DYNAMIC SOARING UAV GLIDERS by Jeffrey H. Koessler

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Authors Koessler, Jeffrey H.

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DYNAMIC SOARING UAV GLIDERS

Jeffrey H. Koessler

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF AEROSPACE & MECHANICAL ENGINEERING

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN AEROSPACE ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

2018

THEUNIVERSITY OF ARIZONA GRADUATE COLLEGE

Date: 01 May 2018

RIZONA

2 STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of the requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder.

SIGNED: Jeffrey H. Koessler

Acknowledgments

This network of mentors, colleagues, friends, and family is simply amazing!

Many thanks to...

...My advisor Prof. Fasel for enduring my arguments and crazy ideas about the dynamics of soaring. ...My committee members for time investment, positive feedback, and most of all: encouragement. ...The University of Arizona and AME department for a dynamic environment of discovery. ...Prof. Sanfelice for turning fireflies into synchronizing oscillators and pointing me to dynamic soaring. ...Prof. Gross for my first flight in real glider, acceleration pushing me down, and my lunch almost up. ...Quinton H, Andrew D, and Bear V for believing in me 10 years ago with letters of recommendation. ...Jini K and Janae G for keeping me progressing on track and through all the administrative hoops. ...My 21 students during 4 years of Senior Design soaring projects, and Nik K for “Go big or go home”. ...The Hybrid Dynamics & Controls Lab group: Malladi B, Sean A, Yuchun L, Jun C, Alex J, Pablo N, and Ryan J for teamwork and lively arguments over a multitude of projects, exams, and problem sets. ...Malladi B for getting us past written & oral quals through countless hours of practice exam sessions. ...Brian H for our 2 AM phone calls on HW sets and projects while our growing families slept unawares. ...Chad R, Alex J, and Chris L for collaborating on projects that really made lasting impressions on me. ...Co-workers Stephen W, Neal C, Tim M, Chris M, Chris W, Drew W, Jason F, Cody M, Cody T, Jon O, Paul M, Jake E , Mike T... and many others for enduring my countless persistent questions, comments, and interruptions... in particular Stephen, fate placing his office near mine, for bearing the brunt of it. ...Marc W for his excellent, and simultaneous, driving and videoing skills during daring flight launches. ...My parents and siblings for continually believing that after every milestone, something better is ahead. ...Our three kids for believing in their dad, and left to wonder what a normal family must be like. ...My dearest Stephanie for countless hours missed while on projects, test flights, research, and deadlines. ...Last but not in the least- Thanks to all the pillars and researchers of dynamic soaring, and now potential collaborators. Acknowledging the contributions of the past- and looking towards a dynamic future!

~ ~ many thanks to all of you ~ ~

To Trevor, Annika, and Gordon- our dear children who have tolerated a silly dad- Whatever paths your futures hold: life is a journey, with a destination.

To all my family who have given unending support and encouragement to press on- I couldn’t have done it without you- thanks for the boundless provisions.

To Stephanie- for allowing this crazy idea to be born 10 years ago, and seeing it through to fruition- I can’t wait for our next chapter in life to begin!

“My life is in the Master’s hands to purify and mould; When tested, tried, I’m satisfied I shall come forth as gold!”

Table of Contents

List of Figures 11

Abstract 15

Nomenclature 16

Chapter 1: Introduction to Dynamic Soaring 17

Chapter Introduction 17

Historical Perspective 17

Static Soaring 20

Dynamic Soaring 24

Flight Experiments 27

Motivation and Novel Contributions 31

Chapter 2: Dynamic Soaring Framework 33

Chapter Introduction 33

Aerodynamic Models 33

Wind Gradient Models 35

Dynamic Soaring Equations of Motion 40

Dynamic Soaring Energy 42

Chapter 3: Using Airspeed for Kinetic Energy 44

Chapter Introduction 44

Heuristic Assumptions Summary 44

HA1- A universal inertial reference does not exist- terrestrial earth is not uniquely “inertial” 45

HA2- An infinite choice of reference frames for EoM analyses and

simulations are valid 46

HA3- Appropriate choice of KE velocity reference applies to aircraft flying in all conditions 47

HA4- By definition Lift cannot influence airspeed, Drag always opposes airspeed 50

HA5- Aircraft cannot distinguish between locally equivalent spatial and

temporal gradients 52

HA6- KE computed from airspeed in a wind-fixed reference frame yields useful energy 54

HA7- Velocity gradients are responsible for useful energy gain in DS, not downwind turns 56

Consequences of Reference Frame Choice 57

C1- Source of Dynamic Soaring Energy 58

C2- Roles of Lift and Drag Forces in Energy Rates 58

C3- Exchange Between Potential Energy and Kinetic Energy Forms 59

C4- Kinetic Energy Rates as Used in Flight Control Algorithms 60

Conclusion 60

Chapter 4: Steady State Analysis 62

Chapter Introduction 62

Definition of Steady State Dynamic Soaring in an Airspeed Reference Context 62

Modeling Assumptions 63

Case 1: Steady State Descent in No Wind 64

Case 2: Steady State Descent in a Horizontal Linear Gradient of Horizontal

Wind 66

Case 3: Steady State Assent in a Vertical Linear Gradient of Horizontal Wind 68

Case 4: Steady State Descent in a Vertical Linear Gradient of Horizontal Wind 71

Case 5: Steady State Flight in Uniform Accelerating Wind 71

Equivalence in Dynamic Soaring Scenarios 73

Dynamic Soaring Sink Polar Surface 73

Chapter 5: Reverse Kinematics Simulation 75

Chapter Introduction 75

Dynamic Soaring Framework Review 75

Typical Dynamic Soaring Simulations 77

Overview of Reverse Kinematics Simulation 79

Three-Dimensional Trajectory Formulation 80

Reverse Kinematics Algorithm 81

Example: Inclined Circular Trajectory 83

Conclusion 86

Chapter 6: Conclusions and Future Research 88

Chapter Introduction 88

Summary of Novel Contributions 89

Future Research and Applications 90

Appendix A: Dynamic Soaring Reference Frame 92

References 99

List of Figures

Figure 1.1: Thermal and orographic lift 17

Figure 1.2: Rayleigh cycle for dynamic soaring 18

Figure 1.3: L-23 Super Blanik with NASA SENIOR ShWOOPIN team 19

Figure 1.4: 2008 Montague Cross Country Challenge teams 20

Figure 1.5: Typical tandem glider 21

Figure 1.6: SBXC glider sink polar 21

Figure 1.7: Sources of thermal lift 22

Figure 1.8: Sources of ridge lift 23

Figure 1.9: Sources of wave lift 23

Figure 1.10: Boundary layer wind profile 24

Figure 1.11: Albatross flight trajectory in boundary layer wind gradient 25

Figure 1.12: Circular hover 26

Figure 1.13: Snaking hover 26

Figure 1.14: Circular zoom 26

Figure 1.15: Circumnavigation 26

Figure 1.16: Valenta Weasel 2.1 m glider 28

Figure 1.17: Valenta Sharon 4.2 m glider 28

Figure 1.18: UAV architecture block diagram 29

Figure 1.19: Autopilot, sensors, telemetry, and control 29

Figure 1.20: Typical screenshot of mission planner GCS real-time telemetry 30

Figure 1.21: Arizona winds aloft 30

Figure 1.22: Prediction for ideal dynamic soaring conditions at test site 30

Figure 1.23: Sharon 4.2 m UAV glider 32

Figure 2.1: Air relative velocity and applied forces 34

Figure 2.2: Circular flow 38

Figure 2.3: Boundary layer wind profile 39

Figure 2.4: Horizontal wind speed vs. height 39

Figure 2.5: Horizontal wind gradient vs. height 39

Figure 2.6: Dynamic soaring forces 40

Figure 3.1: Air-relative velocity and applied forces 46

Figure 3.2: Four forces in flight 50

Figure 3.3: Dynamic soaring forces 53

Figure 4.1: Dynamic soaring forces 63

Figure 4.2a: Linear horizontal gradient (source) 66

Figure 4.2b: Linear horizontal gradient (sink) 66

Figure 4.2c: Radial horizontal gradient (source) 66

Figure 4.2d: Radial horizontal gradient (sink) 66

Figure 4.3: Vertical linear gradient of horizontal wind 68

Figure 4.4: Sink polar shift for heavier weight 70

Figure 4.5: Dynamic soaring sink polar surface 74

Figure 5.1: Air relative velocity and applied forces 76

Figure 5.2: Wind velocity vs height 78

Figure 5.3: Wind gradient vs height 78

Figure 5.4: Inclined circular trajectory 84

Figure 5.5: Bank angle vs trajectory position parameter for inclined circular

trajectory 85

Figure 5.6: Specific energy vs trajectory position parameter for inclined circular trajectory 85

Figure 5.7: Load factor vs trajectory position parameter for inclined circular trajectory 86

Figure A.1: Glider descending in calm wind 92

Figure A.2: Glider ascending through accelerating wind-fixed reference frame 93

Figure A.3: Glider ascending through accelerating wind-fixed reference frame

with addition of rotated DS reference frame notation (red) 94

Figure A.4: Glider descending in rotated DS reference frame 95

Figure A.5: Glider descending in simplified rotated DS reference frame 96

ABSTRACT

Dynamic soaring in the atmospheric boundary layer of a vertical variation of horizontal winds over arbitrary terrain offers small unmanned aerial vehicle gliders the ability to greatly expand mission operations while also extending endurance. Wandering albatrosses and other oceanic birds have provided the original evidence of the ability to exploit vertical wind gradients through their long distance circumnavigation flights using periodic maneuvers with cycles of upwind, crosswind, and downwind phases orchestrated to extract sufficient energy from the environment to allow perpetual flight. Dynamic soaring analysis presents additional challenges to the established theories of aerodynamic flight dynamics, due to the presence of wind vector fields that can vary their magnitude and direction in spatial and temporal dimensions. Preserving a consistency with typical flight analysis including static soaring, an air-relative wind-aligned / wind-fixed reference frame is proposed for dynamic soaring analysis, in particular for the computation of kinetic energy. Novel contributions of this present work include a logical progression of seven heuristic assumptions leading to a singular conclusion regarding the appropriate usage of airspeed for kinetic energy computation. Also, a concept of fundamental equivalence between spatial and temporal gradients experienced by a flight vehicle is presented, including the mechanism by which both generate the same accelerating reference frame and apparent dynamic soaring thrust. A novel reverse kinematics simulation is introduced, built on a parametric trajectory definition and analysis via Frenet-Serret equations. A series of dynamic soaring steady-state scenarios are presented and used to amplify the spatial and temporal gradient equivalence concept. The familiar glider sink polar curve is used in two novel ways: first, the curve shift procedures typically employed to account for changes in glider weight for uniform wind flight are proposed as also applicable to account for the apparent weight change during dynamic frame acceleration; second, the sink polar curve is transformed into a sink polar surface representing a family of curves with the additional dimension of weight inflation ratio. These novel insights and observations are intended to provide a solid foundation for present and future dynamic soaring analysis across a spectrum of interdisciplinary research.

Nomenclature

Symbols 퐴 = altitude parameter, exponential (--) 퐴푅 = aspect ratio (--) 푏 = wing span (m)

퐶퐷 = drag coefficient (--)

퐶퐷0 = parasitic drag coefficient (--)

퐶퐷푖 = induced drag coefficient (--)

퐶퐿 = lift coefficient (--) 퐷 = drag force (N) 푒 = Oswald’s efficiency factor 푔 = gravitational acceleration (m/s2) 퐿 = lift force (N) 푚 = mass (kg) 푛 = load factor (gees) 푆 = planar wing area (m2) 푉 = aircraft speed (m/s) 푊 = wind speed (m/s) 푥 = terrestrial horizontal position (m) 푦 = terrestrial lateral position (m) 푧 = terrestrial vertical position (m) 훾 = flight path angle (radians) 휌 = air density (kg/m3) 휙 = bank angle (radians) 휓 = heading angle (radians)

Subscripts 푎 = air reference 푟푒푓 = reference parameter 푡 = terrestrial reference

Chapter 1: Introduction to Dynamic Soaring

Chapter Introduction This chapter introduces the general concept of soaring and the human fascination with bird soaring flight. Next, the notions of static and dynamic soaring are introduced, along with the main distinctions between the two flight methods. Sources of static soaring lift are presented, along with methods of exploiting these energy sources by manned gliders. Following this is an introductory discussion on dynamic soaring fundamentals and the atmospheric boundary layer wind gradient. Then there is a summary of the challenge of dynamic soaring flight experiments. In the concluding section of this chapter a motivation for this present research is presented, including a catalogue of novel contributions to the dynamic soaring literature.

Historical Perspective For much of recorded history humans have been fascinated with various soaring birds and their ability to sustain flight with apparently minimal effort. Many land-based birds, including common crows, hawks, and buzzards exploit soaring through convective thermals [11], [72], [53], [97]. Other birds utilize ridges and other terrain features in conjunction with horizontal wind flow to extract energy as they transit through local steady-state updrafts in orographic lift [107], [18], illustrated in Figure 1.1. These so-called static soaring techniques require the birds to maintain nearly constant bank and pitch angles, where the rate of energy increase is based primarily on position and orientation within the flow field.

Figure 1.1: Thermal and orographic lift [118]

In contrast, dynamic soaring techniques exploit spatial and temporal variations in the flow field, where the participating birds fly a more complicated trajectory across the flow gradient in such a manner that total energy, comprised of altitude-based potential energy and airspeed-based kinetic energy, is increased [11], [32], [44], [52], [56], and [79]. John Strutt (Lord Baron Rayleigh) was one of the earliest scientists to speculate on the possibility of there being sufficient energy- harvesting potential in an atmospheric boundary layer shear gradient to allow perpetual flight through dynamic soaring cycles. Rayleigh first proposed, in an 1883 Nature journal article [72], a wind structure that we now call a vertical wind gradient as a possible driver for birds’ soaring flight in the complete absence of vertical wind velocity components. He is often regarded as focusing on a step-discontinuity model, where there are two air masses with relative velocity between them, separated by a thin but distinct separation layer, while a bird flies a modified race- track trajectory such that crossings through the separation occur in directions that increase airspeed.

Figure 1.2: Rayleigh cycle for dynamic soaring [12]

While Figure 1.2 includes the addition of a linear wind shear gradient, nevertheless it illustrates the features of the Rayleigh Cycle flight trajectory. The simplification of atmospheric wind gradients into a step discontinuity model was not envisioned by Lord Rayleigh as a hypothesis of actual wind structure, but only introduced as a means to allow relatively simple aerodynamic analysis. Accomplishing this, he goes on to suggest a more plausible wind gradient theory [72]:

In nature there is of course no such abrupt transition as we have just now supposed, but there is usually a continuous increase of velocity with height.

Efforts to accomplish manned dynamic soaring have had limited success, the most notable effort in this regard that of Gordon [60], compiling 88 test sorties in an L-23 Super Blanik glider operated at Rogers dry lakebed in Edwards AFB (Figure 1.3). As will be developed in chapter 2, the atmospheric boundary layer wind gradients have their highest magnitude close to the ground, the same zone where a dynamic soaring glider achieves it highest velocities. High speeds coupled with proximity to surface terrain while simultaneously banking more than 45 degrees in roll and pulling over 4 g’s create a dangerous combination for manned aircraft and their human pilots.

Figure 1.3: L-23 Super Blanik with NASA SENIOR ShWOOPIN team [60]

Advances in control systems, Global Positioning System (GPS), embedded computing, and composite structures have accelerated the development of unmanned air vehicles (UAV) during the last two decades. Collaboration between Allen [77], [48] and Edwards [14], [68], [53], [13] initiated the current revolution in autonomous soaring, culminating in a landmark performance at the 2008 Montague Cross-Country Challenge, a premier annual radio control glider cross country soaring event in northern California. Using a 4.3 m SBXC glider outfitted with a Piccolo II autopilot system supplementing the customary radio control equipment, Edwards and the ALOFT team placed third overall among 12 teams of seasoned model aircraft glider pilot teams, many of whom were full scale pilots as well. They combined the physical implementation of autopilot hardware, a host of sensors, and synthesis of existing UAV inner-loop stability and control

19 algorithms with novel thermal sensing / center estimation / decision logic to fully exploit cross country flight. The ALOFT team aircraft is visible in Figure 1.4, third from right:

Figure 1.4: 2008 Montague Cross Country Challenge teams

Allen and Edwards focused their attention on autonomous thermal soaring, as have other researchers [64], [19], [95], [96], [100], [106]. Conversely Sachs, building on a European heritage of glider soaring research, has generated numerous papers on dynamic soaring over a span of 30 years [56], [65], [44], [41], [25], [110], [119]. Boslough [11] suggested dynamic soaring as a means for creating an array of distributed mobile sensors aboard UAV platforms. This seminal report issued by Sandia National Laboratories in 2002 firmly cemented the concept of using dynamic soaring as a primary means of autonomous aircraft propulsion. Lissaman [79] and Sachs [56] both developed theoretical frameworks for estimating the minimum wind shear gradient necessary to sustain dynamic soaring flight for particular classes of birds and air vehicles.

Static Soaring A typical manned tandem glider is shown in Figure 1.5. The common characteristics of efficient gliders are long slender wings with high span-to-chord aspect ratios (AR), low drag construction, and lightweight structure. Several glider performance measures can be verified through instrumented flights in still air, including vertical sink rates at various steady-state airspeeds. Edwards [68] documents how a sink polar curve is generated from this sink rate data, along with curve-fitting techniques to create a 2nd order approximation as illustrated in Figure 1.6, below. From such a sink polar curve one can determine the maximum lift-to-drag (L/D) ratio and

Figure 1.5: Typical tandem glider minimum sink rate for the glider under consideration. Any point on the sink polar represents a combination of sink rate and horizontal flight speed. Dividing the vertical sink rate by the horizontal velocity is equivalent to the slope of a line drawn from the (0,0) origin of the sink polar

Figure 1.6: SBXC glider sink polar [68] graph to the point of interest on the sink polar (in a “rise over run” sense), taking care to ensure consistent units are used for both vertical and horizontal velocities. Often the goal is to find the point on the sink polar that maximizes the L/D ratio, which is equivalent to minimizing the glide

21 flight path angle below the horizon. One simply finds the point on the sink polar curve that has the least negative angle relative to lines drawn from the graph origin to points on the graph. This happens to also be the unique line that is both tangent to the sink polar curve and also contains the origin (0,0). Another point of interest on the sink polar is the minimum sink rate. This is easily determined by observing the highest point on the curve, indicating the combination of horizontal velocity and sink rate at which the minimum sink rate occurs. The key element of static soaring is the presence of a positive vertical component of air mass velocity. When this positive vertical air velocity exceeds the steady-state negative sink rate of the glider as a function of its relative horizontal airspeed, the glider is said to be soaring, as opposed to simply gliding, which is stable descending flight absent thrust-generating internal propulsion. There are several naturally occurring sources of these vertical air mass velocities, illustrated in Figures 1.7 - 1.9 [120]. The first of these depicts thermal lift sources, often due to uneven solar heating of the surface, creating

Figure 1.7: Sources of thermal lift [120] rising columns or bubbles of air called thermals, further discussed in [32], [80], [9], and [100]. The presence of cumulous clouds can serve as a visual indicator of these thermal locations, and research is ongoing into autonomous vision-based recognition of these lift indicators [88]. Figure

1.8 depicts a hill deflecting a horizontal wind field into a wind with positive vertical components. This type of lifting air mass is often called orographic lift in meteorology and ridge lift in glider

Figure 1.8: Sources of ridge lift [120] soaring [65]. A third type of air mass with positive vertical velocity component is shown in Figure 1.9, called wave lift. While this third type is also associated with a ridge, the distinction is that the

Figure 1.9: Sources of wave lift [120]

23 vertical velocity components are associated with locations in the lee (downwind) side of the ridge, including primary/secondary/tertiary harmonic waves. Exploitation of wave lift is often reserved for experienced glider pilots, while novice pilots exploit thermals and intermediate glider pilots can utilize ridge lift.

Dynamic Soaring While static soaring techniques utilize upward rising air current components to overcome a glider’s steady-state sink rate, dynamic soaring relies on the presence of an air speed gradient, such that the glider experiences an increase in local wind velocity. Chapters 3 and 4 contain discussions regarding the equivalence of this positive wind rate coming as a function of either transitioning through a spatial velocity gradient, versus experiencing the influence of a temporal velocity gradient. Focusing on spatial wind velocity gradients, consider a representation of the atmospheric boundary layer as depicted in Fig. 1.10. This figure shows a purely horizontal wind in the +푥

Figure 1.10: Boundary layer wind profile direction, where the velocity magnitude 푊푥 increases with altitude 푧 above the surface. Several specific wind shear models are discussed in Chapter 2, however they all share the properties of having only horizontal wind velocity components and comprised of monotonically increasing, or discrete steps of, wind speed as a function of height above the surface. The prime example of flying via dynamic soaring in the atmospheric boundary layer comes from nature: the wandering albatross has been documented to fly well over 1,000 km without stopping (Fig. 1.11).

Figure 1.11: Albatross flight trajectory in boundary layer wind gradient

Albatross flight has been the focus of several foundational works in dynamic soaring, including Sachs [41], [44], Barnes [52], [107], and most recently Bonnin [78]. While not a requirement of dynamic soaring that the maneuver trajectory form a complete cycle, it is evident that in order to continue exploiting the wind gradient field in sustained soaring flight the birds often complete periodic cycles of bank, pitch, and heading such that the elevation minimums and maximums are within the useful elements of the boundary layer. While there is abundant literature offering theories and simulations of how wandering albatrosses fly thousands of miles over the southern latitudes circumnavigating Antarctica, Sachs et al [41] present physical flight data from multiple albatross flights using a proprietary hi-fidelity GPS data-logging system attached to the albatrosses in-situ. This approach validates the theory that albatrosses are efficient at exploiting dynamic soaring for both the upwind ascent and downwind descent phases of the flight cycles. The data is of sufficient resolution that detailed cycle geometry is reconstructed, allowing initial theoretical frameworks and simulations to be updated and validated. Barnes [52] extensively catalogued and analyzed several such maneuvers that the albatrosses can utilize for hovering and also transiting downwind, across, and against the prevailing wind, with four examples graphically represented in Figures 1.12 – 1.15. These are classified into two main maneuvers types: spiral and snaking. The spiral maneuvers include a zooming circular turn, circumnavigation, hover circle, and upwind snaking maneuvers include the crosswind snake, snaking hover, and upwind snake, which all show a reversal of heading angle rate throughout the cycles.

Figure 1.12: Circular hover [52] Figure 1.13: Snaking hover [52]

Figure 1.14: Circular zoom [52] Figure 1.15: Circumnavigation [52]

So far this discussion has involved winds in one direction only, the magnitude varying only with height. A special case of another spatial gradient field that has relatively little representation in the dynamic soaring literature is a rotating wind gradient field. Grenestedt et al [26] investigated a relationship between the wind field exponent and the ability of a dragless glider to extract energy. Willoughby [67] presents a one-dimensional model of tangential winds within the hurricane eye:

푟 푛 푊푡 = 푊푚푎푥 ( ) (1.1) 푅푚푎푥

where 푛 is the wind field exponent, 푊푡 is the tangential wind velocity for a given radial location

푟, normalized by the physical maximum eye wall radius 푅푚푎푥 and proportional to the maximum tangential wind velocity 푊푚푎푥. Chapter 2 contains a brief discussion on when this wind shear model degenerates to a simple rotating wind field with no actual wind shear present, highlighting a scenario where appearances of a shear gradient can be misleading.

In contrast to examining spatial gradients, one can consider the effects of temporal velocity gradients, where wind fields change in magnitude as a function of time. One well studied example is the gust-induced wind gradient, whereby a steady-state wind magnitude is perturbed by a time- varying pulse of wind. These gusts may be approximated as sinusoidal pulses, and several authors have generated algorithms for extracting energy from such sources, for example Langelaan [49], Depenbusch [08], and Patel [33].

Flight Experiments While the dynamic soaring literature is ripe with theoretical analysis, simulations, and optimization techniques, there is a profound lack of actual experimental test flight results of dynamic soaring over flat terrain. Sukumar and Selig propose that it is possible under certain conditions to achieve dynamic soaring over flat ground, and predict actual feasibility with pilot-in-the-loop simulations [28]. Bird and Langelaan have combined a wind field estimating algorithm [80] with optimal trajectory generation [31] and path following control [5], [10] to “close the loop” on dynamic soaring [16]. However, Barnes acknowledges the practical difficulty in dynamic soaring over featureless terrain due to lack of a reliable and sufficient wind gradient [107].

The Fédération Aéronautique Internationale (FAI) encourages developments in aeronautics through the establishment of a record documenting system that catalogues and publishes specific performance achievements to give undisputed claim to the record holders. Regarding unmanned aircraft, these records include the F3H radio controlled soaring cross country racing category, using a 5 kg, 4 m class of glider. For nearly 30 years the record for distance in a straight line to

27 declared goal was held by Joe Wurts [2] with a distance of 226.4 km. Recently, John Ellias set the current record of 301.4 km, with a description of the flight experience documented in [1]. The FAI code currently prohibits autonomous systems in its competitions, therefore radio control must be exclusively used with a pilot in command at all times. While these records clearly demonstrate long distance cross country flights via thermal exploitation, the potential exists for much longer distances utilizing dynamic soaring in extreme wind conditions, predicated on autonomous control. Our own experiences in attempting dynamic soaring flight experiments have yielded promising results, while fully autonomous dynamic soaring flights are still on the horizon.

Two high performance gliders have been converted into research aircraft complete with autopilots, instrumentation, telemetry, and electric propulsion systems: a 2.1 m wingspan Weasel glider and a 4.2 m wingspan Sharon glider, both manufactured by Valenta Models in the Czech Republic (Figs. 1.16 and 1.17). These radio controlled gliders were converted into sophisticated research UAVs through the incorporation of 3DR Pixhawk flight computer autopilots along with associated

Figure 1.16: Valenta Weasel 2.1 m glider Figure 1.17: Valenta Sharon 4.2 m glider sensor, actuator, and telemetry systems shown in the system architecture block diagram and corresponding physical components (Figs. 1.18 and 1.19). Also, both gliders have custom electric propulsion systems installed to enable repeated flight operations without needing to land and be re-launched as usual gliders. With physical architecture in place, both UAVs underwent procedures to calibrate the various sensors: pitot-static air data, barometer, GPS, compass, accelerometer and gyro triads, power system voltage/current sensors, and servo position programing. In addition, proper ballasting to achieve desired wing loading and CG placement was performed.

Figure 1.18: UAV architecture block diagram Figure 1.19: Autopilot, sensors, telemetry, and control

A key modification to both the Weasel and Sharon gliders is the incorporation of electric propulsion systems. These systems include 3-phase brushless motors, electronic speed controllers, lithium polymer batteries, and unique motor mounts. In the Weasel 2.1 m glider a nose mount was incorporated, wherein a custom internal motor to fuselage interface was designed and modeled in PTC Creo, then grown via additive manufacturing in a 3D printer. For the Sharon 4.2 m glider the motor is installed in a custom wing-mounted pylon, also modeled in Creo and grown via SLS process in nylon and stainless steel. The pylon mount complicates the installation of the Sharon’s ESC, as the brushless motor’s 3-phase cables are routed inside the pylon, through the wing, and into the fuselage via custom blind mate high-current connectors. The advantage of such a pylon on this larger research aircraft is the ease in removal of propulsion system for conversion to an efficient and aerodynamically clean pure glider. After the glider flight computers have their control loop gains individually tuned for pitch, roll, and yaw during initial flights in calm winds, a second series of flights for each UAV was also performed to characterize the nominal sink rate performance. Telemetry data was collected for various steady-state gliding flights performed at gradually increasing air speeds to develop the sink polar for each aircraft. Initial dynamic soaring flights conducted to date have been under pilot command using the autopilots as flight stabilizers and data telemetry units, with data displayed on the Mission Planner ground control system (GCS) as shown in Figure 1.20. Figure 1.21 shows the average Arizona winds aloft and can be useful to identify flight test locations, while Figure 1.22 illustrates a typical high-wind forecast that would

Figure 1.20: Typical screenshot of mission planner GCS real-time telemetry be ideal for dynamic soaring flight. The latest flight data indicated regions of energy gain attributed to dynamic soaring, while the overall flight trajectories were less than energy neutral. Maximum flight airspeeds were observed on the order of 120 miles/hr (190 km/hr), while achieving a load factor over 6 g.

Figure 1.21: Arizona winds aloft Figure 1.22: Prediction for ideal dynamic soaring conditions at test site

Motivation and Novel Contributions This present research is motivated by a desire to reestablish a sound foundation of kinematics and dynamics to the art of soaring through an atmosphere in motion. Recent advances in trajectory optimization, wind field estimation / mapping, and genetic learning algorithms have brought autonomous dynamic soaring to the cusp of practical implementation. However, it appears that a fundamental understanding of dynamic soaring mechanisms is not universal. It is the sincerest intent of this author to solidify and distill complex dynamics by introducing unique observations and techniques in the following five novel contribution areas:

1) A logical progression of seven heuristic assumptions leading to a singular conclusion: dynamic soaring kinetic energy must always be computed using airspeed as the reference.

2) The equivalence of spatial and temporal gradients as perceived by the aircraft and how an aircraft cannot distinguish between spatial and temporal gradients as it only senses wind velocities and wind rates.

3) Steady state dynamic soaring analysis generalization with application to novel spatial and temporal gradients, presenting mathematical examples of equivalence between the spatial and temporal gradients.

4) DynaSoar reverse kinematics simulation for analysis of explicit parametric dynamic soaring trajectories in Frenet-Serret frame, without requiring input scheduling or aircraft control algorithms.

5) Unification of static and dynamic soaring mechanics through the development of a novel rotated Dynamic Soaring Reference Frame (DSRF), whereby it is shown that unpowered gliders flying in steady-state always fly “downhill”, regardless of whether utilizing static or dynamic soaring.

Figure 1.23: Sharon 4.2 m UAV glider

Chapter 2: Dynamic Soaring Framework

Chapter Introduction This chapter begins with a discussion on the mathematical modeling of gliding aircraft, and the simplifying assumptions used in formulating these aerodynamic models. Next is a section describing the various wind shear models used in the dynamic soaring literature. Then the resulting Equations of Motion (EoM) will be formulated, including novel perspectives on the interpretation of accelerating reference frames, non-uniqueness of the terrestrial inertial frame, and inclusion of so-called fictitious forces. The adoption of an air-relative, wind-fixed (therefore not inertial) reference frame is proposed and used exclusively throughout this work.

Aerodynamic Models A point-mass model reduces the complexity of our formulation, whereby the mass of the glider is assumed to exist at a singular point (i.e. the center of gravity or CG), and therefore inertia terms are zero. While care must be exercised to ensure that roll rates and roll accelerations are not beyond reasonable values for the aircraft under consideration, the literature is in general consensus (for example [28], [52], [24], and [63]) that this is a valid simplification given the moderate time scales and smooth maneuver rates of dynamic soaring. Also, the customary flat earth model is adopted as the dynamic soaring maneuvers occur in localized zones. Additionally, there is a no side-slip (훽 ≈ 0) assumption, since as noted in [69] this can be considered an effect of on-board control system for low level stability and yaw control. Lift (퐿) and drag (퐷) are computed per now-familiar expressions:

1 퐿 = 휌푆퐶 푉2 (2.1) 2 퐿 푎

1 퐷 = 휌푆퐶 푉2 (2.2) 2 퐷 푎

33 where 휌 is the local air density, 푆 is the reference area, and 푉푎 denotes airspeed. In this present context the drag coefficient is assumed to be a function of generated lift, where the drag coefficient

퐶퐷 is related to lift coefficient 퐶퐿 by the quadratic relation:

퐶2 퐶 = 퐶 + 퐿 (2.3) 퐷 퐷0 휋푒퐴푅

In equation (2.3) 퐶퐷0 is the zero-lift drag coefficient, 푒 the Oswald’s efficiency, and 퐴푅 the aspect ratio of the aircraft. With the addition of aircraft mass 푚 and gravitational acceleration 푔, Eqs. (2.1) - (2.3) completely define an aerodynamic model of the point-mass glider, and could be used to accurately generate a theoretical sink polar to illustrate specific glider performance (See Fig. 1.6 for an experimentally-derived sink polar). While some researchers such as Okamoto et al [30] and Akhtar et al [31] use a more complex relationships between 퐶퐷 and 퐶퐿 for a better fit to a specific model, the net effect of Eq. (2.3) is to remove a degree of freedom from the equations of motion, and that is accomplished regardless of specific form. This gliding aircraft model is not presented as a novel construct, but rather as a reference to the most common equations used in the dynamic soaring literature.

Figure 2.1 illustrates the bank angle 휙, the air-relative flight path angle 훾푎, and the air-relative heading angle 휓푎, along with the wind and terrestrial coordinate definitions.

Figure 2.1: Air relative velocity and applied forces [93]

Wind Gradient Models Throughout the literature of dynamic soaring there appear a variety of wind shear models, including simple step discontinuity shear and linear shear models, and more sophisticated exponential shear and logarithmic shear models. The simpler models were employed in early discussions proposing the as-then unverified existence of dynamic soaring as a soaring mechanism of birds [72]. Later, the more sophisticated non-linear models were created to better approximate observed conditions in the atmospheric boundary layer over land [7], and also over an undulating ocean surface [4].

Borrowing construction from [12], we define the aircraft terrestrial position by vector 푃⃗ as:

푥푡

푃⃗ = [푦푡] (2.4)

푧푡 where Lawrance’s usual subscript 푖 has been replaced by 푡 to emphasize that the terrestrial frame of reference is only one of virtually infinite inertial reference frames possible, a topic discussed in more detail in Chapter 3. In anticipation of the exponential and logarithmic shear model introduction, consider the general state of wind 푊⃗⃗⃗ as observed by an aircraft at its terrestrial position 푃⃗ :

푊푥

푊⃗⃗⃗ = [푊푦], (2.5)

푊푧 where the subscripts 푥, 푦, and 푧 represent the orthogonal components of wind 푊⃗⃗⃗ . Then there exists a wind Jacobian matrix denoted 푱푊 and defined by the spatial gradients as:

휕푊푥 휕푊푥 휕푊푥 휕푥 휕푦 휕푧 휕푊 휕푊 휕푊 푱 = 푦 푦 푦 (2.6) 푊 휕푥 휕푦 휕푧 휕푊푧 휕푊푧 휕푊푧 [ 휕푥 휕푦 휕푧 ]

Since 푱푊 will only have non-zero terms when there exist spatial variations in wind, in order for an aircraft or bird to experience these variations in wind speed it needs to possess motion through the gradient, as evidenced in the scalar chain rule example for a vertical variation in horizontal wind:

휕푊 휕푊 휕푃 휕푊 푊̇ = 푥 = 푥 푧 = 푥 푃⃗ ̇ , (2.7) 푥 휕푡 휕푧 휕푡 휕푧 푧 with general matrix form:

휕푊 휕푊 휕푊 푥 푥 푥 푥̇푡 휕푥 휕푦 휕푧 휕푊푦 휕푊푦 휕푊푦 푊̇ = 푦̇ = 푱 푃⃗ ̇ (2.8) 푃 휕푥 휕푦 휕푧 푡 푊 휕푊푧 휕푊푧 휕푊푧 [ 휕푥 휕푦 휕푧 ] [푧푡̇ ]

Bower [15] uses a form similar to equation (2.7) but also adds a temporal component:

휕푊푥 휕푊푥 휕푊푥 푥̇푡 푊̇푥 휕푥 휕푦 휕푧 휕푊푦 휕푊푦 휕푊푦 ̇ 푊̇ = 푦̇ + 푊̇ = 푱 푃⃗ + 푊̇ , (2.9) 푃 휕푥 휕푦 휕푧 푡 푦 푊 휕푊푧 휕푊푧 휕푊푧 [ 휕푥 휕푦 휕푧 ] [푧푡̇ ] [푊̇푧 ] where this present author adds the subscript 푃 to the left side term to avoid confusion between the total wind acceleration seen at the aircraft point 푃⃗ 푡 and a general temporal wind acceleration as a function of time only, expressed in the far-right term.

The wind gradient description in equation (2.8) is more general than necessary for typical dynamic soaring analysis in an atmospheric boundary layer. While Chapter 4 explores steady state behavior

36 in these complex wind gradients, it is customary to reduce the nine terms in 푱푊 down to a single scalar term as illustrated in the following four wind field models describing horizontal wind velocity as a function of height only.

0 ∶ 푧 < 푧푟푒푓 Step discontinuity: 푊푥 = { } (2.10) 푊푟푒푓 : 푧 ≥ 푧푟푒푓

푑푊 푑2푊 Linear shear: 푊 = 푥 푧 ∶ 푥 = 0 (2.11) 푥 푑푧 푑푧2

퐴 −( )푧 푧푟푒푓 Exponential shear: 푊푥 = 푊푟푒푓 (1 − 푒 ) (2.12)

log(푧/푧0) Logarithmic shear: 푊푥 = 푊푟푒푓 (2.13) log(푧푟푒푓/푧0)

One additional wind profile model, representing circular flow within a hurricane and valid from the center to just inside hurricane wall, is the Willoughby [67] model explored by Grenestedt et al in [26]:

푟 푛 Rotating shear: 푊푡 = 푊푚푎푥 ( ) : 0 ≤ 푟 ≤ 푟푤푎푙푙 (2.14) 푅푚푎푥

where 푊푡 is the tangential wind velocity for a given radial location 푟, normalized by the physical maximum eye wall radius 푅푚푎푥 and proportional to the maximum tangential wind velocity 푊푚푎푥. The wind field exponent 푛 is a shaping parameter that is at the heart of the analysis in [26]. Using 2-D notation for wind gradients, as we note that the model in equation (2.14) is independent of 푧, we have the following expression for the special case of 푛 = 1:

푊푥 푦 푊푚푎푥 2 2 푊푥,푦 = [ ] = ( ) [ ] ∶ √푥 + 푦 ≤ 푟푚푎푥 (2.15) 푅푚푎푥 푊푦 푥

A representation of such a circular flow is given in figure 2.2 below. From equations (2.14) and

Figure 2.2: Circular flow

(2.15) it is clear that this flow, with 푛 = 1, contains no shearing winds in a physical sense, rather only winds that are rotating together as a single mass- the rotational analogue of a uniform wind which we understand more intuitively to have no physical shear. While Grenestedt et al prove that there is no dynamic soaring energy gain possible for 푛 = 1 through a closed loop integration of the air-relative kinetic energy rate and application of Stokes’ Theorem, the resultant lack of dynamic energy extraction should come as no surprise from inspection of the wind field itself: only when the rotating wind field exponent 푛 ≠ 1 will the flow contain actual shear between virtual vertical cylinders of uniform velocity tangential wind elements.

For the present one-dimensional gradient dynamic soaring research the logarithmic profile is suitable since it is most applicable to measurements near the surface of the earth as suggested by Stull [17]. However, in most of the analyses the gradient is left in symbolic form, and any gradient model can be used interchangeably. One-dimensional wind gradients are often depicted as per Figure 2.3, below. While such a representation facilitates an understanding of the spatial distribution of horizontal wind speeds, it actually hides an important aspect of the profile- the

38 gradient as a function of altitude. When the wind profile is presented in a form with the independent variable across the horizontal axis, in this case height 푧, it becomes very apparent where the gradient is the largest: when height 푧 approaches zero, that is, closest to the ground (Figs. 2.4 and 2.5).

Figure 2.3: Boundary layer wind profile

In Chapter 1 the manned glider flight experimental work of Gordon [60] was discussed; this present examination of the typical atmospheric boundary layer wind speed and gradient vs. height clearly illustrates the origin of risks involved with exploiting the low-altitude zone where the highest gradients exist.

Figure 2.4: Horizontal wind speed vs. height Figure 2.5: Horizontal wind gradient vs. height

Dynamic Soaring Equations of Motion Using figure 2.1 above for the three dimensional view of a glider in wind field, and also the longitudinal free body diagram below (Fig. 2.6), the equations of motion are derived below.

Figure 2.6: Dynamic soaring forces

With a wind-aligned / wind-fixed formulation, we can form the usual gliding equations as resulting from Newton’s second law and summing up forces for 푉푎̇ , 훾푎̇ , and 휓̇푎:

̇ ̇ 푚푉푎 = −퐷 − 푚푔 sin 훾푎 + 푚푊푉푎 (2.16)

̇ 푚푉푎훾푎̇ = 퐿 cos 휙 − 푚푔 cos 훾푎 + 푚푊훾푎 (2.17)

̇ ̇ 푚푉푎 cos 훾푎 휓푎 = 퐿 sin 휙 + 푚푊휓푎 (2.18)

Note that these last three equations contain wind acceleration terms rather than wind gradient terms, a deliberate emphasis of the notion that the aircraft or bird cannot sense the difference between spatial and temporal wind accelerations at its singular position in space. This concept is discussed further in Chapters 3 and 4. For a construction specific to an atmospheric boundary layer gradient, that is a vertical variation of horizontal wind speeds in +푥 directrion, we have:

푑푧 = 푉 sin 훾 (2.19) 푑푡 푎 푎

휕푊 푊̇ = − 푥 푉 sin 훾 cos 훾 cos 휓 (2.20) 푉푎 휕푧 푎 푎 푎 푎

휕푊 푊̇ = 푥 푉 sin2 훾 cos 휓 (2.21) 훾푎 휕푧 푎 푎 푎

휕푊 푊̇ = − 푥 푉 sin 훾 sin 휓 (2.22) 휓푎 휕푧 푎 푎 푎

And then substituting (2.20) – (2.22) into (2.16) – (2.18) we have:

휕푊 푚푉̇ = −퐷 − 푚푔 sin 훾 − 푚 푥 푉 sin 훾 cos 훾 cos 휓 (2.23) 푎 푎 휕푧 푎 푎 푎 푎

휕푊 푚푉 훾̇ = 퐿 cos 휙 − 푚푔 cos 훾 + 푚 푥 푉 sin2 훾 cos 휓 (2.24) 푎 푎 푎 휕푧 푎 푎 푎

휕푊 푚푉 cos 훾 휓̇ = 퐿 sin 휙 − 푚 푥 푉 sin 훾 sin 휓 (2.25) 푎 푎 푎 휕푧 푎 푎 푎

Squaring and adding equations (2.24) and (2.25), one obtains an expression of lift L as:

휕푊 2 퐿2 = (푚푉 훾̇ + 푚푔 cos 훾 − 푚 푥 푉 sin2 훾 cos 휓 ) + 푎 푎 푎 휕푧 푎 푎 푎

휕푊 2 ( 푚푉 cos 훾 휓̇ + 푚 푥 푉 sin 훾 sin 휓 ) (2.26) 푎 푎 푎 휕푧 푎 푎 푎

Combining these into convenient expressions for bank angle 휙 and load factor 푛 per [29]:

휕푊 휓̇ + 푥 tan 훾 sin 휓 푎 휕푧 푎 푎 휙 = arctan [ 휕푊 ] (2.27) (훾̇ ⁄cos 훾 )+(푔⁄푉 )+ 푥 sin 훾 tan 훾 cos 휓 푎 푎 푎 휕푧 푎 푎 푎

휕푊푥 2 (푉푎⁄푔)훾̇푎+cos 훾푎+( 푉푎⁄푔) sin 훾푎 cos 휓푎 푛 = 휕푧 (2.28) cos 휙

Dynamic Soaring Energy

The total energy 푇퐸 of an aircraft is computed as a sum of potential and kinetic energies, notionally described by:

푇퐸 = 푃퐸 + 퐾퐸 (2.29)

Potential energy 푃퐸 is due to the conservative gravitational potential 푔 and is function of height

푧. Kinetic energy 퐾퐸 is a function of velocity 푉푎. Chapter 3 contains a thorough discussion on why the kinetic energy velocity term should be based on airspeed, and not based on terrestrial frame velocity. It is convenient to focus on specific energy through division of all terms by weight 푚푔, such that 푇퐸, 푃퐸, and 퐾퐸 all correspond to energy heights:

푉2 푇퐸 = (푧 − 푧 ) + 푎 (2.30) 0 2푔

In (2.30) we can see that each term has the units of length, whereby we can consider specific energy to represent a height, independent of the mass involved. The specific energy rate is given by taking the time derivative of (2.30), yielding:

푉 푉̇ 푇퐸̇ = 푧̇ + 푎 푎 (2.31) 푔

It is clear that the first term on the right hand side of (2.31) is simply the vertical velocity (i.e. climb or sink rate). The second right hand side term is slightly more complex- it can be viewed as a product of airspeed and airspeed acceleration (the later expressed in gees through division by 푔). Chapter 3 sheds light on situations where specific energy 푇퐸 and specific energy rate 푇퐸̇ play critical roles in feedback control systems for autonomous soaring gliders. Ignoring drag for the

42 moment, and considering specific energy 푇퐸 to be constant, equation (2.32) reveals the familiar interaction between height and airspeed:

∆(푉2) 0 = ∆푧 + 푎 (2.32) 2푔

An increase in altitude corresponds to a proportional decrease in the square of airspeed, and vice- versa.

Chapter 3: Using Airspeed for Kinetic Energy

Chapter Introduction Over the course of the last decade and a half a discrepancy has appeared in the literature regarding whether one should utilize terrestrial velocity or local airspeed for the computation of kinetic

1 2 energy: 퐾퐸 = 2푚푉 . In most cases this discrepancy is buried in the theoretical sections of conference papers, PhD dissertations, and journal articles. Most recently this disagreement has reached the point where two of the most prominent authorities on dynamic soaring research (G. Sachs and J. Barnes) have published opposing conference papers on the topic. The most direct paper [110] (Sachs & Grüter, 2017) is singularly focused on the dynamic soaring kinetic energy topic and is a pointed rebuttal to [107] (Barnes, 2015). This chapter aims to provide a clear presentation of the kinetic energy velocity reference frame definition problem, remove complicating factors that cloud the central issue, and in the spirit of Einstein’s thought experiments [113], present a series of mental investigations for guidance to logical conclusions based on Galilean relativity. While a definitive response to the kinetic energy debate is not afforded by physics alone, this chapter concludes with a pair of mutually exclusive outcome sets as involatile consequences of the velocity reference frame choice.

Heuristic Assumptions Summary We begin this journey by considering seven heuristic assumptions in aggregate before addressing them individually in detail. With no loss of generality, we assume validity of Newtonian mechanics to our aircraft flying in the atmosphere of a flat-earth model:

HA1- A universal inertial reference does not exist- terrestrial earth is not uniquely “inertial”.

HA2- An infinite choice of reference frames for EoM analyses and simulations are valid.

HA3- Appropriate choice of KE velocity reference applies to aircraft flying in all conditions.

HA4- By definition Lift cannot influence airspeed, Drag always opposes airspeed.

HA5- Aircraft cannot distinguish between locally equivalent spatial and temporal gradients.

HA6- KE computed from airspeed in a wind-fixed reference frame yields useful energy.

HA7- Velocity gradients are responsible for useful energy gain in DS, not downwind turns.

HA1- A universal inertial reference does not exist- terrestrial earth is not uniquely “inertial”. In his treatise “Relativity- The Special and General Theory” [112], Einstein introduces his now- famous picture of a traveler on a railway car dropping a stone to the ground embankment. This dropping stone “misdeed” is also witnessed by a pedestrian on the embankment footpath. In the eyes of the railway traveler, the stone simply falls straight down, accelerating with gravitational attraction to the earth. However, the pedestrian sees the event quite differently: in his frame of reference the stone falls in an approximate parabolic path due to the apparent initial horizontal velocity. The differing perspectives of the stone’s motion from the moving railway car and the ground-based footpath lead to the principle of relativity in a restricted Galilei-Newton sense. As Halliday and Resnick write it in their text [114]:

The laws of physics must have the same form in all inertial reference frames.

They are echoing a phrase by Einstein [112]:

For the physical description of natural processes, neither of the reference-bodies is unique as compared to the other.

Galileo also discussed a similar invariance principle in his 1632 “Dialogue Concerning the Two Chief World Systems” [115], using a ship sailing on smooth seas in contrast to Einstein’s rolling railway carriage. Galileo asserted through his thought experiments that basic laws governing mechanics of motion were valid regardless of choice of valid inertial reference frame. No greater shoulders than those of Galileo, Newton, and Einstein could be stood on in stating that no single reference frame can be declared unique or fundamentally “inertial”. Certainly a ground-based reference system has some conveniences for terrestrial coordinate bookkeeping purposes, but no other unique or fundamental qualities. D’Alembert’s principle further generalizes this concept of invariance to non-inertial (i.e. accelerating) coordinate systems such that they too can be utilized

45 as acceptable dynamic soaring frames of reference when the so-called fictitious forces are included. The dynamic soaring thrust 푚푊̇푥 of Figure 3.3 is such a force in this context.

HA2- An infinite choice of reference frames for EoM analyses and simulations are valid. Lawrance and Sukkarieh provide an excellent discussion of this in [93], including a derivation of 푡 an air-relative-to-terrestrial-transformation matrix 푪푎 from three canonical rotations 푳푥, 푳푦, and

푳푧, about roll 휙, air-relative glide slope 훾푎, and air-relative heading 휓푎, respectively, as indicated below in equation (3.1) and Figure 3.1:

푡 푪푎 = 푳푧(휓푎)푳푦(훾푎)푳푥(휙) (3.1)

Figure 3.1: Air relative velocity and applied forces [93]

Drawing on the conclusion of heuristic assumption HA1 above, using either terrestrial or air- relative reference frames for describing the kinematic evolution of objects in space is perfectly acceptable, and there exist transformations to navigate between these reference frames as desired. However, this should not be interpreted as a license to use arbitrary reference frames to describe velocity magnitudes for use in kinetic energy calculations. Note that this heuristic assumption merely affirms the valid usage of various reference frames for kinematics, dynamics, and simulations, and says nothing about the usage of either ground-fixed or wind-fixed frames for kinetic energy- that important topic is covered in heuristic assumption HA6. It should be additionally noted that there is no controversy among the researchers about proper reference frame

46 usage for generating appropriate equations of motion and for describing the trajectory evolutions given initial conditions. Any valid reference frame chosen, including non-inertial frames, will yield the same forces and therefore the same accelerations as any other frame. This concept of independence between application of reference frames for EoM derivation versus application of reference frames for KE definition cannot be stressed enough, but has not to this author’s knowledge appeared in the dynamic soaring literature.

HA3- Appropriate choice of KE velocity reference applies to aircraft flying in all conditions. This heuristic is included to emphasize that the central theme of correctly choosing the velocity reference used in kinetic energy computation is not strictly limited to dynamic soaring analysis. While dynamic soaring bluntly indicates to us that there are issues with using ground-referenced velocity for kinetic energy computation, making this same KE velocity reference frame choice in non-gradient flying leads to other subtle revelations. A quasi-steady-state glider thought experiment will illustrate this point, with three different scenarios: HA3.a depicts a glider descending in a straight path in calm air, with constant heading angle, glide slope angle, and airspeed; HA3.b depicts the same glider in calm air descending in a circular helix, with constant heading angle rate, glide slope angle, bank angle, and airspeed; HA3.c depicts a glider in the same scenario as HA3.b except that the entire airmass is moving at a constant velocity 푊푥 relative to the ground. Note that only HA3.c includes this constant horizontal wind 푊푥, HA3.a and HA3.b are both calm wind scenarios.

HA3.a Consider a glider descending in a straight path in calm air, with constant heading angle 휓푎, glide slope angle 훾푎, and airspeed 푉푎. As we are usually most interested in the total energy rate, we can derive 푇퐸̇ from the energy Eq. (3.2), taking time derivatives of all terms to arrive at Eq. (3.3):

푉2 푇퐸 = 푃퐸 + 퐾퐸 = (푧 − 푧 ) + 푎 (3.2) 0 2푔

푉 푉̇ 푇퐸̇ = 푧̇ + 푎 푎 (3.3) 푔

Note that the specific energy terms 푇퐸, 푃퐸, and 퐾퐸 have been divided by weight 푚𝑔 so that they all represent energy height.

Under the calm air assumption specified in this scenario, we can see that for both the ground-fixed and wind-fixed reference frames 푉푎 = 푉푡 and 푉푎̇ = 푉̇푡 = 0, therefore for both reference frames the total energy rate is given by:

푇퐸̇ = 푧̇ = 푉푎 sin 훾푎 = 푉푡 sin 훾푡 (3.4)

HA3.b Next consider the glider in calm air descending in a circular helix, with heading angle rate 휓̇푎, glide slope angle 훾푎, bank angle 휙, and airspeed 푉푎 all constant. We find that nothing changes in the energy rate equation for both ground and airspeed reference frames, as the there is no wind in this scenario. In both reference frames the magnitude of velocity is constant, so the total energy rate is simply still equivalent to the altitude sink rate 푧̇, and Eq. (3.4) is still valid in this scenario.

HA3.c Finally consider the glider descending in a circular helix, with heading angle rate

휓̇푎, glide slope angle 훾푎, bank angle 휙, and airspeed 푉푎 all constant as in HA3.b. In this last scenario we also have the entire airmass moving at a constant velocity 푊푥 relative to a ground- fixed reference, the moving airmass thus effectively an alternate inertial reference system. Note that in this final scenario the flight path of the glider is a circular helix only in the wind-fixed reference frame. Also, since in the presence of wind 푊푥 we have 푉푎 ≠ 푉푡, the usage of subscripts 푎 and 푡 are necessary to distinguish parameters referenced in wind-fixed and ground-fixed reference frames, respectively. Examining the energy rate equation in the wind-fixed reference frame, we note that nothing has changed with the inclusion of a steady wind in the transition from scenario HA3.b to HA3.c: all noted parameters (휓̇푎, 훾푎, 휙, and 푉푎) are still constant within, and relative to, that moving air mass. Therefore, the energy rate in the wind-fixed reference frame is still given by:

푇퐸̇ 푎 = 푧̇ = 푉푎 sin 훾푎. (3.5)

For the ground-fixed reference frame the situation is more complicated. Relative to the ground- 2 fixed reference frame we have a scalar expression for 푉푡 from the squared (푥, 푦, 푧) components of 푉푡 used to compute 퐾퐸푡:

2 2 2 2 푉푡 = (푉푎 cos 훾푎 cos 휓푎 + 푊푥) + (푉푎 cos 훾푎 sin 휓푎) + (푉푎 sin 훾푎) . (3.6)

After expansion and algebraic manipulation, we arrive at the following expression for 퐾퐸푡 and

퐾퐸̇ 푡:

1 1 퐾퐸 = 푉2 = (푉2 + 2푉 cos 훾 cos 휓 푊 + 푊2), (3.7) 푡 2푔 푡 2푔 푎 푎 푎 푎 푥 푥

1 퐾퐸̇ = − (푉 cos 훾 푊 휓̇ sin 휓 ). (3.8) 푡 푔 푎 푎 푥 푎 푎

Therefore, we have total energy 푇퐸̇ 푡:

푇퐸̇ 푡 = 푧̇ − 퐾 sin(휓̇푎푡), (3.9) 1 퐾 = ( ) 푉 cos 훾 푊 휓̇ = constant. (3.10) 푔 푎 푎 푥 푎

Note that the energy rate 푇퐸̇ 푡 as a function of time has a constant component equivalent to the net sink rate 푧̇ of the glider in the calm wind scenarios HA3.a and HA3.b, but now also has a sinusoidal component representing a heading-angle phase-dependent term. Clearly a pilot flying the glider in scenario HA3.c does not experience this phenomenon, nor does an inertial measurement unit (IMU) on the glider sense this sinusoid. It is simply an artifact of using a ground-fixed reference for kinetic energy calculations. For systems that rely on accurate estimation of aircraft energy rate this sinusoid can introduce unwanted (and avoidable) low frequency harmonics. An autonomous glider thermal-centering algorithm in [95] and [100] uses 푇퐸̇ and 푇퐸̈ terms, where the authors note anomalies in the flight test results. They used ground-fixed velocity from GPS sensors for KE basis, which would not have accounted for the observed prevailing winds and resulting thermal

49 drift. This may be manifest in some of the graphs presented therein: several show a remarkable heading-angle phase-dependent oscillation that could be attributed to thermal drift-induced energy rate estimation error from KE based on a ground-fixed velocity reference. That brings us back to the central point of this heuristic- that the argument of which velocity frame of reference to use for kinetic energy computation is of general aerodynamic importance, and not just for the study of dynamic soaring.

HA4- By definition Lift cannot influence airspeed, Drag always opposes airspeed. This heuristic centers about the manner in which the single general aerodynamic force is decomposed into lift and drag components, the orientation of such components being defined by

Figure 3.2: Four forces in flight the local air velocity vector. By way of background, consider Figure 3.2, below. From childhood most students of aerodynamics have been saturated with classic images of the four forces acting on an aircraft: Lift, Drag, Weight, and Thrust. While the principled intent of such diagrams is to help beginners understand how airplanes fly, one potential negative effect is the fostering of an idea that lift and drag forces are somehow unique and distinct from one another. In reality they arise as artifacts of our own modeling, for they are simply an orthogonal decomposition of the single resultant aerodynamic force representing the integration of normal and shear pressures acting over the entire surface of an aircraft in flight. By definition we have termed the component of this net aerodynamic force in the air-velocity direction “drag” and the complementary orthogonal component “lift”. Since, by its very definition, drag is always opposing the air velocity vector it must always work to remove kinetic energy from the aircraft, at a rate proportional to drag times airspeed. Likewise, since by definition lift is always orthogonal to the air velocity vector, it cannot contribute to kinetic energy as defined by airspeed. Lissaman made this point very clearly for a no-wind environment in the opening paragraph of [79]:

As a result, drag always does work on the vehicle while lift forces cannot. Gravitational forces perform conservative work on the vehicle... …This is a fundamental consideration.

Unfortunately, Lissaman left this fundamental consideration at the door when he entered into the analysis of aircraft in moving atmospheres, for in dealing with even the simplest uniform horizontal wind he chose to use a terrestrial reference for kinetic energy, thereby arriving at the misleading conclusion mere sentences later:

…any orientation of the vehicle in which the lift vector is inclined in the direction the wind is blowing will enable the wind to do work on the vehicle.

The only possible explanation for such a departure from a previously stated fundamental consideration is the notion that a terrestrial reference is somehow unique and responsible for keeping tally of the total energy state, that perception being the subject of heuristic assumption HA1 above. While it certainly is true that the choice of a terrestrial reference frame will result in constant velocity wind performing work on a glider turning downwind, it should be that very outcome that ought to make one recognize that the ground-fixed reference frame is not appropriate for flying aircraft energy analysis. Per HA3 above, this applies to all forms of flight in moving atmospheres, not limited to dynamic soaring flight analysis. Bonnin makes a similar explicit pronouncement of this unfortunate conclusion regarding lift, drag, and energy due to misaligned terrestrial flight-path and air-velocity vectors in [39] and specifically [78] by declaring::

By working on the expression of the power from aerodynamic forces, it is laid out that the drag can have a component that provides positive work, which is part of a wider resultant force oriented in the wind direction.

We must pause and give [78] credit for providing a list of references representing researchers on both sides of the argument on the proper treatment of terrestrial versus airspeed for kinetic energy. However, even in presenting the list there is an omission of concepts covered in heuristic

51 assumption HA2: that one is free to use terrestrial or air-reference coordinates as a matter of convenience in equations of motion and simulations (neither is “right” nor “wrong”), but independent of the choice of reference frame for equations of motion one ought to only use airspeed as the magnitude of velocity in kinetic energy computation. This concept of independence cannot be stressed enough, but has not, to this author’s knowledge appeared in the literature.

How then do gliders extract energy from the environment, as we witness occurring during static and dynamic soaring? Just as importantly, what do we mean by extracting energy? How do we compute that quantity, and to what phenomena do we attribute its source? The answer to these questions lies in heuristics HA6 and HA7 below, with the definition of useful energy- a phrase used often in this chapter and borrowed from Lawrance [12], echoed as flight kinetic energy by Barnes [52], [105], [107], [108], and [109], but soundly discounted by Sachs and Grüter in [110].

The main point of this heuristic assumption HA4 is to recognize the importance of how lift and drag forces are defined in an air-reference frame, and why this definition preserves the ideal that lift does no work on an air vehicle while drag always dissipates energy. These basic principles ought to guide us in choosing the appropriate frame of reference for an atmosphere in motion.

HA5- Aircraft cannot distinguish between locally equivalent spatial and temporal gradients. The mechanisms of how dynamic soaring actually works have been portrayed as difficult to explain and understand. Part of this difficulty lies in the foundational material of aerodynamic coursework, where little emphasis is placed on analyses within terrestrial-relative wind. In addition, inadequate consideration of classical mechanics and dynamics with respect to aerodynamics compounds the problem. A particularly confusing theme in works explaining dynamic soaring is the introduction of a “tilting lift vector” [29], [36], and [39]. Per the heuristic assumption HA4 above, lift is by definition always orthogonal to the air velocity vector, and in the case of the point-mass model this airspeed is defined at one unique point in space. The fact that this precisely defined lift vector may have an angular rate (dependent on a time-based change of airspeed orientation due to variable wind) has no effect on the instantaneous definition of the lift vector orientation, which is all that a free body diagram can capture in a snapshot of time, per Figure 3.3 below:

Figure 3.3: Dynamic soaring forces

Possibly what this so-called tilted lift vector is attempting to capture is the very real effect of a dynamic soaring force, which is simply and elegantly produced in an accelerating (i.e. non-inertial) airspeed reference frame as follows: in this accelerating frame the mass of the aircraft must be reflected in a “fictitious” or “apparent” force as an inertial resistance to that linear acceleration (see Deitter et al [40] and Barnes [52] with respect to dynamic soaring, Einstein [113] for treatment of topic in general). The magnitude of that linear acceleration is proportional to the wind velocity rate of change with respect to time (i.e. wind acceleration). Figure 3.3 illustrates the most commonly formulated example of wind acceleration in the +푥 direction. Note that there is no representation of a wind gradient in this figure, an intentional omission discussed below.

The existence of a dynamic soaring wind frame acceleration 푊̇푥 in figure 3.3 indicates only that the wind, as experienced by the aircraft at the unique point in space it occupies, is changing its velocity as a function of time, but offers no explanation as to how exactly that wind accelerates. Two examples are presented below, but by no means preclude other combinations:

HA5.a: consider a glider ascending through a vertical variation of horizontal wind, as modeled in the planetary boundary layer so common in the dynamic soaring literature. We would call that boundary layer a spatial gradient, so that the wind frame acceleration 푊̇푥 depends both

53 on the unique point in space the glider occupies, and with what speed the glider is traversing that particular gradient field:

휕푊 휕푊 휕푧 휕푊 휕푊 푊̇ = 푥 = 푥 = 푥 푉 = 푥 푉 sin 훾 (3.10) 푥 휕푡 휕푧 휕푡 휕푧 푧 휕푧 푎 푎

The glider cannot sense how the spatial gradient appears in other point locations, it can only sense the air speed at its own location, which coupled with its own gradient-traversal velocity yields the accelerating wind frame of reference.

HA5.b: Now consider a different scenario, where the wind frame acceleration 푊̇푥 is in fact a product of a temporal gradient field, where a uniform wind is accelerating everywhere with the same magnitude 푊̇푥 in all points in space, specifically including the unique point occupied by the glider. For this scenario the free body diagram would again be given by Figure 3.3, and the corresponding dynamic analysis at that point in time and space would be identical to that at HA5.a.

Extending this novel concept even further, consider a typical dynamic soaring trajectory of a glider flying through a boundary layer gradient (Fig. 1.10). Such a glider is experiencing a complex variation of horizontal wind velocities, which in turn create varying wind frame accelerations. It is clear based on the equivalence of HA5.a and HA5.b above, that one could imagine the boundary layer replaced by a uniform wind that at all points in time happens to accelerate and decelerate precisely in the same magnitude as the glider experiences in the spatial gradient boundary layer version. The glider could fly the exact same trajectory, feel the same forces, and have the same dynamics in both situations. The author believes these two notions of specific and continuous equivalence to be novel in the dynamic soaring literature. It is through these conceptual analogies and equivalences that the field of dynamic soaring can be advanced, with a new realization of why it makes sense to leave a set of equations of motion in a form that does not proscribe the wind acceleration mechanism per se, but only includes the wind acceleration terms in their most basic state, to be substituted for the actual mechanism in each specific implementation.

HA6- KE computed from airspeed in a wind-fixed reference frame yields useful energy.

The first two heuristic assumptions HA1 and HA2 clarified that one really is free to use any reference frame for analyzing the flight of an aircraft, and that no inertial reference is to be favored over another, save for convenience sake. Heuristic assumption HA3 reminded us that we ought to pay attention to choice of kinetic energy frame of reference for all flight analysis, regardless of whether dynamic soaring is the main objective, while heuristic assumption HA4 revisited how the definitions of Lift and Drag depend solely on the air velocity vector. This present heuristic assumption HA6 puts these together to formulate the argument for always choosing airspeed as the basis for kinetic energy computations.

Returning again to the thought experiment scenario in HA3.b (a glider descending in a steady- state helical path in calm air), consider these two new scenarios using different inertial reference frames: HA6.a in which the kinetic energy is described by researchers in a ground station building, and HA6.b in which the kinetic energy is described by researchers in a mobile tracking van rolling at a constant speed and direction. While heuristic assumptions HA1 and HA2 suggest we can correctly do so, few scientists would argue that there is any fundamental benefit to computing 푇퐸 = 푃퐸 + 퐾퐸 from the rolling reference, as it would result in a periodic oscillation similar to HA3.c. One could say that by utilizing kinetic energy velocity from the ground-fixed ground station reference frame, we are computing a kinetic energy that could at any time be converted back to height via the aerodynamic kinetic-to-potential energy exchange (i.e. “pull up / slow down”). This indicates that kinetic energy computed this way has a tangible, useful quality- hence the term useful energy as originally presented in Boslough’s foundational work [11] on applying dynamic soaring techniques to autonomous unmanned aerial vehicles, and more recently amplified by Lawrence [12].

Next move on to the thought experiment scenario in HA3.c (a glider descending in a steady-state helical path in a moving airmass, velocity 푊푥 representing the airmass constant velocity relative to the fixed ground station) and consider these two new scenarios using different inertial reference frames: HA6.c in which the kinetic energy is described by researchers in a ground station building (as in HA6.a), and HA6.d in which the kinetic energy is described by researchers in a mobile tracking van rolling at a constant speed (as in HA6.b) with magnitude 푊푥 to match the moving airmass speed and direction. Note that the mobile tracking van now experiences zero relative wind

55 as its velocity matches the moving airmass velocity, while the ground station remains stationary and experiences the wind 푊푥. It is clear that in HA6.c the glider energy computation will include the same heading angle phase-dependent oscillation originally described in HA3.c and Eqs. (3.9) - (3.10), while in HA6.d the energy computation is the steady form described in HA3.b and Eq. (3.5). Simple reasoning dictates that one must use the stationary ground station building in HA6.a and the mobile tracking van in HA6.d to yield kinetic energies that are both inherently useful, as only in those frames of reference can the quantities of kinetic energy of motion be fully exploited to potential energy of height and vice versa. This is no different than suggesting that a mechanical laboratory on board an extremely smooth train should not use a ground-fixed terrestrial reference frame but rather the local train car-fixed reference frame for dynamic analysis of systems involving springs, masses, gyroscopes, and even indoor flying aircraft for experiments conducted within its closed railcars.

While respected dynamic soaring authorities disagree on whether it is right or wrong to use one frame of reference over another in analyzing dynamic soaring, it turns out neither side of the argument is wrong per se. Galileo, Newton, and Einstein remind us that we can compute total energy in whatever frame of reference suits us, as long as we are consistent. Boslough, Barnes, and Lawrence make the case for using local airspeed due to its ability to represent useful kinetic energy; no such defense can be made for any alternate frame of reference however convenient such alternate frame may be to terrestrial observers.

HA7- Velocity gradients are responsible for useful energy gain in DS, not downwind turns. Once we accept the claim from heuristic assumption HA6 above, the argument about where in a dynamic soaring cycle the energy increase comes from becomes somewhat moot; still, it is worthwhile exploring this in more detail. The equations of motion in either terrestrial or wind reference frames merely reflect the time rate-of-change of state variables as functions of the state variables themselves, in addition to system inputs (i.e. controls and environments), per Eqs. (2.23)- (2.25). When these equations become too complicated for analytic solutions, simulations are run that demonstrate the evolution of the state variables given appropriate models for the environment (i.e. boundary layer wind shear model) and also control inputs (e.g. bank and pitch commands). The simulation output is typically plotted as terrestrial coordinates of the air vehicle versus time,

56 showing a three-dimensional trajectory in space (Fig 5.4). In addition, other state variables (such as bank angle, flight path angle, and pitch angle) are also plotted versus time (Figs 5.5 – 5.7). Furthermore, there are other computed quantities that are of interest, such as load factor, lift, and drag that are also plotted versus time. As dynamic soaring is an interplay between potential and kinetic energies it is therefore typical to see plots of energy and energy rate versus time. The intent is to then make assessments as to where in a dynamic soaring cycle a particular trajectory yields energy increase, energy decrease, or negligible effect. Depending on whether the kinetic energy portion of total energy is computed using ground-fixed frame velocity or wind-fixed frame velocity, different resulting energy rates will be generated as a function of trajectory evolution, leading the interpretations of the results in very different directions. In particular these ground- fixed based results of kinetic energy rates have led some researchers in [39], [78], and [110] to conclude that most of the energy gained in a typical dynamic soaring cycle comes from the upwind to downwind turn phase rather than during the gradient-crossing upwind climb and downwind descent phases. More importantly, it has caused these researchers to question the role of the wind gradient itself, which we understand to be at the very core of dynamic soaring.

It is important to understand that the differing conclusions deduced above are derived from observations based on different frames of reference. Until we put on our relativistic glasses it will be difficult to accept that both energy source explanations are technically correct, and there is no need to further argue that point. However, from the progression of heuristics in this chapter we can now logically conclude that only an air-relative wind-reference frame yields useful kinetic energy results, and as such is the only frame one should use for dynamic soaring energy observations.

Consequences of Reference Frame Choice While the heuristics presented above provide guidance in navigating the choice of appropriate kinetic energy velocity reference frame, this section introduces two mutually exclusive outcome sets as involatile consequences of that reference frame choice. As a definitive mathematical formulation that allows only a singular frame of reference for KE velocity does not exist (see heuristic assumptions HA1 and HA2), the following discussion will ensure that we in the dynamic soaring community recognize that once a KE velocity reference frame choice is made, there are

57 several logically occurring consequences that must be accepted in aggregate; either C1a – C4a are true or C1t – C4t are true. One is not free to choose these consequences à-la-carte.

C1- Source of Dynamic Soaring Energy

C1a: Dynamic soaring energy comes from flight conditions where the glider experiences a wind gradient 푊̇푥 (either by crossing through a spatial gradient or simply experiencing a temporal gradient, see HA5) in such a manner that its total energy increases:

1 1 푇퐸̇ = ℎ̇ + 푉 푉̇ , ℎ̇ + 푉 푉̇ > 0 (3.11) 푔 푎 푎 푔 푎 푎

1 1 ℎ̇ + 푉 푉̇ = 푉 sin 훾 − 푊̇ 푉 cos 훾 cos 휓 (3.12) 푔 푎 푎 푎 푎 푔 푥 푎 푎 푎

C1t: Dynamic soaring energy comes from flight conditions where the lift vector is inclined in a direction such that a component of lift is in the terrestrial-referenced direction of wind:

푳 ∙ 푾풕 > 0, (3.13)

Observe that C1a relies on existence of a wind gradient 푊̇푥, while C1t does not dictate need for wind gradient. C1a implies climbing against the wind and descending with the wind in a prescribed atmospheric boundary layer, while C1t describes the upwind-to-downwind turn in a typical dynamic soaring cycle accomplished at near-constant height and wind velocity (virtual absence of a wind gradient).

C2- Roles of Lift and Drag Forces in Energy Rates Note: Regardless of KE velocity reference frame choice, lift and drag are always computed using wind-fixed velocity.

C2a: In a wind-fixed reference frame, the lift vector is by definition normal to the air velocity vector, and by definition cannot affect the air velocity vector magnitude. The drag vector is by definition opposite the air velocity vector, and therefore by definition is always acting to reduce the air velocity vector magnitude and performing negative work on the air vehicle.

C2t: In a ground-fixed reference frame, depending on flight conditions lift and drag can both perform positive and negative effects on velocity. This includes the acceptability of drag performing positive work on a glider.

C3- Exchange Between Potential Energy and Kinetic Energy Forms

C3a: In a wind-fixed reference frame, the glider’s kinetic energy can always be converted to potential energy (i.e. height) and vice-versa, neglecting inevitable losses due to drag. Kinetic energy measured using airspeed has an inherent utility in this exchange. The change in wind-fixed velocity ∆푉푎 is related to the change in height ∆ℎ by:

∆퐾퐸푎 = ∆푃퐸푎 (3.14)

1 (푉2 − 푉2 ) = ∆ℎ (3.15) 2푔 푎2 푎1

C3t: In a ground-fixed reference frame the glider’s kinetic energy is not directly resulting from its airspeed, and there is no statement that can be made regarding the ability to convert that ground- based KE into a change in height PE. For example, one can envision a situation where a glider flying against the wind has zero ground speed and therefore zero ground-fixed frame KE, but is quite able to exchange some of its airspeed into height. Conversely, one can envision a situation where a glider may be flying at stall speed within a uniformly moving airmass, its heading aligned such that its groundspeed is the sum of the two velocities. While the ground-fixed frame KE suggests that the glider could exchange that “excess” velocity into height, any attempt to due so would result in a stall condition.

C4- Kinetic Energy Rates as Used in Flight Control Algorithms

C4a: In a wind-fixed reference frame the KE rate computation is derived from aerodynamically-relevant airspeed and airspeed rate, which directly affect how the glider reacts to the local wind environment under command of flight control algorithms that utilize 푇퐸̇ , 푇퐸̈ , etc. (see HA3.c).

C4t: In a ground-fixed reference frame the KE rate computation includes non-aerodynamically- relevant components that in most instances involving an atmosphere in motion will result in 푇퐸̇ , 푇퐸̈ , etc. rates being fed into autopilot control loops that do not represent the local aerodynamic conditions at the glider itself.

Conclusion It is clear from the recent literature that the once-benign disagreement on the treatment of dynamic soaring kinetic energy reference frames has widened into an outright clash of ideas. While several works in the dynamic soaring literature indicate that there is a lack of agreement on how to treat the kinetic energy reference frame issue, little has been published to clearly link classical dynamics and aerodynamics with the choice of KE reference frame. The purpose of this chapter is to add clarity to the dynamic soaring literature though the presentation of first-principle-founded heuristic assumptions and related mutually-exclusive consequence sets. This enables a new understanding of the appropriate kinetic energy reference frame and why it is so vital to both off-line analyses and real-time autonomous flight control. In essence what is proposed is a new way of looking at the dynamic soaring problem, where focus is placed on what the aircraft is sensing in its own reference frame.

The difference of opinions on how to handle the kinetic energy dilemma can easily turn personal. This author has attempted to resolve the dispute by “going back to basics” with Galileo, Newton, and Einstein. The conclusion that neither side is in fact wrong has made the investigation all the more interesting. Regardless of where on the discourse various researchers in the past have cast their lots, we must not undervalue the tremendous contributions of these pioneering investigators. This author’s sincere hope is that rather than be divided into competing camps of thought, we can

60 instead unite in a mutual pursuit towards a deeper understanding of the remarkable world in which we live. Analogous to conflicting wind speeds conspiring to create energy-rich wind gradients, conflicting ideas can serve to energize discourse in a positive manner and foster novel approaches to tackling long-standing problems.

Chapter 4: Steady State Analyses

Chapter Introduction This chapter is a departure from traditional dynamic soaring analyses as it focuses on utilizing the equations of motion developed in Chapter 2 for purely steady state behavior. Initially this appears to be an oxymoron- how can an aircraft perform dynamic maneuvers while the system parameters remain in steady state? The resolution to this conundrum is a description of what is meant by steady state dynamic soaring and how it is achieved. After defining what steady state means in this dynamic soaring context, several specific scenarios are examined:

Case 1: Steady State Descent in No Wind Case 2: Steady State Descent in a Horizontal Linear Gradient of Horizontal Wind Case 3: Steady State Ascent in a Vertical Linear Gradient of Horizontal Wind Case 4: Steady State Descent in a Vertical Linear Gradient of Horizontal Wind Case 5: Steady State Flight in Uniform Accelerating Wind

The chapter concludes with a novel and illuminating discussion on equivalence between wind shear steady state scenarios, amplifying heuristic assumption HA5 from chapter 3.

Figure 4.1 applies to the dynamic soaring cases that follow.

Definition of Steady State Dynamic Soaring in an Airspeed Reference Context At the close of chapter 3 it was concluded that an airspeed reference frame is most appropriate for discussing useful energy transfers between the atmosphere and air vehicles. Within this airspeed reference frame context, steady state is defined to mean the following: 1) Constant airspeed 푉푎,

2) Constant air-relative flight path angle 훾푎, and 3) Constant air-relative heading 휓푎. These three constraints are formalized in Eqs. (4.1) – (4.3):

푉푎 = 푐표푛푠푡푎푛푡 (푉푎̇ = 0) (4.1)

훾푎 = 푐표푛푠푡푎푛푡 (훾푎̇ = 0) (4.2)

휓푎 = 푐표푛푠푡푎푛푡 (휓̇푎 = 0) (4.3)

Figure 4.1: Dynamic soaring forces

Note that these conditions will, in some of the wind fields discussed below, introduce what appear to be transient (i.e. non steady state) behaviors in a terrestrial reference frame. To a ground-based observer, several of these cases will include accelerating motion that is not typical of what we consider steady state. They will, however, provide insight into the mechanisms at work in the more typical highly dynamic trajectories flown by dynamic soaring aircraft.

Modeling Assumptions The dynamic soaring modeling assumptions from chapter 2 will be observed in this present discussion as well: a point mass aircraft, a flat earth model, and no side-slip. The dynamic soaring equations of motion are presented again for convenience:

̇ ̇ 푚푉푎 = −퐷 − 푚푔 sin 훾푎 + 푚푊푉푎 (4.4)

̇ 푚푉푎훾푎̇ = 퐿 cos 휙 − 푚푔 cos 훾푎 + 푚푊훾푎 (4.5)

̇ ̇ 푚푉푎 cos 훾푎 휓푎 = 퐿 sin 휙 + 푚푊휓푎 (4.6)

̇ Since the wind time-derivative term 푊⃗⃗⃗ 푡 is defined in a terrestrial reference frame, it must be 푡 translated to the air-relative reference frame via transformation matrix 푪푎:

푡 푪푎 = 푳푧(휓푎)푳푦(훾푎)푳푥(휙). (4.7)

̇ ̇ ̇ Thus we have the following definitions for 푊푉푎, 푊훾푎, and 푊휓푎 as components of a more generalized version of realized wind rate vector to allow for time-varying wind fields in addition to the usual spatial gradients:

푇 cos 휓푎 cos 훾푎 ̇ ⃗⃗⃗ ̇ 푊푉푎 = [sin 휓푎 cos 훾푎 ] 푊푡 (4.8) sin 훾푎

푇 cos 휓푎 sin 훾푎 ̇ ⃗⃗⃗ ̇ 푊훾푎 = [sin 휓푎 sin 훾푎 ] 푊푡 (4.9) − cos 훾푎

푇 sin 휓푎 ̇ ⃗⃗⃗ ̇ 푊휓푎 = [− cos 휓푎] 푊푡 (4.10) 0

Case 1: Steady State Descent in No Wind This is the simplest environment: no wind, and no temporal or spatial gradients. In this situation we consider one possible steady-state flight: descending flight where the air-relative flight path angle 훾푎 < 0, and 훾푎 is a function of airspeed 푉푎 per the aircraft sink polar. The equations of motion can be reduced after setting appropriate terms to zero:

퐷 = −푚푔 sin 훾푎 (4.11)

퐿 = 푚푔 cos 훾푎 (4.12)

Naturally for this case (no wind), air-relative and terrestrial reference frames are equivalent. The subscript 푎 is therefore not needed, but is kept to emphasize that we are interested in air-relative steady state flight. It is worth noting the following expression for the magnitude of the net aerodynamic force:

√퐷2 + 퐿2 = 푚푔 (4.13)

While in this Case 1 it is trivial to note that the net aerodynamic force equals the weight of the glider (as it well should in a typical steady state condition), we will see in Case 3 that this is no longer the case in dynamic soaring flight profiles.

The shallowest flight path angle possible in steady state flight is by definition the maximum 퐿/퐷 condition, easily identified on an aircraft sink polar as the unique point on the curve where a tangent to the curve intersects the graph origin. This occurs when:

퐿 1 휋퐴푅푒 ( ) = √ (4.14) 퐷 푚푎푥 2 퐶퐷_0

The maximum flight path angle (inverted flight not considered) is 훾푎 = 휋/2, or straight down, which also yields the highest airspeed such that 퐿 = 0 and 퐷 = −푚푔. Maximum airspeed is given by:

푚푔 (푉푎)푚푎푥 = 1 (4.15) √ 휌푆퐶 2 퐷_0

The slowest airspeed possible is also termed stall speed, indicated on the flight polar curve as the lowest valid airspeed on the curve, where the required lift coefficient equals the maximum permissible value. Section 3.2 of [12] contains a uniquely detailed and exhaustive analysis of this no-wind gliding flight case. Note that here 휓푎 is arbitrary (fixed) and has no effect on longitudinal flight analysis. In summary, gliding in no wind allows a finite range of flight path angles and corresponding airspeeds, as depicted in a standard sink polar curve.

Case 2: Steady State Descent in a Horizontal Linear Gradient of Horizontal Wind In these spatial wind gradients, no steady state energy-neutral flight is possible. For steady state

Figure 4.2.a: Linear horizontal gradient (source) Figure 4.2.b: Linear horizontal gradient (sink)

Figure 4.2.c: Radial horizontal gradient (source) Figure 4.2.d: Radial horizontal gradient (sink) descent the aircraft glides in such a manner that a constant wind speed is experienced. As these gradients are defined per terrestrial coordinates, the glider’s position in terrestrial frame must be such that the component(s) that appear in the gradient definition are fixed: i.e. the velocity in terrestrial coordinates must be such that there is no component in the linear gradient directions. 휕푊 휕푊 For Figs. 4.2.a and 4.2.b, the gradients are 푥 = 퐾 and 푥 = −퐾 respectively, where 퐾 is a 휕푥 1 휕푥 1 1 constant gradient coefficient. For steady state flight the glider terrestrial velocity must be such that 푥 = 푐표푛푠푡푎푛푡, thus 푥̇ = 0. A simple subset of this Case 2 is when the glider is constrained to fly in the terrestrial 푦 − 푧 plane, for then the glider will experience no wind at all and the steady state analysis reduces to that of Case 1 above. However, if the glider is not constrained to remain in the terrestrial 푦 − 푧 plane, then the condition that must be satisfied is that 푥̇ = 0 in terrestrial coordinates, which occurs only when:

휕푊 푉 cos 훾 cos 휓 = 푥 푥 = 푊 (4.16) 푎 푎 푎 휕푥 푥

This suggests that the glider can have an arbitrary air-relative heading, but must fly with sufficient airspeed such that the component in the terrestrial 푥푡 direction matches the opposing wind speed for that location. In terrestrial terms, the glider will still travel in a single plane, but this time 휋 within a parallel plane offset from the 푦 − 푧 plane, with terrestrial heading 휓 = ± . An 푡 2 interesting limiting case occurs when the glider is constrained within the 푥 − 푧 plane. Given that the requirement to remain within a plane parallel to the 푦 − 푧 plane is still enforced, the intersection of these two planes implies that in a terrestrial frame the glider will descend in a vertical linear trajectory. This is what happens when a glider flies with heading angle 휓푎 = 0 or

휓푎 = 휋, such that to maintain steady state flight it must have a forward component of 푣푎 that matches the wind velocity:

휕푊 푉 cos 훾 = 푥 푥 = 푊 , (4.17) 푎 푎 휕푥 푥 and the vertical velocity is given in both reference frames by:

푧푎̇ = 푧푡̇ = 푉푎 sin 훾푎 (4.18)

Figs. 4.2.c and 4.2.d illustrate two novel variations of a linear gradient: rather than wind speed proportional to distance from a plane (for example the 푦 − 푧 plane in Figs. 4.2.a and 4.2.b) we have wind speed proportional to distance from a vertical axis, with such wind vector radially oriented away or towards the axis. A similar explanation of steady state flight exists for these two spatial wind gradients, except that unless one considers a helical flight path to be steady state, the only possibilities are for terrestrially vertical flight similar to situations explored through equations

(4.16) and (4.17) above. However, the constraint on 휓푎 = 0 표푟 휋 is relaxed, and the glider will have an air-relative heading angle that is equal to (or opposite to, depending on source vs sink gradient) its terrestrial ground position vector angle. It is worth noting that for these examples of terrestrial vertical flight, terrestrial heading angles are not well defined. Also, for the 4.2.c and 4.2.d subsets of Case 2, straight down vertical flight is also considered steady state if it is coincident with the 푧푡 terrestrial axis, and is covered by equation (4.15).

While these are novel excursions into steady state flight possibilities with horizontal linear gradients of horizontal wind, none of them deliver usable energy from the environment. They are also not likely to exist in nature for durations appropriate of steady state flight. The next cases will explore situations where energy can be exchanged between a dynamic atmosphere and an aircraft, and are more likely to exist in nature in a form similar to the models presented, limited only by the altitudes containing such linear gradients.

Case 3: Steady State Ascent in a Vertical Linear Gradient of Horizontal Wind A simple continuous shear model is shown below (Fig. 4.3), representing a linear vertical gradient of horizontal wind, with all wind vectors pointing in +푥푡 direction, as a function of terrestrial height 푧푡 above an arbitrary reference height. Such a model has been used as an approximation to the atmospheric boundary layer by researchers since the beginning of interest in dynamic soaring [11], [25], [27], and [79].

Figure 4.3: Vertical linear gradient of horizontal wind

With these conditions we simplify the general form of equations (4.4) and (4.5):

퐷 = −푚푔 sin 훾푎 + 푚퐾2푉푎 sin 훾푎 cos 훾푎 (4.19) 2 퐿 = 푚푔 cos 훾푎 + 푚퐾2푉푎 sin 훾푎, (4.20)

휕푊 where 퐾 = 푥, the linear wind shear constant, again expressed in terrestrial coordinates. For 2 휕푧 steady-state ascending flight, the glider must follow the dynamic soaring rule [52] by climbing against the wind, that is flight with air-relative velocity in the −푥푡 direction and flight path angle

훾푎 > 0. Following the line of thought from equation (4.13) the net aerodynamic force is now:

2 2 2 2 2 2 √퐷 + 퐿 = 푚√푔 + (퐾2(푉푎 sin 훾푎)) = 푚√푔 + 푊̇푥 (4.21)

Comparing the right-hand most terms of both Eqs. (4.13) and (4.21) yields an interesting result. This is a fundamental aspect of dynamic soaring: even in steady state, the net aerodynamic force exceeds the “resting” weight of the glider. Figure 4.1 clearly shows us why: in terrestrial coordinates the glider is accelerating in the +푥 direction with a magnitude equivalent to the wind frame acceleration:

휕푊 푊̇ = 푥 푉 = 퐾 푉 sin 훾 (4.22) 푥 휕푧 푧 2 푎 푎

In equation (4.21) we have the equivalence of flight in a conservative “gravitational” field with higher magnitude than the 푔 we are accustomed to. However, the gliding flight analysis is very familiar to full-scale glider pilots; it can be recast as flight in our usual gravitational field but with higher apparent weight, such as when water ballast is used to increase the wing loading of a competition glider in certain conditions. As discussed in [68], [47], and [57], one simply scales the known glider sink polar linearly from the origin to represent the new higher weight performance sink polar, per Edwards’ example in [68], shown in Figure 4.4 below.

Figure 4.4: Sink polar shift for heavier weight [68]

Note that the max 퐿/퐷 of the glider is unchanged, it simply occurs at a higher airspeed 푉푎. Also note that the minimum speed (i.e. stall speed) is increased as well, since the wings must generate more lift due to the apparent weight inflation. As far as this author knows, this represents a novel application of the sink polar shift procedure to steady state dynamic soaring. Appendix A further develops this concept of steady-state soaring in a vertical linear gradient of horizontal wind with the introduction of another concept: a unique rotated DS reference-frame facilitating graphical representation of dynamic soaring forces at work, highlighting the relation of static soaring terms such as “max L/D” to the analysis of dynamic soaring.

With this connection to standard glider calm wind performance estimation, flight analysis is very similar to Case 1 above, where the air-relative flight path angle 훾푎 is established by the point on the sink polar the glider is operating under, specifically:

푉푒푟푡푖푐푎푙 푉푒푙표푐푖푡푦 tan 훾 = , (4.23) 푎 퐻표푟푖푧표푛푡푎푙 푉푒푙표푐푖푡푦 understanding that in this case the velocities are of an air-relative sense and not terrestrial. The net energy gain from this dynamic soaring steady-state flight can be expressed by:

푉 푉̇ 퐸̇ = ℎ̇ + 푎 푎 = ℎ̇ = 푉 = 푉 sin 훾 , (4.24) 푔 푧 푎 푎

noting that the 푉푎̇ term vanishes due to the steady state assumption, leaving only the vertical climb rate contribution to energy rate.

While the wind gradient constant 퐾2 does not explicitly appear in Eq. (4.24), from the above discussion it is clear that it manifests itself in the apparent weight Eq. (4.21) and the corresponding shift in the sink polar per Fig. 4.4.

Case 4: Steady State Descent in a Vertical Linear Gradient of Horizontal Wind For steady-state descending flight, the glider must again follow the dynamic soaring rule [52] by descending with the wind, that is flight with ground path velocity in the +푥푡 direction and flight path angle 훾푎 < 0. It is important to note that, while counter-intuitive, this still yields an accelerating air reference frame such that the apparent dynamic soaring thrust points in the forward direction relative to the air vehicle. Once we substitute this into the equations of motion, the results are what we expect: the set of equations (4.19) – (4.24) from Case 3 above apply exactly to Case 4. It is important to note the following distinction between the ability to harvest energy during a steady state flight and the ability to harvest energy during a similar phase of a dynamic soaring cycle. Barnes [12] clearly shows us that anytime we follow the dynamic soaring rule an aircraft will extract energy from the environment (whether sufficient to overcome effects of drag is a more complicated mater).

Case 5: Steady State Flight in Uniform Accelerating Wind Level or climbing flight through an accelerating wind presents another interesting case for further examination. In this scenario, we consider the spatial wind gradient to be zero for our entire space, which defines a uniform, homogenous wind. However, the wind is accelerating along the 푥 direction such that 푊̇푥 = 퐾3 and 푊̇푦 = 푊̇푧 = 0. Since these temporal gradients exist for all space, the aircraft will experience them regardless of terrestrial location. The rate of change of the wind speed is not dependent on the rate at which the glider crosses a typical spatial gradient.

Considering level flight first, we note that 훾푎 = 0, and so we have:

퐷 = 푚푊̇푥 = 푚퐾3 (4.26)

퐿 = 푚푔 (4.27)

Drawing again on the glider’s sink polar curve, it should be noted that the conditions in equations (4.26) and (4.27) can be satisfied for all the values of airspeed presented in the sink polar, assuming that the polar curve is shifted appropriately per discussion in Case 3 and Fig. 4.4. It should be clear that for steady state level flight in this accelerating wind reference frame there is no net gain ̇ or loss of energy (푉푎 is constant per our definition of dynamic soaring steady state, and ℎ = 0 due to level flight condition).

In contrast, the final scenario examined in this Case 5 is climbing into the uniformly accelerating wind. The corresponding equations for this scenario are:

퐷 = −푚푔 sin 훾푎 + 푚퐾3 cos 훾푎 (4.28)

퐿 = 푚푔 cos 훾푎 + 푚퐾3 sin 훾푎, (4.29)

휕푊 where 푥 = 푊̇ = 퐾 as above. We can again determine the magnitude of the net aerodynamic 휕푡 푥 3 force experienced by the glider:

2 2 2 2 2 ̇ 2 √퐷 + 퐿 = 푚√푔 + 퐾3 = 푚√푔 + 푊푥 . (4.30)

The energy rate of this dynamic soaring flight is given by:

푉 푉̇ 퐸̇ = ℎ̇ + 푎 푎 = ℎ̇ = 푉 = 푉 sin 훾 (4.31) 푔 푧 푎 푎

This equation is the same form as (4.24) in Case 3 above. Below is a discussion on this equivalence principle between particular spatial and temporal gradients.

Equivalence in Dynamic Soaring Scenarios Heuristic assumption HA5 in chapter 3 presented the idea that a glider cannot distinguish between flight across a spatial gradient and flight within a temporal gradient, provided that the magnitudes are equivalent. The glider simply cannot resolve the mechanism behind the accelerating wind vector at its singular location, only that the wind is indeed accelerating. Drawing attention to the equivalent final terms in equations (4.21) and (4.30), we note that the glider not only experiences the same airspeed 푉푎 in both cases, but also the very same net aerodynamic forces given appropriate selection of air-relative flight path angle 훾푎. Note again the same term under the right-most radical in equation (4.30), which represents the apparent weight inflation attributed to the laterally accelerating wind field. The energy rate equations (4.24) and (4.31) are also equivalent between Case 3 and Case 5. This highlights a specific dual pair example of temporal and spatial gradient flights, discussed in a more general form in chapter 3. The author believes this to be a unique and novel concept in dynamic soaring analysis.

Dynamic Soaring Sink Polar Surface A final comment on the often-referenced glider sink polar curve: it stands to reason that for the dynamic soaring discussion above one could generate a three-dimensional sink polar surface, adding a “weight inflation ratio” dimension to graphically illustrate the effects of apparent weight inflation defined above. In that case, the independent variables would be airspeed and apparent weight ratio, with sink rate remaining the dependent variable depicted by the graphed surface (Fig. 4.5). Naturally, one could take a “slice” of such a surface for a given load factor, such a slice yielding the familiar sink polar for the given load factor. While this hardly seems novel, the author has not seen such a presentation of three dimensional sink polar data in the literature.

Figure 4.5: Dynamic soaring sink polar surface

In this figure the weight inflation ratio is simply the right hand term from Eq. (4.21), divided by aircraft weight to get a ratio:

√ 2 ̇ 2 √ 2 ̇ 2 푚 푔 +푊푥 푔 +푊푥 푊̇ 2 푤푒𝑖푔ℎ푡 𝑖푛푓푙푎푡𝑖표푛 푟푎푡𝑖표 = = = √1 + 푥 (4.32) 푚푔 푔 푔2

Chapter 5: Reverse Kinematics Simulation

Chapter Introduction While closed-form analytical solutions exist for a small number of simplified dynamic soaring situations, in general the only way to model this complex behavior is through simulation. Many examples of dynamic soaring simulation models exist the literature, as almost every contributed work in this field includes some aspect of simulation. Of specific interest are the works of Sukumar and Selig [28] and [29], and Barnes [52], [107], and [108]. They illustrate in detail how many of these simulations operate: using scheduled inputs based on phase transformations of periodic heading angle. Other researchers have used aircraft flight simulators embedded in dynamic soaring simulations following prescribed flight paths to generate optimal solutions to specific dynamic soaring problems in [60], [62], and [64]. In contrast to these classical simulation methods, this chapter presents a novel simulation architecture for dynamic soaring: parameterized three- dimensional flight trajectories using reverse kinematics, requiring no input scheduling or control law modeling.

Dynamic Soaring Framework Review The dynamic soaring equations of motion (DS EoM) were formulated and discussed at length in chapter 2. The construction specific to an atmospheric boundary layer (a vertical variation of horizontal wind speeds) is reprinted below for convenience:

휕푊 푚푉̇ = −퐷 − 푚𝑔 sin 훾 − 푚 푥 푉 sin 훾 cos 훾 cos 휓 (5.1) 푎 푎 휕푧 푎 푎 푎 푎

휕푊 푚푉 훾̇ = 퐿 cos 휙 − 푚𝑔 cos 훾 + 푚 푥 푉 sin2 훾 cos 휓 (5.2) 푎 푎 푎 휕푧 푎 푎 푎

휕푊 푚푉 cos 훾 휓̇ = 퐿 sin 휙 − 푚 푥 푉 sin 훾 sin 휓 (5.3) 푎 푎 푎 휕푧 푎 푎 푎

75 where 푉푎 is airspeed. Also 퐿 and 퐷 are lift and drag, respectively. In this context 훾푎 is the air- relative flight path angle, 휓푎 is the air-relative heading angle, and 휙 is bank angle (Fig. 5.1).

Figure 5.1: Air relative velocity and applied forces [93]

Lift and drag take their customary form:

1 퐿 = 휌푆퐶 푉2 (5.4) 2 퐿 푎

1 퐷 = 휌푆퐶 푉2 (5.5) 2 퐷 푎

Here 퐶퐿 and 퐶퐷 are the lift and drag coefficients, respectively, 푆 is the reference area, and 휌 the local air density. Implementing a commonly used drag polar approximation introduced earlier for

퐶퐷 as a function of 퐶퐿, we have:

퐶2 퐶 = 퐶 + 퐿 (5.6) 퐷 퐷0 휋푒퐴푅

In equation (5.6) 퐶퐷0 is the zero-lift (i.e. parasitic) drag coefficient, 푒 the Oswald’s efficiency, and 퐴푅 the aspect ratio of the aircraft. From these equations it can be determined that one could use bank angle 휙 and lift coefficient 퐶퐿 as inputs to a simulation, which is analogous to what some

76 full-scale pilots term “bank and yank” control. Implicit in this description is an assumption of no side slip, as discussed in chapter 2, and an abstraction of control surface deflection (ailerons and elevators) to high level roll and pitch commands.

Typical Dynamic Soaring Simulations

Some simulations generate these 휙 and 퐶퐿 inputs as control law output from a flight simulator attempting to control an aircraft to a prescribed path, using guidance laws similar to [5] and [10]. Other simulations carry out input generation by constraining one aircraft state to another, reducing the overall degrees of freedom. For example, [29] revisits the cascading transformations in [52] whereby flight path angle 훾푎 is related to heading angle 휓푎 through free variables 휓1, 휓2, and 휓3:

휋 휓 = 휓 + (5.7) 1 푎 2

휓 휓 휓 = 2휋 [ 1 − int ( 1) ] (5.8) 2 2휋 2휋

휓 휓 = 휋 [1 − cos ( 2) ] (5.9) 3 2

2 훾푎 = 훾1 sin 휓3 + 훾2 sin 휓3 (5.10)

These transformations provide additional shaping or squashing of the resulting trajectory, giving this particular implementation the ability to modify a glider’s dwell time near the surface. The parameter 훾1 is introduced to limit the maximum flight path angle, while 훾2 may be used to differentiate climb and descend flight path angles, all expressed in the air-relative sense.

Regardless of which simulation architecture is implemented, a model of the desired wind gradient is a necessary ingredient. Most simulations in the literature utilize a one-dimensional vertical variation in horizontal wind, as developed in chapter 2. This change in horizontal speed as a 휕푊 function of height can be expressed as 푥. This is often crafted as a function that outputs the 휕푧

77 horizontal wind velocity vector as a function of height 푊푥(푧) per the desired wind shear model. For example, consider the logarithmic profile presented earlier in chapter 2:

log(푧⁄푧0) 푊푥 = 푊푟푒푓 (5.11) log(푧푟푒푓⁄푧0) where 푊푟푒푓 is the reference wind speed, 푧푟푒푓 is the reference height, and 푧0 is a surface roughness parameter. Inspection of equation (5.11) confirms that 푊푥 = 푊푟푒푓 at 푧 = 푧푟푒푓, a condition that all wind models must satisfy. Figures 5.2 and 5.3 depict wind velocity and gradients versus height:

Figure 5.2: Wind velocity vs height

Figure 5.3: Wind gradient vs height

A typical dynamic soaring simulation operates in the following manner:

10: Generate an aircraft model with equations of motion (EoM) that utilize aircraft controls as inputs (i.e. roll and pitch), with resulting lift, drag, and weight force outputs.

20: Generate a wind-field function that outputs a horizontal wind velocity vector based on altitude, following one of several documented atmospheric boundary layer models.

30: Implement an algorithm for scheduling the inputs as functions of cycle phase to reduce EoM degrees of freedom and enforce intended periodic behavior.

40: Run the simulation using the command scheduling relationships and observe the trajectory flown, logging glider coordinates and parameters vs time.

50: Continue running the simulation until trajectories converge within required tolerance band, plot pertinent data from final trajectory.

Overview of Reverse Kinematics Simulation A novel dynamic soaring simulation DynaSoar was initiated for two express purposes: a) to explore the space of feasible trajectories with the goal of extracting optimization trends in a multi- dimensional setting with competing objectives; and b) to determine favorable conditions and performance limits in anticipation of experimental flights with actual dynamic soaring glider UAVs. While traditional simulation methods have been shown to generate useful results, they do not provide a straightforward means to analyze explicit trajectories, but rather rely on the algorithms to virtually fly the aircraft via control inputs, with additional code to determine if the trajectory flown by scheduled controls satisfies a periodicity requirement. See [3]-[6], [8]-[16], [18]-[23], [25]-[33], [36]-[43], [48]-[56], [58]-[65], [68]-[71], [73], [74], [76]-[89], [93]-[102], [107]-[109], and [111]-[112] regarding the use of simulations in nonlinear dynamic systems.

As an alternative solution, consider an explicit three dimensional trajectory of interest, parametrically computed a priori. The simulation simply flies the glider along this trajectory from the initial conditions combined with a requirement to stay on the trajectory at all times. Conceptually, one can consider this simulation as generating the precise glider control inputs to remain on the determined trajectory. In practice, the simulation is running a reverse-dynamics solver: at each time interval the local tangent / normal / binormal unit vectors of the Frenet-Serret orthogonal basis frame are computed, then the required aircraft bank angle and lift coefficient are solved to ensure sum of forces in binormal direction is zero and in normal direction is equal to centripetal acceleration (due to traveling along a trajectory with computed curvature at a known tangential velocity). Residual net forces in the tangent direction determine whether or not the aircraft is gaining or losing energy as dynamic soaring thrust competes with aircraft drag, and these tangential residuals are used by the simulation to compute the instantaneous state derivatives (i.e. accelerations), thus dictating how the simulation proceeds around the trajectory. During execution, the simulation also check real-time for violations of aircraft control limits and general aircraft performance requirements.

Three-Dimensional Trajectory Formulation A trajectory of interest is converted into a parameterized form yielding three equations defining the trajectory as a curve in space, all three as functions of single scalar parameter 푞 that uniquely define any point on the curve, defined by position vector 풓:

풓(푞) = [푥(푞); 푦(푞); 푧(푞)] (5.12)

Using a process similar to that described in [20] and [117], we can generate important differential geometry properties from 풓(푞) based on the Frenet-Serret (F-S) frame equations. In particular, we can define symbolic expressions for the tangent 풕(푞) and normal 풏(푞) vectors of the curve, and the scalar curvature 휅(푞), which equals the reciprocal of instantaneous radius used for centripetal acceleration computation, all derived from the basic definition of parametric first and second derivatives of 풓(푞):

푑풓 풓′(푞) = 풓′ = (5.13) 푑푞

푑2풓 풓′′(푞) = 풓′′ = (5.14) 푑푞2

풓′ 풓′ 풕(푞) = 풕 = = (5.15) |풓′| √풓′⋅ 풓′

풕′ 풓′× (풓′′× 풓′) 풏(푞) = 풏 = = (5.16) |풕′| |풓′| |풓′′× 풓′|

|풓′ × 풓′′| (풓′⋅ 풓′)(풓′′⋅ 풓′′)−(풓′⋅ 풓′′)2 휅(푞) = 휅 = = √ (5.17) |풓′|3 (풓′⋅ 풓′)3

Note that Eq. (5.12) did not specify arc-length parameterization, therefore the expressions contained in Eqs. (5.15) - (5.17) are valid for any appropriate parameter such as phase angle for a circular orbit. Simplified versions of these equations exist for trajectories that have been arc-length parameterized. Also note that the expressions are not a reflection of motion along a trajectory, but of the physical structure of the trajectory itself- that is, time does not explicitly appear in Eqs. (5.12) – (5.17). The notions of tangential velocity and normal / tangential acceleration appear only when the trajectory is combined with a dynamic object moving along it.

Reverse Kinematics Algorithm The unique features of this algorithm begin to manifest themselves in the first step of generating appropriate initial conditions (IC): due to the nature of a parameterized trajectory, the three- dimensional position vector becomes a one-dimensional parameter to be selected. Likewise, the velocity vector is also reduced to a one-dimensional value. The six states of a typical simulation (푥, 푦, 푧, 푥̇, 푦̇, 푧̇) become simply (푞, 푞̇). Once the simulation is running, there are only two variables that need to be solved simultaneously: lift coefficient 퐶퐿 and bank angle 휙. The general idea is to balance the weight, lift, drag, and centripetal acceleration force components within the curve normal plane (spanned by the 풏 and 풃 unit vectors, where 풃 = 풕 × 풏). All remaining components

81 of the same forces are then, by definition of the normal plane, in the curve tangent 풕 direction, and are used to define the acceleration of the glider along the trajectory (i.e. 푞̈). The following general steps are executed within the DynaSoar simulation:

10: Generate a parameterized version of the desired trajectory, such that the three equations representing the (푥, 푦, 푧) position components are all functions of a single parameter.

20: Develop symbolic expressions for 풕, 풏, and 풌. These symbolic equations are also in terms of the same single trajectory parameter.

30: Establish the initial conditions for the simulation run, resulting in (푞, 푞̇). Note that the trajectory imposes a reduction in degrees of freedom for both the position and velocity.

40: Start the sim from the I.C. in step 30. Transform all known and to-be-determined forces into 풏 - 풃 normal plane.

50: Within the 풏 - 풃 normal plane set ∑ 풇푛,푏 = 0, and then solve for 퐶퐿 and 휙. While this is simply 2 equations with two unknowns, the system is highly non-linear.

60: 퐶퐿 and 휙 represent the inputs of a pilot with a yoke- next solve for components of all forces in tangential 풕 direction.

70: Use one of various integrators to update (푞, 푞̇) via (푞̇, 푞̈) resulting from previous step.

80: Return to step 40, and repeat steps 40-70 until exit criteria are reached. These exit criteria can include run time limits, state limits, and other sim interrupts.

90: Generate plots of pertinent data / parameters from final successful trajectory, assuming sufficient simulation time was allowed for dynamic soaring equilibrium (limit cycles).

Example: Inclined Circular Trajectory A simple case study will be examined in order the highlight the DynaSoar simulation architecture. John Strutt, Baron Rayleigh conjectured in 1883 that a simple dynamic soaring circuit could be accomplished via a circular flight path on a plane inclined upwind, such that the highest point in the trajectory occurred at the most upwind location:

For this purpose it is only necessary for him to descend while moving to leeward, and to ascend while moving to windward, the simplest mode of doing which is to describe circles on a plane which inclines downwards to leeward. [72]

A parameterized version of such an inclined circular trajectory can be given by:

푥(푞) 푅 cos 푞 50 cos 푞 풓(푞) = [푦(푞)] = [ 푅 sin 푞 ] = [ 50 sin 푞 ] (5.18) ℎ ℎ + ( ∆)(1 − cos 푞) 푧(푞) 푚𝑖푛 2 20 − 15 cos 푞

where 푅 is the radius of the circle ground trace, ℎ푚𝑖푛 is the minimum height of the trajectory (often referred to as dwell height) and ℎ∆ is the height difference between minimum and maximum elevation. For the range of 푞: 0 ≤ 푞 ≤ 2휋, and values of 푅 = 50 푚, ℎ푚𝑖푛 = 5 푚, and ℎ∆ = 30 푚, we have the trajectory in Figure 5.4, below. In this example, the winds are blowing from negative 푥 to positive 푥 (i.e. – 푥 to +푥).

Figure 5.4: Inclined circular trajectory

The symbolic representations of the 풕(푞) and 풏(푞) vectors, and 휅(푞) scalar are:

−10 sin 푞 √(9 sin2 푞 + 100)

10 cos 푞 풕(푞) = (5.19) √(9 sin2 푞 + 100)

3 sin 푞 [√(9 sin2 푞 + 100)]

−100 cos 푞 √109(9 sin2 푞 + 100)

−√109 sin 푞 풏(푞) = (5.20) √(9 sin2 푞 + 100)

30 cos 푞 [√109(9 sin2 푞 + 100)]

2 109 휅(푞) = √ (5.21) √(9 sin2 푞 + 100)3

For the given trajectory geometry above, and with the boundary layer wind model from eq. (5.11) where 푧0 = .05 푚, 푧푟푒푓 = 20 푚, and 푊푟푒푓 = 15 푚/푠, the following plots illustrate typical output from the DynaSoar reverse kinematics simulation:

Bank Angle:

Figure 5.5: Bank angle vs trajectory position parameter for inclined circular trajectory

Specific Energy Rate:

Figure 5.6: Specific energy vs trajectory position parameter for inclined circular trajectory

Load Factor:

Figure 5.7: Load factor vs trajectory position parameter for inclined circular trajectory

These figures confirm the trends observed in other simulations of the inclined circular trajectory, while absolute values differ due to minor variations in boundary layer model definition and specific aircraft parameters. Note that the horizontal axis represents the position of the glider as expressed by its polar angle about the vertical z axis, and not time, per se. Due to variation of terrestrial velocity during the periodic cycle, the trajectory parameter does not linearly increase with time. Data are typically expressed with this position reference as a means to more clearly understand where the glider is within the dynamic soaring cycle, facilitating an understanding of the relationship between the various figures of merit versus cycle position.

Conclusion A novel simulation architecture was introduced as an alternative to traditional dynamic soaring simulation schemes. Two of these typical simulation methods were discussed briefly: 1) reduction of equation-of-motion degrees of freedom via input scheduling, and 2) trajectory following with control inputs provided by an aircraft simulator using a control system tuned for the aircraft performance. In contrast, the DynaSoar reverse kinematics simulator begins with a parameterized definition of a trajectory of interest, followed by an automated generation of symbolic expressions for the Frenet-Serret frame unit vectors 풕(푞) and 풏(푞) and scalar 휅(푞), all in terms of a chosen parameter 푞. The trajectory itself provides the constraints needed to reduce the equation-of-motion degrees of freedom, and given an appropriate boundary layer wind model and initial conditions, the simulation iteratively solves the lift coefficient and roll angle values needed to maintain the

86 glider on the trajectory. From the solution of roll angle and lift coefficient, the simulator solves a wide range of glider performance metrics of interest at each time step interval. Plots of these outputs with respect to a trajectory phase parameter promote visualization of the dynamic soaring behavior. Using DynaSoar in conjunction with more traditional simulation schemes will increase confidence in the conclusions drawn from the output.

Chapter 6: Conclusions and Future Research

Chapter Introduction From the earliest observations of sea-faring birds soaring in oceanic winds, humans have yearned to gain a deeper understanding of the hidden forces at work providing these birds with a seemingly infinite capacity to soar in the air currents with minimal effort. Barnes [52] and Sachs [44] explore the evolutionary adaptation of the wandering albatross, creating the most capable dynamic soaring system known: features including a shoulder-lock system that keeps the wings open with no muscular effort, flapping only when needed for takeoff and landing; high wing loading to allow penetration against winds; and efficient overall form for drag reduction via streamlined shape and high aspect ratio wings. It is only recently that we have begun to unlock these mysteries through field work, analysis, simulation, and experimentation, with a goal of developing robotic versions of the efficient albatross.

Literally hundreds of journal articles, conference papers, and dissertations have been published covering various aspects of dynamic soaring. The last decade has seen a tremendous growth in international dynamic soaring research, perhaps reinforced by the need to conserve energy resources as we transition from fossil based energy to renewables. This present work has sought to complement the dynamic soaring literature with numerous references to other works equally worthy of examination, while attempting to minimize repetitions of prior contributions. Dynamic soaring has attracted the attention of researchers from many diverse fields including robotics, aerodynamics, controls, optimization, artificial intelligence, and power systems. Mechanics and dynamics are most likely not the subject of expertise for many working in this broad array of interdisciplinary fields. It is therefore imperative that the basis on which their current research is building upon comes from a solid foundation.

Summary of Novel Contributions Repeating a theme from the introductory Chapter 1, this present research is motivated by a desire to reestablish a sound foundation of kinematics and dynamics to the art of soaring through an atmosphere in motion. The novel contributions summarized below are intended to provide fresh insight and clarity to dynamic soaring theory by eliminating cross-coupling of concepts through a fundamental understanding of dynamic soaring mechanisms. This research has produced the following contributions:

1) A logical progression of seven heuristic assumptions leading to a singular conclusion: dynamic soaring kinetic energy must always be computed using airspeed as the reference.

4) DynaSoar reverse kinematics simulation for analysis of explicit parametric dynamic soaring trajectories in the Frenet-Serret frame, without requiring input scheduling or aircraft control algorithms.

Future Research and Applications The implementation of unmanned aerial vehicles (UAV) in industrial, military, commercial, and agricultural domains has undergone tremendous evolution in the last two decades. Improvements in on-board energy storage, efficient propulsion systems, and miniaturized embedded computing systems coupled with advances in navigational receivers and multi-spectral sensors have made the advancing usage of UAV in our daily lives inevitable. Dynamic soaring can be a vital part of extending the mission of the more advanced unmanned aerial systems, enabling near unlimited mission potential when coupled with other architecture features such as solar power, thermal and ridge soaring regenerative flight, and remote recharging systems.

While there is near-consensus in the dynamic soaring literature over the use of point-mass models and no side-slip assumptions in theoretical constructs as discussed in chapter 2, one area of future research would be to determine specifically what effects these underlying assumptions have on the conclusions that have been drawn from these reduced-order models. Ensuring that the roll and pitch rates are within reasonable bounds has been the standard method of ensuring validity of these models, with respect to the simplifications mentioned above. Trajectories that have been optimized by genetic algorithms or other off-line codes to maximize dynamic soaring energy harvesting have so far shown a favoritism toward these low roll and pitch rate flights. More complex combinations of dynamic soaring along with other unique UAV mission constraints may require control inputs that could invalidate the simplifying assumptions, and future research would point to the consequence and resolution of such scenarios.

Given the restriction on dynamic soaring, that of requiring a threshold wind gradient of sufficient magnitude to provide energy neutral flight, there are several areas of interest that can nevertheless supply such gradients. One prime example is that of extreme weather research and monitoring, as presented by Elston et al in [32]. Their VORTEX2 study simulated a UAV to obtain directly measured thermodynamic data from the rear-flank downdraft of a supercell thunderstorm, while harvesting energy along the way through a combination of static, dynamic, and gust soaring.

Another region of interest for future dynamic soaring application is that of soaring in the atmospheric jetstream, at altitudes between 7 and 16 kilometers depending on the type of jetstream. Grenestedt et al [74] and Sachs et al [25] have discussed the feasibility of such flight, with more recent observations from Barnes [107]. Although an ambitious undertaking, dynamic soaring in jetstream wind gradients has great potential to allow perpetual flight, enabling operation similar to a low-earth orbit satellite for reconnaissance, communications, and meteorological sampling. Future studies could determine feasibility of intercontinental transport via dynamic soaring aircraft.

Appendix A: Dynamic Soaring Reference Frame

The dynamic soaring literature has, in general, presented static and dynamic soaring as individual and distinct approaches to gliding flight. While several works have generalized soaring to include arbitrary atmospheric motions, the resulting EoM are typically reformulated to emphasize the individual contributions of static soaring (i.e. thermal soaring), time-varying winds (i.e. gust soaring), and spatially-varying winds (i.e. dynamic soaring). The purpose of this Appendix is to introduce a novel approach to illuminate a convoluted phenomenon- a rotated reference frame whereby glider soaring in dynamic winds can be related back to the fundamental analysis of glider flight in calm winds, specifically the usage of the familiar “sink polar” and corresponding “max

L/D ratio” more commonly reserved for familiar static soaring situations.

Figure A.1: Glider descending in calm wind

Figure A.1 illustrates the simplest free-body-diagram (FBD) of soaring flight: a glider descending within a stationary air mass (i.e. no wind). The net aerodynamic force is decomposed in the usual manner into lift (L) normal to air-relative velocity and drag (D) opposite the air-relative velocity, with weight (mg) acting straight down. It is simple to observe that during steady-state gliding flight the three forces are balanced, while the glider is flying with a negative air-relative flight path angle − 훾푎. It is intuitive that an unpowered glider in steady-state flight must fly “downhill” in this manner.

Figure A.2 depicts the more complicated FBD of a glider experiencing an accelerating air-relative

(wind-fixed) reference frame, such as would occur when crossing through a vertical gradient of horizontal wind velocity. Lift, drag, and weight are depicted in a similar manner as in Fig. A.1.

The accelerating reference frame is captured in the 푊̇푥 term, with corresponding “fictitious” force

DS Thrust = 푚푊̇푥. Again, considering the case when the glider is in steady-state flight, the forces

Figure A.2: Glider ascending through accelerating wind-fixed reference frame

(now four) are balanced. Chapter 4 covers various situations describing how such steady-state behavior could be accomplished. For simplicity, we can assume Case 3 applies to Figures A.2 –

A.5 (Steady State Ascent in a Vertical Linear Gradient of Horizontal Wind). Since the DS Thrust term 푚푊̇푥 arises from an acceleration 푊̇푥 in a similar manner as the weight term 푚𝑔 arises from an acceleration 𝑔, these two forces are vectorially combined to yield a new force 푊푒𝑖𝑔ℎ푡퐷푆, depicted in red at the bottom-left of Figure A.3 The orientation of 푊푒𝑖𝑔ℎ푡퐷푆 defines a rotation angle 휃퐷푆, which is the angle about which the new DS frame (with new 푥퐷푆 – 푧퐷푆 axes) is rotated from the standard 푥 − 푧 reference frame.

There are two key features of the rotated DS reference frame in Figure A.3: 1) it combines the usual 푚𝑔 weight of the glider with the fictitious dynamic soaring thrust force 푚푊̇푥 to create a new

Figure A.3: Glider ascending through accelerating wind-fixed reference frame with addition of rotated DS reference frame notation (red)

94 single dynamic soaring weight with a new definition of “down” in the −푧퐷푆 direction; 2) it creates a new −훾퐷푆 term, which is the dynamic soaring analogue of the −훾푎 term in figure A.1. As 푥퐷푆 is the new horizontal direction in this rotated reference frame, we can see now that the glider is in fact again flying “downhill” as it should in steady-state, with the flightpath angle −훾퐷푆. Figure

A.4 is simply Figure A.3 rotated through 휃퐷푆, such that 푥퐷푆 is now horizontal and 푧퐷푆 is vertical, and the glider is more clearly flying with a “virtual” flightpath angle −훾퐷푆. Figure A.5 removes most of the original dynamic soaring notation of Figure A.4 so that the analogy to Figure A.1 can be easily observed.

Figure A.4: Glider descending in rotated DS reference frame

It is worth noting that the glider pilot in Figure A.3 really would feel as though “up” were in the

+ 푧퐷푆 direction, confirmed by an onboard inclinometer or inertial measurement unit (IMU). The pilot’s perceived weight (and that of the glider too) would be increased by the ratio:

√ 2 ̇ 2 푔 +푊푥 푊̇ 2 푊푒𝑖𝑔ℎ푡 = 푊푒𝑖𝑔ℎ푡 { } = 푊푒𝑖𝑔ℎ푡 {√1 + 푥 } (A.1) 푝푒푟푐푒푖푣푒푑 푔 푔

And to that same glider pilot or IMU, the nose of the glider really would appear to be dropped by the angle 훾퐷푆 below the “perceived” horizon (represented by ± 푥퐷푆), while a visual inspection of

Figure A.5: Glider descending in simplified rotated DS reference frame the actual horizon outside the cockpit would reveal that the glider nose was in fact raised above the horizon by angle 훾푎. All the while, to a terrestrial observer the glider would appear to be rising

96 with a constant velocity 푉푧 and simultaneously accelerating in the lateral + 푥 direction with constant acceleration of magnitude 푊̇푥.

As far as this author knows this is the first depiction of dynamic soaring in this manner, analogous to static soaring. Since nothing has changed in the point-mass model description (see Chapter 2), the glider still follows the same rules regarding lift and drag, lift-to-drag ratio, and flight path angle

훾퐷푆. Most importantly, the maximum lift-to-drag ratio, corresponding to a minimization of dynamic soaring flight path angle 훾퐷푆, is still given by the familiar expression:

퐿 1 휋퐴푅푒 ( ) = √ , (A.2) 퐷 푚푎푥 2 퐶퐷0

remembering that lift to drag ratio is related to flight path angle by the following relation:

1 (훾퐷푆)푚푖푛 = atan { 퐿 } (A.3) ( ) 퐷 푚푎푥

Also, the corresponding airspeed 푉푎 may be calculated similarly as in the case for steady-state static soaring in Figure A.1, remembering to use the perceived weight ratio in Eq. (A.1) for the inflated glider weight. To those familiar with static soaring, this weight inflation ratio represents an analogue to the common practice of adding ballast to a glider to increase its wing loading under certain wind or competition conditions (see discussion at conclusion of Chapter 4).

So as an unpowered glider in steady-state must always fly “downhill”, dynamic soaring simply requires one to redefine the local horizontal and vertical axes appropriately, such that this intuitive rule is naturally enforced. While the steps to transition from Figure A.2 to Figure A.4 may seem complicated, the remarkable analogy of Figure A.1 to Figure A.5 is attractive mainly due to the simplicity and familiarity of these first and last free body diagrams.

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