affect the estimate. Pressure measurements in close vicinity to the true signal are then more valuable, thus the algorithm needs to be altered to give more weight to pressure measurements with more information. To do this, the formation of the measurement array was altered such that the estimate of the wall shape was derived from each location of pressure on the foil body individually, where each estimate could be weighted based on its ability to match the pressure signal. weight that lies between 0 and 1, with the sum of the weights over the i pressure measurements equal to 1. This is implemented as an expansion to the gain update formulation and pressure measurements with low RMSE to be given more preference in determining the estimated shape. This method is detailed in Algorithm 2.

Weighted b-spline UKF
Algorithm 2 Weighted Gain Update procedure Gain Update

Performance Analysis
Previously, error in predictions was discussed based on how the estimates visually compared to one another, however this does not provide a measurable performance of the algorithm. Following a similar analysis as Maertens [46], Table 8 provides a quantifiable error analysis for the standard and weighted measurement methods. The following error metrics are presented in non-dimensional units of chord lengths and percent error: 1. H -The height of the estimated feature off the flat wall.
2. W -The width of the estimated feature.
3. dis -The minimum distance between the foil body and the estimated feature over the full time history of the estimate.

4.
Cen -The estimated location of the center of the feature referenced from the simulation start point.

5.H -
The percent error of the height compared to the true shape.
6.W -The percent error of the width compared to the true shape.
7.dis -The percent error of the minimum distance between the foil and feature compared to the known distance.
8.Cen -The percent error of the center of the feature compared to the true shape.
9. RMSE T -The total RMSE of the spline estimate, including the flat wall.
10. RMSE F -The RMSE of the estimate only in the region close to the feature.
These ten metrics give a good measure of the performance of the method by identifying quantifiable errors about general features, as well as the quality of the overall spline estimate. From Table 8, one can see that the height of the feature, the location of the center of the feature, and the minimum distance between the foil and the feature was well captured for both the standard and the weighted measurement method, which was anticipated based on visual inspection. However, the standard measurement method shows a 44% error in estimation of the width of the feature, much higher than the 12% error in width using the weighted method. Overall, every general error metric is reduced when the weighted average implementation.
For the overall spline estimate, the weighted average method also reduces the local feature RMSE F by a factor of two and reduces the total RMSE T by a factor of three, demonstrating the improvement in estimation using the weighted method.   While the general shape of each pressure time history is nearly matched by the estimate, small deviations from the true values lead to minor adjustments in the control points, which will alter the shape estimate. While this method can estimate general boundary conditions from simulated pressure time histories, in a practical sense, it would be necessary to minimize these error. Downstream errors in the wall estimate will induce errors in upstream estimates that will propagate error in future estimates.
A significant portion of the observed error between pressure measurements appears as a time lag in the estimate. One approach to addressing this error would be slowing the forward speed of the foil, such that changes in the wall geometry are measured more gradually. A second method for addressing the time lag would be to phase shift the pressure measurements based on the lag.

Control Points
It has been demonstrated that the spline curve estimate from the weighted average method provides a good representation of the wall with relatively low error, with reasonable approximation of the true pressure time histories. However, in the weighted estimate of control points locations, visual inspection indicates that some control point locations are better estimated than others. Observing the vertical position of a select number of control points over time allows insight on the performance of the method in specific regions of the shape. Figure 26a shows the weighted average estimate with five labeled control points shown with a red + and labeled P 1 , . . . , P 5 , and Figure 26b shows the estimates of the positions of those five control points over time. The vertical locations of control points are well estimated for P 2 , P 3 , and P 5 . Figure 26b shows how these three control point estimates move towards the true locations and stay close to those positions until they are frozen. However, P 1 and P 4 deviate significantly from their true locations, either diverging from the start or never moving far enough. This could be due to over-fitting the spline with control points, choice of initial conditions, or the control point estimates becoming stuck in a local minimum. In the case of over-fitting, the estimate could benefit from removing excess control points or "thinning" of the spline (Figure 27). In (b), blue Os mean control points are still in the window, red Xs mean the window has passed and that is now the fixed location of the control point. Each subplot in (b) has a black line to indicate the true value of that control point.   Figure 27) it can be seen that the spline estimate comes much closer to the true shape without fluctuations. On the contrary, starting at the second control point (blue dashed-dot line in Figure 27) and removing every other point, the poorly estimated points are kept and the estimate is poor. The specific choice in which control points are removed has a strong effect on the curve.
Since the true shape is unknown in a given estimate, there is no way too determine which control points may be considered poor estimates and a different method is necessary if one wants to improve the estimate of the shape.

A Second Pass
Selection of initial conditions is critically important for any Bayesian estimation problem, as the estimate is predicated on the refining an a priori estimate. Maertens [46] demonstrated that performing a passing the object a second time using the previous estimate as the initial conditions can improve the estimate of fine features.
Following this, the weighted average estimate was used as the initial conditions for a second pass ( Figure 28). The second pass was performed under two circumstances: 1) using the unmodified control points from the weighted average estimate as the initial conditions and 2) using the weighted average estimate as the initial conditions but modifying the three control points known to be poor estimates.

Number of Control Points
Another factor in the developed algorithm is the number of control points estimated in a windowed region of size N near the foil body. A small study was performed to vary this number to see the effect on the solution. Figure 29 shows the true wall shape as well as wall estimates for four different window sizes, indicated with different colors and line styles. In each test, a different number of windowed control points N is used, with values ranging from 19 to 7.

Effect of location and number of pressure measurements
The number of pressure measurements used for estimating the wall shape can also have a significant effect on the resulting estimate. This is important for practical implementation of the algorithm in a physical system. Selecting points along the foil that provided good information about the feature can improve the accuracy of the estimate while lowering the overall computational costs. Figure 30 below   Figure 30 shows that there must be a minimum of two pressure measurements along the length of the foil in order to estimate the shape. Interestingly, it appears that even with a single measurement of pressure, it was possible to determine that a protrusion exists however the estimate of the whole wall appears to be inverted from the true shape. Additionally, as long as there is a measurement of pressure in the wake and near the front of the foil, the feature is distinguished. Estimation of the flat wall regions were similar for four and 16 pressure measurements. While the estimate was more accurate using all measurements of pressure, it is not necessary to use this large number of measurements to estimate the shape. Conversely, it may be necessary to use more measurements of pressure along the length of the body if one wants to recover the exact shape.

Importance of Measurement Noise
Kalman Filters are predicated on the fact that both the measurement and the state are inherently noisy systems. Due to this, the selection of the noise level is critically important for the success of estimation. In the present study, all measurements are generated through simulated systems, and thus the noise on the measurement and state were assumed to be the resolution level of LilyPad and Matlab respectively. For a physical system, it would be necessary to first characterize the noise levels of the individual sensors, as well as in the entire system, to properly model the noise. It may also be necessary to allow the noise on each sensor as a state of the system, as the noise level could change based on the speed of the body, the characteristics of the fluid, or noise from things outside the system.
While this would increase the number of active sigma points, it would guarantee that the noise in the model remains relevant to the physical system.

How α, β, and κ Affect Estimation
The design of a UKF is sensitive to the values chosen for α, β, and κ since they control the approximation of the statistical distributions for the estimated random variables. These values must be "tuned" a priori for a given system to optimize the performance of the estimator. The most important constant is α, as it controls the spread of sigma points around the current estimated state and also affects the magnitude of the weights used in the unscented transform. Small values of α will allow only small variability about the current estimated state which may result in difficulty in capturing the true statistics of the system. Further, when the value of α is small, the weights for the unscented transform are large, which can provide heavy weight to a poor solution. On the contrary, when α is too large, it is possible that the true state of the system will be difficult to estimate due to too much variability allowed in determining sigma points, though the weights will be lower and the risk of applying heavy weight to a poor solution is smaller.   κ also has a small effect on the estimation and it was often suggested in the literature to set κ as 0 or 1 unless the covariance matrix of the state estimate is forced away from being positive definite [42,45]. In Figure 33 one sees that increasing the value of κ can greatly influence the estimate in a negative way. As the value of κ increases, the estimate on both the front and the back of the feature diverges from the true shape. Increasing κ can increase fluctuations in the flat wall estimate, as seen to a severe degree for κ = 5.

Multiple Shapes
The weighted estimation algorithm is tested with multiple shapes to investigate the estimation method under more general circumstances. Here, two features protrude from the wall in the vicinity of the foil, and the distance between the protrusions is varied. In particular there is interest in: 1) how the estimate performs with a wall that has multiple features and 2) how having features close to one another can create ambiguity in the estimate. For comparison, we define a case where the features are close to one another ( Figure 34a) and spread further apart ( Figure 34b). Figure 34 shows the final wall estimates for both cases using the weighted average UKF, with circles indicating the estimated peaks of each feature. Table 9 presents the performance analysis for both the near and far cases.    Figure 34a where the height and width of the overall feature is estimated reasonably well, however the true shape is not distinguishable, with some blending of the two features. This is likely due to the formation of a recirculation region between the peaks of the two features, creating a situation where pressure may not be able to distinguish between solid walls and a streamline between peaks.
In contrast, moving the shapes apart allows far less ambiguity in the wake such that information about the second peak is captured, as seen in Figure 34b. In this example, the a separation distance was approximately one chord length of the foil, such that the second shape was not passed by the foil until the first shape had already been passed with an estimate frozen in place. In this case, the size and location of both features was reasonable, though the second shape is estimated to be wider and shorter than the true feature.
Contrary to the visual inspection in Figure 34, where it appears the estimate of case 2 is better, Table 9 shows that the two cases are similar in terms of error. In case 1, where the two shapes are near to one another, it was difficult to distinguish between the two features in the estimate and the errors were relatively high for both features. Interestingly, when the features are separated as in case 2, the first feature is estimated with relatively low error and the second feature becomes more difficult to estimate. This in somewhat expected, as any errors in the estimate of first feature will propagate in the simulations used to estimate the second feature, inducing additional errors. Overall, while it is possible to distinguish between multiple shapes it becomes more difficult to determine the true shape due propagation of error. Table 9: Error metrics for both multi-shape cases. Height, width, and distance are in units of chord lengths, and the center is defined by the physical location of the center referenced from the start position of the simulations. Each feature is presented separately, while RMSE T was common between both features. The characteristics of the true wall are also presented for comparison.   Table 10, where each divided row corresponds to a previously tested example. Each row of Table 10 presents the true feature characteristics, the original estimate from the UKF, and the estimate after smoothing (in bold text).  The true wall is shown (black) as well as the estimate from the UKF (orange) and the smoothed solution (blue). Circles with colors corresponding to each wall shape are shown to indicate the location of the observed peak(s).    These trends are emphasized in Table 10 where smoothing only improves results for the singles feature weighted method solution. For the standard update formulation, smoothing caused increases in percent errors between 2% and 58%, and increased the total curve RMSE by ∼700%. The weighted average solution shows reduced errors in specific characteristic estimates such as width, the total RMSE, and the feature RMSE, while other error values are increased by a significant margin. The minimum distance error increased from 1% to 34%, the error in the height was changed from 0.3% to 9%, and the error in the estimate of the center was increased from 0.5% to 5%. Further, Table 10 shows that for both cases involving multiple feature, the error either increased or remained approximately the same.

Method
Case 1 and case 2 in Table 10 demonstrate that if the case does not present with low initial error, smoothing can accentuate error rather than improve the solution. While the method is able to reliably recognize the location of features and does a good job of determining shape and size, it is sensitive to the selection of the UKF constants α, β, and κ. It is critically important for the quality of the estimate that an adequate number of pressure measurements, an appropriate window size, and a sufficient number of control points was selected such that the true characteristics of the feature are captured by the estimated signal.
Also demonstrated was the ability of the method to identify multiple features on a wall, though bringing the shapes too close caused the system to estimate a single feature. Here, it was difficult to determine features that were too close due to the region between the peaks of the features over which the wake of the body did not have time to respond to the second feature. Conversely, when there is sufficient spacing between the features such that the first feature is frozen in place during the estimation of the second feature, the estimate of the second feature improves.

CHAPTER 7
Suggested Future Work • Increase the resolution of the simulation in both time and space such that the solution more accurately captures the physics of the system. A simulation resolution on the order of 2 9 or 2 10 with a foil body resolution of at least 20 grid points should provide more information at every time step, as well as finer information that was not available in the previously coarse mesh.
Using this method, it would also be reasonable to measure pressure at more locations along the length of the foil.
• Investigate different estimation windows and how the size and shape of the feature of interest can influence the required window size.
• Test the influence of the constants α, β, and κ for a variety of more generalized shapes than those used in this investigation, it an attempt to determine globally default values for each constant.
• Investigate a more dynamic definition of α such that the spread of the sigma points was determined in-situ at each iteration based on the quality of the previous estimate.
• The error space of the estimate window should be observed to determine the influence each control point (relative to its position in the window) has on each measurement of pressure on the body. This information could be used to modify the weighting step such that each control point in a window estimate corresponding to a specific pressure measurement is weighted on its relative affect on that pressure measurement location, and then the whole window is weighted as previously mentioned.
• Perform an analysis of the propagation of downstream errors in the estimate, in order to quantify the influence of mis-calculated control points on upstream estimates. Errors in the downstream control points influence the location of upstream control points due to induced errors in the shape of the pressure measurement.
• LilyPad can be initialized with a field of vorticity, such that rather than

APPENDIX B
Description of file structures The following is a description of the files relevant to this thesis. Bold names indicate folders, italicized names indicate files. · output dp.txt -The third column to the end are time histories of the change in pressure between a location on the foil and the stagnation pressure. * LilyPad: This is the source files, modify these values and then export a new .exe (or your preferred operating system) for use with the ukf. * LilyPad.exe -An executable from processing to run the LilyPad simulation.
-Full wall: This contains the observations (pressure), corresponding input files, and the ukf results. * input files/test files: has all the input files/pressure outputs · small shape h0p75 furthest splines.txt -The standard case used.
Presented for column and weighted average method.
· small shape h0p75 twoPeaks splines.txt -The multi-feature case for the close shapes.
· TwoSpread shapes h0p75.txt -The multi-feature case for the far shapes. * rolling ukf : all the output files from the ukf · dd-mm-yyyy HH-MM-SS pressure plots: Pressure comparison plots made during method.
· dd-mm-yyyy HH-MM-SS ##CP initialize.txt(.m) -The initialization for a run, ## corresponds to the number of CPs used. xCp, yCp contains the (x,y) control points for the entire wall estimated thus far at the end of each time step.
-SigmaSims: the Sigma simulations referenced by the ukf. Contains files BezWall##, which are files corresponding to the ## th sigma simulation. Each file contains a data folder, a Java lib folder, and the sigma simulation LilyPad.exe executable.
-Single(three) ukf : Results from the ukf before the window.
process ukf bsplines.m -Processed the bspline estimates before the window.
b spline.m -A function for generating a basis spline.
rolling ukf figures( type).m -There are many iterations of this script.
They are for generating figures of different types from the outputs of the ukf depending on what type of figure is wanted.
send shape ukf.m -The original ukf method.
bspline ukf array.m -The windowed ukf with measurement as an array.
bspline ukf struct.m -The windowed ukf with the weighted average measurement formulation.
-SigmaSimsFileCopy.m -Used to copy sigma simulations from the source folder to the sigmaSims folder. This is not worth doing by hand.
• UKF foil: The investigation of mean heave distance with LilyPad.
• UKF Basic: Source material about the UKF.