Aerospace and Electronic Systems Magazine July 2018 - 17

Shraim, Awada, and Youness
and forward flight, preparing to control for aggressive flights, is
presented in [65]. The possibility of mass change or center of mass
displacement during quadrotor flight is well and clearly treated in
[66]. These authors depart from the paradigm that the center of
mass is coincident with the geometric center by explicitly accounting for the center of mass offset in the controller. They adopt a
linear least squares method for estimating the unknown parameters
that change when a quadrotor transports its payloads.
For the purpose of quadrotor controller design, the nonlinear
model of a quadrotor is described in a state space equation form in
different manners as presented in [67], [68].

QUATERNION DIFFERENTIAL EQUATIONS.
When the Euler angle θ closes to 90, then the roll angle will lose its
meaning, and a problem, so-called gimbal lock, will appear. This
problem can be avoided by the use of the quaternion method. This
method offers a mathematical notation allowing representation of
the three-dimensional (3D) rotations of an object in a four-dimensional space. Reference [69] presents a description of a quaternion
dynamics. Reference [70] presents a new quaternion-based feedback control scheme for the attitude stabilization of a quadrotor
aircraft.

Impacts of Linearization
Although widely used, the methods based on linearization have
some drawbacks. Their use is often accompanied by extreme caution. The local linearization (Jacobian) is only valid on point of operation, while the exact linearization is not always possible. Even if
it exists, it is not always advantageous to eliminate all nonlinearity
in a system. Some nonlinearities help to preserve the stability of
the system, and their removal considerably and unnecessarily increases the effort required by the actuator [71]. Therefore, in recent
years researchers are oriented to methods best suited to the nonlinear nature of the systems. One of the major axes of this orientation
is the design based on the direct (or second) method of Lyapunov
(1966).

CONTROL AND CONTROL MISSION
THE LINEAR CONTROL
The quadrotor is a multivariable highly coupled nonlinear system.
The use of linear control for this system consists of a complicated
algebraic manipulation for state variables under certain environmental conditions. For trajectory tracking, the linear control can
be applied only if the trajectory and the flying conditions for the
quadrotor are not complex and difficult. In such cases, the coupled
nature of the system requires high variations in the angular velocities and fast variations in the altitude, which cannot be realized by
such controllers.

Proportional Integral Derivative Controller-PID
Researchers in [72] used a Proportional Integral Derivative (PID)
to stabilize the attitude of the OS4 quadrotor. In order to cover
JULY 2018

large flight trajectories, they linearized the system locally; consequently, gyroscopic effects are eliminated during the design of
the controller. It was concluded that PID is efficient in hovering.
This performance decreases with the presence of high disturbance.
For the Starmac, researchers used PID to control attitude and position, proving that PID is quite sufficient for low speeds with slight
aerodynamic disturbances, such as indoor flight [27]. In [73], researchers tried to control their X4 using a PID controller. They
used PID to perform attitude stabilization and trajectory tracking
by assuming decoupled pitch and yaw movement. This controller
works indoors on low speeds. In [70], [74], authors treated the attitude stabilization problem. They succeeded in providing asymptotic stability without linearizing gyroscopic effects. In [75], flip
movements are accomplished by the quadrotor. Authors used an
intuitive method to predict the behavior of motors during the flip
maneuver. In the rest of motion, the PID was used with a linearized
near-to-hover system. To execute an iteration of the flip, a managing process first uploads a set of parameters to the controller and
then signals. The vehicle then executes the basics of the flip on its
own, ignoring hover controller commands for the duration of the
maneuver. Reference [76] describes a simplified dynamic model
for Javiator and develops altitude and attitude proportional-derivative (PD) controllers for manual control. For a smooth behavior
at the target value, controllers apply the first derivative of current
values and not the first derivative of deviations. These authors
also develop a trajectory controller for autonomous flights, which
follows given waypoints. The used algorithm splits the routes between waypoints into segments of constant velocity or constant
acceleration.

Linear Quadratic Regulator-LQR
In [77], an integral Linear Quadratic Regulator (LQR) strategy
is presented to ensure reliable stability of position, attitude, and
vehicle tracking. The control of altitude is less simple. There are
many factors that affect the loop altitude; these factors do not respond to conventional control techniques. The largest is the effect
of nonlinear friction and destabilization of the four rotors' interaction. Some researchers, as in [78], have shown that an LQR controller can be applied in order to follow a predetermined path reference. The computation time of trajectory tracking is negligible
and can be easily implemented. The disadvantage of this system is
that the constraint satisfaction is only guaranteed for a single trajectory. Therefore, in case of changes in the environment, such as
a moving obstacle, the trajectory tracking is no longer guaranteed,
which may cause a collision or large tracking errors. To solve this
problem, the researchers proposed a control system in two loops:
the outer loop optimizes every step of the quadrotor's model based
predictive controller (MBPC) path and generates an internal loop
LQR to follow trajectory. The MBPC system offers the satisfaction of constraints, but demands significant computing, while the
inner loop LQR does not guarantee the satisfaction of constraints
but is less demanding in term of calculation.
Other research studies use a quadratic linear path follower.
The controller calculates the necessary control signals, generated
by the monitor to a navigation path. It uses a simplified, linear-

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Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2018