# quadrotor modeling and control - carnegie mellon ??2014-12-01quadrotor modeling and control 16-311...

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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

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

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

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

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

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

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.

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

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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