Journal of Bionic Engineering 4 (2007) 271−280
Optimal Tracking Controller Design for a Small Scale Helicopter
Agus Budiyono1, Singgih S. Wibowo2
1. Department of Aeronautics and Astronautics, Institut Teknologi Bandung, Bandung 40132, Indonesia
2. Simundo Simulation Technology Company, Bandung 40134, Indonesia
Abstract
A model helicopter is more difficult to control than its full scale counterpart. This is due to its greater sensitivity to control
inputs and disturbances as well as higher bandwidth of dynamics. This work is focused on designing practical tracking controller
for a small scale helicopter following predefined trajectories. A tracking controller based on optimal control theory is synthesized as a part of the development of an autonomous helicopter. Some issues with regards to control constraints are addressed.
The weighting between state tracking performance and control power expenditure is analyzed. Overall performance of the
control design is evaluated based on its time domain histories of trajectories as well as control inputs.
Keywords: small scale helicopter, optimal control, tracking control, rotorcraft-based UAV
Copyright © 2007, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved.
Nomenclatures
velocity components in x, y and z-body axes system,
u, v, w
fps
u0, v0, w0 values of u, v, w at trim condition, fps
roll, pitch and yaw rates, rad·s−1
p, q, r
roll, pitch and yaw angles, rad
φ, , ψ
lateral deflection of main rotor, rad
δlat
longitudinal deflection of main rotor, rad
δlon
pitch deflection of tail rotor blade, rad
δped
pitch deflection of main rotor blade, rad
δcol
main rotor longitudinal and lateral flapping motions,
a, b
rad
stabilizer bar longitudinal and lateral flapping moc, d
tions, rad
1 Introduction
The 1990s witnessed the pervasive use of classical
control systems for a small scale helicopter[1]. A Single-Input-Single-Output (SISO) Proportional-Derivative
(PD) feedback control system was primarily used in
which its controller parameters are usually tuned empirically. The approach is based on performance measures defined in the frequency and time domains such as
gain and phase margin, bandwidth, peak overshoot,
rising time and settling time. This trial-and-error approach to design an acceptable control system however
Corresponding author: Agus Budiyono
E-mail: agus.budiyono@ae.itb.ac.id
stabilizer and main rotor time constants
τs, τf
Alon, Blon longitudinal input derivatives
Alat, Blat lateral input derivatives
final time
tf
number of state variables
n
d(⋅)
derivative with respect to time
dt
Lagrange multiplier
pT
gain matrix of co-state equation
K(t)
Superscripts
optimal condition
*
lower bound
−
+
upper bound
T
transpose
is not agreeable with more general complex Multi-InputMulti-Output (MIMO) systems with sophisticated performance criteria. To control a model helicopter as a
complex MIMO system, an approach that can synthesize
a control algorithm to make the helicopter meet performance criteria while satisfy some physical constraints
is required. Recent developments in this field of research
include the use of optimal control (Linear Quadratic
Regulator) implemented on a small aerobatic helicopter
designed at MIT[2,3]. Similar approach was also independently developed for a rotor unmanned aerial vehicle
at UC Berkeley[1]. An adaptive high-band-width
Journal of Bionic Engineering (2007) Vol.4 No.4
272
controller for helicopter was synthesized at Georgia
Technology Research Institute[4].
The current paper addresses this challenge using
optimal control theory and reports encouraging preliminary results amenable to its application to multivariable control synthesis for high bandwidth dynamics
of a small scale helicopter. The practical tracking controller is intended to be implemented on the on-board
computer of the model helicopter as a part of its
autonomous system design.
2 Dynamics of a small scale helicopter
The Yamaha R-50 helicopter dynamics model has
been developed at Carnegie Mellon Robotics Institute.
The experimental helicopter is shown in Fig. 1. It uses a
two bladed main rotor with a Bell-Hiller stabilizer bar.
The physical characteristics of the helicopter are summarized in Table 1. The basic linearized equations of
motion for a model helicopter dynamics are derived
from the Newton-Euler equations for a rigid body that
has six degrees of freedom to move in space. The external forces, induding aerodynamic and gravitational
forces, are represented in a stability derivative form. For
simplicity, the control forces produced by the main and
tail rotors are expressed by the multiplication of a control derivative and the associated control input. Ac
cording to Ref. [5], the equations of motion of the model
helicopter are derived and categorized into the following
groups.
2.1 Lateral and longitudinal fuselage dynamics
Using the Newton-Euler equations, the translational and angular fuselage motions of the helicopter can
be derived as
u& = ( w0 q + v0 r ) − gθ + X u u + X a a ,
v& = (−u0 r + w0 p) − gφ + Yv v + Yb b ,
p& = Lu u + Lv v + L + Lb b ,
q& = M u u + M v v + L + M a a .
(1)
(2)
(3)
(4)
The stability derivatives are used to express the external
aerodynamic and gravitational forces and moments. Xa,
Yb denote rotor derivatives and Lb, Ma the flapping and
spring derivatives. They are used to describe the rotor
forces and moments respectively. General aerodynamic
effects are expressed by speed derivatives given as Xu, Yv,
Lu, Lv, Mu, Mv.
2.2 Heaving (vertical) dynamics
The Newton-Euler rigid body equations for the
heaving dynamics is represented by
w& = (−v0 p + u0 q ) + zw w + zcolδ col .
(5)
In the hovering flight, v0 and u0 are obviously zero. Thus,
the centrifugal forces represented by the terms in parentheses are relevant only in cruise flight.
Fig. 1 The experimental R-50 helicopter.
Table 1
Physical parameter of the Yamaha R-50
Rotor speed
850 r·min−1
Tip speed
449 ft·s−1
Dry weight
97 lb
Instrumented
150 lb
Engine
Single cylinder, 2-strokes
Flight autonomy
30 minutes
2.3 Yaw dynamics
The augmented yaw dynamics is approximated as a
first order bare airframe dynamics with a yaw rate
feedback represented by a simple first-order low-pass
filter. The corresponding differential equations used in
the state-space model are also given in appropriate stability derivatives as
r& = N r r + N ped (δ ped − rfb ) ,
r&fb = − K rfb rfb + K r r .
(6)
(7)
2.4 Coupled rotor-stabilizer bar dynamics
The simplified rotor dynamics is represented by
two first-order differential equations for the lateral (b)
Agus Budiyono, Singgih S. Wibowo: Optimal Tracking Controller Design for a Small Scale Helicopter
and longitudinal (a ) flapping motions. In
state-space model, the rotor models are given as
−b − τ f p + Ba a + Blat (δ lat + K d d ) + Blonδ lon
b& =
,
τf
a& =
−a − τ f q + Ab b + Alat δ lat + Alon (δ lon + K c c)
τf
,
the
(8)
(9)
where the following derivatives related to the gearing of
the Bell-mixer are introduced
B
Kd = d ,
(10)
Blat
Kc =
Ac
.
Alon
(11)
The stabilizer bar receives cyclic inputs from the
swash-plate in a similar way as the main blades do. The
equations for the lateral (d ) and longitudinal (c) flapping
motions are
−d − τ s p + Dlatδ lat
d& =
,
(12)
c& =
τs
−c − τ s q + Clon δ lon
τs
.
(13)
2.5 The state-space model of the R-50 dynamics
The state-space model of the helicopter can be assembled from the above set of differential equations in a
matrix form
x& = Ax + Bu ,
and the cost is quadratic
1
J = x (tf )T Hx (tf ) +
2
1 tf
[ x (t )T Q (t ) x (t ) + u(t )T R(t )u(t )]dt ,
2 ∫t0
(16)
where the requirements of the weighting matrices are
given as
H = HT
0,
⎧⎪Q (t ) = Q (t )T ≥ 0
.
⎨
T
⎪⎩ R(t ) = R(t ) ≥ 0
(17)
(18)
There are no other constraints and τf is fixed, i.e. no
terminal constraints and no constraints on u. Note that
there is a terminal weighting of the state x. The physical
interpretation of the problem statement is that it is desired to maintain the state vector close to the origin
without an excessive expenditure of control efforts[6].
The optimal feedback control law can be derived by
identifying the Hamiltonian of the system and using the
Hamilton-Jacoby-Bellman (HJB) equation to ensure the
optimality[7]. The Hamiltonian H = H[x(t), u*(t), J x* , t]
for the above problem is defined as
H=
1
1
x (t )T Q (t ) x (t ) + u(t )T R(t )u(t )
2
2
*
+ J x ( x , t )[ A(t ) x (t ) + B (t )u(t )].
(19)
(14)
where x = {u, v, p, q, φ, , a, b, w, r, rfb, c, d} is the state
vector and u = {δlat, δlon, δped, δcol}T the input vector. The
dynamic matrix A contains the stability derivatives and
the control matrix B contains the input derivatives. The
complete descriptions of the elements of these matrices
are presented in the Appendix.
T
3 Tracking control design
3.1 Linear regulator problem
A linear regulator problem in the optimal control
theory represents a class of problems where the plane
dynamics is linear and the quadratic form of performance criteria is used. The linear dynamics (which can be
time-varying) is
x& (t ) = A(t ) x (t ) + B (t )u(t ) ,
273
(15)
Minimizing with respect to u, it reads
⎧ dH
T
*
⎪⎪ du = u(t ) R(t ) + J x ( x , t ) B (t )
.
(20)
⎨ 2
⎪ d H = R(t ) > 0
⎪⎩ du2
Note that the above stationary condition defines a global
minimum because the function H only depends quadratically on u.
The optimal control is obtained by using the stationary condition and solving for u,
u* (t ) = − R(t ) −1 B (t )T J x* ( x , t )T .
(21)
The Hamiltonian in Eq. (19) can then be written as
1
H = J x* ( x , t ) A(t ) x (t ) + x (t )T Q (t ) x (t )
2
1 *
(22)
− J x ( x , t ) B (t ) R(t )−1 B (t )T J x* ( x , t )T .
2
Journal of Bionic Engineering (2007) Vol.4 No.4
274
Now, in order to ensure the optimality the HJB equation
must be satisfied. The complete HJB equation is
1
J t* ( x , t ) = J x* ( x , t )T A(t ) x (t ) + x (t )T Q (t ) x (t )
2
1 *
− J x ( x , t ) B (t ) R(t )−1 B(t )T J x* ( x , t )T ,
2
(23)
where the boundary condition (in this case, with fixed tf)
is
1
(24)
J * ( x (tf ), tf ) = x (tf )T Hx (tf ) .
2
A quadratic form for the cost is used to verify its validity
using the HJB equation. Letting
1
J * ( x (t ), t ) = x (t )T Hx (t ) ,
(25)
2
thus, the partial derivatives appearing in Eq. (23), can be
written as
J x* ( x , t ) = x (t )T K (t ) ,
(26)
1
dK (t )
J t* ( x , t ) = x (t )T
x (t ) ,
(27)
2
dt
and the HJB equation becomes
1
1
x (t )T K& (t ) x (t ) = x (t )T Q (t ) x (t ) + x (t )T K (t ) A(t ) x (t )
2
2
1
− x (t )T K (t ) B (t ) x (t ) R(t )−1 B (t )T K (t )T x (t ).
2
(28)
The terms appearing in the above equation are scalar.
The transpose of a scalar term is the same as the term.
Since only scalar terms are dealt with the following
relation applies
1
x (t )T K (t ) A(t ) x (t ) = x (t )T K (t ) A(t ) x (t )
2
1
+ x (t )T A(t )T K (t )T x (t ),
(29)
2
and thus the HJB equation becomes
0=
1
x (t )T [ K& (t ) + Q (t ) + K (t ) A(t ) + A(t )T K (t )T
2
− K (t ) B (t ) R(t )−1 B (t )T K (t )T ] x (t ).
(30)
Now, letting K(tf) = H = HT (symmetric), K(t) will be
symmetric from above and thus symmetry of K(tf) will
be retained ∀t , i.e. K(t) = K(t)T (symmetric). K(tf) = H is
the boundary condition for K(t). Since the above equation must be satisfied ∀x(t ) , the following matrix differential equation is obtained,
− K& (t ) = Q (t ) + K (t ) A(t ) + A(t )T K (t )T
− K (t ) B (t ) R(t )−1 B (t )T K (t )T .
(31)
With a finite terminal time specified the entire solution is
a transient and K(t) will be time varying. However if the
terminal time is taken far enough out, the solution for K
and the corresponding feedback gain might tend to be
constants. To get the invariant asymptotic solution, the
differential equation for K& can be integrated into a
steady state solution or the time derivative term is set to
zero, i.e. K& = 0 . This leads to the Algebraic Riccati
Equation (ARE)[6]
Q + KA(t ) + AT K T − KBR −1 B T K T = 0 .
(32)
For K(t) satisfying the above Riccati form matrix quadratic equation, it is required that Eq. (25) is optimal, and
the optimal control in Eq. (21) is now given by
u* (t ) = − R −1 B T Kx (t ) .
(33)
This is the state feedback control law for the continuous
time LQR problem. Note that both forms of Riccati
equation given in Eqs. (31) and (32) do not depend on x
or u and thus the solution for K can be obtained in advance and finally the gain matrix R−1BTK can be stored.
The control can then be obtained in real time by multiplying the stored gain with x(t).
3.2 Path tracking problem formulation
To use the LQR design for path tracking control,
the regulator problem must be recast as a tracking
problem. In a tracking problem, the output y is compared
to a reference signal r. The goal is to drive the error
between the reference and the output to zero. It is
common to add an integrator to the error signal and then
minimize it. An alternative approach would be using the
derivative of the error signal. Assuming perfect measurements, i.e. the sensor matrix is of identity form
yerror = xerror (t ) = xref (t ) − x (t ) .
(34)
Taking the time derivative of the equation yields
x& error (t ) = x& ref (t ) − x& (t ) .
(35)
When the reference is predefined as a constant,
then x& ref (t ) = 0 , and
Agus Budiyono, Singgih S. Wibowo: Optimal Tracking Controller Design for a Small Scale Helicopter
x& error (t ) = − x& (t ) .
(36)
275
A path tracking control law can be designed by using the
following general relation
and collective pitch. The control input limits for the
helicopter used in this study are
−5˚ δ lat 5˚,
(45)
(37)
(47)
x& error (t ) = −η x& (t ) ,
where is an arbitrary constant representing the weight
of the tracking performance in the cost function. In matrix form, the above relation can be written as
⎡ x&error (t ) ⎤ ⎡ −η x& (t ) ⎤
⎢
⎥
(38)
x& error (t ) = ⎢ y& error (t ) ⎥ = ⎢⎢ −η y& (t ) ⎥⎥ .
⎢⎣ z&error (t ) ⎥⎦ ⎢⎣ −η z& (t ) ⎥⎦
Substituting x& = u , y& = v, z& = w , the above equation can
be expressed as
⎡ −η u (t ) ⎤
(39)
x& error (t ) = ⎢⎢ −η v(t ) ⎥⎥ .
⎢⎣ −η w(t ) ⎥⎦
To accommodate the tracking term in the cost function,
the state-space model is augmented as
(40)
x& aug = Aaug xaug (t ) + Baug u(t ) ,
where
xaug = { xerror , x}T ,
⎡03×3
Aaug = ⎢
⎣014×3
−η I 3
03×11 ⎤
⎥,
Α ⎦
(41)
(42)
⎡0 ⎤
Baug = ⎢ 3×4 ⎥ .
(43)
⎣B ⎦
The size of the matrix is associated with the matrix appearing in Eq. (14) with the addition of three augmented
states. When the terminal weighting is not considered,
the performance measure is now
1 tf
J = ∫ [ xaug (t )T Q (t ) xaug (t ) + u(t )T R(t )u(t )]dt , (44)
2 t0
where the determinations of the weighting matrices , Q
and R are empirical.
3.3 Control synthesis in the presence of input
constraints
In a real system, the control inputs are always limited by hard constraints. The limitation of the control
inputs for a typical aircraft, for instance, is governed by
the maximum allowable deflection of its control surfaces.
In the case of helicopters, the control hard limits are
imposed on their lateral and longitudinal cyclic, pedal
δ lon 5˚,
−22˚ δ ped 22˚,
(46)
−10˚
(48)
−5˚
δ col
10˚.
To incorporate control input constraints into the control
synthesis, the Pontryagin’s minimum principle is employed. Essentially, the Hamiltonian of the system is
re-derived to express the presence of input constraints.
The stationary condition is applied to the modified
Hamiltonian to obtain the optimal bounded control[8].
The Hamiltonian for the above tracking problem is
1
1
H = xaug (t )T Q (t ) xaug (t ) + u(t )T R(t )u(t )
2
2
+ pT ( Axaug (t ) + Bu(t )),
(49)
and its derivative with respect to u is
H u = u(t )T R(t ) + p T B .
(50)
The optimal control can be solved by imposing Hu = 0.
This yields
u* (t ) = − R(t )−1 B T p .
(51)
When the control is constrained or bounded, the optimal
control is
(52)
u* (t ) = arg min[u(t )T R(t )u(t ) + p T Bu(t )] .
u ( t )∈U ( t )
In practice, the control elements are penalized individually. It does not make any sense to minimize the
product between u1 and u2, for example. Thus the
weighting matrix R is not usually a full matrix. The
diagonal matrix R was used in this work to simplify the
optimization considerably. For a diagonal R,
m
1
(53)
u* (t ) = arg min[∑ Rii ui2 + pT bi ui ] ,
u ( t )∈U ( t ) i =1 2
1
(54)
ui* (t ) = arg min[ Rii ui2 + pT bi ui ] .
ui ( t )∈U i ( t ) 2
The unbounded solution can thus be written as
u%i = − Rii−1 p T bi ,
and the bounded control requirement, − M i−
can be implemented in the following logic
⎧u%i ≤ − M i− ui* = − M i−
⎪
if ⎨− M i− ≤ u%i ≤ M i+ ui* = u%i .
⎪%
*
+
+
⎩ui ≥ M i ui = M i
(55)
ui
M i+ ,
(56)
276
Journal of Bionic Engineering (2007) Vol.4 No.4
Note that the solution of the bounded control is not as the
same as the solution obtained by imposing the constraints to the unbounded solution. The optimal control
history of the bounded control case cannot be determined by calculating the optimal control history for the
unbounded case and then allowing it to saturate whenever there is a violation of the stipulated boundaries.
4 Numerical simulation
To evaluate the performance of the tracking controller design, numerical simulation is conducted for a
variety of reference trajectories and for different values
of weighting matrices. The simulation is carried out
using Matlab with the data presented in the Appendix.
Since the focus of the design is on the tracking performance, only the data corresponding to cruise is applicable. The effects of the weighting matrices are analyzed based on the evident tradeoff between tracking
performance and control expenditure. The effect of the
weighting matrix on the state is observed by comparing
the tracking performance between two highly separated
values of Q, Q = 0.01I17 and Q = I17. Similar effects will
be observed for position tracking and control weighting
matrices.
Note that in order to be able to physically interpret
the results of the simulation, a coordinate transformation
is needed between body coordinate and local horizon
coordinate system. The transformation matrix between
these two coordinates is given as
cθ cψ
cθ s
− sθ ⎤
⎡
⎢
T = ⎢ sφ sθ cψ − cφ sψ sφ sθ sψ + cφ cψ sφ cθ ⎥⎥ ,
⎢⎣cφ sθ cψ + sφ sψ cφ sθ sψ − sφ cψ cφ cθ ⎥⎦
b
I
Fig. 2 Trajectory tracking performance,
η = 0.01, Q = 0.01I17, R = 0.01I4.
Fig. 3 Velocity tracking performance, η = 0.01,
Q = 0.01I17, R = 0.01I4.
(57)
where c(·) = cos(·), and s(·) = sin(·). With this transformation, the final results of the simulation are presented
in the local horizon coordinate. The corresponding
equations for the positions and the velocities are
⎧⎪[ N , E , A]T = TIb [ x, y, z ]T
.
(58)
⎨
T
T
b
⎪⎩[Vx ,Vy ,Vz ] = TI [u , v, w]
Case 1: Reference: rectangular trajectory as given
in Fig. 2. Velocity tracking performance and control
input expenditure are shown in Fig. 3 and Fig. 4, respectiuely. The weighting matrices are = 0.01, Q =
0.01I17, R = 0.01I4.
Fig. 4 Control input expenditure, η = 0.01,
Q = 0.01I17, R = 0.01I4.
Case 2: Reference: rectangular continued by circular trajectory as given in Fig. 5. Velocity tracking and
control input expenditure are shown in Fig. 6 and Fig. 7,
respectively. The weighting matrices are = 0.01, Q =
0.01I17, R = 0.01I4.
Agus Budiyono, Singgih S. Wibowo: Optimal Tracking Controller Design for a Small Scale Helicopter
Fig. 5 Trajectory tracking, η = 0.01, Q = 0.01I17, R = 0.01I4.
Fig. 6 Velocity tracking, η = 0.01, Q = 0.01I17, R = 0.01I4.
Fig. 7 Control input expenditure, η = 0.01,
Q = 0.01I17, R = 0.01I4.
Case 3: Reference: rectangular trajectory as given
in Fig. 8. Velocity tracking and control input expenditure
are shown in Fig. 9 and Fig. 10, respectively. The
weighting matrices are = 5, Q = I17, R = I4.
277
Fig. 8 Trajectory tracking, η = 5, Q = I17, R = I4.
Fig. 9 Velocity tracking performance, η = 5, Q = I17, R = I4.
Fig. 10 Control input expenditure, η = 5, Q = I17, R = I4.
Case 4: Reference: rectangular continued by circular trajectory as given in Fig. 11. Velocity tracking is
shown in Fig. 12. The weighting matrices are = 5, Q =
I17, R = I4.
278
Journal of Bionic Engineering (2007) Vol.4 No.4
Fig. 11 Trajectory tracking performance, η = 5, Q = I17, R = I4.
trol input for Case 3. The same controller can successfully handle more complex trajectory for Case 4 as
shown in Fig. 11 through Fig. 12. The tracking errors can
be kept minimum while maintaining the required control
input within the stipulated boundaries.
The numerical comparison between Case 1 and
Case 3, as shown in Table 2, represents the tracking
performance comparison to evaluate the effect of the
weighting matrices.
The performance norm is given as the mean square
error (MSE) between the actual and reference trajectories. It is evident that Case 3 (Q = I17) demonstrates
lower tracking error compared to Case 1 (Q = 0.01I17).
Table 2 Tracking error comparison
Tracking error (MSE)
Case
Fig. 12 Velocity tracking performance, η = 5, Q = I17, R = I4.
5 Discussion
Fig. 2 indicates the tracking performance for Case 1,
where the diagonal elements of weighting matrices were
assigned small numerical values. In the calculation of
the optimal control, no constraints of control input were
imposed. It can be observed that towards the end of the
trajectory the tracking performance is degrading especially in the vertical position. The control input history,
in Fig. 4, shows that even though the control expenditure
is not bounded, it fails to provide acceptable tracking
performance. The degradation in the tracking performance is more pronounced for a more complex reference
trajectory in Case 2, as shown in Fig. 5.
In Cases 3 and 4, the weights for tracking and
control elements were increased in the order of magnitudes to observe their effects on the overall performance
of tracking controller. Fig. 8 through Fig. 10 show the
tracking performance and the associated bounded con-
Velocity (fps)
Position (ft)
Vx
Vy
Vz
N
E
A
1
19.8
20.2
3
148
2043
771
3
0.77
1.13
0.06
9.6
29
0.18
The effect of the weighting matrix Q is more pronounced
for the position tracking of E and A. The MSE type
tracking error for higher Q (Case 3) is 0.0142 and that of
the low value of Q (Case 1) is 0.000 233.
Note that in this study, the weights for augmented
states and control element were assigned the same values. This assumption simply means that in the optimization process the control input elements are considered
equally important. However, for the augmented states,
the assumption means that the importance of the states is
not treated evenly.
Overall results indicate that weight assignments
play a significant role in the optimization process, which
determines the tracking performance. To the author’s
knowledge, despite its great influence, the determination
of weighting values has been so far done primarily on an
ad-hoc and case-by-case basis. The determination of the
weighting values for the tracking performance of a ballistic missile involving its range and azimuth angle, for
instance, can be guided by a simple fact that 0.01 radian
error in the azimuth angle can contribute a position error
of 10 km for missiles with a range of 1000 km. More
refined weight scheme using similar approach can be
Agus Budiyono, Singgih S. Wibowo: Optimal Tracking Controller Design for a Small Scale Helicopter
applied in the optimal control design of helicopters. It is
worthy to note that formal treatment of weight assignment for the optimal control methodology exists in the
literature. One can use the pole placement technique in
conjunction with optimal control theory where poles of
the closed loop system are assigned and the weight assignment can be derived from the corresponding
mathematical relation. A more novel technique has been
recently proposed in Ref. [9] in the framework of polynomial approach where formal weight assignment can
be performed in association with integrated control design criteria. The relevant work of analytical weight
selection is presented in Ref. [10].
279
References
[1] Shim H. Hierarchical Flight Control System Synthesis for
Rotorcraft-Based Unmanned Aerial Vehicles, PhD Thesis,
University of California, Berkeley, USA, 2000.
[2] Gavrilets V, Frazzoli E, Mettler B, Piedmonte M, Feron E.
Aggressive maneuvering of small autonomous helicopters:
A human-centered approach. International Journal of Robotics Research, 2001, 20, 795–807.
[3] Gravilets V, Martinos I, Mettler B, Feron E. Control logic for
automated aerobatic flight of miniature helicopter. AIAA
Guidance, Navigation and Control Conference and Exhibit,
Monterey, California, 2002, AIAA-2002-4834.
[4] Corban J E, Calise A J, Prasad J V R, Hur J, Kim N. Flight
evaluation of adaptive high bandwidth control methods for
6 Concluding remarks
unmanned helicopters. AIAA Guidance, Navigation and
A control design methodology based on optimal
control theory was elaborated and applied in the controller of a small scale helicopter model. It has been
demonstrated that the approach neatly handled more
complex design criteria than ones that can be traditionally afforded by classical control design. The overall
design is a part of an ongoing research, design and integration of a small autonomous helicopter[11] where
robust yet practical control algorithm is desired. The
anticipated practical control design criteria include, but
not limited to:
(1) To fly the helicopter from an arbitrary origin to
a specified waypoint in minimum time which characterizes a minimum-time problem;
(2) To bring the helicopter from an arbitrary initial
state to a specified waypoint, with a minimum expenditure of control effort which is a minimum-controleffort problem;
(3) To minimize the deviation of the final state of
the helicopter from its desired waypoint which represents a terminal control problem.
The above design criteria can be conveniently
formulated and incorporated into the cost function. For
future research direction, it will be interesting to explore
if the proposed control technique can maintain acceptable performance in the presence of wind.
Control Conference and Exhibit, Monterey, California, 2002,
AIAA-2002-4441.
[5] Mettler B, Tischler M B, Kanade T. System identification
modeling of a small-scale unmanned rotorcraft for flight
control design. Journal of the American Helicopter Society,
2002, 47, 50–63.
[6] Kirk D E. Optimal Control Theory: An Introduction, Prentice Hall, New Jersey, USA, 1970.
[7] Budiyono A. Principles of Optimal Control with Applications, lecture notes on optimal control engineering, Department of Aeronautics and Astronautics, Bandung Institute
of Technology, 2004.
[8] Velde V. Principles of Optimal Control, lecture notes,
graduate course in optimal control, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 1995.
[9] Manabe S. The coefficient diagram method. 14th IFAC
Symposium on Automatic Control in Aerospace, Seoul, Korea, 1998, 199–210.
[10] Budiyono A, Sudiyanto T. An algebraic approach for the
MIMO control of small scale helicopter. International
Conference on Intelligent Unmanned System, Bali, Indonesia, 2007.
[11] Budiyono A. Design and Development of a Small Autonomous Helicopter for Surveillance Mission, Technical Report,
Department of Aeronautics and Astronautics, Bandung Institute of Technology, 2005.
Journal of Bionic Engineering (2007) Vol.4 No.4
280
Appendix
I The state-space model of R-50 helicopter
The state-space equation describing the R-50 dynamics is
II The model parameters of R-50 helicopter
The model parameters of the helicopter during hover and cruise-flight are summarized in Table 3[5].
Table 3 Control derivatives and time-constants of the Yamaha R-50
Blat
Blon
Alat
Hover
0.14
0.0138
0.0313
−0.1
−45.8
0
Cruise
0.124
0.02
0.0265
−0.0837
−60.3
6.98
B
B
Alon
Zcol
Mcol
τf
hcg
τs
Nped
Dlat
Clon
−3.33
33.1
0.273
−0.259
0
0.0991
0.046
−0.411
0.342
26.4
0.29
−0.225
11.23
0.0589
0.0346
−0.321
0.259
0
Yped
τp
Ncol