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  • Quadrotor Modeling and Control

    16-311 Introduction to RoboticsGuest Lecture on Aerial Robotics

    February 05, 2014

    Nathan Michael

  • Lecture Outline

    Modeling: Dynamic model from first principles Propeller model and force and moments generation

    Control Attitude control (inner loop) Position control (outer loop)

    Current research challenges

  • Develop preliminary concepts required to enable autonomous flight:

    D. Mellinger, N. Michael, and V. Kumar. Trajectory generation and control for precise aggressive maneuvers with quadrotors. Intl. J. Robot. Research, 31(5):664674, Apr. 2012.

    Lecture Objective

    e2e1

    e3

    1. Vehicle model2. Attitude and position control3. Trajectory generation

  • Concept Review

    Quadrotor Model

    Newton-Euler equations:

    total force

    total torque

    mass

    F

    =

    m13 0303 I3

    a

    +

    ! mv! I3!

    moment of inertia

    linear acceleration

    angular acceleration

    angular velocity

    linear velocity

  • Concept Review

    Quadrotor Model

    Rigid transformation:

    rotation translation

    Euler angle parameterization of rotation:

    Reb

    = Rz

    ( )Ry

    ()Rx

    () ZYX (321) form

    pe = Rebpb + re

    e1

    e2

    e3b2

    b3

    re

    Reb

    pb

    b1

  • Concept Review

    Quadrotor Model

    Euler angle parameterization of rotation:

    Reb

    = Rz

    ( )Ry

    ()Rx

    ()

    yaw pitch roll

    Ry() =

    2

    4c 0 s0 1 0

    s 0 c

    3

    5Rx

    () =

    2

    41 0 00 c s0 s c

    3

    5 Rz( ) =

    2

    4c s 0s c 00 0 1

    3

    5

    e1

    e2

    e3b2

    b3

    re

    Reb

    pb

    b1

  • Quadrotor ModelNewton-Euler equations:

    F

    =

    m13 0303 I3

    a

    +

    ! mv! I3!

    f =4X

    i=1

    fi

    Total force:

    along b3Fb =

    2

    400f

    3

    5

    COM

    f1

    f2f3

    f4 b2b3

    b1

    f1

    f2f3

    f4b2

    b3

    b1

    Fe = RebFb mg

    Body:

    Inertial: gravity

    f1

    f2f3

    f4

    e1

    e2

    e3b2

    b3

    b1re

  • Quadrotor ModelNewton-Euler equations:

    F

    =

    m13 0303 I3

    a

    +

    ! mv! I3!

    Total torque: = r FRecall:

    f1

    f2f3

    f4 b2b3

    b1

    f1

    f2f3

    f4b2

    b3

    b1d

    b1 = d (f2 f4)b2 = d (f3 f1)

    b2

    b1

    +4+2

    3

    1 b3 = 1 + 2 3 + 4

    induced moments

    propeller direction of rotation

    f1

    f2f3

    f4

    e1

    e2

    e3b2

    b3

    b1re

  • Quadrotor ModelEquations of motion:

    b1 = d (f2 f4)b2 = d (f3 f1)b3 = 1 + 2 3 + 4

    Fe = RebFb mg

    m13 0303 I3

    a

    +

    ! mv! I3!

    =

    Fe

    =

    RebFb mg

    [b1 , b2 , b3 ]T

    Fb =

    2

    400f

    3

    5

    Motor model:i = cQ!2ifi = cT!

    2i Approximate relationship between propeller

    speeds and generated thrusts and moments

    2

    664

    fb1b2b3

    3

    775 =

    2

    664

    cT cT cT cT0 dcT 0 dcT

    dcT 0 dcT 0cQ cQ cQ cQ

    3

    775

    2

    664

    w21w22w23w24

    3

    775b2

    b1

    +4+2

    3

    1

  • Lecture Outline

    Modeling: Dynamic model from first principles Propeller model and force and moments generation

    Control Attitude control (inner loop) Position control (outer loop)

    Current research challenges

  • Control System Diagram

    R. Mahony, V. Kumar, and P. Corke. Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor. IEEE Robot. Autom. Mag., 19(3):2032, Sept. 2012.

    Recent tutorial on quadrotor control:

    TrajectoryPlanner

    Position Controller

    Motor Controller

    Attitude Controller

    Dynamic Model

    Attitude Planner d

    pd

    Rd

    u1 = fd

    u2 =db1 ,

    db2 ,

    db3

    T

    !i

  • Inner Loop

    Attitude Control

    PD control law:

    u2 = kReR k!e!

    nonlinear e! = ! !d

    Rotation error metric:

    eR =1

    2

    Rd

    TRRTRd

    _

  • Inner Loop

    Attitude Control

    Linearize the nonlinear model about hover:

    R0 = R (0 = 0, 0 = 0, 0)

    Rotation error metric:

    after linearization

    eR =1

    2

    Rd

    TR0 RT0 Rd

    _

    u

    2

    40

    0 0

    3

    5_

    = [, , ]T

    Rd = Rz

    ( 0 + )Ryx (,)

  • Inner Loop

    Attitude Control

    PD control law:

    u2 = kReR k!e!

    e! = ! !d

    eR = [, , ]T

    TrajectoryPlanner

    Position Controller

    Motor Controller

    Attitude Controller

    Dynamic Model

    Attitude Planner d

    pd

    Rd

    u1 = fd

    u2 =db1 ,

    db2 ,

    db3

    T

    !i

  • Outer Loop

    Position Control

    PD control law:

    ea + kdev + kpep = 0

    Linearize the nonlinear model about hover:

    Nominal input: u1 = mg

    TrajectoryPlanner

    Position Controller

    Motor Controller

    Attitude Controller

    Dynamic Model

    Attitude Planner d

    pd

    Rd

    u1 = fd

    u2 =db1 ,

    db2 ,

    db3

    T

    !i

    u2 = 031

  • Outer Loop

    Position Control

    PD control law:

    TrajectoryPlanner

    Position Controller

    Motor Controller

    Attitude Controller

    Dynamic Model

    Attitude Planner d

    pd

    Rd

    u1 = fd

    u2 =db1 ,

    db2 ,

    db3

    T

    !i

    u1 = mbT3

    g + ad +Kdev +Kpep

    How do we pick the gains?ev = v vdep = p pd

  • Lecture Outline

    Modeling: Dynamic model from first principles Propeller model and force and moments generation

    Control Attitude control (inner loop) Position control (outer loop)

    Current research challenges

  • Current Research ChallengesHow should we coordinate multiple robots given network and vehicle limitations?

  • Current Research ChallengesHow do we estimate the vehicle state and localize in an unknown environment using only onboard sensing?

    CameraGPS

    Laser

    IMU

    Barometer

    Cameras

    IMU

  • Current Research ChallengesHow do we estimate the vehicle state and localize in an unknown environment using only onboard sensing?

  • Lecture Summary

    Modeling: Dynamic model from first principles Propeller model and force and

    moments generation

    Control Attitude control (inner loop) Position control (outer loop)

    Current research challenges

    e2e1

    e3

    1. Vehicle model2. Attitude and position control3. Trajectory generation

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