[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control...

Trajectory Control of Miniature Helicopters Using a Unified Nonlinear Optimal Control Technique

Ming Xin1 Department of Aerospace Engineering, Mississippi State University, Starkville, MS 39759

Yunjun Xu2 Department of Mechanical, Materials, and Aerospace Engineering, University of Central Florida, FL 32816

and

Ricky Hopkins3 Nordam Transparency Division, Tulsa, OK 74117

It is always a challenge to design a real-time optimal full flight envelope controller for a miniature helicopter due to the nonlinear, under-actuated, uncertain, and highly coupled nature of its dynamics. This paper integrates the control of translational, rotational, and flapping motions of a simulated miniature aerobatic helicopter in one unified optimal control framework. In particular, a recently developed real-time nonlinear optimal control method, called the -θ D technique, is employed to solve the resultant challenging problem considering the full nonlinear dynamics without gain scheduling techniques and time-scale separations. The uniqueness of the -θ D method is its ability to obtain an approximate analytical solution to the Hamilton-Jacobi-Bellman equation, which leads to a closed-form suboptimal control law. As a result, it can provide a great advantage in real-time implementation without a high computational load. Two complex trajectory tracking scenarios are used to evaluate the control capabilities of the proposed method in full flight envelope. Realistic uncertainties in modeling parameters and wind gust condition are included in the simulation for the purpose of demonstrating the robustness of the proposed control law.

Nomenclature

1 1,a b,a a

= Lateral and longitudinal flapping angles 15.5rad − = Main rotor blade lift curve slope, mr

tra 15.0 rad = Tail rotor blade lift curve slope, −

lon

nomAδ = Nominal longitudinal cyclic to flap gain, 4.2 /rad rad

, ,x y z

lat

nomBδ = Nominal lateral cyclic to flap gain, 4.2ra /d rad

D D DC

0

mr

C C

= Drag coefficients in three directions, 0.1, 0.5, 0.2

DC 0.024

0

tr

= Main rotor blade zero lift drag coefficient,

DC 0.024ht

= Tail rotor blade zero lift drag coefficient,

C 13.0 rad = Horizontal tail lift curve slope, − Lα

1,* Correspondence: Email: [email protected]; Assistant Professor, Department of Aerospace Engineering, Mississippi State University, Starkville, MS 39759.

American Institute of Aeronautics and Astronautics

1

AIAA Guidance, Navigation, and Control Conference2 - 5 August 2010, Toronto, Ontario Canada

AIAA 2010-7872

Copyright © 2010 by Ming Xin. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

mailto:[email protected]

vfLCα

12.0 rad − = Vertical fin lift curve slope,

mrc = Main rotor chord, 0.058m

trc = Tail rotor chord, 0.029 m

mrQC= = Main rotor torque coefficient QC

TC thrust coe

= Thrust coefficient

ma

mrT = Maximum main rotor fficient,

xC 0.0055

g = Gravity acceleration, 9.81 /m s2

mrh = Main rotor height above c. g., m 0.235

trh = Tail rotor height above c. g., 0.08 m

rotI = Rotating inertia of the main ro e,tor blad 2.5mr

Iβ

xxI = Rolling moment of inertia, 0 2.18kg m = Pitching moment of inertia, 20.34 kg m yyI

zzI 2m = Yawing moment of inertia, 0.28kg20.038kg m = Main rotor blade flappin ing ertia,

mrIβ

= Hub torsional stiffness, 54 /N m rad Kβ

Kλ = Main rotor wake intensity factor K = Scaling of flap response to speed variation, 0.2 μ

, ,M = Rolling, pitching, and yawing moments N L

htl = Location of stabilizer bar behind 0. c. g. behi .,

, 71m

trl = Tail rotor location nd c. g 0.91m m = Helicopter mass, 8.2kg

trn = Tail rotor to main rotor gear ratio, 4.66 ,p ,q r = Roll, pitch, and yaw rate

eQ = Engine torque

mrR = Main rotor radius, m 0.775

trR = Tail rotor radius, 0.13m

( )R Θ = Direct cosine matrix fro bodym to inertial coordinates

htS = Horizontal fin area, 20.01m

vfS = Effective vertical fin area, m 20.012 fus

xS = Front fuselage drag area, 20.1m fusyS = Side fuselage drag area, 0.22 2m fus = Vertical fuselage drag area,

= Thrust

20.15m zS

ocity, 0.42 m/s

Tmax

mrT = Maximum main rotor thrust , ,u v w = Body-axis velocities

V∞ = Axial velocity magnitude trV∞ = Axial velocity at the tail rotor

imrV = Induced main rotor velocity, 4.2 m/s

itrV = Induced tail rotor vel, ,p p px y z = Inertial position , ,X Y Z = Body-axis forces


2

( )2/m s r d ped

trYδ = Tail rotor side force coefficient due to pitch, al

5.4 atr

vY 10.06 s− = Tail rotor side force coefficient due to velocity, = Main rotor collective inpcolδ ut

latδ = Lateral cyclic input

lonδ = Longitudinal cyclic input

pe δ d = Pedal input trvfε = Portion of vertical fin exposed to tail rotor i ucnd ed velocity, 0.2

wη = Coefficient of non-ideal wake contraction, 0.9 = Inflow ratio of the main rotor, 0.033 0λ

μ = Advance ratio

zμ = Normal airflow component

nom

= Rotor speed = Nominal main rotor spee 167 /rad s d,Ω

Ω ensity= Atmospheric d , 31.225 /kg m ρ

[ ], ,θ φ ψ= = Euler angles Θσ = Solidity ratio, 0.0361

= Damping time constant for the flapping motion, 0.1s eτSu

=

bscript a = Actual

d = Desired e Engine fus = Fuselage ht = Horizontal tail mr = Main rotor

= Tail rotor trvf = Vertical fin w = Wind

I. Introduction esearch in the topic of unmanned aerial vehicles (UAVs) control is highly motivated and has a very renewed interest as the commercial demand for such vehicles is growing rapidly. UAVs to date, both military and

commercial, have been allowed to operate only within restricted airspace. Recently, the Department of Defense (DOD) has constructed a plan for UAVs to operate within the National Airspace System (NAS) by 2010.1 The Federal Aviation Administration (FAA) has requirements for UAVs to maintain flight qualities that are at least as stringent as those placed upon manned aircraft so that UAVs can be operated safely within the NAS. One of the major requirements facing the incorporation of UAVs into the NAS is the ability to track the desired trajectories pre

R

cisely, no matter the circumstances. Thus, UAVs must be able to quickly maneuver even under adverse environmental conditions or with degraded systems.

As a subset of UAVs, unmanned rotorcrafts have implications for both commercial and military uses. In the International Aerial Robotics Competition (IARC), which began in 1991, the most common vehicle to successfully complete simulated tasks of toxic waste identification and mapping, survivor or hostage detection in volatile environments, and other life-like reconnaissance missions is miniature helicopters.2 Miniature helicopters can offer unique advantages over traditional fixed-wing craft such as vertical takeoff, vertical landing, hover, and movement independent of heading. The aforementioned capabilities lend the miniature helicopter to be better suited than other aircraft for specific missions or environments. For example, hovering may be an advantage in urban settings.

To date, majority of the control techniques applied in rotorcraft attitude and velocity tracking have some forms of linearization and gain scheduling.3-10 Position tracking adds complexity to the problem due to the extra nonlinearities that arise in translating the velocity vector from a body fixed coordinate system to the inertial system.


3

6 Controllers such as PID 11 , H∞10,12 and 2H 4,6 have been designed for linear motions, but the model under control is

only valid in a small region around its trim condition. These controllers are less reliable when the flight path deviates from the predetermined trim conditions. Specific flight conditions such as forward flight or hover have been controlled adequately with linear controllers. However, challenges will arise for linear controllers when significant nonlinearities exist due to the changing aerodynamics over certain operations such as ascending, descending, and turning maneuvers.

To control nonlinear motions of a miniature helicopter, dynamic inversion,14 neural networks,15 and sliding mode control (SMC)16 have been investigated. For example, Zeng14 applied the combined PID and dynamic inversion technique to a simplified/reduced model (four degrees of freedom (DOF)) with adaptive compensators to account for the modeling uncertainties and disturbances. However, only a limited flight envelope that includes hover and forward flight has been shown.

13

17 Oh investigated position tracking using a zero finding algorithm with a reduced dynamics model and assumed that aerodynamic effects do not play a significant role. In addition, only a simple ascending motion is simulated.

When full dynamics are considered, which is an under-actuated system in the case of miniature helicopters, pseudo inverses are required in the input-output feedback linearization and SMC approaches. Numerical errors coming from this inversion will cause internal state instability. The common practice in aircraft control to handle the under-actuated system is to use the inner-outer loop or multi-timescale strategy.15, 18-23 The flight dynamics is separated into a slow mode for translational motion and a fast mode for attitude motion. The control law is designed accordingly in each mode. For example in Ref 23, a chattering free nonlinear controller combined with a zero-finding algorithm has been investigated for three time-scales full envelope flight. Different flight modes have been tested; however, the computational cost is high. In addition, this conventional design paradigm normally considers the robustness but is not optimized because the separate designs cannot take the coupling from the other elements into full consideration and can result in excessive iterations during the integration process. Johnson and Kannan15 proposed a novel Pseudocontrol Hedging approach to enable neural network adaptation to occur in the outer loop without interacting with the attitude dynamics. This method will alleviate in some extent the coupling effect in the two-loop design process.

The primary contribution of this work is to integrate the control of helicopter trajectory, attitude motion, and flapping dynamics in one unified real-time optimal control framework. The conventional separated control design for different timescale modes is avoided so that the overall system performance can be optimized with one single design process. The resultant problem involves highly nonlinear kinematics and dynamics, and thus traditional linear control designs are unsuitable for a precise control. Bogdanov and Wan24 used a similar optimal control formulation with the state-dependent Riccati equation (SDRE) technique and a complex nonlinear compensator. This method (validated through experiments) shows its capability of tracking position over the entire flight envelope with the assumption of a quasi-static flapping motion. The results are promising. However, the SDRE approach is based on an extended linearization of vehicle dynamics and demands more computational resources because it needs to solve the algebraic Riccati equation numerically at every integration step, which involves a complex and iterative process. The computational load will be particularly high when dealing with the full degree-of-freedom dynamics and higher-order systems. In this paper, a recently emerging nonlinear suboptimal control method, called the Dθ − technique,25, 26 is employed to design this in ted nonlinear control law. Based on an approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation, the D

tegraθ − technique provides a closed-form feedback controller.

Compared with the SDRE method, the Dθ − approach offers a great computational advantage for onboard implementation without solving the Riccati equation repetitively at every instant.

Rest of the paper is organized as follows. In Section II, the dynamics model of the miniature helicopter is described. The integrated control law is designed using the proposed Dθ − nonlinear optimal control technique in Section III. Simulation re e concluding remarks are given in Section V.

d and their values for this particular miniature helicopter can be found in the section of nomenclature. A

sults and analysis are presented in Section IV. Som

II. Miniature Helicopter Dynamics The prevailing way of treating aircraft dynamics, especially with respect to miniature helicopters, breaks the

dynamics into different timescales with the assumption that the flapping dynamics are quasi-static.18-22 In this paper the flapping dynamics is explicitly considered and the optimal control formulation does not differentiate these timescales. However, for the convenience of presentation, the dynamics of the helicopter will still be described in terms of translational (slow) motion (6 states), rotational (middle) motion (6 states), and the flapping (fast) motion (2 states). The model has a total of 14 states and is modified according to Gavrilets.27 Note that the explanation of the symbols use


4

ske of such a miniature helicopter is shown in Figure 1, in which some of the state and control variables are illustrated.

tch

Figure 1. Axis system with forces acting on the helicopter

nslationA. Tra al Motion The stat f the translational motion include the inertial position vector [ , , ]T

p pes o px y z and the inertial velocity

vector [ , Tu v expressed in the helicopter’s body frame nema relation betw, ]w, and w ) an

. The ki tic een the body frame velocity d the velocity in the inertial coordin( u , v ate ( px , , and py zp ) is gi

p

ven by

( )p

p

x uy v

wz

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

R Θ

⎣ ⎦

(1)

where [ ], , Tφ θ ψ=Θ are the Euler angles, and

( )cos cos sin sin cos cos sin cos sin cos sin sincos s sin sin sin cos cos cos sin sin sin cos

sin sin cos cos cos

θ ψ φ θ ψ φ ψ φ θ ψ φ ψθ ψ φ θ ψ φ ψ φ θ ψ φ

− +⎡ ⎤⎢ ⎥+ − ψ

θ φ θ φ θ= ⎢ ⎥⎢ ⎥−⎣

R Θ (2)

⎦is the direct cosine matrix from the body coordinate to the inertial coordinate using a 1-2-3 Euler rotation sequence. The translational d

ynamics is governed by

1

1

/ sin /

/ sin cos /fus mr

trfus tr vf ped mr

vr wq X m g a T muv wp ur Y Y Y m g Y b T m

θ

φ θ δ

− + −⎡ ⎤

/ cos cos /ped

fus mrw uq vp Z m g T m

δ

φ θ

⎡ ⎤−⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤= − + + + + + +⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − + +⎣ ⎦ −⎢ ⎥ ⎣ ⎦⎣ ⎦where

(3)

g is the gravitational constant; m is the mass of the miniature helicopter; 1a and 1b are lateral and longitudinal flapping angles, respectively; , ,p q r are ro , and yaw The external forces contain

on the fuselage, ll, p rates.itch

aerody ic forces acting nam fusX , fusY , and fusZ , force n the vertical fin o

( )12vf vf vfY Sρ= − , force fro rotor tor thrust t ional

te

vf trL vfC V vα ∞

+ v m the tail trvY mYtr trv= , main ro Tmr , and he gravitat

force. Deno 2 2tra trV u w∞ = + , tr

vf a vf itr trv v V l rε= − − , and tr a trw w l q K Vλ imr= + − where a wu u u= − , a wv v= − , and

a ww w w= − are relative velocity with respect to the wind, in which, the subscript ‘w’ denotes the wind. imrV is the induced main rotor velocity, whereas V is the induced tail rotor velocity. S is the effective vertical fin area; tr

v

itr vf vfε is the vertical fin area portion tr exposed ced to the indu velocity from the tail rotor; l is the tail rotor hub location


5

behind the center of gravity. The side force from the vertical fin is constrained by ( )( )2 20.5 trvf vf vfY S Vρ ∞≤ + v to

compensate for potential stall of the vertical fin.23,27 The forces acting on the fuselage are given by

0.5 fusfus Dx x aX C S u Vρ ∞= − (4)

0.5 fusfus Dy y aY C Sρ v V∞= − (5)

and ( )0.5 fusfus Dz z a imrZ C S w V Vρ ∞= − + (6)

where the axial velocity magnitude is ( )22 2a a a imrV u v w V∞ = + + + and fus

xS , fusyS , and fus

zS are the front fuselage

drag area, side fuselage drag area, and vertical fuselage drag area, respectively.

Kλ is the wake intensity factor calculated according to

0

1.5

1.5

aimr a i

imr a

af

imr a

ai

imr a

f i

uif V w or g

V w

uK if g

V w

u gV w

elseg g

λ

⎧⎪⎪ ≤ ≤

−⎪⎪⎪⎪⎪= ≥⎨ −⎪⎪⎪ −

−⎪⎪ −⎪⎪⎩

(7)

with tr mr tri

tr

l R Rg

h− −

= (8)

and tr mr trf

tr

l R Rg

h− +

= (9)

where and are the radius of the tail and main rotor. The main rotor engine speed is calculated by solving the collection of equations below iteratively using the zero-finding algorithm once the main rotor thrust is computed from the control law.

trR mrR mrΩ

mrT

( )24 2 20 02 w mr mr z mrRη λ ρπ μ λ μ 0TΩ + − − = (10)

2 2a a

mr mr

u vR

μ+

=Ω

(11)

az

mr mr

wR

μ =Ω

(12)

Here, wη is a coefficient to account for non-ideal and non-uniform wake, 0λ is the inflow ratio, μ is the advance ratio, and zμ is the normal airflow component.

B. Rotational Motion


6

The states in the rotational motion include the angular velocities [ ,rotx , ]Tp q r and Euler angles [ , , ]Tφ θ ψ . The attitude dynamics model is represented by

( ) ( )( )( ) ( )

( )( )

1

1

/ // /

/ / /

// /

ped

ped

trtr ped xx mr mr xxyy zz xx vf tr xx

zz xx yy ht yy mr mr yy

trped tr zzxx yy zz vf tr e zz

h mY I K T h b Iqr I I I L L Ipq pr I I I M I K T h a Ir mY l Ipq I I I N N Q I

δ β

β

δ

δ

δ

⎡ ⎤⎡ ⎤ + +− + +⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥= − + + +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥ −− + + −⎣ ⎦ ⎣ ⎦

(13)

and the kinematic relation between the angular velocities and Euler angular rates are

sin tan cos tancos sinsin sec cos sec

p q rq rq r

φ φ θ φθ φ φψ φ θ φ θ

⎡ ⎤ + + θ⎡ ⎤⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+⎣ ⎦⎣ ⎦

(14)

In this mode, the torques produced by the tail rotor are calculated using and , where . The moments of the vertical fin are derived to be

trtr tr v trL h mY v=

vf vf trL Y h

trtr tr v trN l mY= − v

ptr a tr trv v l r h= − + = and . The

moment from the horizontal tail is vf vf trN Y l= −

ht ht htM Z l= , where ( )0.5 htht ht L a htht htZ S C w

αρ= +

ht

u w w is the lift from the

horizontal stabilizer with the horizontal tail area, htS LCα

the horizontal tail lift curve slope, and

. The lift from the horizontal stabilizer is constrained by htw wa ht imrl q K Vλ= + − ( )2 2ht a ht0.5htZ S u wρ≤ + to

accommodate for stall and Kλ is defined in Eq. (7) with different values depending upon the relation between and . The engine torque is modeled as

imrV

aw

e rot mr tr trQ I r Q n Q= − + + (15)

where the main rotor torque is calculated by and ( )2 3mrmr Q mr mr mrQ C R Rρ π= Ω ( ) 0 2

071

8 3mr

DQ T z

CC C

σλ μ μ⎛ ⎞= − + +⎜ ⎟

⎝ ⎠.

In this equation, 2 mr

mr

cR

σπ

= , and ( )220 0 zμ λ μ= + − tr tr mrn2T wC η λ . Ω = Ω is the tail rotor speed and is the gear

ratio between the main and tail rotor.

trn

C. Flapping Dynamics The states of the fast mode are the flapping angles, , and the control inputs are the collective pitch

control and the lateral and longitudinal cyclic control inputs, . The collective controls the pitch of the main rotor blades, effectively increasing or decreasing the altitude of the miniature helicopter. The cyclic control adjusts the pitch of the main rotor blade disk. The angle of attack and lift generated by each blade is thus dependent upon the position of the blade as it rotates around the main rotor hub.

[ ]1 1, Tf a b=x

u = [ , , Tδ δ δ ]f col lat lon

The flapping dynamics are reorganized as

( )22

10

1

21 1

0

8162 38

823

lon

lat

noma mra a

col lone mr mr e nome e mr mr e mr mr

noma a mr

cole e mr mr

AK uK w uaq sign K Ra a R R

b Bvb K vp KR R

δμμμ

δμμ

μ δ δμ λτ ττ μ σ τ τ

λ δ δτ τ τ τ

⎡ ⎤ ⎛ ⎞⎡ ⎤ Ω+− − + −⎢ ⎥⎢ ⎥ Ω Ω⎡ ⎤ + Ω Ω ⎝ ⎠⎢ ⎢ ⎥ ⎥⎣ ⎦= +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎛ ⎞Ω⎢ ⎥− − − + ⎜ ⎟⎢ ⎥Ω Ω Ω⎣ ⎦ ⎝ ⎠lat

e mr mr e nom

⎜ ⎟⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎦

(16)

⎣where eτ is a time constant for this mode, Kμ is the scaling of the flap response to the speed variation, and is the main rotor blade lift curve slope.

a(sign )μ is defined as

( )11 0

ifsign

ifμ

μμ≥⎧

= ⎨0

− <⎩ (17)


7

lon

nomAδ and are the longitudinal and lateral gains between the cyclic inputs and flapping angles, respectively. lat

nomBδ

D. Uncertainties and Wind Gust Model There exist various uncertainties in the miniature helicopter model and environmental conditions. In order to test

the robustness of the later designed nonlinear controller, different from other approaches,24, 27 the parameter uncertainties and perturbations will be included in the model as shown in Table 1, which contains 45 types of uncertainties. The wind gust may have a significant effect on the miniature helicopter and must be considered. The wind gust model is scaled down from the one given in AC120-28D.28 Specifically, in the x direction, the wind gust model has a mean value of ( )101.258 0.949log /z m s+

y

,a b

and a standard deviation of , where is the altitude of the helicopter. In the direction, the wind gust model has a zero mean and a standard deviation of

. The uncertainties considered in the helicopter model can be categorized as measurement noise, estimation error, or parameter uncertainties and are shown in Table 1. For example, there is 5% noise assumed in the estimated flapping angles ( ), and 5% noise is considered in the main rotor thrust ( ). Also as another example, a 10% noise is considered in the induced main rotor velocity.

0. /m s

T

199 z

0.199 /m s

1 1 mr

Table 1 Uncertainties considered

Parameters Uncertainty Parameters Uncertainty Parameters Uncertainty

1 1,a b 5% pedalδ 5% mra 5%

lon

nomAδ 5% lat

nomBδ 5% mrc 5%

g 0.5% mrh 5% trh 5%

xxI 5% yyI 5% zzI 5%

Kβ 5% Kμ 5% trl 5%

m 0.5% , ,p q r 5% mrR 0.5%

mrT 5% , ,u v w

p

5% , ,w w wu v w 5%

imrV 10% itrV 10% , ,p px y z 5%

ped

trYδ 10% , ,loncol latδ δ δ 5% , ,φ θ ψ 5%

wη 1% 0λ 1% ρ 0.5%

e

5% τ 5% nom Ω

D−III. Unified Nonlinear Optimal Control Design via θ Method

A. Unified Optimal Control Formulation The approach to controlling the miniature helicopter modeled in Section 2 is to formulate the full dynamics of

motion into one unified state-space and optimal control framework. The control design will be simplified via one


8

single process and the overall system performance will be optimized through one cost function. Furthermore, since trajectory control is the focus of this work, the accuracy of the po tion tra king is important. Thus, three integral states of the position components

si cpx , py , and pz , denoted as ix , iy , and iz , are augmented into the state-space to

improve the tracking performance. Therefore, the state variables are chosen to be: ⎤ ∈ℜ⎦ (18)

where

17 11 1

T

i i iv w p q r a b x y zφ θ ψ ×p p px y z u⎡= ⎣x

ix , iy , and atisfy iz s; ;i p i p i px x y y z z= = = (19)

The state-space equation of motion including the full dynamics, (1), (3), (13), (14), (16), and (19) can be written in a

general form as

( ) ( )B= +x f x x u (20) where the control input vector is

5 1mr ped col lon latT δ δ δ δ

T ×⎡ ⎤= ∈ℜ⎣ ⎦u (21) The trajectory tracking problem is formulated as an optimal control problem by minimizing a quadratic cost

function:

( ) 02

1 T TJ Q R dt= +∫ x x u u (22)

The advantage of this unified optimal control formulation is that system concerns such as tracking errors, control limits, angle and angular rate constraints, and velocity constraints, can be easily addressed in one cost function by imposing proper weights on the corresponding variables. In addition, the design and implementation is easier to conduct without much iterative process in the multi-timescale approach, which typically needs multiple processors in implementation. However, Eq. (20) includes 17 highly nonlinear and coupled dynamic equations and is an under-actuated system. Thus, it i es a challenging control problem

∞

mpos . In this work, we will employ a new nonlinear optimal control technique, Dθ − method, to solve this problem.

B. −θ D Nonlinear Suboptimal Control Technique The Dθ − technique25, 26 is a systematic nonlinear optimal control design approach to address a class of

nonlinear time-invariant systems described by ( ) ( )B= +x f x x u (23)

nimize the cThe objective is to find a stabilizing control ost functional u and in the meantime mi

( ) 02

1 T TJ Q R∞

dtu u (24)

where , , , , ,n n n m m n n m mc R R B R R Q R R R× × ×∈Ω ⊂ ∈ ∈ ∈ ∈ ∈x f u ; Assume that c

= +∫ x x

Ω is a comp t subset in R Q is a po ive semi-defi

ac n ; sit nite matrix and R is a positive definite constant matrix med that; It is assu f is of class in x

ongulator problem can be obtained by solving the

Hamilton-Jacobi-Bellman (HJB) partial d

1C cΩ and f(0)=0. The optimal solution to this infinite-horizon nonlinear re

ifferential equation:29

11( ) ( ) ( ) 0T

T TV V VB R B Q−∂ ∂ ∂⎛ ⎞− +⎜ ⎟1

2 2∂ ∂⎝ ⎠∂f x x x x x =

x x (25)

where V(x) is the optimal cost, i.e. x

021( ) min T TV Q R dt

∞

⎡ ⎤= +⎣ ⎦u ∫x x x u u (26)

Assume that V(x) is continuously differentiabl l is given by e and V(x)>0 with V(0)=0. The optimal contro 1 ( )T VR B− ∂

= −∂

u x (27)

he Dx

The HJB equation is extremely difficult to solve in general. T

θ − method gives an approximate closed-form s to the cost function, i.e. solution by introducing perturbation


9

0

1

1 T i Ti

i2J Q D R dtθ

∞∞

=

⎧ ⎫⎡ ⎤= + +⎨ ⎬⎢ ⎥∑∫ x x u (28)

i

⎣ ⎦⎩ ⎭u

where 1

ii

Dθ∞

=∑ is a perturbation series in terms of an instrumental variable θ . The construction of this series will be

discussed afterwards. Rewrite the state equation (23) in a linear-like structure: 25, 26

0 0( )( ) ( ) ( ) AB F A gθ ( )( ) gB θθ θ

⎧ ⎫ ⎧ ⎫⎡ ⎤= + = + + ⎡ ⎤+ +⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭

x xx f x x u x x = x 29) x u u (

where A0 and 0g are constant matrices such that 0 0( , )A g is a controllable pair and [ ] [ ]{ }0 0( ) , ( )A A g g+ +x x is poi

pendent factorization in (29) is exact and does not in ion process. The form of factorization is not unique, 31 which provides extra flexibility in the control design.

The new optimal control problem (28) and (29) can be solved through the perturbed HJB equation

ntwise controllable. Remark 3.1: The state-de volve any linearizat

10

( ) 1( )T T

T Ti

V A VA B R B Q Dθ∞

−∂ ∂ ∂ ⎛⎛ ⎞ ⎡ + + +⎜ ⎟ ⎜⎢ ∑x x x1

1 ( ) 02 2

i

i

V θθ =

⎞⎤ ⎛ ⎞− =⎜ ⎟ ⎟⎥∂ ∂ ∂⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎝ ⎠x x x

x x x (30)

Assuming a power series expansion of 0

( , ) ii

i

V T θ θ∞

=

∂=

∂ ∑ x xx

, the opt al

im control becomes

1

0

( ) ( , )T ii

i

R B T θ θ= − ∑u x x ∞

−

=

x (31)

where ( , )iT θx ( 0, , ,i n= ) is a symmetric matrix and is solved recursively by the following algorithm (32),

whi0

( , ) ii

i

V T θ θ∞

=

∂=ch is obtained by substituting

∂x HJB equation (30) and equating the ∑ x x in the perturbed

coefficients of powers of θ to zero: 1

0 0 0 0 0 0 0 0 0T TT A A T T g R g T Q−+ − + = (32a)

1 1 0 01 0 0 0 0 0 0 0 0 1

( ) ( )( ) ( )

TT T T T A A T

T A g R g T A T g R g Tθ θ

− −− + − = − −x x 1

0 0 0( )TgT g R Tθ

−+x 1

0 0 0 1TT D( )gT R g

θ−+ −

x (32b)

1 12 0 0 0 0 0 0 0( ) (T TT A g R g T A T g R g− −− + − 1 11 1

0 2 0 0 1 0 0 1( ) ( ) ( ) ( ))

T TT TT A A T g gT T g R T T R g T

θ θ θ θ− −= − − + +

x x x x

1 1 1 10 0 1 0 0 1 1 0 0 1 0

( ) ( ) ( ) ( )T TT Tg g g gT R T T g R g T T g R T T R g T D

θ θ θ θ− − − −+ + + +

x x x x

0 2− (32c)

21 1 11 1

0 0 0 0 0 0 0 0 20

( ) ( ) ( ) ( )( ) ( )T Tn

T T T n nn n j

j

T A A T g gT A g R g T A T g R g T T R Tθ θ θ θ

−− − −− −

− −=

− + − = − − +∑x x x xn j

11 1( ) ( )Tn

Tg gT g R R g T−

− −0 0j nθ θ 1 j

0j− −

=⎢ ⎥⎣ ⎦

∑ ⎡ ⎤+ +

x x 11

nT

0 01

j n j nj

T g R g T D−=

−−+ −∑ (32d)

Note that equation (32a) is lgebraic Riccati equ tion and the rest of equations are Lyapun an a a ov equations that are

The pllinear in terms of iT ( 1, ,i n= ).

steps of ap ying the Dθ − algorithm to solve iT recursively are summarized as follows: 1) Solve the algebraic Riccati equation (32a) to o 0 once 0 Tbtain A , 0g , Q and R are determined. Note that the

resu is a positive definite constant matrix under the controllability and observability con . nov equation (32b) to ob

lting ditions2) Solve the Lyapu tain

0T 1( , )T θx . Note that this is a linear algebraic equation in terms

of 1( , )T θx a ty of this equation is that the coefficient matrices 10 0 0 0

TA g R g T−− and ices.

Assume that 00c

nd a unique properare constant matr

0

10

T TR g−0 0 0A T g−

10 0

TA = A − rough linear algebra, equation (32b) can be b o the form of g R g T− .

[ Th rought int

]0 1ˆ vec( ( , )) 1vec ( , , )A T Mθ θx x ere 1( , , )t wh= M tθx includes all the nonlinear terms on the right-hand side

of the equation (32b); 1vec( ( , , ))M tθx d ing the elements of the matrixenotes stack 1( , , )M tθx by rows in a


10

vector form; 0 0

T0

ˆn c c nA = I A A I⊗ + ⊗ is a con atrix and the symbol ⊗ denotes the Kronecker stant m

product. Th tion of us, the resulting solu 1( , )T θx can be written in sed-form expression a clo

[ ]11

ˆvec( ( , )) vec ( , , )T A M tθ θ−=x x . 1 0

3) Solve equations (32c) an 2 , , nT T by following the similar procedure in 2). Since all th of ,T i left-hand side of the equations (32c ) are the same constant

matrices, i.e. 10 0 0 0

TA g R g T−− and 10 0 0TA T g g−

d (32d) forefficients 1,i n= on

0−

e co the d, close solution for each

-32( ,T d-formR )iT θx can be easily obtained

with only one matrix inverse .e. 10A − . operation, i

The p ructed a erturbation matrix , 1,i i n=D is const s follows:

1 0 01 1 0 0 0 0

( ) ( ) ( ) ( )T Tl t TT A A T g gD k e T g R T T R g T

θ θ θ θ− − −⎡ ⎤

= − − + +⎢ ⎥⎣ ⎦

x x x x (33a) 1 10 0

2 1 11 12 2 0 0 1 0 0

( ) ( ) ( ) ( )T Tl t TT A A T g gD k e T g R T T R g T1θ θ

− − −⎡= − − + +⎢

x x x x

θ θ⎣

1 1 10 0 1 0 0 1

11 0 0 1

( ) ( ) ( ) ( )T TT Tg g g gT R T T g R T T R T T g R g T

θ θ θ− − − − ⎤

+ + + +0 0g ⎥θ ⎦

x x x x (33b)

11 11 1

0 00

( ) ( ) ( ) ( )n

T Tnl t Tn n

n n j nj

T A A T g gD k e T g R R g T 1 jθ θ

−− − −− −

θ θ − −=

⎧ ⎡ ⎤⎪= − − + +⎨ ⎢ ⎥⎣ ⎦

∑x x x x

⎪⎩

2 11 1 2

0j n j

j

T R Tθ θ 0 0

1

( ) ( )Tn nT

j nj

g g T g R g T− −

j − −−

=

⎫⎬⎭

∑x x (33c)

wh a ble design parameters. The are chosen such that

− −=

+ +∑ere k nd 0, 1, ,il i n> = are adjustai

iD2 1 1

1 1 1 11 12 0 0 T −1 0 0

0 0 1

( ) ( ) ( ) ( ) ( ) ( )( )T T Ti i i

T Ti ij i j j i j j i j i

j j j

T A A T g g g gT R T T g R R g T g R g T Dθ θ θ θ θ θ

− − −− − − −− −

− − − −= = =

− − + + + + −∑ ∑ ∑x x x x x x

11 11 1

0 0 1( ) ( ) ( ) ( )( )

T TiTi i

i j i jT A A T g gt T g R R g Tε

θ

−− −− −

−0jθ θ θ −

⎧ ⎡ ⎤⎪= − − + +⎨ ⎢ ⎥⎦

∑x x x x

=⎪ ⎣⎩

2 1

1 1 2 0j T 00

( ) ( )Ti iT

j i j i jj

g gT R T g R g Tθ θ

− −− −

− − −=

⎫+ + ⎬∑ ∑x x (34)

where 1j= ⎭

( ) 1 il ti it k eε −= − (35)

iε is chosen to be a small number used to overcome the initial large control gain problem because the state dependent terms A(x) and g(x) on the right-hand side of the equations (32b)-(32d) may cause a large magnitude of

( , )iT θx if the initial states are large. Suppose there is no iε or Di in Eq. (32). For example, in Eq. (3 b), if there exists a cubic term in A(x) and the initia x is large, this large value will be reflected in the solution of 1T . Since 1T and A(x) will be used in solving for 2T in the ensuing equation (32c), this large value will be propagated and amplified. As a result, it causes the large control gain or even in

2l

stability. Therefore, th all number e s im ε is usedsuppr

to ess this large value from propagating in Eqs. (32b)-(32d).

iε is also required in the proof of convergence and stability of the above algorithm. e exponential term 25 Th il te− in ur

and don m

iD lets the pert bation terms in the cost function (28) diminish as time evolves. Remark 3.2: ik and il are design parameters used to modulate system transient responses. Selection of k

il can be atically by applying least square curve-fitting of the maximum singular value of the Di

e syste θ −

solution (ii

T0

i, )θ θ∞

= 25, 26

∑ x with that of the state dependent Riccati equation solution. The detail justification

referred to.

can be


11

Remark 3.3 θ is merely an intermediate variable. The introduction of θ is for the convenience of power series expansion, and it is cancelled when ( , )iT θx multiply iθ in the final control calculations, i.e., equation (31). Note

that in every equation (32) and expression (33), iT iD 1iθ

factor appears linearly on the right-hand side of the

equations. Consequently, 1iθ

will appear linearly in the solution of , i.e. iT iiT

ˆi

Tθ

= where is the solution without iT

1iθ

. When T is multiplied by iiθ in the control equation (31), iθ gets cancelled. The cancellation is also true for

the perturbation series 1

ii

i

Dθ∞

=∑ .

In the Dθ − algorithm, retaining the first three terms and in the control equation (31) has been sufficient to achieve satisfactory performance in the problems that have been solved25, 26, 30. Theoretical results

concerning the convergence of the series

0 1,T T 2T

0

( , ) ii

i

T θ θ∞

=∑ x , closed-loop stability, and optimality of truncating the series

can be referred to25.

D Method C. Trajectory Control Design Using the −θThe problem we are addressing is to control the miniature helicopter to track desired trajectories. The unified

state variables are defined in (18) and the control variables are defined in (21). The optimal control problem is described by Eqs. (20) and (22).

In order to apply the Dθ − method, f(0)=0 needs to be satisfied in order to factorize f(x) into the linear-like structure . If there are state-independent terms that make ( )F x x (0) 0f ≠ , they prevents a direct factorization of

f(x) and they are called bias terms. For example, the terms such as , , and

1C

trY Y+ +/fusX m /fus vfY m⎡ ⎤⎣ ⎦/ cosg cosZ mfus φ θ+ , are bias terms because they may not go to zero when the states become zero. One of the ways

to handle this problem is to augment the system with a stable state ‘s’ satisfying26, 30, 31 ss sλ= − (36)

in which sλ is a positive number. With the augmented state ‘s’, the bias term denoted by b(t) can then be factorized as

( )( ) b tb t ss

⎡ ⎤= ⎢ ⎥⎣ ⎦

1 1a p p i i i ℜ⎣ ⎦

a a

(37)

The introduction of ‘s’ does not change the actual helicopter dynamics. Each time through the controller, the initial value s(0) is used in the state-dependent coefficient matrix and in calculating the control. Usually we set s(0)=1.

Including the augmented state ‘s’, the total state-space contains 18 states and is denoted by: 18 1T

x y z u v w p q r a b x y z sφ θ ψ ×⎡ ⎤= ∈x (38) p

( )F xThe state-dependent coefficients and of the ( )B x Dθ − formulation, Eq. (29), can be written as:


12

( )3 3 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3 3 13 3

48 4_18

3 3 3 3 57 3 1 3 1 3 1 3 1 3 1 3 1 3 3 5_18

6_18

1 3 1 3 7_11 7_12 1 3

1 3 1 3 8_11 8_12

0 0 0 0 0 0 0 0 0 0 0

00 0 0 0 0 0 0 0 0

0 0

0 0 0 0 0 1 a a 0 0 0 00 0 0 0 0 0 a a 0 0

( )a

a aa a

a

F

ω

× × × × × × × × × × ××

× × × × × × × × ×

× × ×

× ×

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

=

R Θ

x

1 3

1 3 1 3 9_11 9_12 1 3

1 3 1 3 10_11 10_12 10_14 1 3 10_18

1 3 1 3 11_10 11_12 11_13 1 3 11_18

1 3 1 3 12_10 12_11 1 3 12_18

1 3 1 3 13_13 1 3 13_18

1 3 1 3

0 00 0 0 0 0 0 a a 0 0 0 00 0 0 0 0 0 a a 0 a 0 a0 0 0 0 0 a 0 a a 0 0 a0 0 0 0 0 a a 0 0 0 0 a0 0 0 0 0 0 1 0 a 0 0 a0 0 0 0 0 1 0 0 0

×

× × ×

× × ×

× × ×

× × ×

× × ×

× ×

−− 14_14 1 3 14_18

3 3 1 3 1 3 3 1

1 3 1 3 1 3

a 0 a0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 s

Iλ

×

× × × ×

× × ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

(39)

where ; 3 3

00

0

r qr p

q pω ×

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

48sina g θθ

= − ; 4 _18fusX

am s

=⋅

; 57sincosa g φθφ

= ; 5 _18fus tr vfY Y Y

am s+ +

=⋅

; 6 _18cos cosfusZ ga

m s sφ θ

= +⋅

; 7_11 sin tana φ θ= ;

7_12 cos tana φ θ= ; 8_11 cosa φ= ; 8_12 sina φ= − ; 9_11 sin seca φ θ= ; 9_12 cos seca φ θ= ; 10_11yy

xx

rIa

I= ; 10 _12

zz

xx

qIa

I= − ;

10 _14xx

Ka

Iβ= ; 10 _18a vf tr

xx

L LI s+

=⋅

; 11_10xx

yy

rIa

I= − ; 11_12

zz

yy

pIa

I= ; 11_13

yy

Ka

Iβ= ; 11_18

ht

yy

Ma

I s=

⋅; 12_10

xx

zz

qIa

I= ; 12 _11

yy

zz

pIa

I= − ;

12 _18vf tr

zz

N Na

I s+ −

=⋅

eQ; 13_13a 1

eτ= − ; ( )

2

13 _18 0

162

8a a

e mr mr

Kτe mr

igna Rτ

⎡ ⎤⎢ ⎥

mr

K wa s

sμ

μ

μμ λ

μ σ= −

uR s+ Ω ⋅ Ω ⋅⎢ ⎥⎣ ⎦

; 14_141

e

aτ

= − ;

14 _18 02a K a

e mr

v

mrR sΩ ⋅μλ τ= − ;

and

3 1 3 1 3 1 3 1 3 1

41

51 52

61

3 1 3 1 3 1 3 1 3 1

10 _1 10 _ 2

11_1

12 _ 2

13 _ 3 13 _ 4

14 _ 3 14 _ 5

0 0 0 0 00 0 0 0

0 0 00 0 0 0

0 0 0 0 0( ) 0 0 0

0 0 0 00 0 0 00 0 00 0 0

0 0 0 0 0

a

bb bb

B b bb

bb bb b

× × × × ×

× × × × ×

4 1 4 1 4 1 4 1 4 1× × × × ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

x (40)

where


13

141

ab

m= − ; 1

51a

bm

= ; ;52 ped

trb Yδ= 611bm

= − ; 110 _1

mr

xx

h bb

I= ; 10 _ 2

ped

trtr

xx

h mYb

Iδ

= ; 111_1

mr

yy

h ab

I= ; 12 _ 2

ped

trtr

zz

mY lb

Iδ

= − ;

13 _ 3

83

a

e mr mr

K ub

Rμ

τ=

Ω;

2

13_ 4lon

nommr

e nom

Ab δ

τ⎛ ⎞Ω

= ⎜ ⎟Ω⎝ ⎠; 14_ 3

83

a

e mr mr

K vb

Rμ

τ=

Ω;

2

14 _ 5lat

nommr

e nom

Bb δ

τ⎛ ⎞Ω

= ⎜ ⎟Ω⎝ ⎠

The 0A , , ( )aA x 0g , and are chosen to be ( )ag x 0 0( ( ))aA F t= x , 0( ) ( ) ( ( ))a a aA F F t−x = x x , 0 0( ( ))ag B t= x , and 0 ))a( ) ( )a a ( (g B B− tx = x x , i.e.

0 00 0

( ) ( ( )) ( ) ( ( ))( ( )) ( ( ))a a a a

a a a aF F t B B t

F t B tθ θθ θ

⎡ − ⎤ ⎡ −⎛ ⎞ ⎛= + + +⎢ ⎥ ⎢⎜ ⎟ ⎜⎝ ⎠ ⎝⎣ ⎦ ⎣

x x x xx x x x

⎤⎞⎥⎟⎠⎦

u

a

(41)

Remark 3.4: The state dependent factorization approach ( ) captures the nonlinearities and dynamic changes in the helicopter model as the states vary. Thus, the optimal control gains based on this live information can be computed adaptively even under uncertain operating conditions.

( ), ( )aF Bx x

The cost function for this problem is chosen to be a quadratic function:

0

12

T Ta aJ Q R

∞dt⎡ ⎤= +⎣ ⎦∫ x x u u (42)

where and Q R are the weighting matrices. After tuning, they are chosen to be:

( )4 4 7 3 5 5 3 3 610 , 2 10 ,10 ,100,100,1,100,100,10 , 200,1, 200,10 ,10 , 2 10 , 2 10 ,1 10 ,0Q diag ⎡ ⎤= × × × ×⎣ ⎦ (43)

( 7 5 51, 200,10 ,10 ,10R diag ⎡= ⎣ )⎤⎦ (44)

where ( )diag ⋅ represents a diagonal matrix. In order to perform the command tracking, the Dθ − controller is implemented as an integral servomechanism

as shown in (45)26, 30. ( ) ( )1

0 1 2( ) , ,Ta a aR B T T Tθ θ−= − + +⎡⎣u x x x track⎤⎦V

]

(45) where

1 1, , , , , , , , , , , , , , , , ,1T

track p c p c p c i c i c i cx x y y z z u v w p q r a b x x dt y y dt z z dtφ θ ψ⎡ ⎤= − − − − − −⎣ ⎦∫ ∫ ∫V

(46)

[ Tc c cx y z is the desired position vector that the helicopter needs to track. , 0T ( )1 ,aT ,θx and ( )2 ,aT θx are

solved from the Dθ − algorithm (32) and (33). Using these three terms is found to be sufficient to produce good tracking performance.

The parameters in the perturbation matrix (33) are chosen to be: k1=1, k2=1, l1=1, and l2=1. Note that we have designed an optimal controller for this highly nonlinear system with 18 state variables. The Dθ − technique is particularly useful for solving this type of high-order nonlinear control problems since the

optimal controller, Eq. (45), can be obtained in a closed-form by virtue of the Dθ − algorithm, Eqs. (32)-(33). This closed-form law facilitates real-time implementations because it does not demand intensive and iterative computations.

IV. Simulation Results and Analysis In this section, the simulation of the position tracking using the Dθ − control is demonstrated. In order to show

the tracking performance and the robustness of the proposed controller, 45 uncertainties and the wind gust, which were defined in the helicopter dynamics section, are included here. The nominal values of the aerodynamic coefficients, geometry parameters, and physical constants are used in the control law design whereas the uncertainties are added in testing scenarios. All the feedback states are assumed to be measurable. However, measurement uncertainties are considered into the control design by adding white Gaussian noises with the standard deviation of 5% of the magnitude of the feedback states as given in Table 1. In addition, when the computed control law is applied to the helicopter dynamics, noises with the level of 5% of the control magnitude are added to represent the actuator uncertainties. Position tracking control over the entire flight envelope includes trajectories in any direction with the speed up to the maximum allowed by the physical capabilities of the helicopter ( 27). 20 /m s

Two simulation scenarios are used below where these trajectories are combinations of ramps, sinusoidal waves, and step inputs in three directions. These complex paths demonstrate a full flight envelope because they require: (1)


14

vertical takeoff; (2) vertical landing; (3) hover; (4) forward, rearward, and sideward flight at a constant altitude; (5) forward, rearward, and sideward flight at varying altitudes; and (6) tracking in all three dimensions simultaneously.

The first desired trajectory starts with a vertical takeoff to an altitude of 3.5 , followed by a 5-second hover, then a figure “8” trajectory in the x-y plane, followed by a 10-second hover and then a vertical landing.

m

Figures 2-7 present the results of the first simulation scenario. Figure 2 gives a three dimensional view of the trajectory and Figure 3 shows the time histories of the x, y and z components of the inertial position vector, respectively. In the figure “8” maneuver, the desired trajectory has the helicopter travel in the negative direction first and it eventually returns to the start of the figure “8”. As can be seen, the tracking in the x and z directions is very precise even though there are two abrupt command changes. The small deviations on the transient y direction occur during transitions between ascending or descending flight and the figure “8” maneuver. The helicopter is either already hovering and then catching up with the figure “8” trajectory, or is trying to halt and resume hovering following the maneuver in these regions. The helicopter shows slight overshoots at these points in the direction because of the abrupt change of the velocity commands.

y

y

In Figure 4, the three Euler angles are shown. A few quick jumps can be observed in the roll and yaw angles due to the start and stop of the figure “8” maneuver. But in between, they show smooth transient responses. The angular velocity responses are shown in Figure 5. The similar abrupt jumps can be seen during the trajectory maneuvers. Oscillations on the curve stem from the uncertainty and random noises in the dynamics model because angular velocities are fast changing variables with larger bandwidth and thus easier to be affected by the noise. Translational velocities are shown in Figure 6 and are for the most part smooth, with a few large jumps occurring in the ‘w’ response, corresponding to the abrupt changes in altitude commands at these moments. Figure 7 shows the flapping angles. The high frequency oscillation is because the flapping mode is the fastest dynamic mode with a large bandwidth and the uncertainties and noises manifest themselves more significantly. But the magnitude is in a reasonable range. The control deflections and main rotor thrust are shown in Figure 8 and Figure 9, respectively. These curves show the similar trend with large jumps occurring at the transition periods of the trajectory. Also the oscillations are due to the uncertainties and noises added in the simulation.

Figure 2: Scenario I: 3-dimensional view of the trajectory


15

Figure 3: Scenario I: time histories of the position tracking

Figure 4: Scenario I: Euler angle responses


16

Figure 5: Scenario I: angular velocity response

Figure 6: Scenario I: translational velocity responses


17

Figure 7: Scenario I: flapping angle responses

Figure 8: Scenario I: control deflections


18

Figure 9: Scenario I: main rotor thrust

In the second simulation scenario, the helicopter also starts with a vertical takeoff to 3.5 and then traverses a large circle in the

mx y− plane while descending and then ascending, as seen from the three dimensional view in

Figure 10. The time histories of the three position components are shown in Figure 11. The miniature helicopter follows the trajectory in a counterclockwise motion. The result is similar to the first scenario with an excellent tracking in the x and z directions and very small deviations in the y direction due to the same reason as the first scenario. Figures 12-15 give the responses of the Euler angles, angular velocities, translational velocities, and flapping angles, respectively. The results are very similar to the corresponding ones in the first scenario. The large jumps occur because of the abrupt changes in the commanded signal. The high frequency oscillations are due to the noises and uncertainties in the simulations. Figures 16 and 17 show the control deflections and main rotor thrust responses, which are also similar to the results in the first scenario.

Figure 10: Scenario II: 3-dimensional view of the trajectory


19

Figure 11: Scenario II: time histories of the position tracking

Figure 12: Scenario II: Euler angle responses


20

Figure 13: Scenario II: angular velocity responses

Figure 14: Scenario II: translational velocity responses


21

Figure 15: Scenario II: flapping angle responses

Figure 16: Scenario II: control deflections


22

Figure 17: Scenario II: main rotor thrust

The above two simulation results demonstrate the capabilities of the unified optimal control design in full

envelope trajectory tracking since all possible helicopter motions are considered. The single optimization formulation facilitates the design process, which is much easier to conduct than the traditional time-scale separation design paradigm.

V. Conclusion In this paper, trajectory control of a miniature helicopter over complex paths was addressed using the proposed Dθ − nonlinear optimal control technique. The formulation was based on a unified optimal control framework, thus

can be used to simplify the design process and optimize the overall system performance. As the most significant advantage, this method produced an approximate closed-form feedback controller, which does not require online computation of the state dependent Riccati equation as with the SDRE technique. The advantage of the control design in this under-actuated miniature helicopter has been demonstrated by complex paths’ tracking including vertical takeoff, vertical landing, hover, forward, rearward, and sideward flight. The robustness of the controller was also validated by taking into account forty-five functional and parametric uncertainties as well as the wind gust in the simulation. Furthermore, as compared with often used time-separation control systems, the proposed Dθ − control will be much easier to design and implement.

Acknowledgement

This work was supported in part by the National Science Foundation CAREER Award ECCS-0846877.

Reference 1Department of Defense, Airspace Integration Plan for Unmanned Aviation, Washington, Office of the Secretary of Defense,

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