Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
A98-37313
AIAA-98-4357
MATHEMATICAL MODELING AND EXPERIMENTAL
IDENTIFICATION OF A MODEL HELICOPTER
S. K. Kirn* and D. M. Tilburyt
Department of Mechanical Engineering and Applied Mechanics
University of Michigan, Ann Arbor, MI 48109-2125
sungk@engin.umich.edu, tilbury@umich.edu
This paper presents a new mathematical model for a model-scale helicopter.
Working from first principles and basic aerodynamics, the equations of motion for
full six degree-of-freedom motion are derived. The control inputs considered are
the four pilot commands from the radio transmitter: roll, pitch, yaw, and thrust.
The model helicopter has a fast time-domain response due to its small size, and is
inherently unstable. A flybar is used to augment the stability of a model helicopter
to make it easier for a pilot to fly. The main contribution of this paper is to model
the interaction between the flybar and the main rotor blade; it is shown how the
flapping of the flybar increases the stability of the model helicopter as well as assists
in its actuation. After the mathematical model is derived, some preliminary system
identification experiments and results are presented. The paper ends with conclusions
and a short description of future work.
Introduction
I
T has been more than 20 years since the first commercial model helicopter was conceived, and since
then, the design has significantly improved. Model
helicopters are now well within the reach of many
hobbyists and are also often used for commercial
purposes, such as crop dusting or sport-event broadcasting. However, model helicopters are inherently
unstable. Even with improved stability augmentation devices, a skilled, experienced pilot is required
to control them during flight.
As a small, dynamically fast, unstable system, a
model helicopter makes an excellent testbed for nonlinear control experiments. As a highly maneuverable machine, it also is an excellent testbed for path
planning algorithms for autonomous robots. The integration of nonlinear control and path planning is
our main interest in this project. As a preliminary
step, this paper describes the new mathematical
model that we have derived for a model helicopter
control system, as well as some preliminary system
identification experiments we have conducted. The
outline of the paper is as follows. First, we briefly review some previous work on helicopter modeling and
control. We then describe the new mathematical
•Ph.D. Student, University of Michigan, AIAA Student
Member
t Assistant Professor, University of Michigan
Copyright O 1998 by Sung K. Kim. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.
model that we have derived, working from first principles and basic aerodynamics. We then describe the
system identification algorithm that we have used
and present our preliminary results. We end with
conclusions and a description of future work.
Previous Work
In the past several years, there have been a number of researchers interested in model helicopter control, and they have had various degree of success.
System Identification
A group at Caltech attempted to capture the
main dynamic features of a model helicopter near
hover with a linear time-invariant model. A MIMO
identification algorithm was used to account for the
significant cross-couplings in the model helicopter.
Bendotti et al.1 set up a model helicopter on a stand
to provide three degrees of freedom: pitch, roll, and
yaw rotations. Translational movements such as x,
y, and z positions were neglected. The identification was carried out using a quadratic weighing factor with an iterative Gauss-Newton algorithm. The
match between simulation and experiment was good
in the pitch and roll directions, but was poor in the
yaw direction due to the actuation asymmetry. Controllers were designed using HCQ and LQG methods
and the linear model that had been identified. Both
controllers showed good response in disturbance rejection and command tracking; the Hx controller
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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
had a faster response.
Another group from Caltech2 did a similar identification and designed a LQR controller. The realtime control and data acquisition ran on the PC,
which hosted the pulsewidth modulated IO board
and Polhemus sensor board.3 Again, the controller
showed good performance only for pitch and roll response. Yaw response was poor for the same reason
as above. Significant improvement on the yaw response was achieved when a separate loop shaping
controller based on a lead-lag design was implemented.
Fuzzy Control
Dr. Sugeno from the Tokyo Institute of Technology4 has had a considerable amount of success in
flying the model helicopter for commercial purposes.
The project's goal was to develop a controller for an
unmanned helicopter that can operate under hostile conditions. The control system was designed
using fuzzy control theory. The integrated control
system, ranging from low level basic flight modes to
high level supervisory control, takes a human voice
as its input. Because human language voice commands are naturally imprecise or 'fuzzy,' the fuzzy
logic framework was a good fit. The primary issue
here was to design a controller that can easily include qualitative information as well as quantitative
information. Some pre-experiment simulation was
done on a Silicon Graphics IRIS workstation and a
PC. The fuzzy controller rules were first constructed
with the help of a actual human pilot's experience
and knowledge. The rules were then tested and reformulated using the simulator. The helicopter was
equipped with various sensors such as camera, gyroscope for 3D acceleration, Doppler speedometer,
magnetic compass, laser altimeter, and GPS.
Unmanned Aerial Vehicle Competition
The annual unmanned aerial vehicle competition
has been organizing a number of universities to complete a task of recognizing and moving an object to
a designated target. So far, the competitors have
typically been concentrating on the sensory issues
such as the image processing, with a lesser emphasis
on the control problems. Among the participating
universities, USC used a concept called "behavior
based control",5'6 which implements a number of
complex tasks with a collection of simple, interacting behaviors in parallel. Their approach seeks to
use a behavioral based approach as a structure to
unify navigation, motor control, and vision. This
approach was inspired by the distributed yet unified control of biological systems. To maintain the
safety of the helicopter, the lower-level behaviors
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may 'negotiate' with a mid-level to satisfy both sets
of behavioral criteria. They claim this approach was
useful in building an integrated control system; an
open issue is the determination of a set of rules and
principles for creating behaviors. The MIT, Boston
University and Draper Lab team7 was successful in
building an autonomous model helicopter designed
to hover, fly around, and recognize five randomly
placed drums during the 1996 International Aerial
Robotics Competition. The system consisted of the
helicopter with various sensors such as GPS, IMU,
altimeter, and compass, as well as a ground control
station, a vision processor, and a safety pilot. The
control system is divided into four closed loops.for
roll, pitch, yaw, and collective/throttle with integrators to eliminate steady state errors. Pre-determined
trim positions were used on those loops. They used
relatively simple control laws to minimize the development time and be flexible to the changes in
helicopter configuration. Again, their research effort
was concentrated more on the sensor issues and the
interactions between the various components than
on the control structure itself.
Other UAV
The Naval Research Lab8 has designed an airplane style UAV which is intended to be a missile
decoy. It has an electric motor for a nose mounted
propeller, three fiber-optic rate gyro sensors for the
pitch, roll, and yaw axes, a barometric altimeter,
and a pitot-static airspeed sensor. All the actuator dynamics are based on linear dynamic equations,
and the single-axis control law is used for attitude
and altitude control. From the launch through the
cruise transition phase, the trajectory is based on a
first-order linear fit to find the desired pitch angle
profile based on 6DOF simulation. The reference
pitch position is time-scheduled to follow the nominal trajectory. Various parameters are based on a
wind tunnel experiment. Although the paper did not
address the issues regarding how the aircraft would
follow the pre-planned trajectory once it reached the
cruise phase in satisfying the specific mission goal,
it demonstrated the potential capability of the craft
as a missile decoy.
In terms of UAV control, Kaminer et al.9 pointed
out that under a shifting wind disturbance, the traditional control scheme, in which the guidance and
control parts are separate, may be inadequate to satisfy the precise tracking and frequent heading change
necessary. To solve this problem, they examined
a guidance and control scheme where the two are
combined, so that the guidance law becomes an integral part of the feedback control system. With
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
such design specifications as zero steady state error
and certain bandwidth requirements, they devised
a state-space coordinate system in which the linearization of the plant along a reference trajectory
is time-invariant. This realization results in trajectories that consist of straight lines, arcs of circles of
constant radii, and any combinations thereof. The
control system itself is an LQR design. Their simulation showed a good result for an aircraft following a
descending helical trajectory. However, the scheme
only applies to certain specific trajectories as mentioned above, so it remains uncertain how useful the
technique could be to a real system.
Sikorsky Aircraft10 has been developing a saucertype VTOL UAV called Cypher to meet various
civil and military requirements. It has two counterrotating, 4 ft long, coaxial, four-bladed main rotors
shrouded by the 6.5 ft diameter main frame. Similar to a helicopter, it uses collective and cyclic pitch
control for movement, powered by a 60 hp rotary
engine. Sensors include radar, gyros, accelerometer, GPS, video camera, etc. To achieve a simple
operator/vehicle interface, the operator is only required to send out basic maneuver commands such
as takeoff, hover, cruise, desired heading, etc. Linear state space models are used to develop control
laws and to determine specifications for servo and
sensor bandwidths. The vehicle successfully demonstrated reconnaissance and surveillance capability,
and is being improved to accomplish such tasks as
mine-deploying and scouting missions. The vehicle
is not yet completely autonomous nor is it capable
of following a pre-determined trajectory.
Other research
At Purdue,11 a student derived the dynamic equation of a model helicopter's vertical motion using
blade element theory. For the experiment, a model
helicopter was affixed on a stand to let it move only
vertically. Uncertain parameters were estimated
by interpolating the results from number of experiments with different parameters. After linearizing
the dynamic model around the hover condition, controllers were designed using full state feedback poleplacement, LQR, and neural network techniques.
All the controllers showed satisfactory performance.
Furuta et al.12 derived a mathematical model of
a model helicopter fixed on a stand, free to rotate
around the pitch, roll, and yaw axis. Conservation
of angular momentum was used to obtain the model.
Realizing that the model helicopter is hard to control
manually, they designed a stable tracking controller
using the state space method. This stabilizing control input is then used to compare and validate the
"Rotor Disk
Fig. 1 The coordinates defined. The orientat.ion
variables, roll, pitch, yaw (<£, 6,0) and the position
variables (x, y, z) are relative to the body frame
(fixed to the helicopter).
derived mathematical model with the experimental
result. They realized that to be able to control the
model helicopter in a full 6DOF situation, they need
to expand the model to other degrees of freedom including the vertical motion. The model they derived
was based on the full-scale helicopter modeling because they modeled a flapping rotor hub with spring,
without explicitly taking the flybar into account.
Azuma13 from University of Tokyo derived a
mathematical model of a rigid rotor system. In this
rotor system, each rotor blade is hinged but is spring
loaded. He carefully derived a equation of motion
using complex variables considering how different
flapping stiffness affects the response of the system.
Since this model does not take the dynamics of the
helicopter as a whole, the validity of the model has
not been fully proven.
Mathematical model
In this section, we will derive the dynamic equations of motion for the model helicopter including
its actuator dynamics. We will use results from rigid
body dynamics,15 as well as basic aerodynamics and
helicopter theory.14'16
Model vs. full scale helicopters
Before deriving the model helicopter dynamics, it
is important to consider the differences between a
full-scale helicopter and a model helicopter. First
of all, a model helicopter has a much faster timedomain response due to its small size. Therefore,
without employing an extra stability augmentation
device, it would be extremely difficult for a human
pilot to control it. A large control gyro with an airfoil, often referred to as a flybar, is almost always
used nowadays to improve the stability characteristic around the pitch and roll axes and to minimize
205
Copyright© 1998,
American Institute of Aeronautics and Astronautics, Inc.
cos ip
sin^
0
the actuator force required. Also, the tail rotor control for the model helicopter is assisted by an electronic gyro to further stabilize the yaw axis. Most
full scale helicopters do not have such a control gyro
on the rotor system. The large inertia of the rotor
and fuselage and the flapping rotor hinge provide
adequate stability.
Secondly, most model helicopters do not have a
flapping hinge on the rotor to maximize the control power. Full scale helicopters often use either a
free flapping rotor hinge or a spring-mounted hinge;
these are usually absent on a model helicopter.
— sin ip
cos ill
0
1 0
0 cos 4>
0 sin (j>
0
0
1
cos 8
0
-sin9
0
1
0
sin 8
0
cosfi
0
~ sin 4>
cos <f>
The cross "x" notation is used to represent the skewsymmetric cross-product matrix. For a vector a =
[d a2 as]T,
ax =
0
as
—03
—as
0
ai
02
—ai
0
Rigid Body Equations
The rotational inertia matrix of the helicopter is
given by
We will model the helicopter as a rigid body moving in space. As shown in Figure I, we use the
variables (x, y, z) to represent the position of the
helicopter in body coordinates. We use the variables (<J),6,i{>) to represent the roll, pitch, and yaw
angles of the helicopter with respect to the body
coordinates. Because it is a rigid body, the helicopter's position and orientation in body coordinates
will always be zero; however, the velocity and acceleration expressions are greatly simplified by using
these coordinates. We will assume the rotor system is completely rigid (there is no aeroelasticity
effect), and that the the airfoil is symmetric and
non-twisted. The aerodynamic interaction between
the rotor and the fuselage is neglected. The aerodynamic expressions are based on 2-D analyses. This
The terms on the right-hand side of the rigid body
equation (1) include both the applied forces and disturbances. The D terms represent drag forces; these
will be treated as disturbances in our model. The
mass of the helicopter is given by m, and the fuselage inertias are Ixx, Iyy, Izz. Terms such as Ixy and
Iyz are zero due to the symmetry of the helicopter
with respect to the x-z plane. Although Ixz is nonzero, because the helicopter is not symmetric with
respect to the x-y plane, it is typically much smaller
than the other terms. We have included it in the
model for completeness. The rotor rotational inertia
is Ir. The rotor angular velocity is £2, and the offset
between the rotor axis and the helicopter's center
of gravity is ir. Usually this offset is expected to
be zero for better handling quality. Nevertheless,
we will assume this quantity is non-zero for generality. The gravitational acceleration constant is g. It
is assumed that the helicopter's center of gravity is
in-line with the rotor axis laterally.
The four independent inputs are T, the net thrust
type of modeling is often called level-1 modeling and
is appropriate for low bandwidth control and to observe the parametric trends for flying qualities and
performance studies.14
The standard rigid body dynamical equation will
be used to model the motion of the helicopter in its
environment. The state vector q = (x,y,z,4>,6,tl;)T
contains the position and orientation information for
the helicopter. Expressing the equation in coordinates gives 15
"1/3x3
0
0
I
generated by the rotor, and M$,Me,M$, the net
9=
moments acting on the helicopter, which are applied
by a pilot. The mechanisms for creating these inputs
will be described in the following section. The torque
applied by the motor, Tm, is related to the thrust T
and cannot be controlled independently. The electronic gyro acts as a damper on the yaw motion; we
will use a simple linear model, Kgi{> for this gyro,
although more sophisticated (PI controlled) gyros
have recently become available.
(1)
Me-'.
'. -
Kg^
The rotation matrix Rjg transforming the inertial
coordinates into body coordinates using yaw-pitchroll (ZYX)
Euler angles is given by
RIB
Model helicopter actuation
A model helicopter moves forward when a pitching moment is first applied and the fuselage is tilted
=
206
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
Clockwise Rotation
Less lift
in this region
More lift
in this region
Vie
Fig. 2 The top view of the helicopter. The lift
distribution on the rotor disk when a forward
cyclic (pitch forward) input is applied. The precession effect will pitch the helicopter forward.
forward. The thrust vector T then gives a forward
component for a forward thrust. A full scale helicopter only requires a forward tilt of the rotor disk
to move forward while the fuselage stays level,16 but
this type of maneuver is not possible with a model
helicopter.
There are four inputs available to the pilot of a
model helicopter. These are physically controlled
by two joysticks on the radio transmitter, each with
two degrees of freedom. The left joystick commands
throttle with collective pitch (up/down) and yaw
(left/right), and the right joystick commands pitch
cyclic (up/down) and roll cyclic (left/right). This
is the most popular configuration used in the U.S.
(type II*). The four values representing the positions of the sticks are encoded in a pulse-width
modulated (PWM) signal, and sent via radio link
to the helicopter. We will use these four positions
(6t,S^,6e,S^) as the inputs to our actuator dynamic
equations because of the way we will control the helicopter later.
The throttle command (St) controls the power to
the main motor (Tm) as well as the collective pitch
(60) of the rotor blades. As the blade pitch increases,
more lift is created, and the rotational motion of the
main rotor blade is converted into vertical thrust.
Usually, the relationship between 00 and 6t is linear
(80 = KSt, where K is some constant). Meanwhile,
an adequate torque Tm is applied to keep Q constant. Sometimes, an electronic throttle governor
is used for this purpose, but mostly a fixed simple
smooth curve determining the necessary functional
relationship between Tm and St is programmed into
the radio transmitter. This curve is traditionally obtained via trial and error. The yaw command
*In some countries, people prefer the yaw and the roll controls switched (type I).
Fig. 3 Vector diagram explaining how the velocity Vg of the hub point g& is calculated. The linear
velocity of the helicopter CG is vbIB and its angular velocity is WJ B . All velocities are measured
in body coordinates. For clarity in the figure,
lateral motions are neglected.
controls the pitch of the tail rotor blade. The tail
rotor on a helicopter is used to counteract the yaw
moment created by the main rotor blade; thus, altering the amount of pitch on the tail rotor can create
more or less total yaw moment for the helicopter.
The pitch and roll commands influence the cyclic
control, varying the cyclic pitch (8cyc} of the rotor
blades around each cycle of rotation, creating different amounts of lift in different regions (as shown in
Figure 2). These differing amounts of thrust create
a moment around the rotor hub, and can thus create
pitch and roll moments on the helicopter.
Before developing the dynamic equations, we introduce some basic aerodynamic terms that will be
required. The advance ratio, /z, and the descent ratio, v, represent the airspeed components parallel to
and perpendicular to the rotor disk respectively.17
They are close to zero when the helicopter is hovering. Both quantities are non-dimensionalized by
R£lJ To define these quantities, we first need to
find the velocity of the hub point with respect to
the inertial coordinate frame represented in body
coordinates. We denote this velocity by vbq. The
angular velocity of the body frame as viewed in the
body frame is ubIB. The velocity of the CG relative
to the inertial frame is vbIB. The coordinate of the
rotor hub point is qb — \ir 0 — hr]T. The constants —hr and tT are the offsets of the rotor from
the helicopter's center of gravity in the x and z directions respectively; the rotor hub is in-line with
t Strictly speaking, R is the rotor span, excluding the rotor
hub length. However, since the model helicopter's rotor blade
remains nearly rigid due to its short length (0.4 to 0.8 meter),
high rotor speed (1300 to 1900 rpm), and hingelesshub design,
R is assumed to be the distance between the rotor axis and
the rotor tip.
207
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
the CG in the y direction. The geometry is sketched
in Figure 3.
v
g
W
=
/B X
+ R•IB
(2)
where,
<j> — ip sin 6
RTIB IB =
0 cos <f> + -ij) cos <j> sin 8
—6 sin <j> + TJJ cos <j> cos 8
x
(3)
The advance ratio, /z, which is the airspeed component parallel to the rotor disk, is the magnitude
of the first two elements of v^, and the descent ratio,
i/, which is the airspeed component perpendicular to
the rotor disk, is the magnitude of the third element
(4)
v
=
control is the mechanism by which the rotor blade's
pitch is changed in a rotation so that an unequal
distribution of the lift applies a moment around the
rotor hub. This moment then provides pitch and roll
attitude control as depicted in Figure 2. The Bellmixer allows the blade pitch to be changed directly
from the cyclic servo actuator. It is fast in response,
but lacks stability. Meanwhile, the Hiller-mixer allows the pitch of the flybar to be changed instead of
the pitch of the blade. The flybar then flaps, and
this flapping motion causes the pitch of the main
blade to change.
There is a direct relationship between the cyclic
input applied to the main blades 6cyc (which is .the
function of stick commands 6s and 6$) and the cyclic
angle of the rotor blades 8cyc. A similar relationship
exists between the cyclic input applied to the flybar
Sfiy and the flapping angle of the flybar /?. The
orientation of the main blade is given by f. Note the
90° phase difference between 6cyc and S/iy, due to
the geometry of the rotor/flybar assembly sketched
in Figure 5.
(7)
(5)
(8)
The rotor solidity, a, is the ratio between the rotor
blade area and the rotor disk area. It indicates how
"solid" the rotor disk is,16 and is taken to be the
number of blades (2) times the area of a rotor blade
(2cR) divided by the area defined by the rotor disk
An important assumption at this point is that the
rotor system does not apply reaction forces back to
the actuators, including the flybar (the flybar is considered to be another actuator to the main blades).
This is equivalent to assuming the actuators are able
to apply infinite amount of forces to the airfoils. We
also neglect the influence of /i and v on the flybar
due to its relatively small wing surface.
When a cyclic input is applied by the pilot, the
flybar creates lift which tilts the flybar disk. The
2c
The inflow ratio, A, is the net value of the descent
ratio v and the induced air velocity, the velocity
of the air through the rotor blade, Vj. It is nondimensionalized by RQ,.
A=-
+ ——
(61
The lift curve slope, a, is the slope of the function
of the lift vs. angle of attack of the main rotor blade.
Flybar dynamics
The dynamics of the rotor and the flybar are the
most significant nonlinearities involved in the creation of the forces and moments on the helicopter.
The actuator dynamics include as states the flapping angle and velocity (/?, /?) of the flybar and the
position and angular velocity (£, Q) of the main rotor blade. As mentioned before, the flybar plays a
major role in augmenting the stability of the helicopter. This system is often called as a Bell-Hiller
mixer, because it takes advantage of two different
cyclic control systems, as shown in Figure 5. Cyclic
208
flybar acts not only as a main blade angle actuator
but also as a stabilizer. If the cyclic input were applied to the main blades only, large control forces on
the cyclic servo actuators would be required.18 By
applying the cyclic control to the flybar and allowing
the flybar to apply a secondary cyclic input to the
main blade, the servo load is significantly reduced.
The flybar is hinged freely on the main axis and
rotates around the main axis. Its angle, /?, is measured with respect to the plane perpendicular to the
main rotor axis. The rotation matrix which relates
the flybar position to the body coordinates of the
helicopter is denoted RBF', as before, the rotation
matrix between the helicopter body frame and the
inertial frame is RIB- The rotation matrix relating the flybar frame to the inertial frame is thus the
product of the two: Rjp = RiBRsFThe pitch angle of the flybar in the helicopter
body coordinates is /?; its yaw angle is £ -j- 90°. The
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
rotational inertia of the flybar / is unified as Ij;
this is a reasonable assumption because most of the
flybar mass is concentrated at the tip region. The
angular velocity of the flybar up involves the velocity of the flybar and the main axis simultaneously; it
can be found by computing RjpRjp and extracting
terms from the elements in Rjp.i5
r-Ri
If
=
I
=
/
r2 mp dr
JRi
(WFX) =
diag [0
If
If ]
T
R IFRip
The external moment applied on the flybar around
the pitch axis is TF, which is the aerodynamic lift
term. To find the total torque, we integrate along
the length of the flybar. Since we will only be interested in the second of these three equations, we will
not consider the first and the third elements in any
detail.
(9)
The lift element dL depends on the angular velocity
of the flybar and its angle of attack OAOA • The angle
of attack of the flybar will be influenced by the pilot
input Sfiy and the second element of the angular
velocity vector of the flybar, denoted up2 .
dL
=
£
BAOA
p(Qr)2a6AOAcdr
Sfly
- ~~~
(10)
WF2
The roll motion of the flybar only affects the lift
created by changing the angle of attack. Because
this lift term is already included into TF, the roll
motion of the flybar is neglected. The yaw motion of
the flybar follows the angle of the main rotor blade.
Once the external force, angular velocity, and inertia
have been defined, the motion of the flybar can be
described using the Euler equation.
=
up
/ up
(12)
The second of these three equations describes the
flapping motion of the flybar. Although the complete expression is quite complex, it can be simplified
by considering the pitch and roll motions of the
helicopter independently. In addition, the small angle approximation is used for /? since it should not
exceed ±25°. For example, if roll motion only is
considered, the second equation of (12) becomes
J3 + cos ^ - 2fi sin £<£ - sin2 £/?<£2 + Q2/? =
- sin^ -
-
cost
(13)
Disturbance
Flybar tip-path plane
Main blade tip-path plane
Fig. 4 The stabilizing effect of the flybar. In a
hovering situation, the flybar angle ft is zero. If a
wind gust or other disturbance knocks the helicopter out of its equilibrium, the flybar, which is
hinged freely, will continue to rotate in the same
inertial plane. Its angle with respect to the main
blade becomes nonzero, and it will help bring the
helicopter back to equilibrium through its action
on the cyclic angle of the main blade.
In the absence of aerodynamic forces and external
moments, the flybar behaves as a gyroscope, maintaining its orientation relative to inertial space,16
as shown in Figure 4. An external disturbance
would upset the helicopter angles 6 and <j>, effectively
changing /?. This nonzero /? acts to apply an appropriate compensation input to the main blade cyclic
control system to stabilize the helicopter.
Actuator dynamics
Once the flybar dynamics have been found, the
rest of the actuator dynamics can be derived. The
thrust generated by the rotor blade is T, and the
moments represent the moments created by the rotor
blade around the roll and the pitch axes are M^ and
Me respectively.
The moment created by a roll cyclic input is derived by summing the forces around a revolution and
along a rotor blade. As per Figure 2, a positive pitch
input 5g produces roll moment M$, but acts as pitch
moment Me due to the precession effect.16
2ir
= 5/7
m Jo
i-R
Jo
r sin £ dL
We assume that the rotor angular velocity is constant; thus £ = Qi. The lift element equation is
similar to the flybar lift used in equation (10).
dLm = -f
ecyc -
cdr
The induced air velocity V{ must be derived empirically.16 However, it is usually of similar magnitude
around the rotor disk, especially in hover or lowvelocity motions. It therefore has little effect on the
net moment on the helicopter; we will assume that
it is negligible in calculating dLm.
From Figure 5, there are two different kinds of
inputs which affect Qcyc: a direct input by the pilot
and an indirect input by the flybar. The geometry
209
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
of the hub linkages indicate that 6cyc will be the
weighted sum of these two inputs.
Jcyc
(14)
"eye
The expression for the rotor thrust T is wellknown in the literature.17
The constant tip loss factor B takes into account the
fact that a finite length rotor blade would lose some
of the lift generated due to the wing tip vortex effect;16 we will use the value B = 0.97.17 The throttle
stick position St also influences the collective pitch
angle B0 through a simple smooth function, KSO . Under normal operating conditions, these two effects
balance each other, and Q is expected to remain
constant.16 Doing so also ensures constant cyclic
control gain. Recall that 00 is the collective pitch
angle, and 6$ and Sg are the roll and pitch inputs
respectively.
Roll servo input
Pitch Servo Input
Main Rotor Blade
Rybar
(16)
The yaw command S^ directly influences the collective pitch of the tail rotor blades 00T, and the
throttle input St is directly coupled both to the motor torque Tm and the collective pitch of the main
rotor blades 00. We model these relationships as linear because their dynamics are fast compared to the
main rotor dynamics.
Oc
Tm
= K6o-St
= KTm-6t
(17)
(18)
(19)
The angular velocity of the tail rotor blades is related to the angular velocity of the main rotor blades
through a constant Kn.
The yaw moment M^ is equal to the thrust of the
tail rotor multiplied by the distance LI between the
main and tail rotor axes. There is no cyclic input
for the tail rotor blades; only a collective pitch angle. Thus, the yaw moment equation is similar to
the thrust equation of the main rotor blade (15),
replacing the terms with the tail rotor equivalents
where appropriate.
—Q0T - -«-AT LT
Fig. 5 The basic structure of the model helicopter's cyclic control system. The flapping angle
of the flybar, /?, is the angle of the flybar with
respect to the body coordinate frame attached
to the rotor hub. It is zero when the flybar is
perpendicular to the rotor axis. Ball joints are
shown as "o", and fixed joints are shown as "•".
The cyclic pitch input to the main rotor blade is
controlled by the combination of the Bell input
from the swashplate and the Hiller input from
the flybar.
The subscripts T indicate that the values pertain
to the tail rotor. To calculate the inflow ratio of
the tail rotor XT, we can use equation(6). First,
equation(2) is used to compute the velocity of the
tail rotor hub in body coordinates; qb must be replaced [— LT 0 0]T which is the coordinate of tail
rotor hub. Equation (5) can then be used to obtain VT- Again, the induced air velocity Vi must be
estimated or obtained empirically.
System Identification
The system identification is performed by isolating the effects of each input, and considering only
a single output. As a first step, we are attempting to identify some of the physical parameters of
the helicopter while restricting to either roll or pitch
motion only. The direct least squares method is
applied to identify the coefficients of the linearized
o
(20)
210
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
experiment
pitch SISO
roll SISO
pitch coupled
roll coupled
Table 1 A sampling of the mean-squared errors between the actual and simulated outputs
using the identified transfer functions Hg^d (z) and
Hied(z). Although the errors are slightly larger
in the coupled experiment, they are still small
enough to give some degree of confidence in the
identified transfer function.
TX for Pilot
Fig. 6 A sketch of the experimental setup for the
system identification of the helicopter's physical
parameters. The helicopter is controlled through
the radio transmitter by a human pilot. The computer is used to record the input data from the
transmitter and the output data from the sensor.
SISO transfer function. The input and output data
are collected while a pilot gives either a roll or pitch
control input (the other inputs are held to zero).
For example, consider the roll motion of the helicopter. Putting together the roll equation from (1)
and the flybar equation (13), it can be seen that the
equation relating the roll input 8$ to the roll angle
<t> will be fourth order. The unknown parameters
in the equation are the inertias Ixx and Ij and the
lift curve slope a; the rotor angular velocity £2 can
be measured directly. Because the input from the
transmitter is held constant over each sample time,
a discrete-time transfer function Hd(z) can be used
to represent the linearized system:
Hd(z) =
z4 + aiz3 +
+ a^z +
mean-squared error
0.0002
0.0010
0.0013
0.0014
(21)
The coefficients a,- , 6,- will be identified using a least
squares method; finding the best match between
a linear SISO transfer function and the measured
input-output data.
Experimental setup
A Polhemus sensor3 is used to measure the position and orientation of the helicopter; full six degreeof-freedom information (a;, y, z, <j>, 0, •&) is available at
50Hz. As shown in Figure 6, the sensor consists of
a board connected to the PC's ISA slot, a transmitter, and a receiver. The transmitter is fixed to the
ceiling, and it sends out a magnetic field via three
orthogonal inductors. The receiver, fixed to the helicopter, senses the strength and the orientation of the
magnetic field and sends this information back to the
PC. For the preliminary experiments described here,
we have fixed the helicopter to a stand to be able to
concentrate on the pitch and roll motion. The input
and output data are taken at 50 Hz, for a total of
approximately 2 minutes duration.
Results
Once the input-output data has been taken and
stored for the isolated roll motion, the system identification algorithm is run to estimate the discretetime transfer function coefficients a,- and 6,-, and
thus the SISO transfer function H$(z}. The initial
condition of the system is also estimated using a
least-squares method. The input data and the estimated transfer function are used to simulate the
output data, and the simulated and actual output
are compared. A reasonable match is achieved, as
shown in Figure 7. The mean-squared error between
the actual output <f>a and the simulated output (j>s
from the estimated transfer function is computed as
follows:
N
error 2 = —
- <t>,(i))2
Typical
4 values for this error on the order of 10
io- .
to
A similar identification is performed for the pitch
motion, and the SISO transfer function Hg(z) is
identified. The input data and the estimated transfer function are used to simulate the output data,
and the simulated and actual output are compared.
Again, a reasonable match is achieved, as shown in
Figure 8.
To determine the validity of the single-axis identification, an experiment was performed in which both
pitch and roll motions were commanded by the pilot. The input-output data was collected, and the
input data was used with the single-input transfer
functions to simulate both the pitch and roll outputs. The results from this experiment are given
in Figure 9. There is some coupling between the
pitch and roll motions, but it does not overwhelm
the dominant single-input effects. A sampling of the
mean-squared errors is given in Table 1.
211
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
Coupled motion with SISO ID
True vs. Simulated output
500
1000 1500 2000
2500
3000
3500
4000
4500
5000
time step
500
1000 1500
2000
2500
3000
3500
4000
4500
5000
time step
1500
2000
3000
time step
Fig. 7 The comparison between the simulated
roll output based on the identified discrete time
transfer function and the actual output from the
experiment, given the identical roll input command history. The initial value for the simulation
was also determined with least squares method
with first 50 points from the experiment.
True vs. Simulated output
0.15
-0.05
1500
2000
time step
Fig. 8
The comparison between the simulated
pitch output based on the identified discrete time
transfer function and the actual output from the
experiment, given the identical pitch input command history. The initial value for the simulation
was also determined with least squares method
using the first 50 points from the experiment.
212
Fig. 9 The system identification result is applied
to the pitch and roll motion simultaneously. The
figure shows some coupled responses through the
discrepancy between the true and the simulated
result.
Conclusions and Future Work
A model helicopter is significantly different than
a full-scale helicopter—it is faster in response, and
lacks stability. A stability augmentation device
called a flybar is commonly used to help the pilot
control the model helicopter. The main contribution of this paper is to model the interaction between
this flybar and the main rotor blade assembly, and
to use these dynamic equations to derive a new and
complete mathematical model for the dynamics of a
model helicopter. Some preliminary system identification experiments were also presented.
We plan to continue the system identification, and
use the results to identify the some of the physical parameters of the helicopter such as the inertias,
aerodynamic coefficients, and other constants (such
as the gains Kg0, etc.). These parameters will be
used to get an accurate nonlinear model for the helicopter dynamics using the equations outlined in
the modeling section of this paper. Further experiments will then be performed to validate the
nonlinear model. Once we have a fairly accurate
nonlinear model, we will begin our feedback control experiments. A short-term goal of this project
is to autonomously hover the helicopter in the lab.
A longer-term goal is to use the complete nonlinear
model to study the interaction between path planning and feedback control for autonomous vehicles,
including the model-scale helicopter.
Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.
Nomenclature
a
OT
B
c
Dptt!/<z
dL, dLm
main rotor lift slope
tail rotor lift slope
tip loss factor
main rotor blade chord length
fuselage profile drag forces
differential lift elements for flybar
and main rotor blade
g
gravitational acceleration
hr
distance between rotor disk and CG,
parallel to rotor axis
I
rotational inertia matrix of flybar
Isxs.
3x3 identity matrix
//
flybar moment of inertia in flapping
IT
rotor moment of inertia around rotor axis
Ixx,yy,zz,xz fuselage rotational moments of inertia
Z
rotational inertia matrix of helicopter
Kg
gyro gain for tail rotor
^Tm
proportional constant relating 5t to Tm
Kea
proportional constant relating St to 80
KeaT
proportional constant relating <5£ to BOT
Kn
proportional constant relating fZ to fir
1T
distance between rotor axis and CG,
perpendicular to rotor axis
LT
distance between tail rotor axis and CG
Li ... L$ linkage lengths in rotor hub assembly
m
helicopter total mass
mp
Flybar mass per length
M<t,:e,ip
net moments on helicopter
R
length of main rotor blade
RT
length of tail rotor blade
RI
distance between rotor axis and flybar tip
f?2
RIB,BF,IF
distance between rotor axis and flybar root
rotation matrices between Inertia!, Body,
T
Tm
Vi
VIB
x, y, z
/3
6cy<:
8fiy
56,64,, 5$
and Flybar frames
net thrust generated by rotor
torque applied by motor
induced air velocity through rotor disk
linear velocity of helicopter
helicopter position coordinates
flybar flapping angle
cyclic input displacement
cyclic input to flybar
roll, pitch, and yaw command input
5t
SAOA
throttle command input
60
60T
angle of attack of main rotor blade
collective pitch angle of main rotor blades
collective pitch angle of tail rotor blades
8cyc
cyclic pitch angle of a main rotor blade
A
AT
y
i/
VT
p
a
ffr
inflow ratio for main rotor
inflow ratio for tail rotor
advance ratio
descent ratio
descent ratio for tail rotor
air density
main rotor solidity
tail rotor solidity
TF
moment applied to flybar
n
helicopter angular position
angular velocity of helicopter
angular velocity of flybar
main rotor angular velocity
tail rotor angular velocity
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213