Design and characterization of a magnetic-field position sensor for feedback control of a micro-aerical vehicle powered by resonant inductive wireless power transfer

A resonant inductive wireless power transfer (WPT) system has been designed to wirelessly power a micro-aerial vehicle (MAV) to extend flight time and eliminate the need for batteries. The WPT system consists of a seven-turn, 19-cm transmit coil that is able to deliver over 15 W of power to a 13-cm receive coil attached to the MAV. Effective wireless power transfer requires the MAV to be positioned within the range of the power transmit coil. This thesis focuses on the design, fabrication, characterization, and implementation of a high-frequency magnetic-field position sensor for closed-loop control of a MAV within the transmit coil. A compact magnetic-field position sensor, with a footprint of 1.9 cm by 1.9 cm and weighing approximately 1.3 grams, is designed for the MAV. The sensor is characterized and calibrated for sensing the displacement of theMAV relative to the power transmit coil. A feedback controller is designed that utilizes the sensor output for position control. Experimental results show that the MAV and sensor system can hover in a stable configuration at approximately 20 cm above the power transmit coil.

DESIGN AND CHARACTERIZATION OF A MAGNETIC-FIELD POSITION SENSOR FOR FEEDBACK CONTROL OF A MICRO-AERIAL VEHICLE POWERED BY RESONANT INDUCTIVE WIRELESS POWER TRANSFER by Erik John Andersen A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering The University of Utah December 2018 c Erik John Andersen 2018 Copyright All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Erik John Andersen has been approved by the following supervisory committee members: Kam K. Leang , Chair(s) 15 Oct 2018 Date Approved Shadrach Roundy , Member 9 Oct 2018 Date Approved Stephen A. Mascaro, Member 15 Oct 2018 Date Approved by Bruce K. Gale , Chair/Dean of the Department/College/School of Mechanical Engineering and by David B. Kieda , Dean of The Graduate School. ABSTRACT A resonant inductive wireless power transfer (WPT) system has been designed to wirelessly power a micro-aerial vehicle (MAV) to extend flight time and eliminate the need for batteries. The WPT system consists of a seven-turn, 19-cm transmit coil that is able to deliver over 15 W of power to a 13-cm receive coil attached to the MAV. Effective wireless power transfer requires the MAV to be positioned within the range of the power transmit coil. This thesis focuses on the design, fabrication, characterization, and implementation of a high-frequency magnetic-field position sensor for closed-loop control of a MAV within the transmit coil. A compact magnetic-field position sensor, with a footprint of 1.9 cm by 1.9 cm and weighing approximately 1.3 grams, is designed for the MAV. The sensor is characterized and calibrated for sensing the displacement of the MAV relative to the power transmit coil. A feedback controller is designed that utilizes the sensor output for position control. Experimental results show that the MAV and sensor system can hover in a stable configuration at approximately 20 cm above the power transmit coil. For all those who helped me along the way CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii NOTATION AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTERS 1. 2. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 BACKGROUND INFORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Magnetic Field Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Hall Effect Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Magnetostrictive Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Inductive-Coil Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Wireless Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Microwave WPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Power Beaming WPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Capacitive Coupling WPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 Inductive and Resonant Inductive Coupling . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 UAVs with WPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Magnetic Field Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Design Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3. WIRELESS POWER TRANSFER SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Wireless Power Transfer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Tuned Natural Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Coupling/Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Impedance Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Coil Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Coil Geometry Modeling for WPT System . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Z-2-Port Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Coil Design and Full WPT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 11 12 13 14 14 14 15 17 DESIGN OF HIGH-FREQUENCY MAGNETIC-FIELD SENSOR FOR POSITION CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1 Common Position Sensors for MAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Examination of the WPT’s Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 High-Frequency Magnetic-Field Sensor Design . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.1 Design of Inductive-Coil Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Sample and Hold Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Dynamics of Inductive-Coil Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Component Selection of Inductive-Coil Sensor . . . . . . . . . . . . . . . . . . . . . 4.4 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Calibration Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Sensor Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Comparison of Commonly Used Position Sensors for MAVs . . . . . . . . . . . . . . 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. CONTROL SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.1 MAV Positioning Using Gradient Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Gradient Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Position Control in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Control System Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Position Control with Motion Capture Cameras . . . . . . . . . . . . . . . . . . . 5.2.2 Position Control Using Inductive-Coil Sensor . . . . . . . . . . . . . . . . . . . . . 5.2.3 Hybrid Position Control Using Cameras and Inductive-Coil Sensors . . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 41 42 45 45 46 46 50 52 53 EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.1 MAV Power Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Wireless Power Transfer System Model Validation . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Coil Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Wirelessly Powering an MAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Hybrid Control System Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Full System Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 25 26 26 27 28 29 29 30 31 64 65 66 67 68 70 70 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 vi LIST OF TABLES 4.1 A comparison of position sensors for MAVs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 R2 and adjusted R2 for the respective fitting functions . . . . . . . . . . . . . . . . . . . . 32 4.3 Mean and max error for each fitting function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 A comparison of position sensors for MAVs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 The gains for the motion capture camera’s position control PID controller. . . . . 55 5.2 The gains for the MAV’s internal PIV controller. . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 The gains for the hybrid position control PID controller. . . . . . . . . . . . . . . . . . . . 55 6.1 Theoretical and experimental values of a 13.7-cm diameter R x coil with 1-mm pitch, two turns, and 1.29-mm diameter wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Theoretical and experimental values of a 15.3-cm diameter R x coil with 2-mm pitch, four turns, and 1.29-mm diameter wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 The average errors between the sensor’s estimated y coordinate of the MAV’s location and the actual y location of the MAV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 NOTATION AND SYMBOLS WPT MAV Tx Rx Pε Pϕ Γ φ θ ψ wireless power transfer micro-aerial vehicles transmit coil receive coil power received at the load of the R x coil power required by the MAV to hover with the R x coil attached figure of merit, equal to PPϕε Euler angle of roll Euler angle of pitch Euler angle of yaw CHAPTER 1 INTRODUCTION Potential application of unmanned aerial vehicles (UAVs) increases every year, as people and companies find new uses for robots that are not tethered to the Earth’s surface. Goldman Sachs has released a report valuing the potential UAV market at $100 billion USD between 2016 and 2020 [1]. However, the utilization of UAVs, especially smaller-sized versions referred to as micro-aerial vehicles (MAVs), is limited by their short flight time. A typical MAV has a flight time less than 20 to 30 minutes depending on design, battery choice, and payload [2] [3]. The challenge of extending a MAV’s flight time lies in the fact that MAVs have been required to physically carry all their power in the form of an energy storage unit, such as a battery or super capacitor. And while recent developments have led to batteries with a higher energy storage density, that approach alone does not resolve the fundamental problem of requiring a hovering object to physically carry its own energy. Attaching a physical tether such as a power line greatly reduces the flight radius of an MAV, along with its possible utility. An alternative solution to this problem is using wireless power transfer (WPT) to enable a MAV to receive power mid-flight, allowing it to continue to operate without interruption for long periods of time. 1.1 Motivation Wireless power transfer frees MAVs from carrying their available energy storage on their frame at all times. Instead, it can carry a small reservoir of power with a battery or super capacitor, allowing it to operate as normal. As the reservoir runs low over the course of the MAV’s operation, a WPT system quickly recharges the MAV’s power storage. Additionally, using a WPT system allows a MAV to remain airborne and operational while recharging. This is key for such applications as environmental monitoring, in which having the MAV land to change its batteries or recharge wastes precious time. This also allows the MAV to operate in hostile environments where physical contact with the MAV 2 is not possible for safety reasons, i.e., around chemical spills/leaks, as is demonstrated in Fig. 1.1. In these situations, physically touching the MAV to change its batteries would place human workers in danger. Furthermore, the MAV may need to be operational for periods of time exceeding the 30 minute limit imposed by its battery. For tasks that include a repetitive or known path, transmit coils can be arranged in ways to ensure the MAV always has the necessary power requirements and can fly continuously without having to pause to recharge. This would remove the necessity of having even a small battery on the MAV, allowing it to carry a heavier payload. This is applicable to factory or warehouse settings, where products are consistently moved from one location to another along a set path. A key aspect in using WPT to power MAVs is ensuring that the MAV is located in the optimal charging zone within the WPT system. Power transfer using WPT systems can vary widely depending on the location and orientation of the receiver relative to the transmitter. Developing a control and sensing scheme based on the WPT will enable the MAV to locate and stay in the optimal charging zone while hovering. 1.1.1 Thesis Objectives The research objectives of this thesis are threefold: first, to build and demonstrate a WPT system for a hovering MAV. The system must be able to provide enough power to enable the MAV to hover over the transmit (Tx ) coil. Second, a WPT sensor will be developed, designed, and implemented that will allow the MAV to sense the WPT field. Using this sensor, the MAV’s position above the transmit coil can be determined. Lastly, a control system will be implemented enabling the MAV to fly and hover above the Tx coil using position feedback from the WPT sensor. The novelty of this thesis lies in building an inductive resonant WPT system strong enough to power a hovering MAV and developing high-frequency magnetic sensors capable of sensing the electromagnetic field (EMF) produced by the WPT system’s transmit coil. While previous works have included demonstrations of powering MAVs using different WPT systems such as lasers and capacitive plates [5] [6], to the best of the author’s knowledge, no published paper exists that describes using an inductive magnetic resonance WPT system to power an MAV until the author’s paper at ICRA 2018 [7]. While there 3 has been a paper discussing a magnetic resonance WPT system for a dynamic load [8], they did not develop and optimize their system especially for a MAV. Additionally, these works do not include a sensing and control scheme enabling the MAV to locate an optimal charging location through sensing of the WPT method. Likewise, building the MAV’s position control law around the WPT system sensing has not been accomplished in these previous papers. The novelty of the WPT sensor lies in developing and characterizing a high-frequency magnetic-field sensor for sensing the magnetic waves produced by the transmit coil at 13.56 MHz. This sensor also is small and light enough to fit on a MAV and can estimate its position accurately enough to enable the MAV to hover over the Tx coil. 4 Figure 1.1. A MAV being recharged while hovering above a WPT charging pad and inspecting a chemical leak. The WPT system gives the MAV a inexhaustible power source allowing it to operate without the need of physical contact in order to recharge its power supply [4]. CHAPTER 2 BACKGROUND INFORMATION Although wireless power has existed for over a century, its popularity has soared over the last decade. Nikola Telsa was one of the pioneers on this scientific frontier who had demonstrated near field inductive, capacitive coupling, and even resonant inductive coupling methods of WPT systems by the turn of the century [9]. For the next century, research in wireless power transfer advanced in waves. One area of particular interest was powering airborne vehicles using WPT systems. In the 1960s, William Brown took advantage of the development of microwave technology and powered a model helicopter wirelessly [10]. Almost fifty years later, in the early 2000s, NASA used lasers to power a small model plane in flight [5]. However research into WPT systems exploded after 2007 when Dr. Marin Soljacic at MIT used a four coil magnetic resonance system to power a 60 W light bulb over 2 meters with an efficiency of 40% [11]. Now WPT systems are starting to enter more and more of everyday life, from charging ports of electric cars, to new smart phones, and even down to simple items such as toothbrushes and electric razors. All WPT system’s underlying physics depends on electromagnetic waves. Being able to sense these waves involves building a sensor capable of detecting either the electrical or magnetic component of each wave. Although magnetic-field sensors are now common in a wide range of applications, sensing the magnetic field produced by a WPT system presents a unique set of criteria. Furthermore, the sensor is constrained by its size and weight, as it must be able to fit on a MAV. 2.1 Magnetic Field Sensing In 1832, the mathematician Carl Friedrich Gauss developed the first magnetometer capable of sensing the strength of Earth’s magnetic field [12]. By measuring the change in oscillation of a magnetized and demagnetized bar suspended by gold fibers, he cal- 6 culated the absolute strength of Earth’s magnetic field and invented the first working magnetometer. Since then, we have developed much more cost-effective, more accurate, and more precise magnetometers. Three of the most common are Hall effect sensors, magnetostrictive sensors, and inductive-coil sensors. 2.1.1 Hall Effect Sensors Hall effect sensors consist of a thin p-type semiconductor sheet called the Hall element [13]. A continuous electric current is applied to the Hall element. As a magnetic field is applied perpendicular to the semiconductor sheet, it induces the charge-carrying components–electrons or holes–to be pushed towards either side of the Hall element. This grouping of charges causes a voltage potential to build up across the semiconductor sheet, which can be measured and correlated to the strength of the magnetic field by V ∝ I × V, (2.1) where V is the voltage produced by the sensor, I is the current in the Hall effect element, and B is the strength of the magnetic field. These types of sensors are cheap, small, have a wide operating range, and can be used to measure static magnetic fields. However, they do not operate well at high frequencies, and can lack precision compared to other magnetometers in measuring an AC magnetic field. 2.1.2 Magnetostrictive Sensors Magnetostriction is the phenomenon where a material expands or contracts in the presence of a magnetic field [14]. This change in volume causes the resistance of the magnetostricive material to change. By arranging magnetostricitive materials with other resistors, typically in a Wheatstone bridge configuration, the change in resistance can be measured and correlated to the strength of the magnetic field. Similar to Hall effect sensors, magnetostricitive sensors are dependent upon the orientation of the magnetostricitive material to that of the magnetic field. These sensors are small, can yield higher precision than their Hall effect counterparts, and are relatively cheap to produce. However, they do have a dead zone on the upper side of their frequency spectrum, making them difficult to use for high-frequency magnetic field sensing. 7 2.1.3 Inductive-Coil Sensor Inductive-coil sensors, also commonly referred to as search coil or pick coil sensors, are among the simplest yet most effective magnetometers [15] [16]. They are made by winding conductive wire into a coil, usually a solenoid, but other geometries also work. As magnetic flux passes through the coil, it induces a voltage in the coil according to Faraday’s law: V = −n · dB dθ = −n · A , dt dt (2.2) where n is the number of turns on the coil, A is the cross-sectional area of the coil, and θ is the flux passing through the coil. Since V is a function of dθ dt , stationary induction coil sensors cannot measure DC static fields, but by circuit design, they are capable of measuring fields as low as several mHz [15]. Additionally, by physically displacing the search coil, it is possible to sense DC magnetic fields. However, search coils sensors are easy to manufacture and design for high-frequency magnetic field sensing. Other types of magnetometers exist but are not practical for use on a MAV. SQUID magnetometers are extremely accurate, but need to be super cooled with liquid nitrogen or helium. Fluxgates magnetometers operate on a similar principle as inductive sensors but are significantly heavier and not practical to fit on a MAV. Caesium vapor magnetometers or similar magnetometers are accurate but in no way practical to fit on a 100 g MAV. 2.2 Wireless Power Transfer There are two distinct areas of WPT: those were the distance is large relative to the wavelength, called far field WPT, were the power transfer is radiative, and those that use electrical or magnetic field coupling where the distance is relatively short, called near field WPT. Each form of WPT has its unique advantages and disadvantages. 2.2.1 Microwave WPT The research into microwaves and using microwaves in WPT was made possible by the development of high-powered microwave emitters during the Second World War. Microwaves are electromagnetic radiation that have a frequency between 300 MHz and 300 GHz. Through the use of a rectifying antenna, or rectenna as coined by its inventor William C. Brown, the electromagnetic radiation is captured and converted into DC power 8 [17]. As previously mentioned, Dr. Brown demonstrated his concept by powering a flying helicopter with an attached rectenna [10]. Microwave WPT is a far field WPT technology, and can be used to transfer power in distances easily exceeding a kilometer. This was demonstrated in 2008 by John Mankins when microwave WPT technology was used to transfer 20 W of power from Maui to the Big Island of Hawaii a distance of over 148 km [18]. And while microwave WPT can be efficient, its efficiency is very dependent upon having large transmitting and receiving antennas. In Dr. Mankins experiments, he achieved an efficiency of less than 0.00001 % due to the insignificantly sized antennas. Additionally, microwave WPT needs a clear line of sight in order to operate. This line of sight issue and large rectenna size made microwave WPT less ideal for use with MAVs. 2.2.2 Power Beaming WPT Another far field WPT method is power beaming using lasers. First, electrical energy is converted into a highly focused, high-powered laser [19]. The laser is then pointed at a photovoltaic cell located a distance away on the receive object. The photovoltaic cell converts the energy from the laser back to electrical DC power. This method of WPT does not rely on magnetic fields or radio frequency, which can interfere with communication and other signals. Additionally, the photovoltaic cells can be much more compact than a rectenna or receive coil. However, conversion efficiency from the photovoltaic cells can be quite low, 19 % for the average commercially made silicon cell [20]. Additionally, power beaming requires line of sight and a high-powered laser is highly unsafe to surrounding humans, animals, and objects. 2.2.3 Capacitive Coupling WPT Compared to far field WPT systems, near field WPT has a lot smaller working range. However, they also have several advantages that far field WPT systems lack, such as higher efficiencies and higher safety factors. Capacitive coupling is one such example of a near field WPT system. With capacitive coupling, the transmitter and receiver are both metallic plates that, along with the air gap between them acting as the dielectric, form a capacitor [21]. Applying an alternating current to the transmit plate will induce an alternating potential in the receive plate through electrostatic induction. When attached to a load, the alternating potential will produce an alternating current that can be rectified and used 9 to power a load. The electric field is largely confined to the area between the plates, which means less electromagnetic noise and interference than inductive coupling. Additionally, the alignment between the transmit and receive plates is less critical than for inductive coupling [22]. Capacitive coupling, however, has a major drawback: by its nature, it results in fairly high electric fields, which can be dangerous to surrounding people or objects. 2.2.4 Inductive and Resonant Inductive Coupling The most common form of near field WPT is that of inductive coupling [22]. Inductive coupling operates on the same principles as a general electric transformer. An alternating electric current is applied to a transmit coil. That coil is loosely coupled to a receive coil through the air gap between them (contrast this with a normal transformer that is strongly coupled to the receive coil through a highly permeable material). The alternating current in the transmit coil generates a magnetic field, which subsequently induces a current in the receive coil that can be rectified and used to power a load. The electromagnetic field is typically lower than the electric field generated by capacitive coupling and therefore much safer for the surrounding people and environment. The efficiency and distance of WPT can be improved by tuning to and operating the system at a single resonant frequency for both inductive and capacitive coupling WPT systems [23]. 2.3 UAVs with WPT As previously mentioned, there have been many cases of using WPT to power aerial vehicles. In 2016, Solace Power powered a UAV using resonant capacitive coupling [6]. The Solace Power team was able to wirelessly transfer 35 W of power at 71% efficiency over a distance of 25 cm. This method of capacitive coupling, as previously mentioned, has several drawbacks. Capacitive coupling uses high electric fields, which can be dangerous to surrounding people/objects. Additionally, the two plates must remain fairly close together, which can be impractical for charging a hovering MAV. Therefore, most attempts to power MAVs use resonant inductive WPT due to its inherit safety and its potential to achieve greater charging distances. Wibotics designed a magnetic resonant system for wirelessly charging a UAV, where the UAV lands on a charging platform [24]. Researchers at Imperial College in London designed their own WPT system, using an 10 inverter specially designed for a dynamic load such as a UAV [8]. They demonstrated this paper by powering a toy UAV over a distance of several centimeters. 2.4 Magnetic Field Guidance In recent years, there have been several demonstrations of using magnetic fields to be a guidance source for a UAV. In [25], they proposed a method for using the Earth’s magnetic field to gain heading and flight information for a UAV. Additionally, using extremely lowfrequency magnetic fields generated by preplaced coils as a guidance source, a mobile robot relying only upon magnetic sensors was able to successfully navigate an obstacle course [26]. A fixed wing aircraft was shown to be able to land on current-carrying wires by sensing the magnetic field [27]. However, in both cases, the magnetic field used was low frequency, less than 100 Hz. With resonant inductive power transfer, the generated magnetic field is at a frequency of 13.56 MHz, much higher than any used in the previous papers. 2.5 Design Choices Based on this background research, it was decided to use resonant inductive WPT. Resonant inductive WPT is the safest form of WPT, and ideal for charging at relatively close distances. Also, resonant inductive WPT is not constrained by a clear line of sight requirement. Operating at resonance increases both the efficiency of the WPT and the distance at which the WPT can occur over non-resonant inductive WPT systems. Additionally, using resonant inductive WPT produces an electromagnetic field that, although difficult, can be sensed through a custom-built inductive coil high-frequency magnetic-field sensor. The inductive-coil sensor was chosen as the magnetic-field sensor because of its high operating frequency range, its low weight and cost, as well as the ease of fabricating it. CHAPTER 3 WIRELESS POWER TRANSFER SYSTEM The physics behind resonant inductive WPT closely resembles that of a standard electric transformer. A typical transformer consists of primary and secondary conductive coils that are both wound around a highly permeable material such as iron. An alternating electrical current carried in the primary coil will induce a voltage in the secondary coil directly proportional to the amount of magnetic flux passing through the secondary coil according to Faraday’s law; = N× dΦ , dt where N is the number of turns in the secondary coil and (3.1) dΦ dt is the flux passing through the secondary coil [28]. Transformers utilize this property to either increase or decrease the voltage in a given circuit. The highly permeable core of transformers directs nearly all the magnetic flux into the secondary coil, leading to high efficiencies in the system. The goal of resonant inductive WPT is to utilize the same physics but without the advantage of a connecting permeable core. Instead, maximum power is transfered by utilizing other variables, such as coil geometry, coupling coefficients, and by operating the system at a high resonant frequency. 3.1 Wireless Power Transfer Theory With resonance inductive wireless power transfer, at least two tuned coils are needed, the transmit (Tx ) coil and the receive (R x ) coil . Both the transmit and receive coils are LC circuits that are tuned to the same resonant frequency. The efficiency of the wireless power transfer (WPT) of two LC resonators is given by; η = 4U 2 ((1 + Rg RL Rs Rd , Rg Rl 2 )2 )( 1 + ) + U Rs Rd (3.2) 12 where R g is the internal resistance of the power supply, R L is the load resistance, Rs and Rd are the parasitic impedance of the Tx and R x coils, and U is defined as wn M U= √ = k Qs Qd Rs Rd (3.3) where M is the mutual inductance, k is the coupling coefficient between the Tx and R x coils, wn is the tuned natural frequency, and Qs /Qd are the quality factors of the transmit and receive coils. Equation (3.2) shows that the overall efficiency of the WPT depends heavily on three factors; ωn , the tuned natural frequency of the system, k, the coupling coefficient, which is related to M, the mutual inductance between the Tx and R x coils, and RL Rg Rd / Rs , the ratio between the load resistance, source resistance, and parasitic impedances of the system. By optimizing these parameters, the overall amount of transferable power will increase in the WPT system. 3.1.1 Tuned Natural Frequency By operating the system at its resonant frequency, the impedance of the coils is at a minimum, leading to the least internal power loss. At resonance, the complex components– inductance and capacitance– of the coil’s impedance cancel each other out, leading to the highest possible power transfer. Since the coil geometry of the transmit and receive coils determines the inductance of the system, a tuning capacitor is added to change the resonant frequency. This tuning capacitor allows the resonant frequency of the system to be chosen by design. According to Eq. (3.3), the higher the operating frequency is, the higher the corresponding quality factor for the system is. Therefore, choosing a higher resonant frequency will lead a to better quality factor, which improves WPT. However, there are several limitations for operating at high frequencies. The United States’ Federal Communications Commission (FCC) has imposed restrictions on operating in the radio frequency (RF) range. Several bandwidths are off limits for civilian use, while several other frequency bands are specifically reserved for scientific/research operation. Additionally, a power amplifier must be used to generate the AC circuit at such high frequencies, and they increase in cost as their frequency or power output increases. Hence, from the limited choices, a frequency of 13.56 MHz was chosen for the following reasons: it is in the allowed frequency bandwidth, it is possible to purchase an inexpensive power amplifier that can operate at this frequency, and it will produce a sufficient quality factor 13 for the WPT system. Tuning the system to this high frequency does lead to obstacles such as higher AC resistances in the Tx and R x coils, very small tuning capacitances for the Tx and R x coils, and high-frequency magnetic fields, which are more difficult to sense. 3.1.2 Coupling/Mutual Inductance The mutual inductance is an expression of the electromagnetic field produced in one coil and the subsequent current induced in the adjacent coil. For a two-coil system, the mutual inductance can be calculated as follows, M=k L1 L2 , (3.4) where k is the coupling coefficient between the two coils, and L1 and L2 are the inductances of the coils. In a two-coil system with a highly permeable core (such as a transformer), the coupling coefficient, k, is close to 1, meaning that nearly all the magnetic flux lines produced in the Tx coil pass through the R x coil. The coupling coefficient is an expression of the strength of the interaction between the two coils. However, without a highly permeable core, k, decreases rapidly as the distance between the coils increases [29]. The coupling coefficient and thereby the mutual inductance of the coils is dependent on the separation distance between the Tx and the R x coils. Optimizing k in effect optimizes the separation distance between the two coils. When the R x coil is far apart from the Tx coil, the system is described as under-coupled. As the distance between the R x coil and the Tx coil decreases, the coupling coefficient increases up to a critical distance. At this critical distance, usually equal to the diameter of the Tx and R x coils, the system is said to be critically-coupled [30]. This critical distance is where the maximum efficiency of a inductive WPT system will occur. If the R x coil becomes too close with the Tx coil, the system will become over-coupled and the efficiency of the WPT will be reduced. When the system is over-coupled the single resonant frequency of the system splits into two different resonant frequencies. This split leads to a decrease in maximum power transfered [30]. One approach to resolve this problem is to ”auto tune” the system so that it operates along one of the two split resonant frequencies when the system is over-coupled. Then the operating resonant frequency of the system is changed as the coupling coefficient changes and approaches the critically-coupled region [31]. This, however, can overly complicate the system and can be impractical to implement. 14 The simpler solution is to operate in the system’s critical coupled region by designing the appropriate coil geometry and corresponding WPT system. 3.1.3 Impedance Matching In order to supply the necessary current at 13.56 MHz, a class E Wibotic’s power amplifier is used. The output impedance of the power amplifier is 50 ohms. This is problematic due to the low impedance of the MAV, which is on the order of 1 ohm. Maximum power transfer theorem states that the maximum power in a circuit will be delivered to the load when its impedance matches the source impedance [28]. In order to resolve this mismatch, an L network is used to impedance match. The L network is a MFJ-939 antenna tuner consisting of a capacitor bank and variable inductor. As the load or source impedance changes, the capacitance and inductance values can be adjusted to compensate for the mismatch in circuit. Additionally, it has been shown that implementing a buck boost DC-DC converter on the R x coil can also help with impedance matching [32]. This DC-DC converter is also needed to ensure that the voltage going into the MAV does not exceed the upper voltage limitation of the MAV (4.0 V) causing over-voltage that will damage the MAV’s microcontroller. 3.2 Coil Geometry The coil geometry of the R x and Tx coils greatly affects the amount of power the WPT system is able to deliver and also the distance at which it is capable of delivering that power. In terms of coil design, two geometries are standardly used for WPT coils, either a planar Archimedes spiral coil or a solenoid. Although both are viable options for transmitting power wirelessly, the flat planar coil geometry was selected since it is better suited for fitting on an MAV. Also it gives better horizontal misalignment WPT than its solenoid counterpart. 3.2.1 Coil Geometry Modeling for WPT System In order to determine the best coil geometry for constructing the WPT system, a mathematical model was created using circuit theory. Circuit theory involves taking the full WPT system and reducing it down to an equivalent circuit model in order to directly observe how different parameters affect the overall system. 15 3.2.2 Z-2-Port Modeling An effective way to model the WPT system is to reduce it to a two-port equivalent circuit as seen in Fig. 3.1. For simplicity, the impedance matching components and the power conditioning circuity was excluded from the WPT equivalent circuit model. The power amplifier is modeled as an ideal AC voltage source, with its impedance labeled as Zs . The Tx coil dynamics is described by Z11 = R1 + jωL Tx + 1 , jωCTx (3.5) where ω is the frequency, R1 is the total impedance of the Tx coil, L Tx is the inductance, and CTx is the overall capacitance of the transmit coil. Likewise, the R x coil’s total impedance is modeled as Z22 = R L + R2 + jωL Rx + 1 . jωCRx (3.6) where ω is the frequency, R2 is the total impedance of the R x coil, Rl is the load impedance, L Rx is the inductance, and CRx is the overall capacitance of the R x coil. It is important to note that the impedances for the two coils include both the DC resistance of copper wire, which is very low, and also the AC resistance, which is significantly higher due to the skin effect. Operating at high frequency causes the currents to be pushed towards the edges of the conductive material, resulting in a smaller passageway. The AC resistance for the coils is estimated to be on the order of 5-8 ohms, compared to the DC resistance, which is two orders of magnitude smaller. Using the two-port method, the effect of the mutual inductance on both the R x and Tx coil is Z12 = Z21 = jωM. (3.7) The current through the Tx coil can then be calculated as I1 = VsZ22 . Z11 Z22 + (ωM)2 (3.8) Likewise, the current passing through the R x coil is I2 = − jVs (ωM ) . Z11 Z22 + (ωM )2 (3.9) where Vs is the voltage supplied from the source. This allows us to calculate the instantaneous power in the load resistor as 16 Pε = I22 R L . (3.10) Using these equations, a mathematical model was constructed in MATLAB in order to optimize the Tx and R x coil design. The goal of the optimization was not to maximize efficiency, but rather to maximize a figure of merit, Γ, which is defined as Pε , Γ= Pϕ (3.11) where Pε is the power received at the load of the R x coil and Pϕ is the power required by the MAV to hover with the R x coil attached. The figure of merit, Γ, is of interest because it provides a suitable unitless value that can be used to compare coil designs. Traditionally, efficiency has been used to measure effectiveness of WPT systems. However, maximizing efficiency is not optimal in this thesis due to the MAV’s dynamics. As the weight of the R x coil increases, the power needed for the MAV to hover also increases. Therefore, a heavier R x coil may lead to a more efficient WPT systems, but it will require more power for the MAV to hover. Meanwhile, a lighter coil may be less efficient, but it will require the MAV to use less power, thus allowing it to more easily hover. The figure of merit, Γ, is thus designed to optimize coil geometries that are lighter and more energy efficient for the MAV to carry while providing the MAV with enough power to fly. A Γ greater or equal to 1 is required for the WPT system in order for the MAV to be able to hover at a given distance. However, the higher the Γ is, the higher the available power in the MAV will be. Having a high Γ allows the MAV to use the excess power to recharge its batteries or potentially hover at a greater distance from the transmit coil. An important note is that optimizing Γ across different R x coils is good for a system with a fixed input power; it is always possible to increase Γ by just increasing the input power into the system. The different coil geometric parameters that were examined are: wire diameter, coil diameter, number of turns for the planar coils, and pitch. These parameters were varied and the resultant designs were compared. In order to visualize the effects of the changing parameters, three-dimensional plots were generated. The two parameters that were varied are plotted on the X and Y axis while the Z axis is either efficiency, required power for the MAV to hover with attached R x coil, the power received by the MAV with the R x coil, or Γ. 17 Figure 3.2 shows that the number of turns on the R x coil has a much bigger impact than the pitch of the coil. Additionally, a smaller coil with fewer turns performs superior to a larger coil with a larger number of turns. Figure 3.3 indicates that lower turned coils with small to medium diameter give the largest Γ and efficiency. Lastly, Fig. 3.4 also demonstrates that a lower number of turns produces a higher Γ, whereas wire diameter has little impact comparatively. 3.3 Coil Design and Full WPT System Based on the modeling, a 19 cm outer diameter transmit coil was built using 14 AWG copper wire. The Tx coil has seven turns with a 5 mm pitch. The diameter was chosen based on the desired range of flight for the MAV, 10-15 cm above the Tx coil. It has previously been determined that the maximum power transfer for an inductive magnetic resonance system occurs at a distance equal to the diameter of the Tx coil [33]. The R x coil was chosen to be a single turn 15 cm loop made out of 16 AWG solid copper wire based on the WPT modeling. The WPT systems modeling showed that a smaller receive coil led to a higher Γ. Additionally in [33], it was shown that a larger Tx coil and smaller R x coil combination work best to increase overall WPT over a greater distance. The full equivalent circuit model for the inductive magnetic resonance WPT system is outlined in Fig. 3.5. The RF power amplifier is modeled as an AC voltage source with an output impedance of 50 ohms. The L-matching network consists of variable capacitors and inductors. The Tx and R x coils are modeled as inductors with a resistor. The tuning capacitor is placed in series with the coils inductance and resistance. The R x is attached to a harmonic rectifier. Finally, the 1 ohm MAV load is preceded by a DC-DC converter to maintain a constant voltage of 3.7 V, which is the level required by the MAV’s microcontroller. 18 Figure 3.1. The equivalent circuit model of a WPT system with a T-equivalent two-port network, where Zin is the impedance seen by the RF power supply and Zout is the impedance seen by the load. The T-network represents the two-port element formed by the coupled Tx and R x coils in the WPT systems.[7] Figure 3.2. Performance of the WPT system as pitch and number of turns are varied: (a) power required to lift the MAV, (b) efficiency of the WPT system, (c) power received at the load (Pε ), and (d) figure of merit Γ. In each plot, the number of R x coil turns and pitch were varied. As the results show, fewer turns with wider spacing between turns results in a greater Γ. [7] 19 Figure 3.3. An example of the relationship between number of R x coil turns and R x coil diameter: (a) power required to lift the MAV, (b) efficiency of the WPT system, (c) Pε , and (d) Γ, according to the modeling. As seen, the plot indicates that there is an optimal region to maximize the ratio of power transferred to power required. [7] Figure 3.4. An example of the effect on the Γ by the number of R x coil turns and R x wire diameter according to the modeling. The results shown here are (a) power required to lift the MAV, (b) efficiency of the WPT system, (c) power received at the load (Pε ), and (d) Γ. These results indicate that the Γ is maximized for small wire diameters and low number of turns. [7] 20 Figure 3.5. The equivalent circuit diagram of the WPT system where Rs represents the source resistance, R Tx and R Rx represent the inherent Ohmic and radiative resistance of the Tx and R x coils, respectively, and CTx and CRx are the tuning capacitors used to make the coil resonate at 13.56 MHz. The inductances of the coils are denoted by L Tx and L Rx . Parasitic capacitances of the coils are neglected. CHAPTER 4 DESIGN OF HIGH-FREQUENCY MAGNETICFIELD SENSOR FOR POSITION CONTROL Developing a controller around the WPT system is preceded by building a sensor capable of sensing the WPT. Since the WPT in question is a resonant inductive WPT system operating at a high frequency, a high-frequency magnetic-field sensor must be designed. By using the high-frequency magnetic-field s ensor t o s ense t he W PT s ystem’s magnetic field, the position of the sensor above the W PT’s T x coil can be c alculated. Additionally, the sensor is constrained by its size and weight since it must fit o n t he M AV a nd be light enough for the MAV to carry it in flight. F urthermore, t he s ensor r eadings must be converted into digital readings on the MAV in order to be transmitted to the CPU with the controller using the MAV’s radio. The MAV’s CPU is limited in both input pins and memory: therefore, the sensor must be able to communicate with the MAV’s CPU effectively in this network. Additionally, the sensor’s readings must be accurate enough to be used for position control in a control law. 4.1 Common Position Sensors for MAV While using the magnetic field produced by a wireless power transfer system to sense the position of a MAV is a novel concept, there exist a variety of general position sensors that are commonly used on MAVs. The two most common are ultrasonic sensors and LIDAR sensors. Ultrasonic sensors emit an ultrasonic acoustic wave that is reflected by physical objects. The distance between the object and the sensor is determined by measuring the time it takes for the ultrasonic wave being broadcast to be reflected b ack t o t he s ensor. The MB1240 XL-MaxSonar-EZ4 ultrasonic sensor is an example of a recommended ultrasonic sensors to be used with MAVs [34]. 22 Laser Detection and Ranging (LIDAR) sensors operate on a similar principle to ultrasonic sensors but using different physics. Instead of emitting an ultrasonic wave, the LIDAR sensor pulses a laser. By measuring the time the light takes to reflect back to the sensor, the distance between it and the target object is determined. The LIDAR-Lite v3 made by Garmin is an example of a small LIDAR built for a MAV [35]. The reported accuracy of the LIDAR and ultrasonic sensors is 25 mm and 10 mm, respectively, and Table 4.1 shows a comparison between the two sensors. Both sensors can sense position only between the sensor and another object, and cannot be used to position the MAV above the Tx coil without calibrated walls or objects at a set, known position around the coil. An objective of this thesis is to build a high-frequency magnetic-field sensor that can be used to estimate its position above the Tx coil by sensing the WPT system’s magnetic field. The sensor should have similar characteristics to these commonly used position sensors in order to be used in a control system to control the MAV’s position. 4.2 Examination of the WPT’s Electromagnetic Field The strength of the WPT’s electromagnetic field above the transmit coil is not uniform but rather varies as a function of distance. Because the EMF changes with position, a magnetic-field sensor can be used to sense the magnetic field at a position above the Tx coil, then correlate that reading to a position estimate for the sensor’s location. In order to gain a better understanding of how the magnetic field changes with distance, the field produced by the WPT system can be examined using Maxwell’s equations [36]. The current driven through the Tx coil by the power amplifier outputs an alternating magnetic field in accordance with Maxwell’s fourth equation, B · d l = μ0 ( Ienc + 0 d dt C S E · n̂da), (4.1) where B is magnetic field vector along an incremental segment of path C, μ0 is the magnetic permeability of free space, I is the current in the Tx coil, 0 is the electric permittivity of free space, and E · n̂da is the electric flux passing through a surface S bounded by C. Since I is AC at a frequency of 13.56 MHz, the corresponding B-field is also alternating its direction in time at the same frequency. In order to sense this B-field, a high-frequency inductive-coil sensor will be designed. 23 The reasons why it was choosen over other magnetic-field sensors are discussed in the next section. The inductive-coil sensor, as its name implies, is created by making an inductor through winding conductive wire into a coil as seen in Fig. 4.1. An inductor is a passive unit that stores energy in its magnetic field. In typical circuits, this energy comes from an electrical current passing through the inductor. However in this case, the energy stored in the inductor’s magnetic field is externally generated by the WPT Tx coil. This external magnetic field induces a voltage, E, in the sensor, according to Maxwell’s third equation, C E · d l = − S ∂ B · n̂da. ∂t (4.2) The electric field, E, produced in the closed loop circuit of the sensor is equal to the integral of the magnetic flux, ∂∂tB , multiplied by the number of turns, n, passing through its closed surface, S, which is the area enclosed by the coil in the sensor. Increasing the magnetic flux, surface area, or number of turns, will cause the corresponding voltage to also increase. The strength of the magnetic field varies as a function of distance (unless the antenna is specially designed to produce a uniform magnetic field such as a Helmholtz coil or the inside of a solenoid). The strength of the magnetic field, B, due to a current-carrying loop at a particular point a defined distance away, r, can be calculated by using the Biot-Sarvat Law, which is defined as dB = μ0 Idlsin(θ ) . 4π r2 (4.3) The strongest B field will be located at the center of the current carrying loop where r is at its minimum for the entire coil. To calculate the strength of the magnetic field along the center of the current loop, r2 can be written in Eq. (4.3) as z2 + R2 where z is the distance along the z axis, or out of the coil plane, and R is the radius of the coil [37]. The coil is assumed to lie in the xy plane. Because the current carrying wire is in the form of a circle, the components of dB perpendicular to the z-axis sum to zero in pairs and only the components of the z-axis need to be considered. Therefore, at a point along the z axis the magnetic field is equal to B = ĵ loop dBcos(θ ) = ĵ μo I 4π loop cos(θ )dl . z2 + R2 (4.4) R, z, and θ are constant for every dl along the wire and are related by cos(θ ) = √ R z2 + R2 , (4.5) 24 which can be substituted into Eq. (4.4) to give R B = ĵ μo I 4π (z2 + R2 ) 32 loop (4.6) dl. Since the wire is in the shape of a loop, loop dl = 2πR, (4.7) and substituting Eq. (4.7) into Eq. (4.6) yields R2 B = μo I ĵ, 2 (z2 + R2 ) 32 (4.8) which gives us the strength of the B-field at any point along the z-axis. For any given xy plane located at z distance away from the center of the loop, the maximum magnetic field is located along the z-axis at (0, 0) in the xy coordinates of that plane and subsequently decreases as the distance away from the z centerline increases. To calculate strength of the B-field for a given point r not on the z axis but rather in the yz plane, the equation becomes B= μ0 IR 4π 2π (zcosφ x̂ + zsinφ ŷ + [ R − ysinφ ]ẑ)) 0 ( R2 + r2 − 2yRsin(φ )) , (4.9) where r = yŷ + zẑ and φ is the angle of rotation in the xy plane [38]. Figure 4.2 is a visualization of the B-field emitted by the WPT system’s Tx coil using the above equations. The contour lines show how the B-field strength drops as the distance from the center of the Tx coil increases. The Tx coil used in this thesis is a six-turn Archimedes’ spiral. A good approximation for this geometry is six concentric single loop current-carrying wires of various radii whose centers are coincident. The smallest loop’s diameter is equal to the inner diameter of the Tx coil and the largest loop’s diameter is equal to the outer coil diameter. The inner coils have diameters equal to the previous coils diameter plus the pitch of the Tx coil. The calculated B-field can be used in Eq. (4.2) to estimate the voltage produced in the high-frequency magnetic-field sensor at a given position above the Tx coil. By mapping the magnetic field with this sensor, the sensor’s reading at a position can be used to determine its location above the Tx coil. Once the location of the MAV is determined, it can be used to control the MAV’s position above the Tx coil. 25 4.3 High-Frequency Magnetic-Field Sensor Design The difficulty of sensing the WPT system’s magnetic field lies in its high frequency of 13.56 MHz. Typical Hall effect sensors have an operating frequency range up to 25 KHz, which is far too slow to accurately capture the full magnetic wave from the Tx coil [39]. Additionally, Hall effect sensors have poor resolution compared to other magnetic-field sensors, since the Hall effect produced by the magnetic-field is very small and difficult to measure without amplifying the signal. Magnetoresistive sensors have a greater resolution than Hall effect sensors and a higher operating range, typically up to 1 MHz [40]. There are techniques that have recently been demonstrated to increase the bandwidth of a magnetoresistive sensor up to 5 MHz [40], but that is still not fast enough to fully capture the signal. Therefore, the chosen WPT sensor is an inductive-coil sensor or search coil sensor due to its greater operating bandwidth. The inductive-coil sensor can sense the magnetic flux before it passes through the R x coil as illustrated in Fig. 4.3. 4.3.1 Design of Inductive-Coil Sensor The operating bandwidth of an inductive-coil sensor is far greater than its Hall effect or magnetoresistive counterparts. Inductive-coil sensors with operating frequency ranges up to 100 MHz can be easily manufactured, an upper frequency limit much higher than either Hall effect or magnetoresistive sensors. The upper bandwidth limit of a search coil is the result of its parasitic capacitance due to self-resonance. Besides having a greater operating bandwidth, inductive-coil sensors have greater resolution over their Hall effect counterparts and can be more easily manufactured and implemented than magneto-restrictive senors. For these reasons, an inductive-coil sensor was chosen to be designed for sensing the high-frequency magnetic waves produced by the WPT system. The voltage induced in the search coil by the WPT system’s magnetic field is a sine wave with a given magnitude in accordance with Eq. (4.2) and at the same frequency as the B-field that produced it, 13.56 MHz. In order to process this analog signal for use in a control law, it must be converted to a digital form. Thus, the search coil output must be attached to a microcontroller equipped with an analog to digital converter. However, most microcontrollers (MCU) can only sample analog inputs up to around 9.6 KHz. Attempting to sample the sinusoidal sensor voltage with a sampling rate 1000 orders of magnitude 26 lower then its frequency would lead to severe aliasing. According to the Nyquist theorem, the sampling rate must be at minimum twice that of the wave frequency, which in this instance would be a sampling rate of at least 27 MHz. Obtaining an MCU cable of such high sampling would be expensive and impracticable to fit on a small MAV. 4.3.2 Sample and Hold Circuit Instead, a sample and hold circuit will be implemented to allow the microcontroller to accurately interpret the signal coming from the sensors. Sample and hold circuits are commonly used in AC to DC conversion because it gives a more accurate portrayal of the continuous analog signal as it is digitized [41]. The full circuit is seen in Fig. 4.4. The sensor is modeled as an inductor, which is connected in series with a diode that acts as a half wave rectifier. A resistor is then placed in series with the diode to regulate the current. Next a capacitor is placed in parallel with the inputs of the analog to digital converter (ADC) of the microcontroller. By placing the capacitor in parallel, the magnitude of the sine wave will be captured and held for measurement at the inputs of the ADC. This will allow the ADC to sample the magnitude of the voltage produced by the B-field at the sensor’s location. As the sensor’s position changes, the flux passing through the sensor increases or decreases depending on the sensor’s movement relative to the center of the Tx coil in accordance with Eq. (4.9). With this change, the corresponding voltage will increase or decrease in line with the higher or lower B-field according to Eq. (4.2). 4.3.3 Dynamics of Inductive-Coil Sensor The dynamics of the inductive-coil sensor can be examined by looking at the state equations. The five states of the system are the voltage of the power amplifier, Vs , the voltage in the tunning capacitor, C1 , and the voltage across the sample and hold capacitor, C2 , as well as the current in the transmit coil, L1 , and the current in the inductive-coil sensor, L2 . Deriving the state equations in matrix form yields the following equations: 27 ⎡ ⎤ ⎡ V̇C1 ⎢ ⎢ İL ⎥ ⎢ ⎢ 1⎥=⎢ ⎣V̇C ⎦ ⎢ 2 ⎣ İL2 −1 2 ( L1 + M L2 ) 1 C1 − Rs 2 ( L1 + M L2 ) −M ( L1 L2 + M 2 ) − MRs ( L1 L2 + M 2 ) 0 0 0 M ( L1 L2 + M 2 ) 0 0 M2 ( L1 L22 + M2 L2 ) ⎤ 0 − 1 L2 ⎤ ⎡ R2 M ⎥ VC1 ⎥ ⎢ IL ⎥ ( L1 L2 + M 2 ) ⎥⎢ 1⎥+ 1 ⎥ ⎣VC ⎦ 2 ⎦ C2 2 M R2 R2 I L 2 − L2 ( L1 L22 + M2 L2 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ 0 1 2 L1 + M L 0 2 M L1 L2 + M 2 (4.10) ⎥ ⎥ ⎥ Vs . ⎥ ⎦ M is the mutual inductance, which is the coupling coefficient relating the amount of magnetic flux generated by the Tx coil passing through the sensor. M is defined as M=κ L1 L2 . (4.11) Using a linear graph technique, the Tx coil and inductive-coil sensor are coupled through a transformer that relates the voltage of one to the change of current in the other, Va = M İb [42]. The state equations reveal that the current in the Tx and sensor are coupled in a highly nonlinear way. The ADC is measuring the output of the third state, VC2 , at a sample rate of 9.6 KHz. 4.3.4 Component Selection of Inductive-Coil Sensor For the sample and hold circuit, a 1 nF capacitor and a 56 ohm resistor were selected. Additionally, the sample and hold circuit helps to filter noise in the inductive-coil sensor. Resonant inductive WPT systems emit a lot of electrical noise into the surrounding area. Because of the high frequency and large power output of the WPT system, a small copper wire can easily pick up EMF noise from the WPT on the order of several hundred millivolts that can induce a lot of noise into the sensor signal. This noise is reduced through the sample and hold circuit’s capacitor and by converting the sensor’s signal from analog to digital immediately with relatively little wiring. The upper and lower rail voltage inputs to the ADC for the chosen sensor’s microcontroller are 0 and 5 V, respectively. In order to ensure the sensor’s reading stays within the inputs, the sensor was calibrated to ensure that at the center of the Tx coil, the maximum reading was less than 5 V within its operating range. This calibration was made by changing the sensor’s diameter and number of turns in the coil in accordance with Eq. (4.2). Another possible method would have been to include a voltage divider, but reducing the 28 weight and number of components on the MAVs is critical to maintaining a low operating power. Therefore, meeting this constraint by the physical sensor design rather than adding more components such as a voltage divider is preferred. The search coil sensor schematic is shown in Fig. 4.4. It is a tightly wound solenoid with a 1 cm diameter made up of ten turns of 0.127 mm (36 AWG) copper wire. The half wave rectifier is a Nexperia PMEG10020ELRX schottky diode. It is attached to a low pass filter consisting of a 1 nF capacitor and a 56 Ω resistor. This is connected to the analog to digital converter, ADC, input for an ATMEL ATmega32u4 microcontroller that is capable of sampling at 9.6 kHz with a resolution of 4.8 mV. As seen in Figure 4.5, the assembled sensor is 19 mm x 20 mm, about the size of a US penny, and weighs 1.3 g. It easily fits on a MAV, as shown in Fig. 4.6. 4.4 Characterization The objective of the sensors is not to necessarily give an accurate reading of the Tx coil’s B-field but rather to give an accurate position reading of the distance between the sensor and the center of the Tx coil. This is done by carefully mapping the magnetic field produced by the WPT. While the Tx coil is emitting its EMF, the sensor, attached to the MAV, was swept over the transmit coil. The sensor was held at a fixed z distance above the Tx coil and moved in a raster pattern covering the xy plane. The MAV was fitted with IR markers that were used with Optitrax cameras to record the sensor’s position. This allowed the sensor value and the MAV’s position to be recorded. A ROS program was written to ensure that the sensor reading, and the position data from the cameras, were time synced. The z distance was then changed and the process repeated until the operating volume above the Tx coil had been sufficiently mapped. A map of the sensor making a single sweep across the center of the Tx coil at a height of 11.5 cm above the coil is generated. At the center of the Tx coil, the value of the sensor is 3.8 V. At the edges, the sensor readings decrease to 2.3 V, indicating that the B-field is approximately 3 5 less then at the center. The shape of the sensor values is parabolic, with a rapidly decreasing slope as the sensor moves to the edge of the coil. This correlates with the expected B-field values at these locations given in Eq. (4.9). The B-field is not perfectly symmetrical because of the geometry of the Tx coil. 29 4.5 Calibration Curve Fitting With the magnetic field recorded, the sensor values needed to be mapped to a position estimation through a fitting function. In order to see the best fitting function, the WPT system’s magnetic field data were fitted with a second, third, and fourth order functions as seen in Fig. 4.7. The R2 and adjusted R2 are listed in Table 4.2. The fourth order fitting function had the highest R2 and adjusted R2 , but all values for every fitting function were above 0.96. Additionally, a look up interpolation table was also created with the WPT system’s magnetic field data. Once the fittings functions were created, the sensor was ”swept” again over the coil, recording position data as well as the sensor reading for some experimental data. The initial starting point of the sensor was varied along the edges of the Tx coil in order to get a better accuracy for the calibration fit of the whole coil. Once the experimental data were collected, the four different fitting functions created on the calibration data were used to estimate the position of the sensor value using the experimental sensor’s readings. The difference between the estimated position and the actual position was then calculated to find the raw error for each of the four fitting functions. The average error for each function is reported in Table 4.3 and shown in Fig. 4.8. Additionally the error was normalized by dividing the difference between the estimated position and the actual position by the diameter of the Tx coil. The diameter of the Tx coil was chosen as the normalization factor because it represents the viable operating range for the sensor. Additionally, Figure 4.9 shows the how the average error at each position over the Tx coil changed using each of the fitting functions. While the higher order fitting functions had less error then their lower order counterparts, the best results where from the interpolation table. 4.6 Sensor Testing The next step was to test the sensor in a simulated flight environment. First a calibration table was created by holding the MAV, with attached sensor and IR markers, at a fixed height, and moving it in a raster pattern in the xy plane above a section of the Tx coil. The position and sensor reading was recorded and stored. Figure 4.10 shows the sensor reading as a function of the y coordinate of the MAV. As the sensor is on the edge of the transmit 30 coil, the sensor readings are low but increase as the MAV approaches the center of the Tx coil in an inverted parabolic function. The highest peak represents the MAV moving along the y axis and passing directly over the center of the Tx coil. As the x distance increased away from the center of transmit coil, the subsequent curves have a lower peak. After the calibration data were collected, the MAV was held at a fixed height above the Tx coil and moved in a random pattern over the Tx coil simulating a flight. The x and z location of the MAV was supplied by the camera, while the y coordinate was predicted by the sensor reading using the interpolation table. Figure 4.11 shows the plot of the actual MAV’s position vs. the predicted value. The sensor on average predicted the position within 0.89 cm of the actual position. Figure 4.12 shows the distribution of errors, which follows a Gaussian distribution, with 68.7 % of of positions being predicted within 1 cm of the actual position and 88.4 % of positions estimates accurate within 2 cm. 4.7 Comparison of Commonly Used Position Sensors for MAVs In order to give context for the position sensing capabilities of the high-frequency magnetic-field sensor, it is directly compared with the MB1240 XL-MaxSonar-EZ4 ultrasonic sensor and LIDAR-Lite v3 in Table 4.4. The operating distance of the other two sensors is much greater then that of the pick-up coil sensor. However, the range of the pick-up coil sensor is a function of the diameter of the Tx coil because it is designed solely to operate within the magnetic-field produced by the transmit coil. Increasing the size of the Tx coil will likewise increase the range of operation for the magnetic-field sensor. The resolution of inductive-coil sensor is 0.069 cm compared with 1 cm for the LIDAR and ultrasonic sensors. The reported accuracy of the LIDAR and ultrasonic sensors is 25 mm and 10 mm, respectively. The average accuracy of the high-frequency magnetic-field sensor is between 8 mm and 16 mm, on par with the other sensors. The custom-built high-frequency magnetic-field sensor vastly outperforms the other two sensors in operating frequency, cost, size, and weight. The ultrasonic and LIDAR sensors typically operate on the order of hertz (although the LIDAR can be run up to 500 Hz, it typically operates at 50 Hz), while the inductive-coil sensor operates much faster on the order of kilohertz. Additionally, the cost of the inductive-coil sensor is lower than both 31 sensors while being much smaller and lighter. Being smaller allows it to more easily fit on a MAV and its low weight means that less power is required from the MAV to hover. Again, these comparisons are meant to give context for the characterization of the inductive-coil sensor relative to other sensors used for positioning in MAVs. While ultrasonic and LIDAR sensors can detect physical objects anywhere, the high-frequency magnetic-field sensor is only able to sense the MAV’s position directly over the Tx coil. Likewise, the ultrasonic and LIDAR sensors could not be used to position the MAVs directly above the Tx coil unless there were physical objects that could be used as reference points within all three Cartesian planes. 4.8 Conclusion An objective of this thesis is to build a high-frequency magnetic-field sensor that can be used to estimate its position within the EMF produced by the WPT system. Starting with Maxwell’s equations, the strength of the magnetic field as a function of distance is examined. For a given xy plane above the Tx coil, the magnetic field is strongest at the center and decreases parabolically as distance away from the origin increases. Because of this, by measuring the magnetic field, the position of the sensor relative to the Tx coil can be estimated. Three different types of magnetic-field sensor designs were examined, and the inductive-coil sensor was chosen as the best for the given WPT because of it’s high resolution and operating bandwidth. A high-frequency magnetic-field inductive-coil sensor was then designed and built to sense the 13.56 MHz magnetic field produced by the WPT system. Using the sensor, the magnetic field was mapped by recording the sensor’s position and readings along xy planes at various z heights above the Tx coil. These data were used by several different fitting functions that, given an input of the sensor reading, would estimate the sensor’s position. The interpolation table had the highest accuracy and was chosen to be used in the final design. 32 Table 4.1. A comparison of position sensors for MAVs. Sensor Operating Distance Resolution Accuracy Operating Frequency Cost Size Weight 1 Ultrasonic 765 cm1 1 cm 10 mm 10 Hz 40 USD 19.9 x 22 x 25.11 mm 5.9 g LIDAR 4000 cm 1 cm 25 mm 50-500 Hz 120 USD 20 x 48 x40 mm 22 g Between 0-20 cm the ultrasonic sensor has a much lower accuracy and resolution. Table 4.2. R2 and adjusted R2 for the respective fitting functions Fitting Function R2 Adjusted R2 2nd Order 0.9644 0.9602 3rd Order 0.9840 0.9810 4th Order 0.9936 0.9919 Table 4.3. Mean and max error for each fitting function. . Function Mean Error (cm) Max Error (cm) Norm. Mean Error (%) Norm. Max Error (%) 2nd Order 0.50 1.95 2.6 10.2 3rd Order 0.60 1.25 3.2 6.57 4th Order 0.42 1.04 2.2 5.47 Look up 0.24 1.01 1.2 5.32 The error was normalized by the diameter of the Tx coil, 19 cm. This normalization factor was chosen because the diameter of the Tx coil represents the expected operating range of the sensor. 33 Table 4.4. A comparison of position sensors for MAVs. Sensor Operating Distance Resolution Accuracy Operating Frequency Cost Size Weight 1 2 Ultrasonic 765 cm1 1 cm 10 mm 10 Hz 40 USD 19.9 x 22 x 25.11 mm 5.9 g LIDAR 4000 cm 1 cm 25 mm 50-500 Hz 120 USD 20 x 48 x40 mm 22 g Inductive Coil 19 cm2 0.069 cm 8-16 mm 9600 Hz 17 USD 19 x 20 mm 1.3 g Between 0-20 cm the ultrasonic sensor has a much lower accuracy and resolution. 19 cm is equal to the diameter of the WPT’s Tx coil. Figure 4.1. An inductive-coil sensor, labeled L2 in the electrical circuit in Fig. 4.4, is created by winding 0.127 mm (36 AWG) copper wire into a coil. The 10 turn inductive-coil sensor has a diameter of 1 cm. The simplicity in manufacturing and its larger operating bandwidth make it more appealing to use over other magnetic-field sensors. 34 Figure 4.2. The estimated B-field norm around the Tx coil using Maxwell’s equations. The maximum B-field occurs around the current-carrying wires. At a relatively short distance away from the wires, the largest B-field is found along the z-axis. In an xy plane a z distance above the Tx coil, the B-field decreases parabolically as a function of distance. 35 Figure 4.3. The magnetic field produced by the Tx varies as a function of distance away from the center of the Tx coil. By using a small inductive-coil sensor, the magnetic flux of the WPT can be measured before it reaches the R x coil. As the x, y, z position of the sensor changes, the magnetic flux passing through increases or decreases. By measuring the B-field and comparing the reading to a magnetic field map, the sensor position within the WPT magnetic field can be determined. Figure 4.4. The circuit diagram of the sensor. The power amplifier is represented as an AC voltage source with a source resistance, Rs . The Tx coil is modeled as a LC tank, L1 and C1 , with some real resistance, R1 . The inductive-coil sensor is represented by an inductor, L2 , in series with some parasitic resistance R p . Then a diode, D1 , functioning as a half wave rectifier is followed by a sample and hold circuit consisting of 56 ohm resistor, R2 , and a 1 nF capacitor, C1 . This is connected to the ADC of the microcontroller. Once converted to digital form, the sensor reading is passed to the MCU of the MAV. 36 Figure 4.5. When fully assembled, the sensor weighs only 1.3 g and is 19 mm x 20 mm. This allows it to be easily mounted and carried by an MAV. Figure 4.6. The MAV is fitted with IR markers so the Optitrax camera system can record its position. Additionally, the high-frequency magnetic-field sensor is attached to the MAV, where its readings are transmitted via the MAV’s radio back to the computer. 37 Figure 4.7. A graph of the experimental data and the different order fitting functions. The sensor was swept over the Tx coil and its values were recorded at 1 cm intervals. The fourth order polynominal fits the data the closet, but all functions have an R2 value of above 0.96. The raw sensor data were inverted to have the sensor reading at the center of the coil be zero. 38 Figure 4.8. A graph of the errors of the different fitting functions at different locations. The center of the Tx coil is at 11 cm. The best fitting function was the look up interpolation table, with the higher order functions generally preforming better then their lower order counterparts. 39 Figure 4.9. A graph of the sensor’s voltage reading when it is swept across the Tx coil at 1 cm increments. The center of the Tx coil is located at 11 cm. The strength of the magnetic field mirrors the parabolic shape of the sensor readings. Figure 4.10. A graph of the sensor’s reading vs. y distance during the creation of the calibration table. The curve with the highest peak is when the sensor is swept directly over the center of the Tx coil. As the x distance increases for each sweep, the peak value of the sensor is lowered. The regions where the sensor values are flat is where the computer lost contact with the MAV’s radio. 40 Figure 4.11. A graph of the MAV’s actual y position vs. the position predicted by the high-frequency magnetic-field sensor using the interpolation table. Between the two black lines lie all the points that were predicted within 1 cm of the actual position. Figure 4.12. The histogram shows that the error between the inductive-coil sensor’s predicted position and actual position follow a Gaussian distribution. Additionally 68.7% of the errors are less than 1 cm. CHAPTER 5 CONTROL SYSTEM Using the high-frequency magnetic-field sensor allows the MAV to estimate its position within the magnetic field produced by the WPT. Since this field is mostly limited to the area directly above the Tx coil’s surface, the high-frequency magnetic-field sensor discussed in the previous chapter can be used as a position sensor only within this confined area. If the position of the MAV is known, a position control system can be created based on the sensor’s readings of the WPT magnetic field. The objective of the position control system is to direct the MAV to hover above the center of the Tx coil, where the WPT is at its greatest strength. An important note is that the control system discussed in this chapter is for controlling the MAV in the xy plane at a fixed height z. The z component of the MAV’s position is not controlled by the inductive-coil sensor but through an independent source, such as another sensor or by using an outside camera configuration. 5.1 MAV Positioning Using Gradient Estimation The high-frequency magnetic-field sensor, using the interpolation look up table of the WPT system’s magnetic field, can be used to estimate the distance between itself and the center of the Tx coil. However, by itself, it cannot give the position in an absolute global coordinate frame. In polar coordinate terms, the inductive-coil sensor gives r, the radial distance from the center of the Tx coil or reference point, but θ, the angle from the reference point, remains unknown. With just one sensor, the MAV could calculate the distance from itself to the center of the Tx coil, but will lack the knowledge of which direction to fly in order to reach the center as seen in Fig. 5.1. However, a combination of inductive-coil sensors can be used to not only allow the MAV to know the distance to the center of the transmit coil, but also the vector along which it should fly to reach the center. This is done by using a combination of high-frequency magnetic-field sensors to estimate the gradient of the magnetic field being produced by the Tx coil. 42 In the previous chapter, it was shown in a given xy plane at a distance z above the Tx coil, the WPT’s magnetic field strength is strongest at the center of the Tx coil and decreases as a monotonic function as distance from the center of the Tx coil increases. A plot of the measured magnetic field vs. distance from the center of the Tx coil at ll.5 cm above the transmit coil is given in Fig. 5.2. Figure 5.2 shows that the strength of the B-field vs. distance appears to be a convex function. Since the magnetic-field strength appears to be convex with respect to distance, the gradient of the magnetic field can be used to identify the direction the MAV should use to fly to the center of the Tx coil. By following the gradient, the MAV will reach the global maximum or the position with the greatest B-field, which is at the center of the Tx coil. The data can also be inverted to have the center of the Tx coil be a global minimum as is common in most optimization problems. In that case, the MAV will follow the negative gradient to the bottom of the well. In order to find the gradient, multiple high-frequency magnetic-field sensors must be implemented. The ideal way to find the exact gradient would be to place a single sensor in the center of the MAV, then have an array of sensors completely encircling the MAV. Each of the sensor’s readings from the outer array can be compared with the center sensor’s reading and the combination with the greatest difference is the vector that the magnetic-field gradient lies on. However, although the magnetic-field sensors designed in the previous chapter are very small and light, implementing dozens of them in order to encircle a MAV is impractical. Instead, a position control scheme can be implemented using only four inductive-coil sensors and utilizing an estimation of the B-field’s gradient instead of using the true gradient. 5.1.1 Gradient Estimation Algorithm The gradient of the WPT’s magnetic field can be estimated by using a combination of four high-frequency magnetic-field sensors. Three of the sensors are placed in a triangular configuration on the MAV with the center of the triangle being coincident with the center of the MAV as seen in Fig 5.3. The fourth sensor is placed at the center of the MAV. The three outer sensors are at an equidistant 30 mm from the center of the MAV. One of the sensors lies on the MAV coordinate frame’s x axis, while the other two are placed such that 43 there exist 120 degree angles between all three sensors. The three outer sensors, called sensor A, B, and C to differentiate them from each other, are used to estimate the gradient of the WPT’s EMF. The fourth sensor, placed at the center of the MAV, is used to calculate the distance between the MAV and the center of the Tx coil. Since the location of the sensor relative to the center of the MAV is known, three unit vectors can be constructed pointing from the center of the MAV to each sensor. These 3 unit vectors are a, b and c, being defined in the MAV’s coordinate frame as, a = −0.5 −0.5 1 , c = , ,b= −0.866 0.866 0 (5.1) with a being the unit vector that points from the center of the MAV to sensor A and likewise for sensors B and C. Once the MAV is within the Tx coil’s magnetic field, the sensor’s readings from all four sensors are recorded. The three outer sensor’s readings are normalized by the maximum sensor reading, which occurs at the center of the Tx coil. Each of the three unit vectors are then multiplied by its corresponding normalized sensor’s reading. For example, if sensor A reading is As and the maximum sensor reading at the center of the Tx coil is Hs , then the will be created by new vector A = As a. A Hs (5.2) will also be calculated in a likewise manner. These three vectors, A, B and C, are B and C still pointing from the center of the MAV to their respective sensors, but are no longer of uniform length. Instead, the sensor closest to the center of the Tx coil now has a vector with the greater magnitude than its counterparts and vice versa. However, all three vectors still have a magnitude less than or equal to one because their readings are normalized by the maximum possible sensor reading. The three vectors are then summed, + B + C, =A Q (5.3) This new vector’s direction is used as an approximate gradient to create a new vector, Q. of the magnetic field. While the approximate gradient most likely differs from the true gradient, by following it, the MAV will reach the center of the Tx coil. If the true gradient of the magnetic field is known, the MAV’s path will follow a straight line to the center of the Tx coil as seen 44 in Fig. 5.5. This is contrasted with the path of an MAV using the approximate gradient illustrated in Fig. 5.6. The MAV’s flight path is no longer a straight line but rather deviates slightly from the actual gradient, causing the MAV to weave its way towards the center. However, at each step the MAV is taken closer towards the center until it reaches the middle of the Tx coil. An analogy for this method is of that of a three legged tripod on a hill. By seeing which legs sit higher or lower then their counterparts, the direction towards the summit can be approximated. Then the tripod is moved incrementally in the determined direction up the slope and the process is repeated. that While the three outer sensors are used to determine the directional vector, Q, dictates the direction of flight for the MAV, the fourth sensor is used to determine the distance between the MAV and the center of the transmit coil. Using the interpolation look up table of the WPT’s magnetic field discussed in the previous chapter, the fourth sensor’s reading can be converted into a radial distance estimate from the sensor to the center of the Tx coil. This radial distance estimate, R, can be used to create a vector from the MAV to the center of the Tx coil with F = R Q . Q (5.4) F is a vector that ideally spans from the center of the MAV, to the center of the Tx coil. The distance between the MAV and center of the Tx coil can be used to scale the motor efforts of the MAV’s flight. The further the MAV is away from the center, the greater the motor’s effort. As the MAV approaches the center of the Tx coil, the error shrinks and likewise, the motor’s effort is decreased as well. Furthermore, the rate of change of the error and the integral of the error can be included in the control law to form a PID controller to help prevent the MAV from overshooting and instead enable it to hover near the center of the Tx coil. The exact control law will be discussed later in this chapter. However, with a combination of four high-frequency magnetic-field sensors and using a magnetic field gradient estimation algorithm, the MAV can identify its location relative to the center of the Tx coil and the path it should fly in order to reach the center of the Tx coil. 45 5.1.2 Model Validation In order to validate the gradient estimation approach, a MATLAB simulation was used. The high-frequency magnetic-field sensor data from the previous chapter were used to construct a mathematical model of the magnetic field at 11.5 cm above the Tx coil. The MAV with sensors was created by placing the three sensors in the equilateral triangle, 30 mm from the center, and the fourth sensor in the middle. In the simulation, the MAV was placed in 10,000 different positions and orientations spanning the area of the Tx coil. The gradient estimation algorithm described in Eq. (5.2) to Eq. (5.4) was implemented, and the approximate gradient was compared with the actual gradient at each of the MAV’s position. Figure 5.7 shows the MAV with sensors flying in a particular location and orientation above the Tx coil. The approximate gradient is plotted from the center sensor pointing to the predicted center of the Tx coil. In this case, the approximate gradient differs from the true gradient by a matter of 1.62 degrees. Figure 5.8 shows the distribution of errors between the actual and approximate gradients. The average error is 2.1 degrees, with the max error being 14.5 degrees. However, 95.34% of all errors are under 5 degrees. As long as the error between the true gradient and the approximate gradient is less than 90 degrees, the MAV flight path will take it closer to the Tx coil and, as seen with the error above, the greatest error is only 14.5 degrees, much less than 90 degrees. This error is acceptable as the directional vector will always lead the MAV to go closer to the center of the Tx coil. 5.1.3 Position Control in Three Dimensions Following the 2D gradient estimation algorithm, the MAV will arrive at the center of the Tx coil irrelevant of its current height because, for a given xy plane, the Tx coil’s B-field is always strongest in the center of the coil. However, trying to extend the magnetic field gradient algorithm to operate in all three dimensions is not ideal because the strongest B-field is at the center of the Tx coil at z distance of zero. Following the 3D B-field gradient in this case would cause the MAV to land on top of the Tx coil. Additionally, by being too close in the z direction actually decreases the WPT because it leads to the coupling coefficient, k, to become over-coupled, causing a huge decrease in the efficiency of the 46 WPT system as was previously discussed in Chapter 3. The position control of the MAV can be extended to three dimensions through use of a priori knowledge of the Tx coils B-field. The B-field of the Tx coil can be determined by using the Biot-Sarvat laws discussed above, or by experimentally mapping the area above the Tx using the high-frequency magnetic-field sensors. The gradient of the B-field changes with respect to height above the Tx coil. Closer to the Tx coil, the gradient is steeper and it becomes shallower as the z distance is increased. If the MAV is close to the Tx coil in the z direction, the difference between the three sensor readings will be much greater than if the MAV is far above the Tx coil. By having this a priori knowledge of the B-field, the inductive-coil sensors on the MAV can use a look up table to estimate the z position of the MAV within the magnetic field based on the inductive-coil sensors’ readings using z = L( As , Bs , Cs ) (5.5) where As , Bs and Cs are the sensor readings from sensor A, B, and C, respectively, and L is the look up table of the Tx coil’s B-field that outputs the position estimate for z. If the MAV is below its desired z position for optimal WPT, the MAV can increase its motor’s throttle commands to move up, and vice versa if the MAV is too far above the Tx coil. 5.2 Control System Implementation As mentioned above, the gradient approach system described in the previous section works in a 2D plane parallel to the Tx coil. Another way to control the MAV in a 3D environment is to use a sensor to control the z direction of the MAV. Using a typical MAV positional sensor, such as LIDAR or ultrasonic, on the z axis of the MAV is useful because usually the ground is below the MAV, providing a constant reference point. However in this case, a motion capture camera system is used. The camera system is first set up to enable the MAV to hover above the Tx coil without any sensor feedback, then it is adapted to work with the high-frequency magnetic-field sensors. 5.2.1 Position Control with Motion Capture Cameras A camera system was set up with four Optitrax motion capture cameras. By placing three IR markers on an MAV, the position of the MAV can be triangulated using the cameras. The four cameras where set up in a square approximately 1.5 meters apart and at a 47 height of 1 meter. A custom ROS PID controller was created that allows the user to input a desired hover point for the MAV. Through tuning of the PID controller, the MAV was able to hover at the desired point within a couple cm’s of the target position, with the MAV drifting in the xy plane being the greatest cause of error. Figure 5.9 shows the block diagram of the full control law. The desired position, 11 cm above the Tx coil for example, is defined in a matrix Xd , ⎡ ⎤ zd ⎢ xd ⎥ ⎥ Xd = ⎢ (5.6) ⎣ yd ⎦ , ψd which contains the desired x, y, and z coordinates and the desired yaw for the MAV. Xd is then compared in a negative feedback loop with the actual position of the MAV supplied by the motion capture cameras, X (t), which is ⎡ ⎤ z(t) ⎢ x (t) ⎥ ⎥ X (t) = ⎢ ⎣ y(t) ⎦ . ψ(t) (5.7) Although the block diagram above illustrates the control law in the Laplace domain, the actual controller used was written in the time domain, hence X (t). The block diagram in Fig. 5.9 is merely included to help illustrate the workings of the MAV’s motion capture camera position controller. The resulting error, E(t), ⎡ ⎤ ⎡ ⎤ ez ( t ) zd − z(t) ⎢ e x (t) ⎥ ⎢ xd − x (t) ⎥ ⎥ ⎢ ⎥ E(t) = ⎢ ⎣ ey ( t ) ⎦ = ⎣ yd − y ( t ) ⎦ , eψ ( t ) ψd − ψ(t) (5.8) is mapped by a PID controller with constant gains into an effort matrix by dE(t) Mt ( t ) + Ki = K p E ( t ) + Kd Θd (t) dt where ⎡ k pz ⎢0 Kp = ⎢ ⎣0 0 and 0 k py 0 0 0 0 k px 0 ⎤ ⎡ k dz 0 ⎢0 0 ⎥ ⎥, K = ⎢ 0 ⎦ d ⎣0 k pψ 0 0 k dy 0 0 0 0 k dx 0 ⎤ φd (t) Θd (t) = ⎣ θd (t) ⎦ . ψd2 (t) E(t)dt, ⎤ ⎡ 0 k iz ⎢0 0 ⎥ ⎥, K = ⎢ 0 ⎦ i ⎣0 k dψ 0 (5.9) 0 k iy 0 0 0 0 kix 0 ⎤ 0 0⎥ ⎥ (5.10) 0⎦ k iψ ⎡ (5.11) 48 The output of the camera position PID controller is twofold, the first of which is Mt , the MAV’s throttle motor command, which is Mt ( t ) = k p z e z ( t ) + k d z dez (t) + k iz dt ez (t)dt. (5.12) Mt is a motor effort command that corresponds to the error in the z axis, ez , and is separated from the second output of the camera’s position PID because it is not passed into the MAV’s internal PIV controller. The second output of the camera’s PID controller is Θd , defined as a matrix ⎡ ⎤ φd (t) Θd (t) = ⎣ θd (t) ⎦ , ψd2 (t) (5.13) consisting of the Euler angles of roll, φ, pitch, θ, and yaw, ψ. Due to the dynamics of the MAV, it cannot move in the xy plane without either pitching or rolling. For example, in order for the MAV to move in the positive x direction, it must first pitch forward. The higher the pitch angle, φ, the faster the MAV can move along its x axis, although φ cannot become too large or approach 90 degrees, or else the MAV will no longer be able to counteract its gravitational force and will fall. Likewise, for the MAV to move along its y axis, it must first roll. A combination of the MAV rolling and pitching is needed for it to move in an off axis direction in the xy plane. Therefore, the camera’s PID controller does not produce a direct motor effort in accordance with the errors in the x, ex (t), and y, ey (t) (which are in meters), but instead maps its effort to a desired pitch, φd (t), and roll angle, θd (t), that will allow the MAV to move in the appropriate direction to reduce the error in its position. The error in the yaw, eψ (t), is given as an angle and therefore does not need to be mapped by the PID controller. The second output of the camera’s PID, Θd (t), contains the desired roll, pitch, and yaw angles that will enable the MAV to fly to its desired position, Xd , and is compared in a negative feedback loop with the estimated Euler angles, Θ̂d (t), from the MAV’s IMU and estimator. As stated previously, the motor’s throttle command, Mt (t), does not go into the MAV’s internal PIV controller and hence is not a part of the inner loop. The resulting errors between the desired Euler angles and the estimated Euler angles, Θ̃(t), defined as ⎡ ⎤ φd (t) − φ(t) (5.14) Θ̃(t) = ⎣ θd (t) − θ (t) ⎦ , ψd2 (t) − ψ(t) 49 are passed into the MAV’s internal PIV controller and mapped into a matrix of motor effort commands, Mc (t), by ⎤ mr ( t ) dΘ̃(t) + KiΘ Θ̃dt, Mc (t) = ⎣m p (t)⎦ = K pΘ Θ̃(t) + KvΘ dt my (t) where ⎡ K pΘ k pφ ⎣ = 0 0 0 k pθ 0 ⎡ ⎤ ⎡ k dφ 0 ⎦ ⎣ 0 , KvΘ = 0 k pψ 0 0 k dθ 0 ⎤ ⎡ 0 k iφ ⎦ ⎣ 0 , Ki Θ = 0 k dψ 0 0 k iθ 0 (5.15) ⎤ 0 0 ⎦. k iψ (5.16) The PIV controller differs from a PID controller because it does not take the derivative of ˙ Like the Θ̃ but instead directly uses the IMU’s measurement of its angular velocity, Θ̂. camera’s PID controller, the MAV’s PIV controller also uses constant gains. The output of the MAV’s PIV controller, Mc , is a matrix containing the motor effort commands for the MAV’s roll, mr , pitch, m p , and yaw, my , angles. It is recombined with the motor’s throttle command, Mt , to form M, which is ⎡ ⎤ mt (t) ⎢ mr ( t ) ⎥ Mt ( t ) ⎥ M(t) = ⎢ ⎣ m p ( t ) ⎦ = Mc ( t ) . my (t) (5.17) However, in order to give the correct motor efforts for each individual motor, M (t) is passed into the motor mixer, which produces I (t) by ⎡ ⎤ ⎡ ⎤ ⎤⎡ mt (t) I1 (t) 1 −1 1 1 ⎢ I2 (t)⎥ ⎢1 −1 −1 −1⎥ ⎢ mr (t) ⎥ ⎥ ⎢ ⎥ ⎥⎢ I (t) = ⎢ ⎣ I3 (t)⎦ = ⎣1 1 −1 1 ⎦ ⎣m p (t)⎦ . I4 (t) my (t) 1 1 1 −1 (5.18) I (t) is a matrix with the individual motor commands for each the four MAV’s motors. As the motor mixers sends I (t) to the MAV’s four motors, the MAV’s dynamics cause it to move, changing its position, X (t), and Euler’s angles, Θ(t). This change in position and orientation is measured and fed back into the loop by the motion capture cameras and the MAV’s IMU/Estimator, respectively. To gain a better look at the operation of the control system, the equations can be rewritten. For example, the motor command for pitch is m p (t) = k pφ φ̃(t) + k vφ where dφ̃(t) + k iφ dt φ̃(t)dt, (5.19) 50 φ̃(t) = (φd (t) − φ̂(t)) = ([k p ex (t) + k d dex (t) + ki dt ex (t) dt] − φ̂(t)) (5.20) and ex (t) = ( xd − x (t)). (5.21) Substituting Eq. (5.20) and Eq. (5.21) into Eq.( 5.19) and grouping together like terms yields m p (t) = d2 a1 da2 + + a3 + dt dt a4 dt + a5 dt dt, (5.22) where a1 = k d k vφ ( xd − x (t)), (5.23) a2 = k d k pφ ( xd − x (t)) + k p k vφ ( xd − x (t)) + k vφ φ̂(t), (5.24) a3 = k p k pφ ( xd − x (t)) + k i k vφ ( xd − x (t)) + k d k iφ ( xd − x (t)) + k pφ φ̂(t), (5.25) a4 = k p k iφ ( xd − x (t)) + k i k pφ ( xd − x (t)) + k iφ φ̂(t) (5.26) a5 = k i k iφ ( xd − x (t)). (5.27) and which displays the actual gains from the error between the desired position input and the actual position of the MAV. The equations for mr (t) and my (t) are of a similar form, but with their respective error and gains used instead of those from m p (t). mz (t) differs from mr (t), my (t), and m p (t) because it does not go through the MAV’s internal PIV control. Its final form with its gains is seen in Eq. (5.12). The values of each the PID’s and PIV’s gains are given in Table 5.1 and Table 5.2, respectively. 5.2.2 Position Control Using Inductive-Coil Sensor In order to control the position of the MAV using the inductive-coil sensors, the MAV position control system based on the camera control system can be modified as seen in Fig. 5.10 to use the inductive-coil sensors instead of the motion capture cameras. The desired position of the MAV is split into two pieces, first the location of MAV in the xy plane, Xd , defined as ⎡ ⎤ xd Xd = ⎣ y d ⎦ , ψd (5.28) 51 which typically would be at the center of the Tx coil. The second is Zd , the desired height of the MAV above the Tx coil, 11 cm for example. Xd is passed into a magnetic field map taken from a calibration table of the WPT’s magnetic field as follows ⎡ ⎤ v d1 ⎢vd ⎥ 2⎥ Vd = ⎢ ⎣ v d ⎦ = L t ( Xd ), 3 v d4 (5.29) were Lt is an interpolation table made from mapping the WPT system’s magnetic field above the Tx coil and vi is the desired inductive-coil sensor reading for each of the MAV’s four sensors. This mapping converts the desired position of the MAV, Xd , to the desired sensors readings at the MAV’s desired position, Vd . Vd is then compared in a negative feedback loop to the current sensors readings, V (t), to produce Ṽ (t) by ⎡ ⎤ v d1 − v 1 ( t ) ⎢ v d − v2 ( t ) ⎥ 2 ⎥ Ṽ (t) = Vd − Vs (t) = ⎢ ⎣ v d − v3 ( t ) ⎦ . 3 v d4 − v 4 ( t ) (5.30) Ṽ (t) is passed into a PID controller with the magnetic-field gradient estimation algorithm. Using Eq. (5.2) to Eq. (5.4), the gradient estimation algorithm estimates the direction and distance between the MAV and its desired position, while the PID controller maps that error to the effort matrices as follows, Θd (t) = K p Ṽ (t) + Kd where ⎡ k pv1 ⎢ 0 Kp = ⎢ ⎣ 0 0 and 0 k pv2 0 0 0 0 k pv2 0 dṼ (t) + Ki dt ⎤ ⎡ 0 k dv1 ⎥ ⎢ 0 ⎥ 0 ,K =⎢ 0 ⎦ d ⎣ 0 k pv4 0 0 k dv2 0 0 0 0 k pv3 0 Ṽ (t)dt, ⎤ ⎡ 0 k iv1 ⎥ ⎢ 0 ⎥ 0 ,K =⎢ 0 ⎦ i ⎣ 0 k dv4 0 ⎤ φd (t) Θd (t) = ⎣ θd (t) ⎦ . ψd2 (t) (5.31) 0 k iv2 0 0 0 0 k pv3 0 ⎤ 0 0 ⎥ ⎥ (5.32) 0 ⎦ k iv4 ⎡ (5.33) This controller, similar to the one used with the motion capture cameras, outputs Θd (t), the MAV’s desired Euler angles that will allow it to fly to the desired position. The desired height, Zd , is compared in a negative feedback loop with the actual height, Z (t), which comes from an outside sensor source, to form Z̃ (t), which is Z̃ (t) = Zd − Z (t). (5.34) 52 Z̃ (t) is then input into a PID controller that maps it to the motor’s throttle command, Mt (t) as follows dez (t) + k iz Mt ( t ) = k p z e z ( t ) + k d z dt ez (t)dt. (5.35) Once Mt (t) and Θd (t) are calculated, the inner loop of the MAV’s PIV controller remains unchanged and follows Eq. (5.11) to Eq. (5.18). As the MAV dynamics cause the position of the MAV to change above the Tx coil, the magnetic flux captured by the inductive-coil sensors changes as well and they are passed back in a negative feedback loop to the desired sensor readings repeating the cycle. 5.2.3 Hybrid Position Control Using Cameras and InductiveCoil Sensors In order to demonstrate a proof of concept for the position control system with the high-frequency magnetic-field sensors, a hybrid control system using the high-frequency magnetic-field sensor and the motion camera system is outlined in Fig. 5.11. The desired position, Xd, is 11 cm above the center of the Tx coil. Two of the three Cartesian coordinates, x, z, and the desired yaw orientation of the MAV are controlled using the motion capture camera system. The third coordinate, y, is determined solely by the high-frequency magnetic-field sensors. The desired ψ, x, and z coordinates are compared in a negative feedback loop with the actual ψ, x, and z coordinates of the MAV supplied by the motion capture cameras to form E1(t), which is ⎡ ⎤ ⎡ ⎤ ez ( t ) zd − z(t) E1 (t) = ⎣ ex (t) ⎦ = ⎣ xd − x (t) ⎦ . eψ ( t ) ψd − ψ(t) (5.36) The y coordinate is not supplied by the motion capture cameras. The high-frequency magnetic-field sensor reading, vs (t), is mapped into an estimate of the MAV’s position y coordinate, ŷ(t), by ŷ(t) = Lt (vs (t)), (5.37) where Lt is the interpolation look up table of the WPT systems Tx coil’s magnetic field. This estimated y coordinate, ŷ(t), is compared with the desired y component, yd in a negative feedback loop to produce ỹ(t) by ỹ(t) = yd − ŷ(t). (5.38) 53 Both error measurements, E1 (t) and ỹ(t), are combined in a single matrix ⎡ ⎤ ez ( t ) ⎢ ex (t) ⎥ E (t) ⎥ Eh ( t ) = 1 =⎢ ⎣ eψ ( t ) ⎦ . ỹ(t) ỹ(t) (5.39) Eh (t) is passed into a PID controller, following Eq. (5.9) to Eq. (5.18) to produce I (t), a matrix of the individual motor commands for the MAV’s four motors. As the dynamics of the MAV cause it to move, its x, and z coordinates as well as its yaw, ψ, is fed back using the motion capture cameras while its y coordinate is estimated by the inductive-coil sensor. The values of the gains used in the hybrid position control system are listed in Table 5.3. The MAV’s PIV gains are unchanged from Table 5.2. The experimental results from the implementation of the hybrid control system are discussed in the next chapter. 5.3 Conclusion In conclusion, the position of the MAV over the Tx coil can be controlled based on the readings from the high-frequency magnetic-field sensors designed in the previous chapter. Attaching only one inductive-coil sensor to the MAV will enable the MAV to sense the WPT’s magnetic field and estimate the distance between itself and the center of the transmit coil. However, having only one sensor will not give the MAV any information about the direction it should fly to reach the center of the Tx coil. A gradient estimation algorithm is described that, through the use of four inductivecoil sensors, will enable the MAV to estimate the gradient of the transmit coil’s magnetic field. Because the magnetic field produced by the WPT system appears convex, by following the gradient, the MAV will reach the center of the Tx coil. Through simulation, it is shown that following the approximate gradient of the magnetic field produced by the gradient estimation algorithm enables the MAV to fly to and hover at the center of the Tx coil. Additionally, three control laws are presented and discussed. The first one allows the MAV to hover at a position based on position feedback from motion capture cameras. The second is a proposed control law that will enable the MAV to navigate to and hover at the center of the Tx coil using four high-frequency magnetic-field sensors and the gradient estimation algorithm. The third is a hybrid control system that uses both the motion capture 54 cameras and the high-frequency magnetic-field sensors. The hybrid system demonstrates a proof of concept for position control of a MAV based on sensing the magnetic field produced by the WPT. The x and z coordinates of the MAV are controlled through the motion capture cameras, while the y coordinate of the MAV is controlled through an inductive-coil sensor and the magnetic field map of the Tx coil. The experimental results from this control law are discussed in the next chapter. 55 Table 5.1. The gains for the motion capture camera’s position control PID controller. z x y ψ k p 12000 45 45 150 k i 1000 5 5 5 k d 11600 25 25 75 Table 5.2. The gains for the MAV’s internal PIV controller. kp k∗i kd φ 2500 2000 2.5 θ 2500 2000 2.5 ψ 700 16.7 0.37 k i * has integration limits of 666, 666, and 360. Table 5.3. The gains for the hybrid position control PID controller. z x y ψ k p 12000 25 17 150 k i 1000 2 4 5 k d 11600 17 15 75 56 Figure 5.1. The contour plot for a WPT system’s magnetic field. The Tx coil is centered at (0,0). The high-frequency magnetic-field sensor, marked as the red circle, is able to sense the strength of the magnetic field and use a look up interpolation table to estimate r, the sensor’s distance from the center of the Tx coil to itself. Although the sensor knows on which level set it lies, it by itself cannot calculate θ, the angle from the center of the Tx coil to itself. 57 Figure 5.2. A plot of the WPT system’s magnetic field vs. distance from the center of the Tx coil. The magnetic field is strongest at the center of the coil, then decreases monotonically as the distance away from the center increases. The data are inverted so the center of the Tx coil will be a global minimum instead of a maximum. The data appear to be a convex function. 58 Figure 5.3. A schematic of the high-frequency magnetic-field sensor layout on the MAV. Inductive-coil sensor A is placed on the MAV’s coordinates frame’s x axis. inductive-coil sensor B is placed at a clockwise rotation of 120 from inductive-coil sensor A, while inductive-coil sensor C is placed at 120 degrees counterclockwise rotation from inductive-coil sensor A. All three inductive-coil sensors are placed equidistant from the center of the MAV, at a distance of 30 mm. Figure 5.4. Panel 1: The three sensors, A, B and C, are are placed in a triangular configuration, each 30 mm from the center of the MAV. Three unit vectors, a, b and c, point from the center of the MAV to each sensor. An example value of a sensor reading is shown next to each sensor. Panel 2: The three unit vectors are scaled by their corresponding which is an sensor values. Panel 3: The three new vectors are summed to produce Q, approximation of the gradient of the magnetic field. 59 Figure 5.5. The contour plot represents a WPT system’s magnetic field. The Tx coil is centered at (0,0). The dots are the positions of the MAV. The different colors represent the MAV’s positions at different times, with the red dot representing the starting position of the MAV. If the true gradient of the WPT’s magnetic field is known, the MAV can move along it towards the center of the Tx coil in a straight line. 60 Figure 5.6. The contour plot represents a WPT system’s magnetic field. The Tx coil is centered at (0,0). The dots are the positions of the MAV. The different colors represent the MAV’s positions at different times, with the red dot representing the starting position of the MAV. Because the true gradient of the WPT system’s magnetic field is unknown, a gradient estimation is used that allows the MAV to move towards the center of the Tx coil. Because of the error in estimating the gradient, the path is not a straight line along the gradient but rather swerves and weaves a little off the true gradient. However, the MAV still arrives at the center of the Tx coil. 61 Figure 5.7. The contour plot represents a WPT system’s magnetic field. The Tx coil is centered at (0,0). The MAV, with three sensors placed in a triangle with the fourth sensor in the middle, is flying within the magnetic field. The sensors and the gradient approximation algorithm are used to produce a vector that points from the MAV to the center of the Tx coil. In this particular position, the approximate gradient differs from the true gradient by 1.62 degrees. 62 Figure 5.8. A histogram of the error between the actual gradient and the estimated gradient from the MATLAB simulation. The average error is 2.1 degrees, with 95% of errors being under 5 degrees. Figure 5.9. The block diagram of the control system enabling the micro-aerial vehicle (MAV) to hover in place using motion capture cameras. The desired position, Xd , is defined as a matrix containing the three Cartesian coordinates and the desired yaw of the robot. Xd is compared with the measured position, X, of the MAV given by the cameras. The error, E, is passed into a PID controller that outputs the desired Euler angles, Θd , and a motor throttle command, Mt . The Euler angles are passed into the MAV’s internal PIV controller, to produce a Mc , a matrix of motor commands, which is recombined with Mt and passed into the motor mixer. The motor mixer then outputs individual motor commands to each of the four MAV’s motors, causing the MAV to move. 63 Figure 5.10. The block diagram of the control system enabling the micro-aerial vehicle (MAV) to hover at the center of the Tx coil using four inductive-coil sensors. The desired position of the MAV in the xy plane, Xd , is mapped to a desired sensor’s readings for each sensor by the magnetic field map of the Tx coil’s B-field. The desired height, Zd , is separate from Xd . Xd is compared with the measured sensor’s readings and the resulting error is input into a PID controller with the gradient estimation algorithm. The PID controller outputs Θd , the Euler angles, which is passed into the internal MAV control loop that follows the same path described in the motion capture control law section. Zd is compared with the actual height, Z, and passed into its own PID controller with outputs Mt , the throttle commands for the MAV’s motors. Mt and Mc , the output of the MAV’s internal PIV controller, are recombined and fed into the motor mixer following the same path as described previously in the camera control section. Figure 5.11. The block diagram of the hybrid control system enabling the micro-aerial vehicle (MAV) to hover above the center of the Tx coil. The desired position of the MAV is split into two matrices. The first, yd , contains the desired y coordinate and the second, xzψd , contains the desired x and z coordinate as well as the desired yaw. yd is compared with the y position estimated by the inductive-coil sensor while xzψd is compared to the position of the MAV supplied by the cameras. The signals are rejoined and passed into the PID controller. The output of the PID controller follows the same path previously discussed in the camera control section. CHAPTER 6 EXPERIMENTAL RESULTS In order to validate the previously discussed models, several experiments are conducted. First, the wireless power transfer system model is validated by constructing a Tx and two R x coils then comparing their experimental WPT transfer efficiency to their theoretical WPT efficiency predicted by the model. Second, the wireless power transfer system is integrated with a custom-built MAV. The MAV takes off, hovers, and lands while receiving all motor power from the WPT system. The hybrid control system, discussed in Chapter 5, is implemented and used to enable a MAV to hover at the center of the WPT system’s Tx coil. The position of the MAV is partially controlled using the custom-made high-frequency magnetic-field sensor to sense the magnetic field produced by the WPT system. Lastly, The full system is then demonstrated, with a flying MAV being powered wirelessly and partially controlling its position by sensing the WPT’s magnetic field with the custom-built high-frequency magnetic-field sensor. 6.1 MAV Power Characterization The chosen MAV to be used with the WPT system is a custom-built quadcopter. The motors and propellers are from a JJRC H98 Drone. The frame is also from the JJRC H98 Drone but is heavily modified in order to reduce weight and attach the R x coil. The Bitcraze Crazyflie 2.0 flight controller was used as the MAV’s microcontroller. In order to characterize this MAV’s power requirements, a thrust test was conducted. The MAV was attached to a support and placed on a scale. The initial mass of the MAV plus the support was recorded. The MAV was connected to a DC power supply. The DC power supply was turned on, and the MAV’s motor’s thrust levels were varied. At different thrust levels, the power supplied by the DC power supply and the mass reading on the scale were recorded. The thrust of the MAV’s motors was determined by subtracting the recorded mass reading of the scale from the mass of the MAV. These values were plotted 65 in order to determine a relationship between the magnitude of the MAV’s thrust and its power requirement. Figure 6.1 is the plot of the MAV’s thrust vs. its power consumption. The custom-built MAV by itself weighs 58 g without the WPT system’s R x coil attached. With the addition of the R x coil, power rectification/conditioning circuitry, and the infrared (IR) markers, the total mass of the MAV is 86 g. At 86 g, the MAV requires about 14 W of power in order to hover. The maximum thrust produced by the MAV is 115 g, which requires around 19 W of power. In order for the MAV to take off from a platform and hover above the Tx coil, the WPT system must be able to deliver more than 14 W of power to the MAV. 6.2 Wireless Power Transfer System Model Validation In order to validate the WPT model developed in Chapter 3, an experiment was designed to compare the theoretical values and efficiency of the WPT system model to the physical WPT system that was fabricated. For these tests, all impedance matching components, AC-DC rectifiers, and DC-DC converters were removed. The WIBOTIC’s power amplifier is placed in series with the Tx coil. The transmit coil is a seven turn, 19 cm outer diameter coil made from 1.63 mm (14 AWG) copper wire. Two different R x coils were constructed. The first is a two turn, 13.7 cm diameter coil with a 1 mm pitch made of 1.29 mm (16 AWG) copper wire. The second is a four turn, 15.3 cm diameter coil with a 2 mm pitch made of 1.29 mm (16 AWG) copper wire. The R x coil is placed 10 cm away from the Tx coil. A 12.5 Ω high-frequency resistor is attached in series with the R x coil acting as the load resistor. The Tx and R x coils are both tuned to a resonant frequency of 13.56 MHz. The theoretical values from the model and the experimental values from the test for the two R x coils are listed in Table 6.1 and Table 6.2. The theoretical inductance, L, of the R x coils was calculated using a modification of Wheeler’s formula [33], which is: L= N 2 ( D0 − N (w + p))2 39.97 , 16D0 + 28N (w + p) 106 (6.1) where N is the number of turns, D0 is the outer diameter of the coil, p is the pitch of the coil, and w is the wire diameter of the coil. Using Eq. (6.1), the theoretical inductance was reasonably close to the measured inductance of the R x coils differing by 16.2% and 4.4%. 66 The larger discrepancy of 21.5% and 25.8% occurred between the theoretical and measured capacitance of the R x coils. The theoretical capacitance, C, was calculated by: C= 1 , (2π f )2 L (6.2) where L is the inductance of the coil and f is the resonant frequency. A possible reason for the discrepancy between the experimental values and the theoretical values given by Eq. (6.2) is parasitic capacitance. Because of the high resonant frequency, 13.56 MHz, the tuning capacitance needed to bring the coil into resonance is on the order of picofarads. The parasitic capacitance, which is neglected in Eq. (6.2), is likely on a similar order of magnitude. When the tuning capacitance is summed with the parasitic capacitance, the resulting resonant frequency of the coil is different than the desired frequency of 13.56 MHz. Therefore, when constructing the R x coils, the first tuning capacitance used was the predicted value from Eq. (6.2). Then using a function generator and an oscilloscope, the resonant frequency of the R x was measured. The tuning capacitance was then adjusted until the resonant frequency of the R x coil was 13.56 Mhz. The WPT system model predicted power transfers of 16.91 W and 15.33 W for the two coils. This model works quite well for the two-turn coil where the experimental power transfer was within 11 % of the predicted value at 15.2 W. The four-turn coil had a much larger deviation, 52.5 %, from its theoretical value. Possible explanations for this deviation could include: use of a coupling coefficient in the model that differed significantly from the actual coefficient, or increased AC resistance in the four-turn coil. The experiments showed that while the WPT model is not a perfect indicator, it can be used as a predictive model to help design a WPT system for an MAV. The experimental results further validated the modeling results discussed in Chapter 3; that is a medium sized R x coil with fewer number of turns yields a higher figure of merit, Γ, for the MAV’s WPT system. Additionally, it showed that the constructed WPT system is capable of delivering enough power to a hovering MAV at 10 cm. 6.2.1 Coil Misalignment The first set of experiments was done with the R x coil directly centered above the Tx coil. This represents the best possible WPT case for the system. It has been shown that as the R x coil becomes more misaligned with the Tx coil, the efficiency of the WPT system 67 decreases [43]. In order to gain insight into how coil misalignment will effect the MAV’s ability to fly using the WPT system, a second experiment was conducted with the two-turn 13.7 cm diameter R x coil. The R x coil with a 12.5 Ω load was placed 10 cm above the center of the Tx coil and the power across the load resistor was recorded. The R x coil was moved horizontally in 1 cm displacement increments and the power across the load resistor was recorded each time. Figure 6.3 shows how the efficiency of the system decreases as the the misalignment between the Tx and R x coil increases. Within 3 cm of horizontal axial displacement, the efficiency of the WPT is fairly constant and above 90%. The consistency in this region is possibly due to the smaller size of the R x coil versus the Tx coil. At 4 cm of displacement, the efficiency drops off significantly and continues to fall as the misalignment between the two coils increases. These results indicate that, if the MAV with the R x coil stays within 4 cm of horizontal misalignment from the center of the Tx coil, then it will receive enough power to hover. 6.3 Wirelessly Powering an MAV With the validation of the WPT system model, the full system can be used to power an MAV in flight. The full WPT transfer system seen in Fig. 6.4 has been implemented experimentally. The operating frequency of 13.56 MHz is achieved through the WIBOTIC RF power amplifier. The L-matching network is an MFJ antenna tuner, which is made up of a variable inductor and a capacitor bank. The Tx coil is placed under a custom-made acrylic platform that allows the MAV to rest 7 cm above the Tx coil. The platform ensures that the MAV’s R x coil will not get too close to the Tx coil, thus leading to over-coupling, which reduces the effectiveness of the WPT system. The R x coil is attached to a custom-built PCB containing the harmonic AC-DC rectifier and DC-DC converter. The PCB is attached to the bottom of the MAV’s frame. The position of the MAV is being controlled through a custom-built ROS PID position controller using OPTIRAX motion capture cameras. The full dynamics of the PID controller using the motion capture cameras controller is discussed in Chapter 5 in Eq. (5.6) to Eq. (5.18). Three IR markers are attached to the top of the MAV’s frame in order to enable the motion capture cameras to triangulate the MAV’s position. Four cameras are set up in an approximately 1.5 x 1 m2 box at a height of 1 m. 68 Figure 6.5 shows a collage of the MAV flying while being powered through the resonant inductive wireless power transfer system. First, the MAV is placed on the raised platform directly centered over the Tx coil. The WPT system is turned on and the MAV is throttled on. The MAV successfully takes off and reaches a hovering height of approximately 15 cm above the Tx coil. The MAV is in flight for 12 seconds before landing on the raised acrylic platform. The MAV is equipped with a small one cell lithium polymer battery to supply power to its MCU, but the motor power is being supplied through the WPT system. The WPT system is outputting 60 W of power measured by a radio wave power meter. The estimated transfer efficiency of the total WPT system when the MAV is flying is around 30% (about 18 W). 6.4 Hybrid Control System Implementation Now that the WPT transfer system has been proven to be able to power a hovering MAV, the next step is to show that the high-frequency magnetic-field sensors can be used to help control the position of an MAV above the Tx coil. A high-frequency magneticfield sensor is attached to a Crayzflie 2.0 MAV. In order to better gage the workings of the hybrid position control system removed from the added dynamics and variables of the WPT system, the Crazyflie 2.0 is powered by an on-board battery. The global Cartesian coordinate frame is defined such that the center of the Tx coil lies at the origin. The desired position of the MAV, Xd , is hovering 7 cm above the center of the Tx coil, which is defined as ⎡ ⎤ ⎡ ⎤ zd 0.7 ⎢ xd ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ Xd = ⎢ ⎣ yd ⎦ = ⎣ 0 ⎦ . ψd 0 (6.3) Xd is given in meters, hence 0.7 m. The hybrid position controller, outlined in Chapter 5, is implemented following Eq. (5.36) to Eq. (5.39) and Eq. (5.9) to Eq. (5.18). Four OTITRAX motion capture cameras are set up in a rectangle approximately 1 x 1.5 m at a height of 1 m. The MAV has three IR markers attached to its frame, allowing the cameras to triangulate its position. The inductive-coil sensor is attached to the bottom of the MAV’s frame at its center. The WPT system is powered on. The MAV’s launching point is the center of the Tx 69 coil. The custom ROS MAV hybrid position controller is launched, causing the MAV to throttle up its motors and fly to the desired height of 7 cm. The MAV’s launch from the Tx coil causes it to be displaced some distance from the origin or center of the Tx coil. Once the launch sequence of the MAV is complete, the hybrid position controller attempts to drive the MAV to Xd , the desired position 7 cm above the center of the Tx coil, using feedback from the motion capture cameras to control its x, z coordinates and its yaw. The y coordinate is controlled with feedback from the inductive coil high-frequency magneticfield sensor as pictured in Fig. 6.6. Sometimes over the course of the experiments, the MAV launch would cause the MAV to fly several centimeters past the edge of the Tx coil. In this situation, the y coordinate is supplied by the camera until the MAV reaches the edge of the Tx coil, at which point the y coordinate is exclusively determined by the high-frequency magnetic-field sensor. The position controller successfully drives the MAV to the center of the Tx , and holds the MAV hovering at a height of 11 cm. The experiment is repeated ten times with the results listed in Table 6.3. The average error between the MAV’s y coordinate and the MAV’s estimated y coordinate using the high-frequency magnetic-field sensor was 1.86 cm. This resulted in an average steady state position error of 2.73 cm between the MAV’s desired y position and actual location. For comparison, when the motion capture camera control system outlined in Chapter 5 was implemented, the average steady state error between the desired position of the MAV and the actual position of the MAV was 0.64 cm or 0.034 when normalized by the diameter of the Tx coil. Although the position of the MAV was controlled at a higher accuracy using just the motion capture cameras, the hybrid control system demonstrated that by sensing the magnetic field of the WPT system, the MAV is capable of controlling its position above the Tx coil. Additionally, although the error in the sensor’s position estimates led to greater positional error, it kept the MAV within the diameter of the Tx coil. The average positional error of 2.73 cm is a small enough displacement that an MAV powered wirelessly would still be able to receive enough power to hover over the Tx coil in those circumstances. 70 6.4.1 Full System Demonstration With the successful demonstration of the hybrid controller, the next step is to implement the hybrid controller with an MAV powered through the WPT system. The inductive coil is attached to the underside of the custom-built MAV’s frame. The MAV, with attached R x coil and power conditioning electronics, is placed on the 7 cm raised acrylic platform centered above the Tx coil. The MAV takes off, again being displaced some distance from the center of the Tx coil. Using the high-frequency magnetic-field sensor to control its y position, the MAV flies to the center of the Tx coil. After hovering for 23 seconds, the MAV lands once again on the platform. This flight is illustrated in Fig. 6.7. This MAV’s flight is similar to the one seen in Fig. 6.5, except that the y positioning of the MAV is controlled through the feedback from the high-frequency magnetic-field sensor. During the MAV’s flight, while attempting to hover at the center of the Tx coil, the MAV oscillates a few centimeters over the center of the Tx coil along the y-axis as pictured in Fig. 6.8. This oscillation is likely due to two factors: the error in the high-frequency magnetic-field sensor’s position estimates and probably the larger factor of having nonoptimal gains in the PID controller. The y position of the MAV during its flight as a function of time is plotted in Fig. 6.9. The y position estimates from the high-frequency magnetic-field sensor are superimposed on the actual y location of the MAV. The position estimates from the high-frequency magnetic-field sensor track the actual position of the MAV fairly well, with an average error between the actual y location of the MAV and the estimated y location from the inductive-coil sensor of 1.61 cm for this flight. 6.5 Conclusion In conclusion, a custom MAV was built and integrated with a WPT system. The power requirements of the MAV were characterized. The MAV, attached with the R x coil and power conditioning circuits, required 15 W of power in order to hover above the Tx coil. The wireless power transfer system model was validated by building/testing two different R x coils and comparing the WPT system model’s predicted values to the experimentally determined values. The lower turned, smaller R x coil had the greatest efficiency and figure of merit. Additionally, it revealed that the fabricated WPT system was sufficient to provide enough power to sustain a hovering MAV. An experiment was conducted to observe how 71 the WPT efficiency changes as the misalignment between the R x and Tx coils varies. This study showed that the WPT system can supply a hovering MAV with enough power as long as the MAV did not exceed a 4 cm horizontal misalignment between the R x and Tx coils. Next, the full WPT system was successfully implemented. Relying on WPT from the Tx coil to power its motors, the MAV successfully took off, hovered for 12 seconds, then proceeded to land. The power transfer efficiency for the WPT system and subsequent power conditioning electronics was estimated to be around 30%. A custom ROS position control system controlled the position of the MAV using four motion capture cameras, enabling it to hover above the Tx coil. The high-frequency magnetic-field sensor was then successfully interfaced with an MAV. The hybrid position control system was implemented that partially controlled the position of the MAV through sensing the magnetic field of the WPT system. This positioning control was accurate enough to enable the hovering MAV to stay within the Tx coil’s diameter. The hybrid position control system was then implemented on a hovering MAV powered through inductive resonant WPT. This demonstration successfully showed that the position of an MAV powered through WPT can be controlled by sensing the WPT system’s magnetic field. 72 Table 6.1. Theoretical and experimental values of a 13.7-cm diameter R x coil with 1-mm pitch, two turns, and 1.29-mm diameter wire. LRx (μH) CRx (pF) P (W) Pin (W) Theoretical 1.19 115.7 16.91 18.07 Experimental 1.40 93.0 15.2 18 % deviation 16.2 21.5 10.65 0.39 The R x coil was placed at a distance of 10 cm from the Tx coil and was attached with a 12.5 Ω load resistor. Table 6.2. Theoretical and experimental values of a 15.3-cm diameter R x coil with 2-mm pitch, four turns, and 1.29-mm diameter wire. LRx (μH) CRx (pF) P (W) Pin (W) Theoretical 4.4 31.5 15.33 16.0 Experimental 4.6 24.3 9 16 % deviation 4.4 25.8 52.5 0 The R x coil was placed at a distance of 10 cm from the Tx coil and was attached with a 12.5 Ω load resistor. 73 Table 6.3. The average errors between the sensor’s estimated y coordinate of the MAV’s location and the actual y location of the MAV. Trial 1 2 3 4 5 6 7 8 9 10 Avg. Ave. Error between Estimated Y and Actual Y coordinate (cm) Ave. Error between the MAV’s Y Position and the Desired Y Position (cm) Normalized* Ave. Error between Estimated Y and Actual Y Coordinate Normalized* Ave. MAV’s Y Coordinate Error 1.71 1.93 2.11 1.75 1.75 1.68 2.09 1.63 2.28 1.62 1.86 3.27 2.43 2.70 3.14 2.46 2.81 2.44 2.65 2.31 3.09 2.73 0.090 0.102 0.111 0.092 0.092 0.088 0.110 0.086 0.120 0.085 0.098 0.172 0.128 0.142 0.165 0.129 0.148 0.128 0.139 0.122 0.163 0.144 *Normalized by dividing the Error by the diameter of the transmit coil, 19.0 cm. 74 Figure 6.1. The plot of the MAV’s thrust vs. power. The data appear to be highly linear. The custom-built MAV with the R x coil and power conditioning circuity attached weighs 86 g. The MAV requires 14 W of power to produce 86 g of thrust. [7] Figure 6.2. Photos of the different coils used in the experiments. Left: The seven turn 19 cm diameter Tx coil. Middle: The four turn 15.3 cm diameter R x coil. Right: The two turn 13.7 cm diameter R x coil. 75 Figure 6.3. The plot of axial misalignment in the R x and Tx coil vs. WPT efficiency. There is a stable region of above 90 % efficiency between 0 and 3 cm of displacement. At 4 cm, the efficiency drops and continues to drop as the misalignment increases. [7] Figure 6.4. The full WPT system with the MAV. The RF power amplifier drives the system at 13.56 MHz. The L-matching network matches the 50 Ω source impedance to the impedance of the rest of the system. The Tx and R x coil are both tuned to the same resonant frequency, 13.56 MHz. The AC-DC rectifier converts the AC power to DC and the DC-DC regulator caps the power into the MAV’s at 3.9 V. [7] 76 Figure 6.5. A working demonstration of the MAV flying using the WPT system. The MAV starts resting on a platform suspended 80 mm above the Tx coil. The MAV takes off, and hovers approximately 150 mm above the Tx coil for 12 seconds before landing on the raised platform.[7] Figure 6.6. The MAV is using the high-frequency magnetic-field sensor to control its position along the y axis (horizontal axis running left to right in the picture frame). Due to error in the sensor’s position estimates, the MAV slowly oscillates about the center of the Tx coil. However, it is able to maintain its position within the Tx coil, and does not drift more then a few centimeters from the center of the Tx coil. 77 Figure 6.7. The wirelessly powered MAV flying with its position controlled by the hybrid control system. The MAV’s y position is controlled through feedback from the inductive– coil sensor, which is sensing the WPT high-frequency magnetic field. The MAV takes off, hovers above the Tx coil for 17 seconds, then lands on the raised acrylic platform. 78 Figure 6.8. The wirelessly powered MAV flying with its position controlled by the hybrid control system. The MAV’s y position is controlled through feedback from the inductive– coil sensor, which is sensing the WPT high-frequency magnetic field. The MAV oscillates about the center of the Tx coil by several centimeters, but manages to stay close enough to the center to receive adequate power for flight by the WPT system. 79 Figure 6.9. The actual y location of the MAV as a function of time during its flight superimposed with the estimated y location from the high-frequency magnetic-field sensor. The inductive-coil sensors is able to estimate the location of the MAV well enough to enable the MAV to hover above the Tx coil. CHAPTER 7 CONCLUSION AND FUTURE WORK The objectives of this thesis were threefold: to build and demonstrate a WPT system designed for powering a hovering MAV, to develop a sensor to allow the MAV to control its position based off the sensing the WPT, and to use that sensor in a control system to enable the MAV to successfully hover over the WPT system’s Tx coil. All three of these objectives were successfully completed. A model of an inductive resonant wireless power transfer system was constructed and experimentally validated. Using that model, a transmit coil and a receive coil were designed to enable sufficient wireless power transfer for a MAV to fly. In order for the MAV to control its position using feedback from the WPT, a custom high-frequency magnetic-field sensor was designed and characterized. An inductive-coil sensor was constructed and a sample and hold circuit was designed that allowed the sensor readings to be accurately converted from an analog signal to a digital signal. This sensor was then successfully integrated onto a MAV. Using the sensor, a map of the WPT system’s magnetic field above the Tx coil was obtained and converted into a interpolation look up table. Using this table, the custom designed inductive-coil sensor was able to calculate its position within 0.8-1.6 cm of the true position value on average. With a high-frequency magnetic-field sensor that can accurately predict the MAV’s location, a custom-made MAV position control system was designed. Utilizing four inductivecoil sensors, the control system would enable the MAV to control its position above the Tx coil. In order to demonstrate this, a proof of concept hybrid control system was built and tested that partially controlled the MAV’s position using the high-frequency magnetic-field sensor. This hybrid position control system used motion capture cameras to control the MAV’s yaw, x, and z coordinates while the MAV’s y coordinate was determined by sensing the WPT system’s magnetic field. The magnetic field sensing was accurate enough to allow 81 the MAV to hover above the center of the Tx coil. 7.1 Future Work Future work on this project includes: improving the WPT system and the position control system based on the WPT. The model of the WPT system could be improved by adding the power conditioning components and their effects on the system. The wireless power transfer between the MAV could be improved by implementing better impedance matching circuity. Additionally, the WPT could be improved by adding iron cores or backings to help direct the magnetic flux. While this thesis explored using a stationary Tx coil to power a hovering MAV, an interesting possibility exists in reversing the WPT system. This would involve the MAV being fitted with a Tx coil that could be used to power sensors that are stationary and spread throughout an area. On the control system side, the next step would be to experimentally implement the controller designed in Chapter 5 that would control the x and y coordinates of the MAV. By adding an additional sensor to control the height of the MAV, a controller could be developed that would enable the MAV to hover without the use of any motion capture cameras. 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