0
Research Papers

Analysis and Control of a Variable-Pitch Quadrotor for Agile Flight

[+] Author and Article Information
Mark Cutler

Aerospace Controls Laboratory,
Department of Aeronautics and Astronautics,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: cutlerm@mit.edu

Jonathan P. How

Richard Maclaurin Professor
of Aeronautics and Astronautics
Aerospace Controls Laboratory,
Department of Aeronautics and Astronautics,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: jhow@mit.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 15, 2014; final manuscript received April 28, 2015; published online July 1, 2015. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(10), 101002 (Jul 01, 2015) (14 pages) Paper No: DS-14-1418; doi: 10.1115/1.4030676 History: Received October 15, 2014

Fixed-pitch quadrotors are popular research and hobby platforms largely due to their mechanical simplicity relative to other hovering aircraft. This simplicity, however, places fundamental limits on the achievable actuator bandwidth and the possible flight maneuvers. This paper shows that many of these limitations can be overcome by utilizing variable-pitch propellers on a quadrotor. A detailed analysis of the potential benefits of variable-pitch propellers over fixed-pitch propellers for a quadrotor is presented. This analysis is supported with experimental testing to show that variable-pitch propellers, in addition to allowing for generation of reverse thrust, substantially increase the maximum rate of thrust change. A nonlinear, quaternion-based control algorithm for controlling the quadrotor is also presented with an accompanying trajectory generation method that finds polynomial minimum-time paths based on actuator saturation levels. The control law and trajectory generation algorithms are implemented on a custom variable-pitch quadrotor. Several flight tests are shown, which highlight the benefits of a variable-pitch quadrotor over a standard fixed-pitch quadrotor for performing aggressive and aerobatic maneuvers.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Amir, M. Y. , and Abbass, V. , 2008, “Modeling of Quadrotor Helicopter Dynamics,” International Conference on Smart Manufacturing Application (ICSMA), Gyeonggi-do, Apr. 9–11, pp. 100–105.
Erginer, B. , and Altug, E. , 2007, “Modeling and PD Control of a Quadrotor VTOL Vehicle,” IEEE Intelligent Vehicles Symposium, Istanbul, Turkey, June 13–15, pp. 894–899.
Alpen, M. , Frick, K. , and Horn, J. , 2009, “Nonlinear Modeling and Position Control of an Industrial Quadrotor With On-Board Attitude Control,” IEEE International Conference on Robotics and Automation, Christchurch, Dec. 10–11, pp. 2329–2334.
Kim, J. , Kang, M. , and Park, S. , 2010, “Accurate Modeling and Robust Hovering Control for a Quad–Rotor VTOL Aircraft,” J. Intell. Rob. Syst., 57(1), pp. 9–26. [CrossRef]
Huang, H. , Hoffmann, G. , Waslander, S. , and Tomlin, C. , 2009, “Aerodynamics and Control of Autonomous Quadrotor Helicopters in Aggressive Maneuvering,” IEEE International Conference on Robotics and Automation, Kobe, Japan, May 12–17, pp. 3277–3282.
Gurdan, D. , Stumpf, J. , Achtelik, M. , Doth, K. M. , Hirzinger, G. , and Rus, D. , 2007, “Energy-Efficient Autonomous Four-Rotor Flying Robot Controlled at 1 kHz,” IEEE International Conference on Robotics and Automation, Rome, Italy, Apr. 10–14, pp. 361–366.
Michael, N. , Mellinger, D. , Lindsey, Q. , and Kumar, V. , 2010, “The GRASP Multiple Micro-UAV Testbed,” IEEE Rob. Autom. Mag., 17(3), pp. 56–65. [CrossRef]
Gillula, J. , Huang, H. , Vitus, M. , and Tomlin, C. , 2010, “Design of Guaranteed Safe Maneuvers Using Reachable Sets: Autonomous Quadrotor Aerobatics in Theory and Practice,” IEEE International Conference on Robotics and Automation, Anchorage, AK, May 3–7, pp. 1649–1654.
Hoffmann, G. , Waslander, S. , and Tomlin, C. , 2008, “Quadrotor Helicopter Trajectory Tracking Control,” AIAA Paper No. 2008-7410.
Ritz, R. , Hehn, M. , Lupashin, S. , and D’Andrea, R. , 2011, “Quadrocopter Performance Benchmarking Using Optimal Control,” IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, Sep. 25–30, pp. 5179–5186.
Mellinger, D. , and Kumar, V. , 2011, “Minimum Snap Trajectory Generation and Control for Quadrotors,” IEEE International Conference on Robotics and Automation, Shanghai, China, May 9–13, pp. 2520–2525.
Shen, S. , Mulgaonkar, Y. , Michael, N. , and Kumar, V. , 2013, “Vision-Based State Estimation and Trajectory Control Towards Aggressive Flight With a Quadrotor,” Robotics: Science and Systems, pp. 1–8.
Pounds, P. , and Mahony, R. , 2009, “Design Principles of Large Quadrotors for Practical Applications,” IEEE International Conference on Robotics and Automation, Kobe, Japan, May 12–17, pp. 3265–3270.
Kushleyev, A. , Mellinger, D. , Powers, C. , and Kumar, V. , 2013, “Towards a Swarm of Agile Micro Quadrotors,” Auton. Rob., 35(4), pp. 287–300. [CrossRef]
Driessens, S. , and Pounds, P. E. , 2013, “Towards a More Efficient Quadrotor Configuration,” IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, Nov. 3–7, pp. 1386–1392.
Borenstein, J. , 1992, “The Hoverbot, an Electrically Powered Flying Robot,” http://www.cs.cmu.edu/∼motionplanning/papers/sbp_papers/integrated1/borenstein_hovercraft.pdf
d’Ambrosio, G. , and Navoni, R. , “HG3 Willy,” You Tube video, 3:21, July 2001, http://youtu.be/M4uXmekZk-4
Michini, B. , Redding, J. , Ure, N. K. , Cutler, M. , and How, J. P. , 2011, “Design and Flight Testing of an Autonomous Variable-Pitch Quadrotor,” IEEE International Conference on Robotics and Automation, Shanghai, China, May 9–13, pp. 2978–2979.
Chen, H. , “Variable-Pitch Quadrotor,” You Tube video, 3:17, July 2011, http://youtu.be/fkSx3fSz0tE
Lupashin, S. , and D’Andrea, R. , 2012, “Adaptive Fast Open-Loop Maneuvers for Quadrocopters,” Auton. Rob., 33(1–2), pp. 89–102. [CrossRef]
Mellinger, D. , Michael, N. , and Kumar, V. , 2012, “Trajectory Generation and Control for Precise Aggressive Maneuvers With Quadrotors,” Int. J. Rob. Res., 31(5), pp. 664–674. [CrossRef]
Muller, M. , Lupashin, S. , and D’Andrea, R. , 2011, “Quadrocopter Ball Juggling,” IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, Sep. 25–30, pp. 5113–5120.
Ritz, R. , Muller, M. , Hehn, M. , and D’Andrea, R. , 2012, “Cooperative Quadrocopter Ball Throwing and Catching,” IEEE/RSJ International Conference on Intelligent Robots and Systems, Vilamoura, Oct. 7–12, pp. 4972–4978.
Hehn, M. , and D’Andrea, R. , 2011, “A Flying Inverted Pendulum,” IEEE International Conference on Robotics and Automation, Shanghai, China, May 9–13. pp. 763–770.
Abbeel, P. , Coates, A. , and Ng, A. Y. , 2010, “Autonomous Helicopter Aerobatics Through Apprenticeship Learning,” Int. J. Rob. Res., 29(13), pp. 1608–1639. [CrossRef]
Cutler, M. , Ure, N. K. , Michini, B. , and How, J. P. , 2011, “Comparison of Fixed and Variable Pitch Actuators for Agile Quadrotors,” AIAA Paper No. 2011-6406.
Cutler, M. , and How, J. P. , 2012, “Actuator Constrained Trajectory Generation and Control for Variable-Pitch Quadrotors,” AIAA Paper No. 2012-4777.
Drela, M. , 2009, “Qprop Users Guide,” http://web.mit.edu/drela/Public/web/qprop/
Hemati, N. , and Leu, M. , 1992, “A Complete Model Characterization of Brushless DC Motors,” IEEE Trans. Ind. Appl., 28(1), pp. 172–180. [CrossRef]
Colton, S. W. , 2010, “Design and Prototyping Methods for Brushless Motors and Motor Control,” Master’s thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA.
Bristeau, P. , Martin, P. , Salaun, E. , and Petit, N. , 2009, “The Role of Propeller Aerodynamics in the Model of a Quadrotor UAV,” European Control Conference, pp. 683–688.
Drela, M. , 1989, “Xfoil: An Analysis and Design System for Low Reynolds Number Airfoils,” Low Reynolds Number Aerodynamics, pp. 1–12, http://web.mit.edu/drela/Public/web/xfoil/
Kuipers, J. B. , 2002, Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality, Princeton University, Princeton, NJ.
Turpin, M. , Michael, N. , and Kumar, V. , 2012, “Trajectory Design and Control for Aggressive Formation Flight With Quadrotors,” Auton. Rob., 33(1–2), pp. 143–156. [CrossRef]
Lupashin, S. , Schollig, A. , Sherback, M. , and D’Andrea, R. , 2010, “A Simple Learning Strategy for High-Speed Quadrocopter Multi-Flips,” IEEE International Conference on Robotics and Automation, Anchorage, AK, May 3–7, pp. 1642–1648.
Hehn, M. , and D’Andrea, R. , 2011, “Quadrocopter Trajectory Generation and Control,” World Congress, Vol. 18, pp. 1485–1491.
Markley, F. , 2002, “Fast Quaternion Attitude Estimation From Two Vector Measurements,” J. Guid., Control, Dyn., 25(2), pp. 411–414. [CrossRef]
Chaturvedi, N. , Sanyal, A. , and McClamroch, N. , 2011, “Rigid-Body Attitude Control,” IEEE Control Syst., 31(3), pp. 30–51. [CrossRef]
Baruh, H. , 1999, Analytical Dynamics, WCB/McGraw-Hill, New York.
Michini, B. , 2009, “Modeling and Adaptive Control of Indoor Unmanned Aerial Vehicles,” Master’s thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge MA.
Wie, B. , and Barba, P. M. , 1985, “Quaternion Feedback for Spacecraft Large Angle Maneuvers,” AIAA J., 8(3), pp. 360–365.
How, J. P. , Frazzoli, E. , and Chowdhary, G. V. , 2015, “Linear Flight Control Techniques for Unmanned Aerial Vehicles,” Handbook of Unmanned Aerial Vehicles, Springer, New York, pp. 529–576.
Girish, C. V. , Emilio, F. , Jonathan, H. P. , and Hugh, L. , 2015, “Nonlinear Flight Control Techniques for Unmanned Aerial Vehicles,” Handbook of Unmanned Aerial Vehicles, Springer, New York, pp. 577–612.
Van Der Merwe, R. , and Wan, E. A. , 2004, “Sigma-Point Kalman Filters for Integrated Navigation,” 60th Annual Meeting of the Institute of Navigation, pp. 641–654.
Valenti, M. , Bethke, B. , Fiore, G. , How, J. P. , and Feron, E. , 2006, “Indoor Multi-Vehicle Flight Testbed for Fault Detection, Isolation, and Recovery,” AIAA Paper No. 2006-6200.
How, J. P. , Bethke, B. , Frank, A. , Dale, D. , and Vian, J. , 2008, “Real-Time Indoor Autonomous Vehicle Test Environment,” IEEE Control Syst. Mag., 28(2), pp. 51–64. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The variable-pitch quadrotor during inverted flight. The symmetry introduced by the variable-pitch propellers allows the quadrotor to fly equally well upright or inverted.

Grahic Jump Location
Fig. 2

Lines of constant thrust (curved lines) and constant motor speed (straight lines), as a function of motor voltage and propeller pitch. The thrust required by the vehicle to hover is indicated by the dark gray line. The addition of variable-pitch actuators has led to a continuum of possible command settings for generating specific thrust settings. Only positive propeller pitch is displayed.

Grahic Jump Location
Fig. 3

Same plot as in Fig. 2 but with both positive and negative pitch shown. One of the fundamental benefits of the variable-pitch actuators for quadrotors is the addition of negative thrust to the flight regime.

Grahic Jump Location
Fig. 4

Lines of constant thrust (horizontal lines) and constant motor speed (vertical lines), as a function of motor power and propeller pitch. The center shaded area roughly indicates areas of maximum efficiency, where the power is minimized for a given thrust value. Only positive propeller pitch is displayed.

Grahic Jump Location
Fig. 5

Simulated thrust and motor speed responses to step increases in motor voltage and propeller pitch (the steps occur at 0.2 s). Changing thrust by varying both motor speed and propeller pitch significantly reduces the effects of motor dynamics, yielding clean, fast changes in thrust. (a) Simulated thrust response to steps in voltage and pitch and (b) simulated motor speed response to steps in voltage and pitch.

Grahic Jump Location
Fig. 6

Experimental thrust and motor speed responses to step increases in motor voltage and propeller pitch. The drop in motor speed after the pitch increase is larger than what was predicted in Fig. 5; however, the shape of the graphs is consistent with the simulated data. When both actuators are used, the motor speed remains essentially constant, showing that the motor dynamics are largely canceled. (a) Experimental thrust response to steps in voltage and pitch and (b) experimental motor speed response to steps in voltage and pitch.

Grahic Jump Location
Fig. 7

Quadrotor model and reference frames. Superscript i denotes the inertial frame and superscript b denotes the body frame.

Grahic Jump Location
Fig. 8

Example path showing the minimum-time optimization. Both paths satisfy the constraints of starting and ending at hover and passing through the five waypoints; however, the optimal time path keeps the motor commands from saturating and completes the path in less time than the one with arbitrary waypoint arrival times. Each path segment is numbered, with the vertical dashed lines showing the time allotted for each segment. (a) Example path and (b) commanded motor values.

Grahic Jump Location
Fig. 9

Two example vertical flight trajectories computed using the optimization routine in Sec. 4.4. Both trajectories have the same upper bound on motor thrust. The variable-pitch trajectory has a negative thrust lower bound, but the fixed-pitch trajectory has a lower bound of near zero. Note that the variable-pitch trajectory is shorter because it decelerates by quickly generating upward thrust.

Grahic Jump Location
Fig. 10

Trajectory generated by imposing a position-free free-fall acceleration condition between two hover waypoints along the x-axis. The small corner in the commanded attitude trajectory comes from not computing new commanded attitudes when the total force command is close to zero. The vehicle goes inverted at the apex of the trajectory by explicitly changing σ(t) from 1 to −1. (a) Flip trajectory and (b) commanded state trajectory values.

Grahic Jump Location
Fig. 11

Simulation results of a 360 deg backflip. The flip is specified using a −90 deg roll constraint before the peak of the trajectory and a 90 deg roll constraint after the peak. The quadrotor starts and ends in hover.

Grahic Jump Location
Fig. 12

ACL Variable-pitch quadrotor. The servos that actuate the variable-pitch propellers are visible under each of the motors. The quadrotor frame measures 0.35 m across.

Grahic Jump Location
Fig. 13

Left: one of the pitch actuation mechanisms on the current version of the variable-pitch quadrotor. Right: a typical swashplate on a remote-controlled helicopter. (a) Variable-pitch actuator and (b) helicopter swashplate.

Grahic Jump Location
Fig. 14

Overview of the software and data flow for the variable-pitch quadrotor. The off-board algorithms run on a desktop personal computer. All off-board communication is handled via the robot operating system.

Grahic Jump Location
Fig. 15

Custom electronics used to perform attitude estimation and control on the variable-pitch quadrotor. The control board (left) mounts on top of the quadrotor and houses a 16-bit microcontroller, three-axis rate gyro, and wireless radio. The power distribution board (right) mounts beneath the control board and distributes power from the battery and signal lines to the ESCs and servos. (a) UberPilot control board and (b) UberPilot power distribution board.

Grahic Jump Location
Fig. 16

Path tracking qualities of the quadrotor. The vehicle is commanded to follow the same path both upright and inverted. Symmetry in the vehicle and propellers allows for similar flight characteristics upright or inverted.

Grahic Jump Location
Fig. 17

Flight data for the variable-pitch quadrotor flying the same trajectory in variable-pitch mode and in fixed-pitch mode. The variable-pitch propellers allow for faster decelerations and better tracking of the position reference command.

Grahic Jump Location
Fig. 18

Variable-pitch quadrotor performing a 180 deg flip by embedding a 90 deg roll constraint at the top of an arc in the X–Z plane

Grahic Jump Location
Fig. 19

Commanded and measured roll and roll rate values from the quadrotor following a flipping maneuver. The measured values come from the on-board rate gyros. The flip takes less than 0.4 s to complete. Snapshots of the quadrotor during the flip are shown in Fig. 18.

Grahic Jump Location
Fig. 20

The quadrotor performing a translating 180 deg flip. The vehicle starts and ends at hover and performs a half back flip in the middle of the path. The vehicle travels forward at nearly 4 m/s during the maneuver.

Grahic Jump Location
Fig. 21

Variable-pitch quadrotor performing a 360 deg translating backflip (simulations of this maneuver are in Fig. 11)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In