UUnniivveerrssiittyy ooff PPeennnnssyyllvvaanniiaa SScchhoollaarrllyyCCoommmmoonnss Publicly Accessible Penn Dissertations 2016 PPaassssiivvee SSttaabbiilliittyy AAnndd AAccttuuaattiioonn OOff MMiiccrroo AAeerriiaall VVeehhiicclleess Matthew Piccoli Piccoli University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Aerospace Engineering Commons, Mechanical Engineering Commons, and the Robotics Commons RReeccoommmmeennddeedd CCiittaattiioonn Piccoli, Matthew Piccoli, "Passive Stability And Actuation Of Micro Aerial Vehicles" (2016). Publicly Accessible Penn Dissertations. 2529. https://repository.upenn.edu/edissertations/2529 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2529 For more information, please contact [email protected]. PPaassssiivvee SSttaabbiilliittyy AAnndd AAccttuuaattiioonn OOff MMiiccrroo AAeerriiaall VVeehhiicclleess AAbbssttrraacctt Micro Aerial Vehicles (MAVs) have increased in popularity in recent years. The most common platform, the quadrotor, has surpassed other MAVs like traditional helicopters and ornithopters in popularity mainly due to their simplicity. Yet the quadrotor design is a century old and was intended to carry people. We set out to design a MAV that is designed specifically to be a MAV, i.e. a vehicle not intended to carry humans as a payload. With this constraint lifted the vehicle can continuously rotate, which would dizzy a human, can sustain larger forces, which would damage a human, or can take advantage of scaling properties, where it may not work at human scale. Furthermore, we aim for simplicity by removing vehicle controllers and reducing the number of actuators, such that the vehicle can be made cost effective, if not disposable. We begin by studying general equations of motion for hovering MAVs. We search for vehicle configurations that exhibit passive stability, allowing the MAV to operate without a controller or actuators to apply control, ideally a single actuator. The analysis suggests two distinct types of passively stabilized MAVs and we create test vehicles for both. With simple hovering achieved, we concentrate on controlled motion with an emphasis on doing so without adding actuators. We find we can attain three degree of freedom control using separation of time scales with our actuator via low frequency for control in the vertical direction and high frequency for control in the horizontal plane. We explore techniques for achieving high frequency actuator control, which also allow the compensation of motor defects, specifically cogging torque. We combine passive stability with the motion control into two vehicles, UNO and Piccolissimo. UNO, the Underactuated-propeller Naturally-stabilized One-motor vehicle, demonstrates the capabilities of simple vehicles by performing maneuvers like conventional quadrotors. Piccolissimo, Italian for “very little”, demonstrates the merits of passive stability and single actuator control by being the smallest, self- powered, controllable MAV. DDeeggrreeee TTyyppee Dissertation DDeeggrreeee NNaammee Doctor of Philosophy (PhD) GGrraadduuaattee GGrroouupp Mechanical Engineering & Applied Mechanics FFiirrsstt AAddvviissoorr Mark Yim KKeeyywwoorrddss Micro Aerial Vehicles, Motor Control, Unmanned Aerial Vehicles SSuubbjjeecctt CCaatteeggoorriieess Aerospace Engineering | Mechanical Engineering | Robotics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2529 PASSIVE STABILITY AND ACTUATION OF MICRO AERIAL VEHICLES Matthew Piccoli A DISSERTATION in Mechanical Engineering and Applied Mechanics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2016 Mark Yim, Ph.D., Supervisor of Dissertation Professor of Mechanical Engineering and Applied Mechanics, Univ. of Pennsylvania Kevin Turner, Ph.D., Graduate Group Chairperson Professor of Mechanical Engineering and Applied Mechanics, Univ. of Pennsylvania Dissertation Committee: Vijay Kumar, Ph.D., The Nemirovsky Family Dean of Penn Engineering; Professor of Mechanical Engineering and Applied Mechanics, Univ. of Pennsylvania Mark Yim, Ph.D., Professor of Mechanical Engineering and Applied Mechanics, Univ. of Pennsylvania Raffaello D’Andrea, Ph.D., Professor of Mechanical and Process Engineering, Eid- geno¨ssische Technische Hochschule Zu¨rich Daniel Koditschek, Ph.D., Alfred Fitler Moore Professor of Electrical and Systems Engineering, Univ. of Pennsylvania Bruce Kothmann, Ph.D., Lecturer of Mechanical Engineering and Applied Mechan- ics, Univ. of Pennsylvania ABSTRACT PASSIVE STABILITY AND ACTUATION OF MICRO AERIAL VEHICLES Matthew Piccoli Mark Yim Micro Aerial Vehicles (MAVs) have increased in popularity in recent years. The most common platform, the quadrotor, has surpassed other MAVs like traditional helicopters and ornithopters in popularity mainly due to their simplicity. Yet the quadrotor design is a century old and was intended to carry people. We set out to design a MAV that is designed specifically to be a MAV, i.e. a vehicle not intended to carry humans as a payload. With this constraint lifted the vehicle can continuously rotate, which would dizzy a human, can sustain larger forces, which would damage a human, or can take advantage of scaling properties, where it may not work at human scale. Furthermore, we aim for simplicity by removing vehicle controllers and reducing the number of actuators, such that the vehicle can be made cost effective, if not disposable. We begin by studying general equations of motion for hovering MAVs. We search for vehicle configurations that exhibit passive stability, allowing the MAV to operate without a controller or actuators to apply control, ideally a single actuator. The analysis suggests two distinct types of passively stabilized MAVs and we create test vehicles for both. Withsimplehoveringachieved, weconcentrateoncontrolledmotionwithanemphasis on doing so without adding actuators. We find we can attain three degree of freedom ii control using separation of time scales with our actuator via low frequency for control in the vertical direction and high frequency for control in the horizontal plane. We explore techniques for achieving high frequency actuator control, which also allow the compensation of motor defects, specifically cogging torque. We combine passive stability with the motion control into two vehicles, UNO and Piccolissimo. UNO, the Underactuated-propeller Naturally-stabilized One-motor vehicle, demonstrates the capabilities of simple vehicles by performing maneuvers like conven- tional quadrotors. Piccolissimo, Italian for very little, demonstrates the merits of passive stabilityandsingleactuatorcontrolbybeingthesmallest,self-powered,controllableMAV. iii Contents Abstract ii 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Low Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Proposed Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Passive Stability 11 2.1 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Simple Stability Simulation . . . . . . . . . . . . . . . . . . . . . . 15 2.1.2 COP Vs. COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 Differential Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Linearized Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Routh-Hurwitz Stability Criterion . . . . . . . . . . . . . . . . . . 21 2.2.2 Aerodynamic With Angular Momentum . . . . . . . . . . . . . . . 22 2.2.3 Aerodynamic Without Angular Momentum . . . . . . . . . . . . . 32 3 Passive Stability Experiments 36 3.1 COP Vs. COM Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Differential Lift Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Actuation Methods 61 4.1 Control Using Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Control Using Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Actuator Authority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5 Motor Control 72 5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Anticogging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.1 Anticogging Proposed Approach . . . . . . . . . . . . . . . . . . . 87 5.2.2 Waveform Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.3 Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 iv 5.2.4 Waveform Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2.5 Ripple Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2.6 Design and Experimental Results . . . . . . . . . . . . . . . . . . . 101 5.2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6 Three DOF Experimental Vehicles 118 6.1 UNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.1 Design and Manufacturing . . . . . . . . . . . . . . . . . . . . . . . 119 6.1.2 Experimental Results and Analysis . . . . . . . . . . . . . . . . . . 127 6.2 Piccolissimo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2.1 Design and Manufacturing . . . . . . . . . . . . . . . . . . . . . . . 131 6.2.2 Experiments and Analysis . . . . . . . . . . . . . . . . . . . . . . . 138 7 Conclusion 143 7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Bibliography 144 v List of Tables 3.1 Tested vehicle configurations . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Predicted and actual stability. Those that do not follow predictions are highlighted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Vehicle costs for a run of 1000 in USD . . . . . . . . . . . . . . . . . . . . 49 3.4 Stability results of various vehicle configurations . . . . . . . . . . . . . . 58 5.1 Anticogging Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1 Speeds of UNO’s body and propeller. All units are rad/sec. . . . . . . . . 128 6.2 Mass distribution of both versions of Piccolissimo in grams. . . . . . . . . 134 6.3 Comparison of Mini and Maneuverable Piccolissimo with commercial mul- ticopters. TheFY805andFY804aremadebyFayee, whiletheCX-STARS is made by Cheerson, and is the source of Mini Piccolissimo’s motor and propeller. The FY804 and the CX-STARS are the current smallest com- mercial self powered MAVs. . . . . . . . . . . . . . . . . . . . . . . . . . . 139 vi List of Figures 1.1 Yim’s Tracking Device with passive stability fins (125) and offset mass for steering (126). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Ahighlevelmapofthetopicsdiscussedinthisthesis. Greenisthemotiva- tion, partially done by Mark Yim. Blue is theory. Orange is experimental hardware not designed by the author. Yellow is experimental hardware created by the author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 FT-Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Varying the linear drag term, a, from a simulated vehicle with differential lift and angular momentum. The top plot shows all six eigenvalues, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Varying the differential lift term, b, from a simulated vehicle with differ- ential lift and angular momentum. The top plot shows all six eigenvalues, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Varying the COP vs. COM term, c, from a simulated vehicle with differ- ential lift and angular momentum. The top plot shows all six eigenvalues, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Varyingtheangulardragterm, d, fromasimulatedvehiclewithdifferential lift and angular momentum. The top plot shows all six eigenvalues, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Varying the angular momentum term, e, from a simulated vehicle with differential lift and angular momentum. The top plot shows all six eigen- values, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values.. . . . . . . . . . . . . . . . . . . . . . 31 2.7 Varying assumptions used in Section 2.2.2, from a simulated vehicle with differential lift and angular momentum. The top plot shows all six eigen- values, while the bottom plot focuses on the two closest to the imaginary axis with positive imaginary values.. . . . . . . . . . . . . . . . . . . . . . 33 3.1 COP Vs. COM Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Side-by-side comparison of the nine tested configurations . . . . . . . . . . 40 3.3 Root locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Configuration a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Configuration b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Configuration c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 vii 3.7 Configuration d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8 Coordinate systems used while computing the equations of motion. . . . 51 3.9 Propeller 1 FT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.10 Wide (left) and tall (right) variants and motor module (center) . . . . . . 55 3.11 Variable replaceable elements . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Coordinate systems: I for inertial, F for flyer, B for body. F and B are located at COM. The forces and torques generated by the motor and propeller are f and τ. The offset between the COM at the body frame and the propeller’s force is o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1 Possible driven states of a brushed motor. Asynchronous configurations are on the left and synchronous configurations are on the right. . . . . . 74 ◦ ◦ 5.2 Possible driven states of a three phase motor with 120 on the left, 180 on the right, and null or motor braking in the center. Arrows represent motor lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Sinusoidal PWM values and the resulting phase voltages with respect to neutral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.4 Trapezoidal PWM values and the resulting phase voltages with respect to neutral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5 RectangularPWMvaluesandtheresultingnormalizedphasevoltageswith respect to neutral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.6 Quadrature PWM values and the resulting normalized phase voltages with respect to the shared line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ◦ 5.7 Space vector patterns for three phase drivers with 180 vectors drawn as black arrows and quadrature vectors drawn in yellow . . . . . . . . . . . . 81 5.8 Nominal and compensated torque ripple ratio vs. price for a set of motors of nominally same size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.9 Diagram of a single half H-bridge inverter connected to one of three phases of a sectioned motor. δ is the high time duty cycle, δ is the time it takes dt for the FETs to switch, and f is the PWM frequency. . . . . . . . . . 89 pwm 5.10 Positionmethodcollecteddatashowingdutycyclerequiredtoholdposition from motor M4 in Table 5.1. This process is described in Section 5.2.6. (a) ◦ A full 360 dataset with forward and backward trials and (b) a magnified section showing difference between forward and reverse. . . . . . . . . . . 91 5.11 Top view of the robotic arm. . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.12 Trajectory of robotic arm with and without anticogging. Cmd is the com- mandedtrajectory, Cogistheactualtrajectorywithoutcompensation, and Anti is the actual trajectory with anticogging enabled. . . . . . . . . . . . 104 5.13 Torque ripple after anticogging versus torque ripple before anticogging for eleven tested motors. Fit line is y = 0.3139x with an R2 = 0.8922. . . . . 108 5.14 Motor M4 RMS torque versus PWM frequency with anticogging disabled. (+) is τ , ( ) is τ , (*) is τ , (.) is τ , (x) is τ , ((cid:3)) is τ , ( ) is res frq cog fr dt cog ◦ ⋄ τ , ( ) is τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 nom,est nom,act △ viii