IEEE Robotics & Automation Magazine - September 2012 - 29

for rotor commutation and high-frequency pulsewidth
modulation (PWM) to control motor voltage. The
simplest systems generally use a direct voltage control of
the motors since steady-state motor speed is proportional to voltage; however; the dynamic response is
second-order due to the mechanical and electrical
dynamics. Improved performance is obtained by incorporating single-input single-output control at the
motor/rotor level
Vi ¼ k(-Ãi À -i ) þ Vff (-Ãi ),

(23)

where Vi is the applied motor voltage, -Ãi is the desired
speed, and the actual motor speed -i can be measured
from the electronic commutation in the embedded
speed controller. This can help to overcome a common
problem where the rotor speed for a given PWM command setting will decrease as the battery voltage
reduces during flight. The significant load torque due to
aerodynamic drag will lead to a tracking error that can
be minimized by high proportional gain (k) and/or a
feedforward term. A positive benefit of the drag
torque is that the system is heavily damped, which
precludes the need for derivative control. The feedforward term Vff (-Ãi ) compensates for the steady-state
PWM associated with a given velocity set point by
incorporating the best available thrust model determined using static thrust tests and possibly including
battery voltage.
The performance of the motor controllers is ultimately
limited by the current that can be supplied from the batteries. This may be a significant limiting factor for smaller
vehicles. Overly aggressive tuning and extreme maneuvers
may cause the voltage bus to drop excessively, reducing
the thrust from other rotors and, in extreme cases, causing
the onboard electronics to brownout. For this reason, it
is common to introduce a saturation, although this
destroys the linearity of the motor/rotor response during
aggressive maneuvers.
Attitude Control
We first consider the design of an exponentially converging controller in SO(3). Given a desired airframe attitude
R? , we want to first develop a measure of the error in rotations. We choose the measure
eR 3 ¼

Á
1À Ã T
(R ) R À RT RÃ ,
2

(24)

which yields a skew-symmetric matrix representing the
axis of rotation required to go from R to RÃ and whose
magnitude is equal to the sine of the angle of rotation.
To derive linear controllers, we linearize the dynamics
about the nominal hover position at which the roll (/) and
pitch (h) are close to zero and the angular velocities are
close to zero. If we write R ¼ A RB as a product of the yaw

rotation A RE (w) and E RB (/, h), which is a composition of
the roll and pitch, we can linearize the rotation about
(w, /, h) ¼ (w0 , 0, 0)
A

RB ¼

A

RE (w0 þ Dw) E RB (D/, Dh)

0

cos w

B
B
¼ B sin w
@

À sin w
cos w

ÀDh

Dh cos w þ D/ sin w

1

C
C
Dh sin w À D/ cos w C,
A

D/

1

where w ¼ w0 þ Dw. If R? ¼ A RB (w0 þ Dw, D/, Dh) and
R ¼ A RB (w0 , 0, 0), (24) gives
0

0

B
eR 3 ¼ B
@ ÀDw
Dh

Dw
0

ÀDh

1

C
D/ C
A,

ÀD/

(25)

0

which, as we expect, corresponds to the error vector
eR ¼ (D/, Dh, Dw)> ,
with components in the body-fixed frame. If the desired
angular velocity vector is zero, we can compute the
proportional and derivative error to obtain the PD control law
u2 ¼ ÀkR eR À kX eX ,

(26)

where kR and kX are positive definite gain matrices. This
controller guarantees stability for small deviations from
the hover position.
To obtain convergence for larger deviations from
the hover position, it is necessary to revert back to (24)
without linearization. This allows us to directly compute
the error on SO(3). By compensating for the nonlinear
inertial terms and by including the correct error term,
we obtain
u2 ¼J(ÀkR eR ÀkX eX )þX 3 JXÀJ(X 3 RT R? X? ÀRT R? X_ ? ):
(27)
This controller is guaranteed to be exponentially
stable for almost any rotation [23]. From a practical
standpoint, it is possible to neglect the last three terms
in the controller and achieve satisfactory performance,
but the correct calculation of the error term is important [24].
Trajectory Control
We now turn our attention to the control of the trajectory along a specified trajectory n? (t). As before, we
first consider linear controllers by linearizing the dynamics about n ¼ n? (t), h ¼ / ¼ 0, w ¼ w? (t), n_ ¼ 0, and
SEPTEMBER 2012

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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29
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2012
