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Traveling sea stars: hydrodynamic interactions and radially-symmetric motion strategies for biomimetic robot design

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Traveling sea stars: hydrodynamic interactions and radially-symmetric motion strategies for biomimetic robot design

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Content TRAVELING SEA STARS: HYDRODYNAMIC INTERACTIONS AND RADIALLY-SYMMETRIC MOTION STRATEGIES FOR BIOMIMETIC ROBOT DESIGN by Mark Hermes A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) December 2021 Copyright 2021 Mark Hermes Contents List of Tables iv List of Figures v Abstract viii 1 Introduction 1 2 Drag and Lift Measurements of Sea Star Models 6 2.1 Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Literature Review: Fluid Dynamics of Morphing RS Bodies . . . . . . . . . . 7 2.3 Drag and Lift Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Effect of aspect ratio and orientation on drag and lift . . . . . . . . . 8 2.3.2 PIV flow visualization and control volume analysis . . . . . . . . . . 11 2.3.3 CFD simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Experimental Drag and Lift Optimization of Shape Morphing Robot 23 3.1 Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Literature Review: Fluid-structure Interaction Optimization . . . . . . . . . 27 3.3 Shape Optimization for a 2DOF Soft Sea Star . . . . . . . . . . . . . . . . . 29 3.4 Shape Optimization for a 7DOF Soft Sea Star . . . . . . . . . . . . . . . . . 34 3.5 Future Optimization Directions . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Tripedal Robot Motion Analysis and Control 38 4.1 Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Literature Review: Radially-symmetric Robotics . . . . . . . . . . . . . . . . 39 4.3 Literature Review: Gait Optimization . . . . . . . . . . . . . . . . . . . . . . 41 4.4 Experimental Approach and Physics-Based Model . . . . . . . . . . . . . . . 43 4.4.1 Robot Design and Characterization . . . . . . . . . . . . . . . . . . . 43 4.4.2 Motion Experiments and Video Tracking . . . . . . . . . . . . . . . . 44 4.4.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.4 Normal Force Simulation Predictions and Measurements . . . . . . . 48 4.5 Parameter Sweep Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.1 Frequency Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ii 4.5.2 Sweep Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.3 Phase Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5.4 Pure Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5.5 Pure Translation: Brittle Star Inspired Rowing & Reverse Rowing . . 55 4.6 Analysis of Rowing Gait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.6.1 Analysis of Translation . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.6.2 A Successful Reverse Rowing Gait . . . . . . . . . . . . . . . . . . . . 62 4.6.3 Non-sinusoidal Gait Optimization . . . . . . . . . . . . . . . . . . . . 64 4.7 Gait Map and Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7.1 Omnidirectional Gait . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.7.2 Gait Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.7.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.8 Path Following Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Conclusion and Future Work 76 References 78 iii List of Tables 2.1 Mean values for lift (F 0 L ) and drag (F 0 d ) per unit length estimated from control volume analyses of planar vector fields obtained from PIV. . . . . . . . . . . 12 2.2 Lengthsandareasforexperimentalmodels. Fortheseastarmodels,L = 19cm correspondstothefrontalwidthwhenoneofthearmsisorientedintotheflow, Θ = 0 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Parametric combinations tested for sinusoidal actuation experiments. . . . . 45 4.2 Relationship between desired translation direction and the gait parameters for each limb. All angles are prescribed in degrees. . . . . . . . . . . . . . . . 68 4.3 Experimental performance in terms of cumulative path following error (Δ, m) and completion time (T c , s). Rows and columns correspond to Fig. 4.22. . . 73 iv List of Figures 1.1 Figure from Giorgio-Serchi [1] depicting how shape change can be used to cancel drag, or generate thrust. . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Mean fluid forces on sea star and spherical dome models. . . . . . . . . . . . 9 2.2 Mean fluid forces on sea star models as a function of orientation. . . . . . . . 10 2.3 2D flow visualization for centerplane of sea star and spherical dome models. . 11 2.4 Comparison of experiment and simulation mean drag and lift forces. . . . . . 13 2.5 Pathline visualizations from CFD simulations. . . . . . . . . . . . . . . . . . 14 2.6 Experimental schematic of load cell measurement system. . . . . . . . . . . . 17 3.1 Summary of crawling strategies observed in animals. . . . . . . . . . . . . . . 26 3.3 Diagram showing examples of gait optimization. . . . . . . . . . . . . . . . . 26 3.2 Summary of swimming strategies observed in animals. . . . . . . . . . . . . . 27 3.4 Schematic of 2DOF shape optimization study. . . . . . . . . . . . . . . . . . 30 3.5 Comparisonofminimumdragandmaximumdownforce(negativelift)obtained using a grid search and the genetic algorithm. . . . . . . . . . . . . . . . . . 33 3.6 Schematic of morphing 7DOF robot. . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Drag force measurements as a function of arm extension for 7DOF robot. . . 36 3.8 Isosurface plots of drag and lift for 3-parameter grid search. . . . . . . . . . 37 4.1 Minimally-actuated tripedal robot demonstrating curve following capabilities. Experimental video: https://youtu.be/F9UxznYtJGM. . . . . . . . . . . . . 39 v 4.2 (a) Top and (b) iso view of the robot. . . . . . . . . . . . . . . . . . . . . . 44 4.3 Simplified representation of tripedal system. . . . . . . . . . . . . . . . . . . 45 4.4 Comparisonofmodelpredictionsandmeasurementsfornormalforceestimates of tripedal robot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Comparison between model predictions and experimental results over a period of 30 s for the frequency variation experiments. . . . . . . . . . . . . . . . . 50 4.6 Comparison between model predictions and experimental results over a period of 30 s for the sweep variation experiments. . . . . . . . . . . . . . . . . . . . 51 4.7 Comparison between model predictions and experimental results over a period of 30 s for the phase variation experiments. . . . . . . . . . . . . . . . . . . . 52 4.8 Illustration of pure rotation for a phase difference of Δψ = 120 ◦ for the high- friction sandpaper-paper contact. . . . . . . . . . . . . . . . . . . . . . . . . 53 4.9 Model predictions of net translational curvature from the tripedal robot loco- motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.10 Comparison between model predictions and experimental observations for reverse-rowing and rowing locomotion over a period of 20 s. . . . . . . . . . . 57 4.11 Measured and predicted robot translation for the rowing configuration with varying sweep angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.12 Model predictions for rowing translation speed ( ¯ U) as a function of friction coefficient for f = 1.0 Hz and varying sweep angles ξ = [20, 30, 40, 50] ◦ . . . . 58 4.13 Simplified geometric representation of single active limb. . . . . . . . . . . . 59 4.14 Simulation (black) vs experiment (red) data of ˙ r x (dashed) and ˙ x (solid) for rowing with ξ = 30 ◦ ,f = 1 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.15 Schematic showing limb length variationl(s,t) for the modified reverse-rowing locomotion strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.16 Predicted translation speeds for rowing and modified reverse-rowing locomo- tion. In all cases, the sweep angle is ξ = 30 ◦ and actuation frequency is f = 1.0 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi 4.17 Comparison of simulation results for tripedal motion using PCHIP and piece- wise sinusoid gait parameterizations. . . . . . . . . . . . . . . . . . . . . . . 66 4.18 Optimal parameterized curve to maximize distance traveled . . . . . . . . . 66 4.19 Predicted and measured paths x(t) for the sandpaper-paper contact for (a) translation gait and (c) rotational gait. . . . . . . . . . . . . . . . . . . . . . 67 4.20 A comparison of the simulated average map,M avg to experimentally collected data points. This map is used to prescribe anα-limb gait parameter,α, given a desired angle of translation, θ. . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.21 Flow chart and diagram showcasing feedback processing and gait selection. . 70 4.22 Tripedal path following in a flow field. . . . . . . . . . . . . . . . . . . . . . 71 vii Abstract The goal of this thesis is to provide a framework for considering Fluid-Structure Inter- actions (FSI) of Radially Symmetric (RS) crawling robots. We separated our investigation into two parts: examining FSI of surface-attached radial symmetries (our focus was sea star- inspired geometries) and developing a stable locomotion platform for moving through flow fields. To study fluid dynamics of RS structures in steady flow, an ATI-gamma force trans- ducer system measured drag and lift forces on 3D-printed sea star and spherical dome models in a water channel facility. Experiments showed that sea star shapes generated downforce by diverting momentum upward through generation of vortices that created upwelling along the center-line plane. Also, downforce magnitude can be tuned by varying geometric param- eters such as aspect ratio (AR). We applied the results of this work to a shape-morphing sea star-inspired silicone robot. A genetic algorithm optimized drag and lift of a 2 degree of freedom (DOF) and a 3 DOF robot. Current research efforts are underway for optimizing a 7 DOF system. In addition to the FSI studies, we created a tripedal crawling system to generalize and minimize complexity of an RS locomotion device. A mathematical model that accurately captured the dynamics of the tripedal robot was developed and used in an image-feedback control scheme. Through simulations, we created a mapping of gait param- eters to translation direction. Physical experiments demonstrated that the robot is robust to fluid force disturbances while path-following using a PI controller based on the gait map. Applications that can benefit from this research include RS robot cyclic control using our gait map strategy, surface mounted features requiring downforce production using our sea star designs, and Experimental Shape Optimization (ESO) for minimization of drag and lift viii forces using our silicone skin morphing robotic platform. We hope that this work will inspire others to explore RS robotics for further research and applications. ix Chapter 1 Introduction Recent efforts in robotic design have focused on developing locomotion strategies inspired by RS organisms. Dominant examples of these robots include swimmers inspired by (or resembling) jellyfish and crawlers inspired by (or resembling) echinoderms. In this thesis, we focus our efforts on crawling translation modes. Advantages that RS crawling robots have demonstrated over comparable Bilaterally Symmetric (BS) designs include omnidirectional translation[2], robustnesstolimbdamageandgaitadaptation[3], enhancedstabilitythrough multi-point substrate contact [4], arm/leg interchangeability [5], and climbing prowess [6]. However, despite these robots being modeled after ocean dwelling organisms and many appli- cations targeting wave-intense benthic environments, none of these robots have developed strategies for adapting to fluid flow-induced force disturbances. This is an oversight that this present work seeks to address. We believe that applying fluid-force mitigation strategies to robots – learned from organisms which have evolved to survive large impact wave events – is an area rich in discovery. In this introduction, we will first pose scenarios where RS crawling robots deployed in underwater environments can benefit from the ability to adapt their shape for enhancing FSI, and we describe ways in which this might be accomplished. We then present a case study of how the sea star, Pisaster Ochraceus, may change its body shape to minimize hydrodynamic forces. We then connect how this ability can be applied to the robot scenarios discussed prior, including how we applied it to our silicone robotics testbed. To motivate why we are interested in RS shape morphing for fluid dynamics, we describe here how hydrodynamic forcing may affect an underwater crawling robot’s locomotion strat- egy in uniform flow and in alternating flow. Using these scenarios, we propose methods to apply shape adaptation to mitigate these challenges. For a robot moving through a steady or unsteady uni-directional flow, a robot may benefit from understanding its limitations, 1 namely, how large an acceleration or velocity it can withstand before dislodgement. Once a threshold velocity or acceleration is detected, the robot can flatten itself or produce a ramp- like effect to create downforce. Though this strategy may impede locomotion, it could also prove to be essential for resisting surface detachment. Indeed, this strategy is not without precedent in nature. The nudibranch Tritonia tetraquetra will flatten itself in high flow envi- ronments, waiting until the intensity reduces to resume feeding [7]. Blue crabs also exhibit similar behavior and will produce different walking gaits depending on flow velocity [8]. For a robot in an alternating flow environment, the robot could change its shape to take advan- tage of the oscillating motion. Giorgio-Serchi et al. outline this possibility in [9, 10], where they expand and contract a sphere at the natural frequency of the system, shown in fig. 1.1. This work demonstrates drag cancellation in a zero-flow fluid environment. However, the idea of cyclic inflation could also be applied as a stability strategy to regulate position or as propulsion to generate thrust in an alternating flow field. Tousethesestrategies, therobotmusthaveawaytodetecttheflowaroundit. Itssensors need to be able to detect either the magnitude of flow velocity or acceleration, as well as the direction of flow relative to the body. Many researchers have developed local sensory solutions inspired by the lateral line sensory system in fish [11, 12, 13, 14]. These systems are pressure-based, which have advantages over cantilever-based sensors in that they are less disrupted by boundary layer variations and are non-intrusive to the flow. The disadvantage is that the reference pressure needs to be calibrated, and this can be problematic in deep water [15]. Kottapalli and Triantafyllou suggest that encapsulating the sensors to replicate the canal neuromasts of fish may solve this issue [16]. Though boundary layer development and flow disruptions from cantilever sensors may be problematic for some robots, others have used them with success [17, 18, 16]. Other sensing methods that have been successful for external probing, like hot-wire anemometry [19] and laser-doppler anemometry [20] may also have potential for local measurement. Once the flow intensity and direction are estimated, the robot then needs to make deci- sions based on sensor information. One strategy is to identify flow structures and use them 2 Figure1.1: Figure from Giorgio-Serchi [1] depicting how shape change can be used to cancel drag, or generate thrust. (a) Shere renderings based on stair-step shape change profile. (b) Temporal shape changes of a sphere in a viscous fluid with relation to initial velocity (effect of damping is cancelled by shape variations). for navigation. Kruusmaa et al. demonstrated this using a robotic fish to follow wake struc- tures, thus reducing drag and increasing swimming efficiency [11]. Nelson and Mohseni used pressure measurements to approximate body forces on an AUV for jet-correction feedback control [21]. Fernandez et al. showed that a pressure-sensor array, coupled with an extended kalman filter estimation algorithm, could identify the shape of an object by circling it [22]. Colvert and Kanso showed that neural networks with limited local sensing can classify vor- tex structures [23]. Though these applications have been restricted to swimming, similar ideas can be used for crawling: a crawling robot may be able to identify a flow-obstructive structure that provides shelter from large forces. A robot may also be able to exploit the reduced velocities in boundary layers by moving further into the developed region, an effect that insects often exploit on moving vehicles. 3 Pisaster Ochraceous sea stars are an example of a crawling animal that changes its shape in an alternating flow field (intertidal zones). However, they do not change their shape to enhance locomotion, as described by the alternating flow example in the previous paragraph, but rather change shape over an extended time period (∼1 month) to possibly minimize dislodgement risk [24]. They accomplish this by increasing their Aspect Ratio (AR) in environments with high wave intensity, and decreasing it in areas of low wave intensity, where AR is the length to width ratio of arms. Effectively, increasing AR makes them skinnier and reduces their height. This natural demonstration of shape morphing for environmental adaptation inspired us to look closer at the fluid dynamics of these sea star shapes. For the research presented in this thesis, we chose to restrict the complexity of shape changingRSroboticsandfocusexclusivelyonsteady, unidirectionalflowoutsideofboundary layers, though we acknowledge that this scope reduction leaves a large body of research left to explore. We began our investigation into fluid dynamics of RS robots by focusing on pentaradial sea star body shapes with dimensions similar to those reported in [24] and quantifying hydrodynamic characteristics to answer the question, "Is there a fluid dynamic- related performance benefit to changing AR?" In chapter 2, we answer this question by measuring hydrodynamic forces on sea star-shaped and spherical dome 3D printed models. We used what we learned from that project and applied it to the construction of a shape optimizing soft shell (silicone) sea star robot presented in chapter 3. We also sought to constructaminimalcomplexityRSlocomotionplatformthattakesadvantageoftheenhanced stability through multi-point contact property, as this is also important for designing robust robots moving in a fluid. We present our work developing this robot and demonstrating fluid force disturbance rejection while path-following in chapter 4. These works spotlight an unstudied branch of RS robotics and provide a foundation for future studies to improve. Because the advantages of crawling underwater robots over swimming robots are associated with fast moving fluid flow and shallow environments, it is important to design these robots to take advantage of FSI rather than fighting them. 4 The contributions of this work are as follows: • From chapter 2, we showed that sea star body-induced downforce can be exploited for surface mounted objects where the attachment strength is limited in the wall-normal direction. • In chapter 3, we demonstrated the feasibility of ESO applied to fluid dynamic applica- tions. One benefit of ESO is the ability to live-tune body shape to new environments or to compensate for system changes such as aging or payload modifications. Experimen- tal design is another area where ESO can be leveraged for fluid-dynamic performance gains. • Insection4.4.4, wedemonstratedthatatripedalrobotcansuccessfullyestimatenormal forces when its legs are in contact with the ground. This knowledge can be used to measure fluid intensity, direction, and forcing while crawling using a well-characterized mapping of body position to drag and lift force. • Insection4.7wedevelopedahigh-levelabstractionofagait-map-with-feedbackcontrol strategy that can be used for direction control of other cyclic locomotors. 5 Chapter 2 Drag and Lift Measurements of Sea Star Models The text presented in this section is adapted from the Scientific Reports publication: Hermes, M., Luhar, M. (2021) Sea stars generate downforce to stay attached to surfaces. Nature Sci Rep 11, 4513. https://doi.org/10.1038/s41598-021-83961-z 2.1 Chapter Introduction In this chapter, we evaluate the hydrodynamics associated with sea star body shapes in turbulent flow conditions via laboratory experiments and CFD simulations. Though natural sea stars can exhibit significant diversity in surface texture, number of arms, and other morphological properties, we limit our study to smooth pentaradial shapes to isolate the effect of arm aspect ratio. Specifically, we consider models of comparable shape and size to adult purple sea stars. Sea stars are known to increase dramatically in size from the early juvenile to the adult stage [25, 26]. The evolution in the hydrodynamic response during this development is outside the scope of the present study. In addition to providing insight into previous biological observations, the present effort has potential applications in fields ranging from flow control to shape optimization for vehicles. The experiments performed in this chapter similarly show downforce generation for sea star models that is associated with an upward deflection of fluid flow around the body. However, downforce is not observed in experiments with spherical domes of comparable size to the sea star models. These observations, together with results from complementary CFD simulations of flow over cones, pyramids, and triangular cylinders, suggest that the RS sea star geometry generates a hydrodynamic response similar to (nearly) two-dimensional triangular cylinders. 6 2.2 Literature Review: Fluid Dynamics of Morphing RS Bodies Intertidal sea stars often function in environments with extreme hydrodynamic loads [27] that can compromise their ability to remain attached to—and move on—surfaces [28]. While behavioral responses such as burrowing into sand or sheltering in rock crevices can help minimize hydrodynamic loads, previous work shows that sea stars also alter body shape in response to flow conditions [29]. This morphological plasticity suggests that sea star body shape and size may play an important hydrodynamic role. Specifically, Hayne and Palmer [29] demonstrated that the arms of the purple sea star (Pisaster ochraceous) narrow and lose mass when transplanted to a more wave-exposed environment. The authors hypothesized that this transformation is a functional response to wave intensity: that by changing shape they are minimizing drag or related hydrodynamic forcing. Further, Computational Fluid Dynamics (CFD) simulations of wave-exposed sea star models showed a decrease in both lift and drag coefficients compared to models of sheltered sea stars, suggesting that the observed shape adaptation conferred a hydrodynamic benefit. Existing research investigating flow over surface-mounted objects has focused on charac- terizing the effect of surface fouling, designing drag-reducing structures, and studying airflow around buildings [30, 31, 32, 33]. A variety of geometries have been considered, including cubes [34, 35], circular cylinders [36], hemispheres [37, 36, 38], pyramids [39, 40], cones [41, 42, 43], and triangular cylinders [44, 45]. Most of these studies provide drag and drag- coefficient estimates. Measurements of lift are rarer, in part because lift forces may be less important for the applications described. However, lift is a useful performance metric for cases in which surface attachment is a concern. Perhaps the closest shapes to the sea star that have been studied are pyramids and triangular cylinders. Measurements made by Ikhwan [39] indicate that pyramids generate lift forces that increase with increasing aspect ratio. On the other hand, Iungo and Baresti [44] show that triangular cylinders generate downforce. This downforce is shown to originate 7 from an upward deflection of the wake behind the cylinders, and its magnitude increases with increasing steepness of the triangular cross-section. 2.3 Drag and Lift Measurements We mounted 3-D printed sea star and spherical dome models to a load cell in a water channel to study the effect of morphology on the mean drag and lift forces (F d ,F L ) generated by these objects across a range of flow speeds (U). The effect of sea star morphology was studied by varying the arm aspect ratio AR over the range of values reported for Pisaster ochraceous[29]; here, AR is defined as the ratio between the arm length measured from the distal tip to the central axis and the arm width measured at the base intersection. We also measured the effect of body orientation with respect to the flow, Θ, for the pentara- dial sea star models using a servo motor positioning system. We supplemented these force measurements with PIV-based flow visualization and control volume analyses. To provide additional qualitative and quantitative insight into the experimental observations, we also pursued CFD simulations for a limited range of geometries. Details regarding model design, experimental apparatus, and analysis methods can be found in section 2.5. 2.3.1 Effect of aspect ratio and orientation on drag and lift Figure 2.1 compares the mean drag and lift generated by three sea star models of varying aspectratioagainstthedragandliftgeneratedbyasphericaldomeofsimilarheightandbase diameter as the sea star models. For these measurements, the sea star models were oriented at Θ = 0 ◦ , i.e., withonelimbpointingintotheoncomingflow. Asexpected, themagnitudeof the drag and lift forces generated by the models increases with increasing freestream velocity. However, the corresponding drag and lift coefficients (C d , C L ; see eqs. (2.2) and (2.3)) show more limited variation with Reynolds number (Re, defined using base diameter; see eq. (2.4)). Perhaps the most striking feature of the results presented in fig. 2.1 is that all sea star models generate downforce (i.e., F L < 0 and C L < 0) while the spherical dome 8 (a) (b) (c) (d) Figure2.1: Mean (a) drag force, (b) lift force, (c) drag coefficient, and (d) lift coefficient values for sea star and spherical dome models shown as functions of freestream velocity (a, b) and Reynolds number (c, d). The sea star models have aspect ratiosAR = 4.0 (red symbols), 2.5 (green symbols), and 1.5 (blue symbols). model generates positive lift forces that are much higher in magnitude. Lift coefficient values for all three sea star models are similar within uncertainty at the lowest Reynolds number. However, measurements made at higher Reynolds numbers suggest that the AR = 1.5 sea star model (blue symbols) generates the least downforce and has the lowest|C L |. Importantly, though the pentaradial sea star models generate downforce, they also incur a drag penalty compared to the spherical dome. The drag coefficients for the sea star models (C d > 0.9) are significantly higher than those for the spherical dome (C d < 0.6). Further, the drag coefficients for the sea star models increase with increasing aspect ratio. 9 0° 36° 18° 0° 36° 18° U ∞ U ∞ (a) (b) Figure 2.2: Drag and lift coefficient values for sea star models for varying orientation angles at flow speed U≈ 0.46ms −1 . Given the pentaradial symmetry of the sea star models, C d and C L for Θ = 36 ◦ to Θ = 72 ◦ can be estimated by mirroring the data shown in this figure. Figure 2.2 shows drag and lift coefficients for all three sea star models as a function of the orientation angle Θ for the highest Reynolds number case shown in fig. 2.1. Consistent with the results from fig. 2.1, there is a monotonic increase in C d as a function of aspect ratio. However, thedragcoefficientvaluesshownoconsistenttrendwithrespecttoorientation. Lift coefficients for all three geometries similarly show no clear trend with respect to orientation, though there is a consistent increase in C L with Θ for the model with AR = 2.5 (green symbols). In general, measured C L values for the models with AR = 4.0 (red symbols) and AR = 2.5 (green symbols) are significantly lower than the values measured for AR = 1.5 (blue symbols). Together, these observations indicate that the drag and lift forces generated by the sea star models are relatively insensitive to orientation, and confirm thatC d and|C L | increase with increasing AR. 10 (a) (b) (c) (e) (e) (f) (d) (e) (f) Figure 2.3: Mean flow visualization from experiments (a)-(c) and CFD simulations (d)-(f) for: AR = 4.0 sea star model (a), (c); AR = 1.5 sea star model (b), (e); and spherical dome (c), (f). The experiments show results for U = 0.47± 0.01ms −1 while the simulations were performed for U = 0.35ms −1 . Panels (a)-(c) show the vector field estimated from PIV while panels (d)-(f) show contours of the mean velocity in the streamwise direction. All panels show the flow field at the central (or median) plane of the models. The sea star models are oriented at Θ = 0 ◦ . Figure created using Ansys Fluent 2019 R2 https://www.ansys.com/. 2.3.2 PIV flow visualization and control volume analysis To provide further insight into the force measurements shown in fig. 2.1, we pursued PIV experiments at freestream velocity U = 0.47± 0.01. Though the fluid forces acting on the objects arise from three-dimensional flow fields, the planar mean flow visualizations shown in fig. 2.3(a)-(c) provide a partial physical explanation for the observed trends in drag and lift. Specifically, the sharp apex creates a distinct separation point for the flow over the sea star models. In contrast, the flow over the spherical dome has a separation point much further down the body, resulting in a smaller wake region compared to the sea star models. Further, the wake behind the high aspect ratio sea star model is larger than the wake behind the low aspect ratio sea star model. These observations are qualitatively consistent with the drag force measurements shown in fig. 2.1(a): the sea star models generate more drag than the spherical dome, and the drag force generated increases with increasing aspect ratio. Importantly, the wakes behind the sea star models clearly show an upward redirection of the 11 flow beyond the apex. An upward redirection of the flow is not observed for the spherical dome due to the delayed separation. These observations provide a qualitative explanation for the lift trends observed in fig. 2.1(b). The downforce experienced by the sea star models is a consequence of the upward redirection of fluid momentum in the wake. Table 2.1: Mean values for lift (F 0 L ) and drag (F 0 d ) per unit length estimated from control volume analyses of planar vector fields obtained from PIV. Model F 0 L [Nm −1 ] F 0 d [Nm −1 ] Sea star, AR = 4.0 -1.32 5.94 Sea star, AR = 1.5 -1.19 4.06 Spherical dome 2.41 3.27 We also used a control volume approach (described in section 2.5) to estimate drag and lift forces per unit length (F 0 d , F 0 L ) from the planar PIV measurements. Table 2.1 lists the mean values for F 0 d and F 0 L obtained after averaging over all PIV frames. Consistent with the load cell measurements of lift shown in fig. 2.1(b), F 0 L is positive for the spherical dome and negative for the two sea star models. In addition, the magnitude of the estimated lift per unit length (|F 0 L |) for the spherical dome is nearly twice that for the sea star models. Both sea star models have similar F 0 L values though the AR = 4.0 model is estimated to generate slightly higher downforce per unit length. Similarly, the estimates forF 0 d are also consistent with the drag measurements shown in fig. 2.1(a). The high aspect ratio sea star model has higherF 0 d than the low aspect ratio model, and the low aspect ratio star has higher F 0 d than the spherical dome. 2.3.3 CFD simulation results To supplement the experiments, we pursued CFD simulations in ANSYS Fluent for a subset of the sea star models, the spherical dome, and several related geometries. The additional shapes (hemisphere, cone, pyramid, triangular prism; see fig. 2.5) were created to have the same frontal area and height as the sea star. For the triangular prism, the streamwise length was set to be similar to the base width of the sea star model, L = 19cm, such that the prism cross-section was identical to the midplane cross-section of the sea star 12 (a) (b) Figure 2.4: (a) Drag and (b) lift coefficients of models for experiments and simulations in a flow with speed 0.35ms −1 . model shown in fig. 2.3(a). These shapes were tested in water flow with an inlet speed of U = 0.35ms −1 . Experimental data for the sea star and spherical dome models shown in fig. 2.1 were linearly interpolated for comparison at this flow speed. As shown in fig. 2.4, the drag and lift coefficients computed from the simulations agree, within uncertainty, with the values obtained in experiments for the AR = 4.0 sea star and spherical dome. Further, as shown in fig. 2.3, the mean flow fields and wake structures obtained in the simulations are in good qualitative agreement with the PIV results. For instance, the vertical extent of the wake region is largest for the AR = 4.0 sea star and smallest for the spherical dome. These observations give us confidence that the CFD simu- lations can reasonably reproduce the flow physics observed in the real world experiments. The simulation results in fig. 2.4(b) also suggest that downforce is not obtained for the pyramid or cone shapes, despite these objects having a sharp apex similar to the sea star models. Downforce is only observed for the triangular prism, and this downforce carries a significant drag penalty. These observations can be explained by considering the pathline visualizations shown in fig. 2.5. For the downforce-producing shapes, the pathline visual- izations show that the streamwise vorticity is negative (blue) on the left and positive (red) 13 Figure 2.5: Pathlines of flow over a spherical dome, cone, pyramid, AR = 1.5 sea star, AR = 4.0 sea star, and triangular prism. The pathlines are colored based on the local streamwise vorticity. Objects are placed in order of descending lift force. Figure created using Ansys Fluent 2019 R2 https://www.ansys.com/. on the right side of the shapes. This results in a significant upwelling of fluid from the cen- tral region of the wake into the freestream. The momentum transport associated with this upwelling flow explains the high drag and negative lift generated by the sea star models and the triangular prism. Pathlines near the apex for the cone and pyramid shapes also show evidence of a similar arrangement in streamwise vorticity. However, pathlines at the base of the pyramid and cone shapes show a reversal in sign for the streamwise vorticity: the stream- wise vorticity is positive (red) on the left and negative (blue) on the right, similar to the flow field observed around the spherical dome. This is indicative of a downwelling flow that transports high momentum fluid from the freestream into the wake and explains the lower drag and positive lift forces experienced by spherical dome, cone, and pyramid shapes. Note that the pathlines and vorticity distributions around the base of the lift-producing spherical dome, cone, and pyramid shapes in fig. 2.5 are consistent with the horseshoe vortex systems typically observed in flows around surface mounted bodies [46, 47, 36]. Visualizations for the sea star bodies do not show evidence of this horseshoe vortex system. 2.4 Discussion The ability to stay attached to surfaces plays an important role in sea star locomotion and survival. The results presented in this study show that pentaradial sea star body shapes 14 generate downforce independent of the incoming flow direction. This downforce could help sea stars avoid hydrodynamic dislodgement. Hayne and Palmer[29] showed that Pisaster ochraceous sea stars exhibit significant mor- phologicalplasticityinresponsetohydrodynamicconditions. Specifically, observationsmade in different environments showed a linear relationship between mean wave speed and sea star aspect ratio. Transplant studies confirmed this trend: sea stars moved into higher energy environments showed an increase in body aspect ratio. One hypothesis proposed to explain this correlation was that the change in body shape may enable sea stars to better resist hydrodynamic forces. Our results show that an increased aspect ratio produces a larger downforce, but this comes at the expense of a larger drag force. In the present study, sea starheightandfrontalareaweremaintainedconstantacrossthedifferentaspectratiostested. In contrast, the observations made by Hayne and Palmer[29] indicate that the increase in sea star aspect ratio is also accompanied by a decrease in height, i.e., sea stars exhibit higher aspect ratios and reduced height in more energetic environments. It is possible that the high aspect ratio body shape generates greater downforce while the reduction in height limits the drag penalty. Sea stars may also prioritize downforce maximization over drag minimization. Per Mar- tinez [48], the following condition, derived from a simple balance of moments, can be used to evaluate the possibility of animal detachment in steady flows: (F d )h (W−B−F L )(L/2) > 1. (2.1) Here F d is the drag force, h is the height of the center of mass, W is the weight of the organism,B is buoyancy,F L is the lift force, andL is the base length defined in Section 2.5. For a sea star of comparable size to the models tested here, L≈ 19cm and h≈ 5cm, the net vertical force, W−B−F L , has approximately double the effect of the horizontal drag force, F d . Thus, the higher downforce may still be beneficial for sea stars staying attached to surfaces despite the drag penalty incurred. 15 The preceding discussion suggests that the pentaradial body shape and morphological plasticity exhibited by sea stars enable them to better resist hydrodynamic loads. Sea urchinsareoftenfoundinthesameenvironmentasseastarsandhaveacomparablebiological adhesion mechanism [49, 50, 51]. Yet it is unlikely that the spiny spheroidal geometry typical of sea urchins leads to downforce generation. A characterization of the hydrodynamic forces acting on sea urchin body shapes may provide additional insight into how these organisms remain attached to surfaces in energetic flow conditions. Although the ability to generate downforce has been observed previously for BS aquatic organisms such as clams [52] and crabs [48], the orientation-independent nature of the down- force observed in this study is unique to the pentaradial sea star geometry. BS organisms must either be passively aligned in the flow to generate downforce, as is the case for clams while swash-riding [52], or actively posturing in the flow, as is the case with crabs in certain flow conditions [48]. In other words, BS organisms require some degree of passive or active reorientation to produce downforce for different flow directions. On the other hand, sea star body shapes produce downforce independent of orientation relative to the flow. Since downforce is not produced by RS spherical domes and cones, we suggest that the pentaradial geometry of sea star bodies is unique in that it generates a hydrodynamic response that is similar to a (nearly) two-dimensional triangular prism but also insensitive to the incoming flow direction. Our observations suggest that the downforce generated by the sea star bodies and the triangular prism arises from the upwelling flow created along the centerline, with the breakdown of the horseshoe vortex system around the base also playing a role. We recognize that the present work has some important limitations. Specifically, we only consider a limited range of (smooth) sea star morphologies in steady flow. Intertidal sea stars exhibit significant diversity in body shape, size, and surface texture [53]. Moreover, the intertidal zone is likely to be dominated by wave-driven unsteady flows in which iner- tial effects (e.g., added mass) can also play a role [27]. Nevertheless, this study presents the first evidence for downforce generation with pentaradial sea star body shapes. This 16 U PIV Field of view (1) (2) (4) (3) 5cm AR = 1.5 AR = 2.5 AR = 4.0 10cm (a) 19cm 5.4cm * 30 ∘ (b) Spherical Dome Figure 2.6: (a) Schematic showing the sea star and spherical dome models tested in the experi- ments. (b) Schematic of load cell-model attachment assembly, including: (1) servo for controlling orientation angle Θ, (2) bearing and load cell coupling mount, (3) ATI Gamma load cell, (4) linear servo for vertical positioning. orientation-independent downforce could have important implications for sea star locomo- tion and survival. 2.5 Methods Flow facility and experiment setup All experiments were performed in a large-scale free surface water channel facility in the Fluid-Structure Interactions laboratory at USC. This facility has a glass-walled test section of length 7.6m, width 0.9m, and depth 0.6m, and is capable of generating flows with freestream velocities up to U≈ 0.6ms −1 . As shown in fig. 2.6(b), 3D-printed sea star and spherical dome models were mounted towards the leading edge a flat plate setup in the water channel. The models were mounted 3cm from the leading edge of the plate to limit boundary layer development and positioned 5mm from the plate surface. This distance was chosen to approximate the height of the tube feet (or podia) below the sea stars. Additional force measurements conducted with the models placed 25mm from the plate surface showed very similar trends to the results presented in section 2.3.1. A positioning system was used to precisely control model orientation Θ with respect to flow and vertical distance with respect 17 to the smooth plate surface. An Arduino and a high torque Hitec servo motor controlled the rotation system. An Actuonix linear servo motor with 100mm stroke controlled the vertical positioning. The models were tested in flows with freestream velocity ranging from roughly U≈ 0.24ms −1 to U≈ 0.47ms −1 . A Laser Doppler Velocimeter (MSE miniLDV) placed 3 m downstream from the end of the flat plate setup was used to monitor the flow speed. The models tested in the experiments were designed using SolidWorks and manufactured from polylactic acid (PLA) using a Prusa i3 3D-printer. Hydrodynamic forces on the models were measured using an ATI Gamma 6-axis load cell. A PIV system comprising a 5W 532nm continuous wave laser and a Phantom VEO high-speed camera was used for flow visualization and control volume analyses. All measurements were made with the models placed below the plate to eliminate free surface effects. A special faring was designed to isolate the positioning system above the flat plate from the flow, thereby ensuring that the forces measured originated from the models alone. Additional details are provided in the subsections below. 3D-printed models The sea star models were created in SolidWorks by circular patterning an arm of specified aspect ratio around the central axis. Each arm profile is formed from a conic line of curvature γ = 0.75, length 10cm and apex height 5cm, as shown in fig. 2.6. Three different sea star models were created by varying the arm width at the base, resulting in models with aspect ratios AR = 4.0, AR = 2.5, and AR = 1.5. Here, the aspect ratio is defined as the ratio between the arm length from distal tip to the central axis (10cm) and the width at the base arm intersections. The spherical dome used in this study is the top slice of a sphere with radius 11cm. The slicing plane was placed at 30 ◦ from the bisecting plane such that the base diameter for the dome was comparable to the frontal width of the sea star models, L = 19cm, and the height of the dome was comparable to the apex height of the sea star, 5.4cm. Frontal and planform 18 areas for the models are shown in table 2.2. All models were designed with a cylindrical clamp at the base that connected with the positioning system. Table 2.2: Lengths and areas for experimental models. For the sea star models, L = 19cm corresponds to the frontal width when one of the arms is oriented into the flow, Θ = 0 ◦ . AR = 4.0 AR = 2.5 AR = 1.5 Spherical dome Length, L [m] 0.19 0.19 0.19 0.19 Planform area, A p [10 −3 m 2 ] 8.9 9.7 16.1 28.4 Frontal area, A f [10 −3 m 2 ] 4.8 4.8 4.8 7.4 Load cell measurements The hydrodynamic drag and lift forces acting on the models were measured using an ATI Gamma 6-axis load cell capable of 0.00625N resolution in lateral forces and 0.0125N resolution in vertical forces. Data from the load cell were logged to a PC using a 16-bit data acquisition system (National Instruments NI PCIe-6321). The sampling rate was set to 5kHz based on load cell manufacturer specifications. For each configuration, force data were collected for 60s, yielding 300,000 samples. Prior to each measurement made in flow, a zero reading was collected to eliminate the effects of model weight, buoyancy, and load cell drift error from the measured hydrodynamic forces. Followingstandardconvention, themeasureddragandliftforceswereconvertedintodrag and liftcoefficients andexpressed as afunction of theReynolds number. Thesedimensionless parameters were calculated using the following relations: C d = 2F d ρ A f U 2 , (2.2) C L = 2F L ρ A p U 2 , (2.3) Re = U L ν , (2.4) where ρ is fluid density, U is freestream velocity, A f is model frontal area, A p is model planform area,L is a characteristic length, andν is the kinematic viscosity. The fluid density 19 and kinematic viscosity were set to values expected for water at ambient temperature 20 ◦ C. Note that the drag coefficient is calculated using frontal area while the lift coefficient is calculated using the planform area. PIV and control volume analysis The two-dimensional, two-component PIV system comprised a 5W 532nm continuous laser and a Phantom VEO 410L high-speed camera fitted with a 50mm f/1.4 Nikon lens. The camera recorded images at a rate of 400Hz and the spatial resolution for the experiments was 0.23mm per pixel. Because the laser sheet was not wide enough to illuminate both the fore and aft sections of the model, we fixed the position of the camera and moved the laser to obtain two sets of images that were then spliced together at the center of the object. Because of this splicing, we only report mean velocities and force estimates obtained after averaging over 997 frame-pairs. We used PIVLAB [54] for background correction and subsequent PIV analyses. We used a multi-pass fast Fourier transform algorithm for the PIV analysis with a final interrogation window of size 32× 32 pixels and 50% overlap. The vector field data obtained from PIV were used to estimate lift and drag forces per unit length using a control volume approach. Specifically, for a fixed control volume and steady state conditions, the hydrodynamic force (F) can be estimated using the following relation Z S ρ(u · n)u dS =F, (2.5) in which S is the control surface bounding the control volume that encompasses the body, u is the velocity vector, n is the outward normal for the control surface, and F is the force imparted on fluid by the body. Though we do not have access to the full three-dimensional flow field from PIV, we can estimate drag and lift forces per unit length acting on the body (F 0 d , F 0 L ) by considering a planar approximation to eq. (2.5). Separating the momentum 20 conservation relation into the streamwise (x) and wall-normal (y) directions,F 0 d andF 0 L can be estimated using: ρ Z H u 2 in dy− Z W v top u top dx− Z H u 2 out dy =F 0 d (2.6) and ρ Z H u in v in dy− Z W v 2 top dx− Z H u out v out e y dy =F 0 L (2.7) where u in and v in are the streamwise and wall-normal velocities at the upstream (inlet) of the control area, u out and v out are the velocities at the downstream (outlet), and u top and v top are the velocities at the upper bounding surface. The height of the control area is H and the width is W. CFD simulation setup ANSYS Fluent was used to simulate the flow field over the sea star and spherical dome models tested in the experiments, as well as several related geometries. These simulations were carried out using a coupled pressure-velocity method with a built-in steady-state k− turbulence model. The coefficients of the turbulence model were set to the default settings. For all models, the outer flow mesh was a coarse hex-dominant grid. For the near-field flow around the object and for three body-lengths downstream from the rear edge of the models, a fine tetrahedral mesh was used. This also ensured that the sharp geometries involved near the apex of the sea star, pyramid, cone, and triangular prism geometries were adequately resolved. Convergence tests indicated that meshes with roughly 800,000 elements provided a reasonable balance between accuracy and speed. For example, doubling the number of mesh elements beyond this value led to changes of≤ 4% in the computed drag and lift forces for the highest aspect ratio sea star model. For all the simulations, the working fluid was assumed to be water at 20 ◦ C. The models were set 5 mm from the bounding floor in the simulation and 3 cm the edge of the plate, 21 consistent with the experiment setup shown in fig. 2.6. The inlet velocity was set at U = 0.35ms −1 , which is roughly the midpoint of the velocity range tested in the experiments. 22 Chapter 3 Experimental Drag and Lift Optimization of Shape Morphing Robot This chapter is partly adapted from a peer-reviewed book chapter: Hermes, M., Ishida, M., Luhar, M., & Tolley, M. T. (2021). Bioinspired Shape-Changing Soft Robots for Under- water Locomotion: Actuation and Optimization for Crawling and Swimming. Bioinspired Sensing, Actuation, and Control in Underwater Soft Robotic Systems, 7-39. 3.1 Chapter Introduction The underwater environment is one of the most hostile environments for both humans and engineered systems. To explore these hard-to-reach regions, engineers have created robust and powerful remotely operated vehicles (ROVs) and autonomous underwater vehi- cles(AUVs). Althoughtheseexistingsystemsarewell-suitedtomovingquicklythroughopen water, they are often noisy, exhibit high energy consumption, and are rigid, thus unable to enter tight confined spaces or interact with fragile objects or organisms [55]. To solve these problems, we take inspiration from nature. Animals are quiet to avoid predation, energy effi- cient to reduce the amount of food required to survive, and can use their bodies’ compliance to squeeze through tight openings. Although traditional robotics systems are capable of fast and precise motions, they are not well-suited to mimic or reproduce biologically-inspired motions because of their lack of flexibility. Soft robotics systems, on the other hand, can cre- ate continuous deformations to produce motions that are highly adaptable to changing and unpredictable environmental situations [56]. For this reason, there is a wide overlap between soft robotics and bioinspired robotics. This chapter compares and contrasts the principles 23 of soft-material shape-changing locomotion with a focus on actuation and optimization for engineering applications. We present two definitions that are instrumental in this chapter. First, soft materials are characterized by low hardness and high elasticity, which can be used for shape-changing behavior. The material properties of the components differentiate soft robotics from tradi- tional rigid robotics. Soft materials have a bulk elastic moduli in the 10 4 − 10 9 Pa range, similar to those of soft biological materials (i.e., muscle, cartilage, etc), whereas materi- als used in traditional robotics have moduli that are generally at or greater than 10 9 Pa (e.g., aluminum, 10 10 Pa). We also include in our definition deformable structures and stiffness-changing mechanisms, both of which partially consist of rigid materials exhibiting soft properties in bulk. Thus, soft robots are systems that create autonomous, controllable motions using actuators made of soft materials or with composite structures made of rigid materials that have macroscopic properties similar to those of soft materials [57]. Some underwater organisms such as sea cucumbers consist of only soft materials, which motivates the use of soft materials for creating robots inspired by biology. Second, shape-changing for locomotion in this work is defined as continuous deformation ofthesoftcomponent(s)interactingwiththesurroundingmedia(i.e., thewaterforswimming systems and the substrate for crawling systems). A robot or organism with a soft interface between the inner body and the fluid environment can change macroscopic shape in one of the following ways: (1) by changing its volume e.g., intaking water and expanding and (2) by changing the geometry of its body without changing its volume e.g., undulating the body or rowing with flippers. However, a rigid system can produce similar categories of shape changes: (1) changing its volume by extending an appendage from an internal geometry and 2) changing the geometry of its body by rotating joints between rigid links. However, these examples would not result in continuous deformation in the surfaces interacting with the surrounding fluid, distinguishing it from soft shape changes. Shape change is a useful mechanism for enhancing or coordinating locomotion through a fluid. We distinguish two primary methods of moving through a fluid: crawling and 24 swimming. The characteristic difference between the two is the presence or absence of interaction with a substrate. Crawling requires a body to be partially attached to semi-solid objects, such as sand or coral. Crawling in nature is accomplished by a synchronous cyclic motion of appendages, exemplified by crustaceans and echinoderms (Fig. 3.1). This strategy has the benefit of continuous attachment to the substrate during locomotion, which reduces the need to control motion perpendicular from the surface e.g., motion vertically through the water column. In addition, if an organism is moving within a low-velocity boundary layer on the seafloor, assuming the effect from shear stress is minimal compared to drag, crawling can reduce the drag and lift forces from moving in flow. These characteristics aid the stability of the locomotion, but comes at the cost of speed. Natural swimming modes are diverse and make use of a range of different hydrodynamic phenomena, including vortex shedding, jetting, and drag-based propulsion (Fig. 3.2). Swim- ming is a faster method of locomotion than crawling, but requires control of all six degrees of freedom (three translational and two rotational) instead of three degrees of freedom required for planar crawling motions (two translational and one rotational). However, legged crawling results in a larger configuration space for the robot than swimming does as crawling usually requires multiple legs with multiple degrees of freedom. In addition, swimming often requires energy expenditure to maintain a position in unobstructed flow, rather than using passive friction with a substrate. In this chapter, we consider how shape change enabled by soft actuators can be used to improve locomotive efficiency with gait-pattern optimization and hydrodynamic force manipulation. There are two aims for applying optimization techniques to soft robots. First, the designer hopes to identify parameters that create the most favorable dynamic sequence or shape change with respect to a goal metric. Second, the designer would like to find this most favorable parameter set most quickly. Striving toward these two objectives is what motivates this chapter. 25 Figure 3.1: Crawling modes observed in animals. Friction-based crawling uses a pushing motion caused by friction between an appendage and the substrate. Adhesion-based crawling creates a separate force, e.g., suction or chemical; to create a pulling motion between the appendage and the substrate. Citations for right-most column: [58], [59] Figure3.3: Diagram showing examples of gait optimization: (a) changing amplitude of oscillating caudal fin controls vortex intensity, which regulates acceleration [64]; (b) shape optimization is used byIshidaetal. [65]forminimizingdragandmaximizingdownforceonawalkingquadrupedalrobot; (c) Colorado et al. [66] implemented shapemorphing usingSMA actuatorson a batrobot where the goal of speed optimization can be achieved through proper gait sequence selection and aerodynamic force control through shape change; (d) learning optimal CPG parameters for multi-actuator fish robot [67]; (e) structural optimization is observed by Joung et al. [68] for shape optimization of an AUV. 26 Figure 3.2: Swimming modes observed in animals. Rear-body undulation and full-body undula- tion generates thrust through vortex shedding. Drag-induced swimming occurs by exploiting direc- tion dependent drag control. Jet-based swimming is seen in organisms that eject water through a cavity. Citations for right-most column: [60], [61], [62], [63] 3.2 Literature Review: Fluid-structure Interaction Optimization The efficacy of underwater locomotion depends to a large extent on drag, as motion can be greatly restricted if a body is not streamlined. Additionally, in moving water, it can be advantageous for a robot to be able to change its shape to minimize drag across different flow conditions(e.g., Reynoldsnumberranges, changesindirection). Rigid, inflexibleroboticsare 27 often incapable of optimizing for such dynamic environmental conditions. Thus, researchers have increasingly been turning to soft materials for shape optimization for drag reduction. In addition to steady-flow forcing, transient hydrodynamic effects associated with shape- change can also result in added lift, drag, or thrust. Such transient effects have not been studied thoroughly yet and could be exploited using soft materials for control or optimization purposes. The field of shape optimization has seen tremendous growth following the development of Computational Fluid Dynamics (CFD) technologies. Adjoint methods for gradient-based functional optimization have been used successfully in numerous applications. These meth- ods define a surface using grid points and a fitting curve or surface. CFD simulation is then used to evaluate and optimize surface parameters. As computing resources improve, so does the quality and speed of these simulations. Despite the great success of shape optimiza- tion through simulation, unmodeled interactions are still substantial enough to mandate additional wind and water tunnel testing. Furthermore, optimization through simulation can also be very computationally intensive particularly when considering flexible structures interacting with high Reynolds number turbulent flows. Thus, for a robot adapting quickly to changing flow conditions, shape optimization would either require applying a known map- ping of shape to hydrodynamic force or using simplified models for estimating hydrodynamic force. The idea of morphing structures being used to tune body shape based on the environment is not new when considering aircraft [69]. However, recent examples demonstrate that some of the actuation technologies discussed earlier can inspire variations in design and create opportunities to generate new capabilities. For example, Han et al. used SMA actuators to control wingtip vortices for adapting to flight conditions [70]. Rodrigue et al. similarly used SMA wires to create a wing-twisting effect [71]. Comparatively fewer studies have been performed for morphing watercraft. Some ideas for morphing torpedo hulls are provided by Rufino et al. [72]. However, because the Reynolds numbers and fluid forces can be very large for traditional watercraft applications, little attention has been given to optimizing 28 flexible structures for hydrodynamic performance. Below, we provide an example of a soft underwater shape-changing robot as inspiration for further research. 3.3 Shape Optimization for a 2DOF Soft Sea Star We present a case study illustrating how shape-change capabilities enabled by soft mate- rials can be used for engineering benefit. In particular, we show that hydrodynamic body forces on a robot in a steady flow environment can be tuned for drag minimization or down- force maximization (negative lift). For this study, we aim only to provide an example of how shape change may be used for hydrodynamic optimization. Rather than focus on the method, we encourage the reader to view the study as a platform for connecting optimization with soft actuation and physical systems. Biological observations show that intertidal sea stars exhibit morphological plasticity in response to changing flow conditions [24]. Specifically, sea stars in wave-exposed regions have narrower arms and smaller frontal area compared to sea stars from sheltered sites. Further, sea stars transplanted from sheltered sites to exposed regions develop narrower arms. One explanation for this morphological plasticity is that the changes in body shape may enable sea stars to resist dislodgment by reducing drag or maximizing downforce, which can enhance friction. Though such shape changes occur on a time scale of several weeks for sea stars, soft robots are only limited by the speed of the actuators. This section presents a short case study on ESO that is inspired by the morphological plasticity observed in sea stars (Figure 3.4). We cast a silicone body similar in size to the 5-arm sea star (Pisaster ochraceous) that exhibitsmorphologicalplasticityinresponsetoflowconditions[24]. Thebodyshapewascon- trolled by pressurizing a hollow chamber with water using a syringe pump (100mL syringe, pump uncertainty±0.01mL) and extending a linear actuator in the center (2cm stroke, ±0.1cm). In other words, the control parameters were body volume and height. As shown 29 Figure 3.4: (a) Schematic logic flow of the optimizer. Sensor readings evaluate performance and a genetic algorithm computes the next iteration of test parameter combinations. (b,d) A sea star inspired morphing body with silicone skin is attached to a load cell in a water tunnel. (c) Images showing the actuation range. For (i) volume is minimal and height is maximal. In (ii) volume is maximal and height is minimal. in Fig. 3.4(b), this shape-morphing body was attached to a load-cell and subjected to hydro- dynamic forcing. A genetic algorithm was used to identify drag-minimizing and downforce- maximizing optima based on the load cell measurements. The experimental approach was inspired by early studies on evolutionary algorithms by Rechenberg in the 1960s [73]. This prior work demonstrated that evolution-based design of both a segmented plate and a 180 ◦ pipe bend for drag minimization can reduce the number of 30 experiments required compared to using a fine grid search. Furthermore, for highly nonlinear function spaces such as a shape-to-drag mapping, a grid- or gradient-based search may only find a local optimum. Evolutionary shape optimization via experiment is particularly well- suited for applications in underwater soft robotics because the behavior of the materials and actuators involved is difficult to model and the fluid-structure interactions associated with deforming surfaces can be equally complex. Development of a physical shape-changing structure eliminates the need for complex modeling or CFD simulation. Instead, the physical system is allowed to interact with the environment and embedded sensor measurements are used to guide shape optimization. Because the method is completely autonomous, we see this method as a practical design tool for reducing test facility and operator time. Theshape-changingrobotwascreatedbycastingsiliconerubber(Smooth-On, Ecoflex00- 30) in 3D printed negative molds (Figure 3.4(b,d)). Two hydraulic syringe-pump actuators were developed using NEMA23 stepper motors to (i) accurately control fluid injection into a deforming cavity and (ii) drive a secondary a 50 mL syringe, which was used as a linear actuator to change height. Thus, volume and height change served as a surrogate for the shape changes observed in nature. The actuation range is illustrated in Figure 3.4(c). The syringe pump actuators were controlled using an Arduino Uno PWM with stepper motor drivers. The dimensions of the base for the morphing body and the height range were guided by biological observations [24]. A water-channel facility with cross-section dimensions 0.9 m x 0.6 m, capable of gener- ating flows at speeds up to 0.6 m/s, was used for the experiments. The morphing structure was mounted to an ATI Gamma load cell (Figure 3.4(b)). The load cell has 1/160 N res- olution in the streamwise (drag) direction and 1/80 N resolution in the wall-normal (lift) direction. The load cell was connected to a National Instruments data acquisition device that interfaced with MATLAB to generate force readings. Minimum measured drag and lift forces were 0.1 N. We used MATLAB as the logic control to interface with the load-cell measurements and actuation system. This logic control is lumped into the genetic algorithm in Figure 3.4(a). 31 We used an 8-bit binary encoding method for each actuator state, and performed a uniform crossover and mutation procedure to evolve populations. We used high probability for muta- tion in the system (10%) so that the parameter space (height and volume) could be explored sufficiently and quickly. The load cell measurements were used to construct a probability density function (PDF) for the population as a function of height and volume. The location of the maximum for this PDF was interpreted as the parameter set that maximized fitness. Though we chose to use a genetic algorithm as a search technique, any stochastic parameter search would be equally applicable. The experiments were run only once for 300 samples. We subsampled with 100 samples (after 100 sample seed-time for GA) to ensure that the probability distribution of the optimized parameter set is mean-stationary, and we found the means to be Mean N∈[100:200] = [33.9, 1.8]mL, Mean N∈[200:300] = [30.7, 1.7]mL, where N is the sample index. Thus, because the means vary by only 3/100 and 1/100 of the respective actuation ranges, we conclude that our process is stationary. Results obtained from the genetic algorithm are compared with those obtained from a grid search in Figure 3.5. For drag minimization, the optimal shape identified by the grid search yielded better performance (i.e., lower drag) than the shape identified by genetic algo- rithm. However, both shapes are qualitatively similar. Moreover, we observed the genetic algorithm still populating the minimal parameter-set space with samples when the experi- ment completed, indicating that additional run-time may result in a more optimal parameter set. Ontheotherhand, fordownforcemaximization, thegeneticalgorithmidentifiedabetter configuration than the grid search. Importantly,thisshortcasestudydemonstratesthatamorphingbodycanbeusedtotune hydrodynamic performance. This tuning can happen autonomously and in near real-time with appropriate controllers. Therefore, in addition to guiding the design of underwater soft robots, similar methods might also be used for active shape control in untethered systems. For example, an underwater robot could stabilize itself by expanding like a pufferfish in response to a measured disturbance. 32 Figure 3.5: Minimum drag and maximum downforce (negative lift) obtained using a grid search and the genetic algorithm (GA). The uncertainty reported is the standard error associated with the force measurement. This work discusses the ways in which the advantages of soft robotics may be used for underwater bioinspired robots. Soft materials support complex motions that closely mimic the crawling or swimming behaviors of underwater animals and create continuous curvatures well-suited for interactions with underwater flow. These soft actuators may be actuated with tendons, fluidpressure, heat, orhighvoltage, producingvariouslevelsofpowerandefficiency. Although friction-based crawling is a common approach to underwater robots, this method of locomotion is greatly influenced by fluid forces such as lift and buoyancy, requiring these crawlers to adjust shape or posture for optimal performance. Limited work on dis- tributed adhesion-based crawling has been completed to-date and we consider this a promis- ing area of research that has the potential to create significantly more robust robots. In the swimming domain, we see that jellyfish jetting produces the highest efficiency and slow- est velocity, whereas squid jetting produces the highest velocity albeit with lower efficiency. These variations of jet propulsion leverage the elastic restoring force of the soft structures more than drag-based swimming or vortex shedding swimming and we think that working 33 to refine the trade-off between efficiency and speed using soft materials is another promising avenue for future development. 3.4 Shape Optimization for a 7DOF Soft Sea Star To provide more realistic complexity for the fluid force-based experimental optimiza- tion problem, we manufactured a 7-DOF sea star-inspired robot fig. 3.6. All actuators are individually driven by a dedicated ball-screw rack stepper motor system. We discovered that there are nontrivial stick-slip dynamics for the arm actuators. Thus, our 1:1 volume driver assumption was inaccurate. We corrected this problem by incorporating image-based feedback through arm position tracking. The limits of our actuation are 5mm for the arm extensors and is 30mm for the height extensor. Our volume actuator varies 300cc. The uncertainty is 1mm for the arms positions based on threshold limits set in software. The height actuator has a resolution of 0.05mm and the volume actuator has a resolution of 0.1mL. For the larger actuators we did not observe stick/slip, so we use the resolution of our stepper motors and convert based on our linear ball-screw assembly dimensions. We first measured the forces resulting from extending individual arms to their limits in fig. 3.7. The purpose of this experiment was to determine whether forces were sufficiently sensitive to individual arm extensions for further testing. We then constrained our arms to move to equivalent positions, thus reducing our parameter space from seven to three- dimensions. This reduction allows us to sweep values and visualize our function space using isosurfaces, shown in fig. 3.8. We can use this grid data to design our optimization hyper parameters. For the following experiments, we set our channel velocity to U = 0.40± 0.03 cms −1 . We obtain mean values from 30 seconds of 6-dimensional force data at 1KHz for each shape configuration tested. The robot was oriented 36 ◦ based on the definition in fig. 2.2 and arm 1 is pointed downstream, see fig. 3.7. Also in fig. 3.7, we show measurements of drag as 34 a function of arm extension where volume and height fixed at their minimum values. The data is normalized by the drag force of the unextended arm configuration (D o = 3.2N) and reported as D/D o . Arm extension at position 1 has the largest effect on the drag measurements, increasing mean drag by 0.3N. The effects of changing the other arms are less pronounced. In addition, we note that symmetric arm pairs 2+5, and 3+4 are equal within uncertainty, as expected. We conclude the results are promising enough to suggest further examination. Measurements from the parametric survey are shown in fig. 3.8 using the same water channel flow speed described in the previous paragraph. Five grid points are tested for arm extension, height extension, and volume inflation for a total of 125 test points. These values arenormalizedbymaximalactuationandarepresentedasA ∗ ,H ∗ ,V ∗ , respectively. Dragand lift forces are normalized by the volume, arm, and height minimized configuration withD o = 3.2N,L o =−0.05N (downforce). As a general trend, drag decreases with decreasing height and increasing volume. A drag minimizing configuration exists near mid-arm extension and zero height extension. Lift is also shown in fig. 3.8 where the normalizing value is in the downforce direction (though small in magnitude): positive L/L o is downforce and negative L/L o is lift in the wall-normal direction. Downforce increases for increasing arm extension and increasing height, and volume change has no noticeable effect. The downforce maximizing configurations are observed to be near maximum A ∗ ,H ∗ values, and the zero crossing of downforce occurs near A ∗ = 0.25. We do note, however, that the standard deviation was approximately three times larger for lift force ( 0.15N), than for drag force ( 0.05N), so we are less certain of the lift relationships. In the next section, we discuss how the ideas of soft actuator shape optimization can be applied to different fields, and we discuss future directions for our 7DOF robotic platform. 35 Figure 3.6: Morphing 7DOF robot. A custom-molded silicone exterior is wrapped around a rigid base to create a waterproof inner chamber. Five syringe actuators are embedded in the arms and one syringe actuator is embedded in the top surface. Figure 3.7: Drag force of flow over robot for U = 0.4ms −1 , where individual arms are extended. 3.5 Future Optimization Directions As demonstrated by the shape changing robots presented in section 3.4, dynamic shape control is a promising field for soft-material underwater robotics. Numerical topology opti- mization has been well explored using CFD analysis and parametric CAD models of rigid structures. However, few physical experiments have been developed for online topology optimization. We describe a case study demonstrating the potential of using soft deforming materials for design optimization, and envision this idea to be applied to an autonomous 36 Figure 3.8: Isosurfaces* of drag, D and lift, L for 125 point grid of shape changes, where D o = 3.2N,L o =−0.05N (downforce). *Surfaces are linearly interpolated beyond the coarse grid for presentation. untethered system. A future direction for topology optimization is to develop a system capable of autonomously evaluating its hydrodynamic performance and adapting to flow conditions in complex environments. This system could be coupled to a swimming propulsor, thereby connecting hydrodynamic and gait optimizations to achieve maximal performance. Building off the advances detailed in this chapter is a step toward creating more effective soft underwater mobile robots. By carefully considering the shape and function of soft bodies and actuators, we aim to design robots that create favorable interactions with flow in variable fluid environments. These soft robots can be used for exploration of underwater environments not suitable for human or ROV traversal and for monitoring and measuring of vulnerable species and habitats. 37 Chapter 4 Tripedal Robot Motion Analysis and Control The open loop experiments described in this chapter are adapted from a submitted manuscript: Hermes, M., Luhar, M. (2021). Frictional Locomotion of a Radially Sym- metric Tripedal Robot, submitted to Journal of Nonlinear Science, and the closed loop path following experiments are from the conference proceeding: Hermes, M., McLaughlin, T., Luhar, M., & Nguyen, Q. (2020). Locomotion and Control of a Friction-Driven Tripedal Robot. arXiv preprint arXiv:2011.07370. (Presented at IEEE ICRA 2021). 4.1 Chapter Introduction The present work seeks to build on the robotics efforts described in section 4.2 and provide insight into the nonlinear dynamics underlying friction-driven sliding (or crawling) locomotion of systems exhibiting triradial symmetry. For this purpose, we have developed an idealized robot with 3 rigid single DOF limbs. We have characterized the motion of this robot in laboratory experiments comprising a systematic parametric sweep involving open-loop sinusoidal actuation of the limbs. We have also developed a simplified physics- based mathematical model that reproduces the experimental observations, and provides insight into the unique locomotive capabilities of this arrangement. Through this effort, we have identified and characterized actuation patterns that lead to pure rotation and pure translation. We also introduce a novel omnidirectional gait for path following with vision- based feedback control. We demonstrate the effectiveness of our model with a physical robot placed in a turbulent wind field. Our novel normal force estimation procedure, cycle-based gait strategy, and disturbance rejection feedback control method can be useful to other RS robots seeking translation control in the presence of unmodeled forces. 38 Figure4.1: Minimally-actuated tripedal robot demonstrating curve following capabilities. Exper- imental video: https://youtu.be/F9UxznYtJGM. The chapter is structured as follows. First, in section 4.2, we discuss prior work into RS robots and where how our platform advances that field. In section 4.3, we discuss ways in which robots with cyclic gait strategies can be controlled and optimized. Robot design and laboratory experiments are described in section 4.4. The physics-based model is developed in section 4.4.3. Experimental results and model predictions are compared in section 4.5. Actuation patterns that lead to pure rotational and translational motion are also evaluated in this section. We then seek an explanation for answering why the symmetric gait sequence generates net translation in section 4.6, where we also identify optimal non-sinusoid gaits using parameterized splines. In section 4.7, we demonstrate path following with flow disturbances. Finally, concluding remarks are presented in section 4.9. 4.2 Literature Review: Radially-symmetric Robotics Previous locomotion studies have primarily focused on bilaterally symmetric body arrangements. This is partly because the majority of land-based animals are bilaterally symmetric. However, RS body arrangements are common in aquatic environments, e.g. Cnidaria (swimming medusae, or jellyfish; [74]) and Echinodermata (crawling organisms such as sea stars, brittle stars, and sea urchins; [75, 76]). We are interested in exploring how these predominantly aquatic body arrangements can be used for amphibious or terres- trial locomotion. As a starting point in this endeavor, this study analyzes friction-driven 39 locomotion of tripedal geometries involving three contact points - the minimum number required for stable motion with RS bodies - via laboratory experiments and the development of simplified mathematical models. The focus on a tripedal geometry allows us to minimize the complexity of the robot used in the experiments and provide simplified analyses for the nonlinear dynamics we observe. RS morphologies offer some benefits over bilaterally symmetric arrangements. First, RS movers can use any limb to define a lead direction [77, 76]. For example, pentaradial brit- tle stars are shown to exhibit bilaterally symmetric motion strategies in which one of the five limbs remains inactive [76]. However, because they are RS, the inactive limb can be reassigned to change direction immediately [76]. In addition to having no front bias, in the case of RS body plans with more than 3 limbs, the mover is robust to limb damage due to redundancy [78, 79, 80, 81]. This advantage may be of use in remote environments where loss of motor function or general damage to a limb does not render the mover immo- bile. Finally, for locomotion in the presence of strong external forces (e.g., aerodynamic or hydrodynamic disturbances), having multiple contact points enables more robust adhesion. This can reduce risk of surface dislodgement and enhance the net adhesive force through distributive attachment [82]. Several recent efforts have considered the development of RS robots. Our previous work describing the development of the tripedal test system used in this study showed that the RS arrangement is controllable using an omnidirectional gait map [2]. This prior effort also highlighted an advantage of being able to define arbitrary lead limbs for RS robots: sharp path curves can easily be followed. The Robotics and Mechanisms Laboratory at UCLA has produced many RS systems: the hexapod robots SiLVIA, HEX, and MARS [83]; a RS quadruped, ALPHRED [5]; and two tripod robots, THALer [84] and STriDER [85]. In general, these robots have multi-actuated legs for walking and interacting with their environments. The tripedal STriDER and THALer systems have also demonstrated application of a novel swinging gait allowing for translation in three directions. Bevly et 40 al. [86] have demonstrated vertical climbing using a RS tri-arm robot. Several other groups have developed hex-symmetric terrestrial [87, 88, 89, 90] or aerial [91] robots. Other research efforts have considered the development of robots with close similarity to pentaradialbrittlestars. Forexample, Watanabeetal. [92]developedadecentralizedcontrol scheme to orchestrate limb motion for a pentaradial system. Kano et al. [3] demonstrated omnidirectional locomotive capabilities using this control scheme on a physical ophiuroid robot. More recently, Kano et al. [78] demonstrated robust locomotion capabilities in a pen- taradial robot by systematically removing appendages and showing the adapted locomotion strategy. This effort built on biological observations that involved removing arms from live brittle stars and observing the resulting locomotion patterns. Lal et al. [4, 93] developed motion control algorithms for a pentaradial robot with multisegmented arms using genetic algorithms. This work showed that a nonintuitive complex writhing motion generated the largest net translation. Given the triradial geometry considered in this study, a related field of research involves controlling the motion of trident robots, which are tripedal robots with wheels at the end effector instead of simple frictional contacts [94, 95, 96, 97]. In general, these studies use non- holonomic kinematic relations to obtain optimal control strategies. Ishikawa et al. [94] have also demonstrated the ability to track sharp path trajectories with a geometric mechanics algorithm. The efforts described in the preceding paragraphs highlight the broad applicability of RS robots, and emphasize the comparative advantages to systems with bilateral symmetry. 4.3 Literature Review: Gait Optimization Inspired by neuro-muscular control, where the brain regulates complex harmonic muscu- lar activity with simplified commands, researchers have studied the use of Central Pattern Generators (CPG) for discovering optimal gait sequences for robotics [98, 99]. CPG gait optimization reduces the high dimensionality associated with finite state machines in time 41 and space. Generally for CPG implementation, a periodic input is assumed to be the opti- mal gait. The parameters associated with a sinusoidal input, i.e. amplitude, phase, and frequency, can then be optimized autonomously by using an on board processor to solve a system of first-order ODEs with limit-cycle behavior. For instance, a stochastic population- based evolutionary algorithm, such as a genetic algorithm or particle swarm algorithm, can be used to select and optimize the parameter combinations to be tested. Because fish undu- lation is well synchronized and approximately sinusoidal, CPG techniques are effective for optimizing gait sequences of soft robots demonstrating undulatory locomotion [100]. An advantage of using CPG gait generation is that the output transition of position commands resulting from discontinuous parameter changes is continuous because the pat- terns are generated by a set of differential equations. These equations can be solved online using microcontrollers [98]. Thus, in a large testing environment, the robot can operate perpetually, given sufficient power resources [100]. A disadvantage of this method is that optimal gait patterns are assumed to be sinusoidal. For complex, nonlinear functions, where actuators do not necessarily have independence, this method may not converge to an optimal solution. Despite the potential disadvantages, many robots still use CPG methods to avoid the complexity of choosing dynamic gait sequences for optimization. Alternatively, there are other studies that have used state transitions instead of CPG methods. For example, Lal et al. [101] used physical model simulations in conjunction with genetic algorithm-based rule selection to optimize gait sequences for a five-legged brittle star inspired robot and then implemented the pattern on a physical system for testing. Such decentralized control approaches may be useful for investigating more complex gait-function spaces. Anothercomponentofdevelopingonlineoptimalshapechangingstrategiesforlocomotion is the performance-evaluation component. For performance goals to be adaptive, the robot must be able to distinguish changes in objective value. Thus, sensing is very important for system feedback. If a system is tethered and operator interaction is involved, an Eulerian frame with object tracking using image feeds may be used to estimate state information 42 relevant to performance evaluation. However, for unmanned systems, the sensing must be in an Lagrangian framework, and thus housed on board. Several researchers have used inertial measurement units, which yield accelerometer, gyroscrope, and compass data to estimate position and velocity [100, 102]. Researchers have also used infrared sensing and video imaging to avoid collision with walls and identify targets [100]. To estimate dynamic fluid interactions, Bernoulli’s principle (or other simplified models) have been used in conjunction with pressure measurements to obtain estimates of flow speed and body forces [103]. 4.4 Experimental Approach and Physics-Based Model In this section, we provide an overview of the robot design (4.4.1), our mathematical model (4.4.3), and normal force validation (4.4.4). 4.4.1 Robot Design and Characterization A top and isoview of the simple tripedal robot used in this study is shown in section 4.4.1. The tripedal robot comprises 7.5cm long 3D-printed polylactic acid (PLA) arms connected to an stereolithography (SLA) resin motor support structure with an effective radius of 5cm. We used Hitec HS-5646 digital servo motors to drive the arms, which were powered by an externalpowersupplyandcontrolledbyanArduinoUno. Thetotalmassofthestructurewas measured to be 888g. We also embedded limbs with 5kg bending-beam load cells (Chenbo CZL635) with an HX-711 amplifier module to measure normal forces with a noise profile of roughly±1.5%FS (approximately±0.7N). We varied the friction at the contact points using polymer laminate and 600-grit sand- paper surfaces. We used paper as the test platform surface. The friction coefficients mea- sured were μ = 0.33± 0.012 for the polymer-paper contact and μ = 0.85± 0.043 for the sandpaper-paper contact. As discussed below, we ignored high-order friction effects for mod- eling simplicity. To ensure that the observed robot motion was purely due to limb actuation, we ensured that the dynamic impact of surface imperfections and the power/control tether 43 a b Figure 4.2: (a) Top and (b) iso view of the robot. were minimized. All experiments were carried out on a smooth, level optimal support table with tilt less than 0.2 ◦ . The tether cable was supported externally to minimize tension. 4.4.2 Motion Experiments and Video Tracking To characterize the dynamic behavior of the tripedal robot, we pursued a systematic experimental evaluation in which we measured the robot motion resulting from prescribed sinusoidal actuation of each limb. Specifically, the angular position of the i−th limb (see figure 4.3) was prescribed as φ i (t) =ξ sin(2πft +iΔψ), (4.1) where i = 1, 2, 3. Here, ξ represents rotation amplitude, f is the oscillation frequency, and Δψ is a temporal phase shift in actuation between the limbs. We refer to ξ as sweep and Δψ as phase for the remainder of this paper. The following ranges were tested for each of the three actuation parameters: f ∈ [0.4, 1.0]Hz, ξ∈ [20 ◦ , 50 ◦ ], and Δψ∈ [20 ◦ , 80 ◦ ]. The ranges for f and ξ were constrained by servomotor limitations and stability considerations. The range of test values for Δψ was 44 Servo motors Bearings 3-D printed legs Signal tether Figure4.3: Simplified representation of tripedal system. x is the position coordinate of the center of mass with respect to a fixed reference frame, ξ is the rotation of the body based on the hinge point of a specified limb, and φ i is the local limb rotation. Table 4.1: Parametric combinations tested for sinusoidal actuation experiments. Frequency variation f = [0.4, 0.6, 0.8, 1.0] Hz ξ = 30 ◦ Δψ = 30 ◦ Sweep variation ξ = [20, 30, 40, 50] ◦ f = 1.0 Hz Δψ = 30 ◦ Phase variation Δψ = [20, 40, 60, 80] ◦ f = 1.0 Hz ξ = 30 ◦ chosen to span the configuration space. Table 4.1 shows the specific parameter combinations tested. In addition to this sinusoidal parametric sweep, we also tested actuation patterns resem- bling locomotion strategies observed for brittle stars named rowing and reverse rowing [76]. Both of these translation modes involve one unpaired or inactive limb. Rowing and reverse rowingaredistinguishedbytheorientationofthisinactivelimbandaregraphicallydescribed in fig. 4.10. In the case of rowing, the inactive limb leads the body forward, whereas for reverse rowing, the inactive limb passively trails the body. For both locomotor modes, kine- matic observations show that the motion of the active limbs is symmetric with respect to the translation axis [76]. Thus, despite the radial body plan, brittle stars effectively employ coordinated, bilaterally symmetric locomotion; radial symmetry is broken by the presence of an inactive limb. The robot was tracked at 30 frames per second from above using a camera mounted on an elevated platform. Because the largest test frequency was 1.2 Hz, the selected frame 45 rate resolves motion completely. The center of mass for the robot in the horizontal plane, x = [x,y], wasdeterminedbyapplyingaMATLABimage-filterscripttotherecordedimages. A reference measurement object was used for pixel-to-centimeter calibration in the motion plane. The relative rotation of the robot,θ, was tracked by finding the maximum correlation between the preceding image, rotated in the range [−3 ◦ , 3 ◦ ], and the frame being considered. The starting position of the robot for each experiment was defined as x = [0, 0] and θ = 0 ◦ . 4.4.3 Mathematical Model We developed a simplified dynamic model for the tripedal robot based on momentum conservation laws with nonlinear friction forces. In our model, we assume rotational and translational intertias for the limbs to be negligible compared to those for the central body. Friction forces are modeled as point forces at the tips of the limbs. The friction force is simplified in our model to be the signum function of the velocity. In other words, there are no stick-slip effects. [104, 105]. We recognize that this friction model constitutes a significant simplification. However, below we show that this simplified model is able to reasonably reproduce experimental observations. The friction force, F i = [F i,x ,F i,y ], acting at the contact point at the end of each limb in the x−y plane, r i = [r i,x ,r i,y ], is modeled as F i =−μN i ˙ r i |˙ r i | . (4.2) Here,N i is the normal force at contact pointi,μ is the measured kinetic friction coefficient, and ˙ r i = [ ˙ r i,x , ˙ r i,y ] is the velocity of the contact point at the end of each limb. Note that r i and ˙ r i can be expressed in terms of the robot state vector, z = [x, ˙ x,θ, ˙ θ], the actuation angles and rotation rates, φ i and ˙ φ i , and the geometric constants, R and l, using simple 46 trigonometric relations (see figure 4.3). Normal forces are computed by combining a vertical force balance (N 1 +N 2 +N 3 =Mg) with torque balances about the robot center of mass:        1 1 1 r 1,x −x r 2,x −x r 3,x −x r 1,y −y r 2,y −y r 3,y −y               N 1 N 2 N 3        =        Mg 0 0        . (4.3) The second and third lines in the equation above ensure that there are no net torques about the robot center-of-mass due to the normal forces. Under the assumptions stated above, conservation laws for linear momentum in the horizontal plane can be expressed as M ¨ x = 3 X i=1 F i,x (4.4) and M ¨ y = 3 X i=1 F i,y , (4.5) respectively. The conservation law for angular momentum can be expressed compactly as J ¨ θ = 3 X i=1 (r i −x)×F i , (4.6) where J represents the second mass moment of inertia, which is estimated from the mass distribution of the central support structure. The system of ordinary differential equations shown in (4.4-4.6) is solved numerically to yield predictions for robot translation (x) and rotation (θ) using the MATLAB ode45 algorithm for an adaptive time-step 4th-order Runge-Kutta solver. Recall that the location of the contact points relative to the robot center, r i −x, depends on the limb angles, φ i (t) (figure 4.3). Therefore, the prescribed actuation angles φ i (t) appear directly in all three conservation laws via the friction terms dependent on r i and ˙ r i . Also, keep in mind that these model simulations involve no tuning parameters. All geometric and dynamic variables 47 Figure 4.4: Times series showing model predictions (black) and experiment measurements (red) for the normal forces acting at each contact point for phase shifts: (a) Δψ = 60 ◦ , (b) Δψ = 90 ◦ , and (c) Δψ = 120 ◦ . The oscillation frequency and sweep angle are set tof = 0.5 Hz andξ = 30 ◦ for the high-friction sandpaper-paper contact. Shaded red regions represent measurement variability. This is defined as one standard deviation for the phase averaging procedure. appearing in the governing equations (e.g., M,J,l,R,μ) are obtained from independent measurements. 4.4.4 Normal Force Simulation Predictions and Measurements As noted earlier, we embedded three load cells into the robot limbs to measure normal forces. Inthissectionwecomparethemeasurednormalforcesagainstpredictionsmadeusing eq. (4.3). Measured normal forces for f = 0.5Hz, ξ = 30 ◦ , and varying phase differences Δψ = 60 ◦ , Δψ = 90 ◦ , and Δψ = 120 ◦ are plotted with simulation data in fig. 4.4. Note that the measured data are phase averaged over 5 oscillation cycles. Though the measurements are qualitatively similar to the predictions, there are some consistent quantitative discrepancies. First, the amplitude of force oscillation for all limbs is larger for the measurements compared to the simulations. In addition, for Δψ = 120 ◦ , the normal force profiles of the legs are not equal with a phase shift. Unlike the simulations, the measured normal forces differ in magnitude. These measurements suggest that the robot weight may not be evenly distributed, which is a source of potential modeling error. This error could, in part, explain some of the discrepancies in trajectories between simulation data and experiment results observed in the next section. Nevertheless, the qualitative agreement 48 between model predictions and measured normal forces is promising, especially keeping in mind the expected±0.7N noise floor in the load cell measurements, which is comparable to the discrepancy between measurements and predictions. 4.5 Parameter Sweep Results A systematic parametric sweep was performed for the sinusoidal actuation described by (4.1). Oscillation frequencies (f), sweep angles (ξ), and relative phases (Δψ) were varied over the ranges shown in Table 4.1. Comparison between experimental tracking results and model predictions for these cases are shown in Sections 4.5.1-4.5.3. Simulation predictions were interpolated at a rate of 30 Hz, consistent with the experimental imaging results. The total time interval for all the results shown in figures 4.5-4.7 below was 30 seconds. Sections 4.5.4-4.5.5 build upon these findings to consider actuation patterns that result in pure rotation and translation. 4.5.1 Frequency Variation Figure 4.5 compares model predictions for robot position with tracking results from experiments with varying oscillation frequency, f. The sweep angle and relative phase shift were maintained constant atξ = 30 ◦ and Δψ = 30 ◦ for these experiments. Model predictions and experimental results both show the presence of small-scale oscillations in robot center- of-mass as well as a larger-scale orbit or revolution. For the low-friction cases shown in (a)-(d), an increase in frequency led to a contraction in the local oscillation length in the experiments, i.e., the small-scale oscillations clustered closer together. For the high-friction cases shown in (e)-(h), increasing frequency did not significantly change the local oscillation scale or the arc radius of the large-scale revolution. Instead, increasing frequency simply led to an increase in the total distance traveled by the robot, i.e., the arc length traversed by the robot increased. 49 Figure 4.5: Comparison between model predictions and experimental results over a period of 30 s for the frequency variation experiments. For all cases, ξ = 30 ◦ and Δψ = 30 ◦ . The actuation frequency is set to be (a,e) f = 0.4 Hz, (b,f) f = 0.6 Hz, (c,g) f = 0.8 Hz, (d,h) f = 1.0 Hz. Panels (a)-(d) correspond to the low-friction tape-paper contact while panels (e)-(f) correspond to the high-friction sandpaper-paper contact. Tracking data from the experiments are shown as thin red lines and the simulation data are shown as a thick black lines. Movie 1 in the supplementary materials provides a side-by-side comparison of experimental results and model predictions. For the high-friction cases, model predictions (black lines) are in good qualitative and quantitative agreement with experimental observations (red lines). For the low-friction case, the model predictions are in qualitative agreement with the experimental results, though the clustering effect is not reproduced as clearly; only a minor reduction in the oscillation scale is observed from panel (a) to panel (d). Note that the discrepancy between model predictions and experiments is greatest for the lowest-frequency cases considered in (a) and (e). This discrepancy could be attributed to the highly-simplified contact model considered here, which does not distinguish between static and kinetic friction. Transitions between static and kinetic friction (i.e., stick-slip dynamics) are expected to be more important at lower actuation frequencies. 4.5.2 Sweep Variation Next, we consider robot motion for cases in which the sweep values were varied system- atically while the oscillation frequency and phase shift were kept constant at f = 1.0 Hz 50 Figure 4.6: Comparison between model predictions and experimental results over a period of 30 s for the sweep variation experiments. For all cases, f = 1.0 Hz and Δψ = 30 ◦ . The sweep angles correspond to (a,e)ξ = 20 ◦ , (b,f)ξ = 30 ◦ , (c,g)ξ = 40 ◦ , (d,h)ξ = 50 ◦ . Panels (a)-(d) correspond to the low-friction tape-paper contact while panels (e)-(f) correspond to the high-friction sandpaper- paper contact. Tracking data from the experiments are shown as thin red lines and the simulation data are shown as a thick black lines. and Δψ = 30 ◦ . Experimental observations and model predictions of robot trajectory for these cases are shown in figure 4.6. For both the low- and high-friction tests, an increase in sweep angle led to an increase in the local oscillation length. For the high-friction cases, the arc radius for the large-scale revolution also decreased slightly with increasing sweep angles. Together, this increase in oscillation length and reduction in arc radius resulted in a lower revolution period. In other words, for the high-friction cases, the robot completed a large scale revolution faster as the sweep angle increased. This is clear from a comparison between the high-sweep case shown in figure 4.6(h), which shows a complete revolution, and the lower-sweep cases shown in figures 4.6(e)-(g), which do not show a complete revolution. The low-friction cases do not show a clear reduction in arc radius, though the total distance traveled by the robot does increase monotonically with increasing sweep angles. Once again, the mathematical model generates predictions in good quantitative agree- ment with the experimental observations for the high-friction cases. For the low-friction contact, the model qualitatively reproduces the observed increase in the local oscillation 51 Figure 4.7: Comparison between model predictions and experimental results over a period of 30 s for the phase variation experiments. For all cases, f = 1.0 Hz and ξ = 30 ◦ . The phase shift between individual limbs is (a,e) Δψ = 20 ◦ , (b,f) Δψ = 40 ◦ , (c,g) Δψ = 60 ◦ , (d,h) Δψ = 80 ◦ . Panels (a)-(d) correspond to the low-friction tape-paper contact while panels (e)-(f) correspond to the high-friction sandpaper-paper contact. Tracking data from the experiments are shown as thin red lines and the simulation data are shown as a thick black lines. scale and distance traveled by the robot with increasing sweep. However, there are devia- tions between the observed trajectories and model predictions. 4.5.3 Phase Variation Finally, figure 4.7 shows the effect of varying phase difference, Δψ, on robot motion at constant oscillation frequency, f = 1.0 Hz, and sweep angle, ξ = 30 ◦ . For both the low-friction and high-friction cases, an increase in the phase shift led to a decrease in the arc radius for the large-scale revolution. This was accompanied by a marked change in the character or shape of the small-scale oscillation patterns. Specifically, the oscillations became sharper and less rounded with increasing phase separation between limbs. This is in contrast to the results shown in figure 4.6, in which the arc radius decreased slightly with increasing sweep, but the rounded profile of the oscillations remained. As with the sweep and frequency variation experiments, there is very good quantitative agreement between model predictions and experimental trajectories for the high-friction cases, but less so for the low-friction cases. 52 Figure 4.8: Illustration of pure rotation for a phase difference of Δψ = 120 ◦ for the high-friction sandpaper-paper contact. The left panel confirms that there is minimal net translation. The right panel shows a monotonic increase in rotation angle. The actuation parameters were f = 1.0 Hz, and ξ = 30 ◦ . Once again, experimental data are shown as thin red lines while the simulation predictions are shown as thick black lines. Movie 2 shows model predictions and experimental results for a case with Δψ = 120 ◦ . 4.5.4 Pure Rotation The observed decrease in arc radius with increasing phase shift observed in figure 4.7 inspired an additional experiment in which the phase shift was set to Δψ = 120 ◦ . Given the triradial symmetry of the robot, for Δψ6= 120 ◦ , the sinusoidal actuation described by (4.1) leads to a constant phase difference between limbs i = (1, 2) and i = (2, 3) but not i = (3, 1). For Δψ = 120 ◦ , the phase difference is constant across all limbs. Figure 4.8 shows the time-variation in the x-location of the robot center of mass (a) and rotation angle θ (b) for sinusoidal actuation with Δψ = 120 ◦ . The observed trajectory shows minimal translation (x≈ 0 m) and a monotonic increase in the rotation angle, i.e., the robot effectively exhibits pure rotation for Δψ = 120 ◦ . Model predictions for translation in thex-direction are similar to the experimental observations. However, they differ in predicting both the average rate and nature of the rotation. The model predicts a pronounced acceleration-deceleration cycle in rotation rate at the actuation frequency (f = 1.0 Hz), whereas the experiment results show a relatively constant rotation rate. 53 Figure 4.9: Curvature for the large-scale revolution (normalized by l = 0.075 m), obtained by fitting a parametric circle to model predictions for cases with ξ = 30 ◦ , f = 1.0 Hz and Δψ = [20 : 350] ◦ for the high-friction sandpaper-paper contact. To provide further insight into the observed rotational — or pivoting — motion, addi- tional simulations were pursued for Δψ = [20, 30,..., 350] ◦ , with the sweep angle and oscil- lated frequency fixed at ξ = 30 ◦ and f = 1.0 Hz. The resulting large-scale revolution was characterized by fitting a circle with radius r and center (a,b) in the horizontal plane to the observed trajectories. Specifically, a quasi-Newton gradient optimization method (with tolerance 10 −6 ) was used in MATLAB to minimize the error function e(a,b,r) = X j (r− q (x j −a) 2 + (y j −b) 2 ) (4.7) for all data points (x j ,y j ) in the predicted trajectories. The normalized curvature obtained from this procedure,l/r, is plotted in figure 4.9 as a function of the phase different Δψ. The predictive curvature is maximum (i.e., fitted radius r is minimum) at Δψ = 120 ◦ and 240 ◦ . As noted earlier, given the triradial symmetry of the robot, Δψ = 120 ◦ ensures a constant phase difference across all three limbs. This is also true for Δψ = 240 ◦ . Interestingly, the model also predicts zero curvature (i.e., r→∞) for Δψ = 180 ◦ . This particular actuation essentiallycorrespondstoarowing motionwithanoscillatingunpairedlimb. Aswediscussin the following section, this actuation results in translational motion with minimal curvature. 54 Model predictions shown in figure 4.9 indicate that the large-scale revolution is driven by the inconsistent phase differences resulting from Δψ 6= 120 ◦ (or 240 ◦ ). The physical mechanism giving rise to this effect is illustrated by figure 4.4, which shows the time-varying normal forces acting at each contact point for phase shifts Δψ = [60, 90, 120] ◦ . As expected, in all cases, the normal forces show periodic variation at the oscillation frequency. However, the cases with Δψ = 60 ◦ and 90 ◦ are characterized by the presence of a single differentiated limb that exhibits a higher-amplitude oscillation (see black lines in figures 4.4(a,b)). This differentiated limb corresponds toi = 2 in (4.1), which maintains the prescribed phase shift, Δψ relative to the other two limbs. Recall that the phase shift between the remaining two limbs,i = 1 andi = 3, is not Δψ except for cases in which Δψ = 120 ◦ (or a multiple thereof). These differences in relative phase give rise to differences in the frictional forces acting at each limb, which drive the large-scale orbital motion exhibited by the robot. For Δψ = 120 ◦ , there is no differentiated limb. As shown in figure 4.4(c), the normal force waveforms are equal in amplitude and exhibit a constant phase difference relative to one another. In this case, the robot effectively rotates in place. 4.5.5 Pure Translation: Brittle Star Inspired Rowing & Reverse Rowing In addition to characterizing the motion resulting from the sinusoidal actuation in (4.1), we were also inspired to reproduce brittle star locomotion patterns observed by Astley [76], which we interpret as the rowing and reverse-rowing motions shown in figure 4.10. Exact replication of the complex kinematics exhibited by brittle stars is impossible with the ide- alized rigid robot. However, the rowing and reverse-rowing motions can be simulated qual- itatively by making one of the robot limbs inactive, and by altering the neutral position and motion of the remaining two active limbs. To simulate rowing, the neutral position of the limbs was unchanged from the natural state in which the limbs are distributed evenly around the unit circle, i.e., separated by 120 ◦ from each other. To simulate reverse-rowing, the neutral position of the active limbs was shifted by 30 ◦ towards the inactive limb. As 55 shown in figure 4.10, this meant that the active limbs were each separated from the inactive limb by 90 ◦ in the neutral position, and from one another by 180 ◦ . For both rowing and reverse rowing, the active limbs were actuated in anti-phase, i.e., with a phase difference of 180 ◦ . Figure 4.10 provides a comparison of experimental tracking results and model predictions forthesimulatedrowingandreverse-rowingmotionswiththefollowingactuationparameters: oscillationfrequencyf = 1.0Hzandsweepangleξ = 30 ◦ . Therowingresultsshowninpanels (b) and (d) show sustained translation in the positivex-direction. In other words, the robot moves in the direction of the inactive limb, which is consistent with biological observations. Model predictions are also in good agreement with the experimental tracking data. In contrast to the rowing locomotion results presented in figure 4.10(b) and (d), the attempt at simulating reverse-rowing was unsuccessful. As shown in panels (a) and (c), the robot effectively oscillated back and forth in place, with no significant net motion relative to the starting point atx = 0. Model predictions also show similar behavior. Thus, simply altering the neutral position of the active limbs does not reproduce the reverse-rowing locomotion observed in nature. This issue is discussed in greater in the following section. To evaluate the effect of varying actuation parameters on net translation speed for the rowing configuration, we pursued additional experiments and model simulations with sweep angles ξ = [20, 30, 40, 50] ◦ and oscillation frequency f = 1.0 Hz. In other words, the oscil- lation amplitude for the active limbs was varied. Figure 4.11 shows the net translation of the robot center-of-mass as a function of time, x(t). In general, increasing sweep led to an increase in the net translation speed. However, maximum sweep angle was limited by robot stability. Forξ≥ 60 ◦ , the active limbs moved beyond the center-of-mass of the robot during the actuation cycle, causing the robot to tip over. A comparison of panels (a)-(d) with panels (e)-(h) also indicates a reduction in translation speed with increasing friction. For all cases, model predictions are in good agreement with the experimental tracking data. The model reproduces the back-and-forth nature of the observed motion, in which the oscillation period 56 Figure 4.10: Comparison between model predictions and experimental observations for reverse- rowing (a,c) and rowing (b,d) locomotion over a period of 20 s. Panels (a,b) show results from the low-frictiontape-paperexperiments, andpanels(c,d)showresultsfromthehigh-frictionsandpaper- paper experiments. Experimental data are shown as thin red lines and simulation data are shown as thick black lines. All datasets correspond to oscillation frequency f = 1.0 Hz and sweep angle ξ = 30 ◦ . Movie 3 showcases low friction rowing from panel (b). Figure 4.11: Measured and predicted robot translation for the rowing configuration with varying sweep angles. The actuation frequency was f = 1.0 Hz in all cases. The sweep angle is set to ξ = 20 ◦ for (a,e),ξ = 30 ◦ for (b,f),ξ = 40 ◦ for (c,g) andξ = 50 ◦ for (d,h). Panels (a)-(d) show data for the low-friction tape-paper contact. Panels (e)-(h) show data for the high-friction sandpaper- paper contact. Experimental data are shown as thin red lines and simulation data are shown as thick black lines. 57 Figure4.12: Modelpredictionsforrowingtranslationspeed( ¯ U)asafunctionoffrictioncoefficient for f = 1.0 Hz and varying sweep angles ξ = [20, 30, 40, 50] ◦ . isdeterminedbytheactuationfrequency. Moreover, thepredictedaveragetranslationspeeds are consistent with the experimental observations. To provide further insight into the observed reduction in net translation speed for the high-friction sandpaper-paper contact, model predictions were carried out with friction coef- ficients μ∈ [0.1, 0.9] and sweep angles ξ = [20, 30, 40, 50] ◦ . The average translation speeds predicted by the model are shown in figure 4.12. Interestingly, for each value of ξ, there is an optimal friction coefficient that maximizes translation speed. Beyond this optimum, translation speed decreases monotonically with increasing friction coefficient This is consis- tent with the experimental observations in figure 4.11, which showed a decrease in average translation speed for the high-friction sandpaper-paper contact. Model predictions also sug- gest that an increase in the sweep angle increases the maximum translation speed and shifts the optimum friction coefficient higher. Physically, the presence of an optimum could be explained by considering the following limiting scenarios. For μ = 0, the active limbs would simply slide back and forth on the substrate without generating any net friction forces (or translation). However, for very high friction coefficients, the inactive limb may effectively act as an anchor that slows the robot down. 58 4.6 Analysis of Rowing Gait 4.6.1 Analysis of Translation ( ) ℓ Figure 4.13: Simplified geometric representation of single active limb. In this section, we pursue a simplified analysis to provide further insight into rowing and reverse rowing locomotion. Specifically, we seek to provide an explanation for why the attempt at replicating reverse rowing was unsuccessful with the rigid-limb tripedal robot used in this study. Without loss of generality, the representation shown in fig. 4.13 can be used to describe the kinematics of one active limb for the rowing and reverse rowing modes considered in Section 4.5.5. The horizontal and vertical components of velocity for the active limb, ˙ r = [ ˙ r x , ˙ r y ], can be written as ˙ r x = ˙ x−` ˙ φ sin(α +φ) = ˙ x−` ˙ φ(sinα cosφ + cosα sinφ), (4.8) ˙ r y =` ˙ φ cos(α +φ), (4.9) where ˙ x = [ ˙ x, ˙ y] represents the velocity for the robot center of mass,α∈ (0,π) represents the constant offset angle for the active limb and φ =ξ sin(ωt) represents the angular actuation with sweep angle ξ and frequency ω = 2πf. Per the model shown in eq. (4.2), the friction force generated by the active limb in the direction of motion is expected to be proportional to F x ∝− ˙ r x q ˙ r 2 x + ˙ r 2 y =− ˙ x−` ˙ φ sin(α +φ) q ˙ x 2 − 2 ˙ x` ˙ φ sin(α +φ) +` 2 ˙ φ 2 . (4.10) 59 Given the symmetric actuation of the active limbs in rowing and reverse rowing locomotion, the other active limb would generate an identical frictional force in the direction of motion. For there to be net motion in the horizontal direction, the cycle-averaged value of the friction force must be non-zero, ¯ F x 6= 0. To evaluate the conditions in which this occurs analytically, we redefine the offeset angle asα =π/2 + and consider the limit in which the actuation angle is small,|φ| 1. Under this assumption, we have sin(α +φ) = sin( π 2 + +φ) = sin( π 2 ) cos( +φ)≈ cos−φ sin (4.11) such that F x ∝− ˙ x−` ˙ φ(cos−φ sin) q ˙ x 2 − 2 ˙ x` ˙ φ(cos−φ sin) +` 2 ˙ φ 2 . (4.12) To make further analytical progress, we assume that the horizontal motion of the center of mass can be expressed as: ˙ x =A`ωξ sinα cos(ωt +B), (4.13) in which A and B represent a scaling factor and phase shift in the motion of the center of mass relative to the leading harmonic in the actuation velocity, ` ˙ φ sinα =`ωξ sinα cos(ωt). Note that the above expression neglects the (small) cycle-averaged motion of the center of mass as well as any motion at higher harmonics. Substituting the assumed expression for ˙ x into the simplified friction model and factoring out `ω sinα yields F x ∝− A cos(ωt +B)− cos(ωt)[cos−ξ sin(ωt) sin] q [A cos(ωt +B)] 2 − 2A cos(ωt +B) cos(ωt)[cos−ξ sin(ωt) sin] + [cos(ωt)] 2 . (4.14) The above expression indicates that the following three conditions are necessary for the generation of a non-zero cycle-averaged friction force: (1) 6= 0, (2) A6= 0, and (3) B6=nπ with n∈Z. This can be shown by considering each of the three conditions separately. 60 (1) If = 0, the friction force expression simplifies to: F x ∝− A cos(ωt +B)− cos(ωt) q [A cos(ωt +B)− cos(ωt)] 2 , (4.15) or F x ∝−sgn(A cos(ωt +B)− cos(ωt)), (4.16) for whichF x = 0 since the signum function for waveforms comprising single Fourier harmon- ics is positive and negative for the same duration of an oscillation cycle. (2) If A = 0, the friction force expression simplifies to: F x ∝ cos(ωt)[cos−ξ sin sin(ωt)] q [cos(ωt)] 2 , (4.17) or F x ∝sgn(cos(ωt)) [cos−ξ sin sin(ωt)], (4.18) sincesgn(cos(ωt)) = 0 andsgn(cos(ωt)) sin(ωt) = 0, the above expression also yieldsF x = 0. (3) If B =nπ, the n∈Z, the friction force expression simplifies to: F x ∝sgn(cos(ωt)) ±A + cos−ξ sin sin(ωt) q A 2 ± 2A cos + 1± 2Aξ sin sin(ωt) . (4.19) It can be shown formally (or via simple numerical tests) that the expression above again yields zero cycle-averaged force, F x = 0. Condition (1) explains why reverse-rowing, which is characterized by an offset angle α =π/2 or = 0 in the present work, is unsuccessful. Physically, the geometric nonlinearity in limb motion that arises forα6=π/2 and gives rise to a second harmonic in horizontal limb velocity, ˙ r x ∝ sin(2ωt), is essential to the generation of a cycle-averaged friction force. The results presented in section 4.6.2 show how this geometric nonlinearity can be reintroduced via a change in limb length over an actuation cycle. 61 Figure 4.14: Simulation (black) vs experiment (red) data of ˙ r x (dashed) and ˙ x (solid) for rowing with ξ = 30 ◦ ,f = 1 Hz. Conditions (2) and (3) indicate that there is no cycle-averaged force if the center of mass is held in place (A = 0) or if the motion of the robot center of mass is exactly in phase or antiphase with the motion of the active limbs. Physically, for dynamics governed by eq. (4.4), these conditions are expected to be satisfied naturally. In other words, we expect non-zero motion of the robot center of mass that is out of phase with the motion of the limbs. Indeed, as shown in fig. 4.14, the leading harmonic in robot x-direction body velocity is described for simulations by scaling factor A≈ 0.30 and phase shift with x-direction limb velocity, B≈ 0.08 [rad]. For experiments, the scaling factor is A≈ 0.57 and the phase shift is B≈ 0.23 [rad]. Note that the experimental velocities were estimated using a first-order derivative of position data obtained from frame-by-frame tracking. 4.6.2 A Successful Reverse Rowing Gait The results obtained in the previous section showed that reverse-rowing is not possible with the robot developed and tested in this study due to condition (2). This suggests that, in general, rowing locomotion is likely a more effective strategy for RS crawling robots with fixed limbs. To test if changing limb length could indeed enable crawling with the reverse-rowing mode, we pursued additional simulations with the mathematical model from section4.4.3. Notethattheresultsdescribedintheprevioussectionsindicatethatthismodel 62 Figure 4.15: Schematic showing limb length variation l(s,t) for the modified reverse-rowing locomotion strategy. The maximum and minimum limb lengths are set at l o = 7.5 cm andl s = 2.5 cm, respectively. The actuation frequency is f = 1.0 Hz. Movie 4 provides an illustrative example of reverse-rowing with varying limb length. Figure 4.16: Predicted translation speeds for rowing and modified reverse-rowing locomotion. In all cases, the sweep angle is ξ = 30 ◦ and actuation frequency is f = 1.0 Hz. generates very useful qualitative and quantitative predictions, even if perfect agreement with the experiments is not observed in all cases. To mimic the reverse-rowing sequence exhibited by brittle stars where arm length varies over a cycle [78], the length of the active limbs in our model was prescribed to vary as follows over an oscillation cycle: l(s,t) = (l o −l s ) 2 (cos(2πft) + 1) +l s . (4.20) The parameter l s represents the minimum limb length during the cycle, and l o = 0.075 m is the maximum limb length. As shown in figure 4.15, this reproduces a breast stroke-like 63 variation in limb length over a cycle. Model predictions for the net translation speed — away from the inactive limb in this case — are shown in figure 4.16 for varying minimum lengths l s = [0.01, 0.03, 0.05]m. For reference, the translation speed for rowing locomotion with rigid arms of length 0.075 m is also reproduced from figure 4.12. The results shown in figure 4.16 confirm that reverse rowing is possible with varying limb length. Further, the translation speed increases monotonically with decreasing l s , i.e., as the relative change in limb length increases. The model also indicates that translation speed for the modified reverse-rowing mode increases initially with increasing friction coefficient, before approaching an asymp- totic limit that depends on l s . This is unlike the results obtained for rowing locomotion, which predict that there is an optimal friction coefficient that maximizes translation speed. Further increases in μ beyond this optimum lead to a decrease in translation speed. These observations suggest that the modified reverse-rowing locomotor mode may be less sensitive to substrate type. In general, model predictions presented in this section indicate that the ability to vary limb length could lead to more robust locomotion capabilities. For robotic systems, variable limb length could be achieved either via the use of soft appendages or by incorporating additional degrees of freedom into rigid limbs. 4.6.3 Non-sinusoidal Gait Optimization The results produced in the preceding sections have focused on attaining a tractable analysis of sinusoidal gait strategies. However, there may exist non-sinusoidal gaits that result in better performance (faster translation) than what was studied. To address this, we revisit the rowing strategy and pursue parametric optimization of a three degree-of- freedom continuous spline, constrained to be zero at the beginning and end of the cycle, and constrained to cross the zero axis only once. A Piece-wise Cubic Hermite Interpolating Polynomial (PCHIP) spline was fit to the parameters. We selected PCHIP because it allows "peakier" curvature, and a more diverse set of allowable functions than other spline methods. We also tested a piecewise-sinusoid (PS) approximation and compared to the PCHIP results. 64 The disadvantage of the PS parameterization is that the function is not differentiable at the zero crossing, so we estimate that derivative using the t + derivative. The benefit of using PS over PCHIP is the reduction in computation time: PCHIP took 18 minutes to compute 1000 trials, PS took less than 1 minute. The parameters are presented as p =p1,p2,p3, where p1 =max(φ ∗ (t)) (4.21) p2 =min(φ ∗ (t)) (4.22) p3 =arg t (φ ∗ (t)) = 0). (4.23) and φ ∗ (t) is the phase normalized by 30 ◦ ∗ π[rads] 180 ◦ . The functional of interest,Q(p), is the end position,x f , of the x-coordinate after 5 cycles (we selected 5 cycles to eliminate transient effects, but still minimize simulation time). Iso- contours of Q are shown in fig. 4.17. Both PCHIP and PS reproduce the same trends, with only slight deviations in magnitude. Thus, for our optimization experiments presented in the following paragraphs, we use the PS curve for gait interpolation. Though the function space is complex, for the area wherep3 is less than 0.5, the trends are as follows: increasing p1 and decreasing p2 lead to larger translation. This intuitively makes sense because we know that increasing sweep, ξ, also increases translation distance. Because the Q-space appeared to be locally convex near the optimum, we applied an adaptive-step first order forward finite difference gradient search algorithm. We found that the functional is multimodal near the optima and the optimal parameter sets are near p = {1,−1, 0.432},{1,−1, 0.455}. Close to these points, Q is non-smooth and difficult to assess in more detail. This may be due to numerical sensitivities for high precision point operations in the ODE45 algorithm. The results show that the optimal gait is almost a perfect sinusoid (not sinusoidal since there are marginal gains to have the crossover less than 0.5). We speculate that moving the crossover point so that there is less time spent on the recovery stroke, the robot is able 65 Figure 4.17: Isosurface plots of performance metric x f for (left) PCHIP and (right) piecewise sinusoid parameterized gait representations. Figure 4.18: Optimal parameterized curve to maximize distance traveled to translate an additional 0.005m per 5 cycles, or about a 5% gain compared to sinusoidal actuation. 4.7 Gait Map and Controller Design In this section, we discuss the development and parametrization of the omnidirectional gait (4.7.1 and 4.7.2) before discussing controller design (4.7.3). 66 Figure 4.19: Predicted and measured paths x(t) for the sandpaper-paper contact for (a) transla- tion gait and (c) rotational gait. For translation, we show the state x vs t in (b), and for rotation, we show state ξ vs t in (d) as these are the states of interest. Experiment paths are traced with blue markers, and simulation paths are traced with red lines. 4.7.1 Omnidirectional Gait The predictive model showed that an omnidirectional gait would allow for immediate translation in any direction. This new gait was a variation of the previously introduced translational gait. As shown in Fig. 4.19(a,b), when two limbs operate in anti-phase to one another, and the third limb is inactive, the robot translates in the direction of the inactive limb. However, simulations show that sinusoidal actuation of the inactive limb produces translation at a non-zero angle relative to the previously inactive limb. This angle of translation, θ D varied depending on the amplitude, α of the previously inactive limb’s sinusoidal motion. In this new gait, the previously inactive limb is defined as the α-limb because the amplitude of its sinusoidal motion is α. The other two limbs operate in anti- phase to one another, with constant amplitude sinusoidal motion. Simulations generated (details in Section 4.7.2) a map of sinusoid amplitude to angle of translation, M (Fig. 4.20). Importantly, because of the radial symmetry of the model, any limb can be made the α-limb, allowing the model to translate in any direction from any initial configuration. The desired direction of motion (θ D ∈ [0, 360] ◦ ) is divided into six zones spanning 60 ◦ of transla- tion angle, as shown in Table 4.2. Forθ D ∈ [0, 60] ◦ , lead limb amplitudeα is calculated using a mapping of θ D to a = [a 1 ,a 2 ,a 3 ] ◦ = [α, 30,−30] ◦ , in which a i represents the oscillation 67 Table 4.2: Relationship between desired translation direction and the gait parameters for each limb. All angles are prescribed in degrees. Zone Desired Translation Angle (θ D ) Gait Parameters (a) 1 0<θ≤ 60 [α, 30,−30] 2 60<θ≤ 120 [−30,−α, 30] 3 120<θ≤ 180 [−30, α, 30] 4 180<θ≤ 240 [30,−30,−α] 5 240<θ≤ 300 [30,−30, α] 6 300<θ≤ 360 [−α, 30,−30] amplitude for limb i and a 1 is the lead (or α) limb. The mapping between θ D ∈ [0, 60] ◦ andα is plotted in Fig. 4.20. This function is combined with different permutations of limb amplitudes to span θ D ∈ [0, 360] ◦ . The outputs of the MATLAB block in Fig. 4.21 are the gait parameters a = [a 1 ,a 2 ,a 3 ] defining the sinusoidal motion of each limb: φ i =a i sin(2πft). (4.24) In the expression above,f is the frequency of limb oscillation, which was set to 1 Hz in both the simulations and the experiments. The limb angle φ i can be visualized in Fig. 4.3. One of the gait parameters, [a 1 ,a 2 ,a 3 ], is set to equal α. The value of θ D is used to determine what index of a is set to the α-value, following the structure shown in Table 4.2. 4.7.2 Gait Map Open loop simulations with α∈ [0, 30] ◦ and with non-α limb amplitudes 30 ◦ generated M (Fig. 4.20). A maximum value of α = 30 ◦ was selected due to limitations of the physical model. Over the course of a trial,α was held constant, and the resulting translation angleθ was measured relative to the model reference frame for each cycle of motion. The θ values of each cycle were then averaged over the entire trial. This averaging produced the average angle of translation, θ avg . There was minimal variation in θ after the first two cycles of motion. 68 Figure 4.20: A comparison of the simulated average map, M avg to experimentally collected data points. This map is used to prescribe an α-limb gait parameter, α, given a desired angle of translation, θ. This procedure was repeated in three different different friction environments: μ = [0.33, 0.59, 0.87]. These values were selected based upon the friction coefficient values tested with the physical robot. The differences in the mappings generated from these three fric- tion coefficients was small. As a result, we assumed a universal mapping, M avg , to provide sufficient accuracy independent of friction magnitude. Fig. 4.20 shows this averaged map obtained using the predictive model (red curve), together with [θ,α] measurements made in experiments using the tripedal robot (blue symbol). 4.7.3 Controller Design Fig. 4.21 provides an overview of the feedback tracking procedure implemented in the experiments. The position vectorx = [x,y,ξ] and trajectorys are inputs to the gait mapping implemented in MATLAB. The inputs to MATLAB, x and s, are used to determine the desired angle of translation, θ D = G(x,s), using the arctangent function. A simple PI controller, shown in the blue MATLAB block of Fig. 4.21, was implemented to compensate for steady-state error. The PI controller outputs an adjusted desired angle of translation, θ PI according the equation, θ PI [k] = (K p + K I T s z− 1 )e[k] +θ D [k] (4.25) 69 Capture image of robot Process image to determine current state Compare current state to goal trajectory Calculate necessary motor command sequence in MATLAB Send commands to motors MotorController Computer Visual Processing Desired Angle Calculation G(x,s) s Gait Selection F(M avg , θ PI ) x[k] MATLAB e[k] θ PI [k] Controller K P + K I T s z − 1 Error Metric H(s,x) PixyCam2 Arduino Arduino ො x ො y Power Supply MATLAB < , > → < > Trajectory MotorController Computer x Visual Processing Figure 4.21: Flow chart (top) and diagram (bottom) showcasing feedback processing and gait selection. wherek is the discrete time variable,e is the error signal,K P is the proportional gain,K I is the integral gain, T s is the duration in seconds, and z is the z-transform operator. An error metric, H(s,x), estimates e by calculating the minimum distance from a line connecting target points and the position measurement. The Gait Selection block uses θ PI , along with the average gait map, M avg to select α and send amplitudes, a, to the motors. Depending on θ PI , one of the six zones listed in Table 4.2 is selected and the mapping M avg is used to calculate the value of α and the gait parameter vector a. 70 Figure 4.22: Four points are used for the robot to trace a square path in the presence of a variable wind flow field where (a)-(c) are without error compensation and (d)-(f) are with PI error compensation. (a),(d) Plots are without wind flow, (b),(e) plots are with 0 ◦ wind flow, and (c),(f) plots are wind flow at 25 ◦ with respect to the x-direction. Blue markers show experimental measurements while red curves show simulation results. 4.8 Path Following Demonstration To illustrate path following capabilities in simulation and experiments, a rectangular path specified by 4 target points was used (see Fig. 4.22). In the physical experiments, we used a surface with an unknown friction coefficient between the low-friction polymer-paper case (μ = 0.33) and the high-friction sandpaper-paper case (μ = 0.88) to demonstrate the friction-independence ofM avg . In addition, an industrial floor fan was used to generate wind flow across the test surface to study the effect of aerodynamic disturbances on path following capabilities. A Protmex 6252A handheld anemometer measured wind speed at both ends of the testing platform. The edge located closest to the fan had a flow speed of 6.4± 0.2ms −1 and the edge furthest from the fan had a flow speed of 4.5± 0.2ms −1 . Thus, there were significant velocity gradients across the surface, making for a complex flow field. In the numerical simulations, aerodynamic drag was modeled as F d = (1/2)ρC d A d v 2 where ρ is the density of air, C d is a drag coefficient, A d is the frontal area of the robot, andv is wind speed. The drag coefficient was assumed to beC d = 1, corresponding roughly to the drag coefficient of a cubic bluff body. The frontal area for the robot was estimated 71 to be A d = 0.02m 2 . The wind speed was set at v = 5.5ms −1 , an average of the velocities measured at each end of the test section. We note that these values were for simulation purposes only to provide a reference for the experiment results. The controller does not use the fluid dynamics of the system for gait selection. The flow is considered to be a general disturbance. This model may differ from the physical system where we assume the flow field is assumed uniform and constant, there are no ground effects, the drag coefficient does not change with position, and there are no lift effects. Proportional gain was K P = 15 deg cm −1 for rapid correction to path deviations, and integral gain was K I = 1 deg cm −1 to reduce steady-state error. We selected these values based on experimental tuning with an update rate of f = 1 Hz (i.e., every actuation cycle for the limbs). From Fig. 4.22(a) it can be seen that, without proportional-integral (PI) error compensation, the robot does not follow a straight line. This is likely a result of error from unmodeled disturbances in M avg , imperfections in robot manufacturing, error in gait execution, imbalance from cable tether, and experiment surface inhomogeneity. Fig. 4.22(d) shows that the PI controller corrects for this unmodeled drift. Fig. 4.22(b),(c) show the effect of wind in the environment on path following without PI error compensation. The robot is particularly sensitive to wind speed angled at 25 ◦ with respect to the x-direction, where deviations from the nominal trajectory are amplified. The PI controller successfully compensates for this in Fig. 4.22(f). Note that the simulation results shown Fig. 4.22 are in reasonable qualitative agreement with the experimental tracks for the cases with background flow. Table 4.3 shows two performance metrics for the tracking experiments: cumulative path following error Δ = P |e| (m), where e is the perpendicular distance to the nominal path, and the completion time T c (s). In all flow environments, Δ is reduced by implementation of the PI controller. For example, Δ is reduced by over 65% for the cases with flow in the x-direction. However, T c increases with added control for both the no flow and angled flow cases. This is because the robot spends time correcting its trajectory rather than going 72 Table 4.3: Experimental performance in terms of cumulative path following error (Δ, m) and completion time (T c , s). Rows and columns correspond to Fig. 4.22. No flow 0 ◦ flow 25 ◦ flow Δ T c Δ T c Δ T c No PI 3.39 16 8.12 39 5.34 21 With PI 1.81 18 2.81 26 2.73 36 directlytothewaypoints. Atime-optimalpathinthepresenceofflowisnotstraightforward, and is a topic of further investigation. Another demonstration of robot path following capability is provided in Fig. 4.1, which shows that complex curved paths can be tracked by placing intermediate target points. For this figure, we show tracking with no error compensation using the standard M avg map. 4.9 Conclusion This study demonstrates that several locomotion strategies are possible for robotic sys- tems with radially-symmetric bodies. Sequential actuation of all limbs can give rise to locomotion along circular orbits with controllable radius. Periodic actuation with a constant phase shift across all limbs (Δψ = 120 ◦ for a tripedal robot) leads to rotation in-place. Pure translation is achieved through the presence of an inactive limb that breaks radial symmetry. The ability to move along prescribed curves, rotate in place, and travel in a straight line establishes a framework for future motion planning efforts for radially-symmetric crawling robots. Using the models developed in this paper, we have successfully demonstrated path following for this robot using image-based feedback control [2]. The primary advantage of radial symmetries highlighted in this work is the omni-directional translation ability. Conse- quently, robots that require instantaneous omni-directional actuation may benefit most from applying the ideas presented here. For systems with rigid limbs and sinusoidal actuation, such as the robot developed here, we found only the so-called rowing locomotor mode observed in nature to be effective at locomotion (we cannot comment on feasibility with non-sinusoidal gaits). For this mode, 73 the hind limbs are actuated in anti-phase with one another and the robot moves in the direction of the inactive forelimb. In section 4.6, we showed via further simplifications to the dynamic model that net translation is only present if certain conditions are met. These conditionsestablishthataphaseshiftbetweentherobotcentervelocityandthelimbvelocity is generated due to inertial effects, and that this phase shift creates a positive net force over a cycle. Reverse-rowing was only feasible for more complicated systems capable of varying limb length over an actuation cycle. Note that the ability to vary limb length ultimately allows biological and engineered systems to vary the position of the contact points relative to the center of mass. This allows for finer control over the normal and, hence, friction forces acting at each contact point. The ability to modulate frictional forces can play an important role in crawling loco- motion. Indeed, previous studies show that snakes try to maximize normal forces over low-friction surfaces through several different mechanisms; this includes lifting parts of their body off the substrate to enhance normal forces [106, 107]. Predictions made using the simple Coulomb friction model indicate the presence of an optimal sweep-dependent friction coefficient that maximizes translation speed for brittle star inspired rowing locomotion. The optimal friction coefficient values were μ < 0.3 for the range of sweep angles tested here. For higher friction coefficients, translation speed decreased. However, model predictions also suggest that this deterioration of performance can be mitigated by switching to a reverse- rowing strategy, in which limb length varies over an actuation cycle. These differences in crawling efficacy may explain the prevalence of both rowing and reverse-rowing locomotion in nature [76]. The relative performance of these rowing and reverse-rowing locomotor modes in terms of speed and energetic efficiency remains to be evaluated. Finally, it must be emphasized that the simple model developed here is very much a start- ing point for further research. More sophisticated frictional models that account for stick- slip dynamics [108], or analytical models that take advantage of scale separation between the timescale associated with the fast periodic actuation and the slow macro-scale motion 74 (i.e., translation, large-scale revolution) might provide significant further insight into crawl- ing dynamics. Models grounded in geometric mechanics [109, 110] that can predict motion from periodic shape-space changes for robotic systems could also be valuable. We have shown that a momentum-conservation model can accurately reproduce motion for a friction driven tripedal robot. Normal forces are accurately predicted by solving an inverse problem from a force/torque balance system of equations. This principle can be extended to other three-point systems where normal forces are of interest. This model facilitated generation of a gait-map which predicted that the robot can translate in any direction independent of the current state. Open loop experiments confirmed this, and, with this mapping, the physical system demonstrated effective path following with an unknown friction coefficient and in a high-speed wind flow environment. The empirical gait-map control strategy illustrated in this work can be applied to other nonlinear, non-holonomic systems where analytical approaches may not be possible. This study shows how a minimally actuated, RS robot can achieve path following by exploiting asymmetrical gait patterns. Future work will target time optimal path following in the presence of background flows, fluid dynamic characterization of the physical robot, and locomotion on heterogeneous sur- faces. 75 Chapter 5 Conclusion and Future Work RS robotics is a rapidly developing field as researchers are finding an increasing number of applications where these machines excel. Some exceptional abilities these robots possess include adapting to appendage damage [3], climbing walls [6], arm/leg exchangeability [5], and enhanced stability through multiple surface contact points [111]. What has been missing from these works has been an understanding of how these robots interact with the fluid that they are immersed in. We sought to learn more about the FSI of radially symmetric shapes, and how we can better design underwater crawling robots to favorably interact with these forces. In this paper, we developed a minimally-actuated RS crawler and successfully modeled the dynamics with a novel normal force estimation procedure. We used this model in a control loop to demonstrate path-following in steady fluid flow environments. To learn how to best mitigate these fluid forces, we studied how sea star shapes interact with flow to mini- mize dislodgement risk. We attached rigid 3D printed models to a force sensor measurement system in a water channel, and we learned that sea star geometries divert momentum upward through the aid of center-line upwelling edge vortices. They also modify their AR to tune the hydrodynamic response, where a higher AR corresponds to larger drag and downforce. We mimicked this force-tuning ability with a morphing silicone robot and a 2-DOF genetic algo- rithm that experimentally identified drag- and lift-minimum configurations. These results provide background and strategies for future researchers to design FSI-enhancing features on crawling robots in fluid flow. Our tripedal robot builds upon the work of trident snake robots, where we extend the robot to move using nonlinear friction forces as opposed to constrained wheel motion. This generalized the dynamics so that other researchers interested in pursuing tripedal robots can apply our methods. Our normal force estimation procedure and gait-map control strategies 76 provide a platform for future RS crawling robot studies to model and control their systems. We also introduced a study into how shape factors affect fluid forces on crawling robots. We confirmed the hypothesis presented by Hayne and Palmer that changing AR has an impact on fluid forces experienced [29]. We showed that the pentaradial sea star geometry produces omnidirectionaldownforce, thuseliminatingtherequirementforwall-normaladhesion. Using soft robotics technologies, we can design shapes that can adapt to flow conditions. We built on the work of Ishida et al., who showed how shape can be manipulated for enhancing locomotion [65]. We also reimagined Ingo Rechenberg’s vision of autonomous experimental optimization [73] applied to soft robotics. Using an evolutionary algorithm, we showed that an optimum search can converge quickly for smooth functionals. This work has applications in optimal design as well as performance enhancement during operation. For the rest of this discussion, we will suggest directions this research can take. One direction is to pursue implementation of our flow estimation algorithm for three arm systems. We were not able to accomplish this task with our current framework because we needed to have a well-defined mapping of robot pose to drag/lift forces. This requires imple- mentation of an FSI-enhancing shell, which would include a redesign of the robot chassis. However, this objective is worth pursuing as it could result a low-cost force measurement probe. Another direction, as described in chapter 1, the work by Serchi [112] is that active shape modification can also enhance the FSI of robots in oscillating flow with shorter char- acteristic timescales. The shape-changing platform presented in this work, however, has much slower actuators so it cannot achieve the frequencies required for this task. A future direction for this project is to develop shape changing robotics with faster actuation (motor- driven rather than hydraulic-driven). This ability would allow researchers more flexibility in exploiting oscillating flows or larger flow structures, thus merging this topic with flow control. Another future direction is to synthesize the chassis/shell of a RS robot. Since mobility is important for underwater applications, a shape-changing system should be unob- trusiveandself-contained(notether). Thisposesmanychallengesandwillrequiresignificant application-specific design. 77 References [1] Francesco Giorgio-Serchi and Gabriel D Weymouth. Underwater soft robotics, the benefit of body-shape variations in aquatic propulsion. In Soft Robotics: Trends, Applications and Challenges, pages 37–46. Springer, 2017. [2] Mark Hermes, Taylor McLaughlin, Mitul Luhar, and Quan Nguyen. Locomotion and control of a friction-driven tripedal robot. In Proceedings of the 2021 IEEE Interna- tional Conference on Robotics and Automation. IEEE, 2021. [3] Takeshi Kano, Shota Suzuki, Wataru Watanabe, and Akio Ishiguro. Ophiuroid robot that self-organizes periodic and non-periodic arm movements. Bioinspiration & biomimetics, 7(3):034001, 2012. [4] Sunil Pranit Lal, Koji Yamada, and Satoshi Endo. Evolving motion control for a modularrobot. InInternational Conference on Innovative Techniques and Applications of Artificial Intelligence, pages 245–258. Springer, 2007. [5] Joshua Hooks, Min Sung Ahn, Jeffrey Yu, Xiaoguang Zhang, Taoyuanmin Zhu, Hosik Chae, and Dennis Hong. Alphred: A multi-modal operations quadruped robot for package delivery applications. IEEE Robotics and Automation Letters, 5(4):5409–5416, 2020. [6] XuanLin,HariKrishnan,YaoSu,andDennisWHong. Multi-limbedrobotverticaltwo wall climbing based on static indeterminacy modeling and feasibility region analysis. In 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 4355–4362. IEEE, 2018. [7] James A Murray, Adam P Jones, Annie C Links, and AO Dennis Willows. Daily tracking of the locomotion of the nudibranch tritonia tetraquetra (pallas= tritonia diomedea bergh) in nature and the influence of water flow on orientation, crawling, and drag. Marine and freshwater behaviour and physiology, 44(5):265–288, 2011. [8] MJ Weissburg, CP James, DL Smee, and DR Webster. Fluid mechanics produces conflicting, constraints during olfactory navigation of blue crabs, callinectes sapidus. Journal of Experimental Biology, 206(1):171–180, 2003. [9] Francesco Giorgio-Serchi and Gabriel D Weymouth. Can added-mass variation act as a thrust force? In OCEANS 2017-Aberdeen, pages 1–4. IEEE, 2017. 78 [10] Francesco Giorgio-Serchi and GD Weymouth. Drag cancellation by added-mass pump- ing. Journal of fluid Mechanics, 798, 2016. [11] Maarja Kruusmaa, Paolo Fiorini, William Megill, Massimo de Vittorio, Otar Akanyeti, Francesco Visentin, Lily Chambers, Hadi El Daou, Maria-Camilla Fiazza, Jaas Ježov, et al. Filose for svenning: A flow sensing bioinspired robot. IEEE Robotics & Automa- tion Magazine, 21(3):51–62, 2014. [12] Guijie Liu, Huanhuan Hao, Tingting Yang, Shuikuan Liu, Mengmeng Wang, Atilla Incecik, andZhixiongLi. Flowfieldperceptionofamovingcarrierbasedonanartificial lateral line system. Sensors, 20(5):1512, 2020. [13] C Abels, A Qualtieri, M De Vittorio, WM Megill, and F Rizzi. A bio-inspired real-time capable artificial lateral line system for freestream flow measurements. Bioinspiration & biomimetics, 11(3):035006, 2016. [14] Christian Meurer, Juan Francisco Fuentes-Pérez, Narcís Palomeras, Marc Carreras, andMaarjaKruusmaa. Differentialpressuresensorspeedometerforautonomousunder- water vehicle velocity estimation. IEEE Journal of Oceanic Engineering, 2019. [15] MichaelSTriantafyllou, GabrielDWeymouth, andJianminMiao. Biomimeticsurvival hydrodynamics and flow sensing. Annual Review of Fluid Mechanics, 48:1–24, 2016. [16] Ajay Giri Prakash Kottapalli, Meghali Bora, Mohsen Asadnia, Jianmin Miao, Subbu S Venkatraman, and Michael Triantafyllou. Nanofibril scaffold assisted mems artificial hydrogelneuromastsforenhancedsensitivityflowsensing. Scientific reports, 6(1):1–12, 2016. [17] Mohsen Asadnia, Ajay Giri Prakash Kottapalli, Reza Haghighi, Audren Cloitre, Pablo Valdivia y Alvarado, Jianmin Miao, and Michael Triantafyllou. Mems sen- sors for assessing flow-related control of an underwater biomimetic robotic stingray. Bioinspiration & biomimetics, 10(3):036008, 2015. [18] Ajay Giri Prakash Kottapalli, Mohsen Asadnia, Jianmin Miao, and Michael Triantafyl- lou. Touchatadistancesensing: lateral-lineinspiredmemsflowsensors. Bioinspiration & biomimetics, 9(4):046011, 2014. [19] Hans H Bruun. Hot-wire anemometry: principles and signal analysis, 1996. [20] H-E Albrecht, Nils Damaschke, Michael Borys, and Cameron Tropea. Laser Doppler and phase Doppler measurement techniques. Springer Science & Business Media, 2013. [21] Kevin Nelson and Kamran Mohseni. Hydrodynamic force decoupling using a dis- tributedsensorysystem. IEEE Robotics and Automation Letters, 5(2):3235–3242, 2020. [22] Vicente I Fernandez, Jeff Dusek, James Schulmeister, Audrey Maertens, Stephen Hou, Karthik S Sekar, Jason Dahl, Jeffrey H Lang, and Michael S Triantafyllou. Pres- sure sensor arrays to optimize the high speed performance of ocean vehicles. In 11th International Conference on Fast Sea Transportation, pages 1–8, 2011. 79 [23] Brendan Colvert, Mohamad Alsalman, and Eva Kanso. Classifying vortex wakes using neural networks. Bioinspiration & biomimetics, 13(2):025003, 2018. [24] Kurtis JR Hayne and A Richard Palmer. Intertidal sea stars (pisaster ochraceus) alter body shape in response to wave action. Journal of Experimental Biology, 216(9):1717– 1725, 2013. [25] JH Orton and JH Fraser. Rate of growth of the common starfish, asterias rubens. Nature, 126(3180):567–567, 1930. [26] Howard M Feder. Growth and predation by the ochre sea star, pisaster ochraceus (brandt), in monterey bay, california. Ophelia, 8(1):161–185, 1970. [27] Brian Helmuth and Mark W Denny. Predicting wave exposure in the rocky intertidal zone: do bigger waves always lead to larger forces? Limnology and Oceanography, 48(3):1338–1345, 2003. [28] Romana Santos, Stanislav Gorb, Val´ erie Jamar, and Patrick Flammang. Adhesion of echinodermtubefeettoroughsurfaces. Journal of Experimental Biology,208(13):2555– 2567, 2005. [29] Kurtis JR Hayne and A Richard Palmer. Intertidal sea stars (pisaster ochraceus) alter body shape in response to wave action. Journal of Experimental Biology, 216(9):1717– 1725, 2013. [30] Hermann Schlichting. Experimental investigation of the problem of surface roughness. NACA Technical Memorandum, 823, 1937. [31] Javier Jim´ enez. Turbulent flows over rough walls. Annu. Rev. Fluid Mech., 36:173–196, 2004. [32] Sighard F Hoerner. Fluid-dynamic drag. Hoerner Fluid Dynamics, 1965. [33] SH Hoerner. Fluid-dynamic lift. Hoerner Fluid Dynamics, 1985. [34] WH Schofield and E Logan. Turbulent shear flow over surface mounted obstacles. Journal of Fluids Engineering, 112(4):376–385, 1990. [35] BarbaraLdaSilva, RajatChakravarty, DavidSumner, andDonaldJBergstrom. Aero- dynamic forces and three-dimensional flow structures in the mean wake of a surface- mounted finite-height square prism. International Journal of Heat and Fluid Flow, 83:108569, 2020. [36] E Savory and N Toy. Hemisphere and hemisphere-cylinders in turbulent boundary layers. Journal of Wind Engineering and Industrial Aerodynamics, 23:345–364, 1986. [37] Seiichi Taniguchi, Hiroshi Sakamoto, Masaru Kiya, and Mikio Arie. Time-averaged aerodynamic forces acting on a hemisphere immersed in a turbulent boundary. Journal of Wind Engineering and Industrial Aerodynamics, 9(3):257–273, 1982. 80 [38] Jens Nikolas Wood, Guillaume De Nayer, Stephan Schmidt, and Michael Breuer. Experimental investigation and large-eddy simulation of the turbulent flow past a smooth and rigid hemisphere. Flow, Turbulence and Combustion, 97(1):79–119, 2016. [39] M Ikhwan and B Ruck. Wind load coefficients for pyramidal buildings. Proc. 12. GALA-Tagung Lasermethoden in der Stromungmesstechnik, B. Ruck, A. Leder, D. Dopheide (Ed.), Karlsruhe, Deutschland, 2004. [40] RJ Martinuzzi and M AbuOmar. Study of the flow around surface-mounted pyramids. Experiments in Fluids, 34(3):379–389, 2003. [41] SB Vosper. Three-dimensional numerical simulations of strongly stratified flow past conical orography. Journal of the Atmospheric Sciences, 57(22):3716–3739, 2000. [42] Tetsushi Okamoto, Miki Yagita, and Shin-ichi Kataoka. Flow past cone placed on flat plate. Bulletin of JSME, 20(141):329–336, 1977. [43] Michael Gaster. Vortex shedding from slender cones at low reynolds numbers. Journal of Fluid Mechanics, 38(3):565–576, 1969. [44] Giacomo Valerio Iungo and Guido Buresti. Experimental investigation on the aerody- namic loads and wake flow features of low aspect-ratio triangular prisms at different wind directions. Journal of Fluids and Structures, 25(7):1119–1135, 2009. [45] DK Heist and FC Gouldin. Turbulent flow normal to a triangular cylinder. Journal of Fluid Mechanics, 331:107–125, 1997. [46] CJ Baker. The laminar horseshoe vortex. Journal of Fluid Mechanics, 95(2):347–367, 1979. [47] CJBaker. Theturbulenthorseshoevortex. Journal of Wind Engineering and Industrial Aerodynamics, 6(1-2):9–23, 1980. [48] MARLENE M Martinez, RJ Full, and MA Koehl. Underwater punting by an intertidal crab: a novel gait revealed by the kinematics of pedestrian locomotion in air versus water. Journal of Experimental Biology, 201(18):2609–2623, 1998. [49] Elise Hennebert, Ruddy Wattiez, Mélanie Demeuldre, Peter Ladurner, Dong Soo Hwang, J Herbert Waite, and Patrick Flammang. Sea star tenacity mediated by a protein that fragments, then aggregates. Proceedings of the National Academy of Sci- ences, 111(17):6317–6322, 2014. [50] Birgit Lengerer, Morgane Algrain, Mathilde Lefevre, Jérôme Delroisse, Elise Hen- nebert, and Patrick Flammang. Interspecies comparison of sea star adhesive proteins. Philosophical Transactions of the Royal Society B, 374(1784):20190195, 2019. [51] Robert Pjeta, Herbert Lindner, Leopold Kremser, Willi Salvenmoser, Daniel Sobral, Peter Ladurner, and Romana Santos. Integrative transcriptome and proteome anal- ysis of the tube foot and adhesive secretions of the sea urchin paracentrotus lividus. International journal of molecular sciences, 21(3):946, 2020. 81 [52] Olaf Ellers. Form and motion of donax variabilis in flow. The Biological Bulletin, 189(2):138–147, 1995. [53] Robert T Paine. Food web complexity and species diversity. The American Naturalist, 100(910):65–75, 1966. [54] William Thielicke and Eize Stamhuis. Pivlab–towards user-friendly, affordable and accurate digital particle image velocimetry in matlab. Journal of Open Research Soft- ware, 2(1), 2014. [55] Thomas E Laidig, Lisa M Krigsman, and Mary M Yoklavich. Reactions of fishes to two underwater survey tools, a manned submersible and a remotely operated vehicle. Fishery Bulletin, 111(1):54–67, 2013. [56] George M Whitesides. Soft robotics. Angewandte Chemie International Edition, 57(16):4258–4273, 2018. [57] Daniela Rus and Michael T Tolley. Design, fabrication and control of soft robots. Nature, 521(7553):467, 2015. [58] Jung-Yup Kim and Bong-Huan Jun. Design of six-legged walking robot, little crabster for underwater walking and operation. Advanced Robotics, 28(2):77–89, 2014. [59] Michael A Bell, Isabella Pestovski, William Scott, Kitty Kumar, Mohammad K Jawed, Derek A Paley, Carmel Majidi, James C Weaver, and Robert J Wood. Echinoderm- inspired tube feet for robust robot locomotion and adhesion. IEEE Robotics and Automation Letters, 3(3):2222–2228, 2018. [60] Robert K Katzschmann, Joseph DelPreto, Robert MacCurdy, and Daniela Rus. Explo- ration of underwater life with an acoustically controlled soft robotic fish. Science Robotics, 3(16):eaar3449, 2018. [61] Caleb Christianson, Nathaniel N Goldberg, Dimitri D Deheyn, Shengqiang Cai, and Michael T Tolley. Translucent soft robots driven by frameless fluid electrode dielectric elastomer actuators. Science Robotics, 3(17):eaat1893, 2018. [62] Robert L Baines, Joran W Booth, Frank E Fish, and Rebecca Kramer-Bottiglio. Toward a bio-inspired variable-stiffness morphing limb for amphibious robot locomo- tion. In 2019 2nd IEEE International Conference on Soft Robotics (RoboSoft), pages 704–710. IEEE, 2019. [63] MatthewARobertson, FilipEfremov, andJamiePaik. Roboscallop: Abivalveinspired swimming robot. IEEE Robotics and Automation Letters, 4(2):2078–2085, 2019. [64] Otar Akanyeti, Joy Putney, Yuzo R Yanagitsuru, George V Lauder, William JStewart, and James C Liao. Accelerating fishes increase propulsive efficiency by modulating vortexringgeometry. Proceedings of the National Academy of Sciences, 114(52):13828– 13833, 2017. 82 [65] Michael Ishida, Dylan Drotman, Benjamin Shih, Mark Hermes, Mitul Luhar, and Michael T Tolley. Morphing structure for changing hydrodynamic characteristics of a soft underwater walking robot. IEEE Robotics and Automation Letters, 4(4):4163– 4169, 2019. [66] Julian Colorado, Antonio Barrientos, Claudio Rossi, and Kenny S Breuer. Biomechan- ics of smart wings in a bat robot: morphing wings using sma actuators. Bioinspiration & biomimetics, 7(3):036006, 2012. [67] Wei Zhao, Junzhi Yu, Yimin Fang, and Long Wang. Development of multi-mode biomimetic robotic fish based on central pattern generator. In 2006 IEEE/RSJ Inter- national Conference on Intelligent Robots and Systems, pages 3891–3896. IEEE, 2006. [68] Tae-Hwan Joung, Karl Sammut, Fangpo He, and Seung-Keon Lee. Shape optimization ofanautonomousunderwatervehiclewithaductedpropellerusingcomputationalfluid dynamicsanalysis. International Journal of Naval Architecture and Ocean Engineering, 4(1):45–57, 2012. [69] SilvestroBarbarino,OnurBilgen,RaficMAjaj, MichaelIFriswell,andDanielJInman. A review of morphing aircraft. Journal of intelligent material systems and structures, 22(9):823–877, 2011. [70] Min-Woo Han, Hugo Rodrigue, Hyung-Il Kim, Sung-Hyuk Song, and Sung-Hoon Ahn. Shape memory alloy/glass fiber woven composite for soft morphing winglets of unmanned aerial vehicles. Composite Structures, 140:202–212, 2016. [71] Hugo Rodrigue, Seunghyun Cho, Min-Woo Han, Binayak Bhandari, Jae-Eul Shim, and Sung-Hoon Ahn. Effect of twist morphing wing segment on aerodynamic performance of uav. Journal of Mechanical Science and Technology, 30(1):229–236, 2016. [72] Russell J Rufino, Timothy F Miller, and Farhan S Gandhi. Morphing hull concepts for unmanned underwater vehicles. In OCEANS 2008, pages 1–11. IEEE, 2008. [73] Ingo Rechenberg. Case studies in evolutionary experimentation and computation. Computer methods in applied mechanics and engineering, 186(2-4):125–140, 2000. [74] J Lane Cameron and Peter V Fankboner. Reproductive biology of the commercial sea cucumber parastichopus californicus (stimpson)(echinodermata: Holothuroidea). ii. observations on the ecology of development, recruitment, and the juvenile life stage. Journal of Experimental Marine Biology and Ecology, 127(1):43–67, 1989. [75] Michaël Manuel. Early evolution of symmetry and polarity in metazoan body plans. Comptes rendus biologies, 332(2-3):184–209, 2009. [76] Henry C Astley. Getting around when you’re round: quantitative analysis of the loco- motion of the blunt-spined brittle star, ophiocoma echinata. Journal of Experimental Biology, 215(11):1923–1929, 2012. 83 [77] Leon J Cole. Direction of locomotion of the starfish (asterias forbesi). Journal of Experimental Zoology, 14(1):1–32, 1913. [78] Takeshi Kano, Eiki Sato, Tatsuya Ono, Hitoshi Aonuma, Yoshiya Matsuzaka, and Akio Ishiguro. A brittle star-like robot capable of immediately adapting to unexpected physical damage. Royal Society open science, 4(12):171200, 2017. [79] Mattias Sköld and Rutger Rosenberg. Arm regeneration frequency in eight species of ophiuroidea (echinodermata) from european sea areas. Journal of Sea Research, 35(4):353–362, 1996. [80] MD Candia Carnevali. Regeneration in echinoderms: repair, regrowth, cloning. Inver- tebrate Survival Journal, 3(1):64–76, 2006. [81] Yoshiya Matsuzaka, Eiki Sato, Takeshi Kano, Hitoshi Aonuma, and Akio Ishiguro. Non-centralizedandfunctionallylocalizednervoussystemofophiuroids: evidencefrom topical anesthetic experiments. Biology open, 6(4):425–438, 2017. [82] Yu I Arshavskii, SM Kashin, NM Litvinova, GN Orlovskii, and AG Fel’dman. Coordi- nation of arm movement during locomotion in ophiurans. Neurophysiology, 8(5):404– 410, 1976. [83] Mark Showalter, Dennis Hong, and Daniel Larimer. Development and comparison of gait generation algorithms for hexapedal robots based on kinematics with considera- tions for workspace. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 43260, pages 1215– 1220, 2008. [84] Jacob Webb, Alexander Leonessa, and Dennis Hong. Gait design and gain-scheduled balance controller of an under-actuated robotic platform. In 2015 IEEE/RSJ Interna- tional Conference on Intelligent Robots and Systems (IROS), pages 5148–5155. IEEE, 2015. [85] Jeremy Heaston, Dennis Hong, Ivette Morazzani, Ping Ren, and Gabriel Goldman. Strider: Self-excited tripedal dynamic experimental robot. In Proceedings 2007 IEEE International Conference on Robotics and Automation, pages 2776–2777. IEEE, 2007. [86] David Bevly, Steven Dubowsky, and Constantinos Mavroidis. A simplified cartesian- computed torque controller for highly geared systems and its application to an exper- imental climbing robot. J. Dyn. Sys., Meas., Control, 122(1):27–32, 2000. [87] Jan Faigl and Petr Čížek. Adaptive locomotion control of hexapod walking robot for traversing rough terrains with position feedback only. Robotics and Autonomous Systems, 116:136–147, 2019. [88] Marek Žák, Jaroslav Rozman, and František V Zbořil. Design and control of 7-dof omni-directional hexapod robot. Open Computer Science, 11(1):80–89, 2021. 84 [89] Zhen Liu, Hong-Chao Zhuang, Hai-Bo Gao, Zong-Quan Deng, and Liang Ding. Static force analysis of foot of electrically driven heavy-duty six-legged robot under tripod gait. Chinese Journal of Mechanical Engineering, 31(1):1–15, 2018. [90] Marko Bjelonic, Navinda Kottege, Timon Homberger, Paulo Borges, Philipp Beckerle, and Margarita Chli. Weaver: Hexapod robot for autonomous navigation on unstruc- tured terrain. Journal of Field Robotics, 35(7):1063–1079, 2018. [91] Markus Ryll, Giuseppe Muscio, Francesco Pierri, Elisabetta Cataldi, Gianluca Antonelli, Fabrizio Caccavale, and Antonio Franchi. 6d physical interaction with a fully actuated aerial robot. In 2017 IEEE International Conference on Robotics and Automation (ICRA), pages 5190–5195. IEEE, 2017. [92] Wataru Watanabe, Takeshi Kano, Shota Suzuki, and Akio Ishiguro. A decentralized control scheme for orchestrating versatile arm movements in ophiuroid omnidirectional locomotion. Journal of the Royal Society Interface, 9(66):102–109, 2012. [93] Sunil Pranit Lal, Koji Yamada, and Satoshi Endo. Emergent motion characteristics of a modular robot through genetic algorithm. In International Conference on Intelligent Computing, pages 225–234. Springer, 2008. [94] Masato Ishikawa, Yuki Minami, and Toshiharu Sugie. Development and control exper- iment of the trident snake robot. IEEE/ASME Transactions on Mechatronics, 15(1):9– 16, 2009. [95] Masato Ishikawa, Pascal Morin, and Claude Samson. Tracking control of the trident snake robot with the transverse function approach. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pages 4137–4143. IEEE, 2009. [96] JanuszJakubiak, KrzysztofTchoń, andMariuszJaniak. Motionplanningofthetrident snake robot: An endogenous configuration space approach. In ROMANSY 18 Robot Design, Dynamics and Control, pages 159–166. Springer, 2010. [97] Hiroaki Yamaguchi. Dynamical analysis of an undulatory wheeled locomotor: a trident steering walker. IFAC Proceedings Volumes, 45(22):157–164, 2012. [98] Auke Jan Ijspeert. Central pattern generators for locomotion control in animals and robots: a review. Neural networks, 21(4):642–653, 2008. [99] C Stefanini, S Orofino, L Manfredi, S Mintchev, S Marrazza, T Assaf, L Capantini, E Sinibaldi, S Grillner, P Wallén, et al. A novel autonomous, bioinspired swimming robot developed by neuroscientists and bioengineers. Bioinspiration & biomimetics, 7(2):025001, 2012. [100] Deniz Korkmaz, Gonca Ozmen Koca, Guoyuan Li, Cafer Bal, Mustafa Ay, and Zuhtu Hakan Akpolat. Locomotion control of a biomimetic robotic fish based on closed loop sensory feedback cpg model. Journal of Marine Engineering & Technology, pages 1–13, 2019. 85 [101] Sunil Pranit Lal, Koji Yamada, and Satoshi Endo. Studies on motion control of a modular robot using cellular automata. In Australasian Joint Conference on Artificial Intelligence, pages 689–698. Springer, 2006. [102] Zhong Shen, Junhan Na, and Zheng Wang. A biomimetic underwater soft robot inspired by cephalopod mollusc. IEEE Robotics and Automation Letters, 2(4):2217– 2223, 2017. [103] Maarja Kruusmaa, Paolo Fiorini, William Megill, Massimo de Vittorio, Otar Akanyeti, Francesco Visentin, Lily Chambers, Hadi El Daou, Maria-Camilla Fiazza, Jaas Ježov, et al. Filose for svenning: A flow sensing bioinspired robot. IEEE Robotics & Automa- tion Magazine, 21(3):51–62, 2014. [104] Henrik Olsson, Karl Johan Åström, Carlos Canudas De Wit, Magnus Gäfvert, and Pablo Lischinsky. Friction models and friction compensation. Eur. J. Control, 4(3):176–195, 1998. [105] Ettore Pennestrì, Valerio Rossi, Pietro Salvini, and Pier Paolo Valentini. Review and comparison of dry friction force models. Nonlinear dynamics, 83(4):1785–1801, 2016. [106] David L Hu, Jasmine Nirody, Terri Scott, and Michael J Shelley. The mechanics of slitheringlocomotion. Proceedings of the National Academy of Sciences,106(25):10081– 10085, 2009. [107] Hamidreza Marvi and David L Hu. Friction enhancement in concertina locomotion of snakes. Journal of the Royal Society Interface, 9(76):3067–3080, 2012. [108] F Heslot, T Baumberger, B Perrin, B Caroli, and C Caroli. Creep, stick-slip, and dry- friction dynamics: Experiments and a heuristic model. Physical review E, 49(6):4973, 1994. [109] Scott D Kelly and Richard M Murray. Geometric phases and robotic locomotion. Journal of Robotic Systems, 12(6):417–431, 1995. [110] Jim Ostrowski and Joel Burdick. The geometric mechanics of undulatory robotic locomotion. The international journal of robotics research, 17(7):683–701, 1998. [111] H Rastgar, HR Naeimi, and M Agheli. Characterization, validation, and stability analysis of maximized reachable workspace of radially symmetric hexapod machines. Mechanism and machine theory, 137:315–335, 2019. [112] Francesco Giorgio Serchi, Andrea Arienti, and Cecilia Laschi. Biomimetic vor- tex propulsion: toward the new paradigm of soft unmanned underwater vehicles. IEEE/ASME Transactions On Mechatronics, 18(2):484–493, 2012. 86

Abstract (if available)

Abstract The goal of this thesis is to provide a framework for considering Fluid-Structure Interactions (FSI) of Radially Symmetric (RS) crawling robots. We separated our investigation into two parts: examining FSI of surface-attached radial symmetries (our focus was sea star-inspired geometries) and developing a stable locomotion platform for moving through flow fields. To study fluid dynamics of RS structures in steady flow, an ATI-gamma force transducer system measured drag and lift forces on 3D-printed sea star and spherical dome models in a water channel facility. Experiments showed that sea star shapes generated downforce by diverting momentum upward through generation of vortices that created upwelling along the center-line plane. Also, downforce magnitude can be tuned by varying geometric parameters such as aspect ratio (AR). We applied the results of this work to a shape-morphing sea star-inspired silicone robot. A genetic algorithm optimized drag and lift of a 2 degree of freedom (DOF) and a 3 DOF robot. Current research efforts are underway for optimizing a 7 DOF system. In addition to the FSI studies, we created a tripedal crawling system to generalize and minimize the complexity of an RS locomotion device. A mathematical model that accurately captured the dynamics of the tripedal robot was developed and used in an image-feedback control scheme. Through simulations, we created a mapping of gait parameters to translation direction. Physical experiments demonstrated that the robot is robust to fluid force disturbances while path-following using a PI controller based on the gait map. Applications that can benefit from this research include RS robot cyclic control using our gait map strategy, surface mounted features requiring downforce production using our sea star designs, and Experimental Shape Optimization (ESO) for minimization of drag and lift forces using our silicone skin morphing robotic platform. We hope that this work will inspire others to explore RS robotics for further research and applications.

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Asset Metadata

Creator Hermes, Mark Anthony (author)

Core Title Traveling sea stars: hydrodynamic interactions and radially-symmetric motion strategies for biomimetic robot design

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2021-12

Publication Date 09/20/2021

Defense Date 08/20/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag evolutionary algorithm,experimental shape optimization,friction-driven motion,OAI-PMH Harvest,radially symmetric locomotion,sea star hydrodynamics,soft robotics

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Culbertson, Heather (committee chair), Kanso, Eva (committee chair), Luhar, Mitul (committee chair), Nguyen, Quan (committee chair)

Creator Email markherm@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15920209

Unique identifier UC15920209

Legacy Identifier etd-HermesMark-10079

Document Type Dissertation

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Rights Hermes, Mark Anthony

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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