IEEE Robotics & Automation Magazine - September 2020 - 98

For many applications, the underactuation property of the
conventional designs has been alleviated by the use of gimbals
to mount sensors onboard. However, the applicability of multirotor UAVs could be
extended if full actuaMultirotor UAVs have been
tion can be achieved. An
example of such applicawidely used due to their
tions is the emerging field
of aerial physical interacunique qualities, such
tion [1], in which UAVs
are not required to act
as vertical takeoff and
just as flying sensors but
as airborne manipulalanding, hovering, and
tors. Several fully actuated multirotor platforms
mechanical simplicity.
were introduced during
the past decade to overcome the underactuation
property of conventional multirotors. Full actuation has been
realized mainly by using fixed propellers with dissimilar orientations, which we refer to as fixed-tilt concepts, and actively tilting the propellers using extra actuators, which we refer to as
variable-tilt concepts.
The problem with fully actuated UAV concepts is that the
optimal rotor configuration is application-dependent, contrary to underactuated concepts, which usually have the orientation of their rotors in a vertical in-plane symmetric
configuration. To achieve full actuation, the orientation and
location of the rotors must be altered, which results in a
wide range of possible configurations. Depending on the
requirements of the application, a vast variety of different
concepts results.

multirotor UAVs based on the mapping matrix. For a more
comprehensive introduction to the topic, readers are referred
to textbooks such as [2] and [3].

Background
In this section, we show how to derive the control-allocation
matrix, which maps the UAV control inputs (that is, the propellers' thrust) to the total aerodynamic wrench applied to the
UAV's body. This is done by defining the reference frames,
followed by a static wrench analysis and the classification of

where o Bpi ! R 3, R Bpi ! SO (3) denotes the position and
orientation of W pi with respect to W B and zt = (0, 0, 1) < .
The explicit time dependence in (1) is for variable-tilt
UAVs, whereas for fixed-tilt UAVs, p i and u i are constants.
In the case of a planar multirotor (that is, a design with
coplanar rotor positions) it is possible to parametrize the orientation matrix R Bpi by three angles such that

Coordinate Frames
First, we introduce the notion of a coordinate frame, which is
represented by the quadruple {W i : o i, xt i, yt i, zt i}, where o i
represents the origin of the frame and (xt i, yt i, zt i) is a
triad of (right) mutually orthonormal basis vectors. Let
{W B : o B, xt B, yt B, zt B} denote a body-fixed frame with oB
attached to the center of mass (CoM); ztB is chosen such that
gravity acts oppositely when the UAV rests on flat ground,
and xt B represents the UAV's forward direction so that,
when the craft aligns to north, yt B points west, as shown
in Figure 1.
Associated with the ith propeller is the frame
{W pi : o p i, xt pi, yt pi, zt pi}, where the origin, o pi, coincides with
the CoM of the ith rotor and zt pi is oriented to the direction of
the generated thrust (that is, normal to the spinning-disk
area). The axis, xt pi, is chosen so that it is collinear to the line
connecting oB to o pi, while yt pi completes the right-oriented triad, as depicted in Figure 1. The propeller frame, W pi,
does not rotate with the propeller; that is, for fixed-tilt
concepts, it is attached to the body. However, for variabletilt concepts, zt pi is always aligned with the variable thrustgeneration direction.
With the aforementioned definitions of the coordinate
frames, the configuration of each rotor/propeller can be
determined by the displacement vector, p i ! R 3, and orientation vector, u i ! S 2, given by

ψB
zpi

ψpi

ypi

East

xpi

Gravity
Figure 1. A schematic view of the reference frames used,
illustrated on a fully actuated hexarotor.

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u i (t) : = R Bpi (t) zt, (1)

R Bpi = R z (} i) R y (b i) R x (a i), (2)

where R k ($) is the standard rotation matrix around the
kth axis, while the angles a i and b i uniquely define the
direction of thrust-generation axis zt pi in W B . The angle
} i denotes the heading, while a i, b i will be referred to,
respectively, as the cant and dihedral angles.

North

p i : = o p i,

SEPTEMBER 2020

Static-Control Wrench Analysis
The aerodynamic wrench produced by the rotors can be
derived through a static analysis. A wrench represents the
generalized force acting on a rigid body and consists of linearforce, f ! R 3, and rotational-torque, x ! R 3, components. A
wrench applied to the origin of W i with its components speci<
<
fied in W i is denoted by W i = (x i , f i ) <. The change of a

IEEE Robotics & Automation Magazine - September 2020

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