AN ABSTRACT of the DISSERTATION of Andrew W. Otto

AN ABSTRACT OF THE DISSERTATION OF

Andrew W. Otto for the degree of Doctor of Philosophy in Robotics and Mechanical Engineering presented on August 7, 2019. Title: Orb Web Vibrations: Modeling, Localization, and Measurement

Abstract approved:

Ross L. Hatton

The orb web is a multipurpose structure—serving as a home for its inhabitant spider, as a prey capture device, and as a physical extension of the spider’s sensing abilities. Biologists are particularly interested in the spider’s ability to locate prey trapped in its web by sensing the vibrations induced from the initial impact and subsequent struggles of an insect. The presented work focuses on the web as a dynamic structure to better understand its role as a sensing tool for the spider by 1) studying vibrations of synthetic, bio-mimetic orb webs in a controlled engineering environment, 2) creating a computational model for orb web vibrations that includes the effects of web geometry, tension, and material composition, 3) proposing a vibration localization framework suitable for synthetic webs, and 4) combining optical flow measurements from high speed video of webs under motion with experimental modal analysis techniques to study vibration in webs of Araneus diadematus. The vibration model is successfully validated against a set of synthetic webs and used to identify cues useful for vibration localization. The performance of these vibration cues is tested on synthetic webs, and several factors influencing the success rate of the proposed vibration localization framework is presented. Novel motion data of orb webs obtained using phase-based optical flow from high speed video is used to characterize the impulse response of webs built by A. diadematus, and the strengths of optical flow for non-contact motion measurement versus standard tools such as the Laser Doppler Vibrometer (LDV) is discussed. c Copyright by Andrew W. Otto August 7, 2019 All Rights Reserved Orb Web Vibrations: Modeling, Localization, and Measurement by Andrew W. Otto

A DISSERTATION submitted to Oregon State University

in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Presented August 7, 2019 Commencement June 2020 Doctor of Philosophy dissertation of Andrew W. Otto presented on August 7,2019

APPROVED:

Major Professor, representing Robotics and Mechanical Engineering

Director, Robotics Program

Dean of the Graduate School

I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request.

Andrew W. Otto, Author ACKNOWLEDGEMENTS

The author expresses sincere gratitude to Prof. Hatton for his guidance, input, and patience throughout my time at Oregon State. Special thanks is given to Prof. Udell for his close collaboration on the SpiderHarp and his excellent musicianship, without which getting my research in front of others would have proved far more diﬃcult. CONTRIBUTION OF AUTHORS

Andrew W. Otto prepared the manuscripts, designed the test fixtures, wrote the computer code, and performed the experiments covered in this dissertation. Ross L. Hatton aided in manuscript preparation and final editing. Damian O. Elias provided feedback on the first manuscript and took part in final editing. TABLE OF CONTENTS

Page

1 Introduction 1

2 Modeling Transverse Vibration in Spider Webs Using Frequency-Based Dy- namic Substructuring 4 2.1 Abstract ...... 5 2.2 Introduction ...... 6 2.3 Related Work ...... 7 2.3.1 Web Vibrations ...... 7 2.3.2 Dynamic Substructuring ...... 9 2.4 Materials and Methods ...... 10 2.4.1 Artiﬁcial Webs ...... 10 2.4.2 Vibration Model ...... 12 2.5 Results and Discussion ...... 20 2.5.1 Model Validation ...... 20 2.5.2 Web Architecture Eﬀects ...... 21 2.5.3 Vibrational Cues for Stimulus Localization ...... 24 2.6 Conclusion ...... 28

3 Bioinspired Vibration Localization in Artiﬁcial Orb Webs 29 3.1 Abstract ...... 30 3.2 Introduction ...... 30 3.3 Materials and Methods ...... 33 3.3.1 Web Construction ...... 33 3.3.2 Testing ...... 34 3.3.3 Vibration Source Location Estimation ...... 36 3.3.4 Localization Algorithm ...... 42 3.4 Results ...... 43 3.4.1 Optimal Parameter Selection ...... 44 3.4.2 Web Architecture Eﬀects ...... 46 3.5 Conclusion ...... 48

4 Video-based Vibration Measurement of Orb Webs 50 4.1 Abstract ...... 51 4.2 Introduction ...... 51 4.3 Materials and Methods ...... 53 4.3.1 Gradient-based Optical Flow ...... 54 4.3.2 Phase-based Optical Flow ...... 56 4.3.3 Output-Only Modal Analysis ...... 62 TABLE OF CONTENTS (Continued)

Page

4.3.4 Experimental Setup ...... 64 4.3.5 Video Processing ...... 65 4.3.6 Post-Processing ...... 65 4.3.7 Natural Frequency and Damping Ratio Estimation ...... 67 4.4 Results ...... 68 4.4.1 Vibrometer and Optical Flow Comparison ...... 68 4.4.2 Frequency Domain Decomposition ...... 73 4.5 Conclusion ...... 76

5 Conclusion 79

References 81 LIST OF FIGURES

Figure Page

2.1 Illustration of the un-tensioned web geometry and parameters used in this study ...... 11 2.2 (a) Stress-strain data for the parachute cord and shock cord used in the study as well as dragline and viscid spider silk adapted from [22], (b) the test stand and artificial web used in this study ...... 12 2.3 (a) Tension distribution and (b) strain distribution output from the web pre-processing step for a sample web ...... 14 2.4 Model (solid, orange) and experimental (dotted, black) frequency response (taken as the output acceleration magnitude divided by the input acceleration magnitude) for different values of point mass (m) and radial pretension (τ), (a) m = 10 g and τ = 115 N, (b) m = 70 g and τ = 115 N, (c) m = 10 g and τ = 160 N, (d) m = 70 g and τ = 160 N ...... 22 2.5 Web architecture effects on web frequency response (accelerance A(ω)), (a-d) web frequency response at low levels, (e-h) web frequency response at high levels, (i) web frequency response at nominal level, (j) input (black) and output (colors) locations for web frequency response 23 2.6 Influence of vibration stimulus location (at left, black dot) on leg frequency responses (given here as accelerance) and spectral centroid ωc (at center) as well as normalized leg energies (at right) for multiple vibration stimulus placements within the capture spiral for a single web architecture ...... 26

3.1 Slack geoemetry and tension distributions of the β = 45◦ web (a–c), and for the β = 0◦ web (d–e) ...... 35 3.2 Artiﬁcial web, test stand, and 3D printed sensor body (located at web center) used in the experiment ...... 36 3.3 Overview of the orientation estimation process, (a) input pluck location (black) and accelerometer placements (colors), (b) acceleration time histories from the applied pluck, (c) clipped section of the acceleration time histories around pluck onset, (d) visualization of the leg RMS values and predicted orientation using the polygon centroid, (e) visualization of the leg correlation values and predicted orientation . . 38 LIST OF FIGURES (Continued)

Figure Page

3.4 Overview of the range estimation process, (a) input pluck location (black) and accelerometer placements (colors), (b) acceleration time histories near pluck onset, (c) power spectral densities of each accelerometer and the mean peak frequency fp, (d) peak frequencies sorted by range from the web center and curve fit to a decaying exponential function ...... 41 3.5 Flowchart of the vibration source localization algorithm ...... 43 3.6 Response surfaces representing total location error from Eq. (3.3) for each of the web treatments, (a) the 0◦ web at 100 N, (b) the 0◦ web at 120 N, (c) the 45◦ web at 100 N, (d) the 45◦ web at 120 N ...... 45 3.7 Estimated vs. actual plots of source orientation θ and range R for each of the web treatments at their corresponding optimal time delays, (a–d) estimated vs. actual curves for orientation θ, (e–h) estimated vs. actual curves for range R. Perfect prediction is indicated by a dashed line in all plots ...... 47 3.8 Boxplots of euclidean distance error for each web treatment and the localization algorithm using optimal values for tθ and tR ...... 48 4.1 Contours of half maximums (a) and a 1D slice (b) of the Gabor filter bank used in this paper ...... 59 4.2 The three webs and their respective 40×40 px ROIs used for comparison with vibrometer recordings, (a) Web 1, (b) Web 2, and (c) Web 3...... 66 4.3 Velocity recordings V (t) from the vibrometer for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i) ...... 69 4.4 Average velocities in the x- and y-axis computed using phase-based optical flow for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i) ...... 70 4.5 Power spectral densities of vibrometer velocity (a, c, e) and average flow velocity in the y-direction (b, d, f) for the three webs tested in the experiment. Peak frequencies are indicated by fn on each plot. . . . . 71 4.6 Vibrometer displacement X(t) computed by numerically integrating the recorded velocity signal for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i) ...... 73 4.7 Average displacements in the x- and y-axis computed by numerically integrating the velocities estimated using phase-based optical flow for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i) ...... 74 4.8 PSD of displacements ...... 75 4.9 Web video frame setup and results for Frequency Domain Decomposi- tion (FDD), ROIs used in optical flow estimation (a), FDD spectrum of singular values (b), the mode shape estimated using FDD from the right singular vectors (d) ...... 78 LIST OF TABLES

Table Page

2.1 Web geometry nomenclature ...... 11

3.1 Accelerometer channel pairings ...... 40 3.2 Optimal time delays for the four web treatments ...... 45

4.1 Natural frequency (fn) and damping ratio (ζ) estimates ...... 72 1 Introduction

Picture the orb-weaving spider: sitting in its web, eﬀectively blind, conserving energy, and patiently waiting for the next meal to pass by and become snared in its web. Once the web successfully traps an insect, the spider must act quickly to prevent the struggling insect from critically damaging its web or escaping entirely. Where in the web is the insect trapped? Is it too large to subdue? Should the spider approach with caution in case the trapped insect is actually a predator in disguise? These questions are all answered very rapidly by the spider, reacting to prey-capture events in its web with great speed and accuracy. This behavior raises several questions relating to the interaction between the web and the spider. Does the orb web augment the spider’s perception? Are there special properties of the spider-and-web system that provide the spider with additional spatial awareness? These are just a few of the questions that have driven research in spider behavior and ecology.

Biologists seek to understand the natural world by making direct observations, performing controlled laboratory experiments, and creating predictive models. To that end, the body of work concerned with orb web vibrations is rich, with publications spanning more than a hundred years and involving numerous fields of study [1, 2]. Past researchers of web vibrations [3, 4] have stated that the end goal of their work is to create a formal physical model of the web that accounts for silk mechanical properties, web support structures, realistic web geometries, and different vibration types—but such a model has not been forthcoming. Rather, researchers have focused on taking experimental measurements of spider webs in motion and creating numerical models (typically using finite elements) to further their understanding of web vibrations. The fact that a detailed, formal model for web vibrations is not available

1 highlights the challenges involved in such an undertaking.

The spider’s web presents many roadblocks to the traditional observe-model-test- repeat framework for conducting a scientiﬁc investigation of web vibrations. In particular, the orb web’s lightweight gossamer structure and intricate mechanical composition complicate both the modeling and measurement steps. In general, the skills required to fully model and characterize the web as a dynamic structure have little overlap with the ﬁeld of biology. Engineers, on the other hand, have a large toolkit at their disposal for working with vibrations of lightweight structures, and can therefore assist biologists with their need for a modeling and testing framework that is suitable for spider webs.

The following work uniﬁes techniques from mechanical engineering, structural dynamics, and computer graphics and applies them to the study of spider web vibrations. This dissertation focuses on the following items:

1. Studying vibrations of synthetic, bio-mimetic orb webs in a controlled engineering environment

2. Creating an accessible model for orb web vibrations that includes features such as arbitrary web geometry, tension distributions, and material composition

3. Studying both natural orb webs and synthetic webs in a vibration localization context

4. Applying experimental modal analysis techniques to orb web vibration data

The overall goal among these four items is addressing the gaps in modeling of spider webs in order to determine how vibrations can be used to estimate the location of a vibration source in web-like structures.

2 The remainder of this dissertation is structured into three separate manuscripts. Chapter 2 describes the development of a model for transverse vibrations in orb webs and the use of an enlarged, artificial spider web for validation of the model. Chapter 3 presents a vibration localization algorithm implemented on the enlarged artificial webs and tests its performance on different web configurations. Chapter 4 details a non-contact, video-based vibration measurement procedure for orb webs using phase-based optical flow and presents novel, full field vibration measurements taken from the webs of Araneus diadematus.

3 2 Modeling Transverse Vibration in Spider Webs Using Frequency-Based Dynamic Substructuring

Andrew W. Otto, Damian O. Elias, and Ross L. Hatton 1 Oregon State University 2 University of California, Berkeley

In: Dynamics of Coupled Structures, Volume 4. Conference Proceedings of the Soci- ety for Experimental Mechanics Series.

4 2.1 Abstract

Orb weaving spiders and their webs have co-evolved into a highly efficient prey capture and retention system. Spiders also use their web as a sensory extension by “listening” to vibrations transmitted across the web such as struggling prey trapped in the web, courtship from potential mates, or the advance of a predator. What information is available to orb weaving spiders from web vibrations, and how might this information be used for localization and identification of cues? To better understand the information available to the spider in its web, we created both physical and computational models of spider webs. Enlarged, artificial webs suitable for physical measurements (1.2 m in diameter) were constructed from two types of parachute cord to mimic the different silks used by spiders in web construction. Accelerometers placed around the center of the web measured the vibration response of the artificial web. We formed a model for large networks of transversely vibrating strings, such as a web, using Frequency Based dynamic Substructuring (FBS). From the FBS model, we generated frequency response functions at locations corresponding to typical foot placements of orb weaving spiders for vibration sources at various points in the web. We explored the influence of web architecture on web frequency response by altering web composition, pre-tension, and geometry. Frequency response features sensitive to stimulus location were also identified from the FBS model. Our research indicates that localization of both bearing and range for a vibration stimulus is possible in the modeled webs.

5 2.2 Introduction

Orb weaving spiders create webs to act as snares for catching prey. Once built, these webs also serve as a substrate to transmit vibrational information to the spider. How do vibrations propagate through the web? How does the spider use the information contained in vibrations to localize signals in its web? To answer these questions, we seek a physical understanding of the web as a dynamic structure.

The spider web is challenging to study due to its small size. If we seek to understand how the architecture of the web—its geometry and composition—influences web vibrations, then we must exert some form of control over its construction. Controlling variations in web architecture; however, is difficult because we cannot synthesize silk and must rely on the measurements taken from webs founds in nature. To overcome this limitation, we have constructed our own enlarged, artificial webs from parachute cord and elastic shock cord. By scaling up the web, we greatly ease the process of taking vibration measurements. Additionally, the web architecture can be freely specified to study the influence of factors such as web geometry, composition, and tension on web vibrations.

We have also developed a dynamic model of the orb web to study transverse web vibrations and to further explore the question of stimulus localization. We focus on transverse vibrations for two reasons: (i) prey impact is most frequently oriented perpendicular to the plane of the web, and (ii) prey impact occurs in the capture spiral portion of the web, and the spiral strands are better impedance-matched to the radial strands in the out-of-plane direction, and therefore transmit more energy to them in these modes. To model the web, we ﬁrst identify web conﬁgurations (geometry and tension distribution) for which the web is in static equilibrium. We then build up a

6 coupled oscillator model of the web as a network of interconnected vibrating strings. From the model, we generate pointwise Frequency Response Functions (FRFs) to describe motion of the web mesh in the frequency domain. We use this information to identify changes in the vibration response of the web due to variations in web architecture. Finally, we propose a pair of vibration cues that can be used to localize the position of an impulse in the web.

2.3 Related Work

This paper draws from research in biology of orb web vibrations as well as the ﬁeld of dynamic substructuring for modeling structural vibrations.

2.3.1 Web Vibrations

Many investigations have focused on understanding the vibration properties of orb webs. Masters [5, 6], and later Landolfa and Barth [3], measured the transmission of different vibrational directions in the orb web (longitudinal, transverse, and lateral), and found that longitudinal vibrations were least attenuated, suggesting that the spider may use longitudinal vibrations for prey localization. Landolfa and Barth [3] also discovered a gradient in vibration amplitudes (in all vibration directions) arriving at the spider’s tarsi that could be used to indicate stimulus direction. Due to the high measured wave speeds, they conclude that the spider would require equally high temporal fidelity to detect time of arrival differences to localize prey (approximately 1.5 ms). This finding suggests the spider may be exploiting the spatial distribution of the vibrations rather than the temporal to solve the localization problem. More recent works have closely studied the shock response and energy absorption of webs [2,

7 7, 8], but there remains a gap in understanding as to how these vibratory signals can be processed to determine prey location.

In addition to experimentally measuring the vibration response of orb webs, researchers have developed computational models for web vibrations. One of the first models was presented by Frohlich and Buskirk [4], which adopted a simplified representation of the web consisting of interconnected strings and lumped masses. Since then, many works have utilized finite element methods to model web vibrations in greater detail and investigate phenomena such as damage tolerance [9], the role of secondary structures [10], energy absorption [8, 11], and aerodynamic forces [12, 13]. Mortimer et al. [2] utilized computer models and experiments with orb webs to demonstrate the extent to which the spider can control various vibration properties of its web. Most recently, Morassi et al. [14] developed a continuous membrane model for the orb web with the goal of generalizing the web’s vibration properties to analytical relationships. While numerical simulations provide detailed web displacement time- histories and analytical models bring generality to the vibration characteristics of orb webs, neither body of work establishes an input-output mapping between a stimulus (e.g. the impact of an insect) and the signals received by the spider.

A primary issue in comparing model results with web measurements lies in determining web pretension. Measuring web tension directly requires highly specialized tools that are not commercially available and are difficult to fabricate [6, 15]. Alternatively, estimating web tension through other experimental means (e.g. wave speed or natural frequencies) becomes a model fitting procedure using physical principles for taut strings. Secondly, validating model results on live webs raises issues of repeatability— spider silk production is highly influenced by environmental conditions, and no two webs will be alike in terms of composition, pre-tension, or geometry. To address

8 these shortcomings, some researchers have turned toward making physical analogs of the orb web for vibration research [16–18]. We follow a similar approach, using an enlarged artiﬁcial web to validate results from our model.

2.3.2 Dynamic Substructuring

The ﬁeld of dynamic substructuring is concerned with assembling the dynamic response of a complete system from individual subcomponents via a coupling procedure derived from the connectivity of the system [19]. One of the major advantages of dynamic substructuring is its ability to handle systems that are too large or complex to model as a whole, and instead build up a reduced model from detailed subcomponent representations. The origins of dynamic substructuring can be traced back to the method of component mode synthesis (CMS) [20, 21].

Substructuring can be performed in many diﬀerent domains (e.g. physical, modal, time, and frequency). Substructuring in the frequency domain is most commonly employed in the experimental analysis of a structure’s dynamics because it provides a straightforward method to generate a model of a complete system through testing of individual subcomponents at their interfaces. In this work, we use Frequency Based Substructuring (FBS) for its convenient algebraic representation of the substructures that comprise a web—strings and masses. Furthermore, modeling in the frequency domain lends itself to the same types of analysis used in the literature for web vibrations.

9 2.4 Materials and Methods

We carried out our investigation into spider web vibrations by creating both physical and computational models of the spider web. Enlarged, artiﬁcial spider webs were created to enable control over web architecture for experimentation. Our computational model was formed using dynamic substructuring methods.

2.4.1 Artiﬁcial Webs

We constructed webs using two different materials to mimic the different types of silk used by spiders during web construction. We selected a stiff, nylon parachute cord for the radial lines of the artificial web and an elastic shock cord for the capture spiral, both having a diameter of 3.2 mm and a mass per unit length of 7 g/m. The stress- strain data for these materials is shown in Figure 2.2(a) along with representative data for dragline (radial) and viscid (capture spiral) silk.

The web geometry consisted of eight symmetric radial members and an Archimedean capture spiral. Figure 2.1 and Table 2.1 describe the parameters used to define web geometry. At the radial-spiral junctions, cable ties were used to clamp the strings together, and these joints were reinforced with cyanoacrylate glue. Webs were built slack (un-tensioned) in a conical shape and then tensioned to desired levels. The angle of the un-tensioned cone (β) was used to control the ratio of spiral to radial member tension. Different levels of radial tension were obtained by further stretching each web from its flattened state.

Webs were housed in an octagonal aluminum frame accommodating webs with a ﬁnal stretched diameter D of 1.22 m (48 in), shown in Figure 2.2(b). Radial lines were terminated into a ratchet and load cell mechanism to measure and adjust the ten-

10 Figure 2.1: Illustration of the un-tensioned web geometry and parameters used in this study

Table 2.1: Web geometry nomenclature Parameter Description

rn Outer radius of un-tensioned spiral

ri Inner radius of un-tensioned spiral β Angle of un-tensioned web cone

ns Number of spiral turns

δs Inter-spiral spacing, δs = (ro − ri)/ns τ Peripheral radial pre-tension D Final diameter of tensioned web sion of peripheral radial members. A voice-coil shaker excited the web to measure transmissibility. Input signals to the shaker were in the form of a swept sine wave of increasing frequency. Three-axis Analog Devices ADXL326 MEMs-type accelerometers (1000 Hz x- and y-axis bandwidth, 500 Hz z-axis bandwidth) measured the vibration input at the shaker as well as the vibration response at various locations in the web. A personal computer and a National Instruments CompactDAQ collected, ﬁltered, and processed signals arriving from the accelerometers. Data acquisition was triggered at the ﬁrst output sample to the shaker and was sampled at 1000 Hz. The

11 accelerometer waveforms were then filtered using a 3rd order Butterworth bandpass filter with lower and upper cutoff frequencies of 5 Hz and 250 Hz, respectively.

Figure 2.2: (a) Stress-strain data for the parachute cord and shock cord used in the study as well as dragline and viscid spider silk adapted from [22], (b) the test stand and artiﬁcial web used in this study

2.4.2 Vibration Model

Our computational model of the spider web has two components. The ﬁrst component computes web pretension based on web boundary conditions and the mechanical properties of the strings in the web. The second component uses Frequency Based dynamic Substructuring (FBS) to assemble the web from individual string and mass elements to generate frequency response functions for points of interest in the web.

2.4.2.1 Preprocessing and Tensioning

The vibration of strings, such as those in a spider’s web, is governed by the amount of tension in each string. Prior to modeling the vibrations in the web we must deter-

12 mine a pre-tensioned web state that satisfies static equilibrium. The tensioned web configuration under equilibrium is obtained using a minimum strain energy approach commonly employed in the analysis of cable roof structures [23]. To begin, a web geometry under zero tension is specified. Then, the tensioned equilibrium geometry is found from the initial un-tensioned geometry following a minimum strain energy approach as the set of nodal displacements x that satisfies

M X Z min `j fj()d j=1 (2.1)

subject to xi = bi, i = 1,...,N,

where is the vector containing element strains, j, M is the total number of string elements in the web mesh, `j is the current length of element j, xi is the nodal coordinate corresponding to constraint i, bi is the constrained value for nodal coordinate xi, N is the total number of nodal constraints, and fj() is the force in element j which is obtained by interpolating the strain in the element into stress-strain data, such as that given in Figure 2.2.

Strain () is expressed in terms of the nodal coordinates using

` − L |x | − L = j j = mn j , (2.2) Lj Lj

where Lj is the initial length of element j and xmn is the vector spanning nodes m and n that deﬁnes element j. The advantage of computing web equilibrium using minimum strain energy is that it handles large displacements as well as nonlinear material properties in a compact manner, both of which are present in the process of web pre-tensioning. Additionally, this method provides a mapping from a pre-speciﬁed, zero-tension geometry (something that can be easily constructed in a physical sys-

13 tem) to a tensioned structure. Solutions of the minimum strain energy problem can be found using standard numerical optimization packages, such as MATLAB’s fmincon. A sample tension and strain distribution is displayed in Figure 2.3.

Figure 2.3: (a) Tension distribution and (b) strain distribution output from the web pre-processing step for a sample web

2.4.2.2 Frequency Based Substructuring

Once the equilibrium geometry and tension are obtained, we use Frequency Based Substructuring (FBS) to model the transverse vibrations of the orb web. Frequency based substructuring computes the dynamics of a structure in the frequency domain by interconnecting smaller “substructures” through equilibrium and compatibility conditions. The method is covered in detail in [19]. We reproduce relevant portions here for convenience.

In general, a dynamic system can be expressed in the frequency domain using the

14 equations

Z(ω)u(ω) = f(ω) + g(ω) (2.3)

Bu(ω) = 0 (2.4)

LTg(ω) = 0, (2.5)

where Z(ω) is a block-diagonal matrix containing individual substructure dynamic stiﬀness matrices, u(ω) is a vector of unknown complex-valued displacements at the interface nodes, f(ω) is the vector of applied external forces, and g(ω) represents the internal connecting forces between substructures. The signed Boolean matrix B is used to enforce compatibility conditions (equal and opposite forces at interfaces), and the Boolean matrix L is used to enforce equilibrium (equal displacements at interfaces).

Using the primal formulation [19], a unique set of interface degrees of freedom (q) are deﬁned and the interface forces are eliminated using the interface equilibrium condition given in Eq. (2.5). The unique interface degrees of freedom q are given by

u = Lq, (2.6)

where the frequency dependence has been dropped for conciseness. Substituting Eq. (2.6) into Eq. (2.3) gives an expression for the harmonic motion of the system in the unique interface degrees of freedom

ZLq = f + g. (2.7)

Noting that the equilibrium condition states LTg = 0 from Eq. (2.5), we can left

15 multiply Eq. (2.7) by LT to eliminate the internal connecting forces g, reducing Eq. (2.7) to LTZLq = LTf, (2.8) which results in the assembled system with the expressions

Zq˜ = ˜f (2.9)

˜ T Z , L ZL (2.10) ˜ T f , L f, (2.11) where Z˜ is the transformed system dynamic stiﬀness matrix and ˜f is the system’s external force vector. The complex-valued harmonic displacements q for an applied harmonic force ˜f at a given degree of freedom can be obtained by solving the linear system of equations for the assembled system of Eq. (2.9) at discrete frequencies ω.

2.4.2.3 Constitutive Web Elements

To model the orb web using the FBS procedure outlined above, we require the dynamic stiﬀness matrix of each substructure in the web. The FBS model requires that individual substructures are represented using free boundary conditions. We model the orb web as a network of interconnected, free-free transverse string substructures, governed by the 1D wave equation

∂2v ∂2v S = µ , (2.12) ∂x2 ∂t2

where v is the transverse displacement of the string, S is the string tension, x is the coordinate along the length of the string, and µ is the density per unit length of

16 the string. A free-free string with a harmonic force F eiωt at the end x = ` has the boundary conditions

∂v S = 0 at x = 0 (2.13) ∂x ∂v S = F eiωt at x = `. (2.14) ∂x

Assuming steady harmonic motion in phase with the driving force, a general solution to the partial diﬀerential equation (PDE) in Eq. (2.12) can be written as [24]

v(x, t) = V (x)eiωt, (2.15)

where V (x) is the amplitude of motion along the length of the string. Substitution of the general solution into Eq. (2.12) yields the ordinary diﬀerential equation (ODE)

d2V S + µω2V = 0. (2.16) dX2

The general solution to this ODE is

V = A cos (λx) + B sin (λx), (2.17)

where λ = ωpµ/S. The constants A and B are obtained by substitution of the boundary conditions and general solution into Eq. (2.12), yielding

F A = − (2.18) Sλ sin (λ`) B = 0, (2.19)

17 which gives the displacement at points along the sting as

F cos (λx) v(x, t) = − eiωt. (2.20) Sλ sin (λ`)

Using Eq. (2.20), the direct receptance (Y``) at the driving point and cross receptance

(Y0`) from the free end to the driving point are given as

csc (λ`) Y = − (2.21) 0` Sλ cot (λ`) Y = − . (2.22) `` Sλ

Making use of reciprocity in the receptances [24], the full substructure receptance matrix Ys for the free-free string can be written as

    Y Y Y Y s  00 0`  00 0` Y =   =   , (2.23) Y`0 Y`` Y0` Y`` from which the substructure dynamic stiffness matrix Zs is obtained by taking the inverse of the receptance matrix, Zs = (Ys)−1. This operation results in an algebraic dynamic stiffness matrix which, with appropriate parameters, can be used to represent each vibrating string in the structure. A similar procedure can be used to obtain the dynamic stiffness matrix for the viscously damped wave equation under harmonic motion, given by the ODE

d2V S + (µω2 − iωc)V = 0, (2.24) dx2

where c is the viscous damping coeﬃcient of the string.

18 We can also ﬁnd the dynamic stiﬀness of point masses to simulate mass-loading of the orb web at points corresponding to measurement (i.e. the mass of an accelerometer) or where the spider and trapped prey might be situated in the web. The equation of motion for an unconstrained point mass subject to harmonic forcing in one dimension is F eiωt = mx,¨ (2.25) where m is the mass of the object. Assuming the point undergoes harmonic motion, the displacement of the point can be written as

x(t) = Xeiωt, (2.26) where X represents the amplitude of motion. Diﬀerentiation of Eq. (2.26) and substitution into Eq. (2.25) results in

F eiωt = −mω2Xeiωt. (2.27)

From Eq. (2.27) the receptance matrix of the point mass can be identiﬁed as

Ys = 1 , (2.28) − mω2 which gives the dynamic stiﬀness matrix by the inverse operation

Zs = (Ys)−1 = −mω2 . (2.29)

String and point mass substructures may be combined using mesh connectivity relationships between individual substructure degrees of freedom. From this connectivity

19 information, the Boolean matrices B and L can be formed for calculation of the assembled system dynamic stiﬀness matrix Z˜. Frequency response functions are obtained for the assembled system by applying unit external forces to a single degree of freedom and solving for the displacement vector q(ω).

2.5 Results and Discussion

To evaluate the Frequency Based Substructuring (FBS) model we compared its output with measurements taken from the artificial web. Then, we use the FBS model to explore influences of web architecture on web frequency response at points around the center of the spider web corresponding to foot placements of a spider. Finally, we identified the presence of cues in the frequency response functions that can be used to localize a vibration stimulus in the web.

2.5.1 Model Validation

To validate our FBS model, we constructed an artificial web and measured its vibration response. The artificial web was created using a 45 degree cone angle and 14 spiral turns. The peripheral radial tension (τ) was held at two different levels, 115 N and 160 N, and a point mass was attached to the center of the web with two values of 10 g and 70 g. Excitation was applied at a single node at the outermost portion of the capture spiral. An accelerometer was located 12 cm away from the web center on the same radial as the excitation to measure web vibration. Mass loading of the web by the accelerometer was included in the FBS model by adding a 5 g mass at the point of measurement.

Experimental and modeled frequency responses are displayed in Figure 2.4. For com-

20 parison, we focused on frequencies up to 50 Hz, which was slightly above the second web resonance. Damping for the FBS model was determined by matching the average Q factor (the ratio of the resonant frequency to the −3 dB width of the resonant peak) at the first resonant frequency in the experimental data across the four web treatments. As shown in Figure 2.4, the model matches the first and second resonant peaks with fair accuracy, and also captures the anti-resonant trough occurring after the first resonance. In general, the model predicts the location of the first resonance and anti-resonance well, but the shape of the frequency response following the second resonance is less well-predicted. The small differences between the model and experimental web are likely due to imperfections in the web-building process, resulting in discrepancies in the final tension distribution as well as overall shape of the capture spiral portion of the web. The model clearly shows the same shifts in frequency response with increasing mass and increasing web tension as the experimental results (i.e. increasing resonant frequency with increasing tension, decreasing resonant frequency with increasing mass).

2.5.2 Web Architecture Eﬀects

To characterize the inﬂuence of web architecture on web vibration, we varied the design parameters of the enlarged web in the FBS model around a “nominal” web conﬁguration comprised of 16 spiral turns, 20 degree cone angle, 100 N peripheral radial tension, and 100 g of mass at the center of the web. One parameter was varied at a time from the nominal condition by selecting parameter levels above and below the nominal condition. Spiral turns were varied by ±8 turns, cone angle was varied by −20 degrees and +25 degrees, peripheral radial tension was varied by ±40 N, and point mass was varied from 10 g at the low level to 1 kg at the high level. Figure 2.5

21 Figure 2.4: Model (solid, orange) and experimental (dotted, black) frequency response (taken as the output acceleration magnitude divided by the input acceleration magnitude) for diﬀerent values of point mass (m) and radial pretension (τ), (a) m = 10 g and τ = 115 N, (b) m = 70 g and τ = 115 N, (c) m = 10 g and τ = 160 N, (d) m = 70 g and τ = 160 N displays frequency responses for each of these parameter variations (a–d, e–f) as well as the nominal web (i). Each plot contains the frequency response function for eight spider leg placements located in a circle 10 cm from the web center on each radial, shown in Figure 2.5(j).

Several trends can be identified from Fig. 5: Increasing the point mass at the center of the web reduced the frequency of the first resonant peak and caused anti-resonances (seen as valleys or troughs) to occur in the frequency responses corresponding to several foot placement locations (Figure 2.5(d) and Figure 2.5(h)). These foot placement locations can be seen as the valleys located just following the first resonant peak, starting with the foot placement closest to the vibration source (orange). The frequency response following the second resonance was marginally affected by increasing the

22 Figure 2.5: Web architecture eﬀects on web frequency response (accelerance A(ω)), (a- d) web frequency response at low levels, (e-h) web frequency response at high levels, (i) web frequency response at nominal level, (j) input (black) and output (colors) locations for web frequency response

23 point mass. Increasing peripheral radial pretension (Figure 2.5(c) and Figure 2.5(g)) had the opposite effect of increasing mass, as it increased the frequency of the first resonant peak and eliminated several anti-resonant troughs that were present at decreased tension. Increasing the cone angle (Figure 2.5(b) and Figure 2.5(f)) of the web (i.e. decreasing the radial-spiral tension ratio) had a more complex effect on the frequency response structure, primarily increasing the number of anti-resonant troughs present following the first resonant peak as well as the location and magnitude of the second resonant peak. Finally, increasing the number of spirals in the web (Figure 2.5(a) and Figure 2.5(e)) reduced the frequency of the first resonance, caused additional anti-resonances between the first two resonant peaks, and increased the magnitude of the second web resonance.

While the parameters were only varied one-at-time from a nominal condition, they can also interact with one another, potentially allowing for a great degree of “tuning” of the web to achieve certain frequency response characteristics. For example, if it was desirable to have low amplitude motion for certain leg placements at speciﬁc frequencies (i.e. anti-resonances), the web could be constructed with a larger cone angle, which is analogous to decreasing the radial-spiral tension ratio, or the radial tension of the web could be decreased. In all, Figure 2.5 demonstrates the complex way in which web architecture can inﬂuence web vibration, and how the vibration response of the web can be altered through changes in web geometry and composition given a particular end-goal.

2.5.3 Vibrational Cues for Stimulus Localization

Our primary interest in modeling spider web vibrations is to better understand the vibrational information presented to the spider in its web due to a stimulus (e.g.

24 prey impact), and how that information can be used to determine the location of the stimulus. To investigate the presence of cues available to the spider for stimulus localization purposes, we generated frequency response functions for varying stimulus bearings and ranges relative to the center of the web using the FBS model. For this investigation, the measurement points of interest on the web were located at a radius of 10 cm from the center of the web on each of the web’s eight radials, referred to here as leg placement locations. A single web architecture was used, composed of the paracord and shock cord in Figure 2.2(a) for a 1.22 m diameter web and deﬁned by 16 spiral turns, 45 degree cone angle, 120 N peripheral radial tension, and a point mass at the web center of 100 g. Stimuli were applied in varying locations within the capture spiral, corresponding to locations where prey is typically trapped. Leg placement locations, stimulus locations, and leg frequency responses are displayed in Figure 2.6.

In terms of stimulus bearing to the spider (i.e. the angle between the spider heading and the stimulus location), prior work has shown that there is a vibration amplitude drop-oﬀ with increasing angular distance from a stimulus present in the webs of orb weaving spiders that can be used as a localization cue [3]. We interpret this cue as the total energy arriving at each leg in the frequency domain (obtained via integrating the frequency response), with legs that are closer to the vibration source receiving more energy. This energy is given as

Z ωn Ei = Ai(ω)dω, (2.30) ω0

where Ei is the energy arriving at leg placement i and Ai(ω) is the frequency response function (accelerance) of the web at leg i. These leg energies are combined with the leg placement angles of the spider (taken counterclockwise from the positive x-axis in

25 Figure 2.6: Inﬂuence of vibration stimulus location (at left, black dot) on leg frequency responses (given here as accelerance) and spectral centroid ωc (at center) as well as normalized leg energies (at right) for multiple vibration stimulus placements within the capture spiral for a single web architecture

45 degree increments for this case, with the web centered at the origin) to obtain a set of polar coordinates representing the energy arriving at each leg, shown at right in Figure 2.6. To obtain a stimulus direction from the leg energy-angle pairs, we form a polygon and calculate its centroid. Stimulus direction is given by the angle (θ in Figure 2.6) of the vector pointing from the origin to the polygon centroid. The length of this vector (r in Figure 2.6) can be interpreted as the conﬁdence in the detected bearing value.

26 The result of applying the bearing cue to the frequency response data in Figure 2.6 demonstrates a clear ability to obtain accurate heading information from leg frequency responses. The radius values for the polygon centroid (r) show that confidence in stimulus heading decreases with increasing range for this specific web architecture, which intuitively agrees with the argument that distant disturbances are more difficult to localize than those that are close due to attenuation by the web.

Additionally, the frequency responses in Figure 2.6 display a shift in frequency that accompanies changes in range of the vibration source – increasing frequency with decreasing range from the center of the web to the stimulus. We used the spectral centroid (ωc) to quantify this frequency shift, deﬁned as

P ωnXn ωc = P , (2.31) Xn

where Xn is the amplitude of the frequency response at bin n of the frequency spectrum and ωn is the frequency value associated with bin n. Here, we use the averaged frequency response of all the leg placements in our calculation of the spectral centroid. For the three stimulus locations with decreasing range to the center of the web in Fig- ure 2.6, the respective spectral centroid values were 52.6 Hz, 70.3 Hz, and 80.6 Hz. The shift to higher frequency spectral centroids can be linked to the shorter length of the spiral member on which the vibration source is placed, i.e. at a given tension, shorter strings have higher natural frequencies. These ﬁndings demonstrate that our web model contains suﬃcient information to localize not only stimulus bearing as suggested by prior work, but also stimulus range.

27 2.6 Conclusion

This work modeled transverse vibrations in structures resembling the webs of orb- weaving spiders through the use of Frequency Based Substructuring (FBS). We validated the FBS model using an enlarged, artificial web instrumented with off-the-shelf MEMs-type accelerometers. Using the FBS model, we showed how different aspects of web architecture (e.g. capture spiral density, radial-spiral tension ratio, web pretension, and center mass) influence web frequency response. We also proposed a method for obtaining stimulus bearing and range cues from web frequency response by calculating the spectral energy at each leg placement location as well as the spectral centroid of the averaged leg spectrums.

Our results highlight the influence web architecture has on web vibrations and that the orb web is a structure with the potential to transmit a great deal of information to the spider. The analysis of possible tradeoffs in localization with respect to web architecture is possible using the methods presented in this work, which will aid in understanding the evolution of the orb web as both a snare and sensory extension of the spider. Moving forward, it will be necessary to confirm if these same localization cues are present in the webs of actual orb weaving spiders, or to what extent the frequency response patterns in silk webs differ from our model. It will also be important to assess if and how localization cues vary across different types of orb webs and how the predicted tradeoffs between localization and prey capture may drive diversity in web types across spiders. In addition, we plan to use our artificial web as a platform to develop refined localization approaches that function in real-time.

28 3 Bioinspired Vibration Localization in Artiﬁcial Orb Webs

Andrew W. Otto1, Damian O. Elias2, Ross L. Hatton1 1 Oregon State University 2 University of California, Berkeley

29 3.1 Abstract

Orb weaving spiders rely on their webs as an extension of their sensory inputs. These spiders make use of information transmitted through web-borne vibrations to identify the location of trapped prey and discriminate between signals generated by potential mates or predators. In this paper, we present a vibration localization procedure implemented on a large, interconnected network of strings inspired by the webs of orb-weaving spiders such as Araneus diadematus to study the problem of vibration localization. Vibration information collected from a spider-like array of accelerometers located at the center of the web structure is used to characterize the performance of our localization procedure. We tested two variations of architectural features available to spiders during web construction: overall radial tension and radial-to-spiral tension ratio. For each web variation, we tuned our localization procedure to minimize the location error for a pluck-type impulse applied to the web structure. We found that large radial-to-spiral tension ratios improve location estimation. Within each web treatment, pluck location errors were least when plucks were located close to the center of the web and greatest at the edges of the web. We discuss how these results relate to the localization problem presented to orb weaving spiders.

3.2 Introduction

Substrate vibrations are used by many animals for both communication and sensing. The study of communication and sensing through mechanical waves is known as biotremology. The substrates used by animals for communication are generally those that dominate their natural habitat—ranging from the ground, plant stems, leaves, sand, water, or a spider’s web [25–27]. The orb weaving spider is unique among these

30 animals because it constructs the substrate on which it resides, giving it detailed control over the sensing properties of its substrate. For this reason, orb weaving spiders have received much attention from biologists regarding their ability to extract information about their surroundings through web-borne vibrations [2, 3, 5, 28, 29].

Spiders sense vibrations using the lyriform organ located on their legs [30, 31]. The lyriform organ, composed of individual slit sensilla arranged in a pattern, is a strain sensing organ that enables the spider to sense the ﬂexion and extension of its legs. Each of the spider’s feet act as a vibration “pickup” where it is attached to the web, forming a spatial array of sensors. Some orb weaving spiders, such as Araneus diadematus, sit at the center of their webs “listening” for signals that might indicate an insect has become trapped in their web or disturbances that indicate the spider should retreat and seek shelter to avoid any threats. When an insect becomes snared in the web, the spider must quickly locate and subdue the insect to prevent it from escaping and causing damage to the web. The process of receiving, processing, and acting on the information contained in web-borne vibrations is what we refer to as vibration localization.

A spatial arrangement of multiple sensors is necessary for estimating the location of a vibration source. In the case of classic time delay of arrival (TDOA) techniques where the wave speed is known and the transmission medium is continuous, multi- lateration requires at least one more sensor than the number of degrees of freedom in the position estimate (i.e. estimating 3D position requires at least four measurements or sensor stations). The greater the number of sensors, the more robust the location estimate. Fortunately, spiders have all eight legs at their disposal for vibration localization purposes, and orb weavers such as Araneus diadematus assume a pose at the web center from which they can monitor vibrations. Unlike engineering

31 applications such as radio signal surveillance where the wave propagation speed of electromagnetic radiation is fixed and the medium is continuous, a spider’s web is a discrete, heterogeneous structure made up of different types of silk held under different tensions [15], which results in circuitous transmission paths and non-uniform wave speeds. For both this reason and the high temporal fidelity that would be required by a spider to measure time of arrival differences at the web center [31], it is unlikely that the spider makes use of solely timing information to locate vibration sources. We then ask—what features of the signals arriving at the center of the web from a vibration source are correlated with location, and how might those features be used in a vibration localization framework?

Measuring web vibrations and interrogating the inﬂuence of substrate properties such as web architecture (i.e. material composition, geometry, and tension) on how vibrations propagate in the web is challenging due to the delicate gossamer structure of spider webs and the small size of silk threads. As a result, researchers commonly turn to computer models to study web vibration [2, 10–12, 16, 32] to gain control over web architecture as well as measurement of arbitrary motions. Alternately to simulation, biomimicry provides a controlled, physical context to explore orb web vibrations. Bioinspired and biomimetic webs have been used in the past for validating computer models [16, 18], sampling prey in a controlled fashion that is representative of what webs actually capture [33, 34], and as inspiration for communication networks [35]. We employ bioinspired webs and spider-like sensor arrays as a means to control for web composition, geometry, and tension to better understand how those factors alter vibration propagation. Additionally, using artiﬁcial spider webs eases the measurement process, as the web can be scaled up to sizes allowing instrumentation with traditional accelerometers.

32 In this work, we present a signal processing procedure for estimating the location of a vibration source in artiﬁcial, bioinspired orb webs. First, we identify cues that are strongly correlated to both the orientation and range of a vibration source from the web center. Next, we describe an algorithm for processing vibration sensor data to estimate source location. Then, we experimentally characterize the performance of the location estimation procedure on a set of two artiﬁcial orb webs. Finally, we discuss the interaction between the performance of the presented vibration localization procedure and a set of architectural parameters available to orb weaving spiders during web construction.

3.3 Materials and Methods

In this section, we describe the construction and testing of artiﬁcial webs for vibration localization. Vibration cues for estimating orientation and range are presented using sample data collected from the artiﬁcial web. Finally, we combine the orientation and range estimation cues into a vibration localization algorithm.

3.3.1 Web Construction

Two artificial webs were constructed to measure and tune the performance of the localization algorithm. Webs were octagonal in geometry with a simplified design (compared to their orb weaver’s counterparts), which consisted of eight radial threads with equal angular spacing of 45◦ and 16 spiral threads with constant spacing of 25 mm. The outer diameter of the webs was 1.22 m. The webs were built slack (zero pretension) in a cone shape, where the angle of the cone was used to control the radial-to-spiral tension ratio once the web was stretched flat. One of the webs was constructed with a cone angle of β = 45◦ to give a low radial-to-spiral tension ratio

33 while the second web had a cone angle of β = 0◦ to give a high radial-to-spiral tension ratio. The slack geometry of the two webs in the experiment is shown in Figure 3.1(a) and (c).

Two types of materials were used for the threads of the webs to mimic the load- elongation properties of spider silk. A stiﬀ nylon parachute cord (#325) was used for the radial members to mimic major ampullate (dragline) silk, while the spiral threads were made from a highly extensible, elastic shock cord (3 mm diameter) to mimic viscid silk. Radial-spiral thread junctions were formed using plastic cable ties and cyanoacrylate glue. Webs were mounted into a rigid aluminum frame with a pulley and load cell arrangement at the attachment points to measure and control peripheral radial tension. For each web, the peripheral radial tension was varied between two levels—100 N and 120 N—resulting in a total of four web treatments. The calculated tension distributions of the webs are shown in Figure 3.1.

3.3.2 Testing

For each web treatment, 100 impulse-like inputs were randomly distributed throughout the web at the midpoints of capture spiral threads. The impulse-like inputs were generated by gripping individual spiral threads between the thumb and forefinger, pulling the thread back, and then releasing the thread. We refer to this type of an impulse as a pluck. Web vibration was measured by an array of eight MEMs-type accelerometers (Analog Devices ADXL 326) mounted on a 3D printed, spider-like ABS plastic sensor body that was attached to the center of the web using alligator clips, shown in Figure 3.2. The accelerometers were oriented such that the Z-axis was normal to the plane of the web. The Z-axis of the accelerometers was sampled at 1 kHz and band-pass filtered using a second-order Butterworth filter with lower and

34 Figure 3.1: Slack geoemetry and tension distributions of the β = 45◦ web (a–c), and for the β = 0◦ web (d–e) upper cutoﬀ frequencies of 1 Hz and 500 Hz, respectively.

Data acquisition used a ring buﬀer, and was set such that data was recorded from one second before and four seconds after pluck detection. Plucks were detected by con- tinuously computing the 2-norm from one sample of each accelerometer and checking that value against a threshold. A pluck event was triggered when the threshold was exceeded, and the system blocked the operator from triggering additional events for another ﬁve seconds. During testing, the operator conducted a complete experiment by repetitively plucking the web in predetermined (randomized) locations, waiting for the data acquisition system between plucks, and repeating until all of the desired plucks were complete. Data acquisition and triggering was performed using LabVIEW.

35 Figure 3.2: Artiﬁcial web, test stand, and 3D printed sensor body (located at web center) used in the experiment

3.3.3 Vibration Source Location Estimation

The task of vibration source localization was split into estimating the orientation and range to a source from the center of the web. A set of heuristically-determined cues were extracted from the vibration measured by the accelerometers. We use sample datasets collected from the accelerometer array for pluck impulses applied in diﬀerent locations to motivate and describe the cues in the following sections.

36 3.3.3.1 Orientation

Figure 3.3 displays an overview of the orientation estimation process. The input pluck location as well as the eight accelerometer placements (1–8) for each sample pluck are indicated in Figure 3.3(a). A five second recording of the accelerometer responses is shown in Figure 3.3(b). Three regions are present in the five second time span. From t = 0 seconds to approximately t = 1 seconds, the web is under no motion. At t = 1 second, the pluck impulse arrives at the accelerometer array, and the web is transient for approximately one more second. From t = 2 seconds until the end of the recording, the web settles into steady state motion and the transient motion from the pluck impulse has decayed. We utilize the transient portion of the signal in forming the cues for orientation estimation, indicated by the dashed lines spanning 100 ms in Figure 3.3(b), which is magnified in Figure 3.3(c).

Orientation estimation was achieved by averaging together the output from two separate cues. The ﬁrst cue, inspired from previous work on vibration localization by spiders [31], makes use of the well-known property that spider webs attenuate vibrations over a distance, which causes diﬀerences in vibration magnitude at each of the spider’s feet. We measured vibration magnitude over the transient portion of the signal by computing the root mean square (RMS) of each accelerometer channel. Then, a polygon was formed in polar coordinates by combining the RMS value of each accelerometer with its angular position on the web. Orientation was estimated by creating a vector from the origin through the centroid of this polygon and taking its angle with respect to the X-axis. This process is depicted in Figure 3.3(d), where the orientation vector is shown in red.

The second cue for estimating orientation searches for symmetry in the accelerometer

37 Figure 3.3: Overview of the orientation estimation process, (a) input pluck location (black) and accelerometer placements (colors), (b) acceleration time histories from the applied pluck, (c) clipped section of the acceleration time histories around pluck onset, (d) visualization of the leg RMS values and predicted orientation using the polygon centroid, (e) visualization of the leg correlation values and predicted orientation

38 recordings. Upon inspection of Figure 3.3(c), several accelerometer “pairings” are visible. The cause for these pairings arises from the circular symmetry present in both the web and the accelerometer array. We have restricted pluck impulses to originate at the midpoints of spiral threads, which means that a pair of radials always equally borders the input pluck location and experiences very similar responses from an impulse. We measured the strength of these pairings using the normalized, zero- lag cross correlation. The normalized zero-lag cross-correlation between two discrete signals x(n) and y(n) is

Rxy(0) Rxy,norm(0) = p , (3.1) Rxx(0)Ryy(0)

where the discrete raw cross-correlation, Rxy(n), with no normalization is given as

∞ X ∗ Rxy(n) = x (m)y(m + n). (3.2) m=−∞

Caclulating Rxy,norm(0) for every accelerometer channel pairing results in a symmetric, 8 × 8 matrix with ones on the diagonal. Speciﬁc sets of pairing strengths can be combined to obtain an estimate for an axis of symmetry in the accelerometer array from an applied impulse. For example, if a pluck originated on a spiral thread between the 0◦ and 45◦ radials (at an orientation of 22.5◦), we would expect the following channels to be strongly correlated, and thus paired: 1–2, 3–8, 4–7, and 5–6. Table 3.1 displays the accelerometer channel pairings for plucks originating at the midpoints of spiral threads and their corresponding orientation angle for the eight-radial webs used in this experiment.

We averaged together the normalized, zero-lag cross correlations for each leg pairing set to obtain a relative estimate of the likelihood that a pluck impulse was applied

39 Table 3.1: Accelerometer channel pairings Orientation Bin (deg) 22.5 67.5 112.5 157.5 1–2 2–3 3–4 4–5 3–8 1–4 2–5 3–6 Accelerometer Pairs 4–7 5–8 1–6 2–7 5–6 6–7 7–8 1–8 along one of the four possible axes of symmetry in Table 3.1. A continuous representation of source orientation was obtained by first reflecting each of the bin scores 180◦, and then using singular value decomposition (SVD) to obtain the principal axis of an ellipse that best captures the variance in the bin scores. The results of this computation are displayed in Figure 3.3(e), where the ellipse has been drawn around the bin scores (black dots) and the vector along the principal axis is shown in red. The direction of the principal axis is selected such that it points in the direction of the accelerometer channel that peaked first in the signal portion shown in Figure 3.3(c). As a final step, the two orientation estimates are combined by averaging together to obtain a single, continuous estimation of vibration source orientation.

3.3.3.2 Range

Figure 3.4 displays an overview of the range estimation process for the same set of three plucks used in Figure 3.3. The transient portion of the signal is displayed in Figure 3.4(b). Vibration source range estimation was achieved by examining the frequency content of the signals arriving at each accelerometer.

Figure 3.4(c) shows the power spectral density (PSD) of the zoomed data in Fig- ure 3.4(b). The power spectrums have a clear peak, indicated by fp, which can be

40 Figure 3.4: Overview of the range estimation process, (a) input pluck location (black) and accelerometer placements (colors), (b) acceleration time histories near pluck onset, (c) power spectral densities of each accelerometer and the mean peak frequency fp, (d) peak frequencies sorted by range from the web center and curve ﬁt to a decaying exponential function

41 used as an estimate of range to the vibration source. The frequency of this peak represents the natural frequency of the thread that was impulsed, with shorter threads having higher frequencies as a result of the roughly constant tension present in the capture spiral, as shown in Figure 3.1.

We were limited to estimating relative range using this scheme, since the peak frequency can only be used to sort the responses. A simple curve ﬁt is possible with training data as shown by the exponential ﬁt in Figure 3.4(d), but for this experiment we resort to sorting the peak frequencies in descending order to obtain a relative range estimate. Therefore, at least two plucks are required in an experiment to obtain any measure of the relative range from the center of the web to the vibration source.

3.3.4 Localization Algorithm

The cues identiﬁed above for estimating orientation and range to a vibration source were incorporated into a localization algorithm. A ﬂowchart of the localization algorithm is shown in Figure 3.5.

The initial part of the algorithm handles detecting plucks and triggering the subsequent processing steps for orientation and range estimation. Both the orientation and range estimation routines were given their own queue of accelerometer data based

on a time delay value from pluck detection, denoted as tθ and tR, respectively. The duration of these delays was a free parameter that we optimized for each web treatment. The vibration source localization algorithm and time delay optimization was implemented in MATLAB.

42 Figure 3.5: Flowchart of the vibration source localization algorithm

3.4 Results

First, we present results from tuning the localization algorithm on each web treatment to select the time delays tθ and tR that minimize the total source location prediction error for a set of test plucks. Then, we make comparisons across web treatments to investigate the inﬂuence of peripheral radial tension and radial-to-spiral tension ratio on the localization algorithm’s performance.

43 3.4.1 Optimal Parameter Selection

The optimal time delays tθ and tR were chosen by varying their values and evaluating the localization algorithm’s performance for a set of pluck events. The cost function used for optimization sums the total euclidean distance error between the actual source location and the predicted source location for a set of plucks, and is given as:

X C(tθ, tR) = (||xi(tθ, tR)||2 − ||ˆxi(tθ, tR)||2) (3.3) i

where xi(tθ, tR) and ˆxi(tθ, tR) are the actual and predicted positions, respectively, which contain the Cartesian coordinates of pluck i for a combination of time de-

lay parameters tθ and tR. The Cartesian position vectors are calculated from the orientation and range values for each pluck using

xi(tθ, tR) = (Ri(tθ, tR) cos θi(tθ, tR),Ri(tθ, tR) sin θi(tθ, tR)) (3.4) ˆ ˆ ˆ ˆ ˆxi(tθ, tR) = (Ri(tθ, tR) cos θi(tθ, tR), Ri(tθ, tR) sin θi(tθ, tR)) (3.5)

ˆ where θi(tθ, tR) is the actual source orientation, θi(tθ, tR) is the estimated source ˆ orientation, Ri(tθ, tR) is the actual source range, and Ri(tθ, tR) is the estimated source range for a choice of parameters tθ and tR.

Due to the presence of multiple local minima in the cost function and the low di- mensionality of the problem, a brute force approach was used to ﬁnd the optimal time delay parameters for each web treatment. The values of tθ and tR were varied over a uniform grid ranging from 0 ms to 200 ms with a grid spacing of 10 ms. The optimal combination of tθ and tR was found as the set of values that produced the minimum value of the cost function over the entire grid. The optimization results are

44 summarized in Table 3.2 and the response surface for each web treatment is shown in Figure 3.6.

Table 3.2: Optimal time delays for the four web treatments Treatment Time Delay Cost

τ (N) β (deg) tθ (ms) tR (ms) C(tθ, tR) 100 0 10 50 5.2 120 0 20 60 5.1 100 45 10 20 24.6 120 45 10 30 27.1

Figure 3.6: Response surfaces representing total location error from Eq. (3.3) for each of the web treatments, (a) the 0◦ web at 100 N, (b) the 0◦ web at 120 N, (c) the 45◦ web at 100 N, (d) the 45◦ web at 120 N

The response surfaces display a few notable features. First, the 0◦ web treatments had

45 lower location estimation error compared to the 45◦ webs, suggesting that a greater radial-to-spiral tension ratio improves source location estimation. Additionally, the response surfaces show that large time delays (i.e. collecting more information) is not beneﬁcial beyond a certain point, with a maximum optimal time delay of 60 ms from the four web treatments. The poor performance of the algorithm with longer time delays is likely caused by reverberations and echoes in the web masking the original signal generated by the plucked thread. Lastly, the optimal time delay for orientation prediction (tθ) was very similar across the four web treatments (10-20 ms), and its low value indicates that only a handful of samples immediately following impulse detection are needed for orientation estimation.

3.4.2 Web Architecture Eﬀects

Next, we explore how the diﬀerent web treatments eﬀect the orientation and range estimation error of the localization algorithm. Figure 3.7 displays the estimated orientation and range plotted against the actual orientation and range for each web treatment’s set of plucks.

In Figure 3.7, the prediction error is concentrated in the range estimation for the high angle webs (β = 45◦), whereas the estimated orientation matches the actual orientation well across all web treatments. The consistent performance of the orientation estimation indicates that it is robust to changes in peripheral radial tension as well as the radial-to-spiral tension ratio. The estimated vs. actual plots for range in Figure 3.7 display an inability to extract the true range in the high angle webs, and therefore a meaningful peak frequency from the power spectrums. The higher spiral tension in the high angle webs likely causes the natural frequency of individual capture spiral threads to be above the fundamental frequency of the full web, thereby

46 Figure 3.7: Estimated vs. actual plots of source orientation θ and range R for each of the web treatments at their corresponding optimal time delays, (a–d) estimated vs. actual curves for orientation θ, (e–h) estimated vs. actual curves for range R. Perfect prediction is indicated by a dashed line in all plots

47 masking it from a peak detection routine.

Boxplots of euclidean distance prediction error for each of the web treatments is shown in Figure 3.8. The boxplots further reinforce the reduced error in the β = 0◦ webs and also show that altering the peripheral radial tension has a small inﬂuence on the average euclidean distance prediction error. Overall, the localization algorithm had the lowest average prediction error on the 120 N, 0◦ web, while the 120 N, 45◦ web had the greatest average prediction error.

Figure 3.8: Boxplots of euclidean distance error for each web treatment and the localization algorithm using optimal values for tθ and tR

3.5 Conclusion

In this work, we have introduced a vibration source localization method implemented on artiﬁcial orb webs inspired by spiders such as Araneus diadematus. The localization procedure estimates both the orientation and range to a vibration source from the center of the web using a combination of features extracted from both time- and

48 frequency-domain information collected by an array of accelerometers distributed about the center of the web.

Using the artiﬁcial webs and sensor platform, the localization procedure was tuned for a set of diﬀerent web tension distributions. We found that larger radial-to-spiral tension ratios resulted in the least location estimation error, and that error was concentrated in the range estimation. Orientation estimation proved to be robust to changes in web tension distribution. These results suggest that the correlation and amplitude based cues used for orientation prediction are reliable across a range of web architectures. Range estimation was proven to be possible using a detected peak frequency, but this estimate’s reliability was strongly linked to the level of tension in the capture spiral threads.

It is important to note that these results are applicable to networks of transversely vibrating strings in a general sense. When considering differences to actual silken spider webs, additional effects such as air drag and electrostatics can become significant due to the ratio between the diameter of the silk threads and their density. Further work is necessary in the vibration testing of silken webs, ideally with the end result of a linear input-output modal model that can be used to map a stimulus to a measurement location on the web. Provided with such a model, exploring the presence of the cues and algorithm described within this work could be confirmed mathematically.

49 4 Video-based Vibration Measurement of Orb Webs

Andrew W. Otto1, Malcolm F. Rosenthal2, Damian O. Elias2, Ross L. Hatton1 1 Oregon State University 2 University of California, Berkeley

50 4.1 Abstract

Studying the vibrations of orb webs presents a challenging measurement task. In this work, we apply phase-based optical flow to video recordings of orb webs under motion to demonstrate its utility for vibration measurement and analysis of lightweight biological structures. First, we describe our implementation of a phase-based optical flow algorithm inspired by recent work in video motion magnification. We then compare motion measurements obtained from video recordings using our phase-based optical flow implementation with those taken from a Laser Doppler Vibrometer (LDV). Using Frequency Domain Decomposition (FDD), we successfully measured the fundamental frequency and damping ratios of vibrating webs with spiders present. We also were able to extract the shape of the first mode in the webs tested sing FDD. We show that the orb webs of Araneus diadematus tested have their first natural frequencies at approximately 2.5 Hz. We propose that the use of phase-based video motion processing is a strong alternative to Laser Doppler Vibrometry when adequate surface continuity and reflectivity are challenging to obtain (as is the case with orb webs), and suggest that this video processing technique be further refined for sound and vibration research in biology.

4.2 Introduction

The webs of orb weaving spiders are well-known to serve as a sensory extension of the spider, providing information on the location of trapped prey, the advance of a predator, and the intentions of a potential mate through web-borne vibrations [36]. Measuring the vibrations of spider webs has been instrumental to understanding their function as a sensorial surface [2–4, 6].

51 However, the orb web presents many challenges for vibration measurement. Early researchers used a variety of ad-hoc methods to take measurements, such as crys- tal phonograph pickup cartridges in [37] and a combination ﬁber-optic bundle and photodetectors in [6]. The arrival of the Laser Doppler Vibrometer (LDV) gave researchers a standard tool to perform non-contact vibration measurement.

While the LDV boasts excellent bandwidth and measurement resolution, it has several drawbacks when used on orb webs, largely due to their discrete structure. First, the laser dot size is of the same order of magnitude as the diameter of silk (2–4 µm), making it very difficult to focus on single strands unless a microscope-based vibrometer is used, which would eliminate the ability to scan measurements across the entire web. The small diameter of silk is compounded by the low density of the material, such that if any spurious air currents are near the web, the silk can easily drift out of focus of the laser. To obtain sufficient signal strength from a laser vibrometer, reflective microbeads or other targets (e.g. fly wings, reflective tape, foam beads) must be added to the structure, but the influence of this mass on the web is generally difficult to account for since the distribution of beads is non-uniform, and in some cases can severely alter web dynamics. Additionally, obtaining full web scans using a laser vibrometer can take a long period of time, as each point must be sequentially scanned during excitation. The long scan duration again compounds with the tendency of the silk to drift due to air currents, potentially causing many points to no longer be in focus, or off the web entirely during testing. Scanning the web is also not feasible when measuring events that are not perfectly repeatable across multiple trials, such as prey-stimulus response in which the spider tends to damage the web while moving across it. Finally, these problems would be insurmountable if the goal was to measure natural web structures in the field.

52 Recent developments in computer graphics have demonstrated the ability to extract motions from video records using a phase-based processing technique [38–40]. This tool has been used for a number of tasks, including video motion magniﬁcation of vibrating structures [41, 42], estimating material properties [43], and sound recon- struction [44]. Its use as a vibration measurement tool has gained interest in several communities (e.g. structural health monitoring and civil infrastructure testing [41]) due to its non-contact, remote nature and dense full-ﬁeld measurement capabilities.

In this work we demonstrate the application of phase-based video motion processing to vibrations of the orb web of Araneus diadematus and propose it as a superior alternative to the laser vibrometer due to its ability to capture web motion without the need to increase the reﬂectivity of the web or focus on individual measurement points on the silk.

4.3 Materials and Methods

We describe a phase-based optical flow method for motion estimation that is a combination of previous work [38, 45, 46] and inspired by recent developments in video motion magnification [39, 47]. A Laser Doppler Vibrometer (LDV) is used as a ground-truth reference of web motion for validation of the motion estimated using our phase-based optical flow implementation, and we make comparisons by estimating the natural frequency and damping ratio of the web by treating it as a single degree-of-freedom mechanical oscillator. We incorporate output-only modal analysis to extract mode shapes from the motion data collected from the video recordings.

53 4.3.1 Gradient-based Optical Flow

Gradient-based optical flow estimation makes use of gradients in image intensity values to estimate velocity vectors that describe the motion between two subsequent image frames. The book chapter by Fleet and Weiss [48] provides an in-depth back- ground on gradient-based optical flow estimation, while an excellent comparison of different optical flow estimation techniques can be found in the paper by Barron et al. [49]. A brief description of the gradient-based optical flow method is provided here.

In the popular Lucas-Kanade optical ﬂow method [50], the gradient constraint equation used to solve for the motion between image frames is given as

Ixu + Iyv + It = 0 (4.1) I , I(x, y, t),

where I(x, y, t) is the image intensity, Ix is the partial derivative of intensity in the horizontal direction, Iy is the partial derivative of intensity in the vertical direction, It is the partial derivative of intensity with respect to time, u is the horizontal velocity component at pixel location (x, y), and v is the vertical velocity component at pixel location (x, y).

In the Lucas-Kanade method, the motion between two subsequent image frames is assumed to be small and approximately constant within a neighborhood around the pixel under consideration. As a result, Eq. (4.1) holds for pixels in a small window centered on the current pixel under consideration, which forms the system of equations

54 Ix(x1, y1, t)u + Iy(x1, y1, t)v + It(x1, y1, t) = 0

Ix(x2, y2, t)u + Iy(x2, y2, t)v + It(x2, y2, t) = 0 (4.2) . .

Ix(xn, yn, t)u + Iy(xn, yn, t)v + It(xn, yn, t) = 0, where xi and yi represent the coordinates of pixel i within the window and t is the current time. This system of equations can be written in matrix form A~u = ~b, where

    Ix(x1, y1, t) Iy(x1, y1, t) It(x1, y1, t)           Ix(x2, y2, t) Iy(x2, y2, t) u It(x2, y2, t)     ~   A =  . .  , ~u =   , b = −  .  . (4.3)  . .  v  .          Ix(xn, yn, t) Iy(xn, yn, t) It(xn, yn, t)

This system is over-determined since there are more equations than unknowns, and a direct solution is not possible. The Lucas-Kanade method employs least-squares to solve for the velocity vector ~u, given as

ATA~u = AT~b, (4.4) where the elements of ATA and AT~b are

  Pn 2 Pn Ix(xi, yi, t) Ix(xi, yi, t)Iy(xi, yi, t) ATA =  i=1 i=1  (4.5) Pn Pn 2  i=1 Ix(xi, yi, t)Iy(xi, yi, t) i=1 Iy(xi, yi, t)   Pn Ix(xi, yi, t)It(xi, yi, t) AT~b = −  i=1  . (4.6) Pn  i=1 Iy(xi, yi, t)It(xi, yi, t)

55 In practice, more weight is given to the pixels closer to the central pixel in the window. The weighted version of the least squares solution then becomes

ATW A~u = ATW~b, (4.7)

where W is an n × n diagonal matrix containing the pixel weights wi. The weights are typically a Gaussian function of the distance from the center pixel. The elements of ATW A and ATW~b are

  Pn 2 Pn wiIx(xi, yi, t) wiIx(xi, yi, t)Iy(xi, yi, t) ATW A =  i=1 i=1  (4.8) Pn Pn 2  i=1 wiIx(xi, yi, t)Iy(xi, yi, t) i=1 wiIy(xi, yi, t)   Pn wiIx(xi, yi, t)It(xi, yi, t) ATW~b = −  i=1  , (4.9) Pn  i=1 wiIy(xi, yi, t)It(xi, yi, t) which, after computing the summations, becomes a simple 2×2 system. The solution to the ﬁnal 2 × 2 system in Eq. (4.4) or Eq. (4.7) can be obtained eﬃciently using Cramer’s rule.

4.3.2 Phase-based Optical Flow

Phase-based optical ﬂow is a variation on the gradient-based method that begins with decomposition of the input image into complex-valued band-pass channels r(x, y, t).

Image intensity in Eq. (4.1) is replaced by phase, φ(x, y, t) , arg[r(x, y, t)]. The phase-based gradient constraint equation is then

φxu + φyv + φt = 0, (4.10)

56 where φx, φy, and φt are the spatio-temporal derivatives of phase, and the dependency on space and time has been dropped for brevity. Solution of the phase-based gradient constraint equation proceeds identically to the normal intensity gradient constraint equation.

Calculating partial derivatives of phase is challenging in practice, since it is only deﬁned on intervals of 2π. However, properties of complex numbers can be leveraged to take partial derivatives and diﬀerences that avoids phase wraparound issues [46]. Partial derivatives of phase in space and time are given by

Im(r (x, y, t)r∗(x, y, t)) φ (x, y, t) = x (4.11) x |r(x, y, t)|2 arg(r(x, y, t + ∆t)r∗(x, y, t)) φ (x, y, t) = (4.12) t ∆t where r∗ is the complex conjugate of r and ∆t is the time period between successive frames in the image sequence.

Complex band-pass images r(x, y, t) can be obtained by filtering the input image with quadrature pair filters [51], such as those used in the complex steerable pyramid [52] or the Riesz transform [53]. In this work, we chose to use a bank of Gabor filters tuned to specific center frequencies and bandwidths in order to isolate different spatial frequencies and orientations. A two-dimensional Gabor filter G(p, q) with unity aspect ratio is defined in the frequency domain as

(p − p )2 (q − q )2 G(p, q) = exp − 0 + 0 , (4.13) σ2 σ2

where (p, q) deﬁnes points on the frequency plane, p0 is the center frequency along the x-axis, q0 is the center frequency along the y-axis, and σ is used to control the

57 bandwidth of the filter. Alternately, the Gabor filter can be considered as a Gaussian function translated to a center frequency given by (p0, q0) with a standard deviation controlled by σ. Gabor filters can also be expressed in polar coordinates by their center frequency k0 and orientation θ, given by

q 2 2 k0 = p0 + q0 (4.14) q θ = arctan 0 , (4.15) p0

which is helpful for understanding which directions (θ) and spatial scales (k0) of the image the ﬁlter is selecting.

The filters used in this paper were designed to select for four orientations distributed evenly about the frequency half-plane and two spatial scales (center frequencies) that overlapped at their half-maximum values, starting with a center frequency of π/2 rad- s/px and successively decreasing by a factor of two to π/4 rads/px. Such a design pattern results in filters with octave bandwidth and −54 dB worth of damping at the zero frequency (DC) components and frequency components above the frequency twice as large as the center frequency. Contours corresponding to the half-maximums and a 1D slice along the positive X-axis of the filters used in the paper are shown in Figure 4.1.

This combination of orientations and spatial scales was found to be sufficient for measuring motion in the video sequences taken for this paper. However, filter bank design is not a one-size-fits-all scenario, and the most effective design will depend on the specific use case. Video resolution, texture, motion magnitude, and noise will all play a role in determining the proper filter bank for computation of image phase and estimation of optical flow.

58 Figure 4.1: Contours of half maximums (a) and a 1D slice (b) of the Gabor ﬁlter bank used in this paper

Filtering was carried out in the frequency domain for its ease of implementation. Input images were first converted to the frequency domain using the fast Fourier transform (FFT), filtered with the eight members of the Gabor filter bank using element-wise multiplication, and then inverse fast Fourier transformed to obtain the complex valued image sub-bands r(x, y, t). We denote image sub-bands and phase components at specific center frequencies and orientations by rk0,θ and φk0,θ, respectively.

Temporal derivatives of phase, φt, were smoothed using an amplitude weighted Gaus- sian blur, as in [39, 47]. Since the amplitude of the filter output can be used as a measure of the reliability of the phase signal, weighting in this manner has the effect of reducing noise in regions with low texture. The blur operation to obtain smoothed ˆ temporal phase differences φt is given as

(A ◦ φ ) ∗ w φˆ = t , (4.16) t A ∗ w where A ≡ |r| is the amplitude of the complex valued sub-band, ◦ denotes element- wise multiplication, and w is a Gaussian convolution kernel. We used a seven-tap ker-

59 1 nel taken from the binomial distribution for w, given as 64 1 6 15 20 15 6 1 .

We computed the spatial derivatives of image sub-bands r(x, y, t) in Equation 4.11

using the same procedure as [45]. First, each sub-band rk0,θ was convolved with a complex kernel based on the four-point central ﬁnite diﬀerence method and adjusted

k0,θ to the center frequency k0 of each sub-band. Then, the result was added to ik0r to account for the oﬀset of the center frequency in the band-passed channel. This procedure is summarized in Equations 4.17 and 4.18.

1 k0,θ k0,θ −i2k −i2k −i2k −i2k k0,θ r = r ∗ −e 0 8e 0 0 −8e 0 e 0 + ik0r (4.17) x 12 1 T k0,θ k0,θ −i2k −i2k −i2k −i2k k0,θ r = r ∗ −e 0 8e 0 0 −8e 0 e 0 + ik0r (4.18) y 12

The final step in estimating the flow vectors for the image sequence is solution of the phase-based gradient constraint given in Equation 4.10 at every pixel in the image using the least-squares method. A slight modification is necessary to the least squares formulation for the phase-based approach, where the elements of ATW A and ATW~b are also summed over all scales and orientations, given as

  P P Pn 2 P P Pn wiφ wiφxφy T  k0 θ i=1 x k0 θ i=1  A W A =   (4.19) P P Pn w φ φ P P Pn w φ2 k0 θ i=1 i x y k0 θ i=1 i y   P P Pn wiφxφt T ~  k0 θ i=1  A W b = −   . (4.20) P P Pn w φ φ k0 θ i=1 i y t

For this work, we set the weighting function W as the 5-tap convolution kernel taken 1 from the binomial distribution ( 16 1 4 6 4 1 ) and used convolution to perform the innermost summation, which sets the pixel neighborhood Ω to 5×5. The solution

60 to the least squares problem is obtained using Cramer’s rule mentioned earlier. The entire process of estimating the ﬂow vector ~u is summarized below.

1. Load an image sequence I(x, y, t)

2. Generate the Gabor ﬁlter bank Gk0,θ to the desired speciﬁcations

3. For every frame in the image sequence, do the following:

(a) For every spatial scale k0 and orientation θ in the Gabor ﬁlter bank, do the following:

i. Filter the image with a member of the Gabor ﬁlter bank to obtain

rk0,θ

ii. Compute the phase and amplitude components using arg(rk0,θ) and

|rk0,θ|

k0,θ iii. Compute the temporal derivative of the phase component φt using Equation 4.12

iv. Smooth the temporal derivative of phase using the amplitude weighted blur in Equation 4.16

k0,θ k0,θ v. Compute rx and ry using Equations 4.17 and 4.18

k0,θ k0,θ vi. Compute φx and φy using Equation 4.11

(b) Compute the elements of ATW A and ATW~b for all pixels in the image using 2D convolution and the weighting kernel W

i. Solve for the motion vector hu, vi by applying Cramer’s rule to matrix form of the least-squares solution

61 ii. Store the motion vector using the (x, y) pixel location as indexing variables

(d) Store the set of motion vectors using the current frame as the indexing variable

4. Return the full set of motion vectors u(x, y, t) and v(x, y, t) for every frame and pixel in the image sequence

4.3.3 Output-Only Modal Analysis

Output-only modal analysis, also referred to as Operational Modal Analysis (OMA), aims to determine the mode shapes, modal frequencies, and modal damping ratios of a vibrating structure without measurement of the input forces [54]. This is necessary in the case of testing spider webs, where measuring any force-based inputs would require a sensor greatly altering the mass of the web or where applying a displacement-based input would introduce artiﬁcial supports to the web structure altering its dynamics. Many methods are available for performing output only modal analysis, such as Stochastic Subspace Identiﬁcation (SSI) [55], Frequency Domain Decomposition (FDD) [56], and Blind Source Separation (BSS) [57, 58]. In this work, we use the Frequency Domain Decomposition method for its intuitive interpretation of spectral data and ease of implementation.

Frequency domain decomposition starts with estimation of the output power spectral density (PSD) matrix Sy(jω), which is the discrete-time Fourier transform of the output covariance sequence [54, 56]

∞ jω∆t X −jωk∆t Sy(e ) = Rke , (4.21) k=−∞

62 where Ri is the output covariance sequence, given as

N−1 1 X R = y yT . (4.22) i N k k−i k=0

The estimated PSD matrix is then decomposed using singular value decomposition at each discrete frequency ωi:

H §y(jωi) = U(jωi)Σ(jωi)U (jωi), (4.23)

where Σ(jωi) is the diagonal matrix containing singular values at a speciﬁc frequency

H ωi, U(jωi) is the matrix containing the left singular vectors, and U (jωi) is the Hermitian matrix of the left singular vectors.

Modal frequencies are identified as peaks in the singular values as a function of frequency. Typically, the first singular value will dominate all others, but at frequencies where multiple modes are present, other singular values may also exhibit peaks. At peaks where a single mode is dominating, the first singular vector provides an estimate of the mode shape ψˆ: ˆ ψ = ui1 , (4.24)

where ui1 is the singular vector at frequency ωi where a peak occurred in the plot of the ﬁrst singular value Si1 . Frequency domain decomposition was applied to the motion data estimated using phase-based optical ﬂow to identify modal frequencies and estimate mode shapes.

63 4.3.4 Experimental Setup

Webs built by adult female Araneus diadematus spiders were used in the study. Spi- ders were housed at room temperature in a laboratory environment on a controlled 12 hour light/dark schedule. The cages used to keep the spiders were made of clear acrylic plastic, approximately 400 mm square and 65 mm deep. During testing, the side covers of the cage were removed, exposing the web on both sides for motion capture and application of excitation. The cages were placed on a vibration-isolated air table during filming and surrounded with a sound-proofing structure to block spurious air currents and noise in the testing room from influencing web motion.

An impulse-like excitation in the transverse direction was applied to the webs using a small puﬀ of air delivered by a pipet. The air puﬀ was applied to a radial thread near the periphery of the web at approximately the 12 o’clock position. The magnitude of the excitation was low enough to not elicit any prey response behavior from the spider.

Videos were recorded using a GoPro HERO 5 Black at 240 frames per second with 720p resolution. The camera was positioned such that web motion occurred primarily along the Y-axis of the image and that the angle between the camera lens and the plane of the web was approximately 30 degrees.

A laser Doppler vibrometer (Polytec PSV-400) was used as a ground-truth measurement of web motion. A single-point measurement was taken from the cephalothorax of the spider. The vibrometer sampling frequency was set to 2560 Hz. Vibrome- ter and optical ﬂow measurements were aligned in time at their maximum recorded velocities.

Three webs were tested, each constructed by a diﬀerent spider. Spiders were located

64 on the hubs of their webs at the time of recording. For each web, the air puﬀ excitation was applied a total of three times. The vibrometer and camera remained in the same position for each of the three runs.

4.3.5 Video Processing

Specific regions of interest (ROIs) were taken from each video recording for optical flow estimation, rather than computing the flow vectors for the entire image frame. These ROIs were treated as virtual accelerometers, resulting in a point measurement of the local motion in the image frame. To compare results with the vibrometer ground- truth measurement, we placed one 40 × 40 px ROI over the spider’s cephalothorax, shown in Figure 4.2.

To showcase the ability to simultaneously measure motion at diﬀerent points in the image using optical ﬂow and its utility for mode shape estimation, we placed 21 ROIs of 20 × 20 px in one of the videos of Web 3, shown in Figure 4.9(a). These ROIs were meshed together for mode shape visualization in Figure 4.9(c).

4.3.6 Post-Processing

Several post processing steps were taken for analysis purposes following optical flow estimation and collection of the LDV velocity data. We used trapezoidal numerical integration to obtain displacements from both the optical flow and LDV data. A pre- integration, low-pass, fourth-order Butterworth filter with cutoff frequency of 60 Hz was used to reduce the influence of signal noise on the integration process. A post- integration, high-pass, fourth-order Butterworth filter with cutoff frequency of 0.5 Hz was used to eliminate drift in the output.

65 Figure 4.2: The three webs and their respective 40×40 px ROIs used for comparison with vibrometer recordings, (a) Web 1, (b) Web 2, and (c) Web 3

66 We reduced the optical flow data to single point measurements within each ROI from the image sequence by averaging the motion vectors. These quantities are referred to using an overbar notation (e.g.u ¯ for the horizontal velocity component). This averaging technique is only valid for sufficiently small regions within the image such that a single signal dominates the estimated motion vectors. The presence of multiple signals can be determined by examining the number of significant singular values in the estimated motion data for a particular ROI. No physical scaling was applied to the optical flow motion vectors, and therefore we present velocity and displacement results in terms of px/s and px, respectively.

4.3.7 Natural Frequency and Damping Ratio Estimation

To validate our phase-based optical ﬂow implementation against the vibrometer, we compared natural frequency and damping ratio estimates. We estimated the natural

frequency, fn, and damping ratio, ζ, of the first vibration mode using two different methods. The first method estimated the natural frequency using the largest peak in the power spectrum of the recorded motion and estimated the damping ratio by fitting an exponential of the form y = Ae−bt to the amplitude envelope of the recorded motion. Damping ratio was calculated from the fitted curve using the relationship

ζ = b/(2πfn). The amplitude envelope was obtained by taking the absolute value of the Hilbert transform of the motion signal.

The second method estimated natural frequency and damping ratio by ﬁtting a curve

to the autocorrelation sequence Ryy(τ) of the recorded motion. A single degree-of- freedom mechanical oscillator excited by a stationary process (e.g. an impulse or

67 white noise) has an autocorrelation sequence of the form [59]

! −2πfnζ|τ| πGfne ζ Ryy(τ) = cos (2πfdτ) + sin (2πfd|τ|) , (4.25) 2 p 2 4ζks 1 − ζ

where G is the input power spectrum (constant in this case), ks is the representative stiffness, fn is the natural frequency, fd is the damped natural frequency, and τ is the autocorrelation time delay. Curve fitting was accomplished using MATLAB’s fit function.

4.4 Results

First, we present comparisons between measurements taken with the laser vibrometer and motion computed using phase-based optical flow. The estimated natural frequencies and damping ratios are compared between the vibrometer and optical flow data for each of the three webs. Then, we show the results of applying Frequency Domain Decomposition (FDD) to optical flow data taken from a video recording of Web 3.

4.4.1 Vibrometer and Optical Flow Comparison

We provide plots of the velocity signals from both the vibrometer and phase-based optical flow recordings for visual comparisons. Figure 4.3 shows the velocity recordings, V (t), from the vibrometer for each of the three webs. For all recordings, the data was clipped to a 10 second duration. The recordings strongly resemble an underdamped, single degree-of-freedom oscillator, with each web having different amounts of damping. Figure 4.4 shows the corresponding average velocity vectors ~u(t) = hx¯(t), y¯(t)i estimated using phase-based optical flow for the 40 × 40 px ROIs of Figure 4.2.

68 Figure 4.3: Velocity recordings V (t) from the vibrometer for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i)

The velocity vectors estimated using phase-based optical ﬂow exhibit a similar underdamped, single degree-of-freedom oscillator pattern to the vibrometer velocity recordings; however, with much greater levels of noise present in the signal. Motion is predominantly in the y-axis as a result of camera placement.

Figure 4.5 displays the power spectrums of the three webs for both the vibrometer recordings and the y-component of the average optical ﬂow velocity vector. The power spectrums were averaged together for the three runs in each of the webs. The natural frequency, fn, is indicated in each spectrum as the frequency of the largest peak. The

69 Figure 4.4: Average velocities in the x- and y-axis computed using phase-based optical flow for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i) difference in signal-to-noise ratio (SNR) is clear from Figure 4.5. The vibrometer velocity data had a SNR of 50 dB while the optical flow velocity data had a SNR of 25 dB. The reduced dynamic range of the optical flow data can be attributed to several factors such as camera sensor noise, low texture in the image, the number of scales and orientations used in the Gabor filter bank, sub-optimal lighting conditions, the size of the ROI used for signal averaging, and the resolution of the camera sensor.

The estimated natural frequencies and damping ratios of the webs using the velocity signals are shown in Table 4.1. Overall, there is strong agreement between the

70 Figure 4.5: Power spectral densities of vibrometer velocity (a, c, e) and average flow velocity in the y-direction (b, d, f) for the three webs tested in the experiment. Peak frequencies are indicated by fn on each plot. estimated parameters using either method (peak picking and logarithmic decrement or autocorrelation fitting) on the vibrometer and optical flow velocity data. These results indicate that the motion estimated using phase-based optical flow is reliable, albeit with reduced dynamic range compared to the vibrometer. The webs each had slightly different natural frequencies and damping ratios. From Figures 4.3 and 4.4, Web 3 clearly had the greatest amount of damping while Web 2 had the least, which is reflected in the damping ratio estimates shown in Table 4.1. Visually inspecting the

71 webs themselves in Figure 4.2, this additional damping is likely a result of the greater number of spiral threads in Web 3 contributing to aerodynamic drag, an observation consistent with prior work [12].

Table 4.1: Natural frequency (fn) and damping ratio (ζ) estimates f (Hz) ζ Web Data n Peak Pick. Autocorr. Log. Dec. Autocorr. Vib. 2.578 2.588 0.033 0.034 1 Flow 2.582 2.585 0.032 0.035 Vib. 2.227 2.245 0.018 0.021 2 Flow 2.241 2.242 0.017 0.022 Vib. 2.656 2.650 0.046 0.051 3 Flow 2.633 2.641 0.041 0.054

We also show displacement signals for both the vibrometer, X(t), and single-point phase-based optical flow data, ~x(t) = hx,¯ y¯i, in Figures 4.6 and 4.7 obtained via numerical integration. Numerical integration has a smoothing effect on the data, and the visual similarities between the vibrometer and optical flow displacement data are more clear than when comparing the velocity data as a result of the reduction in noise.

Figure 4.8 displays the power spectrums for the LDV displacement recording and the y- component of the optical flow displacement data for each of the three webs. The improvement in SNR is noticeable from these plots, with the LDV having a slight improvement to 55 dB while the optical flow displacements had a significant improvement over the velocity data to a SNR of 50 dB. The indicated natural frequencies, fn, are consistent with those taken from the velocity power spectrums.

72 Figure 4.6: Vibrometer displacement X(t) computed by numerically integrating the recorded velocity signal for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i)

4.4.2 Frequency Domain Decomposition

Figure 4.9 displays the ROIs used to collect motion data for FDD, the first three singular values obtained from FDD at each frequency using the y-component of the optical flow velocity data for each ROI, and the single identified mode shape of Web 3 for the indicated frequency at 2.66 Hz. The identified mode shape strongly resembles the first mode of a continuous membrane, where all points on the membrane are in phase and maximum displacement occurs in the center of the structure.

73 Figure 4.7: Average displacements in the x- and y-axis computed by numerically integrating the velocities estimated using phase-based optical ﬂow for Web 1 (a–c), Web 2 (d–f), and Web 3 (g–i)

No other peak frequencies were present in the singular values, making the web’s response essentially that of a single degree-of-freedom structure. The lack of higher- order modes can be attributed to a few diﬀerent factors. First, the mass of the web- spider structure is dominated by the spider, eﬀectively making the web an elastic support for a point mass. Higher-order modes are also damped much more quickly, which makes exciting and measuring those modes challenging. Another reason higher modes were not detected is due to the lack of multiple input locations—large-scale

74 Figure 4.8: PSD of displacements modal testing typically involves a roving input to sufficiently excite the structure. The framerate of the camera and filters applied to the data may also be below the frequencies at which higher order modes occur in the web. The nature of the input excitation may also play a role in the modes that are excited—the air puff has a non-zero duration of interaction with the web that can contribute to a low-frequency bias in the output spectrums. Other methods of non-contact broadband excitation, such as by white noise played over a loudspeaker or controlled bursts of air from a solenoid would more closely resemble a uniform input spectrum to the web.

75 4.5 Conclusion

In this work, we presented a formulation of phase-based optical flow for making motion measurements and performing modal analysis on orb webs built by Araneus diadematus. We validated the measurements from our optical flow technique by comparing natural frequency and damping ratio estimates to those gathered using a Laser Doppler Vibrometer. Output-only modal analysis was performed using Fre- quency Domain Decomposition, and we were able to recover the first mode shape of the spider-and-web combined structure using a single video recording of the web after a small air puff was applied to a radial thread—a feat that would prove quite challenging, if not impossible, using a laser vibrometer.

Additional work is needed in the area of vibration testing of orb webs. The choice of excitation to the web is a critical piece in the experimental setup, and finding a method that meets the requirements of low added mass and broad-band excitation (e.g. white noise, zero mean stationary Gaussian process) is challenging. Excita- tion using short air puffs delivered using solenoids has shown to be promising for lightweight structures [60], and could be adapted for vibration testing of orb webs. Specific to spider webs, other researchers have made use of non-contact electrostatic excitation [61], which may enable finer control of the excitation process vs. the air puff method used in this work. The long term goal of vibration testing of orb webs is to build a modal model that describes the web’s dynamics. Such an experimental model would enable biologists to understand how the web filters signals in the form of an input-output relationship—a model that is currently unavailable in the literature. The techniques presented here are the first step towards such a model because they are both non-contact and multi-point.

76 In this study, we excited and measured transverse web motion. Other vibration directions (i.e. longitudinal and lateral) are also of interest and biologically relevant [2, 29]. Measuring both in-plane and out of plane vibrations using video-based techniques requires at least two cameras operating in a synchronized fashion to generate stereo images, as well as adapting the optical ﬂow estimator to operate in three spatial dimensions. The results presented here were limited to frequencies below 100 Hz due to the framerate of the camera used in the experiment. For future testing with orb webs, we will be moving toward high speed cameras with framerates exceeding 1000 frames per second with the intention to measure higher order mode shapes that may be heavily damped and therefore decay rapidly.

We propose that phase-based optical flow is well-suited for measuring the motion of lightweight biological structures, such as the orb web. In particular, phase-based optical flow is appropriate for lightweight structures because it doesn’t rely on point tracking for motion estimation, is robust to changes in lighting conditions, and it can perform simultaneous, full-field measurements of a structure’s motion in a single test. These features open the door to additional testing of orb webs, such as measuring prey impact events across the entire web as they occur, making measurements of spider webs in the field more accessible, and allowing researchers to measure the vibration response of individual threads to characterize their stiffness and damping. Other potential extensions include using vibration measurements to estimate the tension distribution in the web.

77 Figure 4.9: Web video frame setup and results for Frequency Domain Decomposition (FDD), ROIs used in optical ﬂow estimation (a), FDD spectrum of singular values (b), the mode shape estimated using FDD from the right singular vectors (d)

78 5 Conclusion

This dissertation addressed several challenges in the field of spider web vibration research. Namely, it established a computational model for web vibrations that accounts for arbitrary web geometry, composition, and tension. This model is formulated using a coupled oscillators approach, with the form of an input-output linear model using frequency response functions. A controlled testing environment for studying web vibrations and vibration localization was presented in the form of an enlarged, biomimetic orb web. The enlarged webs, a physical model of the biological system, were used to validate the computational model. An exploration of vibration cues in large, interconnected networks of strings was made possible through vibration testing of different artificial orb webs. Estimation of vibration source location using an array of sensors located at the center of the artificial webs was demonstrated in hard- ware, confirming the potential for spiders to leverage the signals arriving at their feet for the task of locating trapped prey. Finally, the difficulty in obtaining vibration measurements from orb webs was tackled using a phase-based optical flow estimator that was inspired by recent work in computer graphics concerning Eulerian video magnification.

Each of these contributions represent steps along the path to understanding the process by which orb weaving spiders can locate trapped prey and discriminate between different signal types, such as that of a fly or bee. The work presented here is in- tended to be used as building blocks for an experimentally-derived model of orb web vibrations by pairing frequency based substructuring with non-contact vibration measurements taken directly from orb webs (i.e. those obtained from phase-based optical flow). Modelling a full web using frequency based substructuring relies on

79 assembling the full web’s dynamics from smaller substructures, in this case individual silk threads. Different silk types, tensions, and lengths would be needed to form a library of spider silk dynamics that could be used in a frequency based substructuring model. Using such a library, models for webs could be generated with arbitrary geometries and compositions that adhere to the dynamics of actual spider silk, allowing the researcher to investigate questions such as how web geometry influences vibration propagation and compare the vibration response of webs from different species of orb weaving spiders.

The phase-based optical flow technique for vibration measurement covered in this work opens the door for capturing many interesting events and interactions between the spider and its web in far greater detail than what has been possible with the Laser Doppler Vibrometer (LDV). Simultaneous, multi-point measurement of web motion during prey capture is possible using optical flow. Future use of the optical flow technique on spider web vibrations will involve measuring real prey capture events and understanding how energy and information propagate through the web. Ultimately, establishing a link between the evolutionary pressures that have driven differences between the webs of different orb weaving species could be tracked back to the web’s performance in different roles such as damage tolerance, capturing flying insects, and signaling the location of trapped prey.

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